LMFDB - Embedded newform 9920.2.a.cd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9920,2,Mod(1,9920)] 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("9920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9920 = 2^{6} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-1,0,4,0,0,0,7,0,-6,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$79.2115988051$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.20308.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 4x + 12 $ x^4 - x^3 - 8x^2 + 4x + 12
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.80027$ of defining polynomial
Character	$\chi$	$=$	9920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.342376 q^{3} +1.00000 q^{5} -1.04125 q^{7} -2.88278 q^{9} +4.64180 q^{11} -2.95875 q^{13} -0.342376 q^{15} +6.29942 q^{17} -1.11552 q^{19} +0.356500 q^{21} -3.87454 q^{23} +1.00000 q^{25} +2.01412 q^{27} -2.35650 q^{29} -1.00000 q^{31} -1.58924 q^{33} -1.04125 q^{35} -1.30112 q^{37} +1.01300 q^{39} +1.92573 q^{41} -4.29942 q^{43} -2.88278 q^{45} +3.31525 q^{47} -5.91579 q^{49} -2.15677 q^{51} -5.10964 q^{53} +4.64180 q^{55} +0.381928 q^{57} -5.07256 q^{59} -5.31525 q^{61} +3.00170 q^{63} -2.95875 q^{65} -8.64180 q^{67} +1.32655 q^{69} +3.88278 q^{71} +12.1740 q^{73} -0.342376 q^{75} -4.83329 q^{77} +7.12376 q^{79} +7.95875 q^{81} -7.46614 q^{83} +6.29942 q^{85} +0.806810 q^{87} -12.1598 q^{89} +3.08080 q^{91} +0.342376 q^{93} -1.11552 q^{95} -0.915792 q^{97} -13.3813 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} + 7 q^{9} - 6 q^{11} - 16 q^{13} - q^{15} + q^{17} + 5 q^{19} - 2 q^{21} + 4 q^{25} + 5 q^{27} - 6 q^{29} - 4 q^{31} - 12 q^{33} - 9 q^{37} + 6 q^{39} + 13 q^{41} + 7 q^{43} + 7 q^{45}+ \cdots - 52 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 + 7 * q^9 - 6 * q^11 - 16 * q^13 - q^15 + q^17 + 5 * q^19 - 2 * q^21 + 4 * q^25 + 5 * q^27 - 6 * q^29 - 4 * q^31 - 12 * q^33 - 9 * q^37 + 6 * q^39 + 13 * q^41 + 7 * q^43 + 7 * q^45 + 14 * q^47 - 4 * q^49 + 5 * q^51 - 11 * q^53 - 6 * q^55 - 31 * q^57 + 13 * q^59 - 22 * q^61 + 40 * q^63 - 16 * q^65 - 10 * q^67 - 20 * q^69 - 3 * q^71 + 9 * q^73 - q^75 - 8 * q^77 + 16 * q^79 + 36 * q^81 - 17 * q^83 + q^85 - 38 * q^87 - 12 * q^89 - 24 * q^91 + q^93 + 5 * q^95 + 16 * q^97 - 52 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.342376	−0.197671	−0.0988355	−	0.995104i	$-0.531512\pi$
$3$	−0.342376	−0.197671	−0.0988355	+	0.995104i	$0.531512\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.04125	−0.393557	−0.196778	−	0.980448i	$-0.563048\pi$
$7$	−1.04125	−0.393557	−0.196778	+	0.980448i	$0.563048\pi$
$8$	0	0
$9$	−2.88278	−0.960926
$10$	0	0
$11$	4.64180	1.39955	0.699777	−	0.714361i	$-0.253283\pi$
$11$	4.64180	1.39955	0.699777	+	0.714361i	$0.253283\pi$
$12$	0	0
$13$	−2.95875	−0.820609	−0.410304	−	0.911949i	$-0.634578\pi$
$13$	−2.95875	−0.820609	−0.410304	+	0.911949i	$0.634578\pi$
$14$	0	0
$15$	−0.342376	−0.0884012
$16$	0	0
$17$	6.29942	1.52783	0.763917	−	0.645314i	$-0.223274\pi$
$17$	6.29942	1.52783	0.763917	+	0.645314i	$0.223274\pi$
$18$	0	0
$19$	−1.11552	−0.255918	−0.127959	−	0.991779i	$-0.540843\pi$
$19$	−1.11552	−0.255918	−0.127959	+	0.991779i	$0.540843\pi$
$20$	0	0
$21$	0.356500	0.0777948
$22$	0	0
$23$	−3.87454	−0.807897	−0.403949	−	0.914782i	$-0.632363\pi$
$23$	−3.87454	−0.807897	−0.403949	+	0.914782i	$0.632363\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.01412	0.387618
$28$	0	0
$29$	−2.35650	−0.437591	−0.218796	−	0.975771i	$-0.570213\pi$
$29$	−2.35650	−0.437591	−0.218796	+	0.975771i	$0.570213\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.58924	−0.276651
$34$	0	0
$35$	−1.04125	−0.176004
$36$	0	0
$37$	−1.30112	−0.213903	−0.106952	−	0.994264i	$-0.534109\pi$
$37$	−1.30112	−0.213903	−0.106952	+	0.994264i	$0.534109\pi$
$38$	0	0
$39$	1.01300	0.162211
$40$	0	0
$41$	1.92573	0.300749	0.150375	−	0.988629i	$-0.451952\pi$
$41$	1.92573	0.300749	0.150375	+	0.988629i	$0.451952\pi$
$42$	0	0
$43$	−4.29942	−0.655656	−0.327828	−	0.944737i	$-0.606317\pi$
$43$	−4.29942	−0.655656	−0.327828	+	0.944737i	$0.606317\pi$
$44$	0	0
$45$	−2.88278	−0.429739
$46$	0	0
$47$	3.31525	0.483579	0.241789	−	0.970329i	$-0.422266\pi$
$47$	3.31525	0.483579	0.241789	+	0.970329i	$0.422266\pi$
$48$	0	0
$49$	−5.91579	−0.845113
$50$	0	0
$51$	−2.15677	−0.302009
$52$	0	0
$53$	−5.10964	−0.701862	−0.350931	−	0.936401i	$-0.614135\pi$
$53$	−5.10964	−0.701862	−0.350931	+	0.936401i	$0.614135\pi$
$54$	0	0
$55$	4.64180	0.625900
$56$	0	0
$57$	0.381928	0.0505875
$58$	0	0
$59$	−5.07256	−0.660392	−0.330196	−	0.943912i	$-0.607115\pi$
$59$	−5.07256	−0.660392	−0.330196	+	0.943912i	$0.607115\pi$
$60$	0	0
$61$	−5.31525	−0.680548	−0.340274	−	0.940326i	$-0.610520\pi$
$61$	−5.31525	−0.680548	−0.340274	+	0.940326i	$0.610520\pi$
$62$	0	0
$63$	3.00170	0.378179
$64$	0	0
$65$	−2.95875	−0.366987
$66$	0	0
$67$	−8.64180	−1.05576	−0.527882	−	0.849318i	$-0.677014\pi$
$67$	−8.64180	−1.05576	−0.527882	+	0.849318i	$0.677014\pi$
$68$	0	0
$69$	1.32655	0.159698
$70$	0	0
$71$	3.88278	0.460801	0.230401	−	0.973096i	$-0.425996\pi$
$71$	3.88278	0.460801	0.230401	+	0.973096i	$0.425996\pi$
$72$	0	0
$73$	12.1740	1.42485	0.712427	−	0.701746i	$-0.247596\pi$
$73$	12.1740	1.42485	0.712427	+	0.701746i	$0.247596\pi$
$74$	0	0
$75$	−0.342376	−0.0395342
$76$	0	0
$77$	−4.83329	−0.550804
$78$	0	0
$79$	7.12376	0.801486	0.400743	−	0.916191i	$-0.368752\pi$
$79$	7.12376	0.801486	0.400743	+	0.916191i	$0.368752\pi$
$80$	0	0
$81$	7.95875	0.884305
$82$	0	0
$83$	−7.46614	−0.819515	−0.409757	−	0.912195i	$-0.634387\pi$
$83$	−7.46614	−0.819515	−0.409757	+	0.912195i	$0.634387\pi$
$84$	0	0
$85$	6.29942	0.683268
$86$	0	0
$87$	0.806810	0.0864991
$88$	0	0
$89$	−12.1598	−1.28894	−0.644470	−	0.764630i	$-0.722922\pi$
$89$	−12.1598	−1.28894	−0.644470	+	0.764630i	$0.722922\pi$
$90$	0	0
$91$	3.08080	0.322956
$92$	0	0
$93$	0.342376	0.0355028
$94$	0	0
$95$	−1.11552	−0.114450
$96$	0	0
$97$	−0.915792	−0.0929846	−0.0464923	−	0.998919i	$-0.514804\pi$
$97$	−0.915792	−0.0929846	−0.0464923	+	0.998919i	$0.514804\pi$
$98$	0	0
$99$	−13.3813	−1.34487
$100$	0	0
$101$	4.11382	0.409340	0.204670	−	0.978831i	$-0.434388\pi$
$101$	4.11382	0.409340	0.204670	+	0.978831i	$0.434388\pi$
$102$	0	0
$103$	16.9983	1.67489	0.837446	−	0.546520i	$-0.184048\pi$
$103$	16.9983	1.67489	0.837446	+	0.546520i	$0.184048\pi$
$104$	0	0
$105$	0.356500	0.0347909
$106$	0	0
$107$	4.84976	0.468844	0.234422	−	0.972135i	$-0.424680\pi$
$107$	4.84976	0.468844	0.234422	+	0.972135i	$0.424680\pi$
$108$	0	0
$109$	−9.19803	−0.881011	−0.440506	−	0.897750i	$-0.645201\pi$
$109$	−9.19803	−0.881011	−0.440506	+	0.897750i	$0.645201\pi$
$110$	0	0
$111$	0.445474	0.0422825
$112$	0	0
$113$	3.47508	0.326908	0.163454	−	0.986551i	$-0.447736\pi$
$113$	3.47508	0.326908	0.163454	+	0.986551i	$0.447736\pi$
$114$	0	0
$115$	−3.87454	−0.361303
$116$	0	0
$117$	8.52941	0.788544
$118$	0	0
$119$	−6.55929	−0.601289
$120$	0	0
$121$	10.5463	0.958753
$122$	0	0
$123$	−0.659325	−0.0594494
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.8728	−1.14228	−0.571140	−	0.820853i	$-0.693499\pi$
$127$	−12.8728	−1.14228	−0.571140	+	0.820853i	$0.693499\pi$
$128$	0	0
$129$	1.47202	0.129604
$130$	0	0
$131$	15.4668	1.35134	0.675672	−	0.737202i	$-0.263853\pi$
$131$	15.4668	1.35134	0.675672	+	0.737202i	$0.263853\pi$
$132$	0	0
$133$	1.16154	0.100718
$134$	0	0
$135$	2.01412	0.173348
$136$	0	0
$137$	−16.5734	−1.41596	−0.707981	−	0.706231i	$-0.750394\pi$
$137$	−16.5734	−1.41596	−0.707981	+	0.706231i	$0.750394\pi$
$138$	0	0
$139$	21.5259	1.82581	0.912903	−	0.408176i	$-0.133835\pi$
$139$	21.5259	1.82581	0.912903	+	0.408176i	$0.133835\pi$
$140$	0	0
$141$	−1.13506	−0.0955895
$142$	0	0
$143$	−13.7339	−1.14849
$144$	0	0
$145$	−2.35650	−0.195697
$146$	0	0
$147$	2.02543	0.167054
$148$	0	0
$149$	−20.7556	−1.70037	−0.850183	−	0.526487i	$-0.823509\pi$
$149$	−20.7556	−1.70037	−0.850183	+	0.526487i	$0.823509\pi$
$150$	0	0
$151$	18.4451	1.50104	0.750522	−	0.660846i	$-0.229802\pi$
$151$	18.4451	1.50104	0.750522	+	0.660846i	$0.229802\pi$
$152$	0	0
$153$	−18.1598	−1.46814
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−5.64350	−0.450400	−0.225200	−	0.974313i	$-0.572304\pi$
$157$	−5.64350	−0.450400	−0.225200	+	0.974313i	$0.572304\pi$
$158$	0	0
$159$	1.74942	0.138738
$160$	0	0
$161$	4.03438	0.317953
$162$	0	0
$163$	−16.1994	−1.26883	−0.634417	−	0.772991i	$-0.718760\pi$
$163$	−16.1994	−1.26883	−0.634417	+	0.772991i	$0.718760\pi$
$164$	0	0
$165$	−1.58924	−0.123722
$166$	0	0
$167$	−5.14919	−0.398456	−0.199228	−	0.979953i	$-0.563843\pi$
$167$	−5.14919	−0.398456	−0.199228	+	0.979953i	$0.563843\pi$
$168$	0	0
$169$	−4.24582	−0.326601
$170$	0	0
$171$	3.21580	0.245918
$172$	0	0
$173$	−10.7621	−0.818226	−0.409113	−	0.912484i	$-0.634162\pi$
$173$	−10.7621	−0.818226	−0.409113	+	0.912484i	$0.634162\pi$
$174$	0	0
$175$	−1.04125	−0.0787113
$176$	0	0
$177$	1.73673	0.130540
$178$	0	0
$179$	−10.2740	−0.767914	−0.383957	−	0.923351i	$-0.625439\pi$
$179$	−10.2740	−0.767914	−0.383957	+	0.923351i	$0.625439\pi$
$180$	0	0
$181$	−2.53622	−0.188516	−0.0942578	−	0.995548i	$-0.530048\pi$
$181$	−2.53622	−0.188516	−0.0942578	+	0.995548i	$0.530048\pi$
$182$	0	0
$183$	1.81981	0.134525
$184$	0	0
$185$	−1.30112	−0.0956605
$186$	0	0
$187$	29.2406	2.13829
$188$	0	0
$189$	−2.09721	−0.152550
$190$	0	0
$191$	4.45031	0.322013	0.161007	−	0.986953i	$-0.448526\pi$
$191$	4.45031	0.322013	0.161007	+	0.986953i	$0.448526\pi$
$192$	0	0
$193$	−2.71300	−0.195286	−0.0976430	−	0.995222i	$-0.531130\pi$
$193$	−2.71300	−0.195286	−0.0976430	+	0.995222i	$0.531130\pi$
$194$	0	0
$195$	1.01300	0.0725428
$196$	0	0
$197$	5.04125	0.359174	0.179587	−	0.983742i	$-0.442524\pi$
$197$	5.04125	0.359174	0.179587	+	0.983742i	$0.442524\pi$
$198$	0	0
$199$	14.3531	1.01746	0.508732	−	0.860925i	$-0.330114\pi$
$199$	14.3531	1.01746	0.508732	+	0.860925i	$0.330114\pi$
$200$	0	0
$201$	2.95875	0.208694
$202$	0	0
$203$	2.45371	0.172217
$204$	0	0
$205$	1.92573	0.134499
$206$	0	0
$207$	11.1694	0.776330
$208$	0	0
$209$	−5.17802	−0.358171
$210$	0	0
$211$	−11.6288	−0.800559	−0.400280	−	0.916393i	$-0.631087\pi$
$211$	−11.6288	−0.800559	−0.400280	+	0.916393i	$0.631087\pi$
$212$	0	0
$213$	−1.32937	−0.0910870
$214$	0	0
$215$	−4.29942	−0.293218
$216$	0	0
$217$	1.04125	0.0706849
$218$	0	0
$219$	−4.16808	−0.281652
$220$	0	0
$221$	−18.6384	−1.25375
$222$	0	0
$223$	−5.45483	−0.365283	−0.182641	−	0.983180i	$-0.558465\pi$
$223$	−5.45483	−0.365283	−0.182641	+	0.983180i	$0.558465\pi$
$224$	0	0
$225$	−2.88278	−0.192185
$226$	0	0
$227$	−7.97352	−0.529221	−0.264611	−	0.964355i	$-0.585243\pi$
$227$	−7.97352	−0.529221	−0.264611	+	0.964355i	$0.585243\pi$
$228$	0	0
$229$	−22.0197	−1.45510	−0.727550	−	0.686054i	$-0.759341\pi$
$229$	−22.0197	−1.45510	−0.727550	+	0.686054i	$0.759341\pi$
$230$	0	0
$231$	1.65480	0.108878
$232$	0	0
$233$	−17.2723	−1.13155	−0.565773	−	0.824561i	$-0.691422\pi$
$233$	−17.2723	−1.13155	−0.565773	+	0.824561i	$0.691422\pi$
$234$	0	0
$235$	3.31525	0.216263
$236$	0	0
$237$	−2.43901	−0.158430
$238$	0	0
$239$	21.3462	1.38077	0.690386	−	0.723441i	$-0.257441\pi$
$239$	21.3462	1.38077	0.690386	+	0.723441i	$0.257441\pi$
$240$	0	0
$241$	−4.15984	−0.267959	−0.133979	−	0.990984i	$-0.542776\pi$
$241$	−4.15984	−0.267959	−0.133979	+	0.990984i	$0.542776\pi$
$242$	0	0
$243$	−8.76726	−0.562420
$244$	0	0
$245$	−5.91579	−0.377946
$246$	0	0
$247$	3.30054	0.210008
$248$	0	0
$249$	2.55623	0.161994
$250$	0	0
$251$	5.76896	0.364134	0.182067	−	0.983286i	$-0.441721\pi$
$251$	5.76896	0.364134	0.182067	+	0.983286i	$0.441721\pi$
$252$	0	0
$253$	−17.9848	−1.13070
$254$	0	0
$255$	−2.15677	−0.135062
$256$	0	0
$257$	−22.4417	−1.39988	−0.699938	−	0.714203i	$-0.746789\pi$
$257$	−22.4417	−1.39988	−0.699938	+	0.714203i	$0.746789\pi$
$258$	0	0
$259$	1.35480	0.0841831
$260$	0	0
$261$	6.79327	0.420493
$262$	0	0
$263$	−10.1904	−0.628369	−0.314185	−	0.949362i	$-0.601731\pi$
$263$	−10.1904	−0.628369	−0.314185	+	0.949362i	$0.601731\pi$
$264$	0	0
$265$	−5.10964	−0.313882
$266$	0	0
$267$	4.16324	0.254786
$268$	0	0
$269$	−28.0904	−1.71270	−0.856351	−	0.516394i	$-0.827274\pi$
$269$	−28.0904	−1.71270	−0.856351	+	0.516394i	$0.827274\pi$
$270$	0	0
$271$	−0.0429548	−0.00260932	−0.00130466	−	0.999999i	$-0.500415\pi$
$271$	−0.0429548	−0.00260932	−0.00130466	+	0.999999i	$0.500415\pi$
$272$	0	0
$273$	−1.05479	−0.0638391
$274$	0	0
$275$	4.64180	0.279911
$276$	0	0
$277$	21.8288	1.31156	0.655782	−	0.754951i	$-0.272339\pi$
$277$	21.8288	1.31156	0.655782	+	0.754951i	$0.272339\pi$
$278$	0	0
$279$	2.88278	0.172587
$280$	0	0
$281$	−9.20415	−0.549074	−0.274537	−	0.961577i	$-0.588525\pi$
$281$	−9.20415	−0.549074	−0.274537	+	0.961577i	$0.588525\pi$
$282$	0	0
$283$	−6.85317	−0.407379	−0.203689	−	0.979036i	$-0.565293\pi$
$283$	−6.85317	−0.407379	−0.203689	+	0.979036i	$0.565293\pi$
$284$	0	0
$285$	0.381928	0.0226234
$286$	0	0
$287$	−2.00518	−0.118362
$288$	0	0
$289$	22.6827	1.33428
$290$	0	0
$291$	0.313546	0.0183804
$292$	0	0
$293$	16.6101	0.970375	0.485188	−	0.874410i	$-0.338751\pi$
$293$	16.6101	0.970375	0.485188	+	0.874410i	$0.338751\pi$
$294$	0	0
$295$	−5.07256	−0.295336
$296$	0	0
$297$	9.34916	0.542493
$298$	0	0
$299$	11.4638	0.662968
$300$	0	0
$301$	4.47679	0.258038
$302$	0	0
$303$	−1.40847	−0.0809147
$304$	0	0
$305$	−5.31525	−0.304350
$306$	0	0
$307$	16.9553	0.967693	0.483846	−	0.875153i	$-0.339239\pi$
$307$	16.9553	0.967693	0.483846	+	0.875153i	$0.339239\pi$
$308$	0	0
$309$	−5.81981	−0.331078
$310$	0	0
$311$	0.638734	0.0362193	0.0181096	−	0.999836i	$-0.494235\pi$
$311$	0.638734	0.0362193	0.0181096	+	0.999836i	$0.494235\pi$
$312$	0	0
$313$	−18.1141	−1.02387	−0.511934	−	0.859025i	$-0.671071\pi$
$313$	−18.1141	−1.02387	−0.511934	+	0.859025i	$0.671071\pi$
$314$	0	0
$315$	3.00170	0.169127
$316$	0	0
$317$	5.06433	0.284441	0.142220	−	0.989835i	$-0.454576\pi$
$317$	5.06433	0.284441	0.142220	+	0.989835i	$0.454576\pi$
$318$	0	0
$319$	−10.9384	−0.612433
$320$	0	0
$321$	−1.66044	−0.0926770
$322$	0	0
$323$	−7.02713	−0.391000
$324$	0	0
$325$	−2.95875	−0.164122
$326$	0	0
$327$	3.14919	0.174150
$328$	0	0
$329$	−3.45201	−0.190316
$330$	0	0
$331$	−20.0344	−1.10119	−0.550594	−	0.834773i	$-0.685599\pi$
$331$	−20.0344	−1.10119	−0.550594	+	0.834773i	$0.685599\pi$
$332$	0	0
$333$	3.75085	0.205545
$334$	0	0
$335$	−8.64180	−0.472152
$336$	0	0
$337$	−4.86729	−0.265138	−0.132569	−	0.991174i	$-0.542323\pi$
$337$	−4.86729	−0.265138	−0.132569	+	0.991174i	$0.542323\pi$
$338$	0	0
$339$	−1.18979	−0.0646203
$340$	0	0
$341$	−4.64180	−0.251367
$342$	0	0
$343$	13.4486	0.726157
$344$	0	0
$345$	1.32655	0.0714191
$346$	0	0
$347$	−10.1221	−0.543381	−0.271690	−	0.962385i	$-0.587583\pi$
$347$	−10.1221	−0.543381	−0.271690	+	0.962385i	$0.587583\pi$
$348$	0	0
$349$	18.3245	0.980888	0.490444	−	0.871473i	$-0.336835\pi$
$349$	18.3245	0.980888	0.490444	+	0.871473i	$0.336835\pi$
$350$	0	0
$351$	−5.95928	−0.318083
$352$	0	0
$353$	−14.3248	−0.762435	−0.381217	−	0.924485i	$-0.624495\pi$
$353$	−14.3248	−0.762435	−0.381217	+	0.924485i	$0.624495\pi$
$354$	0	0
$355$	3.88278	0.206076
$356$	0	0
$357$	2.24575	0.118857
$358$	0	0
$359$	−12.3197	−0.650207	−0.325104	−	0.945678i	$-0.605399\pi$
$359$	−12.3197	−0.650207	−0.325104	+	0.945678i	$0.605399\pi$
$360$	0	0
$361$	−17.7556	−0.934506
$362$	0	0
$363$	−3.61080	−0.189518
$364$	0	0
$365$	12.1740	0.637214
$366$	0	0
$367$	−24.4706	−1.27735	−0.638676	−	0.769475i	$-0.720518\pi$
$367$	−24.4706	−1.27735	−0.638676	+	0.769475i	$0.720518\pi$
$368$	0	0
$369$	−5.55146	−0.288998
$370$	0	0
$371$	5.32042	0.276223
$372$	0	0
$373$	−26.5276	−1.37355	−0.686775	−	0.726870i	$-0.740974\pi$
$373$	−26.5276	−1.37355	−0.686775	+	0.726870i	$0.740974\pi$
$374$	0	0
$375$	−0.342376	−0.0176802
$376$	0	0
$377$	6.97229	0.359091
$378$	0	0
$379$	12.5528	0.644795	0.322398	−	0.946604i	$-0.395511\pi$
$379$	12.5528	0.644795	0.322398	+	0.946604i	$0.395511\pi$
$380$	0	0
$381$	4.40735	0.225796
$382$	0	0
$383$	−11.5882	−0.592129	−0.296064	−	0.955168i	$-0.595674\pi$
$383$	−11.5882	−0.592129	−0.296064	+	0.955168i	$0.595674\pi$
$384$	0	0
$385$	−4.83329	−0.246327
$386$	0	0
$387$	12.3943	0.630037
$388$	0	0
$389$	−32.5790	−1.65182	−0.825909	−	0.563803i	$-0.809338\pi$
$389$	−32.5790	−1.65182	−0.825909	+	0.563803i	$0.809338\pi$
$390$	0	0
$391$	−24.4074	−1.23433
$392$	0	0
$393$	−5.29548	−0.267122
$394$	0	0
$395$	7.12376	0.358435
$396$	0	0
$397$	−29.8508	−1.49817	−0.749084	−	0.662475i	$-0.769506\pi$
$397$	−29.8508	−1.49817	−0.749084	+	0.662475i	$0.769506\pi$
$398$	0	0
$399$	−0.397683	−0.0199091
$400$	0	0
$401$	−24.9176	−1.24432	−0.622162	−	0.782889i	$-0.713745\pi$
$401$	−24.9176	−1.24432	−0.622162	+	0.782889i	$0.713745\pi$
$402$	0	0
$403$	2.95875	0.147386
$404$	0	0
$405$	7.95875	0.395473
$406$	0	0
$407$	−6.03955	−0.299369
$408$	0	0
$409$	13.8085	0.682787	0.341393	−	0.939920i	$-0.389101\pi$
$409$	13.8085	0.682787	0.341393	+	0.939920i	$0.389101\pi$
$410$	0	0
$411$	5.67435	0.279895
$412$	0	0
$413$	5.28182	0.259902
$414$	0	0
$415$	−7.46614	−0.366498
$416$	0	0
$417$	−7.36997	−0.360909
$418$	0	0
$419$	−21.7799	−1.06402	−0.532009	−	0.846738i	$-0.678563\pi$
$419$	−21.7799	−1.06402	−0.532009	+	0.846738i	$0.678563\pi$
$420$	0	0
$421$	−3.23758	−0.157790	−0.0788949	−	0.996883i	$-0.525139\pi$
$421$	−3.23758	−0.157790	−0.0788949	+	0.996883i	$0.525139\pi$
$422$	0	0
$423$	−9.55712	−0.464683
$424$	0	0
$425$	6.29942	0.305567
$426$	0	0
$427$	5.53452	0.267834
$428$	0	0
$429$	4.70216	0.227023
$430$	0	0
$431$	12.6466	0.609167	0.304583	−	0.952486i	$-0.401483\pi$
$431$	12.6466	0.609167	0.304583	+	0.952486i	$0.401483\pi$
$432$	0	0
$433$	31.9914	1.53741	0.768705	−	0.639604i	$-0.220902\pi$
$433$	31.9914	1.53741	0.768705	+	0.639604i	$0.220902\pi$
$434$	0	0
$435$	0.806810	0.0386836
$436$	0	0
$437$	4.32212	0.206755
$438$	0	0
$439$	13.8432	0.660701	0.330351	−	0.943858i	$-0.392833\pi$
$439$	13.8432	0.660701	0.330351	+	0.943858i	$0.392833\pi$
$440$	0	0
$441$	17.0539	0.812091
$442$	0	0
$443$	−20.3479	−0.966759	−0.483379	−	0.875411i	$-0.660591\pi$
$443$	−20.3479	−0.966759	−0.483379	+	0.875411i	$0.660591\pi$
$444$	0	0
$445$	−12.1598	−0.576432
$446$	0	0
$447$	7.10623	0.336113
$448$	0	0
$449$	8.71640	0.411353	0.205676	−	0.978620i	$-0.434061\pi$
$449$	8.71640	0.411353	0.205676	+	0.978620i	$0.434061\pi$
$450$	0	0
$451$	8.93886	0.420915
$452$	0	0
$453$	−6.31518	−0.296713
$454$	0	0
$455$	3.08080	0.144430
$456$	0	0
$457$	−0.0311855	−0.00145879	−0.000729397	−	1.00000i	$-0.500232\pi$
$457$	−0.0311855	−0.00145879	−0.000729397		1.00000i	$0.500232\pi$
$458$	0	0
$459$	12.6878	0.592217
$460$	0	0
$461$	29.4882	1.37340	0.686700	−	0.726941i	$-0.259059\pi$
$461$	29.4882	1.37340	0.686700	+	0.726941i	$0.259059\pi$
$462$	0	0
$463$	19.1547	0.890196	0.445098	−	0.895482i	$-0.353169\pi$
$463$	19.1547	0.890196	0.445098	+	0.895482i	$0.353169\pi$
$464$	0	0
$465$	0.342376	0.0158773
$466$	0	0
$467$	−23.9367	−1.10766	−0.553829	−	0.832630i	$-0.686834\pi$
$467$	−23.9367	−1.10766	−0.553829	+	0.832630i	$0.686834\pi$
$468$	0	0
$469$	8.99830	0.415503
$470$	0	0
$471$	1.93220	0.0890311
$472$	0	0
$473$	−19.9570	−0.917626
$474$	0	0
$475$	−1.11552	−0.0511836
$476$	0	0
$477$	14.7299	0.674438
$478$	0	0
$479$	−21.4039	−0.977968	−0.488984	−	0.872293i	$-0.662632\pi$
$479$	−21.4039	−0.977968	−0.488984	+	0.872293i	$0.662632\pi$
$480$	0	0
$481$	3.84969	0.175531
$482$	0	0
$483$	−1.38127	−0.0628502
$484$	0	0
$485$	−0.915792	−0.0415840
$486$	0	0
$487$	−26.0763	−1.18163	−0.590815	−	0.806807i	$-0.701194\pi$
$487$	−26.0763	−1.18163	−0.590815	+	0.806807i	$0.701194\pi$
$488$	0	0
$489$	5.54629	0.250812
$490$	0	0
$491$	−11.9367	−0.538696	−0.269348	−	0.963043i	$-0.586808\pi$
$491$	−11.9367	−0.538696	−0.269348	+	0.963043i	$0.586808\pi$
$492$	0	0
$493$	−14.8446	−0.668567
$494$	0	0
$495$	−13.3813	−0.601444
$496$	0	0
$497$	−4.04295	−0.181351
$498$	0	0
$499$	16.5394	0.740406	0.370203	−	0.928951i	$-0.379288\pi$
$499$	16.5394	0.740406	0.370203	+	0.928951i	$0.379288\pi$
$500$	0	0
$501$	1.76296	0.0787632
$502$	0	0
$503$	−16.6418	−0.742021	−0.371011	−	0.928629i	$-0.620989\pi$
$503$	−16.6418	−0.742021	−0.371011	+	0.928629i	$0.620989\pi$
$504$	0	0
$505$	4.11382	0.183062
$506$	0	0
$507$	1.45367	0.0645596
$508$	0	0
$509$	43.0388	1.90766	0.953831	−	0.300345i	$-0.0971018\pi$
$509$	43.0388	1.90766	0.953831	+	0.300345i	$0.0971018\pi$
$510$	0	0
$511$	−12.6762	−0.560761
$512$	0	0
$513$	−2.24680	−0.0991984
$514$	0	0
$515$	16.9983	0.749035
$516$	0	0
$517$	15.3887	0.676795
$518$	0	0
$519$	3.68468	0.161740
$520$	0	0
$521$	37.2138	1.63037	0.815184	−	0.579203i	$-0.196636\pi$
$521$	37.2138	1.63037	0.815184	+	0.579203i	$0.196636\pi$
$522$	0	0
$523$	−23.4085	−1.02358	−0.511791	−	0.859110i	$-0.671018\pi$
$523$	−23.4085	−1.02358	−0.511791	+	0.859110i	$0.671018\pi$
$524$	0	0
$525$	0.356500	0.0155590
$526$	0	0
$527$	−6.29942	−0.274407
$528$	0	0
$529$	−7.98795	−0.347302
$530$	0	0
$531$	14.6231	0.634588
$532$	0	0
$533$	−5.69776	−0.246797
$534$	0	0
$535$	4.84976	0.209674
$536$	0	0
$537$	3.51757	0.151794
$538$	0	0
$539$	−27.4599	−1.18278
$540$	0	0
$541$	−2.38298	−0.102452	−0.0512261	−	0.998687i	$-0.516313\pi$
$541$	−2.38298	−0.102452	−0.0512261	+	0.998687i	$0.516313\pi$
$542$	0	0
$543$	0.868341	0.0372641
$544$	0	0
$545$	−9.19803	−0.394000
$546$	0	0
$547$	−21.4400	−0.916709	−0.458355	−	0.888769i	$-0.651561\pi$
$547$	−21.4400	−0.916709	−0.458355	+	0.888769i	$0.651561\pi$
$548$	0	0
$549$	15.3227	0.653956
$550$	0	0
$551$	2.62872	0.111987
$552$	0	0
$553$	−7.41764	−0.315430
$554$	0	0
$555$	0.445474	0.0189093
$556$	0	0
$557$	12.5276	0.530813	0.265407	−	0.964137i	$-0.414494\pi$
$557$	12.5276	0.530813	0.265407	+	0.964137i	$0.414494\pi$
$558$	0	0
$559$	12.7209	0.538037
$560$	0	0
$561$	−10.0113	−0.422678
$562$	0	0
$563$	10.7361	0.452472	0.226236	−	0.974073i	$-0.427358\pi$
$563$	10.7361	0.452472	0.226236	+	0.974073i	$0.427358\pi$
$564$	0	0
$565$	3.47508	0.146198
$566$	0	0
$567$	−8.28707	−0.348024
$568$	0	0
$569$	1.00783	0.0422504	0.0211252	−	0.999777i	$-0.493275\pi$
$569$	1.00783	0.0422504	0.0211252	+	0.999777i	$0.493275\pi$
$570$	0	0
$571$	−9.88631	−0.413729	−0.206865	−	0.978370i	$-0.566326\pi$
$571$	−9.88631	−0.413729	−0.206865	+	0.978370i	$0.566326\pi$
$572$	0	0
$573$	−1.52368	−0.0636527
$574$	0	0
$575$	−3.87454	−0.161579
$576$	0	0
$577$	22.1564	0.922384	0.461192	−	0.887300i	$-0.347422\pi$
$577$	22.1564	0.922384	0.461192	+	0.887300i	$0.347422\pi$
$578$	0	0
$579$	0.928867	0.0386024
$580$	0	0
$581$	7.77414	0.322526
$582$	0	0
$583$	−23.7179	−0.982295
$584$	0	0
$585$	8.52941	0.352648
$586$	0	0
$587$	16.7526	0.691452	0.345726	−	0.938336i	$-0.387633\pi$
$587$	16.7526	0.691452	0.345726	+	0.938336i	$0.387633\pi$
$588$	0	0
$589$	1.11552	0.0459642
$590$	0	0
$591$	−1.72601	−0.0709984
$592$	0	0
$593$	−21.0526	−0.864525	−0.432262	−	0.901748i	$-0.642285\pi$
$593$	−21.0526	−0.864525	−0.432262	+	0.901748i	$0.642285\pi$
$594$	0	0
$595$	−6.55929	−0.268905
$596$	0	0
$597$	−4.91416	−0.201123
$598$	0	0
$599$	24.1994	0.988760	0.494380	−	0.869246i	$-0.335395\pi$
$599$	24.1994	0.988760	0.494380	+	0.869246i	$0.335395\pi$
$600$	0	0
$601$	−37.0275	−1.51038	−0.755192	−	0.655504i	$-0.772456\pi$
$601$	−37.0275	−1.51038	−0.755192	+	0.655504i	$0.772456\pi$
$602$	0	0
$603$	24.9124	1.01451
$604$	0	0
$605$	10.5463	0.428768
$606$	0	0
$607$	10.9011	0.442461	0.221231	−	0.975222i	$-0.428993\pi$
$607$	10.9011	0.442461	0.221231	+	0.975222i	$0.428993\pi$
$608$	0	0
$609$	−0.840093	−0.0340423
$610$	0	0
$611$	−9.80898	−0.396829
$612$	0	0
$613$	−17.8882	−0.722497	−0.361249	−	0.932469i	$-0.617649\pi$
$613$	−17.8882	−0.722497	−0.361249	+	0.932469i	$0.617649\pi$
$614$	0	0
$615$	−0.659325	−0.0265866
$616$	0	0
$617$	−39.3548	−1.58436	−0.792182	−	0.610285i	$-0.791055\pi$
$617$	−39.3548	−1.58436	−0.792182	+	0.610285i	$0.791055\pi$
$618$	0	0
$619$	19.5429	0.785495	0.392747	−	0.919646i	$-0.371525\pi$
$619$	19.5429	0.785495	0.392747	+	0.919646i	$0.371525\pi$
$620$	0	0
$621$	−7.80380	−0.313156
$622$	0	0
$623$	12.6615	0.507271
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.77283	0.0708000
$628$	0	0
$629$	−8.19632	−0.326809
$630$	0	0
$631$	33.4486	1.33157	0.665784	−	0.746145i	$-0.268097\pi$
$631$	33.4486	1.33157	0.665784	+	0.746145i	$0.268097\pi$
$632$	0	0
$633$	3.98142	0.158247
$634$	0	0
$635$	−12.8728	−0.510843
$636$	0	0
$637$	17.5033	0.693507
$638$	0	0
$639$	−11.1932	−0.442796
$640$	0	0
$641$	7.13506	0.281818	0.140909	−	0.990023i	$-0.454998\pi$
$641$	7.13506	0.281818	0.140909	+	0.990023i	$0.454998\pi$
$642$	0	0
$643$	21.1141	0.832660	0.416330	−	0.909214i	$-0.363316\pi$
$643$	21.1141	0.832660	0.416330	+	0.909214i	$0.363316\pi$
$644$	0	0
$645$	1.47202	0.0579607
$646$	0	0
$647$	1.08185	0.0425320	0.0212660	−	0.999774i	$-0.493230\pi$
$647$	1.08185	0.0425320	0.0212660	+	0.999774i	$0.493230\pi$
$648$	0	0
$649$	−23.5458	−0.924254
$650$	0	0
$651$	−0.356500	−0.0139724
$652$	0	0
$653$	−47.5112	−1.85926	−0.929629	−	0.368497i	$-0.879873\pi$
$653$	−47.5112	−1.85926	−0.929629	+	0.368497i	$0.879873\pi$
$654$	0	0
$655$	15.4668	0.604340
$656$	0	0
$657$	−35.0948	−1.36918
$658$	0	0
$659$	−24.3048	−0.946782	−0.473391	−	0.880852i	$-0.656970\pi$
$659$	−24.3048	−0.946782	−0.473391	+	0.880852i	$0.656970\pi$
$660$	0	0
$661$	−1.23308	−0.0479613	−0.0239806	−	0.999712i	$-0.507634\pi$
$661$	−1.23308	−0.0479613	−0.0239806	+	0.999712i	$0.507634\pi$
$662$	0	0
$663$	6.38134	0.247831
$664$	0	0
$665$	1.16154	0.0450425
$666$	0	0
$667$	9.13035	0.353529
$668$	0	0
$669$	1.86761	0.0722058
$670$	0	0
$671$	−24.6723	−0.952464
$672$	0	0
$673$	−11.0242	−0.424951	−0.212476	−	0.977166i	$-0.568153\pi$
$673$	−11.0242	−0.424951	−0.212476	+	0.977166i	$0.568153\pi$
$674$	0	0
$675$	2.01412	0.0775237
$676$	0	0
$677$	39.3994	1.51424	0.757122	−	0.653274i	$-0.226605\pi$
$677$	39.3994	1.51424	0.757122	+	0.653274i	$0.226605\pi$
$678$	0	0
$679$	0.953571	0.0365947
$680$	0	0
$681$	2.72995	0.104612
$682$	0	0
$683$	42.8213	1.63851	0.819256	−	0.573428i	$-0.194387\pi$
$683$	42.8213	1.63851	0.819256	+	0.573428i	$0.194387\pi$
$684$	0	0
$685$	−16.5734	−0.633238
$686$	0	0
$687$	7.53901	0.287631
$688$	0	0
$689$	15.1181	0.575954
$690$	0	0
$691$	−34.1911	−1.30069	−0.650346	−	0.759638i	$-0.725376\pi$
$691$	−34.1911	−1.30069	−0.650346	+	0.759638i	$0.725376\pi$
$692$	0	0
$693$	13.9333	0.529282
$694$	0	0
$695$	21.5259	0.816525
$696$	0	0
$697$	12.1310	0.459495
$698$	0	0
$699$	5.91362	0.223674
$700$	0	0
$701$	31.6598	1.19577	0.597886	−	0.801581i	$-0.296007\pi$
$701$	31.6598	1.19577	0.597886	+	0.801581i	$0.296007\pi$
$702$	0	0
$703$	1.45143	0.0547417
$704$	0	0
$705$	−1.13506	−0.0427489
$706$	0	0
$707$	−4.28353	−0.161099
$708$	0	0
$709$	19.8198	0.744349	0.372174	−	0.928163i	$-0.378612\pi$
$709$	19.8198	0.744349	0.372174	+	0.928163i	$0.378612\pi$
$710$	0	0
$711$	−20.5362	−0.770168
$712$	0	0
$713$	3.87454	0.145103
$714$	0	0
$715$	−13.7339	−0.513619
$716$	0	0
$717$	−7.30844	−0.272939
$718$	0	0
$719$	−25.7339	−0.959713	−0.479856	−	0.877347i	$-0.659311\pi$
$719$	−25.7339	−0.959713	−0.479856	+	0.877347i	$0.659311\pi$
$720$	0	0
$721$	−17.6995	−0.659165
$722$	0	0
$723$	1.42423	0.0529677
$724$	0	0
$725$	−2.35650	−0.0875182
$726$	0	0
$727$	−24.6909	−0.915734	−0.457867	−	0.889021i	$-0.651386\pi$
$727$	−24.6909	−0.915734	−0.457867	+	0.889021i	$0.651386\pi$
$728$	0	0
$729$	−20.8745	−0.773131
$730$	0	0
$731$	−27.0839	−1.00173
$732$	0	0
$733$	−23.0265	−0.850505	−0.425252	−	0.905075i	$-0.639815\pi$
$733$	−23.0265	−0.850505	−0.425252	+	0.905075i	$0.639815\pi$
$734$	0	0
$735$	2.02543	0.0747090
$736$	0	0
$737$	−40.1135	−1.47760
$738$	0	0
$739$	−27.2176	−1.00121	−0.500607	−	0.865675i	$-0.666890\pi$
$739$	−27.2176	−1.00121	−0.500607	+	0.865675i	$0.666890\pi$
$740$	0	0
$741$	−1.13003	−0.0415126
$742$	0	0
$743$	23.0434	0.845380	0.422690	−	0.906274i	$-0.361086\pi$
$743$	23.0434	0.845380	0.422690	+	0.906274i	$0.361086\pi$
$744$	0	0
$745$	−20.7556	−0.760427
$746$	0	0
$747$	21.5232	0.787493
$748$	0	0
$749$	−5.04983	−0.184517
$750$	0	0
$751$	12.5480	0.457883	0.228941	−	0.973440i	$-0.426474\pi$
$751$	12.5480	0.457883	0.228941	+	0.973440i	$0.426474\pi$
$752$	0	0
$753$	−1.97516	−0.0719787
$754$	0	0
$755$	18.4451	0.671287
$756$	0	0
$757$	0.441361	0.0160415	0.00802077	−	0.999968i	$-0.497447\pi$
$757$	0.441361	0.0160415	0.00802077	+	0.999968i	$0.497447\pi$
$758$	0	0
$759$	6.15758	0.223506
$760$	0	0
$761$	−23.0903	−0.837024	−0.418512	−	0.908211i	$-0.637448\pi$
$761$	−23.0903	−0.837024	−0.418512	+	0.908211i	$0.637448\pi$
$762$	0	0
$763$	9.57747	0.346728
$764$	0	0
$765$	−18.1598	−0.656570
$766$	0	0
$767$	15.0084	0.541923
$768$	0	0
$769$	−16.5906	−0.598272	−0.299136	−	0.954210i	$-0.596699\pi$
$769$	−16.5906	−0.598272	−0.299136	+	0.954210i	$0.596699\pi$
$770$	0	0
$771$	7.68352	0.276715
$772$	0	0
$773$	−50.6033	−1.82008	−0.910038	−	0.414525i	$-0.863948\pi$
$773$	−50.6033	−1.82008	−0.910038	+	0.414525i	$0.863948\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	−0.463851	−0.0166406
$778$	0	0
$779$	−2.14819	−0.0769670
$780$	0	0
$781$	18.0231	0.644916
$782$	0	0
$783$	−4.74628	−0.169618
$784$	0	0
$785$	−5.64350	−0.201425
$786$	0	0
$787$	42.9038	1.52936	0.764678	−	0.644413i	$-0.222898\pi$
$787$	42.9038	1.52936	0.764678	+	0.644413i	$0.222898\pi$
$788$	0	0
$789$	3.48897	0.124210
$790$	0	0
$791$	−3.61844	−0.128657
$792$	0	0
$793$	15.7265	0.558463
$794$	0	0
$795$	1.74942	0.0620455
$796$	0	0
$797$	33.5707	1.18913	0.594567	−	0.804046i	$-0.297324\pi$
$797$	33.5707	1.18913	0.594567	+	0.804046i	$0.297324\pi$
$798$	0	0
$799$	20.8841	0.738828
$800$	0	0
$801$	35.0541	1.23858
$802$	0	0
$803$	56.5091	1.99416
$804$	0	0
$805$	4.03438	0.142193
$806$	0	0
$807$	9.61749	0.338552
$808$	0	0
$809$	35.3774	1.24380	0.621902	−	0.783095i	$-0.286360\pi$
$809$	35.3774	1.24380	0.621902	+	0.783095i	$0.286360\pi$
$810$	0	0
$811$	54.9833	1.93072	0.965362	−	0.260916i	$-0.0840245\pi$
$811$	54.9833	1.93072	0.965362	+	0.260916i	$0.0840245\pi$
$812$	0	0
$813$	0.0147067	0.000515787 0
$814$	0	0
$815$	−16.1994	−0.567440
$816$	0	0
$817$	4.79609	0.167794
$818$	0	0
$819$	−8.88128	−0.310337
$820$	0	0
$821$	−47.1995	−1.64727	−0.823636	−	0.567118i	$-0.808058\pi$
$821$	−47.1995	−1.64727	−0.823636	+	0.567118i	$0.808058\pi$
$822$	0	0
$823$	31.1661	1.08638	0.543192	−	0.839609i	$-0.317216\pi$
$823$	31.1661	1.08638	0.543192	+	0.839609i	$0.317216\pi$
$824$	0	0
$825$	−1.58924	−0.0553303
$826$	0	0
$827$	2.62427	0.0912548	0.0456274	−	0.998959i	$-0.485471\pi$
$827$	2.62427	0.0912548	0.0456274	+	0.998959i	$0.485471\pi$
$828$	0	0
$829$	−30.1876	−1.04846	−0.524230	−	0.851577i	$-0.675647\pi$
$829$	−30.1876	−1.04846	−0.524230	+	0.851577i	$0.675647\pi$
$830$	0	0
$831$	−7.47365	−0.259258
$832$	0	0
$833$	−37.2661	−1.29119
$834$	0	0
$835$	−5.14919	−0.178195
$836$	0	0
$837$	−2.01412	−0.0696183
$838$	0	0
$839$	9.75555	0.336799	0.168399	−	0.985719i	$-0.446140\pi$
$839$	9.75555	0.336799	0.168399	+	0.985719i	$0.446140\pi$
$840$	0	0
$841$	−23.4469	−0.808514
$842$	0	0
$843$	3.15128	0.108536
$844$	0	0
$845$	−4.24582	−0.146061
$846$	0	0
$847$	−10.9814	−0.377324
$848$	0	0
$849$	2.34636	0.0805270
$850$	0	0
$851$	5.04125	0.172812
$852$	0	0
$853$	−39.4547	−1.35090	−0.675452	−	0.737404i	$-0.736052\pi$
$853$	−39.4547	−1.35090	−0.675452	+	0.737404i	$0.736052\pi$
$854$	0	0
$855$	3.21580	0.109978
$856$	0	0
$857$	−10.6006	−0.362110	−0.181055	−	0.983473i	$-0.557951\pi$
$857$	−10.6006	−0.362110	−0.181055	+	0.983473i	$0.557951\pi$
$858$	0	0
$859$	−30.1660	−1.02925	−0.514625	−	0.857416i	$-0.672069\pi$
$859$	−30.1660	−1.02925	−0.514625	+	0.857416i	$0.672069\pi$
$860$	0	0
$861$	0.686525	0.0233967
$862$	0	0
$863$	−11.7962	−0.401546	−0.200773	−	0.979638i	$-0.564345\pi$
$863$	−11.7962	−0.401546	−0.200773	+	0.979638i	$0.564345\pi$
$864$	0	0
$865$	−10.7621	−0.365922
$866$	0	0
$867$	−7.76602	−0.263748
$868$	0	0
$869$	33.0670	1.12172
$870$	0	0
$871$	25.5689	0.866369
$872$	0	0
$873$	2.64003	0.0893513
$874$	0	0
$875$	−1.04125	−0.0352008
$876$	0	0
$877$	−12.3531	−0.417134	−0.208567	−	0.978008i	$-0.566880\pi$
$877$	−12.3531	−0.417134	−0.208567	+	0.978008i	$0.566880\pi$
$878$	0	0
$879$	−5.68692	−0.191815
$880$	0	0
$881$	28.2740	0.952575	0.476288	−	0.879290i	$-0.341982\pi$
$881$	28.2740	0.952575	0.476288	+	0.879290i	$0.341982\pi$
$882$	0	0
$883$	14.7328	0.495798	0.247899	−	0.968786i	$-0.420260\pi$
$883$	14.7328	0.495798	0.247899	+	0.968786i	$0.420260\pi$
$884$	0	0
$885$	1.73673	0.0583794
$886$	0	0
$887$	−26.9237	−0.904009	−0.452005	−	0.892016i	$-0.649291\pi$
$887$	−26.9237	−0.904009	−0.452005	+	0.892016i	$0.649291\pi$
$888$	0	0
$889$	13.4039	0.449552
$890$	0	0
$891$	36.9429	1.23763
$892$	0	0
$893$	−3.69822	−0.123756
$894$	0	0
$895$	−10.2740	−0.343422
$896$	0	0
$897$	−3.92493	−0.131050
$898$	0	0
$899$	2.35650	0.0785937
$900$	0	0
$901$	−32.1877	−1.07233
$902$	0	0
$903$	−1.53275	−0.0510066
$904$	0	0
$905$	−2.53622	−0.0843068
$906$	0	0
$907$	−25.7869	−0.856239	−0.428119	−	0.903722i	$-0.640824\pi$
$907$	−25.7869	−0.856239	−0.428119	+	0.903722i	$0.640824\pi$
$908$	0	0
$909$	−11.8592	−0.393346
$910$	0	0
$911$	−24.4130	−0.808839	−0.404419	−	0.914574i	$-0.632526\pi$
$911$	−24.4130	−0.808839	−0.404419	+	0.914574i	$0.632526\pi$
$912$	0	0
$913$	−34.6563	−1.14696
$914$	0	0
$915$	1.81981	0.0601612
$916$	0	0
$917$	−16.1049	−0.531831
$918$	0	0
$919$	6.86800	0.226554	0.113277	−	0.993563i	$-0.463865\pi$
$919$	6.86800	0.226554	0.113277	+	0.993563i	$0.463865\pi$
$920$	0	0
$921$	−5.80511	−0.191285
$922$	0	0
$923$	−11.4882	−0.378137
$924$	0	0
$925$	−1.30112	−0.0427807
$926$	0	0
$927$	−49.0023	−1.60945
$928$	0	0
$929$	12.6470	0.414934	0.207467	−	0.978242i	$-0.433478\pi$
$929$	12.6470	0.414934	0.207467	+	0.978242i	$0.433478\pi$
$930$	0	0
$931$	6.59918	0.216279
$932$	0	0
$933$	−0.218687	−0.00715950
$934$	0	0
$935$	29.2406	0.956271
$936$	0	0
$937$	42.8873	1.40107	0.700534	−	0.713619i	$-0.252945\pi$
$937$	42.8873	1.40107	0.700534	+	0.713619i	$0.252945\pi$
$938$	0	0
$939$	6.20183	0.202389
$940$	0	0
$941$	−24.2254	−0.789725	−0.394863	−	0.918740i	$-0.629208\pi$
$941$	−24.2254	−0.789725	−0.394863	+	0.918740i	$0.629208\pi$
$942$	0	0
$943$	−7.46133	−0.242974
$944$	0	0
$945$	−2.09721	−0.0682223
$946$	0	0
$947$	−16.2682	−0.528647	−0.264323	−	0.964434i	$-0.585149\pi$
$947$	−16.2682	−0.528647	−0.264323	+	0.964434i	$0.585149\pi$
$948$	0	0
$949$	−36.0197	−1.16925
$950$	0	0
$951$	−1.73391	−0.0562257
$952$	0	0
$953$	19.2891	0.624837	0.312418	−	0.949945i	$-0.398861\pi$
$953$	19.2891	0.624837	0.312418	+	0.949945i	$0.398861\pi$
$954$	0	0
$955$	4.45031	0.144009
$956$	0	0
$957$	3.74505	0.121060
$958$	0	0
$959$	17.2571	0.557261
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−13.9808	−0.450525
$964$	0	0
$965$	−2.71300	−0.0873346
$966$	0	0
$967$	−20.9745	−0.674496	−0.337248	−	0.941416i	$-0.609496\pi$
$967$	−20.9745	−0.674496	−0.337248	+	0.941416i	$0.609496\pi$
$968$	0	0
$969$	2.40592	0.0772894
$970$	0	0
$971$	−16.2572	−0.521718	−0.260859	−	0.965377i	$-0.584006\pi$
$971$	−16.2572	−0.521718	−0.260859	+	0.965377i	$0.584006\pi$
$972$	0	0
$973$	−22.4139	−0.718558
$974$	0	0
$975$	1.01300	0.0324421
$976$	0	0
$977$	12.3649	0.395587	0.197794	−	0.980244i	$-0.436622\pi$
$977$	12.3649	0.395587	0.197794	+	0.980244i	$0.436622\pi$
$978$	0	0
$979$	−56.4435	−1.80394
$980$	0	0
$981$	26.5159	0.846587
$982$	0	0
$983$	−39.4586	−1.25853	−0.629267	−	0.777189i	$-0.716645\pi$
$983$	−39.4586	−1.25853	−0.629267	+	0.777189i	$0.716645\pi$
$984$	0	0
$985$	5.04125	0.160628
$986$	0	0
$987$	1.18189	0.0376199
$988$	0	0
$989$	16.6583	0.529702
$990$	0	0
$991$	55.5390	1.76425	0.882127	−	0.471011i	$-0.156111\pi$
$991$	55.5390	1.76425	0.882127	+	0.471011i	$0.156111\pi$
$992$	0	0
$993$	6.85930	0.217673
$994$	0	0
$995$	14.3531	0.455024
$996$	0	0
$997$	−13.4120	−0.424762	−0.212381	−	0.977187i	$-0.568122\pi$
$997$	−13.4120	−0.424762	−0.212381	+	0.977187i	$0.568122\pi$
$998$	0	0
$999$	−2.62062	−0.0829129

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9920.2.a.cd.1.3		4
4.3	odd	2		9920.2.a.ch.1.2		4
8.3	odd	2		155.2.a.d.1.1	✓	4
8.5	even	2		2480.2.a.z.1.2		4
24.11	even	2		1395.2.a.m.1.4		4
40.3	even	4		775.2.b.e.249.8		8
40.19	odd	2		775.2.a.g.1.4		4
40.27	even	4		775.2.b.e.249.1		8
56.27	even	2		7595.2.a.q.1.1		4
120.59	even	2		6975.2.a.bj.1.1		4
248.123	even	2		4805.2.a.j.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.d.1.1	✓	4	8.3	odd	2
775.2.a.g.1.4		4	40.19	odd	2
775.2.b.e.249.1		8	40.27	even	4
775.2.b.e.249.8		8	40.3	even	4
1395.2.a.m.1.4		4	24.11	even	2
2480.2.a.z.1.2		4	8.5	even	2
4805.2.a.j.1.1		4	248.123	even	2
6975.2.a.bj.1.1		4	120.59	even	2
7595.2.a.q.1.1		4	56.27	even	2
9920.2.a.cd.1.3		4	1.1	even	1	trivial
9920.2.a.ch.1.2		4	4.3	odd	2

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:	\( N \)	\(=\)	\( 9920 = 2^{6} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9920.a (trivial)

\(f(q)\)	\(=\)	\(q-0.342376 q^{3} +1.00000 q^{5} -1.04125 q^{7} -2.88278 q^{9} +4.64180 q^{11} -2.95875 q^{13} -0.342376 q^{15} +6.29942 q^{17} -1.11552 q^{19} +0.356500 q^{21} -3.87454 q^{23} +1.00000 q^{25} +2.01412 q^{27} -2.35650 q^{29} -1.00000 q^{31} -1.58924 q^{33} -1.04125 q^{35} -1.30112 q^{37} +1.01300 q^{39} +1.92573 q^{41} -4.29942 q^{43} -2.88278 q^{45} +3.31525 q^{47} -5.91579 q^{49} -2.15677 q^{51} -5.10964 q^{53} +4.64180 q^{55} +0.381928 q^{57} -5.07256 q^{59} -5.31525 q^{61} +3.00170 q^{63} -2.95875 q^{65} -8.64180 q^{67} +1.32655 q^{69} +3.88278 q^{71} +12.1740 q^{73} -0.342376 q^{75} -4.83329 q^{77} +7.12376 q^{79} +7.95875 q^{81} -7.46614 q^{83} +6.29942 q^{85} +0.806810 q^{87} -12.1598 q^{89} +3.08080 q^{91} +0.342376 q^{93} -1.11552 q^{95} -0.915792 q^{97} -13.3813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} + 7 q^{9} - 6 q^{11} - 16 q^{13} - q^{15} + q^{17} + 5 q^{19} - 2 q^{21} + 4 q^{25} + 5 q^{27} - 6 q^{29} - 4 q^{31} - 12 q^{33} - 9 q^{37} + 6 q^{39} + 13 q^{41} + 7 q^{43} + 7 q^{45}+ \cdots - 52 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 + 7 * q^9 - 6 * q^11 - 16 * q^13 - q^15 + q^17 + 5 * q^19 - 2 * q^21 + 4 * q^25 + 5 * q^27 - 6 * q^29 - 4 * q^31 - 12 * q^33 - 9 * q^37 + 6 * q^39 + 13 * q^41 + 7 * q^43 + 7 * q^45 + 14 * q^47 - 4 * q^49 + 5 * q^51 - 11 * q^53 - 6 * q^55 - 31 * q^57 + 13 * q^59 - 22 * q^61 + 40 * q^63 - 16 * q^65 - 10 * q^67 - 20 * q^69 - 3 * q^71 + 9 * q^73 - q^75 - 8 * q^77 + 16 * q^79 + 36 * q^81 - 17 * q^83 + q^85 - 38 * q^87 - 12 * q^89 - 24 * q^91 + q^93 + 5 * q^95 + 16 * q^97 - 52 * q^99

Introduction

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