LMFDB - Embedded newform 20.7.f.a.13.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,7,Mod(13,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("20.13");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	20.f (of order $4$, 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:	$4.60108167240$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 450x^{3} + 23409x^{2} - 115668x + 285768 $ x^6 - 2x^5 + 2x^4 - 450x^3 + 23409x^2 - 115668*x + 285768
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{7}\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.3
Root			$7.73238 + 7.73238i$ of defining polynomial
Character	$\chi$	$=$	20.13
Dual form			20.7.f.a.17.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(35.6344 + 35.6344i) q^{3} +(-115.493 + 47.8149i) q^{5} +(53.9521 - 53.9521i) q^{7} +1810.62i q^{9} +O(q^{10})$	\(q+(35.6344 + 35.6344i) q^{3} +(-115.493 + 47.8149i) q^{5} +(53.9521 - 53.9521i) q^{7} +1810.62i q^{9} +1895.82 q^{11} +(-954.888 - 954.888i) q^{13} +(-5819.40 - 2411.69i) q^{15} +(3807.14 - 3807.14i) q^{17} -2616.92i q^{19} +3845.11 q^{21} +(1210.32 + 1210.32i) q^{23} +(11052.5 - 11044.6i) q^{25} +(-38543.0 + 38543.0i) q^{27} -26815.4i q^{29} +4635.89 q^{31} +(67556.4 + 67556.4i) q^{33} +(-3651.41 + 8810.83i) q^{35} +(-36098.6 + 36098.6i) q^{37} -68053.7i q^{39} +40561.4 q^{41} +(-74613.4 - 74613.4i) q^{43} +(-86574.7 - 209115. i) q^{45} +(-22549.0 + 22549.0i) q^{47} +111827. i q^{49} +271330. q^{51} +(67918.4 + 67918.4i) q^{53} +(-218955. + 90648.3i) q^{55} +(93252.6 - 93252.6i) q^{57} +35389.7i q^{59} -261549. q^{61} +(97687.0 + 97687.0i) q^{63} +(155941. + 64625.5i) q^{65} +(-301383. + 301383. i) q^{67} +86258.3i q^{69} +274465. q^{71} +(-48363.4 - 48363.4i) q^{73} +(787417. + 280.559i) q^{75} +(102283. - 102283. i) q^{77} -781918. i q^{79} -1.42697e6 q^{81} +(-432869. - 432869. i) q^{83} +(-257662. + 621737. i) q^{85} +(955551. - 955551. i) q^{87} -480927. i q^{89} -103036. q^{91} +(165197. + 165197. i) q^{93} +(125128. + 302238. i) q^{95} +(165268. - 165268. i) q^{97} +3.43261e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7}+O(q^{10}) $ 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7	$ 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7} + 2200 q^{11} + 858 q^{13} - 7768 q^{15} - 3278 q^{17} + 33176 q^{21} + 19984 q^{23} - 24174 q^{25} - 115528 q^{27} + 104976 q^{31} + 177320 q^{33} - 116072 q^{35} - 241554 q^{37} + 351736 q^{41} + 60720 q^{43} - 287846 q^{45} - 355248 q^{47} + 641872 q^{51} + 346526 q^{53} - 310200 q^{55} - 112816 q^{57} - 492888 q^{61} + 2288 q^{63} - 5082 q^{65} - 230304 q^{67} + 174128 q^{71} - 332442 q^{73} + 1855048 q^{75} + 1618760 q^{77} - 3085166 q^{81} - 2190936 q^{83} - 164934 q^{85} + 2614304 q^{87} - 2186976 q^{91} + 242072 q^{93} + 3484184 q^{95} + 3338406 q^{97}+O(q^{100}) $ 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7 + 2200 * q^11 + 858 * q^13 - 7768 * q^15 - 3278 * q^17 + 33176 * q^21 + 19984 * q^23 - 24174 * q^25 - 115528 * q^27 + 104976 * q^31 + 177320 * q^33 - 116072 * q^35 - 241554 * q^37 + 351736 * q^41 + 60720 * q^43 - 287846 * q^45 - 355248 * q^47 + 641872 * q^51 + 346526 * q^53 - 310200 * q^55 - 112816 * q^57 - 492888 * q^61 + 2288 * q^63 - 5082 * q^65 - 230304 * q^67 + 174128 * q^71 - 332442 * q^73 + 1855048 * q^75 + 1618760 * q^77 - 3085166 * q^81 - 2190936 * q^83 - 164934 * q^85 + 2614304 * q^87 - 2186976 * q^91 + 242072 * q^93 + 3484184 * q^95 + 3338406 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	35.6344	+	35.6344i	1.31979	+	1.31979i	0.913937	+	0.405856i	$0.133027\pi$
$3$	35.6344	+	35.6344i	1.31979	+	1.31979i	0.405856	+	0.913937i	$0.366973\pi$
$4$		0			0
$5$	−115.493	+	47.8149i	−0.923948	+	0.382519i
$5$	−115.493	+	47.8149i	−0.923948	+	0.382519i
$6$		0			0
$7$	53.9521	−	53.9521i	0.157295	−	0.157295i	−0.624072	−	0.781367i	$-0.714523\pi$
$7$	53.9521	−	53.9521i	0.157295	−	0.157295i	0.781367	+	0.624072i	$0.214523\pi$
$8$		0			0
$9$			1810.62i			2.48371i
$10$		0			0
$11$	1895.82			1.42436			0.712178	−	0.701999i	$-0.247709\pi$
$11$	1895.82			1.42436			0.712178	+	0.701999i	$0.247709\pi$
$12$		0			0
$13$	−954.888	−	954.888i	−0.434633	−	0.434633i	0.455568	−	0.890201i	$-0.349436\pi$
$13$	−954.888	−	954.888i	−0.434633	−	0.434633i	−0.890201	+	0.455568i	$0.849436\pi$
$14$		0			0
$15$	−5819.40	−	2411.69i	−1.72427	−	0.714574i
$16$		0			0
$17$	3807.14	−	3807.14i	0.774911	−	0.774911i	−0.204050	−	0.978960i	$-0.565410\pi$
$17$	3807.14	−	3807.14i	0.774911	−	0.774911i	0.978960	+	0.204050i	$0.0654105\pi$
$18$		0			0
$19$		−	2616.92i		−	0.381531i	−0.981636	−	0.190766i	$-0.938903\pi$
$19$		−	2616.92i		−	0.381531i	0.981636	−	0.190766i	$-0.0610971\pi$
$20$		0			0
$21$	3845.11			0.415193
$22$		0			0
$23$	1210.32	+	1210.32i	0.0994759	+	0.0994759i	0.755093	−	0.655617i	$-0.227592\pi$
$23$	1210.32	+	1210.32i	0.0994759	+	0.0994759i	−0.655617	+	0.755093i	$0.727592\pi$
$24$		0			0
$25$	11052.5	−	11044.6i	0.707359	−	0.706855i
$26$		0			0
$27$	−38543.0	+	38543.0i	−1.95819	+	1.95819i
$28$		0			0
$29$		−	26815.4i		−	1.09949i	−0.835333	−	0.549744i	$-0.814725\pi$
$29$		−	26815.4i		−	1.09949i	0.835333	−	0.549744i	$-0.185275\pi$
$30$		0			0
$31$	4635.89			0.155614			0.0778069	−	0.996968i	$-0.475208\pi$
$31$	4635.89			0.155614			0.0778069	+	0.996968i	$0.475208\pi$
$32$		0			0
$33$	67556.4	+	67556.4i	1.87986	+	1.87986i
$34$		0			0
$35$	−3651.41	+	8810.83i	−0.0851640	+	0.205500i
$36$		0			0
$37$	−36098.6	+	36098.6i	−0.712665	+	0.712665i	−0.967092	−	0.254427i	$-0.918113\pi$
$37$	−36098.6	+	36098.6i	−0.712665	+	0.712665i	0.254427	+	0.967092i	$0.418113\pi$
$38$		0			0
$39$		−	68053.7i		−	1.14725i
$40$		0			0
$41$	40561.4			0.588520			0.294260	−	0.955725i	$-0.404927\pi$
$41$	40561.4			0.588520			0.294260	+	0.955725i	$0.404927\pi$
$42$		0			0
$43$	−74613.4	−	74613.4i	−0.938451	−	0.938451i	0.0597614	−	0.998213i	$-0.480966\pi$
$43$	−74613.4	−	74613.4i	−0.938451	−	0.938451i	−0.998213	+	0.0597614i	$0.980966\pi$
$44$		0			0
$45$	−86574.7	−	209115.i	−0.950065	−	2.29482i
$46$		0			0
$47$	−22549.0	+	22549.0i	−0.217187	+	0.217187i	−0.807312	−	0.590125i	$-0.799078\pi$
$47$	−22549.0	+	22549.0i	−0.217187	+	0.217187i	0.590125	+	0.807312i	$0.299078\pi$
$48$		0			0
$49$			111827.i			0.950517i
$50$		0			0
$51$	271330.			2.04544
$52$		0			0
$53$	67918.4	+	67918.4i	0.456205	+	0.456205i	0.897407	−	0.441203i	$-0.145448\pi$
$53$	67918.4	+	67918.4i	0.456205	+	0.456205i	−0.441203	+	0.897407i	$0.645448\pi$
$54$		0			0
$55$	−218955.	+	90648.3i	−1.31603	+	0.544843i
$56$		0			0
$57$	93252.6	−	93252.6i	0.503543	−	0.503543i
$58$		0			0
$59$			35389.7i			0.172314i	0.996282	+	0.0861570i	$0.0274587\pi$
$59$			35389.7i			0.172314i	−0.996282	+	0.0861570i	$0.972541\pi$
$60$		0			0
$61$	−261549.			−1.15230			−0.576148	−	0.817345i	$-0.695445\pi$
$61$	−261549.			−1.15230			−0.576148	+	0.817345i	$0.695445\pi$
$62$		0			0
$63$	97687.0	+	97687.0i	0.390674	+	0.390674i
$64$		0			0
$65$	155941.	+	64625.5i	0.567833	+	0.235323i
$66$		0			0
$67$	−301383.	+	301383.i	−1.00206	+	1.00206i	−0.00206356	+	0.999998i	$0.500657\pi$
$67$	−301383.	+	301383.i	−1.00206	+	1.00206i	−0.999998	+	0.00206356i	$0.999343\pi$
$68$		0			0
$69$			86258.3i			0.262575i
$70$		0			0
$71$	274465.			0.766852			0.383426	−	0.923572i	$-0.374744\pi$
$71$	274465.			0.766852			0.383426	+	0.923572i	$0.374744\pi$
$72$		0			0
$73$	−48363.4	−	48363.4i	−0.124322	−	0.124322i	0.642208	−	0.766530i	$-0.278018\pi$
$73$	−48363.4	−	48363.4i	−0.124322	−	0.124322i	−0.766530	+	0.642208i	$0.778018\pi$
$74$		0			0
$75$	787417.	+	280.559i	1.86647	+	0.000665030i
$76$		0			0
$77$	102283.	−	102283.i	0.224044	−	0.224044i
$78$		0			0
$79$		−	781918.i		−	1.58592i	−0.609276	−	0.792958i	$-0.708540\pi$
$79$		−	781918.i		−	1.58592i	0.609276	−	0.792958i	$-0.291460\pi$
$80$		0			0
$81$	−1.42697e6			−2.68510
$82$		0			0
$83$	−432869.	−	432869.i	−0.757047	−	0.757047i	0.218737	−	0.975784i	$-0.429806\pi$
$83$	−432869.	−	432869.i	−0.757047	−	0.757047i	−0.975784	+	0.218737i	$0.929806\pi$
$84$		0			0
$85$	−257662.	+	621737.i	−0.419559	+	1.01239i
$86$		0			0
$87$	955551.	−	955551.i	1.45110	−	1.45110i
$88$		0			0
$89$		−	480927.i		−	0.682196i	−0.940028	−	0.341098i	$-0.889201\pi$
$89$		−	480927.i		−	0.682196i	0.940028	−	0.341098i	$-0.110799\pi$
$90$		0			0
$91$	−103036.			−0.136731
$92$		0			0
$93$	165197.	+	165197.i	0.205378	+	0.205378i
$94$		0			0
$95$	125128.	+	302238.i	0.145943	+	0.352515i
$96$		0			0
$97$	165268.	−	165268.i	0.181082	−	0.181082i	−0.610745	−	0.791827i	$-0.709130\pi$
$97$	165268.	−	165268.i	0.181082	−	0.181082i	0.791827	+	0.610745i	$0.209130\pi$
$98$		0			0
$99$			3.43261e6i			3.53769i
$100$		0			0
$101$	1.97736e6			1.91920			0.959602	−	0.281361i	$-0.0907859\pi$
$101$	1.97736e6			1.91920			0.959602	+	0.281361i	$0.0907859\pi$
$102$		0			0
$103$	−1.00960e6	−	1.00960e6i	−0.923928	−	0.923928i	0.0733759	−	0.997304i	$-0.476623\pi$
$103$	−1.00960e6	−	1.00960e6i	−0.923928	−	0.923928i	−0.997304	+	0.0733759i	$0.976623\pi$
$104$		0			0
$105$	−444085.	+	183853.i	−0.383617	+	0.158819i
$106$		0			0
$107$	153055.	−	153055.i	0.124938	−	0.124938i	−0.641873	−	0.766811i	$-0.721842\pi$
$107$	153055.	−	153055.i	0.124938	−	0.124938i	0.766811	+	0.641873i	$0.221842\pi$
$108$		0			0
$109$			732355.i			0.565512i	0.959192	+	0.282756i	$0.0912487\pi$
$109$			732355.i			0.565512i	−0.959192	+	0.282756i	$0.908751\pi$
$110$		0			0
$111$	−2.57270e6			−1.88114
$112$		0			0
$113$	−175540.	−	175540.i	−0.121658	−	0.121658i	0.643657	−	0.765314i	$-0.277416\pi$
$113$	−175540.	−	175540.i	−0.121658	−	0.121658i	−0.765314	+	0.643657i	$0.777416\pi$
$114$		0			0
$115$	−197656.	−	81913.0i	−0.129962	−	0.0538591i
$116$		0			0
$117$	1.72894e6	−	1.72894e6i	1.07950	−	1.07950i
$118$		0			0
$119$		−	410806.i		−	0.243779i
$120$		0			0
$121$	1.82257e6			1.02879
$122$		0			0
$123$	1.44538e6	+	1.44538e6i	0.776725	+	0.776725i
$124$		0			0
$125$	−748393.	+	1.80405e6i	−0.383177	+	0.923675i
$126$		0			0
$127$	−1.57305e6	+	1.57305e6i	−0.767947	+	0.767947i	−0.977745	−	0.209798i	$-0.932719\pi$
$127$	−1.57305e6	+	1.57305e6i	−0.767947	+	0.767947i	0.209798	+	0.977745i	$0.432719\pi$
$128$		0			0
$129$		−	5.31761e6i		−	2.47712i
$130$		0			0
$131$	−818974.			−0.364298			−0.182149	−	0.983271i	$-0.558305\pi$
$131$	−818974.			−0.364298			−0.182149	+	0.983271i	$0.558305\pi$
$132$		0			0
$133$	−141189.	−	141189.i	−0.0600129	−	0.0600129i
$134$		0			0
$135$	2.60854e6	−	6.29439e6i	1.06022	−	2.55831i
$136$		0			0
$137$	−2.67122e6	+	2.67122e6i	−1.03884	+	1.03884i	−0.0396222	+	0.999215i	$0.512615\pi$
$137$	−2.67122e6	+	2.67122e6i	−1.03884	+	1.03884i	−0.999215	+	0.0396222i	$0.987385\pi$
$138$		0			0
$139$		−	1.75737e6i		−	0.654362i	−0.944962	−	0.327181i	$-0.893901\pi$
$139$		−	1.75737e6i		−	0.654362i	0.944962	−	0.327181i	$-0.106099\pi$
$140$		0			0
$141$	−1.60704e6			−0.573284
$142$		0			0
$143$	−1.81029e6	−	1.81029e6i	−0.619072	−	0.619072i
$144$		0			0
$145$	1.28218e6	+	3.09700e6i	0.420575	+	1.01587i
$146$		0			0
$147$	−3.98490e6	+	3.98490e6i	−1.25449	+	1.25449i
$148$		0			0
$149$			116045.i			0.0350808i	0.999846	+	0.0175404i	$0.00558357\pi$
$149$			116045.i			0.0350808i	−0.999846	+	0.0175404i	$0.994416\pi$
$150$		0			0
$151$	4.90253e6			1.42393			0.711966	−	0.702214i	$-0.247805\pi$
$151$	4.90253e6			1.42393			0.711966	+	0.702214i	$0.247805\pi$
$152$		0			0
$153$	6.89329e6	+	6.89329e6i	1.92465	+	1.92465i
$154$		0			0
$155$	−535415.	+	221665.i	−0.143779	+	0.0595252i
$156$		0			0
$157$	1.98980e6	−	1.98980e6i	0.514176	−	0.514176i	−0.401627	−	0.915803i	$-0.631555\pi$
$157$	1.98980e6	−	1.98980e6i	0.514176	−	0.514176i	0.915803	+	0.401627i	$0.131555\pi$
$158$		0			0
$159$			4.84046e6i			1.20419i
$160$		0			0
$161$	130599.			0.0312941
$162$		0			0
$163$	2.38346e6	+	2.38346e6i	0.550358	+	0.550358i	0.926544	−	0.376186i	$-0.122765\pi$
$163$	2.38346e6	+	2.38346e6i	0.550358	+	0.550358i	−0.376186	+	0.926544i	$0.622765\pi$
$164$		0			0
$165$	−1.10325e7	−	4.57212e6i	−2.45597	−	1.01781i
$166$		0			0
$167$	−1.31144e6	+	1.31144e6i	−0.281579	+	0.281579i	−0.833739	−	0.552159i	$-0.813804\pi$
$167$	−1.31144e6	+	1.31144e6i	−0.281579	+	0.281579i	0.552159	+	0.833739i	$0.313804\pi$
$168$		0			0
$169$		−	3.00319e6i		−	0.622189i
$170$		0			0
$171$	4.73826e6			0.947613
$172$		0			0
$173$	−3.84996e6	−	3.84996e6i	−0.743562	−	0.743562i	0.229699	−	0.973262i	$-0.426226\pi$
$173$	−3.84996e6	−	3.84996e6i	−0.743562	−	0.743562i	−0.973262	+	0.229699i	$0.926226\pi$
$174$		0			0
$175$	424.780	−	1.19219e6i	7.92592e−5	−	0.222449i
$176$		0			0
$177$	−1.26109e6	+	1.26109e6i	−0.227419	+	0.227419i
$178$		0			0
$179$			4.64255e6i			0.809464i	0.914435	+	0.404732i	$0.132635\pi$
$179$			4.64255e6i			0.809464i	−0.914435	+	0.404732i	$0.867365\pi$
$180$		0			0
$181$	2.75799e6			0.465112			0.232556	−	0.972583i	$-0.425291\pi$
$181$	2.75799e6			0.465112			0.232556	+	0.972583i	$0.425291\pi$
$182$		0			0
$183$	−9.32016e6	−	9.32016e6i	−1.52079	−	1.52079i
$184$		0			0
$185$	2.44310e6	−	5.89520e6i	0.385857	−	0.931072i
$186$		0			0
$187$	7.21764e6	−	7.21764e6i	1.10375	−	1.10375i
$188$		0			0
$189$			4.15895e6i			0.616026i
$190$		0			0
$191$	6.44180e6			0.924501			0.462251	−	0.886749i	$-0.347042\pi$
$191$	6.44180e6			0.924501			0.462251	+	0.886749i	$0.347042\pi$
$192$		0			0
$193$	4.82157e6	+	4.82157e6i	0.670682	+	0.670682i	0.957873	−	0.287191i	$-0.0927214\pi$
$193$	4.82157e6	+	4.82157e6i	0.670682	+	0.670682i	−0.287191	+	0.957873i	$0.592721\pi$
$194$		0			0
$195$	3.25398e6	+	7.85976e6i	0.438845	+	1.06000i
$196$		0			0
$197$	666959.	−	666959.i	0.0872369	−	0.0872369i	−0.662142	−	0.749379i	$-0.730352\pi$
$197$	666959.	−	666959.i	0.0872369	−	0.0872369i	0.749379	+	0.662142i	$0.230352\pi$
$198$		0			0
$199$		−	9.63418e6i		−	1.22252i	−0.791430	−	0.611259i	$-0.790663\pi$
$199$		−	9.63418e6i		−	1.22252i	0.791430	−	0.611259i	$-0.209337\pi$
$200$		0			0
$201$	−2.14792e7			−2.64503
$202$		0			0
$203$	−1.44675e6	−	1.44675e6i	−0.172944	−	0.172944i
$204$		0			0
$205$	−4.68458e6	+	1.93944e6i	−0.543762	+	0.225120i
$206$		0			0
$207$	−2.19144e6	+	2.19144e6i	−0.247069	+	0.247069i
$208$		0			0
$209$		−	4.96121e6i		−	0.543437i
$210$		0			0
$211$	−2.80218e6			−0.298296			−0.149148	−	0.988815i	$-0.547653\pi$
$211$	−2.80218e6			−0.298296			−0.149148	+	0.988815i	$0.547653\pi$
$212$		0			0
$213$	9.78039e6	+	9.78039e6i	1.01209	+	1.01209i
$214$		0			0
$215$	1.21850e7	+	5.04973e6i	1.22606	+	0.508105i
$216$		0			0
$217$	250116.	−	250116.i	0.0244773	−	0.0244773i
$218$		0			0
$219$		−	3.44680e6i		−	0.328159i
$220$		0			0
$221$	−7.27077e6			−0.673603
$222$		0			0
$223$	619198.	+	619198.i	0.0558361	+	0.0558361i	0.734473	−	0.678637i	$-0.237429\pi$
$223$	619198.	+	619198.i	0.0558361	+	0.0558361i	−0.678637	+	0.734473i	$0.737429\pi$
$224$		0			0
$225$	1.99976e7	+	2.00119e7i	1.75562	+	1.75687i
$226$		0			0
$227$	−1.18276e7	+	1.18276e7i	−1.01116	+	1.01116i	−0.0112211	+	0.999937i	$0.503572\pi$
$227$	−1.18276e7	+	1.18276e7i	−1.01116	+	1.01116i	−0.999937	+	0.0112211i	$0.996428\pi$
$228$		0			0
$229$			1.98645e7i			1.65414i	0.562100	+	0.827069i	$0.309993\pi$
$229$			1.98645e7i			1.65414i	−0.562100	+	0.827069i	$0.690007\pi$
$230$		0			0
$231$	7.28962e6			0.591383
$232$		0			0
$233$	−8.61338e6	−	8.61338e6i	−0.680935	−	0.680935i	0.279276	−	0.960211i	$-0.409906\pi$
$233$	−8.61338e6	−	8.61338e6i	−0.680935	−	0.680935i	−0.960211	+	0.279276i	$0.909906\pi$
$234$		0			0
$235$	1.52609e6	−	3.68244e6i	0.117591	−	0.283748i
$236$		0			0
$237$	2.78632e7	−	2.78632e7i	2.09308	−	2.09308i
$238$		0			0
$239$			1.42020e7i			1.04029i	0.854077	+	0.520146i	$0.174122\pi$
$239$			1.42020e7i			1.04029i	−0.854077	+	0.520146i	$0.825878\pi$
$240$		0			0
$241$	−1.76895e7			−1.26376			−0.631881	−	0.775065i	$-0.717717\pi$
$241$	−1.76895e7			−1.26376			−0.631881	+	0.775065i	$0.717717\pi$
$242$		0			0
$243$	−2.27514e7	−	2.27514e7i	−1.58558	−	1.58558i
$244$		0			0
$245$	−5.34701e6	−	1.29153e7i	−0.363591	−	0.878228i
$246$		0			0
$247$	−2.49887e6	+	2.49887e6i	−0.165826	+	0.165826i
$248$		0			0
$249$		−	3.08501e7i		−	1.99829i
$250$		0			0
$251$	−7.78236e6			−0.492142			−0.246071	−	0.969252i	$-0.579140\pi$
$251$	−7.78236e6			−0.492142			−0.246071	+	0.969252i	$0.579140\pi$
$252$		0			0
$253$	2.29455e6	+	2.29455e6i	0.141689	+	0.141689i
$254$		0			0
$255$	−3.13369e7	+	1.29736e7i	−1.88988	+	0.782421i
$256$		0			0
$257$	1.39276e7	−	1.39276e7i	0.820499	−	0.820499i	−0.165681	−	0.986179i	$-0.552982\pi$
$257$	1.39276e7	−	1.39276e7i	0.820499	−	0.820499i	0.986179	+	0.165681i	$0.0529821\pi$
$258$		0			0
$259$			3.89519e6i			0.224197i
$260$		0			0
$261$	4.85526e7			2.73081
$262$		0			0
$263$	1.52537e7	+	1.52537e7i	0.838508	+	0.838508i	0.988663	−	0.150154i	$-0.0479770\pi$
$263$	1.52537e7	+	1.52537e7i	0.838508	+	0.838508i	−0.150154	+	0.988663i	$0.547977\pi$
$264$		0			0
$265$	−1.10916e7	−	4.59662e6i	−0.596016	−	0.247002i
$266$		0			0
$267$	1.71375e7	−	1.71375e7i	0.900357	−	0.900357i
$268$		0			0
$269$			3.62826e7i			1.86398i	0.362481	+	0.931991i	$0.381930\pi$
$269$			3.62826e7i			1.86398i	−0.362481	+	0.931991i	$0.618070\pi$
$270$		0			0
$271$	−3.51878e7			−1.76801			−0.884003	−	0.467480i	$-0.845162\pi$
$271$	−3.51878e7			−1.76801			−0.884003	+	0.467480i	$0.845162\pi$
$272$		0			0
$273$	−3.67164e6	−	3.67164e6i	−0.180457	−	0.180457i
$274$		0			0
$275$	2.09535e7	−	2.09386e7i	1.00753	−	1.00681i
$276$		0			0
$277$	−2.06619e7	+	2.06619e7i	−0.972144	+	0.972144i	−0.999622	−	0.0274784i	$-0.991252\pi$
$277$	−2.06619e7	+	2.06619e7i	−0.972144	+	0.972144i	0.0274784	+	0.999622i	$0.491252\pi$
$278$		0			0
$279$			8.39385e6i			0.386499i
$280$		0			0
$281$	2.53204e7			1.14117			0.570587	−	0.821237i	$-0.306716\pi$
$281$	2.53204e7			1.14117			0.570587	+	0.821237i	$0.306716\pi$
$282$		0			0
$283$	−9.14358e6	−	9.14358e6i	−0.403420	−	0.403420i	0.476017	−	0.879436i	$-0.342080\pi$
$283$	−9.14358e6	−	9.14358e6i	−0.403420	−	0.403420i	−0.879436	+	0.476017i	$0.842080\pi$
$284$		0			0
$285$	−6.31120e6	+	1.52289e7i	−0.272632	+	0.657862i
$286$		0			0
$287$	2.18837e6	−	2.18837e6i	0.0925712	−	0.0925712i
$288$		0			0
$289$		−	4.85099e6i		−	0.200973i
$290$		0			0
$291$	1.17785e7			0.477981
$292$		0			0
$293$	−1.89469e7	−	1.89469e7i	−0.753244	−	0.753244i	0.221839	−	0.975083i	$-0.428794\pi$
$293$	−1.89469e7	−	1.89469e7i	−0.753244	−	0.753244i	−0.975083	+	0.221839i	$0.928794\pi$
$294$		0			0
$295$	−1.69215e6	−	4.08727e6i	−0.0659133	−	0.159209i
$296$		0			0
$297$	−7.30705e7	+	7.30705e7i	−2.78916	+	2.78916i
$298$		0			0
$299$		−	2.31145e6i		−	0.0864709i
$300$		0			0
$301$	−8.05111e6			−0.295227
$302$		0			0
$303$	7.04620e7	+	7.04620e7i	2.53295	+	2.53295i
$304$		0			0
$305$	3.02072e7	−	1.25059e7i	1.06466	−	0.440775i
$306$		0			0
$307$	1.88276e7	−	1.88276e7i	0.650699	−	0.650699i	−0.302462	−	0.953161i	$-0.597809\pi$
$307$	1.88276e7	−	1.88276e7i	0.650699	−	0.650699i	0.953161	+	0.302462i	$0.0978086\pi$
$308$		0			0
$309$		−	7.19531e7i		−	2.43879i
$310$		0			0
$311$	214842.			0.00714231			0.00357116	−	0.999994i	$-0.498863\pi$
$311$	214842.			0.00714231			0.00357116	+	0.999994i	$0.498863\pi$
$312$		0			0
$313$	3.74790e7	+	3.74790e7i	1.22223	+	1.22223i	0.966836	+	0.255399i	$0.0822067\pi$
$313$	3.74790e7	+	3.74790e7i	1.22223	+	1.22223i	0.255399	+	0.966836i	$0.417793\pi$
$314$		0			0
$315$	−1.59531e7	−	6.61132e6i	−0.510403	−	0.211522i
$316$		0			0
$317$	−1.23169e7	+	1.23169e7i	−0.386656	+	0.386656i	−0.873493	−	0.486837i	$-0.838151\pi$
$317$	−1.23169e7	+	1.23169e7i	−0.386656	+	0.386656i	0.486837	+	0.873493i	$0.338151\pi$
$318$		0			0
$319$		−	5.08372e7i		−	1.56606i
$320$		0			0
$321$	1.09080e7			0.329785
$322$		0			0
$323$	−9.96298e6	−	9.96298e6i	−0.295653	−	0.295653i
$324$		0			0
$325$	−2.11002e7	−	7518.09i	−0.614663	−	0.000219007i
$326$		0			0
$327$	−2.60970e7	+	2.60970e7i	−0.746359	+	0.746359i
$328$		0			0
$329$			2.43313e6i			0.0683248i
$330$		0			0
$331$	−3.46935e7			−0.956675			−0.478338	−	0.878176i	$-0.658760\pi$
$331$	−3.46935e7			−0.956675			−0.478338	+	0.878176i	$0.658760\pi$
$332$		0			0
$333$	−6.53609e7	−	6.53609e7i	−1.77005	−	1.77005i
$334$		0			0
$335$	2.03972e7	−	4.92184e7i	0.542545	−	1.30916i
$336$		0			0
$337$	3.50117e7	−	3.50117e7i	0.914794	−	0.914794i	−0.0818508	−	0.996645i	$-0.526083\pi$
$337$	3.50117e7	−	3.50117e7i	0.914794	−	0.914794i	0.996645	+	0.0818508i	$0.0260831\pi$
$338$		0			0
$339$		−	1.25105e7i		−	0.321126i
$340$		0			0
$341$	8.78881e6			0.221650
$342$		0			0
$343$	1.23807e7	+	1.23807e7i	0.306806	+	0.306806i
$344$		0			0
$345$	−4.12443e6	−	9.96227e6i	−0.100440	−	0.242606i
$346$		0			0
$347$	−2.37166e7	+	2.37166e7i	−0.567627	+	0.567627i	−0.931463	−	0.363836i	$-0.881467\pi$
$347$	−2.37166e7	+	2.37166e7i	−0.567627	+	0.567627i	0.363836	+	0.931463i	$0.381467\pi$
$348$		0			0
$349$			5.40891e7i			1.27243i	0.771513	+	0.636214i	$0.219500\pi$
$349$			5.40891e7i			1.27243i	−0.771513	+	0.636214i	$0.780500\pi$
$350$		0			0
$351$	7.36085e7			1.70218
$352$		0			0
$353$	−3.36765e7	−	3.36765e7i	−0.765601	−	0.765601i	0.211728	−	0.977329i	$-0.432091\pi$
$353$	−3.36765e7	−	3.36765e7i	−0.765601	−	0.765601i	−0.977329	+	0.211728i	$0.932091\pi$
$354$		0			0
$355$	−3.16989e7	+	1.31235e7i	−0.708531	+	0.293335i
$356$		0			0
$357$	1.46388e7	−	1.46388e7i	0.321738	−	0.321738i
$358$		0			0
$359$			6.68407e7i			1.44463i	0.691562	+	0.722317i	$0.256923\pi$
$359$			6.68407e7i			1.44463i	−0.691562	+	0.722317i	$0.743077\pi$
$360$		0			0
$361$	4.01976e7			0.854434
$362$		0			0
$363$	6.49461e7	+	6.49461e7i	1.35779	+	1.35779i
$364$		0			0
$365$	7.89815e6	+	3.27317e6i	0.162423	+	0.0673116i
$366$		0			0
$367$	1.83094e7	−	1.83094e7i	0.370404	−	0.370404i	−0.497220	−	0.867624i	$-0.665646\pi$
$367$	1.83094e7	−	1.83094e7i	0.370404	−	0.370404i	0.867624	+	0.497220i	$0.165646\pi$
$368$		0			0
$369$			7.34414e7i			1.46171i
$370$		0			0
$371$	7.32868e6			0.143517
$372$		0			0
$373$	3.36562e7	+	3.36562e7i	0.648542	+	0.648542i	0.952641	−	0.304099i	$-0.0983552\pi$
$373$	3.36562e7	+	3.36562e7i	0.648542	+	0.648542i	−0.304099	+	0.952641i	$0.598355\pi$
$374$		0			0
$375$	−9.09549e7	+	3.76178e7i	−1.72477	+	0.713345i
$376$		0			0
$377$	−2.56057e7	+	2.56057e7i	−0.477873	+	0.477873i
$378$		0			0
$379$		−	6.14439e7i		−	1.12866i	−0.825551	−	0.564328i	$-0.809135\pi$
$379$		−	6.14439e7i		−	1.12866i	0.825551	−	0.564328i	$-0.190865\pi$
$380$		0			0
$381$	−1.12109e8			−2.02706
$382$		0			0
$383$	−7.00364e7	−	7.00364e7i	−1.24660	−	1.24660i	−0.957211	−	0.289390i	$-0.906547\pi$
$383$	−7.00364e7	−	7.00364e7i	−1.24660	−	1.24660i	−0.289390	−	0.957211i	$-0.593453\pi$
$384$		0			0
$385$	−6.92240e6	+	1.67037e7i	−0.121304	+	0.292706i
$386$		0			0
$387$	1.35097e8	−	1.35097e8i	2.33084	−	2.33084i
$388$		0			0
$389$			5.13746e7i			0.872770i	0.899760	+	0.436385i	$0.143741\pi$
$389$			5.13746e7i			0.872770i	−0.899760	+	0.436385i	$0.856259\pi$
$390$		0			0
$391$	9.21573e6			0.154170
$392$		0			0
$393$	−2.91837e7	−	2.91837e7i	−0.480798	−	0.480798i
$394$		0			0
$395$	3.73873e7	+	9.03065e7i	0.606643	+	1.46530i
$396$		0			0
$397$	−1.95795e7	+	1.95795e7i	−0.312918	+	0.312918i	−0.846039	−	0.533121i	$-0.821019\pi$
$397$	−1.95795e7	+	1.95795e7i	−0.312918	+	0.312918i	0.533121	+	0.846039i	$0.321019\pi$
$398$		0			0
$399$		−	1.00624e7i		−	0.158409i
$400$		0			0
$401$	3.73749e7			0.579625			0.289813	−	0.957083i	$-0.406407\pi$
$401$	3.73749e7			0.579625			0.289813	+	0.957083i	$0.406407\pi$
$402$		0			0
$403$	−4.42676e6	−	4.42676e6i	−0.0676349	−	0.0676349i
$404$		0			0
$405$	1.64806e8	−	6.82304e7i	2.48089	−	1.02710i
$406$		0			0
$407$	−6.84364e7	+	6.84364e7i	−1.01509	+	1.01509i
$408$		0			0
$409$			1.69502e7i			0.247745i	0.992298	+	0.123873i	$0.0395314\pi$
$409$			1.69502e7i			0.247745i	−0.992298	+	0.123873i	$0.960469\pi$
$410$		0			0
$411$	−1.90374e8			−2.74210
$412$		0			0
$413$	1.90935e6	+	1.90935e6i	0.0271041	+	0.0271041i
$414$		0			0
$415$	7.06912e7	+	2.92960e7i	0.989056	+	0.409887i
$416$		0			0
$417$	6.26227e7	−	6.26227e7i	0.863622	−	0.863622i
$418$		0			0
$419$		−	8.39101e7i		−	1.14070i	−0.821401	−	0.570351i	$-0.806807\pi$
$419$		−	8.39101e7i		−	1.14070i	0.821401	−	0.570351i	$-0.193193\pi$
$420$		0			0
$421$	7.82429e7			1.04857			0.524286	−	0.851542i	$-0.324332\pi$
$421$	7.82429e7			1.04857			0.524286	+	0.851542i	$0.324332\pi$
$422$		0			0
$423$	−4.08277e7	−	4.08277e7i	−0.539429	−	0.539429i
$424$		0			0
$425$	29974.6	−	8.41266e7i	0.000390469	−	1.09589i
$426$		0			0
$427$	−1.41112e7	+	1.41112e7i	−0.181250	+	0.181250i
$428$		0			0
$429$		−	1.29018e8i		−	1.63409i
$430$		0			0
$431$	−3.26426e7			−0.407711			−0.203856	−	0.979001i	$-0.565347\pi$
$431$	−3.26426e7			−0.407711			−0.203856	+	0.979001i	$0.565347\pi$
$432$		0			0
$433$	−1.48344e7	−	1.48344e7i	−0.182728	−	0.182728i	0.609815	−	0.792544i	$-0.291244\pi$
$433$	−1.48344e7	−	1.48344e7i	−0.182728	−	0.182728i	−0.792544	+	0.609815i	$0.791244\pi$
$434$		0			0
$435$	−6.46704e7	+	1.56050e8i	−0.785666	+	1.89581i
$436$		0			0
$437$	3.16732e6	−	3.16732e6i	0.0379532	−	0.0379532i
$438$		0			0
$439$		−	1.17186e8i		−	1.38510i	−0.721371	−	0.692549i	$-0.756488\pi$
$439$		−	1.17186e8i		−	1.38510i	0.721371	−	0.692549i	$-0.243512\pi$
$440$		0			0
$441$	−2.02477e8			−2.36081
$442$		0			0
$443$	−4.20807e7	−	4.20807e7i	−0.484030	−	0.484030i	0.422386	−	0.906416i	$-0.361193\pi$
$443$	−4.20807e7	−	4.20807e7i	−0.484030	−	0.484030i	−0.906416	+	0.422386i	$0.861193\pi$
$444$		0			0
$445$	2.29954e7	+	5.55439e7i	0.260953	+	0.630313i
$446$		0			0
$447$	−4.13521e6	+	4.13521e6i	−0.0462994	+	0.0462994i
$448$		0			0
$449$			9.43574e7i			1.04241i	0.853433	+	0.521203i	$0.174517\pi$
$449$			9.43574e7i			1.04241i	−0.853433	+	0.521203i	$0.825483\pi$
$450$		0			0
$451$	7.68970e7			0.838262
$452$		0			0
$453$	1.74699e8	+	1.74699e8i	1.87930	+	1.87930i
$454$		0			0
$455$	1.19000e7	−	4.92667e6i	0.126332	−	0.0523022i
$456$		0			0
$457$	1.23071e8	−	1.23071e8i	1.28946	−	1.28946i	0.354347	−	0.935114i	$-0.384703\pi$
$457$	1.23071e8	−	1.23071e8i	1.28946	−	1.28946i	0.935114	−	0.354347i	$-0.115297\pi$
$458$		0			0
$459$			2.93477e8i			3.03484i
$460$		0			0
$461$	4.52473e7			0.461838			0.230919	−	0.972973i	$-0.425827\pi$
$461$	4.52473e7			0.461838			0.230919	+	0.972973i	$0.425827\pi$
$462$		0			0
$463$	3.94536e7	+	3.94536e7i	0.397506	+	0.397506i	0.877353	−	0.479846i	$-0.159308\pi$
$463$	3.94536e7	+	3.94536e7i	0.397506	+	0.397506i	−0.479846	+	0.877353i	$0.659308\pi$
$464$		0			0
$465$	−2.69781e7	−	1.11803e7i	−0.268320	−	0.111198i
$466$		0			0
$467$	−3.51660e7	+	3.51660e7i	−0.345281	+	0.345281i	−0.858348	−	0.513067i	$-0.828509\pi$
$467$	−3.51660e7	+	3.51660e7i	−0.345281	+	0.345281i	0.513067	+	0.858348i	$0.328509\pi$
$468$		0			0
$469$			3.25205e7i			0.315238i
$470$		0			0
$471$	1.41811e8			1.35721
$472$		0			0
$473$	−1.41454e8	−	1.41454e8i	−1.33669	−	1.33669i
$474$		0			0
$475$	−2.89029e7	−	2.89235e7i	−0.269687	−	0.269880i
$476$		0			0
$477$	−1.22975e8	+	1.22975e8i	−1.13308	+	1.13308i
$478$		0			0
$479$		−	4.02821e7i		−	0.366527i	−0.983064	−	0.183263i	$-0.941334\pi$
$479$		−	4.02821e7i		−	0.366527i	0.983064	−	0.183263i	$-0.0586661\pi$
$480$		0			0
$481$	6.89402e7			0.619494
$482$		0			0
$483$	4.65382e6	+	4.65382e6i	0.0413017	+	0.0413017i
$484$		0			0
$485$	−1.11851e7	+	2.69897e7i	−0.0980429	+	0.236577i
$486$		0			0
$487$	−2.30857e7	+	2.30857e7i	−0.199874	+	0.199874i	−0.799946	−	0.600072i	$-0.795139\pi$
$487$	−2.30857e7	+	2.30857e7i	−0.199874	+	0.199874i	0.600072	+	0.799946i	$0.295139\pi$
$488$		0			0
$489$			1.69867e8i			1.45272i
$490$		0			0
$491$	1.30781e8			1.10484			0.552420	−	0.833566i	$-0.313704\pi$
$491$	1.30781e8			1.10484			0.552420	+	0.833566i	$0.313704\pi$
$492$		0			0
$493$	−1.02090e8	−	1.02090e8i	−0.852005	−	0.852005i
$494$		0			0
$495$	−1.64130e8	−	3.96444e8i	−1.35323	−	3.26864i
$496$		0			0
$497$	1.48080e7	−	1.48080e7i	0.120622	−	0.120622i
$498$		0			0
$499$		−	5.56395e7i		−	0.447798i	−0.974612	−	0.223899i	$-0.928121\pi$
$499$		−	5.56395e7i		−	0.447798i	0.974612	−	0.223899i	$-0.0718785\pi$
$500$		0			0
$501$	−9.34651e7			−0.743252
$502$		0			0
$503$	5.26610e7	+	5.26610e7i	0.413795	+	0.413795i	0.883058	−	0.469263i	$-0.155480\pi$
$503$	5.26610e7	+	5.26610e7i	0.413795	+	0.413795i	−0.469263	+	0.883058i	$0.655480\pi$
$504$		0			0
$505$	−2.28372e8	+	9.45471e7i	−1.77324	+	0.734132i
$506$		0			0
$507$	1.07017e8	−	1.07017e8i	0.821161	−	0.821161i
$508$		0			0
$509$		−	1.12045e8i		−	0.849650i	−0.905276	−	0.424825i	$-0.860336\pi$
$509$		−	1.12045e8i		−	0.849650i	0.905276	−	0.424825i	$-0.139664\pi$
$510$		0			0
$511$	−5.21862e6			−0.0391105
$512$		0			0
$513$	1.00864e8	+	1.00864e8i	0.747110	+	0.747110i
$514$		0			0
$515$	1.64876e8	+	6.83284e7i	1.20708	+	0.500242i
$516$		0			0
$517$	−4.27488e7	+	4.27488e7i	−0.309352	+	0.309352i
$518$		0			0
$519$		−	2.74382e8i		−	1.96270i
$520$		0			0
$521$	2.03940e8			1.44208			0.721038	−	0.692895i	$-0.243665\pi$
$521$	2.03940e8			1.44208			0.721038	+	0.692895i	$0.243665\pi$
$522$		0			0
$523$	9.98000e7	+	9.98000e7i	0.697631	+	0.697631i	0.963899	−	0.266268i	$-0.0857907\pi$
$523$	9.98000e7	+	9.98000e7i	0.697631	+	0.697631i	−0.266268	+	0.963899i	$0.585791\pi$
$524$		0			0
$525$	4.24980e7	−	4.24677e7i	0.293691	−	0.293481i
$526$		0			0
$527$	1.76495e7	−	1.76495e7i	0.120587	−	0.120587i
$528$		0			0
$529$		−	1.45106e8i		−	0.980209i
$530$		0			0
$531$	−6.40773e7			−0.427977
$532$		0			0
$533$	−3.87316e7	−	3.87316e7i	−0.255790	−	0.255790i
$534$		0			0
$535$	−1.03585e7	+	2.49951e7i	−0.0676452	+	0.163228i
$536$		0			0
$537$	−1.65435e8	+	1.65435e8i	−1.06832	+	1.06832i
$538$		0			0
$539$			2.12004e8i			1.35387i
$540$		0			0
$541$	−2.21560e8			−1.39927			−0.699633	−	0.714502i	$-0.746653\pi$
$541$	−2.21560e8			−1.39927			−0.699633	+	0.714502i	$0.746653\pi$
$542$		0			0
$543$	9.82794e7	+	9.82794e7i	0.613851	+	0.613851i
$544$		0			0
$545$	−3.50174e7	−	8.45822e7i	−0.216319	−	0.522504i
$546$		0			0
$547$	−1.99416e8	+	1.99416e8i	−1.21842	+	1.21842i	−0.250240	+	0.968184i	$0.580510\pi$
$547$	−1.99416e8	+	1.99416e8i	−1.21842	+	1.21842i	−0.968184	+	0.250240i	$0.919490\pi$
$548$		0			0
$549$		−	4.73567e8i		−	2.86197i
$550$		0			0
$551$	−7.01739e7			−0.419489
$552$		0			0
$553$	−4.21862e7	−	4.21862e7i	−0.249456	−	0.249456i
$554$		0			0
$555$	2.97131e8	−	1.23014e8i	1.73807	−	0.719571i
$556$		0			0
$557$	−2.12978e8	+	2.12978e8i	−1.23245	+	1.23245i	−0.269433	+	0.963019i	$0.586836\pi$
$557$	−2.12978e8	+	2.12978e8i	−1.23245	+	1.23245i	−0.963019	+	0.269433i	$0.913164\pi$
$558$		0			0
$559$			1.42495e8i			0.815763i
$560$		0			0
$561$	5.14393e8			2.91344
$562$		0			0
$563$	−8.58027e7	−	8.58027e7i	−0.480812	−	0.480812i	0.424579	−	0.905391i	$-0.360422\pi$
$563$	−8.58027e7	−	8.58027e7i	−0.480812	−	0.480812i	−0.905391	+	0.424579i	$0.860422\pi$
$564$		0			0
$565$	2.86671e7	+	1.18803e7i	0.158942	+	0.0658690i
$566$		0			0
$567$	−7.69881e7	+	7.69881e7i	−0.422352	+	0.422352i
$568$		0			0
$569$			1.16983e8i			0.635019i	0.948255	+	0.317510i	$0.102847\pi$
$569$			1.16983e8i			0.635019i	−0.948255	+	0.317510i	$0.897153\pi$
$570$		0			0
$571$	−2.93057e8			−1.57414			−0.787070	−	0.616864i	$-0.788403\pi$
$571$	−2.93057e8			−1.57414			−0.787070	+	0.616864i	$0.788403\pi$
$572$		0			0
$573$	2.29550e8	+	2.29550e8i	1.22015	+	1.22015i
$574$		0			0
$575$	2.67446e7	+	9529.21i	0.140680	+	5.01249e-5i
$576$		0			0
$577$	4.74773e7	−	4.74773e7i	0.247149	−	0.247149i	−0.572651	−	0.819799i	$-0.694085\pi$
$577$	4.74773e7	−	4.74773e7i	0.247149	−	0.247149i	0.819799	+	0.572651i	$0.194085\pi$
$578$		0			0
$579$			3.43628e8i			1.77032i
$580$		0			0
$581$	−4.67085e7			−0.238159
$582$		0			0
$583$	1.28761e8	+	1.28761e8i	0.649798	+	0.649798i
$584$		0			0
$585$	−1.17012e8	+	2.82351e8i	−0.584473	+	1.41033i
$586$		0			0
$587$	1.77321e8	−	1.77321e8i	0.876690	−	0.876690i	−0.116501	−	0.993191i	$-0.537168\pi$
$587$	1.77321e8	−	1.77321e8i	0.876690	−	0.876690i	0.993191	+	0.116501i	$0.0371677\pi$
$588$		0			0
$589$		−	1.21318e7i		−	0.0593716i
$590$		0			0
$591$	4.75334e7			0.230269
$592$		0			0
$593$	2.39503e8	+	2.39503e8i	1.14854	+	1.14854i	0.986839	+	0.161705i	$0.0516992\pi$
$593$	2.39503e8	+	2.39503e8i	1.14854	+	1.14854i	0.161705	+	0.986839i	$0.448301\pi$
$594$		0			0
$595$	1.96426e7	+	4.74454e7i	0.0932500	+	0.225239i
$596$		0			0
$597$	3.43308e8	−	3.43308e8i	1.61347	−	1.61347i
$598$		0			0
$599$		−	3.12890e8i		−	1.45583i	−0.685665	−	0.727917i	$-0.740489\pi$
$599$		−	3.12890e8i		−	1.45583i	0.685665	−	0.727917i	$-0.259511\pi$
$600$		0			0
$601$	−1.78301e8			−0.821354			−0.410677	−	0.911781i	$-0.634708\pi$
$601$	−1.78301e8			−0.821354			−0.410677	+	0.911781i	$0.634708\pi$
$602$		0			0
$603$	−5.45691e8	−	5.45691e8i	−2.48883	−	2.48883i
$604$		0			0
$605$	−2.10495e8	+	8.71458e7i	−0.950549	+	0.393532i
$606$		0			0
$607$	−8.83561e7	+	8.83561e7i	−0.395067	+	0.395067i	−0.876489	−	0.481422i	$-0.840120\pi$
$607$	−8.83561e7	+	8.83561e7i	−0.395067	+	0.395067i	0.481422	+	0.876489i	$0.340120\pi$
$608$		0			0
$609$		−	1.03108e8i		−	0.456500i
$610$		0			0
$611$	4.30635e7			0.188793
$612$		0			0
$613$	−2.76511e8	−	2.76511e8i	−1.20041	−	1.20041i	−0.974041	−	0.226373i	$-0.927313\pi$
$613$	−2.76511e8	−	2.76511e8i	−1.20041	−	1.20041i	−0.226373	−	0.974041i	$-0.572687\pi$
$614$		0			0
$615$	−2.36043e8	−	9.78214e7i	−1.01476	−	0.420541i
$616$		0			0
$617$	2.19011e8	−	2.19011e8i	0.932420	−	0.932420i	−0.0654371	−	0.997857i	$-0.520844\pi$
$617$	2.19011e8	−	2.19011e8i	0.932420	−	0.932420i	0.997857	+	0.0654371i	$0.0208442\pi$
$618$		0			0
$619$			2.33037e8i			0.982546i	0.871006	+	0.491273i	$0.163468\pi$
$619$			2.33037e8i			0.982546i	−0.871006	+	0.491273i	$0.836532\pi$
$620$		0			0
$621$	−9.32990e7			−0.389585
$622$		0			0
$623$	−2.59470e7	−	2.59470e7i	−0.107306	−	0.107306i
$624$		0			0
$625$	173976.	−	2.44141e8i	0.000712607	−	1.00000i
$626$		0			0
$627$	1.76790e8	−	1.76790e8i	0.717224	−	0.717224i
$628$		0			0
$629$			2.74864e8i			1.10450i
$630$		0			0
$631$	3.76944e8			1.50034			0.750168	−	0.661247i	$-0.229972\pi$
$631$	3.76944e8			1.50034			0.750168	+	0.661247i	$0.229972\pi$
$632$		0			0
$633$	−9.98539e7	−	9.98539e7i	−0.393690	−	0.393690i
$634$		0			0
$635$	1.06462e8	−	2.56892e8i	0.415789	−	1.00330i
$636$		0			0
$637$	1.06783e8	−	1.06783e8i	0.413126	−	0.413126i
$638$		0			0
$639$			4.96952e8i			1.90464i
$640$		0			0
$641$	−1.78061e8			−0.676076			−0.338038	−	0.941132i	$-0.609763\pi$
$641$	−1.78061e8			−0.676076			−0.338038	+	0.941132i	$0.609763\pi$
$642$		0			0
$643$	−1.11035e8	−	1.11035e8i	−0.417663	−	0.417663i	0.466735	−	0.884397i	$-0.345430\pi$
$643$	−1.11035e8	−	1.11035e8i	−0.417663	−	0.417663i	−0.884397	+	0.466735i	$0.845430\pi$
$644$		0			0
$645$	2.54261e8	+	6.14150e8i	0.947546	+	2.28873i
$646$		0			0
$647$	−8.46874e7	+	8.46874e7i	−0.312684	+	0.312684i	−0.845949	−	0.533264i	$-0.820965\pi$
$647$	−8.46874e7	+	8.46874e7i	−0.312684	+	0.312684i	0.533264	+	0.845949i	$0.320965\pi$
$648$		0			0
$649$			6.70924e7i			0.245436i
$650$		0			0
$651$	1.78255e7			0.0646098
$652$		0			0
$653$	1.46524e8	+	1.46524e8i	0.526224	+	0.526224i	0.919444	−	0.393220i	$-0.128639\pi$
$653$	1.46524e8	+	1.46524e8i	0.526224	+	0.526224i	−0.393220	+	0.919444i	$0.628639\pi$
$654$		0			0
$655$	9.45862e7	−	3.91591e7i	0.336592	−	0.139351i
$656$		0			0
$657$	8.75679e7	−	8.75679e7i	0.308780	−	0.308780i
$658$		0			0
$659$		−	3.05403e8i		−	1.06713i	−0.845759	−	0.533565i	$-0.820852\pi$
$659$		−	3.05403e8i		−	1.06713i	0.845759	−	0.533565i	$-0.179148\pi$
$660$		0			0
$661$	1.97303e8			0.683172			0.341586	−	0.939851i	$-0.389036\pi$
$661$	1.97303e8			0.683172			0.341586	+	0.939851i	$0.389036\pi$
$662$		0			0
$663$	−2.59090e8	−	2.59090e8i	−0.889016	−	0.889016i
$664$		0			0
$665$	2.30573e7	+	9.55545e6i	0.0784049	+	0.0324927i
$666$		0			0
$667$	3.24553e7	−	3.24553e7i	0.109373	−	0.109373i
$668$		0			0
$669$			4.41295e7i			0.147384i
$670$		0			0
$671$	−4.95850e8			−1.64128
$672$		0			0
$673$	1.38405e8	+	1.38405e8i	0.454055	+	0.454055i	0.896698	−	0.442643i	$-0.145959\pi$
$673$	1.38405e8	+	1.38405e8i	0.454055	+	0.454055i	−0.442643	+	0.896698i	$0.645959\pi$
$674$		0			0
$675$	−303460.	+	8.51688e8i	−0.000986710	+	2.76929i
$676$		0			0
$677$	3.58335e8	−	3.58335e8i	1.15484	−	1.15484i	0.169274	−	0.985569i	$-0.445858\pi$
$677$	3.58335e8	−	3.58335e8i	1.15484	−	1.15484i	0.985569	−	0.169274i	$-0.0541424\pi$
$678$		0			0
$679$		−	1.78332e7i		−	0.0569665i
$680$		0			0
$681$	−8.42939e8			−2.66904
$682$		0			0
$683$	4.06177e8	+	4.06177e8i	1.27483	+	1.27483i	0.943522	+	0.331311i	$0.107491\pi$
$683$	4.06177e8	+	4.06177e8i	1.27483	+	1.27483i	0.331311	+	0.943522i	$0.392509\pi$
$684$		0			0
$685$	1.80784e8	−	4.36232e8i	0.562456	−	1.35721i
$686$		0			0
$687$	−7.07861e8	+	7.07861e8i	−2.18312	+	2.18312i
$688$		0			0
$689$		−	1.29709e8i		−	0.396563i
$690$		0			0
$691$	−3.30348e8			−1.00124			−0.500620	−	0.865667i	$-0.666894\pi$
$691$	−3.30348e8			−1.00124			−0.500620	+	0.865667i	$0.666894\pi$
$692$		0			0
$693$	1.85197e8	+	1.85197e8i	0.556460	+	0.556460i
$694$		0			0
$695$	8.40282e7	+	2.02964e8i	0.250306	+	0.604596i
$696$		0			0
$697$	1.54423e8	−	1.54423e8i	0.456050	−	0.456050i
$698$		0			0
$699$		−	6.13865e8i		−	1.79739i
$700$		0			0
$701$	9.85520e7			0.286096			0.143048	−	0.989716i	$-0.454310\pi$
$701$	9.85520e7			0.286096			0.143048	+	0.989716i	$0.454310\pi$
$702$		0			0
$703$	9.44673e7	+	9.44673e7i	0.271904	+	0.271904i
$704$		0			0
$705$	1.85603e8	−	7.68404e7i	0.529684	−	0.219292i
$706$		0			0
$707$	1.06683e8	−	1.06683e8i	0.301881	−	0.301881i
$708$		0			0
$709$			2.86500e8i			0.803870i	0.915668	+	0.401935i	$0.131662\pi$
$709$			2.86500e8i			0.803870i	−0.915668	+	0.401935i	$0.868338\pi$
$710$		0			0
$711$	1.41576e9			3.93895
$712$		0			0
$713$	5.61093e6	+	5.61093e6i	0.0154798	+	0.0154798i
$714$		0			0
$715$	2.95636e8	+	1.22518e8i	0.808797	+	0.335183i
$716$		0			0
$717$	−5.06079e8	+	5.06079e8i	−1.37297	+	1.37297i
$718$		0			0
$719$			1.68198e8i			0.452517i	0.974067	+	0.226259i	$0.0726494\pi$
$719$			1.68198e8i			0.452517i	−0.974067	+	0.226259i	$0.927351\pi$
$720$		0			0
$721$	−1.08940e8			−0.290658
$722$		0			0
$723$	−6.30356e8	−	6.30356e8i	−1.66790	−	1.66790i
$724$		0			0
$725$	−2.96166e8	−	2.96377e8i	−0.777178	−	0.777732i
$726$		0			0
$727$	1.47001e8	−	1.47001e8i	0.382575	−	0.382575i	−0.489454	−	0.872029i	$-0.662804\pi$
$727$	1.47001e8	−	1.47001e8i	0.382575	−	0.382575i	0.872029	+	0.489454i	$0.162804\pi$
$728$		0			0
$729$		−	5.81204e8i		−	1.50019i
$730$		0			0
$731$	−5.68127e8			−1.45443
$732$		0			0
$733$	2.20507e8	+	2.20507e8i	0.559899	+	0.559899i	0.929279	−	0.369379i	$-0.120430\pi$
$733$	2.20507e8	+	2.20507e8i	0.559899	+	0.559899i	−0.369379	+	0.929279i	$0.620430\pi$
$734$		0			0
$735$	2.69693e8	−	6.50768e8i	0.679215	−	1.63894i
$736$		0			0
$737$	−5.71367e8	+	5.71367e8i	−1.42729	+	1.42729i
$738$		0			0
$739$		−	2.00165e7i		−	0.0495970i	−0.999692	−	0.0247985i	$-0.992106\pi$
$739$		−	2.00165e7i		−	0.0495970i	0.999692	−	0.0247985i	$-0.00789442\pi$
$740$		0			0
$741$	−1.78091e8			−0.437712
$742$		0			0
$743$	−2.26439e8	−	2.26439e8i	−0.552057	−	0.552057i	0.374977	−	0.927034i	$-0.377651\pi$
$743$	−2.26439e8	−	2.26439e8i	−0.552057	−	0.552057i	−0.927034	+	0.374977i	$0.877651\pi$
$744$		0			0
$745$	−5.54869e6	−	1.34025e7i	−0.0134191	−	0.0324128i
$746$		0			0
$747$	7.83763e8	−	7.83763e8i	1.88028	−	1.88028i
$748$		0			0
$749$		−	1.65153e7i		−	0.0393043i
$750$		0			0
$751$	3.47339e8			0.820037			0.410019	−	0.912077i	$-0.365522\pi$
$751$	3.47339e8			0.820037			0.410019	+	0.912077i	$0.365522\pi$
$752$		0			0
$753$	−2.77320e8	−	2.77320e8i	−0.649526	−	0.649526i
$754$		0			0
$755$	−5.66210e8	+	2.34414e8i	−1.31564	+	0.544681i
$756$		0			0
$757$	1.19148e8	−	1.19148e8i	0.274661	−	0.274661i	−0.556312	−	0.830973i	$-0.687784\pi$
$757$	1.19148e8	−	1.19148e8i	0.274661	−	0.274661i	0.830973	+	0.556312i	$0.187784\pi$
$758$		0			0
$759$			1.63530e8i			0.374001i
$760$		0			0
$761$	−2.72659e8			−0.618680			−0.309340	−	0.950952i	$-0.600108\pi$
$761$	−2.72659e8			−0.618680			−0.309340	+	0.950952i	$0.600108\pi$
$762$		0			0
$763$	3.95121e7	+	3.95121e7i	0.0889522	+	0.0889522i
$764$		0			0
$765$	−1.12573e9	−	4.66528e8i	−2.51449	−	1.04206i
$766$		0			0
$767$	3.37932e7	−	3.37932e7i	0.0748932	−	0.0748932i
$768$		0			0
$769$			4.45104e7i			0.0978774i	0.998802	+	0.0489387i	$0.0155839\pi$
$769$			4.45104e7i			0.0978774i	−0.998802	+	0.0489387i	$0.984416\pi$
$770$		0			0
$771$	9.92606e8			2.16578
$772$		0			0
$773$	3.76356e8	+	3.76356e8i	0.814818	+	0.814818i	0.985352	−	0.170534i	$-0.0545491\pi$
$773$	3.76356e8	+	3.76356e8i	0.814818	+	0.814818i	−0.170534	+	0.985352i	$0.554549\pi$
$774$		0			0
$775$	5.12381e7	−	5.12016e7i	0.110075	−	0.109996i
$776$		0			0
$777$	−1.38803e8	+	1.38803e8i	−0.295894	+	0.295894i
$778$		0			0
$779$		−	1.06146e8i		−	0.224539i
$780$		0			0
$781$	5.20336e8			1.09227
$782$		0			0
$783$	1.03355e9	+	1.03355e9i	2.15300	+	2.15300i
$784$		0			0
$785$	−1.34667e8	+	3.24952e8i	−0.278390	+	0.671753i
$786$		0			0
$787$	−2.30940e8	+	2.30940e8i	−0.473778	+	0.473778i	−0.903135	−	0.429357i	$-0.858740\pi$
$787$	−2.30940e8	+	2.30940e8i	−0.473778	+	0.473778i	0.429357	+	0.903135i	$0.358740\pi$
$788$		0			0
$789$			1.08711e9i			2.21332i
$790$		0			0
$791$	−1.89415e7			−0.0382723
$792$		0			0
$793$	2.49750e8	+	2.49750e8i	0.500826	+	0.500826i
$794$		0			0
$795$	−2.31446e8	−	5.59042e8i	−0.460626	−	1.11261i
$796$		0			0
$797$	2.32522e8	−	2.32522e8i	0.459293	−	0.459293i	−0.439131	−	0.898423i	$-0.644713\pi$
$797$	2.32522e8	−	2.32522e8i	0.459293	−	0.459293i	0.898423	+	0.439131i	$0.144713\pi$
$798$		0			0
$799$			1.71694e8i			0.336601i
$800$		0			0
$801$	8.70777e8			1.69437
$802$		0			0
$803$	−9.16882e7	−	9.16882e7i	−0.177079	−	0.177079i
$804$		0			0
$805$	−1.50833e7	+	6.24458e6i	−0.0289141	+	0.0119706i
$806$		0			0
$807$	−1.29291e9	+	1.29291e9i	−2.46007	+	2.46007i
$808$		0			0
$809$			9.11846e7i			0.172217i	0.996286	+	0.0861085i	$0.0274432\pi$
$809$			9.11846e7i			0.172217i	−0.996286	+	0.0861085i	$0.972557\pi$
$810$		0			0
$811$	−3.05011e8			−0.571812			−0.285906	−	0.958258i	$-0.592294\pi$
$811$	−3.05011e8			−0.571812			−0.285906	+	0.958258i	$0.592294\pi$
$812$		0			0
$813$	−1.25390e9	−	1.25390e9i	−2.33340	−	2.33340i
$814$		0			0
$815$	−3.89239e8	−	1.61309e8i	−0.719024	−	0.297980i
$816$		0			0
$817$	−1.95258e8	+	1.95258e8i	−0.358049	+	0.358049i
$818$		0			0
$819$		−	1.86560e8i		−	0.339600i
$820$		0			0
$821$	4.77855e8			0.863508			0.431754	−	0.901991i	$-0.357895\pi$
$821$	4.77855e8			0.863508			0.431754	+	0.901991i	$0.357895\pi$
$822$		0			0
$823$	−2.56987e7	−	2.56987e7i	−0.0461011	−	0.0461011i	0.683680	−	0.729782i	$-0.260378\pi$
$823$	−2.56987e7	−	2.56987e7i	−0.0461011	−	0.0461011i	−0.729782	+	0.683680i	$0.760378\pi$
$824$		0			0
$825$	1.49280e9	+	531890.i	2.65852	+	0.000947240i
$826$		0			0
$827$	4.48704e8	−	4.48704e8i	0.793310	−	0.793310i	−0.188721	−	0.982031i	$-0.560434\pi$
$827$	4.48704e8	−	4.48704e8i	0.793310	−	0.793310i	0.982031	+	0.188721i	$0.0604340\pi$
$828$		0			0
$829$		−	6.67875e8i		−	1.17228i	−0.810209	−	0.586141i	$-0.800647\pi$
$829$		−	6.67875e8i		−	1.17228i	0.810209	−	0.586141i	$-0.199353\pi$
$830$		0			0
$831$	−1.47255e9			−2.56606
$832$		0			0
$833$	4.25742e8	+	4.25742e8i	0.736565	+	0.736565i
$834$		0			0
$835$	8.87567e7	−	2.14170e8i	0.152455	−	0.367874i
$836$		0			0
$837$	−1.78681e8	+	1.78681e8i	−0.304721	+	0.304721i
$838$		0			0
$839$		−	2.91494e8i		−	0.493564i	−0.969071	−	0.246782i	$-0.920627\pi$
$839$		−	2.91494e8i		−	0.493564i	0.969071	−	0.246782i	$-0.0793732\pi$
$840$		0			0
$841$	−1.24243e8			−0.208874
$842$		0			0
$843$	9.02279e8	+	9.02279e8i	1.50611	+	1.50611i
$844$		0			0
$845$	1.43597e8	+	3.46849e8i	0.237999	+	0.574870i
$846$		0			0
$847$	9.83314e7	−	9.83314e7i	0.161824	−	0.161824i
$848$		0			0
$849$		−	6.51652e8i		−	1.06486i
$850$		0			0
$851$	−8.73820e7			−0.141786
$852$		0			0
$853$	−8.06735e8	−	8.06735e8i	−1.29982	−	1.29982i	−0.928507	−	0.371314i	$-0.878907\pi$
$853$	−8.06735e8	−	8.06735e8i	−1.29982	−	1.29982i	−0.371314	−	0.928507i	$-0.621093\pi$
$854$		0			0
$855$	−5.47238e8	+	2.26559e8i	−0.875544	+	0.362480i
$856$		0			0
$857$	−4.48305e7	+	4.48305e7i	−0.0712248	+	0.0712248i	−0.741822	−	0.670597i	$-0.766038\pi$
$857$	−4.48305e7	+	4.48305e7i	−0.0712248	+	0.0712248i	0.670597	+	0.741822i	$0.266038\pi$
$858$		0			0
$859$			5.21914e8i			0.823417i	0.911316	+	0.411708i	$0.135068\pi$
$859$			5.21914e8i			0.823417i	−0.911316	+	0.411708i	$0.864932\pi$
$860$		0			0
$861$	1.55963e8			0.244350
$862$		0			0
$863$	6.04145e8	+	6.04145e8i	0.939959	+	0.939959i	0.998297	−	0.0583383i	$-0.0185802\pi$
$863$	6.04145e8	+	6.04145e8i	0.939959	+	0.939959i	−0.0583383	+	0.998297i	$0.518580\pi$
$864$		0			0
$865$	6.28730e8	+	2.60560e8i	0.971439	+	0.402586i
$866$		0			0
$867$	1.72862e8	−	1.72862e8i	0.265242	−	0.265242i
$868$		0			0
$869$		−	1.48238e9i		−	2.25891i
$870$		0			0
$871$	5.75574e8			0.871057
$872$		0			0
$873$	2.99239e8	+	2.99239e8i	0.449754	+	0.449754i
$874$		0			0
$875$	5.69551e7	+	1.37710e8i	0.0850175	+	0.205561i
$876$		0			0
$877$	−4.32194e7	+	4.32194e7i	−0.0640737	+	0.0640737i	−0.738418	−	0.674344i	$-0.764427\pi$
$877$	−4.32194e7	+	4.32194e7i	−0.0640737	+	0.0640737i	0.674344	+	0.738418i	$0.264427\pi$
$878$		0			0
$879$		−	1.35032e9i		−	1.98825i
$880$		0			0
$881$	−4.38003e8			−0.640545			−0.320273	−	0.947325i	$-0.603775\pi$
$881$	−4.38003e8			−0.640545			−0.320273	+	0.947325i	$0.603775\pi$
$882$		0			0
$883$	−6.70815e8	−	6.70815e8i	−0.974363	−	0.974363i	0.0253168	−	0.999679i	$-0.491941\pi$
$883$	−6.70815e8	−	6.70815e8i	−0.974363	−	0.974363i	−0.999679	+	0.0253168i	$0.991941\pi$
$884$		0			0
$885$	8.53488e7	−	2.05946e8i	0.123131	−	0.297115i
$886$		0			0
$887$	6.04117e8	−	6.04117e8i	0.865665	−	0.865665i	−0.126324	−	0.991989i	$-0.540318\pi$
$887$	6.04117e8	−	6.04117e8i	0.865665	−	0.865665i	0.991989	+	0.126324i	$0.0403178\pi$
$888$		0			0
$889$			1.69739e8i			0.241588i
$890$		0			0
$891$	−2.70528e9			−3.82453
$892$		0			0
$893$	5.90090e7	+	5.90090e7i	0.0828637	+	0.0828637i
$894$		0			0
$895$	−2.21983e8	−	5.36184e8i	−0.309635	−	0.747902i
$896$		0			0
$897$	8.23670e7	−	8.23670e7i	0.114124	−	0.114124i
$898$		0			0
$899$		−	1.24313e8i		−	0.171096i
$900$		0			0
$901$	5.17149e8			0.707036
$902$		0			0
$903$	−2.86897e8	−	2.86897e8i	−0.389639	−	0.389639i
$904$		0			0
$905$	−3.18530e8	+	1.31873e8i	−0.429739	+	0.177914i
$906$		0			0
$907$	−4.89905e8	+	4.89905e8i	−0.656583	+	0.656583i	−0.954570	−	0.297987i	$-0.903685\pi$
$907$	−4.89905e8	+	4.89905e8i	−0.656583	+	0.656583i	0.297987	+	0.954570i	$0.403685\pi$
$908$		0			0
$909$			3.58025e9i			4.76674i
$910$		0			0
$911$	3.10050e8			0.410087			0.205044	−	0.978753i	$-0.434266\pi$
$911$	3.10050e8			0.410087			0.205044	+	0.978753i	$0.434266\pi$
$912$		0			0
$913$	−8.20642e8	−	8.20642e8i	−1.07830	−	1.07830i
$914$		0			0
$915$	1.52206e9	+	6.30775e8i	1.98687	+	0.823401i
$916$		0			0
$917$	−4.41854e7	+	4.41854e7i	−0.0573022	+	0.0573022i
$918$		0			0
$919$			3.80950e8i			0.490818i	0.969420	+	0.245409i	$0.0789223\pi$
$919$			3.80950e8i			0.490818i	−0.969420	+	0.245409i	$0.921078\pi$
$920$		0			0
$921$	1.34182e9			1.71758
$922$		0			0
$923$	−2.62083e8	−	2.62083e8i	−0.333299	−	0.333299i
$924$		0			0
$925$	−284214.	+	7.97674e8i	−0.000359104	+	1.00786i
$926$		0			0
$927$	1.82801e9	−	1.82801e9i	2.29477	−	2.29477i
$928$		0			0
$929$		−	4.56924e8i		−	0.569897i	−0.958543	−	0.284949i	$-0.908023\pi$
$929$		−	4.56924e8i		−	0.569897i	0.958543	−	0.284949i	$-0.0919765\pi$
$930$		0			0
$931$	2.92644e8			0.362652
$932$		0			0
$933$	7.65578e6	+	7.65578e6i	0.00942637	+	0.00942637i
$934$		0			0
$935$	−4.88480e8	+	1.17870e9i	−0.597601	+	1.44201i
$936$		0			0
$937$	−8.80343e8	+	8.80343e8i	−1.07012	+	1.07012i	−0.0727732	+	0.997349i	$0.523185\pi$
$937$	−8.80343e8	+	8.80343e8i	−1.07012	+	1.07012i	−0.997349	+	0.0727732i	$0.976815\pi$
$938$		0			0
$939$			2.67108e9i			3.22619i
$940$		0			0
$941$	6.71533e8			0.805932			0.402966	−	0.915215i	$-0.367979\pi$
$941$	6.71533e8			0.805932			0.402966	+	0.915215i	$0.367979\pi$
$942$		0			0
$943$	4.90924e7	+	4.90924e7i	0.0585436	+	0.0585436i
$944$		0			0
$945$	−1.98860e8	−	4.80332e8i	−0.235641	−	0.569175i
$946$		0			0
$947$	4.25809e8	−	4.25809e8i	0.501377	−	0.501377i	−0.410489	−	0.911866i	$-0.634642\pi$
$947$	4.25809e8	−	4.25809e8i	0.501377	−	0.501377i	0.911866	+	0.410489i	$0.134642\pi$
$948$		0			0
$949$			9.23633e7i			0.108069i
$950$		0			0
$951$	−8.77814e8			−1.02061
$952$		0			0
$953$	1.76321e8	+	1.76321e8i	0.203716	+	0.203716i	0.801590	−	0.597874i	$-0.203988\pi$
$953$	1.76321e8	+	1.76321e8i	0.203716	+	0.203716i	−0.597874	+	0.801590i	$0.703988\pi$
$954$		0			0
$955$	−7.43986e8	+	3.08014e8i	−0.854191	+	0.353639i
$956$		0			0
$957$	1.81155e9	−	1.81155e9i	2.06688	−	2.06688i
$958$		0			0
$959$			2.88236e8i			0.326807i
$960$		0			0
$961$	−8.66012e8			−0.975784
$962$		0			0
$963$	2.77124e8	+	2.77124e8i	0.310310	+	0.310310i
$964$		0			0
$965$	−7.87403e8	−	3.26317e8i	−0.876224	−	0.363127i
$966$		0			0
$967$	−1.87686e8	+	1.87686e8i	−0.207564	+	0.207564i	−0.803231	−	0.595667i	$-0.796888\pi$
$967$	−1.87686e8	+	1.87686e8i	−0.207564	+	0.207564i	0.595667	+	0.803231i	$0.296888\pi$
$968$		0			0
$969$		−	7.10050e8i		−	0.780401i
$970$		0			0
$971$	6.70884e8			0.732807			0.366404	−	0.930456i	$-0.380589\pi$
$971$	6.70884e8			0.732807			0.366404	+	0.930456i	$0.380589\pi$
$972$		0			0
$973$	−9.48137e7	−	9.48137e7i	−0.102928	−	0.102928i
$974$		0			0
$975$	−7.51627e8	−	7.52162e8i	−0.810939	−	0.811517i
$976$		0			0
$977$	−4.17576e8	+	4.17576e8i	−0.447767	+	0.447767i	−0.894612	−	0.446845i	$-0.852548\pi$
$977$	−4.17576e8	+	4.17576e8i	−0.447767	+	0.447767i	0.446845	+	0.894612i	$0.352548\pi$
$978$		0			0
$979$		−	9.11750e8i		−	0.971690i
$980$		0			0
$981$	−1.32602e9			−1.40457
$982$		0			0
$983$	7.39716e7	+	7.39716e7i	0.0778761	+	0.0778761i	0.744972	−	0.667096i	$-0.232463\pi$
$983$	7.39716e7	+	7.39716e7i	0.0778761	+	0.0778761i	−0.667096	+	0.744972i	$0.732463\pi$
$984$		0			0
$985$	−4.51388e7	+	1.08920e8i	−0.0472326	+	0.113972i
$986$		0			0
$987$	−8.67033e7	+	8.67033e7i	−0.0901746	+	0.0901746i
$988$		0			0
$989$		−	1.80613e8i		−	0.186707i
$990$		0			0
$991$	−3.99604e8			−0.410591			−0.205295	−	0.978700i	$-0.565815\pi$
$991$	−3.99604e8			−0.410591			−0.205295	+	0.978700i	$0.565815\pi$
$992$		0			0
$993$	−1.23628e9	−	1.23628e9i	−1.26261	−	1.26261i
$994$		0			0
$995$	4.60657e8	+	1.11268e9i	0.467636	+	1.12954i
$996$		0			0
$997$	−2.12005e7	+	2.12005e7i	−0.0213924	+	0.0213924i	−0.717722	−	0.696330i	$-0.754815\pi$
$997$	−2.12005e7	+	2.12005e7i	−0.0213924	+	0.0213924i	0.696330	+	0.717722i	$0.254815\pi$
$998$		0			0
$999$		−	2.78270e9i		−	2.79106i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.7.f.a.13.3	✓	6
3.2	odd	2		180.7.l.a.73.3		6
4.3	odd	2		80.7.p.d.33.1		6
5.2	odd	4	inner	20.7.f.a.17.3	yes	6
5.3	odd	4		100.7.f.b.57.1		6
5.4	even	2		100.7.f.b.93.1		6
15.2	even	4		180.7.l.a.37.3		6
20.7	even	4		80.7.p.d.17.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.7.f.a.13.3	✓	6	1.1	even	1	trivial
20.7.f.a.17.3	yes	6	5.2	odd	4	inner
80.7.p.d.17.1		6	20.7	even	4
80.7.p.d.33.1		6	4.3	odd	2
100.7.f.b.57.1		6	5.3	odd	4
100.7.f.b.93.1		6	5.4	even	2
180.7.l.a.37.3		6	15.2	even	4
180.7.l.a.73.3		6	3.2	odd	2

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:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	20.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(35.6344 + 35.6344i) q^{3} +(-115.493 + 47.8149i) q^{5} +(53.9521 - 53.9521i) q^{7} +1810.62i q^{9} +O(q^{10})\)	\(q+(35.6344 + 35.6344i) q^{3} +(-115.493 + 47.8149i) q^{5} +(53.9521 - 53.9521i) q^{7} +1810.62i q^{9} +1895.82 q^{11} +(-954.888 - 954.888i) q^{13} +(-5819.40 - 2411.69i) q^{15} +(3807.14 - 3807.14i) q^{17} -2616.92i q^{19} +3845.11 q^{21} +(1210.32 + 1210.32i) q^{23} +(11052.5 - 11044.6i) q^{25} +(-38543.0 + 38543.0i) q^{27} -26815.4i q^{29} +4635.89 q^{31} +(67556.4 + 67556.4i) q^{33} +(-3651.41 + 8810.83i) q^{35} +(-36098.6 + 36098.6i) q^{37} -68053.7i q^{39} +40561.4 q^{41} +(-74613.4 - 74613.4i) q^{43} +(-86574.7 - 209115. i) q^{45} +(-22549.0 + 22549.0i) q^{47} +111827. i q^{49} +271330. q^{51} +(67918.4 + 67918.4i) q^{53} +(-218955. + 90648.3i) q^{55} +(93252.6 - 93252.6i) q^{57} +35389.7i q^{59} -261549. q^{61} +(97687.0 + 97687.0i) q^{63} +(155941. + 64625.5i) q^{65} +(-301383. + 301383. i) q^{67} +86258.3i q^{69} +274465. q^{71} +(-48363.4 - 48363.4i) q^{73} +(787417. + 280.559i) q^{75} +(102283. - 102283. i) q^{77} -781918. i q^{79} -1.42697e6 q^{81} +(-432869. - 432869. i) q^{83} +(-257662. + 621737. i) q^{85} +(955551. - 955551. i) q^{87} -480927. i q^{89} -103036. q^{91} +(165197. + 165197. i) q^{93} +(125128. + 302238. i) q^{95} +(165268. - 165268. i) q^{97} +3.43261e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7}+O(q^{10}) \) 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7	\( 6 q + 32 q^{3} - 156 q^{5} - 264 q^{7} + 2200 q^{11} + 858 q^{13} - 7768 q^{15} - 3278 q^{17} + 33176 q^{21} + 19984 q^{23} - 24174 q^{25} - 115528 q^{27} + 104976 q^{31} + 177320 q^{33} - 116072 q^{35} - 241554 q^{37} + 351736 q^{41} + 60720 q^{43} - 287846 q^{45} - 355248 q^{47} + 641872 q^{51} + 346526 q^{53} - 310200 q^{55} - 112816 q^{57} - 492888 q^{61} + 2288 q^{63} - 5082 q^{65} - 230304 q^{67} + 174128 q^{71} - 332442 q^{73} + 1855048 q^{75} + 1618760 q^{77} - 3085166 q^{81} - 2190936 q^{83} - 164934 q^{85} + 2614304 q^{87} - 2186976 q^{91} + 242072 q^{93} + 3484184 q^{95} + 3338406 q^{97}+O(q^{100}) \) 6 * q + 32 * q^3 - 156 * q^5 - 264 * q^7 + 2200 * q^11 + 858 * q^13 - 7768 * q^15 - 3278 * q^17 + 33176 * q^21 + 19984 * q^23 - 24174 * q^25 - 115528 * q^27 + 104976 * q^31 + 177320 * q^33 - 116072 * q^35 - 241554 * q^37 + 351736 * q^41 + 60720 * q^43 - 287846 * q^45 - 355248 * q^47 + 641872 * q^51 + 346526 * q^53 - 310200 * q^55 - 112816 * q^57 - 492888 * q^61 + 2288 * q^63 - 5082 * q^65 - 230304 * q^67 + 174128 * q^71 - 332442 * q^73 + 1855048 * q^75 + 1618760 * q^77 - 3085166 * q^81 - 2190936 * q^83 - 164934 * q^85 + 2614304 * q^87 - 2186976 * q^91 + 242072 * q^93 + 3484184 * q^95 + 3338406 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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