LMFDB - Embedded newform 1380.4.a.e.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1380,4,Mod(1,1380)] 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("1380.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-15,0,-25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$81.4226358079$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 278x^{3} + 216x^{2} + 14064x - 33408 $ x^5 - x^4 - 278x^3 + 216x^2 + 14064*x - 33408
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-9.25159$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +21.4978 q^{7} +9.00000 q^{9} +40.0214 q^{11} -13.4882 q^{13} +15.0000 q^{15} +118.546 q^{17} +81.5201 q^{19} -64.4934 q^{21} +23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -98.5712 q^{29} +41.9361 q^{31} -120.064 q^{33} -107.489 q^{35} -60.8785 q^{37} +40.4645 q^{39} +38.6716 q^{41} +141.524 q^{43} -45.0000 q^{45} -212.948 q^{47} +119.156 q^{49} -355.637 q^{51} -290.961 q^{53} -200.107 q^{55} -244.560 q^{57} +734.568 q^{59} -322.242 q^{61} +193.480 q^{63} +67.4408 q^{65} +833.176 q^{67} -69.0000 q^{69} -780.428 q^{71} -201.464 q^{73} -75.0000 q^{75} +860.371 q^{77} +6.04147 q^{79} +81.0000 q^{81} +195.200 q^{83} -592.729 q^{85} +295.713 q^{87} +224.630 q^{89} -289.966 q^{91} -125.808 q^{93} -407.601 q^{95} -877.289 q^{97} +360.192 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 15 q^{3} - 25 q^{5} - 7 q^{7} + 45 q^{9} + 64 q^{11} + 80 q^{13} + 75 q^{15} - 21 q^{17} - 52 q^{19} + 21 q^{21} + 115 q^{23} + 125 q^{25} - 135 q^{27} - 277 q^{29} - 289 q^{31} - 192 q^{33} + 35 q^{35}+ \cdots + 576 q^{99}+O(q^{100}) $ 5 * q - 15 * q^3 - 25 * q^5 - 7 * q^7 + 45 * q^9 + 64 * q^11 + 80 * q^13 + 75 * q^15 - 21 * q^17 - 52 * q^19 + 21 * q^21 + 115 * q^23 + 125 * q^25 - 135 * q^27 - 277 * q^29 - 289 * q^31 - 192 * q^33 + 35 * q^35 - 303 * q^37 - 240 * q^39 - 95 * q^41 - 372 * q^43 - 225 * q^45 - 462 * q^47 - 272 * q^49 + 63 * q^51 - 73 * q^53 - 320 * q^55 + 156 * q^57 + 529 * q^59 + 174 * q^61 - 63 * q^63 - 400 * q^65 - 919 * q^67 - 345 * q^69 - 141 * q^71 + 448 * q^73 - 375 * q^75 - 136 * q^77 - 496 * q^79 + 405 * q^81 + 1219 * q^83 + 105 * q^85 + 831 * q^87 + 1492 * q^89 + 1560 * q^91 + 867 * q^93 + 260 * q^95 + 574 * q^97 + 576 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	21.4978	1.16077	0.580386	−	0.814341i	$-0.302902\pi$
$7$	21.4978	1.16077	0.580386	+	0.814341i	$0.302902\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	40.0214	1.09699	0.548495	−	0.836154i	$-0.315201\pi$
$11$	40.0214	1.09699	0.548495	+	0.836154i	$0.315201\pi$
$12$	0	0
$13$	−13.4882	−0.287765	−0.143882	−	0.989595i	$-0.545959\pi$
$13$	−13.4882	−0.287765	−0.143882	+	0.989595i	$0.545959\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	118.546	1.69127	0.845634	−	0.533763i	$-0.179222\pi$
$17$	118.546	1.69127	0.845634	+	0.533763i	$0.179222\pi$
$18$	0	0
$19$	81.5201	0.984316	0.492158	−	0.870506i	$-0.336208\pi$
$19$	81.5201	0.984316	0.492158	+	0.870506i	$0.336208\pi$
$20$	0	0
$21$	−64.4934	−0.670172
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−98.5712	−0.631179	−0.315590	−	0.948896i	$-0.602202\pi$
$29$	−98.5712	−0.631179	−0.315590	+	0.948896i	$0.602202\pi$
$30$	0	0
$31$	41.9361	0.242966	0.121483	−	0.992594i	$-0.461235\pi$
$31$	41.9361	0.242966	0.121483	+	0.992594i	$0.461235\pi$
$32$	0	0
$33$	−120.064	−0.633348
$34$	0	0
$35$	−107.489	−0.519113
$36$	0	0
$37$	−60.8785	−0.270496	−0.135248	−	0.990812i	$-0.543183\pi$
$37$	−60.8785	−0.270496	−0.135248	+	0.990812i	$0.543183\pi$
$38$	0	0
$39$	40.4645	0.166141
$40$	0	0
$41$	38.6716	0.147305	0.0736523	−	0.997284i	$-0.476534\pi$
$41$	38.6716	0.147305	0.0736523	+	0.997284i	$0.476534\pi$
$42$	0	0
$43$	141.524	0.501912	0.250956	−	0.967998i	$-0.419255\pi$
$43$	141.524	0.501912	0.250956	+	0.967998i	$0.419255\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−212.948	−0.660888	−0.330444	−	0.943826i	$-0.607198\pi$
$47$	−212.948	−0.660888	−0.330444	+	0.943826i	$0.607198\pi$
$48$	0	0
$49$	119.156	0.347392
$50$	0	0
$51$	−355.637	−0.976454
$52$	0	0
$53$	−290.961	−0.754086	−0.377043	−	0.926196i	$-0.623059\pi$
$53$	−290.961	−0.754086	−0.377043	+	0.926196i	$0.623059\pi$
$54$	0	0
$55$	−200.107	−0.490589
$56$	0	0
$57$	−244.560	−0.568295
$58$	0	0
$59$	734.568	1.62089	0.810446	−	0.585813i	$-0.199225\pi$
$59$	734.568	1.62089	0.810446	+	0.585813i	$0.199225\pi$
$60$	0	0
$61$	−322.242	−0.676375	−0.338188	−	0.941079i	$-0.609814\pi$
$61$	−322.242	−0.676375	−0.338188	+	0.941079i	$0.609814\pi$
$62$	0	0
$63$	193.480	0.386924
$64$	0	0
$65$	67.4408	0.128692
$66$	0	0
$67$	833.176	1.51923	0.759617	−	0.650371i	$-0.225387\pi$
$67$	833.176	1.51923	0.759617	+	0.650371i	$0.225387\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	−780.428	−1.30450	−0.652252	−	0.758003i	$-0.726175\pi$
$71$	−780.428	−1.30450	−0.652252	+	0.758003i	$0.726175\pi$
$72$	0	0
$73$	−201.464	−0.323008	−0.161504	−	0.986872i	$-0.551635\pi$
$73$	−201.464	−0.323008	−0.161504	+	0.986872i	$0.551635\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	860.371	1.27336
$78$	0	0
$79$	6.04147	0.00860402	0.00430201	−	0.999991i	$-0.498631\pi$
$79$	6.04147	0.00860402	0.00430201	+	0.999991i	$0.498631\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	195.200	0.258144	0.129072	−	0.991635i	$-0.458800\pi$
$83$	195.200	0.258144	0.129072	+	0.991635i	$0.458800\pi$
$84$	0	0
$85$	−592.729	−0.756358
$86$	0	0
$87$	295.713	0.364412
$88$	0	0
$89$	224.630	0.267536	0.133768	−	0.991013i	$-0.457292\pi$
$89$	224.630	0.267536	0.133768	+	0.991013i	$0.457292\pi$
$90$	0	0
$91$	−289.966	−0.334030
$92$	0	0
$93$	−125.808	−0.140276
$94$	0	0
$95$	−407.601	−0.440199
$96$	0	0
$97$	−877.289	−0.918301	−0.459150	−	0.888359i	$-0.651846\pi$
$97$	−877.289	−0.918301	−0.459150	+	0.888359i	$0.651846\pi$
$98$	0	0
$99$	360.192	0.365663
$100$	0	0
$101$	−96.9440	−0.0955078	−0.0477539	−	0.998859i	$-0.515206\pi$
$101$	−96.9440	−0.0955078	−0.0477539	+	0.998859i	$0.515206\pi$
$102$	0	0
$103$	−841.448	−0.804955	−0.402477	−	0.915430i	$-0.631851\pi$
$103$	−841.448	−0.804955	−0.402477	+	0.915430i	$0.631851\pi$
$104$	0	0
$105$	322.467	0.299710
$106$	0	0
$107$	931.011	0.841161	0.420581	−	0.907255i	$-0.361826\pi$
$107$	931.011	0.841161	0.420581	+	0.907255i	$0.361826\pi$
$108$	0	0
$109$	−121.322	−0.106610	−0.0533051	−	0.998578i	$-0.516976\pi$
$109$	−121.322	−0.106610	−0.0533051	+	0.998578i	$0.516976\pi$
$110$	0	0
$111$	182.636	0.156171
$112$	0	0
$113$	−39.6067	−0.0329724	−0.0164862	−	0.999864i	$-0.505248\pi$
$113$	−39.6067	−0.0329724	−0.0164862	+	0.999864i	$0.505248\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−121.394	−0.0959217
$118$	0	0
$119$	2548.47	1.96318
$120$	0	0
$121$	270.709	0.203388
$122$	0	0
$123$	−116.015	−0.0850464
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1057.70	−0.739024	−0.369512	−	0.929226i	$-0.620475\pi$
$127$	−1057.70	−0.739024	−0.369512	+	0.929226i	$0.620475\pi$
$128$	0	0
$129$	−424.573	−0.289779
$130$	0	0
$131$	1898.22	1.26602	0.633009	−	0.774145i	$-0.281820\pi$
$131$	1898.22	1.26602	0.633009	+	0.774145i	$0.281820\pi$
$132$	0	0
$133$	1752.50	1.14257
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	1927.92	1.20229	0.601143	−	0.799142i	$-0.294712\pi$
$137$	1927.92	1.20229	0.601143	+	0.799142i	$0.294712\pi$
$138$	0	0
$139$	998.157	0.609083	0.304541	−	0.952499i	$-0.401497\pi$
$139$	998.157	0.609083	0.304541	+	0.952499i	$0.401497\pi$
$140$	0	0
$141$	638.845	0.381564
$142$	0	0
$143$	−539.815	−0.315675
$144$	0	0
$145$	492.856	0.282272
$146$	0	0
$147$	−357.467	−0.200567
$148$	0	0
$149$	3050.98	1.67749	0.838746	−	0.544523i	$-0.183289\pi$
$149$	3050.98	1.67749	0.838746	+	0.544523i	$0.183289\pi$
$150$	0	0
$151$	−377.639	−0.203522	−0.101761	−	0.994809i	$-0.532448\pi$
$151$	−377.639	−0.203522	−0.101761	+	0.994809i	$0.532448\pi$
$152$	0	0
$153$	1066.91	0.563756
$154$	0	0
$155$	−209.680	−0.108658
$156$	0	0
$157$	−1032.60	−0.524908	−0.262454	−	0.964944i	$-0.584532\pi$
$157$	−1032.60	−0.524908	−0.262454	+	0.964944i	$0.584532\pi$
$158$	0	0
$159$	872.883	0.435372
$160$	0	0
$161$	494.449	0.242038
$162$	0	0
$163$	2808.91	1.34976	0.674880	−	0.737928i	$-0.264195\pi$
$163$	2808.91	1.34976	0.674880	+	0.737928i	$0.264195\pi$
$164$	0	0
$165$	600.320	0.283242
$166$	0	0
$167$	−1611.25	−0.746598	−0.373299	−	0.927711i	$-0.621773\pi$
$167$	−1611.25	−0.746598	−0.373299	+	0.927711i	$0.621773\pi$
$168$	0	0
$169$	−2015.07	−0.917191
$170$	0	0
$171$	733.681	0.328105
$172$	0	0
$173$	1960.56	0.861608	0.430804	−	0.902445i	$-0.358230\pi$
$173$	1960.56	0.861608	0.430804	+	0.902445i	$0.358230\pi$
$174$	0	0
$175$	537.445	0.232154
$176$	0	0
$177$	−2203.70	−0.935823
$178$	0	0
$179$	3887.25	1.62317	0.811583	−	0.584238i	$-0.198606\pi$
$179$	3887.25	1.62317	0.811583	+	0.584238i	$0.198606\pi$
$180$	0	0
$181$	−620.323	−0.254742	−0.127371	−	0.991855i	$-0.540654\pi$
$181$	−620.323	−0.254742	−0.127371	+	0.991855i	$0.540654\pi$
$182$	0	0
$183$	966.726	0.390505
$184$	0	0
$185$	304.393	0.120970
$186$	0	0
$187$	4744.36	1.85531
$188$	0	0
$189$	−580.441	−0.223391
$190$	0	0
$191$	1527.92	0.578831	0.289415	−	0.957204i	$-0.406539\pi$
$191$	1527.92	0.578831	0.289415	+	0.957204i	$0.406539\pi$
$192$	0	0
$193$	2413.46	0.900129	0.450065	−	0.892996i	$-0.351401\pi$
$193$	2413.46	0.900129	0.450065	+	0.892996i	$0.351401\pi$
$194$	0	0
$195$	−202.323	−0.0743006
$196$	0	0
$197$	3301.20	1.19391	0.596956	−	0.802274i	$-0.296377\pi$
$197$	3301.20	1.19391	0.596956	+	0.802274i	$0.296377\pi$
$198$	0	0
$199$	1700.14	0.605628	0.302814	−	0.953050i	$-0.402074\pi$
$199$	1700.14	0.605628	0.302814	+	0.953050i	$0.402074\pi$
$200$	0	0
$201$	−2499.53	−0.877130
$202$	0	0
$203$	−2119.06	−0.732656
$204$	0	0
$205$	−193.358	−0.0658766
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	3262.55	1.07979
$210$	0	0
$211$	−530.027	−0.172932	−0.0864658	−	0.996255i	$-0.527557\pi$
$211$	−530.027	−0.172932	−0.0864658	+	0.996255i	$0.527557\pi$
$212$	0	0
$213$	2341.28	0.753155
$214$	0	0
$215$	−707.621	−0.224462
$216$	0	0
$217$	901.534	0.282028
$218$	0	0
$219$	604.393	0.186489
$220$	0	0
$221$	−1598.96	−0.486688
$222$	0	0
$223$	−5898.28	−1.77120	−0.885600	−	0.464448i	$-0.846253\pi$
$223$	−5898.28	−1.77120	−0.885600	+	0.464448i	$0.846253\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−2652.03	−0.775425	−0.387712	−	0.921780i	$-0.626735\pi$
$227$	−2652.03	−0.775425	−0.387712	+	0.921780i	$0.626735\pi$
$228$	0	0
$229$	−725.753	−0.209428	−0.104714	−	0.994502i	$-0.533393\pi$
$229$	−725.753	−0.209428	−0.104714	+	0.994502i	$0.533393\pi$
$230$	0	0
$231$	−2581.11	−0.735172
$232$	0	0
$233$	1089.63	0.306369	0.153185	−	0.988198i	$-0.451047\pi$
$233$	1089.63	0.306369	0.153185	+	0.988198i	$0.451047\pi$
$234$	0	0
$235$	1064.74	0.295558
$236$	0	0
$237$	−18.1244	−0.00496754
$238$	0	0
$239$	−4485.55	−1.21400	−0.607000	−	0.794702i	$-0.707627\pi$
$239$	−4485.55	−1.21400	−0.607000	+	0.794702i	$0.707627\pi$
$240$	0	0
$241$	5351.32	1.43033	0.715164	−	0.698957i	$-0.246352\pi$
$241$	5351.32	1.43033	0.715164	+	0.698957i	$0.246352\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−595.778	−0.155359
$246$	0	0
$247$	−1099.56	−0.283252
$248$	0	0
$249$	−585.599	−0.149039
$250$	0	0
$251$	279.307	0.0702379	0.0351190	−	0.999383i	$-0.488819\pi$
$251$	279.307	0.0702379	0.0351190	+	0.999383i	$0.488819\pi$
$252$	0	0
$253$	920.491	0.228738
$254$	0	0
$255$	1778.19	0.436684
$256$	0	0
$257$	3616.08	0.877684	0.438842	−	0.898564i	$-0.355389\pi$
$257$	3616.08	0.877684	0.438842	+	0.898564i	$0.355389\pi$
$258$	0	0
$259$	−1308.75	−0.313985
$260$	0	0
$261$	−887.140	−0.210393
$262$	0	0
$263$	87.5237	0.0205207	0.0102603	−	0.999947i	$-0.496734\pi$
$263$	87.5237	0.0205207	0.0102603	+	0.999947i	$0.496734\pi$
$264$	0	0
$265$	1454.80	0.337237
$266$	0	0
$267$	−673.890	−0.154462
$268$	0	0
$269$	−142.638	−0.0323302	−0.0161651	−	0.999869i	$-0.505146\pi$
$269$	−142.638	−0.0323302	−0.0161651	+	0.999869i	$0.505146\pi$
$270$	0	0
$271$	5707.35	1.27932	0.639662	−	0.768657i	$-0.279075\pi$
$271$	5707.35	1.27932	0.639662	+	0.768657i	$0.279075\pi$
$272$	0	0
$273$	869.898	0.192852
$274$	0	0
$275$	1000.53	0.219398
$276$	0	0
$277$	4660.87	1.01099	0.505495	−	0.862829i	$-0.331310\pi$
$277$	4660.87	1.01099	0.505495	+	0.862829i	$0.331310\pi$
$278$	0	0
$279$	377.425	0.0809886
$280$	0	0
$281$	−368.325	−0.0781937	−0.0390969	−	0.999235i	$-0.512448\pi$
$281$	−368.325	−0.0781937	−0.0390969	+	0.999235i	$0.512448\pi$
$282$	0	0
$283$	−6908.39	−1.45110	−0.725550	−	0.688169i	$-0.758415\pi$
$283$	−6908.39	−1.45110	−0.725550	+	0.688169i	$0.758415\pi$
$284$	0	0
$285$	1222.80	0.254149
$286$	0	0
$287$	831.355	0.170987
$288$	0	0
$289$	9140.10	1.86039
$290$	0	0
$291$	2631.87	0.530181
$292$	0	0
$293$	−5388.52	−1.07440	−0.537202	−	0.843454i	$-0.680519\pi$
$293$	−5388.52	−1.07440	−0.537202	+	0.843454i	$0.680519\pi$
$294$	0	0
$295$	−3672.84	−0.724885
$296$	0	0
$297$	−1080.58	−0.211116
$298$	0	0
$299$	−310.228	−0.0600031
$300$	0	0
$301$	3042.46	0.582606
$302$	0	0
$303$	290.832	0.0551414
$304$	0	0
$305$	1611.21	0.302484
$306$	0	0
$307$	1309.23	0.243394	0.121697	−	0.992567i	$-0.461166\pi$
$307$	1309.23	0.243394	0.121697	+	0.992567i	$0.461166\pi$
$308$	0	0
$309$	2524.34	0.464741
$310$	0	0
$311$	−2818.70	−0.513936	−0.256968	−	0.966420i	$-0.582723\pi$
$311$	−2818.70	−0.513936	−0.256968	+	0.966420i	$0.582723\pi$
$312$	0	0
$313$	4557.17	0.822959	0.411480	−	0.911419i	$-0.365012\pi$
$313$	4557.17	0.822959	0.411480	+	0.911419i	$0.365012\pi$
$314$	0	0
$315$	−967.401	−0.173038
$316$	0	0
$317$	5128.87	0.908726	0.454363	−	0.890817i	$-0.349867\pi$
$317$	5128.87	0.908726	0.454363	+	0.890817i	$0.349867\pi$
$318$	0	0
$319$	−3944.95	−0.692398
$320$	0	0
$321$	−2793.03	−0.485645
$322$	0	0
$323$	9663.87	1.66474
$324$	0	0
$325$	−337.204	−0.0575530
$326$	0	0
$327$	363.965	0.0615514
$328$	0	0
$329$	−4577.92	−0.767140
$330$	0	0
$331$	−4692.68	−0.779253	−0.389627	−	0.920973i	$-0.627396\pi$
$331$	−4692.68	−0.779253	−0.389627	+	0.920973i	$0.627396\pi$
$332$	0	0
$333$	−547.907	−0.0901654
$334$	0	0
$335$	−4165.88	−0.679422
$336$	0	0
$337$	−2685.75	−0.434131	−0.217065	−	0.976157i	$-0.569648\pi$
$337$	−2685.75	−0.434131	−0.217065	+	0.976157i	$0.569648\pi$
$338$	0	0
$339$	118.820	0.0190366
$340$	0	0
$341$	1678.34	0.266531
$342$	0	0
$343$	−4812.16	−0.757529
$344$	0	0
$345$	345.000	0.0538382
$346$	0	0
$347$	2622.61	0.405733	0.202866	−	0.979206i	$-0.434974\pi$
$347$	2622.61	0.405733	0.202866	+	0.979206i	$0.434974\pi$
$348$	0	0
$349$	3220.34	0.493929	0.246964	−	0.969025i	$-0.420567\pi$
$349$	3220.34	0.493929	0.246964	+	0.969025i	$0.420567\pi$
$350$	0	0
$351$	364.181	0.0553804
$352$	0	0
$353$	9441.80	1.42362	0.711808	−	0.702374i	$-0.247877\pi$
$353$	9441.80	1.42362	0.711808	+	0.702374i	$0.247877\pi$
$354$	0	0
$355$	3902.14	0.583391
$356$	0	0
$357$	−7645.42	−1.13344
$358$	0	0
$359$	6494.59	0.954794	0.477397	−	0.878688i	$-0.341580\pi$
$359$	6494.59	0.954794	0.477397	+	0.878688i	$0.341580\pi$
$360$	0	0
$361$	−213.466	−0.0311220
$362$	0	0
$363$	−812.128	−0.117426
$364$	0	0
$365$	1007.32	0.144454
$366$	0	0
$367$	−3609.03	−0.513324	−0.256662	−	0.966501i	$-0.582623\pi$
$367$	−3609.03	−0.513324	−0.256662	+	0.966501i	$0.582623\pi$
$368$	0	0
$369$	348.044	0.0491016
$370$	0	0
$371$	−6255.02	−0.875322
$372$	0	0
$373$	−11718.4	−1.62668	−0.813342	−	0.581786i	$-0.802354\pi$
$373$	−11718.4	−1.62668	−0.813342	+	0.581786i	$0.802354\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	1329.54	0.181631
$378$	0	0
$379$	−951.582	−0.128970	−0.0644848	−	0.997919i	$-0.520540\pi$
$379$	−951.582	−0.128970	−0.0644848	+	0.997919i	$0.520540\pi$
$380$	0	0
$381$	3173.11	0.426676
$382$	0	0
$383$	−9946.57	−1.32701	−0.663507	−	0.748170i	$-0.730933\pi$
$383$	−9946.57	−1.32701	−0.663507	+	0.748170i	$0.730933\pi$
$384$	0	0
$385$	−4301.86	−0.569462
$386$	0	0
$387$	1273.72	0.167304
$388$	0	0
$389$	4140.43	0.539661	0.269831	−	0.962908i	$-0.413032\pi$
$389$	4140.43	0.539661	0.269831	+	0.962908i	$0.413032\pi$
$390$	0	0
$391$	2726.55	0.352654
$392$	0	0
$393$	−5694.66	−0.730936
$394$	0	0
$395$	−30.2073	−0.00384784
$396$	0	0
$397$	−604.549	−0.0764268	−0.0382134	−	0.999270i	$-0.512167\pi$
$397$	−604.549	−0.0764268	−0.0382134	+	0.999270i	$0.512167\pi$
$398$	0	0
$399$	−5257.51	−0.659661
$400$	0	0
$401$	−9552.58	−1.18961	−0.594804	−	0.803871i	$-0.702770\pi$
$401$	−9552.58	−1.18961	−0.594804	+	0.803871i	$0.702770\pi$
$402$	0	0
$403$	−565.641	−0.0699171
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−2436.44	−0.296732
$408$	0	0
$409$	8668.64	1.04801	0.524006	−	0.851715i	$-0.324437\pi$
$409$	8668.64	1.04801	0.524006	+	0.851715i	$0.324437\pi$
$410$	0	0
$411$	−5783.75	−0.694140
$412$	0	0
$413$	15791.6	1.88149
$414$	0	0
$415$	−975.998	−0.115445
$416$	0	0
$417$	−2994.47	−0.351654
$418$	0	0
$419$	2757.79	0.321543	0.160772	−	0.986992i	$-0.448602\pi$
$419$	2757.79	0.321543	0.160772	+	0.986992i	$0.448602\pi$
$420$	0	0
$421$	3996.08	0.462606	0.231303	−	0.972882i	$-0.425701\pi$
$421$	3996.08	0.462606	0.231303	+	0.972882i	$0.425701\pi$
$422$	0	0
$423$	−1916.54	−0.220296
$424$	0	0
$425$	2963.64	0.338254
$426$	0	0
$427$	−6927.50	−0.785117
$428$	0	0
$429$	1619.44	0.182255
$430$	0	0
$431$	−1125.82	−0.125821	−0.0629105	−	0.998019i	$-0.520038\pi$
$431$	−1125.82	−0.125821	−0.0629105	+	0.998019i	$0.520038\pi$
$432$	0	0
$433$	−1549.93	−0.172020	−0.0860102	−	0.996294i	$-0.527412\pi$
$433$	−1549.93	−0.172020	−0.0860102	+	0.996294i	$0.527412\pi$
$434$	0	0
$435$	−1478.57	−0.162970
$436$	0	0
$437$	1874.96	0.205244
$438$	0	0
$439$	5220.43	0.567557	0.283778	−	0.958890i	$-0.408412\pi$
$439$	5220.43	0.567557	0.283778	+	0.958890i	$0.408412\pi$
$440$	0	0
$441$	1072.40	0.115797
$442$	0	0
$443$	−1056.46	−0.113305	−0.0566523	−	0.998394i	$-0.518043\pi$
$443$	−1056.46	−0.113305	−0.0566523	+	0.998394i	$0.518043\pi$
$444$	0	0
$445$	−1123.15	−0.119646
$446$	0	0
$447$	−9152.95	−0.968500
$448$	0	0
$449$	4993.72	0.524874	0.262437	−	0.964949i	$-0.415474\pi$
$449$	4993.72	0.524874	0.262437	+	0.964949i	$0.415474\pi$
$450$	0	0
$451$	1547.69	0.161592
$452$	0	0
$453$	1132.92	0.117504
$454$	0	0
$455$	1449.83	0.149383
$456$	0	0
$457$	−5962.08	−0.610272	−0.305136	−	0.952309i	$-0.598702\pi$
$457$	−5962.08	−0.610272	−0.305136	+	0.952309i	$0.598702\pi$
$458$	0	0
$459$	−3200.74	−0.325485
$460$	0	0
$461$	8576.37	0.866467	0.433234	−	0.901282i	$-0.357373\pi$
$461$	8576.37	0.866467	0.433234	+	0.901282i	$0.357373\pi$
$462$	0	0
$463$	−1485.34	−0.149092	−0.0745458	−	0.997218i	$-0.523751\pi$
$463$	−1485.34	−0.149092	−0.0745458	+	0.997218i	$0.523751\pi$
$464$	0	0
$465$	629.041	0.0627335
$466$	0	0
$467$	−18703.9	−1.85335	−0.926673	−	0.375869i	$-0.877344\pi$
$467$	−18703.9	−1.85335	−0.926673	+	0.375869i	$0.877344\pi$
$468$	0	0
$469$	17911.5	1.76348
$470$	0	0
$471$	3097.80	0.303056
$472$	0	0
$473$	5663.99	0.550593
$474$	0	0
$475$	2038.00	0.196863
$476$	0	0
$477$	−2618.65	−0.251362
$478$	0	0
$479$	11857.0	1.13103	0.565513	−	0.824739i	$-0.308678\pi$
$479$	11857.0	1.13103	0.565513	+	0.824739i	$0.308678\pi$
$480$	0	0
$481$	821.139	0.0778394
$482$	0	0
$483$	−1483.35	−0.139741
$484$	0	0
$485$	4386.44	0.410676
$486$	0	0
$487$	−15502.8	−1.44250	−0.721251	−	0.692674i	$-0.756432\pi$
$487$	−15502.8	−1.44250	−0.721251	+	0.692674i	$0.756432\pi$
$488$	0	0
$489$	−8426.73	−0.779284
$490$	0	0
$491$	18381.3	1.68948	0.844742	−	0.535174i	$-0.179754\pi$
$491$	18381.3	1.68948	0.844742	+	0.535174i	$0.179754\pi$
$492$	0	0
$493$	−11685.2	−1.06749
$494$	0	0
$495$	−1800.96	−0.163530
$496$	0	0
$497$	−16777.5	−1.51423
$498$	0	0
$499$	−17913.9	−1.60709	−0.803545	−	0.595244i	$-0.797055\pi$
$499$	−17913.9	−1.60709	−0.803545	+	0.595244i	$0.797055\pi$
$500$	0	0
$501$	4833.74	0.431049
$502$	0	0
$503$	21272.7	1.88569	0.942846	−	0.333229i	$-0.108138\pi$
$503$	21272.7	1.88569	0.942846	+	0.333229i	$0.108138\pi$
$504$	0	0
$505$	484.720	0.0427124
$506$	0	0
$507$	6045.21	0.529541
$508$	0	0
$509$	19098.2	1.66309	0.831546	−	0.555456i	$-0.187456\pi$
$509$	19098.2	1.66309	0.831546	+	0.555456i	$0.187456\pi$
$510$	0	0
$511$	−4331.04	−0.374939
$512$	0	0
$513$	−2201.04	−0.189432
$514$	0	0
$515$	4207.24	0.359987
$516$	0	0
$517$	−8522.49	−0.724987
$518$	0	0
$519$	−5881.67	−0.497450
$520$	0	0
$521$	−5006.92	−0.421031	−0.210516	−	0.977590i	$-0.567514\pi$
$521$	−5006.92	−0.421031	−0.210516	+	0.977590i	$0.567514\pi$
$522$	0	0
$523$	−6771.52	−0.566152	−0.283076	−	0.959097i	$-0.591355\pi$
$523$	−6771.52	−0.566152	−0.283076	+	0.959097i	$0.591355\pi$
$524$	0	0
$525$	−1612.34	−0.134034
$526$	0	0
$527$	4971.34	0.410921
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	6611.11	0.540297
$532$	0	0
$533$	−521.609	−0.0423891
$534$	0	0
$535$	−4655.06	−0.376179
$536$	0	0
$537$	−11661.7	−0.937135
$538$	0	0
$539$	4768.77	0.381086
$540$	0	0
$541$	−12351.6	−0.981581	−0.490790	−	0.871278i	$-0.663292\pi$
$541$	−12351.6	−0.981581	−0.490790	+	0.871278i	$0.663292\pi$
$542$	0	0
$543$	1860.97	0.147075
$544$	0	0
$545$	606.608	0.0476775
$546$	0	0
$547$	−16491.4	−1.28907	−0.644534	−	0.764575i	$-0.722949\pi$
$547$	−16491.4	−1.28907	−0.644534	+	0.764575i	$0.722949\pi$
$548$	0	0
$549$	−2900.18	−0.225458
$550$	0	0
$551$	−8035.53	−0.621280
$552$	0	0
$553$	129.878	0.00998731
$554$	0	0
$555$	−913.178	−0.0698419
$556$	0	0
$557$	−4078.06	−0.310221	−0.155110	−	0.987897i	$-0.549573\pi$
$557$	−4078.06	−0.310221	−0.155110	+	0.987897i	$0.549573\pi$
$558$	0	0
$559$	−1908.90	−0.144433
$560$	0	0
$561$	−14233.1	−1.07116
$562$	0	0
$563$	3563.37	0.266746	0.133373	−	0.991066i	$-0.457419\pi$
$563$	3563.37	0.266746	0.133373	+	0.991066i	$0.457419\pi$
$564$	0	0
$565$	198.033	0.0147457
$566$	0	0
$567$	1741.32	0.128975
$568$	0	0
$569$	604.432	0.0445327	0.0222663	−	0.999752i	$-0.492912\pi$
$569$	604.432	0.0445327	0.0222663	+	0.999752i	$0.492912\pi$
$570$	0	0
$571$	3149.87	0.230854	0.115427	−	0.993316i	$-0.463176\pi$
$571$	3149.87	0.230854	0.115427	+	0.993316i	$0.463176\pi$
$572$	0	0
$573$	−4583.77	−0.334188
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−11948.7	−0.862100	−0.431050	−	0.902328i	$-0.641857\pi$
$577$	−11948.7	−0.862100	−0.431050	+	0.902328i	$0.641857\pi$
$578$	0	0
$579$	−7240.39	−0.519690
$580$	0	0
$581$	4196.36	0.299646
$582$	0	0
$583$	−11644.7	−0.827225
$584$	0	0
$585$	606.968	0.0428975
$586$	0	0
$587$	−26958.4	−1.89556	−0.947780	−	0.318925i	$-0.896678\pi$
$587$	−26958.4	−1.89556	−0.947780	+	0.318925i	$0.896678\pi$
$588$	0	0
$589$	3418.64	0.239155
$590$	0	0
$591$	−9903.60	−0.689306
$592$	0	0
$593$	−12051.5	−0.834560	−0.417280	−	0.908778i	$-0.637017\pi$
$593$	−12051.5	−0.834560	−0.417280	+	0.908778i	$0.637017\pi$
$594$	0	0
$595$	−12742.4	−0.877960
$596$	0	0
$597$	−5100.43	−0.349660
$598$	0	0
$599$	−7612.90	−0.519290	−0.259645	−	0.965704i	$-0.583606\pi$
$599$	−7612.90	−0.519290	−0.259645	+	0.965704i	$0.583606\pi$
$600$	0	0
$601$	−10181.2	−0.691012	−0.345506	−	0.938417i	$-0.612293\pi$
$601$	−10181.2	−0.691012	−0.345506	+	0.938417i	$0.612293\pi$
$602$	0	0
$603$	7498.58	0.506411
$604$	0	0
$605$	−1353.55	−0.0909579
$606$	0	0
$607$	791.884	0.0529515	0.0264758	−	0.999649i	$-0.491572\pi$
$607$	791.884	0.0529515	0.0264758	+	0.999649i	$0.491572\pi$
$608$	0	0
$609$	6357.19	0.422999
$610$	0	0
$611$	2872.28	0.190180
$612$	0	0
$613$	4128.11	0.271995	0.135997	−	0.990709i	$-0.456576\pi$
$613$	4128.11	0.271995	0.135997	+	0.990709i	$0.456576\pi$
$614$	0	0
$615$	580.074	0.0380339
$616$	0	0
$617$	−18712.3	−1.22096	−0.610478	−	0.792033i	$-0.709023\pi$
$617$	−18712.3	−1.22096	−0.610478	+	0.792033i	$0.709023\pi$
$618$	0	0
$619$	−21238.9	−1.37910	−0.689549	−	0.724239i	$-0.742191\pi$
$619$	−21238.9	−1.37910	−0.689549	+	0.724239i	$0.742191\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	4829.05	0.310549
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−9787.64	−0.623414
$628$	0	0
$629$	−7216.89	−0.457482
$630$	0	0
$631$	2300.68	0.145149	0.0725743	−	0.997363i	$-0.476879\pi$
$631$	2300.68	0.145149	0.0725743	+	0.997363i	$0.476879\pi$
$632$	0	0
$633$	1590.08	0.0998421
$634$	0	0
$635$	5288.52	0.330502
$636$	0	0
$637$	−1607.19	−0.0999673
$638$	0	0
$639$	−7023.85	−0.434834
$640$	0	0
$641$	19451.2	1.19856	0.599279	−	0.800540i	$-0.295454\pi$
$641$	19451.2	1.19856	0.599279	+	0.800540i	$0.295454\pi$
$642$	0	0
$643$	−2860.61	−0.175446	−0.0877228	−	0.996145i	$-0.527959\pi$
$643$	−2860.61	−0.175446	−0.0877228	+	0.996145i	$0.527959\pi$
$644$	0	0
$645$	2122.86	0.129593
$646$	0	0
$647$	10522.9	0.639412	0.319706	−	0.947517i	$-0.396416\pi$
$647$	10522.9	0.639412	0.319706	+	0.947517i	$0.396416\pi$
$648$	0	0
$649$	29398.4	1.77810
$650$	0	0
$651$	−2704.60	−0.162829
$652$	0	0
$653$	22516.1	1.34935	0.674673	−	0.738117i	$-0.264285\pi$
$653$	22516.1	1.34935	0.674673	+	0.738117i	$0.264285\pi$
$654$	0	0
$655$	−9491.10	−0.566180
$656$	0	0
$657$	−1813.18	−0.107669
$658$	0	0
$659$	−24456.3	−1.44565	−0.722824	−	0.691032i	$-0.757156\pi$
$659$	−24456.3	−1.44565	−0.722824	+	0.691032i	$0.757156\pi$
$660$	0	0
$661$	23451.5	1.37997	0.689984	−	0.723825i	$-0.257618\pi$
$661$	23451.5	1.37997	0.689984	+	0.723825i	$0.257618\pi$
$662$	0	0
$663$	4796.89	0.280989
$664$	0	0
$665$	−8762.52	−0.510971
$666$	0	0
$667$	−2267.14	−0.131610
$668$	0	0
$669$	17694.8	1.02260
$670$	0	0
$671$	−12896.6	−0.741977
$672$	0	0
$673$	−32102.4	−1.83871	−0.919357	−	0.393423i	$-0.871291\pi$
$673$	−32102.4	−1.83871	−0.919357	+	0.393423i	$0.871291\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−12100.0	−0.686914	−0.343457	−	0.939168i	$-0.611598\pi$
$677$	−12100.0	−0.686914	−0.343457	+	0.939168i	$0.611598\pi$
$678$	0	0
$679$	−18859.8	−1.06594
$680$	0	0
$681$	7956.09	0.447692
$682$	0	0
$683$	11765.1	0.659121	0.329560	−	0.944134i	$-0.393099\pi$
$683$	11765.1	0.659121	0.329560	+	0.944134i	$0.393099\pi$
$684$	0	0
$685$	−9639.59	−0.537678
$686$	0	0
$687$	2177.26	0.120914
$688$	0	0
$689$	3924.53	0.216999
$690$	0	0
$691$	15847.1	0.872434	0.436217	−	0.899842i	$-0.356318\pi$
$691$	15847.1	0.872434	0.436217	+	0.899842i	$0.356318\pi$
$692$	0	0
$693$	7743.34	0.424452
$694$	0	0
$695$	−4990.78	−0.272390
$696$	0	0
$697$	4584.36	0.249132
$698$	0	0
$699$	−3268.89	−0.176882
$700$	0	0
$701$	32273.3	1.73887	0.869435	−	0.494048i	$-0.164483\pi$
$701$	32273.3	1.73887	0.869435	+	0.494048i	$0.164483\pi$
$702$	0	0
$703$	−4962.82	−0.266254
$704$	0	0
$705$	−3194.23	−0.170640
$706$	0	0
$707$	−2084.08	−0.110863
$708$	0	0
$709$	10737.1	0.568745	0.284372	−	0.958714i	$-0.408215\pi$
$709$	10737.1	0.568745	0.284372	+	0.958714i	$0.408215\pi$
$710$	0	0
$711$	54.3732	0.00286801
$712$	0	0
$713$	964.530	0.0506619
$714$	0	0
$715$	2699.07	0.141174
$716$	0	0
$717$	13456.6	0.700903
$718$	0	0
$719$	35010.8	1.81597	0.907986	−	0.419001i	$-0.137620\pi$
$719$	35010.8	1.81597	0.907986	+	0.419001i	$0.137620\pi$
$720$	0	0
$721$	−18089.3	−0.934369
$722$	0	0
$723$	−16054.0	−0.825800
$724$	0	0
$725$	−2464.28	−0.126236
$726$	0	0
$727$	8597.69	0.438612	0.219306	−	0.975656i	$-0.429621\pi$
$727$	8597.69	0.438612	0.219306	+	0.975656i	$0.429621\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	16777.1	0.848869
$732$	0	0
$733$	−10359.5	−0.522015	−0.261008	−	0.965337i	$-0.584055\pi$
$733$	−10359.5	−0.522015	−0.261008	+	0.965337i	$0.584055\pi$
$734$	0	0
$735$	1787.33	0.0896963
$736$	0	0
$737$	33344.8	1.66658
$738$	0	0
$739$	23336.3	1.16162	0.580811	−	0.814039i	$-0.302736\pi$
$739$	23336.3	1.16162	0.580811	+	0.814039i	$0.302736\pi$
$740$	0	0
$741$	3298.67	0.163535
$742$	0	0
$743$	7241.38	0.357551	0.178776	−	0.983890i	$-0.442786\pi$
$743$	7241.38	0.357551	0.178776	+	0.983890i	$0.442786\pi$
$744$	0	0
$745$	−15254.9	−0.750197
$746$	0	0
$747$	1756.80	0.0860480
$748$	0	0
$749$	20014.7	0.976397
$750$	0	0
$751$	−23882.1	−1.16041	−0.580207	−	0.814469i	$-0.697028\pi$
$751$	−23882.1	−1.16041	−0.580207	+	0.814469i	$0.697028\pi$
$752$	0	0
$753$	−837.922	−0.0405519
$754$	0	0
$755$	1888.20	0.0910178
$756$	0	0
$757$	4701.51	0.225732	0.112866	−	0.993610i	$-0.463997\pi$
$757$	4701.51	0.225732	0.112866	+	0.993610i	$0.463997\pi$
$758$	0	0
$759$	−2761.47	−0.132062
$760$	0	0
$761$	−16379.9	−0.780251	−0.390126	−	0.920762i	$-0.627568\pi$
$761$	−16379.9	−0.780251	−0.390126	+	0.920762i	$0.627568\pi$
$762$	0	0
$763$	−2608.15	−0.123750
$764$	0	0
$765$	−5334.56	−0.252119
$766$	0	0
$767$	−9907.98	−0.466436
$768$	0	0
$769$	7512.71	0.352296	0.176148	−	0.984364i	$-0.443636\pi$
$769$	7512.71	0.352296	0.176148	+	0.984364i	$0.443636\pi$
$770$	0	0
$771$	−10848.2	−0.506731
$772$	0	0
$773$	1729.18	0.0804582	0.0402291	−	0.999190i	$-0.487191\pi$
$773$	1729.18	0.0804582	0.0402291	+	0.999190i	$0.487191\pi$
$774$	0	0
$775$	1048.40	0.0485932
$776$	0	0
$777$	3926.26	0.181279
$778$	0	0
$779$	3152.52	0.144994
$780$	0	0
$781$	−31233.8	−1.43103
$782$	0	0
$783$	2661.42	0.121471
$784$	0	0
$785$	5163.01	0.234746
$786$	0	0
$787$	9897.44	0.448292	0.224146	−	0.974556i	$-0.428041\pi$
$787$	9897.44	0.448292	0.224146	+	0.974556i	$0.428041\pi$
$788$	0	0
$789$	−262.571	−0.0118476
$790$	0	0
$791$	−851.457	−0.0382735
$792$	0	0
$793$	4346.46	0.194637
$794$	0	0
$795$	−4364.41	−0.194704
$796$	0	0
$797$	−33454.4	−1.48685	−0.743423	−	0.668821i	$-0.766799\pi$
$797$	−33454.4	−1.48685	−0.743423	+	0.668821i	$0.766799\pi$
$798$	0	0
$799$	−25244.1	−1.11774
$800$	0	0
$801$	2021.67	0.0891788
$802$	0	0
$803$	−8062.88	−0.354337
$804$	0	0
$805$	−2472.25	−0.108243
$806$	0	0
$807$	427.915	0.0186658
$808$	0	0
$809$	−14860.0	−0.645797	−0.322898	−	0.946434i	$-0.604657\pi$
$809$	−14860.0	−0.645797	−0.322898	+	0.946434i	$0.604657\pi$
$810$	0	0
$811$	19378.0	0.839030	0.419515	−	0.907748i	$-0.362200\pi$
$811$	19378.0	0.839030	0.419515	+	0.907748i	$0.362200\pi$
$812$	0	0
$813$	−17122.0	−0.738618
$814$	0	0
$815$	−14044.5	−0.603631
$816$	0	0
$817$	11537.1	0.494040
$818$	0	0
$819$	−2609.69	−0.111343
$820$	0	0
$821$	−13859.4	−0.589156	−0.294578	−	0.955627i	$-0.595179\pi$
$821$	−13859.4	−0.589156	−0.294578	+	0.955627i	$0.595179\pi$
$822$	0	0
$823$	29132.2	1.23388	0.616941	−	0.787010i	$-0.288372\pi$
$823$	29132.2	1.23388	0.616941	+	0.787010i	$0.288372\pi$
$824$	0	0
$825$	−3001.60	−0.126670
$826$	0	0
$827$	4471.06	0.187998	0.0939988	−	0.995572i	$-0.470035\pi$
$827$	4471.06	0.187998	0.0939988	+	0.995572i	$0.470035\pi$
$828$	0	0
$829$	40745.6	1.70706	0.853530	−	0.521044i	$-0.174457\pi$
$829$	40745.6	1.70706	0.853530	+	0.521044i	$0.174457\pi$
$830$	0	0
$831$	−13982.6	−0.583695
$832$	0	0
$833$	14125.4	0.587534
$834$	0	0
$835$	8056.23	0.333889
$836$	0	0
$837$	−1132.27	−0.0467588
$838$	0	0
$839$	4213.85	0.173395	0.0866974	−	0.996235i	$-0.472369\pi$
$839$	4213.85	0.173395	0.0866974	+	0.996235i	$0.472369\pi$
$840$	0	0
$841$	−14672.7	−0.601613
$842$	0	0
$843$	1104.98	0.0451452
$844$	0	0
$845$	10075.3	0.410180
$846$	0	0
$847$	5819.66	0.236087
$848$	0	0
$849$	20725.2	0.837793
$850$	0	0
$851$	−1400.21	−0.0564024
$852$	0	0
$853$	22015.9	0.883717	0.441858	−	0.897085i	$-0.354319\pi$
$853$	22015.9	0.883717	0.441858	+	0.897085i	$0.354319\pi$
$854$	0	0
$855$	−3668.41	−0.146733
$856$	0	0
$857$	37921.0	1.51150	0.755750	−	0.654860i	$-0.227272\pi$
$857$	37921.0	1.51150	0.755750	+	0.654860i	$0.227272\pi$
$858$	0	0
$859$	3586.49	0.142456	0.0712279	−	0.997460i	$-0.477308\pi$
$859$	3586.49	0.142456	0.0712279	+	0.997460i	$0.477308\pi$
$860$	0	0
$861$	−2494.06	−0.0987195
$862$	0	0
$863$	−10055.7	−0.396642	−0.198321	−	0.980137i	$-0.563549\pi$
$863$	−10055.7	−0.396642	−0.198321	+	0.980137i	$0.563549\pi$
$864$	0	0
$865$	−9802.78	−0.385323
$866$	0	0
$867$	−27420.3	−1.07410
$868$	0	0
$869$	241.788	0.00943853
$870$	0	0
$871$	−11238.0	−0.437182
$872$	0	0
$873$	−7895.60	−0.306100
$874$	0	0
$875$	−2687.23	−0.103823
$876$	0	0
$877$	−24683.7	−0.950410	−0.475205	−	0.879875i	$-0.657626\pi$
$877$	−24683.7	−0.950410	−0.475205	+	0.879875i	$0.657626\pi$
$878$	0	0
$879$	16165.5	0.620308
$880$	0	0
$881$	−27654.5	−1.05755	−0.528776	−	0.848762i	$-0.677349\pi$
$881$	−27654.5	−1.05755	−0.528776	+	0.848762i	$0.677349\pi$
$882$	0	0
$883$	48590.5	1.85187	0.925934	−	0.377686i	$-0.123280\pi$
$883$	48590.5	1.85187	0.925934	+	0.377686i	$0.123280\pi$
$884$	0	0
$885$	11018.5	0.418513
$886$	0	0
$887$	−12129.2	−0.459144	−0.229572	−	0.973292i	$-0.573733\pi$
$887$	−12129.2	−0.459144	−0.229572	+	0.973292i	$0.573733\pi$
$888$	0	0
$889$	−22738.3	−0.857838
$890$	0	0
$891$	3241.73	0.121888
$892$	0	0
$893$	−17359.6	−0.650522
$894$	0	0
$895$	−19436.2	−0.725902
$896$	0	0
$897$	930.684	0.0346428
$898$	0	0
$899$	−4133.69	−0.153355
$900$	0	0
$901$	−34492.2	−1.27536
$902$	0	0
$903$	−9127.38	−0.336368
$904$	0	0
$905$	3101.62	0.113924
$906$	0	0
$907$	15203.3	0.556578	0.278289	−	0.960497i	$-0.410233\pi$
$907$	15203.3	0.556578	0.278289	+	0.960497i	$0.410233\pi$
$908$	0	0
$909$	−872.496	−0.0318359
$910$	0	0
$911$	2197.66	0.0799249	0.0399625	−	0.999201i	$-0.487276\pi$
$911$	2197.66	0.0799249	0.0399625	+	0.999201i	$0.487276\pi$
$912$	0	0
$913$	7812.15	0.283181
$914$	0	0
$915$	−4833.63	−0.174639
$916$	0	0
$917$	40807.6	1.46956
$918$	0	0
$919$	−26364.7	−0.946345	−0.473173	−	0.880970i	$-0.656891\pi$
$919$	−26364.7	−0.946345	−0.473173	+	0.880970i	$0.656891\pi$
$920$	0	0
$921$	−3927.70	−0.140524
$922$	0	0
$923$	10526.5	0.375390
$924$	0	0
$925$	−1521.96	−0.0540993
$926$	0	0
$927$	−7573.03	−0.268318
$928$	0	0
$929$	−9390.85	−0.331651	−0.165825	−	0.986155i	$-0.553029\pi$
$929$	−9390.85	−0.331651	−0.165825	+	0.986155i	$0.553029\pi$
$930$	0	0
$931$	9713.57	0.341944
$932$	0	0
$933$	8456.11	0.296721
$934$	0	0
$935$	−23721.8	−0.829718
$936$	0	0
$937$	5679.69	0.198023	0.0990115	−	0.995086i	$-0.468432\pi$
$937$	5679.69	0.198023	0.0990115	+	0.995086i	$0.468432\pi$
$938$	0	0
$939$	−13671.5	−0.475136
$940$	0	0
$941$	43504.6	1.50713	0.753565	−	0.657373i	$-0.228332\pi$
$941$	43504.6	1.50713	0.753565	+	0.657373i	$0.228332\pi$
$942$	0	0
$943$	889.447	0.0307151
$944$	0	0
$945$	2902.20	0.0999034
$946$	0	0
$947$	−44473.0	−1.52606	−0.763030	−	0.646364i	$-0.776289\pi$
$947$	−44473.0	−1.52606	−0.763030	+	0.646364i	$0.776289\pi$
$948$	0	0
$949$	2717.39	0.0929505
$950$	0	0
$951$	−15386.6	−0.524653
$952$	0	0
$953$	−54537.7	−1.85378	−0.926888	−	0.375339i	$-0.877526\pi$
$953$	−54537.7	−1.85378	−0.926888	+	0.375339i	$0.877526\pi$
$954$	0	0
$955$	−7639.62	−0.258861
$956$	0	0
$957$	11834.9	0.399756
$958$	0	0
$959$	41446.0	1.39558
$960$	0	0
$961$	−28032.4	−0.940968
$962$	0	0
$963$	8379.10	0.280387
$964$	0	0
$965$	−12067.3	−0.402550
$966$	0	0
$967$	−39815.6	−1.32408	−0.662040	−	0.749469i	$-0.730309\pi$
$967$	−39815.6	−1.32408	−0.662040	+	0.749469i	$0.730309\pi$
$968$	0	0
$969$	−28991.6	−0.961140
$970$	0	0
$971$	4449.25	0.147048	0.0735238	−	0.997293i	$-0.476576\pi$
$971$	4449.25	0.147048	0.0735238	+	0.997293i	$0.476576\pi$
$972$	0	0
$973$	21458.2	0.707007
$974$	0	0
$975$	1011.61	0.0332282
$976$	0	0
$977$	9845.49	0.322400	0.161200	−	0.986922i	$-0.448464\pi$
$977$	9845.49	0.322400	0.161200	+	0.986922i	$0.448464\pi$
$978$	0	0
$979$	8990.00	0.293485
$980$	0	0
$981$	−1091.89	−0.0355367
$982$	0	0
$983$	21712.3	0.704490	0.352245	−	0.935908i	$-0.385418\pi$
$983$	21712.3	0.704490	0.352245	+	0.935908i	$0.385418\pi$
$984$	0	0
$985$	−16506.0	−0.533934
$986$	0	0
$987$	13733.8	0.442908
$988$	0	0
$989$	3255.06	0.104656
$990$	0	0
$991$	−31010.4	−0.994024	−0.497012	−	0.867744i	$-0.665570\pi$
$991$	−31010.4	−0.994024	−0.497012	+	0.867744i	$0.665570\pi$
$992$	0	0
$993$	14078.0	0.449902
$994$	0	0
$995$	−8500.72	−0.270845
$996$	0	0
$997$	−27487.3	−0.873151	−0.436575	−	0.899668i	$-0.643809\pi$
$997$	−27487.3	−0.873151	−0.436575	+	0.899668i	$0.643809\pi$
$998$	0	0
$999$	1643.72	0.0520570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.e.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.e.1.5	✓	5	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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +21.4978 q^{7} +9.00000 q^{9} +40.0214 q^{11} -13.4882 q^{13} +15.0000 q^{15} +118.546 q^{17} +81.5201 q^{19} -64.4934 q^{21} +23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -98.5712 q^{29} +41.9361 q^{31} -120.064 q^{33} -107.489 q^{35} -60.8785 q^{37} +40.4645 q^{39} +38.6716 q^{41} +141.524 q^{43} -45.0000 q^{45} -212.948 q^{47} +119.156 q^{49} -355.637 q^{51} -290.961 q^{53} -200.107 q^{55} -244.560 q^{57} +734.568 q^{59} -322.242 q^{61} +193.480 q^{63} +67.4408 q^{65} +833.176 q^{67} -69.0000 q^{69} -780.428 q^{71} -201.464 q^{73} -75.0000 q^{75} +860.371 q^{77} +6.04147 q^{79} +81.0000 q^{81} +195.200 q^{83} -592.729 q^{85} +295.713 q^{87} +224.630 q^{89} -289.966 q^{91} -125.808 q^{93} -407.601 q^{95} -877.289 q^{97} +360.192 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} - 25 q^{5} - 7 q^{7} + 45 q^{9} + 64 q^{11} + 80 q^{13} + 75 q^{15} - 21 q^{17} - 52 q^{19} + 21 q^{21} + 115 q^{23} + 125 q^{25} - 135 q^{27} - 277 q^{29} - 289 q^{31} - 192 q^{33} + 35 q^{35}+ \cdots + 576 q^{99}+O(q^{100}) \) 5 * q - 15 * q^3 - 25 * q^5 - 7 * q^7 + 45 * q^9 + 64 * q^11 + 80 * q^13 + 75 * q^15 - 21 * q^17 - 52 * q^19 + 21 * q^21 + 115 * q^23 + 125 * q^25 - 135 * q^27 - 277 * q^29 - 289 * q^31 - 192 * q^33 + 35 * q^35 - 303 * q^37 - 240 * q^39 - 95 * q^41 - 372 * q^43 - 225 * q^45 - 462 * q^47 - 272 * q^49 + 63 * q^51 - 73 * q^53 - 320 * q^55 + 156 * q^57 + 529 * q^59 + 174 * q^61 - 63 * q^63 - 400 * q^65 - 919 * q^67 - 345 * q^69 - 141 * q^71 + 448 * q^73 - 375 * q^75 - 136 * q^77 - 496 * q^79 + 405 * q^81 + 1219 * q^83 + 105 * q^85 + 831 * q^87 + 1492 * q^89 + 1560 * q^91 + 867 * q^93 + 260 * q^95 + 574 * q^97 + 576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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