LMFDB - Embedded newform 1380.4.a.e.1.3

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			1.3
Root			$15.0456$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} +5.33445 q^{7} +9.00000 q^{9} -19.5786 q^{11} +72.6950 q^{13} +15.0000 q^{15} +99.8318 q^{17} -59.0008 q^{19} -16.0034 q^{21} +23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -54.3168 q^{29} -291.559 q^{31} +58.7357 q^{33} -26.6723 q^{35} +419.042 q^{37} -218.085 q^{39} -307.385 q^{41} -328.152 q^{43} -45.0000 q^{45} +234.235 q^{47} -314.544 q^{49} -299.495 q^{51} +289.308 q^{53} +97.8928 q^{55} +177.003 q^{57} +31.8122 q^{59} +289.303 q^{61} +48.0101 q^{63} -363.475 q^{65} -912.347 q^{67} -69.0000 q^{69} +997.663 q^{71} +293.255 q^{73} -75.0000 q^{75} -104.441 q^{77} +170.804 q^{79} +81.0000 q^{81} +576.631 q^{83} -499.159 q^{85} +162.951 q^{87} +270.177 q^{89} +387.788 q^{91} +874.677 q^{93} +295.004 q^{95} +76.5491 q^{97} -176.207 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	5.33445	0.288033	0.144017	−	0.989575i	$-0.453998\pi$
$7$	5.33445	0.288033	0.144017	+	0.989575i	$0.453998\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−19.5786	−0.536651	−0.268325	−	0.963328i	$-0.586470\pi$
$11$	−19.5786	−0.536651	−0.268325	+	0.963328i	$0.586470\pi$
$12$	0	0
$13$	72.6950	1.55092	0.775460	−	0.631397i	$-0.217518\pi$
$13$	72.6950	1.55092	0.775460	+	0.631397i	$0.217518\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	99.8318	1.42428	0.712140	−	0.702037i	$-0.247726\pi$
$17$	99.8318	1.42428	0.712140	+	0.702037i	$0.247726\pi$
$18$	0	0
$19$	−59.0008	−0.712406	−0.356203	−	0.934409i	$-0.615929\pi$
$19$	−59.0008	−0.712406	−0.356203	+	0.934409i	$0.615929\pi$
$20$	0	0
$21$	−16.0034	−0.166296
$22$	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
$29$	−54.3168	−0.347806	−0.173903	−	0.984763i	$-0.555638\pi$
$29$	−54.3168	−0.347806	−0.173903	+	0.984763i	$0.555638\pi$
$30$	0	0
$31$	−291.559	−1.68921	−0.844605	−	0.535390i	$-0.820165\pi$
$31$	−291.559	−1.68921	−0.844605	+	0.535390i	$0.820165\pi$
$32$	0	0
$33$	58.7357	0.309835
$34$	0	0
$35$	−26.6723	−0.128812
$36$	0	0
$37$	419.042	1.86189	0.930947	−	0.365155i	$-0.118984\pi$
$37$	419.042	1.86189	0.930947	+	0.365155i	$0.118984\pi$
$38$	0	0
$39$	−218.085	−0.895424
$40$	0	0
$41$	−307.385	−1.17087	−0.585434	−	0.810720i	$-0.699076\pi$
$41$	−307.385	−1.17087	−0.585434	+	0.810720i	$0.699076\pi$
$42$	0	0
$43$	−328.152	−1.16378	−0.581891	−	0.813266i	$-0.697687\pi$
$43$	−328.152	−1.16378	−0.581891	+	0.813266i	$0.697687\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	234.235	0.726951	0.363475	−	0.931604i	$-0.381590\pi$
$47$	234.235	0.726951	0.363475	+	0.931604i	$0.381590\pi$
$48$	0	0
$49$	−314.544	−0.917037
$50$	0	0
$51$	−299.495	−0.822309
$52$	0	0
$53$	289.308	0.749802	0.374901	−	0.927065i	$-0.377677\pi$
$53$	289.308	0.749802	0.374901	+	0.927065i	$0.377677\pi$
$54$	0	0
$55$	97.8928	0.239998
$56$	0	0
$57$	177.003	0.411308
$58$	0	0
$59$	31.8122	0.0701965	0.0350982	−	0.999384i	$-0.488826\pi$
$59$	31.8122	0.0701965	0.0350982	+	0.999384i	$0.488826\pi$
$60$	0	0
$61$	289.303	0.607236	0.303618	−	0.952794i	$-0.401805\pi$
$61$	289.303	0.607236	0.303618	+	0.952794i	$0.401805\pi$
$62$	0	0
$63$	48.0101	0.0960111
$64$	0	0
$65$	−363.475	−0.693592
$66$	0	0
$67$	−912.347	−1.66360	−0.831798	−	0.555079i	$-0.812688\pi$
$67$	−912.347	−1.66360	−0.831798	+	0.555079i	$0.812688\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	997.663	1.66762	0.833808	−	0.552054i	$-0.186156\pi$
$71$	997.663	1.66762	0.833808	+	0.552054i	$0.186156\pi$
$72$	0	0
$73$	293.255	0.470177	0.235088	−	0.971974i	$-0.424462\pi$
$73$	293.255	0.470177	0.235088	+	0.971974i	$0.424462\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−104.441	−0.154573
$78$	0	0
$79$	170.804	0.243252	0.121626	−	0.992576i	$-0.461189\pi$
$79$	170.804	0.243252	0.121626	+	0.992576i	$0.461189\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	576.631	0.762572	0.381286	−	0.924457i	$-0.375481\pi$
$83$	576.631	0.762572	0.381286	+	0.924457i	$0.375481\pi$
$84$	0	0
$85$	−499.159	−0.636958
$86$	0	0
$87$	162.951	0.200806
$88$	0	0
$89$	270.177	0.321783	0.160891	−	0.986972i	$-0.448563\pi$
$89$	270.177	0.321783	0.160891	+	0.986972i	$0.448563\pi$
$90$	0	0
$91$	387.788	0.446716
$92$	0	0
$93$	874.677	0.975266
$94$	0	0
$95$	295.004	0.318598
$96$	0	0
$97$	76.5491	0.0801276	0.0400638	−	0.999197i	$-0.487244\pi$
$97$	76.5491	0.0801276	0.0400638	+	0.999197i	$0.487244\pi$
$98$	0	0
$99$	−176.207	−0.178884
$100$	0	0
$101$	1376.73	1.35634	0.678168	−	0.734907i	$-0.262774\pi$
$101$	1376.73	1.35634	0.678168	+	0.734907i	$0.262774\pi$
$102$	0	0
$103$	−892.405	−0.853702	−0.426851	−	0.904322i	$-0.640377\pi$
$103$	−892.405	−0.853702	−0.426851	+	0.904322i	$0.640377\pi$
$104$	0	0
$105$	80.0168	0.0743699
$106$	0	0
$107$	96.5547	0.0872364	0.0436182	−	0.999048i	$-0.486111\pi$
$107$	96.5547	0.0872364	0.0436182	+	0.999048i	$0.486111\pi$
$108$	0	0
$109$	−181.069	−0.159113	−0.0795563	−	0.996830i	$-0.525350\pi$
$109$	−181.069	−0.159113	−0.0795563	+	0.996830i	$0.525350\pi$
$110$	0	0
$111$	−1257.13	−1.07496
$112$	0	0
$113$	−608.456	−0.506538	−0.253269	−	0.967396i	$-0.581506\pi$
$113$	−608.456	−0.506538	−0.253269	+	0.967396i	$0.581506\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	654.255	0.516973
$118$	0	0
$119$	532.548	0.410240
$120$	0	0
$121$	−947.680	−0.712006
$122$	0	0
$123$	922.156	0.676000
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	524.494	0.366467	0.183234	−	0.983069i	$-0.441344\pi$
$127$	524.494	0.366467	0.183234	+	0.983069i	$0.441344\pi$
$128$	0	0
$129$	984.455	0.671910
$130$	0	0
$131$	−413.338	−0.275676	−0.137838	−	0.990455i	$-0.544015\pi$
$131$	−413.338	−0.275676	−0.137838	+	0.990455i	$0.544015\pi$
$132$	0	0
$133$	−314.737	−0.205197
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	2055.63	1.28193	0.640964	−	0.767571i	$-0.278535\pi$
$137$	2055.63	1.28193	0.640964	+	0.767571i	$0.278535\pi$
$138$	0	0
$139$	1692.64	1.03286	0.516431	−	0.856329i	$-0.327260\pi$
$139$	1692.64	1.03286	0.516431	+	0.856329i	$0.327260\pi$
$140$	0	0
$141$	−702.705	−0.419705
$142$	0	0
$143$	−1423.26	−0.832302
$144$	0	0
$145$	271.584	0.155544
$146$	0	0
$147$	943.631	0.529451
$148$	0	0
$149$	−1184.43	−0.651225	−0.325612	−	0.945503i	$-0.605570\pi$
$149$	−1184.43	−0.651225	−0.325612	+	0.945503i	$0.605570\pi$
$150$	0	0
$151$	2879.25	1.55172	0.775860	−	0.630905i	$-0.217316\pi$
$151$	2879.25	1.55172	0.775860	+	0.630905i	$0.217316\pi$
$152$	0	0
$153$	898.486	0.474760
$154$	0	0
$155$	1457.79	0.755438
$156$	0	0
$157$	219.950	0.111808	0.0559042	−	0.998436i	$-0.482196\pi$
$157$	219.950	0.111808	0.0559042	+	0.998436i	$0.482196\pi$
$158$	0	0
$159$	−867.924	−0.432898
$160$	0	0
$161$	122.692	0.0600591
$162$	0	0
$163$	968.636	0.465457	0.232728	−	0.972542i	$-0.425235\pi$
$163$	968.636	0.465457	0.232728	+	0.972542i	$0.425235\pi$
$164$	0	0
$165$	−293.678	−0.138563
$166$	0	0
$167$	1402.89	0.650052	0.325026	−	0.945705i	$-0.394627\pi$
$167$	1402.89	0.650052	0.325026	+	0.945705i	$0.394627\pi$
$168$	0	0
$169$	3087.56	1.40535
$170$	0	0
$171$	−531.008	−0.237469
$172$	0	0
$173$	201.539	0.0885707	0.0442854	−	0.999019i	$-0.485899\pi$
$173$	201.539	0.0885707	0.0442854	+	0.999019i	$0.485899\pi$
$174$	0	0
$175$	133.361	0.0576067
$176$	0	0
$177$	−95.4365	−0.0405280
$178$	0	0
$179$	2685.44	1.12134	0.560668	−	0.828041i	$-0.310545\pi$
$179$	2685.44	1.12134	0.560668	+	0.828041i	$0.310545\pi$
$180$	0	0
$181$	332.053	0.136361	0.0681804	−	0.997673i	$-0.478281\pi$
$181$	332.053	0.136361	0.0681804	+	0.997673i	$0.478281\pi$
$182$	0	0
$183$	−867.908	−0.350588
$184$	0	0
$185$	−2095.21	−0.832664
$186$	0	0
$187$	−1954.56	−0.764341
$188$	0	0
$189$	−144.030	−0.0554320
$190$	0	0
$191$	−726.225	−0.275119	−0.137560	−	0.990493i	$-0.543926\pi$
$191$	−726.225	−0.275119	−0.137560	+	0.990493i	$0.543926\pi$
$192$	0	0
$193$	904.503	0.337345	0.168672	−	0.985672i	$-0.446052\pi$
$193$	904.503	0.337345	0.168672	+	0.985672i	$0.446052\pi$
$194$	0	0
$195$	1090.42	0.400446
$196$	0	0
$197$	940.195	0.340031	0.170016	−	0.985441i	$-0.445618\pi$
$197$	940.195	0.340031	0.170016	+	0.985441i	$0.445618\pi$
$198$	0	0
$199$	2767.34	0.985788	0.492894	−	0.870089i	$-0.335939\pi$
$199$	2767.34	0.985788	0.492894	+	0.870089i	$0.335939\pi$
$200$	0	0
$201$	2737.04	0.960477
$202$	0	0
$203$	−289.751	−0.100180
$204$	0	0
$205$	1536.93	0.523628
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	1155.15	0.382313
$210$	0	0
$211$	4203.62	1.37151	0.685756	−	0.727831i	$-0.259472\pi$
$211$	4203.62	1.37151	0.685756	+	0.727831i	$0.259472\pi$
$212$	0	0
$213$	−2992.99	−0.962799
$214$	0	0
$215$	1640.76	0.520460
$216$	0	0
$217$	−1555.31	−0.486549
$218$	0	0
$219$	−879.765	−0.271457
$220$	0	0
$221$	7257.27	2.20894
$222$	0	0
$223$	3359.72	1.00889	0.504447	−	0.863443i	$-0.331696\pi$
$223$	3359.72	1.00889	0.504447	+	0.863443i	$0.331696\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3051.50	0.892227	0.446113	−	0.894976i	$-0.352808\pi$
$227$	3051.50	0.892227	0.446113	+	0.894976i	$0.352808\pi$
$228$	0	0
$229$	4440.20	1.28129	0.640647	−	0.767835i	$-0.278666\pi$
$229$	4440.20	1.28129	0.640647	+	0.767835i	$0.278666\pi$
$230$	0	0
$231$	313.323	0.0892429
$232$	0	0
$233$	−198.561	−0.0558291	−0.0279146	−	0.999610i	$-0.508887\pi$
$233$	−198.561	−0.0558291	−0.0279146	+	0.999610i	$0.508887\pi$
$234$	0	0
$235$	−1171.17	−0.325102
$236$	0	0
$237$	−512.411	−0.140442
$238$	0	0
$239$	1867.03	0.505306	0.252653	−	0.967557i	$-0.418697\pi$
$239$	1867.03	0.505306	0.252653	+	0.967557i	$0.418697\pi$
$240$	0	0
$241$	−5651.30	−1.51051	−0.755253	−	0.655433i	$-0.772486\pi$
$241$	−5651.30	−1.51051	−0.755253	+	0.655433i	$0.772486\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1572.72	0.410111
$246$	0	0
$247$	−4289.06	−1.10489
$248$	0	0
$249$	−1729.89	−0.440271
$250$	0	0
$251$	3032.60	0.762614	0.381307	−	0.924449i	$-0.375474\pi$
$251$	3032.60	0.762614	0.381307	+	0.924449i	$0.375474\pi$
$252$	0	0
$253$	−450.307	−0.111899
$254$	0	0
$255$	1497.48	0.367748
$256$	0	0
$257$	−3935.36	−0.955178	−0.477589	−	0.878583i	$-0.658489\pi$
$257$	−3935.36	−0.955178	−0.477589	+	0.878583i	$0.658489\pi$
$258$	0	0
$259$	2235.36	0.536287
$260$	0	0
$261$	−488.852	−0.115935
$262$	0	0
$263$	−4481.85	−1.05081	−0.525404	−	0.850853i	$-0.676086\pi$
$263$	−4481.85	−1.05081	−0.525404	+	0.850853i	$0.676086\pi$
$264$	0	0
$265$	−1446.54	−0.335322
$266$	0	0
$267$	−810.530	−0.185781
$268$	0	0
$269$	5375.20	1.21833	0.609167	−	0.793042i	$-0.291504\pi$
$269$	5375.20	1.21833	0.609167	+	0.793042i	$0.291504\pi$
$270$	0	0
$271$	−8412.42	−1.88568	−0.942838	−	0.333252i	$-0.891854\pi$
$271$	−8412.42	−1.88568	−0.942838	+	0.333252i	$0.891854\pi$
$272$	0	0
$273$	−1163.36	−0.257912
$274$	0	0
$275$	−489.464	−0.107330
$276$	0	0
$277$	−4419.84	−0.958708	−0.479354	−	0.877622i	$-0.659129\pi$
$277$	−4419.84	−0.958708	−0.479354	+	0.877622i	$0.659129\pi$
$278$	0	0
$279$	−2624.03	−0.563070
$280$	0	0
$281$	−582.531	−0.123669	−0.0618343	−	0.998086i	$-0.519695\pi$
$281$	−582.531	−0.123669	−0.0618343	+	0.998086i	$0.519695\pi$
$282$	0	0
$283$	5965.41	1.25303	0.626514	−	0.779410i	$-0.284481\pi$
$283$	5965.41	1.25303	0.626514	+	0.779410i	$0.284481\pi$
$284$	0	0
$285$	−885.013	−0.183943
$286$	0	0
$287$	−1639.73	−0.337249
$288$	0	0
$289$	5053.39	1.02858
$290$	0	0
$291$	−229.647	−0.0462617
$292$	0	0
$293$	6917.24	1.37921	0.689607	−	0.724184i	$-0.257783\pi$
$293$	6917.24	1.37921	0.689607	+	0.724184i	$0.257783\pi$
$294$	0	0
$295$	−159.061	−0.0313928
$296$	0	0
$297$	528.621	0.103278
$298$	0	0
$299$	1671.98	0.323389
$300$	0	0
$301$	−1750.51	−0.335208
$302$	0	0
$303$	−4130.19	−0.783081
$304$	0	0
$305$	−1446.51	−0.271564
$306$	0	0
$307$	−4976.19	−0.925102	−0.462551	−	0.886593i	$-0.653066\pi$
$307$	−4976.19	−0.925102	−0.462551	+	0.886593i	$0.653066\pi$
$308$	0	0
$309$	2677.21	0.492885
$310$	0	0
$311$	−1128.66	−0.205789	−0.102894	−	0.994692i	$-0.532810\pi$
$311$	−1128.66	−0.205789	−0.102894	+	0.994692i	$0.532810\pi$
$312$	0	0
$313$	−765.544	−0.138246	−0.0691232	−	0.997608i	$-0.522020\pi$
$313$	−765.544	−0.138246	−0.0691232	+	0.997608i	$0.522020\pi$
$314$	0	0
$315$	−240.050	−0.0429375
$316$	0	0
$317$	9796.20	1.73568	0.867838	−	0.496847i	$-0.165509\pi$
$317$	9796.20	1.73568	0.867838	+	0.496847i	$0.165509\pi$
$318$	0	0
$319$	1063.45	0.186651
$320$	0	0
$321$	−289.664	−0.0503660
$322$	0	0
$323$	−5890.16	−1.01467
$324$	0	0
$325$	1817.37	0.310184
$326$	0	0
$327$	543.207	0.0918637
$328$	0	0
$329$	1249.51	0.209386
$330$	0	0
$331$	−5911.74	−0.981688	−0.490844	−	0.871248i	$-0.663311\pi$
$331$	−5911.74	−0.981688	−0.490844	+	0.871248i	$0.663311\pi$
$332$	0	0
$333$	3771.38	0.620631
$334$	0	0
$335$	4561.73	0.743982
$336$	0	0
$337$	11629.1	1.87976	0.939881	−	0.341502i	$-0.110935\pi$
$337$	11629.1	1.87976	0.939881	+	0.341502i	$0.110935\pi$
$338$	0	0
$339$	1825.37	0.292450
$340$	0	0
$341$	5708.30	0.906516
$342$	0	0
$343$	−3507.63	−0.552170
$344$	0	0
$345$	345.000	0.0538382
$346$	0	0
$347$	−3367.59	−0.520985	−0.260493	−	0.965476i	$-0.583885\pi$
$347$	−3367.59	−0.520985	−0.260493	+	0.965476i	$0.583885\pi$
$348$	0	0
$349$	11221.5	1.72113	0.860566	−	0.509339i	$-0.170110\pi$
$349$	11221.5	1.72113	0.860566	+	0.509339i	$0.170110\pi$
$350$	0	0
$351$	−1962.76	−0.298475
$352$	0	0
$353$	5924.29	0.893253	0.446627	−	0.894720i	$-0.352625\pi$
$353$	5924.29	0.893253	0.446627	+	0.894720i	$0.352625\pi$
$354$	0	0
$355$	−4988.31	−0.745781
$356$	0	0
$357$	−1597.64	−0.236852
$358$	0	0
$359$	2022.52	0.297339	0.148670	−	0.988887i	$-0.452501\pi$
$359$	2022.52	0.297339	0.148670	+	0.988887i	$0.452501\pi$
$360$	0	0
$361$	−3377.90	−0.492477
$362$	0	0
$363$	2843.04	0.411077
$364$	0	0
$365$	−1466.27	−0.210269
$366$	0	0
$367$	−3099.27	−0.440819	−0.220410	−	0.975407i	$-0.570739\pi$
$367$	−3099.27	−0.440819	−0.220410	+	0.975407i	$0.570739\pi$
$368$	0	0
$369$	−2766.47	−0.390289
$370$	0	0
$371$	1543.30	0.215968
$372$	0	0
$373$	7238.01	1.00474	0.502372	−	0.864651i	$-0.332461\pi$
$373$	7238.01	1.00474	0.502372	+	0.864651i	$0.332461\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−3948.56	−0.539420
$378$	0	0
$379$	−3872.61	−0.524862	−0.262431	−	0.964951i	$-0.584524\pi$
$379$	−3872.61	−0.524862	−0.262431	+	0.964951i	$0.584524\pi$
$380$	0	0
$381$	−1573.48	−0.211580
$382$	0	0
$383$	8139.76	1.08596	0.542980	−	0.839746i	$-0.317296\pi$
$383$	8139.76	1.08596	0.542980	+	0.839746i	$0.317296\pi$
$384$	0	0
$385$	522.204	0.0691273
$386$	0	0
$387$	−2953.37	−0.387928
$388$	0	0
$389$	10811.7	1.40919	0.704597	−	0.709607i	$-0.251128\pi$
$389$	10811.7	1.40919	0.704597	+	0.709607i	$0.251128\pi$
$390$	0	0
$391$	2296.13	0.296983
$392$	0	0
$393$	1240.01	0.159161
$394$	0	0
$395$	−854.018	−0.108786
$396$	0	0
$397$	−7220.25	−0.912780	−0.456390	−	0.889780i	$-0.650858\pi$
$397$	−7220.25	−0.912780	−0.456390	+	0.889780i	$0.650858\pi$
$398$	0	0
$399$	944.211	0.118470
$400$	0	0
$401$	−9300.00	−1.15815	−0.579077	−	0.815273i	$-0.696587\pi$
$401$	−9300.00	−1.15815	−0.579077	+	0.815273i	$0.696587\pi$
$402$	0	0
$403$	−21194.9	−2.61983
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−8204.24	−0.999187
$408$	0	0
$409$	−356.093	−0.0430506	−0.0215253	−	0.999768i	$-0.506852\pi$
$409$	−356.093	−0.0430506	−0.0215253	+	0.999768i	$0.506852\pi$
$410$	0	0
$411$	−6166.88	−0.740121
$412$	0	0
$413$	169.700	0.0202189
$414$	0	0
$415$	−2883.15	−0.341033
$416$	0	0
$417$	−5077.91	−0.596323
$418$	0	0
$419$	−1550.06	−0.180729	−0.0903646	−	0.995909i	$-0.528803\pi$
$419$	−1550.06	−0.180729	−0.0903646	+	0.995909i	$0.528803\pi$
$420$	0	0
$421$	−10184.0	−1.17895	−0.589474	−	0.807787i	$-0.700665\pi$
$421$	−10184.0	−1.17895	−0.589474	+	0.807787i	$0.700665\pi$
$422$	0	0
$423$	2108.11	0.242317
$424$	0	0
$425$	2495.80	0.284856
$426$	0	0
$427$	1543.27	0.174904
$428$	0	0
$429$	4269.79	0.480530
$430$	0	0
$431$	12913.8	1.44324	0.721622	−	0.692288i	$-0.243397\pi$
$431$	12913.8	1.44324	0.721622	+	0.692288i	$0.243397\pi$
$432$	0	0
$433$	−542.131	−0.0601689	−0.0300845	−	0.999547i	$-0.509578\pi$
$433$	−542.131	−0.0601689	−0.0300845	+	0.999547i	$0.509578\pi$
$434$	0	0
$435$	−814.753	−0.0898032
$436$	0	0
$437$	−1357.02	−0.148547
$438$	0	0
$439$	−3958.04	−0.430312	−0.215156	−	0.976580i	$-0.569026\pi$
$439$	−3958.04	−0.430312	−0.215156	+	0.976580i	$0.569026\pi$
$440$	0	0
$441$	−2830.89	−0.305679
$442$	0	0
$443$	12253.7	1.31420	0.657101	−	0.753802i	$-0.271782\pi$
$443$	12253.7	1.31420	0.657101	+	0.753802i	$0.271782\pi$
$444$	0	0
$445$	−1350.88	−0.143906
$446$	0	0
$447$	3553.30	0.375985
$448$	0	0
$449$	−9086.50	−0.955052	−0.477526	−	0.878618i	$-0.658466\pi$
$449$	−9086.50	−0.955052	−0.477526	+	0.878618i	$0.658466\pi$
$450$	0	0
$451$	6018.17	0.628347
$452$	0	0
$453$	−8637.74	−0.895886
$454$	0	0
$455$	−1938.94	−0.199778
$456$	0	0
$457$	13643.8	1.39656	0.698282	−	0.715823i	$-0.253948\pi$
$457$	13643.8	1.39656	0.698282	+	0.715823i	$0.253948\pi$
$458$	0	0
$459$	−2695.46	−0.274103
$460$	0	0
$461$	6709.30	0.677837	0.338919	−	0.940816i	$-0.389939\pi$
$461$	6709.30	0.677837	0.338919	+	0.940816i	$0.389939\pi$
$462$	0	0
$463$	1467.75	0.147326	0.0736630	−	0.997283i	$-0.476531\pi$
$463$	1467.75	0.147326	0.0736630	+	0.997283i	$0.476531\pi$
$464$	0	0
$465$	−4373.38	−0.436152
$466$	0	0
$467$	−8630.84	−0.855219	−0.427610	−	0.903963i	$-0.640644\pi$
$467$	−8630.84	−0.855219	−0.427610	+	0.903963i	$0.640644\pi$
$468$	0	0
$469$	−4866.87	−0.479171
$470$	0	0
$471$	−659.850	−0.0645526
$472$	0	0
$473$	6424.74	0.624545
$474$	0	0
$475$	−1475.02	−0.142481
$476$	0	0
$477$	2603.77	0.249934
$478$	0	0
$479$	−8399.32	−0.801200	−0.400600	−	0.916253i	$-0.631198\pi$
$479$	−8399.32	−0.801200	−0.400600	+	0.916253i	$0.631198\pi$
$480$	0	0
$481$	30462.2	2.88765
$482$	0	0
$483$	−368.077	−0.0346751
$484$	0	0
$485$	−382.745	−0.0358342
$486$	0	0
$487$	8900.47	0.828170	0.414085	−	0.910238i	$-0.364102\pi$
$487$	8900.47	0.828170	0.414085	+	0.910238i	$0.364102\pi$
$488$	0	0
$489$	−2905.91	−0.268732
$490$	0	0
$491$	−12601.9	−1.15828	−0.579140	−	0.815228i	$-0.696611\pi$
$491$	−12601.9	−1.15828	−0.579140	+	0.815228i	$0.696611\pi$
$492$	0	0
$493$	−5422.55	−0.495374
$494$	0	0
$495$	881.035	0.0799992
$496$	0	0
$497$	5321.98	0.480329
$498$	0	0
$499$	−12056.8	−1.08164	−0.540819	−	0.841139i	$-0.681886\pi$
$499$	−12056.8	−1.08164	−0.540819	+	0.841139i	$0.681886\pi$
$500$	0	0
$501$	−4208.66	−0.375307
$502$	0	0
$503$	−696.717	−0.0617596	−0.0308798	−	0.999523i	$-0.509831\pi$
$503$	−696.717	−0.0617596	−0.0308798	+	0.999523i	$0.509831\pi$
$504$	0	0
$505$	−6883.66	−0.606572
$506$	0	0
$507$	−9262.67	−0.811380
$508$	0	0
$509$	−10466.9	−0.911466	−0.455733	−	0.890117i	$-0.650623\pi$
$509$	−10466.9	−0.911466	−0.455733	+	0.890117i	$0.650623\pi$
$510$	0	0
$511$	1564.35	0.135427
$512$	0	0
$513$	1593.02	0.137103
$514$	0	0
$515$	4462.02	0.381787
$516$	0	0
$517$	−4585.98	−0.390119
$518$	0	0
$519$	−604.617	−0.0511363
$520$	0	0
$521$	−20482.4	−1.72236	−0.861180	−	0.508300i	$-0.830274\pi$
$521$	−20482.4	−1.72236	−0.861180	+	0.508300i	$0.830274\pi$
$522$	0	0
$523$	−3297.36	−0.275685	−0.137843	−	0.990454i	$-0.544017\pi$
$523$	−3297.36	−0.275685	−0.137843	+	0.990454i	$0.544017\pi$
$524$	0	0
$525$	−400.084	−0.0332592
$526$	0	0
$527$	−29106.9	−2.40591
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	286.310	0.0233988
$532$	0	0
$533$	−22345.4	−1.81592
$534$	0	0
$535$	−482.774	−0.0390133
$536$	0	0
$537$	−8056.31	−0.647403
$538$	0	0
$539$	6158.31	0.492129
$540$	0	0
$541$	6718.73	0.533939	0.266969	−	0.963705i	$-0.413978\pi$
$541$	6718.73	0.533939	0.266969	+	0.963705i	$0.413978\pi$
$542$	0	0
$543$	−996.159	−0.0787280
$544$	0	0
$545$	905.345	0.0711573
$546$	0	0
$547$	−14405.5	−1.12602	−0.563010	−	0.826450i	$-0.690357\pi$
$547$	−14405.5	−1.12602	−0.563010	+	0.826450i	$0.690357\pi$
$548$	0	0
$549$	2603.72	0.202412
$550$	0	0
$551$	3204.74	0.247779
$552$	0	0
$553$	911.144	0.0700647
$554$	0	0
$555$	6285.63	0.480739
$556$	0	0
$557$	711.822	0.0541488	0.0270744	−	0.999633i	$-0.491381\pi$
$557$	711.822	0.0541488	0.0270744	+	0.999633i	$0.491381\pi$
$558$	0	0
$559$	−23855.0	−1.80493
$560$	0	0
$561$	5863.69	0.441293
$562$	0	0
$563$	12247.6	0.916833	0.458416	−	0.888738i	$-0.348417\pi$
$563$	12247.6	0.916833	0.458416	+	0.888738i	$0.348417\pi$
$564$	0	0
$565$	3042.28	0.226531
$566$	0	0
$567$	432.091	0.0320037
$568$	0	0
$569$	15872.1	1.16940	0.584702	−	0.811248i	$-0.301211\pi$
$569$	15872.1	1.16940	0.584702	+	0.811248i	$0.301211\pi$
$570$	0	0
$571$	7197.30	0.527491	0.263746	−	0.964592i	$-0.415042\pi$
$571$	7197.30	0.527491	0.263746	+	0.964592i	$0.415042\pi$
$572$	0	0
$573$	2178.68	0.158840
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−13663.2	−0.985802	−0.492901	−	0.870085i	$-0.664064\pi$
$577$	−13663.2	−0.985802	−0.492901	+	0.870085i	$0.664064\pi$
$578$	0	0
$579$	−2713.51	−0.194766
$580$	0	0
$581$	3076.01	0.219646
$582$	0	0
$583$	−5664.23	−0.402382
$584$	0	0
$585$	−3271.27	−0.231197
$586$	0	0
$587$	2190.97	0.154056	0.0770280	−	0.997029i	$-0.475457\pi$
$587$	2190.97	0.154056	0.0770280	+	0.997029i	$0.475457\pi$
$588$	0	0
$589$	17202.2	1.20340
$590$	0	0
$591$	−2820.58	−0.196317
$592$	0	0
$593$	4052.46	0.280631	0.140316	−	0.990107i	$-0.455188\pi$
$593$	4052.46	0.280631	0.140316	+	0.990107i	$0.455188\pi$
$594$	0	0
$595$	−2662.74	−0.183465
$596$	0	0
$597$	−8302.03	−0.569145
$598$	0	0
$599$	12186.7	0.831277	0.415638	−	0.909530i	$-0.363558\pi$
$599$	12186.7	0.831277	0.415638	+	0.909530i	$0.363558\pi$
$600$	0	0
$601$	2704.86	0.183583	0.0917915	−	0.995778i	$-0.470741\pi$
$601$	2704.86	0.183583	0.0917915	+	0.995778i	$0.470741\pi$
$602$	0	0
$603$	−8211.12	−0.554532
$604$	0	0
$605$	4738.40	0.318419
$606$	0	0
$607$	−19272.0	−1.28868	−0.644338	−	0.764741i	$-0.722867\pi$
$607$	−19272.0	−1.28868	−0.644338	+	0.764741i	$0.722867\pi$
$608$	0	0
$609$	869.252	0.0578388
$610$	0	0
$611$	17027.7	1.12744
$612$	0	0
$613$	−16815.8	−1.10797	−0.553985	−	0.832527i	$-0.686893\pi$
$613$	−16815.8	−1.10797	−0.553985	+	0.832527i	$0.686893\pi$
$614$	0	0
$615$	−4610.78	−0.302317
$616$	0	0
$617$	9291.47	0.606257	0.303128	−	0.952950i	$-0.401969\pi$
$617$	9291.47	0.606257	0.303128	+	0.952950i	$0.401969\pi$
$618$	0	0
$619$	−8105.13	−0.526289	−0.263144	−	0.964756i	$-0.584760\pi$
$619$	−8105.13	−0.526289	−0.263144	+	0.964756i	$0.584760\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	1441.24	0.0926842
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−3465.45	−0.220729
$628$	0	0
$629$	41833.7	2.65186
$630$	0	0
$631$	−9568.39	−0.603663	−0.301832	−	0.953361i	$-0.597598\pi$
$631$	−9568.39	−0.603663	−0.301832	+	0.953361i	$0.597598\pi$
$632$	0	0
$633$	−12610.9	−0.791843
$634$	0	0
$635$	−2622.47	−0.163889
$636$	0	0
$637$	−22865.7	−1.42225
$638$	0	0
$639$	8978.96	0.555872
$640$	0	0
$641$	12140.3	0.748073	0.374036	−	0.927414i	$-0.377974\pi$
$641$	12140.3	0.748073	0.374036	+	0.927414i	$0.377974\pi$
$642$	0	0
$643$	−831.517	−0.0509982	−0.0254991	−	0.999675i	$-0.508117\pi$
$643$	−831.517	−0.0509982	−0.0254991	+	0.999675i	$0.508117\pi$
$644$	0	0
$645$	−4922.28	−0.300487
$646$	0	0
$647$	−19372.5	−1.17714	−0.588572	−	0.808445i	$-0.700310\pi$
$647$	−19372.5	−1.17714	−0.588572	+	0.808445i	$0.700310\pi$
$648$	0	0
$649$	−622.837	−0.0376710
$650$	0	0
$651$	4665.92	0.280909
$652$	0	0
$653$	26157.3	1.56756	0.783779	−	0.621040i	$-0.213290\pi$
$653$	26157.3	1.56756	0.783779	+	0.621040i	$0.213290\pi$
$654$	0	0
$655$	2066.69	0.123286
$656$	0	0
$657$	2639.29	0.156726
$658$	0	0
$659$	−9968.27	−0.589239	−0.294620	−	0.955615i	$-0.595193\pi$
$659$	−9968.27	−0.589239	−0.294620	+	0.955615i	$0.595193\pi$
$660$	0	0
$661$	−25233.7	−1.48484	−0.742420	−	0.669935i	$-0.766322\pi$
$661$	−25233.7	−1.48484	−0.742420	+	0.669935i	$0.766322\pi$
$662$	0	0
$663$	−21771.8	−1.27533
$664$	0	0
$665$	1573.69	0.0917668
$666$	0	0
$667$	−1249.29	−0.0725226
$668$	0	0
$669$	−10079.1	−0.582485
$670$	0	0
$671$	−5664.13	−0.325874
$672$	0	0
$673$	5349.47	0.306399	0.153200	−	0.988195i	$-0.451042\pi$
$673$	5349.47	0.306399	0.153200	+	0.988195i	$0.451042\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	24986.9	1.41850	0.709251	−	0.704956i	$-0.249033\pi$
$677$	24986.9	1.41850	0.709251	+	0.704956i	$0.249033\pi$
$678$	0	0
$679$	408.347	0.0230794
$680$	0	0
$681$	−9154.51	−0.515127
$682$	0	0
$683$	29567.7	1.65648	0.828240	−	0.560374i	$-0.189343\pi$
$683$	29567.7	1.65648	0.828240	+	0.560374i	$0.189343\pi$
$684$	0	0
$685$	−10278.1	−0.573295
$686$	0	0
$687$	−13320.6	−0.739756
$688$	0	0
$689$	21031.2	1.16288
$690$	0	0
$691$	14797.7	0.814663	0.407332	−	0.913280i	$-0.366459\pi$
$691$	14797.7	0.814663	0.407332	+	0.913280i	$0.366459\pi$
$692$	0	0
$693$	−939.968	−0.0515244
$694$	0	0
$695$	−8463.19	−0.461910
$696$	0	0
$697$	−30686.8	−1.66764
$698$	0	0
$699$	595.684	0.0322330
$700$	0	0
$701$	5348.06	0.288151	0.144075	−	0.989567i	$-0.453979\pi$
$701$	5348.06	0.288151	0.144075	+	0.989567i	$0.453979\pi$
$702$	0	0
$703$	−24723.8	−1.32642
$704$	0	0
$705$	3513.52	0.187698
$706$	0	0
$707$	7344.11	0.390670
$708$	0	0
$709$	−13022.5	−0.689803	−0.344901	−	0.938639i	$-0.612088\pi$
$709$	−13022.5	−0.689803	−0.344901	+	0.938639i	$0.612088\pi$
$710$	0	0
$711$	1537.23	0.0810840
$712$	0	0
$713$	−6705.85	−0.352225
$714$	0	0
$715$	7116.32	0.372217
$716$	0	0
$717$	−5601.09	−0.291738
$718$	0	0
$719$	19525.0	1.01274	0.506370	−	0.862317i	$-0.330987\pi$
$719$	19525.0	1.01274	0.506370	+	0.862317i	$0.330987\pi$
$720$	0	0
$721$	−4760.49	−0.245894
$722$	0	0
$723$	16953.9	0.872091
$724$	0	0
$725$	−1357.92	−0.0695613
$726$	0	0
$727$	33424.3	1.70514	0.852571	−	0.522611i	$-0.175042\pi$
$727$	33424.3	1.70514	0.852571	+	0.522611i	$0.175042\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−32760.0	−1.65755
$732$	0	0
$733$	12295.6	0.619576	0.309788	−	0.950806i	$-0.399742\pi$
$733$	12295.6	0.619576	0.309788	+	0.950806i	$0.399742\pi$
$734$	0	0
$735$	−4718.15	−0.236778
$736$	0	0
$737$	17862.4	0.892770
$738$	0	0
$739$	7219.24	0.359356	0.179678	−	0.983726i	$-0.442494\pi$
$739$	7219.24	0.359356	0.179678	+	0.983726i	$0.442494\pi$
$740$	0	0
$741$	12867.2	0.637906
$742$	0	0
$743$	11627.8	0.574137	0.287069	−	0.957910i	$-0.407319\pi$
$743$	11627.8	0.574137	0.287069	+	0.957910i	$0.407319\pi$
$744$	0	0
$745$	5922.16	0.291237
$746$	0	0
$747$	5189.68	0.254191
$748$	0	0
$749$	515.066	0.0251270
$750$	0	0
$751$	−27208.0	−1.32202	−0.661009	−	0.750378i	$-0.729871\pi$
$751$	−27208.0	−1.32202	−0.661009	+	0.750378i	$0.729871\pi$
$752$	0	0
$753$	−9097.80	−0.440295
$754$	0	0
$755$	−14396.2	−0.693950
$756$	0	0
$757$	−6194.66	−0.297422	−0.148711	−	0.988881i	$-0.547512\pi$
$757$	−6194.66	−0.297422	−0.148711	+	0.988881i	$0.547512\pi$
$758$	0	0
$759$	1350.92	0.0646052
$760$	0	0
$761$	−17525.6	−0.834827	−0.417414	−	0.908717i	$-0.637063\pi$
$761$	−17525.6	−0.834827	−0.417414	+	0.908717i	$0.637063\pi$
$762$	0	0
$763$	−965.904	−0.0458297
$764$	0	0
$765$	−4492.43	−0.212319
$766$	0	0
$767$	2312.59	0.108869
$768$	0	0
$769$	−1861.50	−0.0872919	−0.0436459	−	0.999047i	$-0.513897\pi$
$769$	−1861.50	−0.0872919	−0.0436459	+	0.999047i	$0.513897\pi$
$770$	0	0
$771$	11806.1	0.551472
$772$	0	0
$773$	−19051.4	−0.886457	−0.443228	−	0.896409i	$-0.646167\pi$
$773$	−19051.4	−0.886457	−0.443228	+	0.896409i	$0.646167\pi$
$774$	0	0
$775$	−7288.97	−0.337842
$776$	0	0
$777$	−6706.07	−0.309626
$778$	0	0
$779$	18136.0	0.834133
$780$	0	0
$781$	−19532.8	−0.894928
$782$	0	0
$783$	1466.55	0.0669354
$784$	0	0
$785$	−1099.75	−0.0500023
$786$	0	0
$787$	−1393.53	−0.0631180	−0.0315590	−	0.999502i	$-0.510047\pi$
$787$	−1393.53	−0.0631180	−0.0315590	+	0.999502i	$0.510047\pi$
$788$	0	0
$789$	13445.5	0.606685
$790$	0	0
$791$	−3245.78	−0.145900
$792$	0	0
$793$	21030.9	0.941775
$794$	0	0
$795$	4339.62	0.193598
$796$	0	0
$797$	−36281.1	−1.61248	−0.806238	−	0.591591i	$-0.798500\pi$
$797$	−36281.1	−1.61248	−0.806238	+	0.591591i	$0.798500\pi$
$798$	0	0
$799$	23384.1	1.03538
$800$	0	0
$801$	2431.59	0.107261
$802$	0	0
$803$	−5741.51	−0.252321
$804$	0	0
$805$	−613.462	−0.0268592
$806$	0	0
$807$	−16125.6	−0.703405
$808$	0	0
$809$	−2859.49	−0.124270	−0.0621349	−	0.998068i	$-0.519791\pi$
$809$	−2859.49	−0.124270	−0.0621349	+	0.998068i	$0.519791\pi$
$810$	0	0
$811$	−22565.3	−0.977032	−0.488516	−	0.872555i	$-0.662462\pi$
$811$	−22565.3	−0.977032	−0.488516	+	0.872555i	$0.662462\pi$
$812$	0	0
$813$	25237.3	1.08870
$814$	0	0
$815$	−4843.18	−0.208159
$816$	0	0
$817$	19361.2	0.829086
$818$	0	0
$819$	3490.09	0.148905
$820$	0	0
$821$	19088.7	0.811448	0.405724	−	0.913996i	$-0.367019\pi$
$821$	19088.7	0.811448	0.405724	+	0.913996i	$0.367019\pi$
$822$	0	0
$823$	−42310.6	−1.79205	−0.896024	−	0.444006i	$-0.853557\pi$
$823$	−42310.6	−1.79205	−0.896024	+	0.444006i	$0.853557\pi$
$824$	0	0
$825$	1468.39	0.0619671
$826$	0	0
$827$	−23295.0	−0.979499	−0.489749	−	0.871863i	$-0.662912\pi$
$827$	−23295.0	−0.979499	−0.489749	+	0.871863i	$0.662912\pi$
$828$	0	0
$829$	5637.45	0.236184	0.118092	−	0.993003i	$-0.462322\pi$
$829$	5637.45	0.236184	0.118092	+	0.993003i	$0.462322\pi$
$830$	0	0
$831$	13259.5	0.553510
$832$	0	0
$833$	−31401.5	−1.30612
$834$	0	0
$835$	−7014.43	−0.290712
$836$	0	0
$837$	7872.09	0.325089
$838$	0	0
$839$	−6417.98	−0.264092	−0.132046	−	0.991244i	$-0.542155\pi$
$839$	−6417.98	−0.264092	−0.132046	+	0.991244i	$0.542155\pi$
$840$	0	0
$841$	−21438.7	−0.879031
$842$	0	0
$843$	1747.59	0.0714001
$844$	0	0
$845$	−15437.8	−0.628493
$846$	0	0
$847$	−5055.35	−0.205081
$848$	0	0
$849$	−17896.2	−0.723436
$850$	0	0
$851$	9637.96	0.388232
$852$	0	0
$853$	37124.5	1.49018	0.745088	−	0.666966i	$-0.232408\pi$
$853$	37124.5	1.49018	0.745088	+	0.666966i	$0.232408\pi$
$854$	0	0
$855$	2655.04	0.106199
$856$	0	0
$857$	−33366.5	−1.32996	−0.664980	−	0.746861i	$-0.731560\pi$
$857$	−33366.5	−1.32996	−0.664980	+	0.746861i	$0.731560\pi$
$858$	0	0
$859$	−15103.4	−0.599907	−0.299954	−	0.953954i	$-0.596971\pi$
$859$	−15103.4	−0.599907	−0.299954	+	0.953954i	$0.596971\pi$
$860$	0	0
$861$	4919.20	0.194711
$862$	0	0
$863$	37049.4	1.46139	0.730694	−	0.682705i	$-0.239197\pi$
$863$	37049.4	1.46139	0.730694	+	0.682705i	$0.239197\pi$
$864$	0	0
$865$	−1007.70	−0.0396100
$866$	0	0
$867$	−15160.2	−0.593848
$868$	0	0
$869$	−3344.09	−0.130541
$870$	0	0
$871$	−66323.0	−2.58010
$872$	0	0
$873$	688.942	0.0267092
$874$	0	0
$875$	−666.806	−0.0257625
$876$	0	0
$877$	4230.97	0.162907	0.0814536	−	0.996677i	$-0.474044\pi$
$877$	4230.97	0.162907	0.0814536	+	0.996677i	$0.474044\pi$
$878$	0	0
$879$	−20751.7	−0.796290
$880$	0	0
$881$	39217.2	1.49973	0.749863	−	0.661593i	$-0.230119\pi$
$881$	39217.2	1.49973	0.749863	+	0.661593i	$0.230119\pi$
$882$	0	0
$883$	30463.4	1.16101	0.580506	−	0.814256i	$-0.302855\pi$
$883$	30463.4	1.16101	0.580506	+	0.814256i	$0.302855\pi$
$884$	0	0
$885$	477.183	0.0181247
$886$	0	0
$887$	−9984.21	−0.377945	−0.188972	−	0.981982i	$-0.560516\pi$
$887$	−9984.21	−0.377945	−0.188972	+	0.981982i	$0.560516\pi$
$888$	0	0
$889$	2797.89	0.105555
$890$	0	0
$891$	−1585.86	−0.0596279
$892$	0	0
$893$	−13820.1	−0.517884
$894$	0	0
$895$	−13427.2	−0.501476
$896$	0	0
$897$	−5015.95	−0.186709
$898$	0	0
$899$	15836.6	0.587518
$900$	0	0
$901$	28882.1	1.06793
$902$	0	0
$903$	5251.53	0.193533
$904$	0	0
$905$	−1660.27	−0.0609824
$906$	0	0
$907$	17646.0	0.646004	0.323002	−	0.946398i	$-0.395308\pi$
$907$	17646.0	0.646004	0.323002	+	0.946398i	$0.395308\pi$
$908$	0	0
$909$	12390.6	0.452112
$910$	0	0
$911$	7294.52	0.265289	0.132644	−	0.991164i	$-0.457653\pi$
$911$	7294.52	0.265289	0.132644	+	0.991164i	$0.457653\pi$
$912$	0	0
$913$	−11289.6	−0.409235
$914$	0	0
$915$	4339.54	0.156788
$916$	0	0
$917$	−2204.93	−0.0794038
$918$	0	0
$919$	49116.7	1.76301	0.881507	−	0.472172i	$-0.156530\pi$
$919$	49116.7	1.76301	0.881507	+	0.472172i	$0.156530\pi$
$920$	0	0
$921$	14928.6	0.534108
$922$	0	0
$923$	72525.0	2.58634
$924$	0	0
$925$	10476.0	0.372379
$926$	0	0
$927$	−8031.64	−0.284567
$928$	0	0
$929$	11334.4	0.400292	0.200146	−	0.979766i	$-0.435858\pi$
$929$	11334.4	0.400292	0.200146	+	0.979766i	$0.435858\pi$
$930$	0	0
$931$	18558.3	0.653303
$932$	0	0
$933$	3385.97	0.118812
$934$	0	0
$935$	9772.82	0.341824
$936$	0	0
$937$	5734.39	0.199930	0.0999649	−	0.994991i	$-0.468127\pi$
$937$	5734.39	0.199930	0.0999649	+	0.994991i	$0.468127\pi$
$938$	0	0
$939$	2296.63	0.0798165
$940$	0	0
$941$	−5321.13	−0.184340	−0.0921699	−	0.995743i	$-0.529380\pi$
$941$	−5321.13	−0.184340	−0.0921699	+	0.995743i	$0.529380\pi$
$942$	0	0
$943$	−7069.87	−0.244143
$944$	0	0
$945$	720.151	0.0247900
$946$	0	0
$947$	−23428.4	−0.803931	−0.401965	−	0.915655i	$-0.631673\pi$
$947$	−23428.4	−0.803931	−0.401965	+	0.915655i	$0.631673\pi$
$948$	0	0
$949$	21318.2	0.729206
$950$	0	0
$951$	−29388.6	−1.00209
$952$	0	0
$953$	−43819.7	−1.48947	−0.744733	−	0.667363i	$-0.767423\pi$
$953$	−43819.7	−1.48947	−0.744733	+	0.667363i	$0.767423\pi$
$954$	0	0
$955$	3631.13	0.123037
$956$	0	0
$957$	−3190.34	−0.107763
$958$	0	0
$959$	10965.6	0.369238
$960$	0	0
$961$	55215.6	1.85343
$962$	0	0
$963$	868.992	0.0290788
$964$	0	0
$965$	−4522.52	−0.150865
$966$	0	0
$967$	619.308	0.0205953	0.0102976	−	0.999947i	$-0.496722\pi$
$967$	619.308	0.0205953	0.0102976	+	0.999947i	$0.496722\pi$
$968$	0	0
$969$	17670.5	0.585818
$970$	0	0
$971$	48097.4	1.58962	0.794809	−	0.606860i	$-0.207571\pi$
$971$	48097.4	1.58962	0.794809	+	0.606860i	$0.207571\pi$
$972$	0	0
$973$	9029.29	0.297498
$974$	0	0
$975$	−5452.12	−0.179085
$976$	0	0
$977$	12985.3	0.425218	0.212609	−	0.977137i	$-0.431804\pi$
$977$	12985.3	0.425218	0.212609	+	0.977137i	$0.431804\pi$
$978$	0	0
$979$	−5289.67	−0.172685
$980$	0	0
$981$	−1629.62	−0.0530375
$982$	0	0
$983$	−33636.4	−1.09139	−0.545695	−	0.837984i	$-0.683734\pi$
$983$	−33636.4	−1.09139	−0.545695	+	0.837984i	$0.683734\pi$
$984$	0	0
$985$	−4700.97	−0.152067
$986$	0	0
$987$	−3748.54	−0.120889
$988$	0	0
$989$	−7547.49	−0.242666
$990$	0	0
$991$	−30371.4	−0.973541	−0.486771	−	0.873530i	$-0.661825\pi$
$991$	−30371.4	−0.973541	−0.486771	+	0.873530i	$0.661825\pi$
$992$	0	0
$993$	17735.2	0.566778
$994$	0	0
$995$	−13836.7	−0.440858
$996$	0	0
$997$	−21257.1	−0.675243	−0.337622	−	0.941282i	$-0.609622\pi$
$997$	−21257.1	−0.675243	−0.337622	+	0.941282i	$0.609622\pi$
$998$	0	0
$999$	−11314.1	−0.358322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.e.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.e.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} +5.33445 q^{7} +9.00000 q^{9} -19.5786 q^{11} +72.6950 q^{13} +15.0000 q^{15} +99.8318 q^{17} -59.0008 q^{19} -16.0034 q^{21} +23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -54.3168 q^{29} -291.559 q^{31} +58.7357 q^{33} -26.6723 q^{35} +419.042 q^{37} -218.085 q^{39} -307.385 q^{41} -328.152 q^{43} -45.0000 q^{45} +234.235 q^{47} -314.544 q^{49} -299.495 q^{51} +289.308 q^{53} +97.8928 q^{55} +177.003 q^{57} +31.8122 q^{59} +289.303 q^{61} +48.0101 q^{63} -363.475 q^{65} -912.347 q^{67} -69.0000 q^{69} +997.663 q^{71} +293.255 q^{73} -75.0000 q^{75} -104.441 q^{77} +170.804 q^{79} +81.0000 q^{81} +576.631 q^{83} -499.159 q^{85} +162.951 q^{87} +270.177 q^{89} +387.788 q^{91} +874.677 q^{93} +295.004 q^{95} +76.5491 q^{97} -176.207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} - 25 q^{5} - 7 q^{7} + 45 q^{9} + 64 q^{11} + 80 q^{13} + 75 q^{15} - 21 q^{17} - 52 q^{19} + 21 q^{21} + 115 q^{23} + 125 q^{25} - 135 q^{27} - 277 q^{29} - 289 q^{31} - 192 q^{33} + 35 q^{35}+ \cdots + 576 q^{99}+O(q^{100}) \) 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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