LMFDB - Embedded newform 6025.2.a.p.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-1.49203 q^{2} +3.12646 q^{3} +0.226139 q^{4} -4.66475 q^{6} +4.66024 q^{7} +2.64665 q^{8} +6.77474 q^{9} +O(q^{10})$	\(q-1.49203 q^{2} +3.12646 q^{3} +0.226139 q^{4} -4.66475 q^{6} +4.66024 q^{7} +2.64665 q^{8} +6.77474 q^{9} -5.92431 q^{11} +0.707014 q^{12} -4.29102 q^{13} -6.95319 q^{14} -4.40114 q^{16} -5.72527 q^{17} -10.1081 q^{18} -1.23168 q^{19} +14.5700 q^{21} +8.83922 q^{22} -3.63376 q^{23} +8.27462 q^{24} +6.40231 q^{26} +11.8016 q^{27} +1.05386 q^{28} +1.81182 q^{29} -1.11419 q^{31} +1.27332 q^{32} -18.5221 q^{33} +8.54225 q^{34} +1.53203 q^{36} -7.32690 q^{37} +1.83770 q^{38} -13.4157 q^{39} -6.45271 q^{41} -21.7389 q^{42} -4.57155 q^{43} -1.33972 q^{44} +5.42167 q^{46} -7.71296 q^{47} -13.7600 q^{48} +14.7178 q^{49} -17.8998 q^{51} -0.970368 q^{52} +10.1745 q^{53} -17.6082 q^{54} +12.3340 q^{56} -3.85081 q^{57} -2.70329 q^{58} -5.66965 q^{59} +4.76640 q^{61} +1.66239 q^{62} +31.5719 q^{63} +6.90245 q^{64} +27.6354 q^{66} -14.0103 q^{67} -1.29471 q^{68} -11.3608 q^{69} -3.96855 q^{71} +17.9303 q^{72} +8.92040 q^{73} +10.9319 q^{74} -0.278532 q^{76} -27.6087 q^{77} +20.0166 q^{78} +1.77420 q^{79} +16.5729 q^{81} +9.62761 q^{82} -9.93218 q^{83} +3.29485 q^{84} +6.82087 q^{86} +5.66459 q^{87} -15.6795 q^{88} -7.03898 q^{89} -19.9972 q^{91} -0.821736 q^{92} -3.48346 q^{93} +11.5079 q^{94} +3.98098 q^{96} +5.03884 q^{97} -21.9594 q^{98} -40.1356 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.49203	−1.05502	−0.527511	−	0.849548i	$-0.676874\pi$
$2$	−1.49203	−1.05502	−0.527511	+	0.849548i	$0.676874\pi$
$3$	3.12646	1.80506	0.902531	−	0.430626i	$-0.141707\pi$
$3$	3.12646	1.80506	0.902531	+	0.430626i	$0.141707\pi$
$4$	0.226139	0.113070
$5$	0	0
$5$	0	0
$6$	−4.66475	−1.90438
$7$	4.66024	1.76140	0.880702	−	0.473670i	$-0.157071\pi$
$7$	4.66024	1.76140	0.880702	+	0.473670i	$0.157071\pi$
$8$	2.64665	0.935730
$9$	6.77474	2.25825
$10$	0	0
$11$	−5.92431	−1.78625	−0.893123	−	0.449812i	$-0.851491\pi$
$11$	−5.92431	−1.78625	−0.893123	+	0.449812i	$0.851491\pi$
$12$	0.707014	0.204097
$13$	−4.29102	−1.19012	−0.595058	−	0.803683i	$-0.702871\pi$
$13$	−4.29102	−1.19012	−0.595058	+	0.803683i	$0.702871\pi$
$14$	−6.95319	−1.85832
$15$	0	0
$16$	−4.40114	−1.10028
$17$	−5.72527	−1.38858	−0.694291	−	0.719694i	$-0.744282\pi$
$17$	−5.72527	−1.38858	−0.694291	+	0.719694i	$0.744282\pi$
$18$	−10.1081	−2.38250
$19$	−1.23168	−0.282568	−0.141284	−	0.989969i	$-0.545123\pi$
$19$	−1.23168	−0.282568	−0.141284	+	0.989969i	$0.545123\pi$
$20$	0	0
$21$	14.5700	3.17944
$22$	8.83922	1.88453
$23$	−3.63376	−0.757692	−0.378846	−	0.925460i	$-0.623679\pi$
$23$	−3.63376	−0.757692	−0.378846	+	0.925460i	$0.623679\pi$
$24$	8.27462	1.68905
$25$	0	0
$26$	6.40231	1.25560
$27$	11.8016	2.27121
$28$	1.05386	0.199161
$29$	1.81182	0.336447	0.168224	−	0.985749i	$-0.446197\pi$
$29$	1.81182	0.336447	0.168224	+	0.985749i	$0.446197\pi$
$30$	0	0
$31$	−1.11419	−0.200114	−0.100057	−	0.994982i	$-0.531902\pi$
$31$	−1.11419	−0.200114	−0.100057	+	0.994982i	$0.531902\pi$
$32$	1.27332	0.225093
$33$	−18.5221	−3.22428
$34$	8.54225	1.46498
$35$	0	0
$36$	1.53203	0.255339
$37$	−7.32690	−1.20454	−0.602268	−	0.798294i	$-0.705736\pi$
$37$	−7.32690	−1.20454	−0.602268	+	0.798294i	$0.705736\pi$
$38$	1.83770	0.298115
$39$	−13.4157	−2.14823
$40$	0	0
$41$	−6.45271	−1.00774	−0.503872	−	0.863778i	$-0.668092\pi$
$41$	−6.45271	−1.00774	−0.503872	+	0.863778i	$0.668092\pi$
$42$	−21.7389	−3.35438
$43$	−4.57155	−0.697155	−0.348578	−	0.937280i	$-0.613335\pi$
$43$	−4.57155	−0.697155	−0.348578	+	0.937280i	$0.613335\pi$
$44$	−1.33972	−0.201970
$45$	0	0
$46$	5.42167	0.799381
$47$	−7.71296	−1.12505	−0.562525	−	0.826780i	$-0.690170\pi$
$47$	−7.71296	−1.12505	−0.562525	+	0.826780i	$0.690170\pi$
$48$	−13.7600	−1.98608
$49$	14.7178	2.10255
$50$	0	0
$51$	−17.8998	−2.50648
$52$	−0.970368	−0.134566
$53$	10.1745	1.39758	0.698788	−	0.715329i	$-0.253723\pi$
$53$	10.1745	1.39758	0.698788	+	0.715329i	$0.253723\pi$
$54$	−17.6082	−2.39618
$55$	0	0
$56$	12.3340	1.64820
$57$	−3.85081	−0.510052
$58$	−2.70329	−0.354959
$59$	−5.66965	−0.738126	−0.369063	−	0.929404i	$-0.620321\pi$
$59$	−5.66965	−0.738126	−0.369063	+	0.929404i	$0.620321\pi$
$60$	0	0
$61$	4.76640	0.610275	0.305138	−	0.952308i	$-0.401298\pi$
$61$	4.76640	0.610275	0.305138	+	0.952308i	$0.401298\pi$
$62$	1.66239	0.211124
$63$	31.5719	3.97768
$64$	6.90245	0.862807
$65$	0	0
$66$	27.6354	3.40169
$67$	−14.0103	−1.71163	−0.855814	−	0.517283i	$-0.826943\pi$
$67$	−14.0103	−1.71163	−0.855814	+	0.517283i	$0.826943\pi$
$68$	−1.29471	−0.157006
$69$	−11.3608	−1.36768
$70$	0	0
$71$	−3.96855	−0.470980	−0.235490	−	0.971877i	$-0.575669\pi$
$71$	−3.96855	−0.470980	−0.235490	+	0.971877i	$0.575669\pi$
$72$	17.9303	2.11311
$73$	8.92040	1.04405	0.522027	−	0.852929i	$-0.325176\pi$
$73$	8.92040	1.04405	0.522027	+	0.852929i	$0.325176\pi$
$74$	10.9319	1.27081
$75$	0	0
$76$	−0.278532	−0.0319498
$77$	−27.6087	−3.14630
$78$	20.0166	2.26643
$79$	1.77420	0.199614	0.0998068	−	0.995007i	$-0.468178\pi$
$79$	1.77420	0.199614	0.0998068	+	0.995007i	$0.468178\pi$
$80$	0	0
$81$	16.5729	1.84143
$82$	9.62761	1.06319
$83$	−9.93218	−1.09020	−0.545099	−	0.838372i	$-0.683508\pi$
$83$	−9.93218	−1.09020	−0.545099	+	0.838372i	$0.683508\pi$
$84$	3.29485	0.359498
$85$	0	0
$86$	6.82087	0.735514
$87$	5.66459	0.607308
$88$	−15.6795	−1.67144
$89$	−7.03898	−0.746131	−0.373065	−	0.927805i	$-0.621693\pi$
$89$	−7.03898	−0.746131	−0.373065	+	0.927805i	$0.621693\pi$
$90$	0	0
$91$	−19.9972	−2.09627
$92$	−0.821736	−0.0856719
$93$	−3.48346	−0.361217
$94$	11.5079	1.18695
$95$	0	0
$96$	3.98098	0.406307
$97$	5.03884	0.511617	0.255808	−	0.966727i	$-0.417658\pi$
$97$	5.03884	0.511617	0.255808	+	0.966727i	$0.417658\pi$
$98$	−21.9594	−2.21823
$99$	−40.1356	−4.03378
$100$	0	0
$101$	−8.87959	−0.883552	−0.441776	−	0.897125i	$-0.645651\pi$
$101$	−8.87959	−0.883552	−0.441776	+	0.897125i	$0.645651\pi$
$102$	26.7070	2.64438
$103$	3.12126	0.307547	0.153774	−	0.988106i	$-0.450857\pi$
$103$	3.12126	0.307547	0.153774	+	0.988106i	$0.450857\pi$
$104$	−11.3568	−1.11363
$105$	0	0
$106$	−15.1806	−1.47447
$107$	−2.37926	−0.230012	−0.115006	−	0.993365i	$-0.536689\pi$
$107$	−2.37926	−0.230012	−0.115006	+	0.993365i	$0.536689\pi$
$108$	2.66879	0.256805
$109$	−10.9286	−1.04677	−0.523386	−	0.852096i	$-0.675331\pi$
$109$	−10.9286	−1.04677	−0.523386	+	0.852096i	$0.675331\pi$
$110$	0	0
$111$	−22.9072	−2.17426
$112$	−20.5104	−1.93805
$113$	12.3477	1.16157	0.580787	−	0.814055i	$-0.302745\pi$
$113$	12.3477	1.16157	0.580787	+	0.814055i	$0.302745\pi$
$114$	5.74550	0.538116
$115$	0	0
$116$	0.409724	0.0380419
$117$	−29.0706	−2.68757
$118$	8.45926	0.778738
$119$	−26.6811	−2.44585
$120$	0	0
$121$	24.0974	2.19068
$122$	−7.11159	−0.643853
$123$	−20.1741	−1.81904
$124$	−0.251961	−0.0226268
$125$	0	0
$126$	−47.1061	−4.19654
$127$	−8.83493	−0.783974	−0.391987	−	0.919971i	$-0.628212\pi$
$127$	−8.83493	−0.783974	−0.391987	+	0.919971i	$0.628212\pi$
$128$	−12.8453	−1.13537
$129$	−14.2928	−1.25841
$130$	0	0
$131$	10.6930	0.934252	0.467126	−	0.884191i	$-0.345289\pi$
$131$	10.6930	0.934252	0.467126	+	0.884191i	$0.345289\pi$
$132$	−4.18857	−0.364568
$133$	−5.73994	−0.497716
$134$	20.9037	1.80580
$135$	0	0
$136$	−15.1528	−1.29934
$137$	−6.53145	−0.558020	−0.279010	−	0.960288i	$-0.590006\pi$
$137$	−6.53145	−0.558020	−0.279010	+	0.960288i	$0.590006\pi$
$138$	16.9506	1.44293
$139$	−9.68333	−0.821329	−0.410664	−	0.911787i	$-0.634703\pi$
$139$	−9.68333	−0.821329	−0.410664	+	0.911787i	$0.634703\pi$
$140$	0	0
$141$	−24.1142	−2.03079
$142$	5.92117	0.496893
$143$	25.4213	2.12584
$144$	−29.8166	−2.48471
$145$	0	0
$146$	−13.3095	−1.10150
$147$	46.0147	3.79522
$148$	−1.65690	−0.136196
$149$	14.7898	1.21163	0.605814	−	0.795606i	$-0.292847\pi$
$149$	14.7898	1.21163	0.605814	+	0.795606i	$0.292847\pi$
$150$	0	0
$151$	−1.21804	−0.0991228	−0.0495614	−	0.998771i	$-0.515782\pi$
$151$	−1.21804	−0.0991228	−0.0495614	+	0.998771i	$0.515782\pi$
$152$	−3.25983	−0.264407
$153$	−38.7872	−3.13576
$154$	41.1929	3.31942
$155$	0	0
$156$	−3.03381	−0.242899
$157$	9.48425	0.756926	0.378463	−	0.925616i	$-0.376453\pi$
$157$	9.48425	0.756926	0.378463	+	0.925616i	$0.376453\pi$
$158$	−2.64716	−0.210597
$159$	31.8101	2.52271
$160$	0	0
$161$	−16.9342	−1.33460
$162$	−24.7271	−1.94275
$163$	2.02231	0.158400	0.0791999	−	0.996859i	$-0.474763\pi$
$163$	2.02231	0.158400	0.0791999	+	0.996859i	$0.474763\pi$
$164$	−1.45921	−0.113945
$165$	0	0
$166$	14.8191	1.15018
$167$	−12.0440	−0.931992	−0.465996	−	0.884787i	$-0.654304\pi$
$167$	−12.0440	−0.931992	−0.465996	+	0.884787i	$0.654304\pi$
$168$	38.5617	2.97510
$169$	5.41287	0.416375
$170$	0	0
$171$	−8.34434	−0.638107
$172$	−1.03381	−0.0788270
$173$	10.6055	0.806322	0.403161	−	0.915129i	$-0.367911\pi$
$173$	10.6055	0.806322	0.403161	+	0.915129i	$0.367911\pi$
$174$	−8.45171	−0.640723
$175$	0	0
$176$	26.0737	1.96538
$177$	−17.7259	−1.33236
$178$	10.5023	0.787184
$179$	−15.6670	−1.17101	−0.585505	−	0.810669i	$-0.699104\pi$
$179$	−15.6670	−1.17101	−0.585505	+	0.810669i	$0.699104\pi$
$180$	0	0
$181$	6.00433	0.446298	0.223149	−	0.974784i	$-0.428366\pi$
$181$	6.00433	0.446298	0.223149	+	0.974784i	$0.428366\pi$
$182$	29.8363	2.21161
$183$	14.9020	1.10158
$184$	−9.61728	−0.708995
$185$	0	0
$186$	5.19740	0.381092
$187$	33.9183	2.48035
$188$	−1.74420	−0.127209
$189$	54.9981	4.00052
$190$	0	0
$191$	−14.5050	−1.04954	−0.524772	−	0.851243i	$-0.675849\pi$
$191$	−14.5050	−1.04954	−0.524772	+	0.851243i	$0.675849\pi$
$192$	21.5802	1.55742
$193$	3.41996	0.246174	0.123087	−	0.992396i	$-0.460721\pi$
$193$	3.41996	0.246174	0.123087	+	0.992396i	$0.460721\pi$
$194$	−7.51808	−0.539767
$195$	0	0
$196$	3.32827	0.237734
$197$	20.5076	1.46111	0.730554	−	0.682855i	$-0.239262\pi$
$197$	20.5076	1.46111	0.730554	+	0.682855i	$0.239262\pi$
$198$	59.8834	4.25573
$199$	−2.77785	−0.196917	−0.0984583	−	0.995141i	$-0.531391\pi$
$199$	−2.77785	−0.196917	−0.0984583	+	0.995141i	$0.531391\pi$
$200$	0	0
$201$	−43.8026	−3.08959
$202$	13.2486	0.932166
$203$	8.44353	0.592620
$204$	−4.04785	−0.283406
$205$	0	0
$206$	−4.65700	−0.324469
$207$	−24.6178	−1.71105
$208$	18.8854	1.30947
$209$	7.29688	0.504735
$210$	0	0
$211$	22.9148	1.57752	0.788760	−	0.614701i	$-0.210723\pi$
$211$	22.9148	1.57752	0.788760	+	0.614701i	$0.210723\pi$
$212$	2.30085	0.158023
$213$	−12.4075	−0.850147
$214$	3.54991	0.242667
$215$	0	0
$216$	31.2345	2.12524
$217$	−5.19237	−0.352481
$218$	16.3058	1.10437
$219$	27.8893	1.88458
$220$	0	0
$221$	24.5673	1.65257
$222$	34.1782	2.29389
$223$	7.52468	0.503890	0.251945	−	0.967742i	$-0.418930\pi$
$223$	7.52468	0.503890	0.251945	+	0.967742i	$0.418930\pi$
$224$	5.93397	0.396480
$225$	0	0
$226$	−18.4231	−1.22549
$227$	0.950254	0.0630706	0.0315353	−	0.999503i	$-0.489960\pi$
$227$	0.950254	0.0630706	0.0315353	+	0.999503i	$0.489960\pi$
$228$	−0.870818	−0.0576713
$229$	14.6545	0.968396	0.484198	−	0.874958i	$-0.339111\pi$
$229$	14.6545	0.968396	0.484198	+	0.874958i	$0.339111\pi$
$230$	0	0
$231$	−86.3174	−5.67927
$232$	4.79526	0.314824
$233$	−19.1614	−1.25530	−0.627651	−	0.778495i	$-0.715984\pi$
$233$	−19.1614	−1.25530	−0.627651	+	0.778495i	$0.715984\pi$
$234$	43.3740	2.83545
$235$	0	0
$236$	−1.28213	−0.0834595
$237$	5.54697	0.360315
$238$	39.8089	2.58043
$239$	−3.88636	−0.251388	−0.125694	−	0.992069i	$-0.540116\pi$
$239$	−3.88636	−0.251388	−0.125694	+	0.992069i	$0.540116\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−35.9540	−2.31121
$243$	16.4097	1.05268
$244$	1.07787	0.0690035
$245$	0	0
$246$	30.1003	1.91913
$247$	5.28518	0.336288
$248$	−2.94885	−0.187252
$249$	−31.0525	−1.96787
$250$	0	0
$251$	−13.8452	−0.873904	−0.436952	−	0.899485i	$-0.643942\pi$
$251$	−13.8452	−0.873904	−0.436952	+	0.899485i	$0.643942\pi$
$252$	7.13964	0.449755
$253$	21.5275	1.35342
$254$	13.1819	0.827109
$255$	0	0
$256$	5.36056	0.335035
$257$	6.17221	0.385012	0.192506	−	0.981296i	$-0.438338\pi$
$257$	6.17221	0.385012	0.192506	+	0.981296i	$0.438338\pi$
$258$	21.3252	1.32765
$259$	−34.1451	−2.12167
$260$	0	0
$261$	12.2746	0.759781
$262$	−15.9542	−0.985656
$263$	14.8523	0.915835	0.457917	−	0.888995i	$-0.348596\pi$
$263$	14.8523	0.915835	0.457917	+	0.888995i	$0.348596\pi$
$264$	−49.0214	−3.01706
$265$	0	0
$266$	8.56414	0.525101
$267$	−22.0071	−1.34681
$268$	−3.16827	−0.193533
$269$	24.7594	1.50960	0.754802	−	0.655952i	$-0.227733\pi$
$269$	24.7594	1.50960	0.754802	+	0.655952i	$0.227733\pi$
$270$	0	0
$271$	21.9399	1.33275	0.666376	−	0.745616i	$-0.267845\pi$
$271$	21.9399	1.33275	0.666376	+	0.745616i	$0.267845\pi$
$272$	25.1977	1.52784
$273$	−62.5204	−3.78390
$274$	9.74509	0.588723
$275$	0	0
$276$	−2.56912	−0.154643
$277$	17.9537	1.07873	0.539367	−	0.842071i	$-0.318664\pi$
$277$	17.9537	1.07873	0.539367	+	0.842071i	$0.318664\pi$
$278$	14.4478	0.866519
$279$	−7.54832	−0.451906
$280$	0	0
$281$	14.9520	0.891960	0.445980	−	0.895043i	$-0.352855\pi$
$281$	14.9520	0.891960	0.445980	+	0.895043i	$0.352855\pi$
$282$	35.9791	2.14252
$283$	−16.8819	−1.00352	−0.501761	−	0.865006i	$-0.667314\pi$
$283$	−16.8819	−1.00352	−0.501761	+	0.865006i	$0.667314\pi$
$284$	−0.897443	−0.0532534
$285$	0	0
$286$	−37.9293	−2.24281
$287$	−30.0712	−1.77505
$288$	8.62641	0.508316
$289$	15.7787	0.928160
$290$	0	0
$291$	15.7537	0.923500
$292$	2.01725	0.118051
$293$	−16.3651	−0.956061	−0.478030	−	0.878343i	$-0.658649\pi$
$293$	−16.3651	−0.956061	−0.478030	+	0.878343i	$0.658649\pi$
$294$	−68.6550	−4.00404
$295$	0	0
$296$	−19.3917	−1.12712
$297$	−69.9161	−4.05694
$298$	−22.0668	−1.27829
$299$	15.5926	0.901741
$300$	0	0
$301$	−21.3045	−1.22797
$302$	1.81735	0.104577
$303$	−27.7617	−1.59487
$304$	5.42081	0.310905
$305$	0	0
$306$	57.8715	3.30829
$307$	19.0297	1.08608	0.543041	−	0.839706i	$-0.317273\pi$
$307$	19.0297	1.08608	0.543041	+	0.839706i	$0.317273\pi$
$308$	−6.24340	−0.355751
$309$	9.75850	0.555142
$310$	0	0
$311$	−16.3005	−0.924318	−0.462159	−	0.886797i	$-0.652925\pi$
$311$	−16.3005	−0.924318	−0.462159	+	0.886797i	$0.652925\pi$
$312$	−35.5066	−2.01017
$313$	1.71271	0.0968084	0.0484042	−	0.998828i	$-0.484586\pi$
$313$	1.71271	0.0968084	0.0484042	+	0.998828i	$0.484586\pi$
$314$	−14.1507	−0.798573
$315$	0	0
$316$	0.401217	0.0225702
$317$	9.33229	0.524154	0.262077	−	0.965047i	$-0.415593\pi$
$317$	9.33229	0.524154	0.262077	+	0.965047i	$0.415593\pi$
$318$	−47.4615	−2.66151
$319$	−10.7338	−0.600978
$320$	0	0
$321$	−7.43865	−0.415185
$322$	25.2663	1.40803
$323$	7.05172	0.392368
$324$	3.74777	0.208209
$325$	0	0
$326$	−3.01734	−0.167115
$327$	−34.1678	−1.88949
$328$	−17.0780	−0.942977
$329$	−35.9442	−1.98167
$330$	0	0
$331$	12.2007	0.670613	0.335306	−	0.942109i	$-0.391160\pi$
$331$	12.2007	0.670613	0.335306	+	0.942109i	$0.391160\pi$
$332$	−2.24605	−0.123268
$333$	−49.6378	−2.72014
$334$	17.9699	0.983271
$335$	0	0
$336$	−64.1248	−3.49829
$337$	1.32894	0.0723920	0.0361960	−	0.999345i	$-0.488476\pi$
$337$	1.32894	0.0723920	0.0361960	+	0.999345i	$0.488476\pi$
$338$	−8.07614	−0.439284
$339$	38.6046	2.09671
$340$	0	0
$341$	6.60078	0.357452
$342$	12.4500	0.673217
$343$	35.9669	1.94203
$344$	−12.0993	−0.652350
$345$	0	0
$346$	−15.8237	−0.850687
$347$	−2.21075	−0.118679	−0.0593396	−	0.998238i	$-0.518899\pi$
$347$	−2.21075	−0.118679	−0.0593396	+	0.998238i	$0.518899\pi$
$348$	1.28099	0.0686680
$349$	17.8704	0.956580	0.478290	−	0.878202i	$-0.341257\pi$
$349$	17.8704	0.956580	0.478290	+	0.878202i	$0.341257\pi$
$350$	0	0
$351$	−50.6407	−2.70300
$352$	−7.54354	−0.402072
$353$	9.10115	0.484405	0.242203	−	0.970226i	$-0.422130\pi$
$353$	9.10115	0.484405	0.242203	+	0.970226i	$0.422130\pi$
$354$	26.4475	1.40567
$355$	0	0
$356$	−1.59179	−0.0843647
$357$	−83.4174	−4.41492
$358$	23.3756	1.23544
$359$	−23.0382	−1.21591	−0.607956	−	0.793971i	$-0.708010\pi$
$359$	−23.0382	−1.21591	−0.607956	+	0.793971i	$0.708010\pi$
$360$	0	0
$361$	−17.4830	−0.920155
$362$	−8.95861	−0.470854
$363$	75.3396	3.95430
$364$	−4.52214	−0.237025
$365$	0	0
$366$	−22.2341	−1.16219
$367$	32.4970	1.69633	0.848164	−	0.529734i	$-0.177708\pi$
$367$	32.4970	1.69633	0.848164	+	0.529734i	$0.177708\pi$
$368$	15.9927	0.833677
$369$	−43.7154	−2.27573
$370$	0	0
$371$	47.4156	2.46170
$372$	−0.787745	−0.0408427
$373$	35.2363	1.82447	0.912234	−	0.409671i	$-0.134356\pi$
$373$	35.2363	1.82447	0.912234	+	0.409671i	$0.134356\pi$
$374$	−50.6069	−2.61682
$375$	0	0
$376$	−20.4135	−1.05274
$377$	−7.77458	−0.400411
$378$	−82.0585	−4.22063
$379$	−5.18922	−0.266553	−0.133276	−	0.991079i	$-0.542550\pi$
$379$	−5.18922	−0.266553	−0.133276	+	0.991079i	$0.542550\pi$
$380$	0	0
$381$	−27.6221	−1.41512
$382$	21.6418	1.10729
$383$	−5.72356	−0.292460	−0.146230	−	0.989251i	$-0.546714\pi$
$383$	−5.72356	−0.292460	−0.146230	+	0.989251i	$0.546714\pi$
$384$	−40.1602	−2.04942
$385$	0	0
$386$	−5.10267	−0.259719
$387$	−30.9711	−1.57435
$388$	1.13948	0.0578483
$389$	24.8562	1.26026	0.630130	−	0.776490i	$-0.283002\pi$
$389$	24.8562	1.26026	0.630130	+	0.776490i	$0.283002\pi$
$390$	0	0
$391$	20.8043	1.05212
$392$	38.9529	1.96742
$393$	33.4312	1.68638
$394$	−30.5979	−1.54150
$395$	0	0
$396$	−9.07623	−0.456098
$397$	−34.8213	−1.74763	−0.873815	−	0.486259i	$-0.838361\pi$
$397$	−34.8213	−1.74763	−0.873815	+	0.486259i	$0.838361\pi$
$398$	4.14462	0.207751
$399$	−17.9457	−0.898408
$400$	0	0
$401$	−32.1370	−1.60484	−0.802422	−	0.596757i	$-0.796456\pi$
$401$	−32.1370	−1.60484	−0.802422	+	0.596757i	$0.796456\pi$
$402$	65.3545	3.25959
$403$	4.78100	0.238158
$404$	−2.00802	−0.0999028
$405$	0	0
$406$	−12.5980	−0.625227
$407$	43.4068	2.15160
$408$	−47.3745	−2.34539
$409$	−30.4791	−1.50709	−0.753547	−	0.657395i	$-0.771659\pi$
$409$	−30.4791	−1.50709	−0.753547	+	0.657395i	$0.771659\pi$
$410$	0	0
$411$	−20.4203	−1.00726
$412$	0.705839	0.0347742
$413$	−26.4219	−1.30014
$414$	36.7304	1.80520
$415$	0	0
$416$	−5.46384	−0.267887
$417$	−30.2745	−1.48255
$418$	−10.8871	−0.532507
$419$	−14.0542	−0.686592	−0.343296	−	0.939227i	$-0.611543\pi$
$419$	−14.0542	−0.686592	−0.343296	+	0.939227i	$0.611543\pi$
$420$	0	0
$421$	7.96690	0.388283	0.194141	−	0.980974i	$-0.437808\pi$
$421$	7.96690	0.388283	0.194141	+	0.980974i	$0.437808\pi$
$422$	−34.1895	−1.66432
$423$	−52.2533	−2.54064
$424$	26.9283	1.30775
$425$	0	0
$426$	18.5123	0.896923
$427$	22.2126	1.07494
$428$	−0.538043	−0.0260073
$429$	79.4787	3.83727
$430$	0	0
$431$	−39.6560	−1.91016	−0.955080	−	0.296348i	$-0.904231\pi$
$431$	−39.6560	−1.91016	−0.955080	+	0.296348i	$0.904231\pi$
$432$	−51.9403	−2.49898
$433$	3.27667	0.157467	0.0787333	−	0.996896i	$-0.474912\pi$
$433$	3.27667	0.157467	0.0787333	+	0.996896i	$0.474912\pi$
$434$	7.74715	0.371875
$435$	0	0
$436$	−2.47139	−0.118358
$437$	4.47565	0.214099
$438$	−41.6115	−1.98827
$439$	−14.5064	−0.692355	−0.346177	−	0.938169i	$-0.612520\pi$
$439$	−14.5064	−0.692355	−0.346177	+	0.938169i	$0.612520\pi$
$440$	0	0
$441$	99.7094	4.74807
$442$	−36.6550	−1.74350
$443$	−19.5267	−0.927742	−0.463871	−	0.885903i	$-0.653540\pi$
$443$	−19.5267	−0.927742	−0.463871	+	0.885903i	$0.653540\pi$
$444$	−5.18022	−0.245842
$445$	0	0
$446$	−11.2270	−0.531614
$447$	46.2397	2.18706
$448$	32.1671	1.51975
$449$	−2.85475	−0.134724	−0.0673621	−	0.997729i	$-0.521458\pi$
$449$	−2.85475	−0.134724	−0.0673621	+	0.997729i	$0.521458\pi$
$450$	0	0
$451$	38.2279	1.80008
$452$	2.79230	0.131339
$453$	−3.80815	−0.178923
$454$	−1.41780	−0.0665408
$455$	0	0
$456$	−10.1917	−0.477271
$457$	−11.7309	−0.548749	−0.274375	−	0.961623i	$-0.588471\pi$
$457$	−11.7309	−0.548749	−0.274375	+	0.961623i	$0.588471\pi$
$458$	−21.8649	−1.02168
$459$	−67.5671	−3.15376
$460$	0	0
$461$	1.41285	0.0658028	0.0329014	−	0.999459i	$-0.489525\pi$
$461$	1.41285	0.0658028	0.0329014	+	0.999459i	$0.489525\pi$
$462$	128.788	5.99175
$463$	−30.3025	−1.40827	−0.704137	−	0.710064i	$-0.748666\pi$
$463$	−30.3025	−1.40827	−0.704137	+	0.710064i	$0.748666\pi$
$464$	−7.97409	−0.370188
$465$	0	0
$466$	28.5892	1.32437
$467$	10.2416	0.473927	0.236963	−	0.971519i	$-0.423848\pi$
$467$	10.2416	0.473927	0.236963	+	0.971519i	$0.423848\pi$
$468$	−6.57399	−0.303883
$469$	−65.2913	−3.01487
$470$	0	0
$471$	29.6521	1.36630
$472$	−15.0056	−0.690687
$473$	27.0833	1.24529
$474$	−8.27623	−0.380140
$475$	0	0
$476$	−6.03364	−0.276552
$477$	68.9296	3.15607
$478$	5.79855	0.265219
$479$	−16.8364	−0.769275	−0.384638	−	0.923068i	$-0.625674\pi$
$479$	−16.8364	−0.769275	−0.384638	+	0.923068i	$0.625674\pi$
$480$	0	0
$481$	31.4399	1.43354
$482$	−1.49203	−0.0679599
$483$	−52.9441	−2.40904
$484$	5.44937	0.247699
$485$	0	0
$486$	−24.4836	−1.11060
$487$	−19.1209	−0.866452	−0.433226	−	0.901285i	$-0.642625\pi$
$487$	−19.1209	−0.866452	−0.433226	+	0.901285i	$0.642625\pi$
$488$	12.6150	0.571053
$489$	6.32268	0.285921
$490$	0	0
$491$	13.6920	0.617910	0.308955	−	0.951077i	$-0.400021\pi$
$491$	13.6920	0.617910	0.308955	+	0.951077i	$0.400021\pi$
$492$	−4.56216	−0.205678
$493$	−10.3732	−0.467185
$494$	−7.88563	−0.354791
$495$	0	0
$496$	4.90369	0.220182
$497$	−18.4944	−0.829586
$498$	46.3312	2.07615
$499$	41.2184	1.84519	0.922595	−	0.385770i	$-0.126064\pi$
$499$	41.2184	1.84519	0.922595	+	0.385770i	$0.126064\pi$
$500$	0	0
$501$	−37.6550	−1.68230
$502$	20.6575	0.921988
$503$	23.7713	1.05991	0.529956	−	0.848025i	$-0.322209\pi$
$503$	23.7713	1.05991	0.529956	+	0.848025i	$0.322209\pi$
$504$	83.5596	3.72204
$505$	0	0
$506$	−32.1196	−1.42789
$507$	16.9231	0.751582
$508$	−1.99792	−0.0886435
$509$	0.527203	0.0233679	0.0116839	−	0.999932i	$-0.496281\pi$
$509$	0.527203	0.0233679	0.0116839	+	0.999932i	$0.496281\pi$
$510$	0	0
$511$	41.5712	1.83900
$512$	17.6924	0.781903
$513$	−14.5358	−0.641771
$514$	−9.20910	−0.406196
$515$	0	0
$516$	−3.23215	−0.142288
$517$	45.6939	2.00962
$518$	50.9454	2.23841
$519$	33.1577	1.45546
$520$	0	0
$521$	19.7412	0.864876	0.432438	−	0.901664i	$-0.357654\pi$
$521$	19.7412	0.864876	0.432438	+	0.901664i	$0.357654\pi$
$522$	−18.3141	−0.801585
$523$	22.0882	0.965849	0.482925	−	0.875662i	$-0.339574\pi$
$523$	22.0882	0.965849	0.482925	+	0.875662i	$0.339574\pi$
$524$	2.41811	0.105635
$525$	0	0
$526$	−22.1601	−0.966225
$527$	6.37902	0.277874
$528$	81.5183	3.54763
$529$	−9.79577	−0.425903
$530$	0	0
$531$	−38.4104	−1.66687
$532$	−1.29802	−0.0562765
$533$	27.6887	1.19933
$534$	32.8351	1.42091
$535$	0	0
$536$	−37.0802	−1.60162
$537$	−48.9823	−2.11374
$538$	−36.9416	−1.59267
$539$	−87.1929	−3.75567
$540$	0	0
$541$	−7.32264	−0.314825	−0.157412	−	0.987533i	$-0.550315\pi$
$541$	−7.32264	−0.314825	−0.157412	+	0.987533i	$0.550315\pi$
$542$	−32.7348	−1.40608
$543$	18.7723	0.805596
$544$	−7.29010	−0.312561
$545$	0	0
$546$	93.2820	3.99210
$547$	−27.9274	−1.19409	−0.597045	−	0.802208i	$-0.703658\pi$
$547$	−27.9274	−1.19409	−0.597045	+	0.802208i	$0.703658\pi$
$548$	−1.47702	−0.0630950
$549$	32.2911	1.37815
$550$	0	0
$551$	−2.23160	−0.0950692
$552$	−30.0680	−1.27978
$553$	8.26821	0.351600
$554$	−26.7874	−1.13809
$555$	0	0
$556$	−2.18978	−0.0928673
$557$	13.5066	0.572293	0.286146	−	0.958186i	$-0.407626\pi$
$557$	13.5066	0.572293	0.286146	+	0.958186i	$0.407626\pi$
$558$	11.2623	0.476770
$559$	19.6166	0.829696
$560$	0	0
$561$	106.044	4.47718
$562$	−22.3087	−0.941037
$563$	4.30396	0.181390	0.0906951	−	0.995879i	$-0.471091\pi$
$563$	4.30396	0.181390	0.0906951	+	0.995879i	$0.471091\pi$
$564$	−5.45317	−0.229620
$565$	0	0
$566$	25.1882	1.05874
$567$	77.2335	3.24350
$568$	−10.5033	−0.440710
$569$	10.2392	0.429248	0.214624	−	0.976697i	$-0.431147\pi$
$569$	10.2392	0.429248	0.214624	+	0.976697i	$0.431147\pi$
$570$	0	0
$571$	−25.4305	−1.06423	−0.532117	−	0.846671i	$-0.678603\pi$
$571$	−25.4305	−1.06423	−0.532117	+	0.846671i	$0.678603\pi$
$572$	5.74876	0.240368
$573$	−45.3492	−1.89449
$574$	44.8670	1.87271
$575$	0	0
$576$	46.7623	1.94843
$577$	−12.6994	−0.528684	−0.264342	−	0.964429i	$-0.585155\pi$
$577$	−12.6994	−0.528684	−0.264342	+	0.964429i	$0.585155\pi$
$578$	−23.5423	−0.979229
$579$	10.6924	0.444360
$580$	0	0
$581$	−46.2863	−1.92028
$582$	−23.5050	−0.974312
$583$	−60.2769	−2.49641
$584$	23.6091	0.976953
$585$	0	0
$586$	24.4172	1.00866
$587$	37.3627	1.54212	0.771061	−	0.636761i	$-0.219726\pi$
$587$	37.3627	1.54212	0.771061	+	0.636761i	$0.219726\pi$
$588$	10.4057	0.429124
$589$	1.37233	0.0565457
$590$	0	0
$591$	64.1162	2.63739
$592$	32.2467	1.32533
$593$	−20.9359	−0.859736	−0.429868	−	0.902892i	$-0.641440\pi$
$593$	−20.9359	−0.859736	−0.429868	+	0.902892i	$0.641440\pi$
$594$	104.317	4.28016
$595$	0	0
$596$	3.34455	0.136998
$597$	−8.68483	−0.355447
$598$	−23.2645	−0.951356
$599$	−40.5567	−1.65710	−0.828551	−	0.559914i	$-0.810834\pi$
$599$	−40.5567	−1.65710	−0.828551	+	0.559914i	$0.810834\pi$
$600$	0	0
$601$	−21.1739	−0.863700	−0.431850	−	0.901946i	$-0.642139\pi$
$601$	−21.1739	−0.863700	−0.431850	+	0.901946i	$0.642139\pi$
$602$	31.7869	1.29554
$603$	−94.9160	−3.86528
$604$	−0.275447	−0.0112078
$605$	0	0
$606$	41.4211	1.68262
$607$	−27.6387	−1.12182	−0.560910	−	0.827877i	$-0.689549\pi$
$607$	−27.6387	−1.12182	−0.560910	+	0.827877i	$0.689549\pi$
$608$	−1.56833	−0.0636041
$609$	26.3984	1.06972
$610$	0	0
$611$	33.0965	1.33894
$612$	−8.77130	−0.354559
$613$	36.9421	1.49208	0.746039	−	0.665902i	$-0.231953\pi$
$613$	36.9421	1.49208	0.746039	+	0.665902i	$0.231953\pi$
$614$	−28.3928	−1.14584
$615$	0	0
$616$	−73.0704	−2.94409
$617$	40.1279	1.61549	0.807744	−	0.589533i	$-0.200688\pi$
$617$	40.1279	1.61549	0.807744	+	0.589533i	$0.200688\pi$
$618$	−14.5599	−0.585686
$619$	−30.3391	−1.21943	−0.609716	−	0.792620i	$-0.708716\pi$
$619$	−30.3391	−1.21943	−0.609716	+	0.792620i	$0.708716\pi$
$620$	0	0
$621$	−42.8841	−1.72088
$622$	24.3208	0.975175
$623$	−32.8033	−1.31424
$624$	59.0444	2.36367
$625$	0	0
$626$	−2.55541	−0.102135
$627$	22.8134	0.911078
$628$	2.14476	0.0855852
$629$	41.9485	1.67260
$630$	0	0
$631$	−36.0636	−1.43567	−0.717835	−	0.696213i	$-0.754867\pi$
$631$	−36.0636	−1.43567	−0.717835	+	0.696213i	$0.754867\pi$
$632$	4.69569	0.186784
$633$	71.6422	2.84752
$634$	−13.9240	−0.552993
$635$	0	0
$636$	7.19352	0.285241
$637$	−63.1545	−2.50227
$638$	16.0151	0.634044
$639$	−26.8859	−1.06359
$640$	0	0
$641$	40.0078	1.58021	0.790106	−	0.612970i	$-0.210025\pi$
$641$	40.0078	1.58021	0.790106	+	0.612970i	$0.210025\pi$
$642$	11.0987	0.438029
$643$	−23.3695	−0.921604	−0.460802	−	0.887503i	$-0.652438\pi$
$643$	−23.3695	−0.921604	−0.460802	+	0.887503i	$0.652438\pi$
$644$	−3.82948	−0.150903
$645$	0	0
$646$	−10.5214	−0.413957
$647$	−43.5626	−1.71262	−0.856312	−	0.516459i	$-0.827250\pi$
$647$	−43.5626	−1.71262	−0.856312	+	0.516459i	$0.827250\pi$
$648$	43.8625	1.72308
$649$	33.5888	1.31847
$650$	0	0
$651$	−16.2337	−0.636250
$652$	0.457324	0.0179102
$653$	11.8810	0.464939	0.232469	−	0.972604i	$-0.425319\pi$
$653$	11.8810	0.464939	0.232469	+	0.972604i	$0.425319\pi$
$654$	50.9793	1.99345
$655$	0	0
$656$	28.3993	1.10881
$657$	60.4334	2.35773
$658$	53.6297	2.09070
$659$	22.2402	0.866357	0.433178	−	0.901308i	$-0.357392\pi$
$659$	22.2402	0.866357	0.433178	+	0.901308i	$0.357392\pi$
$660$	0	0
$661$	20.7001	0.805140	0.402570	−	0.915389i	$-0.368117\pi$
$661$	20.7001	0.805140	0.402570	+	0.915389i	$0.368117\pi$
$662$	−18.2038	−0.707511
$663$	76.8085	2.98300
$664$	−26.2870	−1.02013
$665$	0	0
$666$	74.0609	2.86980
$667$	−6.58374	−0.254923
$668$	−2.72362	−0.105380
$669$	23.5256	0.909551
$670$	0	0
$671$	−28.2376	−1.09010
$672$	18.5523	0.715671
$673$	−4.14046	−0.159603	−0.0798016	−	0.996811i	$-0.525429\pi$
$673$	−4.14046	−0.159603	−0.0798016	+	0.996811i	$0.525429\pi$
$674$	−1.98281	−0.0763751
$675$	0	0
$676$	1.22406	0.0470793
$677$	−38.9911	−1.49855	−0.749275	−	0.662259i	$-0.769598\pi$
$677$	−38.9911	−1.49855	−0.749275	+	0.662259i	$0.769598\pi$
$678$	−57.5990	−2.21208
$679$	23.4822	0.901164
$680$	0	0
$681$	2.97093	0.113846
$682$	−9.84853	−0.377120
$683$	−20.2200	−0.773696	−0.386848	−	0.922144i	$-0.626436\pi$
$683$	−20.2200	−0.773696	−0.386848	+	0.922144i	$0.626436\pi$
$684$	−1.88698	−0.0721505
$685$	0	0
$686$	−53.6635	−2.04888
$687$	45.8166	1.74801
$688$	20.1200	0.767070
$689$	−43.6590	−1.66328
$690$	0	0
$691$	−10.4772	−0.398570	−0.199285	−	0.979942i	$-0.563862\pi$
$691$	−10.4772	−0.398570	−0.199285	+	0.979942i	$0.563862\pi$
$692$	2.39832	0.0911705
$693$	−187.042	−7.10512
$694$	3.29849	0.125209
$695$	0	0
$696$	14.9922	0.568277
$697$	36.9435	1.39934
$698$	−26.6631	−1.00921
$699$	−59.9072	−2.26590
$700$	0	0
$701$	23.2337	0.877524	0.438762	−	0.898603i	$-0.355417\pi$
$701$	23.2337	0.877524	0.438762	+	0.898603i	$0.355417\pi$
$702$	75.5573	2.85173
$703$	9.02443	0.340363
$704$	−40.8923	−1.54118
$705$	0	0
$706$	−13.5791	−0.511058
$707$	−41.3810	−1.55629
$708$	−4.00852	−0.150650
$709$	−28.6569	−1.07623	−0.538116	−	0.842871i	$-0.680864\pi$
$709$	−28.6569	−1.07623	−0.538116	+	0.842871i	$0.680864\pi$
$710$	0	0
$711$	12.0198	0.450777
$712$	−18.6297	−0.698177
$713$	4.04869	0.151625
$714$	124.461	4.65783
$715$	0	0
$716$	−3.54293	−0.132405
$717$	−12.1505	−0.453770
$718$	34.3736	1.28281
$719$	−5.87954	−0.219270	−0.109635	−	0.993972i	$-0.534968\pi$
$719$	−5.87954	−0.219270	−0.109635	+	0.993972i	$0.534968\pi$
$720$	0	0
$721$	14.5458	0.541715
$722$	26.0850	0.970783
$723$	3.12646	0.116274
$724$	1.35781	0.0504627
$725$	0	0
$726$	−112.409	−4.17187
$727$	−49.0096	−1.81767	−0.908833	−	0.417161i	$-0.863025\pi$
$727$	−49.0096	−1.81767	−0.908833	+	0.417161i	$0.863025\pi$
$728$	−52.9255	−1.96155
$729$	1.58556	0.0587245
$730$	0	0
$731$	26.1734	0.968058
$732$	3.36991	0.124556
$733$	15.4332	0.570040	0.285020	−	0.958522i	$-0.408000\pi$
$733$	15.4332	0.570040	0.285020	+	0.958522i	$0.408000\pi$
$734$	−48.4863	−1.78966
$735$	0	0
$736$	−4.62694	−0.170551
$737$	83.0012	3.05739
$738$	65.2245	2.40095
$739$	−19.0798	−0.701863	−0.350931	−	0.936401i	$-0.614135\pi$
$739$	−19.0798	−0.701863	−0.350931	+	0.936401i	$0.614135\pi$
$740$	0	0
$741$	16.5239	0.607021
$742$	−70.7453	−2.59714
$743$	−28.2323	−1.03574	−0.517871	−	0.855459i	$-0.673275\pi$
$743$	−28.2323	−1.03574	−0.517871	+	0.855459i	$0.673275\pi$
$744$	−9.21947	−0.338002
$745$	0	0
$746$	−52.5735	−1.92485
$747$	−67.2879	−2.46194
$748$	7.67024	0.280452
$749$	−11.0879	−0.405143
$750$	0	0
$751$	10.2127	0.372667	0.186333	−	0.982487i	$-0.440340\pi$
$751$	10.2127	0.372667	0.186333	+	0.982487i	$0.440340\pi$
$752$	33.9458	1.23788
$753$	−43.2866	−1.57745
$754$	11.5999	0.422442
$755$	0	0
$756$	12.4372	0.452337
$757$	−42.5055	−1.54489	−0.772445	−	0.635082i	$-0.780966\pi$
$757$	−42.5055	−1.54489	−0.772445	+	0.635082i	$0.780966\pi$
$758$	7.74245	0.281219
$759$	67.3049	2.44301
$760$	0	0
$761$	−33.2390	−1.20491	−0.602456	−	0.798152i	$-0.705811\pi$
$761$	−33.2390	−1.20491	−0.602456	+	0.798152i	$0.705811\pi$
$762$	41.2128	1.49298
$763$	−50.9299	−1.84379
$764$	−3.28014	−0.118671
$765$	0	0
$766$	8.53970	0.308552
$767$	24.3286	0.878455
$768$	16.7596	0.604759
$769$	24.0517	0.867328	0.433664	−	0.901075i	$-0.357221\pi$
$769$	24.0517	0.867328	0.433664	+	0.901075i	$0.357221\pi$
$770$	0	0
$771$	19.2972	0.694971
$772$	0.773387	0.0278348
$773$	−19.5958	−0.704813	−0.352407	−	0.935847i	$-0.614637\pi$
$773$	−19.5958	−0.704813	−0.352407	+	0.935847i	$0.614637\pi$
$774$	46.2096	1.66097
$775$	0	0
$776$	13.3360	0.478735
$777$	−106.753	−3.82975
$778$	−37.0861	−1.32960
$779$	7.94770	0.284756
$780$	0	0
$781$	23.5109	0.841286
$782$	−31.0405	−1.11001
$783$	21.3824	0.764143
$784$	−64.7752	−2.31340
$785$	0	0
$786$	−49.8802	−1.77917
$787$	14.4340	0.514515	0.257257	−	0.966343i	$-0.417181\pi$
$787$	14.4340	0.514515	0.257257	+	0.966343i	$0.417181\pi$
$788$	4.63757	0.165207
$789$	46.4352	1.65314
$790$	0	0
$791$	57.5432	2.04600
$792$	−106.225	−3.77453
$793$	−20.4527	−0.726298
$794$	51.9542	1.84379
$795$	0	0
$796$	−0.628181	−0.0222653
$797$	−36.2235	−1.28310	−0.641551	−	0.767081i	$-0.721709\pi$
$797$	−36.2235	−1.28310	−0.641551	+	0.767081i	$0.721709\pi$
$798$	26.7754	0.947839
$799$	44.1588	1.56223
$800$	0	0
$801$	−47.6873	−1.68495
$802$	47.9492	1.69314
$803$	−52.8472	−1.86494
$804$	−9.90547	−0.349339
$805$	0	0
$806$	−7.13337	−0.251262
$807$	77.4091	2.72493
$808$	−23.5011	−0.826766
$809$	−33.5485	−1.17950	−0.589751	−	0.807585i	$-0.700774\pi$
$809$	−33.5485	−1.17950	−0.589751	+	0.807585i	$0.700774\pi$
$810$	0	0
$811$	−41.8458	−1.46941	−0.734703	−	0.678389i	$-0.762678\pi$
$811$	−41.8458	−1.46941	−0.734703	+	0.678389i	$0.762678\pi$
$812$	1.90941	0.0670072
$813$	68.5940	2.40570
$814$	−64.7641	−2.26998
$815$	0	0
$816$	78.7796	2.75784
$817$	5.63071	0.196994
$818$	45.4755	1.59002
$819$	−135.476	−4.73390
$820$	0	0
$821$	−29.6138	−1.03353	−0.516765	−	0.856127i	$-0.672864\pi$
$821$	−29.6138	−1.03353	−0.516765	+	0.856127i	$0.672864\pi$
$822$	30.4676	1.06268
$823$	5.60965	0.195540	0.0977701	−	0.995209i	$-0.468829\pi$
$823$	5.60965	0.195540	0.0977701	+	0.995209i	$0.468829\pi$
$824$	8.26088	0.287781
$825$	0	0
$826$	39.4222	1.37167
$827$	13.9106	0.483720	0.241860	−	0.970311i	$-0.422242\pi$
$827$	13.9106	0.483720	0.241860	+	0.970311i	$0.422242\pi$
$828$	−5.56704	−0.193468
$829$	22.9062	0.795567	0.397783	−	0.917479i	$-0.369780\pi$
$829$	22.9062	0.795567	0.397783	+	0.917479i	$0.369780\pi$
$830$	0	0
$831$	56.1315	1.94718
$832$	−29.6186	−1.02684
$833$	−84.2635	−2.91956
$834$	45.1703	1.56412
$835$	0	0
$836$	1.65011	0.0570702
$837$	−13.1491	−0.454500
$838$	20.9692	0.724369
$839$	−5.57070	−0.192322	−0.0961609	−	0.995366i	$-0.530656\pi$
$839$	−5.57070	−0.192322	−0.0961609	+	0.995366i	$0.530656\pi$
$840$	0	0
$841$	−25.7173	−0.886803
$842$	−11.8868	−0.409647
$843$	46.7467	1.61004
$844$	5.18193	0.178369
$845$	0	0
$846$	77.9632	2.68043
$847$	112.300	3.85867
$848$	−44.7794	−1.53773
$849$	−52.7804	−1.81142
$850$	0	0
$851$	26.6242	0.912667
$852$	−2.80582	−0.0961257
$853$	35.8612	1.22786	0.613931	−	0.789359i	$-0.289587\pi$
$853$	35.8612	1.22786	0.613931	+	0.789359i	$0.289587\pi$
$854$	−33.1417	−1.13409
$855$	0	0
$856$	−6.29705	−0.215229
$857$	−0.654119	−0.0223443	−0.0111721	−	0.999938i	$-0.503556\pi$
$857$	−0.654119	−0.0223443	−0.0111721	+	0.999938i	$0.503556\pi$
$858$	−118.584	−4.04840
$859$	2.58305	0.0881324	0.0440662	−	0.999029i	$-0.485969\pi$
$859$	2.58305	0.0881324	0.0440662	+	0.999029i	$0.485969\pi$
$860$	0	0
$861$	−94.0163	−3.20407
$862$	59.1677	2.01526
$863$	52.4627	1.78585	0.892925	−	0.450206i	$-0.148649\pi$
$863$	52.4627	1.78585	0.892925	+	0.450206i	$0.148649\pi$
$864$	15.0272	0.511234
$865$	0	0
$866$	−4.88887	−0.166130
$867$	49.3315	1.67539
$868$	−1.17420	−0.0398549
$869$	−10.5109	−0.356559
$870$	0	0
$871$	60.1184	2.03704
$872$	−28.9242	−0.979496
$873$	34.1368	1.15536
$874$	−6.67778	−0.225879
$875$	0	0
$876$	6.30685	0.213089
$877$	19.1481	0.646586	0.323293	−	0.946299i	$-0.395210\pi$
$877$	19.1481	0.646586	0.323293	+	0.946299i	$0.395210\pi$
$878$	21.6440	0.730449
$879$	−51.1649	−1.72575
$880$	0	0
$881$	−38.6243	−1.30129	−0.650643	−	0.759384i	$-0.725501\pi$
$881$	−38.6243	−1.30129	−0.650643	+	0.759384i	$0.725501\pi$
$882$	−148.769	−5.00931
$883$	40.7541	1.37149	0.685743	−	0.727844i	$-0.259477\pi$
$883$	40.7541	1.37149	0.685743	+	0.727844i	$0.259477\pi$
$884$	5.55562	0.186856
$885$	0	0
$886$	29.1344	0.978788
$887$	−44.2838	−1.48690	−0.743452	−	0.668789i	$-0.766813\pi$
$887$	−44.2838	−1.48690	−0.743452	+	0.668789i	$0.766813\pi$
$888$	−60.6274	−2.03452
$889$	−41.1729	−1.38089
$890$	0	0
$891$	−98.1827	−3.28925
$892$	1.70162	0.0569745
$893$	9.49993	0.317903
$894$	−68.9908	−2.30740
$895$	0	0
$896$	−59.8620	−1.99985
$897$	48.7495	1.62770
$898$	4.25936	0.142137
$899$	−2.01871	−0.0673277
$900$	0	0
$901$	−58.2518	−1.94065
$902$	−57.0369	−1.89912
$903$	−66.6077	−2.21657
$904$	32.6800	1.08692
$905$	0	0
$906$	5.68186	0.188767
$907$	−0.773290	−0.0256767	−0.0128383	−	0.999918i	$-0.504087\pi$
$907$	−0.773290	−0.0256767	−0.0128383	+	0.999918i	$0.504087\pi$
$908$	0.214889	0.00713136
$909$	−60.1569	−1.99528
$910$	0	0
$911$	39.1307	1.29646	0.648228	−	0.761446i	$-0.275510\pi$
$911$	39.1307	1.29646	0.648228	+	0.761446i	$0.275510\pi$
$912$	16.9479	0.561202
$913$	58.8413	1.94736
$914$	17.5028	0.578942
$915$	0	0
$916$	3.31395	0.109496
$917$	49.8319	1.64560
$918$	100.812	3.32729
$919$	24.2527	0.800023	0.400011	−	0.916510i	$-0.369006\pi$
$919$	24.2527	0.800023	0.400011	+	0.916510i	$0.369006\pi$
$920$	0	0
$921$	59.4955	1.96044
$922$	−2.10800	−0.0694233
$923$	17.0291	0.560520
$924$	−19.5197	−0.642152
$925$	0	0
$926$	45.2120	1.48576
$927$	21.1457	0.694517
$928$	2.30703	0.0757320
$929$	4.12264	0.135260	0.0676298	−	0.997710i	$-0.478456\pi$
$929$	4.12264	0.135260	0.0676298	+	0.997710i	$0.478456\pi$
$930$	0	0
$931$	−18.1277	−0.594112
$932$	−4.33313	−0.141936
$933$	−50.9629	−1.66845
$934$	−15.2808	−0.500003
$935$	0	0
$936$	−76.9394	−2.51484
$937$	−8.27592	−0.270362	−0.135181	−	0.990821i	$-0.543162\pi$
$937$	−8.27592	−0.270362	−0.135181	+	0.990821i	$0.543162\pi$
$938$	97.4162	3.18075
$939$	5.35473	0.174745
$940$	0	0
$941$	11.2610	0.367100	0.183550	−	0.983010i	$-0.441241\pi$
$941$	11.2610	0.367100	0.183550	+	0.983010i	$0.441241\pi$
$942$	−44.2417	−1.44147
$943$	23.4476	0.763560
$944$	24.9529	0.812148
$945$	0	0
$946$	−40.4090	−1.31381
$947$	28.2529	0.918096	0.459048	−	0.888411i	$-0.348191\pi$
$947$	28.2529	0.918096	0.459048	+	0.888411i	$0.348191\pi$
$948$	1.25439	0.0407406
$949$	−38.2776	−1.24254
$950$	0	0
$951$	29.1770	0.946130
$952$	−70.6155	−2.28866
$953$	39.0186	1.26394	0.631968	−	0.774995i	$-0.282248\pi$
$953$	39.0186	1.26394	0.631968	+	0.774995i	$0.282248\pi$
$954$	−102.845	−3.32972
$955$	0	0
$956$	−0.878858	−0.0284243
$957$	−33.5588	−1.08480
$958$	25.1203	0.811602
$959$	−30.4381	−0.982899
$960$	0	0
$961$	−29.7586	−0.959955
$962$	−46.9091	−1.51241
$963$	−16.1188	−0.519423
$964$	0.226139	0.00728345
$965$	0	0
$966$	78.9939	2.54159
$967$	39.9098	1.28341	0.641707	−	0.766950i	$-0.278227\pi$
$967$	39.9098	1.28341	0.641707	+	0.766950i	$0.278227\pi$
$968$	63.7773	2.04988
$969$	22.0469	0.708249
$970$	0	0
$971$	50.1082	1.60805	0.804024	−	0.594597i	$-0.202688\pi$
$971$	50.1082	1.60805	0.804024	+	0.594597i	$0.202688\pi$
$972$	3.71087	0.119026
$973$	−45.1266	−1.44669
$974$	28.5289	0.914126
$975$	0	0
$976$	−20.9776	−0.671477
$977$	−42.4558	−1.35828	−0.679141	−	0.734008i	$-0.737648\pi$
$977$	−42.4558	−1.35828	−0.679141	+	0.734008i	$0.737648\pi$
$978$	−9.43359	−0.301653
$979$	41.7011	1.33277
$980$	0	0
$981$	−74.0385	−2.36387
$982$	−20.4288	−0.651908
$983$	18.7770	0.598895	0.299447	−	0.954113i	$-0.403198\pi$
$983$	18.7770	0.598895	0.299447	+	0.954113i	$0.403198\pi$
$984$	−53.3938	−1.70213
$985$	0	0
$986$	15.4771	0.492890
$987$	−112.378	−3.57703
$988$	1.19519	0.0380239
$989$	16.6119	0.528229
$990$	0	0
$991$	−7.86345	−0.249791	−0.124895	−	0.992170i	$-0.539860\pi$
$991$	−7.86345	−0.249791	−0.124895	+	0.992170i	$0.539860\pi$
$992$	−1.41872	−0.0450443
$993$	38.1451	1.21050
$994$	27.5941	0.875230
$995$	0	0
$996$	−7.02219	−0.222507
$997$	4.32111	0.136851	0.0684255	−	0.997656i	$-0.478202\pi$
$997$	4.32111	0.136851	0.0684255	+	0.997656i	$0.478202\pi$
$998$	−61.4989	−1.94671
$999$	−86.4688	−2.73575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.12		46
5.2	odd	4		1205.2.b.c.724.12	✓	46
5.3	odd	4		1205.2.b.c.724.35	yes	46
5.4	even	2	inner	6025.2.a.p.1.35		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.12	✓	46	5.2	odd	4
1205.2.b.c.724.35	yes	46	5.3	odd	4
6025.2.a.p.1.12		46	1.1	even	1	trivial
6025.2.a.p.1.35		46	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.49203 q^{2} +3.12646 q^{3} +0.226139 q^{4} -4.66475 q^{6} +4.66024 q^{7} +2.64665 q^{8} +6.77474 q^{9} +O(q^{10})\)	\(q-1.49203 q^{2} +3.12646 q^{3} +0.226139 q^{4} -4.66475 q^{6} +4.66024 q^{7} +2.64665 q^{8} +6.77474 q^{9} -5.92431 q^{11} +0.707014 q^{12} -4.29102 q^{13} -6.95319 q^{14} -4.40114 q^{16} -5.72527 q^{17} -10.1081 q^{18} -1.23168 q^{19} +14.5700 q^{21} +8.83922 q^{22} -3.63376 q^{23} +8.27462 q^{24} +6.40231 q^{26} +11.8016 q^{27} +1.05386 q^{28} +1.81182 q^{29} -1.11419 q^{31} +1.27332 q^{32} -18.5221 q^{33} +8.54225 q^{34} +1.53203 q^{36} -7.32690 q^{37} +1.83770 q^{38} -13.4157 q^{39} -6.45271 q^{41} -21.7389 q^{42} -4.57155 q^{43} -1.33972 q^{44} +5.42167 q^{46} -7.71296 q^{47} -13.7600 q^{48} +14.7178 q^{49} -17.8998 q^{51} -0.970368 q^{52} +10.1745 q^{53} -17.6082 q^{54} +12.3340 q^{56} -3.85081 q^{57} -2.70329 q^{58} -5.66965 q^{59} +4.76640 q^{61} +1.66239 q^{62} +31.5719 q^{63} +6.90245 q^{64} +27.6354 q^{66} -14.0103 q^{67} -1.29471 q^{68} -11.3608 q^{69} -3.96855 q^{71} +17.9303 q^{72} +8.92040 q^{73} +10.9319 q^{74} -0.278532 q^{76} -27.6087 q^{77} +20.0166 q^{78} +1.77420 q^{79} +16.5729 q^{81} +9.62761 q^{82} -9.93218 q^{83} +3.29485 q^{84} +6.82087 q^{86} +5.66459 q^{87} -15.6795 q^{88} -7.03898 q^{89} -19.9972 q^{91} -0.821736 q^{92} -3.48346 q^{93} +11.5079 q^{94} +3.98098 q^{96} +5.03884 q^{97} -21.9594 q^{98} -40.1356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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