LMFDB - Embedded newform 6025.2.a.p.1.20

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			1.20
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)

$f(q)$	$=$	$q-0.434323 q^{2} -1.75836 q^{3} -1.81136 q^{4} +0.763695 q^{6} +2.56007 q^{7} +1.65536 q^{8} +0.0918227 q^{9} +O(q^{10})$	\(q-0.434323 q^{2} -1.75836 q^{3} -1.81136 q^{4} +0.763695 q^{6} +2.56007 q^{7} +1.65536 q^{8} +0.0918227 q^{9} -2.96950 q^{11} +3.18503 q^{12} -5.29169 q^{13} -1.11190 q^{14} +2.90377 q^{16} +2.85851 q^{17} -0.0398807 q^{18} +6.43901 q^{19} -4.50152 q^{21} +1.28972 q^{22} -7.60122 q^{23} -2.91072 q^{24} +2.29830 q^{26} +5.11362 q^{27} -4.63722 q^{28} -7.01151 q^{29} +3.60166 q^{31} -4.57190 q^{32} +5.22144 q^{33} -1.24152 q^{34} -0.166324 q^{36} +0.899709 q^{37} -2.79661 q^{38} +9.30468 q^{39} +9.31833 q^{41} +1.95511 q^{42} +12.1683 q^{43} +5.37884 q^{44} +3.30138 q^{46} -3.09637 q^{47} -5.10586 q^{48} -0.446048 q^{49} -5.02629 q^{51} +9.58517 q^{52} -3.14521 q^{53} -2.22096 q^{54} +4.23784 q^{56} -11.3221 q^{57} +3.04526 q^{58} -12.8277 q^{59} -6.41135 q^{61} -1.56428 q^{62} +0.235072 q^{63} -3.82185 q^{64} -2.26779 q^{66} +4.29165 q^{67} -5.17781 q^{68} +13.3657 q^{69} -9.51025 q^{71} +0.152000 q^{72} +7.02430 q^{73} -0.390764 q^{74} -11.6634 q^{76} -7.60212 q^{77} -4.04124 q^{78} +10.5199 q^{79} -9.26704 q^{81} -4.04716 q^{82} +16.8677 q^{83} +8.15388 q^{84} -5.28496 q^{86} +12.3288 q^{87} -4.91559 q^{88} -13.4750 q^{89} -13.5471 q^{91} +13.7686 q^{92} -6.33301 q^{93} +1.34483 q^{94} +8.03903 q^{96} -8.44994 q^{97} +0.193729 q^{98} -0.272667 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.434323	−0.307113	−0.153556	−	0.988140i	$-0.549073\pi$
$2$	−0.434323	−0.307113	−0.153556	+	0.988140i	$0.549073\pi$
$3$	−1.75836	−1.01519	−0.507594	−	0.861596i	$-0.669465\pi$
$3$	−1.75836	−1.01519	−0.507594	+	0.861596i	$0.669465\pi$
$4$	−1.81136	−0.905682
$5$	0	0
$5$	0	0
$6$	0.763695	0.311777
$7$	2.56007	0.967615	0.483808	−	0.875174i	$-0.339253\pi$
$7$	2.56007	0.967615	0.483808	+	0.875174i	$0.339253\pi$
$8$	1.65536	0.585259
$9$	0.0918227	0.0306076
$10$	0	0
$11$	−2.96950	−0.895337	−0.447668	−	0.894200i	$-0.647746\pi$
$11$	−2.96950	−0.895337	−0.447668	+	0.894200i	$0.647746\pi$
$12$	3.18503	0.919438
$13$	−5.29169	−1.46765	−0.733825	−	0.679338i	$-0.762267\pi$
$13$	−5.29169	−1.46765	−0.733825	+	0.679338i	$0.762267\pi$
$14$	−1.11190	−0.297167
$15$	0	0
$16$	2.90377	0.725942
$17$	2.85851	0.693291	0.346646	−	0.937996i	$-0.387321\pi$
$17$	2.85851	0.693291	0.346646	+	0.937996i	$0.387321\pi$
$18$	−0.0398807	−0.00939996
$19$	6.43901	1.47721	0.738606	−	0.674138i	$-0.235485\pi$
$19$	6.43901	1.47721	0.738606	+	0.674138i	$0.235485\pi$
$20$	0	0
$21$	−4.50152	−0.982312
$22$	1.28972	0.274969
$23$	−7.60122	−1.58496	−0.792482	−	0.609896i	$-0.791211\pi$
$23$	−7.60122	−1.58496	−0.792482	+	0.609896i	$0.791211\pi$
$24$	−2.91072	−0.594148
$25$	0	0
$26$	2.29830	0.450734
$27$	5.11362	0.984116
$28$	−4.63722	−0.876351
$29$	−7.01151	−1.30201	−0.651003	−	0.759075i	$-0.725651\pi$
$29$	−7.01151	−1.30201	−0.651003	+	0.759075i	$0.725651\pi$
$30$	0	0
$31$	3.60166	0.646878	0.323439	−	0.946249i	$-0.395161\pi$
$31$	3.60166	0.646878	0.323439	+	0.946249i	$0.395161\pi$
$32$	−4.57190	−0.808205
$33$	5.22144	0.908936
$34$	−1.24152	−0.212918
$35$	0	0
$36$	−0.166324	−0.0277207
$37$	0.899709	0.147911	0.0739557	−	0.997262i	$-0.476438\pi$
$37$	0.899709	0.147911	0.0739557	+	0.997262i	$0.476438\pi$
$38$	−2.79661	−0.453670
$39$	9.30468	1.48994
$40$	0	0
$41$	9.31833	1.45528	0.727640	−	0.685960i	$-0.240617\pi$
$41$	9.31833	1.45528	0.727640	+	0.685960i	$0.240617\pi$
$42$	1.95511	0.301680
$43$	12.1683	1.85565	0.927823	−	0.373020i	$-0.121678\pi$
$43$	12.1683	1.85565	0.927823	+	0.373020i	$0.121678\pi$
$44$	5.37884	0.810890
$45$	0	0
$46$	3.30138	0.486762
$47$	−3.09637	−0.451653	−0.225826	−	0.974168i	$-0.572508\pi$
$47$	−3.09637	−0.451653	−0.225826	+	0.974168i	$0.572508\pi$
$48$	−5.10586	−0.736967
$49$	−0.446048	−0.0637212
$50$	0	0
$51$	−5.02629	−0.703821
$52$	9.58517	1.32922
$53$	−3.14521	−0.432028	−0.216014	−	0.976390i	$-0.569306\pi$
$53$	−3.14521	−0.432028	−0.216014	+	0.976390i	$0.569306\pi$
$54$	−2.22096	−0.302234
$55$	0	0
$56$	4.23784	0.566305
$57$	−11.3221	−1.49965
$58$	3.04526	0.399862
$59$	−12.8277	−1.67002	−0.835010	−	0.550235i	$-0.814538\pi$
$59$	−12.8277	−1.67002	−0.835010	+	0.550235i	$0.814538\pi$
$60$	0	0
$61$	−6.41135	−0.820889	−0.410445	−	0.911886i	$-0.634627\pi$
$61$	−6.41135	−0.820889	−0.410445	+	0.911886i	$0.634627\pi$
$62$	−1.56428	−0.198664
$63$	0.235072	0.0296163
$64$	−3.82185	−0.477732
$65$	0	0
$66$	−2.26779	−0.279146
$67$	4.29165	0.524309	0.262154	−	0.965026i	$-0.415567\pi$
$67$	4.29165	0.524309	0.262154	+	0.965026i	$0.415567\pi$
$68$	−5.17781	−0.627901
$69$	13.3657	1.60904
$70$	0	0
$71$	−9.51025	−1.12866	−0.564329	−	0.825550i	$-0.690865\pi$
$71$	−9.51025	−1.12866	−0.564329	+	0.825550i	$0.690865\pi$
$72$	0.152000	0.0179133
$73$	7.02430	0.822132	0.411066	−	0.911606i	$-0.365157\pi$
$73$	7.02430	0.822132	0.411066	+	0.911606i	$0.365157\pi$
$74$	−0.390764	−0.0454254
$75$	0	0
$76$	−11.6634	−1.33788
$77$	−7.60212	−0.866341
$78$	−4.04124	−0.457580
$79$	10.5199	1.18359	0.591793	−	0.806090i	$-0.298420\pi$
$79$	10.5199	1.18359	0.591793	+	0.806090i	$0.298420\pi$
$80$	0	0
$81$	−9.26704	−1.02967
$82$	−4.04716	−0.446935
$83$	16.8677	1.85147	0.925737	−	0.378167i	$-0.123445\pi$
$83$	16.8677	1.85147	0.925737	+	0.378167i	$0.123445\pi$
$84$	8.15388	0.889662
$85$	0	0
$86$	−5.28496	−0.569892
$87$	12.3288	1.32178
$88$	−4.91559	−0.524004
$89$	−13.4750	−1.42835	−0.714174	−	0.699968i	$-0.753197\pi$
$89$	−13.4750	−1.42835	−0.714174	+	0.699968i	$0.753197\pi$
$90$	0	0
$91$	−13.5471	−1.42012
$92$	13.7686	1.43547
$93$	−6.33301	−0.656703
$94$	1.34483	0.138708
$95$	0	0
$96$	8.03903	0.820480
$97$	−8.44994	−0.857961	−0.428981	−	0.903314i	$-0.641127\pi$
$97$	−8.44994	−0.857961	−0.428981	+	0.903314i	$0.641127\pi$
$98$	0.193729	0.0195696
$99$	−0.272667	−0.0274041
$100$	0	0
$101$	6.34736	0.631586	0.315793	−	0.948828i	$-0.397730\pi$
$101$	6.34736	0.631586	0.315793	+	0.948828i	$0.397730\pi$
$102$	2.18303	0.216152
$103$	4.71869	0.464946	0.232473	−	0.972603i	$-0.425318\pi$
$103$	4.71869	0.464946	0.232473	+	0.972603i	$0.425318\pi$
$104$	−8.75966	−0.858955
$105$	0	0
$106$	1.36604	0.132681
$107$	10.6747	1.03197	0.515983	−	0.856599i	$-0.327427\pi$
$107$	10.6747	1.03197	0.515983	+	0.856599i	$0.327427\pi$
$108$	−9.26262	−0.891296
$109$	9.16193	0.877553	0.438777	−	0.898596i	$-0.355412\pi$
$109$	9.16193	0.877553	0.438777	+	0.898596i	$0.355412\pi$
$110$	0	0
$111$	−1.58201	−0.150158
$112$	7.43384	0.702432
$113$	−2.64944	−0.249238	−0.124619	−	0.992205i	$-0.539771\pi$
$113$	−2.64944	−0.249238	−0.124619	+	0.992205i	$0.539771\pi$
$114$	4.91744	0.460561
$115$	0	0
$116$	12.7004	1.17920
$117$	−0.485897	−0.0449212
$118$	5.57135	0.512884
$119$	7.31799	0.670839
$120$	0	0
$121$	−2.18209	−0.198372
$122$	2.78460	0.252105
$123$	−16.3850	−1.47738
$124$	−6.52392	−0.585865
$125$	0	0
$126$	−0.102097	−0.00909555
$127$	18.9576	1.68221	0.841106	−	0.540870i	$-0.181905\pi$
$127$	18.9576	1.68221	0.841106	+	0.540870i	$0.181905\pi$
$128$	10.8037	0.954922
$129$	−21.3962	−1.88383
$130$	0	0
$131$	2.15948	0.188674	0.0943371	−	0.995540i	$-0.469927\pi$
$131$	2.15948	0.188674	0.0943371	+	0.995540i	$0.469927\pi$
$132$	−9.45792	−0.823207
$133$	16.4843	1.42937
$134$	−1.86396	−0.161022
$135$	0	0
$136$	4.73187	0.405755
$137$	14.4702	1.23627	0.618136	−	0.786072i	$-0.287888\pi$
$137$	14.4702	1.23627	0.618136	+	0.786072i	$0.287888\pi$
$138$	−5.80501	−0.494155
$139$	14.1665	1.20159	0.600796	−	0.799403i	$-0.294851\pi$
$139$	14.1665	1.20159	0.600796	+	0.799403i	$0.294851\pi$
$140$	0	0
$141$	5.44453	0.458512
$142$	4.13052	0.346625
$143$	15.7137	1.31404
$144$	0.266632	0.0222193
$145$	0	0
$146$	−3.05081	−0.252487
$147$	0.784313	0.0646890
$148$	−1.62970	−0.133961
$149$	−7.67393	−0.628673	−0.314336	−	0.949312i	$-0.601782\pi$
$149$	−7.67393	−0.628673	−0.314336	+	0.949312i	$0.601782\pi$
$150$	0	0
$151$	12.4626	1.01419	0.507096	−	0.861890i	$-0.330719\pi$
$151$	12.4626	1.01419	0.507096	+	0.861890i	$0.330719\pi$
$152$	10.6589	0.864551
$153$	0.262476	0.0212199
$154$	3.30177	0.266064
$155$	0	0
$156$	−16.8542	−1.34941
$157$	−1.13132	−0.0902890	−0.0451445	−	0.998980i	$-0.514375\pi$
$157$	−1.13132	−0.0902890	−0.0451445	+	0.998980i	$0.514375\pi$
$158$	−4.56905	−0.363494
$159$	5.53041	0.438590
$160$	0	0
$161$	−19.4596	−1.53363
$162$	4.02488	0.316225
$163$	13.0961	1.02577	0.512885	−	0.858458i	$-0.328577\pi$
$163$	13.0961	1.02577	0.512885	+	0.858458i	$0.328577\pi$
$164$	−16.8789	−1.31802
$165$	0	0
$166$	−7.32604	−0.568611
$167$	−20.9567	−1.62168	−0.810840	−	0.585268i	$-0.800989\pi$
$167$	−20.9567	−1.62168	−0.810840	+	0.585268i	$0.800989\pi$
$168$	−7.45164	−0.574907
$169$	15.0020	1.15400
$170$	0	0
$171$	0.591247	0.0452138
$172$	−22.0412	−1.68063
$173$	21.8599	1.66198	0.830989	−	0.556288i	$-0.187775\pi$
$173$	21.8599	1.66198	0.830989	+	0.556288i	$0.187775\pi$
$174$	−5.35466	−0.405935
$175$	0	0
$176$	−8.62272	−0.649962
$177$	22.5556	1.69539
$178$	5.85250	0.438664
$179$	−3.67446	−0.274642	−0.137321	−	0.990527i	$-0.543849\pi$
$179$	−3.67446	−0.274642	−0.137321	+	0.990527i	$0.543849\pi$
$180$	0	0
$181$	5.78781	0.430205	0.215102	−	0.976592i	$-0.430991\pi$
$181$	5.78781	0.430205	0.215102	+	0.976592i	$0.430991\pi$
$182$	5.88381	0.436137
$183$	11.2734	0.833357
$184$	−12.5828	−0.927614
$185$	0	0
$186$	2.75057	0.201682
$187$	−8.48834	−0.620729
$188$	5.60866	0.409054
$189$	13.0912	0.952245
$190$	0	0
$191$	−22.1639	−1.60372	−0.801862	−	0.597509i	$-0.796157\pi$
$191$	−22.1639	−1.60372	−0.801862	+	0.597509i	$0.796157\pi$
$192$	6.72019	0.484988
$193$	2.47506	0.178159	0.0890795	−	0.996025i	$-0.471607\pi$
$193$	2.47506	0.178159	0.0890795	+	0.996025i	$0.471607\pi$
$194$	3.67000	0.263491
$195$	0	0
$196$	0.807956	0.0577111
$197$	−15.5601	−1.10861	−0.554306	−	0.832313i	$-0.687016\pi$
$197$	−15.5601	−1.10861	−0.554306	+	0.832313i	$0.687016\pi$
$198$	0.118426	0.00841613
$199$	−7.37484	−0.522788	−0.261394	−	0.965232i	$-0.584182\pi$
$199$	−7.37484	−0.522788	−0.261394	+	0.965232i	$0.584182\pi$
$200$	0	0
$201$	−7.54626	−0.532272
$202$	−2.75680	−0.193968
$203$	−17.9500	−1.25984
$204$	9.10444	0.637438
$205$	0	0
$206$	−2.04943	−0.142791
$207$	−0.697964	−0.0485118
$208$	−15.3658	−1.06543
$209$	−19.1206	−1.32260
$210$	0	0
$211$	−22.7501	−1.56618	−0.783092	−	0.621905i	$-0.786359\pi$
$211$	−22.7501	−1.56618	−0.783092	+	0.621905i	$0.786359\pi$
$212$	5.69712	0.391280
$213$	16.7224	1.14580
$214$	−4.63628	−0.316929
$215$	0	0
$216$	8.46489	0.575963
$217$	9.22050	0.625929
$218$	−3.97923	−0.269508
$219$	−12.3512	−0.834619
$220$	0	0
$221$	−15.1264	−1.01751
$222$	0.687103	0.0461154
$223$	−27.9178	−1.86952	−0.934759	−	0.355284i	$-0.884384\pi$
$223$	−27.9178	−1.86952	−0.934759	+	0.355284i	$0.884384\pi$
$224$	−11.7044	−0.782031
$225$	0	0
$226$	1.15071	0.0765441
$227$	−13.5560	−0.899742	−0.449871	−	0.893094i	$-0.648530\pi$
$227$	−13.5560	−0.899742	−0.449871	+	0.893094i	$0.648530\pi$
$228$	20.5084	1.35820
$229$	−19.7523	−1.30527	−0.652633	−	0.757674i	$-0.726336\pi$
$229$	−19.7523	−1.30527	−0.652633	+	0.757674i	$0.726336\pi$
$230$	0	0
$231$	13.3672	0.879500
$232$	−11.6066	−0.762010
$233$	15.8232	1.03661	0.518306	−	0.855195i	$-0.326563\pi$
$233$	15.8232	1.03661	0.518306	+	0.855195i	$0.326563\pi$
$234$	0.211036	0.0137959
$235$	0	0
$236$	23.2356	1.51251
$237$	−18.4978	−1.20156
$238$	−3.17837	−0.206023
$239$	−18.9691	−1.22701	−0.613505	−	0.789691i	$-0.710241\pi$
$239$	−18.9691	−1.22701	−0.613505	+	0.789691i	$0.710241\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0.947731	0.0609225
$243$	0.953917	0.0611938
$244$	11.6133	0.743465
$245$	0	0
$246$	7.11636	0.453723
$247$	−34.0733	−2.16803
$248$	5.96206	0.378591
$249$	−29.6595	−1.87960
$250$	0	0
$251$	−4.53587	−0.286302	−0.143151	−	0.989701i	$-0.545723\pi$
$251$	−4.53587	−0.286302	−0.143151	+	0.989701i	$0.545723\pi$
$252$	−0.425801	−0.0268230
$253$	22.5718	1.41908
$254$	−8.23371	−0.516629
$255$	0	0
$256$	2.95141	0.184463
$257$	0.317324	0.0197941	0.00989706	−	0.999951i	$-0.496850\pi$
$257$	0.317324	0.0197941	0.00989706	+	0.999951i	$0.496850\pi$
$258$	9.29286	0.578548
$259$	2.30332	0.143121
$260$	0	0
$261$	−0.643816	−0.0398512
$262$	−0.937909	−0.0579442
$263$	−1.08643	−0.0669920	−0.0334960	−	0.999439i	$-0.510664\pi$
$263$	−1.08643	−0.0669920	−0.0334960	+	0.999439i	$0.510664\pi$
$264$	8.64337	0.531963
$265$	0	0
$266$	−7.15951	−0.438978
$267$	23.6939	1.45004
$268$	−7.77374	−0.474857
$269$	26.1591	1.59495	0.797475	−	0.603352i	$-0.206169\pi$
$269$	26.1591	1.59495	0.797475	+	0.603352i	$0.206169\pi$
$270$	0	0
$271$	−21.8047	−1.32454	−0.662271	−	0.749264i	$-0.730407\pi$
$271$	−21.8047	−1.32454	−0.662271	+	0.749264i	$0.730407\pi$
$272$	8.30045	0.503289
$273$	23.8206	1.44169
$274$	−6.28473	−0.379674
$275$	0	0
$276$	−24.2101	−1.45727
$277$	−10.6259	−0.638447	−0.319224	−	0.947679i	$-0.603422\pi$
$277$	−10.6259	−0.638447	−0.319224	+	0.947679i	$0.603422\pi$
$278$	−6.15285	−0.369024
$279$	0.330714	0.0197993
$280$	0	0
$281$	−11.0206	−0.657431	−0.328716	−	0.944429i	$-0.606616\pi$
$281$	−11.0206	−0.657431	−0.328716	+	0.944429i	$0.606616\pi$
$282$	−2.36468	−0.140815
$283$	−13.0006	−0.772805	−0.386403	−	0.922330i	$-0.626282\pi$
$283$	−13.0006	−0.772805	−0.386403	+	0.922330i	$0.626282\pi$
$284$	17.2265	1.02221
$285$	0	0
$286$	−6.82480	−0.403559
$287$	23.8556	1.40815
$288$	−0.419804	−0.0247372
$289$	−8.82891	−0.519347
$290$	0	0
$291$	14.8580	0.870992
$292$	−12.7236	−0.744590
$293$	10.5751	0.617801	0.308901	−	0.951094i	$-0.400039\pi$
$293$	10.5751	0.617801	0.308901	+	0.951094i	$0.400039\pi$
$294$	−0.340645	−0.0198668
$295$	0	0
$296$	1.48934	0.0865664
$297$	−15.1849	−0.881115
$298$	3.33296	0.193073
$299$	40.2233	2.32617
$300$	0	0
$301$	31.1517	1.79555
$302$	−5.41279	−0.311471
$303$	−11.1609	−0.641179
$304$	18.6974	1.07237
$305$	0	0
$306$	−0.113999	−0.00651691
$307$	−16.6124	−0.948122	−0.474061	−	0.880492i	$-0.657212\pi$
$307$	−16.6124	−0.948122	−0.474061	+	0.880492i	$0.657212\pi$
$308$	13.7702	0.784630
$309$	−8.29714	−0.472008
$310$	0	0
$311$	12.2930	0.697072	0.348536	−	0.937295i	$-0.386679\pi$
$311$	12.2930	0.697072	0.348536	+	0.937295i	$0.386679\pi$
$312$	15.4026	0.872002
$313$	−7.49184	−0.423464	−0.211732	−	0.977328i	$-0.567910\pi$
$313$	−7.49184	−0.423464	−0.211732	+	0.977328i	$0.567910\pi$
$314$	0.491357	0.0277289
$315$	0	0
$316$	−19.0554	−1.07195
$317$	−6.91570	−0.388425	−0.194212	−	0.980960i	$-0.562215\pi$
$317$	−6.91570	−0.388425	−0.194212	+	0.980960i	$0.562215\pi$
$318$	−2.40198	−0.134696
$319$	20.8207	1.16573
$320$	0	0
$321$	−18.7700	−1.04764
$322$	8.45176	0.470998
$323$	18.4060	1.02414
$324$	16.7860	0.932554
$325$	0	0
$326$	−5.68795	−0.315027
$327$	−16.1099	−0.890882
$328$	15.4252	0.851715
$329$	−7.92693	−0.437026
$330$	0	0
$331$	8.37109	0.460117	0.230058	−	0.973177i	$-0.426108\pi$
$331$	8.37109	0.460117	0.230058	+	0.973177i	$0.426108\pi$
$332$	−30.5536	−1.67685
$333$	0.0826137	0.00452720
$334$	9.10198	0.498038
$335$	0	0
$336$	−13.0714	−0.713101
$337$	−20.3399	−1.10798	−0.553992	−	0.832522i	$-0.686896\pi$
$337$	−20.3399	−1.10798	−0.553992	+	0.832522i	$0.686896\pi$
$338$	−6.51570	−0.354407
$339$	4.65866	0.253023
$340$	0	0
$341$	−10.6951	−0.579174
$342$	−0.256792	−0.0138857
$343$	−19.0624	−1.02927
$344$	20.1429	1.08603
$345$	0	0
$346$	−9.49426	−0.510415
$347$	17.5249	0.940787	0.470393	−	0.882457i	$-0.344112\pi$
$347$	17.5249	0.940787	0.470393	+	0.882457i	$0.344112\pi$
$348$	−22.3319	−1.19711
$349$	−17.0839	−0.914481	−0.457241	−	0.889343i	$-0.651162\pi$
$349$	−17.0839	−0.914481	−0.457241	+	0.889343i	$0.651162\pi$
$350$	0	0
$351$	−27.0597	−1.44434
$352$	13.5762	0.723615
$353$	−9.69646	−0.516091	−0.258045	−	0.966133i	$-0.583078\pi$
$353$	−9.69646	−0.516091	−0.258045	+	0.966133i	$0.583078\pi$
$354$	−9.79642	−0.520674
$355$	0	0
$356$	24.4081	1.29363
$357$	−12.8676	−0.681028
$358$	1.59590	0.0843460
$359$	−5.89623	−0.311191	−0.155596	−	0.987821i	$-0.549730\pi$
$359$	−5.89623	−0.311191	−0.155596	+	0.987821i	$0.549730\pi$
$360$	0	0
$361$	22.4609	1.18215
$362$	−2.51378	−0.132121
$363$	3.83690	0.201385
$364$	24.5387	1.28618
$365$	0	0
$366$	−4.89632	−0.255934
$367$	−5.35529	−0.279544	−0.139772	−	0.990184i	$-0.544637\pi$
$367$	−5.35529	−0.279544	−0.139772	+	0.990184i	$0.544637\pi$
$368$	−22.0722	−1.15059
$369$	0.855634	0.0445425
$370$	0	0
$371$	−8.05195	−0.418037
$372$	11.4714	0.594764
$373$	−13.5963	−0.703989	−0.351995	−	0.936002i	$-0.614496\pi$
$373$	−13.5963	−0.703989	−0.351995	+	0.936002i	$0.614496\pi$
$374$	3.68668	0.190634
$375$	0	0
$376$	−5.12562	−0.264334
$377$	37.1028	1.91089
$378$	−5.68581	−0.292447
$379$	16.3033	0.837445	0.418723	−	0.908114i	$-0.362478\pi$
$379$	16.3033	0.837445	0.418723	+	0.908114i	$0.362478\pi$
$380$	0	0
$381$	−33.3342	−1.70776
$382$	9.62629	0.492524
$383$	−28.5533	−1.45901	−0.729503	−	0.683978i	$-0.760248\pi$
$383$	−28.5533	−1.45901	−0.729503	+	0.683978i	$0.760248\pi$
$384$	−18.9968	−0.969426
$385$	0	0
$386$	−1.07498	−0.0547149
$387$	1.11732	0.0567968
$388$	15.3059	0.777040
$389$	3.86930	0.196181	0.0980906	−	0.995177i	$-0.468727\pi$
$389$	3.86930	0.196181	0.0980906	+	0.995177i	$0.468727\pi$
$390$	0	0
$391$	−21.7282	−1.09884
$392$	−0.738372	−0.0372934
$393$	−3.79713	−0.191540
$394$	6.75811	0.340469
$395$	0	0
$396$	0.493899	0.0248194
$397$	3.78760	0.190094	0.0950470	−	0.995473i	$-0.469700\pi$
$397$	3.78760	0.190094	0.0950470	+	0.995473i	$0.469700\pi$
$398$	3.20306	0.160555
$399$	−28.9853	−1.45108
$400$	0	0
$401$	6.33566	0.316388	0.158194	−	0.987408i	$-0.449433\pi$
$401$	6.33566	0.316388	0.158194	+	0.987408i	$0.449433\pi$
$402$	3.27751	0.163467
$403$	−19.0589	−0.949391
$404$	−11.4974	−0.572016
$405$	0	0
$406$	7.79607	0.386913
$407$	−2.67168	−0.132430
$408$	−8.32033	−0.411917
$409$	−28.3807	−1.40334	−0.701668	−	0.712504i	$-0.747561\pi$
$409$	−28.3807	−1.40334	−0.701668	+	0.712504i	$0.747561\pi$
$410$	0	0
$411$	−25.4438	−1.25505
$412$	−8.54726	−0.421093
$413$	−32.8397	−1.61594
$414$	0.303142	0.0148986
$415$	0	0
$416$	24.1931	1.18616
$417$	−24.9099	−1.21984
$418$	8.30453	0.406188
$419$	−14.4457	−0.705717	−0.352859	−	0.935677i	$-0.614790\pi$
$419$	−14.4457	−0.705717	−0.352859	+	0.935677i	$0.614790\pi$
$420$	0	0
$421$	12.0873	0.589097	0.294548	−	0.955637i	$-0.404831\pi$
$421$	12.0873	0.589097	0.294548	+	0.955637i	$0.404831\pi$
$422$	9.88091	0.480995
$423$	−0.284317	−0.0138240
$424$	−5.20646	−0.252848
$425$	0	0
$426$	−7.26293	−0.351890
$427$	−16.4135	−0.794305
$428$	−19.3358	−0.934632
$429$	−27.6302	−1.33400
$430$	0	0
$431$	−14.7881	−0.712316	−0.356158	−	0.934426i	$-0.615913\pi$
$431$	−14.7881	−0.712316	−0.356158	+	0.934426i	$0.615913\pi$
$432$	14.8487	0.714411
$433$	22.3991	1.07643	0.538216	−	0.842807i	$-0.319098\pi$
$433$	22.3991	1.07643	0.538216	+	0.842807i	$0.319098\pi$
$434$	−4.00467	−0.192231
$435$	0	0
$436$	−16.5956	−0.794784
$437$	−48.9443	−2.34133
$438$	5.36442	0.256322
$439$	−34.0058	−1.62301	−0.811504	−	0.584346i	$-0.801351\pi$
$439$	−34.0058	−1.62301	−0.811504	+	0.584346i	$0.801351\pi$
$440$	0	0
$441$	−0.0409574	−0.00195035
$442$	6.56972	0.312490
$443$	11.4103	0.542120	0.271060	−	0.962562i	$-0.412626\pi$
$443$	11.4103	0.542120	0.271060	+	0.962562i	$0.412626\pi$
$444$	2.86560	0.135995
$445$	0	0
$446$	12.1254	0.574152
$447$	13.4935	0.638221
$448$	−9.78421	−0.462260
$449$	−38.0727	−1.79676	−0.898381	−	0.439216i	$-0.855256\pi$
$449$	−38.0727	−1.79676	−0.898381	+	0.439216i	$0.855256\pi$
$450$	0	0
$451$	−27.6708	−1.30297
$452$	4.79909	0.225730
$453$	−21.9137	−1.02960
$454$	5.88767	0.276322
$455$	0	0
$456$	−18.7422	−0.877682
$457$	−12.1182	−0.566865	−0.283433	−	0.958992i	$-0.591473\pi$
$457$	−12.1182	−0.566865	−0.283433	+	0.958992i	$0.591473\pi$
$458$	8.57885	0.400863
$459$	14.6173	0.682279
$460$	0	0
$461$	−14.6145	−0.680667	−0.340334	−	0.940305i	$-0.610540\pi$
$461$	−14.6145	−0.680667	−0.340334	+	0.940305i	$0.610540\pi$
$462$	−5.80570	−0.270105
$463$	−8.87604	−0.412504	−0.206252	−	0.978499i	$-0.566127\pi$
$463$	−8.87604	−0.412504	−0.206252	+	0.978499i	$0.566127\pi$
$464$	−20.3598	−0.945180
$465$	0	0
$466$	−6.87238	−0.318357
$467$	−39.0070	−1.80503	−0.902515	−	0.430658i	$-0.858282\pi$
$467$	−39.0070	−1.80503	−0.902515	+	0.430658i	$0.858282\pi$
$468$	0.880136	0.0406843
$469$	10.9869	0.507329
$470$	0	0
$471$	1.98926	0.0916604
$472$	−21.2344	−0.977394
$473$	−36.1337	−1.66143
$474$	8.03403	0.369015
$475$	0	0
$476$	−13.2555	−0.607567
$477$	−0.288802	−0.0132233
$478$	8.23871	0.376830
$479$	14.0416	0.641578	0.320789	−	0.947151i	$-0.396052\pi$
$479$	14.0416	0.641578	0.320789	+	0.947151i	$0.396052\pi$
$480$	0	0
$481$	−4.76098	−0.217082
$482$	−0.434323	−0.0197829
$483$	34.2170	1.55693
$484$	3.95256	0.179662
$485$	0	0
$486$	−0.414308	−0.0187934
$487$	7.89889	0.357933	0.178966	−	0.983855i	$-0.442725\pi$
$487$	7.89889	0.357933	0.178966	+	0.983855i	$0.442725\pi$
$488$	−10.6131	−0.480433
$489$	−23.0277	−1.04135
$490$	0	0
$491$	41.5169	1.87363	0.936817	−	0.349821i	$-0.113758\pi$
$491$	41.5169	1.87363	0.936817	+	0.349821i	$0.113758\pi$
$492$	29.6791	1.33804
$493$	−20.0425	−0.902669
$494$	14.7988	0.665829
$495$	0	0
$496$	10.4584	0.469595
$497$	−24.3469	−1.09211
$498$	12.8818	0.577247
$499$	28.3650	1.26979	0.634896	−	0.772597i	$-0.281043\pi$
$499$	28.3650	1.26979	0.634896	+	0.772597i	$0.281043\pi$
$500$	0	0
$501$	36.8494	1.64631
$502$	1.97003	0.0879269
$503$	12.1673	0.542513	0.271257	−	0.962507i	$-0.412561\pi$
$503$	12.1673	0.542513	0.271257	+	0.962507i	$0.412561\pi$
$504$	0.389130	0.0173332
$505$	0	0
$506$	−9.80344	−0.435816
$507$	−26.3788	−1.17153
$508$	−34.3391	−1.52355
$509$	−32.1405	−1.42460	−0.712301	−	0.701875i	$-0.752347\pi$
$509$	−32.1405	−1.42460	−0.712301	+	0.701875i	$0.752347\pi$
$510$	0	0
$511$	17.9827	0.795507
$512$	−22.8893	−1.01157
$513$	32.9267	1.45375
$514$	−0.137821	−0.00607902
$515$	0	0
$516$	38.7563	1.70615
$517$	9.19467	0.404381
$518$	−1.00038	−0.0439543
$519$	−38.4376	−1.68722
$520$	0	0
$521$	16.4370	0.720119	0.360059	−	0.932929i	$-0.382756\pi$
$521$	16.4370	0.720119	0.360059	+	0.932929i	$0.382756\pi$
$522$	0.279624	0.0122388
$523$	30.7325	1.34384	0.671918	−	0.740625i	$-0.265471\pi$
$523$	30.7325	1.34384	0.671918	+	0.740625i	$0.265471\pi$
$524$	−3.91160	−0.170879
$525$	0	0
$526$	0.471860	0.0205741
$527$	10.2954	0.448475
$528$	15.1618	0.659834
$529$	34.7785	1.51211
$530$	0	0
$531$	−1.17787	−0.0511152
$532$	−29.8591	−1.29456
$533$	−49.3097	−2.13584
$534$	−10.2908	−0.445326
$535$	0	0
$536$	7.10424	0.306856
$537$	6.46102	0.278813
$538$	−11.3615	−0.489829
$539$	1.32454	0.0570520
$540$	0	0
$541$	−39.9511	−1.71763	−0.858816	−	0.512284i	$-0.828800\pi$
$541$	−39.9511	−1.71763	−0.858816	+	0.512284i	$0.828800\pi$
$542$	9.47029	0.406784
$543$	−10.1770	−0.436739
$544$	−13.0688	−0.560321
$545$	0	0
$546$	−10.3458	−0.442761
$547$	28.4899	1.21814	0.609071	−	0.793116i	$-0.291543\pi$
$547$	28.4899	1.21814	0.609071	+	0.793116i	$0.291543\pi$
$548$	−26.2108	−1.11967
$549$	−0.588707	−0.0251254
$550$	0	0
$551$	−45.1472	−1.92334
$552$	22.1250	0.941703
$553$	26.9318	1.14526
$554$	4.61506	0.196075
$555$	0	0
$556$	−25.6608	−1.08826
$557$	−4.95855	−0.210100	−0.105050	−	0.994467i	$-0.533500\pi$
$557$	−4.95855	−0.210100	−0.105050	+	0.994467i	$0.533500\pi$
$558$	−0.143637	−0.00608063
$559$	−64.3908	−2.72344
$560$	0	0
$561$	14.9255	0.630157
$562$	4.78648	0.201905
$563$	27.7731	1.17050	0.585249	−	0.810853i	$-0.300997\pi$
$563$	27.7731	1.17050	0.585249	+	0.810853i	$0.300997\pi$
$564$	−9.86203	−0.415266
$565$	0	0
$566$	5.64645	0.237338
$567$	−23.7242	−0.996325
$568$	−15.7429	−0.660558
$569$	25.8478	1.08359	0.541797	−	0.840509i	$-0.317744\pi$
$569$	25.8478	1.08359	0.541797	+	0.840509i	$0.317744\pi$
$570$	0	0
$571$	29.8843	1.25062	0.625309	−	0.780378i	$-0.284973\pi$
$571$	29.8843	1.25062	0.625309	+	0.780378i	$0.284973\pi$
$572$	−28.4631	−1.19010
$573$	38.9721	1.62808
$574$	−10.3610	−0.432461
$575$	0	0
$576$	−0.350933	−0.0146222
$577$	25.0417	1.04250	0.521250	−	0.853404i	$-0.325466\pi$
$577$	25.0417	1.04250	0.521250	+	0.853404i	$0.325466\pi$
$578$	3.83460	0.159498
$579$	−4.35205	−0.180865
$580$	0	0
$581$	43.1826	1.79151
$582$	−6.45317	−0.267493
$583$	9.33969	0.386811
$584$	11.6278	0.481160
$585$	0	0
$586$	−4.59299	−0.189735
$587$	−0.613835	−0.0253357	−0.0126678	−	0.999920i	$-0.504032\pi$
$587$	−0.613835	−0.0253357	−0.0126678	+	0.999920i	$0.504032\pi$
$588$	−1.42068	−0.0585877
$589$	23.1912	0.955575
$590$	0	0
$591$	27.3602	1.12545
$592$	2.61255	0.107375
$593$	−23.7785	−0.976467	−0.488233	−	0.872713i	$-0.662358\pi$
$593$	−23.7785	−0.976467	−0.488233	+	0.872713i	$0.662358\pi$
$594$	6.59513	0.270602
$595$	0	0
$596$	13.9003	0.569377
$597$	12.9676	0.530729
$598$	−17.4699	−0.714397
$599$	−36.9243	−1.50869	−0.754343	−	0.656480i	$-0.772044\pi$
$599$	−36.9243	−1.50869	−0.754343	+	0.656480i	$0.772044\pi$
$600$	0	0
$601$	−28.8902	−1.17846	−0.589228	−	0.807967i	$-0.700568\pi$
$601$	−28.8902	−1.17846	−0.589228	+	0.807967i	$0.700568\pi$
$602$	−13.5299	−0.551436
$603$	0.394071	0.0160478
$604$	−22.5743	−0.918535
$605$	0	0
$606$	4.84745	0.196914
$607$	−7.38525	−0.299758	−0.149879	−	0.988704i	$-0.547888\pi$
$607$	−7.38525	−0.299758	−0.149879	+	0.988704i	$0.547888\pi$
$608$	−29.4385	−1.19389
$609$	31.5624	1.27897
$610$	0	0
$611$	16.3850	0.662868
$612$	−0.475440	−0.0192185
$613$	−28.1453	−1.13678	−0.568388	−	0.822760i	$-0.692433\pi$
$613$	−28.1453	−1.13678	−0.568388	+	0.822760i	$0.692433\pi$
$614$	7.21516	0.291180
$615$	0	0
$616$	−12.5843	−0.507034
$617$	1.73238	0.0697430	0.0348715	−	0.999392i	$-0.488898\pi$
$617$	1.73238	0.0697430	0.0348715	+	0.999392i	$0.488898\pi$
$618$	3.60364	0.144960
$619$	−17.2161	−0.691972	−0.345986	−	0.938240i	$-0.612456\pi$
$619$	−17.2161	−0.691972	−0.345986	+	0.938240i	$0.612456\pi$
$620$	0	0
$621$	−38.8697	−1.55979
$622$	−5.33913	−0.214080
$623$	−34.4969	−1.38209
$624$	27.0186	1.08161
$625$	0	0
$626$	3.25388	0.130051
$627$	33.6209	1.34269
$628$	2.04923	0.0817731
$629$	2.57183	0.102546
$630$	0	0
$631$	3.04591	0.121256	0.0606279	−	0.998160i	$-0.480690\pi$
$631$	3.04591	0.121256	0.0606279	+	0.998160i	$0.480690\pi$
$632$	17.4143	0.692704
$633$	40.0029	1.58997
$634$	3.00365	0.119290
$635$	0	0
$636$	−10.0176	−0.397223
$637$	2.36035	0.0935205
$638$	−9.04289	−0.358011
$639$	−0.873256	−0.0345455
$640$	0	0
$641$	−8.46426	−0.334318	−0.167159	−	0.985930i	$-0.553459\pi$
$641$	−8.46426	−0.334318	−0.167159	+	0.985930i	$0.553459\pi$
$642$	8.15224	0.321743
$643$	−3.99210	−0.157433	−0.0787165	−	0.996897i	$-0.525082\pi$
$643$	−3.99210	−0.157433	−0.0787165	+	0.996897i	$0.525082\pi$
$644$	35.2485	1.38898
$645$	0	0
$646$	−7.99415	−0.314525
$647$	14.5637	0.572559	0.286279	−	0.958146i	$-0.407581\pi$
$647$	14.5637	0.572559	0.286279	+	0.958146i	$0.407581\pi$
$648$	−15.3403	−0.602624
$649$	38.0917	1.49523
$650$	0	0
$651$	−16.2129	−0.635436
$652$	−23.7219	−0.929020
$653$	−7.40618	−0.289826	−0.144913	−	0.989444i	$-0.546290\pi$
$653$	−7.40618	−0.289826	−0.144913	+	0.989444i	$0.546290\pi$
$654$	6.99691	0.273601
$655$	0	0
$656$	27.0583	1.05645
$657$	0.644990	0.0251634
$658$	3.44285	0.134216
$659$	−21.7644	−0.847822	−0.423911	−	0.905704i	$-0.639343\pi$
$659$	−21.7644	−0.847822	−0.423911	+	0.905704i	$0.639343\pi$
$660$	0	0
$661$	−3.19111	−0.124120	−0.0620598	−	0.998072i	$-0.519767\pi$
$661$	−3.19111	−0.124120	−0.0620598	+	0.998072i	$0.519767\pi$
$662$	−3.63575	−0.141308
$663$	26.5976	1.03296
$664$	27.9222	1.08359
$665$	0	0
$666$	−0.0358810	−0.00139036
$667$	53.2960	2.06363
$668$	37.9602	1.46873
$669$	49.0896	1.89791
$670$	0	0
$671$	19.0385	0.734973
$672$	20.5805	0.793909
$673$	7.45989	0.287558	0.143779	−	0.989610i	$-0.454075\pi$
$673$	7.45989	0.287558	0.143779	+	0.989610i	$0.454075\pi$
$674$	8.83407	0.340276
$675$	0	0
$676$	−27.1740	−1.04516
$677$	−1.15471	−0.0443790	−0.0221895	−	0.999754i	$-0.507064\pi$
$677$	−1.15471	−0.0443790	−0.0221895	+	0.999754i	$0.507064\pi$
$678$	−2.02336	−0.0777067
$679$	−21.6324	−0.830176
$680$	0	0
$681$	23.8363	0.913407
$682$	4.64514	0.177871
$683$	−36.9117	−1.41239	−0.706193	−	0.708020i	$-0.749589\pi$
$683$	−36.9117	−1.41239	−0.706193	+	0.708020i	$0.749589\pi$
$684$	−1.07096	−0.0409493
$685$	0	0
$686$	8.27923	0.316103
$687$	34.7315	1.32509
$688$	35.3339	1.34709
$689$	16.6435	0.634066
$690$	0	0
$691$	−42.2188	−1.60608	−0.803039	−	0.595926i	$-0.796785\pi$
$691$	−42.2188	−1.60608	−0.803039	+	0.595926i	$0.796785\pi$
$692$	−39.5963	−1.50522
$693$	−0.698046	−0.0265166
$694$	−7.61147	−0.288927
$695$	0	0
$696$	20.4085	0.773584
$697$	26.6366	1.00893
$698$	7.41994	0.280849
$699$	−27.8229	−1.05236
$700$	0	0
$701$	−1.45850	−0.0550869	−0.0275434	−	0.999621i	$-0.508768\pi$
$701$	−1.45850	−0.0550869	−0.0275434	+	0.999621i	$0.508768\pi$
$702$	11.7526	0.443574
$703$	5.79324	0.218496
$704$	11.3490	0.427731
$705$	0	0
$706$	4.21139	0.158498
$707$	16.2497	0.611132
$708$	−40.8565	−1.53548
$709$	38.9217	1.46174	0.730868	−	0.682519i	$-0.239115\pi$
$709$	38.9217	1.46174	0.730868	+	0.682519i	$0.239115\pi$
$710$	0	0
$711$	0.965969	0.0362267
$712$	−22.3060	−0.835953
$713$	−27.3770	−1.02528
$714$	5.58871	0.209152
$715$	0	0
$716$	6.65578	0.248738
$717$	33.3545	1.24565
$718$	2.56087	0.0955707
$719$	−32.6261	−1.21675	−0.608375	−	0.793650i	$-0.708178\pi$
$719$	−32.6261	−1.21675	−0.608375	+	0.793650i	$0.708178\pi$
$720$	0	0
$721$	12.0802	0.449889
$722$	−9.75528	−0.363054
$723$	−1.75836	−0.0653940
$724$	−10.4838	−0.389629
$725$	0	0
$726$	−1.66645	−0.0618478
$727$	−6.49756	−0.240981	−0.120491	−	0.992714i	$-0.538447\pi$
$727$	−6.49756	−0.240981	−0.120491	+	0.992714i	$0.538447\pi$
$728$	−22.4253	−0.831138
$729$	26.1238	0.967547
$730$	0	0
$731$	34.7832	1.28650
$732$	−20.4203	−0.754757
$733$	−36.8654	−1.36165	−0.680827	−	0.732444i	$-0.738379\pi$
$733$	−36.8654	−1.36165	−0.680827	+	0.732444i	$0.738379\pi$
$734$	2.32593	0.0858515
$735$	0	0
$736$	34.7520	1.28097
$737$	−12.7440	−0.469433
$738$	−0.371621	−0.0136796
$739$	−49.1741	−1.80890	−0.904450	−	0.426580i	$-0.859718\pi$
$739$	−49.1741	−1.80890	−0.904450	+	0.426580i	$0.859718\pi$
$740$	0	0
$741$	59.9130	2.20096
$742$	3.49715	0.128384
$743$	22.5144	0.825974	0.412987	−	0.910737i	$-0.364486\pi$
$743$	22.5144	0.825974	0.412987	+	0.910737i	$0.364486\pi$
$744$	−10.4834	−0.384341
$745$	0	0
$746$	5.90518	0.216204
$747$	1.54884	0.0566691
$748$	15.3755	0.562183
$749$	27.3280	0.998545
$750$	0	0
$751$	−12.5774	−0.458957	−0.229479	−	0.973314i	$-0.573702\pi$
$751$	−12.5774	−0.458957	−0.229479	+	0.973314i	$0.573702\pi$
$752$	−8.99115	−0.327873
$753$	7.97569	0.290650
$754$	−16.1146	−0.586858
$755$	0	0
$756$	−23.7129	−0.862431
$757$	2.31525	0.0841491	0.0420745	−	0.999114i	$-0.486603\pi$
$757$	2.31525	0.0841491	0.0420745	+	0.999114i	$0.486603\pi$
$758$	−7.08090	−0.257190
$759$	−39.6893	−1.44063
$760$	0	0
$761$	−47.6708	−1.72807	−0.864033	−	0.503435i	$-0.832069\pi$
$761$	−47.6708	−1.72807	−0.864033	+	0.503435i	$0.832069\pi$
$762$	14.4778	0.524475
$763$	23.4552	0.849134
$764$	40.1469	1.45246
$765$	0	0
$766$	12.4013	0.448079
$767$	67.8800	2.45101
$768$	−5.18964	−0.187265
$769$	−12.2319	−0.441094	−0.220547	−	0.975376i	$-0.570784\pi$
$769$	−12.2319	−0.441094	−0.220547	+	0.975376i	$0.570784\pi$
$770$	0	0
$771$	−0.557969	−0.0200948
$772$	−4.48324	−0.161355
$773$	9.31386	0.334996	0.167498	−	0.985872i	$-0.446431\pi$
$773$	9.31386	0.334996	0.167498	+	0.985872i	$0.446431\pi$
$774$	−0.485279	−0.0174430
$775$	0	0
$776$	−13.9877	−0.502129
$777$	−4.05006	−0.145295
$778$	−1.68052	−0.0602497
$779$	60.0009	2.14976
$780$	0	0
$781$	28.2407	1.01053
$782$	9.43704	0.337468
$783$	−35.8542	−1.28132
$784$	−1.29522	−0.0462579
$785$	0	0
$786$	1.64918	0.0588243
$787$	3.90661	0.139255	0.0696277	−	0.997573i	$-0.477819\pi$
$787$	3.90661	0.139255	0.0696277	+	0.997573i	$0.477819\pi$
$788$	28.1850	1.00405
$789$	1.91033	0.0680095
$790$	0	0
$791$	−6.78274	−0.241166
$792$	−0.451363	−0.0160385
$793$	33.9269	1.20478
$794$	−1.64504	−0.0583802
$795$	0	0
$796$	13.3585	0.473480
$797$	8.80375	0.311845	0.155922	−	0.987769i	$-0.450165\pi$
$797$	8.80375	0.311845	0.155922	+	0.987769i	$0.450165\pi$
$798$	12.5890	0.445645
$799$	−8.85102	−0.313127
$800$	0	0
$801$	−1.23731	−0.0437182
$802$	−2.75172	−0.0971667
$803$	−20.8586	−0.736085
$804$	13.6690	0.482069
$805$	0	0
$806$	8.27771	0.291570
$807$	−45.9971	−1.61917
$808$	10.5072	0.369641
$809$	31.2057	1.09713	0.548567	−	0.836107i	$-0.315174\pi$
$809$	31.2057	1.09713	0.548567	+	0.836107i	$0.315174\pi$
$810$	0	0
$811$	21.8358	0.766759	0.383380	−	0.923591i	$-0.374760\pi$
$811$	21.8358	0.766759	0.383380	+	0.923591i	$0.374760\pi$
$812$	32.5139	1.14101
$813$	38.3405	1.34466
$814$	1.16037	0.0406711
$815$	0	0
$816$	−14.5952	−0.510933
$817$	78.3518	2.74118
$818$	12.3264	0.430982
$819$	−1.24393	−0.0434664
$820$	0	0
$821$	4.07821	0.142331	0.0711653	−	0.997465i	$-0.477328\pi$
$821$	4.07821	0.142331	0.0711653	+	0.997465i	$0.477328\pi$
$822$	11.0508	0.385441
$823$	−22.9341	−0.799432	−0.399716	−	0.916639i	$-0.630891\pi$
$823$	−22.9341	−0.799432	−0.399716	+	0.916639i	$0.630891\pi$
$824$	7.81114	0.272114
$825$	0	0
$826$	14.2630	0.496274
$827$	−31.1806	−1.08425	−0.542127	−	0.840296i	$-0.682381\pi$
$827$	−31.1806	−1.08425	−0.542127	+	0.840296i	$0.682381\pi$
$828$	1.26427	0.0439363
$829$	36.9740	1.28416	0.642080	−	0.766638i	$-0.278072\pi$
$829$	36.9740	1.28416	0.642080	+	0.766638i	$0.278072\pi$
$830$	0	0
$831$	18.6841	0.648144
$832$	20.2241	0.701143
$833$	−1.27504	−0.0441773
$834$	10.8189	0.374629
$835$	0	0
$836$	34.6344	1.19786
$837$	18.4175	0.636603
$838$	6.27408	0.216735
$839$	−1.70060	−0.0587112	−0.0293556	−	0.999569i	$-0.509346\pi$
$839$	−1.70060	−0.0587112	−0.0293556	+	0.999569i	$0.509346\pi$
$840$	0	0
$841$	20.1613	0.695218
$842$	−5.24977	−0.180919
$843$	19.3781	0.667417
$844$	41.2088	1.41847
$845$	0	0
$846$	0.123485	0.00424552
$847$	−5.58630	−0.191948
$848$	−9.13296	−0.313627
$849$	22.8597	0.784543
$850$	0	0
$851$	−6.83889	−0.234434
$852$	−30.2904	−1.03773
$853$	−29.5900	−1.01314	−0.506571	−	0.862198i	$-0.669087\pi$
$853$	−29.5900	−1.01314	−0.506571	+	0.862198i	$0.669087\pi$
$854$	7.12876	0.243941
$855$	0	0
$856$	17.6705	0.603967
$857$	39.2074	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$857$	39.2074	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$858$	12.0004	0.409688
$859$	4.86252	0.165907	0.0829535	−	0.996553i	$-0.473565\pi$
$859$	4.86252	0.165907	0.0829535	+	0.996553i	$0.473565\pi$
$860$	0	0
$861$	−41.9466	−1.42954
$862$	6.42279	0.218761
$863$	32.0662	1.09155	0.545774	−	0.837933i	$-0.316236\pi$
$863$	32.0662	1.09155	0.545774	+	0.837933i	$0.316236\pi$
$864$	−23.3789	−0.795367
$865$	0	0
$866$	−9.72845	−0.330586
$867$	15.5244	0.527236
$868$	−16.7017	−0.566892
$869$	−31.2389	−1.05971
$870$	0	0
$871$	−22.7101	−0.769502
$872$	15.1663	0.513596
$873$	−0.775896	−0.0262601
$874$	21.2576	0.719050
$875$	0	0
$876$	22.3726	0.755899
$877$	−14.5057	−0.489823	−0.244912	−	0.969545i	$-0.578759\pi$
$877$	−14.5057	−0.489823	−0.244912	+	0.969545i	$0.578759\pi$
$878$	14.7695	0.498446
$879$	−18.5947	−0.627185
$880$	0	0
$881$	−23.9591	−0.807202	−0.403601	−	0.914935i	$-0.632242\pi$
$881$	−23.9591	−0.807202	−0.403601	+	0.914935i	$0.632242\pi$
$882$	0.0177887	0.000598977 0
$883$	27.8892	0.938546	0.469273	−	0.883053i	$-0.344516\pi$
$883$	27.8892	0.938546	0.469273	+	0.883053i	$0.344516\pi$
$884$	27.3993	0.921540
$885$	0	0
$886$	−4.95576	−0.166492
$887$	−7.79600	−0.261764	−0.130882	−	0.991398i	$-0.541781\pi$
$887$	−7.79600	−0.261764	−0.130882	+	0.991398i	$0.541781\pi$
$888$	−2.61880	−0.0878812
$889$	48.5327	1.62773
$890$	0	0
$891$	27.5184	0.921902
$892$	50.5694	1.69319
$893$	−19.9376	−0.667186
$894$	−5.86054	−0.196006
$895$	0	0
$896$	27.6582	0.923997
$897$	−70.7269	−2.36150
$898$	16.5358	0.551808
$899$	−25.2531	−0.842238
$900$	0	0
$901$	−8.99062	−0.299521
$902$	12.0180	0.400157
$903$	−54.7758	−1.82282
$904$	−4.38577	−0.145869
$905$	0	0
$906$	9.51762	0.316202
$907$	−10.2519	−0.340408	−0.170204	−	0.985409i	$-0.554443\pi$
$907$	−10.2519	−0.340408	−0.170204	+	0.985409i	$0.554443\pi$
$908$	24.5548	0.814880
$909$	0.582831	0.0193313
$910$	0	0
$911$	−5.42341	−0.179686	−0.0898428	−	0.995956i	$-0.528636\pi$
$911$	−5.42341	−0.179686	−0.0898428	+	0.995956i	$0.528636\pi$
$912$	−32.8767	−1.08866
$913$	−50.0887	−1.65769
$914$	5.26321	0.174091
$915$	0	0
$916$	35.7785	1.18216
$917$	5.52841	0.182564
$918$	−6.34864	−0.209536
$919$	−33.2008	−1.09519	−0.547596	−	0.836743i	$-0.684457\pi$
$919$	−33.2008	−1.09519	−0.547596	+	0.836743i	$0.684457\pi$
$920$	0	0
$921$	29.2106	0.962522
$922$	6.34743	0.209041
$923$	50.3253	1.65648
$924$	−24.2129	−0.796547
$925$	0	0
$926$	3.85506	0.126685
$927$	0.433282	0.0142309
$928$	32.0559	1.05229
$929$	−15.0877	−0.495012	−0.247506	−	0.968886i	$-0.579611\pi$
$929$	−15.0877	−0.495012	−0.247506	+	0.968886i	$0.579611\pi$
$930$	0	0
$931$	−2.87211	−0.0941297
$932$	−28.6616	−0.938842
$933$	−21.6155	−0.707660
$934$	16.9416	0.554348
$935$	0	0
$936$	−0.804335	−0.0262905
$937$	−20.5706	−0.672011	−0.336005	−	0.941860i	$-0.609076\pi$
$937$	−20.5706	−0.672011	−0.336005	+	0.941860i	$0.609076\pi$
$938$	−4.77187	−0.155807
$939$	13.1733	0.429896
$940$	0	0
$941$	55.0850	1.79572	0.897860	−	0.440281i	$-0.145121\pi$
$941$	55.0850	1.79572	0.897860	+	0.440281i	$0.145121\pi$
$942$	−0.863982	−0.0281500
$943$	−70.8307	−2.30656
$944$	−37.2485	−1.21234
$945$	0	0
$946$	15.6937	0.510246
$947$	43.4046	1.41046	0.705230	−	0.708978i	$-0.250844\pi$
$947$	43.4046	1.41046	0.705230	+	0.708978i	$0.250844\pi$
$948$	33.5063	1.08823
$949$	−37.1704	−1.20660
$950$	0	0
$951$	12.1603	0.394324
$952$	12.1139	0.392614
$953$	−1.58717	−0.0514134	−0.0257067	−	0.999670i	$-0.508184\pi$
$953$	−1.58717	−0.0514134	−0.0257067	+	0.999670i	$0.508184\pi$
$954$	0.125433	0.00406105
$955$	0	0
$956$	34.3600	1.11128
$957$	−36.6102	−1.18344
$958$	−6.09859	−0.197037
$959$	37.0447	1.19623
$960$	0	0
$961$	−18.0280	−0.581549
$962$	2.06780	0.0666686
$963$	0.980182	0.0315859
$964$	−1.81136	−0.0583401
$965$	0	0
$966$	−14.8612	−0.478152
$967$	27.9141	0.897657	0.448829	−	0.893618i	$-0.351841\pi$
$967$	27.9141	0.897657	0.448829	+	0.893618i	$0.351841\pi$
$968$	−3.61215	−0.116099
$969$	−32.3643	−1.03969
$970$	0	0
$971$	−48.4758	−1.55566	−0.777832	−	0.628473i	$-0.783680\pi$
$971$	−48.4758	−1.55566	−0.777832	+	0.628473i	$0.783680\pi$
$972$	−1.72789	−0.0554221
$973$	36.2673	1.16268
$974$	−3.43067	−0.109926
$975$	0	0
$976$	−18.6171	−0.595918
$977$	−3.82977	−0.122525	−0.0612626	−	0.998122i	$-0.519513\pi$
$977$	−3.82977	−0.122525	−0.0612626	+	0.998122i	$0.519513\pi$
$978$	10.0015	0.319811
$979$	40.0140	1.27885
$980$	0	0
$981$	0.841272	0.0268598
$982$	−18.0317	−0.575416
$983$	4.09982	0.130764	0.0653820	−	0.997860i	$-0.479173\pi$
$983$	4.09982	0.130764	0.0653820	+	0.997860i	$0.479173\pi$
$984$	−27.1231	−0.864651
$985$	0	0
$986$	8.70491	0.277221
$987$	13.9384	0.443664
$988$	61.7191	1.96355
$989$	−92.4938	−2.94113
$990$	0	0
$991$	−43.7654	−1.39025	−0.695126	−	0.718888i	$-0.744652\pi$
$991$	−43.7654	−1.39025	−0.695126	+	0.718888i	$0.744652\pi$
$992$	−16.4664	−0.522810
$993$	−14.7194	−0.467105
$994$	10.5744	0.335400
$995$	0	0
$996$	53.7242	1.70232
$997$	−16.2468	−0.514542	−0.257271	−	0.966339i	$-0.582823\pi$
$997$	−16.2468	−0.514542	−0.257271	+	0.966339i	$0.582823\pi$
$998$	−12.3196	−0.389969
$999$	4.60077	0.145562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.20		46
5.2	odd	4		1205.2.b.c.724.20	✓	46
5.3	odd	4		1205.2.b.c.724.27	yes	46
5.4	even	2	inner	6025.2.a.p.1.27		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.20	✓	46	5.2	odd	4
1205.2.b.c.724.27	yes	46	5.3	odd	4
6025.2.a.p.1.20		46	1.1	even	1	trivial
6025.2.a.p.1.27		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)

\(f(q)\)	\(=\)	\(q-0.434323 q^{2} -1.75836 q^{3} -1.81136 q^{4} +0.763695 q^{6} +2.56007 q^{7} +1.65536 q^{8} +0.0918227 q^{9} +O(q^{10})\)	\(q-0.434323 q^{2} -1.75836 q^{3} -1.81136 q^{4} +0.763695 q^{6} +2.56007 q^{7} +1.65536 q^{8} +0.0918227 q^{9} -2.96950 q^{11} +3.18503 q^{12} -5.29169 q^{13} -1.11190 q^{14} +2.90377 q^{16} +2.85851 q^{17} -0.0398807 q^{18} +6.43901 q^{19} -4.50152 q^{21} +1.28972 q^{22} -7.60122 q^{23} -2.91072 q^{24} +2.29830 q^{26} +5.11362 q^{27} -4.63722 q^{28} -7.01151 q^{29} +3.60166 q^{31} -4.57190 q^{32} +5.22144 q^{33} -1.24152 q^{34} -0.166324 q^{36} +0.899709 q^{37} -2.79661 q^{38} +9.30468 q^{39} +9.31833 q^{41} +1.95511 q^{42} +12.1683 q^{43} +5.37884 q^{44} +3.30138 q^{46} -3.09637 q^{47} -5.10586 q^{48} -0.446048 q^{49} -5.02629 q^{51} +9.58517 q^{52} -3.14521 q^{53} -2.22096 q^{54} +4.23784 q^{56} -11.3221 q^{57} +3.04526 q^{58} -12.8277 q^{59} -6.41135 q^{61} -1.56428 q^{62} +0.235072 q^{63} -3.82185 q^{64} -2.26779 q^{66} +4.29165 q^{67} -5.17781 q^{68} +13.3657 q^{69} -9.51025 q^{71} +0.152000 q^{72} +7.02430 q^{73} -0.390764 q^{74} -11.6634 q^{76} -7.60212 q^{77} -4.04124 q^{78} +10.5199 q^{79} -9.26704 q^{81} -4.04716 q^{82} +16.8677 q^{83} +8.15388 q^{84} -5.28496 q^{86} +12.3288 q^{87} -4.91559 q^{88} -13.4750 q^{89} -13.5471 q^{91} +13.7686 q^{92} -6.33301 q^{93} +1.34483 q^{94} +8.03903 q^{96} -8.44994 q^{97} +0.193729 q^{98} -0.272667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 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

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Database

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