LMFDB - Embedded newform 6025.2.a.p.1.34

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.34
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.45267 q^{2} -2.41257 q^{3} +0.110263 q^{4} -3.50468 q^{6} +1.25057 q^{7} -2.74517 q^{8} +2.82049 q^{9} +O(q^{10})$	\(q+1.45267 q^{2} -2.41257 q^{3} +0.110263 q^{4} -3.50468 q^{6} +1.25057 q^{7} -2.74517 q^{8} +2.82049 q^{9} +2.46616 q^{11} -0.266016 q^{12} -2.06969 q^{13} +1.81667 q^{14} -4.20837 q^{16} +1.42467 q^{17} +4.09725 q^{18} -0.479813 q^{19} -3.01709 q^{21} +3.58253 q^{22} +2.87475 q^{23} +6.62292 q^{24} -3.00659 q^{26} +0.433079 q^{27} +0.137891 q^{28} -9.26441 q^{29} +0.180654 q^{31} -0.623041 q^{32} -5.94979 q^{33} +2.06958 q^{34} +0.310994 q^{36} -0.824807 q^{37} -0.697012 q^{38} +4.99327 q^{39} +1.12498 q^{41} -4.38285 q^{42} +3.43795 q^{43} +0.271925 q^{44} +4.17607 q^{46} +3.00871 q^{47} +10.1530 q^{48} -5.43607 q^{49} -3.43711 q^{51} -0.228209 q^{52} +12.3733 q^{53} +0.629123 q^{54} -3.43303 q^{56} +1.15758 q^{57} -13.4582 q^{58} +0.433111 q^{59} -3.88745 q^{61} +0.262432 q^{62} +3.52722 q^{63} +7.51166 q^{64} -8.64311 q^{66} -5.24447 q^{67} +0.157087 q^{68} -6.93552 q^{69} +14.3247 q^{71} -7.74273 q^{72} +4.35588 q^{73} -1.19818 q^{74} -0.0529054 q^{76} +3.08411 q^{77} +7.25360 q^{78} -7.28837 q^{79} -9.50630 q^{81} +1.63423 q^{82} +1.09208 q^{83} -0.332672 q^{84} +4.99422 q^{86} +22.3510 q^{87} -6.77004 q^{88} -12.2161 q^{89} -2.58830 q^{91} +0.316977 q^{92} -0.435841 q^{93} +4.37068 q^{94} +1.50313 q^{96} +9.80240 q^{97} -7.89684 q^{98} +6.95579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * 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 $

$2$	1.45267	1.02720	0.513598	−	0.858031i	$-0.328312\pi$
$2$	1.45267	1.02720	0.513598	+	0.858031i	$0.328312\pi$
$3$	−2.41257	−1.39290	−0.696449	−	0.717607i	$-0.745238\pi$
$3$	−2.41257	−1.39290	−0.696449	+	0.717607i	$0.745238\pi$
$4$	0.110263	0.0551313
$5$	0	0
$5$	0	0
$6$	−3.50468	−1.43078
$7$	1.25057	0.472671	0.236336	−	0.971671i	$-0.424054\pi$
$7$	1.25057	0.472671	0.236336	+	0.971671i	$0.424054\pi$
$8$	−2.74517	−0.970565
$9$	2.82049	0.940164
$10$	0	0
$11$	2.46616	0.743576	0.371788	−	0.928318i	$-0.378745\pi$
$11$	2.46616	0.743576	0.371788	+	0.928318i	$0.378745\pi$
$12$	−0.266016	−0.0767922
$13$	−2.06969	−0.574029	−0.287015	−	0.957926i	$-0.592663\pi$
$13$	−2.06969	−0.574029	−0.287015	+	0.957926i	$0.592663\pi$
$14$	1.81667	0.485526
$15$	0	0
$16$	−4.20837	−1.05209
$17$	1.42467	0.345532	0.172766	−	0.984963i	$-0.444729\pi$
$17$	1.42467	0.345532	0.172766	+	0.984963i	$0.444729\pi$
$18$	4.09725	0.965732
$19$	−0.479813	−0.110077	−0.0550384	−	0.998484i	$-0.517528\pi$
$19$	−0.479813	−0.110077	−0.0550384	+	0.998484i	$0.517528\pi$
$20$	0	0
$21$	−3.01709	−0.658383
$22$	3.58253	0.763798
$23$	2.87475	0.599426	0.299713	−	0.954029i	$-0.403109\pi$
$23$	2.87475	0.599426	0.299713	+	0.954029i	$0.403109\pi$
$24$	6.62292	1.35190
$25$	0	0
$26$	−3.00659	−0.589640
$27$	0.433079	0.0833461
$28$	0.137891	0.0260590
$29$	−9.26441	−1.72036	−0.860179	−	0.509992i	$-0.829648\pi$
$29$	−9.26441	−1.72036	−0.860179	+	0.509992i	$0.829648\pi$
$30$	0	0
$31$	0.180654	0.0324465	0.0162232	−	0.999868i	$-0.494836\pi$
$31$	0.180654	0.0324465	0.0162232	+	0.999868i	$0.494836\pi$
$32$	−0.623041	−0.110139
$33$	−5.94979	−1.03573
$34$	2.06958	0.354929
$35$	0	0
$36$	0.310994	0.0518324
$37$	−0.824807	−0.135597	−0.0677987	−	0.997699i	$-0.521598\pi$
$37$	−0.824807	−0.135597	−0.0677987	+	0.997699i	$0.521598\pi$
$38$	−0.697012	−0.113070
$39$	4.99327	0.799564
$40$	0	0
$41$	1.12498	0.175693	0.0878464	−	0.996134i	$-0.472002\pi$
$41$	1.12498	0.175693	0.0878464	+	0.996134i	$0.472002\pi$
$42$	−4.38285	−0.676288
$43$	3.43795	0.524282	0.262141	−	0.965030i	$-0.415571\pi$
$43$	3.43795	0.524282	0.262141	+	0.965030i	$0.415571\pi$
$44$	0.271925	0.0409943
$45$	0	0
$46$	4.17607	0.615728
$47$	3.00871	0.438866	0.219433	−	0.975628i	$-0.429579\pi$
$47$	3.00871	0.438866	0.219433	+	0.975628i	$0.429579\pi$
$48$	10.1530	1.46546
$49$	−5.43607	−0.776582
$50$	0	0
$51$	−3.43711	−0.481291
$52$	−0.228209	−0.0316470
$53$	12.3733	1.69961	0.849804	−	0.527099i	$-0.176720\pi$
$53$	12.3733	1.69961	0.849804	+	0.527099i	$0.176720\pi$
$54$	0.629123	0.0856128
$55$	0	0
$56$	−3.43303	−0.458758
$57$	1.15758	0.153326
$58$	−13.4582	−1.76714
$59$	0.433111	0.0563863	0.0281931	−	0.999602i	$-0.491025\pi$
$59$	0.433111	0.0563863	0.0281931	+	0.999602i	$0.491025\pi$
$60$	0	0
$61$	−3.88745	−0.497737	−0.248868	−	0.968537i	$-0.580059\pi$
$61$	−3.88745	−0.497737	−0.248868	+	0.968537i	$0.580059\pi$
$62$	0.262432	0.0333289
$63$	3.52722	0.444388
$64$	7.51166	0.938957
$65$	0	0
$66$	−8.64311	−1.06389
$67$	−5.24447	−0.640714	−0.320357	−	0.947297i	$-0.603803\pi$
$67$	−5.24447	−0.640714	−0.320357	+	0.947297i	$0.603803\pi$
$68$	0.157087	0.0190496
$69$	−6.93552	−0.834939
$70$	0	0
$71$	14.3247	1.70003	0.850013	−	0.526762i	$-0.176594\pi$
$71$	14.3247	1.70003	0.850013	+	0.526762i	$0.176594\pi$
$72$	−7.74273	−0.912490
$73$	4.35588	0.509818	0.254909	−	0.966965i	$-0.417955\pi$
$73$	4.35588	0.509818	0.254909	+	0.966965i	$0.417955\pi$
$74$	−1.19818	−0.139285
$75$	0	0
$76$	−0.0529054	−0.00606867
$77$	3.08411	0.351467
$78$	7.25360	0.821309
$79$	−7.28837	−0.820005	−0.410003	−	0.912084i	$-0.634472\pi$
$79$	−7.28837	−0.820005	−0.410003	+	0.912084i	$0.634472\pi$
$80$	0	0
$81$	−9.50630	−1.05626
$82$	1.63423	0.180471
$83$	1.09208	0.119871	0.0599357	−	0.998202i	$-0.480910\pi$
$83$	1.09208	0.119871	0.0599357	+	0.998202i	$0.480910\pi$
$84$	−0.332672	−0.0362975
$85$	0	0
$86$	4.99422	0.538541
$87$	22.3510	2.39628
$88$	−6.77004	−0.721689
$89$	−12.2161	−1.29490	−0.647452	−	0.762106i	$-0.724166\pi$
$89$	−12.2161	−1.29490	−0.647452	+	0.762106i	$0.724166\pi$
$90$	0	0
$91$	−2.58830	−0.271327
$92$	0.316977	0.0330471
$93$	−0.435841	−0.0451946
$94$	4.37068	0.450801
$95$	0	0
$96$	1.50313	0.153412
$97$	9.80240	0.995283	0.497642	−	0.867383i	$-0.334199\pi$
$97$	9.80240	0.995283	0.497642	+	0.867383i	$0.334199\pi$
$98$	−7.89684	−0.797702
$99$	6.95579	0.699083
$100$	0	0
$101$	−10.5462	−1.04939	−0.524695	−	0.851291i	$-0.675820\pi$
$101$	−10.5462	−1.04939	−0.524695	+	0.851291i	$0.675820\pi$
$102$	−4.99300	−0.494380
$103$	−7.56058	−0.744966	−0.372483	−	0.928039i	$-0.621494\pi$
$103$	−7.56058	−0.744966	−0.372483	+	0.928039i	$0.621494\pi$
$104$	5.68166	0.557133
$105$	0	0
$106$	17.9744	1.74583
$107$	−4.30799	−0.416469	−0.208234	−	0.978079i	$-0.566772\pi$
$107$	−4.30799	−0.416469	−0.208234	+	0.978079i	$0.566772\pi$
$108$	0.0477524	0.00459498
$109$	−3.03421	−0.290625	−0.145313	−	0.989386i	$-0.546419\pi$
$109$	−3.03421	−0.290625	−0.145313	+	0.989386i	$0.546419\pi$
$110$	0	0
$111$	1.98990	0.188873
$112$	−5.26286	−0.497294
$113$	−5.48438	−0.515927	−0.257964	−	0.966155i	$-0.583051\pi$
$113$	−5.48438	−0.515927	−0.257964	+	0.966155i	$0.583051\pi$
$114$	1.68159	0.157495
$115$	0	0
$116$	−1.02152	−0.0948455
$117$	−5.83755	−0.539681
$118$	0.629169	0.0579197
$119$	1.78165	0.163323
$120$	0	0
$121$	−4.91804	−0.447095
$122$	−5.64720	−0.511273
$123$	−2.71410	−0.244722
$124$	0.0199194	0.00178882
$125$	0	0
$126$	5.12390	0.456474
$127$	−1.82445	−0.161894	−0.0809471	−	0.996718i	$-0.525794\pi$
$127$	−1.82445	−0.161894	−0.0809471	+	0.996718i	$0.525794\pi$
$128$	12.1581	1.07463
$129$	−8.29429	−0.730271
$130$	0	0
$131$	10.7564	0.939789	0.469895	−	0.882722i	$-0.344292\pi$
$131$	10.7564	0.939789	0.469895	+	0.882722i	$0.344292\pi$
$132$	−0.656039	−0.0571009
$133$	−0.600040	−0.0520301
$134$	−7.61851	−0.658139
$135$	0	0
$136$	−3.91096	−0.335362
$137$	−18.0907	−1.54560	−0.772798	−	0.634653i	$-0.781143\pi$
$137$	−18.0907	−1.54560	−0.772798	+	0.634653i	$0.781143\pi$
$138$	−10.0751	−0.857646
$139$	9.24847	0.784445	0.392222	−	0.919870i	$-0.371706\pi$
$139$	9.24847	0.784445	0.392222	+	0.919870i	$0.371706\pi$
$140$	0	0
$141$	−7.25873	−0.611296
$142$	20.8091	1.74626
$143$	−5.10420	−0.426834
$144$	−11.8697	−0.989138
$145$	0	0
$146$	6.32768	0.523683
$147$	13.1149	1.08170
$148$	−0.0909453	−0.00747566
$149$	17.0170	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$149$	17.0170	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$150$	0	0
$151$	−9.18436	−0.747413	−0.373706	−	0.927547i	$-0.621913\pi$
$151$	−9.18436	−0.747413	−0.373706	+	0.927547i	$0.621913\pi$
$152$	1.31717	0.106837
$153$	4.01826	0.324857
$154$	4.48021	0.361025
$155$	0	0
$156$	0.550571	0.0440810
$157$	−0.176439	−0.0140814	−0.00704069	−	0.999975i	$-0.502241\pi$
$157$	−0.176439	−0.0140814	−0.00704069	+	0.999975i	$0.502241\pi$
$158$	−10.5876	−0.842306
$159$	−29.8515	−2.36738
$160$	0	0
$161$	3.59507	0.283331
$162$	−13.8096	−1.08498
$163$	−9.16744	−0.718050	−0.359025	−	0.933328i	$-0.616891\pi$
$163$	−9.16744	−0.718050	−0.359025	+	0.933328i	$0.616891\pi$
$164$	0.124043	0.00968616
$165$	0	0
$166$	1.58644	0.123131
$167$	−22.5003	−1.74112	−0.870562	−	0.492058i	$-0.836245\pi$
$167$	−22.5003	−1.74112	−0.870562	+	0.492058i	$0.836245\pi$
$168$	8.28243	0.639003
$169$	−8.71638	−0.670491
$170$	0	0
$171$	−1.35331	−0.103490
$172$	0.379077	0.0289043
$173$	5.08571	0.386659	0.193330	−	0.981134i	$-0.438071\pi$
$173$	5.08571	0.386659	0.193330	+	0.981134i	$0.438071\pi$
$174$	32.4688	2.46145
$175$	0	0
$176$	−10.3785	−0.782310
$177$	−1.04491	−0.0785403
$178$	−17.7460	−1.33012
$179$	−15.3122	−1.14449	−0.572244	−	0.820084i	$-0.693927\pi$
$179$	−15.3122	−1.14449	−0.572244	+	0.820084i	$0.693927\pi$
$180$	0	0
$181$	−18.4259	−1.36958	−0.684792	−	0.728738i	$-0.740107\pi$
$181$	−18.4259	−1.36958	−0.684792	+	0.728738i	$0.740107\pi$
$182$	−3.75995	−0.278706
$183$	9.37874	0.693297
$184$	−7.89167	−0.581782
$185$	0	0
$186$	−0.633135	−0.0464237
$187$	3.51346	0.256930
$188$	0.331749	0.0241952
$189$	0.541596	0.0393953
$190$	0	0
$191$	−13.7148	−0.992368	−0.496184	−	0.868217i	$-0.665266\pi$
$191$	−13.7148	−0.992368	−0.496184	+	0.868217i	$0.665266\pi$
$192$	−18.1224	−1.30787
$193$	−12.4066	−0.893047	−0.446524	−	0.894772i	$-0.647338\pi$
$193$	−12.4066	−0.893047	−0.446524	+	0.894772i	$0.647338\pi$
$194$	14.2397	1.02235
$195$	0	0
$196$	−0.599395	−0.0428139
$197$	7.72970	0.550718	0.275359	−	0.961341i	$-0.411203\pi$
$197$	7.72970	0.550718	0.275359	+	0.961341i	$0.411203\pi$
$198$	10.1045	0.718095
$199$	−5.15173	−0.365196	−0.182598	−	0.983188i	$-0.558451\pi$
$199$	−5.15173	−0.365196	−0.182598	+	0.983188i	$0.558451\pi$
$200$	0	0
$201$	12.6526	0.892449
$202$	−15.3202	−1.07793
$203$	−11.5858	−0.813164
$204$	−0.378984	−0.0265342
$205$	0	0
$206$	−10.9831	−0.765226
$207$	8.10819	0.563558
$208$	8.71002	0.603931
$209$	−1.18330	−0.0818504
$210$	0	0
$211$	−20.6216	−1.41965	−0.709823	−	0.704380i	$-0.751225\pi$
$211$	−20.6216	−1.41965	−0.709823	+	0.704380i	$0.751225\pi$
$212$	1.36431	0.0937015
$213$	−34.5593	−2.36796
$214$	−6.25810	−0.427795
$215$	0	0
$216$	−1.18888	−0.0808928
$217$	0.225921	0.0153365
$218$	−4.40772	−0.298529
$219$	−10.5089	−0.710124
$220$	0	0
$221$	−2.94862	−0.198346
$222$	2.89068	0.194010
$223$	−0.789803	−0.0528891	−0.0264446	−	0.999650i	$-0.508419\pi$
$223$	−0.789803	−0.0528891	−0.0264446	+	0.999650i	$0.508419\pi$
$224$	−0.779156	−0.0520596
$225$	0	0
$226$	−7.96702	−0.529958
$227$	−1.06248	−0.0705190	−0.0352595	−	0.999378i	$-0.511226\pi$
$227$	−1.06248	−0.0705190	−0.0352595	+	0.999378i	$0.511226\pi$
$228$	0.127638	0.00845303
$229$	18.7377	1.23822	0.619110	−	0.785304i	$-0.287493\pi$
$229$	18.7377	1.23822	0.619110	+	0.785304i	$0.287493\pi$
$230$	0	0
$231$	−7.44063	−0.489558
$232$	25.4324	1.66972
$233$	−20.6141	−1.35047	−0.675237	−	0.737601i	$-0.735959\pi$
$233$	−20.6141	−1.35047	−0.675237	+	0.737601i	$0.735959\pi$
$234$	−8.48005	−0.554358
$235$	0	0
$236$	0.0477559	0.00310865
$237$	17.5837	1.14218
$238$	2.58815	0.167765
$239$	−17.2630	−1.11665	−0.558324	−	0.829623i	$-0.688555\pi$
$239$	−17.2630	−1.11665	−0.558324	+	0.829623i	$0.688555\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−7.14431	−0.459254
$243$	21.6354	1.38791
$244$	−0.428640	−0.0274409
$245$	0	0
$246$	−3.94270	−0.251377
$247$	0.993065	0.0631872
$248$	−0.495927	−0.0314914
$249$	−2.63472	−0.166969
$250$	0	0
$251$	−21.5943	−1.36302	−0.681510	−	0.731809i	$-0.738677\pi$
$251$	−21.5943	−1.36302	−0.681510	+	0.731809i	$0.738677\pi$
$252$	0.388920	0.0244997
$253$	7.08959	0.445719
$254$	−2.65034	−0.166297
$255$	0	0
$256$	2.63841	0.164900
$257$	22.2276	1.38652	0.693261	−	0.720687i	$-0.256173\pi$
$257$	22.2276	1.38652	0.693261	+	0.720687i	$0.256173\pi$
$258$	−12.0489	−0.750132
$259$	−1.03148	−0.0640930
$260$	0	0
$261$	−26.1302	−1.61742
$262$	15.6255	0.965348
$263$	3.89765	0.240339	0.120170	−	0.992753i	$-0.461656\pi$
$263$	3.89765	0.240339	0.120170	+	0.992753i	$0.461656\pi$
$264$	16.3332	1.00524
$265$	0	0
$266$	−0.871663	−0.0534451
$267$	29.4722	1.80367
$268$	−0.578268	−0.0353234
$269$	6.60591	0.402769	0.201385	−	0.979512i	$-0.435456\pi$
$269$	6.60591	0.402769	0.201385	+	0.979512i	$0.435456\pi$
$270$	0	0
$271$	−16.8832	−1.02558	−0.512791	−	0.858513i	$-0.671388\pi$
$271$	−16.8832	−1.02558	−0.512791	+	0.858513i	$0.671388\pi$
$272$	−5.99552	−0.363532
$273$	6.24444	0.377931
$274$	−26.2799	−1.58763
$275$	0	0
$276$	−0.764728	−0.0460312
$277$	27.0001	1.62228	0.811140	−	0.584852i	$-0.198847\pi$
$277$	27.0001	1.62228	0.811140	+	0.584852i	$0.198847\pi$
$278$	13.4350	0.805778
$279$	0.509534	0.0305050
$280$	0	0
$281$	−9.48720	−0.565959	−0.282979	−	0.959126i	$-0.591323\pi$
$281$	−9.48720	−0.565959	−0.282979	+	0.959126i	$0.591323\pi$
$282$	−10.5446	−0.627920
$283$	−13.7557	−0.817690	−0.408845	−	0.912604i	$-0.634068\pi$
$283$	−13.7557	−0.817690	−0.408845	+	0.912604i	$0.634068\pi$
$284$	1.57947	0.0937246
$285$	0	0
$286$	−7.41474	−0.438442
$287$	1.40687	0.0830449
$288$	−1.75728	−0.103549
$289$	−14.9703	−0.880607
$290$	0	0
$291$	−23.6490	−1.38633
$292$	0.480291	0.0281069
$293$	11.4782	0.670561	0.335280	−	0.942118i	$-0.391169\pi$
$293$	11.4782	0.670561	0.335280	+	0.942118i	$0.391169\pi$
$294$	19.0517	1.11112
$295$	0	0
$296$	2.26424	0.131606
$297$	1.06804	0.0619742
$298$	24.7201	1.43200
$299$	−5.94984	−0.344088
$300$	0	0
$301$	4.29940	0.247813
$302$	−13.3419	−0.767739
$303$	25.4435	1.46169
$304$	2.01923	0.115811
$305$	0	0
$306$	5.83722	0.333692
$307$	−11.8355	−0.675485	−0.337743	−	0.941238i	$-0.609663\pi$
$307$	−11.8355	−0.675485	−0.337743	+	0.941238i	$0.609663\pi$
$308$	0.340062	0.0193768
$309$	18.2404	1.03766
$310$	0	0
$311$	−5.53844	−0.314056	−0.157028	−	0.987594i	$-0.550191\pi$
$311$	−5.53844	−0.314056	−0.157028	+	0.987594i	$0.550191\pi$
$312$	−13.7074	−0.776029
$313$	−24.8522	−1.40473	−0.702365	−	0.711817i	$-0.747873\pi$
$313$	−24.8522	−1.40473	−0.702365	+	0.711817i	$0.747873\pi$
$314$	−0.256309	−0.0144643
$315$	0	0
$316$	−0.803634	−0.0452079
$317$	15.8127	0.888130	0.444065	−	0.895994i	$-0.353536\pi$
$317$	15.8127	0.888130	0.444065	+	0.895994i	$0.353536\pi$
$318$	−43.3645	−2.43176
$319$	−22.8475	−1.27922
$320$	0	0
$321$	10.3933	0.580098
$322$	5.22247	0.291037
$323$	−0.683574	−0.0380351
$324$	−1.04819	−0.0582327
$325$	0	0
$326$	−13.3173	−0.737578
$327$	7.32025	0.404811
$328$	−3.08827	−0.170521
$329$	3.76261	0.207439
$330$	0	0
$331$	23.4865	1.29093	0.645467	−	0.763789i	$-0.276663\pi$
$331$	23.4865	1.29093	0.645467	+	0.763789i	$0.276663\pi$
$332$	0.120416	0.00660866
$333$	−2.32636	−0.127484
$334$	−32.6856	−1.78848
$335$	0	0
$336$	12.6970	0.692679
$337$	−11.1318	−0.606385	−0.303193	−	0.952929i	$-0.598053\pi$
$337$	−11.1318	−0.606385	−0.303193	+	0.952929i	$0.598053\pi$
$338$	−12.6621	−0.688725
$339$	13.2315	0.718634
$340$	0	0
$341$	0.445523	0.0241264
$342$	−1.96592	−0.106305
$343$	−15.5522	−0.839739
$344$	−9.43776	−0.508850
$345$	0	0
$346$	7.38788	0.397175
$347$	−4.44891	−0.238830	−0.119415	−	0.992844i	$-0.538102\pi$
$347$	−4.44891	−0.238830	−0.119415	+	0.992844i	$0.538102\pi$
$348$	2.46448	0.132110
$349$	0.902138	0.0482903	0.0241452	−	0.999708i	$-0.492314\pi$
$349$	0.902138	0.0482903	0.0241452	+	0.999708i	$0.492314\pi$
$350$	0	0
$351$	−0.896340	−0.0478431
$352$	−1.53652	−0.0818968
$353$	0.455388	0.0242379	0.0121189	−	0.999927i	$-0.496142\pi$
$353$	0.455388	0.0242379	0.0121189	+	0.999927i	$0.496142\pi$
$354$	−1.51791	−0.0806763
$355$	0	0
$356$	−1.34698	−0.0713897
$357$	−4.29834	−0.227493
$358$	−22.2436	−1.17561
$359$	−8.65154	−0.456611	−0.228305	−	0.973590i	$-0.573318\pi$
$359$	−8.65154	−0.456611	−0.228305	+	0.973590i	$0.573318\pi$
$360$	0	0
$361$	−18.7698	−0.987883
$362$	−26.7668	−1.40683
$363$	11.8651	0.622757
$364$	−0.285392	−0.0149586
$365$	0	0
$366$	13.6243	0.712151
$367$	−23.7645	−1.24050	−0.620249	−	0.784405i	$-0.712969\pi$
$367$	−23.7645	−1.24050	−0.620249	+	0.784405i	$0.712969\pi$
$368$	−12.0980	−0.630651
$369$	3.17300	0.165180
$370$	0	0
$371$	15.4737	0.803355
$372$	−0.0480570	−0.00249164
$373$	7.72073	0.399764	0.199882	−	0.979820i	$-0.435944\pi$
$373$	7.72073	0.399764	0.199882	+	0.979820i	$0.435944\pi$
$374$	5.10391	0.263917
$375$	0	0
$376$	−8.25944	−0.425948
$377$	19.1745	0.987536
$378$	0.786762	0.0404667
$379$	27.7660	1.42624	0.713122	−	0.701040i	$-0.247281\pi$
$379$	27.7660	1.42624	0.713122	+	0.701040i	$0.247281\pi$
$380$	0	0
$381$	4.40162	0.225502
$382$	−19.9231	−1.01936
$383$	2.17131	0.110949	0.0554744	−	0.998460i	$-0.482333\pi$
$383$	2.17131	0.110949	0.0554744	+	0.998460i	$0.482333\pi$
$384$	−29.3322	−1.49685
$385$	0	0
$386$	−18.0228	−0.917334
$387$	9.69670	0.492911
$388$	1.08084	0.0548712
$389$	−16.3022	−0.826556	−0.413278	−	0.910605i	$-0.635616\pi$
$389$	−16.3022	−0.826556	−0.413278	+	0.910605i	$0.635616\pi$
$390$	0	0
$391$	4.09555	0.207121
$392$	14.9230	0.753723
$393$	−25.9505	−1.30903
$394$	11.2287	0.565696
$395$	0	0
$396$	0.766963	0.0385413
$397$	−4.91814	−0.246835	−0.123417	−	0.992355i	$-0.539385\pi$
$397$	−4.91814	−0.246835	−0.123417	+	0.992355i	$0.539385\pi$
$398$	−7.48378	−0.375128
$399$	1.44764	0.0724726
$400$	0	0
$401$	2.97380	0.148504	0.0742522	−	0.997239i	$-0.476343\pi$
$401$	2.97380	0.148504	0.0742522	+	0.997239i	$0.476343\pi$
$402$	18.3802	0.916720
$403$	−0.373899	−0.0186252
$404$	−1.16285	−0.0578542
$405$	0	0
$406$	−16.8304	−0.835278
$407$	−2.03411	−0.100827
$408$	9.43545	0.467125
$409$	−24.3914	−1.20608	−0.603039	−	0.797712i	$-0.706044\pi$
$409$	−24.3914	−1.20608	−0.603039	+	0.797712i	$0.706044\pi$
$410$	0	0
$411$	43.6451	2.15286
$412$	−0.833649	−0.0410709
$413$	0.541636	0.0266522
$414$	11.7786	0.578885
$415$	0	0
$416$	1.28950	0.0632231
$417$	−22.3126	−1.09265
$418$	−1.71895	−0.0840764
$419$	−29.8864	−1.46005	−0.730023	−	0.683422i	$-0.760491\pi$
$419$	−29.8864	−1.46005	−0.730023	+	0.683422i	$0.760491\pi$
$420$	0	0
$421$	−5.05042	−0.246142	−0.123071	−	0.992398i	$-0.539274\pi$
$421$	−5.05042	−0.246142	−0.123071	+	0.992398i	$0.539274\pi$
$422$	−29.9564	−1.45825
$423$	8.48605	0.412606
$424$	−33.9669	−1.64958
$425$	0	0
$426$	−50.2034	−2.43236
$427$	−4.86153	−0.235266
$428$	−0.475010	−0.0229605
$429$	12.3142	0.594537
$430$	0	0
$431$	10.0234	0.482808	0.241404	−	0.970425i	$-0.422392\pi$
$431$	10.0234	0.482808	0.241404	+	0.970425i	$0.422392\pi$
$432$	−1.82256	−0.0876878
$433$	26.9755	1.29636	0.648179	−	0.761488i	$-0.275531\pi$
$433$	26.9755	1.29636	0.648179	+	0.761488i	$0.275531\pi$
$434$	0.328190	0.0157536
$435$	0	0
$436$	−0.334560	−0.0160225
$437$	−1.37934	−0.0659828
$438$	−15.2660	−0.729436
$439$	24.6330	1.17567	0.587835	−	0.808981i	$-0.299980\pi$
$439$	24.6330	1.17567	0.587835	+	0.808981i	$0.299980\pi$
$440$	0	0
$441$	−15.3324	−0.730114
$442$	−4.28339	−0.203740
$443$	10.5621	0.501819	0.250910	−	0.968010i	$-0.419270\pi$
$443$	10.5621	0.501819	0.250910	+	0.968010i	$0.419270\pi$
$444$	0.219412	0.0104128
$445$	0	0
$446$	−1.14733	−0.0543275
$447$	−41.0546	−1.94182
$448$	9.39386	0.443818
$449$	−5.95137	−0.280863	−0.140431	−	0.990090i	$-0.544849\pi$
$449$	−5.95137	−0.280863	−0.140431	+	0.990090i	$0.544849\pi$
$450$	0	0
$451$	2.77439	0.130641
$452$	−0.604722	−0.0284437
$453$	22.1579	1.04107
$454$	−1.54343	−0.0724369
$455$	0	0
$456$	−3.17776	−0.148812
$457$	−8.98781	−0.420432	−0.210216	−	0.977655i	$-0.567417\pi$
$457$	−8.98781	−0.420432	−0.210216	+	0.977655i	$0.567417\pi$
$458$	27.2197	1.27190
$459$	0.616993	0.0287988
$460$	0	0
$461$	17.2131	0.801693	0.400846	−	0.916145i	$-0.368716\pi$
$461$	17.2131	0.801693	0.400846	+	0.916145i	$0.368716\pi$
$462$	−10.8088	−0.502871
$463$	17.5090	0.813711	0.406856	−	0.913492i	$-0.366625\pi$
$463$	17.5090	0.813711	0.406856	+	0.913492i	$0.366625\pi$
$464$	38.9880	1.80997
$465$	0	0
$466$	−29.9456	−1.38720
$467$	4.34976	0.201283	0.100641	−	0.994923i	$-0.467911\pi$
$467$	4.34976	0.201283	0.100641	+	0.994923i	$0.467911\pi$
$468$	−0.643663	−0.0297533
$469$	−6.55858	−0.302847
$470$	0	0
$471$	0.425672	0.0196139
$472$	−1.18897	−0.0547266
$473$	8.47854	0.389844
$474$	25.5434	1.17325
$475$	0	0
$476$	0.196449	0.00900422
$477$	34.8989	1.59791
$478$	−25.0775	−1.14702
$479$	−28.5754	−1.30565	−0.652823	−	0.757511i	$-0.726415\pi$
$479$	−28.5754	−1.30565	−0.652823	+	0.757511i	$0.726415\pi$
$480$	0	0
$481$	1.70710	0.0778369
$482$	1.45267	0.0661675
$483$	−8.67336	−0.394652
$484$	−0.542276	−0.0246489
$485$	0	0
$486$	31.4292	1.42566
$487$	−15.6356	−0.708516	−0.354258	−	0.935148i	$-0.615266\pi$
$487$	−15.6356	−0.708516	−0.354258	+	0.935148i	$0.615266\pi$
$488$	10.6717	0.483086
$489$	22.1171	1.00017
$490$	0	0
$491$	−15.4298	−0.696338	−0.348169	−	0.937432i	$-0.613196\pi$
$491$	−15.4298	−0.696338	−0.348169	+	0.937432i	$0.613196\pi$
$492$	−0.299263	−0.0134918
$493$	−13.1987	−0.594439
$494$	1.44260	0.0649057
$495$	0	0
$496$	−0.760260	−0.0341367
$497$	17.9140	0.803553
$498$	−3.82739	−0.171509
$499$	1.64702	0.0737308	0.0368654	−	0.999320i	$-0.488263\pi$
$499$	1.64702	0.0737308	0.0368654	+	0.999320i	$0.488263\pi$
$500$	0	0
$501$	54.2835	2.42521
$502$	−31.3695	−1.40009
$503$	39.2147	1.74850	0.874249	−	0.485477i	$-0.161354\pi$
$503$	39.2147	1.74850	0.874249	+	0.485477i	$0.161354\pi$
$504$	−9.68283	−0.431308
$505$	0	0
$506$	10.2989	0.457840
$507$	21.0289	0.933925
$508$	−0.201169	−0.00892543
$509$	−1.68682	−0.0747669	−0.0373835	−	0.999301i	$-0.511902\pi$
$509$	−1.68682	−0.0747669	−0.0373835	+	0.999301i	$0.511902\pi$
$510$	0	0
$511$	5.44734	0.240976
$512$	−20.4834	−0.905247
$513$	−0.207797	−0.00917447
$514$	32.2895	1.42423
$515$	0	0
$516$	−0.914549	−0.0402608
$517$	7.41998	0.326330
$518$	−1.49840	−0.0658360
$519$	−12.2696	−0.538577
$520$	0	0
$521$	16.3399	0.715862	0.357931	−	0.933748i	$-0.383482\pi$
$521$	16.3399	0.715862	0.357931	+	0.933748i	$0.383482\pi$
$522$	−37.9586	−1.66140
$523$	−15.6795	−0.685617	−0.342808	−	0.939405i	$-0.611378\pi$
$523$	−15.6795	−0.685617	−0.342808	+	0.939405i	$0.611378\pi$
$524$	1.18603	0.0518118
$525$	0	0
$526$	5.66201	0.246875
$527$	0.257372	0.0112113
$528$	25.0389	1.08968
$529$	−14.7358	−0.640689
$530$	0	0
$531$	1.22159	0.0530123
$532$	−0.0661620	−0.00286848
$533$	−2.32837	−0.100853
$534$	42.8135	1.85272
$535$	0	0
$536$	14.3970	0.621855
$537$	36.9417	1.59415
$538$	9.59623	0.413723
$539$	−13.4062	−0.577448
$540$	0	0
$541$	−3.92084	−0.168570	−0.0842851	−	0.996442i	$-0.526861\pi$
$541$	−3.92084	−0.168570	−0.0842851	+	0.996442i	$0.526861\pi$
$542$	−24.5258	−1.05347
$543$	44.4537	1.90769
$544$	−0.887626	−0.0380566
$545$	0	0
$546$	9.07114	0.388209
$547$	−36.8903	−1.57732	−0.788658	−	0.614832i	$-0.789224\pi$
$547$	−36.8903	−1.57732	−0.788658	+	0.614832i	$0.789224\pi$
$548$	−1.99473	−0.0852106
$549$	−10.9645	−0.467954
$550$	0	0
$551$	4.44519	0.189371
$552$	19.0392	0.810363
$553$	−9.11462	−0.387593
$554$	39.2224	1.66640
$555$	0	0
$556$	1.01976	0.0432474
$557$	−0.602249	−0.0255181	−0.0127591	−	0.999919i	$-0.504061\pi$
$557$	−0.602249	−0.0255181	−0.0127591	+	0.999919i	$0.504061\pi$
$558$	0.740187	0.0313346
$559$	−7.11549	−0.300953
$560$	0	0
$561$	−8.47647	−0.357877
$562$	−13.7818	−0.581350
$563$	7.05189	0.297202	0.148601	−	0.988897i	$-0.452523\pi$
$563$	7.05189	0.297202	0.148601	+	0.988897i	$0.452523\pi$
$564$	−0.800366	−0.0337015
$565$	0	0
$566$	−19.9825	−0.839928
$567$	−11.8883	−0.499262
$568$	−39.3237	−1.64999
$569$	32.0383	1.34311	0.671557	−	0.740953i	$-0.265626\pi$
$569$	32.0383	1.34311	0.671557	+	0.740953i	$0.265626\pi$
$570$	0	0
$571$	27.2459	1.14020	0.570102	−	0.821574i	$-0.306904\pi$
$571$	27.2459	1.14020	0.570102	+	0.821574i	$0.306904\pi$
$572$	−0.562802	−0.0235319
$573$	33.0879	1.38227
$574$	2.04372	0.0853034
$575$	0	0
$576$	21.1866	0.882773
$577$	−2.31753	−0.0964802	−0.0482401	−	0.998836i	$-0.515361\pi$
$577$	−2.31753	−0.0964802	−0.0482401	+	0.998836i	$0.515361\pi$
$578$	−21.7470	−0.904556
$579$	29.9318	1.24392
$580$	0	0
$581$	1.36572	0.0566598
$582$	−34.3543	−1.42403
$583$	30.5146	1.26379
$584$	−11.9577	−0.494811
$585$	0	0
$586$	16.6740	0.688797
$587$	−3.75229	−0.154874	−0.0774369	−	0.996997i	$-0.524674\pi$
$587$	−3.75229	−0.154874	−0.0774369	+	0.996997i	$0.524674\pi$
$588$	1.44608	0.0596354
$589$	−0.0866804	−0.00357160
$590$	0	0
$591$	−18.6484	−0.767094
$592$	3.47109	0.142661
$593$	−34.1569	−1.40266	−0.701328	−	0.712839i	$-0.747409\pi$
$593$	−34.1569	−1.40266	−0.701328	+	0.712839i	$0.747409\pi$
$594$	1.55152	0.0636596
$595$	0	0
$596$	1.87633	0.0768577
$597$	12.4289	0.508681
$598$	−8.64317	−0.353446
$599$	−19.0779	−0.779503	−0.389752	−	0.920920i	$-0.627439\pi$
$599$	−19.0779	−0.779503	−0.389752	+	0.920920i	$0.627439\pi$
$600$	0	0
$601$	22.3859	0.913140	0.456570	−	0.889687i	$-0.349078\pi$
$601$	22.3859	0.913140	0.456570	+	0.889687i	$0.349078\pi$
$602$	6.24562	0.254553
$603$	−14.7920	−0.602376
$604$	−1.01269	−0.0412058
$605$	0	0
$606$	36.9611	1.50144
$607$	39.6985	1.61131	0.805656	−	0.592384i	$-0.201813\pi$
$607$	39.6985	1.61131	0.805656	+	0.592384i	$0.201813\pi$
$608$	0.298943	0.0121237
$609$	27.9515	1.13265
$610$	0	0
$611$	−6.22711	−0.251922
$612$	0.443063	0.0179098
$613$	22.1222	0.893505	0.446753	−	0.894658i	$-0.352580\pi$
$613$	22.1222	0.893505	0.446753	+	0.894658i	$0.352580\pi$
$614$	−17.1931	−0.693856
$615$	0	0
$616$	−8.46642	−0.341122
$617$	−4.59367	−0.184934	−0.0924670	−	0.995716i	$-0.529475\pi$
$617$	−4.59367	−0.184934	−0.0924670	+	0.995716i	$0.529475\pi$
$618$	26.4974	1.06588
$619$	−11.6527	−0.468364	−0.234182	−	0.972193i	$-0.575241\pi$
$619$	−11.6527	−0.468364	−0.234182	+	0.972193i	$0.575241\pi$
$620$	0	0
$621$	1.24499	0.0499598
$622$	−8.04555	−0.322597
$623$	−15.2771	−0.612064
$624$	−21.0135	−0.841215
$625$	0	0
$626$	−36.1022	−1.44293
$627$	2.85479	0.114009
$628$	−0.0194546	−0.000776324 0
$629$	−1.17507	−0.0468533
$630$	0	0
$631$	−3.75293	−0.149402	−0.0747010	−	0.997206i	$-0.523800\pi$
$631$	−3.75293	−0.149402	−0.0747010	+	0.997206i	$0.523800\pi$
$632$	20.0078	0.795869
$633$	49.7509	1.97742
$634$	22.9707	0.912284
$635$	0	0
$636$	−3.29150	−0.130517
$637$	11.2510	0.445781
$638$	−33.1900	−1.31401
$639$	40.4026	1.59830
$640$	0	0
$641$	30.9147	1.22106	0.610529	−	0.791994i	$-0.290957\pi$
$641$	30.9147	1.22106	0.610529	+	0.791994i	$0.290957\pi$
$642$	15.0981	0.595875
$643$	22.1044	0.871712	0.435856	−	0.900017i	$-0.356446\pi$
$643$	22.1044	0.871712	0.435856	+	0.900017i	$0.356446\pi$
$644$	0.396402	0.0156204
$645$	0	0
$646$	−0.993010	−0.0390695
$647$	−32.9443	−1.29517	−0.647586	−	0.761992i	$-0.724222\pi$
$647$	−32.9443	−1.29517	−0.647586	+	0.761992i	$0.724222\pi$
$648$	26.0965	1.02517
$649$	1.06812	0.0419275
$650$	0	0
$651$	−0.545050	−0.0213622
$652$	−1.01083	−0.0395870
$653$	−40.1758	−1.57220	−0.786100	−	0.618100i	$-0.787903\pi$
$653$	−40.1758	−1.57220	−0.786100	+	0.618100i	$0.787903\pi$
$654$	10.6339	0.415820
$655$	0	0
$656$	−4.73434	−0.184845
$657$	12.2857	0.479312
$658$	5.46585	0.213081
$659$	−24.7128	−0.962672	−0.481336	−	0.876536i	$-0.659848\pi$
$659$	−24.7128	−0.962672	−0.481336	+	0.876536i	$0.659848\pi$
$660$	0	0
$661$	12.9928	0.505361	0.252681	−	0.967550i	$-0.418688\pi$
$661$	12.9928	0.505361	0.252681	+	0.967550i	$0.418688\pi$
$662$	34.1182	1.32604
$663$	7.11375	0.276275
$664$	−2.99795	−0.116343
$665$	0	0
$666$	−3.37944	−0.130951
$667$	−26.6328	−1.03123
$668$	−2.48094	−0.0959904
$669$	1.90545	0.0736691
$670$	0	0
$671$	−9.58708	−0.370105
$672$	1.87977	0.0725137
$673$	6.53422	0.251876	0.125938	−	0.992038i	$-0.459806\pi$
$673$	6.53422	0.251876	0.125938	+	0.992038i	$0.459806\pi$
$674$	−16.1708	−0.622876
$675$	0	0
$676$	−0.961090	−0.0369650
$677$	3.97121	0.152626	0.0763130	−	0.997084i	$-0.475685\pi$
$677$	3.97121	0.152626	0.0763130	+	0.997084i	$0.475685\pi$
$678$	19.2210	0.738178
$679$	12.2586	0.470442
$680$	0	0
$681$	2.56330	0.0982258
$682$	0.647200	0.0247826
$683$	28.9357	1.10719	0.553596	−	0.832785i	$-0.313255\pi$
$683$	28.9357	1.10719	0.553596	+	0.832785i	$0.313255\pi$
$684$	−0.149219	−0.00570554
$685$	0	0
$686$	−22.5923	−0.862577
$687$	−45.2060	−1.72471
$688$	−14.4682	−0.551593
$689$	−25.6090	−0.975624
$690$	0	0
$691$	43.7038	1.66257	0.831285	−	0.555847i	$-0.187606\pi$
$691$	43.7038	1.66257	0.831285	+	0.555847i	$0.187606\pi$
$692$	0.560763	0.0213170
$693$	8.69870	0.330436
$694$	−6.46282	−0.245325
$695$	0	0
$696$	−61.3574	−2.32575
$697$	1.60272	0.0607075
$698$	1.31051	0.0496036
$699$	49.7329	1.88107
$700$	0	0
$701$	−25.3881	−0.958894	−0.479447	−	0.877571i	$-0.659163\pi$
$701$	−25.3881	−0.958894	−0.479447	+	0.877571i	$0.659163\pi$
$702$	−1.30209	−0.0491442
$703$	0.395753	0.0149261
$704$	18.5250	0.698186
$705$	0	0
$706$	0.661531	0.0248970
$707$	−13.1888	−0.496016
$708$	−0.115215	−0.00433003
$709$	47.6484	1.78947	0.894737	−	0.446594i	$-0.147363\pi$
$709$	47.6484	1.78947	0.894737	+	0.446594i	$0.147363\pi$
$710$	0	0
$711$	−20.5568	−0.770939
$712$	33.5353	1.25679
$713$	0.519335	0.0194493
$714$	−6.24409	−0.233679
$715$	0	0
$716$	−1.68836	−0.0630970
$717$	41.6481	1.55538
$718$	−12.5679	−0.469028
$719$	−22.1579	−0.826351	−0.413175	−	0.910651i	$-0.635580\pi$
$719$	−22.1579	−0.826351	−0.413175	+	0.910651i	$0.635580\pi$
$720$	0	0
$721$	−9.45504	−0.352124
$722$	−27.2664	−1.01475
$723$	−2.41257	−0.0897244
$724$	−2.03168	−0.0755069
$725$	0	0
$726$	17.2361	0.639693
$727$	11.4060	0.423026	0.211513	−	0.977375i	$-0.432161\pi$
$727$	11.4060	0.423026	0.211513	+	0.977375i	$0.432161\pi$
$728$	7.10532	0.263341
$729$	−23.6779	−0.876961
$730$	0	0
$731$	4.89793	0.181157
$732$	1.03412	0.0382223
$733$	−32.1861	−1.18882	−0.594411	−	0.804162i	$-0.702615\pi$
$733$	−32.1861	−1.18882	−0.594411	+	0.804162i	$0.702615\pi$
$734$	−34.5221	−1.27423
$735$	0	0
$736$	−1.79108	−0.0660202
$737$	−12.9337	−0.476420
$738$	4.60934	0.169672
$739$	6.90721	0.254086	0.127043	−	0.991897i	$-0.459451\pi$
$739$	6.90721	0.254086	0.127043	+	0.991897i	$0.459451\pi$
$740$	0	0
$741$	−2.39584	−0.0880133
$742$	22.4783	0.825203
$743$	31.6170	1.15991	0.579957	−	0.814647i	$-0.303069\pi$
$743$	31.6170	1.15991	0.579957	+	0.814647i	$0.303069\pi$
$744$	1.19646	0.0438643
$745$	0	0
$746$	11.2157	0.410636
$747$	3.08020	0.112699
$748$	0.387403	0.0141649
$749$	−5.38744	−0.196853
$750$	0	0
$751$	−14.3963	−0.525327	−0.262664	−	0.964887i	$-0.584601\pi$
$751$	−14.3963	−0.525327	−0.262664	+	0.964887i	$0.584601\pi$
$752$	−12.6618	−0.461727
$753$	52.0978	1.89855
$754$	27.8543	1.01439
$755$	0	0
$756$	0.0597177	0.00217191
$757$	−31.1923	−1.13370	−0.566852	−	0.823820i	$-0.691839\pi$
$757$	−31.1923	−1.13370	−0.566852	+	0.823820i	$0.691839\pi$
$758$	40.3350	1.46503
$759$	−17.1041	−0.620841
$760$	0	0
$761$	4.10282	0.148727	0.0743636	−	0.997231i	$-0.476307\pi$
$761$	4.10282	0.148727	0.0743636	+	0.997231i	$0.476307\pi$
$762$	6.39412	0.231635
$763$	−3.79450	−0.137370
$764$	−1.51223	−0.0547105
$765$	0	0
$766$	3.15421	0.113966
$767$	−0.896407	−0.0323674
$768$	−6.36534	−0.229689
$769$	−13.7105	−0.494413	−0.247206	−	0.968963i	$-0.579513\pi$
$769$	−13.7105	−0.494413	−0.247206	+	0.968963i	$0.579513\pi$
$770$	0	0
$771$	−53.6257	−1.93128
$772$	−1.36798	−0.0492348
$773$	34.2990	1.23365	0.616825	−	0.787100i	$-0.288418\pi$
$773$	34.2990	1.23365	0.616825	+	0.787100i	$0.288418\pi$
$774$	14.0861	0.506316
$775$	0	0
$776$	−26.9093	−0.965987
$777$	2.48851	0.0892750
$778$	−23.6818	−0.849035
$779$	−0.539781	−0.0193397
$780$	0	0
$781$	35.3270	1.26410
$782$	5.94951	0.212754
$783$	−4.01222	−0.143385
$784$	22.8770	0.817035
$785$	0	0
$786$	−37.6976	−1.34463
$787$	−1.05124	−0.0374728	−0.0187364	−	0.999824i	$-0.505964\pi$
$787$	−1.05124	−0.0374728	−0.0187364	+	0.999824i	$0.505964\pi$
$788$	0.852296	0.0303618
$789$	−9.40335	−0.334768
$790$	0	0
$791$	−6.85861	−0.243864
$792$	−19.0948	−0.678506
$793$	8.04582	0.285716
$794$	−7.14446	−0.253547
$795$	0	0
$796$	−0.568042	−0.0201337
$797$	46.9746	1.66393	0.831963	−	0.554832i	$-0.187217\pi$
$797$	46.9746	1.66393	0.831963	+	0.554832i	$0.187217\pi$
$798$	2.10295	0.0744435
$799$	4.28642	0.151642
$800$	0	0
$801$	−34.4554	−1.21742
$802$	4.31996	0.152543
$803$	10.7423	0.379088
$804$	1.39511	0.0492018
$805$	0	0
$806$	−0.543153	−0.0191318
$807$	−15.9372	−0.561016
$808$	28.9512	1.01850
$809$	−40.7051	−1.43111	−0.715557	−	0.698554i	$-0.753827\pi$
$809$	−40.7051	−1.43111	−0.715557	+	0.698554i	$0.753827\pi$
$810$	0	0
$811$	−9.80946	−0.344457	−0.172228	−	0.985057i	$-0.555097\pi$
$811$	−9.80946	−0.344457	−0.172228	+	0.985057i	$0.555097\pi$
$812$	−1.27748	−0.0448307
$813$	40.7319	1.42853
$814$	−2.95490	−0.103569
$815$	0	0
$816$	14.4646	0.506363
$817$	−1.64957	−0.0577113
$818$	−35.4328	−1.23888
$819$	−7.30026	−0.255092
$820$	0	0
$821$	−7.27965	−0.254061	−0.127031	−	0.991899i	$-0.540545\pi$
$821$	−7.27965	−0.254061	−0.127031	+	0.991899i	$0.540545\pi$
$822$	63.4022	2.21140
$823$	10.9222	0.380725	0.190362	−	0.981714i	$-0.439034\pi$
$823$	10.9222	0.380725	0.190362	+	0.981714i	$0.439034\pi$
$824$	20.7551	0.723039
$825$	0	0
$826$	0.786821	0.0273770
$827$	19.4083	0.674894	0.337447	−	0.941344i	$-0.390437\pi$
$827$	19.4083	0.674894	0.337447	+	0.941344i	$0.390437\pi$
$828$	0.894030	0.0310697
$829$	28.4668	0.988692	0.494346	−	0.869265i	$-0.335408\pi$
$829$	28.4668	0.988692	0.494346	+	0.869265i	$0.335408\pi$
$830$	0	0
$831$	−65.1397	−2.25967
$832$	−15.5468	−0.538989
$833$	−7.74459	−0.268334
$834$	−32.4129	−1.12237
$835$	0	0
$836$	−0.130473	−0.00451252
$837$	0.0782376	0.00270429
$838$	−43.4152	−1.49975
$839$	−19.8074	−0.683826	−0.341913	−	0.939732i	$-0.611075\pi$
$839$	−19.8074	−0.683826	−0.341913	+	0.939732i	$0.611075\pi$
$840$	0	0
$841$	56.8293	1.95963
$842$	−7.33662	−0.252836
$843$	22.8885	0.788322
$844$	−2.27378	−0.0782669
$845$	0	0
$846$	12.3275	0.423827
$847$	−6.15036	−0.211329
$848$	−52.0715	−1.78814
$849$	33.1865	1.13896
$850$	0	0
$851$	−2.37111	−0.0812806
$852$	−3.81059	−0.130549
$853$	8.54067	0.292427	0.146214	−	0.989253i	$-0.453291\pi$
$853$	8.54067	0.292427	0.146214	+	0.989253i	$0.453291\pi$
$854$	−7.06222	−0.241664
$855$	0	0
$856$	11.8262	0.404210
$857$	9.52177	0.325257	0.162629	−	0.986687i	$-0.448003\pi$
$857$	9.52177	0.325257	0.162629	+	0.986687i	$0.448003\pi$
$858$	17.8886	0.610705
$859$	−32.9092	−1.12285	−0.561423	−	0.827529i	$-0.689746\pi$
$859$	−32.9092	−1.12285	−0.561423	+	0.827529i	$0.689746\pi$
$860$	0	0
$861$	−3.39417	−0.115673
$862$	14.5607	0.495939
$863$	−2.93045	−0.0997538	−0.0498769	−	0.998755i	$-0.515883\pi$
$863$	−2.93045	−0.0997538	−0.0498769	+	0.998755i	$0.515883\pi$
$864$	−0.269826	−0.00917967
$865$	0	0
$866$	39.1866	1.33161
$867$	36.1169	1.22660
$868$	0.0249106	0.000845522 0
$869$	−17.9743	−0.609736
$870$	0	0
$871$	10.8544	0.367788
$872$	8.32944	0.282071
$873$	27.6476	0.935729
$874$	−2.00373	−0.0677773
$875$	0	0
$876$	−1.15873	−0.0391500
$877$	−47.5268	−1.60487	−0.802434	−	0.596741i	$-0.796462\pi$
$877$	−47.5268	−1.60487	−0.802434	+	0.596741i	$0.796462\pi$
$878$	35.7838	1.20764
$879$	−27.6918	−0.934022
$880$	0	0
$881$	−14.8772	−0.501227	−0.250614	−	0.968087i	$-0.580632\pi$
$881$	−14.8772	−0.501227	−0.250614	+	0.968087i	$0.580632\pi$
$882$	−22.2730	−0.749970
$883$	18.1229	0.609884	0.304942	−	0.952371i	$-0.401363\pi$
$883$	18.1229	0.609884	0.304942	+	0.952371i	$0.401363\pi$
$884$	−0.325122	−0.0109351
$885$	0	0
$886$	15.3433	0.515467
$887$	−29.7170	−0.997799	−0.498899	−	0.866660i	$-0.666262\pi$
$887$	−29.7170	−0.997799	−0.498899	+	0.866660i	$0.666262\pi$
$888$	−5.46263	−0.183314
$889$	−2.28161	−0.0765227
$890$	0	0
$891$	−23.4441	−0.785407
$892$	−0.0870857	−0.00291584
$893$	−1.44362	−0.0483089
$894$	−59.6390	−1.99463
$895$	0	0
$896$	15.2045	0.507948
$897$	14.3544	0.479279
$898$	−8.64540	−0.288501
$899$	−1.67366	−0.0558196
$900$	0	0
$901$	17.6279	0.587269
$902$	4.03028	0.134194
$903$	−10.3726	−0.345178
$904$	15.0556	0.500741
$905$	0	0
$906$	32.1882	1.06938
$907$	−42.0265	−1.39547	−0.697733	−	0.716358i	$-0.745808\pi$
$907$	−42.0265	−1.39547	−0.697733	+	0.716358i	$0.745808\pi$
$908$	−0.117151	−0.00388780
$909$	−29.7455	−0.986597
$910$	0	0
$911$	−48.7145	−1.61398	−0.806992	−	0.590562i	$-0.798906\pi$
$911$	−48.7145	−1.61398	−0.806992	+	0.590562i	$0.798906\pi$
$912$	−4.87153	−0.161313
$913$	2.69325	0.0891335
$914$	−13.0564	−0.431866
$915$	0	0
$916$	2.06606	0.0682647
$917$	13.4516	0.444211
$918$	0.896290	0.0295820
$919$	14.6475	0.483177	0.241589	−	0.970379i	$-0.422332\pi$
$919$	14.6475	0.483177	0.241589	+	0.970379i	$0.422332\pi$
$920$	0	0
$921$	28.5539	0.940882
$922$	25.0050	0.823495
$923$	−29.6477	−0.975865
$924$	−0.820423	−0.0269899
$925$	0	0
$926$	25.4349	0.835841
$927$	−21.3246	−0.700390
$928$	5.77211	0.189479
$929$	45.7110	1.49973	0.749864	−	0.661592i	$-0.230119\pi$
$929$	45.7110	1.49973	0.749864	+	0.661592i	$0.230119\pi$
$930$	0	0
$931$	2.60830	0.0854836
$932$	−2.27296	−0.0744533
$933$	13.3619	0.437448
$934$	6.31878	0.206757
$935$	0	0
$936$	16.0251	0.523796
$937$	39.7753	1.29940	0.649701	−	0.760190i	$-0.274894\pi$
$937$	39.7753	1.29940	0.649701	+	0.760190i	$0.274894\pi$
$938$	−9.52748	−0.311083
$939$	59.9577	1.95665
$940$	0	0
$941$	18.0359	0.587953	0.293977	−	0.955813i	$-0.405021\pi$
$941$	18.0359	0.587953	0.293977	+	0.955813i	$0.405021\pi$
$942$	0.618362	0.0201473
$943$	3.23404	0.105315
$944$	−1.82269	−0.0593235
$945$	0	0
$946$	12.3166	0.400446
$947$	−34.4377	−1.11908	−0.559538	−	0.828805i	$-0.689021\pi$
$947$	−34.4377	−1.11908	−0.559538	+	0.828805i	$0.689021\pi$
$948$	1.93882	0.0629700
$949$	−9.01534	−0.292650
$950$	0	0
$951$	−38.1493	−1.23707
$952$	−4.89093	−0.158516
$953$	−30.0386	−0.973045	−0.486522	−	0.873668i	$-0.661735\pi$
$953$	−30.0386	−0.973045	−0.486522	+	0.873668i	$0.661735\pi$
$954$	50.6967	1.64137
$955$	0	0
$956$	−1.90346	−0.0615622
$957$	55.1213	1.78182
$958$	−41.5108	−1.34115
$959$	−22.6237	−0.730558
$960$	0	0
$961$	−30.9674	−0.998947
$962$	2.47985	0.0799537
$963$	−12.1506	−0.391549
$964$	0.110263	0.00355132
$965$	0	0
$966$	−12.5996	−0.405384
$967$	47.3839	1.52376	0.761882	−	0.647716i	$-0.224276\pi$
$967$	47.3839	1.52376	0.761882	+	0.647716i	$0.224276\pi$
$968$	13.5009	0.433934
$969$	1.64917	0.0529790
$970$	0	0
$971$	6.81587	0.218732	0.109366	−	0.994002i	$-0.465118\pi$
$971$	6.81587	0.218732	0.109366	+	0.994002i	$0.465118\pi$
$972$	2.38557	0.0765173
$973$	11.5659	0.370784
$974$	−22.7134	−0.727784
$975$	0	0
$976$	16.3598	0.523665
$977$	51.4729	1.64676	0.823382	−	0.567488i	$-0.192085\pi$
$977$	51.4729	1.64676	0.823382	+	0.567488i	$0.192085\pi$
$978$	32.1289	1.02737
$979$	−30.1269	−0.962860
$980$	0	0
$981$	−8.55797	−0.273235
$982$	−22.4145	−0.715275
$983$	17.8746	0.570112	0.285056	−	0.958511i	$-0.407988\pi$
$983$	17.8746	0.570112	0.285056	+	0.958511i	$0.407988\pi$
$984$	7.45067	0.237519
$985$	0	0
$986$	−19.1734	−0.610606
$987$	−9.07756	−0.288942
$988$	0.109498	0.00348359
$989$	9.88323	0.314268
$990$	0	0
$991$	28.7353	0.912805	0.456403	−	0.889773i	$-0.349138\pi$
$991$	28.7353	0.912805	0.456403	+	0.889773i	$0.349138\pi$
$992$	−0.112555	−0.00357363
$993$	−56.6627	−1.79814
$994$	26.0232	0.825407
$995$	0	0
$996$	−0.290511	−0.00920519
$997$	44.7201	1.41630	0.708150	−	0.706062i	$-0.249530\pi$
$997$	44.7201	1.41630	0.708150	+	0.706062i	$0.249530\pi$
$998$	2.39258	0.0757359
$999$	−0.357207	−0.0113015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.34		46
5.2	odd	4		1205.2.b.c.724.34	yes	46
5.3	odd	4		1205.2.b.c.724.13	✓	46
5.4	even	2	inner	6025.2.a.p.1.13		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.13	✓	46	5.3	odd	4
1205.2.b.c.724.34	yes	46	5.2	odd	4
6025.2.a.p.1.13		46	5.4	even	2	inner
6025.2.a.p.1.34		46	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.45267 q^{2} -2.41257 q^{3} +0.110263 q^{4} -3.50468 q^{6} +1.25057 q^{7} -2.74517 q^{8} +2.82049 q^{9} +O(q^{10})\)	\(q+1.45267 q^{2} -2.41257 q^{3} +0.110263 q^{4} -3.50468 q^{6} +1.25057 q^{7} -2.74517 q^{8} +2.82049 q^{9} +2.46616 q^{11} -0.266016 q^{12} -2.06969 q^{13} +1.81667 q^{14} -4.20837 q^{16} +1.42467 q^{17} +4.09725 q^{18} -0.479813 q^{19} -3.01709 q^{21} +3.58253 q^{22} +2.87475 q^{23} +6.62292 q^{24} -3.00659 q^{26} +0.433079 q^{27} +0.137891 q^{28} -9.26441 q^{29} +0.180654 q^{31} -0.623041 q^{32} -5.94979 q^{33} +2.06958 q^{34} +0.310994 q^{36} -0.824807 q^{37} -0.697012 q^{38} +4.99327 q^{39} +1.12498 q^{41} -4.38285 q^{42} +3.43795 q^{43} +0.271925 q^{44} +4.17607 q^{46} +3.00871 q^{47} +10.1530 q^{48} -5.43607 q^{49} -3.43711 q^{51} -0.228209 q^{52} +12.3733 q^{53} +0.629123 q^{54} -3.43303 q^{56} +1.15758 q^{57} -13.4582 q^{58} +0.433111 q^{59} -3.88745 q^{61} +0.262432 q^{62} +3.52722 q^{63} +7.51166 q^{64} -8.64311 q^{66} -5.24447 q^{67} +0.157087 q^{68} -6.93552 q^{69} +14.3247 q^{71} -7.74273 q^{72} +4.35588 q^{73} -1.19818 q^{74} -0.0529054 q^{76} +3.08411 q^{77} +7.25360 q^{78} -7.28837 q^{79} -9.50630 q^{81} +1.63423 q^{82} +1.09208 q^{83} -0.332672 q^{84} +4.99422 q^{86} +22.3510 q^{87} -6.77004 q^{88} -12.2161 q^{89} -2.58830 q^{91} +0.316977 q^{92} -0.435841 q^{93} +4.37068 q^{94} +1.50313 q^{96} +9.80240 q^{97} -7.89684 q^{98} +6.95579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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