LMFDB - Embedded newform 168.10.q.a.121.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,10,Mod(25,168)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("168.25");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	168.q (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$86.5260204755$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18660372 x^{14} - 3458782984 x^{13} + 143123973101310 x^{12} + \cdots + 50\!\cdots\!97 $ x^16 - 18660372x^14 - 3458782984x^13 + 143123973101310x^12 + 48872053104519072x^11 - 578734520240854872804x^10 - 270042331189660586117064x^9 + 1317886932354758017843044627x^8 + 736352464217070243815722699808x^7 - 1673655580870016843616444263455812x^6 - 1020366474648370188416591376208359000x^5 + 1103325097889381327876435664684216492298x^4 + 649904594580478032138799228161609255854736x^3 - 331733869309416601638773034133657913991577560x^2 - 128648437449643500784782173321364984397814457552x + 50061741614807564009961574976312432423689032591597
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{38}\cdot 3^{5}\cdot 5^{2}\cdot 7^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.2
Root			$-1797.08 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	168.121
Dual form			168.10.q.a.25.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-40.5000 - 70.1481i) q^{3} +(-910.790 + 1577.53i) q^{5} +(2936.64 - 5632.92i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})$	\(q+(-40.5000 - 70.1481i) q^{3} +(-910.790 + 1577.53i) q^{5} +(2936.64 - 5632.92i) q^{7} +(-3280.50 + 5681.99i) q^{9} +(36217.5 + 62730.6i) q^{11} +3125.28 q^{13} +147548. q^{15} +(-111243. - 192678. i) q^{17} +(91071.5 - 157740. i) q^{19} +(-514072. + 22133.8i) q^{21} +(69459.7 - 120308. i) q^{23} +(-682514. - 1.18215e6i) q^{25} +531441. q^{27} +1.95635e6 q^{29} +(-164413. - 284771. i) q^{31} +(2.93362e6 - 5.08118e6i) q^{33} +(6.21146e6 + 9.76305e6i) q^{35} +(7.95090e6 - 1.37714e7i) q^{37} +(-126574. - 219232. i) q^{39} -1.59660e6 q^{41} -2.11583e7 q^{43} +(-5.97569e6 - 1.03502e7i) q^{45} +(-2.23762e7 + 3.87567e7i) q^{47} +(-2.31059e7 - 3.30837e7i) q^{49} +(-9.01067e6 + 1.56069e7i) q^{51} +(-8.02367e6 - 1.38974e7i) q^{53} -1.31946e8 q^{55} -1.47536e7 q^{57} +(4.52725e7 + 7.84143e7i) q^{59} +(-2.74389e7 + 4.75255e7i) q^{61} +(2.23726e7 + 3.51647e7i) q^{63} +(-2.84647e6 + 4.93024e6i) q^{65} +(-4.54109e7 - 7.86540e7i) q^{67} -1.12525e7 q^{69} +2.73022e8 q^{71} +(1.08429e8 + 1.87804e8i) q^{73} +(-5.52836e7 + 9.57540e7i) q^{75} +(4.59714e8 - 1.97934e7i) q^{77} +(-2.44675e6 + 4.23789e6i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} +2.20213e8 q^{83} +4.05275e8 q^{85} +(-7.92320e7 - 1.37234e8i) q^{87} +(-3.24985e8 + 5.62890e8i) q^{89} +(9.17781e6 - 1.76044e7i) q^{91} +(-1.33174e7 + 2.30665e7i) q^{93} +(1.65894e8 + 2.87337e8i) q^{95} -7.52839e8 q^{97} -4.75246e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9	$ 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) $ 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9 + 32460 * q^11 + 119048 * q^13 + 31752 * q^15 + 208352 * q^17 + 914588 * q^19 - 428652 * q^21 + 460920 * q^23 - 3040180 * q^25 + 8503056 * q^27 - 16376136 * q^29 - 944064 * q^31 + 2629260 * q^33 - 15546664 * q^35 - 9826516 * q^37 - 4821444 * q^39 + 11449216 * q^41 - 6933624 * q^43 - 1285956 * q^45 + 26549360 * q^47 + 83657504 * q^49 + 16876512 * q^51 - 15354476 * q^53 + 134121944 * q^55 - 148163256 * q^57 + 18404996 * q^59 - 260632792 * q^61 + 35823060 * q^63 + 191461840 * q^65 + 53879788 * q^67 - 74669040 * q^69 - 164207456 * q^71 + 248475540 * q^73 - 246254580 * q^75 + 670121788 * q^77 + 16631256 * q^79 - 344373768 * q^81 - 1138943272 * q^83 - 1690136272 * q^85 + 663233508 * q^87 + 236796360 * q^89 - 1455575212 * q^91 - 76469184 * q^93 + 182450488 * q^95 + 1339799464 * q^97 - 425940120 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−40.5000	−	70.1481i	−0.288675	−	0.500000i
$3$	−40.5000	−	70.1481i	−0.288675	−	0.500000i
$4$		0			0
$5$	−910.790	+	1577.53i	−0.651708	+	1.12879i	0.331000	+	0.943631i	$0.392614\pi$
$5$	−910.790	+	1577.53i	−0.651708	+	1.12879i	−0.982708	+	0.185161i	$0.940719\pi$
$6$		0			0
$7$	2936.64	−	5632.92i	0.462284	−	0.886732i
$7$	2936.64	−	5632.92i	0.462284	−	0.886732i
$8$		0			0
$9$	−3280.50	+	5681.99i	−0.166667	+	0.288675i
$10$		0			0
$11$	36217.5	+	62730.6i	0.745850	+	1.29185i	0.949797	+	0.312868i	$0.101290\pi$
$11$	36217.5	+	62730.6i	0.745850	+	1.29185i	−0.203947	+	0.978982i	$0.565377\pi$
$12$		0			0
$13$	3125.28			0.0303490			0.0151745	−	0.999885i	$-0.495170\pi$
$13$	3125.28			0.0303490			0.0151745	+	0.999885i	$0.495170\pi$
$14$		0			0
$15$	147548.			0.752528
$16$		0			0
$17$	−111243.	−	192678.i	−0.323037	−	0.559516i	0.658076	−	0.752951i	$-0.271370\pi$
$17$	−111243.	−	192678.i	−0.323037	−	0.559516i	−0.981113	+	0.193435i	$0.938037\pi$
$18$		0			0
$19$	91071.5	−	157740.i	0.160321	−	0.277685i	−0.774663	−	0.632375i	$-0.782080\pi$
$19$	91071.5	−	157740.i	0.160321	−	0.277685i	0.934984	+	0.354690i	$0.115414\pi$
$20$		0			0
$21$	−514072.	+	22133.8i	−0.576816	+	0.0248353i
$22$		0			0
$23$	69459.7	−	120308.i	0.0517556	−	0.0896434i	−0.838987	−	0.544152i	$-0.816852\pi$
$23$	69459.7	−	120308.i	0.0517556	−	0.0896434i	0.890743	+	0.454508i	$0.150185\pi$
$24$		0			0
$25$	−682514.	−	1.18215e6i	−0.349447	−	0.605260i
$26$		0			0
$27$	531441.			0.192450
$28$		0			0
$29$	1.95635e6			0.513635			0.256817	−	0.966460i	$-0.417326\pi$
$29$	1.95635e6			0.513635			0.256817	+	0.966460i	$0.417326\pi$
$30$		0			0
$31$	−164413.	−	284771.i	−0.0319748	−	0.0553820i	0.849595	−	0.527435i	$-0.176846\pi$
$31$	−164413.	−	284771.i	−0.0319748	−	0.0553820i	−0.881570	+	0.472053i	$0.843513\pi$
$32$		0			0
$33$	2.93362e6	−	5.08118e6i	0.430617	−	0.745850i
$34$		0			0
$35$	6.21146e6	+	9.76305e6i	0.699661	+	1.09971i
$36$		0			0
$37$	7.95090e6	−	1.37714e7i	0.697442	−	1.20801i	−0.271908	−	0.962323i	$-0.587655\pi$
$37$	7.95090e6	−	1.37714e7i	0.697442	−	1.20801i	0.969350	−	0.245682i	$-0.0790120\pi$
$38$		0			0
$39$	−126574.	−	219232.i	−0.00876099	−	0.0151745i
$40$		0			0
$41$	−1.59660e6			−0.0882405			−0.0441202	−	0.999026i	$-0.514048\pi$
$41$	−1.59660e6			−0.0882405			−0.0441202	+	0.999026i	$0.514048\pi$
$42$		0			0
$43$	−2.11583e7			−0.943786			−0.471893	−	0.881656i	$-0.656429\pi$
$43$	−2.11583e7			−0.943786			−0.471893	+	0.881656i	$0.656429\pi$
$44$		0			0
$45$	−5.97569e6	−	1.03502e7i	−0.217236	−	0.376264i
$46$		0			0
$47$	−2.23762e7	+	3.87567e7i	−0.668876	+	1.15853i	0.309343	+	0.950951i	$0.399891\pi$
$47$	−2.23762e7	+	3.87567e7i	−0.668876	+	1.15853i	−0.978219	+	0.207576i	$0.933442\pi$
$48$		0			0
$49$	−2.31059e7	−	3.30837e7i	−0.572587	−	0.819844i
$50$		0			0
$51$	−9.01067e6	+	1.56069e7i	−0.186505	+	0.323037i
$52$		0			0
$53$	−8.02367e6	−	1.38974e7i	−0.139679	−	0.241931i	0.787696	−	0.616064i	$-0.211274\pi$
$53$	−8.02367e6	−	1.38974e7i	−0.139679	−	0.241931i	−0.927375	+	0.374133i	$0.877940\pi$
$54$		0			0
$55$	−1.31946e8			−1.94431
$56$		0			0
$57$	−1.47536e7			−0.185123
$58$		0			0
$59$	4.52725e7	+	7.84143e7i	0.486408	+	0.842483i	0.999878	−	0.0156242i	$-0.00497354\pi$
$59$	4.52725e7	+	7.84143e7i	0.486408	+	0.842483i	−0.513470	+	0.858108i	$0.671640\pi$
$60$		0			0
$61$	−2.74389e7	+	4.75255e7i	−0.253736	+	0.439484i	−0.964551	−	0.263895i	$-0.914993\pi$
$61$	−2.74389e7	+	4.75255e7i	−0.253736	+	0.439484i	0.710815	+	0.703379i	$0.248326\pi$
$62$		0			0
$63$	2.23726e7	+	3.51647e7i	0.178930	+	0.281239i
$64$		0			0
$65$	−2.84647e6	+	4.93024e6i	−0.0197787	+	0.0342577i
$66$		0			0
$67$	−4.54109e7	−	7.86540e7i	−0.275311	−	0.476853i	0.694903	−	0.719104i	$-0.255447\pi$
$67$	−4.54109e7	−	7.86540e7i	−0.275311	−	0.476853i	−0.970214	+	0.242251i	$0.922114\pi$
$68$		0			0
$69$	−1.12525e7			−0.0597622
$70$		0			0
$71$	2.73022e8			1.27507			0.637536	−	0.770421i	$-0.279954\pi$
$71$	2.73022e8			1.27507			0.637536	+	0.770421i	$0.279954\pi$
$72$		0			0
$73$	1.08429e8	+	1.87804e8i	0.446881	+	0.774021i	0.998181	−	0.0602863i	$-0.0192014\pi$
$73$	1.08429e8	+	1.87804e8i	0.446881	+	0.774021i	−0.551300	+	0.834307i	$0.685868\pi$
$74$		0			0
$75$	−5.52836e7	+	9.57540e7i	−0.201753	+	0.349447i
$76$		0			0
$77$	4.59714e8	−	1.97934e7i	1.49032	−	0.0641669i
$78$		0			0
$79$	−2.44675e6	+	4.23789e6i	−0.00706752	+	0.0122413i	−0.869538	−	0.493867i	$-0.835583\pi$
$79$	−2.44675e6	+	4.23789e6i	−0.00706752	+	0.0122413i	0.862470	+	0.506108i	$0.168916\pi$
$80$		0			0
$81$	−2.15234e7	−	3.72796e7i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	2.20213e8			0.509321			0.254660	−	0.967031i	$-0.418036\pi$
$83$	2.20213e8			0.509321			0.254660	+	0.967031i	$0.418036\pi$
$84$		0			0
$85$	4.05275e8			0.842103
$86$		0			0
$87$	−7.92320e7	−	1.37234e8i	−0.148274	−	0.256817i
$88$		0			0
$89$	−3.24985e8	+	5.62890e8i	−0.549045	+	0.950974i	0.449295	+	0.893384i	$0.351675\pi$
$89$	−3.24985e8	+	5.62890e8i	−0.549045	+	0.950974i	−0.998340	+	0.0575909i	$0.981658\pi$
$90$		0			0
$91$	9.17781e6	−	1.76044e7i	0.0140298	−	0.0269114i
$92$		0			0
$93$	−1.33174e7	+	2.30665e7i	−0.0184607	+	0.0319748i
$94$		0			0
$95$	1.65894e8	+	2.87337e8i	0.208965	+	0.361939i
$96$		0			0
$97$	−7.52839e8			−0.863434			−0.431717	−	0.902009i	$-0.642092\pi$
$97$	−7.52839e8			−0.863434			−0.431717	+	0.902009i	$0.642092\pi$
$98$		0			0
$99$	−4.75246e8			−0.497233
$100$		0			0
$101$	1.02777e9	+	1.78015e9i	0.982766	+	1.70220i	0.651471	+	0.758673i	$0.274152\pi$
$101$	1.02777e9	+	1.78015e9i	0.982766	+	1.70220i	0.331295	+	0.943527i	$0.392515\pi$
$102$		0			0
$103$	−6.25403e8	+	1.08323e9i	−0.547510	+	0.948316i	0.450934	+	0.892557i	$0.351091\pi$
$103$	−6.25403e8	+	1.08323e9i	−0.547510	+	0.948316i	−0.998444	+	0.0557583i	$0.982242\pi$
$104$		0			0
$105$	4.33295e8	−	8.31126e8i	0.347882	−	0.667290i
$106$		0			0
$107$	−1.20764e9	+	2.09169e9i	−0.890655	+	1.54266i	−0.0515638	+	0.998670i	$0.516421\pi$
$107$	−1.20764e9	+	2.09169e9i	−0.890655	+	1.54266i	−0.839091	+	0.543990i	$0.816913\pi$
$108$		0			0
$109$	8.98463e8	+	1.55618e9i	0.609651	+	1.05595i	0.991298	+	0.131638i	$0.0420236\pi$
$109$	8.98463e8	+	1.55618e9i	0.609651	+	1.05595i	−0.381647	+	0.924308i	$0.624643\pi$
$110$		0			0
$111$	−1.28805e9			−0.805337
$112$		0			0
$113$	−1.48078e9			−0.854355			−0.427177	−	0.904168i	$-0.640492\pi$
$113$	−1.48078e9			−0.854355			−0.427177	+	0.904168i	$0.640492\pi$
$114$		0			0
$115$	1.26526e8	+	2.19150e8i	0.0674591	+	0.116843i
$116$		0			0
$117$	−1.02525e7	+	1.77578e7i	−0.00505816	+	0.00876099i
$118$		0			0
$119$	−1.41202e9	+	6.07957e7i	−0.645476	+	0.0277915i
$120$		0			0
$121$	−1.44444e9	+	2.50185e9i	−0.612584	+	1.06103i
$122$		0			0
$123$	6.46622e7	+	1.11998e8i	0.0254728	+	0.0441202i
$124$		0			0
$125$	−1.07127e9			−0.392466
$126$		0			0
$127$	1.10412e9			0.376616			0.188308	−	0.982110i	$-0.439700\pi$
$127$	1.10412e9			0.376616			0.188308	+	0.982110i	$0.439700\pi$
$128$		0			0
$129$	8.56913e8	+	1.48422e9i	0.272447	+	0.471893i
$130$		0			0
$131$	1.50394e9	−	2.60490e9i	0.446179	−	0.772805i	−0.551954	−	0.833874i	$-0.686118\pi$
$131$	1.50394e9	−	2.60490e9i	0.446179	−	0.772805i	0.998134	+	0.0610693i	$0.0194511\pi$
$132$		0			0
$133$	−6.21095e8	−	9.76224e8i	−0.172118	−	0.270531i
$134$		0			0
$135$	−4.84031e8	+	8.38366e8i	−0.125421	+	0.217236i
$136$		0			0
$137$	7.10431e8	+	1.23050e9i	0.172298	+	0.298428i	0.939223	−	0.343308i	$-0.111548\pi$
$137$	7.10431e8	+	1.23050e9i	0.172298	+	0.298428i	−0.766925	+	0.641737i	$0.778214\pi$
$138$		0			0
$139$	1.42949e9			0.324799			0.162400	−	0.986725i	$-0.448077\pi$
$139$	1.42949e9			0.324799			0.162400	+	0.986725i	$0.448077\pi$
$140$		0			0
$141$	3.62494e9			0.772351
$142$		0			0
$143$	1.13190e8	+	1.96051e8i	0.0226358	+	0.0392063i
$144$		0			0
$145$	−1.78182e9	+	3.08620e9i	−0.334740	+	0.579787i
$146$		0			0
$147$	−1.38497e9	+	2.96073e9i	−0.244631	+	0.522962i
$148$		0			0
$149$	−7.06634e8	+	1.22393e9i	−0.117451	+	0.203431i	−0.918757	−	0.394824i	$-0.870806\pi$
$149$	−7.06634e8	+	1.22393e9i	−0.117451	+	0.203431i	0.801306	+	0.598255i	$0.204139\pi$
$150$		0			0
$151$	5.31015e9	+	9.19745e9i	0.831209	+	1.43970i	0.897080	+	0.441869i	$0.145684\pi$
$151$	5.31015e9	+	9.19745e9i	0.831209	+	1.43970i	−0.0658702	+	0.997828i	$0.520982\pi$
$152$		0			0
$153$	1.45973e9			0.215358
$154$		0			0
$155$	5.98982e8			0.0833529
$156$		0			0
$157$	2.82740e9	+	4.89720e9i	0.371398	+	0.643280i	0.989781	−	0.142597i	$-0.0455453\pi$
$157$	2.82740e9	+	4.89720e9i	0.371398	+	0.643280i	−0.618383	+	0.785877i	$0.712212\pi$
$158$		0			0
$159$	−6.49917e8	+	1.12569e9i	−0.0806438	+	0.139679i
$160$		0			0
$161$	−4.73706e8	−	7.44561e8i	−0.0555638	−	0.0873341i
$162$		0			0
$163$	−4.48823e8	+	7.77385e8i	−0.0498002	+	0.0862565i	−0.889851	−	0.456251i	$-0.849192\pi$
$163$	−4.48823e8	+	7.77385e8i	−0.0498002	+	0.0862565i	0.840051	+	0.542508i	$0.182525\pi$
$164$		0			0
$165$	5.34382e9	+	9.25576e9i	0.561273	+	0.972153i
$166$		0			0
$167$	−1.92001e10			−1.91020			−0.955100	−	0.296284i	$-0.904252\pi$
$167$	−1.92001e10			−1.91020			−0.955100	+	0.296284i	$0.904252\pi$
$168$		0			0
$169$	−1.05947e10			−0.999079
$170$		0			0
$171$	5.97520e8	+	1.03493e9i	0.0534404	+	0.0925615i
$172$		0			0
$173$	5.45866e9	−	9.45467e9i	0.463317	−	0.802489i	−0.535807	−	0.844341i	$-0.679992\pi$
$173$	5.45866e9	−	9.45467e9i	0.463317	−	0.802489i	0.999124	+	0.0418520i	$0.0133258\pi$
$174$		0			0
$175$	−8.66324e9	+	3.73003e8i	−0.698247	+	0.0300636i
$176$		0			0
$177$	3.66707e9	−	6.35156e9i	0.280828	−	0.486408i
$178$		0			0
$179$	2.74312e9	+	4.75123e9i	0.199713	+	0.345913i	0.948435	−	0.316970i	$-0.102666\pi$
$179$	2.74312e9	+	4.75123e9i	0.199713	+	0.345913i	−0.748722	+	0.662884i	$0.769332\pi$
$180$		0			0
$181$	−1.95193e10			−1.35180			−0.675899	−	0.736995i	$-0.736244\pi$
$181$	−1.95193e10			−1.35180			−0.675899	+	0.736995i	$0.736244\pi$
$182$		0			0
$183$	4.44510e9			0.292989
$184$		0			0
$185$	1.44832e10	+	2.50856e10i	0.909058	+	1.57453i
$186$		0			0
$187$	8.05788e9	−	1.39567e10i	0.481874	−	0.834630i
$188$		0			0
$189$	1.56065e9	−	2.99356e9i	0.0889666	−	0.170652i
$190$		0			0
$191$	−1.14044e10	+	1.97530e10i	−0.620042	+	1.07394i	0.369435	+	0.929257i	$0.379551\pi$
$191$	−1.14044e10	+	1.97530e10i	−0.620042	+	1.07394i	−0.989477	+	0.144688i	$0.953782\pi$
$192$		0			0
$193$	1.69107e10	+	2.92902e10i	0.877310	+	1.51955i	0.854281	+	0.519811i	$0.173998\pi$
$193$	1.69107e10	+	2.92902e10i	0.877310	+	1.51955i	0.0230293	+	0.999735i	$0.492669\pi$
$194$		0			0
$195$	4.61129e8			0.0228384
$196$		0			0
$197$	4.67993e9			0.221381			0.110691	−	0.993855i	$-0.464694\pi$
$197$	4.67993e9			0.221381			0.110691	+	0.993855i	$0.464694\pi$
$198$		0			0
$199$	−1.15006e10	−	1.99197e10i	−0.519856	−	0.900417i	−0.999734	−	0.0230817i	$-0.992652\pi$
$199$	−1.15006e10	−	1.99197e10i	−0.519856	−	0.900417i	0.479877	−	0.877336i	$-0.340681\pi$
$200$		0			0
$201$	−3.67828e9	+	6.37097e9i	−0.158951	+	0.275311i
$202$		0			0
$203$	5.74508e9	−	1.10199e10i	0.237445	−	0.455456i
$204$		0			0
$205$	1.45416e9	−	2.51869e9i	0.0575070	−	0.0996051i
$206$		0			0
$207$	4.55725e8	+	7.89339e8i	0.0172519	+	0.0298811i
$208$		0			0
$209$	1.31935e10			0.478302
$210$		0			0
$211$	4.13924e10			1.43764			0.718820	−	0.695197i	$-0.244683\pi$
$211$	4.13924e10			1.43764			0.718820	+	0.695197i	$0.244683\pi$
$212$		0			0
$213$	−1.10574e10	−	1.91519e10i	−0.368081	−	0.637536i
$214$		0			0
$215$	1.92708e10	−	3.33780e10i	0.615073	−	1.06534i
$216$		0			0
$217$	−2.08691e9	+	8.98538e7i	−0.0638904	+	0.00275085i
$218$		0			0
$219$	8.78274e9	−	1.52121e10i	0.258007	−	0.446881i
$220$		0			0
$221$	−3.47665e8	−	6.02174e8i	−0.00980383	−	0.0169807i
$222$		0			0
$223$	−7.00667e10			−1.89732			−0.948659	−	0.316302i	$-0.897559\pi$
$223$	−7.00667e10			−1.89732			−0.948659	+	0.316302i	$0.897559\pi$
$224$		0			0
$225$	8.95594e9			0.232965
$226$		0			0
$227$	−7.06267e9	−	1.22329e10i	−0.176544	−	0.305783i	0.764151	−	0.645038i	$-0.223158\pi$
$227$	−7.06267e9	−	1.22329e10i	−0.176544	−	0.305783i	−0.940694	+	0.339255i	$0.889825\pi$
$228$		0			0
$229$	6.77592e9	−	1.17362e10i	0.162820	−	0.282013i	−0.773059	−	0.634334i	$-0.781274\pi$
$229$	6.77592e9	−	1.17362e10i	0.162820	−	0.282013i	0.935879	+	0.352321i	$0.114608\pi$
$230$		0			0
$231$	−2.00069e10	−	3.14464e10i	−0.462302	−	0.726636i
$232$		0			0
$233$	1.13083e10	−	1.95865e10i	0.251359	−	0.435367i	−0.712541	−	0.701630i	$-0.752456\pi$
$233$	1.13083e10	−	1.95865e10i	0.251359	−	0.435367i	0.963900	+	0.266264i	$0.0857892\pi$
$234$		0			0
$235$	−4.07600e10	−	7.05984e10i	−0.871824	−	1.51004i
$236$		0			0
$237$	3.96373e8			0.00816087
$238$		0			0
$239$	−4.01943e9			−0.0796845			−0.0398422	−	0.999206i	$-0.512686\pi$
$239$	−4.01943e9			−0.0796845			−0.0398422	+	0.999206i	$0.512686\pi$
$240$		0			0
$241$	2.51344e10	+	4.35340e10i	0.479945	+	0.831289i	0.999735	−	0.0230046i	$-0.00732324\pi$
$241$	2.51344e10	+	4.35340e10i	0.479945	+	0.831289i	−0.519790	+	0.854294i	$0.673990\pi$
$242$		0			0
$243$	−1.74339e9	+	3.01964e9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	7.32353e10	−	6.31812e9i	1.29859	−	0.112032i
$246$		0			0
$247$	2.84624e8	−	4.92983e8i	0.00486558	−	0.00842744i
$248$		0			0
$249$	−8.91863e9	−	1.54475e10i	−0.147028	−	0.254660i
$250$		0			0
$251$	−3.45717e10			−0.549780			−0.274890	−	0.961476i	$-0.588642\pi$
$251$	−3.45717e10			−0.549780			−0.274890	+	0.961476i	$0.588642\pi$
$252$		0			0
$253$	1.00626e10			0.154408
$254$		0			0
$255$	−1.64137e10	−	2.84293e10i	−0.243094	−	0.421051i
$256$		0			0
$257$	5.19650e10	−	9.00061e10i	0.743040	−	1.28698i	−0.208065	−	0.978115i	$-0.566717\pi$
$257$	5.19650e10	−	9.00061e10i	0.743040	−	1.28698i	0.951105	−	0.308868i	$-0.0999501\pi$
$258$		0			0
$259$	−5.42240e10	−	8.52282e10i	−0.748760	−	1.17689i
$260$		0			0
$261$	−6.41779e9	+	1.11159e10i	−0.0856058	+	0.148274i
$262$		0			0
$263$	−5.89499e10	−	1.02104e11i	−0.759770	−	1.31596i	−0.942968	−	0.332885i	$-0.891978\pi$
$263$	−5.89499e10	−	1.02104e11i	−0.759770	−	1.31596i	0.183197	−	0.983076i	$-0.441355\pi$
$264$		0			0
$265$	2.92315e10			0.364120
$266$		0			0
$267$	5.26476e10			0.633983
$268$		0			0
$269$	4.58451e10	+	7.94061e10i	0.533836	+	0.924632i	0.999219	+	0.0395219i	$0.0125835\pi$
$269$	4.58451e10	+	7.94061e10i	0.533836	+	0.924632i	−0.465382	+	0.885110i	$0.654083\pi$
$270$		0			0
$271$	−8.35007e10	+	1.44628e11i	−0.940434	+	1.62888i	−0.175790	+	0.984428i	$0.556248\pi$
$271$	−8.35007e10	+	1.44628e11i	−0.940434	+	1.62888i	−0.764645	+	0.644452i	$0.777085\pi$
$272$		0			0
$273$	−1.60662e9	+	6.91743e7i	−0.0175058	+	0.000753725i
$274$		0			0
$275$	4.94379e10	−	8.56289e10i	0.521270	−	0.902866i
$276$		0			0
$277$	−4.20062e10	−	7.27568e10i	−0.428700	−	0.742531i	0.568058	−	0.822989i	$-0.307695\pi$
$277$	−4.20062e10	−	7.27568e10i	−0.428700	−	0.742531i	−0.996758	+	0.0804578i	$0.974362\pi$
$278$		0			0
$279$	2.15742e9			0.0213165
$280$		0			0
$281$	6.86050e10			0.656413			0.328206	−	0.944606i	$-0.393556\pi$
$281$	6.86050e10			0.656413			0.328206	+	0.944606i	$0.393556\pi$
$282$		0			0
$283$	−3.72199e10	−	6.44668e10i	−0.344934	−	0.597444i	0.640408	−	0.768035i	$-0.278765\pi$
$283$	−3.72199e10	−	6.44668e10i	−0.344934	−	0.597444i	−0.985342	+	0.170592i	$0.945432\pi$
$284$		0			0
$285$	1.34374e10	−	2.32743e10i	0.120646	−	0.208965i
$286$		0			0
$287$	−4.68863e9	+	8.99350e9i	−0.0407922	+	0.0782456i
$288$		0			0
$289$	3.45440e10	−	5.98320e10i	0.291294	−	0.504537i
$290$		0			0
$291$	3.04900e10	+	5.28102e10i	0.249252	+	0.431717i
$292$		0			0
$293$	−7.29219e10			−0.578035			−0.289017	−	0.957324i	$-0.593329\pi$
$293$	−7.29219e10			−0.578035			−0.289017	+	0.957324i	$0.593329\pi$
$294$		0			0
$295$	−1.64935e11			−1.26798
$296$		0			0
$297$	1.92475e10	+	3.33376e10i	0.143539	+	0.248617i
$298$		0			0
$299$	2.17081e8	−	3.75995e8i	0.00157073	−	0.00272058i
$300$		0			0
$301$	−6.21344e10	+	1.19183e11i	−0.436297	+	0.836885i
$302$		0			0
$303$	8.32494e10	−	1.44192e11i	0.567400	−	0.982766i
$304$		0			0
$305$	−4.99821e10	−	8.65715e10i	−0.330724	−	0.572830i
$306$		0			0
$307$	−6.78395e10			−0.435873			−0.217937	−	0.975963i	$-0.569933\pi$
$307$	−6.78395e10			−0.435873			−0.217937	+	0.975963i	$0.569933\pi$
$308$		0			0
$309$	1.01315e11			0.632210
$310$		0			0
$311$	4.11303e10	+	7.12397e10i	0.249310	+	0.431818i	0.963335	−	0.268303i	$-0.0864628\pi$
$311$	4.11303e10	+	7.12397e10i	0.249310	+	0.431818i	−0.714024	+	0.700121i	$0.753129\pi$
$312$		0			0
$313$	6.01370e10	−	1.04160e11i	0.354154	−	0.613413i	−0.632819	−	0.774300i	$-0.718102\pi$
$313$	6.01370e10	−	1.04160e11i	0.354154	−	0.613413i	0.986973	+	0.160887i	$0.0514354\pi$
$314$		0			0
$315$	−7.58503e10	+	3.26580e9i	−0.434070	+	0.0186892i
$316$		0			0
$317$	3.91974e10	−	6.78919e10i	0.218017	−	0.377617i	−0.736185	−	0.676781i	$-0.763374\pi$
$317$	3.91974e10	−	6.78919e10i	0.218017	−	0.377617i	0.954202	+	0.299164i	$0.0967078\pi$
$318$		0			0
$319$	7.08539e10	+	1.22723e11i	0.383094	+	0.663539i
$320$		0			0
$321$	1.95637e11			1.02844
$322$		0			0
$323$	−4.05242e10			−0.207159
$324$		0			0
$325$	−2.13305e9	−	3.69454e9i	−0.0106054	−	0.0183690i
$326$		0			0
$327$	7.27755e10	−	1.26051e11i	0.351982	−	0.609651i
$328$		0			0
$329$	1.52602e11	+	2.39857e11i	0.718092	+	1.12868i
$330$		0			0
$331$	5.35556e9	−	9.27611e9i	0.0245233	−	0.0424756i	−0.853503	−	0.521087i	$-0.825527\pi$
$331$	5.35556e9	−	9.27611e9i	0.0245233	−	0.0424756i	0.878027	+	0.478612i	$0.158860\pi$
$332$		0			0
$333$	5.21658e10	+	9.03539e10i	0.232481	+	0.402669i
$334$		0			0
$335$	1.65439e11			0.717690
$336$		0			0
$337$	3.56670e10			0.150637			0.0753186	−	0.997160i	$-0.476003\pi$
$337$	3.56670e10			0.150637			0.0753186	+	0.997160i	$0.476003\pi$
$338$		0			0
$339$	5.99717e10	+	1.03874e11i	0.246631	+	0.427177i
$340$		0			0
$341$	1.19092e10	−	2.06274e10i	0.0476968	−	0.0826133i
$342$		0			0
$343$	−2.54211e11	+	3.29991e10i	−0.991680	+	0.128730i
$344$		0			0
$345$	1.02486e10	−	1.77512e10i	0.0389475	−	0.0674591i
$346$		0			0
$347$	−7.09771e10	−	1.22936e11i	−0.262806	−	0.455193i	0.704180	−	0.710021i	$-0.251315\pi$
$347$	−7.09771e10	−	1.22936e11i	−0.262806	−	0.455193i	−0.966986	+	0.254828i	$0.917981\pi$
$348$		0			0
$349$	−5.88829e10			−0.212459			−0.106229	−	0.994342i	$-0.533878\pi$
$349$	−5.88829e10			−0.212459			−0.106229	+	0.994342i	$0.533878\pi$
$350$		0			0
$351$	1.66090e9			0.00584066
$352$		0			0
$353$	6.12541e10	+	1.06095e11i	0.209966	+	0.363672i	0.951704	−	0.307018i	$-0.0993313\pi$
$353$	6.12541e10	+	1.06095e11i	0.209966	+	0.363672i	−0.741738	+	0.670690i	$0.765998\pi$
$354$		0			0
$355$	−2.48665e11	+	4.30701e11i	−0.830975	+	1.43929i
$356$		0			0
$357$	6.14515e10	+	9.65883e10i	0.200228	+	0.314715i
$358$		0			0
$359$	2.71873e11	−	4.70897e11i	0.863854	−	1.49624i	−0.00432676	−	0.999991i	$-0.501377\pi$
$359$	2.71873e11	−	4.70897e11i	0.863854	−	1.49624i	0.868181	−	0.496248i	$-0.165289\pi$
$360$		0			0
$361$	1.44756e11	+	2.50724e11i	0.448594	+	0.776988i
$362$		0			0
$363$	2.33999e11			0.707351
$364$		0			0
$365$	−3.95024e11			−1.16494
$366$		0			0
$367$	−1.37246e11	−	2.37717e11i	−0.394914	−	0.684011i	0.598176	−	0.801365i	$-0.295892\pi$
$367$	−1.37246e11	−	2.37717e11i	−0.394914	−	0.684011i	−0.993090	+	0.117353i	$0.962559\pi$
$368$		0			0
$369$	5.23764e9	−	9.07185e9i	0.0147067	−	0.0254728i
$370$		0			0
$371$	−1.01846e11	+	4.38505e9i	−0.279100	+	0.0120169i
$372$		0			0
$373$	−1.59424e11	+	2.76131e11i	−0.426446	+	0.738626i	−0.996554	−	0.0829434i	$-0.973568\pi$
$373$	−1.59424e11	+	2.76131e11i	−0.426446	+	0.738626i	0.570108	+	0.821570i	$0.306901\pi$
$374$		0			0
$375$	4.33863e10	+	7.51473e10i	0.113295	+	0.196233i
$376$		0			0
$377$	6.11413e9			0.0155883
$378$		0			0
$379$	−4.60128e11			−1.14552			−0.572759	−	0.819724i	$-0.694127\pi$
$379$	−4.60128e11			−1.14552			−0.572759	+	0.819724i	$0.694127\pi$
$380$		0			0
$381$	−4.47167e10	−	7.74517e10i	−0.108720	−	0.188308i
$382$		0			0
$383$	−2.23991e11	+	3.87963e11i	−0.531907	+	0.921290i	0.467399	+	0.884046i	$0.345191\pi$
$383$	−2.23991e11	+	3.87963e11i	−0.531907	+	0.921290i	−0.999306	+	0.0372433i	$0.988142\pi$
$384$		0			0
$385$	−3.87478e11	+	7.43242e11i	−0.898822	+	1.72408i
$386$		0			0
$387$	6.94099e10	−	1.20222e11i	0.157298	−	0.272447i
$388$		0			0
$389$	−2.34281e11	−	4.05786e11i	−0.518756	−	0.898512i	−0.999762	−	0.0217947i	$-0.993062\pi$
$389$	−2.34281e11	−	4.05786e11i	−0.518756	−	0.898512i	0.481007	−	0.876717i	$-0.340271\pi$
$390$		0			0
$391$	−3.09076e10			−0.0668759
$392$		0			0
$393$	−2.43638e11			−0.515203
$394$		0			0
$395$	−4.45695e9	−	7.71966e9i	−0.00921192	−	0.0159555i
$396$		0			0
$397$	−2.42482e11	+	4.19991e11i	−0.489916	+	0.848559i	−0.999933	−	0.0116051i	$-0.996306\pi$
$397$	−2.42482e11	+	4.19991e11i	−0.489916	+	0.848559i	0.510017	+	0.860165i	$0.329639\pi$
$398$		0			0
$399$	−4.33259e10	+	8.31057e10i	−0.0855795	+	0.164154i
$400$		0			0
$401$	−1.91023e11	+	3.30861e11i	−0.368923	+	0.638993i	−0.989397	−	0.145234i	$-0.953607\pi$
$401$	−1.91023e11	+	3.30861e11i	−0.368923	+	0.638993i	0.620475	+	0.784226i	$0.286940\pi$
$402$		0			0
$403$	−5.13836e8	−	8.89990e8i	−0.000970402	−	0.00168079i
$404$		0			0
$405$	7.84130e10			0.144824
$406$		0			0
$407$	1.15185e12			2.08075
$408$		0			0
$409$	4.77735e11	+	8.27461e11i	0.844174	+	1.46215i	0.886337	+	0.463041i	$0.153242\pi$
$409$	4.77735e11	+	8.27461e11i	0.844174	+	1.46215i	−0.0421630	+	0.999111i	$0.513425\pi$
$410$		0			0
$411$	5.75449e10	−	9.96707e10i	0.0994761	−	0.172298i
$412$		0			0
$413$	5.74650e11	−	2.47421e10i	0.971916	−	0.0418466i
$414$		0			0
$415$	−2.00568e11	+	3.47394e11i	−0.331929	+	0.574917i
$416$		0			0
$417$	−5.78944e10	−	1.00276e11i	−0.0937615	−	0.162400i
$418$		0			0
$419$	−3.92894e11			−0.622747			−0.311374	−	0.950288i	$-0.600789\pi$
$419$	−3.92894e11			−0.622747			−0.311374	+	0.950288i	$0.600789\pi$
$420$		0			0
$421$	−1.00761e12			−1.56323			−0.781617	−	0.623759i	$-0.785605\pi$
$421$	−1.00761e12			−1.56323			−0.781617	+	0.623759i	$0.785605\pi$
$422$		0			0
$423$	−1.46810e11	−	2.54283e11i	−0.222959	−	0.386176i
$424$		0			0
$425$	−1.51850e11	+	2.63011e11i	−0.225768	+	0.391042i
$426$		0			0
$427$	1.87129e11	+	2.94126e11i	0.272406	+	0.428162i
$428$		0			0
$429$	9.16838e9	−	1.58801e10i	0.0130688	−	0.0226358i
$430$		0			0
$431$	6.90017e11	+	1.19514e12i	0.963190	+	1.66829i	0.714401	+	0.699736i	$0.246699\pi$
$431$	6.90017e11	+	1.19514e12i	0.963190	+	1.66829i	0.248789	+	0.968558i	$0.419968\pi$
$432$		0			0
$433$	−7.36177e11			−1.00644			−0.503219	−	0.864159i	$-0.667851\pi$
$433$	−7.36177e11			−1.00644			−0.503219	+	0.864159i	$0.667851\pi$
$434$		0			0
$435$	2.88655e11			0.386524
$436$		0			0
$437$	−1.26516e10	−	2.19132e10i	−0.0165951	−	0.0287435i
$438$		0			0
$439$	7.49786e10	−	1.29867e11i	0.0963489	−	0.166881i	−0.813822	−	0.581114i	$-0.802617\pi$
$439$	7.49786e10	−	1.29867e11i	0.0963489	−	0.166881i	0.910171	+	0.414233i	$0.135950\pi$
$440$		0			0
$441$	2.63780e11	−	2.27567e10i	0.332100	−	0.0286508i
$442$		0			0
$443$	−2.39546e11	+	4.14906e11i	−0.295510	+	0.511838i	−0.975103	−	0.221751i	$-0.928823\pi$
$443$	−2.39546e11	+	4.14906e11i	−0.295510	+	0.511838i	0.679594	+	0.733589i	$0.262156\pi$
$444$		0			0
$445$	−5.91986e11	−	1.02535e12i	−0.715635	−	1.23952i
$446$		0			0
$447$	1.14475e11			0.135621
$448$		0			0
$449$	1.41673e12			1.64505			0.822524	−	0.568731i	$-0.192565\pi$
$449$	1.41673e12			1.64505			0.822524	+	0.568731i	$0.192565\pi$
$450$		0			0
$451$	−5.78247e10	−	1.00155e11i	−0.0658142	−	0.113993i
$452$		0			0
$453$	4.30122e11	−	7.44993e11i	0.479899	−	0.831209i
$454$		0			0
$455$	1.94126e10	+	3.05123e10i	0.0212340	+	0.0333751i
$456$		0			0
$457$	3.62907e11	−	6.28573e11i	0.389200	−	0.674113i	−0.603143	−	0.797633i	$-0.706085\pi$
$457$	3.62907e11	−	6.28573e11i	0.389200	−	0.674113i	0.992342	+	0.123520i	$0.0394183\pi$
$458$		0			0
$459$	−5.91190e10	−	1.02397e11i	−0.0621685	−	0.107679i
$460$		0			0
$461$	1.82959e12			1.88668			0.943342	−	0.331821i	$-0.107663\pi$
$461$	1.82959e12			1.88668			0.943342	+	0.331821i	$0.107663\pi$
$462$		0			0
$463$	7.46579e10			0.0755025			0.0377513	−	0.999287i	$-0.487981\pi$
$463$	7.46579e10			0.0755025			0.0377513	+	0.999287i	$0.487981\pi$
$464$		0			0
$465$	−2.42588e10	−	4.20174e10i	−0.0240619	−	0.0416765i
$466$		0			0
$467$	−9.84259e11	+	1.70479e12i	−0.957599	+	1.65861i	−0.229293	+	0.973357i	$0.573641\pi$
$467$	−9.84259e11	+	1.70479e12i	−0.957599	+	1.65861i	−0.728306	+	0.685252i	$0.759692\pi$
$468$		0			0
$469$	−5.76407e11	+	2.48177e10i	−0.550112	+	0.0236855i
$470$		0			0
$471$	2.29020e11	−	3.96674e11i	0.214427	−	0.371398i
$472$		0			0
$473$	−7.66302e11	−	1.32727e12i	−0.703923	−	1.21923i
$474$		0			0
$475$	−2.48630e11			−0.224095
$476$		0			0
$477$	1.05287e11			0.0931195
$478$		0			0
$479$	8.72815e11	+	1.51176e12i	0.757552	+	1.31212i	0.944096	+	0.329672i	$0.106938\pi$
$479$	8.72815e11	+	1.51176e12i	0.757552	+	1.31212i	−0.186544	+	0.982447i	$0.559729\pi$
$480$		0			0
$481$	2.48488e10	−	4.30393e10i	0.0211667	−	0.0366617i
$482$		0			0
$483$	−3.30444e10	+	6.33842e10i	−0.0276271	+	0.0529931i
$484$		0			0
$485$	6.85678e11	−	1.18763e12i	0.562707	−	0.974637i
$486$		0			0
$487$	8.49649e11	+	1.47163e12i	0.684477	+	1.18555i	0.973601	+	0.228258i	$0.0733028\pi$
$487$	8.49649e11	+	1.47163e12i	0.684477	+	1.18555i	−0.289124	+	0.957292i	$0.593364\pi$
$488$		0			0
$489$	7.27094e10			0.0575043
$490$		0			0
$491$	2.19622e12			1.70533			0.852667	−	0.522455i	$-0.174984\pi$
$491$	2.19622e12			1.70533			0.852667	+	0.522455i	$0.174984\pi$
$492$		0			0
$493$	−2.17629e11	−	3.76945e11i	−0.165923	−	0.287387i
$494$		0			0
$495$	4.32849e11	−	7.49717e11i	0.324051	−	0.561273i
$496$		0			0
$497$	8.01766e11	−	1.53791e12i	0.589446	−	1.13065i
$498$		0			0
$499$	−6.57759e11	+	1.13927e12i	−0.474913	+	0.822574i	−0.999587	−	0.0287293i	$-0.990854\pi$
$499$	−6.57759e11	+	1.13927e12i	−0.474913	+	0.822574i	0.524674	+	0.851303i	$0.324187\pi$
$500$		0			0
$501$	7.77603e11	+	1.34685e12i	0.551427	+	0.955100i
$502$		0			0
$503$	−2.24668e12			−1.56490			−0.782448	−	0.622715i	$-0.786029\pi$
$503$	−2.24668e12			−1.56490			−0.782448	+	0.622715i	$0.786029\pi$
$504$		0			0
$505$	−3.74433e12			−2.56191
$506$		0			0
$507$	4.29087e11	+	7.43200e11i	0.288409	+	0.499539i
$508$		0			0
$509$	−3.77925e11	+	6.54585e11i	−0.249560	+	0.432251i	−0.963404	−	0.268054i	$-0.913619\pi$
$509$	−3.77925e11	+	6.54585e11i	−0.249560	+	0.432251i	0.713843	+	0.700305i	$0.246953\pi$
$510$		0			0
$511$	1.37630e12	−	5.92578e10i	0.892935	−	0.0384461i
$512$		0			0
$513$	4.83991e10	−	8.38297e10i	0.0308538	−	0.0534404i
$514$		0			0
$515$	−1.13922e12	−	1.97319e12i	−0.713634	−	1.23605i
$516$		0			0
$517$	−3.24164e12			−1.99552
$518$		0			0
$519$	−8.84302e11			−0.534992
$520$		0			0
$521$	−1.51947e12	−	2.63180e12i	−0.903487	−	1.56489i	−0.822936	−	0.568135i	$-0.807665\pi$
$521$	−1.51947e12	−	2.63180e12i	−0.903487	−	1.56489i	−0.0805513	−	0.996750i	$-0.525668\pi$
$522$		0			0
$523$	7.05213e11	−	1.22146e12i	0.412157	−	0.713877i	−0.582968	−	0.812495i	$-0.698109\pi$
$523$	7.05213e11	−	1.22146e12i	0.412157	−	0.713877i	0.995125	+	0.0986179i	$0.0314422\pi$
$524$		0			0
$525$	3.77027e11	+	5.92603e11i	0.216598	+	0.340445i
$526$		0			0
$527$	−3.65795e10	+	6.33575e10i	−0.0206581	+	0.0357808i
$528$		0			0
$529$	8.90927e11	+	1.54313e12i	0.494643	+	0.856746i
$530$		0			0
$531$	−5.94066e11			−0.324272
$532$		0			0
$533$	−4.98981e9			−0.00267801
$534$		0			0
$535$	−2.19981e12	−	3.81018e12i	−1.16089	−	2.01073i
$536$		0			0
$537$	2.22193e11	−	3.84850e11i	0.115304	−	0.199713i
$538$		0			0
$539$	1.23852e12	−	2.64766e12i	0.632052	−	1.35118i
$540$		0			0
$541$	1.22919e11	−	2.12901e11i	0.0616921	−	0.106854i	−0.833530	−	0.552475i	$-0.813684\pi$
$541$	1.22919e11	−	2.12901e11i	0.0616921	−	0.106854i	0.895222	+	0.445621i	$0.147017\pi$
$542$		0			0
$543$	7.90533e11	+	1.36924e12i	0.390230	+	0.675899i
$544$		0			0
$545$	−3.27325e12			−1.58926
$546$		0			0
$547$	2.73499e12			1.30621			0.653103	−	0.757269i	$-0.273467\pi$
$547$	2.73499e12			1.30621			0.653103	+	0.757269i	$0.273467\pi$
$548$		0			0
$549$	−1.80026e11	−	3.11815e11i	−0.0845787	−	0.146495i
$550$		0			0
$551$	1.78167e11	−	3.08595e11i	0.0823466	−	0.142628i
$552$		0			0
$553$	1.66865e10	+	2.62275e10i	0.00758755	+	0.0119260i
$554$		0			0
$555$	1.17314e12	−	2.03194e12i	0.524845	−	0.909058i
$556$		0			0
$557$	8.86607e10	+	1.53565e11i	0.0390286	+	0.0675995i	0.884880	−	0.465819i	$-0.154240\pi$
$557$	8.86607e10	+	1.53565e11i	0.0390286	+	0.0675995i	−0.845851	+	0.533419i	$0.820907\pi$
$558$		0			0
$559$	−6.61257e10			−0.0286429
$560$		0			0
$561$	−1.30538e12			−0.556420
$562$		0			0
$563$	−7.04115e11	−	1.21956e12i	−0.295363	−	0.511583i	0.679706	−	0.733484i	$-0.262107\pi$
$563$	−7.04115e11	−	1.21956e12i	−0.295363	−	0.511583i	−0.975069	+	0.221901i	$0.928774\pi$
$564$		0			0
$565$	1.34868e12	−	2.33598e12i	0.556790	−	0.964388i
$566$		0			0
$567$	−2.73199e11	+	1.17628e10i	−0.111008	+	0.00477955i
$568$		0			0
$569$	1.43395e12	−	2.48367e12i	0.573492	−	0.993317i	−0.422712	−	0.906264i	$-0.638922\pi$
$569$	1.43395e12	−	2.48367e12i	0.573492	−	0.993317i	0.996204	−	0.0870530i	$-0.0277450\pi$
$570$		0			0
$571$	1.23660e12	+	2.14185e12i	0.486817	+	0.843191i	0.999885	−	0.0151565i	$-0.00482463\pi$
$571$	1.23660e12	+	2.14185e12i	0.486817	+	0.843191i	−0.513068	+	0.858348i	$0.671491\pi$
$572$		0			0
$573$	1.84751e12			0.715963
$574$		0			0
$575$	−1.89629e11			−0.0723434
$576$		0			0
$577$	8.60958e11	+	1.49122e12i	0.323363	+	0.560082i	0.981180	−	0.193097i	$-0.0618531\pi$
$577$	8.60958e11	+	1.49122e12i	0.323363	+	0.560082i	−0.657816	+	0.753178i	$0.728520\pi$
$578$		0			0
$579$	1.36977e12	−	2.37250e12i	0.506515	−	0.877310i
$580$		0			0
$581$	6.46686e11	−	1.24044e12i	0.235451	−	0.451631i
$582$		0			0
$583$	5.81195e11	−	1.00666e12i	0.208359	−	0.360889i
$584$		0			0
$585$	−1.86757e10	−	3.23473e10i	−0.00659289	−	0.0114192i
$586$		0			0
$587$	1.30604e12			0.454030			0.227015	−	0.973891i	$-0.427103\pi$
$587$	1.30604e12			0.454030			0.227015	+	0.973891i	$0.427103\pi$
$588$		0			0
$589$	−5.98932e10			−0.0205050
$590$		0			0
$591$	−1.89537e11	−	3.28288e11i	−0.0639073	−	0.110691i
$592$		0			0
$593$	1.37328e11	−	2.37859e11i	0.0456050	−	0.0789902i	−0.842322	−	0.538975i	$-0.818812\pi$
$593$	1.37328e11	−	2.37859e11i	0.0456050	−	0.0789902i	0.887927	+	0.459985i	$0.152145\pi$
$594$		0			0
$595$	1.19015e12	−	2.28288e12i	0.389291	−	0.746719i
$596$		0			0
$597$	−9.31552e11	+	1.61350e12i	−0.300139	+	0.519856i
$598$		0			0
$599$	2.01232e12	+	3.48544e12i	0.638670	+	1.10621i	0.985725	+	0.168364i	$0.0538485\pi$
$599$	2.01232e12	+	3.48544e12i	0.638670	+	1.10621i	−0.347055	+	0.937845i	$0.612818\pi$
$600$		0			0
$601$	4.81731e12			1.50615			0.753076	−	0.657933i	$-0.228569\pi$
$601$	4.81731e12			1.50615			0.753076	+	0.657933i	$0.228569\pi$
$602$		0			0
$603$	5.95882e11			0.183541
$604$		0			0
$605$	−2.63116e12	−	4.55731e12i	−0.798452	−	1.38296i
$606$		0			0
$607$	1.18734e12	−	2.05653e12i	0.354998	−	0.614875i	−0.632120	−	0.774871i	$-0.717815\pi$
$607$	1.18734e12	−	2.05653e12i	0.354998	−	0.614875i	0.987118	+	0.159996i	$0.0511482\pi$
$608$		0			0
$609$	−1.00570e12	+	4.33014e10i	−0.296273	+	0.0127563i
$610$		0			0
$611$	−6.99318e10	+	1.21125e11i	−0.0202997	+	0.0351601i
$612$		0			0
$613$	8.70800e11	+	1.50827e12i	0.249084	+	0.431427i	0.963272	−	0.268527i	$-0.0865370\pi$
$613$	8.70800e11	+	1.50827e12i	0.249084	+	0.431427i	−0.714188	+	0.699954i	$0.753204\pi$
$614$		0			0
$615$	−2.35575e11			−0.0664034
$616$		0			0
$617$	−2.95601e12			−0.821150			−0.410575	−	0.911827i	$-0.634672\pi$
$617$	−2.95601e12			−0.821150			−0.410575	+	0.911827i	$0.634672\pi$
$618$		0			0
$619$	−2.67336e10	−	4.63039e10i	−0.00731895	−	0.0126768i	0.862343	−	0.506325i	$-0.168996\pi$
$619$	−2.67336e10	−	4.63039e10i	−0.00731895	−	0.0126768i	−0.869662	+	0.493648i	$0.835663\pi$
$620$		0			0
$621$	3.69137e10	−	6.39364e10i	0.00996037	−	0.0172519i
$622$		0			0
$623$	2.21635e12	+	3.48362e12i	0.589444	+	0.926476i
$624$		0			0
$625$	2.30873e12	−	3.99884e12i	0.605221	−	1.04827i
$626$		0			0
$627$	−5.34338e11	−	9.25500e11i	−0.138074	−	0.239151i
$628$		0			0
$629$	−3.53792e12			−0.901198
$630$		0			0
$631$	−6.94940e12			−1.74508			−0.872540	−	0.488543i	$-0.837529\pi$
$631$	−6.94940e12			−1.74508			−0.872540	+	0.488543i	$0.837529\pi$
$632$		0			0
$633$	−1.67639e12	−	2.90360e12i	−0.415011	−	0.718820i
$634$		0			0
$635$	−1.00562e12	+	1.74178e12i	−0.245443	+	0.425121i
$636$		0			0
$637$	−7.22125e10	−	1.03396e11i	−0.0173774	−	0.0248814i
$638$		0			0
$639$	−8.95648e11	+	1.55131e12i	−0.212512	+	0.368081i
$640$		0			0
$641$	−8.65234e11	−	1.49863e12i	−0.202429	−	0.350617i	0.746882	−	0.664957i	$-0.231550\pi$
$641$	−8.65234e11	−	1.49863e12i	−0.202429	−	0.350617i	−0.949311	+	0.314340i	$0.898217\pi$
$642$		0			0
$643$	−2.27237e12			−0.524240			−0.262120	−	0.965035i	$-0.584422\pi$
$643$	−2.27237e12			−0.524240			−0.262120	+	0.965035i	$0.584422\pi$
$644$		0			0
$645$	−3.12187e12			−0.710225
$646$		0			0
$647$	3.81659e12	+	6.61053e12i	0.856262	+	1.48309i	0.875469	+	0.483274i	$0.160552\pi$
$647$	3.81659e12	+	6.61053e12i	0.856262	+	1.48309i	−0.0192072	+	0.999816i	$0.506114\pi$
$648$		0			0
$649$	−3.27932e12	+	5.67994e12i	−0.725575	+	1.25673i
$650$		0			0
$651$	9.08231e10	+	1.42754e11i	0.0198190	+	0.0311511i
$652$		0			0
$653$	−1.72965e12	+	2.99584e12i	−0.372263	+	0.644778i	−0.989913	−	0.141675i	$-0.954751\pi$
$653$	−1.72965e12	+	2.99584e12i	−0.372263	+	0.644778i	0.617651	+	0.786453i	$0.288085\pi$
$654$		0			0
$655$	2.73954e12	+	4.74503e12i	0.581557	+	1.00729i
$656$		0			0
$657$	−1.42280e12			−0.297921
$658$		0			0
$659$	3.17292e10			0.00655353			0.00327676	−	0.999995i	$-0.498957\pi$
$659$	3.17292e10			0.00655353			0.00327676	+	0.999995i	$0.498957\pi$
$660$		0			0
$661$	−1.53990e12	−	2.66718e12i	−0.313751	−	0.543432i	0.665420	−	0.746469i	$-0.268252\pi$
$661$	−1.53990e12	−	2.66718e12i	−0.313751	−	0.543432i	−0.979171	+	0.203037i	$0.934919\pi$
$662$		0			0
$663$	−2.81609e10	+	4.87761e10i	−0.00566024	+	0.00980383i
$664$		0			0
$665$	2.10571e12	−	9.06633e10i	0.417544	−	0.0179777i
$666$		0			0
$667$	1.35887e11	−	2.35363e11i	0.0265835	−	0.0460440i
$668$		0			0
$669$	2.83770e12	+	4.91504e12i	0.547708	+	0.948659i
$670$		0			0
$671$	−3.97507e12			−0.756996
$672$		0			0
$673$	1.00311e13			1.88486			0.942430	−	0.334403i	$-0.108535\pi$
$673$	1.00311e13			1.88486			0.942430	+	0.334403i	$0.108535\pi$
$674$		0			0
$675$	−3.62716e11	−	6.28242e11i	−0.0672511	−	0.116482i
$676$		0			0
$677$	−1.68928e12	+	2.92592e12i	−0.309067	+	0.535320i	−0.978159	−	0.207860i	$-0.933350\pi$
$677$	−1.68928e12	+	2.92592e12i	−0.309067	+	0.535320i	0.669091	+	0.743180i	$0.266684\pi$
$678$		0			0
$679$	−2.21081e12	+	4.24068e12i	−0.399152	+	0.765634i
$680$		0			0
$681$	−5.72076e11	+	9.90865e11i	−0.101928	+	0.176544i
$682$		0			0
$683$	1.34947e11	+	2.33734e11i	0.0237284	+	0.0410988i	0.877646	−	0.479310i	$-0.159113\pi$
$683$	1.34947e11	+	2.33734e11i	0.0237284	+	0.0410988i	−0.853917	+	0.520409i	$0.825780\pi$
$684$		0			0
$685$	−2.58821e12			−0.449151
$686$		0			0
$687$	−1.09770e12			−0.188009
$688$		0			0
$689$	−2.50762e10	−	4.34333e10i	−0.00423912	−	0.00734237i
$690$		0			0
$691$	2.65504e12	−	4.59867e12i	0.443017	−	0.767328i	−0.554895	−	0.831921i	$-0.687241\pi$
$691$	2.65504e12	−	4.59867e12i	0.443017	−	0.767328i	0.997912	+	0.0645927i	$0.0205748\pi$
$692$		0			0
$693$	−1.39563e12	+	2.67702e12i	−0.229863	+	0.440913i
$694$		0			0
$695$	−1.30197e12	+	2.25507e12i	−0.211674	+	0.366631i
$696$		0			0
$697$	1.77610e11	+	3.07630e11i	0.0285049	+	0.0493720i
$698$		0			0
$699$	−1.83194e12			−0.290244
$700$		0			0
$701$	1.19884e13			1.87512			0.937562	−	0.347817i	$-0.113077\pi$
$701$	1.19884e13			1.87512			0.937562	+	0.347817i	$0.113077\pi$
$702$		0			0
$703$	−1.44820e12	−	2.50836e12i	−0.223630	−	0.387338i
$704$		0			0
$705$	−3.30156e12	+	5.71847e12i	−0.503348	+	0.871824i
$706$		0			0
$707$	1.30456e13	−	5.61691e11i	1.96371	−	0.0845493i
$708$		0			0
$709$	4.49200e12	−	7.78038e12i	0.667624	−	1.15636i	−0.310942	−	0.950429i	$-0.600645\pi$
$709$	4.49200e12	−	7.78038e12i	0.667624	−	1.15636i	0.978567	−	0.205930i	$-0.0660221\pi$
$710$		0			0
$711$	−1.60531e10	−	2.78048e10i	−0.00235584	−	0.00408044i
$712$		0			0
$713$	−4.56802e10			−0.00661950
$714$		0			0
$715$	−4.12369e11			−0.0590077
$716$		0			0
$717$	1.62787e11	+	2.81955e11i	0.0230029	+	0.0398422i
$718$		0			0
$719$	−4.45765e12	+	7.72087e12i	−0.622051	+	1.07742i	0.367052	+	0.930200i	$0.380367\pi$
$719$	−4.45765e12	+	7.72087e12i	−0.622051	+	1.07742i	−0.989103	+	0.147223i	$0.952966\pi$
$720$		0			0
$721$	4.26516e12	+	6.70389e12i	0.587796	+	0.923886i
$722$		0			0
$723$	2.03589e12	−	3.52626e12i	0.277096	−	0.479945i
$724$		0			0
$725$	−1.33523e12	−	2.31269e12i	−0.179488	−	0.310883i
$726$		0			0
$727$	2.84770e12			0.378085			0.189042	−	0.981969i	$-0.439462\pi$
$727$	2.84770e12			0.378085			0.189042	+	0.981969i	$0.439462\pi$
$728$		0			0
$729$	2.82430e11			0.0370370
$730$		0			0
$731$	2.35371e12	+	4.07675e12i	0.304878	+	0.528063i
$732$		0			0
$733$	−5.33570e12	+	9.24170e12i	−0.682690	+	1.18245i	0.291467	+	0.956581i	$0.405857\pi$
$733$	−5.33570e12	+	9.24170e12i	−0.682690	+	1.18245i	−0.974157	+	0.225872i	$0.927477\pi$
$734$		0			0
$735$	−3.40923e12	−	4.88143e12i	−0.430887	−	0.616956i
$736$		0			0
$737$	3.28934e12	−	5.69730e12i	0.410681	−	0.711321i
$738$		0			0
$739$	−6.57859e12	−	1.13945e13i	−0.811396	−	1.40538i	−0.911887	−	0.410441i	$-0.865375\pi$
$739$	−6.57859e12	−	1.13945e13i	−0.811396	−	1.40538i	0.100491	−	0.994938i	$-0.467959\pi$
$740$		0			0
$741$	−4.61091e10			−0.00561829
$742$		0			0
$743$	−1.64438e13			−1.97948			−0.989741	−	0.142874i	$-0.954366\pi$
$743$	−1.64438e13			−1.97948			−0.989741	+	0.142874i	$0.954366\pi$
$744$		0			0
$745$	−1.28719e12	−	2.22948e12i	−0.153087	−	0.265155i
$746$		0			0
$747$	−7.22409e11	+	1.25125e12i	−0.0848868	+	0.147028i
$748$		0			0
$749$	8.23592e12	+	1.29451e13i	0.956190	+	1.50292i
$750$		0			0
$751$	1.20956e12	−	2.09502e12i	0.138755	−	0.240330i	−0.788271	−	0.615329i	$-0.789023\pi$
$751$	1.20956e12	−	2.09502e12i	0.138755	−	0.240330i	0.927026	+	0.374998i	$0.122357\pi$
$752$		0			0
$753$	1.40015e12	+	2.42514e12i	0.158708	+	0.274890i
$754$		0			0
$755$	−1.93457e13			−2.16682
$756$		0			0
$757$	−8.14893e11			−0.0901922			−0.0450961	−	0.998983i	$-0.514359\pi$
$757$	−8.14893e11			−0.0901922			−0.0450961	+	0.998983i	$0.514359\pi$
$758$		0			0
$759$	−4.07536e11	−	7.05874e11i	−0.0445737	−	0.0772039i
$760$		0			0
$761$	−3.82612e12	+	6.62704e12i	−0.413550	+	0.716289i	−0.995275	−	0.0970961i	$-0.969045\pi$
$761$	−3.82612e12	+	6.62704e12i	−0.413550	+	0.716289i	0.581725	+	0.813385i	$0.302378\pi$
$762$		0			0
$763$	1.14043e13	−	4.91022e11i	1.21817	−	0.0524495i
$764$		0			0
$765$	−1.32951e12	+	2.30277e12i	−0.140350	+	0.243094i
$766$		0			0
$767$	1.41489e11	+	2.45067e11i	0.0147620	+	0.0255685i
$768$		0			0
$769$	1.44384e13			1.48885			0.744426	−	0.667705i	$-0.232723\pi$
$769$	1.44384e13			1.48885			0.744426	+	0.667705i	$0.232723\pi$
$770$		0			0
$771$	−8.41833e12			−0.857989
$772$		0			0
$773$	3.42001e11	+	5.92363e11i	0.0344524	+	0.0596733i	0.882737	−	0.469867i	$-0.155698\pi$
$773$	3.42001e11	+	5.92363e11i	0.0344524	+	0.0596733i	−0.848285	+	0.529540i	$0.822365\pi$
$774$		0			0
$775$	−2.24428e11	+	3.88720e11i	−0.0223470	+	0.0387061i
$776$		0			0
$777$	−3.78252e12	+	7.25545e12i	−0.372295	+	0.714118i
$778$		0			0
$779$	−1.45404e11	+	2.51848e11i	−0.0141468	+	0.0245030i
$780$		0			0
$781$	9.88816e12	+	1.71268e13i	0.951012	+	1.64720i
$782$		0			0
$783$	1.03968e12			0.0988491
$784$		0			0
$785$	−1.03007e13			−0.968172
$786$		0			0
$787$	−4.47625e12	−	7.75310e12i	−0.415938	−	0.720425i	0.579589	−	0.814909i	$-0.303213\pi$
$787$	−4.47625e12	−	7.75310e12i	−0.415938	−	0.720425i	−0.995526	+	0.0944839i	$0.969880\pi$
$788$		0			0
$789$	−4.77494e12	+	8.27044e12i	−0.438654	+	0.759770i
$790$		0			0
$791$	−4.34852e12	+	8.34113e12i	−0.394955	+	0.757583i
$792$		0			0
$793$	−8.57542e10	+	1.48531e11i	−0.00770062	+	0.0133379i
$794$		0			0
$795$	−1.18388e12	−	2.05053e12i	−0.105112	−	0.182060i
$796$		0			0
$797$	1.46759e12			0.128838			0.0644188	−	0.997923i	$-0.479481\pi$
$797$	1.46759e12			0.128838			0.0644188	+	0.997923i	$0.479481\pi$
$798$		0			0
$799$	9.95676e12			0.864286
$800$		0			0
$801$	−2.13223e12	−	3.69312e12i	−0.183015	−	0.316991i
$802$		0			0
$803$	−7.85404e12	+	1.36036e13i	−0.666612	+	1.15461i
$804$		0			0
$805$	1.60602e12	−	6.91484e10i	0.134793	−	0.00580364i
$806$		0			0
$807$	3.71346e12	−	6.43190e12i	0.308211	−	0.533836i
$808$		0			0
$809$	−3.31865e12	−	5.74808e12i	−0.272392	−	0.471796i	0.697082	−	0.716991i	$-0.254481\pi$
$809$	−3.31865e12	−	5.74808e12i	−0.272392	−	0.471796i	−0.969474	+	0.245195i	$0.921148\pi$
$810$		0			0
$811$	−9.44189e12			−0.766417			−0.383208	−	0.923662i	$-0.625181\pi$
$811$	−9.44189e12			−0.766417			−0.383208	+	0.923662i	$0.625181\pi$
$812$		0			0
$813$	1.35271e13			1.08592
$814$		0			0
$815$	−8.17568e11	−	1.41607e12i	−0.0649104	−	0.112428i
$816$		0			0
$817$	−1.92692e12	+	3.33752e12i	−0.151309	+	0.262075i
$818$		0			0
$819$	6.99205e10	+	1.09900e11i	0.00543034	+	0.00853530i
$820$		0			0
$821$	9.64074e12	−	1.66982e13i	0.740570	−	1.28270i	−0.211666	−	0.977342i	$-0.567889\pi$
$821$	9.64074e12	−	1.66982e13i	0.740570	−	1.28270i	0.952236	−	0.305363i	$-0.0987777\pi$
$822$		0			0
$823$	7.29503e12	+	1.26354e13i	0.554278	+	0.960038i	0.997959	+	0.0638531i	$0.0203389\pi$
$823$	7.29503e12	+	1.26354e13i	0.554278	+	0.960038i	−0.443681	+	0.896185i	$0.646328\pi$
$824$		0			0
$825$	−8.00894e12			−0.601911
$826$		0			0
$827$	−1.44162e13			−1.07170			−0.535852	−	0.844312i	$-0.680009\pi$
$827$	−1.44162e13			−1.07170			−0.535852	+	0.844312i	$0.680009\pi$
$828$		0			0
$829$	4.33612e12	+	7.51039e12i	0.318865	+	0.552290i	0.980251	−	0.197755i	$-0.0633652\pi$
$829$	4.33612e12	+	7.51039e12i	0.318865	+	0.552290i	−0.661387	+	0.750045i	$0.730032\pi$
$830$		0			0
$831$	−3.40250e12	+	5.89330e12i	−0.247510	+	0.428700i
$832$		0			0
$833$	−3.80414e12	+	8.13233e12i	−0.273750	+	0.585211i
$834$		0			0
$835$	1.74872e13	−	3.02888e13i	1.24489	−	2.15622i
$836$		0			0
$837$	−8.73757e10	−	1.51339e11i	−0.00615355	−	0.0106583i
$838$		0			0
$839$	−1.14878e13			−0.800400			−0.400200	−	0.916428i	$-0.631059\pi$
$839$	−1.14878e13			−0.800400			−0.400200	+	0.916428i	$0.631059\pi$
$840$		0			0
$841$	−1.06799e13			−0.736179
$842$		0			0
$843$	−2.77850e12	−	4.81250e12i	−0.189490	−	0.328206i
$844$		0			0
$845$	9.64957e12	−	1.67136e13i	0.651108	−	1.12775i
$846$		0			0
$847$	9.85089e12	+	1.54834e13i	0.657658	+	1.03369i
$848$		0			0
$849$	−3.01481e12	+	5.22181e12i	−0.199148	+	0.344934i
$850$		0			0
$851$	−1.10453e12	−	1.91311e12i	−0.0721931	−	0.125042i
$852$		0			0
$853$	1.44699e13			0.935826			0.467913	−	0.883775i	$-0.345006\pi$
$853$	1.44699e13			0.935826			0.467913	+	0.883775i	$0.345006\pi$
$854$		0			0
$855$	−2.17686e12			−0.139310
$856$		0			0
$857$	−1.23491e13	−	2.13893e13i	−0.782028	−	1.35451i	−0.930759	−	0.365634i	$-0.880852\pi$
$857$	−1.23491e13	−	2.13893e13i	−0.782028	−	1.35451i	0.148731	−	0.988878i	$-0.452481\pi$
$858$		0			0
$859$	8.71941e12	−	1.51025e13i	0.546409	−	0.946408i	−0.452108	−	0.891963i	$-0.649328\pi$
$859$	8.71941e12	−	1.51025e13i	0.546409	−	0.946408i	0.998517	−	0.0544450i	$-0.0173390\pi$
$860$		0			0
$861$	8.20766e11	−	3.53388e10i	0.0508985	−	0.00219148i
$862$		0			0
$863$	−4.90638e12	+	8.49810e12i	−0.301101	+	0.521523i	−0.976386	−	0.216035i	$-0.930688\pi$
$863$	−4.90638e12	+	8.49810e12i	−0.301101	+	0.521523i	0.675284	+	0.737557i	$0.264021\pi$
$864$		0			0
$865$	9.94338e12	+	1.72224e13i	0.603895	+	1.04598i
$866$		0			0
$867$	−5.59613e12			−0.336358
$868$		0			0
$869$	−3.54460e11			−0.0210852
$870$		0			0
$871$	−1.41922e11	−	2.45816e11i	−0.00835540	−	0.0144720i
$872$		0			0
$873$	2.46969e12	−	4.27762e12i	0.143906	−	0.249252i
$874$		0			0
$875$	−3.14592e12	+	6.03436e12i	−0.181431	+	0.348012i
$876$		0			0
$877$	1.44067e13	−	2.49532e13i	0.822371	−	1.42439i	−0.0815411	−	0.996670i	$-0.525984\pi$
$877$	1.44067e13	−	2.49532e13i	0.822371	−	1.42439i	0.903912	−	0.427718i	$-0.140682\pi$
$878$		0			0
$879$	2.95334e12	+	5.11533e12i	0.166864	+	0.289017i
$880$		0			0
$881$	1.29785e13			0.725829			0.362915	−	0.931822i	$-0.381782\pi$
$881$	1.29785e13			0.725829			0.362915	+	0.931822i	$0.381782\pi$
$882$		0			0
$883$	2.40079e13			1.32902			0.664508	−	0.747281i	$-0.268641\pi$
$883$	2.40079e13			1.32902			0.664508	+	0.747281i	$0.268641\pi$
$884$		0			0
$885$	6.67987e12	+	1.15699e13i	0.366036	+	0.633992i
$886$		0			0
$887$	8.27083e12	−	1.43255e13i	0.448635	−	0.777058i	−0.549663	−	0.835387i	$-0.685244\pi$
$887$	8.27083e12	−	1.43255e13i	0.448635	−	0.777058i	0.998297	+	0.0583284i	$0.0185771\pi$
$888$		0			0
$889$	3.24239e12	−	6.21940e12i	0.174103	−	0.333957i
$890$		0			0
$891$	1.55904e12	−	2.70034e12i	0.0828722	−	0.143539i
$892$		0			0
$893$	4.07566e12	+	7.05925e12i	0.214470	+	0.371473i
$894$		0			0
$895$	−9.99364e12			−0.520619
$896$		0			0
$897$	−3.51671e10			−0.00181372
$898$		0			0
$899$	−3.21648e11	−	5.57111e11i	−0.0164234	−	0.0284461i
$900$		0			0
$901$	−1.78515e12	+	3.09197e12i	−0.0902430	+	0.156306i
$902$		0			0
$903$	1.08769e13	−	4.68314e11i	0.544391	−	0.0234392i
$904$		0			0
$905$	1.77780e13	−	3.07924e13i	0.880977	−	1.52590i
$906$		0			0
$907$	−1.75229e13	−	3.03506e13i	−0.859754	−	1.48914i	−0.872164	−	0.489214i	$-0.837284\pi$
$907$	−1.75229e13	−	3.03506e13i	−0.859754	−	1.48914i	0.0124102	−	0.999923i	$-0.496050\pi$
$908$		0			0
$909$	−1.34864e13			−0.655177
$910$		0			0
$911$	−7.91775e12			−0.380863			−0.190432	−	0.981700i	$-0.560989\pi$
$911$	−7.91775e12			−0.380863			−0.190432	+	0.981700i	$0.560989\pi$
$912$		0			0
$913$	7.97556e12	+	1.38141e13i	0.379877	+	0.657966i
$914$		0			0
$915$	−4.04855e12	+	7.01229e12i	−0.190943	+	0.330724i
$916$		0			0
$917$	−1.02567e13	−	1.61212e13i	−0.479009	−	0.752897i
$918$		0			0
$919$	9.65970e12	−	1.67311e13i	0.446728	−	0.773756i	−0.551442	−	0.834213i	$-0.685922\pi$
$919$	9.65970e12	−	1.67311e13i	0.446728	−	0.773756i	0.998171	+	0.0604566i	$0.0192557\pi$
$920$		0			0
$921$	2.74750e12	+	4.75881e12i	0.125826	+	0.217937i
$922$		0			0
$923$	8.53269e11			0.0386971
$924$		0			0
$925$	−2.17064e13			−0.974877
$926$		0			0
$927$	−4.10327e12	−	7.10707e12i	−0.182503	−	0.316105i
$928$		0			0
$929$	6.09269e12	−	1.05529e13i	0.268373	−	0.464835i	−0.700069	−	0.714075i	$-0.746847\pi$
$929$	6.09269e12	−	1.05529e13i	0.268373	−	0.464835i	0.968442	+	0.249240i	$0.0801808\pi$
$930$		0			0
$931$	−7.32292e12	+	6.31760e11i	−0.319456	+	0.0275600i
$932$		0			0
$933$	3.33155e12	−	5.77042e12i	0.143939	−	0.249310i
$934$		0			0
$935$	1.46781e13	+	2.54232e13i	0.628082	+	1.08787i
$936$		0			0
$937$	−2.59851e13			−1.10128			−0.550639	−	0.834744i	$-0.685616\pi$
$937$	−2.59851e13			−1.10128			−0.550639	+	0.834744i	$0.685616\pi$
$938$		0			0
$939$	−9.74220e12			−0.408942
$940$		0			0
$941$	−1.37133e12	−	2.37522e12i	−0.0570150	−	0.0987529i	0.836109	−	0.548563i	$-0.184825\pi$
$941$	−1.37133e12	−	2.37522e12i	−0.0570150	−	0.0987529i	−0.893124	+	0.449810i	$0.851492\pi$
$942$		0			0
$943$	−1.10899e11	+	1.92083e11i	−0.00456694	+	0.00791017i
$944$		0			0
$945$	3.30103e12	+	5.18849e12i	0.134650	+	0.211640i
$946$		0			0
$947$	1.45167e13	−	2.51437e13i	0.586534	−	1.01591i	−0.408149	−	0.912916i	$-0.633825\pi$
$947$	1.45167e13	−	2.51437e13i	0.586534	−	1.01591i	0.994682	−	0.102991i	$-0.0328412\pi$
$948$		0			0
$949$	3.38870e11	+	5.86941e11i	0.0135624	+	0.0234907i
$950$		0			0
$951$	−6.34998e12			−0.251744
$952$		0			0
$953$	−6.42103e12			−0.252166			−0.126083	−	0.992020i	$-0.540241\pi$
$953$	−6.42103e12			−0.252166			−0.126083	+	0.992020i	$0.540241\pi$
$954$		0			0
$955$	−2.07740e13	−	3.59816e13i	−0.808173	−	1.39980i
$956$		0			0
$957$	5.73917e12	−	9.94053e12i	0.221180	−	0.383094i
$958$		0			0
$959$	9.01760e12	−	3.88260e11i	0.344276	−	0.0148231i
$960$		0			0
$961$	1.31657e13	−	2.28037e13i	0.497955	−	0.862484i
$962$		0			0
$963$	−7.92331e12	−	1.37236e13i	−0.296885	−	0.514220i
$964$		0			0
$965$	−6.16083e13			−2.28700
$966$		0			0
$967$	3.50370e13			1.28857			0.644284	−	0.764786i	$-0.277155\pi$
$967$	3.50370e13			1.28857			0.644284	+	0.764786i	$0.277155\pi$
$968$		0			0
$969$	1.64123e12	+	2.84269e12i	0.0598015	+	0.103579i
$970$		0			0
$971$	−9.50504e11	+	1.64632e12i	−0.0343137	+	0.0594331i	−0.882672	−	0.469989i	$-0.844258\pi$
$971$	−9.50504e11	+	1.64632e12i	−0.0343137	+	0.0594331i	0.848359	+	0.529422i	$0.177591\pi$
$972$		0			0
$973$	4.19790e12	−	8.05221e12i	0.150150	−	0.288010i
$974$		0			0
$975$	−1.72777e11	+	2.99258e11i	−0.00612300	+	0.0106054i
$976$		0			0
$977$	1.89596e13	+	3.28389e13i	0.665738	+	1.15309i	0.979085	+	0.203453i	$0.0652163\pi$
$977$	1.89596e13	+	3.28389e13i	0.665738	+	1.15309i	−0.313347	+	0.949639i	$0.601450\pi$
$978$		0			0
$979$	−4.70806e13			−1.63802
$980$		0			0
$981$	−1.17896e13			−0.406434
$982$		0			0
$983$	6.05046e12	+	1.04797e13i	0.206679	+	0.357979i	0.950667	−	0.310215i	$-0.100401\pi$
$983$	6.05046e12	+	1.04797e13i	0.206679	+	0.357979i	−0.743987	+	0.668194i	$0.767068\pi$
$984$		0			0
$985$	−4.26243e12	+	7.38275e12i	−0.144276	+	0.249893i
$986$		0			0
$987$	1.06451e13	−	2.04190e13i	0.357046	−	0.684868i
$988$		0			0
$989$	−1.46965e12	+	2.54551e12i	−0.0488462	+	0.0846041i
$990$		0			0
$991$	−9.12274e12	−	1.58011e13i	−0.300465	−	0.520421i	0.675776	−	0.737107i	$-0.263809\pi$
$991$	−9.12274e12	−	1.58011e13i	−0.300465	−	0.520421i	−0.976241	+	0.216686i	$0.930475\pi$
$992$		0			0
$993$	−8.67601e11			−0.0283171
$994$		0			0
$995$	4.18987e13			1.35518
$996$		0			0
$997$	−1.33125e13	−	2.30580e13i	−0.426710	−	0.739083i	0.569869	−	0.821736i	$-0.306994\pi$
$997$	−1.33125e13	−	2.30580e13i	−0.426710	−	0.739083i	−0.996578	+	0.0826527i	$0.973661\pi$
$998$		0			0
$999$	4.22543e12	−	7.31866e12i	0.134223	−	0.232481i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.10.q.a.121.2	yes	16
7.4	even	3	inner	168.10.q.a.25.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.10.q.a.25.2	✓	16	7.4	even	3	inner
168.10.q.a.121.2	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	168.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-40.5000 - 70.1481i) q^{3} +(-910.790 + 1577.53i) q^{5} +(2936.64 - 5632.92i) q^{7} +(-3280.50 + 5681.99i) q^{9} +O(q^{10})\)	\(q+(-40.5000 - 70.1481i) q^{3} +(-910.790 + 1577.53i) q^{5} +(2936.64 - 5632.92i) q^{7} +(-3280.50 + 5681.99i) q^{9} +(36217.5 + 62730.6i) q^{11} +3125.28 q^{13} +147548. q^{15} +(-111243. - 192678. i) q^{17} +(91071.5 - 157740. i) q^{19} +(-514072. + 22133.8i) q^{21} +(69459.7 - 120308. i) q^{23} +(-682514. - 1.18215e6i) q^{25} +531441. q^{27} +1.95635e6 q^{29} +(-164413. - 284771. i) q^{31} +(2.93362e6 - 5.08118e6i) q^{33} +(6.21146e6 + 9.76305e6i) q^{35} +(7.95090e6 - 1.37714e7i) q^{37} +(-126574. - 219232. i) q^{39} -1.59660e6 q^{41} -2.11583e7 q^{43} +(-5.97569e6 - 1.03502e7i) q^{45} +(-2.23762e7 + 3.87567e7i) q^{47} +(-2.31059e7 - 3.30837e7i) q^{49} +(-9.01067e6 + 1.56069e7i) q^{51} +(-8.02367e6 - 1.38974e7i) q^{53} -1.31946e8 q^{55} -1.47536e7 q^{57} +(4.52725e7 + 7.84143e7i) q^{59} +(-2.74389e7 + 4.75255e7i) q^{61} +(2.23726e7 + 3.51647e7i) q^{63} +(-2.84647e6 + 4.93024e6i) q^{65} +(-4.54109e7 - 7.86540e7i) q^{67} -1.12525e7 q^{69} +2.73022e8 q^{71} +(1.08429e8 + 1.87804e8i) q^{73} +(-5.52836e7 + 9.57540e7i) q^{75} +(4.59714e8 - 1.97934e7i) q^{77} +(-2.44675e6 + 4.23789e6i) q^{79} +(-2.15234e7 - 3.72796e7i) q^{81} +2.20213e8 q^{83} +4.05275e8 q^{85} +(-7.92320e7 - 1.37234e8i) q^{87} +(-3.24985e8 + 5.62890e8i) q^{89} +(9.17781e6 - 1.76044e7i) q^{91} +(-1.33174e7 + 2.30665e7i) q^{93} +(1.65894e8 + 2.87337e8i) q^{95} -7.52839e8 q^{97} -4.75246e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9}+O(q^{10}) \) 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9	\( 16 q - 648 q^{3} - 196 q^{5} - 168 q^{7} - 52488 q^{9} + 32460 q^{11} + 119048 q^{13} + 31752 q^{15} + 208352 q^{17} + 914588 q^{19} - 428652 q^{21} + 460920 q^{23} - 3040180 q^{25} + 8503056 q^{27} - 16376136 q^{29} - 944064 q^{31} + 2629260 q^{33} - 15546664 q^{35} - 9826516 q^{37} - 4821444 q^{39} + 11449216 q^{41} - 6933624 q^{43} - 1285956 q^{45} + 26549360 q^{47} + 83657504 q^{49} + 16876512 q^{51} - 15354476 q^{53} + 134121944 q^{55} - 148163256 q^{57} + 18404996 q^{59} - 260632792 q^{61} + 35823060 q^{63} + 191461840 q^{65} + 53879788 q^{67} - 74669040 q^{69} - 164207456 q^{71} + 248475540 q^{73} - 246254580 q^{75} + 670121788 q^{77} + 16631256 q^{79} - 344373768 q^{81} - 1138943272 q^{83} - 1690136272 q^{85} + 663233508 q^{87} + 236796360 q^{89} - 1455575212 q^{91} - 76469184 q^{93} + 182450488 q^{95} + 1339799464 q^{97} - 425940120 q^{99}+O(q^{100}) \) 16 * q - 648 * q^3 - 196 * q^5 - 168 * q^7 - 52488 * q^9 + 32460 * q^11 + 119048 * q^13 + 31752 * q^15 + 208352 * q^17 + 914588 * q^19 - 428652 * q^21 + 460920 * q^23 - 3040180 * q^25 + 8503056 * q^27 - 16376136 * q^29 - 944064 * q^31 + 2629260 * q^33 - 15546664 * q^35 - 9826516 * q^37 - 4821444 * q^39 + 11449216 * q^41 - 6933624 * q^43 - 1285956 * q^45 + 26549360 * q^47 + 83657504 * q^49 + 16876512 * q^51 - 15354476 * q^53 + 134121944 * q^55 - 148163256 * q^57 + 18404996 * q^59 - 260632792 * q^61 + 35823060 * q^63 + 191461840 * q^65 + 53879788 * q^67 - 74669040 * q^69 - 164207456 * q^71 + 248475540 * q^73 - 246254580 * q^75 + 670121788 * q^77 + 16631256 * q^79 - 344373768 * q^81 - 1138943272 * q^83 - 1690136272 * q^85 + 663233508 * q^87 + 236796360 * q^89 - 1455575212 * q^91 - 76469184 * q^93 + 182450488 * q^95 + 1339799464 * q^97 - 425940120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more