LMFDB - Embedded newform 6025.2.a.p.1.32

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			1.32
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)

$f(q)$	$=$	$q+1.09514 q^{2} +0.223927 q^{3} -0.800662 q^{4} +0.245232 q^{6} +2.26365 q^{7} -3.06713 q^{8} -2.94986 q^{9} +O(q^{10})$	\(q+1.09514 q^{2} +0.223927 q^{3} -0.800662 q^{4} +0.245232 q^{6} +2.26365 q^{7} -3.06713 q^{8} -2.94986 q^{9} -3.16701 q^{11} -0.179290 q^{12} -0.0649874 q^{13} +2.47902 q^{14} -1.75761 q^{16} +4.44087 q^{17} -3.23051 q^{18} +7.82407 q^{19} +0.506893 q^{21} -3.46833 q^{22} -2.53175 q^{23} -0.686812 q^{24} -0.0711705 q^{26} -1.33233 q^{27} -1.81242 q^{28} -6.53411 q^{29} +8.27459 q^{31} +4.20941 q^{32} -0.709178 q^{33} +4.86338 q^{34} +2.36184 q^{36} -2.73984 q^{37} +8.56847 q^{38} -0.0145524 q^{39} -7.52088 q^{41} +0.555120 q^{42} -10.7428 q^{43} +2.53570 q^{44} -2.77262 q^{46} +8.53037 q^{47} -0.393577 q^{48} -1.87588 q^{49} +0.994430 q^{51} +0.0520330 q^{52} +0.0654906 q^{53} -1.45909 q^{54} -6.94290 q^{56} +1.75202 q^{57} -7.15578 q^{58} +11.9066 q^{59} +7.88792 q^{61} +9.06186 q^{62} -6.67745 q^{63} +8.12514 q^{64} -0.776651 q^{66} -13.3617 q^{67} -3.55564 q^{68} -0.566926 q^{69} -8.56086 q^{71} +9.04758 q^{72} -16.6336 q^{73} -3.00052 q^{74} -6.26444 q^{76} -7.16900 q^{77} -0.0159370 q^{78} -0.725508 q^{79} +8.55123 q^{81} -8.23644 q^{82} -6.57206 q^{83} -0.405850 q^{84} -11.7649 q^{86} -1.46316 q^{87} +9.71361 q^{88} -10.8164 q^{89} -0.147109 q^{91} +2.02707 q^{92} +1.85290 q^{93} +9.34197 q^{94} +0.942600 q^{96} -9.64642 q^{97} -2.05436 q^{98} +9.34222 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.09514	0.774383	0.387191	−	0.921999i	$-0.373445\pi$
$2$	1.09514	0.774383	0.387191	+	0.921999i	$0.373445\pi$
$3$	0.223927	0.129284	0.0646421	−	0.997909i	$-0.479409\pi$
$3$	0.223927	0.129284	0.0646421	+	0.997909i	$0.479409\pi$
$4$	−0.800662	−0.400331
$5$	0	0
$5$	0	0
$6$	0.245232	0.100116
$7$	2.26365	0.855580	0.427790	−	0.903878i	$-0.359292\pi$
$7$	2.26365	0.855580	0.427790	+	0.903878i	$0.359292\pi$
$8$	−3.06713	−1.08439
$9$	−2.94986	−0.983286
$10$	0	0
$11$	−3.16701	−0.954889	−0.477444	−	0.878662i	$-0.658437\pi$
$11$	−3.16701	−0.954889	−0.477444	+	0.878662i	$0.658437\pi$
$12$	−0.179290	−0.0517565
$13$	−0.0649874	−0.0180243	−0.00901213	−	0.999959i	$-0.502869\pi$
$13$	−0.0649874	−0.0180243	−0.00901213	+	0.999959i	$0.502869\pi$
$14$	2.47902	0.662546
$15$	0	0
$16$	−1.75761	−0.439404
$17$	4.44087	1.07707	0.538534	−	0.842604i	$-0.318978\pi$
$17$	4.44087	1.07707	0.538534	+	0.842604i	$0.318978\pi$
$18$	−3.23051	−0.761439
$19$	7.82407	1.79496	0.897482	−	0.441050i	$-0.145394\pi$
$19$	7.82407	1.79496	0.897482	+	0.441050i	$0.145394\pi$
$20$	0	0
$21$	0.506893	0.110613
$22$	−3.46833	−0.739450
$23$	−2.53175	−0.527906	−0.263953	−	0.964536i	$-0.585026\pi$
$23$	−2.53175	−0.527906	−0.263953	+	0.964536i	$0.585026\pi$
$24$	−0.686812	−0.140195
$25$	0	0
$26$	−0.0711705	−0.0139577
$27$	−1.33233	−0.256408
$28$	−1.81242	−0.342515
$29$	−6.53411	−1.21335	−0.606677	−	0.794949i	$-0.707498\pi$
$29$	−6.53411	−1.21335	−0.606677	+	0.794949i	$0.707498\pi$
$30$	0	0
$31$	8.27459	1.48616	0.743080	−	0.669202i	$-0.233364\pi$
$31$	8.27459	1.48616	0.743080	+	0.669202i	$0.233364\pi$
$32$	4.20941	0.744126
$33$	−0.709178	−0.123452
$34$	4.86338	0.834063
$35$	0	0
$36$	2.36184	0.393640
$37$	−2.73984	−0.450428	−0.225214	−	0.974309i	$-0.572308\pi$
$37$	−2.73984	−0.450428	−0.225214	+	0.974309i	$0.572308\pi$
$38$	8.56847	1.38999
$39$	−0.0145524	−0.00233025
$40$	0	0
$41$	−7.52088	−1.17456	−0.587282	−	0.809382i	$-0.699802\pi$
$41$	−7.52088	−1.17456	−0.587282	+	0.809382i	$0.699802\pi$
$42$	0.555120	0.0856568
$43$	−10.7428	−1.63827	−0.819134	−	0.573603i	$-0.805545\pi$
$43$	−10.7428	−1.63827	−0.819134	+	0.573603i	$0.805545\pi$
$44$	2.53570	0.382272
$45$	0	0
$46$	−2.77262	−0.408801
$47$	8.53037	1.24428	0.622141	−	0.782905i	$-0.286263\pi$
$47$	8.53037	1.24428	0.622141	+	0.782905i	$0.286263\pi$
$48$	−0.393577	−0.0568080
$49$	−1.87588	−0.267983
$50$	0	0
$51$	0.994430	0.139248
$52$	0.0520330	0.00721567
$53$	0.0654906	0.00899583	0.00449791	−	0.999990i	$-0.498568\pi$
$53$	0.0654906	0.00899583	0.00449791	+	0.999990i	$0.498568\pi$
$54$	−1.45909	−0.198558
$55$	0	0
$56$	−6.94290	−0.927784
$57$	1.75202	0.232061
$58$	−7.15578	−0.939600
$59$	11.9066	1.55011	0.775055	−	0.631894i	$-0.217722\pi$
$59$	11.9066	1.55011	0.775055	+	0.631894i	$0.217722\pi$
$60$	0	0
$61$	7.88792	1.00995	0.504973	−	0.863135i	$-0.331503\pi$
$61$	7.88792	1.00995	0.504973	+	0.863135i	$0.331503\pi$
$62$	9.06186	1.15086
$63$	−6.67745	−0.841279
$64$	8.12514	1.01564
$65$	0	0
$66$	−0.776651	−0.0955992
$67$	−13.3617	−1.63240	−0.816199	−	0.577771i	$-0.803923\pi$
$67$	−13.3617	−1.63240	−0.816199	+	0.577771i	$0.803923\pi$
$68$	−3.55564	−0.431184
$69$	−0.566926	−0.0682499
$70$	0	0
$71$	−8.56086	−1.01599	−0.507994	−	0.861361i	$-0.669613\pi$
$71$	−8.56086	−1.01599	−0.507994	+	0.861361i	$0.669613\pi$
$72$	9.04758	1.06627
$73$	−16.6336	−1.94682	−0.973410	−	0.229068i	$-0.926432\pi$
$73$	−16.6336	−1.94682	−0.973410	+	0.229068i	$0.926432\pi$
$74$	−3.00052	−0.348803
$75$	0	0
$76$	−6.26444	−0.718580
$77$	−7.16900	−0.816984
$78$	−0.0159370	−0.00180451
$79$	−0.725508	−0.0816260	−0.0408130	−	0.999167i	$-0.512995\pi$
$79$	−0.725508	−0.0816260	−0.0408130	+	0.999167i	$0.512995\pi$
$80$	0	0
$81$	8.55123	0.950136
$82$	−8.23644	−0.909562
$83$	−6.57206	−0.721378	−0.360689	−	0.932686i	$-0.617458\pi$
$83$	−6.57206	−0.721378	−0.360689	+	0.932686i	$0.617458\pi$
$84$	−0.405850	−0.0442818
$85$	0	0
$86$	−11.7649	−1.26865
$87$	−1.46316	−0.156868
$88$	9.71361	1.03547
$89$	−10.8164	−1.14654	−0.573269	−	0.819367i	$-0.694325\pi$
$89$	−10.8164	−1.14654	−0.573269	+	0.819367i	$0.694325\pi$
$90$	0	0
$91$	−0.147109	−0.0154212
$92$	2.02707	0.211337
$93$	1.85290	0.192137
$94$	9.34197	0.963551
$95$	0	0
$96$	0.942600	0.0962038
$97$	−9.64642	−0.979445	−0.489723	−	0.871878i	$-0.662902\pi$
$97$	−9.64642	−0.979445	−0.489723	+	0.871878i	$0.662902\pi$
$98$	−2.05436	−0.207521
$99$	9.34222	0.938928
$100$	0	0
$101$	−16.4798	−1.63980	−0.819902	−	0.572504i	$-0.805972\pi$
$101$	−16.4798	−1.63980	−0.819902	+	0.572504i	$0.805972\pi$
$102$	1.08904	0.107831
$103$	10.7964	1.06380	0.531898	−	0.846808i	$-0.321479\pi$
$103$	10.7964	1.06380	0.531898	+	0.846808i	$0.321479\pi$
$104$	0.199324	0.0195454
$105$	0	0
$106$	0.0717216	0.00696622
$107$	1.67365	0.161798	0.0808988	−	0.996722i	$-0.474221\pi$
$107$	1.67365	0.161798	0.0808988	+	0.996722i	$0.474221\pi$
$108$	1.06675	0.102648
$109$	−16.9594	−1.62442	−0.812208	−	0.583368i	$-0.801735\pi$
$109$	−16.9594	−1.62442	−0.812208	+	0.583368i	$0.801735\pi$
$110$	0	0
$111$	−0.613525	−0.0582332
$112$	−3.97863	−0.375945
$113$	−8.64948	−0.813675	−0.406837	−	0.913501i	$-0.633368\pi$
$113$	−8.64948	−0.813675	−0.406837	+	0.913501i	$0.633368\pi$
$114$	1.91871	0.179704
$115$	0	0
$116$	5.23162	0.485743
$117$	0.191703	0.0177230
$118$	13.0394	1.20038
$119$	10.0526	0.921518
$120$	0	0
$121$	−0.970060	−0.0881872
$122$	8.63840	0.782084
$123$	−1.68413	−0.151853
$124$	−6.62515	−0.594956
$125$	0	0
$126$	−7.31276	−0.651472
$127$	6.06109	0.537835	0.268918	−	0.963163i	$-0.413334\pi$
$127$	6.06109	0.537835	0.268918	+	0.963163i	$0.413334\pi$
$128$	0.479360	0.0423699
$129$	−2.40561	−0.211802
$130$	0	0
$131$	−6.54103	−0.571493	−0.285746	−	0.958305i	$-0.592241\pi$
$131$	−6.54103	−0.571493	−0.285746	+	0.958305i	$0.592241\pi$
$132$	0.567812	0.0494217
$133$	17.7110	1.53574
$134$	−14.6330	−1.26410
$135$	0	0
$136$	−13.6207	−1.16797
$137$	−8.14464	−0.695844	−0.347922	−	0.937524i	$-0.613113\pi$
$137$	−8.14464	−0.695844	−0.347922	+	0.937524i	$0.613113\pi$
$138$	−0.620865	−0.0528515
$139$	−6.38850	−0.541865	−0.270933	−	0.962598i	$-0.587332\pi$
$139$	−6.38850	−0.541865	−0.270933	+	0.962598i	$0.587332\pi$
$140$	0	0
$141$	1.91018	0.160866
$142$	−9.37537	−0.786763
$143$	0.205816	0.0172112
$144$	5.18471	0.432059
$145$	0	0
$146$	−18.2162	−1.50758
$147$	−0.420060	−0.0346460
$148$	2.19369	0.180320
$149$	9.29157	0.761195	0.380598	−	0.924741i	$-0.375718\pi$
$149$	9.29157	0.761195	0.380598	+	0.924741i	$0.375718\pi$
$150$	0	0
$151$	−4.44437	−0.361677	−0.180839	−	0.983513i	$-0.557881\pi$
$151$	−4.44437	−0.361677	−0.180839	+	0.983513i	$0.557881\pi$
$152$	−23.9974	−1.94645
$153$	−13.0999	−1.05907
$154$	−7.85108	−0.632658
$155$	0	0
$156$	0.0116516	0.000932873 0
$157$	−0.260484	−0.0207889	−0.0103944	−	0.999946i	$-0.503309\pi$
$157$	−0.260484	−0.0207889	−0.0103944	+	0.999946i	$0.503309\pi$
$158$	−0.794534	−0.0632098
$159$	0.0146651	0.00116302
$160$	0	0
$161$	−5.73099	−0.451666
$162$	9.36481	0.735769
$163$	5.42016	0.424539	0.212270	−	0.977211i	$-0.431914\pi$
$163$	5.42016	0.424539	0.212270	+	0.977211i	$0.431914\pi$
$164$	6.02169	0.470215
$165$	0	0
$166$	−7.19735	−0.558623
$167$	−22.0325	−1.70492	−0.852462	−	0.522789i	$-0.824891\pi$
$167$	−22.0325	−1.70492	−0.852462	+	0.522789i	$0.824891\pi$
$168$	−1.55470	−0.119948
$169$	−12.9958	−0.999675
$170$	0	0
$171$	−23.0799	−1.76496
$172$	8.60139	0.655849
$173$	18.0964	1.37584	0.687922	−	0.725785i	$-0.258523\pi$
$173$	18.0964	1.37584	0.687922	+	0.725785i	$0.258523\pi$
$174$	−1.60237	−0.121476
$175$	0	0
$176$	5.56638	0.419582
$177$	2.66621	0.200405
$178$	−11.8455	−0.887859
$179$	−17.4131	−1.30151	−0.650757	−	0.759286i	$-0.725548\pi$
$179$	−17.4131	−1.30151	−0.650757	+	0.759286i	$0.725548\pi$
$180$	0	0
$181$	6.69828	0.497879	0.248940	−	0.968519i	$-0.419918\pi$
$181$	6.69828	0.497879	0.248940	+	0.968519i	$0.419918\pi$
$182$	−0.161105	−0.0119419
$183$	1.76632	0.130570
$184$	7.76518	0.572457
$185$	0	0
$186$	2.02919	0.148788
$187$	−14.0643	−1.02848
$188$	−6.82995	−0.498125
$189$	−3.01594	−0.219377
$190$	0	0
$191$	18.2205	1.31839	0.659196	−	0.751971i	$-0.270897\pi$
$191$	18.2205	1.31839	0.659196	+	0.751971i	$0.270897\pi$
$192$	1.81944	0.131307
$193$	−5.88201	−0.423396	−0.211698	−	0.977335i	$-0.567899\pi$
$193$	−5.88201	−0.423396	−0.211698	+	0.977335i	$0.567899\pi$
$194$	−10.5642	−0.758466
$195$	0	0
$196$	1.50195	0.107282
$197$	18.3999	1.31094	0.655468	−	0.755223i	$-0.272472\pi$
$197$	18.3999	1.31094	0.655468	+	0.755223i	$0.272472\pi$
$198$	10.2311	0.727090
$199$	11.8695	0.841404	0.420702	−	0.907199i	$-0.361784\pi$
$199$	11.8695	0.841404	0.420702	+	0.907199i	$0.361784\pi$
$200$	0	0
$201$	−2.99206	−0.211043
$202$	−18.0478	−1.26984
$203$	−14.7909	−1.03812
$204$	−0.796203	−0.0557453
$205$	0	0
$206$	11.8235	0.823786
$207$	7.46829	0.519082
$208$	0.114223	0.00791992
$209$	−24.7789	−1.71399
$210$	0	0
$211$	17.3377	1.19358	0.596788	−	0.802399i	$-0.296443\pi$
$211$	17.3377	1.19358	0.596788	+	0.802399i	$0.296443\pi$
$212$	−0.0524359	−0.00360131
$213$	−1.91701	−0.131351
$214$	1.83288	0.125293
$215$	0	0
$216$	4.08643	0.278046
$217$	18.7308	1.27153
$218$	−18.5730	−1.25792
$219$	−3.72472	−0.251693
$220$	0	0
$221$	−0.288600	−0.0194134
$222$	−0.671897	−0.0450948
$223$	−0.685671	−0.0459159	−0.0229580	−	0.999736i	$-0.507308\pi$
$223$	−0.685671	−0.0459159	−0.0229580	+	0.999736i	$0.507308\pi$
$224$	9.52864	0.636659
$225$	0	0
$226$	−9.47242	−0.630096
$227$	−23.7034	−1.57325	−0.786626	−	0.617429i	$-0.788174\pi$
$227$	−23.7034	−1.57325	−0.786626	+	0.617429i	$0.788174\pi$
$228$	−1.40278	−0.0929011
$229$	−18.1670	−1.20051	−0.600254	−	0.799810i	$-0.704934\pi$
$229$	−18.1670	−1.20051	−0.600254	+	0.799810i	$0.704934\pi$
$230$	0	0
$231$	−1.60533	−0.105623
$232$	20.0409	1.31575
$233$	−1.69882	−0.111293	−0.0556466	−	0.998451i	$-0.517722\pi$
$233$	−1.69882	−0.111293	−0.0556466	+	0.998451i	$0.517722\pi$
$234$	0.209943	0.0137244
$235$	0	0
$236$	−9.53318	−0.620557
$237$	−0.162461	−0.0105530
$238$	11.0090	0.713608
$239$	4.67855	0.302631	0.151315	−	0.988486i	$-0.451649\pi$
$239$	4.67855	0.302631	0.151315	+	0.988486i	$0.451649\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−1.06235	−0.0682907
$243$	5.91185	0.379245
$244$	−6.31556	−0.404313
$245$	0	0
$246$	−1.84436	−0.117592
$247$	−0.508466	−0.0323529
$248$	−25.3792	−1.61158
$249$	−1.47166	−0.0932628
$250$	0	0
$251$	−6.44769	−0.406974	−0.203487	−	0.979078i	$-0.565228\pi$
$251$	−6.44769	−0.406974	−0.203487	+	0.979078i	$0.565228\pi$
$252$	5.34638	0.336790
$253$	8.01806	0.504091
$254$	6.63776	0.416490
$255$	0	0
$256$	−15.7253	−0.982831
$257$	−15.0125	−0.936452	−0.468226	−	0.883609i	$-0.655107\pi$
$257$	−15.0125	−0.936452	−0.468226	+	0.883609i	$0.655107\pi$
$258$	−2.63449	−0.164016
$259$	−6.20205	−0.385377
$260$	0	0
$261$	19.2747	1.19307
$262$	−7.16336	−0.442554
$263$	11.1566	0.687945	0.343973	−	0.938980i	$-0.388227\pi$
$263$	11.1566	0.687945	0.343973	+	0.938980i	$0.388227\pi$
$264$	2.17514	0.133871
$265$	0	0
$266$	19.3960	1.18925
$267$	−2.42209	−0.148229
$268$	10.6983	0.653500
$269$	−13.5075	−0.823569	−0.411785	−	0.911281i	$-0.635094\pi$
$269$	−13.5075	−0.823569	−0.411785	+	0.911281i	$0.635094\pi$
$270$	0	0
$271$	13.0987	0.795687	0.397844	−	0.917453i	$-0.369759\pi$
$271$	13.0987	0.795687	0.397844	+	0.917453i	$0.369759\pi$
$272$	−7.80533	−0.473268
$273$	−0.0329416	−0.00199372
$274$	−8.91954	−0.538849
$275$	0	0
$276$	0.453917	0.0273226
$277$	14.4487	0.868135	0.434068	−	0.900880i	$-0.357078\pi$
$277$	14.4487	0.868135	0.434068	+	0.900880i	$0.357078\pi$
$278$	−6.99631	−0.419611
$279$	−24.4089	−1.46132
$280$	0	0
$281$	1.77667	0.105988	0.0529938	−	0.998595i	$-0.483124\pi$
$281$	1.77667	0.105988	0.0529938	+	0.998595i	$0.483124\pi$
$282$	2.09192	0.124572
$283$	18.4509	1.09679	0.548397	−	0.836218i	$-0.315238\pi$
$283$	18.4509	1.09679	0.548397	+	0.836218i	$0.315238\pi$
$284$	6.85436	0.406732
$285$	0	0
$286$	0.225397	0.0133280
$287$	−17.0247	−1.00493
$288$	−12.4172	−0.731688
$289$	2.72130	0.160077
$290$	0	0
$291$	−2.16009	−0.126627
$292$	13.3179	0.779373
$293$	−8.65394	−0.505569	−0.252784	−	0.967523i	$-0.581346\pi$
$293$	−8.65394	−0.505569	−0.252784	+	0.967523i	$0.581346\pi$
$294$	−0.460026	−0.0268293
$295$	0	0
$296$	8.40345	0.488440
$297$	4.21951	0.244841
$298$	10.1756	0.589456
$299$	0.164532	0.00951511
$300$	0	0
$301$	−24.3180	−1.40167
$302$	−4.86722	−0.280077
$303$	−3.69028	−0.212001
$304$	−13.7517	−0.788714
$305$	0	0
$306$	−14.3463	−0.820123
$307$	−6.61143	−0.377334	−0.188667	−	0.982041i	$-0.560417\pi$
$307$	−6.61143	−0.377334	−0.188667	+	0.982041i	$0.560417\pi$
$308$	5.73995	0.327064
$309$	2.41759	0.137532
$310$	0	0
$311$	−19.4206	−1.10124	−0.550622	−	0.834755i	$-0.685609\pi$
$311$	−19.4206	−1.10124	−0.550622	+	0.834755i	$0.685609\pi$
$312$	0.0446341	0.00252691
$313$	19.4518	1.09948	0.549740	−	0.835336i	$-0.314727\pi$
$313$	19.4518	1.09948	0.549740	+	0.835336i	$0.314727\pi$
$314$	−0.285267	−0.0160986
$315$	0	0
$316$	0.580887	0.0326774
$317$	0.792618	0.0445179	0.0222589	−	0.999752i	$-0.492914\pi$
$317$	0.792618	0.0445179	0.0222589	+	0.999752i	$0.492914\pi$
$318$	0.0160604	0.000900622 0
$319$	20.6936	1.15862
$320$	0	0
$321$	0.374775	0.0209179
$322$	−6.27626	−0.349762
$323$	34.7457	1.93330
$324$	−6.84664	−0.380369
$325$	0	0
$326$	5.93584	0.328756
$327$	−3.79767	−0.210011
$328$	23.0675	1.27369
$329$	19.3098	1.06458
$330$	0	0
$331$	14.6988	0.807917	0.403958	−	0.914777i	$-0.367634\pi$
$331$	14.6988	0.807917	0.403958	+	0.914777i	$0.367634\pi$
$332$	5.26201	0.288790
$333$	8.08215	0.442899
$334$	−24.1287	−1.32026
$335$	0	0
$336$	−0.890922	−0.0486038
$337$	−17.3385	−0.944487	−0.472244	−	0.881468i	$-0.656556\pi$
$337$	−17.3385	−0.944487	−0.472244	+	0.881468i	$0.656556\pi$
$338$	−14.2322	−0.774131
$339$	−1.93685	−0.105195
$340$	0	0
$341$	−26.2057	−1.41912
$342$	−25.2758	−1.36676
$343$	−20.0919	−1.08486
$344$	32.9496	1.77652
$345$	0	0
$346$	19.8181	1.06543
$347$	13.9745	0.750188	0.375094	−	0.926987i	$-0.377610\pi$
$347$	13.9745	0.750188	0.375094	+	0.926987i	$0.377610\pi$
$348$	1.17150	0.0627990
$349$	18.2257	0.975600	0.487800	−	0.872955i	$-0.337800\pi$
$349$	18.2257	0.975600	0.487800	+	0.872955i	$0.337800\pi$
$350$	0	0
$351$	0.0865848	0.00462156
$352$	−13.3312	−0.710557
$353$	−25.1550	−1.33886	−0.669432	−	0.742873i	$-0.733463\pi$
$353$	−25.1550	−1.33886	−0.669432	+	0.742873i	$0.733463\pi$
$354$	2.91988	0.155190
$355$	0	0
$356$	8.66030	0.458995
$357$	2.25104	0.119138
$358$	−19.0698	−1.00787
$359$	−6.05193	−0.319409	−0.159704	−	0.987165i	$-0.551054\pi$
$359$	−6.05193	−0.319409	−0.159704	+	0.987165i	$0.551054\pi$
$360$	0	0
$361$	42.2161	2.22190
$362$	7.33558	0.385549
$363$	−0.217222	−0.0114012
$364$	0.117784	0.00617359
$365$	0	0
$366$	1.93437	0.101111
$367$	31.3055	1.63413	0.817067	−	0.576543i	$-0.195599\pi$
$367$	31.3055	1.63413	0.817067	+	0.576543i	$0.195599\pi$
$368$	4.44984	0.231964
$369$	22.1855	1.15493
$370$	0	0
$371$	0.148248	0.00769665
$372$	−1.48355	−0.0769185
$373$	17.5060	0.906424	0.453212	−	0.891403i	$-0.350278\pi$
$373$	17.5060	0.906424	0.453212	+	0.891403i	$0.350278\pi$
$374$	−15.4024	−0.796438
$375$	0	0
$376$	−26.1637	−1.34929
$377$	0.424635	0.0218698
$378$	−3.30288	−0.169882
$379$	0.569958	0.0292768	0.0146384	−	0.999893i	$-0.495340\pi$
$379$	0.569958	0.0292768	0.0146384	+	0.999893i	$0.495340\pi$
$380$	0	0
$381$	1.35724	0.0695336
$382$	19.9541	1.02094
$383$	24.0263	1.22769	0.613843	−	0.789428i	$-0.289623\pi$
$383$	24.0263	1.22769	0.613843	+	0.789428i	$0.289623\pi$
$384$	0.107342	0.00547776
$385$	0	0
$386$	−6.44164	−0.327871
$387$	31.6898	1.61088
$388$	7.72353	0.392103
$389$	−13.5820	−0.688633	−0.344317	−	0.938854i	$-0.611889\pi$
$389$	−13.5820	−0.688633	−0.344317	+	0.938854i	$0.611889\pi$
$390$	0	0
$391$	−11.2432	−0.568591
$392$	5.75356	0.290599
$393$	−1.46471	−0.0738850
$394$	20.1505	1.01517
$395$	0	0
$396$	−7.47997	−0.375882
$397$	9.52930	0.478262	0.239131	−	0.970987i	$-0.423137\pi$
$397$	9.52930	0.478262	0.239131	+	0.970987i	$0.423137\pi$
$398$	12.9988	0.651569
$399$	3.96596	0.198546
$400$	0	0
$401$	9.98608	0.498681	0.249341	−	0.968416i	$-0.419786\pi$
$401$	9.98608	0.498681	0.249341	+	0.968416i	$0.419786\pi$
$402$	−3.27673	−0.163428
$403$	−0.537744	−0.0267869
$404$	13.1948	0.656465
$405$	0	0
$406$	−16.1982	−0.803903
$407$	8.67711	0.430108
$408$	−3.05004	−0.150999
$409$	−1.85693	−0.0918193	−0.0459097	−	0.998946i	$-0.514619\pi$
$409$	−1.85693	−0.0918193	−0.0459097	+	0.998946i	$0.514619\pi$
$410$	0	0
$411$	−1.82380	−0.0899616
$412$	−8.64424	−0.425871
$413$	26.9524	1.32624
$414$	8.17884	0.401968
$415$	0	0
$416$	−0.273559	−0.0134123
$417$	−1.43056	−0.0700546
$418$	−27.1364	−1.32729
$419$	−9.53574	−0.465851	−0.232926	−	0.972495i	$-0.574830\pi$
$419$	−9.53574	−0.465851	−0.232926	+	0.972495i	$0.574830\pi$
$420$	0	0
$421$	−21.1822	−1.03235	−0.516177	−	0.856482i	$-0.672645\pi$
$421$	−21.1822	−1.03235	−0.516177	+	0.856482i	$0.672645\pi$
$422$	18.9872	0.924284
$423$	−25.1634	−1.22348
$424$	−0.200868	−0.00975501
$425$	0	0
$426$	−2.09940	−0.101716
$427$	17.8555	0.864089
$428$	−1.34003	−0.0647726
$429$	0.0460876	0.00222513
$430$	0	0
$431$	−35.7158	−1.72037	−0.860186	−	0.509981i	$-0.829652\pi$
$431$	−35.7158	−1.72037	−0.860186	+	0.509981i	$0.829652\pi$
$432$	2.34173	0.112666
$433$	11.8322	0.568618	0.284309	−	0.958733i	$-0.408236\pi$
$433$	11.8322	0.568618	0.284309	+	0.958733i	$0.408236\pi$
$434$	20.5129	0.984650
$435$	0	0
$436$	13.5788	0.650305
$437$	−19.8086	−0.947572
$438$	−4.07910	−0.194907
$439$	2.11764	0.101069	0.0505347	−	0.998722i	$-0.483907\pi$
$439$	2.11764	0.101069	0.0505347	+	0.998722i	$0.483907\pi$
$440$	0	0
$441$	5.53358	0.263504
$442$	−0.316059	−0.0150334
$443$	26.8075	1.27366	0.636832	−	0.771002i	$-0.280244\pi$
$443$	26.8075	1.27366	0.636832	+	0.771002i	$0.280244\pi$
$444$	0.491226	0.0233126
$445$	0	0
$446$	−0.750907	−0.0355565
$447$	2.08063	0.0984106
$448$	18.3925	0.868963
$449$	−35.6307	−1.68152	−0.840759	−	0.541409i	$-0.817891\pi$
$449$	−35.6307	−1.68152	−0.840759	+	0.541409i	$0.817891\pi$
$450$	0	0
$451$	23.8187	1.12158
$452$	6.92532	0.325739
$453$	−0.995213	−0.0467592
$454$	−25.9587	−1.21830
$455$	0	0
$456$	−5.37366	−0.251645
$457$	−4.37323	−0.204571	−0.102286	−	0.994755i	$-0.532616\pi$
$457$	−4.37323	−0.204571	−0.102286	+	0.994755i	$0.532616\pi$
$458$	−19.8954	−0.929652
$459$	−5.91671	−0.276169
$460$	0	0
$461$	2.32865	0.108456	0.0542281	−	0.998529i	$-0.482730\pi$
$461$	2.32865	0.108456	0.0542281	+	0.998529i	$0.482730\pi$
$462$	−1.75807	−0.0817927
$463$	−24.3683	−1.13249	−0.566246	−	0.824236i	$-0.691605\pi$
$463$	−24.3683	−1.13249	−0.566246	+	0.824236i	$0.691605\pi$
$464$	11.4844	0.533152
$465$	0	0
$466$	−1.86045	−0.0861835
$467$	42.0547	1.94606	0.973030	−	0.230678i	$-0.0740944\pi$
$467$	42.0547	1.94606	0.973030	+	0.230678i	$0.0740944\pi$
$468$	−0.153490	−0.00709507
$469$	−30.2463	−1.39665
$470$	0	0
$471$	−0.0583294	−0.00268768
$472$	−36.5191	−1.68093
$473$	34.0226	1.56436
$474$	−0.177918	−0.00817203
$475$	0	0
$476$	−8.04872	−0.368913
$477$	−0.193188	−0.00884547
$478$	5.12369	0.234352
$479$	−30.3166	−1.38520	−0.692600	−	0.721322i	$-0.743535\pi$
$479$	−30.3166	−1.38520	−0.692600	+	0.721322i	$0.743535\pi$
$480$	0	0
$481$	0.178055	0.00811862
$482$	1.09514	0.0498824
$483$	−1.28332	−0.0583932
$484$	0.776690	0.0353041
$485$	0	0
$486$	6.47432	0.293681
$487$	16.4118	0.743692	0.371846	−	0.928295i	$-0.378725\pi$
$487$	16.4118	0.743692	0.371846	+	0.928295i	$0.378725\pi$
$488$	−24.1932	−1.09518
$489$	1.21372	0.0548863
$490$	0	0
$491$	−0.561056	−0.0253201	−0.0126600	−	0.999920i	$-0.504030\pi$
$491$	−0.561056	−0.0253201	−0.0126600	+	0.999920i	$0.504030\pi$
$492$	1.34842	0.0607914
$493$	−29.0171	−1.30686
$494$	−0.556843	−0.0250535
$495$	0	0
$496$	−14.5435	−0.653024
$497$	−19.3788	−0.869258
$498$	−1.61168	−0.0722211
$499$	−39.1100	−1.75080	−0.875401	−	0.483397i	$-0.839403\pi$
$499$	−39.1100	−1.75080	−0.875401	+	0.483397i	$0.839403\pi$
$500$	0	0
$501$	−4.93366	−0.220420
$502$	−7.06114	−0.315154
$503$	−36.7064	−1.63666	−0.818329	−	0.574750i	$-0.805099\pi$
$503$	−36.7064	−1.63666	−0.818329	+	0.574750i	$0.805099\pi$
$504$	20.4806	0.912277
$505$	0	0
$506$	8.78092	0.390360
$507$	−2.91010	−0.129242
$508$	−4.85289	−0.215312
$509$	−20.8479	−0.924065	−0.462033	−	0.886863i	$-0.652880\pi$
$509$	−20.8479	−0.924065	−0.462033	+	0.886863i	$0.652880\pi$
$510$	0	0
$511$	−37.6528	−1.66566
$512$	−18.1802	−0.803458
$513$	−10.4243	−0.460243
$514$	−16.4408	−0.725172
$515$	0	0
$516$	1.92608	0.0847910
$517$	−27.0158	−1.18815
$518$	−6.79213	−0.298429
$519$	4.05227	0.177875
$520$	0	0
$521$	−23.2776	−1.01981	−0.509904	−	0.860231i	$-0.670319\pi$
$521$	−23.2776	−1.01981	−0.509904	+	0.860231i	$0.670319\pi$
$522$	21.1085	0.923895
$523$	16.3641	0.715552	0.357776	−	0.933807i	$-0.383535\pi$
$523$	16.3641	0.715552	0.357776	+	0.933807i	$0.383535\pi$
$524$	5.23716	0.228786
$525$	0	0
$526$	12.2181	0.532733
$527$	36.7464	1.60070
$528$	1.24646	0.0542453
$529$	−16.5903	−0.721316
$530$	0	0
$531$	−35.1228	−1.52420
$532$	−14.1805	−0.614803
$533$	0.488762	0.0211706
$534$	−2.65253	−0.114786
$535$	0	0
$536$	40.9822	1.77016
$537$	−3.89926	−0.168265
$538$	−14.7927	−0.637758
$539$	5.94093	0.255894
$540$	0	0
$541$	−23.0819	−0.992369	−0.496184	−	0.868217i	$-0.665266\pi$
$541$	−23.0819	−0.992369	−0.496184	+	0.868217i	$0.665266\pi$
$542$	14.3449	0.616166
$543$	1.49993	0.0643680
$544$	18.6934	0.801475
$545$	0	0
$546$	−0.0360758	−0.00154390
$547$	17.3551	0.742049	0.371024	−	0.928623i	$-0.379007\pi$
$547$	17.3551	0.742049	0.371024	+	0.928623i	$0.379007\pi$
$548$	6.52111	0.278568
$549$	−23.2682	−0.993064
$550$	0	0
$551$	−51.1233	−2.17793
$552$	1.73883	0.0740097
$553$	−1.64230	−0.0698376
$554$	15.8233	0.672269
$555$	0	0
$556$	5.11503	0.216926
$557$	−6.28655	−0.266370	−0.133185	−	0.991091i	$-0.542520\pi$
$557$	−6.28655	−0.266370	−0.133185	+	0.991091i	$0.542520\pi$
$558$	−26.7312	−1.13162
$559$	0.698149	0.0295285
$560$	0	0
$561$	−3.14937	−0.132966
$562$	1.94571	0.0820749
$563$	6.26725	0.264133	0.132067	−	0.991241i	$-0.457839\pi$
$563$	6.26725	0.264133	0.132067	+	0.991241i	$0.457839\pi$
$564$	−1.52941	−0.0643997
$565$	0	0
$566$	20.2064	0.849338
$567$	19.3570	0.812917
$568$	26.2572	1.10173
$569$	21.6171	0.906234	0.453117	−	0.891451i	$-0.350312\pi$
$569$	21.6171	0.906234	0.453117	+	0.891451i	$0.350312\pi$
$570$	0	0
$571$	−36.4039	−1.52345	−0.761727	−	0.647898i	$-0.775648\pi$
$571$	−36.4039	−1.52345	−0.761727	+	0.647898i	$0.775648\pi$
$572$	−0.164789	−0.00689017
$573$	4.08007	0.170447
$574$	−18.6444	−0.778203
$575$	0	0
$576$	−23.9680	−0.998666
$577$	31.3479	1.30503	0.652514	−	0.757776i	$-0.273714\pi$
$577$	31.3479	1.30503	0.652514	+	0.757776i	$0.273714\pi$
$578$	2.98022	0.123961
$579$	−1.31714	−0.0547385
$580$	0	0
$581$	−14.8769	−0.617196
$582$	−2.36561	−0.0980577
$583$	−0.207409	−0.00859002
$584$	51.0175	2.11112
$585$	0	0
$586$	−9.47730	−0.391504
$587$	−44.4381	−1.83416	−0.917079	−	0.398705i	$-0.869460\pi$
$587$	−44.4381	−1.83416	−0.917079	+	0.398705i	$0.869460\pi$
$588$	0.336326	0.0138699
$589$	64.7410	2.66761
$590$	0	0
$591$	4.12022	0.169483
$592$	4.81559	0.197920
$593$	−6.62491	−0.272052	−0.136026	−	0.990705i	$-0.543433\pi$
$593$	−6.62491	−0.272052	−0.136026	+	0.990705i	$0.543433\pi$
$594$	4.62096	0.189600
$595$	0	0
$596$	−7.43941	−0.304730
$597$	2.65789	0.108780
$598$	0.180186	0.00736834
$599$	24.8397	1.01492	0.507460	−	0.861675i	$-0.330584\pi$
$599$	24.8397	1.01492	0.507460	+	0.861675i	$0.330584\pi$
$600$	0	0
$601$	−6.70931	−0.273679	−0.136839	−	0.990593i	$-0.543694\pi$
$601$	−6.70931	−0.273679	−0.136839	+	0.990593i	$0.543694\pi$
$602$	−26.6317	−1.08543
$603$	39.4152	1.60511
$604$	3.55844	0.144791
$605$	0	0
$606$	−4.04138	−0.164170
$607$	−18.3020	−0.742857	−0.371428	−	0.928462i	$-0.621132\pi$
$607$	−18.3020	−0.742857	−0.371428	+	0.928462i	$0.621132\pi$
$608$	32.9347	1.33568
$609$	−3.31209	−0.134213
$610$	0	0
$611$	−0.554366	−0.0224273
$612$	10.4886	0.423977
$613$	−26.7397	−1.08000	−0.540002	−	0.841664i	$-0.681577\pi$
$613$	−26.7397	−1.08000	−0.540002	+	0.841664i	$0.681577\pi$
$614$	−7.24045	−0.292201
$615$	0	0
$616$	21.9882	0.885931
$617$	−44.9237	−1.80856	−0.904280	−	0.426939i	$-0.859592\pi$
$617$	−44.9237	−1.80856	−0.904280	+	0.426939i	$0.859592\pi$
$618$	2.64761	0.106503
$619$	34.7225	1.39562	0.697808	−	0.716285i	$-0.254159\pi$
$619$	34.7225	1.39562	0.697808	+	0.716285i	$0.254159\pi$
$620$	0	0
$621$	3.37313	0.135359
$622$	−21.2684	−0.852784
$623$	−24.4846	−0.980954
$624$	0.0255776	0.00102392
$625$	0	0
$626$	21.3025	0.851418
$627$	−5.54866	−0.221592
$628$	0.208560	0.00832244
$629$	−12.1673	−0.485142
$630$	0	0
$631$	−30.3033	−1.20635	−0.603177	−	0.797607i	$-0.706099\pi$
$631$	−30.3033	−1.20635	−0.603177	+	0.797607i	$0.706099\pi$
$632$	2.22522	0.0885146
$633$	3.88237	0.154311
$634$	0.868030	0.0344739
$635$	0	0
$636$	−0.0117418	−0.000465593 0
$637$	0.121909	0.00483019
$638$	22.6624	0.897214
$639$	25.2533	0.999006
$640$	0	0
$641$	44.8929	1.77316	0.886582	−	0.462571i	$-0.153073\pi$
$641$	44.8929	1.77316	0.886582	+	0.462571i	$0.153073\pi$
$642$	0.410432	0.0161984
$643$	−0.746188	−0.0294268	−0.0147134	−	0.999892i	$-0.504684\pi$
$643$	−0.746188	−0.0294268	−0.0147134	+	0.999892i	$0.504684\pi$
$644$	4.58859	0.180816
$645$	0	0
$646$	38.0515	1.49711
$647$	25.5099	1.00290	0.501448	−	0.865188i	$-0.332801\pi$
$647$	25.5099	1.00290	0.501448	+	0.865188i	$0.332801\pi$
$648$	−26.2277	−1.03032
$649$	−37.7084	−1.48018
$650$	0	0
$651$	4.19433	0.164389
$652$	−4.33972	−0.169956
$653$	−28.7035	−1.12325	−0.561627	−	0.827391i	$-0.689824\pi$
$653$	−28.7035	−1.12325	−0.561627	+	0.827391i	$0.689824\pi$
$654$	−4.15899	−0.162629
$655$	0	0
$656$	13.2188	0.516108
$657$	49.0669	1.91428
$658$	21.1470	0.824395
$659$	15.5090	0.604146	0.302073	−	0.953285i	$-0.402321\pi$
$659$	15.5090	0.604146	0.302073	+	0.953285i	$0.402321\pi$
$660$	0	0
$661$	−28.9524	−1.12612	−0.563058	−	0.826417i	$-0.690375\pi$
$661$	−28.9524	−1.12612	−0.563058	+	0.826417i	$0.690375\pi$
$662$	16.0972	0.625637
$663$	−0.0646254	−0.00250984
$664$	20.1573	0.782257
$665$	0	0
$666$	8.85111	0.342973
$667$	16.5427	0.640536
$668$	17.6406	0.682534
$669$	−0.153540	−0.00593620
$670$	0	0
$671$	−24.9811	−0.964385
$672$	2.13372	0.0823100
$673$	35.4119	1.36503	0.682513	−	0.730873i	$-0.260887\pi$
$673$	35.4119	1.36503	0.682513	+	0.730873i	$0.260887\pi$
$674$	−18.9881	−0.731395
$675$	0	0
$676$	10.4052	0.400201
$677$	17.5836	0.675791	0.337896	−	0.941184i	$-0.390285\pi$
$677$	17.5836	0.675791	0.337896	+	0.941184i	$0.390285\pi$
$678$	−2.12113	−0.0814615
$679$	−21.8361	−0.837994
$680$	0	0
$681$	−5.30784	−0.203397
$682$	−28.6990	−1.09894
$683$	−29.9328	−1.14535	−0.572674	−	0.819783i	$-0.694094\pi$
$683$	−29.9328	−1.14535	−0.572674	+	0.819783i	$0.694094\pi$
$684$	18.4792	0.706570
$685$	0	0
$686$	−22.0035	−0.840098
$687$	−4.06807	−0.155207
$688$	18.8818	0.719861
$689$	−0.00425606	−0.000162143 0
$690$	0	0
$691$	13.2291	0.503260	0.251630	−	0.967823i	$-0.419033\pi$
$691$	13.2291	0.503260	0.251630	+	0.967823i	$0.419033\pi$
$692$	−14.4891	−0.550793
$693$	21.1475	0.803328
$694$	15.3040	0.580933
$695$	0	0
$696$	4.48770	0.170106
$697$	−33.3992	−1.26509
$698$	19.9598	0.755488
$699$	−0.380411	−0.0143884
$700$	0	0
$701$	15.9444	0.602214	0.301107	−	0.953590i	$-0.402644\pi$
$701$	15.9444	0.602214	0.301107	+	0.953590i	$0.402644\pi$
$702$	0.0948228	0.00357885
$703$	−21.4367	−0.808502
$704$	−25.7324	−0.969825
$705$	0	0
$706$	−27.5483	−1.03679
$707$	−37.3046	−1.40298
$708$	−2.13474	−0.0802283
$709$	37.1049	1.39351	0.696753	−	0.717312i	$-0.254628\pi$
$709$	37.1049	1.39351	0.696753	+	0.717312i	$0.254628\pi$
$710$	0	0
$711$	2.14014	0.0802617
$712$	33.1753	1.24330
$713$	−20.9492	−0.784552
$714$	2.46521	0.0922583
$715$	0	0
$716$	13.9420	0.521037
$717$	1.04765	0.0391254
$718$	−6.62773	−0.247345
$719$	5.70240	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$719$	5.70240	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$720$	0	0
$721$	24.4392	0.910163
$722$	46.2326	1.72060
$723$	0.223927	0.00832793
$724$	−5.36306	−0.199317
$725$	0	0
$726$	−0.237890	−0.00882891
$727$	−5.81600	−0.215703	−0.107852	−	0.994167i	$-0.534397\pi$
$727$	−5.81600	−0.215703	−0.107852	+	0.994167i	$0.534397\pi$
$728$	0.451201	0.0167226
$729$	−24.3299	−0.901106
$730$	0	0
$731$	−47.7075	−1.76453
$732$	−1.41422	−0.0522712
$733$	43.9492	1.62330	0.811651	−	0.584143i	$-0.198569\pi$
$733$	43.9492	1.62330	0.811651	+	0.584143i	$0.198569\pi$
$734$	34.2840	1.26545
$735$	0	0
$736$	−10.6572	−0.392828
$737$	42.3168	1.55876
$738$	24.2963	0.894359
$739$	−22.5626	−0.829979	−0.414989	−	0.909826i	$-0.636215\pi$
$739$	−22.5626	−0.829979	−0.414989	+	0.909826i	$0.636215\pi$
$740$	0	0
$741$	−0.113859	−0.00418272
$742$	0.162353	0.00596015
$743$	7.18672	0.263655	0.131828	−	0.991273i	$-0.457915\pi$
$743$	7.18672	0.263655	0.131828	+	0.991273i	$0.457915\pi$
$744$	−5.68309	−0.208352
$745$	0	0
$746$	19.1715	0.701919
$747$	19.3866	0.709320
$748$	11.2607	0.411733
$749$	3.78855	0.138431
$750$	0	0
$751$	30.0828	1.09774	0.548868	−	0.835909i	$-0.315059\pi$
$751$	30.0828	1.09774	0.548868	+	0.835909i	$0.315059\pi$
$752$	−14.9931	−0.546742
$753$	−1.44381	−0.0526154
$754$	0.465036	0.0169356
$755$	0	0
$756$	2.41475	0.0878235
$757$	−6.52000	−0.236973	−0.118487	−	0.992956i	$-0.537804\pi$
$757$	−6.52000	−0.236973	−0.118487	+	0.992956i	$0.537804\pi$
$758$	0.624185	0.0226714
$759$	1.79546	0.0651711
$760$	0	0
$761$	14.1010	0.511162	0.255581	−	0.966788i	$-0.417733\pi$
$761$	14.1010	0.511162	0.255581	+	0.966788i	$0.417733\pi$
$762$	1.48637	0.0538456
$763$	−38.3902	−1.38982
$764$	−14.5885	−0.527793
$765$	0	0
$766$	26.3122	0.950699
$767$	−0.773780	−0.0279396
$768$	−3.52132	−0.127065
$769$	18.1800	0.655589	0.327795	−	0.944749i	$-0.393695\pi$
$769$	18.1800	0.655589	0.327795	+	0.944749i	$0.393695\pi$
$770$	0	0
$771$	−3.36169	−0.121068
$772$	4.70950	0.169499
$773$	43.7032	1.57190	0.785948	−	0.618293i	$-0.212175\pi$
$773$	43.7032	1.57190	0.785948	+	0.618293i	$0.212175\pi$
$774$	34.7049	1.24744
$775$	0	0
$776$	29.5868	1.06210
$777$	−1.38881	−0.0498232
$778$	−14.8742	−0.533266
$779$	−58.8439	−2.10830
$780$	0	0
$781$	27.1123	0.970155
$782$	−12.3129	−0.440307
$783$	8.70561	0.311113
$784$	3.29708	0.117753
$785$	0	0
$786$	−1.60407	−0.0572153
$787$	20.3278	0.724608	0.362304	−	0.932060i	$-0.381990\pi$
$787$	20.3278	0.724608	0.362304	+	0.932060i	$0.381990\pi$
$788$	−14.7321	−0.524808
$789$	2.49826	0.0889405
$790$	0	0
$791$	−19.5794	−0.696164
$792$	−28.6538	−1.01817
$793$	−0.512615	−0.0182035
$794$	10.4359	0.370358
$795$	0	0
$796$	−9.50343	−0.336840
$797$	−8.91772	−0.315882	−0.157941	−	0.987449i	$-0.550486\pi$
$797$	−8.91772	−0.315882	−0.157941	+	0.987449i	$0.550486\pi$
$798$	4.34329	0.153751
$799$	37.8822	1.34018
$800$	0	0
$801$	31.9069	1.12737
$802$	10.9362	0.386170
$803$	52.6789	1.85900
$804$	2.39563	0.0844872
$805$	0	0
$806$	−0.588906	−0.0207433
$807$	−3.02470	−0.106475
$808$	50.5457	1.77819
$809$	−23.4279	−0.823680	−0.411840	−	0.911256i	$-0.635114\pi$
$809$	−23.4279	−0.823680	−0.411840	+	0.911256i	$0.635114\pi$
$810$	0	0
$811$	17.8208	0.625774	0.312887	−	0.949790i	$-0.398704\pi$
$811$	17.8208	0.625774	0.312887	+	0.949790i	$0.398704\pi$
$812$	11.8426	0.415592
$813$	2.93314	0.102870
$814$	9.50267	0.333069
$815$	0	0
$816$	−1.74782	−0.0611861
$817$	−84.0527	−2.94063
$818$	−2.03360	−0.0711033
$819$	0.433950	0.0151634
$820$	0	0
$821$	33.2196	1.15937	0.579686	−	0.814840i	$-0.303175\pi$
$821$	33.2196	1.15937	0.579686	+	0.814840i	$0.303175\pi$
$822$	−1.99733	−0.0696647
$823$	13.2155	0.460663	0.230331	−	0.973112i	$-0.426019\pi$
$823$	13.2155	0.460663	0.230331	+	0.973112i	$0.426019\pi$
$824$	−33.1138	−1.15357
$825$	0	0
$826$	29.5168	1.02702
$827$	47.8885	1.66525	0.832623	−	0.553840i	$-0.186838\pi$
$827$	47.8885	1.66525	0.832623	+	0.553840i	$0.186838\pi$
$828$	−5.97958	−0.207805
$829$	50.0186	1.73722	0.868609	−	0.495498i	$-0.165015\pi$
$829$	50.0186	1.73722	0.868609	+	0.495498i	$0.165015\pi$
$830$	0	0
$831$	3.23544	0.112236
$832$	−0.528031	−0.0183062
$833$	−8.33054	−0.288636
$834$	−1.56666	−0.0542491
$835$	0	0
$836$	19.8395	0.686164
$837$	−11.0245	−0.381063
$838$	−10.4430	−0.360747
$839$	−23.0264	−0.794958	−0.397479	−	0.917611i	$-0.630115\pi$
$839$	−23.0264	−0.794958	−0.397479	+	0.917611i	$0.630115\pi$
$840$	0	0
$841$	13.6946	0.472227
$842$	−23.1975	−0.799438
$843$	0.397845	0.0137025
$844$	−13.8816	−0.477826
$845$	0	0
$846$	−27.5575	−0.947446
$847$	−2.19588	−0.0754512
$848$	−0.115107	−0.00395280
$849$	4.13166	0.141798
$850$	0	0
$851$	6.93659	0.237783
$852$	1.53488	0.0525840
$853$	−39.2071	−1.34243	−0.671213	−	0.741265i	$-0.734226\pi$
$853$	−39.2071	−1.34243	−0.671213	+	0.741265i	$0.734226\pi$
$854$	19.5543	0.669135
$855$	0	0
$856$	−5.13328	−0.175452
$857$	−36.7331	−1.25478	−0.627389	−	0.778706i	$-0.715876\pi$
$857$	−36.7331	−1.25478	−0.627389	+	0.778706i	$0.715876\pi$
$858$	0.0504725	0.00172310
$859$	−47.6475	−1.62571	−0.812856	−	0.582465i	$-0.802088\pi$
$859$	−47.6475	−1.62571	−0.812856	+	0.582465i	$0.802088\pi$
$860$	0	0
$861$	−3.81228	−0.129922
$862$	−39.1139	−1.33223
$863$	−35.6174	−1.21243	−0.606215	−	0.795301i	$-0.707313\pi$
$863$	−35.6174	−1.21243	−0.606215	+	0.795301i	$0.707313\pi$
$864$	−5.60834	−0.190800
$865$	0	0
$866$	12.9579	0.440328
$867$	0.609373	0.0206954
$868$	−14.9970	−0.509033
$869$	2.29769	0.0779438
$870$	0	0
$871$	0.868345	0.0294228
$872$	52.0166	1.76150
$873$	28.4556	0.963075
$874$	−21.6932	−0.733784
$875$	0	0
$876$	2.98224	0.100761
$877$	39.1576	1.32226	0.661129	−	0.750272i	$-0.270078\pi$
$877$	39.1576	1.32226	0.661129	+	0.750272i	$0.270078\pi$
$878$	2.31912	0.0782664
$879$	−1.93785	−0.0653621
$880$	0	0
$881$	−30.9027	−1.04114	−0.520569	−	0.853820i	$-0.674280\pi$
$881$	−30.9027	−1.04114	−0.520569	+	0.853820i	$0.674280\pi$
$882$	6.06006	0.204053
$883$	−9.36408	−0.315126	−0.157563	−	0.987509i	$-0.550364\pi$
$883$	−9.36408	−0.315126	−0.157563	+	0.987509i	$0.550364\pi$
$884$	0.231071	0.00777177
$885$	0	0
$886$	29.3581	0.986304
$887$	48.6485	1.63346	0.816728	−	0.577023i	$-0.195786\pi$
$887$	48.6485	1.63346	0.816728	+	0.577023i	$0.195786\pi$
$888$	1.88176	0.0631477
$889$	13.7202	0.460161
$890$	0	0
$891$	−27.0818	−0.907274
$892$	0.548991	0.0183816
$893$	66.7422	2.23344
$894$	2.27859	0.0762074
$895$	0	0
$896$	1.08510	0.0362508
$897$	0.0368431	0.00123015
$898$	−39.0207	−1.30214
$899$	−54.0671	−1.80324
$900$	0	0
$901$	0.290835	0.00968913
$902$	26.0849	0.868531
$903$	−5.44546	−0.181214
$904$	26.5290	0.882343
$905$	0	0
$906$	−1.08990	−0.0362095
$907$	42.3130	1.40498	0.702489	−	0.711694i	$-0.252072\pi$
$907$	42.3130	1.40498	0.702489	+	0.711694i	$0.252072\pi$
$908$	18.9785	0.629822
$909$	48.6131	1.61240
$910$	0	0
$911$	2.03059	0.0672763	0.0336382	−	0.999434i	$-0.489291\pi$
$911$	2.03059	0.0672763	0.0336382	+	0.999434i	$0.489291\pi$
$912$	−3.07938	−0.101968
$913$	20.8138	0.688836
$914$	−4.78931	−0.158416
$915$	0	0
$916$	14.5456	0.480601
$917$	−14.8066	−0.488958
$918$	−6.47965	−0.213860
$919$	−24.5581	−0.810097	−0.405048	−	0.914295i	$-0.632745\pi$
$919$	−24.5581	−0.810097	−0.405048	+	0.914295i	$0.632745\pi$
$920$	0	0
$921$	−1.48048	−0.0487833
$922$	2.55021	0.0839866
$923$	0.556348	0.0183124
$924$	1.28533	0.0422842
$925$	0	0
$926$	−26.6868	−0.876983
$927$	−31.8477	−1.04602
$928$	−27.5048	−0.902888
$929$	6.26058	0.205403	0.102702	−	0.994712i	$-0.467251\pi$
$929$	6.26058	0.205403	0.102702	+	0.994712i	$0.467251\pi$
$930$	0	0
$931$	−14.6770	−0.481020
$932$	1.36018	0.0445541
$933$	−4.34881	−0.142374
$934$	46.0559	1.50700
$935$	0	0
$936$	−0.587979	−0.0192187
$937$	21.1381	0.690552	0.345276	−	0.938501i	$-0.387785\pi$
$937$	21.1381	0.690552	0.345276	+	0.938501i	$0.387785\pi$
$938$	−33.1241	−1.08154
$939$	4.35577	0.142145
$940$	0	0
$941$	−3.32725	−0.108465	−0.0542327	−	0.998528i	$-0.517271\pi$
$941$	−3.32725	−0.108465	−0.0542327	+	0.998528i	$0.517271\pi$
$942$	−0.0638790	−0.00208129
$943$	19.0410	0.620059
$944$	−20.9272	−0.681124
$945$	0	0
$946$	37.2597	1.21142
$947$	3.04446	0.0989318	0.0494659	−	0.998776i	$-0.484248\pi$
$947$	3.04446	0.0989318	0.0494659	+	0.998776i	$0.484248\pi$
$948$	0.130076	0.00422468
$949$	1.08098	0.0350900
$950$	0	0
$951$	0.177489	0.00575546
$952$	−30.8325	−0.999287
$953$	−9.75807	−0.316095	−0.158047	−	0.987432i	$-0.550520\pi$
$953$	−9.75807	−0.316095	−0.158047	+	0.987432i	$0.550520\pi$
$954$	−0.211568	−0.00684978
$955$	0	0
$956$	−3.74594	−0.121152
$957$	4.63385	0.149791
$958$	−33.2010	−1.07267
$959$	−18.4366	−0.595350
$960$	0	0
$961$	37.4688	1.20867
$962$	0.194996	0.00628692
$963$	−4.93702	−0.159093
$964$	−0.800662	−0.0257876
$965$	0	0
$966$	−1.40542	−0.0452187
$967$	8.16066	0.262429	0.131215	−	0.991354i	$-0.458112\pi$
$967$	8.16066	0.262429	0.131215	+	0.991354i	$0.458112\pi$
$968$	2.97529	0.0956296
$969$	7.78049	0.249945
$970$	0	0
$971$	16.0914	0.516398	0.258199	−	0.966092i	$-0.416871\pi$
$971$	16.0914	0.516398	0.258199	+	0.966092i	$0.416871\pi$
$972$	−4.73340	−0.151824
$973$	−14.4613	−0.463609
$974$	17.9733	0.575902
$975$	0	0
$976$	−13.8639	−0.443774
$977$	−23.1058	−0.739220	−0.369610	−	0.929187i	$-0.620509\pi$
$977$	−23.1058	−0.739220	−0.369610	+	0.929187i	$0.620509\pi$
$978$	1.32920	0.0425030
$979$	34.2557	1.09482
$980$	0	0
$981$	50.0278	1.59727
$982$	−0.614436	−0.0196074
$983$	−44.1346	−1.40768	−0.703838	−	0.710361i	$-0.748532\pi$
$983$	−44.1346	−1.40768	−0.703838	+	0.710361i	$0.748532\pi$
$984$	5.16543	0.164668
$985$	0	0
$986$	−31.7779	−1.01201
$987$	4.32398	0.137634
$988$	0.407110	0.0129519
$989$	27.1981	0.864851
$990$	0	0
$991$	−5.42300	−0.172267	−0.0861337	−	0.996284i	$-0.527451\pi$
$991$	−5.42300	−0.172267	−0.0861337	+	0.996284i	$0.527451\pi$
$992$	34.8312	1.10589
$993$	3.29145	0.104451
$994$	−21.2226	−0.673139
$995$	0	0
$996$	1.17830	0.0373360
$997$	56.5705	1.79161	0.895803	−	0.444451i	$-0.146601\pi$
$997$	56.5705	1.79161	0.895803	+	0.444451i	$0.146601\pi$
$998$	−42.8310	−1.35579
$999$	3.65039	0.115493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.32		46
5.2	odd	4		1205.2.b.c.724.32	yes	46
5.3	odd	4		1205.2.b.c.724.15	✓	46
5.4	even	2	inner	6025.2.a.p.1.15		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.15	✓	46	5.3	odd	4
1205.2.b.c.724.32	yes	46	5.2	odd	4
6025.2.a.p.1.15		46	5.4	even	2	inner
6025.2.a.p.1.32		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)

\(f(q)\)	\(=\)	\(q+1.09514 q^{2} +0.223927 q^{3} -0.800662 q^{4} +0.245232 q^{6} +2.26365 q^{7} -3.06713 q^{8} -2.94986 q^{9} +O(q^{10})\)	\(q+1.09514 q^{2} +0.223927 q^{3} -0.800662 q^{4} +0.245232 q^{6} +2.26365 q^{7} -3.06713 q^{8} -2.94986 q^{9} -3.16701 q^{11} -0.179290 q^{12} -0.0649874 q^{13} +2.47902 q^{14} -1.75761 q^{16} +4.44087 q^{17} -3.23051 q^{18} +7.82407 q^{19} +0.506893 q^{21} -3.46833 q^{22} -2.53175 q^{23} -0.686812 q^{24} -0.0711705 q^{26} -1.33233 q^{27} -1.81242 q^{28} -6.53411 q^{29} +8.27459 q^{31} +4.20941 q^{32} -0.709178 q^{33} +4.86338 q^{34} +2.36184 q^{36} -2.73984 q^{37} +8.56847 q^{38} -0.0145524 q^{39} -7.52088 q^{41} +0.555120 q^{42} -10.7428 q^{43} +2.53570 q^{44} -2.77262 q^{46} +8.53037 q^{47} -0.393577 q^{48} -1.87588 q^{49} +0.994430 q^{51} +0.0520330 q^{52} +0.0654906 q^{53} -1.45909 q^{54} -6.94290 q^{56} +1.75202 q^{57} -7.15578 q^{58} +11.9066 q^{59} +7.88792 q^{61} +9.06186 q^{62} -6.67745 q^{63} +8.12514 q^{64} -0.776651 q^{66} -13.3617 q^{67} -3.55564 q^{68} -0.566926 q^{69} -8.56086 q^{71} +9.04758 q^{72} -16.6336 q^{73} -3.00052 q^{74} -6.26444 q^{76} -7.16900 q^{77} -0.0159370 q^{78} -0.725508 q^{79} +8.55123 q^{81} -8.23644 q^{82} -6.57206 q^{83} -0.405850 q^{84} -11.7649 q^{86} -1.46316 q^{87} +9.71361 q^{88} -10.8164 q^{89} -0.147109 q^{91} +2.02707 q^{92} +1.85290 q^{93} +9.34197 q^{94} +0.942600 q^{96} -9.64642 q^{97} -2.05436 q^{98} +9.34222 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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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