LMFDB - Embedded newform 980.2.q.i.949.3

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			949.3
Root			$-1.62968 - 0.586627i$ of defining polynomial
Character	$\chi$	$=$	980.949
Dual form			980.2.q.i.569.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.01607 + 0.586627i) q^{3} +(-1.93649 + 1.11803i) q^{5} +(-0.811738 - 1.40597i) q^{9} +(-3.31174 + 5.73610i) q^{11} -5.64539i q^{13} -2.62348 q^{15} +(-6.92119 - 3.99595i) q^{17} +(2.50000 - 4.33013i) q^{25} -5.42451i q^{27} +0.623475 q^{29} +(-6.72990 + 3.88551i) q^{33} +(3.31174 - 5.73610i) q^{39} +(3.14385 + 1.81510i) q^{45} +(-10.7942 + 6.23202i) q^{47} +(-4.68826 - 8.12031i) q^{51} -14.8105i q^{55} +(6.31174 + 10.9323i) q^{65} -12.0000 q^{71} +(-11.6190 - 6.70820i) q^{73} +(5.08034 - 2.93313i) q^{75} +(7.93521 + 13.7442i) q^{79} +(0.746951 - 1.29376i) q^{81} -8.94427i q^{83} +17.8704 q^{85} +(0.633493 + 0.365747i) q^{87} +12.6849i q^{97} +10.7530 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.01607	+	0.586627i	0.586627	+	0.338689i	0.763763	−	0.645497i	$-0.223350\pi$
$3$	1.01607	+	0.586627i	0.586627	+	0.338689i	−0.177136	+	0.984186i	$0.556683\pi$
$4$		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
$9$	−0.811738	−	1.40597i	−0.270579	−	0.468657i
$10$		0			0
$11$	−3.31174	+	5.73610i	−0.998526	+	1.72950i	−0.452267	+	0.891883i	$0.649385\pi$
$11$	−3.31174	+	5.73610i	−0.998526	+	1.72950i	−0.546259	+	0.837616i	$0.683949\pi$
$12$		0			0
$13$		−	5.64539i		−	1.56575i	−0.622179	−	0.782875i	$-0.713753\pi$
$13$		−	5.64539i		−	1.56575i	0.622179	−	0.782875i	$-0.286247\pi$
$14$		0			0
$15$	−2.62348			−0.677378
$16$		0			0
$17$	−6.92119	−	3.99595i	−1.67863	−	0.969160i	−0.962533	−	0.271163i	$-0.912592\pi$
$17$	−6.92119	−	3.99595i	−1.67863	−	0.969160i	−0.716101	−	0.697997i	$-0.754075\pi$
$18$		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
$27$		−	5.42451i		−	1.04395i
$28$		0			0
$29$	0.623475			0.115776			0.0578882	−	0.998323i	$-0.481563\pi$
$29$	0.623475			0.115776			0.0578882	+	0.998323i	$0.481563\pi$
$30$		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
$33$	−6.72990	+	3.88551i	−1.17153	+	0.676380i
$34$		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
$39$	3.31174	−	5.73610i	0.530302	−	0.918511i
$40$		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
$45$	3.14385	+	1.81510i	0.468657	+	0.270579i
$46$		0			0
$47$	−10.7942	+	6.23202i	−1.57449	+	0.909033i	−0.578884	+	0.815410i	$0.696511\pi$
$47$	−10.7942	+	6.23202i	−1.57449	+	0.909033i	−0.995608	+	0.0936230i	$0.970155\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−4.68826	−	8.12031i	−0.656488	−	1.13707i
$52$		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
$55$		−	14.8105i		−	1.99705i
$56$		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
$65$	6.31174	+	10.9323i	0.782875	+	1.35598i
$66$		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
$75$	5.08034	−	2.93313i	0.586627	−	0.338689i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	7.93521	+	13.7442i	0.892781	+	1.54634i	0.836527	+	0.547926i	$0.184582\pi$
$79$	7.93521	+	13.7442i	0.892781	+	1.54634i	0.0562544	+	0.998416i	$0.482084\pi$
$80$		0			0
$81$	0.746951	−	1.29376i	0.0829945	−	0.143751i
$82$		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
$85$	17.8704			1.93832
$86$		0			0
$87$	0.633493	+	0.365747i	0.0679176	+	0.0392122i
$88$		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
$97$			12.6849i			1.28796i	0.765043	+	0.643979i	$0.222718\pi$
$97$			12.6849i			1.28796i	−0.765043	+	0.643979i	$0.777282\pi$
$98$		0			0
$99$	10.7530			1.08072
$100$		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
$103$	6.72990	−	3.88551i	0.663117	−	0.382851i	−0.130347	−	0.991468i	$-0.541609\pi$
$103$	6.72990	−	3.88551i	0.663117	−	0.382851i	0.793463	+	0.608618i	$0.208276\pi$
$104$		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
$109$	−4.93521	+	8.54804i	−0.472708	+	0.818754i	−0.999512	−	0.0312328i	$-0.990057\pi$
$109$	−4.93521	+	8.54804i	−0.472708	+	0.818754i	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
$121$	−16.4352	−	28.4666i	−1.49411	−	2.58787i
$122$		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
$135$	6.06479	+	10.5045i	0.521974	+	0.904085i
$136$		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
$141$	−14.6235			−1.23152
$142$		0			0
$143$	32.3825	+	18.6961i	2.70796	+	1.56344i
$144$		0			0
$145$	−1.20735	+	0.697067i	−0.100265	+	0.0578882i
$146$		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
$151$	−11.9352	+	20.6724i	−0.971274	+	1.68230i	−0.279554	+	0.960130i	$0.590186\pi$
$151$	−11.9352	+	20.6724i	−0.971274	+	1.68230i	−0.691720	+	0.722166i	$0.743147\pi$
$152$		0			0
$153$			12.9746i			1.04894i
$154$		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
$165$	8.68826	−	15.0485i	0.676380	−	1.17153i
$166$		0			0
$167$			5.42451i			0.419761i	0.977727	+	0.209881i	$0.0673075\pi$
$167$			5.42451i			0.419761i	−0.977727	+	0.209881i	$0.932692\pi$
$168$		0			0
$169$	−18.8704			−1.45157
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.0176	−	7.51571i	0.989709	−	0.571409i	0.0845218	−	0.996422i	$-0.473064\pi$
$173$	13.0176	−	7.51571i	0.989709	−	0.571409i	0.905187	+	0.425013i	$0.139730\pi$
$174$		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
$187$	45.8423	−	26.4671i	3.35232	−	1.93546i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.31174	−	16.1284i	−0.673774	−	1.16701i	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−9.31174	−	16.1284i	−0.673774	−	1.16701i	0.303052	−	0.952974i	$-0.401994\pi$
$192$		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
$195$			14.8105i			1.06060i
$196$		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
$211$	3.87043			0.266451			0.133226	−	0.991086i	$-0.457467\pi$
$211$	3.87043			0.266451			0.133226	+	0.991086i	$0.457467\pi$
$212$		0			0
$213$	−12.1928	−	7.03952i	−0.835438	−	0.482340i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.87043	−	13.6320i	−0.531834	−	0.921164i
$220$		0			0
$221$	−22.5587	+	39.0728i	−1.51746	+	2.62832i
$222$		0			0
$223$			21.8501i			1.46319i	0.681740	+	0.731594i	$0.261223\pi$
$223$			21.8501i			1.46319i	−0.681740	+	0.731594i	$0.738777\pi$
$224$		0			0
$225$	−8.11738			−0.541158
$226$		0			0
$227$	18.5401	+	10.7042i	1.23055	+	0.710460i	0.967145	−	0.254225i	$-0.0818204\pi$
$227$	18.5401	+	10.7042i	1.23055	+	0.710460i	0.263407	+	0.964685i	$0.415154\pi$
$228$		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
$235$	13.9352	−	24.1365i	0.909033	−	1.57449i
$236$		0			0
$237$			18.6200i			1.20950i
$238$		0			0
$239$	21.1174			1.36597			0.682985	−	0.730433i	$-0.260682\pi$
$239$	21.1174			1.36597			0.682985	+	0.730433i	$0.260682\pi$
$240$		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
$243$	−12.5754	+	7.26040i	−0.806711	+	0.465755i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	5.24695	−	9.08799i	0.332512	−	0.575928i
$250$		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
$255$	18.1576	+	10.4833i	1.13707	+	0.656488i
$256$		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
$261$	−0.506098	−	0.876588i	−0.0313267	−	0.0542595i
$262$		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
$275$	16.5587	+	28.6805i	0.998526	+	1.72950i
$276$		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
$281$	11.3765			0.678667			0.339333	−	0.940666i	$-0.389799\pi$
$281$	11.3765			0.678667			0.339333	+	0.940666i	$0.389799\pi$
$282$		0			0
$283$	−16.5080	−	9.53090i	−0.981299	−	0.566553i	−0.0786368	−	0.996903i	$-0.525057\pi$
$283$	−16.5080	−	9.53090i	−0.981299	−	0.566553i	−0.902662	+	0.430350i	$0.858390\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	23.4352	+	40.5910i	1.37854	+	2.38770i
$290$		0			0
$291$	−7.44131	+	12.8887i	−0.436217	+	0.755551i
$292$		0			0
$293$			32.9200i			1.92320i	0.274446	+	0.961602i	$0.411505\pi$
$293$			32.9200i			1.92320i	−0.274446	+	0.961602i	$0.588495\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	31.1155	+	17.9646i	1.80551	+	1.04241i
$298$		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
$307$		−	33.1408i		−	1.89145i	−0.324971	−	0.945724i	$-0.605355\pi$
$307$		−	33.1408i		−	1.89145i	0.324971	−	0.945724i	$-0.394645\pi$
$308$		0			0
$309$	9.11738			0.518669
$310$		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
$313$	30.5417	−	17.6332i	1.72632	−	0.996689i	0.822507	−	0.568755i	$-0.192575\pi$
$313$	30.5417	−	17.6332i	1.72632	−	0.996689i	0.903810	−	0.427934i	$-0.140759\pi$
$314$		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
$319$	−2.06479	+	3.57632i	−0.115606	+	0.200235i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−24.4453	−	14.1135i	−1.35598	−	0.782875i
$326$		0			0
$327$	−10.0290	+	5.79026i	−0.554606	+	0.320202i
$328$		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
$351$	−30.6235			−1.63456
$352$		0			0
$353$	−5.27163	−	3.04357i	−0.280580	−	0.161993i	0.353106	−	0.935583i	$-0.385126\pi$
$353$	−5.27163	−	3.04357i	−0.280580	−	0.161993i	−0.633686	+	0.773590i	$0.718459\pi$
$354$		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
$363$		−	38.5654i		−	2.02416i
$364$		0			0
$365$	30.0000			1.57027
$366$		0			0
$367$	−0.633493	−	0.365747i	−0.0330681	−	0.0190919i	0.483375	−	0.875413i	$-0.339411\pi$
$367$	−0.633493	−	0.365747i	−0.0330681	−	0.0190919i	−0.516443	+	0.856322i	$0.672744\pi$
$368$		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
$375$	−6.55869	+	11.3600i	−0.338689	+	0.586627i
$376$		0			0
$377$		−	3.51976i		−	0.181277i
$378$		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
$389$	12.3117	−	21.3246i	0.624230	−	1.08120i	−0.364459	−	0.931219i	$-0.618746\pi$
$389$	12.3117	−	21.3246i	0.624230	−	1.08120i	0.988689	−	0.149979i	$-0.0479205\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−30.7329	−	17.7437i	−1.54634	−	0.892781i
$396$		0			0
$397$	−17.0819	+	9.86222i	−0.857314	+	0.494971i	−0.863112	−	0.505013i	$-0.831488\pi$
$397$	−17.0819	+	9.86222i	−0.857314	+	0.494971i	0.00579782	+	0.999983i	$0.498154\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−19.5587	−	33.8766i	−0.976714	−	1.69172i	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−19.5587	−	33.8766i	−0.976714	−	1.69172i	−0.302556	−	0.953131i	$-0.597840\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$			3.34047i			0.165989i
$406$		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
$421$	−33.8704			−1.65074			−0.825372	−	0.564590i	$-0.809034\pi$
$421$	−33.8704			−1.65074			−0.825372	+	0.564590i	$0.809034\pi$
$422$		0			0
$423$	17.5241	+	10.1175i	0.852049	+	0.491931i
$424$		0			0
$425$	−34.6059	+	19.9797i	−1.67863	+	0.969160i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	21.9352	+	37.9929i	1.05904	+	1.83431i
$430$		0			0
$431$	17.1822	−	29.7604i	0.827636	−	1.43351i	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	17.1822	−	29.7604i	0.827636	−	1.43351i	0.899888	−	0.436121i	$-0.143648\pi$
$432$		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
$435$	−1.63567			−0.0784245
$436$		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
$447$			7.03952i			0.332958i
$448$		0			0
$449$	−40.3643			−1.90491			−0.952455	−	0.304679i	$-0.901451\pi$
$449$	−40.3643			−1.90491			−0.952455	+	0.304679i	$0.901451\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−24.2540	+	14.0030i	−1.13955	+	0.657920i
$454$		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
$459$	−21.6761	+	37.5440i	−1.01175	+	1.75241i
$460$		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
$467$	−12.4437	+	7.18439i	−0.575827	+	0.332454i	−0.759473	−	0.650538i	$-0.774543\pi$
$467$	−12.4437	+	7.18439i	−0.575827	+	0.332454i	0.183646	+	0.982992i	$0.441210\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−7.87043	−	13.6320i	−0.362650	−	0.628128i
$472$		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
$485$	−14.1822	−	24.5642i	−0.643979	−	1.11540i
$486$		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
$491$	−9.11738			−0.411461			−0.205731	−	0.978609i	$-0.565957\pi$
$491$	−9.11738			−0.411461			−0.205731	+	0.978609i	$0.565957\pi$
$492$		0			0
$493$	−4.31519	−	2.49138i	−0.194346	−	0.112206i
$494$		0			0
$495$	−20.8232	+	12.0223i	−0.935933	+	0.540361i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−17.9352	−	31.0647i	−0.802890	−	1.39065i	−0.917706	−	0.397260i	$-0.869961\pi$
$499$	−17.9352	−	31.0647i	−0.802890	−	1.39065i	0.114816	−	0.993387i	$-0.463372\pi$
$500$		0			0
$501$	−3.18216	+	5.51167i	−0.142169	+	0.246243i
$502$		0			0
$503$			1.61501i			0.0720099i	0.999352	+	0.0360049i	$0.0114632\pi$
$503$			1.61501i			0.0720099i	−0.999352	+	0.0360049i	$0.988537\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−19.1736	−	11.0699i	−0.851531	−	0.491632i
$508$		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
$515$	−8.68826	+	15.0485i	−0.382851	+	0.663117i
$516$		0			0
$517$		−	82.5552i		−	3.63077i
$518$		0			0
$519$	17.6357			0.774120
$520$		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
$537$	24.3856	−	14.0790i	1.05232	−	0.607556i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.06479	+	5.30837i	0.131765	+	0.228225i	0.924357	−	0.381528i	$-0.124602\pi$
$541$	3.06479	+	5.30837i	0.131765	+	0.228225i	−0.792592	+	0.609753i	$0.791269\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	22.0709i		−	0.945415i
$546$		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
$561$	62.1052			2.62208
$562$		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
$573$		−	21.8501i		−	0.912800i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−40.3198	−	23.2786i	−1.67853	−	0.969102i	−0.962597	−	0.270936i	$-0.912667\pi$
$577$	−40.3198	−	23.2786i	−1.67853	−	0.969102i	−0.715936	−	0.698165i	$-0.754000\pi$
$578$		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
$593$	−36.2555	+	20.9321i	−1.48883	+	0.859579i	−0.999919	−	0.0127518i	$-0.995941\pi$
$593$	−36.2555	+	20.9321i	−1.48883	+	0.859579i	−0.488916	+	0.872331i	$0.662608\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.5587	−	39.0728i	0.921723	−	1.59647i	0.124975	−	0.992160i	$-0.460115\pi$
$599$	22.5587	−	39.0728i	0.921723	−	1.59647i	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
$605$	63.6533	+	36.7503i	2.58787	+	1.49411i
$606$		0			0
$607$	−25.0191	+	14.4448i	−1.01549	+	0.586296i	−0.912796	−	0.408416i	$-0.866081\pi$
$607$	−25.0191	+	14.4448i	−1.01549	+	0.586296i	−0.102699	+	0.994712i	$0.532748\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	35.1822	+	60.9373i	1.42332	+	2.46526i
$612$		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
$631$	8.12957			0.323633			0.161817	−	0.986821i	$-0.448265\pi$
$631$	8.12957			0.323633			0.161817	+	0.986821i	$0.448265\pi$
$632$		0			0
$633$	3.93261	+	2.27050i	0.156307	+	0.0902441i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	9.74085	+	16.8717i	0.385342	+	0.667432i
$640$		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
$643$			40.1804i			1.58456i	0.610158	+	0.792279i	$0.291106\pi$
$643$			40.1804i			1.58456i	−0.610158	+	0.792279i	$0.708894\pi$
$644$		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
$657$			21.7812i			0.849766i
$658$		0			0
$659$	−30.6235			−1.19292			−0.596461	−	0.802642i	$-0.703427\pi$
$659$	−30.6235			−1.19292			−0.596461	+	0.802642i	$0.703427\pi$
$660$		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
$663$	−45.8423	+	26.4671i	−1.78037	+	1.02790i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−12.8178	+	22.2011i	−0.495566	+	0.858346i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−23.4888	−	13.5613i	−0.904085	−	0.521974i
$676$		0			0
$677$	16.3167	−	9.42046i	0.627102	−	0.362058i	−0.152527	−	0.988299i	$-0.548741\pi$
$677$	16.3167	−	9.42046i	0.627102	−	0.362058i	0.779629	+	0.626242i	$0.215408\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	12.5587	+	21.7523i	0.481250	+	0.833549i
$682$		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
$701$	52.3643			1.97777			0.988887	−	0.148671i	$-0.0474996\pi$
$701$	52.3643			1.97777			0.988887	+	0.148671i	$0.0474996\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	28.3182	−	16.3495i	1.06653	−	0.615759i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−22.8056	−	39.5005i	−0.856484	−	1.48347i	−0.875262	−	0.483650i	$-0.839311\pi$
$709$	−22.8056	−	39.5005i	−0.856484	−	1.48347i	0.0187779	−	0.999824i	$-0.494022\pi$
$710$		0			0
$711$	12.8826	−	22.3134i	0.483136	−	0.836816i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−83.6113			−3.12688
$716$		0			0
$717$	21.4567	+	12.3880i	0.801315	+	0.462639i
$718$		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
$725$	1.55869	−	2.69973i	0.0578882	−	0.100265i
$726$		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
$729$	−21.5183			−0.796974
$730$		0			0
$731$		0			0
$732$		0			0
$733$	46.4162	−	26.7984i	1.71442	−	0.989821i	0.786051	−	0.618161i	$-0.212122\pi$
$733$	46.4162	−	26.7984i	1.71442	−	0.989821i	0.928369	−	0.371660i	$-0.121211\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	25.8056	−	44.6967i	0.949276	−	1.64419i	0.202321	−	0.979319i	$-0.435152\pi$
$739$	25.8056	−	44.6967i	0.949276	−	1.64419i	0.746955	−	0.664875i	$-0.231515\pi$
$740$		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
$747$	−12.5754	+	7.26040i	−0.460109	+	0.265644i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	19.8056	+	34.3044i	0.722718	+	1.25178i	0.959906	+	0.280321i	$0.0904408\pi$
$751$	19.8056	+	34.3044i	0.722718	+	1.25178i	−0.237188	+	0.971464i	$0.576226\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	53.3759i		−	1.94255i
$756$		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
$765$	−14.5061	−	25.1253i	−0.524469	−	0.908407i
$766$		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
$771$	−5.24695			−0.188964
$772$		0			0
$773$	40.7023	+	23.4995i	1.46396	+	0.845218i	0.999191	−	0.0402129i	$-0.0128036\pi$
$773$	40.7023	+	23.4995i	1.46396	+	0.845218i	0.464770	+	0.885431i	$0.346137\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	39.7409	−	68.8332i	1.42204	−	2.46304i
$782$		0			0
$783$		−	3.38205i		−	0.120865i
$784$		0			0
$785$	30.0000			1.07075
$786$		0			0
$787$	−48.2570	−	27.8612i	−1.72018	−	0.993145i	−0.918535	−	0.395340i	$-0.870627\pi$
$787$	−48.2570	−	27.8612i	−1.72018	−	0.993145i	−0.801642	−	0.597804i	$-0.796040\pi$
$788$		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
$797$			34.8247i			1.23355i	0.787138	+	0.616777i	$0.211562\pi$
$797$			34.8247i			1.23355i	−0.787138	+	0.616777i	$0.788438\pi$
$798$		0			0
$799$	99.6113			3.52399
$800$		0			0
$801$		0			0
$802$		0			0
$803$	76.9578	−	44.4316i	2.71578	−	1.56796i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.18216	+	14.1719i	−0.287670	+	0.498258i	−0.973253	−	0.229736i	$-0.926214\pi$
$809$	−8.18216	+	14.1719i	−0.287670	+	0.498258i	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
$821$	11.6883	+	20.2447i	0.407923	+	0.706544i	0.994657	−	0.103236i	$-0.0329198\pi$
$821$	11.6883	+	20.2447i	0.407923	+	0.706544i	−0.586734	+	0.809780i	$0.699586\pi$
$822$		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
$825$			38.8551i			1.35276i
$826$		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
$835$	−6.06479	−	10.5045i	−0.209881	−	0.363524i
$836$		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
$841$	−28.6113			−0.986596
$842$		0			0
$843$	11.5593	+	6.67378i	0.398124	+	0.229857i
$844$		0			0
$845$	36.5424	−	21.0978i	1.25710	−	0.725786i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−11.1822	−	19.3681i	−0.383771	−	0.664711i
$850$		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
$865$	−16.8056	+	29.1082i	−0.571409	+	0.989709i
$866$		0			0
$867$			54.9909i			1.86759i
$868$		0			0
$869$	−105.117			−3.56586
$870$		0			0
$871$		0			0
$872$		0			0
$873$	17.8346	−	10.2968i	0.603610	−	0.348495i
$874$		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
$879$	−19.3117	+	33.4489i	−0.651369	+	1.12820i
$880$		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
$891$	4.94741	+	8.56917i	0.165744	+	0.287078i
$892$		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
$913$	51.3052	+	29.6211i	1.69795	+	0.980315i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−15.8056	−	27.3762i	−0.521380	−	0.903057i	−0.999691	−	0.0248659i	$-0.992084\pi$
$919$	−15.8056	−	27.3762i	−0.521380	−	0.903057i	0.478311	−	0.878191i	$-0.341249\pi$
$920$		0			0
$921$	19.4413	−	33.6733i	0.640613	−	1.10957i
$922$		0			0
$923$			67.7447i			2.22984i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−10.9258	−	6.30803i	−0.358851	−	0.207183i
$928$		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
$935$	−59.1822	+	102.507i	−1.93546	+	3.35232i
$936$		0			0
$937$			49.3455i			1.61205i	0.591883	+	0.806024i	$0.298385\pi$
$937$			49.3455i			1.61205i	−0.591883	+	0.806024i	$0.701615\pi$
$938$		0			0
$939$	41.3765			1.35027
$940$		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
$949$	−37.8704	+	65.5935i	−1.22933	+	2.12926i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	36.0642	+	20.8217i	1.16701	+	0.673774i
$956$		0			0
$957$	−4.19593	+	2.42252i	−0.135635	+	0.0783089i
$958$		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
$975$	−16.5587	−	28.6805i	−0.530302	−	0.918511i
$976$		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
$981$	16.0244			0.511620
$982$		0			0
$983$	−29.0834	−	16.7913i	−0.927616	−	0.535559i	−0.0415592	−	0.999136i	$-0.513233\pi$
$983$	−29.0834	−	16.7913i	−0.927616	−	0.535559i	−0.886057	+	0.463577i	$0.846566\pi$
$984$		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
$993$		−	9.38603i		−	0.297857i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	12.2524	+	7.07395i	0.388039	+	0.224034i	0.681310	−	0.731995i	$-0.261411\pi$
$997$	12.2524	+	7.07395i	0.388039	+	0.224034i	−0.293271	+	0.956029i	$0.594744\pi$
$998$		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.949.3		8
5.4	even	2	inner	980.2.q.i.949.2		8
7.2	even	3	inner	980.2.q.i.569.2		8
7.3	odd	6		980.2.e.d.589.3	yes	4
7.4	even	3		980.2.e.d.589.2	✓	4
7.5	odd	6	inner	980.2.q.i.569.3		8
7.6	odd	2	inner	980.2.q.i.949.2		8
35.3	even	12		4900.2.a.bj.1.2		4
35.4	even	6		980.2.e.d.589.3	yes	4
35.9	even	6	inner	980.2.q.i.569.3		8
35.17	even	12		4900.2.a.bj.1.3		4
35.18	odd	12		4900.2.a.bj.1.3		4
35.19	odd	6	inner	980.2.q.i.569.2		8
35.24	odd	6		980.2.e.d.589.2	✓	4
35.32	odd	12		4900.2.a.bj.1.2		4
35.34	odd	2	CM	980.2.q.i.949.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.2.e.d.589.2	✓	4	7.4	even	3
980.2.e.d.589.2	✓	4	35.24	odd	6
980.2.e.d.589.3	yes	4	7.3	odd	6
980.2.e.d.589.3	yes	4	35.4	even	6
980.2.q.i.569.2		8	7.2	even	3	inner
980.2.q.i.569.2		8	35.19	odd	6	inner
980.2.q.i.569.3		8	7.5	odd	6	inner
980.2.q.i.569.3		8	35.9	even	6	inner
980.2.q.i.949.2		8	5.4	even	2	inner
980.2.q.i.949.2		8	7.6	odd	2	inner
980.2.q.i.949.3		8	1.1	even	1	trivial
980.2.q.i.949.3		8	35.34	odd	2	CM
4900.2.a.bj.1.2		4	35.3	even	12
4900.2.a.bj.1.2		4	35.32	odd	12
4900.2.a.bj.1.3		4	35.17	even	12
4900.2.a.bj.1.3		4	35.18	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)

\(f(q)\)	\(=\)	\(q+(1.01607 + 0.586627i) q^{3} +(-1.93649 + 1.11803i) q^{5} +(-0.811738 - 1.40597i) q^{9} +(-3.31174 + 5.73610i) q^{11} -5.64539i q^{13} -2.62348 q^{15} +(-6.92119 - 3.99595i) q^{17} +(2.50000 - 4.33013i) q^{25} -5.42451i q^{27} +0.623475 q^{29} +(-6.72990 + 3.88551i) q^{33} +(3.31174 - 5.73610i) q^{39} +(3.14385 + 1.81510i) q^{45} +(-10.7942 + 6.23202i) q^{47} +(-4.68826 - 8.12031i) q^{51} -14.8105i q^{55} +(6.31174 + 10.9323i) q^{65} -12.0000 q^{71} +(-11.6190 - 6.70820i) q^{73} +(5.08034 - 2.93313i) q^{75} +(7.93521 + 13.7442i) q^{79} +(0.746951 - 1.29376i) q^{81} -8.94427i q^{83} +17.8704 q^{85} +(0.633493 + 0.365747i) q^{87} +12.6849i q^{97} +10.7530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{9} - 6 q^{11} + 20 q^{15} + 20 q^{25} - 36 q^{29} + 6 q^{39} - 58 q^{51} + 30 q^{65} - 96 q^{71} + 2 q^{79} - 76 q^{81} + 20 q^{85} + 168 q^{99}+O(q^{100}) \) 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

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