LMFDB - Embedded newform 6025.2.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11193 q^{2} +3.26338 q^{3} +2.46023 q^{4} -6.89202 q^{6} +1.21893 q^{7} -0.971967 q^{8} +7.64967 q^{9} +O(q^{10})$	\(q-2.11193 q^{2} +3.26338 q^{3} +2.46023 q^{4} -6.89202 q^{6} +1.21893 q^{7} -0.971967 q^{8} +7.64967 q^{9} -0.657173 q^{11} +8.02866 q^{12} +3.70129 q^{13} -2.57429 q^{14} -2.86773 q^{16} -4.59640 q^{17} -16.1555 q^{18} +5.68126 q^{19} +3.97783 q^{21} +1.38790 q^{22} +8.39925 q^{23} -3.17190 q^{24} -7.81684 q^{26} +15.1736 q^{27} +2.99884 q^{28} +9.06763 q^{29} +4.34253 q^{31} +8.00037 q^{32} -2.14461 q^{33} +9.70726 q^{34} +18.8199 q^{36} -11.6103 q^{37} -11.9984 q^{38} +12.0787 q^{39} +1.50674 q^{41} -8.40088 q^{42} +6.16317 q^{43} -1.61680 q^{44} -17.7386 q^{46} +0.492488 q^{47} -9.35851 q^{48} -5.51421 q^{49} -14.9998 q^{51} +9.10601 q^{52} -1.85887 q^{53} -32.0456 q^{54} -1.18476 q^{56} +18.5401 q^{57} -19.1502 q^{58} -5.83532 q^{59} -14.7560 q^{61} -9.17110 q^{62} +9.32440 q^{63} -11.1607 q^{64} +4.52925 q^{66} +9.03835 q^{67} -11.3082 q^{68} +27.4100 q^{69} +7.93167 q^{71} -7.43522 q^{72} -9.57092 q^{73} +24.5202 q^{74} +13.9772 q^{76} -0.801048 q^{77} -25.5093 q^{78} -16.3001 q^{79} +26.5684 q^{81} -3.18212 q^{82} +3.31730 q^{83} +9.78637 q^{84} -13.0161 q^{86} +29.5912 q^{87} +0.638751 q^{88} -4.76514 q^{89} +4.51160 q^{91} +20.6641 q^{92} +14.1713 q^{93} -1.04010 q^{94} +26.1083 q^{96} +6.06806 q^{97} +11.6456 q^{98} -5.02716 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.11193	−1.49336	−0.746678	−	0.665185i	$-0.768353\pi$
$2$	−2.11193	−1.49336	−0.746678	+	0.665185i	$0.768353\pi$
$3$	3.26338	1.88411	0.942057	−	0.335452i	$-0.108889\pi$
$3$	3.26338	1.88411	0.942057	+	0.335452i	$0.108889\pi$
$4$	2.46023	1.23011
$5$	0	0
$5$	0	0
$6$	−6.89202	−2.81366
$7$	1.21893	0.460712	0.230356	−	0.973106i	$-0.426011\pi$
$7$	1.21893	0.460712	0.230356	+	0.973106i	$0.426011\pi$
$8$	−0.971967	−0.343642
$9$	7.64967	2.54989
$10$	0	0
$11$	−0.657173	−0.198145	−0.0990726	−	0.995080i	$-0.531588\pi$
$11$	−0.657173	−0.198145	−0.0990726	+	0.995080i	$0.531588\pi$
$12$	8.02866	2.31768
$13$	3.70129	1.02655	0.513276	−	0.858224i	$-0.328432\pi$
$13$	3.70129	1.02655	0.513276	+	0.858224i	$0.328432\pi$
$14$	−2.57429	−0.688007
$15$	0	0
$16$	−2.86773	−0.716934
$17$	−4.59640	−1.11479	−0.557396	−	0.830247i	$-0.688199\pi$
$17$	−4.59640	−1.11479	−0.557396	+	0.830247i	$0.688199\pi$
$18$	−16.1555	−3.80789
$19$	5.68126	1.30337	0.651685	−	0.758490i	$-0.274063\pi$
$19$	5.68126	1.30337	0.651685	+	0.758490i	$0.274063\pi$
$20$	0	0
$21$	3.97783	0.868034
$22$	1.38790	0.295901
$23$	8.39925	1.75136	0.875682	−	0.482888i	$-0.160412\pi$
$23$	8.39925	1.75136	0.875682	+	0.482888i	$0.160412\pi$
$24$	−3.17190	−0.647461
$25$	0	0
$26$	−7.81684	−1.53301
$27$	15.1736	2.92017
$28$	2.99884	0.566728
$29$	9.06763	1.68382	0.841909	−	0.539620i	$-0.181432\pi$
$29$	9.06763	1.68382	0.841909	+	0.539620i	$0.181432\pi$
$30$	0	0
$31$	4.34253	0.779942	0.389971	−	0.920827i	$-0.372485\pi$
$31$	4.34253	0.779942	0.389971	+	0.920827i	$0.372485\pi$
$32$	8.00037	1.41428
$33$	−2.14461	−0.373328
$34$	9.70726	1.66478
$35$	0	0
$36$	18.8199	3.13665
$37$	−11.6103	−1.90873	−0.954364	−	0.298646i	$-0.903465\pi$
$37$	−11.6103	−1.90873	−0.954364	+	0.298646i	$0.903465\pi$
$38$	−11.9984	−1.94640
$39$	12.0787	1.93414
$40$	0	0
$41$	1.50674	0.235313	0.117656	−	0.993054i	$-0.462462\pi$
$41$	1.50674	0.235313	0.117656	+	0.993054i	$0.462462\pi$
$42$	−8.40088	−1.29628
$43$	6.16317	0.939874	0.469937	−	0.882700i	$-0.344277\pi$
$43$	6.16317	0.939874	0.469937	+	0.882700i	$0.344277\pi$
$44$	−1.61680	−0.243741
$45$	0	0
$46$	−17.7386	−2.61541
$47$	0.492488	0.0718368	0.0359184	−	0.999355i	$-0.488564\pi$
$47$	0.492488	0.0718368	0.0359184	+	0.999355i	$0.488564\pi$
$48$	−9.35851	−1.35079
$49$	−5.51421	−0.787745
$50$	0	0
$51$	−14.9998	−2.10039
$52$	9.10601	1.26278
$53$	−1.85887	−0.255336	−0.127668	−	0.991817i	$-0.540749\pi$
$53$	−1.85887	−0.255336	−0.127668	+	0.991817i	$0.540749\pi$
$54$	−32.0456	−4.36085
$55$	0	0
$56$	−1.18476	−0.158320
$57$	18.5401	2.45570
$58$	−19.1502	−2.51454
$59$	−5.83532	−0.759694	−0.379847	−	0.925049i	$-0.624023\pi$
$59$	−5.83532	−0.759694	−0.379847	+	0.925049i	$0.624023\pi$
$60$	0	0
$61$	−14.7560	−1.88931	−0.944655	−	0.328065i	$-0.893603\pi$
$61$	−14.7560	−1.88931	−0.944655	+	0.328065i	$0.893603\pi$
$62$	−9.17110	−1.16473
$63$	9.32440	1.17476
$64$	−11.1607	−1.39509
$65$	0	0
$66$	4.52925	0.557512
$67$	9.03835	1.10421	0.552105	−	0.833775i	$-0.313825\pi$
$67$	9.03835	1.10421	0.552105	+	0.833775i	$0.313825\pi$
$68$	−11.3082	−1.37132
$69$	27.4100	3.29977
$70$	0	0
$71$	7.93167	0.941316	0.470658	−	0.882316i	$-0.344017\pi$
$71$	7.93167	0.941316	0.470658	+	0.882316i	$0.344017\pi$
$72$	−7.43522	−0.876249
$73$	−9.57092	−1.12019	−0.560096	−	0.828428i	$-0.689236\pi$
$73$	−9.57092	−1.12019	−0.560096	+	0.828428i	$0.689236\pi$
$74$	24.5202	2.85041
$75$	0	0
$76$	13.9772	1.60329
$77$	−0.801048	−0.0912878
$78$	−25.5093	−2.88836
$79$	−16.3001	−1.83390	−0.916951	−	0.398999i	$-0.869358\pi$
$79$	−16.3001	−1.83390	−0.916951	+	0.398999i	$0.869358\pi$
$80$	0	0
$81$	26.5684	2.95204
$82$	−3.18212	−0.351406
$83$	3.31730	0.364121	0.182061	−	0.983287i	$-0.441723\pi$
$83$	3.31730	0.364121	0.182061	+	0.983287i	$0.441723\pi$
$84$	9.78637	1.06778
$85$	0	0
$86$	−13.0161	−1.40357
$87$	29.5912	3.17251
$88$	0.638751	0.0680910
$89$	−4.76514	−0.505104	−0.252552	−	0.967583i	$-0.581270\pi$
$89$	−4.76514	−0.505104	−0.252552	+	0.967583i	$0.581270\pi$
$90$	0	0
$91$	4.51160	0.472945
$92$	20.6641	2.15438
$93$	14.1713	1.46950
$94$	−1.04010	−0.107278
$95$	0	0
$96$	26.1083	2.66467
$97$	6.06806	0.616119	0.308059	−	0.951367i	$-0.400320\pi$
$97$	6.06806	0.616119	0.308059	+	0.951367i	$0.400320\pi$
$98$	11.6456	1.17638
$99$	−5.02716	−0.505248
$100$	0	0
$101$	15.1722	1.50969	0.754845	−	0.655903i	$-0.227712\pi$
$101$	15.1722	1.50969	0.754845	+	0.655903i	$0.227712\pi$
$102$	31.6785	3.13664
$103$	−10.2417	−1.00915	−0.504573	−	0.863369i	$-0.668350\pi$
$103$	−10.2417	−1.00915	−0.504573	+	0.863369i	$0.668350\pi$
$104$	−3.59753	−0.352767
$105$	0	0
$106$	3.92580	0.381308
$107$	−10.8855	−1.05234	−0.526168	−	0.850380i	$-0.676372\pi$
$107$	−10.8855	−1.05234	−0.526168	+	0.850380i	$0.676372\pi$
$108$	37.3306	3.59214
$109$	−15.3782	−1.47297	−0.736484	−	0.676455i	$-0.763515\pi$
$109$	−15.3782	−1.47297	−0.736484	+	0.676455i	$0.763515\pi$
$110$	0	0
$111$	−37.8890	−3.59626
$112$	−3.49556	−0.330300
$113$	16.5356	1.55554	0.777768	−	0.628552i	$-0.216352\pi$
$113$	16.5356	1.55554	0.777768	+	0.628552i	$0.216352\pi$
$114$	−39.1553	−3.66723
$115$	0	0
$116$	22.3084	2.07129
$117$	28.3136	2.61759
$118$	12.3238	1.13449
$119$	−5.60269	−0.513597
$120$	0	0
$121$	−10.5681	−0.960738
$122$	31.1635	2.82141
$123$	4.91706	0.443356
$124$	10.6836	0.959417
$125$	0	0
$126$	−19.6924	−1.75434
$127$	−14.8873	−1.32103	−0.660517	−	0.750811i	$-0.729663\pi$
$127$	−14.8873	−1.32103	−0.660517	+	0.750811i	$0.729663\pi$
$128$	7.56986	0.669088
$129$	20.1128	1.77083
$130$	0	0
$131$	−4.13692	−0.361444	−0.180722	−	0.983534i	$-0.557844\pi$
$131$	−4.13692	−0.361444	−0.180722	+	0.983534i	$0.557844\pi$
$132$	−5.27622	−0.459236
$133$	6.92505	0.600478
$134$	−19.0883	−1.64898
$135$	0	0
$136$	4.46755	0.383089
$137$	−7.75840	−0.662845	−0.331423	−	0.943482i	$-0.607529\pi$
$137$	−7.75840	−0.662845	−0.331423	+	0.943482i	$0.607529\pi$
$138$	−57.8878	−4.92773
$139$	9.07808	0.769993	0.384996	−	0.922918i	$-0.374203\pi$
$139$	9.07808	0.769993	0.384996	+	0.922918i	$0.374203\pi$
$140$	0	0
$141$	1.60718	0.135349
$142$	−16.7511	−1.40572
$143$	−2.43239	−0.203406
$144$	−21.9372	−1.82810
$145$	0	0
$146$	20.2131	1.67285
$147$	−17.9950	−1.48420
$148$	−28.5641	−2.34795
$149$	3.72705	0.305332	0.152666	−	0.988278i	$-0.451214\pi$
$149$	3.72705	0.305332	0.152666	+	0.988278i	$0.451214\pi$
$150$	0	0
$151$	−5.44293	−0.442939	−0.221470	−	0.975167i	$-0.571085\pi$
$151$	−5.44293	−0.442939	−0.221470	+	0.975167i	$0.571085\pi$
$152$	−5.52199	−0.447893
$153$	−35.1609	−2.84259
$154$	1.69175	0.136325
$155$	0	0
$156$	29.7164	2.37921
$157$	10.8998	0.869898	0.434949	−	0.900455i	$-0.356766\pi$
$157$	10.8998	0.869898	0.434949	+	0.900455i	$0.356766\pi$
$158$	34.4246	2.73867
$159$	−6.06622	−0.481082
$160$	0	0
$161$	10.2381	0.806874
$162$	−56.1104	−4.40845
$163$	7.52799	0.589638	0.294819	−	0.955553i	$-0.404741\pi$
$163$	7.52799	0.589638	0.294819	+	0.955553i	$0.404741\pi$
$164$	3.70692	0.289462
$165$	0	0
$166$	−7.00589	−0.543763
$167$	12.0732	0.934256	0.467128	−	0.884190i	$-0.345289\pi$
$167$	12.0732	0.934256	0.467128	+	0.884190i	$0.345289\pi$
$168$	−3.86632	−0.298293
$169$	0.699515	0.0538088
$170$	0	0
$171$	43.4597	3.32345
$172$	15.1628	1.15615
$173$	4.96913	0.377796	0.188898	−	0.981997i	$-0.439508\pi$
$173$	4.96913	0.377796	0.188898	+	0.981997i	$0.439508\pi$
$174$	−62.4943	−4.73768
$175$	0	0
$176$	1.88460	0.142057
$177$	−19.0429	−1.43135
$178$	10.0636	0.754301
$179$	−20.1380	−1.50519	−0.752594	−	0.658485i	$-0.771198\pi$
$179$	−20.1380	−1.50519	−0.752594	+	0.658485i	$0.771198\pi$
$180$	0	0
$181$	22.8289	1.69686	0.848428	−	0.529311i	$-0.177550\pi$
$181$	22.8289	1.69686	0.848428	+	0.529311i	$0.177550\pi$
$182$	−9.52817	−0.706275
$183$	−48.1544	−3.55968
$184$	−8.16379	−0.601842
$185$	0	0
$186$	−29.9288	−2.19449
$187$	3.02063	0.220890
$188$	1.21163	0.0883674
$189$	18.4956	1.34536
$190$	0	0
$191$	−15.1503	−1.09623	−0.548117	−	0.836402i	$-0.684655\pi$
$191$	−15.1503	−1.09623	−0.548117	+	0.836402i	$0.684655\pi$
$192$	−36.4217	−2.62851
$193$	13.2647	0.954816	0.477408	−	0.878682i	$-0.341576\pi$
$193$	13.2647	0.954816	0.477408	+	0.878682i	$0.341576\pi$
$194$	−12.8153	−0.920085
$195$	0	0
$196$	−13.5662	−0.969016
$197$	19.8564	1.41471	0.707356	−	0.706857i	$-0.249888\pi$
$197$	19.8564	1.41471	0.707356	+	0.706857i	$0.249888\pi$
$198$	10.6170	0.754516
$199$	−19.4482	−1.37865	−0.689324	−	0.724453i	$-0.742092\pi$
$199$	−19.4482	−1.37865	−0.689324	+	0.724453i	$0.742092\pi$
$200$	0	0
$201$	29.4956	2.08046
$202$	−32.0425	−2.25451
$203$	11.0528	0.775755
$204$	−36.9030	−2.58372
$205$	0	0
$206$	21.6297	1.50701
$207$	64.2514	4.46578
$208$	−10.6143	−0.735970
$209$	−3.73357	−0.258256
$210$	0	0
$211$	−16.4002	−1.12903	−0.564517	−	0.825421i	$-0.690938\pi$
$211$	−16.4002	−1.12903	−0.564517	+	0.825421i	$0.690938\pi$
$212$	−4.57325	−0.314092
$213$	25.8841	1.77355
$214$	22.9893	1.57151
$215$	0	0
$216$	−14.7483	−1.00349
$217$	5.29324	0.359328
$218$	32.4777	2.19967
$219$	−31.2336	−2.11057
$220$	0	0
$221$	−17.0126	−1.14439
$222$	80.0187	5.37050
$223$	3.13981	0.210257	0.105128	−	0.994459i	$-0.466475\pi$
$223$	3.13981	0.210257	0.105128	+	0.994459i	$0.466475\pi$
$224$	9.75189	0.651575
$225$	0	0
$226$	−34.9219	−2.32297
$227$	18.8877	1.25362	0.626809	−	0.779173i	$-0.284360\pi$
$227$	18.8877	1.25362	0.626809	+	0.779173i	$0.284360\pi$
$228$	45.6129	3.02079
$229$	25.4403	1.68114	0.840570	−	0.541704i	$-0.182220\pi$
$229$	25.4403	1.68114	0.840570	+	0.541704i	$0.182220\pi$
$230$	0	0
$231$	−2.61412	−0.171997
$232$	−8.81344	−0.578631
$233$	11.2416	0.736463	0.368231	−	0.929734i	$-0.379963\pi$
$233$	11.2416	0.736463	0.368231	+	0.929734i	$0.379963\pi$
$234$	−59.7962	−3.90900
$235$	0	0
$236$	−14.3562	−0.934510
$237$	−53.1934	−3.45528
$238$	11.8325	0.766984
$239$	−20.2884	−1.31235	−0.656175	−	0.754609i	$-0.727827\pi$
$239$	−20.2884	−1.31235	−0.656175	+	0.754609i	$0.727827\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	22.3191	1.43473
$243$	41.1819	2.64182
$244$	−36.3031	−2.32407
$245$	0	0
$246$	−10.3845	−0.662089
$247$	21.0279	1.33798
$248$	−4.22080	−0.268021
$249$	10.8256	0.686046
$250$	0	0
$251$	11.6711	0.736675	0.368338	−	0.929692i	$-0.379927\pi$
$251$	11.6711	0.736675	0.368338	+	0.929692i	$0.379927\pi$
$252$	22.9401	1.44509
$253$	−5.51976	−0.347024
$254$	31.4409	1.97278
$255$	0	0
$256$	6.33446	0.395904
$257$	15.3825	0.959533	0.479766	−	0.877396i	$-0.340721\pi$
$257$	15.3825	0.959533	0.479766	+	0.877396i	$0.340721\pi$
$258$	−42.4767	−2.64448
$259$	−14.1522	−0.879373
$260$	0	0
$261$	69.3644	4.29355
$262$	8.73687	0.539765
$263$	−9.95790	−0.614030	−0.307015	−	0.951705i	$-0.599330\pi$
$263$	−9.95790	−0.614030	−0.307015	+	0.951705i	$0.599330\pi$
$264$	2.08449	0.128291
$265$	0	0
$266$	−14.6252	−0.896727
$267$	−15.5505	−0.951674
$268$	22.2364	1.35830
$269$	−20.4397	−1.24623	−0.623116	−	0.782129i	$-0.714134\pi$
$269$	−20.4397	−1.24623	−0.623116	+	0.782129i	$0.714134\pi$
$270$	0	0
$271$	−13.3434	−0.810554	−0.405277	−	0.914194i	$-0.632825\pi$
$271$	−13.3434	−0.810554	−0.405277	+	0.914194i	$0.632825\pi$
$272$	13.1813	0.799231
$273$	14.7231	0.891082
$274$	16.3852	0.989864
$275$	0	0
$276$	67.4347	4.05909
$277$	2.91467	0.175126	0.0875629	−	0.996159i	$-0.472092\pi$
$277$	2.91467	0.175126	0.0875629	+	0.996159i	$0.472092\pi$
$278$	−19.1722	−1.14987
$279$	33.2189	1.98876
$280$	0	0
$281$	−15.5780	−0.929308	−0.464654	−	0.885492i	$-0.653821\pi$
$281$	−15.5780	−0.929308	−0.464654	+	0.885492i	$0.653821\pi$
$282$	−3.39424	−0.202124
$283$	2.81294	0.167212	0.0836058	−	0.996499i	$-0.473356\pi$
$283$	2.81294	0.167212	0.0836058	+	0.996499i	$0.473356\pi$
$284$	19.5137	1.15793
$285$	0	0
$286$	5.13702	0.303758
$287$	1.83661	0.108411
$288$	61.2002	3.60626
$289$	4.12690	0.242759
$290$	0	0
$291$	19.8024	1.16084
$292$	−23.5466	−1.37796
$293$	17.7083	1.03453	0.517265	−	0.855826i	$-0.326950\pi$
$293$	17.7083	1.03453	0.517265	+	0.855826i	$0.326950\pi$
$294$	38.0041	2.21644
$295$	0	0
$296$	11.2849	0.655919
$297$	−9.97171	−0.578617
$298$	−7.87125	−0.455969
$299$	31.0880	1.79787
$300$	0	0
$301$	7.51246	0.433011
$302$	11.4951	0.661466
$303$	49.5127	2.84443
$304$	−16.2923	−0.934429
$305$	0	0
$306$	74.2573	4.24500
$307$	−3.06111	−0.174707	−0.0873533	−	0.996177i	$-0.527841\pi$
$307$	−3.06111	−0.174707	−0.0873533	+	0.996177i	$0.527841\pi$
$308$	−1.97076	−0.112294
$309$	−33.4226	−1.90135
$310$	0	0
$311$	0.872426	0.0494707	0.0247354	−	0.999694i	$-0.492126\pi$
$311$	0.872426	0.0494707	0.0247354	+	0.999694i	$0.492126\pi$
$312$	−11.7401	−0.664653
$313$	19.4197	1.09767	0.548834	−	0.835931i	$-0.315072\pi$
$313$	19.4197	1.09767	0.548834	+	0.835931i	$0.315072\pi$
$314$	−23.0195	−1.29907
$315$	0	0
$316$	−40.1019	−2.25591
$317$	−20.9802	−1.17836	−0.589182	−	0.808000i	$-0.700550\pi$
$317$	−20.9802	−1.17836	−0.589182	+	0.808000i	$0.700550\pi$
$318$	12.8114	0.718427
$319$	−5.95901	−0.333640
$320$	0	0
$321$	−35.5234	−1.98272
$322$	−21.6221	−1.20495
$323$	−26.1133	−1.45298
$324$	65.3643	3.63135
$325$	0	0
$326$	−15.8986	−0.880540
$327$	−50.1851	−2.77524
$328$	−1.46450	−0.0808634
$329$	0.600308	0.0330961
$330$	0	0
$331$	−13.1682	−0.723789	−0.361895	−	0.932219i	$-0.617870\pi$
$331$	−13.1682	−0.723789	−0.361895	+	0.932219i	$0.617870\pi$
$332$	8.16132	0.447910
$333$	−88.8152	−4.86704
$334$	−25.4978	−1.39518
$335$	0	0
$336$	−11.4074	−0.622323
$337$	5.29417	0.288392	0.144196	−	0.989549i	$-0.453940\pi$
$337$	5.29417	0.288392	0.144196	+	0.989549i	$0.453940\pi$
$338$	−1.47732	−0.0803558
$339$	53.9619	2.93081
$340$	0	0
$341$	−2.85380	−0.154542
$342$	−91.7836	−4.96309
$343$	−15.2539	−0.823635
$344$	−5.99039	−0.322980
$345$	0	0
$346$	−10.4944	−0.564184
$347$	9.61126	0.515960	0.257980	−	0.966150i	$-0.416943\pi$
$347$	9.61126	0.515960	0.257980	+	0.966150i	$0.416943\pi$
$348$	72.8010	3.90254
$349$	17.0472	0.912517	0.456258	−	0.889847i	$-0.349189\pi$
$349$	17.0472	0.912517	0.456258	+	0.889847i	$0.349189\pi$
$350$	0	0
$351$	56.1620	2.99770
$352$	−5.25763	−0.280233
$353$	−0.920491	−0.0489928	−0.0244964	−	0.999700i	$-0.507798\pi$
$353$	−0.920491	−0.0489928	−0.0244964	+	0.999700i	$0.507798\pi$
$354$	40.2171	2.13752
$355$	0	0
$356$	−11.7233	−0.621336
$357$	−18.2837	−0.967676
$358$	42.5300	2.24778
$359$	0.786068	0.0414871	0.0207435	−	0.999785i	$-0.493397\pi$
$359$	0.786068	0.0414871	0.0207435	+	0.999785i	$0.493397\pi$
$360$	0	0
$361$	13.2767	0.698772
$362$	−48.2128	−2.53401
$363$	−34.4878	−1.81014
$364$	11.0996	0.581776
$365$	0	0
$366$	101.699	5.31587
$367$	3.73156	0.194786	0.0973930	−	0.995246i	$-0.468950\pi$
$367$	3.73156	0.194786	0.0973930	+	0.995246i	$0.468950\pi$
$368$	−24.0868	−1.25561
$369$	11.5260	0.600022
$370$	0	0
$371$	−2.26583	−0.117636
$372$	34.8647	1.80765
$373$	−25.1118	−1.30024	−0.650120	−	0.759831i	$-0.725282\pi$
$373$	−25.1118	−1.30024	−0.650120	+	0.759831i	$0.725282\pi$
$374$	−6.37935	−0.329868
$375$	0	0
$376$	−0.478682	−0.0246861
$377$	33.5619	1.72853
$378$	−39.0613	−2.00910
$379$	−2.41224	−0.123908	−0.0619542	−	0.998079i	$-0.519733\pi$
$379$	−2.41224	−0.123908	−0.0619542	+	0.998079i	$0.519733\pi$
$380$	0	0
$381$	−48.5830	−2.48898
$382$	31.9962	1.63707
$383$	−8.18201	−0.418081	−0.209041	−	0.977907i	$-0.567034\pi$
$383$	−8.18201	−0.418081	−0.209041	+	0.977907i	$0.567034\pi$
$384$	24.7034	1.26064
$385$	0	0
$386$	−28.0141	−1.42588
$387$	47.1462	2.39657
$388$	14.9288	0.757896
$389$	−5.03827	−0.255451	−0.127725	−	0.991810i	$-0.540768\pi$
$389$	−5.03827	−0.255451	−0.127725	+	0.991810i	$0.540768\pi$
$390$	0	0
$391$	−38.6063	−1.95240
$392$	5.35963	0.270702
$393$	−13.5004	−0.681003
$394$	−41.9353	−2.11267
$395$	0	0
$396$	−12.3679	−0.621513
$397$	3.09902	0.155536	0.0777678	−	0.996972i	$-0.475221\pi$
$397$	3.09902	0.155536	0.0777678	+	0.996972i	$0.475221\pi$
$398$	41.0732	2.05881
$399$	22.5991	1.13137
$400$	0	0
$401$	−2.73436	−0.136547	−0.0682737	−	0.997667i	$-0.521749\pi$
$401$	−2.73436	−0.136547	−0.0682737	+	0.997667i	$0.521749\pi$
$402$	−62.2925	−3.10687
$403$	16.0729	0.800651
$404$	37.3271	1.85709
$405$	0	0
$406$	−23.3427	−1.15848
$407$	7.63000	0.378205
$408$	14.5793	0.721784
$409$	3.08172	0.152381	0.0761905	−	0.997093i	$-0.475724\pi$
$409$	3.08172	0.152381	0.0761905	+	0.997093i	$0.475724\pi$
$410$	0	0
$411$	−25.3186	−1.24888
$412$	−25.1970	−1.24136
$413$	−7.11284	−0.350000
$414$	−135.694	−6.66901
$415$	0	0
$416$	29.6117	1.45183
$417$	29.6252	1.45075
$418$	7.88502	0.385669
$419$	5.66460	0.276734	0.138367	−	0.990381i	$-0.455815\pi$
$419$	5.66460	0.276734	0.138367	+	0.990381i	$0.455815\pi$
$420$	0	0
$421$	−19.9719	−0.973372	−0.486686	−	0.873577i	$-0.661794\pi$
$421$	−19.9719	−0.973372	−0.486686	+	0.873577i	$0.661794\pi$
$422$	34.6359	1.68605
$423$	3.76737	0.183176
$424$	1.80676	0.0877442
$425$	0	0
$426$	−54.6652	−2.64854
$427$	−17.9865	−0.870428
$428$	−26.7807	−1.29449
$429$	−7.93781	−0.383241
$430$	0	0
$431$	−23.9528	−1.15377	−0.576884	−	0.816826i	$-0.695731\pi$
$431$	−23.9528	−1.15377	−0.576884	+	0.816826i	$0.695731\pi$
$432$	−43.5140	−2.09357
$433$	13.0061	0.625034	0.312517	−	0.949912i	$-0.398828\pi$
$433$	13.0061	0.625034	0.312517	+	0.949912i	$0.398828\pi$
$434$	−11.1789	−0.536605
$435$	0	0
$436$	−37.8340	−1.81192
$437$	47.7183	2.28267
$438$	65.9630	3.15183
$439$	31.4301	1.50008	0.750040	−	0.661393i	$-0.230034\pi$
$439$	31.4301	1.50008	0.750040	+	0.661393i	$0.230034\pi$
$440$	0	0
$441$	−42.1819	−2.00866
$442$	35.9293	1.70898
$443$	−0.661286	−0.0314187	−0.0157093	−	0.999877i	$-0.505001\pi$
$443$	−0.661286	−0.0314187	−0.0157093	+	0.999877i	$0.505001\pi$
$444$	−93.2155	−4.42381
$445$	0	0
$446$	−6.63104	−0.313989
$447$	12.1628	0.575280
$448$	−13.6041	−0.642735
$449$	25.4307	1.20015	0.600073	−	0.799945i	$-0.295138\pi$
$449$	25.4307	1.20015	0.600073	+	0.799945i	$0.295138\pi$
$450$	0	0
$451$	−0.990188	−0.0466261
$452$	40.6813	1.91349
$453$	−17.7624	−0.834549
$454$	−39.8893	−1.87210
$455$	0	0
$456$	−18.0204	−0.843881
$457$	−12.7885	−0.598219	−0.299110	−	0.954219i	$-0.596690\pi$
$457$	−12.7885	−0.598219	−0.299110	+	0.954219i	$0.596690\pi$
$458$	−53.7279	−2.51054
$459$	−69.7441	−3.25538
$460$	0	0
$461$	41.5641	1.93583	0.967916	−	0.251275i	$-0.0808497\pi$
$461$	41.5641	1.93583	0.967916	+	0.251275i	$0.0808497\pi$
$462$	5.52084	0.256853
$463$	9.13828	0.424692	0.212346	−	0.977195i	$-0.431890\pi$
$463$	9.13828	0.424692	0.212346	+	0.977195i	$0.431890\pi$
$464$	−26.0036	−1.20719
$465$	0	0
$466$	−23.7415	−1.09980
$467$	−33.9846	−1.57262	−0.786311	−	0.617831i	$-0.788012\pi$
$467$	−33.9846	−1.57262	−0.786311	+	0.617831i	$0.788012\pi$
$468$	69.6579	3.21994
$469$	11.0171	0.508723
$470$	0	0
$471$	35.5702	1.63899
$472$	5.67174	0.261063
$473$	−4.05027	−0.186232
$474$	112.341	5.15997
$475$	0	0
$476$	−13.7839	−0.631783
$477$	−14.2198	−0.651078
$478$	42.8476	1.95981
$479$	26.1818	1.19628	0.598139	−	0.801392i	$-0.295907\pi$
$479$	26.1818	1.19628	0.598139	+	0.801392i	$0.295907\pi$
$480$	0	0
$481$	−42.9732	−1.95941
$482$	−2.11193	−0.0961956
$483$	33.4108	1.52024
$484$	−26.0000	−1.18182
$485$	0	0
$486$	−86.9731	−3.94518
$487$	15.1633	0.687113	0.343556	−	0.939132i	$-0.388368\pi$
$487$	15.1633	0.687113	0.343556	+	0.939132i	$0.388368\pi$
$488$	14.3423	0.649247
$489$	24.5667	1.11095
$490$	0	0
$491$	−5.88298	−0.265495	−0.132748	−	0.991150i	$-0.542380\pi$
$491$	−5.88298	−0.265495	−0.132748	+	0.991150i	$0.542380\pi$
$492$	12.0971	0.545379
$493$	−41.6785	−1.87710
$494$	−44.4095	−1.99808
$495$	0	0
$496$	−12.4532	−0.559166
$497$	9.66814	0.433675
$498$	−22.8629	−1.02451
$499$	11.1782	0.500406	0.250203	−	0.968193i	$-0.419503\pi$
$499$	11.1782	0.500406	0.250203	+	0.968193i	$0.419503\pi$
$500$	0	0
$501$	39.3996	1.76025
$502$	−24.6486	−1.10012
$503$	−31.9201	−1.42325	−0.711624	−	0.702560i	$-0.752040\pi$
$503$	−31.9201	−1.42325	−0.711624	+	0.702560i	$0.752040\pi$
$504$	−9.06301	−0.403698
$505$	0	0
$506$	11.6573	0.518231
$507$	2.28279	0.101382
$508$	−36.6262	−1.62502
$509$	29.9255	1.32642	0.663212	−	0.748432i	$-0.269193\pi$
$509$	29.9255	1.32642	0.663212	+	0.748432i	$0.269193\pi$
$510$	0	0
$511$	−11.6663	−0.516086
$512$	−28.5176	−1.26031
$513$	86.2053	3.80606
$514$	−32.4867	−1.43292
$515$	0	0
$516$	49.4820	2.17832
$517$	−0.323650	−0.0142341
$518$	29.8883	1.31322
$519$	16.2162	0.711811
$520$	0	0
$521$	−2.37025	−0.103842	−0.0519212	−	0.998651i	$-0.516534\pi$
$521$	−2.37025	−0.103842	−0.0519212	+	0.998651i	$0.516534\pi$
$522$	−146.492	−6.41180
$523$	−17.7021	−0.774058	−0.387029	−	0.922068i	$-0.626499\pi$
$523$	−17.7021	−0.774058	−0.387029	+	0.922068i	$0.626499\pi$
$524$	−10.1778	−0.444618
$525$	0	0
$526$	21.0303	0.916966
$527$	−19.9600	−0.869472
$528$	6.15017	0.267652
$529$	47.5473	2.06728
$530$	0	0
$531$	−44.6383	−1.93714
$532$	17.0372	0.738656
$533$	5.57687	0.241561
$534$	32.8415	1.42119
$535$	0	0
$536$	−8.78497	−0.379453
$537$	−65.7181	−2.83595
$538$	43.1672	1.86107
$539$	3.62379	0.156088
$540$	0	0
$541$	−14.9668	−0.643471	−0.321736	−	0.946830i	$-0.604266\pi$
$541$	−14.9668	−0.643471	−0.321736	+	0.946830i	$0.604266\pi$
$542$	28.1803	1.21045
$543$	74.4993	3.19707
$544$	−36.7729	−1.57663
$545$	0	0
$546$	−31.0941	−1.33070
$547$	−15.8306	−0.676868	−0.338434	−	0.940990i	$-0.609897\pi$
$547$	−15.8306	−0.676868	−0.338434	+	0.940990i	$0.609897\pi$
$548$	−19.0874	−0.815375
$549$	−112.878	−4.81753
$550$	0	0
$551$	51.5156	2.19464
$552$	−26.6416	−1.13394
$553$	−19.8686	−0.844901
$554$	−6.15557	−0.261525
$555$	0	0
$556$	22.3341	0.947179
$557$	0.766512	0.0324782	0.0162391	−	0.999868i	$-0.494831\pi$
$557$	0.766512	0.0324782	0.0162391	+	0.999868i	$0.494831\pi$
$558$	−70.1559	−2.96993
$559$	22.8116	0.964830
$560$	0	0
$561$	9.85748	0.416183
$562$	32.8997	1.38779
$563$	−6.03346	−0.254280	−0.127140	−	0.991885i	$-0.540580\pi$
$563$	−6.03346	−0.254280	−0.127140	+	0.991885i	$0.540580\pi$
$564$	3.95402	0.166494
$565$	0	0
$566$	−5.94071	−0.249707
$567$	32.3850	1.36004
$568$	−7.70932	−0.323476
$569$	−28.6120	−1.19948	−0.599740	−	0.800195i	$-0.704729\pi$
$569$	−28.6120	−1.19948	−0.599740	+	0.800195i	$0.704729\pi$
$570$	0	0
$571$	−19.7548	−0.826711	−0.413355	−	0.910570i	$-0.635643\pi$
$571$	−19.7548	−0.826711	−0.413355	+	0.910570i	$0.635643\pi$
$572$	−5.98422	−0.250213
$573$	−49.4411	−2.06543
$574$	−3.87877	−0.161897
$575$	0	0
$576$	−85.3758	−3.55732
$577$	−43.4208	−1.80763	−0.903815	−	0.427923i	$-0.859245\pi$
$577$	−43.4208	−1.80763	−0.903815	+	0.427923i	$0.859245\pi$
$578$	−8.71571	−0.362526
$579$	43.2879	1.79898
$580$	0	0
$581$	4.04355	0.167755
$582$	−41.8212	−1.73355
$583$	1.22160	0.0505936
$584$	9.30262	0.384945
$585$	0	0
$586$	−37.3986	−1.54492
$587$	−38.4912	−1.58870	−0.794351	−	0.607459i	$-0.792189\pi$
$587$	−38.4912	−1.58870	−0.794351	+	0.607459i	$0.792189\pi$
$588$	−44.2718	−1.82574
$589$	24.6710	1.01655
$590$	0	0
$591$	64.7992	2.66548
$592$	33.2954	1.36843
$593$	2.61739	0.107483	0.0537416	−	0.998555i	$-0.482885\pi$
$593$	2.61739	0.107483	0.0537416	+	0.998555i	$0.482885\pi$
$594$	21.0595	0.864082
$595$	0	0
$596$	9.16939	0.375593
$597$	−63.4670	−2.59753
$598$	−65.6555	−2.68485
$599$	2.83028	0.115642	0.0578210	−	0.998327i	$-0.481585\pi$
$599$	2.83028	0.115642	0.0578210	+	0.998327i	$0.481585\pi$
$600$	0	0
$601$	−31.1749	−1.27165	−0.635826	−	0.771833i	$-0.719340\pi$
$601$	−31.1749	−1.27165	−0.635826	+	0.771833i	$0.719340\pi$
$602$	−15.8658	−0.646640
$603$	69.1403	2.81561
$604$	−13.3908	−0.544866
$605$	0	0
$606$	−104.567	−4.24775
$607$	40.0892	1.62717	0.813586	−	0.581445i	$-0.197512\pi$
$607$	40.0892	1.62717	0.813586	+	0.581445i	$0.197512\pi$
$608$	45.4522	1.84333
$609$	36.0695	1.46161
$610$	0	0
$611$	1.82284	0.0737442
$612$	−86.5039	−3.49671
$613$	−20.0038	−0.807944	−0.403972	−	0.914771i	$-0.632371\pi$
$613$	−20.0038	−0.807944	−0.403972	+	0.914771i	$0.632371\pi$
$614$	6.46483	0.260899
$615$	0	0
$616$	0.778592	0.0313704
$617$	0.0960630	0.00386735	0.00193368	−	0.999998i	$-0.499384\pi$
$617$	0.0960630	0.00386735	0.00193368	+	0.999998i	$0.499384\pi$
$618$	70.5861	2.83939
$619$	1.20680	0.0485055	0.0242527	−	0.999706i	$-0.492279\pi$
$619$	1.20680	0.0485055	0.0242527	+	0.999706i	$0.492279\pi$
$620$	0	0
$621$	127.447	5.11428
$622$	−1.84250	−0.0738774
$623$	−5.80837	−0.232707
$624$	−34.6385	−1.38665
$625$	0	0
$626$	−41.0130	−1.63921
$627$	−12.1841	−0.486585
$628$	26.8160	1.07007
$629$	53.3658	2.12783
$630$	0	0
$631$	19.6660	0.782890	0.391445	−	0.920202i	$-0.371975\pi$
$631$	19.6660	0.782890	0.391445	+	0.920202i	$0.371975\pi$
$632$	15.8431	0.630206
$633$	−53.5200	−2.12723
$634$	44.3086	1.75972
$635$	0	0
$636$	−14.9243	−0.591786
$637$	−20.4097	−0.808661
$638$	12.5850	0.498244
$639$	60.6746	2.40025
$640$	0	0
$641$	−11.7106	−0.462541	−0.231271	−	0.972889i	$-0.574288\pi$
$641$	−11.7106	−0.462541	−0.231271	+	0.972889i	$0.574288\pi$
$642$	75.0228	2.96091
$643$	28.9151	1.14030	0.570149	−	0.821541i	$-0.306885\pi$
$643$	28.9151	1.14030	0.570149	+	0.821541i	$0.306885\pi$
$644$	25.1880	0.992547
$645$	0	0
$646$	55.1494	2.16982
$647$	−16.1258	−0.633971	−0.316986	−	0.948430i	$-0.602671\pi$
$647$	−16.1258	−0.633971	−0.316986	+	0.948430i	$0.602671\pi$
$648$	−25.8236	−1.01445
$649$	3.83482	0.150530
$650$	0	0
$651$	17.2739	0.677016
$652$	18.5206	0.725322
$653$	−20.0791	−0.785756	−0.392878	−	0.919591i	$-0.628521\pi$
$653$	−20.0791	−0.785756	−0.392878	+	0.919591i	$0.628521\pi$
$654$	105.987	4.14443
$655$	0	0
$656$	−4.32092	−0.168704
$657$	−73.2143	−2.85636
$658$	−1.26781	−0.0494242
$659$	−24.8026	−0.966173	−0.483086	−	0.875573i	$-0.660484\pi$
$659$	−24.8026	−0.966173	−0.483086	+	0.875573i	$0.660484\pi$
$660$	0	0
$661$	−24.2424	−0.942920	−0.471460	−	0.881888i	$-0.656273\pi$
$661$	−24.2424	−0.942920	−0.471460	+	0.881888i	$0.656273\pi$
$662$	27.8102	1.08088
$663$	−55.5186	−2.15616
$664$	−3.22431	−0.125127
$665$	0	0
$666$	187.571	7.26823
$667$	76.1613	2.94898
$668$	29.7029	1.14924
$669$	10.2464	0.396148
$670$	0	0
$671$	9.69724	0.374358
$672$	31.8241	1.22764
$673$	−8.98411	−0.346312	−0.173156	−	0.984894i	$-0.555397\pi$
$673$	−8.98411	−0.346312	−0.173156	+	0.984894i	$0.555397\pi$
$674$	−11.1809	−0.430672
$675$	0	0
$676$	1.72097	0.0661910
$677$	42.7808	1.64420	0.822101	−	0.569342i	$-0.192802\pi$
$677$	42.7808	1.64420	0.822101	+	0.569342i	$0.192802\pi$
$678$	−113.963	−4.37674
$679$	7.39654	0.283853
$680$	0	0
$681$	61.6377	2.36196
$682$	6.02700	0.230786
$683$	45.2538	1.73159	0.865793	−	0.500402i	$-0.166814\pi$
$683$	45.2538	1.73159	0.865793	+	0.500402i	$0.166814\pi$
$684$	106.921	4.08822
$685$	0	0
$686$	32.2152	1.22998
$687$	83.0213	3.16746
$688$	−17.6743	−0.673827
$689$	−6.88022	−0.262116
$690$	0	0
$691$	−1.69289	−0.0644006	−0.0322003	−	0.999481i	$-0.510251\pi$
$691$	−1.69289	−0.0644006	−0.0322003	+	0.999481i	$0.510251\pi$
$692$	12.2252	0.464732
$693$	−6.12775	−0.232774
$694$	−20.2983	−0.770512
$695$	0	0
$696$	−28.7616	−1.09021
$697$	−6.92557	−0.262325
$698$	−36.0024	−1.36271
$699$	36.6857	1.38758
$700$	0	0
$701$	23.5389	0.889054	0.444527	−	0.895765i	$-0.353372\pi$
$701$	23.5389	0.889054	0.444527	+	0.895765i	$0.353372\pi$
$702$	−118.610	−4.47664
$703$	−65.9613	−2.48778
$704$	7.33453	0.276430
$705$	0	0
$706$	1.94401	0.0731637
$707$	18.4938	0.695532
$708$	−46.8498	−1.76072
$709$	5.00731	0.188054	0.0940268	−	0.995570i	$-0.470026\pi$
$709$	5.00731	0.188054	0.0940268	+	0.995570i	$0.470026\pi$
$710$	0	0
$711$	−124.690	−4.67625
$712$	4.63156	0.173575
$713$	36.4740	1.36596
$714$	38.6138	1.44509
$715$	0	0
$716$	−49.5442	−1.85155
$717$	−66.2089	−2.47262
$718$	−1.66012	−0.0619550
$719$	−11.2959	−0.421266	−0.210633	−	0.977565i	$-0.567552\pi$
$719$	−11.2959	−0.421266	−0.210633	+	0.977565i	$0.567552\pi$
$720$	0	0
$721$	−12.4839	−0.464926
$722$	−28.0393	−1.04352
$723$	3.26338	0.121367
$724$	56.1642	2.08733
$725$	0	0
$726$	72.8357	2.70319
$727$	−7.51782	−0.278820	−0.139410	−	0.990235i	$-0.544521\pi$
$727$	−7.51782	−0.278820	−0.139410	+	0.990235i	$0.544521\pi$
$728$	−4.38513	−0.162524
$729$	54.6871	2.02545
$730$	0	0
$731$	−28.3284	−1.04776
$732$	−118.471	−4.37881
$733$	6.40154	0.236446	0.118223	−	0.992987i	$-0.462280\pi$
$733$	6.40154	0.236446	0.118223	+	0.992987i	$0.462280\pi$
$734$	−7.88078	−0.290885
$735$	0	0
$736$	67.1971	2.47692
$737$	−5.93976	−0.218794
$738$	−24.3421	−0.896046
$739$	29.0507	1.06865	0.534323	−	0.845281i	$-0.320567\pi$
$739$	29.0507	1.06865	0.534323	+	0.845281i	$0.320567\pi$
$740$	0	0
$741$	68.6222	2.52090
$742$	4.78527	0.175673
$743$	−37.3862	−1.37157	−0.685784	−	0.727806i	$-0.740540\pi$
$743$	−37.3862	−1.37157	−0.685784	+	0.727806i	$0.740540\pi$
$744$	−13.7741	−0.504982
$745$	0	0
$746$	53.0343	1.94172
$747$	25.3762	0.928468
$748$	7.43144	0.271720
$749$	−13.2686	−0.484824
$750$	0	0
$751$	18.8923	0.689390	0.344695	−	0.938715i	$-0.387982\pi$
$751$	18.8923	0.689390	0.344695	+	0.938715i	$0.387982\pi$
$752$	−1.41233	−0.0515022
$753$	38.0874	1.38798
$754$	−70.8802	−2.58131
$755$	0	0
$756$	45.5034	1.65494
$757$	−13.9997	−0.508827	−0.254414	−	0.967095i	$-0.581882\pi$
$757$	−13.9997	−0.508827	−0.254414	+	0.967095i	$0.581882\pi$
$758$	5.09447	0.185040
$759$	−18.0131	−0.653834
$760$	0	0
$761$	31.7104	1.14950	0.574750	−	0.818329i	$-0.305099\pi$
$761$	31.7104	1.14950	0.574750	+	0.818329i	$0.305099\pi$
$762$	102.604	3.71693
$763$	−18.7450	−0.678614
$764$	−37.2731	−1.34849
$765$	0	0
$766$	17.2798	0.624344
$767$	−21.5982	−0.779865
$768$	20.6718	0.745928
$769$	−16.0164	−0.577566	−0.288783	−	0.957395i	$-0.593251\pi$
$769$	−16.0164	−0.577566	−0.288783	+	0.957395i	$0.593251\pi$
$770$	0	0
$771$	50.1989	1.80787
$772$	32.6343	1.17453
$773$	39.1819	1.40927	0.704637	−	0.709568i	$-0.251110\pi$
$773$	39.1819	1.40927	0.704637	+	0.709568i	$0.251110\pi$
$774$	−99.5692	−3.57894
$775$	0	0
$776$	−5.89796	−0.211724
$777$	−46.1840	−1.65684
$778$	10.6405	0.381479
$779$	8.56016	0.306700
$780$	0	0
$781$	−5.21248	−0.186517
$782$	81.5336	2.91564
$783$	137.589	4.91703
$784$	15.8133	0.564761
$785$	0	0
$786$	28.5117	1.01698
$787$	−38.6665	−1.37831	−0.689157	−	0.724612i	$-0.742019\pi$
$787$	−38.6665	−1.37831	−0.689157	+	0.724612i	$0.742019\pi$
$788$	48.8514	1.74026
$789$	−32.4964	−1.15690
$790$	0	0
$791$	20.1557	0.716653
$792$	4.88623	0.173625
$793$	−54.6161	−1.93948
$794$	−6.54491	−0.232270
$795$	0	0
$796$	−47.8470	−1.69589
$797$	47.3510	1.67726	0.838630	−	0.544701i	$-0.183357\pi$
$797$	47.3510	1.67726	0.838630	+	0.544701i	$0.183357\pi$
$798$	−47.7276	−1.68954
$799$	−2.26367	−0.0800830
$800$	0	0
$801$	−36.4518	−1.28796
$802$	5.77476	0.203914
$803$	6.28975	0.221961
$804$	72.5659	2.55920
$805$	0	0
$806$	−33.9449	−1.19566
$807$	−66.7026	−2.34804
$808$	−14.7469	−0.518793
$809$	−31.6409	−1.11244	−0.556218	−	0.831037i	$-0.687748\pi$
$809$	−31.6409	−1.11244	−0.556218	+	0.831037i	$0.687748\pi$
$810$	0	0
$811$	−17.7690	−0.623955	−0.311977	−	0.950090i	$-0.600991\pi$
$811$	−17.7690	−0.623955	−0.311977	+	0.950090i	$0.600991\pi$
$812$	27.1924	0.954267
$813$	−43.5446	−1.52718
$814$	−16.1140	−0.564795
$815$	0	0
$816$	43.0155	1.50584
$817$	35.0145	1.22500
$818$	−6.50836	−0.227559
$819$	34.5123	1.20596
$820$	0	0
$821$	33.4743	1.16826	0.584131	−	0.811659i	$-0.301435\pi$
$821$	33.4743	1.16826	0.584131	+	0.811659i	$0.301435\pi$
$822$	53.4711	1.86502
$823$	45.7771	1.59569	0.797845	−	0.602863i	$-0.205974\pi$
$823$	45.7771	1.59569	0.797845	+	0.602863i	$0.205974\pi$
$824$	9.95461	0.346785
$825$	0	0
$826$	15.0218	0.522675
$827$	−52.8806	−1.83884	−0.919420	−	0.393278i	$-0.871341\pi$
$827$	−52.8806	−1.83884	−0.919420	+	0.393278i	$0.871341\pi$
$828$	158.073	5.49342
$829$	−6.32956	−0.219835	−0.109917	−	0.993941i	$-0.535059\pi$
$829$	−6.32956	−0.219835	−0.109917	+	0.993941i	$0.535059\pi$
$830$	0	0
$831$	9.51169	0.329957
$832$	−41.3090	−1.43213
$833$	25.3455	0.878171
$834$	−62.5663	−2.16649
$835$	0	0
$836$	−9.18543	−0.317685
$837$	65.8920	2.27756
$838$	−11.9632	−0.413262
$839$	17.9793	0.620715	0.310357	−	0.950620i	$-0.399551\pi$
$839$	17.9793	0.620715	0.310357	+	0.950620i	$0.399551\pi$
$840$	0	0
$841$	53.2220	1.83524
$842$	42.1792	1.45359
$843$	−50.8371	−1.75092
$844$	−40.3482	−1.38884
$845$	0	0
$846$	−7.95640	−0.273547
$847$	−12.8818	−0.442624
$848$	5.33076	0.183059
$849$	9.17968	0.315046
$850$	0	0
$851$	−97.5181	−3.34288
$852$	63.6807	2.18166
$853$	35.0208	1.19909	0.599544	−	0.800341i	$-0.295348\pi$
$853$	35.0208	1.19909	0.599544	+	0.800341i	$0.295348\pi$
$854$	37.9861	1.29986
$855$	0	0
$856$	10.5803	0.361627
$857$	43.9920	1.50274	0.751369	−	0.659882i	$-0.229394\pi$
$857$	43.9920	1.50274	0.751369	+	0.659882i	$0.229394\pi$
$858$	16.7641	0.572315
$859$	−2.81742	−0.0961290	−0.0480645	−	0.998844i	$-0.515305\pi$
$859$	−2.81742	−0.0961290	−0.0480645	+	0.998844i	$0.515305\pi$
$860$	0	0
$861$	5.99355	0.204260
$862$	50.5866	1.72299
$863$	46.3014	1.57612	0.788059	−	0.615600i	$-0.211086\pi$
$863$	46.3014	1.57612	0.788059	+	0.615600i	$0.211086\pi$
$864$	121.395	4.12993
$865$	0	0
$866$	−27.4679	−0.933398
$867$	13.4677	0.457386
$868$	13.0226	0.442015
$869$	10.7120	0.363379
$870$	0	0
$871$	33.4535	1.13353
$872$	14.9471	0.506174
$873$	46.4187	1.57103
$874$	−100.777	−3.40885
$875$	0	0
$876$	−76.8417	−2.59624
$877$	−10.5420	−0.355976	−0.177988	−	0.984033i	$-0.556959\pi$
$877$	−10.5420	−0.355976	−0.177988	+	0.984033i	$0.556959\pi$
$878$	−66.3781	−2.24015
$879$	57.7889	1.94917
$880$	0	0
$881$	−9.15838	−0.308554	−0.154277	−	0.988028i	$-0.549305\pi$
$881$	−9.15838	−0.308554	−0.154277	+	0.988028i	$0.549305\pi$
$882$	89.0850	2.99965
$883$	35.0866	1.18076	0.590378	−	0.807127i	$-0.298979\pi$
$883$	35.0866	1.18076	0.590378	+	0.807127i	$0.298979\pi$
$884$	−41.8549	−1.40773
$885$	0	0
$886$	1.39659	0.0469192
$887$	−35.2186	−1.18252	−0.591262	−	0.806480i	$-0.701370\pi$
$887$	−35.2186	−1.18252	−0.591262	+	0.806480i	$0.701370\pi$
$888$	36.8268	1.23583
$889$	−18.1466	−0.608616
$890$	0	0
$891$	−17.4600	−0.584933
$892$	7.72464	0.258640
$893$	2.79795	0.0936299
$894$	−25.6869	−0.859098
$895$	0	0
$896$	9.22712	0.308257
$897$	101.452	3.38739
$898$	−53.7076	−1.79225
$899$	39.3765	1.31328
$900$	0	0
$901$	8.54413	0.284646
$902$	2.09120	0.0696294
$903$	24.5160	0.815843
$904$	−16.0720	−0.534547
$905$	0	0
$906$	37.5128	1.24628
$907$	18.3533	0.609413	0.304706	−	0.952446i	$-0.401442\pi$
$907$	18.3533	0.609413	0.304706	+	0.952446i	$0.401442\pi$
$908$	46.4680	1.54209
$909$	116.062	3.84954
$910$	0	0
$911$	19.6809	0.652058	0.326029	−	0.945360i	$-0.394289\pi$
$911$	19.6809	0.652058	0.326029	+	0.945360i	$0.394289\pi$
$912$	−53.1681	−1.76057
$913$	−2.18004	−0.0721488
$914$	27.0083	0.893354
$915$	0	0
$916$	62.5888	2.06799
$917$	−5.04261	−0.166522
$918$	147.294	4.86144
$919$	−6.57880	−0.217014	−0.108507	−	0.994096i	$-0.534607\pi$
$919$	−6.57880	−0.217014	−0.108507	+	0.994096i	$0.534607\pi$
$920$	0	0
$921$	−9.98956	−0.329167
$922$	−87.7802	−2.89089
$923$	29.3574	0.966309
$924$	−6.43134	−0.211576
$925$	0	0
$926$	−19.2994	−0.634217
$927$	−78.3457	−2.57321
$928$	72.5445	2.38139
$929$	−19.4495	−0.638119	−0.319059	−	0.947735i	$-0.603367\pi$
$929$	−19.4495	−0.638119	−0.319059	+	0.947735i	$0.603367\pi$
$930$	0	0
$931$	−31.3276	−1.02672
$932$	27.6569	0.905933
$933$	2.84706	0.0932085
$934$	71.7730	2.34849
$935$	0	0
$936$	−27.5199	−0.899515
$937$	19.3381	0.631749	0.315875	−	0.948801i	$-0.397702\pi$
$937$	19.3381	0.631749	0.315875	+	0.948801i	$0.397702\pi$
$938$	−23.2673	−0.759704
$939$	63.3740	2.06813
$940$	0	0
$941$	−25.4600	−0.829972	−0.414986	−	0.909828i	$-0.636214\pi$
$941$	−25.4600	−0.829972	−0.414986	+	0.909828i	$0.636214\pi$
$942$	−75.1216	−2.44759
$943$	12.6555	0.412118
$944$	16.7342	0.544650
$945$	0	0
$946$	8.55386	0.278110
$947$	29.6923	0.964870	0.482435	−	0.875932i	$-0.339752\pi$
$947$	29.6923	0.964870	0.482435	+	0.875932i	$0.339752\pi$
$948$	−130.868	−4.25039
$949$	−35.4247	−1.14993
$950$	0	0
$951$	−68.4664	−2.22017
$952$	5.44563	0.176494
$953$	−46.8480	−1.51756	−0.758778	−	0.651350i	$-0.774203\pi$
$953$	−46.8480	−1.51756	−0.758778	+	0.651350i	$0.774203\pi$
$954$	30.0311	0.972292
$955$	0	0
$956$	−49.9142	−1.61434
$957$	−19.4465	−0.628617
$958$	−55.2941	−1.78647
$959$	−9.45694	−0.305381
$960$	0	0
$961$	−12.1424	−0.391691
$962$	90.7561	2.92610
$963$	−83.2701	−2.68334
$964$	2.46023	0.0792386
$965$	0	0
$966$	−70.5611	−2.27027
$967$	39.0522	1.25583	0.627917	−	0.778280i	$-0.283908\pi$
$967$	39.0522	1.25583	0.627917	+	0.778280i	$0.283908\pi$
$968$	10.2719	0.330150
$969$	−85.2178	−2.73759
$970$	0	0
$971$	40.9727	1.31488	0.657439	−	0.753508i	$-0.271640\pi$
$971$	40.9727	1.31488	0.657439	+	0.753508i	$0.271640\pi$
$972$	101.317	3.24974
$973$	11.0655	0.354745
$974$	−32.0237	−1.02610
$975$	0	0
$976$	42.3162	1.35451
$977$	1.82911	0.0585183	0.0292591	−	0.999572i	$-0.490685\pi$
$977$	1.82911	0.0585183	0.0292591	+	0.999572i	$0.490685\pi$
$978$	−51.8831	−1.65904
$979$	3.13152	0.100084
$980$	0	0
$981$	−117.638	−3.75591
$982$	12.4244	0.396479
$983$	11.4733	0.365940	0.182970	−	0.983118i	$-0.441429\pi$
$983$	11.4733	0.365940	0.182970	+	0.983118i	$0.441429\pi$
$984$	−4.77922	−0.152356
$985$	0	0
$986$	88.0218	2.80319
$987$	1.95903	0.0623568
$988$	51.7335	1.64586
$989$	51.7660	1.64606
$990$	0	0
$991$	24.9387	0.792204	0.396102	−	0.918207i	$-0.370363\pi$
$991$	24.9387	0.792204	0.396102	+	0.918207i	$0.370363\pi$
$992$	34.7419	1.10306
$993$	−42.9728	−1.36370
$994$	−20.4184	−0.647632
$995$	0	0
$996$	26.6335	0.843915
$997$	−28.9030	−0.915366	−0.457683	−	0.889116i	$-0.651320\pi$
$997$	−28.9030	−0.915366	−0.457683	+	0.889116i	$0.651320\pi$
$998$	−23.6076	−0.747284
$999$	−176.171	−5.57381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.n.1.5	yes	40
5.4	even	2		6025.2.a.m.1.36	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.36	✓	40	5.4	even	2
6025.2.a.n.1.5	yes	40	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)

\(f(q)\)	\(=\)	\(q-2.11193 q^{2} +3.26338 q^{3} +2.46023 q^{4} -6.89202 q^{6} +1.21893 q^{7} -0.971967 q^{8} +7.64967 q^{9} +O(q^{10})\)	\(q-2.11193 q^{2} +3.26338 q^{3} +2.46023 q^{4} -6.89202 q^{6} +1.21893 q^{7} -0.971967 q^{8} +7.64967 q^{9} -0.657173 q^{11} +8.02866 q^{12} +3.70129 q^{13} -2.57429 q^{14} -2.86773 q^{16} -4.59640 q^{17} -16.1555 q^{18} +5.68126 q^{19} +3.97783 q^{21} +1.38790 q^{22} +8.39925 q^{23} -3.17190 q^{24} -7.81684 q^{26} +15.1736 q^{27} +2.99884 q^{28} +9.06763 q^{29} +4.34253 q^{31} +8.00037 q^{32} -2.14461 q^{33} +9.70726 q^{34} +18.8199 q^{36} -11.6103 q^{37} -11.9984 q^{38} +12.0787 q^{39} +1.50674 q^{41} -8.40088 q^{42} +6.16317 q^{43} -1.61680 q^{44} -17.7386 q^{46} +0.492488 q^{47} -9.35851 q^{48} -5.51421 q^{49} -14.9998 q^{51} +9.10601 q^{52} -1.85887 q^{53} -32.0456 q^{54} -1.18476 q^{56} +18.5401 q^{57} -19.1502 q^{58} -5.83532 q^{59} -14.7560 q^{61} -9.17110 q^{62} +9.32440 q^{63} -11.1607 q^{64} +4.52925 q^{66} +9.03835 q^{67} -11.3082 q^{68} +27.4100 q^{69} +7.93167 q^{71} -7.43522 q^{72} -9.57092 q^{73} +24.5202 q^{74} +13.9772 q^{76} -0.801048 q^{77} -25.5093 q^{78} -16.3001 q^{79} +26.5684 q^{81} -3.18212 q^{82} +3.31730 q^{83} +9.78637 q^{84} -13.0161 q^{86} +29.5912 q^{87} +0.638751 q^{88} -4.76514 q^{89} +4.51160 q^{91} +20.6641 q^{92} +14.1713 q^{93} -1.04010 q^{94} +26.1083 q^{96} +6.06806 q^{97} +11.6456 q^{98} -5.02716 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 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

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