LMFDB - Embedded newform 7440.2.a.bn.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-3,0,2,0,3,0,-2,0,2,0,3,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1860)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.86620$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.34889 q^{7} +1.00000 q^{9} -4.69779 q^{11} -2.38350 q^{13} +1.00000 q^{15} -1.21509 q^{17} -1.03461 q^{19} +1.34889 q^{21} -6.24970 q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.59859 q^{29} -1.00000 q^{31} +4.69779 q^{33} +1.34889 q^{35} +5.41811 q^{37} +2.38350 q^{39} +1.30221 q^{41} -5.73240 q^{43} -1.00000 q^{45} -3.48270 q^{47} -5.18048 q^{49} +1.21509 q^{51} +2.51730 q^{53} +4.69779 q^{55} +1.03461 q^{57} -7.86620 q^{59} -5.46479 q^{61} -1.34889 q^{63} +2.38350 q^{65} +1.41811 q^{67} +6.24970 q^{69} -3.79698 q^{71} -1.41811 q^{73} -1.00000 q^{75} +6.33682 q^{77} +1.81952 q^{79} +1.00000 q^{81} -13.2151 q^{83} +1.21509 q^{85} +5.59859 q^{87} -2.20302 q^{89} +3.21509 q^{91} +1.00000 q^{93} +1.03461 q^{95} +6.69779 q^{97} -4.69779 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{15} + 10 q^{17} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 16 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 3 * q^15 + 10 * q^17 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 16 * q^41 - 2 * q^43 - 3 * q^45 - 12 * q^47 - 5 * q^49 - 10 * q^51 + 6 * q^53 + 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^63 - 2 * q^65 - 8 * q^67 + 2 * q^69 - 10 * q^71 + 8 * q^73 - 3 * q^75 + 28 * q^77 + 16 * q^79 + 3 * q^81 - 26 * q^83 - 10 * q^85 - 6 * q^87 - 8 * q^89 - 4 * q^91 + 3 * q^93 + 8 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.34889	−0.509834	−0.254917	−	0.966963i	$-0.582048\pi$
$7$	−1.34889	−0.509834	−0.254917	+	0.966963i	$0.582048\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.69779	−1.41644	−0.708218	−	0.705994i	$-0.750501\pi$
$11$	−4.69779	−1.41644	−0.708218	+	0.705994i	$0.750501\pi$
$12$	0	0
$13$	−2.38350	−0.661065	−0.330532	−	0.943795i	$-0.607228\pi$
$13$	−2.38350	−0.661065	−0.330532	+	0.943795i	$0.607228\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.21509	−0.294703	−0.147352	−	0.989084i	$-0.547075\pi$
$17$	−1.21509	−0.294703	−0.147352	+	0.989084i	$0.547075\pi$
$18$	0	0
$19$	−1.03461	−0.237355	−0.118678	−	0.992933i	$-0.537866\pi$
$19$	−1.03461	−0.237355	−0.118678	+	0.992933i	$0.537866\pi$
$20$	0	0
$21$	1.34889	0.294353
$22$	0	0
$23$	−6.24970	−1.30315	−0.651576	−	0.758583i	$-0.725892\pi$
$23$	−6.24970	−1.30315	−0.651576	+	0.758583i	$0.725892\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.59859	−1.03963	−0.519816	−	0.854278i	$-0.674000\pi$
$29$	−5.59859	−1.03963	−0.519816	+	0.854278i	$0.674000\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	4.69779	0.817780
$34$	0	0
$35$	1.34889	0.228005
$36$	0	0
$37$	5.41811	0.890732	0.445366	−	0.895349i	$-0.353074\pi$
$37$	5.41811	0.890732	0.445366	+	0.895349i	$0.353074\pi$
$38$	0	0
$39$	2.38350	0.381666
$40$	0	0
$41$	1.30221	0.203371	0.101686	−	0.994817i	$-0.467576\pi$
$41$	1.30221	0.203371	0.101686	+	0.994817i	$0.467576\pi$
$42$	0	0
$43$	−5.73240	−0.874182	−0.437091	−	0.899417i	$-0.643991\pi$
$43$	−5.73240	−0.874182	−0.437091	+	0.899417i	$0.643991\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−3.48270	−0.508003	−0.254002	−	0.967204i	$-0.581747\pi$
$47$	−3.48270	−0.508003	−0.254002	+	0.967204i	$0.581747\pi$
$48$	0	0
$49$	−5.18048	−0.740069
$50$	0	0
$51$	1.21509	0.170147
$52$	0	0
$53$	2.51730	0.345778	0.172889	−	0.984941i	$-0.444690\pi$
$53$	2.51730	0.345778	0.172889	+	0.984941i	$0.444690\pi$
$54$	0	0
$55$	4.69779	0.633450
$56$	0	0
$57$	1.03461	0.137037
$58$	0	0
$59$	−7.86620	−1.02409	−0.512046	−	0.858958i	$-0.671112\pi$
$59$	−7.86620	−1.02409	−0.512046	+	0.858958i	$0.671112\pi$
$60$	0	0
$61$	−5.46479	−0.699695	−0.349848	−	0.936807i	$-0.613767\pi$
$61$	−5.46479	−0.699695	−0.349848	+	0.936807i	$0.613767\pi$
$62$	0	0
$63$	−1.34889	−0.169945
$64$	0	0
$65$	2.38350	0.295637
$66$	0	0
$67$	1.41811	0.173250	0.0866249	−	0.996241i	$-0.472392\pi$
$67$	1.41811	0.173250	0.0866249	+	0.996241i	$0.472392\pi$
$68$	0	0
$69$	6.24970	0.752376
$70$	0	0
$71$	−3.79698	−0.450619	−0.225309	−	0.974287i	$-0.572339\pi$
$71$	−3.79698	−0.450619	−0.225309	+	0.974287i	$0.572339\pi$
$72$	0	0
$73$	−1.41811	−0.165977	−0.0829886	−	0.996550i	$-0.526447\pi$
$73$	−1.41811	−0.165977	−0.0829886	+	0.996550i	$0.526447\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	6.33682	0.722148
$78$	0	0
$79$	1.81952	0.204711	0.102356	−	0.994748i	$-0.467362\pi$
$79$	1.81952	0.204711	0.102356	+	0.994748i	$0.467362\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.2151	−1.45054	−0.725272	−	0.688462i	$-0.758286\pi$
$83$	−13.2151	−1.45054	−0.725272	+	0.688462i	$0.758286\pi$
$84$	0	0
$85$	1.21509	0.131795
$86$	0	0
$87$	5.59859	0.600232
$88$	0	0
$89$	−2.20302	−0.233519	−0.116760	−	0.993160i	$-0.537251\pi$
$89$	−2.20302	−0.233519	−0.116760	+	0.993160i	$0.537251\pi$
$90$	0	0
$91$	3.21509	0.337033
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	1.03461	0.106149
$96$	0	0
$97$	6.69779	0.680057	0.340029	−	0.940415i	$-0.389563\pi$
$97$	6.69779	0.680057	0.340029	+	0.940415i	$0.389563\pi$
$98$	0	0
$99$	−4.69779	−0.472146
$100$	0	0
$101$	13.1280	1.30628	0.653141	−	0.757236i	$-0.273451\pi$
$101$	13.1280	1.30628	0.653141	+	0.757236i	$0.273451\pi$
$102$	0	0
$103$	16.4769	1.62351	0.811757	−	0.583995i	$-0.198511\pi$
$103$	16.4769	1.62351	0.811757	+	0.583995i	$0.198511\pi$
$104$	0	0
$105$	−1.34889	−0.131639
$106$	0	0
$107$	−4.78491	−0.462574	−0.231287	−	0.972886i	$-0.574294\pi$
$107$	−4.78491	−0.462574	−0.231287	+	0.972886i	$0.574294\pi$
$108$	0	0
$109$	0.947489	0.0907530	0.0453765	−	0.998970i	$-0.485551\pi$
$109$	0.947489	0.0907530	0.0453765	+	0.998970i	$0.485551\pi$
$110$	0	0
$111$	−5.41811	−0.514264
$112$	0	0
$113$	17.1972	1.61778	0.808888	−	0.587963i	$-0.200070\pi$
$113$	17.1972	1.61778	0.808888	+	0.587963i	$0.200070\pi$
$114$	0	0
$115$	6.24970	0.582788
$116$	0	0
$117$	−2.38350	−0.220355
$118$	0	0
$119$	1.63903	0.150250
$120$	0	0
$121$	11.0692	1.00629
$122$	0	0
$123$	−1.30221	−0.117416
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.16258	−0.191898	−0.0959490	−	0.995386i	$-0.530589\pi$
$127$	−2.16258	−0.191898	−0.0959490	+	0.995386i	$0.530589\pi$
$128$	0	0
$129$	5.73240	0.504709
$130$	0	0
$131$	−10.9700	−0.958455	−0.479228	−	0.877691i	$-0.659083\pi$
$131$	−10.9700	−0.958455	−0.479228	+	0.877691i	$0.659083\pi$
$132$	0	0
$133$	1.39558	0.121012
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−8.67989	−0.741573	−0.370786	−	0.928718i	$-0.620912\pi$
$137$	−8.67989	−0.741573	−0.370786	+	0.928718i	$0.620912\pi$
$138$	0	0
$139$	−4.76700	−0.404332	−0.202166	−	0.979351i	$-0.564798\pi$
$139$	−4.76700	−0.404332	−0.202166	+	0.979351i	$0.564798\pi$
$140$	0	0
$141$	3.48270	0.293296
$142$	0	0
$143$	11.1972	0.936356
$144$	0	0
$145$	5.59859	0.464938
$146$	0	0
$147$	5.18048	0.427279
$148$	0	0
$149$	−10.6978	−0.876397	−0.438198	−	0.898878i	$-0.644383\pi$
$149$	−10.6978	−0.876397	−0.438198	+	0.898878i	$0.644383\pi$
$150$	0	0
$151$	−5.01671	−0.408254	−0.204127	−	0.978944i	$-0.565435\pi$
$151$	−5.01671	−0.408254	−0.204127	+	0.978944i	$0.565435\pi$
$152$	0	0
$153$	−1.21509	−0.0982344
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−5.80161	−0.463019	−0.231510	−	0.972833i	$-0.574366\pi$
$157$	−5.80161	−0.463019	−0.231510	+	0.972833i	$0.574366\pi$
$158$	0	0
$159$	−2.51730	−0.199635
$160$	0	0
$161$	8.43018	0.664392
$162$	0	0
$163$	10.3143	0.807877	0.403939	−	0.914786i	$-0.367641\pi$
$163$	10.3143	0.807877	0.403939	+	0.914786i	$0.367641\pi$
$164$	0	0
$165$	−4.69779	−0.365722
$166$	0	0
$167$	5.66318	0.438230	0.219115	−	0.975699i	$-0.429683\pi$
$167$	5.66318	0.438230	0.219115	+	0.975699i	$0.429683\pi$
$168$	0	0
$169$	−7.31892	−0.562994
$170$	0	0
$171$	−1.03461	−0.0791185
$172$	0	0
$173$	−11.6632	−0.886735	−0.443368	−	0.896340i	$-0.646216\pi$
$173$	−11.6632	−0.886735	−0.443368	+	0.896340i	$0.646216\pi$
$174$	0	0
$175$	−1.34889	−0.101967
$176$	0	0
$177$	7.86620	0.591260
$178$	0	0
$179$	11.8258	0.883899	0.441949	−	0.897040i	$-0.354287\pi$
$179$	11.8258	0.883899	0.441949	+	0.897040i	$0.354287\pi$
$180$	0	0
$181$	−14.8604	−1.10456	−0.552281	−	0.833658i	$-0.686243\pi$
$181$	−14.8604	−1.10456	−0.552281	+	0.833658i	$0.686243\pi$
$182$	0	0
$183$	5.46479	0.403969
$184$	0	0
$185$	−5.41811	−0.398347
$186$	0	0
$187$	5.70825	0.417428
$188$	0	0
$189$	1.34889	0.0981176
$190$	0	0
$191$	5.93541	0.429472	0.214736	−	0.976672i	$-0.431111\pi$
$191$	5.93541	0.429472	0.214736	+	0.976672i	$0.431111\pi$
$192$	0	0
$193$	−3.19719	−0.230139	−0.115069	−	0.993357i	$-0.536709\pi$
$193$	−3.19719	−0.230139	−0.115069	+	0.993357i	$0.536709\pi$
$194$	0	0
$195$	−2.38350	−0.170686
$196$	0	0
$197$	18.7491	1.33582	0.667909	−	0.744243i	$-0.267189\pi$
$197$	18.7491	1.33582	0.667909	+	0.744243i	$0.267189\pi$
$198$	0	0
$199$	21.9129	1.55336	0.776681	−	0.629894i	$-0.216901\pi$
$199$	21.9129	1.55336	0.776681	+	0.629894i	$0.216901\pi$
$200$	0	0
$201$	−1.41811	−0.100026
$202$	0	0
$203$	7.55191	0.530040
$204$	0	0
$205$	−1.30221	−0.0909504
$206$	0	0
$207$	−6.24970	−0.434384
$208$	0	0
$209$	4.86037	0.336199
$210$	0	0
$211$	−1.66318	−0.114498	−0.0572490	−	0.998360i	$-0.518233\pi$
$211$	−1.66318	−0.114498	−0.0572490	+	0.998360i	$0.518233\pi$
$212$	0	0
$213$	3.79698	0.260165
$214$	0	0
$215$	5.73240	0.390946
$216$	0	0
$217$	1.34889	0.0915689
$218$	0	0
$219$	1.41811	0.0958270
$220$	0	0
$221$	2.89618	0.194818
$222$	0	0
$223$	−5.10382	−0.341777	−0.170889	−	0.985290i	$-0.554664\pi$
$223$	−5.10382	−0.341777	−0.170889	+	0.985290i	$0.554664\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−2.85412	−0.189435	−0.0947174	−	0.995504i	$-0.530195\pi$
$227$	−2.85412	−0.189435	−0.0947174	+	0.995504i	$0.530195\pi$
$228$	0	0
$229$	−18.0934	−1.19564	−0.597822	−	0.801629i	$-0.703967\pi$
$229$	−18.0934	−1.19564	−0.597822	+	0.801629i	$0.703967\pi$
$230$	0	0
$231$	−6.33682	−0.416932
$232$	0	0
$233$	−9.01671	−0.590704	−0.295352	−	0.955389i	$-0.595437\pi$
$233$	−9.01671	−0.590704	−0.295352	+	0.955389i	$0.595437\pi$
$234$	0	0
$235$	3.48270	0.227186
$236$	0	0
$237$	−1.81952	−0.118190
$238$	0	0
$239$	−13.9642	−0.903269	−0.451634	−	0.892203i	$-0.649159\pi$
$239$	−13.9642	−0.903269	−0.451634	+	0.892203i	$0.649159\pi$
$240$	0	0
$241$	13.1972	0.850106	0.425053	−	0.905169i	$-0.360256\pi$
$241$	13.1972	0.850106	0.425053	+	0.905169i	$0.360256\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.18048	0.330969
$246$	0	0
$247$	2.46599	0.156907
$248$	0	0
$249$	13.2151	0.837472
$250$	0	0
$251$	4.36097	0.275262	0.137631	−	0.990484i	$-0.456051\pi$
$251$	4.36097	0.275262	0.137631	+	0.990484i	$0.456051\pi$
$252$	0	0
$253$	29.3598	1.84583
$254$	0	0
$255$	−1.21509	−0.0760920
$256$	0	0
$257$	11.6211	0.724906	0.362453	−	0.932002i	$-0.381939\pi$
$257$	11.6211	0.724906	0.362453	+	0.932002i	$0.381939\pi$
$258$	0	0
$259$	−7.30846	−0.454125
$260$	0	0
$261$	−5.59859	−0.346544
$262$	0	0
$263$	6.16258	0.380001	0.190001	−	0.981784i	$-0.439151\pi$
$263$	6.16258	0.380001	0.190001	+	0.981784i	$0.439151\pi$
$264$	0	0
$265$	−2.51730	−0.154637
$266$	0	0
$267$	2.20302	0.134823
$268$	0	0
$269$	−22.4590	−1.36935	−0.684674	−	0.728850i	$-0.740055\pi$
$269$	−22.4590	−1.36935	−0.684674	+	0.728850i	$0.740055\pi$
$270$	0	0
$271$	22.8362	1.38720	0.693601	−	0.720360i	$-0.256023\pi$
$271$	22.8362	1.38720	0.693601	+	0.720360i	$0.256023\pi$
$272$	0	0
$273$	−3.21509	−0.194586
$274$	0	0
$275$	−4.69779	−0.283287
$276$	0	0
$277$	14.1851	0.852301	0.426150	−	0.904652i	$-0.359869\pi$
$277$	14.1851	0.852301	0.426150	+	0.904652i	$0.359869\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	12.8362	0.765745	0.382872	−	0.923801i	$-0.374935\pi$
$281$	12.8362	0.765745	0.382872	+	0.923801i	$0.374935\pi$
$282$	0	0
$283$	−5.70986	−0.339416	−0.169708	−	0.985494i	$-0.554282\pi$
$283$	−5.70986	−0.339416	−0.169708	+	0.985494i	$0.554282\pi$
$284$	0	0
$285$	−1.03461	−0.0612849
$286$	0	0
$287$	−1.75655	−0.103686
$288$	0	0
$289$	−15.5236	−0.913150
$290$	0	0
$291$	−6.69779	−0.392631
$292$	0	0
$293$	8.01790	0.468411	0.234205	−	0.972187i	$-0.424751\pi$
$293$	8.01790	0.468411	0.234205	+	0.972187i	$0.424751\pi$
$294$	0	0
$295$	7.86620	0.457988
$296$	0	0
$297$	4.69779	0.272593
$298$	0	0
$299$	14.8962	0.861468
$300$	0	0
$301$	7.73240	0.445688
$302$	0	0
$303$	−13.1280	−0.754182
$304$	0	0
$305$	5.46479	0.312913
$306$	0	0
$307$	18.7445	1.06980	0.534902	−	0.844914i	$-0.320349\pi$
$307$	18.7445	1.06980	0.534902	+	0.844914i	$0.320349\pi$
$308$	0	0
$309$	−16.4769	−0.937336
$310$	0	0
$311$	19.2260	1.09020	0.545102	−	0.838370i	$-0.316491\pi$
$311$	19.2260	1.09020	0.545102	+	0.838370i	$0.316491\pi$
$312$	0	0
$313$	14.5219	0.820828	0.410414	−	0.911899i	$-0.365384\pi$
$313$	14.5219	0.820828	0.410414	+	0.911899i	$0.365384\pi$
$314$	0	0
$315$	1.34889	0.0760016
$316$	0	0
$317$	0.378872	0.0212796	0.0106398	−	0.999943i	$-0.496613\pi$
$317$	0.378872	0.0212796	0.0106398	+	0.999943i	$0.496613\pi$
$318$	0	0
$319$	26.3010	1.47257
$320$	0	0
$321$	4.78491	0.267067
$322$	0	0
$323$	1.25714	0.0699494
$324$	0	0
$325$	−2.38350	−0.132213
$326$	0	0
$327$	−0.947489	−0.0523963
$328$	0	0
$329$	4.69779	0.258997
$330$	0	0
$331$	−17.1459	−0.942423	−0.471211	−	0.882020i	$-0.656183\pi$
$331$	−17.1459	−0.942423	−0.471211	+	0.882020i	$0.656183\pi$
$332$	0	0
$333$	5.41811	0.296911
$334$	0	0
$335$	−1.41811	−0.0774796
$336$	0	0
$337$	−2.04668	−0.111490	−0.0557450	−	0.998445i	$-0.517753\pi$
$337$	−2.04668	−0.111490	−0.0557450	+	0.998445i	$0.517753\pi$
$338$	0	0
$339$	−17.1972	−0.934023
$340$	0	0
$341$	4.69779	0.254400
$342$	0	0
$343$	16.4302	0.887147
$344$	0	0
$345$	−6.24970	−0.336473
$346$	0	0
$347$	9.26641	0.497447	0.248723	−	0.968575i	$-0.419989\pi$
$347$	9.26641	0.497447	0.248723	+	0.968575i	$0.419989\pi$
$348$	0	0
$349$	−3.41348	−0.182719	−0.0913597	−	0.995818i	$-0.529121\pi$
$349$	−3.41348	−0.182719	−0.0913597	+	0.995818i	$0.529121\pi$
$350$	0	0
$351$	2.38350	0.127222
$352$	0	0
$353$	−20.6799	−1.10068	−0.550340	−	0.834941i	$-0.685502\pi$
$353$	−20.6799	−1.10068	−0.550340	+	0.834941i	$0.685502\pi$
$354$	0	0
$355$	3.79698	0.201523
$356$	0	0
$357$	−1.63903	−0.0867467
$358$	0	0
$359$	33.6227	1.77454	0.887270	−	0.461250i	$-0.152599\pi$
$359$	33.6227	1.77454	0.887270	+	0.461250i	$0.152599\pi$
$360$	0	0
$361$	−17.9296	−0.943662
$362$	0	0
$363$	−11.0692	−0.580983
$364$	0	0
$365$	1.41811	0.0742273
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	1.30221	0.0677904
$370$	0	0
$371$	−3.39558	−0.176290
$372$	0	0
$373$	35.2664	1.82603	0.913013	−	0.407931i	$-0.133750\pi$
$373$	35.2664	1.82603	0.913013	+	0.407931i	$0.133750\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	13.3443	0.687265
$378$	0	0
$379$	19.1038	0.981298	0.490649	−	0.871357i	$-0.336760\pi$
$379$	19.1038	0.981298	0.490649	+	0.871357i	$0.336760\pi$
$380$	0	0
$381$	2.16258	0.110792
$382$	0	0
$383$	−20.0062	−1.02227	−0.511136	−	0.859500i	$-0.670775\pi$
$383$	−20.0062	−1.02227	−0.511136	+	0.859500i	$0.670775\pi$
$384$	0	0
$385$	−6.33682	−0.322954
$386$	0	0
$387$	−5.73240	−0.291394
$388$	0	0
$389$	−13.7036	−0.694801	−0.347400	−	0.937717i	$-0.612936\pi$
$389$	−13.7036	−0.694801	−0.347400	+	0.937717i	$0.612936\pi$
$390$	0	0
$391$	7.59396	0.384043
$392$	0	0
$393$	10.9700	0.553364
$394$	0	0
$395$	−1.81952	−0.0915498
$396$	0	0
$397$	−27.3598	−1.37315	−0.686574	−	0.727060i	$-0.740886\pi$
$397$	−27.3598	−1.37315	−0.686574	+	0.727060i	$0.740886\pi$
$398$	0	0
$399$	−1.39558	−0.0698662
$400$	0	0
$401$	−9.79698	−0.489238	−0.244619	−	0.969619i	$-0.578663\pi$
$401$	−9.79698	−0.489238	−0.244619	+	0.969619i	$0.578663\pi$
$402$	0	0
$403$	2.38350	0.118731
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−25.4531	−1.26167
$408$	0	0
$409$	16.2559	0.803805	0.401902	−	0.915683i	$-0.368349\pi$
$409$	16.2559	0.803805	0.401902	+	0.915683i	$0.368349\pi$
$410$	0	0
$411$	8.67989	0.428147
$412$	0	0
$413$	10.6107	0.522117
$414$	0	0
$415$	13.2151	0.648703
$416$	0	0
$417$	4.76700	0.233441
$418$	0	0
$419$	25.6920	1.25513	0.627567	−	0.778562i	$-0.284051\pi$
$419$	25.6920	1.25513	0.627567	+	0.778562i	$0.284051\pi$
$420$	0	0
$421$	−12.8541	−0.626472	−0.313236	−	0.949675i	$-0.601413\pi$
$421$	−12.8541	−0.626472	−0.313236	+	0.949675i	$0.601413\pi$
$422$	0	0
$423$	−3.48270	−0.169334
$424$	0	0
$425$	−1.21509	−0.0589406
$426$	0	0
$427$	7.37143	0.356728
$428$	0	0
$429$	−11.1972	−0.540605
$430$	0	0
$431$	−36.8891	−1.77689	−0.888444	−	0.458985i	$-0.848213\pi$
$431$	−36.8891	−1.77689	−0.888444	+	0.458985i	$0.848213\pi$
$432$	0	0
$433$	0.512673	0.0246375	0.0123188	−	0.999924i	$-0.496079\pi$
$433$	0.512673	0.0246375	0.0123188	+	0.999924i	$0.496079\pi$
$434$	0	0
$435$	−5.59859	−0.268432
$436$	0	0
$437$	6.46599	0.309310
$438$	0	0
$439$	24.7219	1.17991	0.589957	−	0.807435i	$-0.299145\pi$
$439$	24.7219	1.17991	0.589957	+	0.807435i	$0.299145\pi$
$440$	0	0
$441$	−5.18048	−0.246690
$442$	0	0
$443$	−35.4018	−1.68199	−0.840996	−	0.541042i	$-0.818030\pi$
$443$	−35.4018	−1.68199	−0.840996	+	0.541042i	$0.818030\pi$
$444$	0	0
$445$	2.20302	0.104433
$446$	0	0
$447$	10.6978	0.505988
$448$	0	0
$449$	−8.53984	−0.403020	−0.201510	−	0.979486i	$-0.564585\pi$
$449$	−8.53984	−0.403020	−0.201510	+	0.979486i	$0.564585\pi$
$450$	0	0
$451$	−6.11751	−0.288063
$452$	0	0
$453$	5.01671	0.235705
$454$	0	0
$455$	−3.21509	−0.150726
$456$	0	0
$457$	−23.3131	−1.09054	−0.545270	−	0.838260i	$-0.683573\pi$
$457$	−23.3131	−1.09054	−0.545270	+	0.838260i	$0.683573\pi$
$458$	0	0
$459$	1.21509	0.0567157
$460$	0	0
$461$	−28.7958	−1.34115	−0.670577	−	0.741840i	$-0.733953\pi$
$461$	−28.7958	−1.34115	−0.670577	+	0.741840i	$0.733953\pi$
$462$	0	0
$463$	18.7670	0.872177	0.436088	−	0.899904i	$-0.356364\pi$
$463$	18.7670	0.872177	0.436088	+	0.899904i	$0.356364\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	3.48270	0.161160	0.0805800	−	0.996748i	$-0.474323\pi$
$467$	3.48270	0.161160	0.0805800	+	0.996748i	$0.474323\pi$
$468$	0	0
$469$	−1.91288	−0.0883286
$470$	0	0
$471$	5.80161	0.267324
$472$	0	0
$473$	26.9296	1.23822
$474$	0	0
$475$	−1.03461	−0.0474711
$476$	0	0
$477$	2.51730	0.115259
$478$	0	0
$479$	7.02998	0.321208	0.160604	−	0.987019i	$-0.448656\pi$
$479$	7.02998	0.321208	0.160604	+	0.987019i	$0.448656\pi$
$480$	0	0
$481$	−12.9141	−0.588831
$482$	0	0
$483$	−8.43018	−0.383587
$484$	0	0
$485$	−6.69779	−0.304131
$486$	0	0
$487$	16.4543	0.745617	0.372809	−	0.927908i	$-0.378395\pi$
$487$	16.4543	0.745617	0.372809	+	0.927908i	$0.378395\pi$
$488$	0	0
$489$	−10.3143	−0.466428
$490$	0	0
$491$	11.1972	0.505322	0.252661	−	0.967555i	$-0.418694\pi$
$491$	11.1972	0.505322	0.252661	+	0.967555i	$0.418694\pi$
$492$	0	0
$493$	6.80281	0.306383
$494$	0	0
$495$	4.69779	0.211150
$496$	0	0
$497$	5.12173	0.229741
$498$	0	0
$499$	2.74286	0.122787	0.0613935	−	0.998114i	$-0.480446\pi$
$499$	2.74286	0.122787	0.0613935	+	0.998114i	$0.480446\pi$
$500$	0	0
$501$	−5.66318	−0.253012
$502$	0	0
$503$	15.9821	0.712606	0.356303	−	0.934370i	$-0.384037\pi$
$503$	15.9821	0.712606	0.356303	+	0.934370i	$0.384037\pi$
$504$	0	0
$505$	−13.1280	−0.584187
$506$	0	0
$507$	7.31892	0.325045
$508$	0	0
$509$	9.49357	0.420795	0.210398	−	0.977616i	$-0.432524\pi$
$509$	9.49357	0.420795	0.210398	+	0.977616i	$0.432524\pi$
$510$	0	0
$511$	1.91288	0.0846209
$512$	0	0
$513$	1.03461	0.0456791
$514$	0	0
$515$	−16.4769	−0.726058
$516$	0	0
$517$	16.3610	0.719555
$518$	0	0
$519$	11.6632	0.511957
$520$	0	0
$521$	6.83622	0.299500	0.149750	−	0.988724i	$-0.452153\pi$
$521$	6.83622	0.299500	0.149750	+	0.988724i	$0.452153\pi$
$522$	0	0
$523$	27.1972	1.18925	0.594625	−	0.804003i	$-0.297301\pi$
$523$	27.1972	1.18925	0.594625	+	0.804003i	$0.297301\pi$
$524$	0	0
$525$	1.34889	0.0588706
$526$	0	0
$527$	1.21509	0.0529303
$528$	0	0
$529$	16.0588	0.698207
$530$	0	0
$531$	−7.86620	−0.341364
$532$	0	0
$533$	−3.10382	−0.134442
$534$	0	0
$535$	4.78491	0.206870
$536$	0	0
$537$	−11.8258	−0.510319
$538$	0	0
$539$	24.3368	1.04826
$540$	0	0
$541$	41.6787	1.79191	0.895953	−	0.444148i	$-0.146494\pi$
$541$	41.6787	1.79191	0.895953	+	0.444148i	$0.146494\pi$
$542$	0	0
$543$	14.8604	0.637720
$544$	0	0
$545$	−0.947489	−0.0405860
$546$	0	0
$547$	19.0362	0.813930	0.406965	−	0.913444i	$-0.366587\pi$
$547$	19.0362	0.813930	0.406965	+	0.913444i	$0.366587\pi$
$548$	0	0
$549$	−5.46479	−0.233232
$550$	0	0
$551$	5.79235	0.246762
$552$	0	0
$553$	−2.45433	−0.104369
$554$	0	0
$555$	5.41811	0.229986
$556$	0	0
$557$	36.2831	1.53736	0.768682	−	0.639631i	$-0.220913\pi$
$557$	36.2831	1.53736	0.768682	+	0.639631i	$0.220913\pi$
$558$	0	0
$559$	13.6632	0.577891
$560$	0	0
$561$	−5.70825	−0.241002
$562$	0	0
$563$	−24.0755	−1.01466	−0.507330	−	0.861752i	$-0.669367\pi$
$563$	−24.0755	−1.01466	−0.507330	+	0.861752i	$0.669367\pi$
$564$	0	0
$565$	−17.1972	−0.723491
$566$	0	0
$567$	−1.34889	−0.0566482
$568$	0	0
$569$	−11.4360	−0.479423	−0.239711	−	0.970844i	$-0.577053\pi$
$569$	−11.4360	−0.479423	−0.239711	+	0.970844i	$0.577053\pi$
$570$	0	0
$571$	−24.2318	−1.01407	−0.507035	−	0.861926i	$-0.669258\pi$
$571$	−24.2318	−1.01407	−0.507035	+	0.861926i	$0.669258\pi$
$572$	0	0
$573$	−5.93541	−0.247955
$574$	0	0
$575$	−6.24970	−0.260631
$576$	0	0
$577$	1.53401	0.0638616	0.0319308	−	0.999490i	$-0.489834\pi$
$577$	1.53401	0.0638616	0.0319308	+	0.999490i	$0.489834\pi$
$578$	0	0
$579$	3.19719	0.132871
$580$	0	0
$581$	17.8258	0.739537
$582$	0	0
$583$	−11.8258	−0.489773
$584$	0	0
$585$	2.38350	0.0985457
$586$	0	0
$587$	18.1626	0.749650	0.374825	−	0.927096i	$-0.377703\pi$
$587$	18.1626	0.749650	0.374825	+	0.927096i	$0.377703\pi$
$588$	0	0
$589$	1.03461	0.0426303
$590$	0	0
$591$	−18.7491	−0.771235
$592$	0	0
$593$	−19.6274	−0.806000	−0.403000	−	0.915200i	$-0.632032\pi$
$593$	−19.6274	−0.806000	−0.403000	+	0.915200i	$0.632032\pi$
$594$	0	0
$595$	−1.63903	−0.0671937
$596$	0	0
$597$	−21.9129	−0.896835
$598$	0	0
$599$	0.609054	0.0248853	0.0124426	−	0.999923i	$-0.496039\pi$
$599$	0.609054	0.0248853	0.0124426	+	0.999923i	$0.496039\pi$
$600$	0	0
$601$	−23.6274	−0.963781	−0.481890	−	0.876232i	$-0.660050\pi$
$601$	−23.6274	−0.963781	−0.481890	+	0.876232i	$0.660050\pi$
$602$	0	0
$603$	1.41811	0.0577499
$604$	0	0
$605$	−11.0692	−0.450028
$606$	0	0
$607$	41.1505	1.67025	0.835124	−	0.550062i	$-0.185396\pi$
$607$	41.1505	1.67025	0.835124	+	0.550062i	$0.185396\pi$
$608$	0	0
$609$	−7.55191	−0.306019
$610$	0	0
$611$	8.30101	0.335823
$612$	0	0
$613$	45.8966	1.85375	0.926873	−	0.375375i	$-0.122486\pi$
$613$	45.8966	1.85375	0.926873	+	0.375375i	$0.122486\pi$
$614$	0	0
$615$	1.30221	0.0525102
$616$	0	0
$617$	46.8604	1.88653	0.943264	−	0.332044i	$-0.107738\pi$
$617$	46.8604	1.88653	0.943264	+	0.332044i	$0.107738\pi$
$618$	0	0
$619$	−11.8529	−0.476409	−0.238205	−	0.971215i	$-0.576559\pi$
$619$	−11.8529	−0.476409	−0.238205	+	0.971215i	$0.576559\pi$
$620$	0	0
$621$	6.24970	0.250792
$622$	0	0
$623$	2.97164	0.119056
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.86037	−0.194104
$628$	0	0
$629$	−6.58351	−0.262502
$630$	0	0
$631$	17.2571	0.686996	0.343498	−	0.939153i	$-0.388388\pi$
$631$	17.2571	0.686996	0.343498	+	0.939153i	$0.388388\pi$
$632$	0	0
$633$	1.66318	0.0661055
$634$	0	0
$635$	2.16258	0.0858194
$636$	0	0
$637$	12.3477	0.489234
$638$	0	0
$639$	−3.79698	−0.150206
$640$	0	0
$641$	−8.15795	−0.322220	−0.161110	−	0.986936i	$-0.551507\pi$
$641$	−8.15795	−0.322220	−0.161110	+	0.986936i	$0.551507\pi$
$642$	0	0
$643$	−7.19719	−0.283829	−0.141915	−	0.989879i	$-0.545326\pi$
$643$	−7.19719	−0.283829	−0.141915	+	0.989879i	$0.545326\pi$
$644$	0	0
$645$	−5.73240	−0.225713
$646$	0	0
$647$	15.6873	0.616733	0.308366	−	0.951268i	$-0.400218\pi$
$647$	15.6873	0.616733	0.308366	+	0.951268i	$0.400218\pi$
$648$	0	0
$649$	36.9537	1.45056
$650$	0	0
$651$	−1.34889	−0.0528673
$652$	0	0
$653$	−37.8771	−1.48224	−0.741122	−	0.671370i	$-0.765706\pi$
$653$	−37.8771	−1.48224	−0.741122	+	0.671370i	$0.765706\pi$
$654$	0	0
$655$	10.9700	0.428634
$656$	0	0
$657$	−1.41811	−0.0553258
$658$	0	0
$659$	46.3298	1.80475	0.902376	−	0.430949i	$-0.141821\pi$
$659$	46.3298	1.80475	0.902376	+	0.430949i	$0.141821\pi$
$660$	0	0
$661$	−9.19719	−0.357729	−0.178865	−	0.983874i	$-0.557242\pi$
$661$	−9.19719	−0.357729	−0.178865	+	0.983874i	$0.557242\pi$
$662$	0	0
$663$	−2.89618	−0.112478
$664$	0	0
$665$	−1.39558	−0.0541181
$666$	0	0
$667$	34.9895	1.35480
$668$	0	0
$669$	5.10382	0.197325
$670$	0	0
$671$	25.6724	0.991074
$672$	0	0
$673$	−20.4169	−0.787014	−0.393507	−	0.919322i	$-0.628738\pi$
$673$	−20.4169	−0.787014	−0.393507	+	0.919322i	$0.628738\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	15.7744	0.606261	0.303131	−	0.952949i	$-0.401968\pi$
$677$	15.7744	0.606261	0.303131	+	0.952949i	$0.401968\pi$
$678$	0	0
$679$	−9.03461	−0.346716
$680$	0	0
$681$	2.85412	0.109370
$682$	0	0
$683$	−11.4920	−0.439728	−0.219864	−	0.975531i	$-0.570561\pi$
$683$	−11.4920	−0.439728	−0.219864	+	0.975531i	$0.570561\pi$
$684$	0	0
$685$	8.67989	0.331641
$686$	0	0
$687$	18.0934	0.690305
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−13.1638	−0.500774	−0.250387	−	0.968146i	$-0.580558\pi$
$691$	−13.1638	−0.500774	−0.250387	+	0.968146i	$0.580558\pi$
$692$	0	0
$693$	6.33682	0.240716
$694$	0	0
$695$	4.76700	0.180823
$696$	0	0
$697$	−1.58231	−0.0599342
$698$	0	0
$699$	9.01671	0.341043
$700$	0	0
$701$	−33.1372	−1.25158	−0.625788	−	0.779993i	$-0.715223\pi$
$701$	−33.1372	−1.25158	−0.625788	+	0.779993i	$0.715223\pi$
$702$	0	0
$703$	−5.60562	−0.211420
$704$	0	0
$705$	−3.48270	−0.131166
$706$	0	0
$707$	−17.7082	−0.665987
$708$	0	0
$709$	6.86037	0.257647	0.128823	−	0.991668i	$-0.458880\pi$
$709$	6.86037	0.257647	0.128823	+	0.991668i	$0.458880\pi$
$710$	0	0
$711$	1.81952	0.0682372
$712$	0	0
$713$	6.24970	0.234053
$714$	0	0
$715$	−11.1972	−0.418751
$716$	0	0
$717$	13.9642	0.521502
$718$	0	0
$719$	9.90663	0.369455	0.184728	−	0.982790i	$-0.440860\pi$
$719$	9.90663	0.369455	0.184728	+	0.982790i	$0.440860\pi$
$720$	0	0
$721$	−22.2256	−0.827723
$722$	0	0
$723$	−13.1972	−0.490809
$724$	0	0
$725$	−5.59859	−0.207927
$726$	0	0
$727$	−15.4423	−0.572722	−0.286361	−	0.958122i	$-0.592446\pi$
$727$	−15.4423	−0.572722	−0.286361	+	0.958122i	$0.592446\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.96539	0.257624
$732$	0	0
$733$	40.9296	1.51177	0.755884	−	0.654705i	$-0.227207\pi$
$733$	40.9296	1.51177	0.755884	+	0.654705i	$0.227207\pi$
$734$	0	0
$735$	−5.18048	−0.191085
$736$	0	0
$737$	−6.66198	−0.245397
$738$	0	0
$739$	−50.4123	−1.85445	−0.927223	−	0.374510i	$-0.877811\pi$
$739$	−50.4123	−1.85445	−0.927223	+	0.374510i	$0.877811\pi$
$740$	0	0
$741$	−2.46599	−0.0905904
$742$	0	0
$743$	−21.3505	−0.783274	−0.391637	−	0.920120i	$-0.628091\pi$
$743$	−21.3505	−0.783274	−0.391637	+	0.920120i	$0.628091\pi$
$744$	0	0
$745$	10.6978	0.391937
$746$	0	0
$747$	−13.2151	−0.483515
$748$	0	0
$749$	6.45433	0.235836
$750$	0	0
$751$	−32.1960	−1.17485	−0.587424	−	0.809279i	$-0.699858\pi$
$751$	−32.1960	−1.17485	−0.587424	+	0.809279i	$0.699858\pi$
$752$	0	0
$753$	−4.36097	−0.158923
$754$	0	0
$755$	5.01671	0.182577
$756$	0	0
$757$	28.2334	1.02616	0.513080	−	0.858341i	$-0.328504\pi$
$757$	28.2334	1.02616	0.513080	+	0.858341i	$0.328504\pi$
$758$	0	0
$759$	−29.3598	−1.06569
$760$	0	0
$761$	−33.9145	−1.22940	−0.614700	−	0.788761i	$-0.710723\pi$
$761$	−33.9145	−1.22940	−0.614700	+	0.788761i	$0.710723\pi$
$762$	0	0
$763$	−1.27806	−0.0462690
$764$	0	0
$765$	1.21509	0.0439318
$766$	0	0
$767$	18.7491	0.676991
$768$	0	0
$769$	36.5990	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$769$	36.5990	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$770$	0	0
$771$	−11.6211	−0.418525
$772$	0	0
$773$	−33.2634	−1.19640	−0.598201	−	0.801346i	$-0.704117\pi$
$773$	−33.2634	−1.19640	−0.598201	+	0.801346i	$0.704117\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	7.30846	0.262189
$778$	0	0
$779$	−1.34728	−0.0482713
$780$	0	0
$781$	17.8374	0.638273
$782$	0	0
$783$	5.59859	0.200077
$784$	0	0
$785$	5.80161	0.207068
$786$	0	0
$787$	−16.2559	−0.579462	−0.289731	−	0.957108i	$-0.593566\pi$
$787$	−16.2559	−0.579462	−0.289731	+	0.957108i	$0.593566\pi$
$788$	0	0
$789$	−6.16258	−0.219394
$790$	0	0
$791$	−23.1972	−0.824797
$792$	0	0
$793$	13.0253	0.462544
$794$	0	0
$795$	2.51730	0.0892796
$796$	0	0
$797$	24.7157	0.875475	0.437737	−	0.899103i	$-0.355780\pi$
$797$	24.7157	0.875475	0.437737	+	0.899103i	$0.355780\pi$
$798$	0	0
$799$	4.23180	0.149710
$800$	0	0
$801$	−2.20302	−0.0778398
$802$	0	0
$803$	6.66198	0.235096
$804$	0	0
$805$	−8.43018	−0.297125
$806$	0	0
$807$	22.4590	0.790593
$808$	0	0
$809$	19.8211	0.696874	0.348437	−	0.937332i	$-0.386713\pi$
$809$	19.8211	0.696874	0.348437	+	0.937332i	$0.386713\pi$
$810$	0	0
$811$	−11.2330	−0.394444	−0.197222	−	0.980359i	$-0.563192\pi$
$811$	−11.2330	−0.394444	−0.197222	+	0.980359i	$0.563192\pi$
$812$	0	0
$813$	−22.8362	−0.800901
$814$	0	0
$815$	−10.3143	−0.361294
$816$	0	0
$817$	5.93078	0.207492
$818$	0	0
$819$	3.21509	0.112344
$820$	0	0
$821$	22.9700	0.801659	0.400830	−	0.916153i	$-0.368722\pi$
$821$	22.9700	0.801659	0.400830	+	0.916153i	$0.368722\pi$
$822$	0	0
$823$	−0.697788	−0.0243234	−0.0121617	−	0.999926i	$-0.503871\pi$
$823$	−0.697788	−0.0243234	−0.0121617	+	0.999926i	$0.503871\pi$
$824$	0	0
$825$	4.69779	0.163556
$826$	0	0
$827$	−30.0755	−1.04583	−0.522913	−	0.852386i	$-0.675155\pi$
$827$	−30.0755	−1.04583	−0.522913	+	0.852386i	$0.675155\pi$
$828$	0	0
$829$	−16.6620	−0.578695	−0.289347	−	0.957224i	$-0.593438\pi$
$829$	−16.6620	−0.578695	−0.289347	+	0.957224i	$0.593438\pi$
$830$	0	0
$831$	−14.1851	−0.492076
$832$	0	0
$833$	6.29477	0.218101
$834$	0	0
$835$	−5.66318	−0.195982
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−16.9584	−0.585468	−0.292734	−	0.956194i	$-0.594565\pi$
$839$	−16.9584	−0.585468	−0.292734	+	0.956194i	$0.594565\pi$
$840$	0	0
$841$	2.34426	0.0808367
$842$	0	0
$843$	−12.8362	−0.442103
$844$	0	0
$845$	7.31892	0.251778
$846$	0	0
$847$	−14.9312	−0.513042
$848$	0	0
$849$	5.70986	0.195962
$850$	0	0
$851$	−33.8616	−1.16076
$852$	0	0
$853$	−5.43138	−0.185967	−0.0929835	−	0.995668i	$-0.529640\pi$
$853$	−5.43138	−0.185967	−0.0929835	+	0.995668i	$0.529640\pi$
$854$	0	0
$855$	1.03461	0.0353829
$856$	0	0
$857$	−11.6632	−0.398407	−0.199203	−	0.979958i	$-0.563835\pi$
$857$	−11.6632	−0.398407	−0.199203	+	0.979958i	$0.563835\pi$
$858$	0	0
$859$	−54.9779	−1.87582	−0.937911	−	0.346877i	$-0.887242\pi$
$859$	−54.9779	−1.87582	−0.937911	+	0.346877i	$0.887242\pi$
$860$	0	0
$861$	1.75655	0.0598629
$862$	0	0
$863$	−20.4636	−0.696589	−0.348294	−	0.937385i	$-0.613239\pi$
$863$	−20.4636	−0.696589	−0.348294	+	0.937385i	$0.613239\pi$
$864$	0	0
$865$	11.6632	0.396560
$866$	0	0
$867$	15.5236	0.527207
$868$	0	0
$869$	−8.54770	−0.289961
$870$	0	0
$871$	−3.38007	−0.114529
$872$	0	0
$873$	6.69779	0.226686
$874$	0	0
$875$	1.34889	0.0456009
$876$	0	0
$877$	−33.1855	−1.12060	−0.560298	−	0.828291i	$-0.689313\pi$
$877$	−33.1855	−1.12060	−0.560298	+	0.828291i	$0.689313\pi$
$878$	0	0
$879$	−8.01790	−0.270437
$880$	0	0
$881$	−5.30684	−0.178792	−0.0893960	−	0.995996i	$-0.528494\pi$
$881$	−5.30684	−0.178792	−0.0893960	+	0.995996i	$0.528494\pi$
$882$	0	0
$883$	−29.6032	−0.996228	−0.498114	−	0.867112i	$-0.665974\pi$
$883$	−29.6032	−0.996228	−0.498114	+	0.867112i	$0.665974\pi$
$884$	0	0
$885$	−7.86620	−0.264419
$886$	0	0
$887$	−47.2727	−1.58726	−0.793630	−	0.608401i	$-0.791811\pi$
$887$	−47.2727	−1.58726	−0.793630	+	0.608401i	$0.791811\pi$
$888$	0	0
$889$	2.91709	0.0978362
$890$	0	0
$891$	−4.69779	−0.157382
$892$	0	0
$893$	3.60323	0.120577
$894$	0	0
$895$	−11.8258	−0.395292
$896$	0	0
$897$	−14.8962	−0.497369
$898$	0	0
$899$	5.59859	0.186724
$900$	0	0
$901$	−3.05876	−0.101902
$902$	0	0
$903$	−7.73240	−0.257318
$904$	0	0
$905$	14.8604	0.493975
$906$	0	0
$907$	20.5010	0.680725	0.340363	−	0.940294i	$-0.389450\pi$
$907$	20.5010	0.680725	0.340363	+	0.940294i	$0.389450\pi$
$908$	0	0
$909$	13.1280	0.435427
$910$	0	0
$911$	−7.62737	−0.252706	−0.126353	−	0.991985i	$-0.540327\pi$
$911$	−7.62737	−0.252706	−0.126353	+	0.991985i	$0.540327\pi$
$912$	0	0
$913$	62.0817	2.05460
$914$	0	0
$915$	−5.46479	−0.180660
$916$	0	0
$917$	14.7974	0.488653
$918$	0	0
$919$	6.92959	0.228586	0.114293	−	0.993447i	$-0.463540\pi$
$919$	6.92959	0.228586	0.114293	+	0.993447i	$0.463540\pi$
$920$	0	0
$921$	−18.7445	−0.617651
$922$	0	0
$923$	9.05012	0.297888
$924$	0	0
$925$	5.41811	0.178146
$926$	0	0
$927$	16.4769	0.541171
$928$	0	0
$929$	−39.6678	−1.30146	−0.650729	−	0.759310i	$-0.725537\pi$
$929$	−39.6678	−1.30146	−0.650729	+	0.759310i	$0.725537\pi$
$930$	0	0
$931$	5.35977	0.175659
$932$	0	0
$933$	−19.2260	−0.629430
$934$	0	0
$935$	−5.70825	−0.186680
$936$	0	0
$937$	31.6966	1.03548	0.517741	−	0.855537i	$-0.326773\pi$
$937$	31.6966	1.03548	0.517741	+	0.855537i	$0.326773\pi$
$938$	0	0
$939$	−14.5219	−0.473905
$940$	0	0
$941$	−36.8557	−1.20146	−0.600731	−	0.799451i	$-0.705124\pi$
$941$	−36.8557	−1.20146	−0.600731	+	0.799451i	$0.705124\pi$
$942$	0	0
$943$	−8.13843	−0.265024
$944$	0	0
$945$	−1.34889	−0.0438795
$946$	0	0
$947$	34.6107	1.12470	0.562348	−	0.826901i	$-0.309898\pi$
$947$	34.6107	1.12470	0.562348	+	0.826901i	$0.309898\pi$
$948$	0	0
$949$	3.38007	0.109722
$950$	0	0
$951$	−0.378872	−0.0122858
$952$	0	0
$953$	4.14468	0.134259	0.0671297	−	0.997744i	$-0.478616\pi$
$953$	4.14468	0.134259	0.0671297	+	0.997744i	$0.478616\pi$
$954$	0	0
$955$	−5.93541	−0.192066
$956$	0	0
$957$	−26.3010	−0.850191
$958$	0	0
$959$	11.7082	0.378079
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−4.78491	−0.154191
$964$	0	0
$965$	3.19719	0.102921
$966$	0	0
$967$	24.7553	0.796078	0.398039	−	0.917368i	$-0.369691\pi$
$967$	24.7553	0.796078	0.398039	+	0.917368i	$0.369691\pi$
$968$	0	0
$969$	−1.25714	−0.0403853
$970$	0	0
$971$	−39.9930	−1.28344	−0.641718	−	0.766941i	$-0.721778\pi$
$971$	−39.9930	−1.28344	−0.641718	+	0.766941i	$0.721778\pi$
$972$	0	0
$973$	6.43018	0.206142
$974$	0	0
$975$	2.38350	0.0763332
$976$	0	0
$977$	10.8604	0.347454	0.173727	−	0.984794i	$-0.444419\pi$
$977$	10.8604	0.347454	0.173727	+	0.984794i	$0.444419\pi$
$978$	0	0
$979$	10.3493	0.330765
$980$	0	0
$981$	0.947489	0.0302510
$982$	0	0
$983$	−5.03461	−0.160579	−0.0802895	−	0.996772i	$-0.525584\pi$
$983$	−5.03461	−0.160579	−0.0802895	+	0.996772i	$0.525584\pi$
$984$	0	0
$985$	−18.7491	−0.597396
$986$	0	0
$987$	−4.69779	−0.149532
$988$	0	0
$989$	35.8258	1.13919
$990$	0	0
$991$	−55.0230	−1.74786	−0.873931	−	0.486050i	$-0.838437\pi$
$991$	−55.0230	−1.74786	−0.873931	+	0.486050i	$0.838437\pi$
$992$	0	0
$993$	17.1459	0.544108
$994$	0	0
$995$	−21.9129	−0.694685
$996$	0	0
$997$	5.40723	0.171249	0.0856244	−	0.996327i	$-0.472711\pi$
$997$	5.40723	0.171249	0.0856244	+	0.996327i	$0.472711\pi$
$998$	0	0
$999$	−5.41811	−0.171421

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bn.1.1		3
4.3	odd	2		1860.2.a.g.1.3	✓	3
12.11	even	2		5580.2.a.j.1.3		3
20.3	even	4		9300.2.g.q.3349.4		6
20.7	even	4		9300.2.g.q.3349.3		6
20.19	odd	2		9300.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.3	✓	3	4.3	odd	2
5580.2.a.j.1.3		3	12.11	even	2
7440.2.a.bn.1.1		3	1.1	even	1	trivial
9300.2.a.u.1.1		3	20.19	odd	2
9300.2.g.q.3349.3		6	20.7	even	4
9300.2.g.q.3349.4		6	20.3	even	4

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.34889 q^{7} +1.00000 q^{9} -4.69779 q^{11} -2.38350 q^{13} +1.00000 q^{15} -1.21509 q^{17} -1.03461 q^{19} +1.34889 q^{21} -6.24970 q^{23} +1.00000 q^{25} -1.00000 q^{27} -5.59859 q^{29} -1.00000 q^{31} +4.69779 q^{33} +1.34889 q^{35} +5.41811 q^{37} +2.38350 q^{39} +1.30221 q^{41} -5.73240 q^{43} -1.00000 q^{45} -3.48270 q^{47} -5.18048 q^{49} +1.21509 q^{51} +2.51730 q^{53} +4.69779 q^{55} +1.03461 q^{57} -7.86620 q^{59} -5.46479 q^{61} -1.34889 q^{63} +2.38350 q^{65} +1.41811 q^{67} +6.24970 q^{69} -3.79698 q^{71} -1.41811 q^{73} -1.00000 q^{75} +6.33682 q^{77} +1.81952 q^{79} +1.00000 q^{81} -13.2151 q^{83} +1.21509 q^{85} +5.59859 q^{87} -2.20302 q^{89} +3.21509 q^{91} +1.00000 q^{93} +1.03461 q^{95} +6.69779 q^{97} -4.69779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 3 q^{15} + 10 q^{17} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{29} - 3 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 16 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 3 * q^15 + 10 * q^17 - 2 * q^21 - 2 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^29 - 3 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 16 * q^41 - 2 * q^43 - 3 * q^45 - 12 * q^47 - 5 * q^49 - 10 * q^51 + 6 * q^53 + 2 * q^55 - 16 * q^59 + 14 * q^61 + 2 * q^63 - 2 * q^65 - 8 * q^67 + 2 * q^69 - 10 * q^71 + 8 * q^73 - 3 * q^75 + 28 * q^77 + 16 * q^79 + 3 * q^81 - 26 * q^83 - 10 * q^85 - 6 * q^87 - 8 * q^89 - 4 * q^91 + 3 * q^93 + 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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