LMFDB - Embedded newform 624.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.13916.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 6 $ x^3 - x^2 - 16*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.388065$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -20.1466 q^{5} +19.6988 q^{7} +9.00000 q^{9} -13.5523 q^{11} +13.0000 q^{13} +60.4397 q^{15} +116.377 q^{17} -68.1169 q^{19} -59.0964 q^{21} -122.377 q^{23} +280.884 q^{25} -27.0000 q^{27} +204.193 q^{29} +194.603 q^{31} +40.6568 q^{33} -396.863 q^{35} -142.879 q^{37} -39.0000 q^{39} -175.051 q^{41} +219.490 q^{43} -181.319 q^{45} -236.413 q^{47} +45.0432 q^{49} -349.132 q^{51} -628.790 q^{53} +273.031 q^{55} +204.351 q^{57} -446.014 q^{59} +224.470 q^{61} +177.289 q^{63} -261.905 q^{65} -165.328 q^{67} +367.132 q^{69} -902.979 q^{71} -15.1121 q^{73} -842.651 q^{75} -266.963 q^{77} -670.013 q^{79} +81.0000 q^{81} -1040.50 q^{83} -2344.60 q^{85} -612.579 q^{87} -562.139 q^{89} +256.085 q^{91} -583.809 q^{93} +1372.32 q^{95} +1648.53 q^{97} -121.970 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 4 q^{5} - 6 q^{7} + 27 q^{9} - 32 q^{11} + 39 q^{13} + 12 q^{15} + 158 q^{17} - 70 q^{19} + 18 q^{21} - 176 q^{23} + 209 q^{25} - 81 q^{27} + 222 q^{29} - 54 q^{31} + 96 q^{33} - 496 q^{35}+ \cdots - 288 q^{99}+O(q^{100}) $ 3 * q - 9 * q^3 - 4 * q^5 - 6 * q^7 + 27 * q^9 - 32 * q^11 + 39 * q^13 + 12 * q^15 + 158 * q^17 - 70 * q^19 + 18 * q^21 - 176 * q^23 + 209 * q^25 - 81 * q^27 + 222 * q^29 - 54 * q^31 + 96 * q^33 - 496 * q^35 - 90 * q^37 - 117 * q^39 + 104 * q^41 - 140 * q^43 - 36 * q^45 - 328 * q^47 - 65 * q^49 - 474 * q^51 - 358 * q^53 - 32 * q^55 + 210 * q^57 - 1060 * q^59 - 186 * q^61 - 54 * q^63 - 52 * q^65 - 354 * q^67 + 528 * q^69 - 1692 * q^71 - 974 * q^73 - 627 * q^75 - 384 * q^77 - 776 * q^79 + 243 * q^81 - 2340 * q^83 - 2824 * q^85 - 666 * q^87 - 684 * q^89 - 78 * q^91 + 162 * q^93 - 742 * q^97 - 288 * 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
$5$	−20.1466	−1.80196	−0.900981	−	0.433858i	$-0.857152\pi$
$5$	−20.1466	−1.80196	−0.900981	+	0.433858i	$0.857152\pi$
$6$	0	0
$7$	19.6988	1.06364	0.531818	−	0.846859i	$-0.321509\pi$
$7$	19.6988	1.06364	0.531818	+	0.846859i	$0.321509\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−13.5523	−0.371469	−0.185735	−	0.982600i	$-0.559466\pi$
$11$	−13.5523	−0.371469	−0.185735	+	0.982600i	$0.559466\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	60.4397	1.04036
$16$	0	0
$17$	116.377	1.66033	0.830165	−	0.557518i	$-0.188246\pi$
$17$	116.377	1.66033	0.830165	+	0.557518i	$0.188246\pi$
$18$	0	0
$19$	−68.1169	−0.822478	−0.411239	−	0.911528i	$-0.634904\pi$
$19$	−68.1169	−0.822478	−0.411239	+	0.911528i	$0.634904\pi$
$20$	0	0
$21$	−59.0964	−0.614090
$22$	0	0
$23$	−122.377	−1.10945	−0.554726	−	0.832033i	$-0.687177\pi$
$23$	−122.377	−1.10945	−0.554726	+	0.832033i	$0.687177\pi$
$24$	0	0
$25$	280.884	2.24707
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	204.193	1.30751	0.653753	−	0.756708i	$-0.273194\pi$
$29$	204.193	1.30751	0.653753	+	0.756708i	$0.273194\pi$
$30$	0	0
$31$	194.603	1.12747	0.563737	−	0.825954i	$-0.309363\pi$
$31$	194.603	1.12747	0.563737	+	0.825954i	$0.309363\pi$
$32$	0	0
$33$	40.6568	0.214468
$34$	0	0
$35$	−396.863	−1.91663
$36$	0	0
$37$	−142.879	−0.634844	−0.317422	−	0.948284i	$-0.602817\pi$
$37$	−142.879	−0.634844	−0.317422	+	0.948284i	$0.602817\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−175.051	−0.666788	−0.333394	−	0.942788i	$-0.608194\pi$
$41$	−175.051	−0.666788	−0.333394	+	0.942788i	$0.608194\pi$
$42$	0	0
$43$	219.490	0.778417	0.389209	−	0.921150i	$-0.372748\pi$
$43$	219.490	0.778417	0.389209	+	0.921150i	$0.372748\pi$
$44$	0	0
$45$	−181.319	−0.600654
$46$	0	0
$47$	−236.413	−0.733711	−0.366855	−	0.930278i	$-0.619566\pi$
$47$	−236.413	−0.733711	−0.366855	+	0.930278i	$0.619566\pi$
$48$	0	0
$49$	45.0432	0.131321
$50$	0	0
$51$	−349.132	−0.958592
$52$	0	0
$53$	−628.790	−1.62964	−0.814820	−	0.579714i	$-0.803164\pi$
$53$	−628.790	−1.62964	−0.814820	+	0.579714i	$0.803164\pi$
$54$	0	0
$55$	273.031	0.669373
$56$	0	0
$57$	204.351	0.474858
$58$	0	0
$59$	−446.014	−0.984170	−0.492085	−	0.870547i	$-0.663765\pi$
$59$	−446.014	−0.984170	−0.492085	+	0.870547i	$0.663765\pi$
$60$	0	0
$61$	224.470	0.471154	0.235577	−	0.971856i	$-0.424302\pi$
$61$	224.470	0.471154	0.235577	+	0.971856i	$0.424302\pi$
$62$	0	0
$63$	177.289	0.354545
$64$	0	0
$65$	−261.905	−0.499774
$66$	0	0
$67$	−165.328	−0.301463	−0.150731	−	0.988575i	$-0.548163\pi$
$67$	−165.328	−0.301463	−0.150731	+	0.988575i	$0.548163\pi$
$68$	0	0
$69$	367.132	0.640543
$70$	0	0
$71$	−902.979	−1.50935	−0.754675	−	0.656098i	$-0.772206\pi$
$71$	−902.979	−1.50935	−0.754675	+	0.656098i	$0.772206\pi$
$72$	0	0
$73$	−15.1121	−0.0242293	−0.0121147	−	0.999927i	$-0.503856\pi$
$73$	−15.1121	−0.0242293	−0.0121147	+	0.999927i	$0.503856\pi$
$74$	0	0
$75$	−842.651	−1.29735
$76$	0	0
$77$	−266.963	−0.395108
$78$	0	0
$79$	−670.013	−0.954207	−0.477104	−	0.878847i	$-0.658313\pi$
$79$	−670.013	−0.954207	−0.477104	+	0.878847i	$0.658313\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1040.50	−1.37602	−0.688010	−	0.725701i	$-0.741515\pi$
$83$	−1040.50	−1.37602	−0.688010	+	0.725701i	$0.741515\pi$
$84$	0	0
$85$	−2344.60	−2.99185
$86$	0	0
$87$	−612.579	−0.754889
$88$	0	0
$89$	−562.139	−0.669512	−0.334756	−	0.942305i	$-0.608654\pi$
$89$	−562.139	−0.669512	−0.334756	+	0.942305i	$0.608654\pi$
$90$	0	0
$91$	256.085	0.295000
$92$	0	0
$93$	−583.809	−0.650948
$94$	0	0
$95$	1372.32	1.48207
$96$	0	0
$97$	1648.53	1.72560	0.862798	−	0.505548i	$-0.168710\pi$
$97$	1648.53	1.72560	0.862798	+	0.505548i	$0.168710\pi$
$98$	0	0
$99$	−121.970	−0.123823
$100$	0	0
$101$	397.182	0.391298	0.195649	−	0.980674i	$-0.437319\pi$
$101$	397.182	0.391298	0.195649	+	0.980674i	$0.437319\pi$
$102$	0	0
$103$	−1642.17	−1.57095	−0.785476	−	0.618892i	$-0.787582\pi$
$103$	−1642.17	−1.57095	−0.785476	+	0.618892i	$0.787582\pi$
$104$	0	0
$105$	1190.59	1.10657
$106$	0	0
$107$	644.674	0.582458	0.291229	−	0.956653i	$-0.405936\pi$
$107$	644.674	0.582458	0.291229	+	0.956653i	$0.405936\pi$
$108$	0	0
$109$	−297.565	−0.261482	−0.130741	−	0.991417i	$-0.541736\pi$
$109$	−297.565	−0.261482	−0.130741	+	0.991417i	$0.541736\pi$
$110$	0	0
$111$	428.638	0.366527
$112$	0	0
$113$	1902.40	1.58374	0.791872	−	0.610687i	$-0.209107\pi$
$113$	1902.40	1.58374	0.791872	+	0.610687i	$0.209107\pi$
$114$	0	0
$115$	2465.48	1.99919
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	2292.49	1.76599
$120$	0	0
$121$	−1147.34	−0.862011
$122$	0	0
$123$	525.152	0.384970
$124$	0	0
$125$	−3140.52	−2.24717
$126$	0	0
$127$	−175.822	−0.122848	−0.0614240	−	0.998112i	$-0.519564\pi$
$127$	−175.822	−0.122848	−0.0614240	+	0.998112i	$0.519564\pi$
$128$	0	0
$129$	−658.471	−0.449420
$130$	0	0
$131$	−2195.26	−1.46413	−0.732066	−	0.681234i	$-0.761444\pi$
$131$	−2195.26	−1.46413	−0.732066	+	0.681234i	$0.761444\pi$
$132$	0	0
$133$	−1341.82	−0.874817
$134$	0	0
$135$	543.957	0.346788
$136$	0	0
$137$	−927.288	−0.578274	−0.289137	−	0.957288i	$-0.593368\pi$
$137$	−927.288	−0.578274	−0.289137	+	0.957288i	$0.593368\pi$
$138$	0	0
$139$	2911.57	1.77666	0.888331	−	0.459205i	$-0.151866\pi$
$139$	2911.57	1.77666	0.888331	+	0.459205i	$0.151866\pi$
$140$	0	0
$141$	709.239	0.423608
$142$	0	0
$143$	−176.179	−0.103027
$144$	0	0
$145$	−4113.78	−2.35608
$146$	0	0
$147$	−135.130	−0.0758183
$148$	0	0
$149$	3078.29	1.69251	0.846254	−	0.532780i	$-0.178853\pi$
$149$	3078.29	1.69251	0.846254	+	0.532780i	$0.178853\pi$
$150$	0	0
$151$	866.444	0.466955	0.233478	−	0.972362i	$-0.424989\pi$
$151$	866.444	0.466955	0.233478	+	0.972362i	$0.424989\pi$
$152$	0	0
$153$	1047.39	0.553443
$154$	0	0
$155$	−3920.58	−2.03167
$156$	0	0
$157$	−1545.89	−0.785831	−0.392915	−	0.919575i	$-0.628533\pi$
$157$	−1545.89	−0.785831	−0.392915	+	0.919575i	$0.628533\pi$
$158$	0	0
$159$	1886.37	0.940873
$160$	0	0
$161$	−2410.68	−1.18005
$162$	0	0
$163$	−1635.09	−0.785707	−0.392854	−	0.919601i	$-0.628512\pi$
$163$	−1635.09	−0.785707	−0.392854	+	0.919601i	$0.628512\pi$
$164$	0	0
$165$	−819.094	−0.386463
$166$	0	0
$167$	−2260.09	−1.04725	−0.523626	−	0.851948i	$-0.675421\pi$
$167$	−2260.09	−1.04725	−0.523626	+	0.851948i	$0.675421\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−613.052	−0.274159
$172$	0	0
$173$	−3079.96	−1.35355	−0.676777	−	0.736188i	$-0.736624\pi$
$173$	−3079.96	−1.35355	−0.676777	+	0.736188i	$0.736624\pi$
$174$	0	0
$175$	5533.07	2.39006
$176$	0	0
$177$	1338.04	0.568211
$178$	0	0
$179$	1564.49	0.653272	0.326636	−	0.945150i	$-0.394085\pi$
$179$	1564.49	0.653272	0.326636	+	0.945150i	$0.394085\pi$
$180$	0	0
$181$	641.793	0.263559	0.131779	−	0.991279i	$-0.457931\pi$
$181$	641.793	0.263559	0.131779	+	0.991279i	$0.457931\pi$
$182$	0	0
$183$	−673.409	−0.272021
$184$	0	0
$185$	2878.53	1.14396
$186$	0	0
$187$	−1577.17	−0.616761
$188$	0	0
$189$	−531.868	−0.204697
$190$	0	0
$191$	−4000.90	−1.51568	−0.757841	−	0.652439i	$-0.773746\pi$
$191$	−4000.90	−1.51568	−0.757841	+	0.652439i	$0.773746\pi$
$192$	0	0
$193$	−4255.28	−1.58706	−0.793528	−	0.608533i	$-0.791758\pi$
$193$	−4255.28	−1.58706	−0.793528	+	0.608533i	$0.791758\pi$
$194$	0	0
$195$	785.716	0.288545
$196$	0	0
$197$	3774.59	1.36512	0.682559	−	0.730831i	$-0.260867\pi$
$197$	3774.59	1.36512	0.682559	+	0.730831i	$0.260867\pi$
$198$	0	0
$199$	−2071.28	−0.737835	−0.368918	−	0.929462i	$-0.620272\pi$
$199$	−2071.28	−0.737835	−0.368918	+	0.929462i	$0.620272\pi$
$200$	0	0
$201$	495.984	0.174050
$202$	0	0
$203$	4022.36	1.39071
$204$	0	0
$205$	3526.67	1.20153
$206$	0	0
$207$	−1101.39	−0.369817
$208$	0	0
$209$	923.138	0.305525
$210$	0	0
$211$	2753.26	0.898304	0.449152	−	0.893455i	$-0.351726\pi$
$211$	2753.26	0.898304	0.449152	+	0.893455i	$0.351726\pi$
$212$	0	0
$213$	2708.94	0.871424
$214$	0	0
$215$	−4421.97	−1.40268
$216$	0	0
$217$	3833.45	1.19922
$218$	0	0
$219$	45.3364	0.0139888
$220$	0	0
$221$	1512.90	0.460493
$222$	0	0
$223$	−3994.02	−1.19937	−0.599685	−	0.800236i	$-0.704707\pi$
$223$	−3994.02	−1.19937	−0.599685	+	0.800236i	$0.704707\pi$
$224$	0	0
$225$	2527.95	0.749023
$226$	0	0
$227$	−1797.20	−0.525480	−0.262740	−	0.964867i	$-0.584626\pi$
$227$	−1797.20	−0.525480	−0.262740	+	0.964867i	$0.584626\pi$
$228$	0	0
$229$	−5171.16	−1.49223	−0.746114	−	0.665819i	$-0.768082\pi$
$229$	−5171.16	−1.49223	−0.746114	+	0.665819i	$0.768082\pi$
$230$	0	0
$231$	800.890	0.228116
$232$	0	0
$233$	−4501.40	−1.26565	−0.632826	−	0.774294i	$-0.718105\pi$
$233$	−4501.40	−1.26565	−0.632826	+	0.774294i	$0.718105\pi$
$234$	0	0
$235$	4762.91	1.32212
$236$	0	0
$237$	2010.04	0.550912
$238$	0	0
$239$	6948.26	1.88053	0.940263	−	0.340450i	$-0.110579\pi$
$239$	6948.26	1.88053	0.940263	+	0.340450i	$0.110579\pi$
$240$	0	0
$241$	1484.32	0.396737	0.198368	−	0.980128i	$-0.436436\pi$
$241$	1484.32	0.396737	0.198368	+	0.980128i	$0.436436\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−907.465	−0.236636
$246$	0	0
$247$	−885.519	−0.228114
$248$	0	0
$249$	3121.50	0.794445
$250$	0	0
$251$	2840.45	0.714293	0.357147	−	0.934048i	$-0.383750\pi$
$251$	2840.45	0.714293	0.357147	+	0.934048i	$0.383750\pi$
$252$	0	0
$253$	1658.49	0.412127
$254$	0	0
$255$	7033.80	1.72735
$256$	0	0
$257$	−1500.98	−0.364313	−0.182156	−	0.983270i	$-0.558308\pi$
$257$	−1500.98	−0.364313	−0.182156	+	0.983270i	$0.558308\pi$
$258$	0	0
$259$	−2814.55	−0.675242
$260$	0	0
$261$	1837.74	0.435835
$262$	0	0
$263$	690.021	0.161781	0.0808907	−	0.996723i	$-0.474224\pi$
$263$	690.021	0.161781	0.0808907	+	0.996723i	$0.474224\pi$
$264$	0	0
$265$	12667.9	2.93655
$266$	0	0
$267$	1686.42	0.386543
$268$	0	0
$269$	507.641	0.115061	0.0575305	−	0.998344i	$-0.481677\pi$
$269$	507.641	0.115061	0.0575305	+	0.998344i	$0.481677\pi$
$270$	0	0
$271$	1528.40	0.342598	0.171299	−	0.985219i	$-0.445204\pi$
$271$	1528.40	0.342598	0.171299	+	0.985219i	$0.445204\pi$
$272$	0	0
$273$	−768.254	−0.170318
$274$	0	0
$275$	−3806.61	−0.834717
$276$	0	0
$277$	3886.11	0.842939	0.421469	−	0.906843i	$-0.361515\pi$
$277$	3886.11	0.842939	0.421469	+	0.906843i	$0.361515\pi$
$278$	0	0
$279$	1751.43	0.375825
$280$	0	0
$281$	2705.38	0.574339	0.287170	−	0.957880i	$-0.407286\pi$
$281$	2705.38	0.574339	0.287170	+	0.957880i	$0.407286\pi$
$282$	0	0
$283$	2944.62	0.618514	0.309257	−	0.950978i	$-0.399920\pi$
$283$	2944.62	0.618514	0.309257	+	0.950978i	$0.399920\pi$
$284$	0	0
$285$	−4116.96	−0.855676
$286$	0	0
$287$	−3448.29	−0.709220
$288$	0	0
$289$	8630.65	1.75670
$290$	0	0
$291$	−4945.59	−0.996274
$292$	0	0
$293$	−3835.89	−0.764830	−0.382415	−	0.923991i	$-0.624908\pi$
$293$	−3835.89	−0.764830	−0.382415	+	0.923991i	$0.624908\pi$
$294$	0	0
$295$	8985.63	1.77344
$296$	0	0
$297$	365.911	0.0714893
$298$	0	0
$299$	−1590.90	−0.307707
$300$	0	0
$301$	4323.70	0.827953
$302$	0	0
$303$	−1191.55	−0.225916
$304$	0	0
$305$	−4522.29	−0.849002
$306$	0	0
$307$	−4068.20	−0.756301	−0.378151	−	0.925744i	$-0.623440\pi$
$307$	−4068.20	−0.756301	−0.378151	+	0.925744i	$0.623440\pi$
$308$	0	0
$309$	4926.52	0.906990
$310$	0	0
$311$	−7154.18	−1.30443	−0.652213	−	0.758036i	$-0.726159\pi$
$311$	−7154.18	−1.30443	−0.652213	+	0.758036i	$0.726159\pi$
$312$	0	0
$313$	6439.37	1.16286	0.581429	−	0.813597i	$-0.302494\pi$
$313$	6439.37	1.16286	0.581429	+	0.813597i	$0.302494\pi$
$314$	0	0
$315$	−3571.77	−0.638877
$316$	0	0
$317$	−7721.21	−1.36803	−0.684016	−	0.729467i	$-0.739768\pi$
$317$	−7721.21	−1.36803	−0.684016	+	0.729467i	$0.739768\pi$
$318$	0	0
$319$	−2767.27	−0.485698
$320$	0	0
$321$	−1934.02	−0.336282
$322$	0	0
$323$	−7927.25	−1.36559
$324$	0	0
$325$	3651.49	0.623225
$326$	0	0
$327$	892.694	0.150967
$328$	0	0
$329$	−4657.06	−0.780401
$330$	0	0
$331$	2940.41	0.488276	0.244138	−	0.969740i	$-0.421495\pi$
$331$	2940.41	0.488276	0.244138	+	0.969740i	$0.421495\pi$
$332$	0	0
$333$	−1285.91	−0.211615
$334$	0	0
$335$	3330.79	0.543225
$336$	0	0
$337$	5321.56	0.860189	0.430094	−	0.902784i	$-0.358480\pi$
$337$	5321.56	0.860189	0.430094	+	0.902784i	$0.358480\pi$
$338$	0	0
$339$	−5707.21	−0.914375
$340$	0	0
$341$	−2637.31	−0.418822
$342$	0	0
$343$	−5869.40	−0.923958
$344$	0	0
$345$	−7396.43	−1.15423
$346$	0	0
$347$	10398.3	1.60868	0.804339	−	0.594170i	$-0.202519\pi$
$347$	10398.3	1.60868	0.804339	+	0.594170i	$0.202519\pi$
$348$	0	0
$349$	−2289.87	−0.351214	−0.175607	−	0.984460i	$-0.556189\pi$
$349$	−2289.87	−0.351214	−0.175607	+	0.984460i	$0.556189\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	11216.7	1.69124	0.845619	−	0.533787i	$-0.179231\pi$
$353$	11216.7	1.69124	0.845619	+	0.533787i	$0.179231\pi$
$354$	0	0
$355$	18191.9	2.71979
$356$	0	0
$357$	−6877.48	−1.01959
$358$	0	0
$359$	−1447.78	−0.212844	−0.106422	−	0.994321i	$-0.533939\pi$
$359$	−1447.78	−0.212844	−0.106422	+	0.994321i	$0.533939\pi$
$360$	0	0
$361$	−2219.09	−0.323530
$362$	0	0
$363$	3442.01	0.497682
$364$	0	0
$365$	304.457	0.0436603
$366$	0	0
$367$	−6365.59	−0.905398	−0.452699	−	0.891663i	$-0.649539\pi$
$367$	−6365.59	−0.905398	−0.452699	+	0.891663i	$0.649539\pi$
$368$	0	0
$369$	−1575.46	−0.222263
$370$	0	0
$371$	−12386.4	−1.73334
$372$	0	0
$373$	−9805.65	−1.36117	−0.680586	−	0.732668i	$-0.738275\pi$
$373$	−9805.65	−1.36117	−0.680586	+	0.732668i	$0.738275\pi$
$374$	0	0
$375$	9421.55	1.29740
$376$	0	0
$377$	2654.51	0.362637
$378$	0	0
$379$	−7337.60	−0.994478	−0.497239	−	0.867614i	$-0.665653\pi$
$379$	−7337.60	−0.994478	−0.497239	+	0.867614i	$0.665653\pi$
$380$	0	0
$381$	527.467	0.0709263
$382$	0	0
$383$	3729.09	0.497514	0.248757	−	0.968566i	$-0.419978\pi$
$383$	3729.09	0.497514	0.248757	+	0.968566i	$0.419978\pi$
$384$	0	0
$385$	5378.39	0.711969
$386$	0	0
$387$	1975.41	0.259472
$388$	0	0
$389$	−13546.0	−1.76558	−0.882788	−	0.469771i	$-0.844336\pi$
$389$	−13546.0	−1.76558	−0.882788	+	0.469771i	$0.844336\pi$
$390$	0	0
$391$	−14241.9	−1.84206
$392$	0	0
$393$	6585.79	0.845316
$394$	0	0
$395$	13498.5	1.71945
$396$	0	0
$397$	−11119.4	−1.40570	−0.702852	−	0.711336i	$-0.748090\pi$
$397$	−11119.4	−1.40570	−0.702852	+	0.711336i	$0.748090\pi$
$398$	0	0
$399$	4025.46	0.505076
$400$	0	0
$401$	−2852.32	−0.355207	−0.177603	−	0.984102i	$-0.556834\pi$
$401$	−2852.32	−0.355207	−0.177603	+	0.984102i	$0.556834\pi$
$402$	0	0
$403$	2529.84	0.312705
$404$	0	0
$405$	−1631.87	−0.200218
$406$	0	0
$407$	1936.34	0.235825
$408$	0	0
$409$	−6656.75	−0.804781	−0.402390	−	0.915468i	$-0.631821\pi$
$409$	−6656.75	−0.804781	−0.402390	+	0.915468i	$0.631821\pi$
$410$	0	0
$411$	2781.86	0.333867
$412$	0	0
$413$	−8785.94	−1.04680
$414$	0	0
$415$	20962.5	2.47954
$416$	0	0
$417$	−8734.70	−1.02576
$418$	0	0
$419$	−10198.0	−1.18903	−0.594514	−	0.804085i	$-0.702656\pi$
$419$	−10198.0	−1.18903	−0.594514	+	0.804085i	$0.702656\pi$
$420$	0	0
$421$	−4463.77	−0.516747	−0.258374	−	0.966045i	$-0.583187\pi$
$421$	−4463.77	−0.516747	−0.258374	+	0.966045i	$0.583187\pi$
$422$	0	0
$423$	−2127.72	−0.244570
$424$	0	0
$425$	32688.4	3.73088
$426$	0	0
$427$	4421.79	0.501137
$428$	0	0
$429$	528.538	0.0594827
$430$	0	0
$431$	−3656.22	−0.408617	−0.204309	−	0.978907i	$-0.565495\pi$
$431$	−3656.22	−0.408617	−0.204309	+	0.978907i	$0.565495\pi$
$432$	0	0
$433$	−2816.59	−0.312602	−0.156301	−	0.987709i	$-0.549957\pi$
$433$	−2816.59	−0.312602	−0.156301	+	0.987709i	$0.549957\pi$
$434$	0	0
$435$	12341.3	1.36028
$436$	0	0
$437$	8335.95	0.912500
$438$	0	0
$439$	−11396.3	−1.23899	−0.619496	−	0.785000i	$-0.712663\pi$
$439$	−11396.3	−1.23899	−0.619496	+	0.785000i	$0.712663\pi$
$440$	0	0
$441$	405.389	0.0437737
$442$	0	0
$443$	4245.96	0.455376	0.227688	−	0.973734i	$-0.426883\pi$
$443$	4245.96	0.455376	0.227688	+	0.973734i	$0.426883\pi$
$444$	0	0
$445$	11325.2	1.20644
$446$	0	0
$447$	−9234.88	−0.977170
$448$	0	0
$449$	−2021.21	−0.212443	−0.106222	−	0.994342i	$-0.533875\pi$
$449$	−2021.21	−0.212443	−0.106222	+	0.994342i	$0.533875\pi$
$450$	0	0
$451$	2372.33	0.247691
$452$	0	0
$453$	−2599.33	−0.269597
$454$	0	0
$455$	−5159.22	−0.531578
$456$	0	0
$457$	13740.0	1.40641	0.703206	−	0.710986i	$-0.251751\pi$
$457$	13740.0	1.40641	0.703206	+	0.710986i	$0.251751\pi$
$458$	0	0
$459$	−3142.18	−0.319531
$460$	0	0
$461$	1519.14	0.153478	0.0767392	−	0.997051i	$-0.475549\pi$
$461$	1519.14	0.153478	0.0767392	+	0.997051i	$0.475549\pi$
$462$	0	0
$463$	1108.49	0.111265	0.0556325	−	0.998451i	$-0.482282\pi$
$463$	1108.49	0.111265	0.0556325	+	0.998451i	$0.482282\pi$
$464$	0	0
$465$	11761.7	1.17298
$466$	0	0
$467$	10147.9	1.00554	0.502771	−	0.864420i	$-0.332314\pi$
$467$	10147.9	1.00554	0.502771	+	0.864420i	$0.332314\pi$
$468$	0	0
$469$	−3256.76	−0.320647
$470$	0	0
$471$	4637.67	0.453700
$472$	0	0
$473$	−2974.59	−0.289158
$474$	0	0
$475$	−19132.9	−1.84816
$476$	0	0
$477$	−5659.11	−0.543213
$478$	0	0
$479$	14277.3	1.36190	0.680948	−	0.732332i	$-0.261568\pi$
$479$	14277.3	1.36190	0.680948	+	0.732332i	$0.261568\pi$
$480$	0	0
$481$	−1857.43	−0.176074
$482$	0	0
$483$	7232.05	0.681304
$484$	0	0
$485$	−33212.2	−3.10946
$486$	0	0
$487$	2138.76	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$487$	2138.76	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$488$	0	0
$489$	4905.28	0.453628
$490$	0	0
$491$	−511.287	−0.0469940	−0.0234970	−	0.999724i	$-0.507480\pi$
$491$	−511.287	−0.0469940	−0.0234970	+	0.999724i	$0.507480\pi$
$492$	0	0
$493$	23763.4	2.17089
$494$	0	0
$495$	2457.28	0.223124
$496$	0	0
$497$	−17787.6	−1.60540
$498$	0	0
$499$	484.443	0.0434602	0.0217301	−	0.999764i	$-0.493083\pi$
$499$	484.443	0.0434602	0.0217301	+	0.999764i	$0.493083\pi$
$500$	0	0
$501$	6780.27	0.604631
$502$	0	0
$503$	−20675.7	−1.83277	−0.916386	−	0.400295i	$-0.868908\pi$
$503$	−20675.7	−1.83277	−0.916386	+	0.400295i	$0.868908\pi$
$504$	0	0
$505$	−8001.85	−0.705104
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−15364.4	−1.33795	−0.668975	−	0.743285i	$-0.733266\pi$
$509$	−15364.4	−1.33795	−0.668975	+	0.743285i	$0.733266\pi$
$510$	0	0
$511$	−297.691	−0.0257712
$512$	0	0
$513$	1839.16	0.158286
$514$	0	0
$515$	33084.1	2.83080
$516$	0	0
$517$	3203.93	0.272551
$518$	0	0
$519$	9239.87	0.781475
$520$	0	0
$521$	−13318.8	−1.11997	−0.559986	−	0.828502i	$-0.689193\pi$
$521$	−13318.8	−1.11997	−0.559986	+	0.828502i	$0.689193\pi$
$522$	0	0
$523$	−12904.9	−1.07895	−0.539476	−	0.842001i	$-0.681378\pi$
$523$	−12904.9	−1.07895	−0.539476	+	0.842001i	$0.681378\pi$
$524$	0	0
$525$	−16599.2	−1.37990
$526$	0	0
$527$	22647.3	1.87198
$528$	0	0
$529$	2809.17	0.230885
$530$	0	0
$531$	−4014.12	−0.328057
$532$	0	0
$533$	−2275.66	−0.184934
$534$	0	0
$535$	−12988.0	−1.04957
$536$	0	0
$537$	−4693.48	−0.377167
$538$	0	0
$539$	−610.437	−0.0487818
$540$	0	0
$541$	−5767.47	−0.458342	−0.229171	−	0.973386i	$-0.573602\pi$
$541$	−5767.47	−0.458342	−0.229171	+	0.973386i	$0.573602\pi$
$542$	0	0
$543$	−1925.38	−0.152166
$544$	0	0
$545$	5994.91	0.471181
$546$	0	0
$547$	1804.58	0.141057	0.0705286	−	0.997510i	$-0.477531\pi$
$547$	1804.58	0.141057	0.0705286	+	0.997510i	$0.477531\pi$
$548$	0	0
$549$	2020.23	0.157051
$550$	0	0
$551$	−13909.0	−1.07539
$552$	0	0
$553$	−13198.5	−1.01493
$554$	0	0
$555$	−8635.58	−0.660468
$556$	0	0
$557$	−3897.95	−0.296519	−0.148260	−	0.988948i	$-0.547367\pi$
$557$	−3897.95	−0.296519	−0.148260	+	0.988948i	$0.547367\pi$
$558$	0	0
$559$	2853.37	0.215894
$560$	0	0
$561$	4731.52	0.356087
$562$	0	0
$563$	18806.2	1.40779	0.703894	−	0.710305i	$-0.251443\pi$
$563$	18806.2	1.40779	0.703894	+	0.710305i	$0.251443\pi$
$564$	0	0
$565$	−38326.9	−2.85385
$566$	0	0
$567$	1595.60	0.118182
$568$	0	0
$569$	24680.0	1.81835	0.909175	−	0.416415i	$-0.136714\pi$
$569$	24680.0	1.81835	0.909175	+	0.416415i	$0.136714\pi$
$570$	0	0
$571$	−23437.7	−1.71775	−0.858877	−	0.512182i	$-0.828837\pi$
$571$	−23437.7	−1.71775	−0.858877	+	0.512182i	$0.828837\pi$
$572$	0	0
$573$	12002.7	0.875079
$574$	0	0
$575$	−34373.7	−2.49302
$576$	0	0
$577$	10814.1	0.780240	0.390120	−	0.920764i	$-0.372434\pi$
$577$	10814.1	0.780240	0.390120	+	0.920764i	$0.372434\pi$
$578$	0	0
$579$	12765.8	0.916288
$580$	0	0
$581$	−20496.6	−1.46358
$582$	0	0
$583$	8521.52	0.605361
$584$	0	0
$585$	−2357.15	−0.166591
$586$	0	0
$587$	−27148.1	−1.90889	−0.954447	−	0.298379i	$-0.903554\pi$
$587$	−27148.1	−1.90889	−0.954447	+	0.298379i	$0.903554\pi$
$588$	0	0
$589$	−13255.7	−0.927323
$590$	0	0
$591$	−11323.8	−0.788151
$592$	0	0
$593$	4549.19	0.315030	0.157515	−	0.987517i	$-0.449652\pi$
$593$	4549.19	0.315030	0.157515	+	0.987517i	$0.449652\pi$
$594$	0	0
$595$	−46185.8	−3.18224
$596$	0	0
$597$	6213.84	0.425989
$598$	0	0
$599$	−2875.51	−0.196144	−0.0980720	−	0.995179i	$-0.531268\pi$
$599$	−2875.51	−0.196144	−0.0980720	+	0.995179i	$0.531268\pi$
$600$	0	0
$601$	−22636.2	−1.53636	−0.768179	−	0.640235i	$-0.778837\pi$
$601$	−22636.2	−1.53636	−0.768179	+	0.640235i	$0.778837\pi$
$602$	0	0
$603$	−1487.95	−0.100488
$604$	0	0
$605$	23114.9	1.55331
$606$	0	0
$607$	3458.51	0.231263	0.115631	−	0.993292i	$-0.463111\pi$
$607$	3458.51	0.231263	0.115631	+	0.993292i	$0.463111\pi$
$608$	0	0
$609$	−12067.1	−0.802927
$610$	0	0
$611$	−3073.37	−0.203495
$612$	0	0
$613$	−18029.5	−1.18794	−0.593969	−	0.804488i	$-0.702440\pi$
$613$	−18029.5	−1.18794	−0.593969	+	0.804488i	$0.702440\pi$
$614$	0	0
$615$	−10580.0	−0.693702
$616$	0	0
$617$	5410.04	0.352999	0.176499	−	0.984301i	$-0.443523\pi$
$617$	5410.04	0.352999	0.176499	+	0.984301i	$0.443523\pi$
$618$	0	0
$619$	2133.47	0.138532	0.0692661	−	0.997598i	$-0.477934\pi$
$619$	2133.47	0.138532	0.0692661	+	0.997598i	$0.477934\pi$
$620$	0	0
$621$	3304.18	0.213514
$622$	0	0
$623$	−11073.5	−0.712117
$624$	0	0
$625$	28160.1	1.80225
$626$	0	0
$627$	−2769.41	−0.176395
$628$	0	0
$629$	−16627.9	−1.05405
$630$	0	0
$631$	−20370.5	−1.28516	−0.642580	−	0.766219i	$-0.722136\pi$
$631$	−20370.5	−1.28516	−0.642580	+	0.766219i	$0.722136\pi$
$632$	0	0
$633$	−8259.78	−0.518636
$634$	0	0
$635$	3542.21	0.221367
$636$	0	0
$637$	585.561	0.0364220
$638$	0	0
$639$	−8126.81	−0.503117
$640$	0	0
$641$	−14112.2	−0.869573	−0.434787	−	0.900534i	$-0.643176\pi$
$641$	−14112.2	−0.869573	−0.434787	+	0.900534i	$0.643176\pi$
$642$	0	0
$643$	−20309.8	−1.24563	−0.622815	−	0.782369i	$-0.714011\pi$
$643$	−20309.8	−1.24563	−0.622815	+	0.782369i	$0.714011\pi$
$644$	0	0
$645$	13265.9	0.809837
$646$	0	0
$647$	−12214.8	−0.742212	−0.371106	−	0.928590i	$-0.621021\pi$
$647$	−12214.8	−0.742212	−0.371106	+	0.928590i	$0.621021\pi$
$648$	0	0
$649$	6044.49	0.365589
$650$	0	0
$651$	−11500.3	−0.692371
$652$	0	0
$653$	31803.3	1.90591	0.952955	−	0.303111i	$-0.0980253\pi$
$653$	31803.3	1.90591	0.952955	+	0.303111i	$0.0980253\pi$
$654$	0	0
$655$	44227.0	2.63831
$656$	0	0
$657$	−136.009	−0.00807644
$658$	0	0
$659$	21545.8	1.27360	0.636802	−	0.771027i	$-0.280257\pi$
$659$	21545.8	1.27360	0.636802	+	0.771027i	$0.280257\pi$
$660$	0	0
$661$	26775.5	1.57556	0.787781	−	0.615956i	$-0.211230\pi$
$661$	26775.5	1.57556	0.787781	+	0.615956i	$0.211230\pi$
$662$	0	0
$663$	−4538.71	−0.265866
$664$	0	0
$665$	27033.1	1.57639
$666$	0	0
$667$	−24988.5	−1.45062
$668$	0	0
$669$	11982.1	0.692457
$670$	0	0
$671$	−3042.07	−0.175019
$672$	0	0
$673$	15091.5	0.864388	0.432194	−	0.901781i	$-0.357740\pi$
$673$	15091.5	0.864388	0.432194	+	0.901781i	$0.357740\pi$
$674$	0	0
$675$	−7583.86	−0.432449
$676$	0	0
$677$	17169.2	0.974689	0.487345	−	0.873210i	$-0.337966\pi$
$677$	17169.2	0.974689	0.487345	+	0.873210i	$0.337966\pi$
$678$	0	0
$679$	32474.1	1.83541
$680$	0	0
$681$	5391.59	0.303386
$682$	0	0
$683$	−1579.31	−0.0884781	−0.0442391	−	0.999021i	$-0.514086\pi$
$683$	−1579.31	−0.0884781	−0.0442391	+	0.999021i	$0.514086\pi$
$684$	0	0
$685$	18681.7	1.04203
$686$	0	0
$687$	15513.5	0.861538
$688$	0	0
$689$	−8174.27	−0.451981
$690$	0	0
$691$	2423.59	0.133427	0.0667134	−	0.997772i	$-0.478749\pi$
$691$	2423.59	0.133427	0.0667134	+	0.997772i	$0.478749\pi$
$692$	0	0
$693$	−2402.67	−0.131703
$694$	0	0
$695$	−58658.0	−3.20148
$696$	0	0
$697$	−20371.9	−1.10709
$698$	0	0
$699$	13504.2	0.730724
$700$	0	0
$701$	7893.64	0.425305	0.212652	−	0.977128i	$-0.431790\pi$
$701$	7893.64	0.425305	0.212652	+	0.977128i	$0.431790\pi$
$702$	0	0
$703$	9732.49	0.522145
$704$	0	0
$705$	−14288.7	−0.763326
$706$	0	0
$707$	7824.01	0.416198
$708$	0	0
$709$	−11523.5	−0.610399	−0.305199	−	0.952288i	$-0.598723\pi$
$709$	−11523.5	−0.610399	−0.305199	+	0.952288i	$0.598723\pi$
$710$	0	0
$711$	−6030.12	−0.318069
$712$	0	0
$713$	−23814.9	−1.25088
$714$	0	0
$715$	3549.41	0.185651
$716$	0	0
$717$	−20844.8	−1.08572
$718$	0	0
$719$	4029.94	0.209028	0.104514	−	0.994523i	$-0.466671\pi$
$719$	4029.94	0.209028	0.104514	+	0.994523i	$0.466671\pi$
$720$	0	0
$721$	−32348.9	−1.67092
$722$	0	0
$723$	−4452.97	−0.229056
$724$	0	0
$725$	57354.4	2.93806
$726$	0	0
$727$	36930.6	1.88402	0.942009	−	0.335588i	$-0.108935\pi$
$727$	36930.6	1.88402	0.942009	+	0.335588i	$0.108935\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	25543.7	1.29243
$732$	0	0
$733$	14378.9	0.724552	0.362276	−	0.932071i	$-0.382000\pi$
$733$	14378.9	0.724552	0.362276	+	0.932071i	$0.382000\pi$
$734$	0	0
$735$	2722.39	0.136622
$736$	0	0
$737$	2240.57	0.111984
$738$	0	0
$739$	30657.0	1.52603	0.763015	−	0.646381i	$-0.223718\pi$
$739$	30657.0	1.52603	0.763015	+	0.646381i	$0.223718\pi$
$740$	0	0
$741$	2656.56	0.131702
$742$	0	0
$743$	31072.6	1.53424	0.767122	−	0.641501i	$-0.221688\pi$
$743$	31072.6	1.53424	0.767122	+	0.641501i	$0.221688\pi$
$744$	0	0
$745$	−62017.0	−3.04983
$746$	0	0
$747$	−9364.50	−0.458673
$748$	0	0
$749$	12699.3	0.619524
$750$	0	0
$751$	−149.995	−0.00728816	−0.00364408	−	0.999993i	$-0.501160\pi$
$751$	−149.995	−0.00728816	−0.00364408	+	0.999993i	$0.501160\pi$
$752$	0	0
$753$	−8521.35	−0.412397
$754$	0	0
$755$	−17455.9	−0.841436
$756$	0	0
$757$	13482.2	0.647318	0.323659	−	0.946174i	$-0.395087\pi$
$757$	13482.2	0.647318	0.323659	+	0.946174i	$0.395087\pi$
$758$	0	0
$759$	−4975.46	−0.237942
$760$	0	0
$761$	11123.3	0.529857	0.264928	−	0.964268i	$-0.414652\pi$
$761$	11123.3	0.529857	0.264928	+	0.964268i	$0.414652\pi$
$762$	0	0
$763$	−5861.67	−0.278122
$764$	0	0
$765$	−21101.4	−0.997284
$766$	0	0
$767$	−5798.18	−0.272960
$768$	0	0
$769$	2527.85	0.118539	0.0592695	−	0.998242i	$-0.481123\pi$
$769$	2527.85	0.118539	0.0592695	+	0.998242i	$0.481123\pi$
$770$	0	0
$771$	4502.93	0.210336
$772$	0	0
$773$	9690.00	0.450873	0.225437	−	0.974258i	$-0.427619\pi$
$773$	9690.00	0.450873	0.225437	+	0.974258i	$0.427619\pi$
$774$	0	0
$775$	54660.8	2.53351
$776$	0	0
$777$	8443.66	0.389851
$778$	0	0
$779$	11923.9	0.548419
$780$	0	0
$781$	12237.4	0.560677
$782$	0	0
$783$	−5513.21	−0.251630
$784$	0	0
$785$	31144.3	1.41604
$786$	0	0
$787$	−34825.8	−1.57739	−0.788694	−	0.614786i	$-0.789243\pi$
$787$	−34825.8	−1.57739	−0.788694	+	0.614786i	$0.789243\pi$
$788$	0	0
$789$	−2070.06	−0.0934045
$790$	0	0
$791$	37475.1	1.68453
$792$	0	0
$793$	2918.11	0.130675
$794$	0	0
$795$	−38003.8	−1.69542
$796$	0	0
$797$	11789.0	0.523950	0.261975	−	0.965075i	$-0.415626\pi$
$797$	11789.0	0.523950	0.261975	+	0.965075i	$0.415626\pi$
$798$	0	0
$799$	−27513.1	−1.21820
$800$	0	0
$801$	−5059.25	−0.223171
$802$	0	0
$803$	204.803	0.00900044
$804$	0	0
$805$	48567.0	2.12641
$806$	0	0
$807$	−1522.92	−0.0664305
$808$	0	0
$809$	−31672.3	−1.37644	−0.688220	−	0.725502i	$-0.741608\pi$
$809$	−31672.3	−1.37644	−0.688220	+	0.725502i	$0.741608\pi$
$810$	0	0
$811$	978.791	0.0423798	0.0211899	−	0.999775i	$-0.493255\pi$
$811$	978.791	0.0423798	0.0211899	+	0.999775i	$0.493255\pi$
$812$	0	0
$813$	−4585.21	−0.197799
$814$	0	0
$815$	32941.5	1.41582
$816$	0	0
$817$	−14951.0	−0.640231
$818$	0	0
$819$	2304.76	0.0983332
$820$	0	0
$821$	14144.2	0.601261	0.300630	−	0.953741i	$-0.402803\pi$
$821$	14144.2	0.601261	0.300630	+	0.953741i	$0.402803\pi$
$822$	0	0
$823$	29145.9	1.23446	0.617232	−	0.786781i	$-0.288254\pi$
$823$	29145.9	1.23446	0.617232	+	0.786781i	$0.288254\pi$
$824$	0	0
$825$	11419.8	0.481924
$826$	0	0
$827$	41540.2	1.74667	0.873333	−	0.487123i	$-0.161954\pi$
$827$	41540.2	1.74667	0.873333	+	0.487123i	$0.161954\pi$
$828$	0	0
$829$	−23632.8	−0.990111	−0.495056	−	0.868861i	$-0.664852\pi$
$829$	−23632.8	−0.990111	−0.495056	+	0.868861i	$0.664852\pi$
$830$	0	0
$831$	−11658.3	−0.486671
$832$	0	0
$833$	5242.00	0.218037
$834$	0	0
$835$	45533.0	1.88711
$836$	0	0
$837$	−5254.28	−0.216983
$838$	0	0
$839$	4136.95	0.170230	0.0851152	−	0.996371i	$-0.472874\pi$
$839$	4136.95	0.170230	0.0851152	+	0.996371i	$0.472874\pi$
$840$	0	0
$841$	17305.7	0.709571
$842$	0	0
$843$	−8116.13	−0.331595
$844$	0	0
$845$	−3404.77	−0.138612
$846$	0	0
$847$	−22601.2	−0.916866
$848$	0	0
$849$	−8833.86	−0.357099
$850$	0	0
$851$	17485.2	0.704329
$852$	0	0
$853$	−17049.0	−0.684345	−0.342173	−	0.939637i	$-0.611163\pi$
$853$	−17049.0	−0.684345	−0.342173	+	0.939637i	$0.611163\pi$
$854$	0	0
$855$	12350.9	0.494025
$856$	0	0
$857$	−692.763	−0.0276130	−0.0138065	−	0.999905i	$-0.504395\pi$
$857$	−692.763	−0.0276130	−0.0138065	+	0.999905i	$0.504395\pi$
$858$	0	0
$859$	18591.1	0.738441	0.369220	−	0.929342i	$-0.379625\pi$
$859$	18591.1	0.738441	0.369220	+	0.929342i	$0.379625\pi$
$860$	0	0
$861$	10344.9	0.409468
$862$	0	0
$863$	−13759.6	−0.542738	−0.271369	−	0.962475i	$-0.587476\pi$
$863$	−13759.6	−0.542738	−0.271369	+	0.962475i	$0.587476\pi$
$864$	0	0
$865$	62050.5	2.43905
$866$	0	0
$867$	−25891.9	−1.01423
$868$	0	0
$869$	9080.19	0.354458
$870$	0	0
$871$	−2149.26	−0.0836108
$872$	0	0
$873$	14836.8	0.575199
$874$	0	0
$875$	−61864.5	−2.39017
$876$	0	0
$877$	10418.1	0.401133	0.200566	−	0.979680i	$-0.435722\pi$
$877$	10418.1	0.401133	0.200566	+	0.979680i	$0.435722\pi$
$878$	0	0
$879$	11507.7	0.441575
$880$	0	0
$881$	−10012.1	−0.382880	−0.191440	−	0.981504i	$-0.561316\pi$
$881$	−10012.1	−0.382880	−0.191440	+	0.981504i	$0.561316\pi$
$882$	0	0
$883$	18276.9	0.696565	0.348283	−	0.937390i	$-0.386765\pi$
$883$	18276.9	0.696565	0.348283	+	0.937390i	$0.386765\pi$
$884$	0	0
$885$	−26956.9	−1.02389
$886$	0	0
$887$	−8044.35	−0.304513	−0.152256	−	0.988341i	$-0.548654\pi$
$887$	−8044.35	−0.304513	−0.152256	+	0.988341i	$0.548654\pi$
$888$	0	0
$889$	−3463.49	−0.130666
$890$	0	0
$891$	−1097.73	−0.0412743
$892$	0	0
$893$	16103.7	0.603461
$894$	0	0
$895$	−31519.1	−1.17717
$896$	0	0
$897$	4772.71	0.177655
$898$	0	0
$899$	39736.5	1.47418
$900$	0	0
$901$	−73176.8	−2.70574
$902$	0	0
$903$	−12971.1	−0.478019
$904$	0	0
$905$	−12929.9	−0.474923
$906$	0	0
$907$	−36799.1	−1.34718	−0.673592	−	0.739104i	$-0.735249\pi$
$907$	−36799.1	−1.34718	−0.673592	+	0.739104i	$0.735249\pi$
$908$	0	0
$909$	3574.64	0.130433
$910$	0	0
$911$	4530.89	0.164781	0.0823904	−	0.996600i	$-0.473745\pi$
$911$	4530.89	0.164781	0.0823904	+	0.996600i	$0.473745\pi$
$912$	0	0
$913$	14101.1	0.511149
$914$	0	0
$915$	13566.9	0.490172
$916$	0	0
$917$	−43244.1	−1.55730
$918$	0	0
$919$	31538.8	1.13207	0.566033	−	0.824382i	$-0.308477\pi$
$919$	31538.8	1.13207	0.566033	+	0.824382i	$0.308477\pi$
$920$	0	0
$921$	12204.6	0.436651
$922$	0	0
$923$	−11738.7	−0.418619
$924$	0	0
$925$	−40132.5	−1.42654
$926$	0	0
$927$	−14779.6	−0.523651
$928$	0	0
$929$	−39177.6	−1.38361	−0.691805	−	0.722084i	$-0.743184\pi$
$929$	−39177.6	−1.38361	−0.691805	+	0.722084i	$0.743184\pi$
$930$	0	0
$931$	−3068.20	−0.108009
$932$	0	0
$933$	21462.6	0.753111
$934$	0	0
$935$	31774.6	1.11138
$936$	0	0
$937$	−25066.2	−0.873936	−0.436968	−	0.899477i	$-0.643948\pi$
$937$	−25066.2	−0.873936	−0.436968	+	0.899477i	$0.643948\pi$
$938$	0	0
$939$	−19318.1	−0.671377
$940$	0	0
$941$	−18774.7	−0.650412	−0.325206	−	0.945643i	$-0.605434\pi$
$941$	−18774.7	−0.650412	−0.325206	+	0.945643i	$0.605434\pi$
$942$	0	0
$943$	21422.2	0.739770
$944$	0	0
$945$	10715.3	0.368856
$946$	0	0
$947$	45833.2	1.57273	0.786367	−	0.617760i	$-0.211960\pi$
$947$	45833.2	1.57273	0.786367	+	0.617760i	$0.211960\pi$
$948$	0	0
$949$	−196.458	−0.00672000
$950$	0	0
$951$	23163.6	0.789834
$952$	0	0
$953$	33232.5	1.12960	0.564798	−	0.825229i	$-0.308954\pi$
$953$	33232.5	1.12960	0.564798	+	0.825229i	$0.308954\pi$
$954$	0	0
$955$	80604.4	2.73120
$956$	0	0
$957$	8301.82	0.280418
$958$	0	0
$959$	−18266.5	−0.615073
$960$	0	0
$961$	8079.27	0.271198
$962$	0	0
$963$	5802.07	0.194153
$964$	0	0
$965$	85729.3	2.85982
$966$	0	0
$967$	18054.3	0.600400	0.300200	−	0.953876i	$-0.402947\pi$
$967$	18054.3	0.600400	0.300200	+	0.953876i	$0.402947\pi$
$968$	0	0
$969$	23781.8	0.788421
$970$	0	0
$971$	26222.9	0.866667	0.433333	−	0.901234i	$-0.357337\pi$
$971$	26222.9	0.866667	0.433333	+	0.901234i	$0.357337\pi$
$972$	0	0
$973$	57354.4	1.88972
$974$	0	0
$975$	−10954.5	−0.359819
$976$	0	0
$977$	−5968.90	−0.195457	−0.0977287	−	0.995213i	$-0.531158\pi$
$977$	−5968.90	−0.195457	−0.0977287	+	0.995213i	$0.531158\pi$
$978$	0	0
$979$	7618.25	0.248703
$980$	0	0
$981$	−2678.08	−0.0871607
$982$	0	0
$983$	−22537.8	−0.731277	−0.365639	−	0.930757i	$-0.619149\pi$
$983$	−22537.8	−0.731277	−0.365639	+	0.930757i	$0.619149\pi$
$984$	0	0
$985$	−76044.9	−2.45989
$986$	0	0
$987$	13971.2	0.450565
$988$	0	0
$989$	−26860.6	−0.863617
$990$	0	0
$991$	−11681.6	−0.374447	−0.187223	−	0.982317i	$-0.559949\pi$
$991$	−11681.6	−0.374447	−0.187223	+	0.982317i	$0.559949\pi$
$992$	0	0
$993$	−8821.22	−0.281906
$994$	0	0
$995$	41729.2	1.32955
$996$	0	0
$997$	−23095.6	−0.733645	−0.366822	−	0.930291i	$-0.619554\pi$
$997$	−23095.6	−0.733645	−0.366822	+	0.930291i	$0.619554\pi$
$998$	0	0
$999$	3857.74	0.122176

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.s.1.1		3
3.2	odd	2		1872.4.a.bl.1.3		3
4.3	odd	2		312.4.a.h.1.1	✓	3
8.3	odd	2		2496.4.a.bm.1.3		3
8.5	even	2		2496.4.a.bq.1.3		3
12.11	even	2		936.4.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.h.1.1	✓	3	4.3	odd	2
624.4.a.s.1.1		3	1.1	even	1	trivial
936.4.a.l.1.3		3	12.11	even	2
1872.4.a.bl.1.3		3	3.2	odd	2
2496.4.a.bm.1.3		3	8.3	odd	2
2496.4.a.bq.1.3		3	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -20.1466 q^{5} +19.6988 q^{7} +9.00000 q^{9} -13.5523 q^{11} +13.0000 q^{13} +60.4397 q^{15} +116.377 q^{17} -68.1169 q^{19} -59.0964 q^{21} -122.377 q^{23} +280.884 q^{25} -27.0000 q^{27} +204.193 q^{29} +194.603 q^{31} +40.6568 q^{33} -396.863 q^{35} -142.879 q^{37} -39.0000 q^{39} -175.051 q^{41} +219.490 q^{43} -181.319 q^{45} -236.413 q^{47} +45.0432 q^{49} -349.132 q^{51} -628.790 q^{53} +273.031 q^{55} +204.351 q^{57} -446.014 q^{59} +224.470 q^{61} +177.289 q^{63} -261.905 q^{65} -165.328 q^{67} +367.132 q^{69} -902.979 q^{71} -15.1121 q^{73} -842.651 q^{75} -266.963 q^{77} -670.013 q^{79} +81.0000 q^{81} -1040.50 q^{83} -2344.60 q^{85} -612.579 q^{87} -562.139 q^{89} +256.085 q^{91} -583.809 q^{93} +1372.32 q^{95} +1648.53 q^{97} -121.970 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 4 q^{5} - 6 q^{7} + 27 q^{9} - 32 q^{11} + 39 q^{13} + 12 q^{15} + 158 q^{17} - 70 q^{19} + 18 q^{21} - 176 q^{23} + 209 q^{25} - 81 q^{27} + 222 q^{29} - 54 q^{31} + 96 q^{33} - 496 q^{35}+ \cdots - 288 q^{99}+O(q^{100}) \) 3 * q - 9 * q^3 - 4 * q^5 - 6 * q^7 + 27 * q^9 - 32 * q^11 + 39 * q^13 + 12 * q^15 + 158 * q^17 - 70 * q^19 + 18 * q^21 - 176 * q^23 + 209 * q^25 - 81 * q^27 + 222 * q^29 - 54 * q^31 + 96 * q^33 - 496 * q^35 - 90 * q^37 - 117 * q^39 + 104 * q^41 - 140 * q^43 - 36 * q^45 - 328 * q^47 - 65 * q^49 - 474 * q^51 - 358 * q^53 - 32 * q^55 + 210 * q^57 - 1060 * q^59 - 186 * q^61 - 54 * q^63 - 52 * q^65 - 354 * q^67 + 528 * q^69 - 1692 * q^71 - 974 * q^73 - 627 * q^75 - 384 * q^77 - 776 * q^79 + 243 * q^81 - 2340 * q^83 - 2824 * q^85 - 666 * q^87 - 684 * q^89 - 78 * q^91 + 162 * q^93 - 742 * q^97 - 288 * q^99

Introduction

L-functions

Modular forms

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