LMFDB - Embedded newform 6025.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$25$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.63003 q^{2} +1.40060 q^{3} +4.91705 q^{4} -3.68362 q^{6} -1.05429 q^{7} -7.67192 q^{8} -1.03832 q^{9} +O(q^{10})$	\(q-2.63003 q^{2} +1.40060 q^{3} +4.91705 q^{4} -3.68362 q^{6} -1.05429 q^{7} -7.67192 q^{8} -1.03832 q^{9} +0.340120 q^{11} +6.88683 q^{12} -6.67580 q^{13} +2.77280 q^{14} +10.3433 q^{16} -1.88405 q^{17} +2.73080 q^{18} -0.838634 q^{19} -1.47663 q^{21} -0.894526 q^{22} -0.613843 q^{23} -10.7453 q^{24} +17.5576 q^{26} -5.65607 q^{27} -5.18398 q^{28} +1.63368 q^{29} +6.39503 q^{31} -11.8593 q^{32} +0.476373 q^{33} +4.95511 q^{34} -5.10545 q^{36} +2.44693 q^{37} +2.20563 q^{38} -9.35014 q^{39} -3.67966 q^{41} +3.88359 q^{42} +4.17929 q^{43} +1.67239 q^{44} +1.61443 q^{46} -0.850277 q^{47} +14.4868 q^{48} -5.88848 q^{49} -2.63881 q^{51} -32.8253 q^{52} +7.75319 q^{53} +14.8756 q^{54} +8.08840 q^{56} -1.17459 q^{57} -4.29664 q^{58} +2.57044 q^{59} -1.92112 q^{61} -16.8191 q^{62} +1.09468 q^{63} +10.5037 q^{64} -1.25287 q^{66} -8.14434 q^{67} -9.26398 q^{68} -0.859749 q^{69} -7.56392 q^{71} +7.96589 q^{72} +6.53710 q^{73} -6.43550 q^{74} -4.12360 q^{76} -0.358584 q^{77} +24.5911 q^{78} -0.258974 q^{79} -4.80695 q^{81} +9.67761 q^{82} +5.90935 q^{83} -7.26069 q^{84} -10.9916 q^{86} +2.28814 q^{87} -2.60938 q^{88} +3.73938 q^{89} +7.03821 q^{91} -3.01830 q^{92} +8.95688 q^{93} +2.23625 q^{94} -16.6101 q^{96} -17.9225 q^{97} +15.4869 q^{98} -0.353153 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * 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 $

$2$	−2.63003	−1.85971	−0.929855	−	0.367925i	$-0.880068\pi$
$2$	−2.63003	−1.85971	−0.929855	+	0.367925i	$0.880068\pi$
$3$	1.40060	0.808637	0.404319	−	0.914618i	$-0.367509\pi$
$3$	1.40060	0.808637	0.404319	+	0.914618i	$0.367509\pi$
$4$	4.91705	2.45852
$5$	0	0
$5$	0	0
$6$	−3.68362	−1.50383
$7$	−1.05429	−0.398483	−0.199241	−	0.979950i	$-0.563848\pi$
$7$	−1.05429	−0.398483	−0.199241	+	0.979950i	$0.563848\pi$
$8$	−7.67192	−2.71243
$9$	−1.03832	−0.346106
$10$	0	0
$11$	0.340120	0.102550	0.0512751	−	0.998685i	$-0.483671\pi$
$11$	0.340120	0.102550	0.0512751	+	0.998685i	$0.483671\pi$
$12$	6.88683	1.98806
$13$	−6.67580	−1.85153	−0.925767	−	0.378094i	$-0.876580\pi$
$13$	−6.67580	−1.85153	−0.925767	+	0.378094i	$0.876580\pi$
$14$	2.77280	0.741063
$15$	0	0
$16$	10.3433	2.58582
$17$	−1.88405	−0.456950	−0.228475	−	0.973550i	$-0.573374\pi$
$17$	−1.88405	−0.456950	−0.228475	+	0.973550i	$0.573374\pi$
$18$	2.73080	0.643656
$19$	−0.838634	−0.192396	−0.0961979	−	0.995362i	$-0.530668\pi$
$19$	−0.838634	−0.192396	−0.0961979	+	0.995362i	$0.530668\pi$
$20$	0	0
$21$	−1.47663	−0.322228
$22$	−0.894526	−0.190714
$23$	−0.613843	−0.127995	−0.0639976	−	0.997950i	$-0.520385\pi$
$23$	−0.613843	−0.127995	−0.0639976	+	0.997950i	$0.520385\pi$
$24$	−10.7453	−2.19338
$25$	0	0
$26$	17.5576	3.44332
$27$	−5.65607	−1.08851
$28$	−5.18398	−0.979680
$29$	1.63368	0.303368	0.151684	−	0.988429i	$-0.451530\pi$
$29$	1.63368	0.303368	0.151684	+	0.988429i	$0.451530\pi$
$30$	0	0
$31$	6.39503	1.14858	0.574290	−	0.818652i	$-0.305278\pi$
$31$	6.39503	1.14858	0.574290	+	0.818652i	$0.305278\pi$
$32$	−11.8593	−2.09644
$33$	0.476373	0.0829259
$34$	4.95511	0.849795
$35$	0	0
$36$	−5.10545	−0.850909
$37$	2.44693	0.402273	0.201137	−	0.979563i	$-0.435536\pi$
$37$	2.44693	0.402273	0.201137	+	0.979563i	$0.435536\pi$
$38$	2.20563	0.357800
$39$	−9.35014	−1.49722
$40$	0	0
$41$	−3.67966	−0.574666	−0.287333	−	0.957831i	$-0.592769\pi$
$41$	−3.67966	−0.574666	−0.287333	+	0.957831i	$0.592769\pi$
$42$	3.88359	0.599251
$43$	4.17929	0.637335	0.318668	−	0.947866i	$-0.396765\pi$
$43$	4.17929	0.637335	0.318668	+	0.947866i	$0.396765\pi$
$44$	1.67239	0.252122
$45$	0	0
$46$	1.61443	0.238034
$47$	−0.850277	−0.124026	−0.0620128	−	0.998075i	$-0.519752\pi$
$47$	−0.850277	−0.124026	−0.0620128	+	0.998075i	$0.519752\pi$
$48$	14.4868	2.09099
$49$	−5.88848	−0.841211
$50$	0	0
$51$	−2.63881	−0.369507
$52$	−32.8253	−4.55204
$53$	7.75319	1.06498	0.532491	−	0.846435i	$-0.321256\pi$
$53$	7.75319	1.06498	0.532491	+	0.846435i	$0.321256\pi$
$54$	14.8756	2.02432
$55$	0	0
$56$	8.08840	1.08086
$57$	−1.17459	−0.155578
$58$	−4.29664	−0.564176
$59$	2.57044	0.334642	0.167321	−	0.985902i	$-0.446488\pi$
$59$	2.57044	0.334642	0.167321	+	0.985902i	$0.446488\pi$
$60$	0	0
$61$	−1.92112	−0.245975	−0.122987	−	0.992408i	$-0.539247\pi$
$61$	−1.92112	−0.245975	−0.122987	+	0.992408i	$0.539247\pi$
$62$	−16.8191	−2.13603
$63$	1.09468	0.137917
$64$	10.5037	1.31296
$65$	0	0
$66$	−1.25287	−0.154218
$67$	−8.14434	−0.994989	−0.497495	−	0.867467i	$-0.665747\pi$
$67$	−8.14434	−0.994989	−0.497495	+	0.867467i	$0.665747\pi$
$68$	−9.26398	−1.12342
$69$	−0.859749	−0.103502
$70$	0	0
$71$	−7.56392	−0.897672	−0.448836	−	0.893614i	$-0.648161\pi$
$71$	−7.56392	−0.897672	−0.448836	+	0.893614i	$0.648161\pi$
$72$	7.96589	0.938789
$73$	6.53710	0.765109	0.382555	−	0.923933i	$-0.375044\pi$
$73$	6.53710	0.765109	0.382555	+	0.923933i	$0.375044\pi$
$74$	−6.43550	−0.748112
$75$	0	0
$76$	−4.12360	−0.473010
$77$	−0.358584	−0.0408645
$78$	24.5911	2.78440
$79$	−0.258974	−0.0291368	−0.0145684	−	0.999894i	$-0.504637\pi$
$79$	−0.258974	−0.0291368	−0.0145684	+	0.999894i	$0.504637\pi$
$80$	0	0
$81$	−4.80695	−0.534105
$82$	9.67761	1.06871
$83$	5.90935	0.648635	0.324317	−	0.945948i	$-0.394865\pi$
$83$	5.90935	0.648635	0.324317	+	0.945948i	$0.394865\pi$
$84$	−7.26069	−0.792206
$85$	0	0
$86$	−10.9916	−1.18526
$87$	2.28814	0.245314
$88$	−2.60938	−0.278161
$89$	3.73938	0.396373	0.198187	−	0.980164i	$-0.436495\pi$
$89$	3.73938	0.396373	0.198187	+	0.980164i	$0.436495\pi$
$90$	0	0
$91$	7.03821	0.737805
$92$	−3.01830	−0.314679
$93$	8.95688	0.928785
$94$	2.23625	0.230652
$95$	0	0
$96$	−16.6101	−1.69526
$97$	−17.9225	−1.81975	−0.909877	−	0.414879i	$-0.863824\pi$
$97$	−17.9225	−1.81975	−0.909877	+	0.414879i	$0.863824\pi$
$98$	15.4869	1.56441
$99$	−0.353153	−0.0354932
$100$	0	0
$101$	−0.845554	−0.0841358	−0.0420679	−	0.999115i	$-0.513395\pi$
$101$	−0.845554	−0.0841358	−0.0420679	+	0.999115i	$0.513395\pi$
$102$	6.94014	0.687176
$103$	−0.956502	−0.0942469	−0.0471235	−	0.998889i	$-0.515005\pi$
$103$	−0.956502	−0.0942469	−0.0471235	+	0.998889i	$0.515005\pi$
$104$	51.2163	5.02217
$105$	0	0
$106$	−20.3911	−1.98056
$107$	6.25907	0.605087	0.302544	−	0.953136i	$-0.402164\pi$
$107$	6.25907	0.605087	0.302544	+	0.953136i	$0.402164\pi$
$108$	−27.8112	−2.67613
$109$	−12.6376	−1.21047	−0.605233	−	0.796048i	$-0.706920\pi$
$109$	−12.6376	−1.21047	−0.605233	+	0.796048i	$0.706920\pi$
$110$	0	0
$111$	3.42718	0.325293
$112$	−10.9048	−1.03040
$113$	8.31845	0.782534	0.391267	−	0.920277i	$-0.372037\pi$
$113$	8.31845	0.782534	0.391267	+	0.920277i	$0.372037\pi$
$114$	3.08921	0.289331
$115$	0	0
$116$	8.03291	0.745837
$117$	6.93160	0.640826
$118$	−6.76032	−0.622338
$119$	1.98633	0.182087
$120$	0	0
$121$	−10.8843	−0.989483
$122$	5.05261	0.457442
$123$	−5.15373	−0.464697
$124$	31.4447	2.82381
$125$	0	0
$126$	−2.87905	−0.256486
$127$	−4.71260	−0.418176	−0.209088	−	0.977897i	$-0.567049\pi$
$127$	−4.71260	−0.418176	−0.209088	+	0.977897i	$0.567049\pi$
$128$	−3.90639	−0.345279
$129$	5.85352	0.515373
$130$	0	0
$131$	−13.5244	−1.18163	−0.590814	−	0.806808i	$-0.701193\pi$
$131$	−13.5244	−1.18163	−0.590814	+	0.806808i	$0.701193\pi$
$132$	2.34235	0.203875
$133$	0.884160	0.0766664
$134$	21.4198	1.85039
$135$	0	0
$136$	14.4543	1.23945
$137$	12.2503	1.04661	0.523306	−	0.852145i	$-0.324698\pi$
$137$	12.2503	1.04661	0.523306	+	0.852145i	$0.324698\pi$
$138$	2.26117	0.192483
$139$	−11.7399	−0.995762	−0.497881	−	0.867245i	$-0.665888\pi$
$139$	−11.7399	−0.995762	−0.497881	+	0.867245i	$0.665888\pi$
$140$	0	0
$141$	−1.19090	−0.100292
$142$	19.8933	1.66941
$143$	−2.27058	−0.189875
$144$	−10.7396	−0.894967
$145$	0	0
$146$	−17.1928	−1.42288
$147$	−8.24741	−0.680235
$148$	12.0317	0.988999
$149$	16.7360	1.37107	0.685533	−	0.728042i	$-0.259569\pi$
$149$	16.7360	1.37107	0.685533	+	0.728042i	$0.259569\pi$
$150$	0	0
$151$	5.05216	0.411139	0.205570	−	0.978642i	$-0.434095\pi$
$151$	5.05216	0.411139	0.205570	+	0.978642i	$0.434095\pi$
$152$	6.43393	0.521861
$153$	1.95624	0.158153
$154$	0.943087	0.0759961
$155$	0	0
$156$	−45.9751	−3.68095
$157$	−2.40755	−0.192144	−0.0960718	−	0.995374i	$-0.530628\pi$
$157$	−2.40755	−0.192144	−0.0960718	+	0.995374i	$0.530628\pi$
$158$	0.681109	0.0541861
$159$	10.8591	0.861185
$160$	0	0
$161$	0.647166	0.0510039
$162$	12.6424	0.993282
$163$	22.4228	1.75629	0.878145	−	0.478395i	$-0.158781\pi$
$163$	22.4228	1.75629	0.878145	+	0.478395i	$0.158781\pi$
$164$	−18.0931	−1.41283
$165$	0	0
$166$	−15.5417	−1.20627
$167$	−2.55035	−0.197352	−0.0986760	−	0.995120i	$-0.531461\pi$
$167$	−2.55035	−0.197352	−0.0986760	+	0.995120i	$0.531461\pi$
$168$	11.3286	0.874023
$169$	31.5663	2.42818
$170$	0	0
$171$	0.870767	0.0665892
$172$	20.5498	1.56691
$173$	14.3207	1.08878	0.544391	−	0.838832i	$-0.316761\pi$
$173$	14.3207	1.08878	0.544391	+	0.838832i	$0.316761\pi$
$174$	−6.01787	−0.456214
$175$	0	0
$176$	3.51796	0.265176
$177$	3.60016	0.270604
$178$	−9.83467	−0.737139
$179$	4.89603	0.365946	0.182973	−	0.983118i	$-0.441428\pi$
$179$	4.89603	0.365946	0.182973	+	0.983118i	$0.441428\pi$
$180$	0	0
$181$	11.3118	0.840797	0.420398	−	0.907340i	$-0.361890\pi$
$181$	11.3118	0.840797	0.420398	+	0.907340i	$0.361890\pi$
$182$	−18.5107	−1.37210
$183$	−2.69073	−0.198904
$184$	4.70936	0.347179
$185$	0	0
$186$	−23.5569	−1.72727
$187$	−0.640805	−0.0468603
$188$	−4.18086	−0.304920
$189$	5.96312	0.433753
$190$	0	0
$191$	22.7536	1.64640	0.823198	−	0.567755i	$-0.192188\pi$
$191$	22.7536	1.64640	0.823198	+	0.567755i	$0.192188\pi$
$192$	14.7114	1.06171
$193$	26.7507	1.92556	0.962780	−	0.270287i	$-0.0871186\pi$
$193$	26.7507	1.92556	0.962780	+	0.270287i	$0.0871186\pi$
$194$	47.1367	3.38421
$195$	0	0
$196$	−28.9540	−2.06814
$197$	10.3126	0.734746	0.367373	−	0.930074i	$-0.380257\pi$
$197$	10.3126	0.734746	0.367373	+	0.930074i	$0.380257\pi$
$198$	0.928801	0.0660070
$199$	7.83878	0.555676	0.277838	−	0.960628i	$-0.410382\pi$
$199$	7.83878	0.555676	0.277838	+	0.960628i	$0.410382\pi$
$200$	0	0
$201$	−11.4070	−0.804585
$202$	2.22383	0.156468
$203$	−1.72237	−0.120887
$204$	−12.9751	−0.908442
$205$	0	0
$206$	2.51563	0.175272
$207$	0.637364	0.0442998
$208$	−69.0497	−4.78773
$209$	−0.285236	−0.0197302
$210$	0	0
$211$	22.0494	1.51795	0.758973	−	0.651122i	$-0.225702\pi$
$211$	22.0494	1.51795	0.758973	+	0.651122i	$0.225702\pi$
$212$	38.1228	2.61829
$213$	−10.5940	−0.725891
$214$	−16.4615	−1.12529
$215$	0	0
$216$	43.3929	2.95252
$217$	−6.74219	−0.457690
$218$	33.2374	2.25112
$219$	9.15587	0.618696
$220$	0	0
$221$	12.5776	0.846058
$222$	−9.01357	−0.604951
$223$	−13.3306	−0.892680	−0.446340	−	0.894863i	$-0.647273\pi$
$223$	−13.3306	−0.892680	−0.446340	+	0.894863i	$0.647273\pi$
$224$	12.5031	0.835396
$225$	0	0
$226$	−21.8778	−1.45529
$227$	0.973112	0.0645877	0.0322939	−	0.999478i	$-0.489719\pi$
$227$	0.973112	0.0645877	0.0322939	+	0.999478i	$0.489719\pi$
$228$	−5.77552	−0.382493
$229$	23.3499	1.54301	0.771504	−	0.636225i	$-0.219505\pi$
$229$	23.3499	1.54301	0.771504	+	0.636225i	$0.219505\pi$
$230$	0	0
$231$	−0.502233	−0.0330445
$232$	−12.5335	−0.822865
$233$	8.88083	0.581802	0.290901	−	0.956753i	$-0.406045\pi$
$233$	8.88083	0.581802	0.290901	+	0.956753i	$0.406045\pi$
$234$	−18.2303	−1.19175
$235$	0	0
$236$	12.6390	0.822727
$237$	−0.362719	−0.0235611
$238$	−5.22411	−0.338629
$239$	−26.7991	−1.73349	−0.866746	−	0.498750i	$-0.833792\pi$
$239$	−26.7991	−1.73349	−0.866746	+	0.498750i	$0.833792\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	28.6261	1.84015
$243$	10.2356	0.656614
$244$	−9.44626	−0.604735
$245$	0	0
$246$	13.5545	0.864201
$247$	5.59855	0.356227
$248$	−49.0622	−3.11545
$249$	8.27664	0.524510
$250$	0	0
$251$	27.0375	1.70659	0.853296	−	0.521427i	$-0.174600\pi$
$251$	27.0375	1.70659	0.853296	+	0.521427i	$0.174600\pi$
$252$	5.38261	0.339073
$253$	−0.208781	−0.0131259
$254$	12.3943	0.777686
$255$	0	0
$256$	−10.7334	−0.670838
$257$	14.2022	0.885909	0.442954	−	0.896544i	$-0.353930\pi$
$257$	14.2022	0.885909	0.442954	+	0.896544i	$0.353930\pi$
$258$	−15.3949	−0.958445
$259$	−2.57977	−0.160299
$260$	0	0
$261$	−1.69628	−0.104997
$262$	35.5694	2.19749
$263$	6.25591	0.385756	0.192878	−	0.981223i	$-0.438218\pi$
$263$	6.25591	0.385756	0.192878	+	0.981223i	$0.438218\pi$
$264$	−3.65470	−0.224931
$265$	0	0
$266$	−2.32537	−0.142577
$267$	5.23737	0.320522
$268$	−40.0461	−2.44621
$269$	−18.2012	−1.10975	−0.554873	−	0.831935i	$-0.687233\pi$
$269$	−18.2012	−1.10975	−0.554873	+	0.831935i	$0.687233\pi$
$270$	0	0
$271$	2.18669	0.132832	0.0664160	−	0.997792i	$-0.478844\pi$
$271$	2.18669	0.132832	0.0664160	+	0.997792i	$0.478844\pi$
$272$	−19.4873	−1.18159
$273$	9.85772	0.596616
$274$	−32.2186	−1.94640
$275$	0	0
$276$	−4.22743	−0.254461
$277$	−30.7385	−1.84690	−0.923449	−	0.383720i	$-0.874643\pi$
$277$	−30.7385	−1.84690	−0.923449	+	0.383720i	$0.874643\pi$
$278$	30.8762	1.85183
$279$	−6.64006	−0.397530
$280$	0	0
$281$	−11.4664	−0.684029	−0.342014	−	0.939695i	$-0.611109\pi$
$281$	−11.4664	−0.684029	−0.342014	+	0.939695i	$0.611109\pi$
$282$	3.13210	0.186514
$283$	7.18393	0.427040	0.213520	−	0.976939i	$-0.431507\pi$
$283$	7.18393	0.427040	0.213520	+	0.976939i	$0.431507\pi$
$284$	−37.1922	−2.20695
$285$	0	0
$286$	5.97168	0.353113
$287$	3.87941	0.228995
$288$	12.3137	0.725590
$289$	−13.4503	−0.791197
$290$	0	0
$291$	−25.1023	−1.47152
$292$	32.1432	1.88104
$293$	−14.3751	−0.839805	−0.419903	−	0.907569i	$-0.637936\pi$
$293$	−14.3751	−0.839805	−0.419903	+	0.907569i	$0.637936\pi$
$294$	21.6909	1.26504
$295$	0	0
$296$	−18.7727	−1.09114
$297$	−1.92374	−0.111627
$298$	−44.0161	−2.54979
$299$	4.09790	0.236987
$300$	0	0
$301$	−4.40617	−0.253967
$302$	−13.2873	−0.764600
$303$	−1.18428	−0.0680354
$304$	−8.67422	−0.497501
$305$	0	0
$306$	−5.14498	−0.294119
$307$	−27.1770	−1.55107	−0.775537	−	0.631302i	$-0.782521\pi$
$307$	−27.1770	−1.55107	−0.775537	+	0.631302i	$0.782521\pi$
$308$	−1.76318	−0.100466
$309$	−1.33968	−0.0762116
$310$	0	0
$311$	2.64295	0.149868	0.0749341	−	0.997188i	$-0.476125\pi$
$311$	2.64295	0.149868	0.0749341	+	0.997188i	$0.476125\pi$
$312$	71.7335	4.06111
$313$	−1.22280	−0.0691167	−0.0345584	−	0.999403i	$-0.511002\pi$
$313$	−1.22280	−0.0691167	−0.0345584	+	0.999403i	$0.511002\pi$
$314$	6.33193	0.357332
$315$	0	0
$316$	−1.27339	−0.0716337
$317$	7.27855	0.408804	0.204402	−	0.978887i	$-0.434475\pi$
$317$	7.27855	0.408804	0.204402	+	0.978887i	$0.434475\pi$
$318$	−28.5598	−1.60155
$319$	0.555649	0.0311104
$320$	0	0
$321$	8.76646	0.489296
$322$	−1.70207	−0.0948524
$323$	1.58003	0.0879152
$324$	−23.6360	−1.31311
$325$	0	0
$326$	−58.9726	−3.26619
$327$	−17.7003	−0.978828
$328$	28.2301	1.55874
$329$	0.896436	0.0494221
$330$	0	0
$331$	21.7820	1.19725	0.598624	−	0.801030i	$-0.295714\pi$
$331$	21.7820	1.19725	0.598624	+	0.801030i	$0.295714\pi$
$332$	29.0565	1.59469
$333$	−2.54069	−0.139229
$334$	6.70749	0.367018
$335$	0	0
$336$	−15.2732	−0.833224
$337$	10.3985	0.566444	0.283222	−	0.959054i	$-0.408597\pi$
$337$	10.3985	0.566444	0.283222	+	0.959054i	$0.408597\pi$
$338$	−83.0204	−4.51571
$339$	11.6508	0.632786
$340$	0	0
$341$	2.17508	0.117787
$342$	−2.29014	−0.123837
$343$	13.5881	0.733691
$344$	−32.0632	−1.72873
$345$	0	0
$346$	−37.6638	−2.02482
$347$	33.3363	1.78959	0.894795	−	0.446478i	$-0.147322\pi$
$347$	33.3363	1.78959	0.894795	+	0.446478i	$0.147322\pi$
$348$	11.2509	0.603111
$349$	−1.05667	−0.0565625	−0.0282813	−	0.999600i	$-0.509003\pi$
$349$	−1.05667	−0.0565625	−0.0282813	+	0.999600i	$0.509003\pi$
$350$	0	0
$351$	37.7588	2.01542
$352$	−4.03358	−0.214990
$353$	20.8908	1.11190	0.555952	−	0.831214i	$-0.312354\pi$
$353$	20.8908	1.11190	0.555952	+	0.831214i	$0.312354\pi$
$354$	−9.46852	−0.503246
$355$	0	0
$356$	18.3867	0.974493
$357$	2.78206	0.147242
$358$	−12.8767	−0.680554
$359$	−4.92712	−0.260043	−0.130022	−	0.991511i	$-0.541505\pi$
$359$	−4.92712	−0.260043	−0.130022	+	0.991511i	$0.541505\pi$
$360$	0	0
$361$	−18.2967	−0.962984
$362$	−29.7503	−1.56364
$363$	−15.2446	−0.800133
$364$	34.6072	1.81391
$365$	0	0
$366$	7.07669	0.369904
$367$	−17.5814	−0.917741	−0.458870	−	0.888503i	$-0.651746\pi$
$367$	−17.5814	−0.917741	−0.458870	+	0.888503i	$0.651746\pi$
$368$	−6.34915	−0.330972
$369$	3.82065	0.198895
$370$	0	0
$371$	−8.17408	−0.424377
$372$	44.0414	2.28344
$373$	5.62638	0.291323	0.145661	−	0.989334i	$-0.453469\pi$
$373$	5.62638	0.291323	0.145661	+	0.989334i	$0.453469\pi$
$374$	1.68533	0.0871466
$375$	0	0
$376$	6.52326	0.336412
$377$	−10.9062	−0.561695
$378$	−15.6832	−0.806655
$379$	33.1203	1.70128	0.850639	−	0.525751i	$-0.176215\pi$
$379$	33.1203	1.70128	0.850639	+	0.525751i	$0.176215\pi$
$380$	0	0
$381$	−6.60047	−0.338152
$382$	−59.8427	−3.06182
$383$	−12.0598	−0.616229	−0.308115	−	0.951349i	$-0.599698\pi$
$383$	−12.0598	−0.616229	−0.308115	+	0.951349i	$0.599698\pi$
$384$	−5.47129	−0.279206
$385$	0	0
$386$	−70.3552	−3.58098
$387$	−4.33942	−0.220585
$388$	−88.1258	−4.47391
$389$	31.1477	1.57925	0.789625	−	0.613590i	$-0.210275\pi$
$389$	31.1477	1.57925	0.789625	+	0.613590i	$0.210275\pi$
$390$	0	0
$391$	1.15651	0.0584874
$392$	45.1760	2.28173
$393$	−18.9422	−0.955509
$394$	−27.1226	−1.36641
$395$	0	0
$396$	−1.73647	−0.0872608
$397$	16.5885	0.832553	0.416277	−	0.909238i	$-0.363335\pi$
$397$	16.5885	0.832553	0.416277	+	0.909238i	$0.363335\pi$
$398$	−20.6162	−1.03340
$399$	1.23836	0.0619953
$400$	0	0
$401$	32.2622	1.61110	0.805549	−	0.592529i	$-0.201870\pi$
$401$	32.2622	1.61110	0.805549	+	0.592529i	$0.201870\pi$
$402$	30.0006	1.49630
$403$	−42.6919	−2.12664
$404$	−4.15763	−0.206850
$405$	0	0
$406$	4.52988	0.224814
$407$	0.832252	0.0412532
$408$	20.2447	1.00226
$409$	−20.5807	−1.01765	−0.508825	−	0.860870i	$-0.669920\pi$
$409$	−20.5807	−1.01765	−0.508825	+	0.860870i	$0.669920\pi$
$410$	0	0
$411$	17.1578	0.846330
$412$	−4.70317	−0.231708
$413$	−2.70998	−0.133349
$414$	−1.67628	−0.0823849
$415$	0	0
$416$	79.1702	3.88164
$417$	−16.4429	−0.805210
$418$	0.750180	0.0366925
$419$	−25.8936	−1.26499	−0.632493	−	0.774566i	$-0.717968\pi$
$419$	−25.8936	−1.26499	−0.632493	+	0.774566i	$0.717968\pi$
$420$	0	0
$421$	25.0218	1.21949	0.609743	−	0.792599i	$-0.291273\pi$
$421$	25.0218	1.21949	0.609743	+	0.792599i	$0.291273\pi$
$422$	−57.9906	−2.82294
$423$	0.882857	0.0429260
$424$	−59.4819	−2.88870
$425$	0	0
$426$	27.8626	1.34995
$427$	2.02541	0.0980166
$428$	30.7762	1.48762
$429$	−3.18017	−0.153540
$430$	0	0
$431$	−6.85922	−0.330397	−0.165199	−	0.986260i	$-0.552826\pi$
$431$	−6.85922	−0.330397	−0.165199	+	0.986260i	$0.552826\pi$
$432$	−58.5023	−2.81469
$433$	24.6281	1.18355	0.591776	−	0.806103i	$-0.298427\pi$
$433$	24.6281	1.18355	0.591776	+	0.806103i	$0.298427\pi$
$434$	17.7321	0.851170
$435$	0	0
$436$	−62.1399	−2.97596
$437$	0.514790	0.0246257
$438$	−24.0802	−1.15060
$439$	−33.4693	−1.59740	−0.798701	−	0.601729i	$-0.794479\pi$
$439$	−33.4693	−1.59740	−0.798701	+	0.601729i	$0.794479\pi$
$440$	0	0
$441$	6.11411	0.291148
$442$	−33.0794	−1.57342
$443$	−21.0492	−1.00008	−0.500039	−	0.866003i	$-0.666681\pi$
$443$	−21.0492	−1.00008	−0.500039	+	0.866003i	$0.666681\pi$
$444$	16.8516	0.799742
$445$	0	0
$446$	35.0598	1.66013
$447$	23.4405	1.10870
$448$	−11.0739	−0.523191
$449$	20.9547	0.988913	0.494456	−	0.869202i	$-0.335367\pi$
$449$	20.9547	0.988913	0.494456	+	0.869202i	$0.335367\pi$
$450$	0	0
$451$	−1.25153	−0.0589321
$452$	40.9022	1.92388
$453$	7.07607	0.332463
$454$	−2.55931	−0.120114
$455$	0	0
$456$	9.01137	0.421996
$457$	−7.68098	−0.359301	−0.179650	−	0.983731i	$-0.557497\pi$
$457$	−7.68098	−0.359301	−0.179650	+	0.983731i	$0.557497\pi$
$458$	−61.4110	−2.86955
$459$	10.6563	0.497395
$460$	0	0
$461$	11.9447	0.556319	0.278160	−	0.960535i	$-0.410276\pi$
$461$	11.9447	0.556319	0.278160	+	0.960535i	$0.410276\pi$
$462$	1.32089	0.0614533
$463$	−27.7857	−1.29131	−0.645655	−	0.763630i	$-0.723415\pi$
$463$	−27.7857	−1.29131	−0.645655	+	0.763630i	$0.723415\pi$
$464$	16.8977	0.784454
$465$	0	0
$466$	−23.3568	−1.08198
$467$	0.688082	0.0318407	0.0159203	−	0.999873i	$-0.494932\pi$
$467$	0.688082	0.0318407	0.0159203	+	0.999873i	$0.494932\pi$
$468$	34.0830	1.57549
$469$	8.58646	0.396486
$470$	0	0
$471$	−3.37202	−0.155375
$472$	−19.7202	−0.907696
$473$	1.42146	0.0653588
$474$	0.953962	0.0438169
$475$	0	0
$476$	9.76689	0.447665
$477$	−8.05027	−0.368596
$478$	70.4825	3.22379
$479$	31.3059	1.43040	0.715202	−	0.698918i	$-0.246335\pi$
$479$	31.3059	1.43040	0.715202	+	0.698918i	$0.246335\pi$
$480$	0	0
$481$	−16.3352	−0.744823
$482$	−2.63003	−0.119795
$483$	0.906422	0.0412436
$484$	−53.5187	−2.43267
$485$	0	0
$486$	−26.9199	−1.22111
$487$	−19.6517	−0.890505	−0.445253	−	0.895405i	$-0.646886\pi$
$487$	−19.6517	−0.890505	−0.445253	+	0.895405i	$0.646886\pi$
$488$	14.7387	0.667190
$489$	31.4054	1.42020
$490$	0	0
$491$	−1.74379	−0.0786963	−0.0393481	−	0.999226i	$-0.512528\pi$
$491$	−1.74379	−0.0786963	−0.0393481	+	0.999226i	$0.512528\pi$
$492$	−25.3412	−1.14247
$493$	−3.07795	−0.138624
$494$	−14.7244	−0.662480
$495$	0	0
$496$	66.1455	2.97002
$497$	7.97453	0.357707
$498$	−21.7678	−0.975438
$499$	12.1886	0.545637	0.272818	−	0.962066i	$-0.412044\pi$
$499$	12.1886	0.545637	0.272818	+	0.962066i	$0.412044\pi$
$500$	0	0
$501$	−3.57202	−0.159586
$502$	−71.1094	−3.17377
$503$	−3.81849	−0.170258	−0.0851289	−	0.996370i	$-0.527130\pi$
$503$	−3.81849	−0.170258	−0.0851289	+	0.996370i	$0.527130\pi$
$504$	−8.39832	−0.374091
$505$	0	0
$506$	0.549099	0.0244104
$507$	44.2119	1.96352
$508$	−23.1721	−1.02809
$509$	27.0780	1.20021	0.600106	−	0.799921i	$-0.295125\pi$
$509$	27.0780	1.20021	0.600106	+	0.799921i	$0.295125\pi$
$510$	0	0
$511$	−6.89197	−0.304883
$512$	36.0420	1.59284
$513$	4.74337	0.209425
$514$	−37.3522	−1.64753
$515$	0	0
$516$	28.7820	1.26706
$517$	−0.289197	−0.0127188
$518$	6.78486	0.298110
$519$	20.0576	0.880430
$520$	0	0
$521$	42.2190	1.84965	0.924823	−	0.380397i	$-0.124213\pi$
$521$	42.2190	1.84965	0.924823	+	0.380397i	$0.124213\pi$
$522$	4.46127	0.195264
$523$	−11.7103	−0.512057	−0.256028	−	0.966669i	$-0.582414\pi$
$523$	−11.7103	−0.512057	−0.256028	+	0.966669i	$0.582414\pi$
$524$	−66.4999	−2.90506
$525$	0	0
$526$	−16.4532	−0.717394
$527$	−12.0486	−0.524844
$528$	4.92726	0.214431
$529$	−22.6232	−0.983617
$530$	0	0
$531$	−2.66893	−0.115822
$532$	4.34746	0.188486
$533$	24.5647	1.06401
$534$	−13.7744	−0.596078
$535$	0	0
$536$	62.4827	2.69884
$537$	6.85738	0.295918
$538$	47.8696	2.06380
$539$	−2.00279	−0.0862664
$540$	0	0
$541$	−45.8157	−1.96977	−0.984886	−	0.173204i	$-0.944588\pi$
$541$	−45.8157	−1.96977	−0.984886	+	0.173204i	$0.944588\pi$
$542$	−5.75106	−0.247029
$543$	15.8433	0.679900
$544$	22.3435	0.957969
$545$	0	0
$546$	−25.9261	−1.10953
$547$	−9.27779	−0.396690	−0.198345	−	0.980132i	$-0.563557\pi$
$547$	−9.27779	−0.396690	−0.198345	+	0.980132i	$0.563557\pi$
$548$	60.2353	2.57312
$549$	1.99473	0.0851332
$550$	0	0
$551$	−1.37006	−0.0583666
$552$	6.59593	0.280742
$553$	0.273033	0.0116105
$554$	80.8432	3.43470
$555$	0	0
$556$	−57.7255	−2.44811
$557$	22.4092	0.949507	0.474753	−	0.880119i	$-0.342537\pi$
$557$	22.4092	0.949507	0.474753	+	0.880119i	$0.342537\pi$
$558$	17.4636	0.739291
$559$	−27.9001	−1.18005
$560$	0	0
$561$	−0.897512	−0.0378930
$562$	30.1570	1.27210
$563$	−29.0486	−1.22425	−0.612126	−	0.790760i	$-0.709686\pi$
$563$	−29.0486	−1.22425	−0.612126	+	0.790760i	$0.709686\pi$
$564$	−5.85571	−0.246570
$565$	0	0
$566$	−18.8939	−0.794171
$567$	5.06790	0.212832
$568$	58.0298	2.43488
$569$	−27.4610	−1.15123	−0.575613	−	0.817722i	$-0.695237\pi$
$569$	−27.4610	−1.15123	−0.575613	+	0.817722i	$0.695237\pi$
$570$	0	0
$571$	−43.0161	−1.80017	−0.900083	−	0.435718i	$-0.856495\pi$
$571$	−43.0161	−1.80017	−0.900083	+	0.435718i	$0.856495\pi$
$572$	−11.1645	−0.466813
$573$	31.8688	1.33134
$574$	−10.2030	−0.425864
$575$	0	0
$576$	−10.9061	−0.454422
$577$	3.73661	0.155557	0.0777785	−	0.996971i	$-0.475217\pi$
$577$	3.73661	0.155557	0.0777785	+	0.996971i	$0.475217\pi$
$578$	35.3748	1.47140
$579$	37.4671	1.55708
$580$	0	0
$581$	−6.23014	−0.258470
$582$	66.0197	2.73660
$583$	2.63702	0.109214
$584$	−50.1521	−2.07531
$585$	0	0
$586$	37.8070	1.56179
$587$	−29.2099	−1.20562	−0.602812	−	0.797884i	$-0.705953\pi$
$587$	−29.2099	−1.20562	−0.602812	+	0.797884i	$0.705953\pi$
$588$	−40.5529	−1.67237
$589$	−5.36308	−0.220982
$590$	0	0
$591$	14.4439	0.594143
$592$	25.3093	1.04021
$593$	34.1877	1.40392	0.701961	−	0.712215i	$-0.252308\pi$
$593$	34.1877	1.40392	0.701961	+	0.712215i	$0.252308\pi$
$594$	5.05950	0.207594
$595$	0	0
$596$	82.2917	3.37080
$597$	10.9790	0.449341
$598$	−10.7776	−0.440728
$599$	21.0328	0.859378	0.429689	−	0.902977i	$-0.358623\pi$
$599$	21.0328	0.859378	0.429689	+	0.902977i	$0.358623\pi$
$600$	0	0
$601$	−27.0695	−1.10419	−0.552093	−	0.833783i	$-0.686171\pi$
$601$	−27.0695	−1.10419	−0.552093	+	0.833783i	$0.686171\pi$
$602$	11.5883	0.472306
$603$	8.45640	0.344371
$604$	24.8417	1.01080
$605$	0	0
$606$	3.11470	0.126526
$607$	−37.6086	−1.52649	−0.763244	−	0.646111i	$-0.776394\pi$
$607$	−37.6086	−1.52649	−0.763244	+	0.646111i	$0.776394\pi$
$608$	9.94558	0.403347
$609$	−2.41235	−0.0977535
$610$	0	0
$611$	5.67628	0.229638
$612$	9.61895	0.388823
$613$	1.33214	0.0538046	0.0269023	−	0.999638i	$-0.491436\pi$
$613$	1.33214	0.0538046	0.0269023	+	0.999638i	$0.491436\pi$
$614$	71.4763	2.88455
$615$	0	0
$616$	2.75103	0.110842
$617$	18.6238	0.749767	0.374884	−	0.927072i	$-0.377683\pi$
$617$	18.6238	0.749767	0.374884	+	0.927072i	$0.377683\pi$
$618$	3.52339	0.141732
$619$	27.4064	1.10156	0.550778	−	0.834651i	$-0.314331\pi$
$619$	27.4064	1.10156	0.550778	+	0.834651i	$0.314331\pi$
$620$	0	0
$621$	3.47194	0.139324
$622$	−6.95105	−0.278712
$623$	−3.94237	−0.157948
$624$	−96.7111	−3.87154
$625$	0	0
$626$	3.21600	0.128537
$627$	−0.399502	−0.0159546
$628$	−11.8381	−0.472390
$629$	−4.61015	−0.183819
$630$	0	0
$631$	27.0454	1.07666	0.538330	−	0.842734i	$-0.319056\pi$
$631$	27.0454	1.07666	0.538330	+	0.842734i	$0.319056\pi$
$632$	1.98683	0.0790318
$633$	30.8825	1.22747
$634$	−19.1428	−0.760257
$635$	0	0
$636$	53.3949	2.11724
$637$	39.3103	1.55753
$638$	−1.46137	−0.0578563
$639$	7.85374	0.310689
$640$	0	0
$641$	−25.3918	−1.00292	−0.501458	−	0.865182i	$-0.667203\pi$
$641$	−25.3918	−1.00292	−0.501458	+	0.865182i	$0.667203\pi$
$642$	−23.0560	−0.909950
$643$	36.4163	1.43612	0.718059	−	0.695982i	$-0.245031\pi$
$643$	36.4163	1.43612	0.718059	+	0.695982i	$0.245031\pi$
$644$	3.18215	0.125394
$645$	0	0
$646$	−4.15552	−0.163497
$647$	30.1565	1.18557	0.592786	−	0.805360i	$-0.298028\pi$
$647$	30.1565	1.18557	0.592786	+	0.805360i	$0.298028\pi$
$648$	36.8785	1.44873
$649$	0.874258	0.0343176
$650$	0	0
$651$	−9.44312	−0.370105
$652$	110.254	4.31788
$653$	13.0434	0.510428	0.255214	−	0.966885i	$-0.417854\pi$
$653$	13.0434	0.510428	0.255214	+	0.966885i	$0.417854\pi$
$654$	46.5523	1.82034
$655$	0	0
$656$	−38.0597	−1.48598
$657$	−6.78758	−0.264809
$658$	−2.35765	−0.0919108
$659$	23.8553	0.929270	0.464635	−	0.885502i	$-0.346186\pi$
$659$	23.8553	0.929270	0.464635	+	0.885502i	$0.346186\pi$
$660$	0	0
$661$	21.4176	0.833047	0.416524	−	0.909125i	$-0.363248\pi$
$661$	21.4176	0.833047	0.416524	+	0.909125i	$0.363248\pi$
$662$	−57.2873	−2.22654
$663$	17.6161	0.684155
$664$	−45.3361	−1.75938
$665$	0	0
$666$	6.68209	0.258926
$667$	−1.00283	−0.0388296
$668$	−12.5402	−0.485195
$669$	−18.6708	−0.721855
$670$	0	0
$671$	−0.653413	−0.0252247
$672$	17.5118	0.675533
$673$	33.7918	1.30258	0.651289	−	0.758830i	$-0.274229\pi$
$673$	33.7918	1.30258	0.651289	+	0.758830i	$0.274229\pi$
$674$	−27.3484	−1.05342
$675$	0	0
$676$	155.213	5.96974
$677$	33.1742	1.27499	0.637495	−	0.770455i	$-0.279971\pi$
$677$	33.1742	1.27499	0.637495	+	0.770455i	$0.279971\pi$
$678$	−30.6420	−1.17680
$679$	18.8954	0.725140
$680$	0	0
$681$	1.36294	0.0522280
$682$	−5.72052	−0.219050
$683$	−38.5145	−1.47372	−0.736858	−	0.676047i	$-0.763691\pi$
$683$	−38.5145	−1.47372	−0.736858	+	0.676047i	$0.763691\pi$
$684$	4.28161	0.163711
$685$	0	0
$686$	−35.7372	−1.36445
$687$	32.7040	1.24773
$688$	43.2275	1.64803
$689$	−51.7588	−1.97185
$690$	0	0
$691$	13.4298	0.510894	0.255447	−	0.966823i	$-0.417777\pi$
$691$	13.4298	0.510894	0.255447	+	0.966823i	$0.417777\pi$
$692$	70.4155	2.67680
$693$	0.372324	0.0141434
$694$	−87.6755	−3.32812
$695$	0	0
$696$	−17.5544	−0.665399
$697$	6.93267	0.262594
$698$	2.77908	0.105190
$699$	12.4385	0.470467
$700$	0	0
$701$	40.6852	1.53666	0.768330	−	0.640054i	$-0.221088\pi$
$701$	40.6852	1.53666	0.768330	+	0.640054i	$0.221088\pi$
$702$	−99.3067	−3.74809
$703$	−2.05208	−0.0773957
$704$	3.57251	0.134644
$705$	0	0
$706$	−54.9434	−2.06782
$707$	0.891456	0.0335267
$708$	17.7022	0.665288
$709$	−4.38707	−0.164760	−0.0823800	−	0.996601i	$-0.526252\pi$
$709$	−4.38707	−0.164760	−0.0823800	+	0.996601i	$0.526252\pi$
$710$	0	0
$711$	0.268897	0.0100844
$712$	−28.6882	−1.07514
$713$	−3.92554	−0.147013
$714$	−7.31689	−0.273828
$715$	0	0
$716$	24.0740	0.899688
$717$	−37.5349	−1.40177
$718$	12.9585	0.483605
$719$	−11.4502	−0.427021	−0.213511	−	0.976941i	$-0.568490\pi$
$719$	−11.4502	−0.427021	−0.213511	+	0.976941i	$0.568490\pi$
$720$	0	0
$721$	1.00843	0.0375558
$722$	48.1208	1.79087
$723$	1.40060	0.0520889
$724$	55.6205	2.06712
$725$	0	0
$726$	40.0937	1.48802
$727$	36.1917	1.34228	0.671138	−	0.741333i	$-0.265806\pi$
$727$	36.1917	1.34228	0.671138	+	0.741333i	$0.265806\pi$
$728$	−53.9966	−2.00125
$729$	28.7568	1.06507
$730$	0	0
$731$	−7.87400	−0.291230
$732$	−13.2304	−0.489011
$733$	27.6382	1.02084	0.510420	−	0.859925i	$-0.329490\pi$
$733$	27.6382	1.02084	0.510420	+	0.859925i	$0.329490\pi$
$734$	46.2395	1.70673
$735$	0	0
$736$	7.27973	0.268335
$737$	−2.77005	−0.102036
$738$	−10.0484	−0.369888
$739$	−15.7570	−0.579630	−0.289815	−	0.957083i	$-0.593594\pi$
$739$	−15.7570	−0.579630	−0.289815	+	0.957083i	$0.593594\pi$
$740$	0	0
$741$	7.84134	0.288059
$742$	21.4981	0.789219
$743$	15.0656	0.552702	0.276351	−	0.961057i	$-0.410875\pi$
$743$	15.0656	0.552702	0.276351	+	0.961057i	$0.410875\pi$
$744$	−68.7165	−2.51927
$745$	0	0
$746$	−14.7975	−0.541776
$747$	−6.13577	−0.224496
$748$	−3.15087	−0.115207
$749$	−6.59885	−0.241117
$750$	0	0
$751$	52.9339	1.93158	0.965792	−	0.259317i	$-0.0834975\pi$
$751$	52.9339	1.93158	0.965792	+	0.259317i	$0.0834975\pi$
$752$	−8.79465	−0.320708
$753$	37.8688	1.38001
$754$	28.6835	1.04459
$755$	0	0
$756$	29.3209	1.06639
$757$	−23.7838	−0.864436	−0.432218	−	0.901769i	$-0.642269\pi$
$757$	−23.7838	−0.864436	−0.432218	+	0.901769i	$0.642269\pi$
$758$	−87.1074	−3.16388
$759$	−0.292418	−0.0106141
$760$	0	0
$761$	−13.4064	−0.485981	−0.242990	−	0.970029i	$-0.578128\pi$
$761$	−13.4064	−0.485981	−0.242990	+	0.970029i	$0.578128\pi$
$762$	17.3594	0.628866
$763$	13.3237	0.482350
$764$	111.881	4.04770
$765$	0	0
$766$	31.7177	1.14601
$767$	−17.1597	−0.619602
$768$	−15.0332	−0.542465
$769$	5.63293	0.203129	0.101564	−	0.994829i	$-0.467615\pi$
$769$	5.63293	0.203129	0.101564	+	0.994829i	$0.467615\pi$
$770$	0	0
$771$	19.8916	0.716379
$772$	131.535	4.73404
$773$	−25.2946	−0.909782	−0.454891	−	0.890547i	$-0.650322\pi$
$773$	−25.2946	−0.909782	−0.454891	+	0.890547i	$0.650322\pi$
$774$	11.4128	0.410225
$775$	0	0
$776$	137.500	4.93596
$777$	−3.61323	−0.129624
$778$	−81.9193	−2.93695
$779$	3.08589	0.110563
$780$	0	0
$781$	−2.57264	−0.0920564
$782$	−3.04166	−0.108770
$783$	−9.24023	−0.330219
$784$	−60.9062	−2.17522
$785$	0	0
$786$	49.8186	1.77697
$787$	−0.967855	−0.0345003	−0.0172501	−	0.999851i	$-0.505491\pi$
$787$	−0.967855	−0.0345003	−0.0172501	+	0.999851i	$0.505491\pi$
$788$	50.7078	1.80639
$789$	8.76203	0.311937
$790$	0	0
$791$	−8.77003	−0.311826
$792$	2.70936	0.0962729
$793$	12.8250	0.455430
$794$	−43.6282	−1.54831
$795$	0	0
$796$	38.5437	1.36614
$797$	14.8849	0.527252	0.263626	−	0.964625i	$-0.415082\pi$
$797$	14.8849	0.527252	0.263626	+	0.964625i	$0.415082\pi$
$798$	−3.25691	−0.115293
$799$	1.60197	0.0566735
$800$	0	0
$801$	−3.88266	−0.137187
$802$	−84.8506	−2.99618
$803$	2.22340	0.0784621
$804$	−56.0886	−1.97809
$805$	0	0
$806$	112.281	3.95493
$807$	−25.4926	−0.897381
$808$	6.48703	0.228213
$809$	0.815220	0.0286616	0.0143308	−	0.999897i	$-0.495438\pi$
$809$	0.815220	0.0286616	0.0143308	+	0.999897i	$0.495438\pi$
$810$	0	0
$811$	1.64228	0.0576682	0.0288341	−	0.999584i	$-0.490821\pi$
$811$	1.64228	0.0576682	0.0288341	+	0.999584i	$0.490821\pi$
$812$	−8.46898	−0.297203
$813$	3.06268	0.107413
$814$	−2.18885	−0.0767190
$815$	0	0
$816$	−27.2939	−0.955478
$817$	−3.50489	−0.122621
$818$	54.1278	1.89253
$819$	−7.30789	−0.255358
$820$	0	0
$821$	−22.1336	−0.772467	−0.386234	−	0.922401i	$-0.626224\pi$
$821$	−22.1336	−0.772467	−0.386234	+	0.922401i	$0.626224\pi$
$822$	−45.1254	−1.57393
$823$	4.14878	0.144617	0.0723087	−	0.997382i	$-0.476963\pi$
$823$	4.14878	0.144617	0.0723087	+	0.997382i	$0.476963\pi$
$824$	7.33821	0.255639
$825$	0	0
$826$	7.12732	0.247991
$827$	−25.2438	−0.877813	−0.438907	−	0.898533i	$-0.644634\pi$
$827$	−25.2438	−0.877813	−0.438907	+	0.898533i	$0.644634\pi$
$828$	3.13395	0.108912
$829$	4.46911	0.155219	0.0776093	−	0.996984i	$-0.475271\pi$
$829$	4.46911	0.155219	0.0776093	+	0.996984i	$0.475271\pi$
$830$	0	0
$831$	−43.0524	−1.49347
$832$	−70.1204	−2.43099
$833$	11.0942	0.384391
$834$	43.2452	1.49746
$835$	0	0
$836$	−1.40252	−0.0485072
$837$	−36.1707	−1.25024
$838$	68.1009	2.35251
$839$	22.4342	0.774515	0.387258	−	0.921972i	$-0.373422\pi$
$839$	22.4342	0.774515	0.387258	+	0.921972i	$0.373422\pi$
$840$	0	0
$841$	−26.3311	−0.907968
$842$	−65.8080	−2.26789
$843$	−16.0599	−0.553131
$844$	108.418	3.73191
$845$	0	0
$846$	−2.32194	−0.0798299
$847$	11.4752	0.394292
$848$	80.1934	2.75385
$849$	10.0618	0.345321
$850$	0	0
$851$	−1.50203	−0.0514890
$852$	−52.0914	−1.78462
$853$	29.4281	1.00760	0.503800	−	0.863821i	$-0.331935\pi$
$853$	29.4281	1.00760	0.503800	+	0.863821i	$0.331935\pi$
$854$	−5.32690	−0.182283
$855$	0	0
$856$	−48.0191	−1.64126
$857$	−46.8759	−1.60125	−0.800625	−	0.599166i	$-0.795499\pi$
$857$	−46.8759	−1.60125	−0.800625	+	0.599166i	$0.795499\pi$
$858$	8.36394	0.285540
$859$	−6.14715	−0.209738	−0.104869	−	0.994486i	$-0.533442\pi$
$859$	−6.14715	−0.209738	−0.104869	+	0.994486i	$0.533442\pi$
$860$	0	0
$861$	5.43351	0.185174
$862$	18.0399	0.614443
$863$	23.1181	0.786950	0.393475	−	0.919335i	$-0.371273\pi$
$863$	23.1181	0.786950	0.393475	+	0.919335i	$0.371273\pi$
$864$	67.0769	2.28200
$865$	0	0
$866$	−64.7726	−2.20106
$867$	−18.8386	−0.639791
$868$	−33.1517	−1.12524
$869$	−0.0880823	−0.00298799
$870$	0	0
$871$	54.3700	1.84226
$872$	96.9550	3.28331
$873$	18.6092	0.629827
$874$	−1.35391	−0.0457967
$875$	0	0
$876$	45.0198	1.52108
$877$	−15.5761	−0.525969	−0.262984	−	0.964800i	$-0.584707\pi$
$877$	−15.5761	−0.525969	−0.262984	+	0.964800i	$0.584707\pi$
$878$	88.0251	2.97070
$879$	−20.1338	−0.679098
$880$	0	0
$881$	−39.6132	−1.33460	−0.667302	−	0.744787i	$-0.732551\pi$
$881$	−39.6132	−1.33460	−0.667302	+	0.744787i	$0.732551\pi$
$882$	−16.0803	−0.541451
$883$	−42.6806	−1.43632	−0.718158	−	0.695880i	$-0.755015\pi$
$883$	−42.6806	−1.43632	−0.718158	+	0.695880i	$0.755015\pi$
$884$	61.8445	2.08006
$885$	0	0
$886$	55.3600	1.85986
$887$	25.9509	0.871345	0.435672	−	0.900105i	$-0.356511\pi$
$887$	25.9509	0.871345	0.435672	+	0.900105i	$0.356511\pi$
$888$	−26.2930	−0.882337
$889$	4.96843	0.166636
$890$	0	0
$891$	−1.63494	−0.0547726
$892$	−65.5470	−2.19468
$893$	0.713071	0.0238620
$894$	−61.6491	−2.06185
$895$	0	0
$896$	4.11845	0.137588
$897$	5.73952	0.191637
$898$	−55.1114	−1.83909
$899$	10.4475	0.348442
$900$	0	0
$901$	−14.6074	−0.486644
$902$	3.29155	0.109597
$903$	−6.17128	−0.205367
$904$	−63.8185	−2.12257
$905$	0	0
$906$	−18.6103	−0.618284
$907$	−5.90063	−0.195927	−0.0979636	−	0.995190i	$-0.531233\pi$
$907$	−5.90063	−0.195927	−0.0979636	+	0.995190i	$0.531233\pi$
$908$	4.78484	0.158790
$909$	0.877953	0.0291199
$910$	0	0
$911$	−50.2483	−1.66480	−0.832399	−	0.554176i	$-0.813033\pi$
$911$	−50.2483	−1.66480	−0.832399	+	0.554176i	$0.813033\pi$
$912$	−12.1491	−0.402298
$913$	2.00989	0.0665176
$914$	20.2012	0.668196
$915$	0	0
$916$	114.813	3.79352
$917$	14.2585	0.470859
$918$	−28.0265	−0.925011
$919$	−14.8216	−0.488921	−0.244460	−	0.969659i	$-0.578611\pi$
$919$	−14.8216	−0.488921	−0.244460	+	0.969659i	$0.578611\pi$
$920$	0	0
$921$	−38.0641	−1.25426
$922$	−31.4148	−1.03459
$923$	50.4952	1.66207
$924$	−2.46951	−0.0812408
$925$	0	0
$926$	73.0771	2.40146
$927$	0.993152	0.0326194
$928$	−19.3743	−0.635993
$929$	−14.1382	−0.463858	−0.231929	−	0.972733i	$-0.574504\pi$
$929$	−14.1382	−0.463858	−0.231929	+	0.972733i	$0.574504\pi$
$930$	0	0
$931$	4.93828	0.161845
$932$	43.6675	1.43038
$933$	3.70173	0.121189
$934$	−1.80968	−0.0592144
$935$	0	0
$936$	−53.1787	−1.73820
$937$	−55.6047	−1.81653	−0.908263	−	0.418399i	$-0.862591\pi$
$937$	−55.6047	−1.81653	−0.908263	+	0.418399i	$0.862591\pi$
$938$	−22.5826	−0.737349
$939$	−1.71265	−0.0558904
$940$	0	0
$941$	−49.0980	−1.60055	−0.800274	−	0.599635i	$-0.795313\pi$
$941$	−49.0980	−1.60055	−0.800274	+	0.599635i	$0.795313\pi$
$942$	8.86851	0.288952
$943$	2.25873	0.0735545
$944$	26.5868	0.865325
$945$	0	0
$946$	−3.73848	−0.121549
$947$	50.8425	1.65216	0.826079	−	0.563554i	$-0.190566\pi$
$947$	50.8425	1.65216	0.826079	+	0.563554i	$0.190566\pi$
$948$	−1.78351	−0.0579257
$949$	−43.6404	−1.41663
$950$	0	0
$951$	10.1943	0.330574
$952$	−15.2390	−0.493898
$953$	22.7876	0.738161	0.369081	−	0.929397i	$-0.379673\pi$
$953$	22.7876	0.738161	0.369081	+	0.929397i	$0.379673\pi$
$954$	21.1724	0.685483
$955$	0	0
$956$	−131.773	−4.26183
$957$	0.778243	0.0251570
$958$	−82.3354	−2.66014
$959$	−12.9153	−0.417057
$960$	0	0
$961$	9.89637	0.319238
$962$	42.9621	1.38516
$963$	−6.49890	−0.209424
$964$	4.91705	0.158368
$965$	0	0
$966$	−2.38392	−0.0767012
$967$	25.3877	0.816413	0.408207	−	0.912890i	$-0.366154\pi$
$967$	25.3877	0.816413	0.408207	+	0.912890i	$0.366154\pi$
$968$	83.5037	2.68391
$969$	2.21299	0.0710915
$970$	0	0
$971$	−33.7404	−1.08278	−0.541391	−	0.840771i	$-0.682102\pi$
$971$	−33.7404	−1.08278	−0.541391	+	0.840771i	$0.682102\pi$
$972$	50.3289	1.61430
$973$	12.3772	0.396794
$974$	51.6846	1.65608
$975$	0	0
$976$	−19.8707	−0.636046
$977$	−5.59644	−0.179046	−0.0895230	−	0.995985i	$-0.528534\pi$
$977$	−5.59644	−0.179046	−0.0895230	+	0.995985i	$0.528534\pi$
$978$	−82.5971	−2.64116
$979$	1.27184	0.0406481
$980$	0	0
$981$	13.1219	0.418949
$982$	4.58623	0.146352
$983$	44.6820	1.42513	0.712567	−	0.701604i	$-0.247532\pi$
$983$	44.6820	1.42513	0.712567	+	0.701604i	$0.247532\pi$
$984$	39.5391	1.26046
$985$	0	0
$986$	8.09509	0.257800
$987$	1.25555	0.0399646
$988$	27.5284	0.875794
$989$	−2.56543	−0.0815759
$990$	0	0
$991$	39.6401	1.25921	0.629605	−	0.776915i	$-0.283217\pi$
$991$	39.6401	1.25921	0.629605	+	0.776915i	$0.283217\pi$
$992$	−75.8404	−2.40793
$993$	30.5079	0.968140
$994$	−20.9732	−0.665231
$995$	0	0
$996$	40.6966	1.28952
$997$	15.8185	0.500976	0.250488	−	0.968120i	$-0.419409\pi$
$997$	15.8185	0.500976	0.250488	+	0.968120i	$0.419409\pi$
$998$	−32.0564	−1.01473
$999$	−13.8400	−0.437879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.k.1.1		25
5.4	even	2		1205.2.a.d.1.25	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.d.1.25	✓	25	5.4	even	2
6025.2.a.k.1.1		25	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)

\(f(q)\)	\(=\)	\(q-2.63003 q^{2} +1.40060 q^{3} +4.91705 q^{4} -3.68362 q^{6} -1.05429 q^{7} -7.67192 q^{8} -1.03832 q^{9} +O(q^{10})\)	\(q-2.63003 q^{2} +1.40060 q^{3} +4.91705 q^{4} -3.68362 q^{6} -1.05429 q^{7} -7.67192 q^{8} -1.03832 q^{9} +0.340120 q^{11} +6.88683 q^{12} -6.67580 q^{13} +2.77280 q^{14} +10.3433 q^{16} -1.88405 q^{17} +2.73080 q^{18} -0.838634 q^{19} -1.47663 q^{21} -0.894526 q^{22} -0.613843 q^{23} -10.7453 q^{24} +17.5576 q^{26} -5.65607 q^{27} -5.18398 q^{28} +1.63368 q^{29} +6.39503 q^{31} -11.8593 q^{32} +0.476373 q^{33} +4.95511 q^{34} -5.10545 q^{36} +2.44693 q^{37} +2.20563 q^{38} -9.35014 q^{39} -3.67966 q^{41} +3.88359 q^{42} +4.17929 q^{43} +1.67239 q^{44} +1.61443 q^{46} -0.850277 q^{47} +14.4868 q^{48} -5.88848 q^{49} -2.63881 q^{51} -32.8253 q^{52} +7.75319 q^{53} +14.8756 q^{54} +8.08840 q^{56} -1.17459 q^{57} -4.29664 q^{58} +2.57044 q^{59} -1.92112 q^{61} -16.8191 q^{62} +1.09468 q^{63} +10.5037 q^{64} -1.25287 q^{66} -8.14434 q^{67} -9.26398 q^{68} -0.859749 q^{69} -7.56392 q^{71} +7.96589 q^{72} +6.53710 q^{73} -6.43550 q^{74} -4.12360 q^{76} -0.358584 q^{77} +24.5911 q^{78} -0.258974 q^{79} -4.80695 q^{81} +9.67761 q^{82} +5.90935 q^{83} -7.26069 q^{84} -10.9916 q^{86} +2.28814 q^{87} -2.60938 q^{88} +3.73938 q^{89} +7.03821 q^{91} -3.01830 q^{92} +8.95688 q^{93} +2.23625 q^{94} -16.6101 q^{96} -17.9225 q^{97} +15.4869 q^{98} -0.353153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Introduction

L-functions

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Database

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