LMFDB - Embedded newform 6025.2.a.p.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.36215 q^{2} +2.05253 q^{3} +3.57974 q^{4} -4.84838 q^{6} +3.07728 q^{7} -3.73158 q^{8} +1.21289 q^{9} +O(q^{10})$	\(q-2.36215 q^{2} +2.05253 q^{3} +3.57974 q^{4} -4.84838 q^{6} +3.07728 q^{7} -3.73158 q^{8} +1.21289 q^{9} +0.942447 q^{11} +7.34753 q^{12} -1.99403 q^{13} -7.26899 q^{14} +1.65507 q^{16} -3.11539 q^{17} -2.86502 q^{18} -6.33222 q^{19} +6.31621 q^{21} -2.22620 q^{22} -7.91084 q^{23} -7.65919 q^{24} +4.71020 q^{26} -3.66811 q^{27} +11.0159 q^{28} -1.63432 q^{29} +6.96040 q^{31} +3.55366 q^{32} +1.93440 q^{33} +7.35900 q^{34} +4.34182 q^{36} +0.864280 q^{37} +14.9576 q^{38} -4.09282 q^{39} +1.70732 q^{41} -14.9198 q^{42} +10.2266 q^{43} +3.37372 q^{44} +18.6866 q^{46} +6.49786 q^{47} +3.39707 q^{48} +2.46965 q^{49} -6.39443 q^{51} -7.13813 q^{52} -2.20905 q^{53} +8.66461 q^{54} -11.4831 q^{56} -12.9971 q^{57} +3.86050 q^{58} -7.69694 q^{59} -3.70322 q^{61} -16.4415 q^{62} +3.73239 q^{63} -11.7044 q^{64} -4.56935 q^{66} +7.96470 q^{67} -11.1523 q^{68} -16.2373 q^{69} -7.02031 q^{71} -4.52599 q^{72} +6.13583 q^{73} -2.04156 q^{74} -22.6677 q^{76} +2.90017 q^{77} +9.66784 q^{78} -14.0583 q^{79} -11.1676 q^{81} -4.03294 q^{82} -3.98789 q^{83} +22.6104 q^{84} -24.1566 q^{86} -3.35449 q^{87} -3.51682 q^{88} -11.7782 q^{89} -6.13620 q^{91} -28.3188 q^{92} +14.2864 q^{93} -15.3489 q^{94} +7.29399 q^{96} +17.6694 q^{97} -5.83367 q^{98} +1.14308 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.36215	−1.67029	−0.835145	−	0.550029i	$-0.814617\pi$
$2$	−2.36215	−1.67029	−0.835145	+	0.550029i	$0.814617\pi$
$3$	2.05253	1.18503	0.592515	−	0.805559i	$-0.298135\pi$
$3$	2.05253	1.18503	0.592515	+	0.805559i	$0.298135\pi$
$4$	3.57974	1.78987
$5$	0	0
$5$	0	0
$6$	−4.84838	−1.97934
$7$	3.07728	1.16310	0.581551	−	0.813510i	$-0.302446\pi$
$7$	3.07728	1.16310	0.581551	+	0.813510i	$0.302446\pi$
$8$	−3.73158	−1.31931
$9$	1.21289	0.404296
$10$	0	0
$11$	0.942447	0.284159	0.142079	−	0.989855i	$-0.454621\pi$
$11$	0.942447	0.284159	0.142079	+	0.989855i	$0.454621\pi$
$12$	7.34753	2.12105
$13$	−1.99403	−0.553046	−0.276523	−	0.961007i	$-0.589182\pi$
$13$	−1.99403	−0.553046	−0.276523	+	0.961007i	$0.589182\pi$
$14$	−7.26899	−1.94272
$15$	0	0
$16$	1.65507	0.413766
$17$	−3.11539	−0.755592	−0.377796	−	0.925889i	$-0.623318\pi$
$17$	−3.11539	−0.755592	−0.377796	+	0.925889i	$0.623318\pi$
$18$	−2.86502	−0.675291
$19$	−6.33222	−1.45271	−0.726355	−	0.687320i	$-0.758787\pi$
$19$	−6.33222	−1.45271	−0.726355	+	0.687320i	$0.758787\pi$
$20$	0	0
$21$	6.31621	1.37831
$22$	−2.22620	−0.474627
$23$	−7.91084	−1.64952	−0.824762	−	0.565479i	$-0.808691\pi$
$23$	−7.91084	−1.64952	−0.824762	+	0.565479i	$0.808691\pi$
$24$	−7.65919	−1.56343
$25$	0	0
$26$	4.71020	0.923747
$27$	−3.66811	−0.705927
$28$	11.0159	2.08180
$29$	−1.63432	−0.303485	−0.151742	−	0.988420i	$-0.548488\pi$
$29$	−1.63432	−0.303485	−0.151742	+	0.988420i	$0.548488\pi$
$30$	0	0
$31$	6.96040	1.25012	0.625062	−	0.780575i	$-0.285074\pi$
$31$	6.96040	1.25012	0.625062	+	0.780575i	$0.285074\pi$
$32$	3.55366	0.628203
$33$	1.93440	0.336736
$34$	7.35900	1.26206
$35$	0	0
$36$	4.34182	0.723637
$37$	0.864280	0.142087	0.0710434	−	0.997473i	$-0.477367\pi$
$37$	0.864280	0.142087	0.0710434	+	0.997473i	$0.477367\pi$
$38$	14.9576	2.42645
$39$	−4.09282	−0.655375
$40$	0	0
$41$	1.70732	0.266639	0.133319	−	0.991073i	$-0.457436\pi$
$41$	1.70732	0.266639	0.133319	+	0.991073i	$0.457436\pi$
$42$	−14.9198	−2.30218
$43$	10.2266	1.55953	0.779767	−	0.626069i	$-0.215337\pi$
$43$	10.2266	1.55953	0.779767	+	0.626069i	$0.215337\pi$
$44$	3.37372	0.508607
$45$	0	0
$46$	18.6866	2.75519
$47$	6.49786	0.947811	0.473905	−	0.880576i	$-0.342844\pi$
$47$	6.49786	0.947811	0.473905	+	0.880576i	$0.342844\pi$
$48$	3.39707	0.490326
$49$	2.46965	0.352807
$50$	0	0
$51$	−6.39443	−0.895399
$52$	−7.13813	−0.989880
$53$	−2.20905	−0.303436	−0.151718	−	0.988424i	$-0.548481\pi$
$53$	−2.20905	−0.303436	−0.151718	+	0.988424i	$0.548481\pi$
$54$	8.66461	1.17910
$55$	0	0
$56$	−11.4831	−1.53450
$57$	−12.9971	−1.72150
$58$	3.86050	0.506908
$59$	−7.69694	−1.00206	−0.501028	−	0.865431i	$-0.667045\pi$
$59$	−7.69694	−1.00206	−0.501028	+	0.865431i	$0.667045\pi$
$60$	0	0
$61$	−3.70322	−0.474149	−0.237075	−	0.971491i	$-0.576189\pi$
$61$	−3.70322	−0.474149	−0.237075	+	0.971491i	$0.576189\pi$
$62$	−16.4415	−2.08807
$63$	3.73239	0.470237
$64$	−11.7044	−1.46305
$65$	0	0
$66$	−4.56935	−0.562448
$67$	7.96470	0.973043	0.486521	−	0.873669i	$-0.338266\pi$
$67$	7.96470	0.973043	0.486521	+	0.873669i	$0.338266\pi$
$68$	−11.1523	−1.35241
$69$	−16.2373	−1.95474
$70$	0	0
$71$	−7.02031	−0.833157	−0.416579	−	0.909100i	$-0.636771\pi$
$71$	−7.02031	−0.833157	−0.416579	+	0.909100i	$0.636771\pi$
$72$	−4.52599	−0.533393
$73$	6.13583	0.718144	0.359072	−	0.933310i	$-0.383093\pi$
$73$	6.13583	0.718144	0.359072	+	0.933310i	$0.383093\pi$
$74$	−2.04156	−0.237326
$75$	0	0
$76$	−22.6677	−2.60016
$77$	2.90017	0.330505
$78$	9.66784	1.09467
$79$	−14.0583	−1.58168	−0.790842	−	0.612021i	$-0.790357\pi$
$79$	−14.0583	−1.58168	−0.790842	+	0.612021i	$0.790357\pi$
$80$	0	0
$81$	−11.1676	−1.24084
$82$	−4.03294	−0.445364
$83$	−3.98789	−0.437728	−0.218864	−	0.975755i	$-0.570235\pi$
$83$	−3.98789	−0.437728	−0.218864	+	0.975755i	$0.570235\pi$
$84$	22.6104	2.46700
$85$	0	0
$86$	−24.1566	−2.60488
$87$	−3.35449	−0.359639
$88$	−3.51682	−0.374894
$89$	−11.7782	−1.24849	−0.624245	−	0.781229i	$-0.714593\pi$
$89$	−11.7782	−1.24849	−0.624245	+	0.781229i	$0.714593\pi$
$90$	0	0
$91$	−6.13620	−0.643249
$92$	−28.3188	−2.95244
$93$	14.2864	1.48144
$94$	−15.3489	−1.58312
$95$	0	0
$96$	7.29399	0.744440
$97$	17.6694	1.79405	0.897026	−	0.441978i	$-0.145723\pi$
$97$	17.6694	1.79405	0.897026	+	0.441978i	$0.145723\pi$
$98$	−5.83367	−0.589290
$99$	1.14308	0.114884
$100$	0	0
$101$	9.05634	0.901140	0.450570	−	0.892741i	$-0.351221\pi$
$101$	9.05634	0.901140	0.450570	+	0.892741i	$0.351221\pi$
$102$	15.1046	1.49558
$103$	−2.20110	−0.216881	−0.108440	−	0.994103i	$-0.534586\pi$
$103$	−2.20110	−0.216881	−0.108440	+	0.994103i	$0.534586\pi$
$104$	7.44090	0.729641
$105$	0	0
$106$	5.21809	0.506826
$107$	7.08595	0.685024	0.342512	−	0.939513i	$-0.388722\pi$
$107$	7.08595	0.685024	0.342512	+	0.939513i	$0.388722\pi$
$108$	−13.1309	−1.26352
$109$	−7.57661	−0.725708	−0.362854	−	0.931846i	$-0.618198\pi$
$109$	−7.57661	−0.725708	−0.362854	+	0.931846i	$0.618198\pi$
$110$	0	0
$111$	1.77396	0.168377
$112$	5.09310	0.481253
$113$	−17.7967	−1.67418	−0.837088	−	0.547069i	$-0.815744\pi$
$113$	−17.7967	−1.67418	−0.837088	+	0.547069i	$0.815744\pi$
$114$	30.7010	2.87541
$115$	0	0
$116$	−5.85043	−0.543199
$117$	−2.41854	−0.223594
$118$	18.1813	1.67373
$119$	−9.58691	−0.878831
$120$	0	0
$121$	−10.1118	−0.919254
$122$	8.74756	0.791967
$123$	3.50433	0.315975
$124$	24.9164	2.23756
$125$	0	0
$126$	−8.81646	−0.785433
$127$	−19.4716	−1.72782	−0.863912	−	0.503642i	$-0.831993\pi$
$127$	−19.4716	−1.72782	−0.863912	+	0.503642i	$0.831993\pi$
$128$	20.5402	1.81551
$129$	20.9903	1.84810
$130$	0	0
$131$	−8.30848	−0.725915	−0.362958	−	0.931806i	$-0.618233\pi$
$131$	−8.30848	−0.725915	−0.362958	+	0.931806i	$0.618233\pi$
$132$	6.92466	0.602714
$133$	−19.4860	−1.68965
$134$	−18.8138	−1.62526
$135$	0	0
$136$	11.6253	0.996863
$137$	4.26661	0.364521	0.182261	−	0.983250i	$-0.441659\pi$
$137$	4.26661	0.364521	0.182261	+	0.983250i	$0.441659\pi$
$138$	38.3548	3.26498
$139$	−4.25699	−0.361073	−0.180537	−	0.983568i	$-0.557783\pi$
$139$	−4.25699	−0.361073	−0.180537	+	0.983568i	$0.557783\pi$
$140$	0	0
$141$	13.3371	1.12318
$142$	16.5830	1.39162
$143$	−1.87927	−0.157153
$144$	2.00741	0.167284
$145$	0	0
$146$	−14.4937	−1.19951
$147$	5.06903	0.418086
$148$	3.09390	0.254317
$149$	6.80166	0.557214	0.278607	−	0.960405i	$-0.410127\pi$
$149$	6.80166	0.557214	0.278607	+	0.960405i	$0.410127\pi$
$150$	0	0
$151$	−4.11661	−0.335005	−0.167503	−	0.985872i	$-0.553570\pi$
$151$	−4.11661	−0.335005	−0.167503	+	0.985872i	$0.553570\pi$
$152$	23.6292	1.91658
$153$	−3.77861	−0.305483
$154$	−6.85064	−0.552040
$155$	0	0
$156$	−14.6512	−1.17304
$157$	−19.2876	−1.53931	−0.769657	−	0.638457i	$-0.779573\pi$
$157$	−19.2876	−1.53931	−0.769657	+	0.638457i	$0.779573\pi$
$158$	33.2078	2.64187
$159$	−4.53414	−0.359580
$160$	0	0
$161$	−24.3439	−1.91857
$162$	26.3794	2.07256
$163$	1.77142	0.138749	0.0693743	−	0.997591i	$-0.477900\pi$
$163$	1.77142	0.138749	0.0693743	+	0.997591i	$0.477900\pi$
$164$	6.11176	0.477249
$165$	0	0
$166$	9.41999	0.731133
$167$	−0.0398907	−0.00308683	−0.00154342	−	0.999999i	$-0.500491\pi$
$167$	−0.0398907	−0.00308683	−0.00154342	+	0.999999i	$0.500491\pi$
$168$	−23.5695	−1.81842
$169$	−9.02383	−0.694141
$170$	0	0
$171$	−7.68027	−0.587325
$172$	36.6084	2.79137
$173$	−16.1574	−1.22842	−0.614210	−	0.789142i	$-0.710525\pi$
$173$	−16.1574	−1.22842	−0.614210	+	0.789142i	$0.710525\pi$
$174$	7.92379	0.600701
$175$	0	0
$176$	1.55981	0.117575
$177$	−15.7982	−1.18747
$178$	27.8219	2.08534
$179$	0.727604	0.0543837	0.0271918	−	0.999630i	$-0.491344\pi$
$179$	0.727604	0.0543837	0.0271918	+	0.999630i	$0.491344\pi$
$180$	0	0
$181$	13.7237	1.02008	0.510039	−	0.860151i	$-0.329631\pi$
$181$	13.7237	1.02008	0.510039	+	0.860151i	$0.329631\pi$
$182$	14.4946	1.07441
$183$	−7.60098	−0.561881
$184$	29.5200	2.17624
$185$	0	0
$186$	−33.7467	−2.47443
$187$	−2.93609	−0.214708
$188$	23.2607	1.69646
$189$	−11.2878	−0.821066
$190$	0	0
$191$	−12.2809	−0.888612	−0.444306	−	0.895875i	$-0.646550\pi$
$191$	−12.2809	−0.888612	−0.444306	+	0.895875i	$0.646550\pi$
$192$	−24.0236	−1.73376
$193$	9.92702	0.714563	0.357281	−	0.933997i	$-0.383704\pi$
$193$	9.92702	0.714563	0.357281	+	0.933997i	$0.383704\pi$
$194$	−41.7376	−2.99659
$195$	0	0
$196$	8.84070	0.631478
$197$	−12.1160	−0.863229	−0.431614	−	0.902058i	$-0.642056\pi$
$197$	−12.1160	−0.863229	−0.431614	+	0.902058i	$0.642056\pi$
$198$	−2.70013	−0.191890
$199$	13.7540	0.974995	0.487498	−	0.873124i	$-0.337910\pi$
$199$	13.7540	0.974995	0.487498	+	0.873124i	$0.337910\pi$
$200$	0	0
$201$	16.3478	1.15308
$202$	−21.3924	−1.50517
$203$	−5.02925	−0.352984
$204$	−22.8904	−1.60265
$205$	0	0
$206$	5.19932	0.362254
$207$	−9.59496	−0.666896
$208$	−3.30026	−0.228832
$209$	−5.96778	−0.412800
$210$	0	0
$211$	−18.0235	−1.24079	−0.620396	−	0.784289i	$-0.713028\pi$
$211$	−18.0235	−1.24079	−0.620396	+	0.784289i	$0.713028\pi$
$212$	−7.90781	−0.543111
$213$	−14.4094	−0.987316
$214$	−16.7381	−1.14419
$215$	0	0
$216$	13.6878	0.931339
$217$	21.4191	1.45402
$218$	17.8971	1.21214
$219$	12.5940	0.851022
$220$	0	0
$221$	6.21219	0.417877
$222$	−4.19036	−0.281239
$223$	27.9985	1.87492	0.937458	−	0.348098i	$-0.113172\pi$
$223$	27.9985	1.87492	0.937458	+	0.348098i	$0.113172\pi$
$224$	10.9356	0.730665
$225$	0	0
$226$	42.0385	2.79636
$227$	−12.7878	−0.848755	−0.424378	−	0.905485i	$-0.639507\pi$
$227$	−12.7878	−0.848755	−0.424378	+	0.905485i	$0.639507\pi$
$228$	−46.5262	−3.08127
$229$	−15.9201	−1.05203	−0.526015	−	0.850475i	$-0.676314\pi$
$229$	−15.9201	−1.05203	−0.526015	+	0.850475i	$0.676314\pi$
$230$	0	0
$231$	5.95270	0.391659
$232$	6.09859	0.400392
$233$	20.1234	1.31833	0.659165	−	0.751999i	$-0.270910\pi$
$233$	20.1234	1.31833	0.659165	+	0.751999i	$0.270910\pi$
$234$	5.71295	0.373467
$235$	0	0
$236$	−27.5531	−1.79355
$237$	−28.8551	−1.87434
$238$	22.6457	1.46790
$239$	4.68343	0.302946	0.151473	−	0.988461i	$-0.451598\pi$
$239$	4.68343	0.302946	0.151473	+	0.988461i	$0.451598\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	23.8855	1.53542
$243$	−11.9175	−0.764506
$244$	−13.2566	−0.848665
$245$	0	0
$246$	−8.27774	−0.527770
$247$	12.6267	0.803415
$248$	−25.9733	−1.64931
$249$	−8.18528	−0.518721
$250$	0	0
$251$	−23.0713	−1.45625	−0.728123	−	0.685447i	$-0.759607\pi$
$251$	−23.0713	−1.45625	−0.728123	+	0.685447i	$0.759607\pi$
$252$	13.3610	0.841664
$253$	−7.45555	−0.468727
$254$	45.9948	2.88597
$255$	0	0
$256$	−25.1102	−1.56939
$257$	−11.6020	−0.723715	−0.361857	−	0.932233i	$-0.617857\pi$
$257$	−11.6020	−0.723715	−0.361857	+	0.932233i	$0.617857\pi$
$258$	−49.5822	−3.08686
$259$	2.65963	0.165261
$260$	0	0
$261$	−1.98224	−0.122698
$262$	19.6259	1.21249
$263$	−15.3744	−0.948024	−0.474012	−	0.880518i	$-0.657195\pi$
$263$	−15.3744	−0.948024	−0.474012	+	0.880518i	$0.657195\pi$
$264$	−7.21838	−0.444261
$265$	0	0
$266$	46.0288	2.82221
$267$	−24.1752	−1.47950
$268$	28.5116	1.74162
$269$	9.03287	0.550744	0.275372	−	0.961338i	$-0.411199\pi$
$269$	9.03287	0.550744	0.275372	+	0.961338i	$0.411199\pi$
$270$	0	0
$271$	−25.0727	−1.52306	−0.761528	−	0.648132i	$-0.775550\pi$
$271$	−25.0727	−1.52306	−0.761528	+	0.648132i	$0.775550\pi$
$272$	−5.15617	−0.312639
$273$	−12.5947	−0.762269
$274$	−10.0784	−0.608856
$275$	0	0
$276$	−58.1252	−3.49873
$277$	10.5495	0.633855	0.316928	−	0.948450i	$-0.397349\pi$
$277$	10.5495	0.633855	0.316928	+	0.948450i	$0.397349\pi$
$278$	10.0556	0.603097
$279$	8.44218	0.505420
$280$	0	0
$281$	16.2094	0.966971	0.483485	−	0.875352i	$-0.339371\pi$
$281$	16.2094	0.966971	0.483485	+	0.875352i	$0.339371\pi$
$282$	−31.5041	−1.87604
$283$	8.03684	0.477741	0.238870	−	0.971051i	$-0.423223\pi$
$283$	8.03684	0.477741	0.238870	+	0.971051i	$0.423223\pi$
$284$	−25.1309	−1.49124
$285$	0	0
$286$	4.43912	0.262491
$287$	5.25390	0.310128
$288$	4.31018	0.253980
$289$	−7.29437	−0.429081
$290$	0	0
$291$	36.2669	2.12600
$292$	21.9647	1.28539
$293$	−26.1918	−1.53014	−0.765072	−	0.643945i	$-0.777297\pi$
$293$	−26.1918	−1.53014	−0.765072	+	0.643945i	$0.777297\pi$
$294$	−11.9738	−0.698326
$295$	0	0
$296$	−3.22513	−0.187457
$297$	−3.45700	−0.200595
$298$	−16.0665	−0.930709
$299$	15.7745	0.912262
$300$	0	0
$301$	31.4700	1.81390
$302$	9.72405	0.559556
$303$	18.5884	1.06788
$304$	−10.4802	−0.601083
$305$	0	0
$306$	8.92564	0.510245
$307$	−4.41238	−0.251828	−0.125914	−	0.992041i	$-0.540186\pi$
$307$	−4.41238	−0.251828	−0.125914	+	0.992041i	$0.540186\pi$
$308$	10.3819	0.591562
$309$	−4.51783	−0.257010
$310$	0	0
$311$	−13.2563	−0.751693	−0.375847	−	0.926682i	$-0.622648\pi$
$311$	−13.2563	−0.751693	−0.375847	+	0.926682i	$0.622648\pi$
$312$	15.2727	0.864646
$313$	−22.1456	−1.25174	−0.625871	−	0.779927i	$-0.715256\pi$
$313$	−22.1456	−1.25174	−0.625871	+	0.779927i	$0.715256\pi$
$314$	45.5601	2.57110
$315$	0	0
$316$	−50.3251	−2.83101
$317$	−3.01320	−0.169238	−0.0846190	−	0.996413i	$-0.526967\pi$
$317$	−3.01320	−0.169238	−0.0846190	+	0.996413i	$0.526967\pi$
$318$	10.7103	0.600604
$319$	−1.54026	−0.0862378
$320$	0	0
$321$	14.5441	0.811774
$322$	57.5038	3.20456
$323$	19.7273	1.09766
$324$	−39.9770	−2.22094
$325$	0	0
$326$	−4.18436	−0.231750
$327$	−15.5512	−0.859985
$328$	−6.37101	−0.351780
$329$	19.9957	1.10240
$330$	0	0
$331$	9.41939	0.517736	0.258868	−	0.965913i	$-0.416650\pi$
$331$	9.41939	0.517736	0.258868	+	0.965913i	$0.416650\pi$
$332$	−14.2756	−0.783477
$333$	1.04827	0.0574451
$334$	0.0942276	0.00515591
$335$	0	0
$336$	10.4537	0.570299
$337$	17.0177	0.927011	0.463505	−	0.886094i	$-0.346591\pi$
$337$	17.0177	0.927011	0.463505	+	0.886094i	$0.346591\pi$
$338$	21.3156	1.15942
$339$	−36.5284	−1.98395
$340$	0	0
$341$	6.55981	0.355234
$342$	18.1419	0.981003
$343$	−13.9412	−0.752752
$344$	−38.1612	−2.05752
$345$	0	0
$346$	38.1661	2.05182
$347$	−23.6966	−1.27210	−0.636049	−	0.771649i	$-0.719432\pi$
$347$	−23.6966	−1.27210	−0.636049	+	0.771649i	$0.719432\pi$
$348$	−12.0082	−0.643707
$349$	4.05897	0.217272	0.108636	−	0.994082i	$-0.465352\pi$
$349$	4.05897	0.217272	0.108636	+	0.994082i	$0.465352\pi$
$350$	0	0
$351$	7.31433	0.390410
$352$	3.34913	0.178509
$353$	−15.8788	−0.845144	−0.422572	−	0.906329i	$-0.638873\pi$
$353$	−15.8788	−0.845144	−0.422572	+	0.906329i	$0.638873\pi$
$354$	37.3177	1.98342
$355$	0	0
$356$	−42.1630	−2.23463
$357$	−19.6774	−1.04144
$358$	−1.71871	−0.0908366
$359$	−4.02791	−0.212585	−0.106293	−	0.994335i	$-0.533898\pi$
$359$	−4.02791	−0.212585	−0.106293	+	0.994335i	$0.533898\pi$
$360$	0	0
$361$	21.0970	1.11037
$362$	−32.4175	−1.70383
$363$	−20.7548	−1.08934
$364$	−21.9660	−1.15133
$365$	0	0
$366$	17.9546	0.938504
$367$	38.1519	1.99151	0.995757	−	0.0920267i	$-0.0293345\pi$
$367$	38.1519	1.99151	0.995757	+	0.0920267i	$0.0293345\pi$
$368$	−13.0930	−0.682518
$369$	2.07079	0.107801
$370$	0	0
$371$	−6.79785	−0.352927
$372$	51.1418	2.65158
$373$	20.3518	1.05377	0.526887	−	0.849935i	$-0.323359\pi$
$373$	20.3518	1.05377	0.526887	+	0.849935i	$0.323359\pi$
$374$	6.93547	0.358625
$375$	0	0
$376$	−24.2473	−1.25046
$377$	3.25888	0.167841
$378$	26.6634	1.37142
$379$	−30.2111	−1.55184	−0.775919	−	0.630833i	$-0.782713\pi$
$379$	−30.2111	−1.55184	−0.775919	+	0.630833i	$0.782713\pi$
$380$	0	0
$381$	−39.9661	−2.04752
$382$	29.0092	1.48424
$383$	−23.4619	−1.19885	−0.599425	−	0.800431i	$-0.704604\pi$
$383$	−23.4619	−1.19885	−0.599425	+	0.800431i	$0.704604\pi$
$384$	42.1594	2.15144
$385$	0	0
$386$	−23.4491	−1.19353
$387$	12.4037	0.630513
$388$	63.2517	3.21112
$389$	0.990019	0.0501959	0.0250980	−	0.999685i	$-0.492010\pi$
$389$	0.990019	0.0501959	0.0250980	+	0.999685i	$0.492010\pi$
$390$	0	0
$391$	24.6453	1.24637
$392$	−9.21569	−0.465463
$393$	−17.0534	−0.860231
$394$	28.6198	1.44184
$395$	0	0
$396$	4.09194	0.205628
$397$	4.90685	0.246268	0.123134	−	0.992390i	$-0.460706\pi$
$397$	4.90685	0.246268	0.123134	+	0.992390i	$0.460706\pi$
$398$	−32.4890	−1.62853
$399$	−39.9956	−2.00229
$400$	0	0
$401$	−30.6157	−1.52888	−0.764439	−	0.644696i	$-0.776984\pi$
$401$	−30.6157	−1.52888	−0.764439	+	0.644696i	$0.776984\pi$
$402$	−38.6159	−1.92599
$403$	−13.8793	−0.691376
$404$	32.4194	1.61292
$405$	0	0
$406$	11.8798	0.589586
$407$	0.814538	0.0403752
$408$	23.8613	1.18131
$409$	−4.00516	−0.198043	−0.0990213	−	0.995085i	$-0.531571\pi$
$409$	−4.00516	−0.198043	−0.0990213	+	0.995085i	$0.531571\pi$
$410$	0	0
$411$	8.75735	0.431968
$412$	−7.87937	−0.388189
$413$	−23.6856	−1.16549
$414$	22.6647	1.11391
$415$	0	0
$416$	−7.08611	−0.347425
$417$	−8.73761	−0.427883
$418$	14.0968	0.689496
$419$	31.4086	1.53441	0.767206	−	0.641401i	$-0.221646\pi$
$419$	31.4086	1.53441	0.767206	+	0.641401i	$0.221646\pi$
$420$	0	0
$421$	39.1136	1.90628	0.953141	−	0.302526i	$-0.0978298\pi$
$421$	39.1136	1.90628	0.953141	+	0.302526i	$0.0978298\pi$
$422$	42.5743	2.07248
$423$	7.88117	0.383196
$424$	8.24323	0.400327
$425$	0	0
$426$	34.0372	1.64911
$427$	−11.3958	−0.551484
$428$	25.3659	1.22610
$429$	−3.85727	−0.186231
$430$	0	0
$431$	7.68918	0.370375	0.185188	−	0.982703i	$-0.440711\pi$
$431$	7.68918	0.370375	0.185188	+	0.982703i	$0.440711\pi$
$432$	−6.07096	−0.292089
$433$	4.67921	0.224869	0.112434	−	0.993659i	$-0.464135\pi$
$433$	4.67921	0.224869	0.112434	+	0.993659i	$0.464135\pi$
$434$	−50.5951	−2.42864
$435$	0	0
$436$	−27.1223	−1.29892
$437$	50.0932	2.39628
$438$	−29.7488	−1.42145
$439$	14.7770	0.705270	0.352635	−	0.935761i	$-0.385286\pi$
$439$	14.7770	0.705270	0.352635	+	0.935761i	$0.385286\pi$
$440$	0	0
$441$	2.99540	0.142638
$442$	−14.6741	−0.697976
$443$	−3.71495	−0.176503	−0.0882513	−	0.996098i	$-0.528128\pi$
$443$	−3.71495	−0.176503	−0.0882513	+	0.996098i	$0.528128\pi$
$444$	6.35033	0.301373
$445$	0	0
$446$	−66.1365	−3.13165
$447$	13.9606	0.660315
$448$	−36.0177	−1.70168
$449$	−23.6261	−1.11498	−0.557492	−	0.830183i	$-0.688236\pi$
$449$	−23.6261	−1.11498	−0.557492	+	0.830183i	$0.688236\pi$
$450$	0	0
$451$	1.60906	0.0757676
$452$	−63.7077	−2.99656
$453$	−8.44948	−0.396991
$454$	30.2066	1.41767
$455$	0	0
$456$	48.4997	2.27120
$457$	16.0363	0.750146	0.375073	−	0.926995i	$-0.377618\pi$
$457$	16.0363	0.750146	0.375073	+	0.926995i	$0.377618\pi$
$458$	37.6056	1.75720
$459$	11.4276	0.533393
$460$	0	0
$461$	23.0734	1.07464	0.537319	−	0.843379i	$-0.319437\pi$
$461$	23.0734	1.07464	0.537319	+	0.843379i	$0.319437\pi$
$462$	−14.0612	−0.654184
$463$	34.0986	1.58470	0.792349	−	0.610068i	$-0.208858\pi$
$463$	34.0986	1.58470	0.792349	+	0.610068i	$0.208858\pi$
$464$	−2.70490	−0.125572
$465$	0	0
$466$	−47.5345	−2.20199
$467$	15.2453	0.705467	0.352734	−	0.935724i	$-0.385252\pi$
$467$	15.2453	0.705467	0.352734	+	0.935724i	$0.385252\pi$
$468$	−8.65774	−0.400204
$469$	24.5096	1.13175
$470$	0	0
$471$	−39.5883	−1.82413
$472$	28.7218	1.32203
$473$	9.63798	0.443155
$474$	68.1601	3.13070
$475$	0	0
$476$	−34.3187	−1.57299
$477$	−2.67932	−0.122678
$478$	−11.0630	−0.506008
$479$	41.7622	1.90816	0.954082	−	0.299546i	$-0.0968351\pi$
$479$	41.7622	1.90816	0.954082	+	0.299546i	$0.0968351\pi$
$480$	0	0
$481$	−1.72340	−0.0785805
$482$	−2.36215	−0.107593
$483$	−49.9666	−2.27356
$484$	−36.1976	−1.64535
$485$	0	0
$486$	28.1508	1.27695
$487$	33.0645	1.49830	0.749148	−	0.662402i	$-0.230463\pi$
$487$	33.0645	1.49830	0.749148	+	0.662402i	$0.230463\pi$
$488$	13.8189	0.625551
$489$	3.63590	0.164421
$490$	0	0
$491$	−12.6839	−0.572415	−0.286208	−	0.958168i	$-0.592395\pi$
$491$	−12.6839	−0.572415	−0.286208	+	0.958168i	$0.592395\pi$
$492$	12.5446	0.565554
$493$	5.09153	0.229311
$494$	−29.8260	−1.34194
$495$	0	0
$496$	11.5199	0.517260
$497$	−21.6035	−0.969047
$498$	19.3348	0.866415
$499$	−15.4405	−0.691213	−0.345606	−	0.938380i	$-0.612327\pi$
$499$	−15.4405	−0.691213	−0.345606	+	0.938380i	$0.612327\pi$
$500$	0	0
$501$	−0.0818768	−0.00365799
$502$	54.4977	2.43235
$503$	26.3951	1.17690	0.588450	−	0.808534i	$-0.299738\pi$
$503$	26.3951	1.17690	0.588450	+	0.808534i	$0.299738\pi$
$504$	−13.9277	−0.620390
$505$	0	0
$506$	17.6111	0.782910
$507$	−18.5217	−0.822577
$508$	−69.7033	−3.09258
$509$	13.4761	0.597318	0.298659	−	0.954360i	$-0.403461\pi$
$509$	13.4761	0.597318	0.298659	+	0.954360i	$0.403461\pi$
$510$	0	0
$511$	18.8817	0.835275
$512$	18.2336	0.805817
$513$	23.2272	1.02551
$514$	27.4057	1.20881
$515$	0	0
$516$	75.1399	3.30785
$517$	6.12389	0.269328
$518$	−6.28244	−0.276035
$519$	−33.1635	−1.45572
$520$	0	0
$521$	−23.7045	−1.03851	−0.519256	−	0.854619i	$-0.673791\pi$
$521$	−23.7045	−1.03851	−0.519256	+	0.854619i	$0.673791\pi$
$522$	4.68235	0.204941
$523$	−3.98543	−0.174271	−0.0871354	−	0.996196i	$-0.527771\pi$
$523$	−3.98543	−0.174271	−0.0871354	+	0.996196i	$0.527771\pi$
$524$	−29.7422	−1.29929
$525$	0	0
$526$	36.3165	1.58348
$527$	−21.6843	−0.944584
$528$	3.20156	0.139330
$529$	39.5814	1.72093
$530$	0	0
$531$	−9.33553	−0.405127
$532$	−69.7548	−3.02426
$533$	−3.40445	−0.147463
$534$	57.1053	2.47119
$535$	0	0
$536$	−29.7209	−1.28375
$537$	1.49343	0.0644463
$538$	−21.3370	−0.919902
$539$	2.32751	0.100253
$540$	0	0
$541$	14.3140	0.615405	0.307703	−	0.951483i	$-0.400440\pi$
$541$	14.3140	0.615405	0.307703	+	0.951483i	$0.400440\pi$
$542$	59.2253	2.54394
$543$	28.1684	1.20882
$544$	−11.0710	−0.474666
$545$	0	0
$546$	29.7506	1.27321
$547$	25.1893	1.07702	0.538508	−	0.842621i	$-0.318988\pi$
$547$	25.1893	1.07702	0.538508	+	0.842621i	$0.318988\pi$
$548$	15.2734	0.652446
$549$	−4.49159	−0.191696
$550$	0	0
$551$	10.3488	0.440876
$552$	60.5907	2.57891
$553$	−43.2613	−1.83966
$554$	−24.9194	−1.05872
$555$	0	0
$556$	−15.2389	−0.646275
$557$	−16.4450	−0.696798	−0.348399	−	0.937346i	$-0.613275\pi$
$557$	−16.4450	−0.696798	−0.348399	+	0.937346i	$0.613275\pi$
$558$	−19.9417	−0.844199
$559$	−20.3921	−0.862494
$560$	0	0
$561$	−6.02641	−0.254435
$562$	−38.2890	−1.61512
$563$	14.8796	0.627100	0.313550	−	0.949572i	$-0.398482\pi$
$563$	14.8796	0.627100	0.313550	+	0.949572i	$0.398482\pi$
$564$	47.7433	2.01035
$565$	0	0
$566$	−18.9842	−0.797966
$567$	−34.3657	−1.44322
$568$	26.1969	1.09920
$569$	−37.4832	−1.57138	−0.785689	−	0.618622i	$-0.787691\pi$
$569$	−37.4832	−1.57138	−0.785689	+	0.618622i	$0.787691\pi$
$570$	0	0
$571$	1.58161	0.0661885	0.0330943	−	0.999452i	$-0.489464\pi$
$571$	1.58161	0.0661885	0.0330943	+	0.999452i	$0.489464\pi$
$572$	−6.72731	−0.281283
$573$	−25.2069	−1.05303
$574$	−12.4105	−0.518004
$575$	0	0
$576$	−14.1961	−0.591504
$577$	−37.7122	−1.56998	−0.784990	−	0.619508i	$-0.787332\pi$
$577$	−37.7122	−1.56998	−0.784990	+	0.619508i	$0.787332\pi$
$578$	17.2304	0.716689
$579$	20.3755	0.846778
$580$	0	0
$581$	−12.2719	−0.509123
$582$	−85.6678	−3.55105
$583$	−2.08191	−0.0862238
$584$	−22.8963	−0.947457
$585$	0	0
$586$	61.8690	2.55578
$587$	−27.6567	−1.14151	−0.570756	−	0.821120i	$-0.693350\pi$
$587$	−27.6567	−1.14151	−0.570756	+	0.821120i	$0.693350\pi$
$588$	18.1458	0.748321
$589$	−44.0748	−1.81607
$590$	0	0
$591$	−24.8685	−1.02295
$592$	1.43044	0.0587907
$593$	18.1987	0.747329	0.373665	−	0.927564i	$-0.378101\pi$
$593$	18.1987	0.747329	0.373665	+	0.927564i	$0.378101\pi$
$594$	8.16593	0.335052
$595$	0	0
$596$	24.3482	0.997341
$597$	28.2305	1.15540
$598$	−37.2617	−1.52374
$599$	−7.12390	−0.291075	−0.145537	−	0.989353i	$-0.546491\pi$
$599$	−7.12390	−0.291075	−0.145537	+	0.989353i	$0.546491\pi$
$600$	0	0
$601$	11.3425	0.462672	0.231336	−	0.972874i	$-0.425690\pi$
$601$	11.3425	0.462672	0.231336	+	0.972874i	$0.425690\pi$
$602$	−74.3367	−3.02974
$603$	9.66028	0.393397
$604$	−14.7364	−0.599616
$605$	0	0
$606$	−43.9086	−1.78367
$607$	−9.88248	−0.401117	−0.200559	−	0.979682i	$-0.564276\pi$
$607$	−9.88248	−0.401117	−0.200559	+	0.979682i	$0.564276\pi$
$608$	−22.5025	−0.912598
$609$	−10.3227	−0.418297
$610$	0	0
$611$	−12.9570	−0.524182
$612$	−13.5265	−0.546775
$613$	6.32265	0.255369	0.127685	−	0.991815i	$-0.459245\pi$
$613$	6.32265	0.255369	0.127685	+	0.991815i	$0.459245\pi$
$614$	10.4227	0.420626
$615$	0	0
$616$	−10.8222	−0.436040
$617$	−28.4003	−1.14335	−0.571676	−	0.820480i	$-0.693707\pi$
$617$	−28.4003	−1.14335	−0.571676	+	0.820480i	$0.693707\pi$
$618$	10.6718	0.429282
$619$	14.3962	0.578633	0.289316	−	0.957234i	$-0.406572\pi$
$619$	14.3962	0.578633	0.289316	+	0.957234i	$0.406572\pi$
$620$	0	0
$621$	29.0178	1.16444
$622$	31.3132	1.25555
$623$	−36.2449	−1.45212
$624$	−6.77388	−0.271172
$625$	0	0
$626$	52.3111	2.09077
$627$	−12.2491	−0.489180
$628$	−69.0445	−2.75517
$629$	−2.69257	−0.107360
$630$	0	0
$631$	−16.4378	−0.654379	−0.327190	−	0.944959i	$-0.606102\pi$
$631$	−16.4378	−0.654379	−0.327190	+	0.944959i	$0.606102\pi$
$632$	52.4597	2.08674
$633$	−36.9939	−1.47038
$634$	7.11761	0.282677
$635$	0	0
$636$	−16.2310	−0.643602
$637$	−4.92456	−0.195118
$638$	3.63831	0.144042
$639$	−8.51484	−0.336842
$640$	0	0
$641$	9.38495	0.370683	0.185342	−	0.982674i	$-0.440661\pi$
$641$	9.38495	0.370683	0.185342	+	0.982674i	$0.440661\pi$
$642$	−34.3554	−1.35590
$643$	−2.90484	−0.114556	−0.0572778	−	0.998358i	$-0.518242\pi$
$643$	−2.90484	−0.114556	−0.0572778	+	0.998358i	$0.518242\pi$
$644$	−87.1448	−3.43399
$645$	0	0
$646$	−46.5988	−1.83341
$647$	−16.2707	−0.639667	−0.319833	−	0.947474i	$-0.603627\pi$
$647$	−16.2707	−0.639667	−0.319833	+	0.947474i	$0.603627\pi$
$648$	41.6727	1.63706
$649$	−7.25396	−0.284743
$650$	0	0
$651$	43.9634	1.72306
$652$	6.34124	0.248342
$653$	−11.7812	−0.461035	−0.230517	−	0.973068i	$-0.574042\pi$
$653$	−11.7812	−0.461035	−0.230517	+	0.973068i	$0.574042\pi$
$654$	36.7343	1.43643
$655$	0	0
$656$	2.82573	0.110326
$657$	7.44207	0.290343
$658$	−47.2329	−1.84133
$659$	47.0426	1.83252	0.916260	−	0.400583i	$-0.131193\pi$
$659$	47.0426	1.83252	0.916260	+	0.400583i	$0.131193\pi$
$660$	0	0
$661$	−37.3779	−1.45383	−0.726916	−	0.686726i	$-0.759047\pi$
$661$	−37.3779	−1.45383	−0.726916	+	0.686726i	$0.759047\pi$
$662$	−22.2500	−0.864770
$663$	12.7507	0.495197
$664$	14.8812	0.577501
$665$	0	0
$666$	−2.47618	−0.0959500
$667$	12.9288	0.500606
$668$	−0.142798	−0.00552503
$669$	57.4678	2.22183
$670$	0	0
$671$	−3.49009	−0.134733
$672$	22.4456	0.865860
$673$	37.3014	1.43786	0.718931	−	0.695081i	$-0.244632\pi$
$673$	37.3014	1.43786	0.718931	+	0.695081i	$0.244632\pi$
$674$	−40.1982	−1.54838
$675$	0	0
$676$	−32.3030	−1.24242
$677$	−21.0921	−0.810636	−0.405318	−	0.914176i	$-0.632839\pi$
$677$	−21.0921	−0.810636	−0.405318	+	0.914176i	$0.632839\pi$
$678$	86.2854	3.31377
$679$	54.3735	2.08667
$680$	0	0
$681$	−26.2473	−1.00580
$682$	−15.4952	−0.593343
$683$	30.1078	1.15204	0.576022	−	0.817434i	$-0.304604\pi$
$683$	30.1078	1.15204	0.576022	+	0.817434i	$0.304604\pi$
$684$	−27.4934	−1.05124
$685$	0	0
$686$	32.9311	1.25731
$687$	−32.6765	−1.24669
$688$	16.9256	0.645283
$689$	4.40491	0.167814
$690$	0	0
$691$	−22.9001	−0.871162	−0.435581	−	0.900149i	$-0.643457\pi$
$691$	−22.9001	−0.871162	−0.435581	+	0.900149i	$0.643457\pi$
$692$	−57.8391	−2.19871
$693$	3.51758	0.133622
$694$	55.9748	2.12477
$695$	0	0
$696$	12.5175	0.474476
$697$	−5.31896	−0.201470
$698$	−9.58790	−0.362907
$699$	41.3040	1.56226
$700$	0	0
$701$	−12.7502	−0.481567	−0.240784	−	0.970579i	$-0.577404\pi$
$701$	−12.7502	−0.481567	−0.240784	+	0.970579i	$0.577404\pi$
$702$	−17.2775	−0.652098
$703$	−5.47281	−0.206411
$704$	−11.0308	−0.415738
$705$	0	0
$706$	37.5081	1.41164
$707$	27.8689	1.04812
$708$	−56.5536	−2.12541
$709$	−30.5429	−1.14706	−0.573531	−	0.819184i	$-0.694427\pi$
$709$	−30.5429	−1.14706	−0.573531	+	0.819184i	$0.694427\pi$
$710$	0	0
$711$	−17.0511	−0.639468
$712$	43.9514	1.64715
$713$	−55.0626	−2.06211
$714$	46.4810	1.73951
$715$	0	0
$716$	2.60463	0.0973398
$717$	9.61289	0.359000
$718$	9.51453	0.355079
$719$	−4.70422	−0.175438	−0.0877190	−	0.996145i	$-0.527958\pi$
$719$	−4.70422	−0.175438	−0.0877190	+	0.996145i	$0.527958\pi$
$720$	0	0
$721$	−6.77340	−0.252255
$722$	−49.8342	−1.85464
$723$	2.05253	0.0763345
$724$	49.1275	1.82581
$725$	0	0
$726$	49.0259	1.81952
$727$	−34.0042	−1.26114	−0.630572	−	0.776130i	$-0.717180\pi$
$727$	−34.0042	−1.26114	−0.630572	+	0.776130i	$0.717180\pi$
$728$	22.8977	0.848646
$729$	9.04171	0.334878
$730$	0	0
$731$	−31.8597	−1.17837
$732$	−27.2096	−1.00569
$733$	−16.4990	−0.609406	−0.304703	−	0.952447i	$-0.598557\pi$
$733$	−16.4990	−0.609406	−0.304703	+	0.952447i	$0.598557\pi$
$734$	−90.1204	−3.32641
$735$	0	0
$736$	−28.1124	−1.03624
$737$	7.50631	0.276498
$738$	−4.89150	−0.180059
$739$	41.0735	1.51091	0.755456	−	0.655199i	$-0.227415\pi$
$739$	41.0735	1.51091	0.755456	+	0.655199i	$0.227415\pi$
$740$	0	0
$741$	25.9166	0.952071
$742$	16.0575	0.589490
$743$	−6.32830	−0.232163	−0.116081	−	0.993240i	$-0.537033\pi$
$743$	−6.32830	−0.232163	−0.116081	+	0.993240i	$0.537033\pi$
$744$	−53.3110	−1.95448
$745$	0	0
$746$	−48.0739	−1.76011
$747$	−4.83687	−0.176972
$748$	−10.5104	−0.384299
$749$	21.8054	0.796753
$750$	0	0
$751$	53.2850	1.94440	0.972199	−	0.234157i	$-0.0752331\pi$
$751$	53.2850	1.94440	0.972199	+	0.234157i	$0.0752331\pi$
$752$	10.7544	0.392172
$753$	−47.3545	−1.72569
$754$	−7.69796	−0.280343
$755$	0	0
$756$	−40.4074	−1.46960
$757$	−38.5378	−1.40068	−0.700340	−	0.713810i	$-0.746968\pi$
$757$	−38.5378	−1.40068	−0.700340	+	0.713810i	$0.746968\pi$
$758$	71.3630	2.59202
$759$	−15.3028	−0.555455
$760$	0	0
$761$	−16.2682	−0.589723	−0.294861	−	0.955540i	$-0.595273\pi$
$761$	−16.2682	−0.589723	−0.294861	+	0.955540i	$0.595273\pi$
$762$	94.4057	3.41996
$763$	−23.3153	−0.844072
$764$	−43.9623	−1.59050
$765$	0	0
$766$	55.4206	2.00243
$767$	15.3480	0.554183
$768$	−51.5394	−1.85977
$769$	−35.5516	−1.28202	−0.641012	−	0.767531i	$-0.721485\pi$
$769$	−35.5516	−1.28202	−0.641012	+	0.767531i	$0.721485\pi$
$770$	0	0
$771$	−23.8135	−0.857623
$772$	35.5362	1.27898
$773$	0.807556	0.0290458	0.0145229	−	0.999895i	$-0.495377\pi$
$773$	0.807556	0.0290458	0.0145229	+	0.999895i	$0.495377\pi$
$774$	−29.2993	−1.05314
$775$	0	0
$776$	−65.9347	−2.36692
$777$	5.45898	0.195840
$778$	−2.33857	−0.0838418
$779$	−10.8111	−0.387349
$780$	0	0
$781$	−6.61627	−0.236749
$782$	−58.2159	−2.08180
$783$	5.99484	0.214238
$784$	4.08743	0.145980
$785$	0	0
$786$	40.2827	1.43684
$787$	13.8726	0.494504	0.247252	−	0.968951i	$-0.420472\pi$
$787$	13.8726	0.494504	0.247252	+	0.968951i	$0.420472\pi$
$788$	−43.3721	−1.54507
$789$	−31.5564	−1.12344
$790$	0	0
$791$	−54.7655	−1.94724
$792$	−4.26551	−0.151568
$793$	7.38435	0.262226
$794$	−11.5907	−0.411338
$795$	0	0
$796$	49.2358	1.74512
$797$	3.24320	0.114880	0.0574400	−	0.998349i	$-0.481706\pi$
$797$	3.24320	0.114880	0.0574400	+	0.998349i	$0.481706\pi$
$798$	94.4756	3.34440
$799$	−20.2433	−0.716158
$800$	0	0
$801$	−14.2857	−0.504759
$802$	72.3189	2.55367
$803$	5.78269	0.204067
$804$	58.5209	2.06387
$805$	0	0
$806$	32.7849	1.15480
$807$	18.5403	0.652648
$808$	−33.7945	−1.18889
$809$	54.4503	1.91437	0.957184	−	0.289479i	$-0.0934819\pi$
$809$	54.4503	1.91437	0.957184	+	0.289479i	$0.0934819\pi$
$810$	0	0
$811$	−20.6483	−0.725059	−0.362530	−	0.931972i	$-0.618087\pi$
$811$	−20.6483	−0.725059	−0.362530	+	0.931972i	$0.618087\pi$
$812$	−18.0034	−0.631796
$813$	−51.4624	−1.80487
$814$	−1.92406	−0.0674383
$815$	0	0
$816$	−10.5832	−0.370486
$817$	−64.7567	−2.26555
$818$	9.46079	0.330789
$819$	−7.44252	−0.260063
$820$	0	0
$821$	20.0000	0.698006	0.349003	−	0.937122i	$-0.386520\pi$
$821$	20.0000	0.698006	0.349003	+	0.937122i	$0.386520\pi$
$822$	−20.6862	−0.721513
$823$	25.3107	0.882277	0.441138	−	0.897439i	$-0.354575\pi$
$823$	25.3107	0.882277	0.441138	+	0.897439i	$0.354575\pi$
$824$	8.21359	0.286134
$825$	0	0
$826$	55.9490	1.94671
$827$	28.9282	1.00593	0.502966	−	0.864306i	$-0.332242\pi$
$827$	28.9282	1.00593	0.502966	+	0.864306i	$0.332242\pi$
$828$	−34.3475	−1.19366
$829$	10.9469	0.380201	0.190100	−	0.981765i	$-0.439119\pi$
$829$	10.9469	0.380201	0.190100	+	0.981765i	$0.439119\pi$
$830$	0	0
$831$	21.6531	0.751137
$832$	23.3390	0.809133
$833$	−7.69390	−0.266578
$834$	20.6395	0.714689
$835$	0	0
$836$	−21.3631	−0.738859
$837$	−25.5315	−0.882497
$838$	−74.1918	−2.56291
$839$	16.6348	0.574298	0.287149	−	0.957886i	$-0.407293\pi$
$839$	16.6348	0.574298	0.287149	+	0.957886i	$0.407293\pi$
$840$	0	0
$841$	−26.3290	−0.907897
$842$	−92.3922	−3.18405
$843$	33.2703	1.14589
$844$	−64.5196	−2.22086
$845$	0	0
$846$	−18.6165	−0.640048
$847$	−31.1168	−1.06919
$848$	−3.65611	−0.125552
$849$	16.4959	0.566137
$850$	0	0
$851$	−6.83719	−0.234376
$852$	−51.5820	−1.76717
$853$	−1.31481	−0.0450182	−0.0225091	−	0.999747i	$-0.507165\pi$
$853$	−1.31481	−0.0450182	−0.0225091	+	0.999747i	$0.507165\pi$
$854$	26.9187	0.921138
$855$	0	0
$856$	−26.4418	−0.903762
$857$	−51.6669	−1.76491	−0.882454	−	0.470399i	$-0.844110\pi$
$857$	−51.6669	−1.76491	−0.882454	+	0.470399i	$0.844110\pi$
$858$	9.11143	0.311059
$859$	−39.7084	−1.35483	−0.677417	−	0.735599i	$-0.736901\pi$
$859$	−39.7084	−1.35483	−0.677417	+	0.735599i	$0.736901\pi$
$860$	0	0
$861$	10.7838	0.367511
$862$	−18.1630	−0.618634
$863$	28.5307	0.971196	0.485598	−	0.874182i	$-0.338602\pi$
$863$	28.5307	0.971196	0.485598	+	0.874182i	$0.338602\pi$
$864$	−13.0352	−0.443466
$865$	0	0
$866$	−11.0530	−0.375596
$867$	−14.9719	−0.508473
$868$	76.6748	2.60251
$869$	−13.2492	−0.449449
$870$	0	0
$871$	−15.8819	−0.538137
$872$	28.2727	0.957436
$873$	21.4309	0.725327
$874$	−118.327	−4.00249
$875$	0	0
$876$	45.0832	1.52322
$877$	4.60926	0.155644	0.0778219	−	0.996967i	$-0.475203\pi$
$877$	4.60926	0.155644	0.0778219	+	0.996967i	$0.475203\pi$
$878$	−34.9055	−1.17801
$879$	−53.7596	−1.81327
$880$	0	0
$881$	−14.5627	−0.490630	−0.245315	−	0.969443i	$-0.578891\pi$
$881$	−14.5627	−0.490630	−0.245315	+	0.969443i	$0.578891\pi$
$882$	−7.07559	−0.238247
$883$	−4.65262	−0.156573	−0.0782866	−	0.996931i	$-0.524945\pi$
$883$	−4.65262	−0.156573	−0.0782866	+	0.996931i	$0.524945\pi$
$884$	22.2380	0.747946
$885$	0	0
$886$	8.77526	0.294810
$887$	8.77283	0.294563	0.147281	−	0.989095i	$-0.452948\pi$
$887$	8.77283	0.294563	0.147281	+	0.989095i	$0.452948\pi$
$888$	−6.61969	−0.222142
$889$	−59.9195	−2.00964
$890$	0	0
$891$	−10.5248	−0.352595
$892$	100.227	3.35586
$893$	−41.1459	−1.37689
$894$	−32.9771	−1.10292
$895$	0	0
$896$	63.2079	2.11163
$897$	32.3776	1.08106
$898$	55.8083	1.86235
$899$	−11.3755	−0.379394
$900$	0	0
$901$	6.88203	0.229274
$902$	−3.80083	−0.126554
$903$	64.5931	2.14952
$904$	66.4100	2.20876
$905$	0	0
$906$	19.9589	0.663091
$907$	−17.5415	−0.582457	−0.291229	−	0.956653i	$-0.594064\pi$
$907$	−17.5415	−0.582457	−0.291229	+	0.956653i	$0.594064\pi$
$908$	−45.7770	−1.51916
$909$	10.9843	0.364327
$910$	0	0
$911$	−26.2299	−0.869034	−0.434517	−	0.900664i	$-0.643081\pi$
$911$	−26.2299	−0.869034	−0.434517	+	0.900664i	$0.643081\pi$
$912$	−21.5110	−0.712301
$913$	−3.75838	−0.124384
$914$	−37.8801	−1.25296
$915$	0	0
$916$	−56.9898	−1.88300
$917$	−25.5675	−0.844314
$918$	−26.9936	−0.890921
$919$	−26.0740	−0.860101	−0.430051	−	0.902805i	$-0.641504\pi$
$919$	−26.0740	−0.860101	−0.430051	+	0.902805i	$0.641504\pi$
$920$	0	0
$921$	−9.05655	−0.298424
$922$	−54.5029	−1.79496
$923$	13.9987	0.460774
$924$	21.3091	0.701019
$925$	0	0
$926$	−80.5460	−2.64691
$927$	−2.66969	−0.0876840
$928$	−5.80780	−0.190650
$929$	22.2068	0.728582	0.364291	−	0.931285i	$-0.381311\pi$
$929$	22.2068	0.728582	0.364291	+	0.931285i	$0.381311\pi$
$930$	0	0
$931$	−15.6383	−0.512526
$932$	72.0367	2.35964
$933$	−27.2089	−0.890779
$934$	−36.0116	−1.17834
$935$	0	0
$936$	9.02498	0.294991
$937$	−39.4539	−1.28890	−0.644452	−	0.764645i	$-0.722914\pi$
$937$	−39.4539	−1.28890	−0.644452	+	0.764645i	$0.722914\pi$
$938$	−57.8953	−1.89035
$939$	−45.4545	−1.48335
$940$	0	0
$941$	7.12527	0.232277	0.116139	−	0.993233i	$-0.462948\pi$
$941$	7.12527	0.232277	0.116139	+	0.993233i	$0.462948\pi$
$942$	93.5135	3.04683
$943$	−13.5063	−0.439827
$944$	−12.7389	−0.414617
$945$	0	0
$946$	−22.7663	−0.740198
$947$	47.2038	1.53392	0.766959	−	0.641696i	$-0.221769\pi$
$947$	47.2038	1.53392	0.766959	+	0.641696i	$0.221769\pi$
$948$	−103.294	−3.35483
$949$	−12.2350	−0.397166
$950$	0	0
$951$	−6.18468	−0.200552
$952$	35.7744	1.15945
$953$	−6.73058	−0.218025	−0.109013	−	0.994040i	$-0.534769\pi$
$953$	−6.73058	−0.218025	−0.109013	+	0.994040i	$0.534769\pi$
$954$	6.32896	0.204908
$955$	0	0
$956$	16.7655	0.542234
$957$	−3.16143	−0.102194
$958$	−98.6485	−3.18719
$959$	13.1296	0.423975
$960$	0	0
$961$	17.4472	0.562812
$962$	4.07094	0.131252
$963$	8.59446	0.276952
$964$	3.57974	0.115296
$965$	0	0
$966$	118.028	3.79750
$967$	51.9356	1.67014	0.835069	−	0.550146i	$-0.185428\pi$
$967$	51.9356	1.67014	0.835069	+	0.550146i	$0.185428\pi$
$968$	37.7330	1.21278
$969$	40.4909	1.30076
$970$	0	0
$971$	21.5150	0.690451	0.345225	−	0.938520i	$-0.387803\pi$
$971$	21.5150	0.690451	0.345225	+	0.938520i	$0.387803\pi$
$972$	−42.6615	−1.36837
$973$	−13.1000	−0.419965
$974$	−78.1033	−2.50259
$975$	0	0
$976$	−6.12908	−0.196187
$977$	1.00409	0.0321238	0.0160619	−	0.999871i	$-0.494887\pi$
$977$	1.00409	0.0321238	0.0160619	+	0.999871i	$0.494887\pi$
$978$	−8.58854	−0.274631
$979$	−11.1004	−0.354769
$980$	0	0
$981$	−9.18958	−0.293401
$982$	29.9612	0.956100
$983$	29.7602	0.949203	0.474601	−	0.880201i	$-0.342592\pi$
$983$	29.7602	0.949203	0.474601	+	0.880201i	$0.342592\pi$
$984$	−13.0767	−0.416870
$985$	0	0
$986$	−12.0269	−0.383016
$987$	41.0419	1.30638
$988$	45.2002	1.43801
$989$	−80.9007	−2.57249
$990$	0	0
$991$	−17.2504	−0.547976	−0.273988	−	0.961733i	$-0.588343\pi$
$991$	−17.2504	−0.547976	−0.273988	+	0.961733i	$0.588343\pi$
$992$	24.7349	0.785333
$993$	19.3336	0.613533
$994$	51.0305	1.61859
$995$	0	0
$996$	−29.3012	−0.928444
$997$	36.3908	1.15251	0.576254	−	0.817271i	$-0.304514\pi$
$997$	36.3908	1.15251	0.576254	+	0.817271i	$0.304514\pi$
$998$	36.4728	1.15453
$999$	−3.17027	−0.100303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.5		46
5.2	odd	4		1205.2.b.c.724.5	✓	46
5.3	odd	4		1205.2.b.c.724.42	yes	46
5.4	even	2	inner	6025.2.a.p.1.42		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.5	✓	46	5.2	odd	4
1205.2.b.c.724.42	yes	46	5.3	odd	4
6025.2.a.p.1.5		46	1.1	even	1	trivial
6025.2.a.p.1.42		46	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)

\(f(q)\)	\(=\)	\(q-2.36215 q^{2} +2.05253 q^{3} +3.57974 q^{4} -4.84838 q^{6} +3.07728 q^{7} -3.73158 q^{8} +1.21289 q^{9} +O(q^{10})\)	\(q-2.36215 q^{2} +2.05253 q^{3} +3.57974 q^{4} -4.84838 q^{6} +3.07728 q^{7} -3.73158 q^{8} +1.21289 q^{9} +0.942447 q^{11} +7.34753 q^{12} -1.99403 q^{13} -7.26899 q^{14} +1.65507 q^{16} -3.11539 q^{17} -2.86502 q^{18} -6.33222 q^{19} +6.31621 q^{21} -2.22620 q^{22} -7.91084 q^{23} -7.65919 q^{24} +4.71020 q^{26} -3.66811 q^{27} +11.0159 q^{28} -1.63432 q^{29} +6.96040 q^{31} +3.55366 q^{32} +1.93440 q^{33} +7.35900 q^{34} +4.34182 q^{36} +0.864280 q^{37} +14.9576 q^{38} -4.09282 q^{39} +1.70732 q^{41} -14.9198 q^{42} +10.2266 q^{43} +3.37372 q^{44} +18.6866 q^{46} +6.49786 q^{47} +3.39707 q^{48} +2.46965 q^{49} -6.39443 q^{51} -7.13813 q^{52} -2.20905 q^{53} +8.66461 q^{54} -11.4831 q^{56} -12.9971 q^{57} +3.86050 q^{58} -7.69694 q^{59} -3.70322 q^{61} -16.4415 q^{62} +3.73239 q^{63} -11.7044 q^{64} -4.56935 q^{66} +7.96470 q^{67} -11.1523 q^{68} -16.2373 q^{69} -7.02031 q^{71} -4.52599 q^{72} +6.13583 q^{73} -2.04156 q^{74} -22.6677 q^{76} +2.90017 q^{77} +9.66784 q^{78} -14.0583 q^{79} -11.1676 q^{81} -4.03294 q^{82} -3.98789 q^{83} +22.6104 q^{84} -24.1566 q^{86} -3.35449 q^{87} -3.51682 q^{88} -11.7782 q^{89} -6.13620 q^{91} -28.3188 q^{92} +14.2864 q^{93} -15.3489 q^{94} +7.29399 q^{96} +17.6694 q^{97} -5.83367 q^{98} +1.14308 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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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