LMFDB - Embedded newform 1872.4.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, 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("1872.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.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:	$110.451575531$
Analytic rank:	$0$
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			1.3
Root			$-0.388065$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.1466 q^{5} +19.6988 q^{7} +13.5523 q^{11} +13.0000 q^{13} -116.377 q^{17} -68.1169 q^{19} +122.377 q^{23} +280.884 q^{25} -204.193 q^{29} +194.603 q^{31} +396.863 q^{35} -142.879 q^{37} +175.051 q^{41} +219.490 q^{43} +236.413 q^{47} +45.0432 q^{49} +628.790 q^{53} +273.031 q^{55} +446.014 q^{59} +224.470 q^{61} +261.905 q^{65} -165.328 q^{67} +902.979 q^{71} -15.1121 q^{73} +266.963 q^{77} -670.013 q^{79} +1040.50 q^{83} -2344.60 q^{85} +562.139 q^{89} +256.085 q^{91} -1372.32 q^{95} +1648.53 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{5} - 6 q^{7} + 32 q^{11} + 39 q^{13} - 158 q^{17} - 70 q^{19} + 176 q^{23} + 209 q^{25} - 222 q^{29} - 54 q^{31} + 496 q^{35} - 90 q^{37} - 104 q^{41} - 140 q^{43} + 328 q^{47} - 65 q^{49} + 358 q^{53}+ \cdots - 742 q^{97}+O(q^{100}) $ 3 * q + 4 * q^5 - 6 * q^7 + 32 * q^11 + 39 * q^13 - 158 * q^17 - 70 * q^19 + 176 * q^23 + 209 * q^25 - 222 * q^29 - 54 * q^31 + 496 * q^35 - 90 * q^37 - 104 * q^41 - 140 * q^43 + 328 * q^47 - 65 * q^49 + 358 * q^53 - 32 * q^55 + 1060 * q^59 - 186 * q^61 + 52 * q^65 - 354 * q^67 + 1692 * q^71 - 974 * q^73 + 384 * q^77 - 776 * q^79 + 2340 * q^83 - 2824 * q^85 + 684 * q^89 - 78 * q^91 - 742 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	20.1466	1.80196	0.900981	−	0.433858i	$-0.142848\pi$
$5$	20.1466	1.80196	0.900981	+	0.433858i	$0.142848\pi$
$6$	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
$9$	0	0
$10$	0	0
$11$	13.5523	0.371469	0.185735	−	0.982600i	$-0.440534\pi$
$11$	13.5523	0.371469	0.185735	+	0.982600i	$0.440534\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−116.377	−1.66033	−0.830165	−	0.557518i	$-0.811754\pi$
$17$	−116.377	−1.66033	−0.830165	+	0.557518i	$0.811754\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	122.377	1.10945	0.554726	−	0.832033i	$-0.312823\pi$
$23$	122.377	1.10945	0.554726	+	0.832033i	$0.312823\pi$
$24$	0	0
$25$	280.884	2.24707
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−204.193	−1.30751	−0.653753	−	0.756708i	$-0.726806\pi$
$29$	−204.193	−1.30751	−0.653753	+	0.756708i	$0.726806\pi$
$30$	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
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	175.051	0.666788	0.333394	−	0.942788i	$-0.391806\pi$
$41$	175.051	0.666788	0.333394	+	0.942788i	$0.391806\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	236.413	0.733711	0.366855	−	0.930278i	$-0.380434\pi$
$47$	236.413	0.733711	0.366855	+	0.930278i	$0.380434\pi$
$48$	0	0
$49$	45.0432	0.131321
$50$	0	0
$51$	0	0
$52$	0	0
$53$	628.790	1.62964	0.814820	−	0.579714i	$-0.196836\pi$
$53$	628.790	1.62964	0.814820	+	0.579714i	$0.196836\pi$
$54$	0	0
$55$	273.031	0.669373
$56$	0	0
$57$	0	0
$58$	0	0
$59$	446.014	0.984170	0.492085	−	0.870547i	$-0.336235\pi$
$59$	446.014	0.984170	0.492085	+	0.870547i	$0.336235\pi$
$60$	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
$63$	0	0
$64$	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
$69$	0	0
$70$	0	0
$71$	902.979	1.50935	0.754675	−	0.656098i	$-0.227794\pi$
$71$	902.979	1.50935	0.754675	+	0.656098i	$0.227794\pi$
$72$	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
$75$	0	0
$76$	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
$81$	0	0
$82$	0	0
$83$	1040.50	1.37602	0.688010	−	0.725701i	$-0.258485\pi$
$83$	1040.50	1.37602	0.688010	+	0.725701i	$0.258485\pi$
$84$	0	0
$85$	−2344.60	−2.99185
$86$	0	0
$87$	0	0
$88$	0	0
$89$	562.139	0.669512	0.334756	−	0.942305i	$-0.391346\pi$
$89$	562.139	0.669512	0.334756	+	0.942305i	$0.391346\pi$
$90$	0	0
$91$	256.085	0.295000
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	−397.182	−0.391298	−0.195649	−	0.980674i	$-0.562681\pi$
$101$	−397.182	−0.391298	−0.195649	+	0.980674i	$0.562681\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	−644.674	−0.582458	−0.291229	−	0.956653i	$-0.594064\pi$
$107$	−644.674	−0.582458	−0.291229	+	0.956653i	$0.594064\pi$
$108$	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
$111$	0	0
$112$	0	0
$113$	−1902.40	−1.58374	−0.791872	−	0.610687i	$-0.790893\pi$
$113$	−1902.40	−1.58374	−0.791872	+	0.610687i	$0.790893\pi$
$114$	0	0
$115$	2465.48	1.99919
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2292.49	−1.76599
$120$	0	0
$121$	−1147.34	−0.862011
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	0	0
$131$	2195.26	1.46413	0.732066	−	0.681234i	$-0.238556\pi$
$131$	2195.26	1.46413	0.732066	+	0.681234i	$0.238556\pi$
$132$	0	0
$133$	−1341.82	−0.874817
$134$	0	0
$135$	0	0
$136$	0	0
$137$	927.288	0.578274	0.289137	−	0.957288i	$-0.406632\pi$
$137$	927.288	0.578274	0.289137	+	0.957288i	$0.406632\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	176.179	0.103027
$144$	0	0
$145$	−4113.78	−2.35608
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3078.29	−1.69251	−0.846254	−	0.532780i	$-0.821147\pi$
$149$	−3078.29	−1.69251	−0.846254	+	0.532780i	$0.821147\pi$
$150$	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
$153$	0	0
$154$	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
$159$	0	0
$160$	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
$165$	0	0
$166$	0	0
$167$	2260.09	1.04725	0.523626	−	0.851948i	$-0.324579\pi$
$167$	2260.09	1.04725	0.523626	+	0.851948i	$0.324579\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3079.96	1.35355	0.676777	−	0.736188i	$-0.263376\pi$
$173$	3079.96	1.35355	0.676777	+	0.736188i	$0.263376\pi$
$174$	0	0
$175$	5533.07	2.39006
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1564.49	−0.653272	−0.326636	−	0.945150i	$-0.605915\pi$
$179$	−1564.49	−0.653272	−0.326636	+	0.945150i	$0.605915\pi$
$180$	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
$183$	0	0
$184$	0	0
$185$	−2878.53	−1.14396
$186$	0	0
$187$	−1577.17	−0.616761
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4000.90	1.51568	0.757841	−	0.652439i	$-0.226254\pi$
$191$	4000.90	1.51568	0.757841	+	0.652439i	$0.226254\pi$
$192$	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
$195$	0	0
$196$	0	0
$197$	−3774.59	−1.36512	−0.682559	−	0.730831i	$-0.739133\pi$
$197$	−3774.59	−1.36512	−0.682559	+	0.730831i	$0.739133\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	−4022.36	−1.39071
$204$	0	0
$205$	3526.67	1.20153
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	4421.97	1.40268
$216$	0	0
$217$	3833.45	1.19922
$218$	0	0
$219$	0	0
$220$	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
$225$	0	0
$226$	0	0
$227$	1797.20	0.525480	0.262740	−	0.964867i	$-0.415374\pi$
$227$	1797.20	0.525480	0.262740	+	0.964867i	$0.415374\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	4501.40	1.26565	0.632826	−	0.774294i	$-0.281895\pi$
$233$	4501.40	1.26565	0.632826	+	0.774294i	$0.281895\pi$
$234$	0	0
$235$	4762.91	1.32212
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6948.26	−1.88053	−0.940263	−	0.340450i	$-0.889421\pi$
$239$	−6948.26	−1.88053	−0.940263	+	0.340450i	$0.889421\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	907.465	0.236636
$246$	0	0
$247$	−885.519	−0.228114
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2840.45	−0.714293	−0.357147	−	0.934048i	$-0.616250\pi$
$251$	−2840.45	−0.714293	−0.357147	+	0.934048i	$0.616250\pi$
$252$	0	0
$253$	1658.49	0.412127
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1500.98	0.364313	0.182156	−	0.983270i	$-0.441692\pi$
$257$	1500.98	0.364313	0.182156	+	0.983270i	$0.441692\pi$
$258$	0	0
$259$	−2814.55	−0.675242
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−690.021	−0.161781	−0.0808907	−	0.996723i	$-0.525776\pi$
$263$	−690.021	−0.161781	−0.0808907	+	0.996723i	$0.525776\pi$
$264$	0	0
$265$	12667.9	2.93655
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−507.641	−0.115061	−0.0575305	−	0.998344i	$-0.518323\pi$
$269$	−507.641	−0.115061	−0.0575305	+	0.998344i	$0.518323\pi$
$270$	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
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	−2705.38	−0.574339	−0.287170	−	0.957880i	$-0.592714\pi$
$281$	−2705.38	−0.574339	−0.287170	+	0.957880i	$0.592714\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	3448.29	0.709220
$288$	0	0
$289$	8630.65	1.75670
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3835.89	0.764830	0.382415	−	0.923991i	$-0.375092\pi$
$293$	3835.89	0.764830	0.382415	+	0.923991i	$0.375092\pi$
$294$	0	0
$295$	8985.63	1.77344
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1590.90	0.307707
$300$	0	0
$301$	4323.70	0.827953
$302$	0	0
$303$	0	0
$304$	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
$309$	0	0
$310$	0	0
$311$	7154.18	1.30443	0.652213	−	0.758036i	$-0.273841\pi$
$311$	7154.18	1.30443	0.652213	+	0.758036i	$0.273841\pi$
$312$	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
$315$	0	0
$316$	0	0
$317$	7721.21	1.36803	0.684016	−	0.729467i	$-0.260232\pi$
$317$	7721.21	1.36803	0.684016	+	0.729467i	$0.260232\pi$
$318$	0	0
$319$	−2767.27	−0.485698
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7927.25	1.36559
$324$	0	0
$325$	3651.49	0.623225
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	2637.31	0.418822
$342$	0	0
$343$	−5869.40	−0.923958
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10398.3	−1.60868	−0.804339	−	0.594170i	$-0.797481\pi$
$347$	−10398.3	−1.60868	−0.804339	+	0.594170i	$0.797481\pi$
$348$	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
$351$	0	0
$352$	0	0
$353$	−11216.7	−1.69124	−0.845619	−	0.533787i	$-0.820769\pi$
$353$	−11216.7	−1.69124	−0.845619	+	0.533787i	$0.820769\pi$
$354$	0	0
$355$	18191.9	2.71979
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1447.78	0.212844	0.106422	−	0.994321i	$-0.466061\pi$
$359$	1447.78	0.212844	0.106422	+	0.994321i	$0.466061\pi$
$360$	0	0
$361$	−2219.09	−0.323530
$362$	0	0
$363$	0	0
$364$	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
$369$	0	0
$370$	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
$375$	0	0
$376$	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
$381$	0	0
$382$	0	0
$383$	−3729.09	−0.497514	−0.248757	−	0.968566i	$-0.580022\pi$
$383$	−3729.09	−0.497514	−0.248757	+	0.968566i	$0.580022\pi$
$384$	0	0
$385$	5378.39	0.711969
$386$	0	0
$387$	0	0
$388$	0	0
$389$	13546.0	1.76558	0.882788	−	0.469771i	$-0.155664\pi$
$389$	13546.0	1.76558	0.882788	+	0.469771i	$0.155664\pi$
$390$	0	0
$391$	−14241.9	−1.84206
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	2852.32	0.355207	0.177603	−	0.984102i	$-0.443166\pi$
$401$	2852.32	0.355207	0.177603	+	0.984102i	$0.443166\pi$
$402$	0	0
$403$	2529.84	0.312705
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	8785.94	1.04680
$414$	0	0
$415$	20962.5	2.47954
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10198.0	1.18903	0.594514	−	0.804085i	$-0.297344\pi$
$419$	10198.0	1.18903	0.594514	+	0.804085i	$0.297344\pi$
$420$	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
$423$	0	0
$424$	0	0
$425$	−32688.4	−3.73088
$426$	0	0
$427$	4421.79	0.501137
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3656.22	0.408617	0.204309	−	0.978907i	$-0.434505\pi$
$431$	3656.22	0.408617	0.204309	+	0.978907i	$0.434505\pi$
$432$	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
$435$	0	0
$436$	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
$441$	0	0
$442$	0	0
$443$	−4245.96	−0.455376	−0.227688	−	0.973734i	$-0.573117\pi$
$443$	−4245.96	−0.455376	−0.227688	+	0.973734i	$0.573117\pi$
$444$	0	0
$445$	11325.2	1.20644
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2021.21	0.212443	0.106222	−	0.994342i	$-0.466125\pi$
$449$	2021.21	0.212443	0.106222	+	0.994342i	$0.466125\pi$
$450$	0	0
$451$	2372.33	0.247691
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	−1519.14	−0.153478	−0.0767392	−	0.997051i	$-0.524451\pi$
$461$	−1519.14	−0.153478	−0.0767392	+	0.997051i	$0.524451\pi$
$462$	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
$465$	0	0
$466$	0	0
$467$	−10147.9	−1.00554	−0.502771	−	0.864420i	$-0.667686\pi$
$467$	−10147.9	−1.00554	−0.502771	+	0.864420i	$0.667686\pi$
$468$	0	0
$469$	−3256.76	−0.320647
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2974.59	0.289158
$474$	0	0
$475$	−19132.9	−1.84816
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14277.3	−1.36190	−0.680948	−	0.732332i	$-0.738432\pi$
$479$	−14277.3	−1.36190	−0.680948	+	0.732332i	$0.738432\pi$
$480$	0	0
$481$	−1857.43	−0.176074
$482$	0	0
$483$	0	0
$484$	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
$489$	0	0
$490$	0	0
$491$	511.287	0.0469940	0.0234970	−	0.999724i	$-0.492520\pi$
$491$	511.287	0.0469940	0.0234970	+	0.999724i	$0.492520\pi$
$492$	0	0
$493$	23763.4	2.17089
$494$	0	0
$495$	0	0
$496$	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
$501$	0	0
$502$	0	0
$503$	20675.7	1.83277	0.916386	−	0.400295i	$-0.131092\pi$
$503$	20675.7	1.83277	0.916386	+	0.400295i	$0.131092\pi$
$504$	0	0
$505$	−8001.85	−0.705104
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15364.4	1.33795	0.668975	−	0.743285i	$-0.266734\pi$
$509$	15364.4	1.33795	0.668975	+	0.743285i	$0.266734\pi$
$510$	0	0
$511$	−297.691	−0.0257712
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−33084.1	−2.83080
$516$	0	0
$517$	3203.93	0.272551
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13318.8	1.11997	0.559986	−	0.828502i	$-0.310807\pi$
$521$	13318.8	1.11997	0.559986	+	0.828502i	$0.310807\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	−22647.3	−1.87198
$528$	0	0
$529$	2809.17	0.230885
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2275.66	0.184934
$534$	0	0
$535$	−12988.0	−1.04957
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	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
$549$	0	0
$550$	0	0
$551$	13909.0	1.07539
$552$	0	0
$553$	−13198.5	−1.01493
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3897.95	0.296519	0.148260	−	0.988948i	$-0.452633\pi$
$557$	3897.95	0.296519	0.148260	+	0.988948i	$0.452633\pi$
$558$	0	0
$559$	2853.37	0.215894
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18806.2	−1.40779	−0.703894	−	0.710305i	$-0.748557\pi$
$563$	−18806.2	−1.40779	−0.703894	+	0.710305i	$0.748557\pi$
$564$	0	0
$565$	−38326.9	−2.85385
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24680.0	−1.81835	−0.909175	−	0.416415i	$-0.863286\pi$
$569$	−24680.0	−1.81835	−0.909175	+	0.416415i	$0.863286\pi$
$570$	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
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	20496.6	1.46358
$582$	0	0
$583$	8521.52	0.605361
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27148.1	1.90889	0.954447	−	0.298379i	$-0.0964459\pi$
$587$	27148.1	1.90889	0.954447	+	0.298379i	$0.0964459\pi$
$588$	0	0
$589$	−13255.7	−0.927323
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4549.19	−0.315030	−0.157515	−	0.987517i	$-0.550348\pi$
$593$	−4549.19	−0.315030	−0.157515	+	0.987517i	$0.550348\pi$
$594$	0	0
$595$	−46185.8	−3.18224
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2875.51	0.196144	0.0980720	−	0.995179i	$-0.468732\pi$
$599$	2875.51	0.196144	0.0980720	+	0.995179i	$0.468732\pi$
$600$	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
$603$	0	0
$604$	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
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	−5410.04	−0.352999	−0.176499	−	0.984301i	$-0.556477\pi$
$617$	−5410.04	−0.352999	−0.176499	+	0.984301i	$0.556477\pi$
$618$	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
$621$	0	0
$622$	0	0
$623$	11073.5	0.712117
$624$	0	0
$625$	28160.1	1.80225
$626$	0	0
$627$	0	0
$628$	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
$633$	0	0
$634$	0	0
$635$	−3542.21	−0.221367
$636$	0	0
$637$	585.561	0.0364220
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14112.2	0.869573	0.434787	−	0.900534i	$-0.356824\pi$
$641$	14112.2	0.869573	0.434787	+	0.900534i	$0.356824\pi$
$642$	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
$645$	0	0
$646$	0	0
$647$	12214.8	0.742212	0.371106	−	0.928590i	$-0.378979\pi$
$647$	12214.8	0.742212	0.371106	+	0.928590i	$0.378979\pi$
$648$	0	0
$649$	6044.49	0.365589
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31803.3	−1.90591	−0.952955	−	0.303111i	$-0.901975\pi$
$653$	−31803.3	−1.90591	−0.952955	+	0.303111i	$0.901975\pi$
$654$	0	0
$655$	44227.0	2.63831
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21545.8	−1.27360	−0.636802	−	0.771027i	$-0.719743\pi$
$659$	−21545.8	−1.27360	−0.636802	+	0.771027i	$0.719743\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	−27033.1	−1.57639
$666$	0	0
$667$	−24988.5	−1.45062
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	−17169.2	−0.974689	−0.487345	−	0.873210i	$-0.662034\pi$
$677$	−17169.2	−0.974689	−0.487345	+	0.873210i	$0.662034\pi$
$678$	0	0
$679$	32474.1	1.83541
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1579.31	0.0884781	0.0442391	−	0.999021i	$-0.485914\pi$
$683$	1579.31	0.0884781	0.0442391	+	0.999021i	$0.485914\pi$
$684$	0	0
$685$	18681.7	1.04203
$686$	0	0
$687$	0	0
$688$	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
$693$	0	0
$694$	0	0
$695$	58658.0	3.20148
$696$	0	0
$697$	−20371.9	−1.10709
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7893.64	−0.425305	−0.212652	−	0.977128i	$-0.568210\pi$
$701$	−7893.64	−0.425305	−0.212652	+	0.977128i	$0.568210\pi$
$702$	0	0
$703$	9732.49	0.522145
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	23814.9	1.25088
$714$	0	0
$715$	3549.41	0.185651
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4029.94	−0.209028	−0.104514	−	0.994523i	$-0.533329\pi$
$719$	−4029.94	−0.209028	−0.104514	+	0.994523i	$0.533329\pi$
$720$	0	0
$721$	−32348.9	−1.67092
$722$	0	0
$723$	0	0
$724$	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
$729$	0	0
$730$	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
$735$	0	0
$736$	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
$741$	0	0
$742$	0	0
$743$	−31072.6	−1.53424	−0.767122	−	0.641501i	$-0.778312\pi$
$743$	−31072.6	−1.53424	−0.767122	+	0.641501i	$0.778312\pi$
$744$	0	0
$745$	−62017.0	−3.04983
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	−11123.3	−0.529857	−0.264928	−	0.964268i	$-0.585348\pi$
$761$	−11123.3	−0.529857	−0.264928	+	0.964268i	$0.585348\pi$
$762$	0	0
$763$	−5861.67	−0.278122
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	−9690.00	−0.450873	−0.225437	−	0.974258i	$-0.572381\pi$
$773$	−9690.00	−0.450873	−0.225437	+	0.974258i	$0.572381\pi$
$774$	0	0
$775$	54660.8	2.53351
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11923.9	−0.548419
$780$	0	0
$781$	12237.4	0.560677
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	−37475.1	−1.68453
$792$	0	0
$793$	2918.11	0.130675
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11789.0	−0.523950	−0.261975	−	0.965075i	$-0.584374\pi$
$797$	−11789.0	−0.523950	−0.261975	+	0.965075i	$0.584374\pi$
$798$	0	0
$799$	−27513.1	−1.21820
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−204.803	−0.00900044
$804$	0	0
$805$	48567.0	2.12641
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31672.3	1.37644	0.688220	−	0.725502i	$-0.258392\pi$
$809$	31672.3	1.37644	0.688220	+	0.725502i	$0.258392\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	−32941.5	−1.41582
$816$	0	0
$817$	−14951.0	−0.640231
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14144.2	−0.601261	−0.300630	−	0.953741i	$-0.597197\pi$
$821$	−14144.2	−0.601261	−0.300630	+	0.953741i	$0.597197\pi$
$822$	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
$825$	0	0
$826$	0	0
$827$	−41540.2	−1.74667	−0.873333	−	0.487123i	$-0.838046\pi$
$827$	−41540.2	−1.74667	−0.873333	+	0.487123i	$0.838046\pi$
$828$	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
$831$	0	0
$832$	0	0
$833$	−5242.00	−0.218037
$834$	0	0
$835$	45533.0	1.88711
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4136.95	−0.170230	−0.0851152	−	0.996371i	$-0.527126\pi$
$839$	−4136.95	−0.170230	−0.0851152	+	0.996371i	$0.527126\pi$
$840$	0	0
$841$	17305.7	0.709571
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3404.77	0.138612
$846$	0	0
$847$	−22601.2	−0.916866
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	692.763	0.0276130	0.0138065	−	0.999905i	$-0.495605\pi$
$857$	692.763	0.0276130	0.0138065	+	0.999905i	$0.495605\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	13759.6	0.542738	0.271369	−	0.962475i	$-0.412524\pi$
$863$	13759.6	0.542738	0.271369	+	0.962475i	$0.412524\pi$
$864$	0	0
$865$	62050.5	2.43905
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9080.19	−0.354458
$870$	0	0
$871$	−2149.26	−0.0836108
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	10012.1	0.382880	0.191440	−	0.981504i	$-0.438684\pi$
$881$	10012.1	0.382880	0.191440	+	0.981504i	$0.438684\pi$
$882$	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
$885$	0	0
$886$	0	0
$887$	8044.35	0.304513	0.152256	−	0.988341i	$-0.451346\pi$
$887$	8044.35	0.304513	0.152256	+	0.988341i	$0.451346\pi$
$888$	0	0
$889$	−3463.49	−0.130666
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−16103.7	−0.603461
$894$	0	0
$895$	−31519.1	−1.17717
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−39736.5	−1.47418
$900$	0	0
$901$	−73176.8	−2.70574
$902$	0	0
$903$	0	0
$904$	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
$909$	0	0
$910$	0	0
$911$	−4530.89	−0.164781	−0.0823904	−	0.996600i	$-0.526255\pi$
$911$	−4530.89	−0.164781	−0.0823904	+	0.996600i	$0.526255\pi$
$912$	0	0
$913$	14101.1	0.511149
$914$	0	0
$915$	0	0
$916$	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
$921$	0	0
$922$	0	0
$923$	11738.7	0.418619
$924$	0	0
$925$	−40132.5	−1.42654
$926$	0	0
$927$	0	0
$928$	0	0
$929$	39177.6	1.38361	0.691805	−	0.722084i	$-0.256816\pi$
$929$	39177.6	1.38361	0.691805	+	0.722084i	$0.256816\pi$
$930$	0	0
$931$	−3068.20	−0.108009
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	18774.7	0.650412	0.325206	−	0.945643i	$-0.394566\pi$
$941$	18774.7	0.650412	0.325206	+	0.945643i	$0.394566\pi$
$942$	0	0
$943$	21422.2	0.739770
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45833.2	−1.57273	−0.786367	−	0.617760i	$-0.788040\pi$
$947$	−45833.2	−1.57273	−0.786367	+	0.617760i	$0.788040\pi$
$948$	0	0
$949$	−196.458	−0.00672000
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−33232.5	−1.12960	−0.564798	−	0.825229i	$-0.691046\pi$
$953$	−33232.5	−1.12960	−0.564798	+	0.825229i	$0.691046\pi$
$954$	0	0
$955$	80604.4	2.73120
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18266.5	0.615073
$960$	0	0
$961$	8079.27	0.271198
$962$	0	0
$963$	0	0
$964$	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
$969$	0	0
$970$	0	0
$971$	−26222.9	−0.866667	−0.433333	−	0.901234i	$-0.642663\pi$
$971$	−26222.9	−0.866667	−0.433333	+	0.901234i	$0.642663\pi$
$972$	0	0
$973$	57354.4	1.88972
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5968.90	0.195457	0.0977287	−	0.995213i	$-0.468842\pi$
$977$	5968.90	0.195457	0.0977287	+	0.995213i	$0.468842\pi$
$978$	0	0
$979$	7618.25	0.248703
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22537.8	0.731277	0.365639	−	0.930757i	$-0.380851\pi$
$983$	22537.8	0.731277	0.365639	+	0.930757i	$0.380851\pi$
$984$	0	0
$985$	−76044.9	−2.45989
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.h.1.1	✓	3	12.11	even	2
624.4.a.s.1.1		3	3.2	odd	2
936.4.a.l.1.3		3	4.3	odd	2
1872.4.a.bl.1.3		3	1.1	even	1	trivial
2496.4.a.bm.1.3		3	24.11	even	2
2496.4.a.bq.1.3		3	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+20.1466 q^{5} +19.6988 q^{7} +13.5523 q^{11} +13.0000 q^{13} -116.377 q^{17} -68.1169 q^{19} +122.377 q^{23} +280.884 q^{25} -204.193 q^{29} +194.603 q^{31} +396.863 q^{35} -142.879 q^{37} +175.051 q^{41} +219.490 q^{43} +236.413 q^{47} +45.0432 q^{49} +628.790 q^{53} +273.031 q^{55} +446.014 q^{59} +224.470 q^{61} +261.905 q^{65} -165.328 q^{67} +902.979 q^{71} -15.1121 q^{73} +266.963 q^{77} -670.013 q^{79} +1040.50 q^{83} -2344.60 q^{85} +562.139 q^{89} +256.085 q^{91} -1372.32 q^{95} +1648.53 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{5} - 6 q^{7} + 32 q^{11} + 39 q^{13} - 158 q^{17} - 70 q^{19} + 176 q^{23} + 209 q^{25} - 222 q^{29} - 54 q^{31} + 496 q^{35} - 90 q^{37} - 104 q^{41} - 140 q^{43} + 328 q^{47} - 65 q^{49} + 358 q^{53}+ \cdots - 742 q^{97}+O(q^{100}) \) 3 * q + 4 * q^5 - 6 * q^7 + 32 * q^11 + 39 * q^13 - 158 * q^17 - 70 * q^19 + 176 * q^23 + 209 * q^25 - 222 * q^29 - 54 * q^31 + 496 * q^35 - 90 * q^37 - 104 * q^41 - 140 * q^43 + 328 * q^47 - 65 * q^49 + 358 * q^53 - 32 * q^55 + 1060 * q^59 - 186 * q^61 + 52 * q^65 - 354 * q^67 + 1692 * q^71 - 974 * q^73 + 384 * q^77 - 776 * q^79 + 2340 * q^83 - 2824 * q^85 + 684 * q^89 - 78 * q^91 - 742 * q^97

Introduction

L-functions

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