LMFDB - Embedded newform 2548.2.g.a.2157.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2548,2,Mod(2157,2548)] 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("2548.2157"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2548, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2548 = 2^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2548.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$20.3458824350$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.65712.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 3 $ x^4 + 7*x^2 + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.1
Root			$0.677214i$ of defining polynomial
Character	$\chi$	$=$	2548.2157
Dual form			2548.2.g.a.2157.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.75270i q^{5} -2.00000 q^{9} -3.75270i q^{11} +(-3.54138 - 0.677214i) q^{13} +3.75270i q^{15} +8.08276 q^{17} -5.10713i q^{19} +3.00000 q^{23} -9.08276 q^{25} +5.00000 q^{27} -5.10713i q^{31} +3.75270i q^{33} -2.39827i q^{37} +(3.54138 + 0.677214i) q^{39} -7.50540i q^{41} -9.08276 q^{43} +7.50540i q^{45} +3.75270i q^{47} -8.08276 q^{51} -2.08276 q^{53} -14.0828 q^{55} +5.10713i q^{57} -11.2581i q^{59} +6.08276 q^{61} +(-2.54138 + 13.2897i) q^{65} +6.46156i q^{67} -3.00000 q^{69} +4.06328i q^{71} +8.54925i q^{73} +9.08276 q^{75} -10.0828 q^{79} +1.00000 q^{81} +4.06328i q^{83} -30.3322i q^{85} +11.2581i q^{89} +5.10713i q^{93} -19.1655 q^{95} +11.5687i q^{97} +7.50540i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 8 q^{9} - 2 q^{13} + 8 q^{17} + 12 q^{23} - 12 q^{25} + 20 q^{27} + 2 q^{39} - 12 q^{43} - 8 q^{51} + 16 q^{53} - 32 q^{55} + 2 q^{65} - 12 q^{69} + 12 q^{75} - 16 q^{79} + 4 q^{81} - 28 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 - 8 * q^9 - 2 * q^13 + 8 * q^17 + 12 * q^23 - 12 * q^25 + 20 * q^27 + 2 * q^39 - 12 * q^43 - 8 * q^51 + 16 * q^53 - 32 * q^55 + 2 * q^65 - 12 * q^69 + 12 * q^75 - 16 * q^79 + 4 * q^81 - 28 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$.

$n$	$197$	$885$	$1275$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350			−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000			−0.577350			−0.288675	+	0.957427i	$0.593215\pi$
$4$		0			0
$5$		−	3.75270i		−	1.67826i	−0.543932	−	0.839129i	$-0.683065\pi$
$5$		−	3.75270i		−	1.67826i	0.543932	−	0.839129i	$-0.316935\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$		−	3.75270i		−	1.13148i	−0.824583	−	0.565741i	$-0.808590\pi$
$11$		−	3.75270i		−	1.13148i	0.824583	−	0.565741i	$-0.191410\pi$
$12$		0			0
$13$	−3.54138	−	0.677214i	−0.982202	−	0.187825i
$13$	−3.54138	−	0.677214i	−0.982202	−	0.187825i
$14$		0			0
$15$			3.75270i			0.968943i
$16$		0			0
$17$	8.08276			1.96036			0.980179	−	0.198114i	$-0.0634817\pi$
$17$	8.08276			1.96036			0.980179	+	0.198114i	$0.0634817\pi$
$18$		0			0
$19$		−	5.10713i		−	1.17166i	−0.810436	−	0.585828i	$-0.800769\pi$
$19$		−	5.10713i		−	1.17166i	0.810436	−	0.585828i	$-0.199231\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.00000			0.625543			0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000			0.625543			0.312772	+	0.949828i	$0.398743\pi$
$24$		0			0
$25$	−9.08276			−1.81655
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		−	5.10713i		−	0.917267i	−0.888625	−	0.458634i	$-0.848339\pi$
$31$		−	5.10713i		−	0.917267i	0.888625	−	0.458634i	$-0.151661\pi$
$32$		0			0
$33$			3.75270i			0.653261i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	2.39827i		−	0.394274i	−0.980376	−	0.197137i	$-0.936836\pi$
$37$		−	2.39827i		−	0.394274i	0.980376	−	0.197137i	$-0.0631643\pi$
$38$		0			0
$39$	3.54138	+	0.677214i	0.567075	+	0.108441i
$40$		0			0
$41$		−	7.50540i		−	1.17215i	−0.810258	−	0.586073i	$-0.800673\pi$
$41$		−	7.50540i		−	1.17215i	0.810258	−	0.586073i	$-0.199327\pi$
$42$		0			0
$43$	−9.08276			−1.38511			−0.692554	−	0.721366i	$-0.743515\pi$
$43$	−9.08276			−1.38511			−0.692554	+	0.721366i	$0.743515\pi$
$44$		0			0
$45$			7.50540i			1.11884i
$46$		0			0
$47$			3.75270i			0.547388i	0.961817	+	0.273694i	$0.0882455\pi$
$47$			3.75270i			0.547388i	−0.961817	+	0.273694i	$0.911755\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−8.08276			−1.13181
$52$		0			0
$53$	−2.08276			−0.286089			−0.143045	−	0.989716i	$-0.545689\pi$
$53$	−2.08276			−0.286089			−0.143045	+	0.989716i	$0.545689\pi$
$54$		0			0
$55$	−14.0828			−1.89892
$56$		0			0
$57$			5.10713i			0.676456i
$58$		0			0
$59$		−	11.2581i		−	1.46568i	−0.680401	−	0.732840i	$-0.738194\pi$
$59$		−	11.2581i		−	1.46568i	0.680401	−	0.732840i	$-0.261806\pi$
$60$		0			0
$61$	6.08276			0.778818			0.389409	−	0.921065i	$-0.372679\pi$
$61$	6.08276			0.778818			0.389409	+	0.921065i	$0.372679\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.54138	+	13.2897i	−0.315220	+	1.64839i
$66$		0			0
$67$			6.46156i			0.789405i	0.918809	+	0.394702i	$0.129152\pi$
$67$			6.46156i			0.789405i	−0.918809	+	0.394702i	$0.870848\pi$
$68$		0			0
$69$	−3.00000			−0.361158
$70$		0			0
$71$			4.06328i			0.482223i	0.970497	+	0.241112i	$0.0775120\pi$
$71$			4.06328i			0.482223i	−0.970497	+	0.241112i	$0.922488\pi$
$72$		0			0
$73$			8.54925i			1.00061i	0.865848	+	0.500307i	$0.166779\pi$
$73$			8.54925i			1.00061i	−0.865848	+	0.500307i	$0.833221\pi$
$74$		0			0
$75$	9.08276			1.04879
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−10.0828			−1.13440			−0.567200	−	0.823580i	$-0.691973\pi$
$79$	−10.0828			−1.13440			−0.567200	+	0.823580i	$0.691973\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			4.06328i			0.446003i	0.974818	+	0.223002i	$0.0715855\pi$
$83$			4.06328i			0.446003i	−0.974818	+	0.223002i	$0.928414\pi$
$84$		0			0
$85$		−	30.3322i		−	3.28999i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			11.2581i			1.19336i	0.802481	+	0.596678i	$0.203513\pi$
$89$			11.2581i			1.19336i	−0.802481	+	0.596678i	$0.796487\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			5.10713i			0.529585i
$94$		0			0
$95$	−19.1655			−1.96634
$96$		0			0
$97$			11.5687i			1.17462i	0.809361	+	0.587311i	$0.199813\pi$
$97$			11.5687i			1.17462i	−0.809361	+	0.587311i	$0.800187\pi$
$98$		0			0
$99$			7.50540i			0.754321i
$100$		0			0
$101$	8.08276			0.804265			0.402132	−	0.915581i	$-0.368269\pi$
$101$	8.08276			0.804265			0.402132	+	0.915581i	$0.368269\pi$
$102$		0			0
$103$	−12.0828			−1.19055			−0.595275	−	0.803522i	$-0.702957\pi$
$103$	−12.0828			−1.19055			−0.595275	+	0.803522i	$0.702957\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.1655			−1.27276			−0.636380	−	0.771376i	$-0.719569\pi$
$107$	−13.1655			−1.27276			−0.636380	+	0.771376i	$0.719569\pi$
$108$		0			0
$109$			9.90367i			0.948600i	0.880363	+	0.474300i	$0.157299\pi$
$109$			9.90367i			0.948600i	−0.880363	+	0.474300i	$0.842701\pi$
$110$		0			0
$111$			2.39827i			0.227634i
$112$		0			0
$113$	−0.917237			−0.0862864			−0.0431432	−	0.999069i	$-0.513737\pi$
$113$	−0.917237			−0.0862864			−0.0431432	+	0.999069i	$0.513737\pi$
$114$		0			0
$115$		−	11.2581i		−	1.04982i
$116$		0			0
$117$	7.08276	+	1.35443i	0.654802	+	0.125217i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.08276			−0.280251
$122$		0			0
$123$			7.50540i			0.676739i
$124$		0			0
$125$			15.3214i			1.37039i
$126$		0			0
$127$	10.0000			0.887357			0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000			0.887357			0.443678	+	0.896186i	$0.353673\pi$
$128$		0			0
$129$	9.08276			0.799693
$130$		0			0
$131$	15.0000			1.31056			0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000			1.31056			0.655278	+	0.755388i	$0.272551\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		−	18.7635i		−	1.61491i
$136$		0			0
$137$		−	7.19482i		−	0.614695i	−0.951597	−	0.307347i	$-0.900559\pi$
$137$		−	7.19482i		−	0.614695i	0.951597	−	0.307347i	$-0.0994414\pi$
$138$		0			0
$139$	−8.16553			−0.692591			−0.346295	−	0.938126i	$-0.612561\pi$
$139$	−8.16553			−0.692591			−0.346295	+	0.938126i	$0.612561\pi$
$140$		0			0
$141$		−	3.75270i		−	0.316034i
$142$		0			0
$143$	−2.54138	+	13.2897i	−0.212521	+	1.11134i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	11.2581i		−	0.922300i	−0.887322	−	0.461150i	$-0.847437\pi$
$149$		−	11.2581i		−	0.922300i	0.887322	−	0.461150i	$-0.152563\pi$
$150$		0			0
$151$		−	3.75270i		−	0.305390i	−0.988273	−	0.152695i	$-0.951205\pi$
$151$		−	3.75270i		−	0.305390i	0.988273	−	0.152695i	$-0.0487953\pi$
$152$		0			0
$153$	−16.1655			−1.30691
$154$		0			0
$155$	−19.1655			−1.53941
$156$		0			0
$157$	10.0828			0.804692			0.402346	−	0.915488i	$-0.368195\pi$
$157$	10.0828			0.804692			0.402346	+	0.915488i	$0.368195\pi$
$158$		0			0
$159$	2.08276			0.165174
$160$		0			0
$161$		0			0
$162$		0			0
$163$			0.310583i			0.0243267i	0.999926	+	0.0121634i	$0.00387182\pi$
$163$			0.310583i			0.0243267i	−0.999926	+	0.0121634i	$0.996128\pi$
$164$		0			0
$165$	14.0828			1.09634
$166$		0			0
$167$			7.50540i			0.580785i	0.956908	+	0.290393i	$0.0937859\pi$
$167$			7.50540i			0.580785i	−0.956908	+	0.290393i	$0.906214\pi$
$168$		0			0
$169$	12.0828	+	4.79655i	0.929443	+	0.368965i
$170$		0			0
$171$			10.2143i			0.781104i
$172$		0			0
$173$	−20.0828			−1.52686			−0.763432	−	0.645888i	$-0.776487\pi$
$173$	−20.0828			−1.52686			−0.763432	+	0.645888i	$0.776487\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			11.2581i			0.846211i
$178$		0			0
$179$	−3.00000			−0.224231			−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000			−0.224231			−0.112115	+	0.993695i	$0.535763\pi$
$180$		0			0
$181$	2.91724			0.216837			0.108418	−	0.994105i	$-0.465421\pi$
$181$	2.91724			0.216837			0.108418	+	0.994105i	$0.465421\pi$
$182$		0			0
$183$	−6.08276			−0.449651
$184$		0			0
$185$	−9.00000			−0.661693
$186$		0			0
$187$		−	30.3322i		−	2.21811i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.16553			−0.518479			−0.259240	−	0.965813i	$-0.583472\pi$
$191$	−7.16553			−0.518479			−0.259240	+	0.965813i	$0.583472\pi$
$192$		0			0
$193$			6.46156i			0.465113i	0.972583	+	0.232557i	$0.0747091\pi$
$193$			6.46156i			0.465113i	−0.972583	+	0.232557i	$0.925291\pi$
$194$		0			0
$195$	2.54138	−	13.2897i	0.181992	−	0.951698i
$196$		0			0
$197$			25.9583i			1.84945i	0.380631	+	0.924727i	$0.375707\pi$
$197$			25.9583i			1.84945i	−0.380631	+	0.924727i	$0.624293\pi$
$198$		0			0
$199$	−21.1655			−1.50038			−0.750192	−	0.661220i	$-0.770039\pi$
$199$	−21.1655			−1.50038			−0.750192	+	0.661220i	$0.770039\pi$
$200$		0			0
$201$		−	6.46156i		−	0.455763i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−28.1655			−1.96717
$206$		0			0
$207$	−6.00000			−0.417029
$208$		0			0
$209$	−19.1655			−1.32571
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		−	4.06328i		−	0.278412i
$214$		0			0
$215$			34.0849i			2.32457i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	8.54925i		−	0.577705i
$220$		0			0
$221$	−28.6241	−	5.47376i	−1.92547	−	0.368205i
$222$		0			0
$223$		−	19.0741i		−	1.27730i	−0.769499	−	0.638648i	$-0.779494\pi$
$223$		−	19.0741i		−	1.27730i	0.769499	−	0.638648i	$-0.220506\pi$
$224$		0			0
$225$	18.1655			1.21104
$226$		0			0
$227$		−	15.3214i		−	1.01692i	−0.861087	−	0.508458i	$-0.830216\pi$
$227$		−	15.3214i		−	1.01692i	0.861087	−	0.508458i	$-0.169784\pi$
$228$		0			0
$229$		−	27.6233i		−	1.82540i	−0.408629	−	0.912701i	$-0.633993\pi$
$229$		−	27.6233i		−	1.82540i	0.408629	−	0.912701i	$-0.366007\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.00000			−0.196537			−0.0982683	−	0.995160i	$-0.531330\pi$
$233$	−3.00000			−0.196537			−0.0982683	+	0.995160i	$0.531330\pi$
$234$		0			0
$235$	14.0828			0.918658
$236$		0			0
$237$	10.0828			0.654946
$238$		0			0
$239$			15.6320i			1.01115i	0.862783	+	0.505574i	$0.168719\pi$
$239$			15.6320i			1.01115i	−0.862783	+	0.505574i	$0.831281\pi$
$240$		0			0
$241$		−	22.2056i		−	1.43039i	−0.698925	−	0.715195i	$-0.746338\pi$
$241$		−	22.2056i		−	1.43039i	0.698925	−	0.715195i	$-0.253662\pi$
$242$		0			0
$243$	−16.0000			−1.02640
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.45862	+	18.0863i	−0.220067	+	1.15080i
$248$		0			0
$249$		−	4.06328i		−	0.257500i
$250$		0			0
$251$	−11.0828			−0.699538			−0.349769	−	0.936836i	$-0.613740\pi$
$251$	−11.0828			−0.699538			−0.349769	+	0.936836i	$0.613740\pi$
$252$		0			0
$253$		−	11.2581i		−	0.707791i
$254$		0			0
$255$			30.3322i			1.89948i
$256$		0			0
$257$	15.9172			0.992890			0.496445	−	0.868068i	$-0.334638\pi$
$257$	15.9172			0.992890			0.496445	+	0.868068i	$0.334638\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.83447			0.298106			0.149053	−	0.988829i	$-0.452377\pi$
$263$	4.83447			0.298106			0.149053	+	0.988829i	$0.452377\pi$
$264$		0			0
$265$			7.81598i			0.480132i
$266$		0			0
$267$		−	11.2581i		−	0.688985i
$268$		0			0
$269$	−3.91724			−0.238838			−0.119419	−	0.992844i	$-0.538103\pi$
$269$	−3.91724			−0.238838			−0.119419	+	0.992844i	$0.538103\pi$
$270$		0			0
$271$			3.75270i			0.227960i	0.993483	+	0.113980i	$0.0363600\pi$
$271$			3.75270i			0.227960i	−0.993483	+	0.113980i	$0.963640\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			34.0849i			2.05540i
$276$		0			0
$277$	15.1655			0.911208			0.455604	−	0.890183i	$-0.349423\pi$
$277$	15.1655			0.911208			0.455604	+	0.890183i	$0.349423\pi$
$278$		0			0
$279$			10.2143i			0.611512i
$280$		0			0
$281$			19.0741i			1.13786i	0.822384	+	0.568932i	$0.192643\pi$
$281$			19.0741i			1.13786i	−0.822384	+	0.568932i	$0.807357\pi$
$282$		0			0
$283$	−8.24829			−0.490310			−0.245155	−	0.969484i	$-0.578839\pi$
$283$	−8.24829			−0.490310			−0.245155	+	0.969484i	$0.578839\pi$
$284$		0			0
$285$	19.1655			1.13527
$286$		0			0
$287$		0			0
$288$		0			0
$289$	48.3311			2.84300
$290$		0			0
$291$		−	11.5687i		−	0.678168i
$292$		0			0
$293$		−	23.1374i		−	1.35170i	−0.737039	−	0.675850i	$-0.763777\pi$
$293$		−	23.1374i		−	1.35170i	0.737039	−	0.675850i	$-0.236223\pi$
$294$		0			0
$295$	−42.2483			−2.45979
$296$		0			0
$297$		−	18.7635i		−	1.08877i
$298$		0			0
$299$	−10.6241	−	2.03164i	−0.614410	−	0.117493i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−8.08276			−0.464343
$304$		0			0
$305$		−	22.8268i		−	1.30706i
$306$		0			0
$307$		−	23.8706i		−	1.36237i	−0.732112	−	0.681184i	$-0.761465\pi$
$307$		−	23.8706i		−	1.36237i	0.732112	−	0.681184i	$-0.238535\pi$
$308$		0			0
$309$	12.0828			0.687364
$310$		0			0
$311$	3.91724			0.222126			0.111063	−	0.993813i	$-0.464574\pi$
$311$	3.91724			0.222126			0.111063	+	0.993813i	$0.464574\pi$
$312$		0			0
$313$	13.0000			0.734803			0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000			0.734803			0.367402	+	0.930062i	$0.380247\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	15.3214i		−	0.860535i	−0.902701	−	0.430267i	$-0.858419\pi$
$317$		−	15.3214i		−	0.860535i	0.902701	−	0.430267i	$-0.141581\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	13.1655			0.734828
$322$		0			0
$323$		−	41.2797i		−	2.29686i
$324$		0			0
$325$	32.1655	+	6.15097i	1.78422	+	0.341195i
$326$		0			0
$327$		−	9.90367i		−	0.547674i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	12.6125i		−	0.693247i	−0.938004	−	0.346624i	$-0.887328\pi$
$331$		−	12.6125i		−	0.693247i	0.938004	−	0.346624i	$-0.112672\pi$
$332$		0			0
$333$			4.79655i			0.262849i
$334$		0			0
$335$	24.2483			1.32483
$336$		0			0
$337$	18.3311			0.998556			0.499278	−	0.866442i	$-0.333599\pi$
$337$	18.3311			0.998556			0.499278	+	0.866442i	$0.333599\pi$
$338$		0			0
$339$	0.917237			0.0498175
$340$		0			0
$341$	−19.1655			−1.03787
$342$		0			0
$343$		0			0
$344$		0			0
$345$			11.2581i			0.606116i
$346$		0			0
$347$	−18.2483			−0.979619			−0.489810	−	0.871829i	$-0.662934\pi$
$347$	−18.2483			−0.979619			−0.489810	+	0.871829i	$0.662934\pi$
$348$		0			0
$349$		−	7.50540i		−	0.401755i	−0.979616	−	0.200877i	$-0.935621\pi$
$349$		−	7.50540i		−	0.401755i	0.979616	−	0.200877i	$-0.0643793\pi$
$350$		0			0
$351$	−17.7069	−	3.38607i	−0.945125	−	0.180735i
$352$		0			0
$353$		−	0.310583i		−	0.0165307i	−0.999966	−	0.00826533i	$-0.997369\pi$
$353$		−	0.310583i		−	0.0165307i	0.999966	−	0.00826533i	$-0.00263097\pi$
$354$		0			0
$355$	15.2483			0.809295
$356$		0			0
$357$		0			0
$358$		0			0
$359$			11.2581i			0.594180i	0.954849	+	0.297090i	$0.0960161\pi$
$359$			11.2581i			0.594180i	−0.954849	+	0.297090i	$0.903984\pi$
$360$		0			0
$361$	−7.08276			−0.372777
$362$		0			0
$363$	3.08276			0.161803
$364$		0			0
$365$	32.0828			1.67929
$366$		0			0
$367$	−2.83447			−0.147958			−0.0739792	−	0.997260i	$-0.523570\pi$
$367$	−2.83447			−0.147958			−0.0739792	+	0.997260i	$0.523570\pi$
$368$		0			0
$369$			15.0108i			0.781431i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	22.2483			1.15197			0.575986	−	0.817459i	$-0.304618\pi$
$373$	22.2483			1.15197			0.575986	+	0.817459i	$0.304618\pi$
$374$		0			0
$375$		−	15.3214i		−	0.791193i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	10.2143i		−	0.524671i	−0.964977	−	0.262336i	$-0.915507\pi$
$379$		−	10.2143i		−	0.524671i	0.964977	−	0.262336i	$-0.0844927\pi$
$380$		0			0
$381$	−10.0000			−0.512316
$382$		0			0
$383$			11.8793i			0.607002i	0.952831	+	0.303501i	$0.0981556\pi$
$383$			11.8793i			0.607002i	−0.952831	+	0.303501i	$0.901844\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	18.1655			0.923406
$388$		0			0
$389$	7.16553			0.363307			0.181653	−	0.983363i	$-0.441855\pi$
$389$	7.16553			0.363307			0.181653	+	0.983363i	$0.441855\pi$
$390$		0			0
$391$	24.2483			1.22629
$392$		0			0
$393$	−15.0000			−0.756650
$394$		0			0
$395$			37.8376i			1.90382i
$396$		0			0
$397$			9.90367i			0.497051i	0.968625	+	0.248526i	$0.0799460\pi$
$397$			9.90367i			0.497051i	−0.968625	+	0.248526i	$0.920054\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	30.9534i		−	1.54574i	−0.634566	−	0.772868i	$-0.718821\pi$
$401$		−	30.9534i		−	1.54574i	0.634566	−	0.772868i	$-0.281179\pi$
$402$		0			0
$403$	−3.45862	+	18.0863i	−0.172286	+	0.900942i
$404$		0			0
$405$		−	3.75270i		−	0.186473i
$406$		0			0
$407$	−9.00000			−0.446113
$408$		0			0
$409$		−	6.46156i		−	0.319503i	−0.987157	−	0.159752i	$-0.948931\pi$
$409$		−	6.46156i		−	0.319503i	0.987157	−	0.159752i	$-0.0510694\pi$
$410$		0			0
$411$			7.19482i			0.354894i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	15.2483			0.748509
$416$		0			0
$417$	8.16553			0.399867
$418$		0			0
$419$	−1.83447			−0.0896200			−0.0448100	−	0.998996i	$-0.514268\pi$
$419$	−1.83447			−0.0896200			−0.0448100	+	0.998996i	$0.514268\pi$
$420$		0			0
$421$			21.1618i			1.03136i	0.856781	+	0.515681i	$0.172461\pi$
$421$			21.1618i			1.03136i	−0.856781	+	0.515681i	$0.827539\pi$
$422$		0			0
$423$		−	7.50540i		−	0.364925i
$424$		0			0
$425$	−73.4138			−3.56109
$426$		0			0
$427$		0			0
$428$		0			0
$429$	2.54138	−	13.2897i	0.122699	−	0.641635i
$430$		0			0
$431$		−	3.13153i		−	0.150841i	−0.997152	−	0.0754204i	$-0.975970\pi$
$431$		−	3.13153i		−	0.150841i	0.997152	−	0.0754204i	$-0.0240299\pi$
$432$		0			0
$433$	−8.00000			−0.384455			−0.192228	−	0.981350i	$-0.561571\pi$
$433$	−8.00000			−0.384455			−0.192228	+	0.981350i	$0.561571\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	15.3214i		−	0.732921i
$438$		0			0
$439$	6.83447			0.326192			0.163096	−	0.986610i	$-0.447852\pi$
$439$	6.83447			0.326192			0.163096	+	0.986610i	$0.447852\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.2483			0.581934			0.290967	−	0.956733i	$-0.406023\pi$
$443$	12.2483			0.581934			0.290967	+	0.956733i	$0.406023\pi$
$444$		0			0
$445$	42.2483			2.00276
$446$		0			0
$447$			11.2581i			0.532490i
$448$		0			0
$449$		−	26.5795i		−	1.25436i	−0.778873	−	0.627182i	$-0.784208\pi$
$449$		−	26.5795i		−	1.25436i	0.778873	−	0.627182i	$-0.215792\pi$
$450$		0			0
$451$	−28.1655			−1.32626
$452$		0			0
$453$			3.75270i			0.176317i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			5.84039i			0.273202i	0.990626	+	0.136601i	$0.0436178\pi$
$457$			5.84039i			0.273202i	−0.990626	+	0.136601i	$0.956382\pi$
$458$		0			0
$459$	40.4138			1.88636
$460$		0			0
$461$			18.4529i			0.859438i	0.902963	+	0.429719i	$0.141387\pi$
$461$			18.4529i			0.859438i	−0.902963	+	0.429719i	$0.858613\pi$
$462$		0			0
$463$			32.7305i			1.52111i	0.649271	+	0.760557i	$0.275074\pi$
$463$			32.7305i			1.52111i	−0.649271	+	0.760557i	$0.724926\pi$
$464$		0			0
$465$	19.1655			0.888780
$466$		0			0
$467$	7.16553			0.331581			0.165790	−	0.986161i	$-0.446982\pi$
$467$	7.16553			0.331581			0.165790	+	0.986161i	$0.446982\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−10.0828			−0.464589
$472$		0			0
$473$			34.0849i			1.56722i
$474$		0			0
$475$			46.3868i			2.12837i
$476$		0			0
$477$	4.16553			0.190726
$478$		0			0
$479$		−	11.2581i		−	0.514396i	−0.966359	−	0.257198i	$-0.917201\pi$
$479$		−	11.2581i		−	0.514396i	0.966359	−	0.257198i	$-0.0827992\pi$
$480$		0			0
$481$	−1.62414	+	8.49320i	−0.0740546	+	0.387257i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	43.4138			1.97132
$486$		0			0
$487$			8.54925i			0.387403i	0.981060	+	0.193702i	$0.0620494\pi$
$487$			8.54925i			0.387403i	−0.981060	+	0.193702i	$0.937951\pi$
$488$		0			0
$489$		−	0.310583i		−	0.0140450i
$490$		0			0
$491$	2.75171			0.124183			0.0620915	−	0.998070i	$-0.480223\pi$
$491$	2.75171			0.124183			0.0620915	+	0.998070i	$0.480223\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	28.1655			1.26595
$496$		0			0
$497$		0			0
$498$		0			0
$499$			31.0655i			1.39068i	0.718681	+	0.695340i	$0.244746\pi$
$499$			31.0655i			1.39068i	−0.718681	+	0.695340i	$0.755254\pi$
$500$		0			0
$501$		−	7.50540i		−	0.335317i
$502$		0			0
$503$	32.3311			1.44157			0.720785	−	0.693159i	$-0.243781\pi$
$503$	32.3311			1.44157			0.720785	+	0.693159i	$0.243781\pi$
$504$		0			0
$505$		−	30.3322i		−	1.34976i
$506$		0			0
$507$	−12.0828	−	4.79655i	−0.536614	−	0.213022i
$508$		0			0
$509$			11.8793i			0.526539i	0.964722	+	0.263270i	$0.0848009\pi$
$509$			11.8793i			0.526539i	−0.964722	+	0.263270i	$0.915199\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	25.5356i		−	1.12743i
$514$		0			0
$515$			45.3430i			1.99805i
$516$		0			0
$517$	14.0828			0.619359
$518$		0			0
$519$	20.0828			0.881535
$520$		0			0
$521$	−14.0828			−0.616977			−0.308489	−	0.951228i	$-0.599823\pi$
$521$	−14.0828			−0.616977			−0.308489	+	0.951228i	$0.599823\pi$
$522$		0			0
$523$	4.08276			0.178527			0.0892634	−	0.996008i	$-0.471549\pi$
$523$	4.08276			0.178527			0.0892634	+	0.996008i	$0.471549\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	41.2797i		−	1.79817i
$528$		0			0
$529$	−14.0000			−0.608696
$530$		0			0
$531$			22.5162i			0.977120i
$532$		0			0
$533$	−5.08276	+	26.5795i	−0.220159	+	1.15129i
$534$		0			0
$535$			49.4063i			2.13602i
$536$		0			0
$537$	3.00000			0.129460
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	27.6233i		−	1.18762i	−0.804605	−	0.593810i	$-0.797623\pi$
$541$		−	27.6233i		−	1.18762i	0.804605	−	0.593810i	$-0.202377\pi$
$542$		0			0
$543$	−2.91724			−0.125191
$544$		0			0
$545$	37.1655			1.59200
$546$		0			0
$547$	41.2483			1.76365			0.881825	−	0.471577i	$-0.156315\pi$
$547$	41.2483			1.76365			0.881825	+	0.471577i	$0.156315\pi$
$548$		0			0
$549$	−12.1655			−0.519212
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	9.00000			0.382029
$556$		0			0
$557$			22.8268i			0.967202i	0.875289	+	0.483601i	$0.160671\pi$
$557$			22.8268i			0.967202i	−0.875289	+	0.483601i	$0.839329\pi$
$558$		0			0
$559$	32.1655	+	6.15097i	1.36046	+	0.260158i
$560$		0			0
$561$			30.3322i			1.28063i
$562$		0			0
$563$	31.1655			1.31347			0.656735	−	0.754121i	$-0.271937\pi$
$563$	31.1655			1.31347			0.656735	+	0.754121i	$0.271937\pi$
$564$		0			0
$565$			3.44212i			0.144811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.24829			0.261942			0.130971	−	0.991386i	$-0.458191\pi$
$569$	6.24829			0.261942			0.130971	+	0.991386i	$0.458191\pi$
$570$		0			0
$571$	9.33105			0.390492			0.195246	−	0.980754i	$-0.437449\pi$
$571$	9.33105			0.390492			0.195246	+	0.980754i	$0.437449\pi$
$572$		0			0
$573$	7.16553			0.299344
$574$		0			0
$575$	−27.2483			−1.13633
$576$		0			0
$577$		−	2.39827i		−	0.0998414i	−0.998753	−	0.0499207i	$-0.984103\pi$
$577$		−	2.39827i		−	0.0998414i	0.998753	−	0.0499207i	$-0.0158969\pi$
$578$		0			0
$579$		−	6.46156i		−	0.268533i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			7.81598i			0.323705i
$584$		0			0
$585$	5.08276	−	26.5795i	0.210146	−	1.09893i
$586$		0			0
$587$			10.9475i			0.451852i	0.974144	+	0.225926i	$0.0725408\pi$
$587$			10.9475i			0.451852i	−0.974144	+	0.225926i	$0.927459\pi$
$588$		0			0
$589$	−26.0828			−1.07472
$590$		0			0
$591$		−	25.9583i		−	1.06778i
$592$		0			0
$593$		−	33.1531i		−	1.36144i	−0.732545	−	0.680718i	$-0.761668\pi$
$593$		−	33.1531i		−	1.36144i	0.732545	−	0.680718i	$-0.238332\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	21.1655			0.866247
$598$		0			0
$599$	−18.2483			−0.745605			−0.372802	−	0.927911i	$-0.621603\pi$
$599$	−18.2483			−0.745605			−0.372802	+	0.927911i	$0.621603\pi$
$600$		0			0
$601$	−24.1655			−0.985732			−0.492866	−	0.870105i	$-0.664051\pi$
$601$	−24.1655			−0.985732			−0.492866	+	0.870105i	$0.664051\pi$
$602$		0			0
$603$		−	12.9231i		−	0.526270i
$604$		0			0
$605$			11.5687i			0.470334i
$606$		0			0
$607$	−14.8345			−0.602113			−0.301056	−	0.953606i	$-0.597339\pi$
$607$	−14.8345			−0.602113			−0.301056	+	0.953606i	$0.597339\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.54138	−	13.2897i	0.102813	−	0.537645i
$612$		0			0
$613$			10.6369i			0.429622i	0.976656	+	0.214811i	$0.0689135\pi$
$613$			10.6369i			0.429622i	−0.976656	+	0.214811i	$0.931086\pi$
$614$		0			0
$615$	28.1655			1.13574
$616$		0			0
$617$		−	15.6320i		−	0.629319i	−0.949205	−	0.314660i	$-0.898110\pi$
$617$		−	15.6320i		−	0.629319i	0.949205	−	0.314660i	$-0.101890\pi$
$618$		0			0
$619$		−	11.9914i		−	0.481974i	−0.970528	−	0.240987i	$-0.922529\pi$
$619$		−	11.9914i		−	0.481974i	0.970528	−	0.240987i	$-0.0774711\pi$
$620$		0			0
$621$	15.0000			0.601929
$622$		0			0
$623$		0			0
$624$		0			0
$625$	12.0828			0.483311
$626$		0			0
$627$	19.1655			0.765397
$628$		0			0
$629$		−	19.3847i		−	0.772917i
$630$		0			0
$631$			23.1374i			0.921084i	0.887638	+	0.460542i	$0.152345\pi$
$631$			23.1374i			0.921084i	−0.887638	+	0.460542i	$0.847655\pi$
$632$		0			0
$633$	−4.00000			−0.158986
$634$		0			0
$635$		−	37.5270i		−	1.48921i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	8.12657i		−	0.321482i
$640$		0			0
$641$	−1.16553			−0.0460355			−0.0230177	−	0.999735i	$-0.507327\pi$
$641$	−1.16553			−0.0460355			−0.0230177	+	0.999735i	$0.507327\pi$
$642$		0			0
$643$		−	47.7413i		−	1.88273i	−0.337388	−	0.941366i	$-0.609543\pi$
$643$		−	47.7413i		−	1.88273i	0.337388	−	0.941366i	$-0.390457\pi$
$644$		0			0
$645$		−	34.0849i		−	1.34209i
$646$		0			0
$647$	36.2483			1.42507			0.712534	−	0.701638i	$-0.247548\pi$
$647$	36.2483			1.42507			0.712534	+	0.701638i	$0.247548\pi$
$648$		0			0
$649$	−42.2483			−1.65839
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−14.0828			−0.551101			−0.275551	−	0.961287i	$-0.588860\pi$
$653$	−14.0828			−0.551101			−0.275551	+	0.961287i	$0.588860\pi$
$654$		0			0
$655$		−	56.2905i		−	2.19945i
$656$		0			0
$657$		−	17.0985i		−	0.667076i
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$			21.4724i			0.835178i	0.908636	+	0.417589i	$0.137125\pi$
$661$			21.4724i			0.835178i	−0.908636	+	0.417589i	$0.862875\pi$
$662$		0			0
$663$	28.6241	+	5.47376i	1.11167	+	0.212583i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			19.0741i			0.737447i
$670$		0			0
$671$		−	22.8268i		−	0.881218i
$672$		0			0
$673$	32.0000			1.23351			0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000			1.23351			0.616755	+	0.787155i	$0.288447\pi$
$674$		0			0
$675$	−45.4138			−1.74798
$676$		0			0
$677$	−11.3311			−0.435488			−0.217744	−	0.976006i	$-0.569870\pi$
$677$	−11.3311			−0.435488			−0.217744	+	0.976006i	$0.569870\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			15.3214i			0.587117i
$682$		0			0
$683$		−	38.4588i		−	1.47158i	−0.677208	−	0.735792i	$-0.736810\pi$
$683$		−	38.4588i		−	1.47158i	0.677208	−	0.735792i	$-0.263190\pi$
$684$		0			0
$685$	−27.0000			−1.03162
$686$		0			0
$687$			27.6233i			1.05390i
$688$		0			0
$689$	7.37586	+	1.41048i	0.280998	+	0.0537349i
$690$		0			0
$691$		−	13.2337i		−	0.503434i	−0.967801	−	0.251717i	$-0.919005\pi$
$691$		−	13.2337i		−	0.503434i	0.967801	−	0.251717i	$-0.0809951\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			30.6428i			1.16235i
$696$		0			0
$697$		−	60.6644i		−	2.29783i
$698$		0			0
$699$	3.00000			0.113470
$700$		0			0
$701$	−27.2483			−1.02915			−0.514577	−	0.857444i	$-0.672051\pi$
$701$	−27.2483			−1.02915			−0.514577	+	0.857444i	$0.672051\pi$
$702$		0			0
$703$	−12.2483			−0.461953
$704$		0			0
$705$	−14.0828			−0.530388
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	15.9426i		−	0.598735i	−0.954138	−	0.299368i	$-0.903224\pi$
$709$		−	15.9426i		−	0.598735i	0.954138	−	0.299368i	$-0.0967757\pi$
$710$		0			0
$711$	20.1655			0.756266
$712$		0			0
$713$		−	15.3214i		−	0.573790i
$714$		0			0
$715$	49.8724	+	9.53704i	1.86512	+	0.356665i
$716$		0			0
$717$		−	15.6320i		−	0.583787i
$718$		0			0
$719$	−34.4138			−1.28342			−0.641709	−	0.766948i	$-0.721774\pi$
$719$	−34.4138			−1.28342			−0.641709	+	0.766948i	$0.721774\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			22.2056i			0.825836i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−42.1655			−1.56383			−0.781916	−	0.623383i	$-0.785758\pi$
$727$	−42.1655			−1.56383			−0.781916	+	0.623383i	$0.785758\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−73.4138			−2.71531
$732$		0			0
$733$			43.2553i			1.59767i	0.601550	+	0.798836i	$0.294550\pi$
$733$			43.2553i			1.59767i	−0.601550	+	0.798836i	$0.705450\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.2483			0.893197
$738$		0			0
$739$		−	44.6097i		−	1.64100i	−0.571650	−	0.820498i	$-0.693696\pi$
$739$		−	44.6097i		−	1.64100i	0.571650	−	0.820498i	$-0.306304\pi$
$740$		0			0
$741$	3.45862	−	18.0863i	0.127056	−	0.664416i
$742$		0			0
$743$		−	40.9691i		−	1.50301i	−0.659727	−	0.751506i	$-0.729328\pi$
$743$		−	40.9691i		−	1.50301i	0.659727	−	0.751506i	$-0.270672\pi$
$744$		0			0
$745$	−42.2483			−1.54786
$746$		0			0
$747$		−	8.12657i		−	0.297336i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	11.0000			0.401396			0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000			0.401396			0.200698	+	0.979653i	$0.435679\pi$
$752$		0			0
$753$	11.0828			0.403878
$754$		0			0
$755$	−14.0828			−0.512524
$756$		0			0
$757$	33.5793			1.22046			0.610231	−	0.792224i	$-0.291077\pi$
$757$	33.5793			1.22046			0.610231	+	0.792224i	$0.291077\pi$
$758$		0			0
$759$			11.2581i			0.408643i
$760$		0			0
$761$		−	33.7743i		−	1.22432i	−0.790735	−	0.612159i	$-0.790301\pi$
$761$		−	33.7743i		−	1.22432i	0.790735	−	0.612159i	$-0.209699\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$			60.6644i			2.19333i
$766$		0			0
$767$	−7.62414	+	39.8692i	−0.275292	+	1.43959i
$768$		0			0
$769$			24.4918i			0.883197i	0.897213	+	0.441598i	$0.145588\pi$
$769$			24.4918i			0.883197i	−0.897213	+	0.441598i	$0.854412\pi$
$770$		0			0
$771$	−15.9172			−0.573245
$772$		0			0
$773$			26.8901i			0.967169i	0.875298	+	0.483584i	$0.160665\pi$
$773$			26.8901i			0.967169i	−0.875298	+	0.483584i	$0.839335\pi$
$774$		0			0
$775$			46.3868i			1.66626i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−38.3311			−1.37335
$780$		0			0
$781$	15.2483			0.545627
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	37.8376i		−	1.35048i
$786$		0			0
$787$		−	8.43715i		−	0.300752i	−0.988629	−	0.150376i	$-0.951952\pi$
$787$		−	8.43715i		−	0.300752i	0.988629	−	0.150376i	$-0.0480484\pi$
$788$		0			0
$789$	−4.83447			−0.172112
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−21.5414	−	4.11933i	−0.764957	−	0.146282i
$794$		0			0
$795$		−	7.81598i		−	0.277204i
$796$		0			0
$797$	28.1655			0.997674			0.498837	−	0.866696i	$-0.333761\pi$
$797$	28.1655			0.997674			0.498837	+	0.866696i	$0.333761\pi$
$798$		0			0
$799$			30.3322i			1.07308i
$800$		0			0
$801$		−	22.5162i		−	0.795571i
$802$		0			0
$803$	32.0828			1.13218
$804$		0			0
$805$		0			0
$806$		0			0
$807$	3.91724			0.137893
$808$		0			0
$809$	27.9172			0.981518			0.490759	−	0.871295i	$-0.336720\pi$
$809$	27.9172			0.981518			0.490759	+	0.871295i	$0.336720\pi$
$810$		0			0
$811$		−	12.1899i		−	0.428044i	−0.976829	−	0.214022i	$-0.931344\pi$
$811$		−	12.1899i		−	0.428044i	0.976829	−	0.214022i	$-0.0686564\pi$
$812$		0			0
$813$		−	3.75270i		−	0.131613i
$814$		0			0
$815$	1.16553			0.0408266
$816$		0			0
$817$			46.3868i			1.62287i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			23.4480i			0.818339i	0.912458	+	0.409170i	$0.134182\pi$
$821$			23.4480i			0.818339i	−0.912458	+	0.409170i	$0.865818\pi$
$822$		0			0
$823$	−15.3311			−0.534407			−0.267203	−	0.963640i	$-0.586100\pi$
$823$	−15.3311			−0.534407			−0.267203	+	0.963640i	$0.586100\pi$
$824$		0			0
$825$		−	34.0849i		−	1.18668i
$826$		0			0
$827$		−	8.12657i		−	0.282588i	−0.989968	−	0.141294i	$-0.954874\pi$
$827$		−	8.12657i		−	0.282588i	0.989968	−	0.141294i	$-0.0451264\pi$
$828$		0			0
$829$	3.75171			0.130302			0.0651512	−	0.997875i	$-0.479247\pi$
$829$	3.75171			0.130302			0.0651512	+	0.997875i	$0.479247\pi$
$830$		0			0
$831$	−15.1655			−0.526086
$832$		0			0
$833$		0			0
$834$		0			0
$835$	28.1655			0.974708
$836$		0			0
$837$		−	25.5356i		−	0.882641i
$838$		0			0
$839$		−	10.9475i		−	0.377950i	−0.981982	−	0.188975i	$-0.939483\pi$
$839$		−	10.9475i		−	0.377950i	0.981982	−	0.188975i	$-0.0605166\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		−	19.0741i		−	0.656946i
$844$		0			0
$845$	18.0000	−	45.3430i	0.619219	−	1.55985i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	8.24829			0.283080
$850$		0			0
$851$		−	7.19482i		−	0.246635i
$852$		0			0
$853$		−	2.08769i		−	0.0714811i	−0.999361	−	0.0357406i	$-0.988621\pi$
$853$		−	2.08769i		−	0.0714811i	0.999361	−	0.0357406i	$-0.0113790\pi$
$854$		0			0
$855$	38.3311			1.31089
$856$		0			0
$857$	37.1655			1.26955			0.634775	−	0.772697i	$-0.281093\pi$
$857$	37.1655			1.26955			0.634775	+	0.772697i	$0.281093\pi$
$858$		0			0
$859$	31.0000			1.05771			0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000			1.05771			0.528853	+	0.848713i	$0.322622\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	7.19482i		−	0.244914i	−0.992474	−	0.122457i	$-0.960923\pi$
$863$		−	7.19482i		−	0.244914i	0.992474	−	0.122457i	$-0.0390774\pi$
$864$		0			0
$865$			75.3646i			2.56247i
$866$		0			0
$867$	−48.3311			−1.64141
$868$		0			0
$869$			37.8376i			1.28355i
$870$		0			0
$871$	4.37586	−	22.8828i	0.148270	−	0.775355i
$872$		0			0
$873$		−	23.1374i		−	0.783081i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			22.2056i			0.749831i	0.927059	+	0.374915i	$0.122328\pi$
$877$			22.2056i			0.749831i	−0.927059	+	0.374915i	$0.877672\pi$
$878$		0			0
$879$			23.1374i			0.780404i
$880$		0			0
$881$	17.0828			0.575533			0.287766	−	0.957701i	$-0.407087\pi$
$881$	17.0828			0.575533			0.287766	+	0.957701i	$0.407087\pi$
$882$		0			0
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$		0			0
$885$	42.2483			1.42016
$886$		0			0
$887$	−44.0828			−1.48015			−0.740077	−	0.672522i	$-0.765211\pi$
$887$	−44.0828			−1.48015			−0.740077	+	0.672522i	$0.765211\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	3.75270i		−	0.125720i
$892$		0			0
$893$	19.1655			0.641350
$894$		0			0
$895$			11.2581i			0.376317i
$896$		0			0
$897$	10.6241	+	2.03164i	0.354730	+	0.0678345i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−16.8345			−0.560838
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	10.9475i		−	0.363908i
$906$		0			0
$907$	−12.8345			−0.426162			−0.213081	−	0.977035i	$-0.568350\pi$
$907$	−12.8345			−0.426162			−0.213081	+	0.977035i	$0.568350\pi$
$908$		0			0
$909$	−16.1655			−0.536177
$910$		0			0
$911$	−46.6621			−1.54598			−0.772992	−	0.634416i	$-0.781241\pi$
$911$	−46.6621			−1.54598			−0.772992	+	0.634416i	$0.781241\pi$
$912$		0			0
$913$	15.2483			0.504645
$914$		0			0
$915$			22.8268i			0.754630i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	3.75171			0.123758			0.0618788	−	0.998084i	$-0.480291\pi$
$919$	3.75171			0.123758			0.0618788	+	0.998084i	$0.480291\pi$
$920$		0			0
$921$			23.8706i			0.786564i
$922$		0			0
$923$	2.75171	−	14.3896i	0.0905737	−	0.473641i
$924$		0			0
$925$			21.7829i			0.716219i
$926$		0			0
$927$	24.1655			0.793700
$928$		0			0
$929$			35.0166i			1.14886i	0.818554	+	0.574429i	$0.194776\pi$
$929$			35.0166i			1.14886i	−0.818554	+	0.574429i	$0.805224\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−3.91724			−0.128245
$934$		0			0
$935$	−113.828			−3.72256
$936$		0			0
$937$	−23.2483			−0.759488			−0.379744	−	0.925092i	$-0.623988\pi$
$937$	−23.2483			−0.759488			−0.379744	+	0.925092i	$0.623988\pi$
$938$		0			0
$939$	−13.0000			−0.424239
$940$		0			0
$941$		−	29.7110i		−	0.968552i	−0.874915	−	0.484276i	$-0.839083\pi$
$941$		−	29.7110i		−	0.968552i	0.874915	−	0.484276i	$-0.160917\pi$
$942$		0			0
$943$		−	22.5162i		−	0.733228i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	7.81598i		−	0.253985i	−0.991904	−	0.126993i	$-0.959468\pi$
$947$		−	7.81598i		−	0.253985i	0.991904	−	0.126993i	$-0.0405325\pi$
$948$		0			0
$949$	5.78967	−	30.2761i	0.187941	−	0.982805i
$950$		0			0
$951$			15.3214i			0.496830i
$952$		0			0
$953$	18.0000			0.583077			0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000			0.583077			0.291539	+	0.956559i	$0.405833\pi$
$954$		0			0
$955$			26.8901i			0.870142i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	4.91724			0.158621
$962$		0			0
$963$	26.3311			0.848506
$964$		0			0
$965$	24.2483			0.780580
$966$		0			0
$967$		−	52.4257i		−	1.68590i	−0.537994	−	0.842949i	$-0.680818\pi$
$967$		−	52.4257i		−	1.68590i	0.537994	−	0.842949i	$-0.319182\pi$
$968$		0			0
$969$			41.2797i			1.32610i
$970$		0			0
$971$	−34.8345			−1.11789			−0.558946	−	0.829204i	$-0.688794\pi$
$971$	−34.8345			−1.11789			−0.558946	+	0.829204i	$0.688794\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−32.1655	−	6.15097i	−1.03012	−	0.196989i
$976$		0			0
$977$			11.2581i			0.360179i	0.983650	+	0.180089i	$0.0576387\pi$
$977$			11.2581i			0.360179i	−0.983650	+	0.180089i	$0.942361\pi$
$978$		0			0
$979$	42.2483			1.35026
$980$		0			0
$981$		−	19.8073i		−	0.632400i
$982$		0			0
$983$		−	41.9009i		−	1.33643i	−0.743968	−	0.668215i	$-0.767059\pi$
$983$		−	41.9009i		−	1.33643i	0.743968	−	0.668215i	$-0.232941\pi$
$984$		0			0
$985$	97.4138			3.10386
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−27.2483			−0.866445
$990$		0			0
$991$	6.83447			0.217104			0.108552	−	0.994091i	$-0.465379\pi$
$991$	6.83447			0.217104			0.108552	+	0.994091i	$0.465379\pi$
$992$		0			0
$993$			12.6125i			0.400247i
$994$		0			0
$995$			79.4279i			2.51803i
$996$		0			0
$997$	−54.0828			−1.71282			−0.856409	−	0.516298i	$-0.827310\pi$
$997$	−54.0828			−1.71282			−0.856409	+	0.516298i	$0.827310\pi$
$998$		0			0
$999$		−	11.9914i		−	0.379390i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2548.2.g.a.2157.1		4
7.2	even	3		364.2.y.a.25.1	✓	8
7.3	odd	6		2548.2.y.a.961.1		8
7.4	even	3		364.2.y.a.233.4	yes	8
7.5	odd	6		2548.2.y.a.753.4		8
7.6	odd	2		2548.2.g.d.2157.4		4
13.12	even	2	inner	2548.2.g.a.2157.4		4
21.2	odd	6		3276.2.gv.d.1117.4		8
21.11	odd	6		3276.2.gv.d.2053.1		8
91.12	odd	6		2548.2.y.a.753.1		8
91.25	even	6		364.2.y.a.233.1	yes	8
91.38	odd	6		2548.2.y.a.961.4		8
91.51	even	6		364.2.y.a.25.4	yes	8
91.90	odd	2		2548.2.g.d.2157.1		4
273.116	odd	6		3276.2.gv.d.2053.4		8
273.233	odd	6		3276.2.gv.d.1117.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.y.a.25.1	✓	8	7.2	even	3
364.2.y.a.25.4	yes	8	91.51	even	6
364.2.y.a.233.1	yes	8	91.25	even	6
364.2.y.a.233.4	yes	8	7.4	even	3
2548.2.g.a.2157.1		4	1.1	even	1	trivial
2548.2.g.a.2157.4		4	13.12	even	2	inner
2548.2.g.d.2157.1		4	91.90	odd	2
2548.2.g.d.2157.4		4	7.6	odd	2
2548.2.y.a.753.1		8	91.12	odd	6
2548.2.y.a.753.4		8	7.5	odd	6
2548.2.y.a.961.1		8	7.3	odd	6
2548.2.y.a.961.4		8	91.38	odd	6
3276.2.gv.d.1117.1		8	273.233	odd	6
3276.2.gv.d.1117.4		8	21.2	odd	6
3276.2.gv.d.2053.1		8	21.11	odd	6
3276.2.gv.d.2053.4		8	273.116	odd	6

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:	\( N \)	\(=\)	\( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2548.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.75270i q^{5} -2.00000 q^{9} -3.75270i q^{11} +(-3.54138 - 0.677214i) q^{13} +3.75270i q^{15} +8.08276 q^{17} -5.10713i q^{19} +3.00000 q^{23} -9.08276 q^{25} +5.00000 q^{27} -5.10713i q^{31} +3.75270i q^{33} -2.39827i q^{37} +(3.54138 + 0.677214i) q^{39} -7.50540i q^{41} -9.08276 q^{43} +7.50540i q^{45} +3.75270i q^{47} -8.08276 q^{51} -2.08276 q^{53} -14.0828 q^{55} +5.10713i q^{57} -11.2581i q^{59} +6.08276 q^{61} +(-2.54138 + 13.2897i) q^{65} +6.46156i q^{67} -3.00000 q^{69} +4.06328i q^{71} +8.54925i q^{73} +9.08276 q^{75} -10.0828 q^{79} +1.00000 q^{81} +4.06328i q^{83} -30.3322i q^{85} +11.2581i q^{89} +5.10713i q^{93} -19.1655 q^{95} +11.5687i q^{97} +7.50540i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 8 q^{9} - 2 q^{13} + 8 q^{17} + 12 q^{23} - 12 q^{25} + 20 q^{27} + 2 q^{39} - 12 q^{43} - 8 q^{51} + 16 q^{53} - 32 q^{55} + 2 q^{65} - 12 q^{69} + 12 q^{75} - 16 q^{79} + 4 q^{81} - 28 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 8 * q^9 - 2 * q^13 + 8 * q^17 + 12 * q^23 - 12 * q^25 + 20 * q^27 + 2 * q^39 - 12 * q^43 - 8 * q^51 + 16 * q^53 - 32 * q^55 + 2 * q^65 - 12 * q^69 + 12 * q^75 - 16 * q^79 + 4 * q^81 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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