LMFDB - Embedded newform 6025.2.a.q.1.11

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:	$0$
Dimension:	$66$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
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)

$f(q)$	$=$	$q-2.20997 q^{2} -2.26852 q^{3} +2.88397 q^{4} +5.01336 q^{6} +0.372373 q^{7} -1.95355 q^{8} +2.14617 q^{9} +O(q^{10})$	\(q-2.20997 q^{2} -2.26852 q^{3} +2.88397 q^{4} +5.01336 q^{6} +0.372373 q^{7} -1.95355 q^{8} +2.14617 q^{9} +3.78368 q^{11} -6.54234 q^{12} +3.03526 q^{13} -0.822934 q^{14} -1.45065 q^{16} +0.702265 q^{17} -4.74298 q^{18} +4.26314 q^{19} -0.844736 q^{21} -8.36182 q^{22} -7.07928 q^{23} +4.43167 q^{24} -6.70783 q^{26} +1.93692 q^{27} +1.07391 q^{28} +2.22462 q^{29} +8.12419 q^{31} +7.11300 q^{32} -8.58334 q^{33} -1.55199 q^{34} +6.18950 q^{36} +0.768032 q^{37} -9.42143 q^{38} -6.88553 q^{39} +11.3969 q^{41} +1.86684 q^{42} -10.1749 q^{43} +10.9120 q^{44} +15.6450 q^{46} +5.84261 q^{47} +3.29083 q^{48} -6.86134 q^{49} -1.59310 q^{51} +8.75359 q^{52} +7.70807 q^{53} -4.28054 q^{54} -0.727451 q^{56} -9.67102 q^{57} -4.91634 q^{58} +14.0213 q^{59} +10.5586 q^{61} -17.9542 q^{62} +0.799178 q^{63} -12.8182 q^{64} +18.9689 q^{66} +12.7576 q^{67} +2.02531 q^{68} +16.0595 q^{69} -0.694776 q^{71} -4.19266 q^{72} -9.45754 q^{73} -1.69733 q^{74} +12.2948 q^{76} +1.40894 q^{77} +15.2168 q^{78} +4.15383 q^{79} -10.8325 q^{81} -25.1868 q^{82} +11.0345 q^{83} -2.43619 q^{84} +22.4862 q^{86} -5.04658 q^{87} -7.39161 q^{88} -17.5969 q^{89} +1.13025 q^{91} -20.4165 q^{92} -18.4299 q^{93} -12.9120 q^{94} -16.1360 q^{96} +16.6822 q^{97} +15.1634 q^{98} +8.12043 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * 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 $

$2$	−2.20997	−1.56269	−0.781343	−	0.624102i	$-0.785465\pi$
$2$	−2.20997	−1.56269	−0.781343	+	0.624102i	$0.785465\pi$
$3$	−2.26852	−1.30973	−0.654865	−	0.755746i	$-0.727274\pi$
$3$	−2.26852	−1.30973	−0.654865	+	0.755746i	$0.727274\pi$
$4$	2.88397	1.44199
$5$	0	0
$5$	0	0
$6$	5.01336	2.04669
$7$	0.372373	0.140744	0.0703720	−	0.997521i	$-0.477581\pi$
$7$	0.372373	0.140744	0.0703720	+	0.997521i	$0.477581\pi$
$8$	−1.95355	−0.690685
$9$	2.14617	0.715391
$10$	0	0
$11$	3.78368	1.14082	0.570411	−	0.821359i	$-0.306784\pi$
$11$	3.78368	1.14082	0.570411	+	0.821359i	$0.306784\pi$
$12$	−6.54234	−1.88861
$13$	3.03526	0.841828	0.420914	−	0.907100i	$-0.361709\pi$
$13$	3.03526	0.841828	0.420914	+	0.907100i	$0.361709\pi$
$14$	−0.822934	−0.219938
$15$	0	0
$16$	−1.45065	−0.362663
$17$	0.702265	0.170324	0.0851622	−	0.996367i	$-0.472859\pi$
$17$	0.702265	0.170324	0.0851622	+	0.996367i	$0.472859\pi$
$18$	−4.74298	−1.11793
$19$	4.26314	0.978033	0.489016	−	0.872275i	$-0.337356\pi$
$19$	4.26314	0.978033	0.489016	+	0.872275i	$0.337356\pi$
$20$	0	0
$21$	−0.844736	−0.184336
$22$	−8.36182	−1.78275
$23$	−7.07928	−1.47613	−0.738066	−	0.674728i	$-0.764261\pi$
$23$	−7.07928	−1.47613	−0.738066	+	0.674728i	$0.764261\pi$
$24$	4.43167	0.904610
$25$	0	0
$26$	−6.70783	−1.31551
$27$	1.93692	0.372761
$28$	1.07391	0.202951
$29$	2.22462	0.413101	0.206551	−	0.978436i	$-0.433776\pi$
$29$	2.22462	0.413101	0.206551	+	0.978436i	$0.433776\pi$
$30$	0	0
$31$	8.12419	1.45915	0.729573	−	0.683902i	$-0.239719\pi$
$31$	8.12419	1.45915	0.729573	+	0.683902i	$0.239719\pi$
$32$	7.11300	1.25741
$33$	−8.58334	−1.49417
$34$	−1.55199	−0.266163
$35$	0	0
$36$	6.18950	1.03158
$37$	0.768032	0.126264	0.0631319	−	0.998005i	$-0.479891\pi$
$37$	0.768032	0.126264	0.0631319	+	0.998005i	$0.479891\pi$
$38$	−9.42143	−1.52836
$39$	−6.88553	−1.10257
$40$	0	0
$41$	11.3969	1.77990	0.889949	−	0.456061i	$-0.150740\pi$
$41$	11.3969	1.77990	0.889949	+	0.456061i	$0.150740\pi$
$42$	1.86684	0.288060
$43$	−10.1749	−1.55165	−0.775827	−	0.630946i	$-0.782667\pi$
$43$	−10.1749	−1.55165	−0.775827	+	0.630946i	$0.782667\pi$
$44$	10.9120	1.64505
$45$	0	0
$46$	15.6450	2.30673
$47$	5.84261	0.852233	0.426116	−	0.904668i	$-0.359881\pi$
$47$	5.84261	0.852233	0.426116	+	0.904668i	$0.359881\pi$
$48$	3.29083	0.474990
$49$	−6.86134	−0.980191
$50$	0	0
$51$	−1.59310	−0.223079
$52$	8.75359	1.21390
$53$	7.70807	1.05878	0.529392	−	0.848377i	$-0.322420\pi$
$53$	7.70807	1.05878	0.529392	+	0.848377i	$0.322420\pi$
$54$	−4.28054	−0.582508
$55$	0	0
$56$	−0.727451	−0.0972097
$57$	−9.67102	−1.28096
$58$	−4.91634	−0.645547
$59$	14.0213	1.82541	0.912706	−	0.408616i	$-0.133988\pi$
$59$	14.0213	1.82541	0.912706	+	0.408616i	$0.133988\pi$
$60$	0	0
$61$	10.5586	1.35188	0.675942	−	0.736955i	$-0.263737\pi$
$61$	10.5586	1.35188	0.675942	+	0.736955i	$0.263737\pi$
$62$	−17.9542	−2.28019
$63$	0.799178	0.100687
$64$	−12.8182	−1.60228
$65$	0	0
$66$	18.9689	2.33491
$67$	12.7576	1.55859	0.779296	−	0.626656i	$-0.215577\pi$
$67$	12.7576	1.55859	0.779296	+	0.626656i	$0.215577\pi$
$68$	2.02531	0.245605
$69$	16.0595	1.93333
$70$	0	0
$71$	−0.694776	−0.0824547	−0.0412274	−	0.999150i	$-0.513127\pi$
$71$	−0.694776	−0.0824547	−0.0412274	+	0.999150i	$0.513127\pi$
$72$	−4.19266	−0.494110
$73$	−9.45754	−1.10692	−0.553460	−	0.832875i	$-0.686693\pi$
$73$	−9.45754	−1.10692	−0.553460	+	0.832875i	$0.686693\pi$
$74$	−1.69733	−0.197311
$75$	0	0
$76$	12.2948	1.41031
$77$	1.40894	0.160564
$78$	15.2168	1.72297
$79$	4.15383	0.467342	0.233671	−	0.972316i	$-0.424926\pi$
$79$	4.15383	0.467342	0.233671	+	0.972316i	$0.424926\pi$
$80$	0	0
$81$	−10.8325	−1.20361
$82$	−25.1868	−2.78142
$83$	11.0345	1.21120	0.605599	−	0.795770i	$-0.292934\pi$
$83$	11.0345	1.21120	0.605599	+	0.795770i	$0.292934\pi$
$84$	−2.43619	−0.265811
$85$	0	0
$86$	22.4862	2.42475
$87$	−5.04658	−0.541051
$88$	−7.39161	−0.787949
$89$	−17.5969	−1.86527	−0.932633	−	0.360827i	$-0.882494\pi$
$89$	−17.5969	−1.86527	−0.932633	+	0.360827i	$0.882494\pi$
$90$	0	0
$91$	1.13025	0.118482
$92$	−20.4165	−2.12856
$93$	−18.4299	−1.91109
$94$	−12.9120	−1.33177
$95$	0	0
$96$	−16.1360	−1.64687
$97$	16.6822	1.69382	0.846911	−	0.531734i	$-0.178460\pi$
$97$	16.6822	1.69382	0.846911	+	0.531734i	$0.178460\pi$
$98$	15.1634	1.53173
$99$	8.12043	0.816134
$100$	0	0
$101$	13.7376	1.36694	0.683470	−	0.729979i	$-0.260470\pi$
$101$	13.7376	1.36694	0.683470	+	0.729979i	$0.260470\pi$
$102$	3.52071	0.348602
$103$	1.25499	0.123658	0.0618290	−	0.998087i	$-0.480307\pi$
$103$	1.25499	0.123658	0.0618290	+	0.998087i	$0.480307\pi$
$104$	−5.92953	−0.581438
$105$	0	0
$106$	−17.0346	−1.65455
$107$	−1.64792	−0.159310	−0.0796552	−	0.996822i	$-0.525382\pi$
$107$	−1.64792	−0.159310	−0.0796552	+	0.996822i	$0.525382\pi$
$108$	5.58603	0.537516
$109$	−9.08309	−0.870002	−0.435001	−	0.900430i	$-0.643252\pi$
$109$	−9.08309	−0.870002	−0.435001	+	0.900430i	$0.643252\pi$
$110$	0	0
$111$	−1.74230	−0.165371
$112$	−0.540183	−0.0510425
$113$	8.30222	0.781007	0.390504	−	0.920601i	$-0.372301\pi$
$113$	8.30222	0.781007	0.390504	+	0.920601i	$0.372301\pi$
$114$	21.3727	2.00173
$115$	0	0
$116$	6.41573	0.595686
$117$	6.51418	0.602236
$118$	−30.9866	−2.85255
$119$	0.261505	0.0239721
$120$	0	0
$121$	3.31622	0.301475
$122$	−23.3341	−2.11257
$123$	−25.8541	−2.33118
$124$	23.4299	2.10407
$125$	0	0
$126$	−1.76616	−0.157342
$127$	6.19396	0.549625	0.274813	−	0.961498i	$-0.411384\pi$
$127$	6.19396	0.549625	0.274813	+	0.961498i	$0.411384\pi$
$128$	14.1019	1.24644
$129$	23.0819	2.03225
$130$	0	0
$131$	−12.0088	−1.04922	−0.524609	−	0.851344i	$-0.675788\pi$
$131$	−12.0088	−1.04922	−0.524609	+	0.851344i	$0.675788\pi$
$132$	−24.7541	−2.15457
$133$	1.58748	0.137652
$134$	−28.1940	−2.43559
$135$	0	0
$136$	−1.37191	−0.117640
$137$	2.38223	0.203527	0.101764	−	0.994809i	$-0.467551\pi$
$137$	2.38223	0.203527	0.101764	+	0.994809i	$0.467551\pi$
$138$	−35.4910	−3.02119
$139$	3.03084	0.257073	0.128536	−	0.991705i	$-0.458972\pi$
$139$	3.03084	0.257073	0.128536	+	0.991705i	$0.458972\pi$
$140$	0	0
$141$	−13.2541	−1.11619
$142$	1.53543	0.128851
$143$	11.4844	0.960376
$144$	−3.11335	−0.259445
$145$	0	0
$146$	20.9009	1.72977
$147$	15.5651	1.28379
$148$	2.21498	0.182071
$149$	13.9560	1.14332	0.571659	−	0.820491i	$-0.306300\pi$
$149$	13.9560	1.14332	0.571659	+	0.820491i	$0.306300\pi$
$150$	0	0
$151$	−19.8038	−1.61161	−0.805804	−	0.592182i	$-0.798267\pi$
$151$	−19.8038	−1.61161	−0.805804	+	0.592182i	$0.798267\pi$
$152$	−8.32828	−0.675512
$153$	1.50718	0.121848
$154$	−3.11372	−0.250911
$155$	0	0
$156$	−19.8577	−1.58989
$157$	−4.69657	−0.374827	−0.187414	−	0.982281i	$-0.560010\pi$
$157$	−4.69657	−0.374827	−0.187414	+	0.982281i	$0.560010\pi$
$158$	−9.17984	−0.730309
$159$	−17.4859	−1.38672
$160$	0	0
$161$	−2.63614	−0.207757
$162$	23.9394	1.88086
$163$	−12.2047	−0.955943	−0.477972	−	0.878375i	$-0.658628\pi$
$163$	−12.2047	−0.955943	−0.477972	+	0.878375i	$0.658628\pi$
$164$	32.8684	2.56659
$165$	0	0
$166$	−24.3860	−1.89272
$167$	8.56021	0.662409	0.331204	−	0.943559i	$-0.392545\pi$
$167$	8.56021	0.662409	0.331204	+	0.943559i	$0.392545\pi$
$168$	1.65024	0.127318
$169$	−3.78722	−0.291325
$170$	0	0
$171$	9.14944	0.699675
$172$	−29.3440	−2.23746
$173$	11.8122	0.898063	0.449032	−	0.893516i	$-0.351769\pi$
$173$	11.8122	0.898063	0.449032	+	0.893516i	$0.351769\pi$
$174$	11.1528	0.845492
$175$	0	0
$176$	−5.48879	−0.413733
$177$	−31.8075	−2.39080
$178$	38.8886	2.91482
$179$	−10.2287	−0.764527	−0.382263	−	0.924053i	$-0.624855\pi$
$179$	−10.2287	−0.764527	−0.382263	+	0.924053i	$0.624855\pi$
$180$	0	0
$181$	3.35297	0.249224	0.124612	−	0.992206i	$-0.460231\pi$
$181$	3.35297	0.249224	0.124612	+	0.992206i	$0.460231\pi$
$182$	−2.49782	−0.185150
$183$	−23.9523	−1.77060
$184$	13.8298	1.01954
$185$	0	0
$186$	40.7295	2.98643
$187$	2.65715	0.194310
$188$	16.8499	1.22891
$189$	0.721259	0.0524638
$190$	0	0
$191$	−20.1631	−1.45895	−0.729477	−	0.684006i	$-0.760236\pi$
$191$	−20.1631	−1.45895	−0.729477	+	0.684006i	$0.760236\pi$
$192$	29.0784	2.09855
$193$	18.7859	1.35224	0.676118	−	0.736793i	$-0.263661\pi$
$193$	18.7859	1.35224	0.676118	+	0.736793i	$0.263661\pi$
$194$	−36.8672	−2.64691
$195$	0	0
$196$	−19.7879	−1.41342
$197$	−22.2950	−1.58845	−0.794225	−	0.607624i	$-0.792123\pi$
$197$	−22.2950	−1.58845	−0.794225	+	0.607624i	$0.792123\pi$
$198$	−17.9459	−1.27536
$199$	−8.52382	−0.604237	−0.302119	−	0.953270i	$-0.597694\pi$
$199$	−8.52382	−0.604237	−0.302119	+	0.953270i	$0.597694\pi$
$200$	0	0
$201$	−28.9409	−2.04133
$202$	−30.3596	−2.13610
$203$	0.828388	0.0581415
$204$	−4.59446	−0.321676
$205$	0	0
$206$	−2.77349	−0.193239
$207$	−15.1934	−1.05601
$208$	−4.40309	−0.305300
$209$	16.1304	1.11576
$210$	0	0
$211$	−14.3197	−0.985807	−0.492903	−	0.870084i	$-0.664064\pi$
$211$	−14.3197	−0.985807	−0.492903	+	0.870084i	$0.664064\pi$
$212$	22.2299	1.52675
$213$	1.57611	0.107993
$214$	3.64185	0.248952
$215$	0	0
$216$	−3.78388	−0.257460
$217$	3.02523	0.205366
$218$	20.0734	1.35954
$219$	21.4546	1.44977
$220$	0	0
$221$	2.13155	0.143384
$222$	3.85042	0.258423
$223$	−6.98392	−0.467678	−0.233839	−	0.972275i	$-0.575129\pi$
$223$	−6.98392	−0.467678	−0.233839	+	0.972275i	$0.575129\pi$
$224$	2.64869	0.176973
$225$	0	0
$226$	−18.3477	−1.22047
$227$	−3.06063	−0.203141	−0.101571	−	0.994828i	$-0.532387\pi$
$227$	−3.06063	−0.203141	−0.101571	+	0.994828i	$0.532387\pi$
$228$	−27.8909	−1.84712
$229$	−2.01475	−0.133139	−0.0665694	−	0.997782i	$-0.521205\pi$
$229$	−2.01475	−0.133139	−0.0665694	+	0.997782i	$0.521205\pi$
$230$	0	0
$231$	−3.19621	−0.210295
$232$	−4.34591	−0.285323
$233$	−1.73468	−0.113642	−0.0568212	−	0.998384i	$-0.518096\pi$
$233$	−1.73468	−0.113642	−0.0568212	+	0.998384i	$0.518096\pi$
$234$	−14.3962	−0.941106
$235$	0	0
$236$	40.4369	2.63222
$237$	−9.42303	−0.612092
$238$	−0.577918	−0.0374609
$239$	0.0847133	0.00547965	0.00273982	−	0.999996i	$-0.499128\pi$
$239$	0.0847133	0.00547965	0.00273982	+	0.999996i	$0.499128\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−7.32875	−0.471110
$243$	18.7629	1.20364
$244$	30.4506	1.94940
$245$	0	0
$246$	57.1368	3.64291
$247$	12.9397	0.823336
$248$	−15.8710	−1.00781
$249$	−25.0320	−1.58634
$250$	0	0
$251$	24.0806	1.51995	0.759976	−	0.649951i	$-0.225211\pi$
$251$	24.0806	1.51995	0.759976	+	0.649951i	$0.225211\pi$
$252$	2.30481	0.145189
$253$	−26.7857	−1.68400
$254$	−13.6885	−0.858891
$255$	0	0
$256$	−5.52835	−0.345522
$257$	6.09887	0.380437	0.190219	−	0.981742i	$-0.439080\pi$
$257$	6.09887	0.380437	0.190219	+	0.981742i	$0.439080\pi$
$258$	−51.0103	−3.17576
$259$	0.285995	0.0177709
$260$	0	0
$261$	4.77441	0.295529
$262$	26.5392	1.63960
$263$	10.2746	0.633562	0.316781	−	0.948499i	$-0.397398\pi$
$263$	10.2746	0.633562	0.316781	+	0.948499i	$0.397398\pi$
$264$	16.7680	1.03200
$265$	0	0
$266$	−3.50829	−0.215107
$267$	39.9188	2.44299
$268$	36.7926	2.24747
$269$	−19.5761	−1.19358	−0.596788	−	0.802399i	$-0.703557\pi$
$269$	−19.5761	−1.19358	−0.596788	+	0.802399i	$0.703557\pi$
$270$	0	0
$271$	14.7957	0.898773	0.449386	−	0.893337i	$-0.351643\pi$
$271$	14.7957	0.898773	0.449386	+	0.893337i	$0.351643\pi$
$272$	−1.01874	−0.0617703
$273$	−2.56399	−0.155180
$274$	−5.26465	−0.318049
$275$	0	0
$276$	46.3151	2.78784
$277$	−10.5979	−0.636763	−0.318382	−	0.947963i	$-0.603139\pi$
$277$	−10.5979	−0.636763	−0.318382	+	0.947963i	$0.603139\pi$
$278$	−6.69807	−0.401724
$279$	17.4359	1.04386
$280$	0	0
$281$	−19.7017	−1.17531	−0.587653	−	0.809113i	$-0.699948\pi$
$281$	−19.7017	−1.17531	−0.587653	+	0.809113i	$0.699948\pi$
$282$	29.2911	1.74426
$283$	17.1408	1.01892	0.509459	−	0.860495i	$-0.329846\pi$
$283$	17.1408	1.01892	0.509459	+	0.860495i	$0.329846\pi$
$284$	−2.00371	−0.118899
$285$	0	0
$286$	−25.3803	−1.50077
$287$	4.24390	0.250510
$288$	15.2657	0.899541
$289$	−16.5068	−0.970990
$290$	0	0
$291$	−37.8439	−2.21845
$292$	−27.2753	−1.59616
$293$	3.80087	0.222049	0.111025	−	0.993818i	$-0.464587\pi$
$293$	3.80087	0.222049	0.111025	+	0.993818i	$0.464587\pi$
$294$	−34.3983	−2.00615
$295$	0	0
$296$	−1.50039	−0.0872085
$297$	7.32869	0.425254
$298$	−30.8423	−1.78665
$299$	−21.4874	−1.24265
$300$	0	0
$301$	−3.78885	−0.218386
$302$	43.7658	2.51844
$303$	−31.1639	−1.79032
$304$	−6.18433	−0.354696
$305$	0	0
$306$	−3.33083	−0.190411
$307$	14.5018	0.827664	0.413832	−	0.910353i	$-0.364190\pi$
$307$	14.5018	0.827664	0.413832	+	0.910353i	$0.364190\pi$
$308$	4.06335	0.231531
$309$	−2.84697	−0.161958
$310$	0	0
$311$	−5.65261	−0.320530	−0.160265	−	0.987074i	$-0.551235\pi$
$311$	−5.65261	−0.320530	−0.160265	+	0.987074i	$0.551235\pi$
$312$	13.4512	0.761527
$313$	−23.3042	−1.31723	−0.658616	−	0.752479i	$-0.728858\pi$
$313$	−23.3042	−1.31723	−0.658616	+	0.752479i	$0.728858\pi$
$314$	10.3793	0.585737
$315$	0	0
$316$	11.9795	0.673901
$317$	−17.3855	−0.976465	−0.488232	−	0.872714i	$-0.662358\pi$
$317$	−17.3855	−0.976465	−0.488232	+	0.872714i	$0.662358\pi$
$318$	38.6433	2.16701
$319$	8.41724	0.471275
$320$	0	0
$321$	3.73833	0.208653
$322$	5.82579	0.324658
$323$	2.99386	0.166583
$324$	−31.2405	−1.73558
$325$	0	0
$326$	26.9720	1.49384
$327$	20.6052	1.13947
$328$	−22.2644	−1.22935
$329$	2.17563	0.119947
$330$	0	0
$331$	5.19902	0.285764	0.142882	−	0.989740i	$-0.454363\pi$
$331$	5.19902	0.285764	0.142882	+	0.989740i	$0.454363\pi$
$332$	31.8233	1.74653
$333$	1.64833	0.0903279
$334$	−18.9178	−1.03514
$335$	0	0
$336$	1.22542	0.0668519
$337$	−17.0208	−0.927183	−0.463591	−	0.886049i	$-0.653439\pi$
$337$	−17.0208	−0.927183	−0.463591	+	0.886049i	$0.653439\pi$
$338$	8.36965	0.455249
$339$	−18.8337	−1.02291
$340$	0	0
$341$	30.7393	1.66463
$342$	−20.2200	−1.09337
$343$	−5.16159	−0.278700
$344$	19.8771	1.07170
$345$	0	0
$346$	−26.1046	−1.40339
$347$	3.82926	0.205565	0.102783	−	0.994704i	$-0.467225\pi$
$347$	3.82926	0.205565	0.102783	+	0.994704i	$0.467225\pi$
$348$	−14.5542	−0.780187
$349$	−12.6129	−0.675151	−0.337576	−	0.941298i	$-0.609607\pi$
$349$	−12.6129	−0.675151	−0.337576	+	0.941298i	$0.609607\pi$
$350$	0	0
$351$	5.87906	0.313801
$352$	26.9133	1.43448
$353$	−18.6132	−0.990680	−0.495340	−	0.868699i	$-0.664956\pi$
$353$	−18.6132	−0.990680	−0.495340	+	0.868699i	$0.664956\pi$
$354$	70.2936	3.73606
$355$	0	0
$356$	−50.7489	−2.68969
$357$	−0.593228	−0.0313970
$358$	22.6051	1.19471
$359$	1.47759	0.0779841	0.0389921	−	0.999240i	$-0.487585\pi$
$359$	1.47759	0.0779841	0.0389921	+	0.999240i	$0.487585\pi$
$360$	0	0
$361$	−0.825596	−0.0434524
$362$	−7.40997	−0.389459
$363$	−7.52291	−0.394850
$364$	3.25960	0.170850
$365$	0	0
$366$	52.9338	2.76689
$367$	−15.9466	−0.832408	−0.416204	−	0.909271i	$-0.636640\pi$
$367$	−15.9466	−0.832408	−0.416204	+	0.909271i	$0.636640\pi$
$368$	10.2696	0.535338
$369$	24.4597	1.27332
$370$	0	0
$371$	2.87028	0.149018
$372$	−53.1512	−2.75576
$373$	−35.9531	−1.86158	−0.930791	−	0.365551i	$-0.880881\pi$
$373$	−35.9531	−1.86158	−0.930791	+	0.365551i	$0.880881\pi$
$374$	−5.87222	−0.303645
$375$	0	0
$376$	−11.4139	−0.588624
$377$	6.75228	0.347760
$378$	−1.59396	−0.0819845
$379$	16.8578	0.865927	0.432963	−	0.901411i	$-0.357468\pi$
$379$	16.8578	0.865927	0.432963	+	0.901411i	$0.357468\pi$
$380$	0	0
$381$	−14.0511	−0.719860
$382$	44.5600	2.27989
$383$	−10.3324	−0.527963	−0.263982	−	0.964528i	$-0.585036\pi$
$383$	−10.3324	−0.527963	−0.263982	+	0.964528i	$0.585036\pi$
$384$	−31.9904	−1.63250
$385$	0	0
$386$	−41.5162	−2.11312
$387$	−21.8370	−1.11004
$388$	48.1110	2.44247
$389$	35.3884	1.79426	0.897131	−	0.441764i	$-0.145647\pi$
$389$	35.3884	1.79426	0.897131	+	0.441764i	$0.145647\pi$
$390$	0	0
$391$	−4.97154	−0.251421
$392$	13.4040	0.677003
$393$	27.2423	1.37419
$394$	49.2712	2.48225
$395$	0	0
$396$	23.4191	1.17685
$397$	−6.32475	−0.317430	−0.158715	−	0.987324i	$-0.550735\pi$
$397$	−6.32475	−0.317430	−0.158715	+	0.987324i	$0.550735\pi$
$398$	18.8374	0.944233
$399$	−3.60123	−0.180287
$400$	0	0
$401$	−29.2556	−1.46095	−0.730477	−	0.682937i	$-0.760702\pi$
$401$	−29.2556	−1.46095	−0.730477	+	0.682937i	$0.760702\pi$
$402$	63.9586	3.18996
$403$	24.6590	1.22835
$404$	39.6188	1.97111
$405$	0	0
$406$	−1.83071	−0.0908568
$407$	2.90599	0.144044
$408$	3.11221	0.154077
$409$	−6.96607	−0.344450	−0.172225	−	0.985058i	$-0.555096\pi$
$409$	−6.96607	−0.344450	−0.172225	+	0.985058i	$0.555096\pi$
$410$	0	0
$411$	−5.40412	−0.266566
$412$	3.61936	0.178313
$413$	5.22114	0.256916
$414$	33.5769	1.65021
$415$	0	0
$416$	21.5898	1.05853
$417$	−6.87552	−0.336696
$418$	−35.6476	−1.74358
$419$	−11.9149	−0.582079	−0.291039	−	0.956711i	$-0.594001\pi$
$419$	−11.9149	−0.582079	−0.291039	+	0.956711i	$0.594001\pi$
$420$	0	0
$421$	30.4399	1.48355	0.741776	−	0.670648i	$-0.233984\pi$
$421$	30.4399	1.48355	0.741776	+	0.670648i	$0.233984\pi$
$422$	31.6461	1.54051
$423$	12.5393	0.609680
$424$	−15.0581	−0.731287
$425$	0	0
$426$	−3.48316	−0.168760
$427$	3.93172	0.190269
$428$	−4.75255	−0.229723
$429$	−26.0526	−1.25783
$430$	0	0
$431$	16.3542	0.787756	0.393878	−	0.919163i	$-0.371133\pi$
$431$	16.3542	0.787756	0.393878	+	0.919163i	$0.371133\pi$
$432$	−2.80980	−0.135186
$433$	−3.31562	−0.159338	−0.0796692	−	0.996821i	$-0.525386\pi$
$433$	−3.31562	−0.159338	−0.0796692	+	0.996821i	$0.525386\pi$
$434$	−6.68567	−0.320923
$435$	0	0
$436$	−26.1954	−1.25453
$437$	−30.1800	−1.44371
$438$	−47.4140	−2.26553
$439$	−1.51016	−0.0720759	−0.0360379	−	0.999350i	$-0.511474\pi$
$439$	−1.51016	−0.0720759	−0.0360379	+	0.999350i	$0.511474\pi$
$440$	0	0
$441$	−14.7256	−0.701220
$442$	−4.71067	−0.224064
$443$	36.1571	1.71787	0.858937	−	0.512082i	$-0.171125\pi$
$443$	36.1571	1.71787	0.858937	+	0.512082i	$0.171125\pi$
$444$	−5.02473	−0.238463
$445$	0	0
$446$	15.4343	0.730834
$447$	−31.6594	−1.49744
$448$	−4.77316	−0.225511
$449$	9.67351	0.456521	0.228260	−	0.973600i	$-0.426696\pi$
$449$	9.67351	0.456521	0.228260	+	0.973600i	$0.426696\pi$
$450$	0	0
$451$	43.1222	2.03055
$452$	23.9434	1.12620
$453$	44.9252	2.11077
$454$	6.76391	0.317446
$455$	0	0
$456$	18.8928	0.884738
$457$	−28.9905	−1.35612	−0.678060	−	0.735007i	$-0.737179\pi$
$457$	−28.9905	−1.35612	−0.678060	+	0.735007i	$0.737179\pi$
$458$	4.45255	0.208054
$459$	1.36023	0.0634903
$460$	0	0
$461$	−14.7354	−0.686295	−0.343148	−	0.939281i	$-0.611493\pi$
$461$	−14.7354	−0.686295	−0.343148	+	0.939281i	$0.611493\pi$
$462$	7.06353	0.328625
$463$	−28.9590	−1.34584	−0.672921	−	0.739715i	$-0.734960\pi$
$463$	−28.9590	−1.34584	−0.672921	+	0.739715i	$0.734960\pi$
$464$	−3.22714	−0.149816
$465$	0	0
$466$	3.83358	0.177587
$467$	36.7827	1.70210	0.851051	−	0.525084i	$-0.175966\pi$
$467$	36.7827	1.70210	0.851051	+	0.525084i	$0.175966\pi$
$468$	18.7867	0.868416
$469$	4.75060	0.219362
$470$	0	0
$471$	10.6543	0.490922
$472$	−27.3913	−1.26079
$473$	−38.4984	−1.77016
$474$	20.8246	0.956507
$475$	0	0
$476$	0.754173	0.0345675
$477$	16.5428	0.757445
$478$	−0.187214	−0.00856297
$479$	5.89590	0.269390	0.134695	−	0.990887i	$-0.456994\pi$
$479$	5.89590	0.269390	0.134695	+	0.990887i	$0.456994\pi$
$480$	0	0
$481$	2.33117	0.106292
$482$	2.20997	0.100661
$483$	5.98012	0.272105
$484$	9.56389	0.434722
$485$	0	0
$486$	−41.4654	−1.88091
$487$	21.8628	0.990700	0.495350	−	0.868694i	$-0.335040\pi$
$487$	21.8628	0.990700	0.495350	+	0.868694i	$0.335040\pi$
$488$	−20.6267	−0.933726
$489$	27.6865	1.25203
$490$	0	0
$491$	18.4329	0.831863	0.415931	−	0.909396i	$-0.363456\pi$
$491$	18.4329	0.831863	0.415931	+	0.909396i	$0.363456\pi$
$492$	−74.5624	−3.36153
$493$	1.56227	0.0703612
$494$	−28.5964	−1.28661
$495$	0	0
$496$	−11.7854	−0.529178
$497$	−0.258716	−0.0116050
$498$	55.3201	2.47895
$499$	4.73483	0.211960	0.105980	−	0.994368i	$-0.466202\pi$
$499$	4.73483	0.211960	0.105980	+	0.994368i	$0.466202\pi$
$500$	0	0
$501$	−19.4190	−0.867576
$502$	−53.2173	−2.37521
$503$	29.0515	1.29534	0.647670	−	0.761921i	$-0.275743\pi$
$503$	29.0515	1.29534	0.647670	+	0.761921i	$0.275743\pi$
$504$	−1.56123	−0.0695429
$505$	0	0
$506$	59.1957	2.63157
$507$	8.59138	0.381557
$508$	17.8632	0.792552
$509$	5.09179	0.225690	0.112845	−	0.993613i	$-0.464004\pi$
$509$	5.09179	0.225690	0.112845	+	0.993613i	$0.464004\pi$
$510$	0	0
$511$	−3.52173	−0.155792
$512$	−15.9863	−0.706502
$513$	8.25738	0.364572
$514$	−13.4783	−0.594504
$515$	0	0
$516$	66.5675	2.93047
$517$	22.1066	0.972246
$518$	−0.632040	−0.0277703
$519$	−26.7961	−1.17622
$520$	0	0
$521$	18.0399	0.790342	0.395171	−	0.918608i	$-0.370685\pi$
$521$	18.0399	0.790342	0.395171	+	0.918608i	$0.370685\pi$
$522$	−10.5513	−0.461818
$523$	−19.6243	−0.858111	−0.429056	−	0.903278i	$-0.641154\pi$
$523$	−19.6243	−0.858111	−0.429056	+	0.903278i	$0.641154\pi$
$524$	−34.6332	−1.51296
$525$	0	0
$526$	−22.7067	−0.990058
$527$	5.70533	0.248528
$528$	12.4514	0.541879
$529$	27.1163	1.17897
$530$	0	0
$531$	30.0920	1.30588
$532$	4.57825	0.198492
$533$	34.5925	1.49837
$534$	−88.2194	−3.81763
$535$	0	0
$536$	−24.9227	−1.07650
$537$	23.2039	1.00132
$538$	43.2627	1.86519
$539$	−25.9611	−1.11822
$540$	0	0
$541$	20.5064	0.881639	0.440820	−	0.897596i	$-0.354688\pi$
$541$	20.5064	0.881639	0.440820	+	0.897596i	$0.354688\pi$
$542$	−32.6980	−1.40450
$543$	−7.60628	−0.326417
$544$	4.99521	0.214168
$545$	0	0
$546$	5.66634	0.242497
$547$	2.72035	0.116314	0.0581568	−	0.998307i	$-0.481478\pi$
$547$	2.72035	0.116314	0.0581568	+	0.998307i	$0.481478\pi$
$548$	6.87027	0.293484
$549$	22.6605	0.967125
$550$	0	0
$551$	9.48387	0.404026
$552$	−31.3730	−1.33533
$553$	1.54678	0.0657756
$554$	23.4209	0.995061
$555$	0	0
$556$	8.74086	0.370695
$557$	−21.3629	−0.905175	−0.452588	−	0.891720i	$-0.649499\pi$
$557$	−21.3629	−0.905175	−0.452588	+	0.891720i	$0.649499\pi$
$558$	−38.5328	−1.63123
$559$	−30.8833	−1.30623
$560$	0	0
$561$	−6.02778	−0.254493
$562$	43.5402	1.83663
$563$	39.4767	1.66374	0.831872	−	0.554967i	$-0.187269\pi$
$563$	39.4767	1.66374	0.831872	+	0.554967i	$0.187269\pi$
$564$	−38.2244	−1.60954
$565$	0	0
$566$	−37.8808	−1.59225
$567$	−4.03372	−0.169400
$568$	1.35728	0.0569502
$569$	46.0974	1.93250	0.966252	−	0.257598i	$-0.0829309\pi$
$569$	46.0974	1.93250	0.966252	+	0.257598i	$0.0829309\pi$
$570$	0	0
$571$	5.50019	0.230176	0.115088	−	0.993355i	$-0.463285\pi$
$571$	5.50019	0.230176	0.115088	+	0.993355i	$0.463285\pi$
$572$	33.1208	1.38485
$573$	45.7404	1.91083
$574$	−9.37891	−0.391468
$575$	0	0
$576$	−27.5101	−1.14625
$577$	18.5505	0.772266	0.386133	−	0.922443i	$-0.373811\pi$
$577$	18.5505	0.772266	0.386133	+	0.922443i	$0.373811\pi$
$578$	36.4796	1.51735
$579$	−42.6161	−1.77106
$580$	0	0
$581$	4.10897	0.170469
$582$	83.6339	3.46674
$583$	29.1649	1.20788
$584$	18.4758	0.764534
$585$	0	0
$586$	−8.39982	−0.346993
$587$	20.0073	0.825789	0.412895	−	0.910779i	$-0.364518\pi$
$587$	20.0073	0.825789	0.412895	+	0.910779i	$0.364518\pi$
$588$	44.8892	1.85120
$589$	34.6346	1.42709
$590$	0	0
$591$	50.5765	2.08044
$592$	−1.11415	−0.0457911
$593$	25.2242	1.03583	0.517916	−	0.855432i	$-0.326708\pi$
$593$	25.2242	1.03583	0.517916	+	0.855432i	$0.326708\pi$
$594$	−16.1962	−0.664538
$595$	0	0
$596$	40.2487	1.64865
$597$	19.3364	0.791387
$598$	47.4866	1.94187
$599$	−20.8882	−0.853469	−0.426735	−	0.904377i	$-0.640336\pi$
$599$	−20.8882	−0.853469	−0.426735	+	0.904377i	$0.640336\pi$
$600$	0	0
$601$	12.6800	0.517227	0.258614	−	0.965981i	$-0.416734\pi$
$601$	12.6800	0.517227	0.258614	+	0.965981i	$0.416734\pi$
$602$	8.37325	0.341268
$603$	27.3801	1.11500
$604$	−57.1135	−2.32392
$605$	0	0
$606$	68.8714	2.79771
$607$	7.80257	0.316697	0.158348	−	0.987383i	$-0.449383\pi$
$607$	7.80257	0.316697	0.158348	+	0.987383i	$0.449383\pi$
$608$	30.3237	1.22979
$609$	−1.87921	−0.0761496
$610$	0	0
$611$	17.7338	0.717434
$612$	4.34667	0.175704
$613$	11.9022	0.480724	0.240362	−	0.970683i	$-0.422734\pi$
$613$	11.9022	0.480724	0.240362	+	0.970683i	$0.422734\pi$
$614$	−32.0487	−1.29338
$615$	0	0
$616$	−2.75244	−0.110899
$617$	44.3848	1.78686	0.893432	−	0.449199i	$-0.148291\pi$
$617$	44.3848	1.78686	0.893432	+	0.449199i	$0.148291\pi$
$618$	6.29172	0.253090
$619$	15.8942	0.638842	0.319421	−	0.947613i	$-0.396512\pi$
$619$	15.8942	0.638842	0.319421	+	0.947613i	$0.396512\pi$
$620$	0	0
$621$	−13.7120	−0.550245
$622$	12.4921	0.500888
$623$	−6.55261	−0.262525
$624$	9.98850	0.399860
$625$	0	0
$626$	51.5017	2.05842
$627$	−36.5920	−1.46134
$628$	−13.5448	−0.540496
$629$	0.539362	0.0215058
$630$	0	0
$631$	−38.0169	−1.51343	−0.756714	−	0.653746i	$-0.773196\pi$
$631$	−38.0169	−1.51343	−0.756714	+	0.653746i	$0.773196\pi$
$632$	−8.11472	−0.322786
$633$	32.4844	1.29114
$634$	38.4214	1.52591
$635$	0	0
$636$	−50.4288	−1.99963
$637$	−20.8259	−0.825153
$638$	−18.6018	−0.736454
$639$	−1.49111	−0.0589874
$640$	0	0
$641$	−0.557890	−0.0220353	−0.0110177	−	0.999939i	$-0.503507\pi$
$641$	−0.557890	−0.0220353	−0.0110177	+	0.999939i	$0.503507\pi$
$642$	−8.26161	−0.326060
$643$	29.3830	1.15875	0.579377	−	0.815060i	$-0.303296\pi$
$643$	29.3830	1.15875	0.579377	+	0.815060i	$0.303296\pi$
$644$	−7.60255	−0.299582
$645$	0	0
$646$	−6.61634	−0.260316
$647$	−2.00527	−0.0788355	−0.0394177	−	0.999223i	$-0.512550\pi$
$647$	−2.00527	−0.0788355	−0.0394177	+	0.999223i	$0.512550\pi$
$648$	21.1618	0.831313
$649$	53.0519	2.08247
$650$	0	0
$651$	−6.86279	−0.268974
$652$	−35.1979	−1.37846
$653$	−14.6250	−0.572319	−0.286160	−	0.958182i	$-0.592379\pi$
$653$	−14.6250	−0.572319	−0.286160	+	0.958182i	$0.592379\pi$
$654$	−45.5368	−1.78063
$655$	0	0
$656$	−16.5329	−0.645502
$657$	−20.2975	−0.791881
$658$	−4.80809	−0.187439
$659$	−18.2521	−0.710999	−0.355500	−	0.934676i	$-0.615689\pi$
$659$	−18.2521	−0.710999	−0.355500	+	0.934676i	$0.615689\pi$
$660$	0	0
$661$	16.5367	0.643203	0.321602	−	0.946875i	$-0.395779\pi$
$661$	16.5367	0.643203	0.321602	+	0.946875i	$0.395779\pi$
$662$	−11.4897	−0.446559
$663$	−4.83547	−0.187794
$664$	−21.5565	−0.836556
$665$	0	0
$666$	−3.64276	−0.141154
$667$	−15.7487	−0.609792
$668$	24.6874	0.955184
$669$	15.8431	0.612531
$670$	0	0
$671$	39.9502	1.54226
$672$	−6.00860	−0.231787
$673$	−37.1550	−1.43222	−0.716110	−	0.697988i	$-0.754079\pi$
$673$	−37.1550	−1.43222	−0.716110	+	0.697988i	$0.754079\pi$
$674$	37.6155	1.44889
$675$	0	0
$676$	−10.9222	−0.420086
$677$	50.4114	1.93747	0.968733	−	0.248104i	$-0.0798074\pi$
$677$	50.4114	1.93747	0.968733	+	0.248104i	$0.0798074\pi$
$678$	41.6220	1.59848
$679$	6.21201	0.238395
$680$	0	0
$681$	6.94310	0.266060
$682$	−67.9330	−2.60129
$683$	−27.1344	−1.03827	−0.519135	−	0.854692i	$-0.673746\pi$
$683$	−27.1344	−1.03827	−0.519135	+	0.854692i	$0.673746\pi$
$684$	26.3867	1.00892
$685$	0	0
$686$	11.4070	0.435520
$687$	4.57051	0.174376
$688$	14.7602	0.562727
$689$	23.3960	0.891315
$690$	0	0
$691$	−42.9496	−1.63388	−0.816939	−	0.576724i	$-0.804331\pi$
$691$	−42.9496	−1.63388	−0.816939	+	0.576724i	$0.804331\pi$
$692$	34.0660	1.29499
$693$	3.02383	0.114866
$694$	−8.46255	−0.321234
$695$	0	0
$696$	9.85877	0.373696
$697$	8.00365	0.303160
$698$	27.8741	1.05505
$699$	3.93514	0.148841
$700$	0	0
$701$	23.2165	0.876875	0.438437	−	0.898762i	$-0.355532\pi$
$701$	23.2165	0.876875	0.438437	+	0.898762i	$0.355532\pi$
$702$	−12.9925	−0.490372
$703$	3.27423	0.123490
$704$	−48.5000	−1.82791
$705$	0	0
$706$	41.1346	1.54812
$707$	5.11551	0.192388
$708$	−91.7319	−3.44749
$709$	−35.9993	−1.35198	−0.675991	−	0.736910i	$-0.736284\pi$
$709$	−35.9993	−1.35198	−0.675991	+	0.736910i	$0.736284\pi$
$710$	0	0
$711$	8.91483	0.334332
$712$	34.3764	1.28831
$713$	−57.5134	−2.15389
$714$	1.31102	0.0490636
$715$	0	0
$716$	−29.4992	−1.10244
$717$	−0.192174	−0.00717686
$718$	−3.26543	−0.121865
$719$	30.5664	1.13993	0.569967	−	0.821668i	$-0.306956\pi$
$719$	30.5664	1.13993	0.569967	+	0.821668i	$0.306956\pi$
$720$	0	0
$721$	0.467325	0.0174041
$722$	1.82454	0.0679025
$723$	2.26852	0.0843671
$724$	9.66988	0.359378
$725$	0	0
$726$	16.6254	0.617027
$727$	−37.2636	−1.38203	−0.691015	−	0.722840i	$-0.742836\pi$
$727$	−37.2636	−1.38203	−0.691015	+	0.722840i	$0.742836\pi$
$728$	−2.20800	−0.0818339
$729$	−10.0665	−0.372833
$730$	0	0
$731$	−7.14546	−0.264284
$732$	−69.0776	−2.55318
$733$	22.9111	0.846243	0.423121	−	0.906073i	$-0.360934\pi$
$733$	22.9111	0.846243	0.423121	+	0.906073i	$0.360934\pi$
$734$	35.2416	1.30079
$735$	0	0
$736$	−50.3549	−1.85611
$737$	48.2708	1.77808
$738$	−54.0553	−1.98980
$739$	49.9652	1.83800	0.919000	−	0.394258i	$-0.128998\pi$
$739$	49.9652	1.83800	0.919000	+	0.394258i	$0.128998\pi$
$740$	0	0
$741$	−29.3540	−1.07835
$742$	−6.34324	−0.232868
$743$	−6.00868	−0.220437	−0.110218	−	0.993907i	$-0.535155\pi$
$743$	−6.00868	−0.220437	−0.110218	+	0.993907i	$0.535155\pi$
$744$	36.0037	1.31996
$745$	0	0
$746$	79.4554	2.90907
$747$	23.6820	0.866479
$748$	7.66313	0.280192
$749$	−0.613641	−0.0224220
$750$	0	0
$751$	4.02821	0.146992	0.0734958	−	0.997296i	$-0.476584\pi$
$751$	4.02821	0.146992	0.0734958	+	0.997296i	$0.476584\pi$
$752$	−8.47559	−0.309073
$753$	−54.6272	−1.99073
$754$	−14.9223	−0.543440
$755$	0	0
$756$	2.08009	0.0756521
$757$	−35.6364	−1.29523	−0.647614	−	0.761969i	$-0.724233\pi$
$757$	−35.6364	−1.29523	−0.647614	+	0.761969i	$0.724233\pi$
$758$	−37.2552	−1.35317
$759$	60.7639	2.20559
$760$	0	0
$761$	−33.5375	−1.21573	−0.607867	−	0.794039i	$-0.707975\pi$
$761$	−33.5375	−1.21573	−0.607867	+	0.794039i	$0.707975\pi$
$762$	31.0526	1.12492
$763$	−3.38230	−0.122448
$764$	−58.1499	−2.10379
$765$	0	0
$766$	22.8344	0.825040
$767$	42.5581	1.53668
$768$	12.5412	0.452540
$769$	−43.4481	−1.56678	−0.783389	−	0.621532i	$-0.786511\pi$
$769$	−43.4481	−1.56678	−0.783389	+	0.621532i	$0.786511\pi$
$770$	0	0
$771$	−13.8354	−0.498270
$772$	54.1779	1.94991
$773$	23.6950	0.852248	0.426124	−	0.904665i	$-0.359879\pi$
$773$	23.6950	0.852248	0.426124	+	0.904665i	$0.359879\pi$
$774$	48.2592	1.73464
$775$	0	0
$776$	−32.5896	−1.16990
$777$	−0.648784	−0.0232750
$778$	−78.2073	−2.80387
$779$	48.5867	1.74080
$780$	0	0
$781$	−2.62881	−0.0940662
$782$	10.9869	0.392893
$783$	4.30891	0.153988
$784$	9.95340	0.355479
$785$	0	0
$786$	−60.2046	−2.14743
$787$	18.5145	0.659969	0.329985	−	0.943986i	$-0.392956\pi$
$787$	18.5145	0.659969	0.329985	+	0.943986i	$0.392956\pi$
$788$	−64.2980	−2.29052
$789$	−23.3082	−0.829794
$790$	0	0
$791$	3.09153	0.109922
$792$	−15.8637	−0.563691
$793$	32.0479	1.13805
$794$	13.9775	0.496043
$795$	0	0
$796$	−24.5824	−0.871302
$797$	−46.9694	−1.66374	−0.831871	−	0.554970i	$-0.812730\pi$
$797$	−46.9694	−1.66374	−0.831871	+	0.554970i	$0.812730\pi$
$798$	7.95861	0.281732
$799$	4.10306	0.145156
$800$	0	0
$801$	−37.7659	−1.33439
$802$	64.6540	2.28301
$803$	−35.7843	−1.26280
$804$	−83.4648	−2.94358
$805$	0	0
$806$	−54.4956	−1.91953
$807$	44.4088	1.56326
$808$	−26.8371	−0.944125
$809$	−0.668110	−0.0234895	−0.0117448	−	0.999931i	$-0.503739\pi$
$809$	−0.668110	−0.0234895	−0.0117448	+	0.999931i	$0.503739\pi$
$810$	0	0
$811$	20.2312	0.710414	0.355207	−	0.934788i	$-0.384410\pi$
$811$	20.2312	0.710414	0.355207	+	0.934788i	$0.384410\pi$
$812$	2.38905	0.0838392
$813$	−33.5642	−1.17715
$814$	−6.42215	−0.225096
$815$	0	0
$816$	2.31103	0.0809023
$817$	−43.3770	−1.51757
$818$	15.3948	0.538267
$819$	2.42571	0.0847611
$820$	0	0
$821$	−14.4859	−0.505563	−0.252781	−	0.967523i	$-0.581345\pi$
$821$	−14.4859	−0.505563	−0.252781	+	0.967523i	$0.581345\pi$
$822$	11.9430	0.416558
$823$	16.6518	0.580445	0.290223	−	0.956959i	$-0.406271\pi$
$823$	16.6518	0.580445	0.290223	+	0.956959i	$0.406271\pi$
$824$	−2.45169	−0.0854087
$825$	0	0
$826$	−11.5386	−0.401478
$827$	−7.02125	−0.244153	−0.122076	−	0.992521i	$-0.538955\pi$
$827$	−7.02125	−0.244153	−0.122076	+	0.992521i	$0.538955\pi$
$828$	−43.8172	−1.52275
$829$	−16.9465	−0.588576	−0.294288	−	0.955717i	$-0.595083\pi$
$829$	−16.9465	−0.588576	−0.294288	+	0.955717i	$0.595083\pi$
$830$	0	0
$831$	24.0414	0.833987
$832$	−38.9066	−1.34884
$833$	−4.81848	−0.166950
$834$	15.1947	0.526149
$835$	0	0
$836$	46.5195	1.60891
$837$	15.7359	0.543913
$838$	26.3315	0.909606
$839$	−4.22287	−0.145790	−0.0728949	−	0.997340i	$-0.523224\pi$
$839$	−4.22287	−0.145790	−0.0728949	+	0.997340i	$0.523224\pi$
$840$	0	0
$841$	−24.0511	−0.829347
$842$	−67.2714	−2.31833
$843$	44.6937	1.53933
$844$	−41.2975	−1.42152
$845$	0	0
$846$	−27.7114	−0.952737
$847$	1.23487	0.0424307
$848$	−11.1817	−0.383982
$849$	−38.8843	−1.33451
$850$	0	0
$851$	−5.43712	−0.186382
$852$	4.54546	0.155725
$853$	−13.6874	−0.468648	−0.234324	−	0.972159i	$-0.575288\pi$
$853$	−13.6874	−0.468648	−0.234324	+	0.972159i	$0.575288\pi$
$854$	−8.68899	−0.297331
$855$	0	0
$856$	3.21930	0.110033
$857$	16.7973	0.573786	0.286893	−	0.957963i	$-0.407378\pi$
$857$	16.7973	0.573786	0.286893	+	0.957963i	$0.407378\pi$
$858$	57.5756	1.96560
$859$	2.95778	0.100918	0.0504591	−	0.998726i	$-0.483932\pi$
$859$	2.95778	0.100918	0.0504591	+	0.998726i	$0.483932\pi$
$860$	0	0
$861$	−9.62737	−0.328100
$862$	−36.1424	−1.23102
$863$	−28.1942	−0.959741	−0.479871	−	0.877339i	$-0.659316\pi$
$863$	−28.1942	−0.959741	−0.479871	+	0.877339i	$0.659316\pi$
$864$	13.7773	0.468714
$865$	0	0
$866$	7.32742	0.248996
$867$	37.4460	1.27173
$868$	8.72468	0.296135
$869$	15.7168	0.533154
$870$	0	0
$871$	38.7227	1.31207
$872$	17.7443	0.600898
$873$	35.8029	1.21174
$874$	66.6970	2.25606
$875$	0	0
$876$	61.8744	2.09054
$877$	21.7682	0.735059	0.367529	−	0.930012i	$-0.380204\pi$
$877$	21.7682	0.735059	0.367529	+	0.930012i	$0.380204\pi$
$878$	3.33740	0.112632
$879$	−8.62235	−0.290825
$880$	0	0
$881$	17.4794	0.588897	0.294449	−	0.955667i	$-0.404864\pi$
$881$	17.4794	0.588897	0.294449	+	0.955667i	$0.404864\pi$
$882$	32.5432	1.09579
$883$	1.08094	0.0363766	0.0181883	−	0.999835i	$-0.494210\pi$
$883$	1.08094	0.0363766	0.0181883	+	0.999835i	$0.494210\pi$
$884$	6.14734	0.206758
$885$	0	0
$886$	−79.9061	−2.68450
$887$	−27.8526	−0.935200	−0.467600	−	0.883940i	$-0.654881\pi$
$887$	−27.8526	−0.935200	−0.467600	+	0.883940i	$0.654881\pi$
$888$	3.40366	0.114219
$889$	2.30647	0.0773564
$890$	0	0
$891$	−40.9865	−1.37310
$892$	−20.1414	−0.674385
$893$	24.9079	0.833511
$894$	69.9663	2.34002
$895$	0	0
$896$	5.25117	0.175429
$897$	48.7446	1.62754
$898$	−21.3782	−0.713399
$899$	18.0732	0.602775
$900$	0	0
$901$	5.41311	0.180337
$902$	−95.2989	−3.17310
$903$	8.59508	0.286026
$904$	−16.2188	−0.539430
$905$	0	0
$906$	−99.2834	−3.29847
$907$	−26.3979	−0.876529	−0.438265	−	0.898846i	$-0.644407\pi$
$907$	−26.3979	−0.876529	−0.438265	+	0.898846i	$0.644407\pi$
$908$	−8.82678	−0.292927
$909$	29.4832	0.977896
$910$	0	0
$911$	0.713573	0.0236417	0.0118209	−	0.999930i	$-0.496237\pi$
$911$	0.713573	0.0236417	0.0118209	+	0.999930i	$0.496237\pi$
$912$	14.0293	0.464555
$913$	41.7511	1.38176
$914$	64.0682	2.11919
$915$	0	0
$916$	−5.81049	−0.191984
$917$	−4.47177	−0.147671
$918$	−3.00608	−0.0992153
$919$	43.1004	1.42175	0.710876	−	0.703317i	$-0.248299\pi$
$919$	43.1004	1.42175	0.710876	+	0.703317i	$0.248299\pi$
$920$	0	0
$921$	−32.8977	−1.08402
$922$	32.5648	1.07246
$923$	−2.10882	−0.0694127
$924$	−9.21777	−0.303242
$925$	0	0
$926$	63.9987	2.10313
$927$	2.69343	0.0884638
$928$	15.8237	0.519438
$929$	6.29923	0.206671	0.103336	−	0.994647i	$-0.467048\pi$
$929$	6.29923	0.206671	0.103336	+	0.994647i	$0.467048\pi$
$930$	0	0
$931$	−29.2509	−0.958659
$932$	−5.00276	−0.163871
$933$	12.8231	0.419808
$934$	−81.2887	−2.65985
$935$	0	0
$936$	−12.7258	−0.415956
$937$	−17.1954	−0.561751	−0.280875	−	0.959744i	$-0.590625\pi$
$937$	−17.1954	−0.561751	−0.280875	+	0.959744i	$0.590625\pi$
$938$	−10.4987	−0.342794
$939$	52.8660	1.72522
$940$	0	0
$941$	−15.5318	−0.506322	−0.253161	−	0.967424i	$-0.581470\pi$
$941$	−15.5318	−0.506322	−0.253161	+	0.967424i	$0.581470\pi$
$942$	−23.5456	−0.767157
$943$	−80.6819	−2.62737
$944$	−20.3399	−0.662009
$945$	0	0
$946$	85.0804	2.76620
$947$	−43.2607	−1.40578	−0.702892	−	0.711297i	$-0.748108\pi$
$947$	−43.2607	−1.40578	−0.702892	+	0.711297i	$0.748108\pi$
$948$	−27.1758	−0.882628
$949$	−28.7060	−0.931838
$950$	0	0
$951$	39.4392	1.27890
$952$	−0.510863	−0.0165572
$953$	−10.3314	−0.334666	−0.167333	−	0.985900i	$-0.553516\pi$
$953$	−10.3314	−0.334666	−0.167333	+	0.985900i	$0.553516\pi$
$954$	−36.5592	−1.18365
$955$	0	0
$956$	0.244311	0.00790158
$957$	−19.0947	−0.617242
$958$	−13.0298	−0.420973
$959$	0.887078	0.0286452
$960$	0	0
$961$	35.0024	1.12911
$962$	−5.15183	−0.166102
$963$	−3.53672	−0.113969
$964$	−2.88397	−0.0928865
$965$	0	0
$966$	−13.2159	−0.425215
$967$	−15.9401	−0.512598	−0.256299	−	0.966598i	$-0.582503\pi$
$967$	−15.9401	−0.512598	−0.256299	+	0.966598i	$0.582503\pi$
$968$	−6.47841	−0.208224
$969$	−6.79162	−0.218178
$970$	0	0
$971$	21.6770	0.695647	0.347824	−	0.937560i	$-0.386921\pi$
$971$	21.6770	0.695647	0.347824	+	0.937560i	$0.386921\pi$
$972$	54.1116	1.73563
$973$	1.12861	0.0361814
$974$	−48.3162	−1.54815
$975$	0	0
$976$	−15.3168	−0.490278
$977$	48.6376	1.55605	0.778027	−	0.628231i	$-0.216221\pi$
$977$	48.6376	1.55605	0.778027	+	0.628231i	$0.216221\pi$
$978$	−61.1864	−1.95652
$979$	−66.5809	−2.12794
$980$	0	0
$981$	−19.4939	−0.622392
$982$	−40.7361	−1.29994
$983$	53.0177	1.69100	0.845500	−	0.533975i	$-0.179302\pi$
$983$	53.0177	1.69100	0.845500	+	0.533975i	$0.179302\pi$
$984$	50.5073	1.61011
$985$	0	0
$986$	−3.45257	−0.109952
$987$	−4.93546	−0.157098
$988$	37.3178	1.18724
$989$	72.0308	2.29045
$990$	0	0
$991$	34.2607	1.08833	0.544164	−	0.838979i	$-0.316847\pi$
$991$	34.2607	1.08833	0.544164	+	0.838979i	$0.316847\pi$
$992$	57.7873	1.83475
$993$	−11.7941	−0.374273
$994$	0.571755	0.0181350
$995$	0	0
$996$	−72.1917	−2.28748
$997$	49.2198	1.55881	0.779404	−	0.626522i	$-0.215522\pi$
$997$	49.2198	1.55881	0.779404	+	0.626522i	$0.215522\pi$
$998$	−10.4638	−0.331227
$999$	1.48762	0.0470662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.11		66
5.2	odd	4		1205.2.b.d.724.11	✓	66
5.3	odd	4		1205.2.b.d.724.56	yes	66
5.4	even	2	inner	6025.2.a.q.1.56		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.11	✓	66	5.2	odd	4
1205.2.b.d.724.56	yes	66	5.3	odd	4
6025.2.a.q.1.11		66	1.1	even	1	trivial
6025.2.a.q.1.56		66	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.20997 q^{2} -2.26852 q^{3} +2.88397 q^{4} +5.01336 q^{6} +0.372373 q^{7} -1.95355 q^{8} +2.14617 q^{9} +O(q^{10})\)	\(q-2.20997 q^{2} -2.26852 q^{3} +2.88397 q^{4} +5.01336 q^{6} +0.372373 q^{7} -1.95355 q^{8} +2.14617 q^{9} +3.78368 q^{11} -6.54234 q^{12} +3.03526 q^{13} -0.822934 q^{14} -1.45065 q^{16} +0.702265 q^{17} -4.74298 q^{18} +4.26314 q^{19} -0.844736 q^{21} -8.36182 q^{22} -7.07928 q^{23} +4.43167 q^{24} -6.70783 q^{26} +1.93692 q^{27} +1.07391 q^{28} +2.22462 q^{29} +8.12419 q^{31} +7.11300 q^{32} -8.58334 q^{33} -1.55199 q^{34} +6.18950 q^{36} +0.768032 q^{37} -9.42143 q^{38} -6.88553 q^{39} +11.3969 q^{41} +1.86684 q^{42} -10.1749 q^{43} +10.9120 q^{44} +15.6450 q^{46} +5.84261 q^{47} +3.29083 q^{48} -6.86134 q^{49} -1.59310 q^{51} +8.75359 q^{52} +7.70807 q^{53} -4.28054 q^{54} -0.727451 q^{56} -9.67102 q^{57} -4.91634 q^{58} +14.0213 q^{59} +10.5586 q^{61} -17.9542 q^{62} +0.799178 q^{63} -12.8182 q^{64} +18.9689 q^{66} +12.7576 q^{67} +2.02531 q^{68} +16.0595 q^{69} -0.694776 q^{71} -4.19266 q^{72} -9.45754 q^{73} -1.69733 q^{74} +12.2948 q^{76} +1.40894 q^{77} +15.2168 q^{78} +4.15383 q^{79} -10.8325 q^{81} -25.1868 q^{82} +11.0345 q^{83} -2.43619 q^{84} +22.4862 q^{86} -5.04658 q^{87} -7.39161 q^{88} -17.5969 q^{89} +1.13025 q^{91} -20.4165 q^{92} -18.4299 q^{93} -12.9120 q^{94} -16.1360 q^{96} +16.6822 q^{97} +15.1634 q^{98} +8.12043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more