LMFDB - Embedded newform 4024.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28143 q^{3} -3.08568 q^{5} +0.442605 q^{7} +7.76776 q^{9} +O(q^{10})$	$q-3.28143 q^{3} -3.08568 q^{5} +0.442605 q^{7} +7.76776 q^{9} +1.35583 q^{11} -2.73678 q^{13} +10.1254 q^{15} +4.78655 q^{17} +2.61153 q^{19} -1.45238 q^{21} +3.60709 q^{23} +4.52140 q^{25} -15.6451 q^{27} -5.20932 q^{29} -0.413942 q^{31} -4.44907 q^{33} -1.36574 q^{35} +0.790740 q^{37} +8.98054 q^{39} -2.32855 q^{41} +4.33765 q^{43} -23.9688 q^{45} -1.17067 q^{47} -6.80410 q^{49} -15.7067 q^{51} -5.33153 q^{53} -4.18366 q^{55} -8.56954 q^{57} +7.03781 q^{59} -6.81666 q^{61} +3.43805 q^{63} +8.44481 q^{65} -9.76211 q^{67} -11.8364 q^{69} -4.57493 q^{71} +4.48228 q^{73} -14.8366 q^{75} +0.600099 q^{77} +15.1812 q^{79} +28.0348 q^{81} -1.79783 q^{83} -14.7697 q^{85} +17.0940 q^{87} -7.04174 q^{89} -1.21131 q^{91} +1.35832 q^{93} -8.05833 q^{95} -13.8492 q^{97} +10.5318 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−3.28143	−1.89453	−0.947266	−	0.320447i	$-0.896167\pi$
$3$	−3.28143	−1.89453	−0.947266	+	0.320447i	$0.896167\pi$
$4$	0	0
$5$	−3.08568	−1.37996	−0.689978	−	0.723830i	$-0.742380\pi$
$5$	−3.08568	−1.37996	−0.689978	+	0.723830i	$0.742380\pi$
$6$	0	0
$7$	0.442605	0.167289	0.0836445	−	0.996496i	$-0.473344\pi$
$7$	0.442605	0.167289	0.0836445	+	0.996496i	$0.473344\pi$
$8$	0	0
$9$	7.76776	2.58925
$10$	0	0
$11$	1.35583	0.408799	0.204400	−	0.978888i	$-0.434476\pi$
$11$	1.35583	0.408799	0.204400	+	0.978888i	$0.434476\pi$
$12$	0	0
$13$	−2.73678	−0.759046	−0.379523	−	0.925182i	$-0.623912\pi$
$13$	−2.73678	−0.759046	−0.379523	+	0.925182i	$0.623912\pi$
$14$	0	0
$15$	10.1254	2.61437
$16$	0	0
$17$	4.78655	1.16091	0.580454	−	0.814293i	$-0.302875\pi$
$17$	4.78655	1.16091	0.580454	+	0.814293i	$0.302875\pi$
$18$	0	0
$19$	2.61153	0.599126	0.299563	−	0.954077i	$-0.403159\pi$
$19$	2.61153	0.599126	0.299563	+	0.954077i	$0.403159\pi$
$20$	0	0
$21$	−1.45238	−0.316934
$22$	0	0
$23$	3.60709	0.752130	0.376065	−	0.926593i	$-0.377277\pi$
$23$	3.60709	0.752130	0.376065	+	0.926593i	$0.377277\pi$
$24$	0	0
$25$	4.52140	0.904280
$26$	0	0
$27$	−15.6451	−3.01089
$28$	0	0
$29$	−5.20932	−0.967346	−0.483673	−	0.875249i	$-0.660698\pi$
$29$	−5.20932	−0.967346	−0.483673	+	0.875249i	$0.660698\pi$
$30$	0	0
$31$	−0.413942	−0.0743462	−0.0371731	−	0.999309i	$-0.511835\pi$
$31$	−0.413942	−0.0743462	−0.0371731	+	0.999309i	$0.511835\pi$
$32$	0	0
$33$	−4.44907	−0.774484
$34$	0	0
$35$	−1.36574	−0.230851
$36$	0	0
$37$	0.790740	0.129997	0.0649984	−	0.997885i	$-0.479296\pi$
$37$	0.790740	0.129997	0.0649984	+	0.997885i	$0.479296\pi$
$38$	0	0
$39$	8.98054	1.43804
$40$	0	0
$41$	−2.32855	−0.363658	−0.181829	−	0.983330i	$-0.558202\pi$
$41$	−2.32855	−0.363658	−0.181829	+	0.983330i	$0.558202\pi$
$42$	0	0
$43$	4.33765	0.661486	0.330743	−	0.943721i	$-0.392701\pi$
$43$	4.33765	0.661486	0.330743	+	0.943721i	$0.392701\pi$
$44$	0	0
$45$	−23.9688	−3.57306
$46$	0	0
$47$	−1.17067	−0.170760	−0.0853799	−	0.996348i	$-0.527210\pi$
$47$	−1.17067	−0.170760	−0.0853799	+	0.996348i	$0.527210\pi$
$48$	0	0
$49$	−6.80410	−0.972014
$50$	0	0
$51$	−15.7067	−2.19938
$52$	0	0
$53$	−5.33153	−0.732342	−0.366171	−	0.930548i	$-0.619331\pi$
$53$	−5.33153	−0.732342	−0.366171	+	0.930548i	$0.619331\pi$
$54$	0	0
$55$	−4.18366	−0.564125
$56$	0	0
$57$	−8.56954	−1.13506
$58$	0	0
$59$	7.03781	0.916245	0.458123	−	0.888889i	$-0.348522\pi$
$59$	7.03781	0.916245	0.458123	+	0.888889i	$0.348522\pi$
$60$	0	0
$61$	−6.81666	−0.872785	−0.436392	−	0.899757i	$-0.643744\pi$
$61$	−6.81666	−0.872785	−0.436392	+	0.899757i	$0.643744\pi$
$62$	0	0
$63$	3.43805	0.433154
$64$	0	0
$65$	8.44481	1.04745
$66$	0	0
$67$	−9.76211	−1.19263	−0.596316	−	0.802750i	$-0.703369\pi$
$67$	−9.76211	−1.19263	−0.596316	+	0.802750i	$0.703369\pi$
$68$	0	0
$69$	−11.8364	−1.42494
$70$	0	0
$71$	−4.57493	−0.542944	−0.271472	−	0.962446i	$-0.587510\pi$
$71$	−4.57493	−0.542944	−0.271472	+	0.962446i	$0.587510\pi$
$72$	0	0
$73$	4.48228	0.524611	0.262306	−	0.964985i	$-0.415517\pi$
$73$	4.48228	0.524611	0.262306	+	0.964985i	$0.415517\pi$
$74$	0	0
$75$	−14.8366	−1.71319
$76$	0	0
$77$	0.600099	0.0683876
$78$	0	0
$79$	15.1812	1.70801	0.854007	−	0.520262i	$-0.174166\pi$
$79$	15.1812	1.70801	0.854007	+	0.520262i	$0.174166\pi$
$80$	0	0
$81$	28.0348	3.11498
$82$	0	0
$83$	−1.79783	−0.197337	−0.0986685	−	0.995120i	$-0.531458\pi$
$83$	−1.79783	−0.197337	−0.0986685	+	0.995120i	$0.531458\pi$
$84$	0	0
$85$	−14.7697	−1.60200
$86$	0	0
$87$	17.0940	1.83267
$88$	0	0
$89$	−7.04174	−0.746423	−0.373211	−	0.927746i	$-0.621743\pi$
$89$	−7.04174	−0.746423	−0.373211	+	0.927746i	$0.621743\pi$
$90$	0	0
$91$	−1.21131	−0.126980
$92$	0	0
$93$	1.35832	0.140851
$94$	0	0
$95$	−8.05833	−0.826767
$96$	0	0
$97$	−13.8492	−1.40617	−0.703087	−	0.711104i	$-0.748196\pi$
$97$	−13.8492	−1.40617	−0.703087	+	0.711104i	$0.748196\pi$
$98$	0	0
$99$	10.5318	1.05849
$100$	0	0
$101$	−1.60295	−0.159500	−0.0797499	−	0.996815i	$-0.525412\pi$
$101$	−1.60295	−0.159500	−0.0797499	+	0.996815i	$0.525412\pi$
$102$	0	0
$103$	−16.5793	−1.63360	−0.816802	−	0.576918i	$-0.804255\pi$
$103$	−16.5793	−1.63360	−0.816802	+	0.576918i	$0.804255\pi$
$104$	0	0
$105$	4.48156	0.437356
$106$	0	0
$107$	2.05821	0.198974	0.0994872	−	0.995039i	$-0.468280\pi$
$107$	2.05821	0.198974	0.0994872	+	0.995039i	$0.468280\pi$
$108$	0	0
$109$	13.0045	1.24560	0.622802	−	0.782379i	$-0.285994\pi$
$109$	13.0045	1.24560	0.622802	+	0.782379i	$0.285994\pi$
$110$	0	0
$111$	−2.59475	−0.246283
$112$	0	0
$113$	7.10480	0.668363	0.334181	−	0.942509i	$-0.391540\pi$
$113$	7.10480	0.668363	0.334181	+	0.942509i	$0.391540\pi$
$114$	0	0
$115$	−11.1303	−1.03791
$116$	0	0
$117$	−21.2586	−1.96536
$118$	0	0
$119$	2.11855	0.194207
$120$	0	0
$121$	−9.16171	−0.832883
$122$	0	0
$123$	7.64097	0.688963
$124$	0	0
$125$	1.47681	0.132090
$126$	0	0
$127$	14.3220	1.27087	0.635434	−	0.772155i	$-0.280821\pi$
$127$	14.3220	1.27087	0.635434	+	0.772155i	$0.280821\pi$
$128$	0	0
$129$	−14.2337	−1.25321
$130$	0	0
$131$	21.4358	1.87286	0.936429	−	0.350857i	$-0.114110\pi$
$131$	21.4358	1.87286	0.936429	+	0.350857i	$0.114110\pi$
$132$	0	0
$133$	1.15588	0.100227
$134$	0	0
$135$	48.2756	4.15490
$136$	0	0
$137$	7.21995	0.616842	0.308421	−	0.951250i	$-0.400199\pi$
$137$	7.21995	0.616842	0.308421	+	0.951250i	$0.400199\pi$
$138$	0	0
$139$	6.70870	0.569025	0.284512	−	0.958672i	$-0.408168\pi$
$139$	6.70870	0.569025	0.284512	+	0.958672i	$0.408168\pi$
$140$	0	0
$141$	3.84147	0.323510
$142$	0	0
$143$	−3.71062	−0.310297
$144$	0	0
$145$	16.0743	1.33490
$146$	0	0
$147$	22.3272	1.84151
$148$	0	0
$149$	7.46934	0.611912	0.305956	−	0.952046i	$-0.401024\pi$
$149$	7.46934	0.611912	0.305956	+	0.952046i	$0.401024\pi$
$150$	0	0
$151$	−1.81980	−0.148093	−0.0740466	−	0.997255i	$-0.523591\pi$
$151$	−1.81980	−0.148093	−0.0740466	+	0.997255i	$0.523591\pi$
$152$	0	0
$153$	37.1808	3.00589
$154$	0	0
$155$	1.27729	0.102595
$156$	0	0
$157$	−17.6017	−1.40477	−0.702384	−	0.711798i	$-0.747881\pi$
$157$	−17.6017	−1.40477	−0.702384	+	0.711798i	$0.747881\pi$
$158$	0	0
$159$	17.4950	1.38745
$160$	0	0
$161$	1.59652	0.125823
$162$	0	0
$163$	11.3873	0.891920	0.445960	−	0.895053i	$-0.352862\pi$
$163$	11.3873	0.891920	0.445960	+	0.895053i	$0.352862\pi$
$164$	0	0
$165$	13.7284	1.06875
$166$	0	0
$167$	−22.3331	−1.72819	−0.864093	−	0.503332i	$-0.832107\pi$
$167$	−22.3331	−1.72819	−0.864093	+	0.503332i	$0.832107\pi$
$168$	0	0
$169$	−5.51005	−0.423850
$170$	0	0
$171$	20.2857	1.55129
$172$	0	0
$173$	−20.8726	−1.58691	−0.793456	−	0.608628i	$-0.791720\pi$
$173$	−20.8726	−1.58691	−0.793456	+	0.608628i	$0.791720\pi$
$174$	0	0
$175$	2.00119	0.151276
$176$	0	0
$177$	−23.0941	−1.73586
$178$	0	0
$179$	13.7125	1.02492	0.512461	−	0.858711i	$-0.328734\pi$
$179$	13.7125	1.02492	0.512461	+	0.858711i	$0.328734\pi$
$180$	0	0
$181$	−4.74659	−0.352811	−0.176406	−	0.984318i	$-0.556447\pi$
$181$	−4.74659	−0.352811	−0.176406	+	0.984318i	$0.556447\pi$
$182$	0	0
$183$	22.3684	1.65352
$184$	0	0
$185$	−2.43997	−0.179390
$186$	0	0
$187$	6.48976	0.474578
$188$	0	0
$189$	−6.92458	−0.503689
$190$	0	0
$191$	7.06306	0.511065	0.255533	−	0.966800i	$-0.417749\pi$
$191$	7.06306	0.511065	0.255533	+	0.966800i	$0.417749\pi$
$192$	0	0
$193$	6.23904	0.449096	0.224548	−	0.974463i	$-0.427909\pi$
$193$	6.23904	0.449096	0.224548	+	0.974463i	$0.427909\pi$
$194$	0	0
$195$	−27.7110	−1.98443
$196$	0	0
$197$	−1.36501	−0.0972529	−0.0486264	−	0.998817i	$-0.515484\pi$
$197$	−1.36501	−0.0972529	−0.0486264	+	0.998817i	$0.515484\pi$
$198$	0	0
$199$	20.3430	1.44208	0.721040	−	0.692894i	$-0.243665\pi$
$199$	20.3430	1.44208	0.721040	+	0.692894i	$0.243665\pi$
$200$	0	0
$201$	32.0337	2.25948
$202$	0	0
$203$	−2.30567	−0.161826
$204$	0	0
$205$	7.18515	0.501833
$206$	0	0
$207$	28.0190	1.94746
$208$	0	0
$209$	3.54080	0.244922
$210$	0	0
$211$	−2.08108	−0.143267	−0.0716337	−	0.997431i	$-0.522821\pi$
$211$	−2.08108	−0.143267	−0.0716337	+	0.997431i	$0.522821\pi$
$212$	0	0
$213$	15.0123	1.02863
$214$	0	0
$215$	−13.3846	−0.912822
$216$	0	0
$217$	−0.183213	−0.0124373
$218$	0	0
$219$	−14.7083	−0.993893
$220$	0	0
$221$	−13.0997	−0.881182
$222$	0	0
$223$	−4.60655	−0.308477	−0.154239	−	0.988034i	$-0.549292\pi$
$223$	−4.60655	−0.308477	−0.154239	+	0.988034i	$0.549292\pi$
$224$	0	0
$225$	35.1211	2.34141
$226$	0	0
$227$	3.94295	0.261703	0.130851	−	0.991402i	$-0.458229\pi$
$227$	3.94295	0.261703	0.130851	+	0.991402i	$0.458229\pi$
$228$	0	0
$229$	−8.62003	−0.569628	−0.284814	−	0.958583i	$-0.591932\pi$
$229$	−8.62003	−0.569628	−0.284814	+	0.958583i	$0.591932\pi$
$230$	0	0
$231$	−1.96918	−0.129563
$232$	0	0
$233$	12.9498	0.848368	0.424184	−	0.905576i	$-0.360561\pi$
$233$	12.9498	0.848368	0.424184	+	0.905576i	$0.360561\pi$
$234$	0	0
$235$	3.61231	0.235641
$236$	0	0
$237$	−49.8159	−3.23589
$238$	0	0
$239$	7.12663	0.460983	0.230492	−	0.973074i	$-0.425967\pi$
$239$	7.12663	0.460983	0.230492	+	0.973074i	$0.425967\pi$
$240$	0	0
$241$	7.29738	0.470066	0.235033	−	0.971987i	$-0.424480\pi$
$241$	7.29738	0.470066	0.235033	+	0.971987i	$0.424480\pi$
$242$	0	0
$243$	−45.0591	−2.89054
$244$	0	0
$245$	20.9953	1.34134
$246$	0	0
$247$	−7.14717	−0.454764
$248$	0	0
$249$	5.89943	0.373861
$250$	0	0
$251$	11.6318	0.734194	0.367097	−	0.930183i	$-0.380352\pi$
$251$	11.6318	0.734194	0.367097	+	0.930183i	$0.380352\pi$
$252$	0	0
$253$	4.89061	0.307470
$254$	0	0
$255$	48.4658	3.03505
$256$	0	0
$257$	19.4121	1.21090	0.605448	−	0.795885i	$-0.292994\pi$
$257$	19.4121	1.21090	0.605448	+	0.795885i	$0.292994\pi$
$258$	0	0
$259$	0.349985	0.0217470
$260$	0	0
$261$	−40.4647	−2.50470
$262$	0	0
$263$	−18.0589	−1.11356	−0.556779	−	0.830661i	$-0.687963\pi$
$263$	−18.0589	−1.11356	−0.556779	+	0.830661i	$0.687963\pi$
$264$	0	0
$265$	16.4514	1.01060
$266$	0	0
$267$	23.1069	1.41412
$268$	0	0
$269$	13.4317	0.818948	0.409474	−	0.912322i	$-0.365712\pi$
$269$	13.4317	0.818948	0.409474	+	0.912322i	$0.365712\pi$
$270$	0	0
$271$	−9.23641	−0.561072	−0.280536	−	0.959843i	$-0.590512\pi$
$271$	−9.23641	−0.561072	−0.280536	+	0.959843i	$0.590512\pi$
$272$	0	0
$273$	3.97483	0.240568
$274$	0	0
$275$	6.13026	0.369669
$276$	0	0
$277$	12.1968	0.732833	0.366416	−	0.930451i	$-0.380585\pi$
$277$	12.1968	0.732833	0.366416	+	0.930451i	$0.380585\pi$
$278$	0	0
$279$	−3.21540	−0.192501
$280$	0	0
$281$	17.0539	1.01735	0.508674	−	0.860959i	$-0.330136\pi$
$281$	17.0539	1.01735	0.508674	+	0.860959i	$0.330136\pi$
$282$	0	0
$283$	−9.77243	−0.580910	−0.290455	−	0.956889i	$-0.593807\pi$
$283$	−9.77243	−0.580910	−0.290455	+	0.956889i	$0.593807\pi$
$284$	0	0
$285$	26.4428	1.56634
$286$	0	0
$287$	−1.03063	−0.0608360
$288$	0	0
$289$	5.91104	0.347708
$290$	0	0
$291$	45.4452	2.66404
$292$	0	0
$293$	16.1334	0.942523	0.471261	−	0.881994i	$-0.343799\pi$
$293$	16.1334	0.942523	0.471261	+	0.881994i	$0.343799\pi$
$294$	0	0
$295$	−21.7164	−1.26438
$296$	0	0
$297$	−21.2121	−1.23085
$298$	0	0
$299$	−9.87180	−0.570901
$300$	0	0
$301$	1.91987	0.110659
$302$	0	0
$303$	5.25997	0.302178
$304$	0	0
$305$	21.0340	1.20440
$306$	0	0
$307$	14.9500	0.853241	0.426621	−	0.904431i	$-0.359704\pi$
$307$	14.9500	0.853241	0.426621	+	0.904431i	$0.359704\pi$
$308$	0	0
$309$	54.4037	3.09492
$310$	0	0
$311$	26.4740	1.50120	0.750602	−	0.660755i	$-0.229764\pi$
$311$	26.4740	1.50120	0.750602	+	0.660755i	$0.229764\pi$
$312$	0	0
$313$	4.42421	0.250071	0.125035	−	0.992152i	$-0.460096\pi$
$313$	4.42421	0.250071	0.125035	+	0.992152i	$0.460096\pi$
$314$	0	0
$315$	−10.6087	−0.597733
$316$	0	0
$317$	2.46196	0.138277	0.0691386	−	0.997607i	$-0.477975\pi$
$317$	2.46196	0.138277	0.0691386	+	0.997607i	$0.477975\pi$
$318$	0	0
$319$	−7.06297	−0.395450
$320$	0	0
$321$	−6.75386	−0.376964
$322$	0	0
$323$	12.5002	0.695530
$324$	0	0
$325$	−12.3741	−0.686389
$326$	0	0
$327$	−42.6733	−2.35984
$328$	0	0
$329$	−0.518145	−0.0285662
$330$	0	0
$331$	33.6829	1.85138	0.925689	−	0.378287i	$-0.123487\pi$
$331$	33.6829	1.85138	0.925689	+	0.378287i	$0.123487\pi$
$332$	0	0
$333$	6.14228	0.336595
$334$	0	0
$335$	30.1227	1.64578
$336$	0	0
$337$	−26.5623	−1.44694	−0.723470	−	0.690355i	$-0.757454\pi$
$337$	−26.5623	−1.44694	−0.723470	+	0.690355i	$0.757454\pi$
$338$	0	0
$339$	−23.3139	−1.26624
$340$	0	0
$341$	−0.561237	−0.0303927
$342$	0	0
$343$	−6.10976	−0.329896
$344$	0	0
$345$	36.5233	1.96635
$346$	0	0
$347$	−7.82926	−0.420297	−0.210148	−	0.977669i	$-0.567395\pi$
$347$	−7.82926	−0.420297	−0.210148	+	0.977669i	$0.567395\pi$
$348$	0	0
$349$	11.6817	0.625306	0.312653	−	0.949867i	$-0.398782\pi$
$349$	11.6817	0.625306	0.312653	+	0.949867i	$0.398782\pi$
$350$	0	0
$351$	42.8171	2.28541
$352$	0	0
$353$	−16.1862	−0.861503	−0.430752	−	0.902470i	$-0.641752\pi$
$353$	−16.1862	−0.861503	−0.430752	+	0.902470i	$0.641752\pi$
$354$	0	0
$355$	14.1168	0.749239
$356$	0	0
$357$	−6.95187	−0.367932
$358$	0	0
$359$	5.40994	0.285526	0.142763	−	0.989757i	$-0.454401\pi$
$359$	5.40994	0.285526	0.142763	+	0.989757i	$0.454401\pi$
$360$	0	0
$361$	−12.1799	−0.641048
$362$	0	0
$363$	30.0635	1.57792
$364$	0	0
$365$	−13.8309	−0.723940
$366$	0	0
$367$	−3.30285	−0.172408	−0.0862038	−	0.996278i	$-0.527474\pi$
$367$	−3.30285	−0.172408	−0.0862038	+	0.996278i	$0.527474\pi$
$368$	0	0
$369$	−18.0876	−0.941604
$370$	0	0
$371$	−2.35976	−0.122513
$372$	0	0
$373$	4.35120	0.225296	0.112648	−	0.993635i	$-0.464067\pi$
$373$	4.35120	0.225296	0.112648	+	0.993635i	$0.464067\pi$
$374$	0	0
$375$	−4.84605	−0.250249
$376$	0	0
$377$	14.2567	0.734260
$378$	0	0
$379$	4.44827	0.228493	0.114246	−	0.993452i	$-0.463555\pi$
$379$	4.44827	0.228493	0.114246	+	0.993452i	$0.463555\pi$
$380$	0	0
$381$	−46.9964	−2.40770
$382$	0	0
$383$	10.4681	0.534896	0.267448	−	0.963572i	$-0.413820\pi$
$383$	10.4681	0.534896	0.267448	+	0.963572i	$0.413820\pi$
$384$	0	0
$385$	−1.85171	−0.0943719
$386$	0	0
$387$	33.6939	1.71276
$388$	0	0
$389$	−28.1326	−1.42638	−0.713191	−	0.700970i	$-0.752751\pi$
$389$	−28.1326	−1.42638	−0.713191	+	0.700970i	$0.752751\pi$
$390$	0	0
$391$	17.2655	0.873154
$392$	0	0
$393$	−70.3401	−3.54819
$394$	0	0
$395$	−46.8441	−2.35698
$396$	0	0
$397$	−4.20333	−0.210959	−0.105480	−	0.994421i	$-0.533638\pi$
$397$	−4.20333	−0.210959	−0.105480	+	0.994421i	$0.533638\pi$
$398$	0	0
$399$	−3.79292	−0.189884
$400$	0	0
$401$	8.25124	0.412047	0.206024	−	0.978547i	$-0.433948\pi$
$401$	8.25124	0.412047	0.206024	+	0.978547i	$0.433948\pi$
$402$	0	0
$403$	1.13287	0.0564322
$404$	0	0
$405$	−86.5065	−4.29854
$406$	0	0
$407$	1.07211	0.0531426
$408$	0	0
$409$	23.2905	1.15164	0.575820	−	0.817576i	$-0.304683\pi$
$409$	23.2905	1.15164	0.575820	+	0.817576i	$0.304683\pi$
$410$	0	0
$411$	−23.6917	−1.16863
$412$	0	0
$413$	3.11497	0.153278
$414$	0	0
$415$	5.54751	0.272316
$416$	0	0
$417$	−22.0141	−1.07804
$418$	0	0
$419$	28.3483	1.38490	0.692452	−	0.721464i	$-0.256531\pi$
$419$	28.3483	1.38490	0.692452	+	0.721464i	$0.256531\pi$
$420$	0	0
$421$	−37.1755	−1.81182	−0.905912	−	0.423465i	$-0.860814\pi$
$421$	−37.1755	−1.81182	−0.905912	+	0.423465i	$0.860814\pi$
$422$	0	0
$423$	−9.09349	−0.442141
$424$	0	0
$425$	21.6419	1.04979
$426$	0	0
$427$	−3.01709	−0.146007
$428$	0	0
$429$	12.1761	0.587868
$430$	0	0
$431$	7.78276	0.374882	0.187441	−	0.982276i	$-0.439981\pi$
$431$	7.78276	0.374882	0.187441	+	0.982276i	$0.439981\pi$
$432$	0	0
$433$	10.9620	0.526800	0.263400	−	0.964687i	$-0.415156\pi$
$433$	10.9620	0.526800	0.263400	+	0.964687i	$0.415156\pi$
$434$	0	0
$435$	−52.7465	−2.52900
$436$	0	0
$437$	9.42002	0.450621
$438$	0	0
$439$	−30.3709	−1.44952	−0.724762	−	0.689000i	$-0.758050\pi$
$439$	−30.3709	−1.44952	−0.724762	+	0.689000i	$0.758050\pi$
$440$	0	0
$441$	−52.8526	−2.51679
$442$	0	0
$443$	−0.183732	−0.00872938	−0.00436469	−	0.999990i	$-0.501389\pi$
$443$	−0.183732	−0.00872938	−0.00436469	+	0.999990i	$0.501389\pi$
$444$	0	0
$445$	21.7285	1.03003
$446$	0	0
$447$	−24.5101	−1.15929
$448$	0	0
$449$	−16.4438	−0.776032	−0.388016	−	0.921653i	$-0.626839\pi$
$449$	−16.4438	−0.776032	−0.388016	+	0.921653i	$0.626839\pi$
$450$	0	0
$451$	−3.15713	−0.148663
$452$	0	0
$453$	5.97154	0.280568
$454$	0	0
$455$	3.73772	0.175227
$456$	0	0
$457$	−35.2619	−1.64948	−0.824741	−	0.565510i	$-0.808679\pi$
$457$	−35.2619	−1.64948	−0.824741	+	0.565510i	$0.808679\pi$
$458$	0	0
$459$	−74.8858	−3.49537
$460$	0	0
$461$	31.8676	1.48422	0.742111	−	0.670276i	$-0.233824\pi$
$461$	31.8676	1.48422	0.742111	+	0.670276i	$0.233824\pi$
$462$	0	0
$463$	−2.75486	−0.128029	−0.0640147	−	0.997949i	$-0.520390\pi$
$463$	−2.75486	−0.128029	−0.0640147	+	0.997949i	$0.520390\pi$
$464$	0	0
$465$	−4.19134	−0.194369
$466$	0	0
$467$	−11.1559	−0.516233	−0.258117	−	0.966114i	$-0.583102\pi$
$467$	−11.1559	−0.516233	−0.258117	+	0.966114i	$0.583102\pi$
$468$	0	0
$469$	−4.32076	−0.199514
$470$	0	0
$471$	57.7587	2.66138
$472$	0	0
$473$	5.88114	0.270415
$474$	0	0
$475$	11.8078	0.541777
$476$	0	0
$477$	−41.4141	−1.89622
$478$	0	0
$479$	22.6504	1.03492	0.517462	−	0.855706i	$-0.326877\pi$
$479$	22.6504	1.03492	0.517462	+	0.855706i	$0.326877\pi$
$480$	0	0
$481$	−2.16408	−0.0986735
$482$	0	0
$483$	−5.23885	−0.238376
$484$	0	0
$485$	42.7342	1.94046
$486$	0	0
$487$	−31.0435	−1.40671	−0.703357	−	0.710836i	$-0.748317\pi$
$487$	−31.0435	−1.40671	−0.703357	+	0.710836i	$0.748317\pi$
$488$	0	0
$489$	−37.3665	−1.68977
$490$	0	0
$491$	12.6874	0.572574	0.286287	−	0.958144i	$-0.407579\pi$
$491$	12.6874	0.572574	0.286287	+	0.958144i	$0.407579\pi$
$492$	0	0
$493$	−24.9346	−1.12300
$494$	0	0
$495$	−32.4977	−1.46066
$496$	0	0
$497$	−2.02489	−0.0908286
$498$	0	0
$499$	18.2535	0.817138	0.408569	−	0.912727i	$-0.366028\pi$
$499$	18.2535	0.817138	0.408569	+	0.912727i	$0.366028\pi$
$500$	0	0
$501$	73.2844	3.27411
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	4.94619	0.220103
$506$	0	0
$507$	18.0808	0.802997
$508$	0	0
$509$	−16.8311	−0.746026	−0.373013	−	0.927826i	$-0.621675\pi$
$509$	−16.8311	−0.746026	−0.373013	+	0.927826i	$0.621675\pi$
$510$	0	0
$511$	1.98388	0.0877616
$512$	0	0
$513$	−40.8575	−1.80390
$514$	0	0
$515$	51.1583	2.25430
$516$	0	0
$517$	−1.58724	−0.0698065
$518$	0	0
$519$	68.4918	3.00646
$520$	0	0
$521$	33.7106	1.47689	0.738444	−	0.674315i	$-0.235561\pi$
$521$	33.7106	1.47689	0.738444	+	0.674315i	$0.235561\pi$
$522$	0	0
$523$	−15.0218	−0.656859	−0.328430	−	0.944528i	$-0.606519\pi$
$523$	−15.0218	−0.656859	−0.328430	+	0.944528i	$0.606519\pi$
$524$	0	0
$525$	−6.56677	−0.286597
$526$	0	0
$527$	−1.98135	−0.0863091
$528$	0	0
$529$	−9.98891	−0.434300
$530$	0	0
$531$	54.6681	2.37239
$532$	0	0
$533$	6.37272	0.276033
$534$	0	0
$535$	−6.35096	−0.274576
$536$	0	0
$537$	−44.9966	−1.94175
$538$	0	0
$539$	−9.22523	−0.397359
$540$	0	0
$541$	42.9411	1.84618	0.923092	−	0.384580i	$-0.125654\pi$
$541$	42.9411	1.84618	0.923092	+	0.384580i	$0.125654\pi$
$542$	0	0
$543$	15.5756	0.668413
$544$	0	0
$545$	−40.1276	−1.71888
$546$	0	0
$547$	−17.4084	−0.744330	−0.372165	−	0.928167i	$-0.621384\pi$
$547$	−17.4084	−0.744330	−0.372165	+	0.928167i	$0.621384\pi$
$548$	0	0
$549$	−52.9502	−2.25986
$550$	0	0
$551$	−13.6043	−0.579562
$552$	0	0
$553$	6.71926	0.285732
$554$	0	0
$555$	8.00657	0.339860
$556$	0	0
$557$	−14.7686	−0.625766	−0.312883	−	0.949792i	$-0.601295\pi$
$557$	−14.7686	−0.625766	−0.312883	+	0.949792i	$0.601295\pi$
$558$	0	0
$559$	−11.8712	−0.502098
$560$	0	0
$561$	−21.2957	−0.899104
$562$	0	0
$563$	−8.48794	−0.357724	−0.178862	−	0.983874i	$-0.557242\pi$
$563$	−8.48794	−0.357724	−0.178862	+	0.983874i	$0.557242\pi$
$564$	0	0
$565$	−21.9231	−0.922312
$566$	0	0
$567$	12.4084	0.521102
$568$	0	0
$569$	20.4423	0.856986	0.428493	−	0.903545i	$-0.359045\pi$
$569$	20.4423	0.856986	0.428493	+	0.903545i	$0.359045\pi$
$570$	0	0
$571$	−38.8737	−1.62681	−0.813406	−	0.581696i	$-0.802389\pi$
$571$	−38.8737	−1.62681	−0.813406	+	0.581696i	$0.802389\pi$
$572$	0	0
$573$	−23.1769	−0.968230
$574$	0	0
$575$	16.3091	0.680136
$576$	0	0
$577$	19.0198	0.791805	0.395903	−	0.918292i	$-0.370432\pi$
$577$	19.0198	0.791805	0.395903	+	0.918292i	$0.370432\pi$
$578$	0	0
$579$	−20.4730	−0.850827
$580$	0	0
$581$	−0.795727	−0.0330123
$582$	0	0
$583$	−7.22867	−0.299381
$584$	0	0
$585$	65.5973	2.71211
$586$	0	0
$587$	−36.3854	−1.50179	−0.750893	−	0.660424i	$-0.770377\pi$
$587$	−36.3854	−1.50179	−0.750893	+	0.660424i	$0.770377\pi$
$588$	0	0
$589$	−1.08102	−0.0445427
$590$	0	0
$591$	4.47918	0.184249
$592$	0	0
$593$	23.8017	0.977419	0.488710	−	0.872447i	$-0.337468\pi$
$593$	23.8017	0.977419	0.488710	+	0.872447i	$0.337468\pi$
$594$	0	0
$595$	−6.53716	−0.267997
$596$	0	0
$597$	−66.7542	−2.73207
$598$	0	0
$599$	8.37519	0.342201	0.171101	−	0.985254i	$-0.445268\pi$
$599$	8.37519	0.342201	0.171101	+	0.985254i	$0.445268\pi$
$600$	0	0
$601$	−30.1214	−1.22868	−0.614339	−	0.789042i	$-0.710577\pi$
$601$	−30.1214	−1.22868	−0.614339	+	0.789042i	$0.710577\pi$
$602$	0	0
$603$	−75.8298	−3.08803
$604$	0	0
$605$	28.2701	1.14934
$606$	0	0
$607$	−39.3800	−1.59839	−0.799193	−	0.601075i	$-0.794739\pi$
$607$	−39.3800	−1.59839	−0.799193	+	0.601075i	$0.794739\pi$
$608$	0	0
$609$	7.56589	0.306585
$610$	0	0
$611$	3.20387	0.129615
$612$	0	0
$613$	−15.2598	−0.616337	−0.308169	−	0.951332i	$-0.599716\pi$
$613$	−15.2598	−0.616337	−0.308169	+	0.951332i	$0.599716\pi$
$614$	0	0
$615$	−23.5775	−0.950738
$616$	0	0
$617$	46.0041	1.85205	0.926027	−	0.377457i	$-0.123201\pi$
$617$	46.0041	1.85205	0.926027	+	0.377457i	$0.123201\pi$
$618$	0	0
$619$	−28.7344	−1.15493	−0.577466	−	0.816415i	$-0.695958\pi$
$619$	−28.7344	−1.15493	−0.577466	+	0.816415i	$0.695958\pi$
$620$	0	0
$621$	−56.4331	−2.26458
$622$	0	0
$623$	−3.11671	−0.124868
$624$	0	0
$625$	−27.1639	−1.08656
$626$	0	0
$627$	−11.6189	−0.464013
$628$	0	0
$629$	3.78491	0.150914
$630$	0	0
$631$	27.9138	1.11123	0.555615	−	0.831440i	$-0.312483\pi$
$631$	27.9138	1.11123	0.555615	+	0.831440i	$0.312483\pi$
$632$	0	0
$633$	6.82891	0.271425
$634$	0	0
$635$	−44.1929	−1.75374
$636$	0	0
$637$	18.6213	0.737803
$638$	0	0
$639$	−35.5370	−1.40582
$640$	0	0
$641$	48.8944	1.93121	0.965606	−	0.260009i	$-0.0837255\pi$
$641$	48.8944	1.93121	0.965606	+	0.260009i	$0.0837255\pi$
$642$	0	0
$643$	35.5485	1.40190	0.700948	−	0.713213i	$-0.252761\pi$
$643$	35.5485	1.40190	0.700948	+	0.713213i	$0.252761\pi$
$644$	0	0
$645$	43.9206	1.72937
$646$	0	0
$647$	−16.7222	−0.657416	−0.328708	−	0.944432i	$-0.606613\pi$
$647$	−16.7222	−0.657416	−0.328708	+	0.944432i	$0.606613\pi$
$648$	0	0
$649$	9.54211	0.374560
$650$	0	0
$651$	0.601200	0.0235629
$652$	0	0
$653$	19.4933	0.762833	0.381416	−	0.924403i	$-0.375436\pi$
$653$	19.4933	0.762833	0.381416	+	0.924403i	$0.375436\pi$
$654$	0	0
$655$	−66.1441	−2.58446
$656$	0	0
$657$	34.8173	1.35835
$658$	0	0
$659$	16.9068	0.658597	0.329298	−	0.944226i	$-0.393188\pi$
$659$	16.9068	0.658597	0.329298	+	0.944226i	$0.393188\pi$
$660$	0	0
$661$	48.9068	1.90225	0.951127	−	0.308799i	$-0.0999269\pi$
$661$	48.9068	1.90225	0.951127	+	0.308799i	$0.0999269\pi$
$662$	0	0
$663$	42.9858	1.66943
$664$	0	0
$665$	−3.56666	−0.138309
$666$	0	0
$667$	−18.7905	−0.727570
$668$	0	0
$669$	15.1161	0.584420
$670$	0	0
$671$	−9.24226	−0.356794
$672$	0	0
$673$	33.3065	1.28387	0.641936	−	0.766759i	$-0.278132\pi$
$673$	33.3065	1.28387	0.641936	+	0.766759i	$0.278132\pi$
$674$	0	0
$675$	−70.7376	−2.72269
$676$	0	0
$677$	21.3778	0.821617	0.410809	−	0.911722i	$-0.365246\pi$
$677$	21.3778	0.821617	0.410809	+	0.911722i	$0.365246\pi$
$678$	0	0
$679$	−6.12973	−0.235237
$680$	0	0
$681$	−12.9385	−0.495804
$682$	0	0
$683$	15.0330	0.575222	0.287611	−	0.957747i	$-0.407139\pi$
$683$	15.0330	0.575222	0.287611	+	0.957747i	$0.407139\pi$
$684$	0	0
$685$	−22.2784	−0.851215
$686$	0	0
$687$	28.2860	1.07918
$688$	0	0
$689$	14.5912	0.555881
$690$	0	0
$691$	17.9517	0.682916	0.341458	−	0.939897i	$-0.389079\pi$
$691$	17.9517	0.682916	0.341458	+	0.939897i	$0.389079\pi$
$692$	0	0
$693$	4.66143	0.177073
$694$	0	0
$695$	−20.7009	−0.785229
$696$	0	0
$697$	−11.1457	−0.422174
$698$	0	0
$699$	−42.4938	−1.60726
$700$	0	0
$701$	51.7186	1.95339	0.976693	−	0.214642i	$-0.0688586\pi$
$701$	51.7186	1.95339	0.976693	+	0.214642i	$0.0688586\pi$
$702$	0	0
$703$	2.06504	0.0778844
$704$	0	0
$705$	−11.8535	−0.446430
$706$	0	0
$707$	−0.709475	−0.0266826
$708$	0	0
$709$	−2.68167	−0.100712	−0.0503562	−	0.998731i	$-0.516036\pi$
$709$	−2.68167	−0.100712	−0.0503562	+	0.998731i	$0.516036\pi$
$710$	0	0
$711$	117.924	4.42248
$712$	0	0
$713$	−1.49313	−0.0559180
$714$	0	0
$715$	11.4498	0.428197
$716$	0	0
$717$	−23.3855	−0.873348
$718$	0	0
$719$	26.1438	0.974998	0.487499	−	0.873124i	$-0.337909\pi$
$719$	26.1438	0.974998	0.487499	+	0.873124i	$0.337909\pi$
$720$	0	0
$721$	−7.33807	−0.273284
$722$	0	0
$723$	−23.9458	−0.890555
$724$	0	0
$725$	−23.5534	−0.874751
$726$	0	0
$727$	44.4199	1.64744	0.823721	−	0.566995i	$-0.191894\pi$
$727$	44.4199	1.64744	0.823721	+	0.566995i	$0.191894\pi$
$728$	0	0
$729$	63.7536	2.36125
$730$	0	0
$731$	20.7624	0.767925
$732$	0	0
$733$	−31.1905	−1.15205	−0.576023	−	0.817433i	$-0.695396\pi$
$733$	−31.1905	−1.15205	−0.576023	+	0.817433i	$0.695396\pi$
$734$	0	0
$735$	−68.8944	−2.54121
$736$	0	0
$737$	−13.2358	−0.487547
$738$	0	0
$739$	−39.2965	−1.44555	−0.722773	−	0.691085i	$-0.757133\pi$
$739$	−39.2965	−1.44555	−0.722773	+	0.691085i	$0.757133\pi$
$740$	0	0
$741$	23.4529	0.861565
$742$	0	0
$743$	35.6731	1.30872	0.654359	−	0.756184i	$-0.272938\pi$
$743$	35.6731	1.30872	0.654359	+	0.756184i	$0.272938\pi$
$744$	0	0
$745$	−23.0480	−0.844412
$746$	0	0
$747$	−13.9651	−0.510956
$748$	0	0
$749$	0.910973	0.0332862
$750$	0	0
$751$	1.10526	0.0403317	0.0201658	−	0.999797i	$-0.493581\pi$
$751$	1.10526	0.0403317	0.0201658	+	0.999797i	$0.493581\pi$
$752$	0	0
$753$	−38.1690	−1.39095
$754$	0	0
$755$	5.61532	0.204362
$756$	0	0
$757$	40.0109	1.45422	0.727110	−	0.686521i	$-0.240863\pi$
$757$	40.0109	1.45422	0.727110	+	0.686521i	$0.240863\pi$
$758$	0	0
$759$	−16.0482	−0.582512
$760$	0	0
$761$	−15.3197	−0.555338	−0.277669	−	0.960677i	$-0.589562\pi$
$761$	−15.3197	−0.555338	−0.277669	+	0.960677i	$0.589562\pi$
$762$	0	0
$763$	5.75585	0.208376
$764$	0	0
$765$	−114.728	−4.14799
$766$	0	0
$767$	−19.2609	−0.695472
$768$	0	0
$769$	−4.21507	−0.151999	−0.0759996	−	0.997108i	$-0.524215\pi$
$769$	−4.21507	−0.151999	−0.0759996	+	0.997108i	$0.524215\pi$
$770$	0	0
$771$	−63.6995	−2.29408
$772$	0	0
$773$	41.0607	1.47685	0.738425	−	0.674336i	$-0.235570\pi$
$773$	41.0607	1.47685	0.738425	+	0.674336i	$0.235570\pi$
$774$	0	0
$775$	−1.87160	−0.0672298
$776$	0	0
$777$	−1.14845	−0.0412005
$778$	0	0
$779$	−6.08108	−0.217877
$780$	0	0
$781$	−6.20285	−0.221955
$782$	0	0
$783$	81.5001	2.91258
$784$	0	0
$785$	54.3132	1.93852
$786$	0	0
$787$	6.72032	0.239553	0.119777	−	0.992801i	$-0.461782\pi$
$787$	6.72032	0.239553	0.119777	+	0.992801i	$0.461782\pi$
$788$	0	0
$789$	59.2589	2.10967
$790$	0	0
$791$	3.14462	0.111810
$792$	0	0
$793$	18.6557	0.662483
$794$	0	0
$795$	−53.9840	−1.91462
$796$	0	0
$797$	−9.82444	−0.348000	−0.174000	−	0.984746i	$-0.555669\pi$
$797$	−9.82444	−0.348000	−0.174000	+	0.984746i	$0.555669\pi$
$798$	0	0
$799$	−5.60347	−0.198237
$800$	0	0
$801$	−54.6986	−1.93268
$802$	0	0
$803$	6.07723	0.214461
$804$	0	0
$805$	−4.92633	−0.173630
$806$	0	0
$807$	−44.0753	−1.55152
$808$	0	0
$809$	−1.32478	−0.0465768	−0.0232884	−	0.999729i	$-0.507414\pi$
$809$	−1.32478	−0.0465768	−0.0232884	+	0.999729i	$0.507414\pi$
$810$	0	0
$811$	27.9985	0.983159	0.491579	−	0.870833i	$-0.336420\pi$
$811$	27.9985	0.983159	0.491579	+	0.870833i	$0.336420\pi$
$812$	0	0
$813$	30.3086	1.06297
$814$	0	0
$815$	−35.1374	−1.23081
$816$	0	0
$817$	11.3279	0.396313
$818$	0	0
$819$	−9.40918	−0.328783
$820$	0	0
$821$	24.6410	0.859977	0.429989	−	0.902834i	$-0.358518\pi$
$821$	24.6410	0.859977	0.429989	+	0.902834i	$0.358518\pi$
$822$	0	0
$823$	47.6538	1.66111	0.830554	−	0.556938i	$-0.188024\pi$
$823$	47.6538	1.66111	0.830554	+	0.556938i	$0.188024\pi$
$824$	0	0
$825$	−20.1160	−0.700350
$826$	0	0
$827$	24.3419	0.846452	0.423226	−	0.906024i	$-0.360898\pi$
$827$	24.3419	0.846452	0.423226	+	0.906024i	$0.360898\pi$
$828$	0	0
$829$	5.77932	0.200724	0.100362	−	0.994951i	$-0.468000\pi$
$829$	5.77932	0.200724	0.100362	+	0.994951i	$0.468000\pi$
$830$	0	0
$831$	−40.0228	−1.38838
$832$	0	0
$833$	−32.5682	−1.12842
$834$	0	0
$835$	68.9127	2.38482
$836$	0	0
$837$	6.47615	0.223849
$838$	0	0
$839$	−28.6175	−0.987986	−0.493993	−	0.869466i	$-0.664463\pi$
$839$	−28.6175	−0.987986	−0.493993	+	0.869466i	$0.664463\pi$
$840$	0	0
$841$	−1.86301	−0.0642418
$842$	0	0
$843$	−55.9610	−1.92740
$844$	0	0
$845$	17.0022	0.584894
$846$	0	0
$847$	−4.05502	−0.139332
$848$	0	0
$849$	32.0675	1.10055
$850$	0	0
$851$	2.85227	0.0977745
$852$	0	0
$853$	36.1576	1.23801	0.619007	−	0.785386i	$-0.287535\pi$
$853$	36.1576	1.23801	0.619007	+	0.785386i	$0.287535\pi$
$854$	0	0
$855$	−62.5952	−2.14071
$856$	0	0
$857$	−10.4361	−0.356490	−0.178245	−	0.983986i	$-0.557042\pi$
$857$	−10.4361	−0.356490	−0.178245	+	0.983986i	$0.557042\pi$
$858$	0	0
$859$	25.5072	0.870296	0.435148	−	0.900359i	$-0.356696\pi$
$859$	25.5072	0.870296	0.435148	+	0.900359i	$0.356696\pi$
$860$	0	0
$861$	3.38193	0.115256
$862$	0	0
$863$	−29.8134	−1.01486	−0.507430	−	0.861693i	$-0.669404\pi$
$863$	−29.8134	−1.01486	−0.507430	+	0.861693i	$0.669404\pi$
$864$	0	0
$865$	64.4060	2.18987
$866$	0	0
$867$	−19.3966	−0.658744
$868$	0	0
$869$	20.5831	0.698235
$870$	0	0
$871$	26.7167	0.905262
$872$	0	0
$873$	−107.577	−3.64094
$874$	0	0
$875$	0.653644	0.0220972
$876$	0	0
$877$	−0.513062	−0.0173249	−0.00866244	−	0.999962i	$-0.502757\pi$
$877$	−0.513062	−0.0173249	−0.00866244	+	0.999962i	$0.502757\pi$
$878$	0	0
$879$	−52.9406	−1.78564
$880$	0	0
$881$	27.5668	0.928750	0.464375	−	0.885639i	$-0.346279\pi$
$881$	27.5668	0.928750	0.464375	+	0.885639i	$0.346279\pi$
$882$	0	0
$883$	−16.3382	−0.549825	−0.274912	−	0.961469i	$-0.588649\pi$
$883$	−16.3382	−0.549825	−0.274912	+	0.961469i	$0.588649\pi$
$884$	0	0
$885$	71.2608	2.39541
$886$	0	0
$887$	−24.9419	−0.837467	−0.418733	−	0.908109i	$-0.637526\pi$
$887$	−24.9419	−0.837467	−0.418733	+	0.908109i	$0.637526\pi$
$888$	0	0
$889$	6.33897	0.212602
$890$	0	0
$891$	38.0106	1.27340
$892$	0	0
$893$	−3.05724	−0.102307
$894$	0	0
$895$	−42.3124	−1.41435
$896$	0	0
$897$	32.3936	1.08159
$898$	0	0
$899$	2.15636	0.0719185
$900$	0	0
$901$	−25.5196	−0.850182
$902$	0	0
$903$	−6.29991	−0.209648
$904$	0	0
$905$	14.6464	0.486864
$906$	0	0
$907$	1.70224	0.0565219	0.0282609	−	0.999601i	$-0.491003\pi$
$907$	1.70224	0.0565219	0.0282609	+	0.999601i	$0.491003\pi$
$908$	0	0
$909$	−12.4514	−0.412986
$910$	0	0
$911$	34.4831	1.14248	0.571239	−	0.820784i	$-0.306463\pi$
$911$	34.4831	1.14248	0.571239	+	0.820784i	$0.306463\pi$
$912$	0	0
$913$	−2.43755	−0.0806712
$914$	0	0
$915$	−69.0216	−2.28178
$916$	0	0
$917$	9.48761	0.313308
$918$	0	0
$919$	33.1805	1.09452	0.547262	−	0.836961i	$-0.315670\pi$
$919$	33.1805	1.09452	0.547262	+	0.836961i	$0.315670\pi$
$920$	0	0
$921$	−49.0573	−1.61649
$922$	0	0
$923$	12.5206	0.412120
$924$	0	0
$925$	3.57525	0.117553
$926$	0	0
$927$	−128.784	−4.22982
$928$	0	0
$929$	24.2559	0.795811	0.397905	−	0.917426i	$-0.369737\pi$
$929$	24.2559	0.795811	0.397905	+	0.917426i	$0.369737\pi$
$930$	0	0
$931$	−17.7691	−0.582359
$932$	0	0
$933$	−86.8725	−2.84408
$934$	0	0
$935$	−20.0253	−0.654898
$936$	0	0
$937$	−36.4820	−1.19181	−0.595907	−	0.803053i	$-0.703207\pi$
$937$	−36.4820	−1.19181	−0.595907	+	0.803053i	$0.703207\pi$
$938$	0	0
$939$	−14.5177	−0.473768
$940$	0	0
$941$	−56.3376	−1.83655	−0.918277	−	0.395938i	$-0.870419\pi$
$941$	−56.3376	−1.83655	−0.918277	+	0.395938i	$0.870419\pi$
$942$	0	0
$943$	−8.39929	−0.273518
$944$	0	0
$945$	21.3670	0.695069
$946$	0	0
$947$	17.6051	0.572090	0.286045	−	0.958216i	$-0.407659\pi$
$947$	17.6051	0.572090	0.286045	+	0.958216i	$0.407659\pi$
$948$	0	0
$949$	−12.2670	−0.398204
$950$	0	0
$951$	−8.07873	−0.261971
$952$	0	0
$953$	30.7308	0.995469	0.497734	−	0.867330i	$-0.334165\pi$
$953$	30.7308	0.995469	0.497734	+	0.867330i	$0.334165\pi$
$954$	0	0
$955$	−21.7943	−0.705248
$956$	0	0
$957$	23.1766	0.749194
$958$	0	0
$959$	3.19559	0.103191
$960$	0	0
$961$	−30.8287	−0.994473
$962$	0	0
$963$	15.9877	0.515195
$964$	0	0
$965$	−19.2517	−0.619733
$966$	0	0
$967$	13.9603	0.448934	0.224467	−	0.974482i	$-0.427936\pi$
$967$	13.9603	0.448934	0.224467	+	0.974482i	$0.427936\pi$
$968$	0	0
$969$	−41.0185	−1.31770
$970$	0	0
$971$	−52.5728	−1.68714	−0.843571	−	0.537018i	$-0.819551\pi$
$971$	−52.5728	−1.68714	−0.843571	+	0.537018i	$0.819551\pi$
$972$	0	0
$973$	2.96930	0.0951915
$974$	0	0
$975$	40.6046	1.30039
$976$	0	0
$977$	57.7331	1.84704	0.923522	−	0.383545i	$-0.125297\pi$
$977$	57.7331	1.84704	0.923522	+	0.383545i	$0.125297\pi$
$978$	0	0
$979$	−9.54743	−0.305137
$980$	0	0
$981$	101.016	3.22519
$982$	0	0
$983$	−35.5474	−1.13378	−0.566892	−	0.823792i	$-0.691855\pi$
$983$	−35.5474	−1.13378	−0.566892	+	0.823792i	$0.691855\pi$
$984$	0	0
$985$	4.21198	0.134205
$986$	0	0
$987$	1.70025	0.0541197
$988$	0	0
$989$	15.6463	0.497524
$990$	0	0
$991$	−27.1321	−0.861879	−0.430940	−	0.902381i	$-0.641818\pi$
$991$	−27.1321	−0.861879	−0.430940	+	0.902381i	$0.641818\pi$
$992$	0	0
$993$	−110.528	−3.50749
$994$	0	0
$995$	−62.7720	−1.99001
$996$	0	0
$997$	−12.1675	−0.385350	−0.192675	−	0.981263i	$-0.561716\pi$
$997$	−12.1675	−0.385350	−0.192675	+	0.981263i	$0.561716\pi$
$998$	0	0
$999$	−12.3712	−0.391407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.1	✓	33
4.3	odd	2		8048.2.a.x.1.33		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.1	✓	33	1.1	even	1	trivial
8048.2.a.x.1.33		33	4.3	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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.28143 q^{3} -3.08568 q^{5} +0.442605 q^{7} +7.76776 q^{9} +O(q^{10})\)	\(q-3.28143 q^{3} -3.08568 q^{5} +0.442605 q^{7} +7.76776 q^{9} +1.35583 q^{11} -2.73678 q^{13} +10.1254 q^{15} +4.78655 q^{17} +2.61153 q^{19} -1.45238 q^{21} +3.60709 q^{23} +4.52140 q^{25} -15.6451 q^{27} -5.20932 q^{29} -0.413942 q^{31} -4.44907 q^{33} -1.36574 q^{35} +0.790740 q^{37} +8.98054 q^{39} -2.32855 q^{41} +4.33765 q^{43} -23.9688 q^{45} -1.17067 q^{47} -6.80410 q^{49} -15.7067 q^{51} -5.33153 q^{53} -4.18366 q^{55} -8.56954 q^{57} +7.03781 q^{59} -6.81666 q^{61} +3.43805 q^{63} +8.44481 q^{65} -9.76211 q^{67} -11.8364 q^{69} -4.57493 q^{71} +4.48228 q^{73} -14.8366 q^{75} +0.600099 q^{77} +15.1812 q^{79} +28.0348 q^{81} -1.79783 q^{83} -14.7697 q^{85} +17.0940 q^{87} -7.04174 q^{89} -1.21131 q^{91} +1.35832 q^{93} -8.05833 q^{95} -13.8492 q^{97} +10.5318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more