LMFDB - Embedded newform 252.9.d.c.181.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(181,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.181");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.d (of order $2$, degree $1$, 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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2 x^{7} + 229371 x^{6} - 1944986 x^{5} + 13911336469 x^{4} - 182147783748 x^{3} + \cdots + 45\!\cdots\!56 $ x^8 - 2x^7 + 229371x^6 - 1944986x^5 + 13911336469x^4 - 182147783748x^3 + 180227670127764x^2 - 1401469286341824*x + 451631142832842756
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.7
Root			$-3.81971 - 370.997i$ of defining polynomial
Character	$\chi$	$=$	252.181
Dual form			252.9.d.c.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1071.65i q^{5} +(-436.993 - 2360.90i) q^{7} +O(q^{10})$	$q+1071.65i q^{5} +(-436.993 - 2360.90i) q^{7} +10581.1 q^{11} +24672.2i q^{13} +63685.7i q^{17} -136002. i q^{19} +340275. q^{23} -757811. q^{25} +967918. q^{29} +94428.3i q^{31} +(2.53006e6 - 468304. i) q^{35} +733395. q^{37} +2.00732e6i q^{41} +5.98286e6 q^{43} +1.51713e6i q^{47} +(-5.38287e6 + 2.06339e6i) q^{49} -3.99981e6 q^{53} +1.13393e7i q^{55} -9.18726e6i q^{59} -2.02054e7i q^{61} -2.64400e7 q^{65} -4.55149e6 q^{67} -2.39953e7 q^{71} +3.93925e7i q^{73} +(-4.62388e6 - 2.49810e7i) q^{77} -7.44371e6 q^{79} +1.51689e7i q^{83} -6.82488e7 q^{85} +3.77542e7i q^{89} +(5.82485e7 - 1.07816e7i) q^{91} +1.45747e8 q^{95} +8.92820e7i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6076 q^{7}+O(q^{10}) $ 8 * q + 6076 * q^7	$ 8 q + 6076 q^{7} - 2154400 q^{25} + 2495096 q^{37} + 18340232 q^{43} - 13983424 q^{49} + 12320248 q^{67} + 53237896 q^{79} - 79347240 q^{85} + 257358192 q^{91}+O(q^{100}) $ 8 * q + 6076 * q^7 - 2154400 * q^25 + 2495096 * q^37 + 18340232 * q^43 - 13983424 * q^49 + 12320248 * q^67 + 53237896 * q^79 - 79347240 * q^85 + 257358192 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			1071.65i			1.71464i	0.514782	+	0.857321i	$0.327873\pi$
$5$			1071.65i			1.71464i	−0.514782	+	0.857321i	$0.672127\pi$
$6$		0			0
$7$	−436.993	−	2360.90i	−0.182005	−	0.983298i
$7$	−436.993	−	2360.90i	−0.182005	−	0.983298i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10581.1			0.722706			0.361353	−	0.932429i	$-0.382315\pi$
$11$	10581.1			0.722706			0.361353	+	0.932429i	$0.382315\pi$
$12$		0			0
$13$			24672.2i			0.863842i	0.901911	+	0.431921i	$0.142164\pi$
$13$			24672.2i			0.863842i	−0.901911	+	0.431921i	$0.857836\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			63685.7i			0.762511i	0.924470	+	0.381255i	$0.124508\pi$
$17$			63685.7i			0.762511i	−0.924470	+	0.381255i	$0.875492\pi$
$18$		0			0
$19$		−	136002.i		−	1.04359i	−0.853070	−	0.521796i	$-0.825262\pi$
$19$		−	136002.i		−	1.04359i	0.853070	−	0.521796i	$-0.174738\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	340275.			1.21596			0.607979	−	0.793953i	$-0.291981\pi$
$23$	340275.			1.21596			0.607979	+	0.793953i	$0.291981\pi$
$24$		0			0
$25$	−757811.			−1.94000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	967918.			1.36851			0.684253	−	0.729245i	$-0.260128\pi$
$29$	967918.			1.36851			0.684253	+	0.729245i	$0.260128\pi$
$30$		0			0
$31$			94428.3i			0.102248i	0.998692	+	0.0511241i	$0.0162804\pi$
$31$			94428.3i			0.102248i	−0.998692	+	0.0511241i	$0.983720\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.53006e6	−	468304.i	1.68600	−	0.312073i
$36$		0			0
$37$	733395.			0.391319			0.195659	−	0.980672i	$-0.437315\pi$
$37$	733395.			0.391319			0.195659	+	0.980672i	$0.437315\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			2.00732e6i			0.710365i	0.934797	+	0.355183i	$0.115581\pi$
$41$			2.00732e6i			0.710365i	−0.934797	+	0.355183i	$0.884419\pi$
$42$		0			0
$43$	5.98286e6			1.74999			0.874993	−	0.484135i	$-0.160866\pi$
$43$	5.98286e6			1.74999			0.874993	+	0.484135i	$0.160866\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			1.51713e6i			0.310908i	0.987843	+	0.155454i	$0.0496841\pi$
$47$			1.51713e6i			0.310908i	−0.987843	+	0.155454i	$0.950316\pi$
$48$		0			0
$49$	−5.38287e6	+	2.06339e6i	−0.933749	+	0.357930i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.99981e6			−0.506916			−0.253458	−	0.967346i	$-0.581568\pi$
$53$	−3.99981e6			−0.506916			−0.253458	+	0.967346i	$0.581568\pi$
$54$		0			0
$55$			1.13393e7i			1.23918i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	9.18726e6i		−	0.758190i	−0.925358	−	0.379095i	$-0.876235\pi$
$59$		−	9.18726e6i		−	0.758190i	0.925358	−	0.379095i	$-0.123765\pi$
$60$		0			0
$61$		−	2.02054e7i		−	1.45931i	−0.683813	−	0.729657i	$-0.739680\pi$
$61$		−	2.02054e7i		−	1.45931i	0.683813	−	0.729657i	$-0.260320\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.64400e7			−1.48118
$66$		0			0
$67$	−4.55149e6			−0.225868			−0.112934	−	0.993603i	$-0.536025\pi$
$67$	−4.55149e6			−0.225868			−0.112934	+	0.993603i	$0.536025\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.39953e7			−0.944263			−0.472132	−	0.881528i	$-0.656515\pi$
$71$	−2.39953e7			−0.944263			−0.472132	+	0.881528i	$0.656515\pi$
$72$		0			0
$73$			3.93925e7i			1.38715i	0.720386	+	0.693574i	$0.243965\pi$
$73$			3.93925e7i			1.38715i	−0.720386	+	0.693574i	$0.756035\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.62388e6	−	2.49810e7i	−0.131536	−	0.710635i
$78$		0			0
$79$	−7.44371e6			−0.191109			−0.0955545	−	0.995424i	$-0.530462\pi$
$79$	−7.44371e6			−0.191109			−0.0955545	+	0.995424i	$0.530462\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			1.51689e7i			0.319625i	0.987147	+	0.159813i	$0.0510890\pi$
$83$			1.51689e7i			0.319625i	−0.987147	+	0.159813i	$0.948911\pi$
$84$		0			0
$85$	−6.82488e7			−1.30743
$86$		0			0
$87$		0			0
$88$		0			0
$89$			3.77542e7i			0.601735i	0.953666	+	0.300868i	$0.0972762\pi$
$89$			3.77542e7i			0.601735i	−0.953666	+	0.300868i	$0.902724\pi$
$90$		0			0
$91$	5.82485e7	−	1.07816e7i	0.849414	−	0.157223i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.45747e8			1.78939
$96$		0			0
$97$			8.92820e7i			1.00850i	0.863557	+	0.504251i	$0.168231\pi$
$97$			8.92820e7i			1.00850i	−0.863557	+	0.504251i	$0.831769\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			4.02581e7i			0.386872i	0.981113	+	0.193436i	$0.0619632\pi$
$101$			4.02581e7i			0.386872i	−0.981113	+	0.193436i	$0.938037\pi$
$102$		0			0
$103$			1.16646e8i			1.03638i	0.855264	+	0.518192i	$0.173395\pi$
$103$			1.16646e8i			1.03638i	−0.855264	+	0.518192i	$0.826605\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.01941e8			1.54059			0.770297	−	0.637685i	$-0.220108\pi$
$107$	2.01941e8			1.54059			0.770297	+	0.637685i	$0.220108\pi$
$108$		0			0
$109$	9.21391e7			0.652736			0.326368	−	0.945243i	$-0.394175\pi$
$109$	9.21391e7			0.652736			0.326368	+	0.945243i	$0.394175\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.47904e8			−0.907125			−0.453563	−	0.891224i	$-0.649847\pi$
$113$	−1.47904e8			−0.907125			−0.453563	+	0.891224i	$0.649847\pi$
$114$		0			0
$115$			3.64656e8i			2.08493i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.50355e8	−	2.78302e7i	0.749775	−	0.138781i
$120$		0			0
$121$	−1.02399e8			−0.477697
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3.93495e8i		−	1.61176i
$126$		0			0
$127$	9.15585e7			0.351952			0.175976	−	0.984394i	$-0.443692\pi$
$127$	9.15585e7			0.351952			0.175976	+	0.984394i	$0.443692\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			3.92078e8i			1.33134i	0.746248	+	0.665668i	$0.231853\pi$
$131$			3.92078e8i			1.33134i	−0.746248	+	0.665668i	$0.768147\pi$
$132$		0			0
$133$	−3.21087e8	+	5.94319e7i	−1.02616	+	0.189939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.11204e8			−1.16728			−0.583639	−	0.812013i	$-0.698372\pi$
$137$	−4.11204e8			−1.16728			−0.583639	+	0.812013i	$0.698372\pi$
$138$		0			0
$139$			3.29853e8i			0.883612i	0.897111	+	0.441806i	$0.145662\pi$
$139$			3.29853e8i			0.883612i	−0.897111	+	0.441806i	$0.854338\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.61060e8i			0.624304i
$144$		0			0
$145$			1.03727e9i			2.34650i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.82348e8			−1.79017			−0.895087	−	0.445892i	$-0.852886\pi$
$149$	−8.82348e8			−1.79017			−0.895087	+	0.445892i	$0.852886\pi$
$150$		0			0
$151$	−9.58744e8			−1.84414			−0.922072	−	0.387017i	$-0.873505\pi$
$151$	−9.58744e8			−1.84414			−0.922072	+	0.387017i	$0.873505\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.01194e8			−0.175319
$156$		0			0
$157$			3.02276e8i			0.497514i	0.968566	+	0.248757i	$0.0800220\pi$
$157$			3.02276e8i			0.497514i	−0.968566	+	0.248757i	$0.919978\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.48698e8	−	8.03354e8i	−0.221310	−	1.19565i
$162$		0			0
$163$	3.20474e8			0.453986			0.226993	−	0.973896i	$-0.427111\pi$
$163$	3.20474e8			0.453986			0.226993	+	0.973896i	$0.427111\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.22231e9i		−	1.57151i	−0.618540	−	0.785753i	$-0.712276\pi$
$167$		−	1.22231e9i		−	1.57151i	0.618540	−	0.785753i	$-0.287724\pi$
$168$		0			0
$169$	2.07013e8			0.253776
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1.69146e9i			1.88833i	0.329473	+	0.944165i	$0.393129\pi$
$173$			1.69146e9i			1.88833i	−0.329473	+	0.944165i	$0.606871\pi$
$174$		0			0
$175$	3.31158e8	+	1.78911e9i	0.353089	+	1.90759i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.28023e8			0.903954			0.451977	−	0.892030i	$-0.350719\pi$
$179$	9.28023e8			0.903954			0.451977	+	0.892030i	$0.350719\pi$
$180$		0			0
$181$			1.94295e9i			1.81029i	0.425105	+	0.905144i	$0.360237\pi$
$181$			1.94295e9i			1.81029i	−0.425105	+	0.905144i	$0.639763\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			7.85943e8i			0.670972i
$186$		0			0
$187$			6.73867e8i			0.551071i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.11035e9			0.834305			0.417152	−	0.908837i	$-0.363028\pi$
$191$	1.11035e9			0.834305			0.417152	+	0.908837i	$0.363028\pi$
$192$		0			0
$193$	1.88156e9			1.35609			0.678047	−	0.735019i	$-0.262827\pi$
$193$	1.88156e9			1.35609			0.678047	+	0.735019i	$0.262827\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.85779e9			1.89743			0.948715	−	0.316133i	$-0.102384\pi$
$197$	2.85779e9			1.89743			0.948715	+	0.316133i	$0.102384\pi$
$198$		0			0
$199$			8.58663e8i			0.547533i	0.961796	+	0.273767i	$0.0882696\pi$
$199$			8.58663e8i			0.547533i	−0.961796	+	0.273767i	$0.911730\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.22974e8	−	2.28516e9i	−0.249075	−	1.34565i
$204$		0			0
$205$	−2.15115e9			−1.21802
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	1.43905e9i		−	0.754209i
$210$		0			0
$211$	−3.18684e9			−1.60779			−0.803897	−	0.594769i	$-0.797243\pi$
$211$	−3.18684e9			−1.60779			−0.803897	+	0.594769i	$0.797243\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			6.41153e9i			3.00060i
$216$		0			0
$217$	2.22936e8	−	4.12645e7i	0.100540	−	0.0186096i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.57127e9			−0.658689
$222$		0			0
$223$			7.41102e8i			0.299681i	0.988710	+	0.149840i	$0.0478760\pi$
$223$			7.41102e8i			0.299681i	−0.988710	+	0.149840i	$0.952124\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	7.75636e8i		−	0.292116i	−0.989276	−	0.146058i	$-0.953341\pi$
$227$		−	7.75636e8i		−	0.292116i	0.989276	−	0.146058i	$-0.0466585\pi$
$228$		0			0
$229$			1.44246e9i			0.524521i	0.964997	+	0.262261i	$0.0844680\pi$
$229$			1.44246e9i			0.524521i	−0.964997	+	0.262261i	$0.915532\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.18218e9			1.07969			0.539847	−	0.841763i	$-0.318482\pi$
$233$	3.18218e9			1.07969			0.539847	+	0.841763i	$0.318482\pi$
$234$		0			0
$235$	−1.62584e9			−0.533096
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.01977e9			−1.53848			−0.769240	−	0.638960i	$-0.779365\pi$
$239$	−5.01977e9			−1.53848			−0.769240	+	0.638960i	$0.779365\pi$
$240$		0			0
$241$		−	1.09821e8i		−	0.0325549i	−0.999868	−	0.0162774i	$-0.994819\pi$
$241$		−	1.09821e8i		−	0.0325549i	0.999868	−	0.0162774i	$-0.00518150\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.21124e9	−	5.76856e9i	−0.613721	−	1.60104i
$246$		0			0
$247$	3.35547e9			0.901499
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	7.07598e9i		−	1.78276i	−0.453261	−	0.891378i	$-0.649739\pi$
$251$		−	7.07598e9i		−	1.78276i	0.453261	−	0.891378i	$-0.350261\pi$
$252$		0			0
$253$	3.60049e9			0.878779
$254$		0			0
$255$		0			0
$256$		0			0
$257$			5.92292e9i			1.35770i	0.734278	+	0.678849i	$0.237521\pi$
$257$			5.92292e9i			1.35770i	−0.734278	+	0.678849i	$0.762479\pi$
$258$		0			0
$259$	−3.20488e8	−	1.73147e9i	−0.0712219	−	0.384783i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.58926e9			−1.37725			−0.688626	−	0.725116i	$-0.741786\pi$
$263$	−6.58926e9			−1.37725			−0.688626	+	0.725116i	$0.741786\pi$
$264$		0			0
$265$		−	4.28641e9i		−	0.869180i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			6.23482e9i			1.19073i	0.803454	+	0.595367i	$0.202993\pi$
$269$			6.23482e9i			1.19073i	−0.803454	+	0.595367i	$0.797007\pi$
$270$		0			0
$271$		−	4.05326e9i		−	0.751498i	−0.926722	−	0.375749i	$-0.877386\pi$
$271$		−	4.05326e9i		−	0.751498i	0.926722	−	0.375749i	$-0.122614\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.01850e9			−1.40205
$276$		0			0
$277$	−4.65609e9			−0.790865			−0.395433	−	0.918495i	$-0.629405\pi$
$277$	−4.65609e9			−0.790865			−0.395433	+	0.918495i	$0.629405\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.83530e9			0.615141			0.307570	−	0.951525i	$-0.400484\pi$
$281$	3.83530e9			0.615141			0.307570	+	0.951525i	$0.400484\pi$
$282$		0			0
$283$			4.32516e9i			0.674306i	0.941450	+	0.337153i	$0.109464\pi$
$283$			4.32516e9i			0.674306i	−0.941450	+	0.337153i	$0.890536\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.73908e9	−	8.77186e8i	0.698500	−	0.129290i
$288$		0			0
$289$	2.91989e9			0.418577
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	6.99484e9i		−	0.949089i	−0.880231	−	0.474545i	$-0.842613\pi$
$293$		−	6.99484e9i		−	0.949089i	0.880231	−	0.474545i	$-0.157387\pi$
$294$		0			0
$295$	9.84554e9			1.30002
$296$		0			0
$297$		0			0
$298$		0			0
$299$			8.39533e9i			1.05040i
$300$		0			0
$301$	−2.61447e9	−	1.41249e10i	−0.318506	−	1.72076i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.16532e10			2.50220
$306$		0			0
$307$		−	8.11196e9i		−	0.913214i	−0.889669	−	0.456607i	$-0.849065\pi$
$307$		−	8.11196e9i		−	0.913214i	0.889669	−	0.456607i	$-0.150935\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			1.49867e9i			0.160201i	0.996787	+	0.0801006i	$0.0255242\pi$
$311$			1.49867e9i			0.160201i	−0.996787	+	0.0801006i	$0.974476\pi$
$312$		0			0
$313$		−	9.72355e9i		−	1.01309i	−0.862214	−	0.506545i	$-0.830923\pi$
$313$		−	9.72355e9i		−	1.01309i	0.862214	−	0.506545i	$-0.169077\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.16325e8			−0.0511312			−0.0255656	−	0.999673i	$-0.508139\pi$
$317$	−5.16325e8			−0.0511312			−0.0255656	+	0.999673i	$0.508139\pi$
$318$		0			0
$319$	1.02417e10			0.989027
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.66137e9			0.795750
$324$		0			0
$325$		−	1.86969e10i		−	1.67585i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.58180e9	−	6.62977e8i	0.305715	−	0.0565868i
$330$		0			0
$331$	−1.13625e10			−0.946590			−0.473295	−	0.880904i	$-0.656936\pi$
$331$	−1.13625e10			−0.946590			−0.473295	+	0.880904i	$0.656936\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	4.87761e9i		−	0.387282i
$336$		0			0
$337$	−1.90839e10			−1.47961			−0.739805	−	0.672822i	$-0.765082\pi$
$337$	−1.90839e10			−1.47961			−0.739805	+	0.672822i	$0.765082\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			9.99158e8i			0.0738953i
$342$		0			0
$343$	7.22374e9	+	1.18067e10i	0.521898	+	0.853008i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.18230e9			−0.426415			−0.213207	−	0.977007i	$-0.568391\pi$
$347$	−6.18230e9			−0.426415			−0.213207	+	0.977007i	$0.568391\pi$
$348$		0			0
$349$		−	1.57447e10i		−	1.06129i	−0.847595	−	0.530643i	$-0.821950\pi$
$349$		−	1.57447e10i		−	1.06129i	0.847595	−	0.530643i	$-0.178050\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.96364e9i		−	0.190865i	−0.995436	−	0.0954327i	$-0.969577\pi$
$353$		−	2.96364e9i		−	0.190865i	0.995436	−	0.0954327i	$-0.0304235\pi$
$354$		0			0
$355$		−	2.57146e10i		−	1.61907i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.13412e10			0.682782			0.341391	−	0.939921i	$-0.389102\pi$
$359$	1.13412e10			0.682782			0.341391	+	0.939921i	$0.389102\pi$
$360$		0			0
$361$	−1.51295e9			−0.0890833
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.22151e10			−2.37846
$366$		0			0
$367$		−	2.34457e10i		−	1.29241i	−0.763164	−	0.646204i	$-0.776355\pi$
$367$		−	2.34457e10i		−	1.29241i	0.763164	−	0.646204i	$-0.223645\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.74789e9	+	9.44315e9i	0.0922612	+	0.498450i
$372$		0			0
$373$	3.30201e10			1.70586			0.852930	−	0.522026i	$-0.174824\pi$
$373$	3.30201e10			1.70586			0.852930	+	0.522026i	$0.174824\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2.38807e10i			1.18217i
$378$		0			0
$379$	−6.57661e9			−0.318746			−0.159373	−	0.987218i	$-0.550947\pi$
$379$	−6.57661e9			−0.318746			−0.159373	+	0.987218i	$0.550947\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.31609e10i		−	1.54110i	−0.637379	−	0.770550i	$-0.719982\pi$
$383$		−	3.31609e10i		−	1.54110i	0.637379	−	0.770550i	$-0.280018\pi$
$384$		0			0
$385$	2.67709e10	−	4.95519e9i	1.21848	−	0.225537i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.63950e10			−1.15272			−0.576359	−	0.817197i	$-0.695527\pi$
$389$	−2.63950e10			−1.15272			−0.576359	+	0.817197i	$0.695527\pi$
$390$		0			0
$391$			2.16706e10i			0.927181i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	7.97707e9i		−	0.327684i
$396$		0			0
$397$			4.49632e9i			0.181007i	0.995896	+	0.0905034i	$0.0288476\pi$
$397$			4.49632e9i			0.181007i	−0.995896	+	0.0905034i	$0.971152\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.56656e10			−0.605855			−0.302928	−	0.953014i	$-0.597964\pi$
$401$	−1.56656e10			−0.605855			−0.302928	+	0.953014i	$0.597964\pi$
$402$		0			0
$403$	−2.32975e9			−0.0883263
$404$		0			0
$405$		0			0
$406$		0			0
$407$	7.76014e9			0.282808
$408$		0			0
$409$		−	8.48977e9i		−	0.303391i	−0.988427	−	0.151695i	$-0.951527\pi$
$409$		−	8.48977e9i		−	0.303391i	0.988427	−	0.151695i	$-0.0484733\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.16902e10	+	4.01477e9i	−0.745527	+	0.137994i
$414$		0			0
$415$	−1.62558e10			−0.548043
$416$		0			0
$417$		0			0
$418$		0			0
$419$			4.39173e10i			1.42488i	0.701731	+	0.712442i	$0.252411\pi$
$419$			4.39173e10i			1.42488i	−0.701731	+	0.712442i	$0.747589\pi$
$420$		0			0
$421$	4.09583e10			1.30381			0.651903	−	0.758302i	$-0.273971\pi$
$421$	4.09583e10			1.30381			0.651903	+	0.758302i	$0.273971\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	4.82617e10i		−	1.47927i
$426$		0			0
$427$	−4.77030e10	+	8.82964e9i	−1.43494	+	0.265602i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.31374e10			0.380716			0.190358	−	0.981715i	$-0.439035\pi$
$431$	1.31374e10			0.380716			0.190358	+	0.981715i	$0.439035\pi$
$432$		0			0
$433$		−	6.11013e10i		−	1.73820i	−0.494638	−	0.869099i	$-0.664699\pi$
$433$		−	6.11013e10i		−	1.73820i	0.494638	−	0.869099i	$-0.335301\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	4.62780e10i		−	1.26896i
$438$		0			0
$439$			5.25752e10i			1.41554i	0.706442	+	0.707771i	$0.250299\pi$
$439$			5.25752e10i			1.41554i	−0.706442	+	0.707771i	$0.749701\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.52730e10			1.43515			0.717576	−	0.696480i	$-0.245252\pi$
$443$	5.52730e10			1.43515			0.717576	+	0.696480i	$0.245252\pi$
$444$		0			0
$445$	−4.04594e10			−1.03176
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.08377e10			1.49688			0.748440	−	0.663202i	$-0.230803\pi$
$449$	6.08377e10			1.49688			0.748440	+	0.663202i	$0.230803\pi$
$450$		0			0
$451$			2.12397e10i			0.513385i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.15541e10	+	6.24221e10i	0.269582	+	1.45644i
$456$		0			0
$457$	−1.75283e10			−0.401860			−0.200930	−	0.979606i	$-0.564396\pi$
$457$	−1.75283e10			−0.401860			−0.200930	+	0.979606i	$0.564396\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			7.14631e10i			1.58226i	0.611648	+	0.791130i	$0.290507\pi$
$461$			7.14631e10i			1.58226i	−0.611648	+	0.791130i	$0.709493\pi$
$462$		0			0
$463$	−4.51271e9			−0.0982004			−0.0491002	−	0.998794i	$-0.515635\pi$
$463$	−4.51271e9			−0.0982004			−0.0491002	+	0.998794i	$0.515635\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	5.62414e10i		−	1.18247i	−0.806501	−	0.591233i	$-0.798641\pi$
$467$		−	5.62414e10i		−	1.18247i	0.806501	−	0.591233i	$-0.201359\pi$
$468$		0			0
$469$	1.98897e9	+	1.07456e10i	0.0411090	+	0.222095i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.33054e10			1.26473
$474$		0			0
$475$			1.03064e11i			2.02456i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			3.56246e10i			0.676719i	0.941017	+	0.338359i	$0.109872\pi$
$479$			3.56246e10i			0.676719i	−0.941017	+	0.338359i	$0.890128\pi$
$480$		0			0
$481$			1.80945e10i			0.338038i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.56792e10			−1.72922
$486$		0			0
$487$	−5.09256e10			−0.905359			−0.452679	−	0.891673i	$-0.649532\pi$
$487$	−5.09256e10			−0.905359			−0.452679	+	0.891673i	$0.649532\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.24005e10			1.07365			0.536825	−	0.843694i	$-0.319624\pi$
$491$	6.24005e10			1.07365			0.536825	+	0.843694i	$0.319624\pi$
$492$		0			0
$493$			6.16425e10i			1.04350i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.04858e10	+	5.66505e10i	0.171860	+	0.928492i
$498$		0			0
$499$	9.04114e10			1.45821			0.729106	−	0.684401i	$-0.239936\pi$
$499$	9.04114e10			1.45821			0.729106	+	0.684401i	$0.239936\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	3.50934e10i		−	0.548219i	−0.961698	−	0.274109i	$-0.911617\pi$
$503$		−	3.50934e10i		−	0.548219i	0.961698	−	0.274109i	$-0.0883831\pi$
$504$		0			0
$505$	−4.31426e10			−0.663347
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	9.94040e10i		−	1.48092i	−0.672098	−	0.740462i	$-0.734607\pi$
$509$		−	9.94040e10i		−	1.48092i	0.672098	−	0.740462i	$-0.265393\pi$
$510$		0			0
$511$	9.30018e10	−	1.72143e10i	1.36398	−	0.252467i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.25004e11			−1.77703
$516$		0			0
$517$			1.60530e10i			0.224695i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	5.21734e10i		−	0.708106i	−0.935225	−	0.354053i	$-0.884803\pi$
$521$		−	5.21734e10i		−	0.708106i	0.935225	−	0.354053i	$-0.115197\pi$
$522$		0			0
$523$			1.15036e11i			1.53755i	0.639522	+	0.768773i	$0.279132\pi$
$523$			1.15036e11i			1.53755i	−0.639522	+	0.768773i	$0.720868\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.01373e9			−0.0779653
$528$		0			0
$529$	3.74759e10			0.478552
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.95250e10			−0.613643
$534$		0			0
$535$			2.16410e11i			2.64157i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.69569e10	+	2.18330e10i	−0.674825	+	0.258678i
$540$		0			0
$541$	−3.48673e10			−0.407033			−0.203517	−	0.979072i	$-0.565237\pi$
$541$	−3.48673e10			−0.407033			−0.203517	+	0.979072i	$0.565237\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			9.87409e10i			1.11921i
$546$		0			0
$547$	−1.32416e11			−1.47908			−0.739540	−	0.673113i	$-0.764957\pi$
$547$	−1.32416e11			−1.47908			−0.739540	+	0.673113i	$0.764957\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1.31639e11i		−	1.42816i
$552$		0			0
$553$	3.25285e9	+	1.75738e10i	0.0347828	+	0.187917i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−7.05214e10			−0.732656			−0.366328	−	0.930486i	$-0.619385\pi$
$557$	−7.05214e10			−0.732656			−0.366328	+	0.930486i	$0.619385\pi$
$558$		0			0
$559$			1.47610e11i			1.51171i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			7.67216e10i			0.763632i	0.924238	+	0.381816i	$0.124701\pi$
$563$			7.67216e10i			0.763632i	−0.924238	+	0.381816i	$0.875299\pi$
$564$		0			0
$565$		−	1.58502e11i		−	1.55540i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.53682e11			1.46613			0.733067	−	0.680156i	$-0.238088\pi$
$569$	1.53682e11			1.46613			0.733067	+	0.680156i	$0.238088\pi$
$570$		0			0
$571$	−9.69767e10			−0.912269			−0.456135	−	0.889911i	$-0.650766\pi$
$571$	−9.69767e10			−0.912269			−0.456135	+	0.889911i	$0.650766\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.57864e11			−2.35895
$576$		0			0
$577$		−	1.00096e11i		−	0.903049i	−0.892259	−	0.451525i	$-0.850880\pi$
$577$		−	1.00096e11i		−	0.903049i	0.892259	−	0.451525i	$-0.149120\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.58122e10	−	6.62870e9i	0.314287	−	0.0581733i
$582$		0			0
$583$	−4.23226e10			−0.366351
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.00055e11i		−	0.842728i	−0.906892	−	0.421364i	$-0.861551\pi$
$587$		−	1.00055e11i		−	0.842728i	0.906892	−	0.421364i	$-0.138449\pi$
$588$		0			0
$589$	1.28424e10			0.106705
$590$		0			0
$591$		0			0
$592$		0			0
$593$			2.05395e11i			1.66100i	0.557017	+	0.830501i	$0.311946\pi$
$593$			2.05395e11i			1.66100i	−0.557017	+	0.830501i	$0.688054\pi$
$594$		0			0
$595$	2.98243e10	+	1.61128e11i	0.237959	+	1.28560i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.46446e11			1.13755			0.568776	−	0.822492i	$-0.307417\pi$
$599$	1.46446e11			1.13755			0.568776	+	0.822492i	$0.307417\pi$
$600$		0			0
$601$			1.44783e10i			0.110974i	0.998459	+	0.0554869i	$0.0176711\pi$
$601$			1.44783e10i			0.110974i	−0.998459	+	0.0554869i	$0.982329\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	1.09735e11i		−	0.819079i
$606$		0			0
$607$			9.26418e10i			0.682421i	0.939987	+	0.341211i	$0.110837\pi$
$607$			9.26418e10i			0.682421i	−0.939987	+	0.341211i	$0.889163\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.74310e10			−0.268576
$612$		0			0
$613$	−5.36935e10			−0.380259			−0.190130	−	0.981759i	$-0.560891\pi$
$613$	−5.36935e10			−0.380259			−0.190130	+	0.981759i	$0.560891\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.61826e11			1.80665			0.903323	−	0.428962i	$-0.141120\pi$
$617$	2.61826e11			1.80665			0.903323	+	0.428962i	$0.141120\pi$
$618$		0			0
$619$		−	1.54542e10i		−	0.105265i	−0.998614	−	0.0526323i	$-0.983239\pi$
$619$		−	1.54542e10i		−	0.105265i	0.998614	−	0.0526323i	$-0.0167611\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.91338e10	−	1.64983e10i	0.591685	−	0.109519i
$624$		0			0
$625$	1.25670e11			0.823590
$626$		0			0
$627$		0			0
$628$		0			0
$629$			4.67067e10i			0.298385i
$630$		0			0
$631$	1.70767e11			1.07718			0.538588	−	0.842569i	$-0.318958\pi$
$631$	1.70767e11			1.07718			0.538588	+	0.842569i	$0.318958\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			9.81188e10i			0.603472i
$636$		0			0
$637$	−5.09084e10	−	1.32807e11i	−0.309195	−	0.806612i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3.09185e11			1.83142			0.915708	−	0.401845i	$-0.131631\pi$
$641$	3.09185e11			1.83142			0.915708	+	0.401845i	$0.131631\pi$
$642$		0			0
$643$		−	1.68796e10i		−	0.0987456i	−0.998780	−	0.0493728i	$-0.984278\pi$
$643$		−	1.68796e10i		−	0.0987456i	0.998780	−	0.0493728i	$-0.0157222\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.14934e11i			0.655893i	0.944696	+	0.327947i	$0.106357\pi$
$647$			1.14934e11i			0.655893i	−0.944696	+	0.327947i	$0.893643\pi$
$648$		0			0
$649$		−	9.72116e10i		−	0.547948i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.00660e10			0.110359			0.0551797	−	0.998476i	$-0.482427\pi$
$653$	2.00660e10			0.110359			0.0551797	+	0.998476i	$0.482427\pi$
$654$		0			0
$655$	−4.20171e11			−2.28276
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.04521e10			−0.320531			−0.160265	−	0.987074i	$-0.551235\pi$
$659$	−6.04521e10			−0.320531			−0.160265	+	0.987074i	$0.551235\pi$
$660$		0			0
$661$		−	1.36602e11i		−	0.715567i	−0.933805	−	0.357783i	$-0.883533\pi$
$661$		−	1.36602e11i		−	0.715567i	0.933805	−	0.357783i	$-0.116467\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.36903e10	−	3.44093e11i	−0.325677	−	1.75950i
$666$		0			0
$667$	3.29358e11			1.66404
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	2.13796e11i		−	1.05465i
$672$		0			0
$673$	8.00609e10			0.390266			0.195133	−	0.980777i	$-0.437486\pi$
$673$	8.00609e10			0.390266			0.195133	+	0.980777i	$0.437486\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			3.07291e11i			1.46284i	0.681930	+	0.731418i	$0.261141\pi$
$677$			3.07291e11i			1.46284i	−0.681930	+	0.731418i	$0.738859\pi$
$678$		0			0
$679$	2.10786e11	−	3.90156e10i	0.991658	−	0.183552i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.86033e10			−0.0854882			−0.0427441	−	0.999086i	$-0.513610\pi$
$683$	−1.86033e10			−0.0854882			−0.0427441	+	0.999086i	$0.513610\pi$
$684$		0			0
$685$		−	4.40667e11i		−	2.00147i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	9.86842e10i		−	0.437896i
$690$		0			0
$691$			2.32933e11i			1.02169i	0.859673	+	0.510845i	$0.170668\pi$
$691$			2.32933e11i			1.02169i	−0.859673	+	0.510845i	$0.829332\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.53488e11			−1.51508
$696$		0			0
$697$	−1.27838e11			−0.541661
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.24988e11			−0.931722			−0.465861	−	0.884858i	$-0.654255\pi$
$701$	−2.24988e11			−0.931722			−0.465861	+	0.884858i	$0.654255\pi$
$702$		0			0
$703$		−	9.97430e10i		−	0.408377i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	9.50452e10	−	1.75925e10i	0.380411	−	0.0704126i
$708$		0			0
$709$	3.33098e11			1.31822			0.659109	−	0.752047i	$-0.270933\pi$
$709$	3.33098e11			1.31822			0.659109	+	0.752047i	$0.270933\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.21316e10i			0.124329i
$714$		0			0
$715$	−2.79765e11			−1.07046
$716$		0			0
$717$		0			0
$718$		0			0
$719$			3.44582e11i			1.28937i	0.764449	+	0.644684i	$0.223011\pi$
$719$			3.44582e11i			1.28937i	−0.764449	+	0.644684i	$0.776989\pi$
$720$		0			0
$721$	2.75389e11	−	5.09735e10i	1.01907	−	0.188627i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.33499e11			−2.65490
$726$		0			0
$727$		−	2.17564e11i		−	0.778841i	−0.921060	−	0.389421i	$-0.872675\pi$
$727$		−	2.17564e11i		−	0.778841i	0.921060	−	0.389421i	$-0.127325\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			3.81022e11i			1.33438i
$732$		0			0
$733$		−	3.55371e11i		−	1.23102i	−0.788128	−	0.615511i	$-0.788950\pi$
$733$		−	3.55371e11i		−	1.23102i	0.788128	−	0.615511i	$-0.211050\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.81599e10			−0.163236
$738$		0			0
$739$	1.02474e11			0.343588			0.171794	−	0.985133i	$-0.445044\pi$
$739$	1.02474e11			0.343588			0.171794	+	0.985133i	$0.445044\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.25856e11			1.06923			0.534614	−	0.845096i	$-0.320457\pi$
$743$	3.25856e11			1.06923			0.534614	+	0.845096i	$0.320457\pi$
$744$		0			0
$745$		−	9.45570e11i		−	3.06951i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.82467e10	−	4.76761e11i	−0.280395	−	1.51486i
$750$		0			0
$751$	−4.85707e11			−1.52691			−0.763457	−	0.645858i	$-0.776500\pi$
$751$	−4.85707e11			−1.52691			−0.763457	+	0.645858i	$0.776500\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1.02744e12i		−	3.16205i
$756$		0			0
$757$	−3.42261e11			−1.04226			−0.521128	−	0.853479i	$-0.674489\pi$
$757$	−3.42261e11			−1.04226			−0.521128	+	0.853479i	$0.674489\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			3.32098e11i			0.990210i	0.868833	+	0.495105i	$0.164870\pi$
$761$			3.32098e11i			0.990210i	−0.868833	+	0.495105i	$0.835130\pi$
$762$		0			0
$763$	−4.02642e10	−	2.17531e11i	−0.118801	−	0.641834i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.26670e11			0.654957
$768$		0			0
$769$		−	3.52075e10i		−	0.100677i	−0.998732	−	0.0503384i	$-0.983970\pi$
$769$		−	3.52075e10i		−	0.100677i	0.998732	−	0.0503384i	$-0.0160300\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1.04549e11i		−	0.292820i	−0.989224	−	0.146410i	$-0.953228\pi$
$773$		−	1.04549e11i		−	0.292820i	0.989224	−	0.146410i	$-0.0467719\pi$
$774$		0			0
$775$		−	7.15588e10i		−	0.198361i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.73000e11			0.741331
$780$		0			0
$781$	−2.53898e11			−0.682424
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.23935e11			−0.853058
$786$		0			0
$787$			6.08147e11i			1.58529i	0.609682	+	0.792646i	$0.291297\pi$
$787$			6.08147e11i			1.58529i	−0.609682	+	0.792646i	$0.708703\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.46332e10	+	3.49187e11i	0.165101	+	0.891974i
$792$		0			0
$793$	4.98513e11			1.26062
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.76664e11i			0.437841i	0.975743	+	0.218920i	$0.0702535\pi$
$797$			1.76664e11i			0.437841i	−0.975743	+	0.218920i	$0.929747\pi$
$798$		0			0
$799$	−9.66197e10			−0.237071
$800$		0			0
$801$		0			0
$802$		0			0
$803$			4.16818e11i			1.00250i
$804$		0			0
$805$	8.60915e11	−	1.59352e11i	2.05011	−	0.379467i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.14399e11			0.267071			0.133536	−	0.991044i	$-0.457367\pi$
$809$	1.14399e11			0.267071			0.133536	+	0.991044i	$0.457367\pi$
$810$		0			0
$811$		−	4.13662e11i		−	0.956230i	−0.878297	−	0.478115i	$-0.841320\pi$
$811$		−	4.13662e11i		−	0.956230i	0.878297	−	0.478115i	$-0.158680\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			3.43436e11i			0.778423i
$816$		0			0
$817$		−	8.13680e11i		−	1.82627i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.20987e11			1.58692			0.793459	−	0.608623i	$-0.208278\pi$
$821$	7.20987e11			1.58692			0.793459	+	0.608623i	$0.208278\pi$
$822$		0			0
$823$	−5.28868e11			−1.15278			−0.576392	−	0.817174i	$-0.695540\pi$
$823$	−5.28868e11			−1.15278			−0.576392	+	0.817174i	$0.695540\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.45686e11			1.80795			0.903977	−	0.427582i	$-0.140635\pi$
$827$	8.45686e11			1.80795			0.903977	+	0.427582i	$0.140635\pi$
$828$		0			0
$829$			2.55650e11i			0.541288i	0.962680	+	0.270644i	$0.0872366\pi$
$829$			2.55650e11i			0.541288i	−0.962680	+	0.270644i	$0.912763\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.31409e11	−	3.42812e11i	−0.272925	−	0.711993i
$834$		0			0
$835$	1.30989e12			2.69457
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	4.54287e11i		−	0.916817i	−0.888742	−	0.458409i	$-0.848420\pi$
$839$		−	4.54287e11i		−	0.916817i	0.888742	−	0.458409i	$-0.151580\pi$
$840$		0			0
$841$	4.36620e11			0.872809
$842$		0			0
$843$		0			0
$844$		0			0
$845$			2.21846e11i			0.435136i
$846$		0			0
$847$	4.47475e10	+	2.41752e11i	0.0869430	+	0.469718i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.49556e11			0.475827
$852$		0			0
$853$			6.04456e11i			1.14174i	0.821039	+	0.570871i	$0.193395\pi$
$853$			6.04456e11i			1.14174i	−0.821039	+	0.570871i	$0.806605\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	7.79880e10i		−	0.144579i	−0.997384	−	0.0722894i	$-0.976969\pi$
$857$		−	7.79880e10i		−	0.144579i	0.997384	−	0.0722894i	$-0.0230305\pi$
$858$		0			0
$859$			1.42565e11i			0.261843i	0.991393	+	0.130922i	$0.0417936\pi$
$859$			1.42565e11i			0.261843i	−0.991393	+	0.130922i	$0.958206\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5.37615e11			0.969233			0.484616	−	0.874727i	$-0.338959\pi$
$863$	5.37615e11			0.969233			0.484616	+	0.874727i	$0.338959\pi$
$864$		0			0
$865$	−1.81266e12			−3.23781
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.87629e10			−0.138116
$870$		0			0
$871$		−	1.12295e11i		−	0.195114i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.29002e11	+	1.71955e11i	−1.58484	+	0.293347i
$876$		0			0
$877$	1.81440e11			0.306715			0.153357	−	0.988171i	$-0.450991\pi$
$877$	1.81440e11			0.306715			0.153357	+	0.988171i	$0.450991\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			9.93727e11i			1.64954i	0.565467	+	0.824771i	$0.308696\pi$
$881$			9.93727e11i			1.64954i	−0.565467	+	0.824771i	$0.691304\pi$
$882$		0			0
$883$	−1.60514e10			−0.0264040			−0.0132020	−	0.999913i	$-0.504202\pi$
$883$	−1.60514e10			−0.0264040			−0.0132020	+	0.999913i	$0.504202\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	6.10193e11i		−	0.985763i	−0.870096	−	0.492882i	$-0.835944\pi$
$887$		−	6.10193e11i		−	0.985763i	0.870096	−	0.492882i	$-0.164056\pi$
$888$		0			0
$889$	−4.00105e10	−	2.16160e11i	−0.0640570	−	0.346074i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.06333e11			0.324461
$894$		0			0
$895$			9.94516e11i			1.54996i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			9.13989e10i			0.139927i
$900$		0			0
$901$		−	2.54731e11i		−	0.386529i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.08217e12			−3.10400
$906$		0			0
$907$	−5.22879e11			−0.772630			−0.386315	−	0.922367i	$-0.626252\pi$
$907$	−5.22879e11			−0.772630			−0.386315	+	0.922367i	$0.626252\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−5.98988e11			−0.869650			−0.434825	−	0.900515i	$-0.643190\pi$
$911$	−5.98988e11			−0.869650			−0.434825	+	0.900515i	$0.643190\pi$
$912$		0			0
$913$			1.60504e11i			0.230995i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.25657e11	−	1.71336e11i	1.30910	−	0.242309i
$918$		0			0
$919$	2.21058e10			0.0309916			0.0154958	−	0.999880i	$-0.495067\pi$
$919$	2.21058e10			0.0309916			0.0154958	+	0.999880i	$0.495067\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	5.92017e11i		−	0.815694i
$924$		0			0
$925$	−5.55775e11			−0.759157
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	9.56340e11i		−	1.28395i	−0.766724	−	0.641977i	$-0.778115\pi$
$929$		−	9.56340e11i		−	1.28395i	0.766724	−	0.641977i	$-0.221885\pi$
$930$		0			0
$931$	2.80625e11	+	7.32081e11i	0.373532	+	0.974452i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7.22150e11			−0.944889
$936$		0			0
$937$		−	6.08996e11i		−	0.790052i	−0.918670	−	0.395026i	$-0.870736\pi$
$937$		−	6.08996e11i		−	0.790052i	0.918670	−	0.395026i	$-0.129264\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.11301e12i		−	1.41952i	−0.704445	−	0.709759i	$-0.748804\pi$
$941$		−	1.11301e12i		−	1.41952i	0.704445	−	0.709759i	$-0.251196\pi$
$942$		0			0
$943$			6.83041e11i			0.863774i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.71686e11			−0.959491			−0.479745	−	0.877408i	$-0.659271\pi$
$947$	−7.71686e11			−0.959491			−0.479745	+	0.877408i	$0.659271\pi$
$948$		0			0
$949$	−9.71901e11			−1.19828
$950$		0			0
$951$		0			0
$952$		0			0
$953$	2.10736e10			0.0255485			0.0127743	−	0.999918i	$-0.495934\pi$
$953$	2.10736e10			0.0255485			0.0127743	+	0.999918i	$0.495934\pi$
$954$		0			0
$955$			1.18990e12i			1.43053i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.79693e11	+	9.70810e11i	0.212450	+	1.14778i
$960$		0			0
$961$	8.43974e11			0.989545
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2.01638e12i			2.32521i
$966$		0			0
$967$	−2.86699e11			−0.327884			−0.163942	−	0.986470i	$-0.552421\pi$
$967$	−2.86699e11			−0.327884			−0.163942	+	0.986470i	$0.552421\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	5.09523e11i		−	0.573175i	−0.958054	−	0.286587i	$-0.907479\pi$
$971$		−	5.09523e11i		−	0.573175i	0.958054	−	0.286587i	$-0.0925209\pi$
$972$		0			0
$973$	7.78750e11	−	1.44144e11i	0.868854	−	0.160822i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.89697e11			−0.647218			−0.323609	−	0.946191i	$-0.604896\pi$
$977$	−5.89697e11			−0.647218			−0.323609	+	0.946191i	$0.604896\pi$
$978$		0			0
$979$			3.99482e11i			0.434877i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			3.49357e11i			0.374159i	0.982345	+	0.187079i	$0.0599021\pi$
$983$			3.49357e11i			0.374159i	−0.982345	+	0.187079i	$0.940098\pi$
$984$		0			0
$985$			3.06256e12i			3.25341i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.03581e12			2.12791
$990$		0			0
$991$	−9.86530e11			−1.02286			−0.511429	−	0.859325i	$-0.670884\pi$
$991$	−9.86530e11			−1.02286			−0.511429	+	0.859325i	$0.670884\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.20188e11			−0.938824
$996$		0			0
$997$			7.46139e11i			0.755160i	0.925977	+	0.377580i	$0.123244\pi$
$997$			7.46139e11i			0.755160i	−0.925977	+	0.377580i	$0.876756\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.d.c.181.7	yes	8
3.2	odd	2	inner	252.9.d.c.181.1	✓	8
7.6	odd	2	inner	252.9.d.c.181.2	yes	8
21.20	even	2	inner	252.9.d.c.181.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.d.c.181.1	✓	8	3.2	odd	2	inner
252.9.d.c.181.2	yes	8	7.6	odd	2	inner
252.9.d.c.181.7	yes	8	1.1	even	1	trivial
252.9.d.c.181.8	yes	8	21.20	even	2	inner

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1071.65i q^{5} +(-436.993 - 2360.90i) q^{7} +O(q^{10})\)	\(q+1071.65i q^{5} +(-436.993 - 2360.90i) q^{7} +10581.1 q^{11} +24672.2i q^{13} +63685.7i q^{17} -136002. i q^{19} +340275. q^{23} -757811. q^{25} +967918. q^{29} +94428.3i q^{31} +(2.53006e6 - 468304. i) q^{35} +733395. q^{37} +2.00732e6i q^{41} +5.98286e6 q^{43} +1.51713e6i q^{47} +(-5.38287e6 + 2.06339e6i) q^{49} -3.99981e6 q^{53} +1.13393e7i q^{55} -9.18726e6i q^{59} -2.02054e7i q^{61} -2.64400e7 q^{65} -4.55149e6 q^{67} -2.39953e7 q^{71} +3.93925e7i q^{73} +(-4.62388e6 - 2.49810e7i) q^{77} -7.44371e6 q^{79} +1.51689e7i q^{83} -6.82488e7 q^{85} +3.77542e7i q^{89} +(5.82485e7 - 1.07816e7i) q^{91} +1.45747e8 q^{95} +8.92820e7i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6076 q^{7}+O(q^{10}) \) 8 * q + 6076 * q^7	\( 8 q + 6076 q^{7} - 2154400 q^{25} + 2495096 q^{37} + 18340232 q^{43} - 13983424 q^{49} + 12320248 q^{67} + 53237896 q^{79} - 79347240 q^{85} + 257358192 q^{91}+O(q^{100}) \) 8 * q + 6076 * q^7 - 2154400 * q^25 + 2495096 * q^37 + 18340232 * q^43 - 13983424 * q^49 + 12320248 * q^67 + 53237896 * q^79 - 79347240 * q^85 + 257358192 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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