LMFDB - Embedded newform 6025.2.a.l.1.7

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			1.7
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)

$f(q)$	$=$	$q-2.40563 q^{2} -2.30634 q^{3} +3.78704 q^{4} +5.54820 q^{6} +4.72754 q^{7} -4.29895 q^{8} +2.31922 q^{9} +O(q^{10})$	\(q-2.40563 q^{2} -2.30634 q^{3} +3.78704 q^{4} +5.54820 q^{6} +4.72754 q^{7} -4.29895 q^{8} +2.31922 q^{9} +1.32782 q^{11} -8.73422 q^{12} -2.94680 q^{13} -11.3727 q^{14} +2.76760 q^{16} -2.94196 q^{17} -5.57918 q^{18} +5.65404 q^{19} -10.9033 q^{21} -3.19423 q^{22} -5.25516 q^{23} +9.91487 q^{24} +7.08890 q^{26} +1.57011 q^{27} +17.9034 q^{28} +6.80665 q^{29} +4.16015 q^{31} +1.94010 q^{32} -3.06240 q^{33} +7.07726 q^{34} +8.78299 q^{36} +1.74440 q^{37} -13.6015 q^{38} +6.79633 q^{39} -7.98498 q^{41} +26.2293 q^{42} +0.0704942 q^{43} +5.02850 q^{44} +12.6420 q^{46} -3.91004 q^{47} -6.38303 q^{48} +15.3496 q^{49} +6.78517 q^{51} -11.1596 q^{52} -8.86887 q^{53} -3.77710 q^{54} -20.3235 q^{56} -13.0402 q^{57} -16.3743 q^{58} -4.11944 q^{59} -1.61770 q^{61} -10.0078 q^{62} +10.9642 q^{63} -10.2024 q^{64} +7.36700 q^{66} -10.3246 q^{67} -11.1413 q^{68} +12.1202 q^{69} +9.46274 q^{71} -9.97023 q^{72} -3.02808 q^{73} -4.19639 q^{74} +21.4121 q^{76} +6.27731 q^{77} -16.3494 q^{78} -7.08151 q^{79} -10.5789 q^{81} +19.2089 q^{82} +1.44306 q^{83} -41.2914 q^{84} -0.169583 q^{86} -15.6985 q^{87} -5.70823 q^{88} -12.6805 q^{89} -13.9311 q^{91} -19.9015 q^{92} -9.59474 q^{93} +9.40610 q^{94} -4.47454 q^{96} -6.93859 q^{97} -36.9255 q^{98} +3.07950 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	$ 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) $ 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * 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 $

$2$	−2.40563	−1.70104	−0.850518	−	0.525947i	$-0.823711\pi$
$2$	−2.40563	−1.70104	−0.850518	+	0.525947i	$0.823711\pi$
$3$	−2.30634	−1.33157	−0.665784	−	0.746144i	$-0.731903\pi$
$3$	−2.30634	−1.33157	−0.665784	+	0.746144i	$0.731903\pi$
$4$	3.78704	1.89352
$5$	0	0
$5$	0	0
$6$	5.54820	2.26504
$7$	4.72754	1.78684	0.893421	−	0.449221i	$-0.148298\pi$
$7$	4.72754	1.78684	0.893421	+	0.449221i	$0.148298\pi$
$8$	−4.29895	−1.51991
$9$	2.31922	0.773074
$10$	0	0
$11$	1.32782	0.400352	0.200176	−	0.979760i	$-0.435849\pi$
$11$	1.32782	0.400352	0.200176	+	0.979760i	$0.435849\pi$
$12$	−8.73422	−2.52135
$13$	−2.94680	−0.817295	−0.408647	−	0.912692i	$-0.633999\pi$
$13$	−2.94680	−0.817295	−0.408647	+	0.912692i	$0.633999\pi$
$14$	−11.3727	−3.03948
$15$	0	0
$16$	2.76760	0.691900
$17$	−2.94196	−0.713530	−0.356765	−	0.934194i	$-0.616120\pi$
$17$	−2.94196	−0.713530	−0.356765	+	0.934194i	$0.616120\pi$
$18$	−5.57918	−1.31503
$19$	5.65404	1.29713	0.648563	−	0.761161i	$-0.275371\pi$
$19$	5.65404	1.29713	0.648563	+	0.761161i	$0.275371\pi$
$20$	0	0
$21$	−10.9033	−2.37930
$22$	−3.19423	−0.681013
$23$	−5.25516	−1.09578	−0.547888	−	0.836551i	$-0.684568\pi$
$23$	−5.25516	−1.09578	−0.547888	+	0.836551i	$0.684568\pi$
$24$	9.91487	2.02386
$25$	0	0
$26$	7.08890	1.39025
$27$	1.57011	0.302168
$28$	17.9034	3.38342
$29$	6.80665	1.26396	0.631982	−	0.774983i	$-0.282242\pi$
$29$	6.80665	1.26396	0.631982	+	0.774983i	$0.282242\pi$
$30$	0	0
$31$	4.16015	0.747185	0.373593	−	0.927593i	$-0.378126\pi$
$31$	4.16015	0.747185	0.373593	+	0.927593i	$0.378126\pi$
$32$	1.94010	0.342964
$33$	−3.06240	−0.533096
$34$	7.07726	1.21374
$35$	0	0
$36$	8.78299	1.46383
$37$	1.74440	0.286778	0.143389	−	0.989666i	$-0.454200\pi$
$37$	1.74440	0.286778	0.143389	+	0.989666i	$0.454200\pi$
$38$	−13.6015	−2.20646
$39$	6.79633	1.08828
$40$	0	0
$41$	−7.98498	−1.24704	−0.623522	−	0.781806i	$-0.714299\pi$
$41$	−7.98498	−1.24704	−0.623522	+	0.781806i	$0.714299\pi$
$42$	26.2293	4.04728
$43$	0.0704942	0.0107503	0.00537513	−	0.999986i	$-0.498289\pi$
$43$	0.0704942	0.0107503	0.00537513	+	0.999986i	$0.498289\pi$
$44$	5.02850	0.758075
$45$	0	0
$46$	12.6420	1.86395
$47$	−3.91004	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$47$	−3.91004	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$48$	−6.38303	−0.921311
$49$	15.3496	2.19280
$50$	0	0
$51$	6.78517	0.950114
$52$	−11.1596	−1.54756
$53$	−8.86887	−1.21823	−0.609116	−	0.793081i	$-0.708476\pi$
$53$	−8.86887	−1.21823	−0.609116	+	0.793081i	$0.708476\pi$
$54$	−3.77710	−0.513998
$55$	0	0
$56$	−20.3235	−2.71584
$57$	−13.0402	−1.72721
$58$	−16.3743	−2.15005
$59$	−4.11944	−0.536305	−0.268153	−	0.963376i	$-0.586413\pi$
$59$	−4.11944	−0.536305	−0.268153	+	0.963376i	$0.586413\pi$
$60$	0	0
$61$	−1.61770	−0.207125	−0.103562	−	0.994623i	$-0.533024\pi$
$61$	−1.61770	−0.207125	−0.103562	+	0.994623i	$0.533024\pi$
$62$	−10.0078	−1.27099
$63$	10.9642	1.38136
$64$	−10.2024	−1.27529
$65$	0	0
$66$	7.36700	0.906815
$67$	−10.3246	−1.26135	−0.630674	−	0.776048i	$-0.717221\pi$
$67$	−10.3246	−1.26135	−0.630674	+	0.776048i	$0.717221\pi$
$68$	−11.1413	−1.35108
$69$	12.1202	1.45910
$70$	0	0
$71$	9.46274	1.12302	0.561510	−	0.827470i	$-0.310220\pi$
$71$	9.46274	1.12302	0.561510	+	0.827470i	$0.310220\pi$
$72$	−9.97023	−1.17500
$73$	−3.02808	−0.354410	−0.177205	−	0.984174i	$-0.556706\pi$
$73$	−3.02808	−0.354410	−0.177205	+	0.984174i	$0.556706\pi$
$74$	−4.19639	−0.487820
$75$	0	0
$76$	21.4121	2.45613
$77$	6.27731	0.715366
$78$	−16.3494	−1.85121
$79$	−7.08151	−0.796732	−0.398366	−	0.917227i	$-0.630423\pi$
$79$	−7.08151	−0.796732	−0.398366	+	0.917227i	$0.630423\pi$
$80$	0	0
$81$	−10.5789	−1.17543
$82$	19.2089	2.12127
$83$	1.44306	0.158396	0.0791981	−	0.996859i	$-0.474764\pi$
$83$	1.44306	0.158396	0.0791981	+	0.996859i	$0.474764\pi$
$84$	−41.2914	−4.50526
$85$	0	0
$86$	−0.169583	−0.0182866
$87$	−15.6985	−1.68305
$88$	−5.70823	−0.608499
$89$	−12.6805	−1.34413	−0.672064	−	0.740493i	$-0.734592\pi$
$89$	−12.6805	−1.34413	−0.672064	+	0.740493i	$0.734592\pi$
$90$	0	0
$91$	−13.9311	−1.46038
$92$	−19.9015	−2.07488
$93$	−9.59474	−0.994928
$94$	9.40610	0.970165
$95$	0	0
$96$	−4.47454	−0.456681
$97$	−6.93859	−0.704507	−0.352253	−	0.935905i	$-0.614584\pi$
$97$	−6.93859	−0.704507	−0.352253	+	0.935905i	$0.614584\pi$
$98$	−36.9255	−3.73004
$99$	3.07950	0.309502
$100$	0	0
$101$	−18.8185	−1.87251	−0.936253	−	0.351327i	$-0.885731\pi$
$101$	−18.8185	−1.87251	−0.936253	+	0.351327i	$0.885731\pi$
$102$	−16.3226	−1.61618
$103$	17.9424	1.76792	0.883959	−	0.467565i	$-0.154869\pi$
$103$	17.9424	1.76792	0.883959	+	0.467565i	$0.154869\pi$
$104$	12.6682	1.24221
$105$	0	0
$106$	21.3352	2.07226
$107$	−2.45995	−0.237812	−0.118906	−	0.992906i	$-0.537939\pi$
$107$	−2.45995	−0.237812	−0.118906	+	0.992906i	$0.537939\pi$
$108$	5.94607	0.572160
$109$	−12.7884	−1.22490	−0.612452	−	0.790508i	$-0.709817\pi$
$109$	−12.7884	−1.22490	−0.612452	+	0.790508i	$0.709817\pi$
$110$	0	0
$111$	−4.02320	−0.381865
$112$	13.0839	1.23631
$113$	0.133542	0.0125626	0.00628129	−	0.999980i	$-0.498001\pi$
$113$	0.133542	0.0125626	0.00628129	+	0.999980i	$0.498001\pi$
$114$	31.3697	2.93805
$115$	0	0
$116$	25.7771	2.39334
$117$	−6.83428	−0.631829
$118$	9.90983	0.912274
$119$	−13.9082	−1.27497
$120$	0	0
$121$	−9.23690	−0.839718
$122$	3.89158	0.352327
$123$	18.4161	1.66052
$124$	15.7547	1.41481
$125$	0	0
$126$	−26.3758	−2.34974
$127$	14.5868	1.29437	0.647186	−	0.762332i	$-0.275946\pi$
$127$	14.5868	1.29437	0.647186	+	0.762332i	$0.275946\pi$
$128$	20.6629	1.82636
$129$	−0.162584	−0.0143147
$130$	0	0
$131$	0.406951	0.0355555	0.0177777	−	0.999842i	$-0.494341\pi$
$131$	0.406951	0.0355555	0.0177777	+	0.999842i	$0.494341\pi$
$132$	−11.5975	−1.00943
$133$	26.7297	2.31776
$134$	24.8371	2.14560
$135$	0	0
$136$	12.6474	1.08450
$137$	−3.35142	−0.286331	−0.143166	−	0.989699i	$-0.545728\pi$
$137$	−3.35142	−0.286331	−0.143166	+	0.989699i	$0.545728\pi$
$138$	−29.1567	−2.48198
$139$	−11.3326	−0.961215	−0.480608	−	0.876936i	$-0.659584\pi$
$139$	−11.3326	−0.961215	−0.480608	+	0.876936i	$0.659584\pi$
$140$	0	0
$141$	9.01790	0.759444
$142$	−22.7638	−1.91030
$143$	−3.91281	−0.327206
$144$	6.41867	0.534890
$145$	0	0
$146$	7.28444	0.602864
$147$	−35.4015	−2.91987
$148$	6.60613	0.543021
$149$	15.4180	1.26309	0.631545	−	0.775340i	$-0.282421\pi$
$149$	15.4180	1.26309	0.631545	+	0.775340i	$0.282421\pi$
$150$	0	0
$151$	10.4825	0.853057	0.426528	−	0.904474i	$-0.359736\pi$
$151$	10.4825	0.853057	0.426528	+	0.904474i	$0.359736\pi$
$152$	−24.3064	−1.97151
$153$	−6.82306	−0.551612
$154$	−15.1009	−1.21686
$155$	0	0
$156$	25.7380	2.06069
$157$	−15.3632	−1.22612	−0.613059	−	0.790037i	$-0.710061\pi$
$157$	−15.3632	−1.22612	−0.613059	+	0.790037i	$0.710061\pi$
$158$	17.0355	1.35527
$159$	20.4547	1.62216
$160$	0	0
$161$	−24.8440	−1.95798
$162$	25.4488	1.99945
$163$	−0.207992	−0.0162912	−0.00814559	−	0.999967i	$-0.502593\pi$
$163$	−0.207992	−0.0162912	−0.00814559	+	0.999967i	$0.502593\pi$
$164$	−30.2394	−2.36130
$165$	0	0
$166$	−3.47146	−0.269437
$167$	12.8901	0.997466	0.498733	−	0.866756i	$-0.333799\pi$
$167$	12.8901	0.997466	0.498733	+	0.866756i	$0.333799\pi$
$168$	46.8729	3.61632
$169$	−4.31638	−0.332029
$170$	0	0
$171$	13.1130	1.00277
$172$	0.266964	0.0203559
$173$	0.133377	0.0101404	0.00507022	−	0.999987i	$-0.498386\pi$
$173$	0.133377	0.0101404	0.00507022	+	0.999987i	$0.498386\pi$
$174$	37.7647	2.86293
$175$	0	0
$176$	3.67487	0.277003
$177$	9.50084	0.714127
$178$	30.5045	2.28641
$179$	6.05186	0.452337	0.226169	−	0.974088i	$-0.427380\pi$
$179$	6.05186	0.452337	0.226169	+	0.974088i	$0.427380\pi$
$180$	0	0
$181$	6.63179	0.492937	0.246469	−	0.969151i	$-0.420730\pi$
$181$	6.63179	0.492937	0.246469	+	0.969151i	$0.420730\pi$
$182$	33.5130	2.48415
$183$	3.73097	0.275801
$184$	22.5917	1.66548
$185$	0	0
$186$	23.0814	1.69241
$187$	−3.90639	−0.285663
$188$	−14.8075	−1.07995
$189$	7.42275	0.539926
$190$	0	0
$191$	−20.4813	−1.48197	−0.740987	−	0.671519i	$-0.765642\pi$
$191$	−20.4813	−1.48197	−0.740987	+	0.671519i	$0.765642\pi$
$192$	23.5301	1.69814
$193$	6.74976	0.485859	0.242929	−	0.970044i	$-0.421892\pi$
$193$	6.74976	0.485859	0.242929	+	0.970044i	$0.421892\pi$
$194$	16.6916	1.19839
$195$	0	0
$196$	58.1297	4.15212
$197$	1.32739	0.0945725	0.0472862	−	0.998881i	$-0.484943\pi$
$197$	1.32739	0.0945725	0.0472862	+	0.998881i	$0.484943\pi$
$198$	−7.40814	−0.526473
$199$	20.9284	1.48357	0.741786	−	0.670636i	$-0.233979\pi$
$199$	20.9284	1.48357	0.741786	+	0.670636i	$0.233979\pi$
$200$	0	0
$201$	23.8120	1.67957
$202$	45.2702	3.18520
$203$	32.1787	2.25850
$204$	25.6957	1.79906
$205$	0	0
$206$	−43.1627	−3.00729
$207$	−12.1879	−0.847116
$208$	−8.15555	−0.565486
$209$	7.50753	0.519307
$210$	0	0
$211$	−12.6669	−0.872025	−0.436012	−	0.899941i	$-0.643610\pi$
$211$	−12.6669	−0.872025	−0.436012	+	0.899941i	$0.643610\pi$
$212$	−33.5868	−2.30675
$213$	−21.8243	−1.49538
$214$	5.91772	0.404527
$215$	0	0
$216$	−6.74982	−0.459267
$217$	19.6673	1.33510
$218$	30.7641	2.08360
$219$	6.98380	0.471921
$220$	0	0
$221$	8.66936	0.583164
$222$	9.67831	0.649566
$223$	−11.6909	−0.782878	−0.391439	−	0.920204i	$-0.628023\pi$
$223$	−11.6909	−0.782878	−0.391439	+	0.920204i	$0.628023\pi$
$224$	9.17190	0.612823
$225$	0	0
$226$	−0.321252	−0.0213694
$227$	13.4494	0.892665	0.446333	−	0.894867i	$-0.352730\pi$
$227$	13.4494	0.892665	0.446333	+	0.894867i	$0.352730\pi$
$228$	−49.3836	−3.27051
$229$	−12.8436	−0.848729	−0.424365	−	0.905491i	$-0.639502\pi$
$229$	−12.8436	−0.848729	−0.424365	+	0.905491i	$0.639502\pi$
$230$	0	0
$231$	−14.4776	−0.952559
$232$	−29.2615	−1.92111
$233$	−20.7534	−1.35960	−0.679802	−	0.733396i	$-0.737934\pi$
$233$	−20.7534	−1.35960	−0.679802	+	0.733396i	$0.737934\pi$
$234$	16.4407	1.07476
$235$	0	0
$236$	−15.6005	−1.01550
$237$	16.3324	1.06090
$238$	33.4580	2.16876
$239$	−11.3779	−0.735976	−0.367988	−	0.929831i	$-0.619953\pi$
$239$	−11.3779	−0.735976	−0.367988	+	0.929831i	$0.619953\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	22.2205	1.42839
$243$	19.6882	1.26300
$244$	−6.12628	−0.392195
$245$	0	0
$246$	−44.3023	−2.82461
$247$	−16.6613	−1.06013
$248$	−17.8843	−1.13565
$249$	−3.32819	−0.210915
$250$	0	0
$251$	−27.7851	−1.75378	−0.876889	−	0.480693i	$-0.840385\pi$
$251$	−27.7851	−1.75378	−0.876889	+	0.480693i	$0.840385\pi$
$252$	41.5219	2.61564
$253$	−6.97790	−0.438697
$254$	−35.0905	−2.20177
$255$	0	0
$256$	−29.3024	−1.83140
$257$	−18.9517	−1.18218	−0.591088	−	0.806607i	$-0.701301\pi$
$257$	−18.9517	−1.18218	−0.591088	+	0.806607i	$0.701301\pi$
$258$	0.391116	0.0243498
$259$	8.24674	0.512427
$260$	0	0
$261$	15.7861	0.977137
$262$	−0.978972	−0.0604811
$263$	4.77721	0.294576	0.147288	−	0.989094i	$-0.452946\pi$
$263$	4.77721	0.294576	0.147288	+	0.989094i	$0.452946\pi$
$264$	13.1651	0.810258
$265$	0	0
$266$	−64.3016	−3.94259
$267$	29.2455	1.78980
$268$	−39.0996	−2.38839
$269$	16.2176	0.988803	0.494402	−	0.869234i	$-0.335387\pi$
$269$	16.2176	0.988803	0.494402	+	0.869234i	$0.335387\pi$
$270$	0	0
$271$	16.7434	1.01709	0.508545	−	0.861035i	$-0.330183\pi$
$271$	16.7434	1.01709	0.508545	+	0.861035i	$0.330183\pi$
$272$	−8.14216	−0.493691
$273$	32.1299	1.94459
$274$	8.06226	0.487059
$275$	0	0
$276$	45.8997	2.76284
$277$	−21.5420	−1.29433	−0.647166	−	0.762349i	$-0.724046\pi$
$277$	−21.5420	−1.29433	−0.647166	+	0.762349i	$0.724046\pi$
$278$	27.2619	1.63506
$279$	9.64831	0.577629
$280$	0	0
$281$	−29.3958	−1.75361	−0.876804	−	0.480848i	$-0.840329\pi$
$281$	−29.3958	−1.75361	−0.876804	+	0.480848i	$0.840329\pi$
$282$	−21.6937	−1.29184
$283$	12.0990	0.719208	0.359604	−	0.933105i	$-0.382912\pi$
$283$	12.0990	0.719208	0.359604	+	0.933105i	$0.382912\pi$
$284$	35.8358	2.12646
$285$	0	0
$286$	9.41276	0.556588
$287$	−37.7493	−2.22827
$288$	4.49952	0.265137
$289$	−8.34487	−0.490875
$290$	0	0
$291$	16.0028	0.938099
$292$	−11.4675	−0.671083
$293$	18.4228	1.07627	0.538135	−	0.842859i	$-0.319129\pi$
$293$	18.4228	1.07627	0.538135	+	0.842859i	$0.319129\pi$
$294$	85.1628	4.96680
$295$	0	0
$296$	−7.49911	−0.435877
$297$	2.08482	0.120973
$298$	−37.0899	−2.14856
$299$	15.4859	0.895573
$300$	0	0
$301$	0.333264	0.0192090
$302$	−25.2171	−1.45108
$303$	43.4018	2.49337
$304$	15.6481	0.897480
$305$	0	0
$306$	16.4137	0.938311
$307$	−25.3609	−1.44743	−0.723713	−	0.690101i	$-0.757566\pi$
$307$	−25.3609	−1.44743	−0.723713	+	0.690101i	$0.757566\pi$
$308$	23.7724	1.35456
$309$	−41.3813	−2.35410
$310$	0	0
$311$	27.2345	1.54433	0.772165	−	0.635423i	$-0.219174\pi$
$311$	27.2345	1.54433	0.772165	+	0.635423i	$0.219174\pi$
$312$	−29.2171	−1.65409
$313$	19.6214	1.10907	0.554534	−	0.832161i	$-0.312896\pi$
$313$	19.6214	1.10907	0.554534	+	0.832161i	$0.312896\pi$
$314$	36.9582	2.08567
$315$	0	0
$316$	−26.8180	−1.50863
$317$	22.8576	1.28381	0.641904	−	0.766785i	$-0.278145\pi$
$317$	22.8576	1.28381	0.641904	+	0.766785i	$0.278145\pi$
$318$	−49.2063	−2.75935
$319$	9.03800	0.506031
$320$	0	0
$321$	5.67348	0.316663
$322$	59.7653	3.33059
$323$	−16.6340	−0.925538
$324$	−40.0626	−2.22570
$325$	0	0
$326$	0.500351	0.0277119
$327$	29.4944	1.63104
$328$	34.3271	1.89539
$329$	−18.4849	−1.01910
$330$	0	0
$331$	−9.11714	−0.501124	−0.250562	−	0.968101i	$-0.580615\pi$
$331$	−9.11714	−0.501124	−0.250562	+	0.968101i	$0.580615\pi$
$332$	5.46492	0.299926
$333$	4.04566	0.221701
$334$	−31.0088	−1.69673
$335$	0	0
$336$	−30.1760	−1.64624
$337$	11.3738	0.619570	0.309785	−	0.950807i	$-0.399743\pi$
$337$	11.3738	0.619570	0.309785	+	0.950807i	$0.399743\pi$
$338$	10.3836	0.564794
$339$	−0.307994	−0.0167279
$340$	0	0
$341$	5.52392	0.299137
$342$	−31.5449	−1.70575
$343$	39.4732	2.13135
$344$	−0.303051	−0.0163394
$345$	0	0
$346$	−0.320854	−0.0172492
$347$	−24.3274	−1.30596	−0.652981	−	0.757374i	$-0.726482\pi$
$347$	−24.3274	−1.30596	−0.652981	+	0.757374i	$0.726482\pi$
$348$	−59.4508	−3.18690
$349$	14.3278	0.766950	0.383475	−	0.923551i	$-0.374727\pi$
$349$	14.3278	0.766950	0.383475	+	0.923551i	$0.374727\pi$
$350$	0	0
$351$	−4.62679	−0.246960
$352$	2.57610	0.137307
$353$	−0.141543	−0.00753355	−0.00376678	−	0.999993i	$-0.501199\pi$
$353$	−0.141543	−0.00753355	−0.00376678	+	0.999993i	$0.501199\pi$
$354$	−22.8555	−1.21476
$355$	0	0
$356$	−48.0215	−2.54513
$357$	32.0772	1.69770
$358$	−14.5585	−0.769442
$359$	4.52913	0.239038	0.119519	−	0.992832i	$-0.461865\pi$
$359$	4.52913	0.239038	0.119519	+	0.992832i	$0.461865\pi$
$360$	0	0
$361$	12.9681	0.682533
$362$	−15.9536	−0.838504
$363$	21.3035	1.11814
$364$	−52.7577	−2.76525
$365$	0	0
$366$	−8.97531	−0.469147
$367$	22.2728	1.16263	0.581314	−	0.813679i	$-0.302539\pi$
$367$	22.2728	1.16263	0.581314	+	0.813679i	$0.302539\pi$
$368$	−14.5442	−0.758167
$369$	−18.5189	−0.964057
$370$	0	0
$371$	−41.9279	−2.17679
$372$	−36.3357	−1.88392
$373$	1.87062	0.0968570	0.0484285	−	0.998827i	$-0.484579\pi$
$373$	1.87062	0.0968570	0.0484285	+	0.998827i	$0.484579\pi$
$374$	9.39731	0.485923
$375$	0	0
$376$	16.8091	0.866863
$377$	−20.0578	−1.03303
$378$	−17.8564	−0.918432
$379$	−5.28038	−0.271235	−0.135617	−	0.990761i	$-0.543302\pi$
$379$	−5.28038	−0.271235	−0.135617	+	0.990761i	$0.543302\pi$
$380$	0	0
$381$	−33.6423	−1.72355
$382$	49.2703	2.52089
$383$	−20.9183	−1.06888	−0.534438	−	0.845208i	$-0.679477\pi$
$383$	−20.9183	−1.06888	−0.534438	+	0.845208i	$0.679477\pi$
$384$	−47.6556	−2.43192
$385$	0	0
$386$	−16.2374	−0.826463
$387$	0.163492	0.00831075
$388$	−26.2767	−1.33400
$389$	23.9935	1.21652	0.608260	−	0.793738i	$-0.291868\pi$
$389$	23.9935	1.21652	0.608260	+	0.793738i	$0.291868\pi$
$390$	0	0
$391$	15.4605	0.781870
$392$	−65.9873	−3.33286
$393$	−0.938569	−0.0473445
$394$	−3.19320	−0.160871
$395$	0	0
$396$	11.6622	0.586048
$397$	28.0387	1.40722	0.703612	−	0.710584i	$-0.251569\pi$
$397$	28.0387	1.40722	0.703612	+	0.710584i	$0.251569\pi$
$398$	−50.3459	−2.52361
$399$	−61.6478	−3.08625
$400$	0	0
$401$	34.8926	1.74245	0.871226	−	0.490883i	$-0.163326\pi$
$401$	34.8926	1.74245	0.871226	+	0.490883i	$0.163326\pi$
$402$	−57.2828	−2.85701
$403$	−12.2591	−0.610670
$404$	−71.2662	−3.54563
$405$	0	0
$406$	−77.4100	−3.84179
$407$	2.31625	0.114812
$408$	−29.1691	−1.44409
$409$	−2.48502	−0.122876	−0.0614382	−	0.998111i	$-0.519569\pi$
$409$	−2.48502	−0.122876	−0.0614382	+	0.998111i	$0.519569\pi$
$410$	0	0
$411$	7.72952	0.381269
$412$	67.9486	3.34759
$413$	−19.4748	−0.958292
$414$	29.3195	1.44097
$415$	0	0
$416$	−5.71708	−0.280303
$417$	26.1368	1.27992
$418$	−18.0603	−0.883359
$419$	8.17378	0.399315	0.199658	−	0.979866i	$-0.436017\pi$
$419$	8.17378	0.399315	0.199658	+	0.979866i	$0.436017\pi$
$420$	0	0
$421$	30.8969	1.50582	0.752910	−	0.658123i	$-0.228649\pi$
$421$	30.8969	1.50582	0.752910	+	0.658123i	$0.228649\pi$
$422$	30.4718	1.48334
$423$	−9.06825	−0.440914
$424$	38.1269	1.85160
$425$	0	0
$426$	52.5012	2.54369
$427$	−7.64772	−0.370099
$428$	−9.31592	−0.450302
$429$	9.02429	0.435697
$430$	0	0
$431$	−32.9490	−1.58710	−0.793548	−	0.608508i	$-0.791768\pi$
$431$	−32.9490	−1.58710	−0.793548	+	0.608508i	$0.791768\pi$
$432$	4.34543	0.209070
$433$	−33.9380	−1.63096	−0.815479	−	0.578787i	$-0.803526\pi$
$433$	−33.9380	−1.63096	−0.815479	+	0.578787i	$0.803526\pi$
$434$	−47.3121	−2.27105
$435$	0	0
$436$	−48.4301	−2.31938
$437$	−29.7129	−1.42136
$438$	−16.8004	−0.802755
$439$	33.3383	1.59115	0.795574	−	0.605856i	$-0.207169\pi$
$439$	33.3383	1.59115	0.795574	+	0.605856i	$0.207169\pi$
$440$	0	0
$441$	35.5992	1.69520
$442$	−20.8553	−0.991983
$443$	19.9173	0.946299	0.473150	−	0.880982i	$-0.343117\pi$
$443$	19.9173	0.946299	0.473150	+	0.880982i	$0.343117\pi$
$444$	−15.2360	−0.723069
$445$	0	0
$446$	28.1239	1.33170
$447$	−35.5591	−1.68189
$448$	−48.2320	−2.27875
$449$	−1.66324	−0.0784933	−0.0392467	−	0.999230i	$-0.512496\pi$
$449$	−1.66324	−0.0784933	−0.0392467	+	0.999230i	$0.512496\pi$
$450$	0	0
$451$	−10.6026	−0.499257
$452$	0.505729	0.0237875
$453$	−24.1763	−1.13590
$454$	−32.3541	−1.51846
$455$	0	0
$456$	56.0590	2.62520
$457$	−28.8367	−1.34892	−0.674462	−	0.738309i	$-0.735624\pi$
$457$	−28.8367	−1.34892	−0.674462	+	0.738309i	$0.735624\pi$
$458$	30.8969	1.44372
$459$	−4.61920	−0.215606
$460$	0	0
$461$	0.136263	0.00634638	0.00317319	−	0.999995i	$-0.498990\pi$
$461$	0.136263	0.00634638	0.00317319	+	0.999995i	$0.498990\pi$
$462$	34.8278	1.62034
$463$	−15.6633	−0.727936	−0.363968	−	0.931412i	$-0.618578\pi$
$463$	−15.6633	−0.727936	−0.363968	+	0.931412i	$0.618578\pi$
$464$	18.8381	0.874536
$465$	0	0
$466$	49.9250	2.31273
$467$	27.9541	1.29356	0.646781	−	0.762676i	$-0.276115\pi$
$467$	27.9541	1.29356	0.646781	+	0.762676i	$0.276115\pi$
$468$	−25.8817	−1.19638
$469$	−48.8098	−2.25383
$470$	0	0
$471$	35.4329	1.63266
$472$	17.7093	0.815136
$473$	0.0936035	0.00430389
$474$	−39.2896	−1.80463
$475$	0	0
$476$	−52.6710	−2.41417
$477$	−20.5689	−0.941784
$478$	27.3710	1.25192
$479$	−0.171192	−0.00782195	−0.00391097	−	0.999992i	$-0.501245\pi$
$479$	−0.171192	−0.00782195	−0.00391097	+	0.999992i	$0.501245\pi$
$480$	0	0
$481$	−5.14041	−0.234382
$482$	2.40563	0.109573
$483$	57.2988	2.60718
$484$	−34.9805	−1.59002
$485$	0	0
$486$	−47.3625	−2.14840
$487$	−22.1449	−1.00348	−0.501740	−	0.865018i	$-0.667307\pi$
$487$	−22.1449	−1.00348	−0.501740	+	0.865018i	$0.667307\pi$
$488$	6.95441	0.314811
$489$	0.479701	0.0216928
$490$	0	0
$491$	−15.3838	−0.694262	−0.347131	−	0.937817i	$-0.612844\pi$
$491$	−15.3838	−0.694262	−0.347131	+	0.937817i	$0.612844\pi$
$492$	69.7426	3.14424
$493$	−20.0249	−0.901876
$494$	40.0809	1.80332
$495$	0	0
$496$	11.5136	0.516977
$497$	44.7355	2.00666
$498$	8.00638	0.358774
$499$	20.5081	0.918070	0.459035	−	0.888418i	$-0.348195\pi$
$499$	20.5081	0.918070	0.459035	+	0.888418i	$0.348195\pi$
$500$	0	0
$501$	−29.7290	−1.32819
$502$	66.8405	2.98324
$503$	−37.3306	−1.66449	−0.832245	−	0.554408i	$-0.812945\pi$
$503$	−37.3306	−1.66449	−0.832245	+	0.554408i	$0.812945\pi$
$504$	−47.1346	−2.09954
$505$	0	0
$506$	16.7862	0.746238
$507$	9.95506	0.442120
$508$	55.2410	2.45092
$509$	−13.3633	−0.592317	−0.296158	−	0.955139i	$-0.595706\pi$
$509$	−13.3633	−0.592317	−0.296158	+	0.955139i	$0.595706\pi$
$510$	0	0
$511$	−14.3154	−0.633275
$512$	29.1650	1.28892
$513$	8.87745	0.391949
$514$	45.5908	2.01092
$515$	0	0
$516$	−0.615712	−0.0271052
$517$	−5.19182	−0.228336
$518$	−19.8386	−0.871657
$519$	−0.307612	−0.0135027
$520$	0	0
$521$	−42.5766	−1.86531	−0.932657	−	0.360763i	$-0.882516\pi$
$521$	−42.5766	−1.86531	−0.932657	+	0.360763i	$0.882516\pi$
$522$	−37.9756	−1.66215
$523$	37.7767	1.65186	0.825931	−	0.563772i	$-0.190650\pi$
$523$	37.7767	1.65186	0.825931	+	0.563772i	$0.190650\pi$
$524$	1.54114	0.0673250
$525$	0	0
$526$	−11.4922	−0.501083
$527$	−12.2390	−0.533139
$528$	−8.47551	−0.368849
$529$	4.61671	0.200727
$530$	0	0
$531$	−9.55389	−0.414604
$532$	101.226	4.38872
$533$	23.5301	1.01920
$534$	−70.3538	−3.04451
$535$	0	0
$536$	44.3849	1.91714
$537$	−13.9577	−0.602318
$538$	−39.0135	−1.68199
$539$	20.3815	0.877894
$540$	0	0
$541$	−25.1169	−1.07986	−0.539930	−	0.841710i	$-0.681549\pi$
$541$	−25.1169	−1.07986	−0.539930	+	0.841710i	$0.681549\pi$
$542$	−40.2784	−1.73011
$543$	−15.2952	−0.656380
$544$	−5.70770	−0.244715
$545$	0	0
$546$	−77.2926	−3.30782
$547$	19.2551	0.823288	0.411644	−	0.911345i	$-0.364955\pi$
$547$	19.2551	0.823288	0.411644	+	0.911345i	$0.364955\pi$
$548$	−12.6920	−0.542174
$549$	−3.75180	−0.160123
$550$	0	0
$551$	38.4851	1.63952
$552$	−52.1042	−2.21770
$553$	−33.4781	−1.42363
$554$	51.8220	2.20171
$555$	0	0
$556$	−42.9169	−1.82008
$557$	16.9523	0.718291	0.359146	−	0.933282i	$-0.383068\pi$
$557$	16.9523	0.718291	0.359146	+	0.933282i	$0.383068\pi$
$558$	−23.2102	−0.982568
$559$	−0.207732	−0.00878614
$560$	0	0
$561$	9.00947	0.380380
$562$	70.7154	2.98295
$563$	−7.83487	−0.330200	−0.165100	−	0.986277i	$-0.552795\pi$
$563$	−7.83487	−0.330200	−0.165100	+	0.986277i	$0.552795\pi$
$564$	34.1512	1.43802
$565$	0	0
$566$	−29.1056	−1.22340
$567$	−50.0120	−2.10031
$568$	−40.6799	−1.70689
$569$	−28.9497	−1.21364	−0.606818	−	0.794841i	$-0.707554\pi$
$569$	−28.9497	−1.21364	−0.606818	+	0.794841i	$0.707554\pi$
$570$	0	0
$571$	−25.2259	−1.05567	−0.527835	−	0.849347i	$-0.676996\pi$
$571$	−25.2259	−1.05567	−0.527835	+	0.849347i	$0.676996\pi$
$572$	−14.8180	−0.619571
$573$	47.2369	1.97335
$574$	90.8107	3.79037
$575$	0	0
$576$	−23.6615	−0.985897
$577$	−26.7054	−1.11176	−0.555880	−	0.831262i	$-0.687619\pi$
$577$	−26.7054	−1.11176	−0.555880	+	0.831262i	$0.687619\pi$
$578$	20.0746	0.834995
$579$	−15.5673	−0.646954
$580$	0	0
$581$	6.82211	0.283029
$582$	−38.4967	−1.59574
$583$	−11.7762	−0.487722
$584$	13.0176	0.538672
$585$	0	0
$586$	−44.3183	−1.83077
$587$	−5.38865	−0.222413	−0.111207	−	0.993797i	$-0.535472\pi$
$587$	−5.38865	−0.222413	−0.111207	+	0.993797i	$0.535472\pi$
$588$	−134.067	−5.52883
$589$	23.5216	0.969192
$590$	0	0
$591$	−3.06141	−0.125930
$592$	4.82781	0.198422
$593$	−12.1302	−0.498127	−0.249064	−	0.968487i	$-0.580123\pi$
$593$	−12.1302	−0.498127	−0.249064	+	0.968487i	$0.580123\pi$
$594$	−5.01529	−0.205780
$595$	0	0
$596$	58.3885	2.39169
$597$	−48.2680	−1.97548
$598$	−37.2533	−1.52340
$599$	−21.0578	−0.860399	−0.430199	−	0.902734i	$-0.641557\pi$
$599$	−21.0578	−0.860399	−0.430199	+	0.902734i	$0.641557\pi$
$600$	0	0
$601$	19.9819	0.815077	0.407538	−	0.913188i	$-0.366387\pi$
$601$	19.9819	0.815077	0.407538	+	0.913188i	$0.366387\pi$
$602$	−0.801709	−0.0326752
$603$	−23.9450	−0.975115
$604$	39.6978	1.61528
$605$	0	0
$606$	−104.409	−4.24131
$607$	34.3498	1.39421	0.697107	−	0.716967i	$-0.254470\pi$
$607$	34.3498	1.39421	0.697107	+	0.716967i	$0.254470\pi$
$608$	10.9694	0.444868
$609$	−74.2152	−3.00735
$610$	0	0
$611$	11.5221	0.466134
$612$	−25.8392	−1.04449
$613$	31.2031	1.26028	0.630141	−	0.776481i	$-0.282997\pi$
$613$	31.2031	1.26028	0.630141	+	0.776481i	$0.282997\pi$
$614$	61.0090	2.46212
$615$	0	0
$616$	−26.9859	−1.08729
$617$	41.3869	1.66618	0.833088	−	0.553141i	$-0.186571\pi$
$617$	41.3869	1.66618	0.833088	+	0.553141i	$0.186571\pi$
$618$	99.5481	4.00441
$619$	−17.4237	−0.700319	−0.350159	−	0.936690i	$-0.613873\pi$
$619$	−17.4237	−0.700319	−0.350159	+	0.936690i	$0.613873\pi$
$620$	0	0
$621$	−8.25117	−0.331108
$622$	−65.5162	−2.62696
$623$	−59.9474	−2.40174
$624$	18.8095	0.752983
$625$	0	0
$626$	−47.2018	−1.88657
$627$	−17.3149	−0.691492
$628$	−58.1811	−2.32168
$629$	−5.13197	−0.204625
$630$	0	0
$631$	−31.2579	−1.24436	−0.622178	−	0.782876i	$-0.713752\pi$
$631$	−31.2579	−1.24436	−0.622178	+	0.782876i	$0.713752\pi$
$632$	30.4431	1.21096
$633$	29.2142	1.16116
$634$	−54.9868	−2.18380
$635$	0	0
$636$	77.4626	3.07159
$637$	−45.2322	−1.79217
$638$	−21.7420	−0.860776
$639$	21.9462	0.868178
$640$	0	0
$641$	−39.5889	−1.56367	−0.781834	−	0.623487i	$-0.785715\pi$
$641$	−39.5889	−1.56367	−0.781834	+	0.623487i	$0.785715\pi$
$642$	−13.6483	−0.538655
$643$	11.5131	0.454034	0.227017	−	0.973891i	$-0.427103\pi$
$643$	11.5131	0.454034	0.227017	+	0.973891i	$0.427103\pi$
$644$	−94.0852	−3.70747
$645$	0	0
$646$	40.0151	1.57437
$647$	−15.3373	−0.602970	−0.301485	−	0.953471i	$-0.597482\pi$
$647$	−15.3373	−0.602970	−0.301485	+	0.953471i	$0.597482\pi$
$648$	45.4781	1.78655
$649$	−5.46986	−0.214711
$650$	0	0
$651$	−45.3595	−1.77778
$652$	−0.787674	−0.0308477
$653$	−0.216647	−0.00847803	−0.00423902	−	0.999991i	$-0.501349\pi$
$653$	−0.216647	−0.00847803	−0.00423902	+	0.999991i	$0.501349\pi$
$654$	−70.9525	−2.77446
$655$	0	0
$656$	−22.0992	−0.862829
$657$	−7.02280	−0.273985
$658$	44.4677	1.73353
$659$	37.3726	1.45583	0.727914	−	0.685668i	$-0.240490\pi$
$659$	37.3726	1.45583	0.727914	+	0.685668i	$0.240490\pi$
$660$	0	0
$661$	17.3069	0.673162	0.336581	−	0.941655i	$-0.390729\pi$
$661$	17.3069	0.673162	0.336581	+	0.941655i	$0.390729\pi$
$662$	21.9324	0.852429
$663$	−19.9945	−0.776523
$664$	−6.20364	−0.240748
$665$	0	0
$666$	−9.73235	−0.377121
$667$	−35.7701	−1.38502
$668$	48.8154	1.88872
$669$	26.9632	1.04246
$670$	0	0
$671$	−2.14801	−0.0829229
$672$	−21.1535	−0.816016
$673$	38.5861	1.48739	0.743693	−	0.668522i	$-0.233073\pi$
$673$	38.5861	1.48739	0.743693	+	0.668522i	$0.233073\pi$
$674$	−27.3611	−1.05391
$675$	0	0
$676$	−16.3463	−0.628704
$677$	16.1678	0.621380	0.310690	−	0.950511i	$-0.399440\pi$
$677$	16.1678	0.621380	0.310690	+	0.950511i	$0.399440\pi$
$678$	0.740918	0.0284548
$679$	−32.8024	−1.25884
$680$	0	0
$681$	−31.0188	−1.18864
$682$	−13.2885	−0.508843
$683$	−32.0829	−1.22762	−0.613809	−	0.789454i	$-0.710364\pi$
$683$	−32.0829	−1.22762	−0.613809	+	0.789454i	$0.710364\pi$
$684$	49.6593	1.89877
$685$	0	0
$686$	−94.9577	−3.62550
$687$	29.6218	1.13014
$688$	0.195100	0.00743810
$689$	26.1348	0.995655
$690$	0	0
$691$	24.0908	0.916457	0.458228	−	0.888835i	$-0.348484\pi$
$691$	24.0908	0.916457	0.458228	+	0.888835i	$0.348484\pi$
$692$	0.505103	0.0192011
$693$	14.5585	0.553031
$694$	58.5226	2.22149
$695$	0	0
$696$	67.4871	2.55809
$697$	23.4915	0.889804
$698$	−34.4674	−1.30461
$699$	47.8646	1.81040
$700$	0	0
$701$	−32.7835	−1.23822	−0.619109	−	0.785305i	$-0.712506\pi$
$701$	−32.7835	−1.23822	−0.619109	+	0.785305i	$0.712506\pi$
$702$	11.1303	0.420087
$703$	9.86293	0.371987
$704$	−13.5469	−0.510567
$705$	0	0
$706$	0.340499	0.0128148
$707$	−88.9650	−3.34587
$708$	35.9801	1.35221
$709$	−16.8810	−0.633980	−0.316990	−	0.948429i	$-0.602672\pi$
$709$	−16.8810	−0.633980	−0.316990	+	0.948429i	$0.602672\pi$
$710$	0	0
$711$	−16.4236	−0.615932
$712$	54.5128	2.04295
$713$	−21.8623	−0.818748
$714$	−77.1657	−2.88785
$715$	0	0
$716$	22.9186	0.856510
$717$	26.2414	0.980002
$718$	−10.8954	−0.406612
$719$	−8.98696	−0.335157	−0.167579	−	0.985859i	$-0.553595\pi$
$719$	−8.98696	−0.335157	−0.167579	+	0.985859i	$0.553595\pi$
$720$	0	0
$721$	84.8234	3.15899
$722$	−31.1965	−1.16101
$723$	2.30634	0.0857739
$724$	25.1149	0.933387
$725$	0	0
$726$	−51.2482	−1.90200
$727$	−4.15815	−0.154217	−0.0771087	−	0.997023i	$-0.524569\pi$
$727$	−4.15815	−0.154217	−0.0771087	+	0.997023i	$0.524569\pi$
$728$	59.8892	2.21964
$729$	−13.6711	−0.506338
$730$	0	0
$731$	−0.207391	−0.00767064
$732$	14.1293	0.522235
$733$	−41.7610	−1.54248	−0.771238	−	0.636547i	$-0.780362\pi$
$733$	−41.7610	−1.54248	−0.771238	+	0.636547i	$0.780362\pi$
$734$	−53.5800	−1.97767
$735$	0	0
$736$	−10.1955	−0.375812
$737$	−13.7092	−0.504983
$738$	44.5497	1.63990
$739$	11.3944	0.419150	0.209575	−	0.977793i	$-0.432792\pi$
$739$	11.3944	0.419150	0.209575	+	0.977793i	$0.432792\pi$
$740$	0	0
$741$	38.4267	1.41164
$742$	100.863	3.70279
$743$	−43.8096	−1.60722	−0.803610	−	0.595157i	$-0.797090\pi$
$743$	−43.8096	−1.60722	−0.803610	+	0.595157i	$0.797090\pi$
$744$	41.2473	1.51220
$745$	0	0
$746$	−4.50001	−0.164757
$747$	3.34677	0.122452
$748$	−14.7936	−0.540909
$749$	−11.6295	−0.424933
$750$	0	0
$751$	−0.441834	−0.0161228	−0.00806138	−	0.999968i	$-0.502566\pi$
$751$	−0.441834	−0.0161228	−0.00806138	+	0.999968i	$0.502566\pi$
$752$	−10.8214	−0.394617
$753$	64.0819	2.33528
$754$	48.2517	1.75722
$755$	0	0
$756$	28.1103	1.02236
$757$	7.34650	0.267013	0.133506	−	0.991048i	$-0.457376\pi$
$757$	7.34650	0.267013	0.133506	+	0.991048i	$0.457376\pi$
$758$	12.7026	0.461380
$759$	16.0934	0.584154
$760$	0	0
$761$	−31.8692	−1.15526	−0.577629	−	0.816299i	$-0.696022\pi$
$761$	−31.8692	−1.15526	−0.577629	+	0.816299i	$0.696022\pi$
$762$	80.9307	2.93181
$763$	−60.4575	−2.18871
$764$	−77.5635	−2.80615
$765$	0	0
$766$	50.3217	1.81820
$767$	12.1392	0.438319
$768$	67.5814	2.43864
$769$	2.41714	0.0871644	0.0435822	−	0.999050i	$-0.486123\pi$
$769$	2.41714	0.0871644	0.0435822	+	0.999050i	$0.486123\pi$
$770$	0	0
$771$	43.7092	1.57415
$772$	25.5616	0.919983
$773$	16.5954	0.596893	0.298447	−	0.954426i	$-0.403531\pi$
$773$	16.5954	0.596893	0.298447	+	0.954426i	$0.403531\pi$
$774$	−0.393300	−0.0141369
$775$	0	0
$776$	29.8287	1.07079
$777$	−19.0198	−0.682332
$778$	−57.7195	−2.06934
$779$	−45.1474	−1.61757
$780$	0	0
$781$	12.5648	0.449604
$782$	−37.1921	−1.32999
$783$	10.6872	0.381929
$784$	42.4816	1.51720
$785$	0	0
$786$	2.25785	0.0805347
$787$	−44.1195	−1.57269	−0.786344	−	0.617789i	$-0.788029\pi$
$787$	−44.1195	−1.57269	−0.786344	+	0.617789i	$0.788029\pi$
$788$	5.02687	0.179075
$789$	−11.0179	−0.392247
$790$	0	0
$791$	0.631325	0.0224473
$792$	−13.2386	−0.470415
$793$	4.76703	0.169282
$794$	−67.4508	−2.39374
$795$	0	0
$796$	79.2566	2.80918
$797$	38.4507	1.36199	0.680997	−	0.732286i	$-0.261547\pi$
$797$	38.4507	1.36199	0.680997	+	0.732286i	$0.261547\pi$
$798$	148.302	5.24982
$799$	11.5032	0.406953
$800$	0	0
$801$	−29.4088	−1.03911
$802$	−83.9385	−2.96397
$803$	−4.02074	−0.141889
$804$	90.1771	3.18030
$805$	0	0
$806$	29.4909	1.03877
$807$	−37.4033	−1.31666
$808$	80.8997	2.84604
$809$	9.56285	0.336212	0.168106	−	0.985769i	$-0.446235\pi$
$809$	9.56285	0.336212	0.168106	+	0.985769i	$0.446235\pi$
$810$	0	0
$811$	3.73686	0.131219	0.0656094	−	0.997845i	$-0.479101\pi$
$811$	3.73686	0.131219	0.0656094	+	0.997845i	$0.479101\pi$
$812$	121.862	4.27652
$813$	−38.6161	−1.35433
$814$	−5.57204	−0.195300
$815$	0	0
$816$	18.7786	0.657384
$817$	0.398577	0.0139444
$818$	5.97803	0.209017
$819$	−32.3093	−1.12898
$820$	0	0
$821$	38.6326	1.34829	0.674143	−	0.738601i	$-0.264513\pi$
$821$	38.6326	1.34829	0.674143	+	0.738601i	$0.264513\pi$
$822$	−18.5944	−0.648553
$823$	−35.4745	−1.23657	−0.618283	−	0.785956i	$-0.712171\pi$
$823$	−35.4745	−1.23657	−0.618283	+	0.785956i	$0.712171\pi$
$824$	−77.1336	−2.68707
$825$	0	0
$826$	46.8491	1.63009
$827$	21.2917	0.740386	0.370193	−	0.928955i	$-0.379292\pi$
$827$	21.2917	0.740386	0.370193	+	0.928955i	$0.379292\pi$
$828$	−46.1560	−1.60403
$829$	21.0520	0.731166	0.365583	−	0.930779i	$-0.380870\pi$
$829$	21.0520	0.731166	0.365583	+	0.930779i	$0.380870\pi$
$830$	0	0
$831$	49.6832	1.72349
$832$	30.0643	1.04229
$833$	−45.1580	−1.56463
$834$	−62.8753	−2.17720
$835$	0	0
$836$	28.4313	0.983318
$837$	6.53189	0.225775
$838$	−19.6631	−0.679249
$839$	−0.594892	−0.0205379	−0.0102690	−	0.999947i	$-0.503269\pi$
$839$	−0.594892	−0.0205379	−0.0102690	+	0.999947i	$0.503269\pi$
$840$	0	0
$841$	17.3305	0.597604
$842$	−74.3263	−2.56145
$843$	67.7969	2.33505
$844$	−47.9700	−1.65120
$845$	0	0
$846$	21.8148	0.750009
$847$	−43.6678	−1.50044
$848$	−24.5455	−0.842895
$849$	−27.9044	−0.957675
$850$	0	0
$851$	−9.16713	−0.314245
$852$	−82.6497	−2.83153
$853$	20.1002	0.688218	0.344109	−	0.938930i	$-0.388181\pi$
$853$	20.1002	0.688218	0.344109	+	0.938930i	$0.388181\pi$
$854$	18.3976	0.629552
$855$	0	0
$856$	10.5752	0.361453
$857$	0.377390	0.0128914	0.00644570	−	0.999979i	$-0.497948\pi$
$857$	0.377390	0.0128914	0.00644570	+	0.999979i	$0.497948\pi$
$858$	−21.7091	−0.741135
$859$	−6.22435	−0.212372	−0.106186	−	0.994346i	$-0.533864\pi$
$859$	−6.22435	−0.212372	−0.106186	+	0.994346i	$0.533864\pi$
$860$	0	0
$861$	87.0629	2.96709
$862$	79.2629	2.69971
$863$	14.6126	0.497418	0.248709	−	0.968578i	$-0.419994\pi$
$863$	14.6126	0.497418	0.248709	+	0.968578i	$0.419994\pi$
$864$	3.04617	0.103633
$865$	0	0
$866$	81.6423	2.77432
$867$	19.2461	0.653633
$868$	74.4808	2.52804
$869$	−9.40295	−0.318973
$870$	0	0
$871$	30.4244	1.03089
$872$	54.9766	1.86174
$873$	−16.0921	−0.544636
$874$	71.4781	2.41778
$875$	0	0
$876$	26.4479	0.893593
$877$	−19.1127	−0.645391	−0.322695	−	0.946503i	$-0.604589\pi$
$877$	−19.1127	−0.645391	−0.322695	+	0.946503i	$0.604589\pi$
$878$	−80.1994	−2.70660
$879$	−42.4892	−1.43313
$880$	0	0
$881$	0.0301670	0.00101635	0.000508176	−	1.00000i	$-0.499838\pi$
$881$	0.0301670	0.00101635	0.000508176		1.00000i	$0.499838\pi$
$882$	−85.6384	−2.88359
$883$	−42.9653	−1.44590	−0.722949	−	0.690902i	$-0.757214\pi$
$883$	−42.9653	−1.44590	−0.722949	+	0.690902i	$0.757214\pi$
$884$	32.8312	1.10423
$885$	0	0
$886$	−47.9136	−1.60969
$887$	−38.8144	−1.30326	−0.651631	−	0.758536i	$-0.725915\pi$
$887$	−38.8144	−1.30326	−0.651631	+	0.758536i	$0.725915\pi$
$888$	17.2955	0.580400
$889$	68.9598	2.31284
$890$	0	0
$891$	−14.0468	−0.470586
$892$	−44.2738	−1.48240
$893$	−22.1075	−0.739800
$894$	85.5420	2.86095
$895$	0	0
$896$	97.6845	3.26341
$897$	−35.7158	−1.19252
$898$	4.00114	0.133520
$899$	28.3167	0.944415
$900$	0	0
$901$	26.0919	0.869246
$902$	25.5059	0.849253
$903$	−0.768622	−0.0255781
$904$	−0.574091	−0.0190940
$905$	0	0
$906$	58.1592	1.93221
$907$	−32.0312	−1.06358	−0.531790	−	0.846876i	$-0.678480\pi$
$907$	−32.0312	−1.06358	−0.531790	+	0.846876i	$0.678480\pi$
$908$	50.9333	1.69028
$909$	−43.6442	−1.44759
$910$	0	0
$911$	48.1477	1.59520	0.797602	−	0.603184i	$-0.206101\pi$
$911$	48.1477	1.59520	0.797602	+	0.603184i	$0.206101\pi$
$912$	−36.0899	−1.19506
$913$	1.91612	0.0634142
$914$	69.3704	2.29457
$915$	0	0
$916$	−48.6392	−1.60709
$917$	1.92388	0.0635320
$918$	11.1121	0.366753
$919$	−49.8370	−1.64397	−0.821985	−	0.569509i	$-0.807134\pi$
$919$	−49.8370	−1.64397	−0.821985	+	0.569509i	$0.807134\pi$
$920$	0	0
$921$	58.4911	1.92735
$922$	−0.327797	−0.0107954
$923$	−27.8848	−0.917839
$924$	−54.8274	−1.80369
$925$	0	0
$926$	37.6801	1.23824
$927$	41.6124	1.36673
$928$	13.2056	0.433495
$929$	−37.3382	−1.22503	−0.612513	−	0.790461i	$-0.709841\pi$
$929$	−37.3382	−1.22503	−0.612513	+	0.790461i	$0.709841\pi$
$930$	0	0
$931$	86.7873	2.84434
$932$	−78.5941	−2.57444
$933$	−62.8122	−2.05638
$934$	−67.2472	−2.20040
$935$	0	0
$936$	29.3803	0.960324
$937$	−5.34704	−0.174680	−0.0873401	−	0.996179i	$-0.527837\pi$
$937$	−5.34704	−0.174680	−0.0873401	+	0.996179i	$0.527837\pi$
$938$	117.418	3.83384
$939$	−45.2538	−1.47680
$940$	0	0
$941$	−47.7673	−1.55717	−0.778585	−	0.627539i	$-0.784062\pi$
$941$	−47.7673	−1.55717	−0.778585	+	0.627539i	$0.784062\pi$
$942$	−85.2383	−2.77721
$943$	41.9623	1.36648
$944$	−11.4009	−0.371069
$945$	0	0
$946$	−0.225175	−0.00732107
$947$	−10.4114	−0.338324	−0.169162	−	0.985588i	$-0.554106\pi$
$947$	−10.4114	−0.338324	−0.169162	+	0.985588i	$0.554106\pi$
$948$	61.8514	2.00884
$949$	8.92315	0.289658
$950$	0	0
$951$	−52.7174	−1.70948
$952$	59.7908	1.93783
$953$	−57.7309	−1.87009	−0.935044	−	0.354531i	$-0.884641\pi$
$953$	−57.7309	−1.87009	−0.935044	+	0.354531i	$0.884641\pi$
$954$	49.4810	1.60201
$955$	0	0
$956$	−43.0886	−1.39358
$957$	−20.8447	−0.673814
$958$	0.411823	0.0133054
$959$	−15.8440	−0.511628
$960$	0	0
$961$	−13.6931	−0.441714
$962$	12.3659	0.398693
$963$	−5.70516	−0.183846
$964$	−3.78704	−0.121972
$965$	0	0
$966$	−137.839	−4.43491
$967$	−55.4561	−1.78335	−0.891673	−	0.452679i	$-0.850468\pi$
$967$	−55.4561	−1.78335	−0.891673	+	0.452679i	$0.850468\pi$
$968$	39.7090	1.27630
$969$	38.3636	1.23242
$970$	0	0
$971$	−36.9603	−1.18611	−0.593056	−	0.805161i	$-0.702079\pi$
$971$	−36.9603	−1.18611	−0.593056	+	0.805161i	$0.702079\pi$
$972$	74.5600	2.39151
$973$	−53.5751	−1.71754
$974$	53.2724	1.70696
$975$	0	0
$976$	−4.47713	−0.143310
$977$	35.8731	1.14768	0.573841	−	0.818967i	$-0.305453\pi$
$977$	35.8731	1.14768	0.573841	+	0.818967i	$0.305453\pi$
$978$	−1.15398	−0.0369003
$979$	−16.8374	−0.538124
$980$	0	0
$981$	−29.6591	−0.946941
$982$	37.0077	1.18096
$983$	0.0939877	0.00299774	0.00149887	−	0.999999i	$-0.499523\pi$
$983$	0.0939877	0.00299774	0.00149887	+	0.999999i	$0.499523\pi$
$984$	−79.1700	−2.52385
$985$	0	0
$986$	48.1724	1.53412
$987$	42.6325	1.35701
$988$	−63.0970	−2.00738
$989$	−0.370458	−0.0117799
$990$	0	0
$991$	−23.9749	−0.761588	−0.380794	−	0.924660i	$-0.624349\pi$
$991$	−23.9749	−0.761588	−0.380794	+	0.924660i	$0.624349\pi$
$992$	8.07111	0.256258
$993$	21.0273	0.667280
$994$	−107.617	−3.41340
$995$	0	0
$996$	−12.6040	−0.399372
$997$	−14.9154	−0.472376	−0.236188	−	0.971707i	$-0.575898\pi$
$997$	−14.9154	−0.472376	−0.236188	+	0.971707i	$0.575898\pi$
$998$	−49.3349	−1.56167
$999$	2.73890	0.0866551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.l.1.7	✓	40
5.4	even	2		6025.2.a.o.1.34	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.l.1.7	✓	40	1.1	even	1	trivial
6025.2.a.o.1.34	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)

\(f(q)\)	\(=\)	\(q-2.40563 q^{2} -2.30634 q^{3} +3.78704 q^{4} +5.54820 q^{6} +4.72754 q^{7} -4.29895 q^{8} +2.31922 q^{9} +O(q^{10})\)	\(q-2.40563 q^{2} -2.30634 q^{3} +3.78704 q^{4} +5.54820 q^{6} +4.72754 q^{7} -4.29895 q^{8} +2.31922 q^{9} +1.32782 q^{11} -8.73422 q^{12} -2.94680 q^{13} -11.3727 q^{14} +2.76760 q^{16} -2.94196 q^{17} -5.57918 q^{18} +5.65404 q^{19} -10.9033 q^{21} -3.19423 q^{22} -5.25516 q^{23} +9.91487 q^{24} +7.08890 q^{26} +1.57011 q^{27} +17.9034 q^{28} +6.80665 q^{29} +4.16015 q^{31} +1.94010 q^{32} -3.06240 q^{33} +7.07726 q^{34} +8.78299 q^{36} +1.74440 q^{37} -13.6015 q^{38} +6.79633 q^{39} -7.98498 q^{41} +26.2293 q^{42} +0.0704942 q^{43} +5.02850 q^{44} +12.6420 q^{46} -3.91004 q^{47} -6.38303 q^{48} +15.3496 q^{49} +6.78517 q^{51} -11.1596 q^{52} -8.86887 q^{53} -3.77710 q^{54} -20.3235 q^{56} -13.0402 q^{57} -16.3743 q^{58} -4.11944 q^{59} -1.61770 q^{61} -10.0078 q^{62} +10.9642 q^{63} -10.2024 q^{64} +7.36700 q^{66} -10.3246 q^{67} -11.1413 q^{68} +12.1202 q^{69} +9.46274 q^{71} -9.97023 q^{72} -3.02808 q^{73} -4.19639 q^{74} +21.4121 q^{76} +6.27731 q^{77} -16.3494 q^{78} -7.08151 q^{79} -10.5789 q^{81} +19.2089 q^{82} +1.44306 q^{83} -41.2914 q^{84} -0.169583 q^{86} -15.6985 q^{87} -5.70823 q^{88} -12.6805 q^{89} -13.9311 q^{91} -19.9015 q^{92} -9.59474 q^{93} +9.40610 q^{94} -4.47454 q^{96} -6.93859 q^{97} -36.9255 q^{98} +3.07950 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9	\( 40 q - 11 q^{2} - 8 q^{3} + 41 q^{4} + 3 q^{6} - 16 q^{7} - 33 q^{8} + 38 q^{9} + q^{11} - 14 q^{12} - 9 q^{13} - q^{14} + 43 q^{16} - 12 q^{17} - 42 q^{18} + 2 q^{21} - 5 q^{22} - 77 q^{23} - 2 q^{24} + 2 q^{26} - 38 q^{27} - 42 q^{28} + 2 q^{29} + q^{31} - 72 q^{32} - 20 q^{33} + 5 q^{34} + 32 q^{36} - 28 q^{37} - 23 q^{38} - 2 q^{39} - 2 q^{41} - 37 q^{42} - 31 q^{43} + 3 q^{44} + 14 q^{46} - 96 q^{47} - 13 q^{48} + 40 q^{49} - 10 q^{51} - 42 q^{52} - 54 q^{53} + 4 q^{54} - 15 q^{56} - 37 q^{57} - 27 q^{58} + q^{59} + 5 q^{61} - 39 q^{62} - 70 q^{63} + 65 q^{64} - 52 q^{66} - 34 q^{67} - 52 q^{68} + 21 q^{69} - 9 q^{71} - 70 q^{72} - 25 q^{73} + 22 q^{74} - 47 q^{76} - 54 q^{77} - 58 q^{78} + 13 q^{79} + 12 q^{81} + 5 q^{82} - 63 q^{83} + 95 q^{84} - 18 q^{86} - 47 q^{87} - 13 q^{88} + 19 q^{89} - 31 q^{91} - 137 q^{92} - 52 q^{93} + 120 q^{94} - 49 q^{96} - 36 q^{97} - 64 q^{98} + 16 q^{99}+O(q^{100}) \) 40 * q - 11 * q^2 - 8 * q^3 + 41 * q^4 + 3 * q^6 - 16 * q^7 - 33 * q^8 + 38 * q^9 + q^11 - 14 * q^12 - 9 * q^13 - q^14 + 43 * q^16 - 12 * q^17 - 42 * q^18 + 2 * q^21 - 5 * q^22 - 77 * q^23 - 2 * q^24 + 2 * q^26 - 38 * q^27 - 42 * q^28 + 2 * q^29 + q^31 - 72 * q^32 - 20 * q^33 + 5 * q^34 + 32 * q^36 - 28 * q^37 - 23 * q^38 - 2 * q^39 - 2 * q^41 - 37 * q^42 - 31 * q^43 + 3 * q^44 + 14 * q^46 - 96 * q^47 - 13 * q^48 + 40 * q^49 - 10 * q^51 - 42 * q^52 - 54 * q^53 + 4 * q^54 - 15 * q^56 - 37 * q^57 - 27 * q^58 + q^59 + 5 * q^61 - 39 * q^62 - 70 * q^63 + 65 * q^64 - 52 * q^66 - 34 * q^67 - 52 * q^68 + 21 * q^69 - 9 * q^71 - 70 * q^72 - 25 * q^73 + 22 * q^74 - 47 * q^76 - 54 * q^77 - 58 * q^78 + 13 * q^79 + 12 * q^81 + 5 * q^82 - 63 * q^83 + 95 * q^84 - 18 * q^86 - 47 * q^87 - 13 * q^88 + 19 * q^89 - 31 * q^91 - 137 * q^92 - 52 * q^93 + 120 * q^94 - 49 * q^96 - 36 * q^97 - 64 * q^98 + 16 * q^99

Introduction

L-functions

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