LMFDB - Embedded newform 2912.2.c.a.1457.29

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1457,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.1457"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.29
Character	$\chi$	$=$	2912.1457
Dual form			2912.2.c.a.1457.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.38465i q^{3} -1.59665i q^{5} +1.00000 q^{7} -2.68654 q^{9} +4.18960i q^{11} -1.00000i q^{13} +3.80744 q^{15} +0.570087 q^{17} +2.53707i q^{19} +2.38465i q^{21} -0.0190668 q^{23} +2.45072 q^{25} +0.747485i q^{27} +0.742125i q^{29} +3.88210 q^{31} -9.99073 q^{33} -1.59665i q^{35} +1.77977i q^{37} +2.38465 q^{39} -8.08770 q^{41} -11.7998i q^{43} +4.28946i q^{45} +5.33750 q^{47} +1.00000 q^{49} +1.35946i q^{51} +4.60925i q^{53} +6.68932 q^{55} -6.05001 q^{57} +7.12700i q^{59} +10.6962i q^{61} -2.68654 q^{63} -1.59665 q^{65} +8.45532i q^{67} -0.0454676i q^{69} +0.425314 q^{71} +8.18795 q^{73} +5.84409i q^{75} +4.18960i q^{77} -1.30570 q^{79} -9.84212 q^{81} +11.3255i q^{83} -0.910228i q^{85} -1.76971 q^{87} -1.64900 q^{89} -1.00000i q^{91} +9.25743i q^{93} +4.05081 q^{95} -4.79975 q^{97} -11.2556i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) $ 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	2.38465i	1.37678i	0.725342	+	0.688388i	$0.241682\pi$
$3$	2.38465i	1.37678i	−0.725342	+	0.688388i	$0.758318\pi$
$4$	0	0
$5$	− 1.59665i	− 0.714043i	−0.934096	−	0.357021i	$-0.883792\pi$
$5$	− 1.59665i	− 0.714043i	0.934096	−	0.357021i	$-0.116208\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.68654	−0.895514
$10$	0	0
$11$	4.18960i	1.26321i	0.775289	+	0.631607i	$0.217604\pi$
$11$	4.18960i	1.26321i	−0.775289	+	0.631607i	$0.782396\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	3.80744	0.983077
$16$	0	0
$17$	0.570087	0.138266	0.0691332	−	0.997607i	$-0.477977\pi$
$17$	0.570087	0.138266	0.0691332	+	0.997607i	$0.477977\pi$
$18$	0	0
$19$	2.53707i	0.582043i	0.956716	+	0.291022i	$0.0939952\pi$
$19$	2.53707i	0.582043i	−0.956716	+	0.291022i	$0.906005\pi$
$20$	0	0
$21$	2.38465i	0.520373i
$22$	0	0
$23$	−0.0190668	−0.00397570	−0.00198785	−	0.999998i	$-0.500633\pi$
$23$	−0.0190668	−0.00397570	−0.00198785	+	0.999998i	$0.500633\pi$
$24$	0	0
$25$	2.45072	0.490143
$26$	0	0
$27$	0.747485i	0.143854i
$28$	0	0
$29$	0.742125i	0.137809i	0.997623	+	0.0689046i	$0.0219504\pi$
$29$	0.742125i	0.137809i	−0.997623	+	0.0689046i	$0.978050\pi$
$30$	0	0
$31$	3.88210	0.697245	0.348623	−	0.937263i	$-0.386650\pi$
$31$	3.88210	0.697245	0.348623	+	0.937263i	$0.386650\pi$
$32$	0	0
$33$	−9.99073	−1.73916
$34$	0	0
$35$	− 1.59665i	− 0.269883i
$36$	0	0
$37$	1.77977i	0.292593i	0.989241	+	0.146296i	$0.0467353\pi$
$37$	1.77977i	0.292593i	−0.989241	+	0.146296i	$0.953265\pi$
$38$	0	0
$39$	2.38465	0.381849
$40$	0	0
$41$	−8.08770	−1.26309	−0.631543	−	0.775341i	$-0.717578\pi$
$41$	−8.08770	−1.26309	−0.631543	+	0.775341i	$0.717578\pi$
$42$	0	0
$43$	− 11.7998i	− 1.79946i	−0.436450	−	0.899729i	$-0.643764\pi$
$43$	− 11.7998i	− 1.79946i	0.436450	−	0.899729i	$-0.356236\pi$
$44$	0	0
$45$	4.28946i	0.639435i
$46$	0	0
$47$	5.33750	0.778555	0.389277	−	0.921121i	$-0.372725\pi$
$47$	5.33750	0.778555	0.389277	+	0.921121i	$0.372725\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.35946i	0.190362i
$52$	0	0
$53$	4.60925i	0.633130i	0.948571	+	0.316565i	$0.102530\pi$
$53$	4.60925i	0.633130i	−0.948571	+	0.316565i	$0.897470\pi$
$54$	0	0
$55$	6.68932	0.901988
$56$	0	0
$57$	−6.05001	−0.801344
$58$	0	0
$59$	7.12700i	0.927857i	0.885873	+	0.463928i	$0.153561\pi$
$59$	7.12700i	0.927857i	−0.885873	+	0.463928i	$0.846439\pi$
$60$	0	0
$61$	10.6962i	1.36950i	0.728777	+	0.684751i	$0.240089\pi$
$61$	10.6962i	1.36950i	−0.728777	+	0.684751i	$0.759911\pi$
$62$	0	0
$63$	−2.68654	−0.338473
$64$	0	0
$65$	−1.59665	−0.198040
$66$	0	0
$67$	8.45532i	1.03298i	0.856293	+	0.516491i	$0.172762\pi$
$67$	8.45532i	1.03298i	−0.856293	+	0.516491i	$0.827238\pi$
$68$	0	0
$69$	− 0.0454676i	− 0.00547365i
$70$	0	0
$71$	0.425314	0.0504754	0.0252377	−	0.999681i	$-0.491966\pi$
$71$	0.425314	0.0504754	0.0252377	+	0.999681i	$0.491966\pi$
$72$	0	0
$73$	8.18795	0.958327	0.479164	−	0.877726i	$-0.340940\pi$
$73$	8.18795	0.958327	0.479164	+	0.877726i	$0.340940\pi$
$74$	0	0
$75$	5.84409i	0.674818i
$76$	0	0
$77$	4.18960i	0.477450i
$78$	0	0
$79$	−1.30570	−0.146902	−0.0734512	−	0.997299i	$-0.523401\pi$
$79$	−1.30570	−0.146902	−0.0734512	+	0.997299i	$0.523401\pi$
$80$	0	0
$81$	−9.84212	−1.09357
$82$	0	0
$83$	11.3255i	1.24313i	0.783361	+	0.621567i	$0.213504\pi$
$83$	11.3255i	1.24313i	−0.783361	+	0.621567i	$0.786496\pi$
$84$	0	0
$85$	− 0.910228i	− 0.0987281i
$86$	0	0
$87$	−1.76971	−0.189733
$88$	0	0
$89$	−1.64900	−0.174794	−0.0873970	−	0.996174i	$-0.527855\pi$
$89$	−1.64900	−0.174794	−0.0873970	+	0.996174i	$0.527855\pi$
$90$	0	0
$91$	− 1.00000i	− 0.104828i
$92$	0	0
$93$	9.25743i	0.959951i
$94$	0	0
$95$	4.05081	0.415604
$96$	0	0
$97$	−4.79975	−0.487341	−0.243670	−	0.969858i	$-0.578352\pi$
$97$	−4.79975	−0.487341	−0.243670	+	0.969858i	$0.578352\pi$
$98$	0	0
$99$	− 11.2556i	− 1.13123i
$100$	0	0
$101$	− 7.48152i	− 0.744439i	−0.928145	−	0.372220i	$-0.878597\pi$
$101$	− 7.48152i	− 0.744439i	0.928145	−	0.372220i	$-0.121403\pi$
$102$	0	0
$103$	−11.8173	−1.16439	−0.582197	−	0.813048i	$-0.697807\pi$
$103$	−11.8173	−1.16439	−0.582197	+	0.813048i	$0.697807\pi$
$104$	0	0
$105$	3.80744	0.371568
$106$	0	0
$107$	16.5805i	1.60289i	0.598066	+	0.801447i	$0.295936\pi$
$107$	16.5805i	1.60289i	−0.598066	+	0.801447i	$0.704064\pi$
$108$	0	0
$109$	11.9418i	1.14381i	0.820318	+	0.571907i	$0.193796\pi$
$109$	11.9418i	1.14381i	−0.820318	+	0.571907i	$0.806204\pi$
$110$	0	0
$111$	−4.24413	−0.402835
$112$	0	0
$113$	5.32589	0.501017	0.250509	−	0.968114i	$-0.419402\pi$
$113$	5.32589	0.501017	0.250509	+	0.968114i	$0.419402\pi$
$114$	0	0
$115$	0.0304429i	0.00283882i
$116$	0	0
$117$	2.68654i	0.248371i
$118$	0	0
$119$	0.570087	0.0522598
$120$	0	0
$121$	−6.55279	−0.595708
$122$	0	0
$123$	− 19.2863i	− 1.73899i
$124$	0	0
$125$	− 11.8962i	− 1.06403i
$126$	0	0
$127$	−15.1726	−1.34635	−0.673176	−	0.739483i	$-0.735070\pi$
$127$	−15.1726	−1.34635	−0.673176	+	0.739483i	$0.735070\pi$
$128$	0	0
$129$	28.1384	2.47745
$130$	0	0
$131$	1.62340i	0.141837i	0.997482	+	0.0709184i	$0.0225930\pi$
$131$	1.62340i	0.141837i	−0.997482	+	0.0709184i	$0.977407\pi$
$132$	0	0
$133$	2.53707i	0.219992i
$134$	0	0
$135$	1.19347	0.102718
$136$	0	0
$137$	−9.53449	−0.814587	−0.407293	−	0.913297i	$-0.633527\pi$
$137$	−9.53449	−0.814587	−0.407293	+	0.913297i	$0.633527\pi$
$138$	0	0
$139$	12.2351i	1.03776i	0.854846	+	0.518882i	$0.173652\pi$
$139$	12.2351i	1.03776i	−0.854846	+	0.518882i	$0.826348\pi$
$140$	0	0
$141$	12.7281i	1.07190i
$142$	0	0
$143$	4.18960	0.350352
$144$	0	0
$145$	1.18491	0.0984017
$146$	0	0
$147$	2.38465i	0.196682i
$148$	0	0
$149$	1.97177i	0.161534i	0.996733	+	0.0807669i	$0.0257369\pi$
$149$	1.97177i	0.161534i	−0.996733	+	0.0807669i	$0.974263\pi$
$150$	0	0
$151$	16.5785	1.34914	0.674571	−	0.738210i	$-0.264328\pi$
$151$	16.5785	1.34914	0.674571	+	0.738210i	$0.264328\pi$
$152$	0	0
$153$	−1.53156	−0.123819
$154$	0	0
$155$	− 6.19834i	− 0.497863i
$156$	0	0
$157$	7.21761i	0.576028i	0.957626	+	0.288014i	$0.0929950\pi$
$157$	7.21761i	0.576028i	−0.957626	+	0.288014i	$0.907005\pi$
$158$	0	0
$159$	−10.9914	−0.871678
$160$	0	0
$161$	−0.0190668	−0.00150267
$162$	0	0
$163$	− 20.8322i	− 1.63170i	−0.578260	−	0.815852i	$-0.696268\pi$
$163$	− 20.8322i	− 1.63170i	0.578260	−	0.815852i	$-0.303732\pi$
$164$	0	0
$165$	15.9517i	1.24184i
$166$	0	0
$167$	−16.5728	−1.28245	−0.641223	−	0.767355i	$-0.721572\pi$
$167$	−16.5728	−1.28245	−0.641223	+	0.767355i	$0.721572\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 6.81594i	− 0.521228i
$172$	0	0
$173$	− 3.48473i	− 0.264939i	−0.991187	−	0.132469i	$-0.957709\pi$
$173$	− 3.48473i	− 0.264939i	0.991187	−	0.132469i	$-0.0422906\pi$
$174$	0	0
$175$	2.45072	0.185257
$176$	0	0
$177$	−16.9954	−1.27745
$178$	0	0
$179$	22.1527i	1.65577i	0.560900	+	0.827884i	$0.310455\pi$
$179$	22.1527i	1.65577i	−0.560900	+	0.827884i	$0.689545\pi$
$180$	0	0
$181$	− 16.9513i	− 1.25998i	−0.776604	−	0.629989i	$-0.783059\pi$
$181$	− 16.9513i	− 1.25998i	0.776604	−	0.629989i	$-0.216941\pi$
$182$	0	0
$183$	−25.5066	−1.88550
$184$	0	0
$185$	2.84167	0.208924
$186$	0	0
$187$	2.38844i	0.174660i
$188$	0	0
$189$	0.747485i	0.0543715i
$190$	0	0
$191$	21.6345	1.56542	0.782708	−	0.622389i	$-0.213838\pi$
$191$	21.6345	1.56542	0.782708	+	0.622389i	$0.213838\pi$
$192$	0	0
$193$	16.1797	1.16464	0.582319	−	0.812960i	$-0.302145\pi$
$193$	16.1797	1.16464	0.582319	+	0.812960i	$0.302145\pi$
$194$	0	0
$195$	− 3.80744i	− 0.272657i
$196$	0	0
$197$	− 15.2317i	− 1.08521i	−0.839987	−	0.542607i	$-0.817437\pi$
$197$	− 15.2317i	− 1.08521i	0.839987	−	0.542607i	$-0.182563\pi$
$198$	0	0
$199$	−6.21008	−0.440221	−0.220110	−	0.975475i	$-0.570642\pi$
$199$	−6.21008	−0.440221	−0.220110	+	0.975475i	$0.570642\pi$
$200$	0	0
$201$	−20.1630	−1.42219
$202$	0	0
$203$	0.742125i	0.0520870i
$204$	0	0
$205$	12.9132i	0.901898i
$206$	0	0
$207$	0.0512237	0.00356030
$208$	0	0
$209$	−10.6293	−0.735245
$210$	0	0
$211$	− 5.94685i	− 0.409398i	−0.978825	−	0.204699i	$-0.934378\pi$
$211$	− 5.94685i	− 0.409398i	0.978825	−	0.204699i	$-0.0656215\pi$
$212$	0	0
$213$	1.01422i	0.0694934i
$214$	0	0
$215$	−18.8402	−1.28489
$216$	0	0
$217$	3.88210	0.263534
$218$	0	0
$219$	19.5254i	1.31940i
$220$	0	0
$221$	− 0.570087i	− 0.0383482i
$222$	0	0
$223$	−9.96356	−0.667209	−0.333605	−	0.942713i	$-0.608265\pi$
$223$	−9.96356	−0.667209	−0.333605	+	0.942713i	$0.608265\pi$
$224$	0	0
$225$	−6.58395	−0.438930
$226$	0	0
$227$	− 10.4441i	− 0.693198i	−0.938013	−	0.346599i	$-0.887337\pi$
$227$	− 10.4441i	− 0.693198i	0.938013	−	0.346599i	$-0.112663\pi$
$228$	0	0
$229$	− 26.5939i	− 1.75737i	−0.477398	−	0.878687i	$-0.658420\pi$
$229$	− 26.5939i	− 1.75737i	0.477398	−	0.878687i	$-0.341580\pi$
$230$	0	0
$231$	−9.99073	−0.657342
$232$	0	0
$233$	14.4199	0.944676	0.472338	−	0.881417i	$-0.343410\pi$
$233$	14.4199	0.944676	0.472338	+	0.881417i	$0.343410\pi$
$234$	0	0
$235$	− 8.52211i	− 0.555921i
$236$	0	0
$237$	− 3.11363i	− 0.202252i
$238$	0	0
$239$	−1.16899	−0.0756154	−0.0378077	−	0.999285i	$-0.512037\pi$
$239$	−1.16899	−0.0756154	−0.0378077	+	0.999285i	$0.512037\pi$
$240$	0	0
$241$	17.6819	1.13899	0.569495	−	0.821995i	$-0.307139\pi$
$241$	17.6819	1.13899	0.569495	+	0.821995i	$0.307139\pi$
$242$	0	0
$243$	− 21.2275i	− 1.36175i
$244$	0	0
$245$	− 1.59665i	− 0.102006i
$246$	0	0
$247$	2.53707	0.161430
$248$	0	0
$249$	−27.0073	−1.71152
$250$	0	0
$251$	15.3853i	0.971109i	0.874206	+	0.485554i	$0.161382\pi$
$251$	15.3853i	0.971109i	−0.874206	+	0.485554i	$0.838618\pi$
$252$	0	0
$253$	− 0.0798823i	− 0.00502216i
$254$	0	0
$255$	2.17057	0.135927
$256$	0	0
$257$	−3.88506	−0.242343	−0.121172	−	0.992632i	$-0.538665\pi$
$257$	−3.88506	−0.242343	−0.121172	+	0.992632i	$0.538665\pi$
$258$	0	0
$259$	1.77977i	0.110590i
$260$	0	0
$261$	− 1.99375i	− 0.123410i
$262$	0	0
$263$	−21.8664	−1.34834	−0.674169	−	0.738577i	$-0.735498\pi$
$263$	−21.8664	−1.34834	−0.674169	+	0.738577i	$0.735498\pi$
$264$	0	0
$265$	7.35936	0.452082
$266$	0	0
$267$	− 3.93229i	− 0.240652i
$268$	0	0
$269$	28.9652i	1.76604i	0.469338	+	0.883019i	$0.344493\pi$
$269$	28.9652i	1.76604i	−0.469338	+	0.883019i	$0.655507\pi$
$270$	0	0
$271$	−16.7574	−1.01794	−0.508971	−	0.860784i	$-0.669974\pi$
$271$	−16.7574	−1.01794	−0.508971	+	0.860784i	$0.669974\pi$
$272$	0	0
$273$	2.38465	0.144325
$274$	0	0
$275$	10.2675i	0.619155i
$276$	0	0
$277$	− 19.3546i	− 1.16290i	−0.813581	−	0.581451i	$-0.802485\pi$
$277$	− 19.3546i	− 1.16290i	0.813581	−	0.581451i	$-0.197515\pi$
$278$	0	0
$279$	−10.4294	−0.624393
$280$	0	0
$281$	2.77097	0.165302	0.0826510	−	0.996579i	$-0.473661\pi$
$281$	2.77097	0.165302	0.0826510	+	0.996579i	$0.473661\pi$
$282$	0	0
$283$	27.6707i	1.64485i	0.568873	+	0.822426i	$0.307380\pi$
$283$	27.6707i	1.64485i	−0.568873	+	0.822426i	$0.692620\pi$
$284$	0	0
$285$	9.65974i	0.572194i
$286$	0	0
$287$	−8.08770	−0.477402
$288$	0	0
$289$	−16.6750	−0.980882
$290$	0	0
$291$	− 11.4457i	− 0.670960i
$292$	0	0
$293$	− 4.72248i	− 0.275890i	−0.990440	−	0.137945i	$-0.955950\pi$
$293$	− 4.72248i	− 0.275890i	0.990440	−	0.137945i	$-0.0440498\pi$
$294$	0	0
$295$	11.3793	0.662529
$296$	0	0
$297$	−3.13167	−0.181718
$298$	0	0
$299$	0.0190668i	0.00110266i
$300$	0	0
$301$	− 11.7998i	− 0.680131i
$302$	0	0
$303$	17.8408	1.02493
$304$	0	0
$305$	17.0780	0.977883
$306$	0	0
$307$	20.2156i	1.15376i	0.816827	+	0.576882i	$0.195731\pi$
$307$	20.2156i	1.15376i	−0.816827	+	0.576882i	$0.804269\pi$
$308$	0	0
$309$	− 28.1801i	− 1.60311i
$310$	0	0
$311$	−33.4197	−1.89505	−0.947527	−	0.319675i	$-0.896426\pi$
$311$	−33.4197	−1.89505	−0.947527	+	0.319675i	$0.896426\pi$
$312$	0	0
$313$	−29.8694	−1.68832	−0.844159	−	0.536093i	$-0.819900\pi$
$313$	−29.8694	−1.68832	−0.844159	+	0.536093i	$0.819900\pi$
$314$	0	0
$315$	4.28946i	0.241684i
$316$	0	0
$317$	− 2.84875i	− 0.160002i	−0.996795	−	0.0800009i	$-0.974508\pi$
$317$	− 2.84875i	− 0.160002i	0.996795	−	0.0800009i	$-0.0254923\pi$
$318$	0	0
$319$	−3.10921	−0.174082
$320$	0	0
$321$	−39.5385	−2.20683
$322$	0	0
$323$	1.44635i	0.0804770i
$324$	0	0
$325$	− 2.45072i	− 0.135941i
$326$	0	0
$327$	−28.4769	−1.57478
$328$	0	0
$329$	5.33750	0.294266
$330$	0	0
$331$	− 17.6436i	− 0.969778i	−0.874576	−	0.484889i	$-0.838860\pi$
$331$	− 17.6436i	− 0.969778i	0.874576	−	0.484889i	$-0.161140\pi$
$332$	0	0
$333$	− 4.78144i	− 0.262021i
$334$	0	0
$335$	13.5002	0.737593
$336$	0	0
$337$	21.7156	1.18292	0.591462	−	0.806333i	$-0.298551\pi$
$337$	21.7156	1.18292	0.591462	+	0.806333i	$0.298551\pi$
$338$	0	0
$339$	12.7004i	0.689789i
$340$	0	0
$341$	16.2644i	0.880769i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.0725957	−0.00390842
$346$	0	0
$347$	1.68632i	0.0905264i	0.998975	+	0.0452632i	$0.0144127\pi$
$347$	1.68632i	0.0905264i	−0.998975	+	0.0452632i	$0.985587\pi$
$348$	0	0
$349$	− 5.49984i	− 0.294400i	−0.989107	−	0.147200i	$-0.952974\pi$
$349$	− 5.49984i	− 0.294400i	0.989107	−	0.147200i	$-0.0470260\pi$
$350$	0	0
$351$	0.747485	0.0398978
$352$	0	0
$353$	5.12477	0.272764	0.136382	−	0.990656i	$-0.456453\pi$
$353$	5.12477	0.272764	0.136382	+	0.990656i	$0.456453\pi$
$354$	0	0
$355$	− 0.679076i	− 0.0360416i
$356$	0	0
$357$	1.35946i	0.0719500i
$358$	0	0
$359$	28.3506	1.49629	0.748144	−	0.663537i	$-0.230945\pi$
$359$	28.3506	1.49629	0.748144	+	0.663537i	$0.230945\pi$
$360$	0	0
$361$	12.5633	0.661225
$362$	0	0
$363$	− 15.6261i	− 0.820157i
$364$	0	0
$365$	− 13.0733i	− 0.684287i
$366$	0	0
$367$	24.8090	1.29502	0.647509	−	0.762058i	$-0.275811\pi$
$367$	24.8090	1.29502	0.647509	+	0.762058i	$0.275811\pi$
$368$	0	0
$369$	21.7279	1.13111
$370$	0	0
$371$	4.60925i	0.239301i
$372$	0	0
$373$	34.3736i	1.77980i	0.456159	+	0.889898i	$0.349225\pi$
$373$	34.3736i	1.77980i	−0.456159	+	0.889898i	$0.650775\pi$
$374$	0	0
$375$	28.3682	1.46493
$376$	0	0
$377$	0.742125	0.0382214
$378$	0	0
$379$	− 3.46978i	− 0.178231i	−0.996021	−	0.0891153i	$-0.971596\pi$
$379$	− 3.46978i	− 0.178231i	0.996021	−	0.0891153i	$-0.0284039\pi$
$380$	0	0
$381$	− 36.1813i	− 1.85362i
$382$	0	0
$383$	30.3382	1.55021	0.775104	−	0.631834i	$-0.217698\pi$
$383$	30.3382	1.55021	0.775104	+	0.631834i	$0.217698\pi$
$384$	0	0
$385$	6.68932	0.340919
$386$	0	0
$387$	31.7007i	1.61144i
$388$	0	0
$389$	− 6.57349i	− 0.333289i	−0.986017	−	0.166645i	$-0.946707\pi$
$389$	− 6.57349i	− 0.333289i	0.986017	−	0.166645i	$-0.0532933\pi$
$390$	0	0
$391$	−0.0108697	−0.000549706 0
$392$	0	0
$393$	−3.87123	−0.195278
$394$	0	0
$395$	2.08474i	0.104895i
$396$	0	0
$397$	− 17.5876i	− 0.882698i	−0.897336	−	0.441349i	$-0.854500\pi$
$397$	− 17.5876i	− 0.882698i	0.897336	−	0.441349i	$-0.145500\pi$
$398$	0	0
$399$	−6.05001	−0.302880
$400$	0	0
$401$	13.1238	0.655370	0.327685	−	0.944787i	$-0.393732\pi$
$401$	13.1238	0.655370	0.327685	+	0.944787i	$0.393732\pi$
$402$	0	0
$403$	− 3.88210i	− 0.193381i
$404$	0	0
$405$	15.7144i	0.780855i
$406$	0	0
$407$	−7.45654	−0.369607
$408$	0	0
$409$	−8.55038	−0.422789	−0.211394	−	0.977401i	$-0.567800\pi$
$409$	−8.55038	−0.422789	−0.211394	+	0.977401i	$0.567800\pi$
$410$	0	0
$411$	− 22.7364i	− 1.12150i
$412$	0	0
$413$	7.12700i	0.350697i
$414$	0	0
$415$	18.0828	0.887651
$416$	0	0
$417$	−29.1763	−1.42877
$418$	0	0
$419$	− 0.544846i	− 0.0266175i	−0.999911	−	0.0133087i	$-0.995764\pi$
$419$	− 0.544846i	− 0.0266175i	0.999911	−	0.0133087i	$-0.00423643\pi$
$420$	0	0
$421$	− 34.7806i	− 1.69510i	−0.530715	−	0.847550i	$-0.678077\pi$
$421$	− 34.7806i	− 1.69510i	0.530715	−	0.847550i	$-0.321923\pi$
$422$	0	0
$423$	−14.3394	−0.697207
$424$	0	0
$425$	1.39712	0.0677703
$426$	0	0
$427$	10.6962i	0.517623i
$428$	0	0
$429$	9.99073i	0.482357i
$430$	0	0
$431$	40.6679	1.95890	0.979451	−	0.201682i	$-0.0646408\pi$
$431$	40.6679	1.95890	0.979451	+	0.201682i	$0.0646408\pi$
$432$	0	0
$433$	39.7133	1.90850	0.954251	−	0.299008i	$-0.0966556\pi$
$433$	39.7133	1.90850	0.954251	+	0.299008i	$0.0966556\pi$
$434$	0	0
$435$	2.82560i	0.135477i
$436$	0	0
$437$	− 0.0483737i	− 0.00231403i
$438$	0	0
$439$	14.2217	0.678766	0.339383	−	0.940648i	$-0.389782\pi$
$439$	14.2217	0.678766	0.339383	+	0.940648i	$0.389782\pi$
$440$	0	0
$441$	−2.68654	−0.127931
$442$	0	0
$443$	19.4745i	0.925262i	0.886551	+	0.462631i	$0.153094\pi$
$443$	19.4745i	0.925262i	−0.886551	+	0.462631i	$0.846906\pi$
$444$	0	0
$445$	2.63288i	0.124810i
$446$	0	0
$447$	−4.70198	−0.222396
$448$	0	0
$449$	13.7969	0.651114	0.325557	−	0.945522i	$-0.394448\pi$
$449$	13.7969	0.651114	0.325557	+	0.945522i	$0.394448\pi$
$450$	0	0
$451$	− 33.8843i	− 1.59555i
$452$	0	0
$453$	39.5340i	1.85747i
$454$	0	0
$455$	−1.59665	−0.0748520
$456$	0	0
$457$	41.3406	1.93383	0.966915	−	0.255100i	$-0.0821084\pi$
$457$	41.3406	1.93383	0.966915	+	0.255100i	$0.0821084\pi$
$458$	0	0
$459$	0.426131i	0.0198901i
$460$	0	0
$461$	3.00550i	0.139980i	0.997548	+	0.0699900i	$0.0222967\pi$
$461$	3.00550i	0.139980i	−0.997548	+	0.0699900i	$0.977703\pi$
$462$	0	0
$463$	33.4122	1.55280	0.776399	−	0.630241i	$-0.217044\pi$
$463$	33.4122	1.55280	0.776399	+	0.630241i	$0.217044\pi$
$464$	0	0
$465$	14.7809	0.685446
$466$	0	0
$467$	− 35.0509i	− 1.62196i	−0.585071	−	0.810982i	$-0.698934\pi$
$467$	− 35.0509i	− 1.62196i	0.585071	−	0.810982i	$-0.301066\pi$
$468$	0	0
$469$	8.45532i	0.390431i
$470$	0	0
$471$	−17.2115	−0.793062
$472$	0	0
$473$	49.4366	2.27310
$474$	0	0
$475$	6.21763i	0.285285i
$476$	0	0
$477$	− 12.3830i	− 0.566977i
$478$	0	0
$479$	−16.9518	−0.774550	−0.387275	−	0.921964i	$-0.626584\pi$
$479$	−16.9518	−0.774550	−0.387275	+	0.921964i	$0.626584\pi$
$480$	0	0
$481$	1.77977	0.0811507
$482$	0	0
$483$	− 0.0454676i	− 0.00206885i
$484$	0	0
$485$	7.66351i	0.347982i
$486$	0	0
$487$	5.22200	0.236631	0.118316	−	0.992976i	$-0.462251\pi$
$487$	5.22200	0.236631	0.118316	+	0.992976i	$0.462251\pi$
$488$	0	0
$489$	49.6775	2.24649
$490$	0	0
$491$	9.41493i	0.424890i	0.977173	+	0.212445i	$0.0681426\pi$
$491$	9.41493i	0.424890i	−0.977173	+	0.212445i	$0.931857\pi$
$492$	0	0
$493$	0.423076i	0.0190544i
$494$	0	0
$495$	−17.9712	−0.807743
$496$	0	0
$497$	0.425314	0.0190779
$498$	0	0
$499$	− 26.2053i	− 1.17311i	−0.809910	−	0.586555i	$-0.800484\pi$
$499$	− 26.2053i	− 1.17311i	0.809910	−	0.586555i	$-0.199516\pi$
$500$	0	0
$501$	− 39.5204i	− 1.76564i
$502$	0	0
$503$	−16.3145	−0.727427	−0.363713	−	0.931511i	$-0.618491\pi$
$503$	−16.3145	−0.727427	−0.363713	+	0.931511i	$0.618491\pi$
$504$	0	0
$505$	−11.9454	−0.531561
$506$	0	0
$507$	− 2.38465i	− 0.105906i
$508$	0	0
$509$	28.9494i	1.28316i	0.767056	+	0.641580i	$0.221721\pi$
$509$	28.9494i	1.28316i	−0.767056	+	0.641580i	$0.778279\pi$
$510$	0	0
$511$	8.18795	0.362214
$512$	0	0
$513$	−1.89642	−0.0837290
$514$	0	0
$515$	18.8681i	0.831426i
$516$	0	0
$517$	22.3620i	0.983481i
$518$	0	0
$519$	8.30985	0.364762
$520$	0	0
$521$	4.82104	0.211213	0.105607	−	0.994408i	$-0.466322\pi$
$521$	4.82104	0.211213	0.105607	+	0.994408i	$0.466322\pi$
$522$	0	0
$523$	− 41.0621i	− 1.79552i	−0.440483	−	0.897761i	$-0.645193\pi$
$523$	− 41.0621i	− 1.79552i	0.440483	−	0.897761i	$-0.354807\pi$
$524$	0	0
$525$	5.84409i	0.255057i
$526$	0	0
$527$	2.21313	0.0964055
$528$	0	0
$529$	−22.9996	−0.999984
$530$	0	0
$531$	− 19.1470i	− 0.830909i
$532$	0	0
$533$	8.08770i	0.350317i
$534$	0	0
$535$	26.4732	1.14453
$536$	0	0
$537$	−52.8263	−2.27962
$538$	0	0
$539$	4.18960i	0.180459i
$540$	0	0
$541$	− 21.6909i	− 0.932563i	−0.884636	−	0.466281i	$-0.845593\pi$
$541$	− 21.6909i	− 0.932563i	0.884636	−	0.466281i	$-0.154407\pi$
$542$	0	0
$543$	40.4228	1.73471
$544$	0	0
$545$	19.0668	0.816732
$546$	0	0
$547$	− 1.00864i	− 0.0431261i	−0.999767	−	0.0215631i	$-0.993136\pi$
$547$	− 1.00864i	− 0.0431261i	0.999767	−	0.0215631i	$-0.00686427\pi$
$548$	0	0
$549$	− 28.7357i	− 1.22641i
$550$	0	0
$551$	−1.88282	−0.0802110
$552$	0	0
$553$	−1.30570	−0.0555239
$554$	0	0
$555$	6.77638i	0.287641i
$556$	0	0
$557$	36.2601i	1.53639i	0.640215	+	0.768196i	$0.278845\pi$
$557$	36.2601i	1.53639i	−0.640215	+	0.768196i	$0.721155\pi$
$558$	0	0
$559$	−11.7998	−0.499080
$560$	0	0
$561$	−5.69558	−0.240468
$562$	0	0
$563$	− 6.62119i	− 0.279050i	−0.990219	−	0.139525i	$-0.955442\pi$
$563$	− 6.62119i	− 0.279050i	0.990219	−	0.139525i	$-0.0445576\pi$
$564$	0	0
$565$	− 8.50356i	− 0.357748i
$566$	0	0
$567$	−9.84212	−0.413330
$568$	0	0
$569$	−7.93150	−0.332506	−0.166253	−	0.986083i	$-0.553167\pi$
$569$	−7.93150	−0.332506	−0.166253	+	0.986083i	$0.553167\pi$
$570$	0	0
$571$	− 12.5240i	− 0.524113i	−0.965053	−	0.262056i	$-0.915599\pi$
$571$	− 12.5240i	− 0.524113i	0.965053	−	0.262056i	$-0.0844006\pi$
$572$	0	0
$573$	51.5906i	2.15523i
$574$	0	0
$575$	−0.0467273	−0.00194866
$576$	0	0
$577$	0.359975	0.0149860	0.00749298	−	0.999972i	$-0.497615\pi$
$577$	0.359975	0.0149860	0.00749298	+	0.999972i	$0.497615\pi$
$578$	0	0
$579$	38.5828i	1.60345i
$580$	0	0
$581$	11.3255i	0.469861i
$582$	0	0
$583$	−19.3110	−0.799778
$584$	0	0
$585$	4.28946	0.177347
$586$	0	0
$587$	− 5.69179i	− 0.234925i	−0.993077	−	0.117463i	$-0.962524\pi$
$587$	− 5.69179i	− 0.234925i	0.993077	−	0.117463i	$-0.0374760\pi$
$588$	0	0
$589$	9.84914i	0.405827i
$590$	0	0
$591$	36.3222	1.49410
$592$	0	0
$593$	7.86945	0.323160	0.161580	−	0.986860i	$-0.448341\pi$
$593$	7.86945	0.323160	0.161580	+	0.986860i	$0.448341\pi$
$594$	0	0
$595$	− 0.910228i	− 0.0373157i
$596$	0	0
$597$	− 14.8088i	− 0.606085i
$598$	0	0
$599$	19.6972	0.804804	0.402402	−	0.915463i	$-0.368175\pi$
$599$	19.6972	0.804804	0.402402	+	0.915463i	$0.368175\pi$
$600$	0	0
$601$	−12.1355	−0.495018	−0.247509	−	0.968886i	$-0.579612\pi$
$601$	−12.1355	−0.495018	−0.247509	+	0.968886i	$0.579612\pi$
$602$	0	0
$603$	− 22.7156i	− 0.925050i
$604$	0	0
$605$	10.4625i	0.425361i
$606$	0	0
$607$	4.16567	0.169080	0.0845398	−	0.996420i	$-0.473058\pi$
$607$	4.16567	0.169080	0.0845398	+	0.996420i	$0.473058\pi$
$608$	0	0
$609$	−1.76971	−0.0717122
$610$	0	0
$611$	− 5.33750i	− 0.215932i
$612$	0	0
$613$	10.8143i	0.436785i	0.975861	+	0.218392i	$0.0700813\pi$
$613$	10.8143i	0.436785i	−0.975861	+	0.218392i	$0.929919\pi$
$614$	0	0
$615$	−30.7934	−1.24171
$616$	0	0
$617$	39.6347	1.59563	0.797817	−	0.602899i	$-0.205988\pi$
$617$	39.6347	1.59563	0.797817	+	0.602899i	$0.205988\pi$
$618$	0	0
$619$	− 29.0915i	− 1.16928i	−0.811291	−	0.584642i	$-0.801235\pi$
$619$	− 29.0915i	− 1.16928i	0.811291	−	0.584642i	$-0.198765\pi$
$620$	0	0
$621$	− 0.0142521i	0 0.000571918i
$622$	0	0
$623$	−1.64900	−0.0660659
$624$	0	0
$625$	−6.74042	−0.269617
$626$	0	0
$627$	− 25.3472i	− 1.01227i
$628$	0	0
$629$	1.01462i	0.0404557i
$630$	0	0
$631$	8.48512	0.337787	0.168894	−	0.985634i	$-0.445981\pi$
$631$	8.48512	0.337787	0.168894	+	0.985634i	$0.445981\pi$
$632$	0	0
$633$	14.1811	0.563650
$634$	0	0
$635$	24.2253i	0.961352i
$636$	0	0
$637$	− 1.00000i	− 0.0396214i
$638$	0	0
$639$	−1.14262	−0.0452015
$640$	0	0
$641$	7.62094	0.301009	0.150504	−	0.988609i	$-0.451910\pi$
$641$	7.62094	0.301009	0.150504	+	0.988609i	$0.451910\pi$
$642$	0	0
$643$	− 6.34543i	− 0.250239i	−0.992142	−	0.125120i	$-0.960069\pi$
$643$	− 6.34543i	− 0.250239i	0.992142	−	0.125120i	$-0.0399314\pi$
$644$	0	0
$645$	− 44.9272i	− 1.76901i
$646$	0	0
$647$	8.51589	0.334794	0.167397	−	0.985890i	$-0.446464\pi$
$647$	8.51589	0.334794	0.167397	+	0.985890i	$0.446464\pi$
$648$	0	0
$649$	−29.8593	−1.17208
$650$	0	0
$651$	9.25743i	0.362827i
$652$	0	0
$653$	− 13.5010i	− 0.528334i	−0.964477	−	0.264167i	$-0.914903\pi$
$653$	− 13.5010i	− 0.528334i	0.964477	−	0.264167i	$-0.0850971\pi$
$654$	0	0
$655$	2.59199	0.101277
$656$	0	0
$657$	−21.9973	−0.858196
$658$	0	0
$659$	17.4197i	0.678574i	0.940683	+	0.339287i	$0.110186\pi$
$659$	17.4197i	0.678574i	−0.940683	+	0.339287i	$0.889814\pi$
$660$	0	0
$661$	− 13.2692i	− 0.516112i	−0.966130	−	0.258056i	$-0.916918\pi$
$661$	− 13.2692i	− 0.516112i	0.966130	−	0.258056i	$-0.0830819\pi$
$662$	0	0
$663$	1.35946	0.0527969
$664$	0	0
$665$	4.05081	0.157083
$666$	0	0
$667$	− 0.0141499i	0 0.000547888i
$668$	0	0
$669$	− 23.7596i	− 0.918598i
$670$	0	0
$671$	−44.8127	−1.72997
$672$	0	0
$673$	−2.84846	−0.109800	−0.0549001	−	0.998492i	$-0.517484\pi$
$673$	−2.84846	−0.109800	−0.0549001	+	0.998492i	$0.517484\pi$
$674$	0	0
$675$	1.83187i	0.0705088i
$676$	0	0
$677$	− 10.2692i	− 0.394677i	−0.980335	−	0.197338i	$-0.936770\pi$
$677$	− 10.2692i	− 0.394677i	0.980335	−	0.197338i	$-0.0632297\pi$
$678$	0	0
$679$	−4.79975	−0.184198
$680$	0	0
$681$	24.9054	0.954379
$682$	0	0
$683$	− 32.8742i	− 1.25790i	−0.777447	−	0.628949i	$-0.783485\pi$
$683$	− 32.8742i	− 1.25790i	0.777447	−	0.628949i	$-0.216515\pi$
$684$	0	0
$685$	15.2232i	0.581650i
$686$	0	0
$687$	63.4171	2.41951
$688$	0	0
$689$	4.60925	0.175599
$690$	0	0
$691$	27.5365i	1.04754i	0.851860	+	0.523769i	$0.175474\pi$
$691$	27.5365i	1.04754i	−0.851860	+	0.523769i	$0.824526\pi$
$692$	0	0
$693$	− 11.2556i	− 0.427563i
$694$	0	0
$695$	19.5351	0.741008
$696$	0	0
$697$	−4.61069	−0.174642
$698$	0	0
$699$	34.3863i	1.30061i
$700$	0	0
$701$	− 12.0330i	− 0.454480i	−0.973839	−	0.227240i	$-0.927030\pi$
$701$	− 12.0330i	− 0.454480i	0.973839	−	0.227240i	$-0.0729701\pi$
$702$	0	0
$703$	−4.51541	−0.170302
$704$	0	0
$705$	20.3222	0.765379
$706$	0	0
$707$	− 7.48152i	− 0.281372i
$708$	0	0
$709$	− 13.3957i	− 0.503085i	−0.967846	−	0.251543i	$-0.919062\pi$
$709$	− 13.3957i	− 0.503085i	0.967846	−	0.251543i	$-0.0809379\pi$
$710$	0	0
$711$	3.50781	0.131553
$712$	0	0
$713$	−0.0740191	−0.00277204
$714$	0	0
$715$	− 6.68932i	− 0.250167i
$716$	0	0
$717$	− 2.78762i	− 0.104105i
$718$	0	0
$719$	−19.5596	−0.729452	−0.364726	−	0.931115i	$-0.618837\pi$
$719$	−19.5596	−0.729452	−0.364726	+	0.931115i	$0.618837\pi$
$720$	0	0
$721$	−11.8173	−0.440099
$722$	0	0
$723$	42.1650i	1.56813i
$724$	0	0
$725$	1.81874i	0.0675462i
$726$	0	0
$727$	1.11125	0.0412139	0.0206069	−	0.999788i	$-0.493440\pi$
$727$	1.11125	0.0412139	0.0206069	+	0.999788i	$0.493440\pi$
$728$	0	0
$729$	21.0938	0.781252
$730$	0	0
$731$	− 6.72693i	− 0.248804i
$732$	0	0
$733$	− 9.82787i	− 0.363001i	−0.983391	−	0.181500i	$-0.941905\pi$
$733$	− 9.82787i	− 0.363001i	0.983391	−	0.181500i	$-0.0580954\pi$
$734$	0	0
$735$	3.80744	0.140440
$736$	0	0
$737$	−35.4245	−1.30488
$738$	0	0
$739$	14.9997i	0.551773i	0.961190	+	0.275887i	$0.0889714\pi$
$739$	14.9997i	0.551773i	−0.961190	+	0.275887i	$0.911029\pi$
$740$	0	0
$741$	6.05001i	0.222253i
$742$	0	0
$743$	9.74460	0.357495	0.178747	−	0.983895i	$-0.442796\pi$
$743$	9.74460	0.357495	0.178747	+	0.983895i	$0.442796\pi$
$744$	0	0
$745$	3.14823	0.115342
$746$	0	0
$747$	− 30.4264i	− 1.11325i
$748$	0	0
$749$	16.5805i	0.605837i
$750$	0	0
$751$	8.70157	0.317525	0.158762	−	0.987317i	$-0.449250\pi$
$751$	8.70157	0.317525	0.158762	+	0.987317i	$0.449250\pi$
$752$	0	0
$753$	−36.6884	−1.33700
$754$	0	0
$755$	− 26.4701i	− 0.963345i
$756$	0	0
$757$	− 15.2448i	− 0.554082i	−0.960858	−	0.277041i	$-0.910646\pi$
$757$	− 15.2448i	− 0.554082i	0.960858	−	0.277041i	$-0.0893539\pi$
$758$	0	0
$759$	0.190491	0.00691439
$760$	0	0
$761$	−43.2546	−1.56798	−0.783990	−	0.620774i	$-0.786819\pi$
$761$	−43.2546	−1.56798	−0.783990	+	0.620774i	$0.786819\pi$
$762$	0	0
$763$	11.9418i	0.432321i
$764$	0	0
$765$	2.44537i	0.0884124i
$766$	0	0
$767$	7.12700	0.257341
$768$	0	0
$769$	39.5036	1.42454	0.712269	−	0.701907i	$-0.247668\pi$
$769$	39.5036	1.42454	0.712269	+	0.701907i	$0.247668\pi$
$770$	0	0
$771$	− 9.26450i	− 0.333653i
$772$	0	0
$773$	− 42.8504i	− 1.54122i	−0.637306	−	0.770611i	$-0.719951\pi$
$773$	− 42.8504i	− 1.54122i	0.637306	−	0.770611i	$-0.280049\pi$
$774$	0	0
$775$	9.51391	0.341750
$776$	0	0
$777$	−4.24413	−0.152257
$778$	0	0
$779$	− 20.5190i	− 0.735171i
$780$	0	0
$781$	1.78190i	0.0637612i
$782$	0	0
$783$	−0.554727	−0.0198243
$784$	0	0
$785$	11.5240	0.411309
$786$	0	0
$787$	− 21.9628i	− 0.782891i	−0.920201	−	0.391446i	$-0.871975\pi$
$787$	− 21.9628i	− 0.782891i	0.920201	−	0.391446i	$-0.128025\pi$
$788$	0	0
$789$	− 52.1436i	− 1.85636i
$790$	0	0
$791$	5.32589	0.189367
$792$	0	0
$793$	10.6962	0.379832
$794$	0	0
$795$	17.5495i	0.622416i
$796$	0	0
$797$	43.1481i	1.52838i	0.644990	+	0.764191i	$0.276862\pi$
$797$	43.1481i	1.52838i	−0.644990	+	0.764191i	$0.723138\pi$
$798$	0	0
$799$	3.04284	0.107648
$800$	0	0
$801$	4.43012	0.156530
$802$	0	0
$803$	34.3043i	1.21057i
$804$	0	0
$805$	0.0304429i	0.00107297i
$806$	0	0
$807$	−69.0717	−2.43144
$808$	0	0
$809$	17.1306	0.602280	0.301140	−	0.953580i	$-0.402633\pi$
$809$	17.1306	0.602280	0.301140	+	0.953580i	$0.402633\pi$
$810$	0	0
$811$	13.1707i	0.462486i	0.972896	+	0.231243i	$0.0742793\pi$
$811$	13.1707i	0.462486i	−0.972896	+	0.231243i	$0.925721\pi$
$812$	0	0
$813$	− 39.9606i	− 1.40148i
$814$	0	0
$815$	−33.2617	−1.16511
$816$	0	0
$817$	29.9370	1.04736
$818$	0	0
$819$	2.68654i	0.0938754i
$820$	0	0
$821$	− 9.58055i	− 0.334364i	−0.985926	−	0.167182i	$-0.946533\pi$
$821$	− 9.58055i	− 0.334364i	0.985926	−	0.167182i	$-0.0534667\pi$
$822$	0	0
$823$	36.0229	1.25568	0.627839	−	0.778343i	$-0.283940\pi$
$823$	36.0229	1.25568	0.627839	+	0.778343i	$0.283940\pi$
$824$	0	0
$825$	−24.4844	−0.852439
$826$	0	0
$827$	− 20.6959i	− 0.719668i	−0.933016	−	0.359834i	$-0.882833\pi$
$827$	− 20.6959i	− 0.719668i	0.933016	−	0.359834i	$-0.117167\pi$
$828$	0	0
$829$	− 6.63330i	− 0.230384i	−0.993343	−	0.115192i	$-0.963252\pi$
$829$	− 6.63330i	− 0.230384i	0.993343	−	0.115192i	$-0.0367483\pi$
$830$	0	0
$831$	46.1538	1.60106
$832$	0	0
$833$	0.570087	0.0197523
$834$	0	0
$835$	26.4610i	0.915721i
$836$	0	0
$837$	2.90181i	0.100301i
$838$	0	0
$839$	2.63463	0.0909576	0.0454788	−	0.998965i	$-0.485519\pi$
$839$	2.63463	0.0909576	0.0454788	+	0.998965i	$0.485519\pi$
$840$	0	0
$841$	28.4493	0.981009
$842$	0	0
$843$	6.60778i	0.227584i
$844$	0	0
$845$	1.59665i	0.0549264i
$846$	0	0
$847$	−6.55279	−0.225156
$848$	0	0
$849$	−65.9848	−2.26459
$850$	0	0
$851$	− 0.0339345i	− 0.00116326i
$852$	0	0
$853$	− 17.6477i	− 0.604247i	−0.953269	−	0.302123i	$-0.902305\pi$
$853$	− 17.6477i	− 0.604247i	0.953269	−	0.302123i	$-0.0976955\pi$
$854$	0	0
$855$	−10.8827	−0.372179
$856$	0	0
$857$	5.05243	0.172588	0.0862939	−	0.996270i	$-0.472498\pi$
$857$	5.05243	0.172588	0.0862939	+	0.996270i	$0.472498\pi$
$858$	0	0
$859$	39.4750i	1.34687i	0.739247	+	0.673434i	$0.235182\pi$
$859$	39.4750i	1.34687i	−0.739247	+	0.673434i	$0.764818\pi$
$860$	0	0
$861$	− 19.2863i	− 0.657276i
$862$	0	0
$863$	−2.70602	−0.0921141	−0.0460571	−	0.998939i	$-0.514666\pi$
$863$	−2.70602	−0.0921141	−0.0460571	+	0.998939i	$0.514666\pi$
$864$	0	0
$865$	−5.56388	−0.189178
$866$	0	0
$867$	− 39.7640i	− 1.35046i
$868$	0	0
$869$	− 5.47035i	− 0.185569i
$870$	0	0
$871$	8.45532	0.286498
$872$	0	0
$873$	12.8947	0.436421
$874$	0	0
$875$	− 11.8962i	− 0.402164i
$876$	0	0
$877$	39.6508i	1.33891i	0.742852	+	0.669456i	$0.233473\pi$
$877$	39.6508i	1.33891i	−0.742852	+	0.669456i	$0.766527\pi$
$878$	0	0
$879$	11.2614	0.379839
$880$	0	0
$881$	−43.1406	−1.45344	−0.726721	−	0.686932i	$-0.758957\pi$
$881$	−43.1406	−1.45344	−0.726721	+	0.686932i	$0.758957\pi$
$882$	0	0
$883$	− 56.2653i	− 1.89348i	−0.322001	−	0.946739i	$-0.604356\pi$
$883$	− 56.2653i	− 1.89348i	0.322001	−	0.946739i	$-0.395644\pi$
$884$	0	0
$885$	27.1357i	0.912155i
$886$	0	0
$887$	−11.6364	−0.390712	−0.195356	−	0.980732i	$-0.562586\pi$
$887$	−11.6364	−0.390712	−0.195356	+	0.980732i	$0.562586\pi$
$888$	0	0
$889$	−15.1726	−0.508873
$890$	0	0
$891$	− 41.2346i	− 1.38141i
$892$	0	0
$893$	13.5416i	0.453153i
$894$	0	0
$895$	35.3700	1.18229
$896$	0	0
$897$	−0.0454676	−0.00151812
$898$	0	0
$899$	2.88100i	0.0960868i
$900$	0	0
$901$	2.62768i	0.0875406i
$902$	0	0
$903$	28.1384	0.936388
$904$	0	0
$905$	−27.0652	−0.899678
$906$	0	0
$907$	− 36.5641i	− 1.21409i	−0.794667	−	0.607046i	$-0.792355\pi$
$907$	− 36.5641i	− 1.21409i	0.794667	−	0.607046i	$-0.207645\pi$
$908$	0	0
$909$	20.0994i	0.666656i
$910$	0	0
$911$	−40.3066	−1.33542	−0.667709	−	0.744423i	$-0.732725\pi$
$911$	−40.3066	−1.33542	−0.667709	+	0.744423i	$0.732725\pi$
$912$	0	0
$913$	−47.4494	−1.57034
$914$	0	0
$915$	40.7250i	1.34633i
$916$	0	0
$917$	1.62340i	0.0536092i
$918$	0	0
$919$	−34.0077	−1.12181	−0.560905	−	0.827880i	$-0.689547\pi$
$919$	−34.0077	−1.12181	−0.560905	+	0.827880i	$0.689547\pi$
$920$	0	0
$921$	−48.2070	−1.58848
$922$	0	0
$923$	− 0.425314i	− 0.0139994i
$924$	0	0
$925$	4.36172i	0.143412i
$926$	0	0
$927$	31.7477	1.04273
$928$	0	0
$929$	−23.9803	−0.786769	−0.393385	−	0.919374i	$-0.628696\pi$
$929$	−23.9803	−0.786769	−0.393385	+	0.919374i	$0.628696\pi$
$930$	0	0
$931$	2.53707i	0.0831491i
$932$	0	0
$933$	− 79.6941i	− 2.60907i
$934$	0	0
$935$	3.81349	0.124715
$936$	0	0
$937$	−33.4215	−1.09183	−0.545917	−	0.837840i	$-0.683818\pi$
$937$	−33.4215	−1.09183	−0.545917	+	0.837840i	$0.683818\pi$
$938$	0	0
$939$	− 71.2280i	− 2.32444i
$940$	0	0
$941$	− 24.8842i	− 0.811201i	−0.914050	−	0.405600i	$-0.867062\pi$
$941$	− 24.8842i	− 0.811201i	0.914050	−	0.405600i	$-0.132938\pi$
$942$	0	0
$943$	0.154206	0.00502165
$944$	0	0
$945$	1.19347	0.0388236
$946$	0	0
$947$	− 40.0196i	− 1.30046i	−0.759737	−	0.650231i	$-0.774672\pi$
$947$	− 40.0196i	− 1.30046i	0.759737	−	0.650231i	$-0.225328\pi$
$948$	0	0
$949$	− 8.18795i	− 0.265792i
$950$	0	0
$951$	6.79327	0.220287
$952$	0	0
$953$	45.4900	1.47356	0.736782	−	0.676130i	$-0.236344\pi$
$953$	45.4900	1.47356	0.736782	+	0.676130i	$0.236344\pi$
$954$	0	0
$955$	− 34.5427i	− 1.11777i
$956$	0	0
$957$	− 7.41437i	− 0.239673i
$958$	0	0
$959$	−9.53449	−0.307885
$960$	0	0
$961$	−15.9293	−0.513849
$962$	0	0
$963$	− 44.5441i	− 1.43541i
$964$	0	0
$965$	− 25.8333i	− 0.831602i
$966$	0	0
$967$	60.9357	1.95956	0.979779	−	0.200081i	$-0.0641205\pi$
$967$	60.9357	1.95956	0.979779	+	0.200081i	$0.0641205\pi$
$968$	0	0
$969$	−3.44903	−0.110799
$970$	0	0
$971$	40.4125i	1.29690i	0.761257	+	0.648450i	$0.224582\pi$
$971$	40.4125i	1.29690i	−0.761257	+	0.648450i	$0.775418\pi$
$972$	0	0
$973$	12.2351i	0.392238i
$974$	0	0
$975$	5.84409	0.187161
$976$	0	0
$977$	−19.0379	−0.609076	−0.304538	−	0.952500i	$-0.598502\pi$
$977$	−19.0379	−0.609076	−0.304538	+	0.952500i	$0.598502\pi$
$978$	0	0
$979$	− 6.90867i	− 0.220802i
$980$	0	0
$981$	− 32.0821i	− 1.02430i
$982$	0	0
$983$	−29.8442	−0.951883	−0.475942	−	0.879477i	$-0.657893\pi$
$983$	−29.8442	−0.951883	−0.475942	+	0.879477i	$0.657893\pi$
$984$	0	0
$985$	−24.3197	−0.774889
$986$	0	0
$987$	12.7281i	0.405139i
$988$	0	0
$989$	0.224985i	0.00715410i
$990$	0	0
$991$	−49.4336	−1.57031	−0.785154	−	0.619300i	$-0.787416\pi$
$991$	−49.4336	−1.57031	−0.785154	+	0.619300i	$0.787416\pi$
$992$	0	0
$993$	42.0737	1.33517
$994$	0	0
$995$	9.91530i	0.314336i
$996$	0	0
$997$	9.50971i	0.301176i	0.988597	+	0.150588i	$0.0481166\pi$
$997$	9.50971i	0.301176i	−0.988597	+	0.150588i	$0.951883\pi$
$998$	0	0
$999$	−1.33035	−0.0420905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.c.a.1457.29		34
4.3	odd	2		728.2.c.a.365.6	yes	34
8.3	odd	2		728.2.c.a.365.5	✓	34
8.5	even	2	inner	2912.2.c.a.1457.6		34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.c.a.365.5	✓	34	8.3	odd	2
728.2.c.a.365.6	yes	34	4.3	odd	2
2912.2.c.a.1457.6		34	8.5	even	2	inner
2912.2.c.a.1457.29		34	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.38465i q^{3} -1.59665i q^{5} +1.00000 q^{7} -2.68654 q^{9} +4.18960i q^{11} -1.00000i q^{13} +3.80744 q^{15} +0.570087 q^{17} +2.53707i q^{19} +2.38465i q^{21} -0.0190668 q^{23} +2.45072 q^{25} +0.747485i q^{27} +0.742125i q^{29} +3.88210 q^{31} -9.99073 q^{33} -1.59665i q^{35} +1.77977i q^{37} +2.38465 q^{39} -8.08770 q^{41} -11.7998i q^{43} +4.28946i q^{45} +5.33750 q^{47} +1.00000 q^{49} +1.35946i q^{51} +4.60925i q^{53} +6.68932 q^{55} -6.05001 q^{57} +7.12700i q^{59} +10.6962i q^{61} -2.68654 q^{63} -1.59665 q^{65} +8.45532i q^{67} -0.0454676i q^{69} +0.425314 q^{71} +8.18795 q^{73} +5.84409i q^{75} +4.18960i q^{77} -1.30570 q^{79} -9.84212 q^{81} +11.3255i q^{83} -0.910228i q^{85} -1.76971 q^{87} -1.64900 q^{89} -1.00000i q^{91} +9.25743i q^{93} +4.05081 q^{95} -4.79975 q^{97} -11.2556i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 34 q^{7} - 26 q^{9} + 8 q^{15} - 20 q^{17} + 20 q^{23} - 22 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} + 8 q^{41} + 34 q^{49} + 32 q^{55} + 8 q^{57} - 26 q^{63} - 20 q^{65} - 64 q^{71} - 20 q^{79}+ \cdots + 56 q^{87}+O(q^{100}) \) 34 * q + 34 * q^7 - 26 * q^9 + 8 * q^15 - 20 * q^17 + 20 * q^23 - 22 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 + 8 * q^41 + 34 * q^49 + 32 * q^55 + 8 * q^57 - 26 * q^63 - 20 * q^65 - 64 * q^71 - 20 * q^79 - 6 * q^81 + 56 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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