LMFDB - Embedded newform 980.2.q.i.949.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,2,Mod(569,980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("980.569"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.31116960000.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{6} - 8x^{4} + 9x^{2} + 81 $ x^8 + x^6 - 8x^4 + 9x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			949.1
Root			$-0.306808 + 1.70466i$ of defining polynomial
Character	$\chi$	$=$	980.949
Dual form			980.2.q.i.569.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.95256 - 1.70466i) q^{3} +(-1.93649 + 1.11803i) q^{5} +(4.31174 + 7.46815i) q^{9} +(1.81174 - 3.13802i) q^{11} -1.06281i q^{13} +7.62348 q^{15} +(4.98469 + 2.87791i) q^{17} +(2.50000 - 4.33013i) q^{25} -19.1722i q^{27} -9.62348 q^{29} +(-10.6985 + 6.17680i) q^{33} +(-1.81174 + 3.13802i) q^{39} +(-16.6993 - 9.64134i) q^{45} +(1.11171 - 0.641847i) q^{47} +(-9.81174 - 16.9944i) q^{51} +8.10234i q^{55} +(1.18826 + 2.05813i) q^{65} -12.0000 q^{71} +(-11.6190 - 6.70820i) q^{73} +(-14.7628 + 8.52330i) q^{75} +(-7.43521 - 12.8782i) q^{79} +(-19.7470 + 34.2027i) q^{81} -8.94427i q^{83} -12.8704 q^{85} +(28.4139 + 16.4048i) q^{87} -19.3931i q^{97} +31.2470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{9} - 6 q^{11} + 20 q^{15} + 20 q^{25} - 36 q^{29} + 6 q^{39} - 58 q^{51} + 30 q^{65} - 96 q^{71} + 2 q^{79} - 76 q^{81} + 20 q^{85} + 168 q^{99}+O(q^{100}) $ 8 * q + 14 * q^9 - 6 * q^11 + 20 * q^15 + 20 * q^25 - 36 * q^29 + 6 * q^39 - 58 * q^51 + 30 * q^65 - 96 * q^71 + 2 * q^79 - 76 * q^81 + 20 * q^85 + 168 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$

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
$3$	−2.95256	−	1.70466i	−1.70466	−	0.984186i	−0.940898	−	0.338689i	$-0.890016\pi$
$3$	−2.95256	−	1.70466i	−1.70466	−	0.984186i	−0.763763	−	0.645497i	$-0.776650\pi$
$4$		0			0
$5$	−1.93649	+	1.11803i	−0.866025	+	0.500000i
$5$	−1.93649	+	1.11803i	−0.866025	+	0.500000i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	4.31174	+	7.46815i	1.43725	+	2.48938i
$10$		0			0
$11$	1.81174	−	3.13802i	0.546259	−	0.946149i	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	1.81174	−	3.13802i	0.546259	−	0.946149i	0.998526	−	0.0542666i	$-0.0172821\pi$
$12$		0			0
$13$		−	1.06281i		−	0.294772i	−0.989079	−	0.147386i	$-0.952914\pi$
$13$		−	1.06281i		−	0.294772i	0.989079	−	0.147386i	$-0.0470859\pi$
$14$		0			0
$15$	7.62348			1.96837
$16$		0			0
$17$	4.98469	+	2.87791i	1.20897	+	0.697997i	0.962533	−	0.271163i	$-0.0874083\pi$
$17$	4.98469	+	2.87791i	1.20897	+	0.697997i	0.246433	+	0.969160i	$0.420742\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		−	19.1722i		−	3.68970i
$28$		0			0
$29$	−9.62348			−1.78703			−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−9.62348			−1.78703			−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$	−10.6985	+	6.17680i	−1.86237	+	1.07524i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$	−1.81174	+	3.13802i	−0.290110	+	0.502486i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−16.6993	−	9.64134i	−2.48938	−	1.43725i
$46$		0			0
$47$	1.11171	−	0.641847i	0.162160	−	0.0936230i	−0.416724	−	0.909033i	$-0.636822\pi$
$47$	1.11171	−	0.641847i	0.162160	−	0.0936230i	0.578884	+	0.815410i	$0.303489\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−9.81174	−	16.9944i	−1.37392	−	2.37970i
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$			8.10234i			1.09252i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.18826	+	2.05813i	0.147386	+	0.255280i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	−11.6190	−	6.70820i	−1.35990	−	0.785136i	−0.370286	−	0.928918i	$-0.620740\pi$
$73$	−11.6190	−	6.70820i	−1.35990	−	0.785136i	−0.989609	+	0.143782i	$0.954074\pi$
$74$		0			0
$75$	−14.7628	+	8.52330i	−1.70466	+	0.984186i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−7.43521	−	12.8782i	−0.836527	−	1.44891i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−7.43521	−	12.8782i	−0.836527	−	1.44891i	0.0562544	−	0.998416i	$-0.482084\pi$
$80$		0			0
$81$	−19.7470	+	34.2027i	−2.19411	+	3.80030i
$82$		0			0
$83$		−	8.94427i		−	0.981761i	−0.871227	−	0.490881i	$-0.836675\pi$
$83$		−	8.94427i		−	0.981761i	0.871227	−	0.490881i	$-0.163325\pi$
$84$		0			0
$85$	−12.8704			−1.39599
$86$		0			0
$87$	28.4139	+	16.4048i	3.04629	+	1.75878i
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	19.3931i		−	1.96907i	−0.175180	−	0.984536i	$-0.556051\pi$
$97$		−	19.3931i		−	1.96907i	0.175180	−	0.984536i	$-0.443949\pi$
$98$		0			0
$99$	31.2470			3.14044
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	10.6985	−	6.17680i	1.05416	−	0.608618i	0.130347	−	0.991468i	$-0.458391\pi$
$103$	10.6985	−	6.17680i	1.05416	−	0.608618i	0.923810	+	0.382851i	$0.125058\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	10.4352	−	18.0743i	0.999512	−	1.73121i	0.472708	−	0.881219i	$-0.343277\pi$
$109$	10.4352	−	18.0743i	0.999512	−	1.73121i	0.526804	−	0.849987i	$-0.323390\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.93725	−	4.58258i	0.733799	−	0.423659i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.06479	−	1.84427i	−0.0967988	−	0.167660i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	21.4352	+	37.1269i	1.84485	+	3.19537i
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−4.37652			−0.368570
$142$		0			0
$143$	−3.33513	−	1.92554i	−0.278898	−	0.161022i
$144$		0			0
$145$	18.6358	−	10.7594i	1.54762	−	0.893517i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	3.43521	−	5.94996i	0.279554	−	0.484201i	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	3.43521	−	5.94996i	0.279554	−	0.484201i	0.971274	+	0.237964i	$0.0764802\pi$
$152$		0			0
$153$			49.6353i			4.01277i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−11.6190	−	6.70820i	−0.927293	−	0.535373i	−0.0413387	−	0.999145i	$-0.513162\pi$
$157$	−11.6190	−	6.70820i	−0.927293	−	0.535373i	−0.885954	+	0.463772i	$0.846496\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$	13.8117	−	23.9226i	1.07524	−	1.86237i
$166$		0			0
$167$			19.1722i			1.48359i	0.670625	+	0.741796i	$0.266026\pi$
$167$			19.1722i			1.48359i	−0.670625	+	0.741796i	$0.733974\pi$
$168$		0			0
$169$	11.8704			0.913110
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−22.7001	+	13.1059i	−1.72585	+	0.996422i	−0.820666	+	0.571409i	$0.806397\pi$
$173$	−22.7001	+	13.1059i	−1.72585	+	0.996422i	−0.905187	+	0.425013i	$0.860270\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000	−	20.7846i	0.896922	−	1.55351i	0.0655145	−	0.997852i	$-0.479131\pi$
$179$	12.0000	−	20.7846i	0.896922	−	1.55351i	0.831408	−	0.555663i	$-0.187536\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	18.0619	−	10.4281i	1.32082	−	0.762575i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.18826	−	7.25428i	−0.303052	−	0.524902i	0.673774	−	0.738938i	$-0.264672\pi$
$191$	−4.18826	−	7.25428i	−0.303052	−	0.524902i	−0.976826	+	0.214036i	$0.931339\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		−	8.10234i		−	0.580220i
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−26.8704			−1.84984			−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−26.8704			−1.84984			−0.924918	+	0.380166i	$0.875867\pi$
$212$		0			0
$213$	35.4307	+	20.4559i	2.42767	+	1.40162i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	22.8704	+	39.6127i	1.54544	+	2.67678i
$220$		0			0
$221$	3.05869	−	5.29780i	0.205750	−	0.356369i
$222$		0			0
$223$		−	28.5583i		−	1.91240i	−0.292709	−	0.956202i	$-0.594557\pi$
$223$		−	28.5583i		−	1.91240i	0.292709	−	0.956202i	$-0.405443\pi$
$224$		0			0
$225$	43.1174			2.87449
$226$		0			0
$227$	6.63426	+	3.83029i	0.440331	+	0.254225i	0.703738	−	0.710460i	$-0.251513\pi$
$227$	6.63426	+	3.83029i	0.440331	+	0.254225i	−0.263407	+	0.964685i	$0.584846\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$	−1.43521	+	2.48586i	−0.0936230	+	0.162160i
$236$		0			0
$237$			50.6981i			3.29319i
$238$		0			0
$239$	−30.1174			−1.94813			−0.974066	−	0.226266i	$-0.927348\pi$
$239$	−30.1174			−1.94813			−0.974066	+	0.226266i	$0.927348\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$		0			0
$243$	66.7972	−	38.5654i	4.28504	−	2.47397i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−15.2470	+	26.4085i	−0.966236	+	1.67357i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	38.0007	+	21.9397i	2.37970	+	1.37392i
$256$		0			0
$257$	−3.87298	+	2.23607i	−0.241590	+	0.139482i	−0.615907	−	0.787819i	$-0.711210\pi$
$257$	−3.87298	+	2.23607i	−0.241590	+	0.139482i	0.374317	+	0.927301i	$0.377877\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−41.4939	−	71.8695i	−2.56841	−	4.44861i
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.05869	−	15.6901i	−0.546259	−	0.946149i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	21.6235			1.28995			0.644974	−	0.764204i	$-0.276868\pi$
$281$	21.6235			1.28995			0.644974	+	0.764204i	$0.276868\pi$
$282$		0			0
$283$	−12.5394	−	7.23961i	−0.745388	−	0.430350i	0.0786368	−	0.996903i	$-0.474943\pi$
$283$	−12.5394	−	7.23961i	−0.745388	−	0.430350i	−0.824025	+	0.566553i	$0.808277\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.06479	+	13.9686i	0.474399	+	0.821684i
$290$		0			0
$291$	−33.0587	+	57.2593i	−1.93793	+	3.35660i
$292$		0			0
$293$		−	8.32322i		−	0.486248i	−0.969995	−	0.243124i	$-0.921828\pi$
$293$		−	8.32322i		−	0.486248i	0.969995	−	0.243124i	$-0.0781721\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−60.1629	−	34.7351i	−3.49101	−	2.01553i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			26.4326i			1.50859i	0.656535	+	0.754295i	$0.272021\pi$
$307$			26.4326i			1.50859i	−0.656535	+	0.754295i	$0.727979\pi$
$308$		0			0
$309$	−42.1174			−2.39597
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−13.1132	+	7.57093i	−0.741204	+	0.427934i	−0.822507	−	0.568755i	$-0.807425\pi$
$313$	−13.1132	+	7.57093i	−0.741204	+	0.427934i	0.0813030	+	0.996689i	$0.474092\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	−17.4352	+	30.1987i	−0.976185	+	1.69080i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−4.60212	−	2.65704i	−0.255280	−	0.147386i
$326$		0			0
$327$	−61.6212	+	35.5770i	−3.40766	+	1.96741i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$	−20.3765			−1.08762
$352$		0			0
$353$	30.4460	+	17.5780i	1.62048	+	0.935583i	0.986792	+	0.161993i	$0.0517922\pi$
$353$	30.4460	+	17.5780i	1.62048	+	0.935583i	0.633686	+	0.773590i	$0.281541\pi$
$354$		0			0
$355$	23.2379	−	13.4164i	1.23334	−	0.712069i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.0000	−	31.1769i	−0.950004	−	1.64545i	−0.745409	−	0.666608i	$-0.767746\pi$
$359$	−18.0000	−	31.1769i	−0.950004	−	1.64545i	−0.204595	−	0.978847i	$-0.565588\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$		0			0
$363$			7.26040i			0.381072i
$364$		0			0
$365$	30.0000			1.57027
$366$		0			0
$367$	−28.4139	−	16.4048i	−1.48319	−	0.856322i	−0.483375	−	0.875413i	$-0.660589\pi$
$367$	−28.4139	−	16.4048i	−1.48319	−	0.856322i	−0.999818	+	0.0190919i	$0.993923\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	19.0587	−	33.0106i	0.984186	−	1.70466i
$376$		0			0
$377$			10.2280i			0.526767i
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−30.9839	+	17.8885i	−1.58320	+	0.914062i	−0.588813	+	0.808269i	$0.700405\pi$
$383$	−30.9839	+	17.8885i	−1.58320	+	0.914062i	−0.994388	+	0.105793i	$0.966262\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.18826	−	12.4504i	0.364459	−	0.631262i	−0.624230	−	0.781241i	$-0.714587\pi$
$389$	7.18826	−	12.4504i	0.364459	−	0.631262i	0.988689	+	0.149979i	$0.0479205\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	28.7965	+	16.6256i	1.44891	+	0.836527i
$396$		0			0
$397$	34.5103	−	19.9245i	1.73202	−	0.999983i	0.863112	−	0.505013i	$-0.168512\pi$
$397$	34.5103	−	19.9245i	1.73202	−	0.999983i	0.868910	−	0.494971i	$-0.164821\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.05869	+	10.4940i	0.302556	+	0.524043i	0.976714	−	0.214544i	$-0.0688266\pi$
$401$	6.05869	+	10.4940i	0.302556	+	0.524043i	−0.674158	+	0.738587i	$0.735493\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	88.3110i		−	4.38821i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.0000	+	17.3205i	0.490881	+	0.850230i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−3.12957			−0.152526			−0.0762630	−	0.997088i	$-0.524299\pi$
$421$	−3.12957			−0.152526			−0.0762630	+	0.997088i	$0.524299\pi$
$422$		0			0
$423$	9.58681	+	5.53495i	0.466127	+	0.269118i
$424$		0			0
$425$	24.9235	−	14.3896i	1.20897	−	0.697997i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	6.56479	+	11.3705i	0.316951	+	0.548975i
$430$		0			0
$431$	−18.6822	+	32.3585i	−0.899888	+	1.55865i	−0.0722525	+	0.997386i	$0.523019\pi$
$431$	−18.6822	+	32.3585i	−0.899888	+	1.55865i	−0.827636	+	0.561266i	$0.810315\pi$
$432$		0			0
$433$		−	40.2492i		−	1.93425i	−0.254293	−	0.967127i	$-0.581843\pi$
$433$		−	40.2492i		−	1.93425i	0.254293	−	0.967127i	$-0.418157\pi$
$434$		0			0
$435$	−73.3643			−3.51755
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	20.4559i		−	0.967532i
$448$		0			0
$449$	31.3643			1.48017			0.740087	−	0.672511i	$-0.234784\pi$
$449$	31.3643			1.48017			0.740087	+	0.672511i	$0.234784\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−20.2853	+	11.7117i	−0.953088	+	0.550266i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$	55.1761	−	95.5677i	2.57540	−	4.46072i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−24.3496	+	14.0583i	−1.12677	+	0.650538i	−0.943119	−	0.332454i	$-0.892123\pi$
$467$	−24.3496	+	14.0583i	−1.12677	+	0.650538i	−0.183646	+	0.982992i	$0.558790\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	22.8704	+	39.6127i	1.05381	+	1.82526i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	21.6822	+	37.5546i	0.984536	+	1.70527i
$486$		0			0
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	42.1174			1.90073			0.950365	−	0.311136i	$-0.100710\pi$
$491$	42.1174			1.90073			0.950365	+	0.311136i	$0.100710\pi$
$492$		0			0
$493$	−47.9701	−	27.6955i	−2.16046	−	1.24734i
$494$		0			0
$495$	−60.5095	+	34.9352i	−2.71970	+	1.57022i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.56479	−	4.44234i	−0.114816	−	0.198867i	0.802890	−	0.596127i	$-0.203294\pi$
$499$	−2.56479	−	4.44234i	−0.114816	−	0.198867i	−0.917706	+	0.397260i	$0.869961\pi$
$500$		0			0
$501$	32.6822	−	56.6072i	1.46013	−	2.52902i
$502$		0			0
$503$		−	39.6282i		−	1.76693i	−0.468495	−	0.883466i	$-0.655203\pi$
$503$		−	39.6282i		−	1.76693i	0.468495	−	0.883466i	$-0.344797\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−35.0481	−	20.2351i	−1.55654	−	0.898670i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.8117	+	23.9226i	−0.608618	+	1.05416i
$516$		0			0
$517$		−	4.65143i		−	0.204570i
$518$		0			0
$519$	89.3643			3.92266
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	23.2379	−	13.4164i	1.01612	−	0.586659i	0.103144	−	0.994666i	$-0.467110\pi$
$523$	23.2379	−	13.4164i	1.01612	−	0.586659i	0.912978	+	0.408008i	$0.133776\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−70.8614	+	40.9119i	−3.05790	+	1.76548i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	18.4352	+	31.9307i	0.792592	+	1.37281i	0.924357	+	0.381528i	$0.124602\pi$
$541$	18.4352	+	31.9307i	0.792592	+	1.37281i	−0.131765	+	0.991281i	$0.542065\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			46.6677i			1.99902i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−71.1052			−3.00206
$562$		0			0
$563$	−38.7298	−	22.3607i	−1.63227	−	0.942390i	−0.983392	−	0.181496i	$-0.941906\pi$
$563$	−38.7298	−	22.3607i	−1.63227	−	0.942390i	−0.648876	−	0.760894i	$-0.724761\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$		0			0
$573$			28.5583i			1.19304i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	11.2724	+	6.50812i	0.469276	+	0.270936i	0.715936	−	0.698165i	$-0.246000\pi$
$577$	11.2724	+	6.50812i	0.469276	+	0.270936i	−0.246661	+	0.969102i	$0.579333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−10.2470	+	17.7482i	−0.423659	+	0.733799i
$586$		0			0
$587$		−	8.94427i		−	0.369170i	−0.982817	−	0.184585i	$-0.940906\pi$
$587$		−	8.94427i		−	0.369170i	0.982817	−	0.184585i	$-0.0590940\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.537850	+	0.310528i	−0.0220868	+	0.0127518i	−0.511003	−	0.859579i	$-0.670726\pi$
$593$	−0.537850	+	0.310528i	−0.0220868	+	0.0127518i	0.488916	+	0.872331i	$0.337392\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.05869	+	5.29780i	−0.124975	+	0.216462i	−0.921723	−	0.387849i	$-0.873218\pi$
$599$	−3.05869	+	5.29780i	−0.124975	+	0.216462i	0.796748	+	0.604311i	$0.206552\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.12390	+	2.38094i	0.167660	+	0.0967988i
$606$		0			0
$607$	42.4475	−	24.5071i	1.72289	−	0.994712i	0.810097	−	0.586296i	$-0.199414\pi$
$607$	42.4475	−	24.5071i	1.72289	−	0.994712i	0.912796	−	0.408416i	$-0.133919\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.682164	−	1.18154i	−0.0275974	−	0.0478001i
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	38.8704			1.54741			0.773704	−	0.633548i	$-0.218402\pi$
$631$	38.8704			1.54741			0.773704	+	0.633548i	$0.218402\pi$
$632$		0			0
$633$	79.3365	+	45.8050i	3.15334	+	1.82058i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−51.7409	−	89.6178i	−2.04684	−	3.54522i
$640$		0			0
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	−0.987200	−	0.159489i	$-0.949015\pi$
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	0.631721	+	0.775196i	$0.282349\pi$
$642$		0			0
$643$		−	46.8886i		−	1.84910i	−0.381055	−	0.924552i	$-0.624439\pi$
$643$		−	46.8886i		−	1.84910i	0.381055	−	0.924552i	$-0.375561\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	15.4919	+	8.94427i	0.609051	+	0.351636i	0.772594	−	0.634901i	$-0.218959\pi$
$647$	15.4919	+	8.94427i	0.609051	+	0.351636i	−0.163543	+	0.986536i	$0.552292\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	115.696i		−	4.51373i
$658$		0			0
$659$	−20.3765			−0.793757			−0.396878	−	0.917871i	$-0.629907\pi$
$659$	−20.3765			−0.793757			−0.396878	+	0.917871i	$0.629907\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$		0			0
$663$	−18.0619	+	10.4281i	−0.701467	+	0.404992i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−48.6822	+	84.3200i	−1.88216	+	3.26000i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−83.0182	−	47.9306i	−3.19537	−	1.84485i
$676$		0			0
$677$	28.2226	−	16.2943i	1.08468	−	0.626242i	0.152527	−	0.988299i	$-0.451259\pi$
$677$	28.2226	−	16.2943i	1.08468	−	0.626242i	0.932156	+	0.362058i	$0.117926\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−13.0587	−	22.6183i	−0.500410	−	0.866736i
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−19.3643			−0.731381			−0.365690	−	0.930737i	$-0.619167\pi$
$701$	−19.3643			−0.731381			−0.365690	+	0.930737i	$0.619167\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	8.47510	−	4.89310i	0.319191	−	0.184285i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	23.3056	+	40.3666i	0.875262	+	1.51600i	0.856484	+	0.516174i	$0.172644\pi$
$709$	23.3056	+	40.3666i	0.875262	+	1.51600i	0.0187779	+	0.999824i	$0.494022\pi$
$710$		0			0
$711$	64.1174	−	111.055i	2.40459	−	4.16487i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	8.61128			0.322044
$716$		0			0
$717$	88.9233	+	51.3399i	3.32090	+	1.91732i
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−24.0587	+	41.6709i	−0.893517	+	1.54762i
$726$		0			0
$727$			53.6656i			1.99035i	0.0981255	+	0.995174i	$0.468715\pi$
$727$			53.6656i			1.99035i	−0.0981255	+	0.995174i	$0.531285\pi$
$728$		0			0
$729$	−144.482			−5.35117
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−28.9877	+	16.7361i	−1.07069	+	0.618161i	−0.928369	−	0.371660i	$-0.878789\pi$
$733$	−28.9877	+	16.7361i	−1.07069	+	0.618161i	−0.142318	+	0.989821i	$0.545455\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−20.3056	+	35.1704i	−0.746955	+	1.29376i	0.202321	+	0.979319i	$0.435152\pi$
$739$	−20.3056	+	35.1704i	−0.746955	+	1.29376i	−0.949276	+	0.314445i	$0.898182\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	−11.6190	−	6.70820i	−0.425685	−	0.245770i
$746$		0			0
$747$	66.7972	−	38.5654i	2.44398	−	1.41103i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.3056	−	45.5627i	−0.959906	−	1.66261i	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	−26.3056	−	45.5627i	−0.959906	−	1.66261i	−0.237188	−	0.971464i	$-0.576226\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			15.3627i			0.559107i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−55.4939	−	96.1183i	−2.00639	−	3.47516i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	15.2470			0.549106
$772$		0			0
$773$	−42.6388	−	24.6175i	−1.53361	−	0.885431i	−0.999191	−	0.0402129i	$-0.987196\pi$
$773$	−42.6388	−	24.6175i	−1.53361	−	0.885431i	−0.534421	−	0.845218i	$-0.679470\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−21.7409	+	37.6563i	−0.777949	+	1.34745i
$782$		0			0
$783$			184.504i			6.59362i
$784$		0			0
$785$	30.0000			1.07075
$786$		0			0
$787$	19.2096	+	11.0907i	0.684750	+	0.395340i	0.801642	−	0.597804i	$-0.203960\pi$
$787$	19.2096	+	11.0907i	0.684750	+	0.395340i	−0.116892	+	0.993145i	$0.537293\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			21.0770i			0.746585i	0.927714	+	0.373293i	$0.121771\pi$
$797$			21.0770i			0.746585i	−0.927714	+	0.373293i	$0.878229\pi$
$798$		0			0
$799$	7.38872			0.261394
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−42.1010	+	24.3070i	−1.48571	+	0.857776i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.6822	−	47.9469i	0.973253	−	1.68572i	0.287670	−	0.957730i	$-0.407120\pi$
$809$	27.6822	−	47.9469i	0.973253	−	1.68572i	0.685583	−	0.727994i	$-0.259547\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	16.8117	+	29.1188i	0.586734	+	1.01625i	0.994657	+	0.103236i	$0.0329198\pi$
$821$	16.8117	+	29.1188i	0.586734	+	1.01625i	−0.407923	+	0.913016i	$0.633747\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$			61.7680i			2.15048i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$829$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−21.4352	−	37.1269i	−0.741796	−	1.28483i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	63.6113			2.19349
$842$		0			0
$843$	−63.8446	−	36.8607i	−2.19892	−	1.26955i
$844$		0			0
$845$	−22.9870	+	13.2715i	−0.790776	+	0.456555i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	24.6822	+	42.7508i	0.847090	+	1.46720i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	40.2492i		−	1.37811i	−0.724710	−	0.689054i	$-0.758026\pi$
$853$		−	40.2492i		−	1.37811i	0.724710	−	0.689054i	$-0.241974\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	42.6028	+	24.5967i	1.45528	+	0.840209i	0.998774	−	0.0495090i	$-0.0157656\pi$
$857$	42.6028	+	24.5967i	1.45528	+	0.840209i	0.456511	+	0.889718i	$0.349099\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$	29.3056	−	50.7589i	0.996422	−	1.72585i
$866$		0			0
$867$		−	54.9909i		−	1.86759i
$868$		0			0
$869$	−53.8826			−1.82784
$870$		0			0
$871$		0			0
$872$		0			0
$873$	144.831	−	83.6180i	4.90178	−	2.83004i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	−14.1883	+	24.5748i	−0.478558	+	0.828887i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−30.9839	+	17.8885i	−1.04034	+	0.600639i	−0.919929	−	0.392086i	$-0.871754\pi$
$887$	−30.9839	+	17.8885i	−1.04034	+	0.600639i	−0.120408	+	0.992725i	$0.538420\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	71.5526	+	123.933i	2.39710	+	4.15190i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			53.6656i			1.79384i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$	−28.0673	−	16.2047i	−0.928893	−	0.536296i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	30.3056	+	52.4909i	0.999691	+	1.73152i	0.521380	+	0.853325i	$0.325417\pi$
$919$	30.3056	+	52.4909i	0.999691	+	1.73152i	0.478311	+	0.878191i	$0.341249\pi$
$920$		0			0
$921$	45.0587	−	78.0439i	1.48473	−	2.57164i
$922$		0			0
$923$			12.7538i			0.419795i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	92.2585	+	53.2655i	3.03017	+	1.74947i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−23.3178	+	40.3877i	−0.762575	+	1.32082i
$936$		0			0
$937$		−	56.0537i		−	1.83120i	−0.402096	−	0.915598i	$-0.631718\pi$
$937$		−	56.0537i		−	1.83120i	0.402096	−	0.915598i	$-0.368282\pi$
$938$		0			0
$939$	51.6235			1.68467
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−7.12957	+	12.3488i	−0.231436	+	0.400858i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	16.2211	+	9.36524i	0.524902	+	0.303052i
$956$		0			0
$957$	102.957	−	59.4423i	3.32813	−	1.92150i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	9.05869	+	15.6901i	0.290110	+	0.502486i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	179.976			5.74618
$982$		0			0
$983$	54.2578	+	31.3257i	1.73055	+	0.999136i	0.886057	+	0.463577i	$0.153434\pi$
$983$	54.2578	+	31.3257i	1.73055	+	0.999136i	0.844498	+	0.535559i	$0.179899\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	26.0000	−	45.0333i	0.825917	−	1.43053i	−0.0752991	−	0.997161i	$-0.523991\pi$
$991$	26.0000	−	45.0333i	0.825917	−	1.43053i	0.901216	−	0.433370i	$-0.142676\pi$
$992$		0			0
$993$			27.2746i			0.865532i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	40.0328	+	23.1130i	1.26785	+	0.731995i	0.974581	−	0.224034i	$-0.0719228\pi$
$997$	40.0328	+	23.1130i	1.26785	+	0.731995i	0.293271	+	0.956029i	$0.405256\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.q.i.949.1		8
5.4	even	2	inner	980.2.q.i.949.4		8
7.2	even	3	inner	980.2.q.i.569.4		8
7.3	odd	6		980.2.e.d.589.1	✓	4
7.4	even	3		980.2.e.d.589.4	yes	4
7.5	odd	6	inner	980.2.q.i.569.1		8
7.6	odd	2	inner	980.2.q.i.949.4		8
35.3	even	12		4900.2.a.bj.1.4		4
35.4	even	6		980.2.e.d.589.1	✓	4
35.9	even	6	inner	980.2.q.i.569.1		8
35.17	even	12		4900.2.a.bj.1.1		4
35.18	odd	12		4900.2.a.bj.1.1		4
35.19	odd	6	inner	980.2.q.i.569.4		8
35.24	odd	6		980.2.e.d.589.4	yes	4
35.32	odd	12		4900.2.a.bj.1.4		4
35.34	odd	2	CM	980.2.q.i.949.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.e.d.589.1	✓	4	7.3	odd	6
980.2.e.d.589.1	✓	4	35.4	even	6
980.2.e.d.589.4	yes	4	7.4	even	3
980.2.e.d.589.4	yes	4	35.24	odd	6
980.2.q.i.569.1		8	7.5	odd	6	inner
980.2.q.i.569.1		8	35.9	even	6	inner
980.2.q.i.569.4		8	7.2	even	3	inner
980.2.q.i.569.4		8	35.19	odd	6	inner
980.2.q.i.949.1		8	1.1	even	1	trivial
980.2.q.i.949.1		8	35.34	odd	2	CM
980.2.q.i.949.4		8	5.4	even	2	inner
980.2.q.i.949.4		8	7.6	odd	2	inner
4900.2.a.bj.1.1		4	35.17	even	12
4900.2.a.bj.1.1		4	35.18	odd	12
4900.2.a.bj.1.4		4	35.3	even	12
4900.2.a.bj.1.4		4	35.32	odd	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.95256 - 1.70466i) q^{3} +(-1.93649 + 1.11803i) q^{5} +(4.31174 + 7.46815i) q^{9} +(1.81174 - 3.13802i) q^{11} -1.06281i q^{13} +7.62348 q^{15} +(4.98469 + 2.87791i) q^{17} +(2.50000 - 4.33013i) q^{25} -19.1722i q^{27} -9.62348 q^{29} +(-10.6985 + 6.17680i) q^{33} +(-1.81174 + 3.13802i) q^{39} +(-16.6993 - 9.64134i) q^{45} +(1.11171 - 0.641847i) q^{47} +(-9.81174 - 16.9944i) q^{51} +8.10234i q^{55} +(1.18826 + 2.05813i) q^{65} -12.0000 q^{71} +(-11.6190 - 6.70820i) q^{73} +(-14.7628 + 8.52330i) q^{75} +(-7.43521 - 12.8782i) q^{79} +(-19.7470 + 34.2027i) q^{81} -8.94427i q^{83} -12.8704 q^{85} +(28.4139 + 16.4048i) q^{87} -19.3931i q^{97} +31.2470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{9} - 6 q^{11} + 20 q^{15} + 20 q^{25} - 36 q^{29} + 6 q^{39} - 58 q^{51} + 30 q^{65} - 96 q^{71} + 2 q^{79} - 76 q^{81} + 20 q^{85} + 168 q^{99}+O(q^{100}) \) 8 * q + 14 * q^9 - 6 * q^11 + 20 * q^15 + 20 * q^25 - 36 * q^29 + 6 * q^39 - 58 * q^51 + 30 * q^65 - 96 * q^71 + 2 * q^79 - 76 * q^81 + 20 * q^85 + 168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more