LMFDB - Embedded newform 6025.2.a.m.1.10

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:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.98624 q^{2} +2.93351 q^{3} +1.94516 q^{4} -5.82666 q^{6} -3.87755 q^{7} +0.108919 q^{8} +5.60547 q^{9} +O(q^{10})$	\(q-1.98624 q^{2} +2.93351 q^{3} +1.94516 q^{4} -5.82666 q^{6} -3.87755 q^{7} +0.108919 q^{8} +5.60547 q^{9} +5.43632 q^{11} +5.70615 q^{12} +4.51589 q^{13} +7.70177 q^{14} -4.10667 q^{16} -3.56474 q^{17} -11.1338 q^{18} -4.95200 q^{19} -11.3748 q^{21} -10.7979 q^{22} -8.91648 q^{23} +0.319515 q^{24} -8.96966 q^{26} +7.64316 q^{27} -7.54248 q^{28} +2.56121 q^{29} -1.87389 q^{31} +7.93900 q^{32} +15.9475 q^{33} +7.08045 q^{34} +10.9035 q^{36} -8.26816 q^{37} +9.83588 q^{38} +13.2474 q^{39} +1.60127 q^{41} +22.5932 q^{42} -10.9508 q^{43} +10.5745 q^{44} +17.7103 q^{46} -0.538956 q^{47} -12.0469 q^{48} +8.03542 q^{49} -10.4572 q^{51} +8.78415 q^{52} +6.76785 q^{53} -15.1812 q^{54} -0.422340 q^{56} -14.5267 q^{57} -5.08719 q^{58} -7.99400 q^{59} -6.30204 q^{61} +3.72201 q^{62} -21.7355 q^{63} -7.55546 q^{64} -31.6756 q^{66} +6.40319 q^{67} -6.93401 q^{68} -26.1566 q^{69} -15.2258 q^{71} +0.610543 q^{72} -8.98213 q^{73} +16.4226 q^{74} -9.63245 q^{76} -21.0796 q^{77} -26.3126 q^{78} -5.39894 q^{79} +5.60486 q^{81} -3.18051 q^{82} +10.5577 q^{83} -22.1259 q^{84} +21.7510 q^{86} +7.51333 q^{87} +0.592120 q^{88} +10.9001 q^{89} -17.5106 q^{91} -17.3440 q^{92} -5.49708 q^{93} +1.07050 q^{94} +23.2891 q^{96} -3.42286 q^{97} -15.9603 q^{98} +30.4731 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9	$ 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9} + q^{11} - 26 q^{12} - 11 q^{13} - q^{14} + 43 q^{16} - 20 q^{17} - 18 q^{18} + 2 q^{21} - 23 q^{22} - 79 q^{23} - 2 q^{24} + 2 q^{26} - 26 q^{27} - 30 q^{28} + 2 q^{29} + q^{31} - 68 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 16 q^{37} - 45 q^{38} - 2 q^{39} - 2 q^{41} - 19 q^{42} - 25 q^{43} + 3 q^{44} + 14 q^{46} - 88 q^{47} - 75 q^{48} + 40 q^{49} - 10 q^{51} - 18 q^{52} - 34 q^{53} + 4 q^{54} - 15 q^{56} - 51 q^{57} - 53 q^{58} + q^{59} + 9 q^{61} - 39 q^{62} - 110 q^{63} + 17 q^{64} + 26 q^{66} - 30 q^{67} - 44 q^{68} - 7 q^{69} + 5 q^{71} - 18 q^{72} - 23 q^{73} - 18 q^{74} + 43 q^{76} - 30 q^{77} - 46 q^{78} + 5 q^{79} + 44 q^{81} - 5 q^{82} - 65 q^{83} - 65 q^{84} + 40 q^{86} - 33 q^{87} - 71 q^{88} - 9 q^{89} + q^{91} - 117 q^{92} - 68 q^{93} - 72 q^{94} + 83 q^{96} + 8 q^{97} - 76 q^{98} - 40 q^{99}+O(q^{100}) $ 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9 + q^11 - 26 * q^12 - 11 * q^13 - q^14 + 43 * q^16 - 20 * q^17 - 18 * q^18 + 2 * q^21 - 23 * q^22 - 79 * q^23 - 2 * q^24 + 2 * q^26 - 26 * q^27 - 30 * q^28 + 2 * q^29 + q^31 - 68 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 16 * q^37 - 45 * q^38 - 2 * q^39 - 2 * q^41 - 19 * q^42 - 25 * q^43 + 3 * q^44 + 14 * q^46 - 88 * q^47 - 75 * q^48 + 40 * q^49 - 10 * q^51 - 18 * q^52 - 34 * q^53 + 4 * q^54 - 15 * q^56 - 51 * q^57 - 53 * q^58 + q^59 + 9 * q^61 - 39 * q^62 - 110 * q^63 + 17 * q^64 + 26 * q^66 - 30 * q^67 - 44 * q^68 - 7 * q^69 + 5 * q^71 - 18 * q^72 - 23 * q^73 - 18 * q^74 + 43 * q^76 - 30 * q^77 - 46 * q^78 + 5 * q^79 + 44 * q^81 - 5 * q^82 - 65 * q^83 - 65 * q^84 + 40 * q^86 - 33 * q^87 - 71 * q^88 - 9 * q^89 + q^91 - 117 * q^92 - 68 * q^93 - 72 * q^94 + 83 * q^96 + 8 * q^97 - 76 * q^98 - 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.98624	−1.40449	−0.702243	−	0.711937i	$-0.747818\pi$
$2$	−1.98624	−1.40449	−0.702243	+	0.711937i	$0.747818\pi$
$3$	2.93351	1.69366	0.846831	−	0.531863i	$-0.178508\pi$
$3$	2.93351	1.69366	0.846831	+	0.531863i	$0.178508\pi$
$4$	1.94516	0.972582
$5$	0	0
$5$	0	0
$6$	−5.82666	−2.37872
$7$	−3.87755	−1.46558	−0.732789	−	0.680456i	$-0.761782\pi$
$7$	−3.87755	−1.46558	−0.732789	+	0.680456i	$0.761782\pi$
$8$	0.108919	0.0385087
$9$	5.60547	1.86849
$10$	0	0
$11$	5.43632	1.63911	0.819556	−	0.572999i	$-0.194220\pi$
$11$	5.43632	1.63911	0.819556	+	0.572999i	$0.194220\pi$
$12$	5.70615	1.64722
$13$	4.51589	1.25248	0.626242	−	0.779629i	$-0.284592\pi$
$13$	4.51589	1.25248	0.626242	+	0.779629i	$0.284592\pi$
$14$	7.70177	2.05838
$15$	0	0
$16$	−4.10667	−1.02667
$17$	−3.56474	−0.864577	−0.432289	−	0.901735i	$-0.642294\pi$
$17$	−3.56474	−0.864577	−0.432289	+	0.901735i	$0.642294\pi$
$18$	−11.1338	−2.62427
$19$	−4.95200	−1.13607	−0.568034	−	0.823005i	$-0.692296\pi$
$19$	−4.95200	−1.13607	−0.568034	+	0.823005i	$0.692296\pi$
$20$	0	0
$21$	−11.3748	−2.48219
$22$	−10.7979	−2.30211
$23$	−8.91648	−1.85922	−0.929608	−	0.368551i	$-0.879854\pi$
$23$	−8.91648	−1.85922	−0.929608	+	0.368551i	$0.879854\pi$
$24$	0.319515	0.0652208
$25$	0	0
$26$	−8.96966	−1.75910
$27$	7.64316	1.47093
$28$	−7.54248	−1.42539
$29$	2.56121	0.475605	0.237802	−	0.971314i	$-0.423573\pi$
$29$	2.56121	0.475605	0.237802	+	0.971314i	$0.423573\pi$
$30$	0	0
$31$	−1.87389	−0.336561	−0.168281	−	0.985739i	$-0.553822\pi$
$31$	−1.87389	−0.336561	−0.168281	+	0.985739i	$0.553822\pi$
$32$	7.93900	1.40343
$33$	15.9475	2.77610
$34$	7.08045	1.21429
$35$	0	0
$36$	10.9035	1.81726
$37$	−8.26816	−1.35928	−0.679639	−	0.733547i	$-0.737863\pi$
$37$	−8.26816	−1.35928	−0.679639	+	0.733547i	$0.737863\pi$
$38$	9.83588	1.59559
$39$	13.2474	2.12128
$40$	0	0
$41$	1.60127	0.250076	0.125038	−	0.992152i	$-0.460095\pi$
$41$	1.60127	0.250076	0.125038	+	0.992152i	$0.460095\pi$
$42$	22.5932	3.48620
$43$	−10.9508	−1.66998	−0.834991	−	0.550263i	$-0.814527\pi$
$43$	−10.9508	−1.66998	−0.834991	+	0.550263i	$0.814527\pi$
$44$	10.5745	1.59417
$45$	0	0
$46$	17.7103	2.61124
$47$	−0.538956	−0.0786148	−0.0393074	−	0.999227i	$-0.512515\pi$
$47$	−0.538956	−0.0786148	−0.0393074	+	0.999227i	$0.512515\pi$
$48$	−12.0469	−1.73883
$49$	8.03542	1.14792
$50$	0	0
$51$	−10.4572	−1.46430
$52$	8.78415	1.21814
$53$	6.76785	0.929636	0.464818	−	0.885406i	$-0.346120\pi$
$53$	6.76785	0.929636	0.464818	+	0.885406i	$0.346120\pi$
$54$	−15.1812	−2.06590
$55$	0	0
$56$	−0.422340	−0.0564376
$57$	−14.5267	−1.92411
$58$	−5.08719	−0.667980
$59$	−7.99400	−1.04073	−0.520365	−	0.853944i	$-0.674204\pi$
$59$	−7.99400	−1.04073	−0.520365	+	0.853944i	$0.674204\pi$
$60$	0	0
$61$	−6.30204	−0.806894	−0.403447	−	0.915003i	$-0.632188\pi$
$61$	−6.30204	−0.806894	−0.403447	+	0.915003i	$0.632188\pi$
$62$	3.72201	0.472696
$63$	−21.7355	−2.73842
$64$	−7.55546	−0.944432
$65$	0	0
$66$	−31.6756	−3.89900
$67$	6.40319	0.782274	0.391137	−	0.920332i	$-0.372082\pi$
$67$	6.40319	0.782274	0.391137	+	0.920332i	$0.372082\pi$
$68$	−6.93401	−0.840872
$69$	−26.1566	−3.14888
$70$	0	0
$71$	−15.2258	−1.80698	−0.903488	−	0.428614i	$-0.859002\pi$
$71$	−15.2258	−1.80698	−0.903488	+	0.428614i	$0.859002\pi$
$72$	0.610543	0.0719532
$73$	−8.98213	−1.05128	−0.525640	−	0.850707i	$-0.676174\pi$
$73$	−8.98213	−1.05128	−0.525640	+	0.850707i	$0.676174\pi$
$74$	16.4226	1.90909
$75$	0	0
$76$	−9.63245	−1.10492
$77$	−21.0796	−2.40225
$78$	−26.3126	−2.97931
$79$	−5.39894	−0.607428	−0.303714	−	0.952763i	$-0.598227\pi$
$79$	−5.39894	−0.607428	−0.303714	+	0.952763i	$0.598227\pi$
$80$	0	0
$81$	5.60486	0.622762
$82$	−3.18051	−0.351229
$83$	10.5577	1.15886	0.579428	−	0.815024i	$-0.303276\pi$
$83$	10.5577	1.15886	0.579428	+	0.815024i	$0.303276\pi$
$84$	−22.1259	−2.41413
$85$	0	0
$86$	21.7510	2.34547
$87$	7.51333	0.805513
$88$	0.592120	0.0631202
$89$	10.9001	1.15541	0.577704	−	0.816246i	$-0.303949\pi$
$89$	10.9001	1.15541	0.577704	+	0.816246i	$0.303949\pi$
$90$	0	0
$91$	−17.5106	−1.83561
$92$	−17.3440	−1.80824
$93$	−5.49708	−0.570021
$94$	1.07050	0.110413
$95$	0	0
$96$	23.2891	2.37694
$97$	−3.42286	−0.347539	−0.173769	−	0.984786i	$-0.555595\pi$
$97$	−3.42286	−0.347539	−0.173769	+	0.984786i	$0.555595\pi$
$98$	−15.9603	−1.61223
$99$	30.4731	3.06266
$100$	0	0
$101$	0.970244	0.0965429	0.0482715	−	0.998834i	$-0.484629\pi$
$101$	0.970244	0.0965429	0.0482715	+	0.998834i	$0.484629\pi$
$102$	20.7705	2.05659
$103$	−17.2509	−1.69978	−0.849890	−	0.526960i	$-0.823332\pi$
$103$	−17.2509	−1.69978	−0.849890	+	0.526960i	$0.823332\pi$
$104$	0.491867	0.0482316
$105$	0	0
$106$	−13.4426	−1.30566
$107$	1.63982	0.158527	0.0792636	−	0.996854i	$-0.474743\pi$
$107$	1.63982	0.158527	0.0792636	+	0.996854i	$0.474743\pi$
$108$	14.8672	1.43060
$109$	6.29189	0.602653	0.301327	−	0.953521i	$-0.402571\pi$
$109$	6.29189	0.602653	0.301327	+	0.953521i	$0.402571\pi$
$110$	0	0
$111$	−24.2547	−2.30216
$112$	15.9238	1.50466
$113$	−9.81404	−0.923228	−0.461614	−	0.887081i	$-0.652729\pi$
$113$	−9.81404	−0.923228	−0.461614	+	0.887081i	$0.652729\pi$
$114$	28.8536	2.70239
$115$	0	0
$116$	4.98197	0.462564
$117$	25.3137	2.34025
$118$	15.8780	1.46169
$119$	13.8225	1.26710
$120$	0	0
$121$	18.5536	1.68669
$122$	12.5174	1.13327
$123$	4.69733	0.423544
$124$	−3.64503	−0.327333
$125$	0	0
$126$	43.1720	3.84607
$127$	−0.747436	−0.0663242	−0.0331621	−	0.999450i	$-0.510558\pi$
$127$	−0.747436	−0.0663242	−0.0331621	+	0.999450i	$0.510558\pi$
$128$	−0.871026	−0.0769885
$129$	−32.1243	−2.82838
$130$	0	0
$131$	−7.06461	−0.617238	−0.308619	−	0.951186i	$-0.599867\pi$
$131$	−7.06461	−0.617238	−0.308619	+	0.951186i	$0.599867\pi$
$132$	31.0205	2.69999
$133$	19.2017	1.66499
$134$	−12.7183	−1.09869
$135$	0	0
$136$	−0.388269	−0.0332938
$137$	−8.32374	−0.711145	−0.355573	−	0.934649i	$-0.615714\pi$
$137$	−8.32374	−0.711145	−0.355573	+	0.934649i	$0.615714\pi$
$138$	51.9533	4.42256
$139$	−4.03004	−0.341823	−0.170912	−	0.985286i	$-0.554671\pi$
$139$	−4.03004	−0.341823	−0.170912	+	0.985286i	$0.554671\pi$
$140$	0	0
$141$	−1.58103	−0.133147
$142$	30.2422	2.53787
$143$	24.5498	2.05296
$144$	−23.0198	−1.91832
$145$	0	0
$146$	17.8407	1.47651
$147$	23.5720	1.94418
$148$	−16.0829	−1.32201
$149$	13.5027	1.10618	0.553091	−	0.833121i	$-0.313448\pi$
$149$	13.5027	1.10618	0.553091	+	0.833121i	$0.313448\pi$
$150$	0	0
$151$	14.4658	1.17721	0.588607	−	0.808420i	$-0.299677\pi$
$151$	14.4658	1.17721	0.588607	+	0.808420i	$0.299677\pi$
$152$	−0.539368	−0.0437485
$153$	−19.9820	−1.61545
$154$	41.8693	3.37392
$155$	0	0
$156$	25.7684	2.06312
$157$	−1.44367	−0.115218	−0.0576089	−	0.998339i	$-0.518348\pi$
$157$	−1.44367	−0.115218	−0.0576089	+	0.998339i	$0.518348\pi$
$158$	10.7236	0.853124
$159$	19.8535	1.57449
$160$	0	0
$161$	34.5741	2.72482
$162$	−11.1326	−0.874661
$163$	−10.3677	−0.812064	−0.406032	−	0.913859i	$-0.633088\pi$
$163$	−10.3677	−0.812064	−0.406032	+	0.913859i	$0.633088\pi$
$164$	3.11473	0.243219
$165$	0	0
$166$	−20.9701	−1.62760
$167$	−20.8554	−1.61384	−0.806920	−	0.590661i	$-0.798867\pi$
$167$	−20.8554	−1.61384	−0.806920	+	0.590661i	$0.798867\pi$
$168$	−1.23894	−0.0955861
$169$	7.39329	0.568715
$170$	0	0
$171$	−27.7583	−2.12273
$172$	−21.3011	−1.62419
$173$	17.8780	1.35924	0.679621	−	0.733563i	$-0.262144\pi$
$173$	17.8780	1.35924	0.679621	+	0.733563i	$0.262144\pi$
$174$	−14.9233	−1.13133
$175$	0	0
$176$	−22.3252	−1.68282
$177$	−23.4505	−1.76265
$178$	−21.6502	−1.62275
$179$	9.33051	0.697395	0.348698	−	0.937235i	$-0.386624\pi$
$179$	9.33051	0.697395	0.348698	+	0.937235i	$0.386624\pi$
$180$	0	0
$181$	15.0060	1.11539	0.557695	−	0.830046i	$-0.311686\pi$
$181$	15.0060	1.11539	0.557695	+	0.830046i	$0.311686\pi$
$182$	34.7804	2.57809
$183$	−18.4871	−1.36661
$184$	−0.971176	−0.0715960
$185$	0	0
$186$	10.9185	0.800587
$187$	−19.3791	−1.41714
$188$	−1.04836	−0.0764593
$189$	−29.6368	−2.15576
$190$	0	0
$191$	12.3478	0.893453	0.446726	−	0.894671i	$-0.352590\pi$
$191$	12.3478	0.893453	0.446726	+	0.894671i	$0.352590\pi$
$192$	−22.1640	−1.59955
$193$	5.16306	0.371645	0.185823	−	0.982583i	$-0.440505\pi$
$193$	5.16306	0.371645	0.185823	+	0.982583i	$0.440505\pi$
$194$	6.79863	0.488113
$195$	0	0
$196$	15.6302	1.11644
$197$	−2.21283	−0.157658	−0.0788289	−	0.996888i	$-0.525118\pi$
$197$	−2.21283	−0.157658	−0.0788289	+	0.996888i	$0.525118\pi$
$198$	−60.5270	−4.30147
$199$	8.99087	0.637346	0.318673	−	0.947865i	$-0.396763\pi$
$199$	8.99087	0.637346	0.318673	+	0.947865i	$0.396763\pi$
$200$	0	0
$201$	18.7838	1.32491
$202$	−1.92714	−0.135593
$203$	−9.93123	−0.697036
$204$	−20.3410	−1.42415
$205$	0	0
$206$	34.2644	2.38732
$207$	−49.9811	−3.47392
$208$	−18.5453	−1.28588
$209$	−26.9207	−1.86214
$210$	0	0
$211$	−5.78841	−0.398490	−0.199245	−	0.979950i	$-0.563849\pi$
$211$	−5.78841	−0.398490	−0.199245	+	0.979950i	$0.563849\pi$
$212$	13.1646	0.904147
$213$	−44.6651	−3.06040
$214$	−3.25708	−0.222649
$215$	0	0
$216$	0.832486	0.0566435
$217$	7.26613	0.493257
$218$	−12.4972	−0.846418
$219$	−26.3492	−1.78051
$220$	0	0
$221$	−16.0980	−1.08287
$222$	48.1758	3.23335
$223$	4.23873	0.283847	0.141923	−	0.989878i	$-0.454671\pi$
$223$	4.23873	0.283847	0.141923	+	0.989878i	$0.454671\pi$
$224$	−30.7839	−2.05684
$225$	0	0
$226$	19.4931	1.29666
$227$	−13.5718	−0.900792	−0.450396	−	0.892829i	$-0.648717\pi$
$227$	−13.5718	−0.900792	−0.450396	+	0.892829i	$0.648717\pi$
$228$	−28.2569	−1.87136
$229$	−5.88333	−0.388781	−0.194391	−	0.980924i	$-0.562273\pi$
$229$	−5.88333	−0.388781	−0.194391	+	0.980924i	$0.562273\pi$
$230$	0	0
$231$	−61.8373	−4.06859
$232$	0.278965	0.0183149
$233$	12.0619	0.790199	0.395099	−	0.918638i	$-0.370710\pi$
$233$	12.0619	0.790199	0.395099	+	0.918638i	$0.370710\pi$
$234$	−50.2792	−3.28685
$235$	0	0
$236$	−15.5496	−1.01220
$237$	−15.8378	−1.02878
$238$	−27.4548	−1.77963
$239$	−26.8308	−1.73554	−0.867769	−	0.496967i	$-0.834447\pi$
$239$	−26.8308	−1.73554	−0.867769	+	0.496967i	$0.834447\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−36.8520	−2.36893
$243$	−6.48757	−0.416178
$244$	−12.2585	−0.784770
$245$	0	0
$246$	−9.33005	−0.594862
$247$	−22.3627	−1.42291
$248$	−0.204103	−0.0129606
$249$	30.9710	1.96271
$250$	0	0
$251$	14.0119	0.884421	0.442210	−	0.896911i	$-0.354194\pi$
$251$	14.0119	0.884421	0.442210	+	0.896911i	$0.354194\pi$
$252$	−42.2791	−2.66333
$253$	−48.4729	−3.04746
$254$	1.48459	0.0931514
$255$	0	0
$256$	16.8410	1.05256
$257$	3.14158	0.195966	0.0979831	−	0.995188i	$-0.468761\pi$
$257$	3.14158	0.195966	0.0979831	+	0.995188i	$0.468761\pi$
$258$	63.8066	3.97243
$259$	32.0602	1.99213
$260$	0	0
$261$	14.3568	0.888662
$262$	14.0320	0.866902
$263$	−20.4977	−1.26394	−0.631970	−	0.774993i	$-0.717753\pi$
$263$	−20.4977	−1.26394	−0.631970	+	0.774993i	$0.717753\pi$
$264$	1.73699	0.106904
$265$	0	0
$266$	−38.1392	−2.33846
$267$	31.9755	1.95687
$268$	12.4552	0.760825
$269$	21.5895	1.31633	0.658166	−	0.752873i	$-0.271332\pi$
$269$	21.5895	1.31633	0.658166	+	0.752873i	$0.271332\pi$
$270$	0	0
$271$	30.3487	1.84355	0.921777	−	0.387721i	$-0.126737\pi$
$271$	30.3487	1.84355	0.921777	+	0.387721i	$0.126737\pi$
$272$	14.6392	0.887632
$273$	−51.3675	−3.10890
$274$	16.5330	0.998794
$275$	0	0
$276$	−50.8788	−3.06254
$277$	−28.3808	−1.70523	−0.852617	−	0.522536i	$-0.824986\pi$
$277$	−28.3808	−1.70523	−0.852617	+	0.522536i	$0.824986\pi$
$278$	8.00464	0.480086
$279$	−10.5041	−0.628861
$280$	0	0
$281$	18.5214	1.10489	0.552446	−	0.833549i	$-0.313695\pi$
$281$	18.5214	1.10489	0.552446	+	0.833549i	$0.313695\pi$
$282$	3.14031	0.187003
$283$	15.6962	0.933045	0.466523	−	0.884509i	$-0.345507\pi$
$283$	15.6962	0.933045	0.466523	+	0.884509i	$0.345507\pi$
$284$	−29.6168	−1.75743
$285$	0	0
$286$	−48.7620	−2.88336
$287$	−6.20901	−0.366506
$288$	44.5018	2.62229
$289$	−4.29261	−0.252507
$290$	0	0
$291$	−10.0410	−0.588613
$292$	−17.4717	−1.02245
$293$	19.4528	1.13644	0.568221	−	0.822876i	$-0.307632\pi$
$293$	19.4528	1.13644	0.568221	+	0.822876i	$0.307632\pi$
$294$	−46.8197	−2.73058
$295$	0	0
$296$	−0.900561	−0.0523441
$297$	41.5507	2.41101
$298$	−26.8196	−1.55362
$299$	−40.2659	−2.32864
$300$	0	0
$301$	42.4623	2.44749
$302$	−28.7327	−1.65338
$303$	2.84622	0.163511
$304$	20.3362	1.16636
$305$	0	0
$306$	39.6892	2.26888
$307$	−18.5628	−1.05944	−0.529718	−	0.848174i	$-0.677702\pi$
$307$	−18.5628	−1.05944	−0.529718	+	0.848174i	$0.677702\pi$
$308$	−41.0033	−2.33638
$309$	−50.6056	−2.87885
$310$	0	0
$311$	−25.1593	−1.42665	−0.713326	−	0.700833i	$-0.752812\pi$
$311$	−25.1593	−1.42665	−0.713326	+	0.700833i	$0.752812\pi$
$312$	1.44290	0.0816879
$313$	22.7874	1.28802	0.644010	−	0.765017i	$-0.277269\pi$
$313$	22.7874	1.28802	0.644010	+	0.765017i	$0.277269\pi$
$314$	2.86749	0.161822
$315$	0	0
$316$	−10.5018	−0.590773
$317$	−16.8654	−0.947253	−0.473627	−	0.880726i	$-0.657055\pi$
$317$	−16.8654	−0.947253	−0.473627	+	0.880726i	$0.657055\pi$
$318$	−39.4340	−2.21135
$319$	13.9236	0.779570
$320$	0	0
$321$	4.81042	0.268492
$322$	−68.6727	−3.82698
$323$	17.6526	0.982218
$324$	10.9024	0.605687
$325$	0	0
$326$	20.5929	1.14053
$327$	18.4573	1.02069
$328$	0.174409	0.00963012
$329$	2.08983	0.115216
$330$	0	0
$331$	34.9055	1.91858	0.959289	−	0.282425i	$-0.0911388\pi$
$331$	34.9055	1.91858	0.959289	+	0.282425i	$0.0911388\pi$
$332$	20.5364	1.12708
$333$	−46.3469	−2.53979
$334$	41.4239	2.26662
$335$	0	0
$336$	46.7127	2.54838
$337$	7.07149	0.385208	0.192604	−	0.981277i	$-0.438307\pi$
$337$	7.07149	0.385208	0.192604	+	0.981277i	$0.438307\pi$
$338$	−14.6849	−0.798752
$339$	−28.7896	−1.56363
$340$	0	0
$341$	−10.1871	−0.551662
$342$	55.1347	2.98134
$343$	−4.01492	−0.216785
$344$	−1.19275	−0.0643089
$345$	0	0
$346$	−35.5101	−1.90904
$347$	33.0795	1.77580	0.887901	−	0.460035i	$-0.152163\pi$
$347$	33.0795	1.77580	0.887901	+	0.460035i	$0.152163\pi$
$348$	14.6147	0.783427
$349$	−22.7013	−1.21517	−0.607587	−	0.794253i	$-0.707862\pi$
$349$	−22.7013	−1.21517	−0.607587	+	0.794253i	$0.707862\pi$
$350$	0	0
$351$	34.5157	1.84231
$352$	43.1590	2.30038
$353$	−20.3867	−1.08508	−0.542538	−	0.840031i	$-0.682536\pi$
$353$	−20.3867	−1.08508	−0.542538	+	0.840031i	$0.682536\pi$
$354$	46.5783	2.47561
$355$	0	0
$356$	21.2025	1.12373
$357$	40.5484	2.14605
$358$	−18.5327	−0.979482
$359$	−24.8228	−1.31010	−0.655049	−	0.755586i	$-0.727352\pi$
$359$	−24.8228	−1.31010	−0.655049	+	0.755586i	$0.727352\pi$
$360$	0	0
$361$	5.52233	0.290649
$362$	−29.8056	−1.56655
$363$	54.4271	2.85668
$364$	−34.0610	−1.78528
$365$	0	0
$366$	36.7199	1.91938
$367$	−18.2538	−0.952843	−0.476422	−	0.879217i	$-0.658066\pi$
$367$	−18.2538	−0.952843	−0.476422	+	0.879217i	$0.658066\pi$
$368$	36.6170	1.90879
$369$	8.97586	0.467265
$370$	0	0
$371$	−26.2427	−1.36245
$372$	−10.6927	−0.554392
$373$	33.2495	1.72159	0.860796	−	0.508951i	$-0.169966\pi$
$373$	33.2495	1.72159	0.860796	+	0.508951i	$0.169966\pi$
$374$	38.4916	1.99035
$375$	0	0
$376$	−0.0587026	−0.00302736
$377$	11.5661	0.595687
$378$	58.8658	3.02773
$379$	6.47197	0.332443	0.166221	−	0.986088i	$-0.446843\pi$
$379$	6.47197	0.332443	0.166221	+	0.986088i	$0.446843\pi$
$380$	0	0
$381$	−2.19261	−0.112331
$382$	−24.5257	−1.25484
$383$	8.33312	0.425802	0.212901	−	0.977074i	$-0.431709\pi$
$383$	8.33312	0.425802	0.212901	+	0.977074i	$0.431709\pi$
$384$	−2.55516	−0.130393
$385$	0	0
$386$	−10.2551	−0.521971
$387$	−61.3844	−3.12034
$388$	−6.65802	−0.338010
$389$	−4.68009	−0.237290	−0.118645	−	0.992937i	$-0.537855\pi$
$389$	−4.68009	−0.237290	−0.118645	+	0.992937i	$0.537855\pi$
$390$	0	0
$391$	31.7850	1.60743
$392$	0.875212	0.0442049
$393$	−20.7241	−1.04539
$394$	4.39522	0.221428
$395$	0	0
$396$	59.2752	2.97869
$397$	−28.7097	−1.44090	−0.720449	−	0.693508i	$-0.756064\pi$
$397$	−28.7097	−1.44090	−0.720449	+	0.693508i	$0.756064\pi$
$398$	−17.8581	−0.895144
$399$	56.3282	2.81994
$400$	0	0
$401$	−18.7554	−0.936600	−0.468300	−	0.883570i	$-0.655133\pi$
$401$	−18.7554	−0.936600	−0.468300	+	0.883570i	$0.655133\pi$
$402$	−37.3092	−1.86081
$403$	−8.46231	−0.421538
$404$	1.88728	0.0938959
$405$	0	0
$406$	19.7258	0.978977
$407$	−44.9484	−2.22801
$408$	−1.13899	−0.0563884
$409$	−11.8347	−0.585186	−0.292593	−	0.956237i	$-0.594518\pi$
$409$	−11.8347	−0.585186	−0.292593	+	0.956237i	$0.594518\pi$
$410$	0	0
$411$	−24.4178	−1.20444
$412$	−33.5558	−1.65317
$413$	30.9972	1.52527
$414$	99.2745	4.87908
$415$	0	0
$416$	35.8517	1.75777
$417$	−11.8221	−0.578933
$418$	53.4710	2.61535
$419$	24.9971	1.22119	0.610593	−	0.791944i	$-0.290931\pi$
$419$	24.9971	1.22119	0.610593	+	0.791944i	$0.290931\pi$
$420$	0	0
$421$	−40.3191	−1.96503	−0.982515	−	0.186182i	$-0.940388\pi$
$421$	−40.3191	−1.96503	−0.982515	+	0.186182i	$0.940388\pi$
$422$	11.4972	0.559674
$423$	−3.02110	−0.146891
$424$	0.737149	0.0357991
$425$	0	0
$426$	88.7158	4.29830
$427$	24.4365	1.18257
$428$	3.18972	0.154181
$429$	72.0172	3.47702
$430$	0	0
$431$	−38.3225	−1.84593	−0.922965	−	0.384883i	$-0.874242\pi$
$431$	−38.3225	−1.84593	−0.922965	+	0.384883i	$0.874242\pi$
$432$	−31.3879	−1.51015
$433$	−11.7086	−0.562677	−0.281339	−	0.959609i	$-0.590778\pi$
$433$	−11.7086	−0.562677	−0.281339	+	0.959609i	$0.590778\pi$
$434$	−14.4323	−0.692772
$435$	0	0
$436$	12.2387	0.586130
$437$	44.1544	2.11219
$438$	52.3358	2.50070
$439$	−21.0713	−1.00568	−0.502840	−	0.864379i	$-0.667712\pi$
$439$	−21.0713	−1.00568	−0.502840	+	0.864379i	$0.667712\pi$
$440$	0	0
$441$	45.0423	2.14487
$442$	31.9745	1.52087
$443$	−35.7235	−1.69728	−0.848638	−	0.528974i	$-0.822577\pi$
$443$	−35.7235	−1.69728	−0.848638	+	0.528974i	$0.822577\pi$
$444$	−47.1794	−2.23903
$445$	0	0
$446$	−8.41915	−0.398659
$447$	39.6102	1.87350
$448$	29.2967	1.38414
$449$	16.5801	0.782462	0.391231	−	0.920293i	$-0.372049\pi$
$449$	16.5801	0.782462	0.391231	+	0.920293i	$0.372049\pi$
$450$	0	0
$451$	8.70501	0.409903
$452$	−19.0899	−0.897914
$453$	42.4356	1.99380
$454$	26.9569	1.26515
$455$	0	0
$456$	−1.58224	−0.0740952
$457$	14.2167	0.665029	0.332515	−	0.943098i	$-0.392103\pi$
$457$	14.2167	0.665029	0.332515	+	0.943098i	$0.392103\pi$
$458$	11.6857	0.546038
$459$	−27.2459	−1.27173
$460$	0	0
$461$	−29.7262	−1.38449	−0.692245	−	0.721663i	$-0.743378\pi$
$461$	−29.7262	−1.38449	−0.692245	+	0.721663i	$0.743378\pi$
$462$	122.824	5.71428
$463$	−16.6345	−0.773069	−0.386535	−	0.922275i	$-0.626328\pi$
$463$	−16.6345	−0.773069	−0.386535	+	0.922275i	$0.626328\pi$
$464$	−10.5180	−0.488287
$465$	0	0
$466$	−23.9578	−1.10982
$467$	−14.0306	−0.649260	−0.324630	−	0.945841i	$-0.605240\pi$
$467$	−14.0306	−0.649260	−0.324630	+	0.945841i	$0.605240\pi$
$468$	49.2393	2.27609
$469$	−24.8287	−1.14648
$470$	0	0
$471$	−4.23503	−0.195140
$472$	−0.870700	−0.0400772
$473$	−59.5321	−2.73729
$474$	31.4578	1.44490
$475$	0	0
$476$	26.8870	1.23236
$477$	37.9370	1.73701
$478$	53.2924	2.43754
$479$	7.38971	0.337645	0.168822	−	0.985647i	$-0.446004\pi$
$479$	7.38971	0.337645	0.168822	+	0.985647i	$0.446004\pi$
$480$	0	0
$481$	−37.3381	−1.70247
$482$	−1.98624	−0.0904709
$483$	101.424	4.61493
$484$	36.0898	1.64044
$485$	0	0
$486$	12.8859	0.584516
$487$	−37.6880	−1.70780	−0.853902	−	0.520433i	$-0.825770\pi$
$487$	−37.6880	−1.70780	−0.853902	+	0.520433i	$0.825770\pi$
$488$	−0.686414	−0.0310725
$489$	−30.4139	−1.37536
$490$	0	0
$491$	18.9491	0.855162	0.427581	−	0.903977i	$-0.359366\pi$
$491$	18.9491	0.855162	0.427581	+	0.903977i	$0.359366\pi$
$492$	9.13708	0.411931
$493$	−9.13005	−0.411197
$494$	44.4178	1.99845
$495$	0	0
$496$	7.69546	0.345536
$497$	59.0390	2.64826
$498$	−61.5160	−2.75660
$499$	−0.653039	−0.0292340	−0.0146170	−	0.999893i	$-0.504653\pi$
$499$	−0.653039	−0.0292340	−0.0146170	+	0.999893i	$0.504653\pi$
$500$	0	0
$501$	−61.1795	−2.73330
$502$	−27.8310	−1.24216
$503$	−18.2091	−0.811905	−0.405953	−	0.913894i	$-0.633060\pi$
$503$	−18.2091	−0.811905	−0.405953	+	0.913894i	$0.633060\pi$
$504$	−2.36741	−0.105453
$505$	0	0
$506$	96.2789	4.28012
$507$	21.6883	0.963210
$508$	−1.45388	−0.0645057
$509$	−30.1462	−1.33621	−0.668104	−	0.744068i	$-0.732894\pi$
$509$	−30.1462	−1.33621	−0.668104	+	0.744068i	$0.732894\pi$
$510$	0	0
$511$	34.8287	1.54073
$512$	−31.7082	−1.40132
$513$	−37.8489	−1.67107
$514$	−6.23994	−0.275232
$515$	0	0
$516$	−62.4870	−2.75084
$517$	−2.92994	−0.128858
$518$	−63.6794	−2.79791
$519$	52.4454	2.30210
$520$	0	0
$521$	−3.89716	−0.170738	−0.0853688	−	0.996349i	$-0.527207\pi$
$521$	−3.89716	−0.170738	−0.0853688	+	0.996349i	$0.527207\pi$
$522$	−28.5161	−1.24811
$523$	32.4111	1.41724	0.708620	−	0.705591i	$-0.249318\pi$
$523$	32.4111	1.41724	0.708620	+	0.705591i	$0.249318\pi$
$524$	−13.7418	−0.600314
$525$	0	0
$526$	40.7133	1.77519
$527$	6.67995	0.290983
$528$	−65.4910	−2.85013
$529$	56.5037	2.45668
$530$	0	0
$531$	−44.8101	−1.94459
$532$	37.3504	1.61934
$533$	7.23116	0.313216
$534$	−63.5112	−2.74840
$535$	0	0
$536$	0.697430	0.0301244
$537$	27.3711	1.18115
$538$	−42.8819	−1.84877
$539$	43.6832	1.88157
$540$	0	0
$541$	−17.5981	−0.756603	−0.378301	−	0.925682i	$-0.623492\pi$
$541$	−17.5981	−0.756603	−0.378301	+	0.925682i	$0.623492\pi$
$542$	−60.2800	−2.58925
$543$	44.0203	1.88909
$544$	−28.3005	−1.21337
$545$	0	0
$546$	102.028	4.36641
$547$	34.2404	1.46401	0.732006	−	0.681298i	$-0.238584\pi$
$547$	34.2404	1.46401	0.732006	+	0.681298i	$0.238584\pi$
$548$	−16.1910	−0.691647
$549$	−35.3259	−1.50767
$550$	0	0
$551$	−12.6831	−0.540319
$552$	−2.84895	−0.121259
$553$	20.9347	0.890233
$554$	56.3711	2.39498
$555$	0	0
$556$	−7.83908	−0.332451
$557$	6.43152	0.272512	0.136256	−	0.990674i	$-0.456493\pi$
$557$	6.43152	0.272512	0.136256	+	0.990674i	$0.456493\pi$
$558$	20.8636	0.883227
$559$	−49.4527	−2.09163
$560$	0	0
$561$	−56.8487	−2.40015
$562$	−36.7879	−1.55181
$563$	−14.0474	−0.592026	−0.296013	−	0.955184i	$-0.595657\pi$
$563$	−14.0474	−0.592026	−0.296013	+	0.955184i	$0.595657\pi$
$564$	−3.07536	−0.129496
$565$	0	0
$566$	−31.1766	−1.31045
$567$	−21.7331	−0.912706
$568$	−1.65839	−0.0695844
$569$	−21.2540	−0.891015	−0.445508	−	0.895278i	$-0.646977\pi$
$569$	−21.2540	−0.891015	−0.445508	+	0.895278i	$0.646977\pi$
$570$	0	0
$571$	−9.79489	−0.409903	−0.204952	−	0.978772i	$-0.565704\pi$
$571$	−9.79489	−0.409903	−0.204952	+	0.978772i	$0.565704\pi$
$572$	47.7535	1.99667
$573$	36.2223	1.51321
$574$	12.3326	0.514753
$575$	0	0
$576$	−42.3519	−1.76466
$577$	2.46709	0.102706	0.0513532	−	0.998681i	$-0.483647\pi$
$577$	2.46709	0.102706	0.0513532	+	0.998681i	$0.483647\pi$
$578$	8.52617	0.354642
$579$	15.1459	0.629441
$580$	0	0
$581$	−40.9380	−1.69839
$582$	19.9438	0.826699
$583$	36.7922	1.52378
$584$	−0.978327	−0.0404834
$585$	0	0
$586$	−38.6379	−1.59612
$587$	−31.4948	−1.29993	−0.649965	−	0.759964i	$-0.725216\pi$
$587$	−31.4948	−1.29993	−0.649965	+	0.759964i	$0.725216\pi$
$588$	45.8513	1.89088
$589$	9.27953	0.382356
$590$	0	0
$591$	−6.49136	−0.267019
$592$	33.9546	1.39552
$593$	21.8589	0.897637	0.448818	−	0.893623i	$-0.351845\pi$
$593$	21.8589	0.897637	0.448818	+	0.893623i	$0.351845\pi$
$594$	−82.5297	−3.38624
$595$	0	0
$596$	26.2649	1.07585
$597$	26.3748	1.07945
$598$	79.9778	3.27054
$599$	−33.0013	−1.34840	−0.674199	−	0.738550i	$-0.735511\pi$
$599$	−33.0013	−1.34840	−0.674199	+	0.738550i	$0.735511\pi$
$600$	0	0
$601$	23.9772	0.978051	0.489025	−	0.872270i	$-0.337353\pi$
$601$	23.9772	0.978051	0.489025	+	0.872270i	$0.337353\pi$
$602$	−84.3406	−3.43746
$603$	35.8929	1.46167
$604$	28.1384	1.14494
$605$	0	0
$606$	−5.65328	−0.229649
$607$	0.694914	0.0282057	0.0141028	−	0.999901i	$-0.495511\pi$
$607$	0.694914	0.0282057	0.0141028	+	0.999901i	$0.495511\pi$
$608$	−39.3140	−1.59439
$609$	−29.1333	−1.18054
$610$	0	0
$611$	−2.43387	−0.0984637
$612$	−38.8683	−1.57116
$613$	31.6306	1.27755	0.638775	−	0.769394i	$-0.279442\pi$
$613$	31.6306	1.27755	0.638775	+	0.769394i	$0.279442\pi$
$614$	36.8703	1.48796
$615$	0	0
$616$	−2.29598	−0.0925075
$617$	14.8903	0.599462	0.299731	−	0.954024i	$-0.403103\pi$
$617$	14.8903	0.599462	0.299731	+	0.954024i	$0.403103\pi$
$618$	100.515	4.04331
$619$	33.9882	1.36610	0.683051	−	0.730371i	$-0.260653\pi$
$619$	33.9882	1.36610	0.683051	+	0.730371i	$0.260653\pi$
$620$	0	0
$621$	−68.1501	−2.73477
$622$	49.9724	2.00371
$623$	−42.2657	−1.69334
$624$	−54.4027	−2.17785
$625$	0	0
$626$	−45.2613	−1.80901
$627$	−78.9720	−3.15384
$628$	−2.80818	−0.112059
$629$	29.4739	1.17520
$630$	0	0
$631$	−14.1877	−0.564803	−0.282402	−	0.959296i	$-0.591131\pi$
$631$	−14.1877	−0.564803	−0.282402	+	0.959296i	$0.591131\pi$
$632$	−0.588048	−0.0233913
$633$	−16.9803	−0.674908
$634$	33.4987	1.33040
$635$	0	0
$636$	38.6184	1.53132
$637$	36.2871	1.43775
$638$	−27.6556	−1.09489
$639$	−85.3480	−3.37631
$640$	0	0
$641$	0.661663	0.0261341	0.0130671	−	0.999915i	$-0.495841\pi$
$641$	0.661663	0.0261341	0.0130671	+	0.999915i	$0.495841\pi$
$642$	−9.55467	−0.377093
$643$	8.27801	0.326453	0.163226	−	0.986589i	$-0.447810\pi$
$643$	8.27801	0.326453	0.163226	+	0.986589i	$0.447810\pi$
$644$	67.2524	2.65011
$645$	0	0
$646$	−35.0624	−1.37951
$647$	−11.5359	−0.453522	−0.226761	−	0.973950i	$-0.572814\pi$
$647$	−11.5359	−0.453522	−0.226761	+	0.973950i	$0.572814\pi$
$648$	0.610477	0.0239818
$649$	−43.4580	−1.70587
$650$	0	0
$651$	21.3152	0.835410
$652$	−20.1670	−0.789799
$653$	15.7264	0.615422	0.307711	−	0.951480i	$-0.400437\pi$
$653$	15.7264	0.615422	0.307711	+	0.951480i	$0.400437\pi$
$654$	−36.6607	−1.43355
$655$	0	0
$656$	−6.57588	−0.256745
$657$	−50.3490	−1.96430
$658$	−4.15091	−0.161819
$659$	−2.84552	−0.110846	−0.0554228	−	0.998463i	$-0.517651\pi$
$659$	−2.84552	−0.110846	−0.0554228	+	0.998463i	$0.517651\pi$
$660$	0	0
$661$	48.6598	1.89265	0.946323	−	0.323222i	$-0.104766\pi$
$661$	48.6598	1.89265	0.946323	+	0.323222i	$0.104766\pi$
$662$	−69.3308	−2.69462
$663$	−47.2236	−1.83401
$664$	1.14993	0.0446261
$665$	0	0
$666$	92.0562	3.56711
$667$	−22.8370	−0.884251
$668$	−40.5672	−1.56959
$669$	12.4344	0.480740
$670$	0	0
$671$	−34.2599	−1.32259
$672$	−90.3048	−3.48358
$673$	38.8479	1.49748	0.748738	−	0.662866i	$-0.230660\pi$
$673$	38.8479	1.49748	0.748738	+	0.662866i	$0.230660\pi$
$674$	−14.0457	−0.541020
$675$	0	0
$676$	14.3812	0.553121
$677$	43.1514	1.65844	0.829222	−	0.558919i	$-0.188784\pi$
$677$	43.1514	1.65844	0.829222	+	0.558919i	$0.188784\pi$
$678$	57.1831	2.19610
$679$	13.2723	0.509345
$680$	0	0
$681$	−39.8130	−1.52564
$682$	20.2341	0.774802
$683$	5.62695	0.215309	0.107655	−	0.994188i	$-0.465666\pi$
$683$	5.62695	0.215309	0.107655	+	0.994188i	$0.465666\pi$
$684$	−53.9944	−2.06453
$685$	0	0
$686$	7.97460	0.304472
$687$	−17.2588	−0.658464
$688$	44.9713	1.71452
$689$	30.5629	1.16435
$690$	0	0
$691$	−15.1423	−0.576041	−0.288020	−	0.957624i	$-0.592997\pi$
$691$	−15.1423	−0.576041	−0.288020	+	0.957624i	$0.592997\pi$
$692$	34.7757	1.32197
$693$	−118.161	−4.48857
$694$	−65.7040	−2.49409
$695$	0	0
$696$	0.818346	0.0310193
$697$	−5.70811	−0.216210
$698$	45.0903	1.70669
$699$	35.3836	1.33833
$700$	0	0
$701$	−26.7358	−1.00980	−0.504899	−	0.863178i	$-0.668470\pi$
$701$	−26.7358	−1.00980	−0.504899	+	0.863178i	$0.668470\pi$
$702$	−68.5566	−2.58750
$703$	40.9440	1.54423
$704$	−41.0739	−1.54803
$705$	0	0
$706$	40.4930	1.52397
$707$	−3.76218	−0.141491
$708$	−45.6150	−1.71432
$709$	19.3807	0.727857	0.363929	−	0.931427i	$-0.381435\pi$
$709$	19.3807	0.727857	0.363929	+	0.931427i	$0.381435\pi$
$710$	0	0
$711$	−30.2636	−1.13497
$712$	1.18723	0.0444933
$713$	16.7085	0.625740
$714$	−80.5389	−3.01409
$715$	0	0
$716$	18.1494	0.678274
$717$	−78.7083	−2.93941
$718$	49.3042	1.84002
$719$	−39.6526	−1.47879	−0.739396	−	0.673271i	$-0.764889\pi$
$719$	−39.6526	−1.47879	−0.739396	+	0.673271i	$0.764889\pi$
$720$	0	0
$721$	66.8912	2.49116
$722$	−10.9687	−0.408212
$723$	2.93351	0.109098
$724$	29.1892	1.08481
$725$	0	0
$726$	−108.106	−4.01217
$727$	−45.2495	−1.67821	−0.839105	−	0.543970i	$-0.816921\pi$
$727$	−45.2495	−1.67821	−0.839105	+	0.543970i	$0.816921\pi$
$728$	−1.90724	−0.0706871
$729$	−35.8459	−1.32763
$730$	0	0
$731$	39.0368	1.44383
$732$	−35.9604	−1.32914
$733$	−18.5883	−0.686574	−0.343287	−	0.939231i	$-0.611540\pi$
$733$	−18.5883	−0.686574	−0.343287	+	0.939231i	$0.611540\pi$
$734$	36.2566	1.33826
$735$	0	0
$736$	−70.7880	−2.60928
$737$	34.8098	1.28224
$738$	−17.8282	−0.656267
$739$	5.85210	0.215273	0.107637	−	0.994190i	$-0.465672\pi$
$739$	5.85210	0.215273	0.107637	+	0.994190i	$0.465672\pi$
$740$	0	0
$741$	−65.6012	−2.40992
$742$	52.1244	1.91355
$743$	−10.0765	−0.369672	−0.184836	−	0.982769i	$-0.559175\pi$
$743$	−10.0765	−0.369672	−0.184836	+	0.982769i	$0.559175\pi$
$744$	−0.598738	−0.0219508
$745$	0	0
$746$	−66.0415	−2.41795
$747$	59.1807	2.16531
$748$	−37.6955	−1.37828
$749$	−6.35849	−0.232334
$750$	0	0
$751$	−1.62585	−0.0593280	−0.0296640	−	0.999560i	$-0.509444\pi$
$751$	−1.62585	−0.0593280	−0.0296640	+	0.999560i	$0.509444\pi$
$752$	2.21331	0.0807112
$753$	41.1039	1.49791
$754$	−22.9732	−0.836634
$755$	0	0
$756$	−57.6483	−2.09665
$757$	3.59438	0.130640	0.0653200	−	0.997864i	$-0.479193\pi$
$757$	3.59438	0.130640	0.0653200	+	0.997864i	$0.479193\pi$
$758$	−12.8549	−0.466911
$759$	−142.196	−5.16137
$760$	0	0
$761$	35.8363	1.29906	0.649532	−	0.760335i	$-0.274965\pi$
$761$	35.8363	1.29906	0.649532	+	0.760335i	$0.274965\pi$
$762$	4.35506	0.157767
$763$	−24.3971	−0.883235
$764$	24.0184	0.868956
$765$	0	0
$766$	−16.5516	−0.598034
$767$	−36.1001	−1.30350
$768$	49.4032	1.78268
$769$	−45.5522	−1.64265	−0.821327	−	0.570457i	$-0.806766\pi$
$769$	−45.5522	−1.64265	−0.821327	+	0.570457i	$0.806766\pi$
$770$	0	0
$771$	9.21584	0.331900
$772$	10.0430	0.361455
$773$	−21.4939	−0.773082	−0.386541	−	0.922272i	$-0.626330\pi$
$773$	−21.4939	−0.773082	−0.386541	+	0.922272i	$0.626330\pi$
$774$	121.924	4.38248
$775$	0	0
$776$	−0.372815	−0.0133833
$777$	94.0490	3.37399
$778$	9.29579	0.333270
$779$	−7.92949	−0.284103
$780$	0	0
$781$	−82.7726	−2.96184
$782$	−63.1327	−2.25762
$783$	19.5757	0.699580
$784$	−32.9988	−1.17853
$785$	0	0
$786$	41.1631	1.46824
$787$	39.0117	1.39062	0.695309	−	0.718711i	$-0.255268\pi$
$787$	39.0117	1.39062	0.695309	+	0.718711i	$0.255268\pi$
$788$	−4.30432	−0.153335
$789$	−60.1300	−2.14069
$790$	0	0
$791$	38.0545	1.35306
$792$	3.31911	0.117939
$793$	−28.4594	−1.01062
$794$	57.0244	2.02372
$795$	0	0
$796$	17.4887	0.619871
$797$	−27.0551	−0.958340	−0.479170	−	0.877722i	$-0.659062\pi$
$797$	−27.0551	−0.958340	−0.479170	+	0.877722i	$0.659062\pi$
$798$	−111.882	−3.96056
$799$	1.92124	0.0679685
$800$	0	0
$801$	61.1001	2.15887
$802$	37.2528	1.31544
$803$	−48.8298	−1.72317
$804$	36.5376	1.28858
$805$	0	0
$806$	16.8082	0.592044
$807$	63.3328	2.22942
$808$	0.105678	0.00371775
$809$	−3.74501	−0.131667	−0.0658337	−	0.997831i	$-0.520971\pi$
$809$	−3.74501	−0.131667	−0.0658337	+	0.997831i	$0.520971\pi$
$810$	0	0
$811$	23.8097	0.836073	0.418036	−	0.908430i	$-0.362718\pi$
$811$	23.8097	0.836073	0.418036	+	0.908430i	$0.362718\pi$
$812$	−19.3179	−0.677924
$813$	89.0282	3.12236
$814$	89.2784	3.12921
$815$	0	0
$816$	42.9442	1.50335
$817$	54.2284	1.89721
$818$	23.5065	0.821886
$819$	−98.1552	−3.42982
$820$	0	0
$821$	9.44339	0.329577	0.164788	−	0.986329i	$-0.447306\pi$
$821$	9.44339	0.329577	0.164788	+	0.986329i	$0.447306\pi$
$822$	48.4996	1.69162
$823$	−22.6151	−0.788314	−0.394157	−	0.919043i	$-0.628963\pi$
$823$	−22.6151	−0.788314	−0.394157	+	0.919043i	$0.628963\pi$
$824$	−1.87895	−0.0654564
$825$	0	0
$826$	−61.5679	−2.14222
$827$	30.2635	1.05237	0.526183	−	0.850372i	$-0.323623\pi$
$827$	30.2635	1.05237	0.526183	+	0.850372i	$0.323623\pi$
$828$	−97.2213	−3.37867
$829$	29.8682	1.03736	0.518682	−	0.854967i	$-0.326423\pi$
$829$	29.8682	1.03736	0.518682	+	0.854967i	$0.326423\pi$
$830$	0	0
$831$	−83.2552	−2.88809
$832$	−34.1196	−1.18289
$833$	−28.6442	−0.992463
$834$	23.4817	0.813104
$835$	0	0
$836$	−52.3651	−1.81109
$837$	−14.3225	−0.495057
$838$	−49.6503	−1.71514
$839$	6.20080	0.214075	0.107038	−	0.994255i	$-0.465863\pi$
$839$	6.20080	0.214075	0.107038	+	0.994255i	$0.465863\pi$
$840$	0	0
$841$	−22.4402	−0.773800
$842$	80.0835	2.75986
$843$	54.3326	1.87131
$844$	−11.2594	−0.387564
$845$	0	0
$846$	6.00064	0.206306
$847$	−71.9426	−2.47198
$848$	−27.7933	−0.954426
$849$	46.0451	1.58026
$850$	0	0
$851$	73.7229	2.52719
$852$	−86.8810	−2.97649
$853$	−6.31693	−0.216288	−0.108144	−	0.994135i	$-0.534491\pi$
$853$	−6.31693	−0.216288	−0.108144	+	0.994135i	$0.534491\pi$
$854$	−48.5369	−1.66090
$855$	0	0
$856$	0.178608	0.00610469
$857$	−37.5338	−1.28213	−0.641065	−	0.767486i	$-0.721507\pi$
$857$	−37.5338	−1.28213	−0.641065	+	0.767486i	$0.721507\pi$
$858$	−143.044	−4.88343
$859$	−0.322062	−0.0109886	−0.00549432	−	0.999985i	$-0.501749\pi$
$859$	−0.322062	−0.0109886	−0.00549432	+	0.999985i	$0.501749\pi$
$860$	0	0
$861$	−18.2142	−0.620737
$862$	76.1178	2.59258
$863$	−27.0433	−0.920566	−0.460283	−	0.887772i	$-0.652252\pi$
$863$	−27.0433	−0.920566	−0.460283	+	0.887772i	$0.652252\pi$
$864$	60.6790	2.06434
$865$	0	0
$866$	23.2560	0.790273
$867$	−12.5924	−0.427661
$868$	14.1338	0.479733
$869$	−29.3504	−0.995643
$870$	0	0
$871$	28.9161	0.979785
$872$	0.685307	0.0232074
$873$	−19.1867	−0.649372
$874$	−87.7015	−2.96655
$875$	0	0
$876$	−51.2534	−1.73169
$877$	−9.62045	−0.324859	−0.162430	−	0.986720i	$-0.551933\pi$
$877$	−9.62045	−0.324859	−0.162430	+	0.986720i	$0.551933\pi$
$878$	41.8528	1.41246
$879$	57.0648	1.92475
$880$	0	0
$881$	0.517032	0.0174193	0.00870963	−	0.999962i	$-0.497228\pi$
$881$	0.517032	0.0174193	0.00870963	+	0.999962i	$0.497228\pi$
$882$	−89.4650	−3.01244
$883$	19.8807	0.669037	0.334519	−	0.942389i	$-0.391426\pi$
$883$	19.8807	0.669037	0.334519	+	0.942389i	$0.391426\pi$
$884$	−31.3132	−1.05318
$885$	0	0
$886$	70.9556	2.38380
$887$	17.2279	0.578456	0.289228	−	0.957260i	$-0.406601\pi$
$887$	17.2279	0.578456	0.289228	+	0.957260i	$0.406601\pi$
$888$	−2.64180	−0.0886531
$889$	2.89822	0.0972033
$890$	0	0
$891$	30.4698	1.02078
$892$	8.24503	0.276064
$893$	2.66891	0.0893117
$894$	−78.6755	−2.63130
$895$	0	0
$896$	3.37745	0.112833
$897$	−118.120	−3.94392
$898$	−32.9321	−1.09896
$899$	−4.79944	−0.160070
$900$	0	0
$901$	−24.1256	−0.803742
$902$	−17.2903	−0.575703
$903$	124.564	4.14522
$904$	−1.06894	−0.0355523
$905$	0	0
$906$	−84.2875	−2.80027
$907$	−16.6575	−0.553104	−0.276552	−	0.960999i	$-0.589192\pi$
$907$	−16.6575	−0.553104	−0.276552	+	0.960999i	$0.589192\pi$
$908$	−26.3994	−0.876094
$909$	5.43867	0.180389
$910$	0	0
$911$	27.5693	0.913411	0.456706	−	0.889618i	$-0.349029\pi$
$911$	27.5693	0.913411	0.456706	+	0.889618i	$0.349029\pi$
$912$	59.6565	1.97542
$913$	57.3949	1.89949
$914$	−28.2378	−0.934024
$915$	0	0
$916$	−11.4440	−0.378121
$917$	27.3934	0.904610
$918$	54.1170	1.78613
$919$	28.4662	0.939012	0.469506	−	0.882929i	$-0.344432\pi$
$919$	28.4662	0.939012	0.469506	+	0.882929i	$0.344432\pi$
$920$	0	0
$921$	−54.4542	−1.79433
$922$	59.0436	1.94450
$923$	−68.7583	−2.26321
$924$	−120.284	−3.95704
$925$	0	0
$926$	33.0401	1.08577
$927$	−96.6992	−3.17602
$928$	20.3334	0.667478
$929$	−40.8486	−1.34020	−0.670100	−	0.742271i	$-0.733749\pi$
$929$	−40.8486	−1.34020	−0.670100	+	0.742271i	$0.733749\pi$
$930$	0	0
$931$	−39.7914	−1.30411
$932$	23.4623	0.768533
$933$	−73.8049	−2.41626
$934$	27.8682	0.911877
$935$	0	0
$936$	2.75715	0.0901202
$937$	11.1714	0.364953	0.182476	−	0.983210i	$-0.441589\pi$
$937$	11.1714	0.364953	0.182476	+	0.983210i	$0.441589\pi$
$938$	49.3159	1.61022
$939$	66.8470	2.18147
$940$	0	0
$941$	43.0290	1.40271	0.701353	−	0.712814i	$-0.252580\pi$
$941$	43.0290	1.40271	0.701353	+	0.712814i	$0.252580\pi$
$942$	8.41180	0.274071
$943$	−14.2777	−0.464945
$944$	32.8287	1.06848
$945$	0	0
$946$	118.245	3.84449
$947$	−12.2469	−0.397970	−0.198985	−	0.980003i	$-0.563764\pi$
$947$	−12.2469	−0.397970	−0.198985	+	0.980003i	$0.563764\pi$
$948$	−30.8071	−1.00057
$949$	−40.5624	−1.31671
$950$	0	0
$951$	−49.4747	−1.60433
$952$	1.50553	0.0487946
$953$	−3.64133	−0.117954	−0.0589771	−	0.998259i	$-0.518784\pi$
$953$	−3.64133	−0.117954	−0.0589771	+	0.998259i	$0.518784\pi$
$954$	−75.3521	−2.43961
$955$	0	0
$956$	−52.1902	−1.68795
$957$	40.8449	1.32033
$958$	−14.6778	−0.474217
$959$	32.2758	1.04224
$960$	0	0
$961$	−27.4885	−0.886726
$962$	74.1626	2.39110
$963$	9.19195	0.296206
$964$	1.94516	0.0626495
$965$	0	0
$966$	−201.452	−6.48161
$967$	58.7347	1.88878	0.944390	−	0.328828i	$-0.106654\pi$
$967$	58.7347	1.88878	0.944390	+	0.328828i	$0.106654\pi$
$968$	2.02084	0.0649523
$969$	51.7841	1.66354
$970$	0	0
$971$	31.9698	1.02596	0.512980	−	0.858400i	$-0.328541\pi$
$971$	31.9698	1.02596	0.512980	+	0.858400i	$0.328541\pi$
$972$	−12.6194	−0.404767
$973$	15.6267	0.500969
$974$	74.8575	2.39859
$975$	0	0
$976$	25.8804	0.828411
$977$	55.9385	1.78963	0.894816	−	0.446436i	$-0.147307\pi$
$977$	55.9385	1.78963	0.894816	+	0.446436i	$0.147307\pi$
$978$	60.4093	1.93168
$979$	59.2564	1.89384
$980$	0	0
$981$	35.2690	1.12605
$982$	−37.6376	−1.20106
$983$	−17.5972	−0.561264	−0.280632	−	0.959815i	$-0.590544\pi$
$983$	−17.5972	−0.561264	−0.280632	+	0.959815i	$0.590544\pi$
$984$	0.511630	0.0163102
$985$	0	0
$986$	18.1345	0.577520
$987$	6.13053	0.195137
$988$	−43.4991	−1.38389
$989$	97.6427	3.10486
$990$	0	0
$991$	21.5494	0.684539	0.342269	−	0.939602i	$-0.388804\pi$
$991$	21.5494	0.684539	0.342269	+	0.939602i	$0.388804\pi$
$992$	−14.8769	−0.472341
$993$	102.395	3.24942
$994$	−117.266	−3.71945
$995$	0	0
$996$	60.2437	1.90889
$997$	24.2979	0.769522	0.384761	−	0.923016i	$-0.374284\pi$
$997$	24.2979	0.769522	0.384761	+	0.923016i	$0.374284\pi$
$998$	1.29709	0.0410588
$999$	−63.1949	−1.99940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.m.1.10	✓	40
5.4	even	2		6025.2.a.n.1.31	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.10	✓	40	1.1	even	1	trivial
6025.2.a.n.1.31	yes	40	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.98624 q^{2} +2.93351 q^{3} +1.94516 q^{4} -5.82666 q^{6} -3.87755 q^{7} +0.108919 q^{8} +5.60547 q^{9} +O(q^{10})\)	\(q-1.98624 q^{2} +2.93351 q^{3} +1.94516 q^{4} -5.82666 q^{6} -3.87755 q^{7} +0.108919 q^{8} +5.60547 q^{9} +5.43632 q^{11} +5.70615 q^{12} +4.51589 q^{13} +7.70177 q^{14} -4.10667 q^{16} -3.56474 q^{17} -11.1338 q^{18} -4.95200 q^{19} -11.3748 q^{21} -10.7979 q^{22} -8.91648 q^{23} +0.319515 q^{24} -8.96966 q^{26} +7.64316 q^{27} -7.54248 q^{28} +2.56121 q^{29} -1.87389 q^{31} +7.93900 q^{32} +15.9475 q^{33} +7.08045 q^{34} +10.9035 q^{36} -8.26816 q^{37} +9.83588 q^{38} +13.2474 q^{39} +1.60127 q^{41} +22.5932 q^{42} -10.9508 q^{43} +10.5745 q^{44} +17.7103 q^{46} -0.538956 q^{47} -12.0469 q^{48} +8.03542 q^{49} -10.4572 q^{51} +8.78415 q^{52} +6.76785 q^{53} -15.1812 q^{54} -0.422340 q^{56} -14.5267 q^{57} -5.08719 q^{58} -7.99400 q^{59} -6.30204 q^{61} +3.72201 q^{62} -21.7355 q^{63} -7.55546 q^{64} -31.6756 q^{66} +6.40319 q^{67} -6.93401 q^{68} -26.1566 q^{69} -15.2258 q^{71} +0.610543 q^{72} -8.98213 q^{73} +16.4226 q^{74} -9.63245 q^{76} -21.0796 q^{77} -26.3126 q^{78} -5.39894 q^{79} +5.60486 q^{81} -3.18051 q^{82} +10.5577 q^{83} -22.1259 q^{84} +21.7510 q^{86} +7.51333 q^{87} +0.592120 q^{88} +10.9001 q^{89} -17.5106 q^{91} -17.3440 q^{92} -5.49708 q^{93} +1.07050 q^{94} +23.2891 q^{96} -3.42286 q^{97} -15.9603 q^{98} +30.4731 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9	\( 40 q - 9 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 20 q^{7} - 27 q^{8} + 38 q^{9} + q^{11} - 26 q^{12} - 11 q^{13} - q^{14} + 43 q^{16} - 20 q^{17} - 18 q^{18} + 2 q^{21} - 23 q^{22} - 79 q^{23} - 2 q^{24} + 2 q^{26} - 26 q^{27} - 30 q^{28} + 2 q^{29} + q^{31} - 68 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 16 q^{37} - 45 q^{38} - 2 q^{39} - 2 q^{41} - 19 q^{42} - 25 q^{43} + 3 q^{44} + 14 q^{46} - 88 q^{47} - 75 q^{48} + 40 q^{49} - 10 q^{51} - 18 q^{52} - 34 q^{53} + 4 q^{54} - 15 q^{56} - 51 q^{57} - 53 q^{58} + q^{59} + 9 q^{61} - 39 q^{62} - 110 q^{63} + 17 q^{64} + 26 q^{66} - 30 q^{67} - 44 q^{68} - 7 q^{69} + 5 q^{71} - 18 q^{72} - 23 q^{73} - 18 q^{74} + 43 q^{76} - 30 q^{77} - 46 q^{78} + 5 q^{79} + 44 q^{81} - 5 q^{82} - 65 q^{83} - 65 q^{84} + 40 q^{86} - 33 q^{87} - 71 q^{88} - 9 q^{89} + q^{91} - 117 q^{92} - 68 q^{93} - 72 q^{94} + 83 q^{96} + 8 q^{97} - 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 20 * q^7 - 27 * q^8 + 38 * q^9 + q^11 - 26 * q^12 - 11 * q^13 - q^14 + 43 * q^16 - 20 * q^17 - 18 * q^18 + 2 * q^21 - 23 * q^22 - 79 * q^23 - 2 * q^24 + 2 * q^26 - 26 * q^27 - 30 * q^28 + 2 * q^29 + q^31 - 68 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 16 * q^37 - 45 * q^38 - 2 * q^39 - 2 * q^41 - 19 * q^42 - 25 * q^43 + 3 * q^44 + 14 * q^46 - 88 * q^47 - 75 * q^48 + 40 * q^49 - 10 * q^51 - 18 * q^52 - 34 * q^53 + 4 * q^54 - 15 * q^56 - 51 * q^57 - 53 * q^58 + q^59 + 9 * q^61 - 39 * q^62 - 110 * q^63 + 17 * q^64 + 26 * q^66 - 30 * q^67 - 44 * q^68 - 7 * q^69 + 5 * q^71 - 18 * q^72 - 23 * q^73 - 18 * q^74 + 43 * q^76 - 30 * q^77 - 46 * q^78 + 5 * q^79 + 44 * q^81 - 5 * q^82 - 65 * q^83 - 65 * q^84 + 40 * q^86 - 33 * q^87 - 71 * q^88 - 9 * q^89 + q^91 - 117 * q^92 - 68 * q^93 - 72 * q^94 + 83 * q^96 + 8 * q^97 - 76 * q^98 - 40 * q^99

Introduction

L-functions

Modular forms

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Database

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