LMFDB - Embedded newform 6025.2.a.k.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q+1.07842 q^{2} +2.50766 q^{3} -0.837004 q^{4} +2.70432 q^{6} -4.80516 q^{7} -3.05949 q^{8} +3.28838 q^{9} +O(q^{10})$	\(q+1.07842 q^{2} +2.50766 q^{3} -0.837004 q^{4} +2.70432 q^{6} -4.80516 q^{7} -3.05949 q^{8} +3.28838 q^{9} -2.54972 q^{11} -2.09893 q^{12} -4.01681 q^{13} -5.18199 q^{14} -1.62542 q^{16} +4.78408 q^{17} +3.54627 q^{18} +3.43896 q^{19} -12.0497 q^{21} -2.74967 q^{22} +4.75312 q^{23} -7.67218 q^{24} -4.33182 q^{26} +0.723167 q^{27} +4.02193 q^{28} +3.27057 q^{29} +0.120151 q^{31} +4.36609 q^{32} -6.39383 q^{33} +5.15926 q^{34} -2.75239 q^{36} +5.16560 q^{37} +3.70865 q^{38} -10.0728 q^{39} +1.60720 q^{41} -12.9947 q^{42} -0.519723 q^{43} +2.13412 q^{44} +5.12587 q^{46} +13.0869 q^{47} -4.07600 q^{48} +16.0895 q^{49} +11.9969 q^{51} +3.36209 q^{52} +0.643264 q^{53} +0.779879 q^{54} +14.7013 q^{56} +8.62376 q^{57} +3.52706 q^{58} +11.5370 q^{59} -1.34252 q^{61} +0.129574 q^{62} -15.8012 q^{63} +7.95933 q^{64} -6.89525 q^{66} -5.79910 q^{67} -4.00429 q^{68} +11.9192 q^{69} +4.79983 q^{71} -10.0608 q^{72} -2.89581 q^{73} +5.57070 q^{74} -2.87842 q^{76} +12.2518 q^{77} -10.8628 q^{78} +7.95574 q^{79} -8.05169 q^{81} +1.73325 q^{82} -12.1315 q^{83} +10.0857 q^{84} -0.560481 q^{86} +8.20150 q^{87} +7.80083 q^{88} +3.69334 q^{89} +19.3014 q^{91} -3.97838 q^{92} +0.301298 q^{93} +14.1133 q^{94} +10.9487 q^{96} +5.96171 q^{97} +17.3513 q^{98} -8.38444 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	$ 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) $ 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.07842	0.762560	0.381280	−	0.924460i	$-0.375483\pi$
$2$	1.07842	0.762560	0.381280	+	0.924460i	$0.375483\pi$
$3$	2.50766	1.44780	0.723900	−	0.689904i	$-0.242347\pi$
$3$	2.50766	1.44780	0.723900	+	0.689904i	$0.242347\pi$
$4$	−0.837004	−0.418502
$5$	0	0
$5$	0	0
$6$	2.70432	1.10404
$7$	−4.80516	−1.81618	−0.908089	−	0.418777i	$-0.862459\pi$
$7$	−4.80516	−1.81618	−0.908089	+	0.418777i	$0.862459\pi$
$8$	−3.05949	−1.08169
$9$	3.28838	1.09613
$10$	0	0
$11$	−2.54972	−0.768768	−0.384384	−	0.923173i	$-0.625586\pi$
$11$	−2.54972	−0.768768	−0.384384	+	0.923173i	$0.625586\pi$
$12$	−2.09893	−0.605908
$13$	−4.01681	−1.11406	−0.557032	−	0.830491i	$-0.688060\pi$
$13$	−4.01681	−1.11406	−0.557032	+	0.830491i	$0.688060\pi$
$14$	−5.18199	−1.38494
$15$	0	0
$16$	−1.62542	−0.406354
$17$	4.78408	1.16031	0.580155	−	0.814506i	$-0.302992\pi$
$17$	4.78408	1.16031	0.580155	+	0.814506i	$0.302992\pi$
$18$	3.54627	0.835863
$19$	3.43896	0.788952	0.394476	−	0.918906i	$-0.370926\pi$
$19$	3.43896	0.788952	0.394476	+	0.918906i	$0.370926\pi$
$20$	0	0
$21$	−12.0497	−2.62946
$22$	−2.74967	−0.586232
$23$	4.75312	0.991093	0.495547	−	0.868581i	$-0.334968\pi$
$23$	4.75312	0.991093	0.495547	+	0.868581i	$0.334968\pi$
$24$	−7.67218	−1.56608
$25$	0	0
$26$	−4.33182	−0.849541
$27$	0.723167	0.139173
$28$	4.02193	0.760074
$29$	3.27057	0.607330	0.303665	−	0.952779i	$-0.401790\pi$
$29$	3.27057	0.607330	0.303665	+	0.952779i	$0.401790\pi$
$30$	0	0
$31$	0.120151	0.0215798	0.0107899	−	0.999942i	$-0.496565\pi$
$31$	0.120151	0.0215798	0.0107899	+	0.999942i	$0.496565\pi$
$32$	4.36609	0.771824
$33$	−6.39383	−1.11302
$34$	5.15926	0.884806
$35$	0	0
$36$	−2.75239	−0.458732
$37$	5.16560	0.849220	0.424610	−	0.905376i	$-0.360411\pi$
$37$	5.16560	0.849220	0.424610	+	0.905376i	$0.360411\pi$
$38$	3.70865	0.601623
$39$	−10.0728	−1.61294
$40$	0	0
$41$	1.60720	0.251003	0.125502	−	0.992093i	$-0.459946\pi$
$41$	1.60720	0.251003	0.125502	+	0.992093i	$0.459946\pi$
$42$	−12.9947	−2.00512
$43$	−0.519723	−0.0792570	−0.0396285	−	0.999214i	$-0.512617\pi$
$43$	−0.519723	−0.0792570	−0.0396285	+	0.999214i	$0.512617\pi$
$44$	2.13412	0.321731
$45$	0	0
$46$	5.12587	0.755768
$47$	13.0869	1.90893	0.954463	−	0.298328i	$-0.0964289\pi$
$47$	13.0869	1.90893	0.954463	+	0.298328i	$0.0964289\pi$
$48$	−4.07600	−0.588320
$49$	16.0895	2.29850
$50$	0	0
$51$	11.9969	1.67990
$52$	3.36209	0.466238
$53$	0.643264	0.0883591	0.0441795	−	0.999024i	$-0.485933\pi$
$53$	0.643264	0.0883591	0.0441795	+	0.999024i	$0.485933\pi$
$54$	0.779879	0.106128
$55$	0	0
$56$	14.7013	1.96455
$57$	8.62376	1.14224
$58$	3.52706	0.463126
$59$	11.5370	1.50199	0.750997	−	0.660305i	$-0.229573\pi$
$59$	11.5370	1.50199	0.750997	+	0.660305i	$0.229573\pi$
$60$	0	0
$61$	−1.34252	−0.171892	−0.0859458	−	0.996300i	$-0.527391\pi$
$61$	−1.34252	−0.171892	−0.0859458	+	0.996300i	$0.527391\pi$
$62$	0.129574	0.0164559
$63$	−15.8012	−1.99076
$64$	7.95933	0.994916
$65$	0	0
$66$	−6.89525	−0.848747
$67$	−5.79910	−0.708473	−0.354237	−	0.935156i	$-0.615259\pi$
$67$	−5.79910	−0.708473	−0.354237	+	0.935156i	$0.615259\pi$
$68$	−4.00429	−0.485592
$69$	11.9192	1.43491
$70$	0	0
$71$	4.79983	0.569635	0.284818	−	0.958582i	$-0.408067\pi$
$71$	4.79983	0.569635	0.284818	+	0.958582i	$0.408067\pi$
$72$	−10.0608	−1.18567
$73$	−2.89581	−0.338928	−0.169464	−	0.985536i	$-0.554204\pi$
$73$	−2.89581	−0.338928	−0.169464	+	0.985536i	$0.554204\pi$
$74$	5.57070	0.647581
$75$	0	0
$76$	−2.87842	−0.330178
$77$	12.2518	1.39622
$78$	−10.8628	−1.22997
$79$	7.95574	0.895091	0.447545	−	0.894261i	$-0.352298\pi$
$79$	7.95574	0.895091	0.447545	+	0.894261i	$0.352298\pi$
$80$	0	0
$81$	−8.05169	−0.894632
$82$	1.73325	0.191405
$83$	−12.1315	−1.33160	−0.665800	−	0.746130i	$-0.731910\pi$
$83$	−12.1315	−1.33160	−0.665800	+	0.746130i	$0.731910\pi$
$84$	10.0857	1.10044
$85$	0	0
$86$	−0.560481	−0.0604382
$87$	8.20150	0.879293
$88$	7.80083	0.831571
$89$	3.69334	0.391493	0.195747	−	0.980654i	$-0.437287\pi$
$89$	3.69334	0.391493	0.195747	+	0.980654i	$0.437287\pi$
$90$	0	0
$91$	19.3014	2.02334
$92$	−3.97838	−0.414775
$93$	0.301298	0.0312432
$94$	14.1133	1.45567
$95$	0	0
$96$	10.9487	1.11745
$97$	5.96171	0.605320	0.302660	−	0.953099i	$-0.402125\pi$
$97$	5.96171	0.605320	0.302660	+	0.953099i	$0.402125\pi$
$98$	17.3513	1.75275
$99$	−8.38444	−0.842668
$100$	0	0
$101$	−5.80431	−0.577550	−0.288775	−	0.957397i	$-0.593248\pi$
$101$	−5.80431	−0.577550	−0.288775	+	0.957397i	$0.593248\pi$
$102$	12.9377	1.28102
$103$	13.0785	1.28866	0.644330	−	0.764748i	$-0.277136\pi$
$103$	13.0785	1.28866	0.644330	+	0.764748i	$0.277136\pi$
$104$	12.2894	1.20508
$105$	0	0
$106$	0.693710	0.0673791
$107$	−17.3029	−1.67274	−0.836368	−	0.548168i	$-0.815326\pi$
$107$	−17.3029	−1.67274	−0.836368	+	0.548168i	$0.815326\pi$
$108$	−0.605293	−0.0582444
$109$	10.3623	0.992532	0.496266	−	0.868170i	$-0.334704\pi$
$109$	10.3623	0.992532	0.496266	+	0.868170i	$0.334704\pi$
$110$	0	0
$111$	12.9536	1.22950
$112$	7.81038	0.738011
$113$	−20.2434	−1.90434	−0.952170	−	0.305569i	$-0.901153\pi$
$113$	−20.2434	−1.90434	−0.952170	+	0.305569i	$0.901153\pi$
$114$	9.30006	0.871030
$115$	0	0
$116$	−2.73748	−0.254169
$117$	−13.2088	−1.22116
$118$	12.4418	1.14536
$119$	−22.9882	−2.10733
$120$	0	0
$121$	−4.49895	−0.408996
$122$	−1.44780	−0.131078
$123$	4.03033	0.363402
$124$	−0.100567	−0.00903117
$125$	0	0
$126$	−17.0404	−1.51808
$127$	−3.43011	−0.304373	−0.152187	−	0.988352i	$-0.548631\pi$
$127$	−3.43011	−0.304373	−0.152187	+	0.988352i	$0.548631\pi$
$128$	−0.148667	−0.0131404
$129$	−1.30329	−0.114748
$130$	0	0
$131$	3.21030	0.280485	0.140242	−	0.990117i	$-0.455212\pi$
$131$	3.21030	0.280485	0.140242	+	0.990117i	$0.455212\pi$
$132$	5.35166	0.465802
$133$	−16.5247	−1.43288
$134$	−6.25389	−0.540253
$135$	0	0
$136$	−14.6368	−1.25510
$137$	−3.20498	−0.273820	−0.136910	−	0.990584i	$-0.543717\pi$
$137$	−3.20498	−0.273820	−0.136910	+	0.990584i	$0.543717\pi$
$138$	12.8540	1.09420
$139$	−9.45423	−0.801897	−0.400948	−	0.916101i	$-0.631319\pi$
$139$	−9.45423	−0.801897	−0.400948	+	0.916101i	$0.631319\pi$
$140$	0	0
$141$	32.8177	2.76375
$142$	5.17625	0.434381
$143$	10.2417	0.856457
$144$	−5.34499	−0.445416
$145$	0	0
$146$	−3.12290	−0.258453
$147$	40.3471	3.32777
$148$	−4.32363	−0.355400
$149$	−4.97835	−0.407843	−0.203921	−	0.978987i	$-0.565369\pi$
$149$	−4.97835	−0.407843	−0.203921	+	0.978987i	$0.565369\pi$
$150$	0	0
$151$	−16.0560	−1.30662	−0.653311	−	0.757090i	$-0.726620\pi$
$151$	−16.0560	−1.30662	−0.653311	+	0.757090i	$0.726620\pi$
$152$	−10.5215	−0.853403
$153$	15.7319	1.27185
$154$	13.2126	1.06470
$155$	0	0
$156$	8.43099	0.675020
$157$	14.7944	1.18072	0.590361	−	0.807139i	$-0.298985\pi$
$157$	14.7944	1.18072	0.590361	+	0.807139i	$0.298985\pi$
$158$	8.57965	0.682560
$159$	1.61309	0.127926
$160$	0	0
$161$	−22.8395	−1.80000
$162$	−8.68312	−0.682211
$163$	−0.579678	−0.0454039	−0.0227019	−	0.999742i	$-0.507227\pi$
$163$	−0.579678	−0.0454039	−0.0227019	+	0.999742i	$0.507227\pi$
$164$	−1.34524	−0.105045
$165$	0	0
$166$	−13.0828	−1.01543
$167$	15.1849	1.17504	0.587520	−	0.809209i	$-0.300104\pi$
$167$	15.1849	1.17504	0.587520	+	0.809209i	$0.300104\pi$
$168$	36.8660	2.84427
$169$	3.13479	0.241138
$170$	0	0
$171$	11.3086	0.864791
$172$	0.435010	0.0331692
$173$	−5.03569	−0.382856	−0.191428	−	0.981507i	$-0.561312\pi$
$173$	−5.03569	−0.382856	−0.191428	+	0.981507i	$0.561312\pi$
$174$	8.84468	0.670514
$175$	0	0
$176$	4.14435	0.312392
$177$	28.9310	2.17459
$178$	3.98298	0.298537
$179$	16.7514	1.25205	0.626027	−	0.779801i	$-0.284680\pi$
$179$	16.7514	1.25205	0.626027	+	0.779801i	$0.284680\pi$
$180$	0	0
$181$	25.1948	1.87272	0.936358	−	0.351048i	$-0.114175\pi$
$181$	25.1948	1.87272	0.936358	+	0.351048i	$0.114175\pi$
$182$	20.8151	1.54292
$183$	−3.36658	−0.248865
$184$	−14.5421	−1.07206
$185$	0	0
$186$	0.324927	0.0238248
$187$	−12.1980	−0.892009
$188$	−10.9538	−0.798890
$189$	−3.47493	−0.252764
$190$	0	0
$191$	19.1075	1.38257	0.691285	−	0.722582i	$-0.257045\pi$
$191$	19.1075	1.38257	0.691285	+	0.722582i	$0.257045\pi$
$192$	19.9593	1.44044
$193$	9.17275	0.660269	0.330135	−	0.943934i	$-0.392906\pi$
$193$	9.17275	0.660269	0.330135	+	0.943934i	$0.392906\pi$
$194$	6.42924	0.461593
$195$	0	0
$196$	−13.4670	−0.961928
$197$	8.07783	0.575522	0.287761	−	0.957702i	$-0.407089\pi$
$197$	8.07783	0.575522	0.287761	+	0.957702i	$0.407089\pi$
$198$	−9.04197	−0.642585
$199$	18.7622	1.33001	0.665007	−	0.746837i	$-0.268429\pi$
$199$	18.7622	1.33001	0.665007	+	0.746837i	$0.268429\pi$
$200$	0	0
$201$	−14.5422	−1.02573
$202$	−6.25950	−0.440417
$203$	−15.7156	−1.10302
$204$	−10.0414	−0.703041
$205$	0	0
$206$	14.1041	0.982681
$207$	15.6301	1.08636
$208$	6.52899	0.452704
$209$	−8.76837	−0.606521
$210$	0	0
$211$	9.15176	0.630033	0.315017	−	0.949086i	$-0.397990\pi$
$211$	9.15176	0.630033	0.315017	+	0.949086i	$0.397990\pi$
$212$	−0.538414	−0.0369784
$213$	12.0364	0.824719
$214$	−18.6599	−1.27556
$215$	0	0
$216$	−2.21252	−0.150543
$217$	−0.577344	−0.0391927
$218$	11.1750	0.756866
$219$	−7.26171	−0.490701
$220$	0	0
$221$	−19.2168	−1.29266
$222$	13.9695	0.937569
$223$	−19.4803	−1.30450	−0.652248	−	0.758006i	$-0.726174\pi$
$223$	−19.4803	−1.30450	−0.652248	+	0.758006i	$0.726174\pi$
$224$	−20.9798	−1.40177
$225$	0	0
$226$	−21.8310	−1.45217
$227$	−15.1311	−1.00429	−0.502143	−	0.864785i	$-0.667455\pi$
$227$	−15.1311	−1.00429	−0.502143	+	0.864785i	$0.667455\pi$
$228$	−7.21812	−0.478032
$229$	4.13834	0.273469	0.136735	−	0.990608i	$-0.456339\pi$
$229$	4.13834	0.273469	0.136735	+	0.990608i	$0.456339\pi$
$230$	0	0
$231$	30.7233	2.02145
$232$	−10.0063	−0.656944
$233$	8.69368	0.569542	0.284771	−	0.958596i	$-0.408082\pi$
$233$	8.69368	0.569542	0.284771	+	0.958596i	$0.408082\pi$
$234$	−14.2447	−0.931205
$235$	0	0
$236$	−9.65655	−0.628588
$237$	19.9503	1.29591
$238$	−24.7911	−1.60697
$239$	10.1517	0.656662	0.328331	−	0.944563i	$-0.393514\pi$
$239$	10.1517	0.656662	0.328331	+	0.944563i	$0.393514\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−4.85177	−0.311884
$243$	−22.3604	−1.43442
$244$	1.12369	0.0719370
$245$	0	0
$246$	4.34640	0.277116
$247$	−13.8137	−0.878942
$248$	−0.367601	−0.0233427
$249$	−30.4216	−1.92789
$250$	0	0
$251$	20.4883	1.29321	0.646604	−	0.762826i	$-0.276189\pi$
$251$	20.4883	1.29321	0.646604	+	0.762826i	$0.276189\pi$
$252$	13.2257	0.833138
$253$	−12.1191	−0.761921
$254$	−3.69911	−0.232103
$255$	0	0
$256$	−16.0790	−1.00494
$257$	−29.7842	−1.85789	−0.928944	−	0.370221i	$-0.879282\pi$
$257$	−29.7842	−1.85789	−0.928944	+	0.370221i	$0.879282\pi$
$258$	−1.40550	−0.0875025
$259$	−24.8215	−1.54233
$260$	0	0
$261$	10.7549	0.665711
$262$	3.46206	0.213887
$263$	−6.41282	−0.395432	−0.197716	−	0.980259i	$-0.563352\pi$
$263$	−6.41282	−0.395432	−0.197716	+	0.980259i	$0.563352\pi$
$264$	19.5619	1.20395
$265$	0	0
$266$	−17.8207	−1.09265
$267$	9.26166	0.566804
$268$	4.85387	0.296497
$269$	−9.22471	−0.562440	−0.281220	−	0.959643i	$-0.590739\pi$
$269$	−9.22471	−0.562440	−0.281220	+	0.959643i	$0.590739\pi$
$270$	0	0
$271$	−5.92565	−0.359957	−0.179979	−	0.983670i	$-0.557603\pi$
$271$	−5.92565	−0.359957	−0.179979	+	0.983670i	$0.557603\pi$
$272$	−7.77612	−0.471497
$273$	48.4015	2.92939
$274$	−3.45632	−0.208804
$275$	0	0
$276$	−9.97644	−0.600511
$277$	−32.3453	−1.94344	−0.971720	−	0.236136i	$-0.924119\pi$
$277$	−32.3453	−1.94344	−0.971720	+	0.236136i	$0.924119\pi$
$278$	−10.1957	−0.611495
$279$	0.395102	0.0236542
$280$	0	0
$281$	32.1305	1.91675	0.958374	−	0.285517i	$-0.0921653\pi$
$281$	32.1305	1.91675	0.958374	+	0.285517i	$0.0921653\pi$
$282$	35.3913	2.10752
$283$	−18.4052	−1.09407	−0.547036	−	0.837109i	$-0.684244\pi$
$283$	−18.4052	−1.09407	−0.547036	+	0.837109i	$0.684244\pi$
$284$	−4.01748	−0.238394
$285$	0	0
$286$	11.0449	0.653100
$287$	−7.72286	−0.455866
$288$	14.3574	0.846017
$289$	5.88742	0.346319
$290$	0	0
$291$	14.9500	0.876383
$292$	2.42380	0.141842
$293$	−1.18085	−0.0689858	−0.0344929	−	0.999405i	$-0.510982\pi$
$293$	−1.18085	−0.0689858	−0.0344929	+	0.999405i	$0.510982\pi$
$294$	43.5113	2.53763
$295$	0	0
$296$	−15.8041	−0.918595
$297$	−1.84387	−0.106992
$298$	−5.36877	−0.311005
$299$	−19.0924	−1.10414
$300$	0	0
$301$	2.49735	0.143945
$302$	−17.3152	−0.996377
$303$	−14.5553	−0.836177
$304$	−5.58974	−0.320594
$305$	0	0
$306$	16.9656	0.969860
$307$	5.51224	0.314600	0.157300	−	0.987551i	$-0.449721\pi$
$307$	5.51224	0.314600	0.157300	+	0.987551i	$0.449721\pi$
$308$	−10.2548	−0.584321
$309$	32.7964	1.86572
$310$	0	0
$311$	17.1089	0.970154	0.485077	−	0.874471i	$-0.338791\pi$
$311$	17.1089	0.970154	0.485077	+	0.874471i	$0.338791\pi$
$312$	30.8177	1.74471
$313$	14.3901	0.813374	0.406687	−	0.913567i	$-0.366684\pi$
$313$	14.3901	0.813374	0.406687	+	0.913567i	$0.366684\pi$
$314$	15.9546	0.900372
$315$	0	0
$316$	−6.65899	−0.374597
$317$	32.6096	1.83154	0.915770	−	0.401703i	$-0.131581\pi$
$317$	32.6096	1.83154	0.915770	+	0.401703i	$0.131581\pi$
$318$	1.73959	0.0975515
$319$	−8.33902	−0.466896
$320$	0	0
$321$	−43.3899	−2.42179
$322$	−24.6306	−1.37261
$323$	16.4523	0.915428
$324$	6.73930	0.374405
$325$	0	0
$326$	−0.625138	−0.0346232
$327$	25.9853	1.43699
$328$	−4.91722	−0.271508
$329$	−62.8848	−3.46695
$330$	0	0
$331$	33.3531	1.83325	0.916626	−	0.399747i	$-0.130902\pi$
$331$	33.3531	1.83325	0.916626	+	0.399747i	$0.130902\pi$
$332$	10.1541	0.557278
$333$	16.9865	0.930853
$334$	16.3757	0.896039
$335$	0	0
$336$	19.5858	1.06849
$337$	24.0184	1.30836	0.654182	−	0.756337i	$-0.273013\pi$
$337$	24.0184	1.30836	0.654182	+	0.756337i	$0.273013\pi$
$338$	3.38063	0.183882
$339$	−50.7637	−2.75710
$340$	0	0
$341$	−0.306351	−0.0165898
$342$	12.1955	0.659455
$343$	−43.6765	−2.35831
$344$	1.59009	0.0857317
$345$	0	0
$346$	−5.43060	−0.291951
$347$	11.7752	0.632128	0.316064	−	0.948738i	$-0.397639\pi$
$347$	11.7752	0.632128	0.316064	+	0.948738i	$0.397639\pi$
$348$	−6.86469	−0.367986
$349$	33.0569	1.76949	0.884747	−	0.466071i	$-0.154331\pi$
$349$	33.0569	1.76949	0.884747	+	0.466071i	$0.154331\pi$
$350$	0	0
$351$	−2.90483	−0.155048
$352$	−11.1323	−0.593353
$353$	11.3030	0.601599	0.300800	−	0.953687i	$-0.402746\pi$
$353$	11.3030	0.601599	0.300800	+	0.953687i	$0.402746\pi$
$354$	31.1999	1.65826
$355$	0	0
$356$	−3.09134	−0.163841
$357$	−57.6468	−3.05099
$358$	18.0650	0.954767
$359$	−6.34738	−0.335002	−0.167501	−	0.985872i	$-0.553570\pi$
$359$	−6.34738	−0.335002	−0.167501	+	0.985872i	$0.553570\pi$
$360$	0	0
$361$	−7.17355	−0.377556
$362$	27.1707	1.42806
$363$	−11.2819	−0.592144
$364$	−16.1554	−0.846771
$365$	0	0
$366$	−3.63060	−0.189774
$367$	−28.9485	−1.51110	−0.755550	−	0.655091i	$-0.772630\pi$
$367$	−28.9485	−1.51110	−0.755550	+	0.655091i	$0.772630\pi$
$368$	−7.72579	−0.402735
$369$	5.28510	0.275131
$370$	0	0
$371$	−3.09098	−0.160476
$372$	−0.252188	−0.0130753
$373$	21.9656	1.13734	0.568668	−	0.822567i	$-0.307459\pi$
$373$	21.9656	1.13734	0.568668	+	0.822567i	$0.307459\pi$
$374$	−13.1546	−0.680211
$375$	0	0
$376$	−40.0394	−2.06487
$377$	−13.1373	−0.676604
$378$	−3.74744	−0.192748
$379$	15.6753	0.805185	0.402592	−	0.915379i	$-0.368109\pi$
$379$	15.6753	0.805185	0.402592	+	0.915379i	$0.368109\pi$
$380$	0	0
$381$	−8.60157	−0.440672
$382$	20.6060	1.05429
$383$	18.2196	0.930978	0.465489	−	0.885054i	$-0.345879\pi$
$383$	18.2196	0.930978	0.465489	+	0.885054i	$0.345879\pi$
$384$	−0.372806	−0.0190247
$385$	0	0
$386$	9.89210	0.503495
$387$	−1.70905	−0.0868758
$388$	−4.98998	−0.253328
$389$	−26.4788	−1.34253	−0.671264	−	0.741218i	$-0.734248\pi$
$389$	−26.4788	−1.34253	−0.671264	+	0.741218i	$0.734248\pi$
$390$	0	0
$391$	22.7393	1.14998
$392$	−49.2257	−2.48627
$393$	8.05035	0.406086
$394$	8.71132	0.438870
$395$	0	0
$396$	7.01781	0.352658
$397$	−1.83493	−0.0920926	−0.0460463	−	0.998939i	$-0.514662\pi$
$397$	−1.83493	−0.0920926	−0.0460463	+	0.998939i	$0.514662\pi$
$398$	20.2335	1.01422
$399$	−41.4385	−2.07452
$400$	0	0
$401$	−15.5981	−0.778934	−0.389467	−	0.921040i	$-0.627341\pi$
$401$	−15.5981	−0.778934	−0.389467	+	0.921040i	$0.627341\pi$
$402$	−15.6826	−0.782179
$403$	−0.482624	−0.0240412
$404$	4.85823	0.241706
$405$	0	0
$406$	−16.9481	−0.841118
$407$	−13.1708	−0.652853
$408$	−36.7043	−1.81713
$409$	37.8820	1.87315	0.936573	−	0.350472i	$-0.113979\pi$
$409$	37.8820	1.87315	0.936573	+	0.350472i	$0.113979\pi$
$410$	0	0
$411$	−8.03700	−0.396436
$412$	−10.9467	−0.539307
$413$	−55.4373	−2.72789
$414$	16.8558	0.828418
$415$	0	0
$416$	−17.5378	−0.859861
$417$	−23.7080	−1.16099
$418$	−9.45601	−0.462509
$419$	−7.70410	−0.376370	−0.188185	−	0.982134i	$-0.560260\pi$
$419$	−7.70410	−0.376370	−0.188185	+	0.982134i	$0.560260\pi$
$420$	0	0
$421$	−26.4377	−1.28850	−0.644248	−	0.764817i	$-0.722830\pi$
$421$	−26.4377	−1.28850	−0.644248	+	0.764817i	$0.722830\pi$
$422$	9.86947	0.480438
$423$	43.0349	2.09243
$424$	−1.96806	−0.0955774
$425$	0	0
$426$	12.9803	0.628898
$427$	6.45100	0.312186
$428$	14.4826	0.700044
$429$	25.6828	1.23998
$430$	0	0
$431$	4.87980	0.235052	0.117526	−	0.993070i	$-0.462504\pi$
$431$	4.87980	0.235052	0.117526	+	0.993070i	$0.462504\pi$
$432$	−1.17545	−0.0565537
$433$	26.3543	1.26651	0.633253	−	0.773945i	$-0.281719\pi$
$433$	26.3543	1.26651	0.633253	+	0.773945i	$0.281719\pi$
$434$	−0.622621	−0.0298868
$435$	0	0
$436$	−8.67332	−0.415377
$437$	16.3458	0.781925
$438$	−7.83119	−0.374189
$439$	27.5279	1.31384	0.656918	−	0.753962i	$-0.271860\pi$
$439$	27.5279	1.31384	0.656918	+	0.753962i	$0.271860\pi$
$440$	0	0
$441$	52.9085	2.51945
$442$	−20.7238	−0.985730
$443$	−20.4664	−0.972390	−0.486195	−	0.873850i	$-0.661616\pi$
$443$	−20.4664	−0.972390	−0.486195	+	0.873850i	$0.661616\pi$
$444$	−10.8422	−0.514549
$445$	0	0
$446$	−21.0080	−0.994757
$447$	−12.4840	−0.590475
$448$	−38.2458	−1.80694
$449$	−21.9440	−1.03560	−0.517800	−	0.855502i	$-0.673249\pi$
$449$	−21.9440	−1.03560	−0.517800	+	0.855502i	$0.673249\pi$
$450$	0	0
$451$	−4.09791	−0.192963
$452$	16.9438	0.796970
$453$	−40.2631	−1.89173
$454$	−16.3177	−0.765828
$455$	0	0
$456$	−26.3843	−1.23556
$457$	−3.85294	−0.180233	−0.0901166	−	0.995931i	$-0.528724\pi$
$457$	−3.85294	−0.180233	−0.0901166	+	0.995931i	$0.528724\pi$
$458$	4.46288	0.208537
$459$	3.45969	0.161484
$460$	0	0
$461$	−0.951406	−0.0443114	−0.0221557	−	0.999755i	$-0.507053\pi$
$461$	−0.951406	−0.0443114	−0.0221557	+	0.999755i	$0.507053\pi$
$462$	33.1328	1.54148
$463$	35.0465	1.62875	0.814375	−	0.580339i	$-0.197080\pi$
$463$	35.0465	1.62875	0.814375	+	0.580339i	$0.197080\pi$
$464$	−5.31604	−0.246791
$465$	0	0
$466$	9.37546	0.434310
$467$	−26.0706	−1.20640	−0.603202	−	0.797588i	$-0.706109\pi$
$467$	−26.0706	−1.20640	−0.603202	+	0.797588i	$0.706109\pi$
$468$	11.0558	0.511056
$469$	27.8656	1.28671
$470$	0	0
$471$	37.0994	1.70945
$472$	−35.2975	−1.62470
$473$	1.32515	0.0609302
$474$	21.5149	0.988212
$475$	0	0
$476$	19.2413	0.881921
$477$	2.11530	0.0968528
$478$	10.9479	0.500744
$479$	28.6631	1.30965	0.654825	−	0.755781i	$-0.272742\pi$
$479$	28.6631	1.30965	0.654825	+	0.755781i	$0.272742\pi$
$480$	0	0
$481$	−20.7493	−0.946085
$482$	1.07842	0.0491208
$483$	−57.2737	−2.60604
$484$	3.76564	0.171166
$485$	0	0
$486$	−24.1140	−1.09383
$487$	20.0009	0.906328	0.453164	−	0.891427i	$-0.350295\pi$
$487$	20.0009	0.906328	0.453164	+	0.891427i	$0.350295\pi$
$488$	4.10742	0.185934
$489$	−1.45364	−0.0657358
$490$	0	0
$491$	−21.0792	−0.951290	−0.475645	−	0.879637i	$-0.657785\pi$
$491$	−21.0792	−0.951290	−0.475645	+	0.879637i	$0.657785\pi$
$492$	−3.37340	−0.152085
$493$	15.6467	0.704691
$494$	−14.8970	−0.670246
$495$	0	0
$496$	−0.195295	−0.00876902
$497$	−23.0639	−1.03456
$498$	−32.8074	−1.47013
$499$	15.0119	0.672027	0.336013	−	0.941857i	$-0.390921\pi$
$499$	15.0119	0.672027	0.336013	+	0.941857i	$0.390921\pi$
$500$	0	0
$501$	38.0786	1.70122
$502$	22.0950	0.986149
$503$	−14.9139	−0.664977	−0.332488	−	0.943107i	$-0.607888\pi$
$503$	−14.9139	−0.664977	−0.332488	+	0.943107i	$0.607888\pi$
$504$	48.3436	2.15339
$505$	0	0
$506$	−13.0695	−0.581010
$507$	7.86101	0.349120
$508$	2.87102	0.127381
$509$	9.23904	0.409513	0.204757	−	0.978813i	$-0.434360\pi$
$509$	9.23904	0.409513	0.204757	+	0.978813i	$0.434360\pi$
$510$	0	0
$511$	13.9148	0.615554
$512$	−17.0426	−0.753184
$513$	2.48694	0.109801
$514$	−32.1200	−1.41675
$515$	0	0
$516$	1.09086	0.0480224
$517$	−33.3680	−1.46752
$518$	−26.7681	−1.17612
$519$	−12.6278	−0.554300
$520$	0	0
$521$	−21.5818	−0.945516	−0.472758	−	0.881192i	$-0.656741\pi$
$521$	−21.5818	−0.945516	−0.472758	+	0.881192i	$0.656741\pi$
$522$	11.5983	0.507645
$523$	−2.96336	−0.129579	−0.0647893	−	0.997899i	$-0.520638\pi$
$523$	−2.96336	−0.129579	−0.0647893	+	0.997899i	$0.520638\pi$
$524$	−2.68703	−0.117384
$525$	0	0
$526$	−6.91574	−0.301540
$527$	0.574812	0.0250392
$528$	10.3926	0.452281
$529$	−0.407881	−0.0177340
$530$	0	0
$531$	37.9382	1.64638
$532$	13.8313	0.599662
$533$	−6.45584	−0.279633
$534$	9.98799	0.432222
$535$	0	0
$536$	17.7423	0.766351
$537$	42.0068	1.81273
$538$	−9.94813	−0.428895
$539$	−41.0237	−1.76702
$540$	0	0
$541$	1.70205	0.0731769	0.0365884	−	0.999330i	$-0.488351\pi$
$541$	1.70205	0.0731769	0.0365884	+	0.999330i	$0.488351\pi$
$542$	−6.39035	−0.274489
$543$	63.1801	2.71132
$544$	20.8877	0.895555
$545$	0	0
$546$	52.1973	2.23384
$547$	17.2566	0.737841	0.368920	−	0.929461i	$-0.379727\pi$
$547$	17.2566	0.737841	0.368920	+	0.929461i	$0.379727\pi$
$548$	2.68258	0.114594
$549$	−4.41471	−0.188415
$550$	0	0
$551$	11.2474	0.479154
$552$	−36.4667	−1.55213
$553$	−38.2286	−1.62564
$554$	−34.8819	−1.48199
$555$	0	0
$556$	7.91322	0.335595
$557$	13.7956	0.584540	0.292270	−	0.956336i	$-0.405589\pi$
$557$	13.7956	0.584540	0.292270	+	0.956336i	$0.405589\pi$
$558$	0.426088	0.0180377
$559$	2.08763	0.0882973
$560$	0	0
$561$	−30.5886	−1.29145
$562$	34.6503	1.46164
$563$	−37.7006	−1.58889	−0.794445	−	0.607337i	$-0.792238\pi$
$563$	−37.7006	−1.58889	−0.794445	+	0.607337i	$0.792238\pi$
$564$	−27.4685	−1.15663
$565$	0	0
$566$	−19.8485	−0.834296
$567$	38.6896	1.62481
$568$	−14.6850	−0.616171
$569$	40.6430	1.70384	0.851922	−	0.523669i	$-0.175437\pi$
$569$	40.6430	1.70384	0.851922	+	0.523669i	$0.175437\pi$
$570$	0	0
$571$	−4.71960	−0.197509	−0.0987545	−	0.995112i	$-0.531486\pi$
$571$	−4.71960	−0.197509	−0.0987545	+	0.995112i	$0.531486\pi$
$572$	−8.57237	−0.358429
$573$	47.9152	2.00169
$574$	−8.32851	−0.347625
$575$	0	0
$576$	26.1733	1.09055
$577$	−13.4787	−0.561126	−0.280563	−	0.959836i	$-0.590521\pi$
$577$	−13.4787	−0.561126	−0.280563	+	0.959836i	$0.590521\pi$
$578$	6.34913	0.264089
$579$	23.0022	0.955938
$580$	0	0
$581$	58.2936	2.41842
$582$	16.1224	0.668295
$583$	−1.64014	−0.0679276
$584$	8.85969	0.366616
$585$	0	0
$586$	−1.27345	−0.0526058
$587$	43.1950	1.78285	0.891424	−	0.453171i	$-0.149707\pi$
$587$	43.1950	1.78285	0.891424	+	0.453171i	$0.149707\pi$
$588$	−33.7707	−1.39268
$589$	0.413195	0.0170254
$590$	0	0
$591$	20.2565	0.833241
$592$	−8.39625	−0.345084
$593$	−9.80411	−0.402607	−0.201303	−	0.979529i	$-0.564518\pi$
$593$	−9.80411	−0.402607	−0.201303	+	0.979529i	$0.564518\pi$
$594$	−1.98847	−0.0815879
$595$	0	0
$596$	4.16690	0.170683
$597$	47.0492	1.92560
$598$	−20.5897	−0.841974
$599$	17.1419	0.700401	0.350200	−	0.936675i	$-0.386113\pi$
$599$	17.1419	0.700401	0.350200	+	0.936675i	$0.386113\pi$
$600$	0	0
$601$	−6.97026	−0.284323	−0.142161	−	0.989843i	$-0.545405\pi$
$601$	−6.97026	−0.284323	−0.142161	+	0.989843i	$0.545405\pi$
$602$	2.69320	0.109767
$603$	−19.0697	−0.776577
$604$	13.4390	0.546824
$605$	0	0
$606$	−15.6967	−0.637636
$607$	36.3521	1.47548	0.737742	−	0.675082i	$-0.235892\pi$
$607$	36.3521	1.47548	0.737742	+	0.675082i	$0.235892\pi$
$608$	15.0148	0.608931
$609$	−39.4095	−1.59695
$610$	0	0
$611$	−52.5678	−2.12667
$612$	−13.1677	−0.532271
$613$	−34.3645	−1.38797	−0.693984	−	0.719990i	$-0.744146\pi$
$613$	−34.3645	−1.38797	−0.693984	+	0.719990i	$0.744146\pi$
$614$	5.94452	0.239901
$615$	0	0
$616$	−37.4842	−1.51028
$617$	32.1838	1.29567	0.647835	−	0.761781i	$-0.275675\pi$
$617$	32.1838	1.29567	0.647835	+	0.761781i	$0.275675\pi$
$618$	35.3684	1.42273
$619$	−2.80668	−0.112810	−0.0564051	−	0.998408i	$-0.517964\pi$
$619$	−2.80668	−0.112810	−0.0564051	+	0.998408i	$0.517964\pi$
$620$	0	0
$621$	3.43730	0.137934
$622$	18.4506	0.739801
$623$	−17.7471	−0.711022
$624$	16.3725	0.655426
$625$	0	0
$626$	15.5186	0.620247
$627$	−21.9881	−0.878121
$628$	−12.3830	−0.494135
$629$	24.7127	0.985358
$630$	0	0
$631$	38.7699	1.54340	0.771702	−	0.635984i	$-0.219406\pi$
$631$	38.7699	1.54340	0.771702	+	0.635984i	$0.219406\pi$
$632$	−24.3405	−0.968213
$633$	22.9496	0.912163
$634$	35.1670	1.39666
$635$	0	0
$636$	−1.35016	−0.0535374
$637$	−64.6286	−2.56068
$638$	−8.99299	−0.356036
$639$	15.7837	0.624393
$640$	0	0
$641$	−49.3843	−1.95056	−0.975282	−	0.220965i	$-0.929079\pi$
$641$	−49.3843	−1.95056	−0.975282	+	0.220965i	$0.929079\pi$
$642$	−46.7927	−1.84676
$643$	29.3153	1.15608	0.578042	−	0.816007i	$-0.303817\pi$
$643$	29.3153	1.15608	0.578042	+	0.816007i	$0.303817\pi$
$644$	19.1167	0.753304
$645$	0	0
$646$	17.7425	0.698069
$647$	−35.3376	−1.38926	−0.694632	−	0.719366i	$-0.744433\pi$
$647$	−35.3376	−1.38926	−0.694632	+	0.719366i	$0.744433\pi$
$648$	24.6341	0.967717
$649$	−29.4162	−1.15469
$650$	0	0
$651$	−1.44779	−0.0567432
$652$	0.485193	0.0190016
$653$	21.5875	0.844786	0.422393	−	0.906413i	$-0.361190\pi$
$653$	21.5875	0.844786	0.422393	+	0.906413i	$0.361190\pi$
$654$	28.0231	1.09579
$655$	0	0
$656$	−2.61237	−0.101996
$657$	−9.52252	−0.371509
$658$	−67.8164	−2.64376
$659$	−12.3378	−0.480612	−0.240306	−	0.970697i	$-0.577248\pi$
$659$	−12.3378	−0.480612	−0.240306	+	0.970697i	$0.577248\pi$
$660$	0	0
$661$	−18.7075	−0.727636	−0.363818	−	0.931470i	$-0.618527\pi$
$661$	−18.7075	−0.727636	−0.363818	+	0.931470i	$0.618527\pi$
$662$	35.9687	1.39796
$663$	−48.1892	−1.87151
$664$	37.1161	1.44038
$665$	0	0
$666$	18.3186	0.709832
$667$	15.5454	0.601921
$668$	−12.7098	−0.491757
$669$	−48.8500	−1.88865
$670$	0	0
$671$	3.42303	0.132145
$672$	−52.6102	−2.02948
$673$	−14.3252	−0.552195	−0.276098	−	0.961130i	$-0.589041\pi$
$673$	−14.3252	−0.552195	−0.276098	+	0.961130i	$0.589041\pi$
$674$	25.9019	0.997706
$675$	0	0
$676$	−2.62384	−0.100917
$677$	−25.1039	−0.964822	−0.482411	−	0.875945i	$-0.660239\pi$
$677$	−25.1039	−0.964822	−0.482411	+	0.875945i	$0.660239\pi$
$678$	−54.7447	−2.10246
$679$	−28.6469	−1.09937
$680$	0	0
$681$	−37.9437	−1.45401
$682$	−0.330376	−0.0126507
$683$	−19.8239	−0.758542	−0.379271	−	0.925286i	$-0.623825\pi$
$683$	−19.8239	−0.758542	−0.379271	+	0.925286i	$0.623825\pi$
$684$	−9.46536	−0.361917
$685$	0	0
$686$	−47.1018	−1.79835
$687$	10.3776	0.395929
$688$	0.844766	0.0322064
$689$	−2.58387	−0.0984376
$690$	0	0
$691$	−14.7969	−0.562901	−0.281450	−	0.959576i	$-0.590815\pi$
$691$	−14.7969	−0.562901	−0.281450	+	0.959576i	$0.590815\pi$
$692$	4.21489	0.160226
$693$	40.2885	1.53043
$694$	12.6987	0.482036
$695$	0	0
$696$	−25.0924	−0.951125
$697$	7.68899	0.291241
$698$	35.6493	1.34935
$699$	21.8008	0.824583
$700$	0	0
$701$	−47.4777	−1.79321	−0.896603	−	0.442835i	$-0.853973\pi$
$701$	−47.4777	−1.79321	−0.896603	+	0.442835i	$0.853973\pi$
$702$	−3.13263	−0.118234
$703$	17.7643	0.669993
$704$	−20.2940	−0.764860
$705$	0	0
$706$	12.1894	0.458756
$707$	27.8906	1.04893
$708$	−24.2154	−0.910070
$709$	18.1055	0.679968	0.339984	−	0.940431i	$-0.389578\pi$
$709$	18.1055	0.679968	0.339984	+	0.940431i	$0.389578\pi$
$710$	0	0
$711$	26.1615	0.981133
$712$	−11.2997	−0.423476
$713$	0.571092	0.0213876
$714$	−62.1676	−2.32657
$715$	0	0
$716$	−14.0209	−0.523987
$717$	25.4572	0.950715
$718$	−6.84516	−0.255459
$719$	17.2972	0.645075	0.322538	−	0.946557i	$-0.395464\pi$
$719$	17.2972	0.645075	0.322538	+	0.946557i	$0.395464\pi$
$720$	0	0
$721$	−62.8441	−2.34044
$722$	−7.73613	−0.287909
$723$	2.50766	0.0932611
$724$	−21.0882	−0.783735
$725$	0	0
$726$	−12.1666	−0.451546
$727$	6.31820	0.234329	0.117165	−	0.993113i	$-0.462620\pi$
$727$	6.31820	0.234329	0.117165	+	0.993113i	$0.462620\pi$
$728$	−59.0525	−2.18863
$729$	−31.9174	−1.18213
$730$	0	0
$731$	−2.48640	−0.0919627
$732$	2.81784	0.104150
$733$	−9.07444	−0.335172	−0.167586	−	0.985857i	$-0.553597\pi$
$733$	−9.07444	−0.335172	−0.167586	+	0.985857i	$0.553597\pi$
$734$	−31.2188	−1.15231
$735$	0	0
$736$	20.7526	0.764949
$737$	14.7861	0.544651
$738$	5.69957	0.209804
$739$	37.4739	1.37850	0.689249	−	0.724525i	$-0.257941\pi$
$739$	37.4739	1.37850	0.689249	+	0.724525i	$0.257941\pi$
$740$	0	0
$741$	−34.6400	−1.27253
$742$	−3.33339	−0.122372
$743$	39.6922	1.45616	0.728082	−	0.685490i	$-0.240412\pi$
$743$	39.6922	1.45616	0.728082	+	0.685490i	$0.240412\pi$
$744$	−0.921820	−0.0337955
$745$	0	0
$746$	23.6882	0.867287
$747$	−39.8929	−1.45960
$748$	10.2098	0.373308
$749$	83.1432	3.03799
$750$	0	0
$751$	−39.0013	−1.42318	−0.711589	−	0.702596i	$-0.752024\pi$
$751$	−39.0013	−1.42318	−0.711589	+	0.702596i	$0.752024\pi$
$752$	−21.2717	−0.775700
$753$	51.3777	1.87231
$754$	−14.1675	−0.515951
$755$	0	0
$756$	2.90853	0.105782
$757$	23.4161	0.851074	0.425537	−	0.904941i	$-0.360085\pi$
$757$	23.4161	0.851074	0.425537	+	0.904941i	$0.360085\pi$
$758$	16.9046	0.614002
$759$	−30.3906	−1.10311
$760$	0	0
$761$	53.7690	1.94912	0.974562	−	0.224119i	$-0.0719504\pi$
$761$	53.7690	1.94912	0.974562	+	0.224119i	$0.0719504\pi$
$762$	−9.27613	−0.336039
$763$	−49.7927	−1.80262
$764$	−15.9931	−0.578609
$765$	0	0
$766$	19.6484	0.709927
$767$	−46.3421	−1.67332
$768$	−40.3207	−1.45495
$769$	7.34773	0.264966	0.132483	−	0.991185i	$-0.457705\pi$
$769$	7.34773	0.264966	0.132483	+	0.991185i	$0.457705\pi$
$770$	0	0
$771$	−74.6888	−2.68985
$772$	−7.67763	−0.276324
$773$	1.82047	0.0654777	0.0327388	−	0.999464i	$-0.489577\pi$
$773$	1.82047	0.0654777	0.0327388	+	0.999464i	$0.489577\pi$
$774$	−1.84308	−0.0662480
$775$	0	0
$776$	−18.2398	−0.654770
$777$	−62.2441	−2.23299
$778$	−28.5554	−1.02376
$779$	5.52711	0.198029
$780$	0	0
$781$	−12.2382	−0.437917
$782$	24.5226	0.876925
$783$	2.36517	0.0845242
$784$	−26.1522	−0.934006
$785$	0	0
$786$	8.68168	0.309665
$787$	7.64010	0.272340	0.136170	−	0.990685i	$-0.456521\pi$
$787$	7.64010	0.272340	0.136170	+	0.990685i	$0.456521\pi$
$788$	−6.76118	−0.240857
$789$	−16.0812	−0.572506
$790$	0	0
$791$	97.2727	3.45862
$792$	25.6521	0.911508
$793$	5.39264	0.191498
$794$	−1.97883	−0.0702262
$795$	0	0
$796$	−15.7040	−0.556613
$797$	40.6367	1.43943	0.719713	−	0.694272i	$-0.244273\pi$
$797$	40.6367	1.43943	0.719713	+	0.694272i	$0.244273\pi$
$798$	−44.6882	−1.58195
$799$	62.6090	2.21495
$800$	0	0
$801$	12.1451	0.429127
$802$	−16.8214	−0.593984
$803$	7.38348	0.260557
$804$	12.1719	0.429269
$805$	0	0
$806$	−0.520473	−0.0183329
$807$	−23.1325	−0.814302
$808$	17.7582	0.624732
$809$	−36.8019	−1.29389	−0.646943	−	0.762538i	$-0.723953\pi$
$809$	−36.8019	−1.29389	−0.646943	+	0.762538i	$0.723953\pi$
$810$	0	0
$811$	4.46256	0.156702	0.0783509	−	0.996926i	$-0.475035\pi$
$811$	4.46256	0.156702	0.0783509	+	0.996926i	$0.475035\pi$
$812$	13.1540	0.461616
$813$	−14.8595	−0.521147
$814$	−14.2037	−0.497840
$815$	0	0
$816$	−19.4999	−0.682633
$817$	−1.78731	−0.0625299
$818$	40.8529	1.42839
$819$	63.4704	2.21784
$820$	0	0
$821$	12.9560	0.452166	0.226083	−	0.974108i	$-0.427408\pi$
$821$	12.9560	0.452166	0.226083	+	0.974108i	$0.427408\pi$
$822$	−8.66729	−0.302306
$823$	52.4066	1.82678	0.913389	−	0.407088i	$-0.133456\pi$
$823$	52.4066	1.82678	0.913389	+	0.407088i	$0.133456\pi$
$824$	−40.0134	−1.39393
$825$	0	0
$826$	−59.7848	−2.08018
$827$	0.501068	0.0174239	0.00871193	−	0.999962i	$-0.497227\pi$
$827$	0.501068	0.0174239	0.00871193	+	0.999962i	$0.497227\pi$
$828$	−13.0824	−0.454646
$829$	−26.6149	−0.924374	−0.462187	−	0.886783i	$-0.652935\pi$
$829$	−26.6149	−0.924374	−0.462187	+	0.886783i	$0.652935\pi$
$830$	0	0
$831$	−81.1111	−2.81371
$832$	−31.9711	−1.10840
$833$	76.9735	2.66697
$834$	−25.5673	−0.885322
$835$	0	0
$836$	7.33916	0.253830
$837$	0.0868892	0.00300333
$838$	−8.30828	−0.287005
$839$	9.39377	0.324309	0.162155	−	0.986765i	$-0.448156\pi$
$839$	9.39377	0.324309	0.162155	+	0.986765i	$0.448156\pi$
$840$	0	0
$841$	−18.3034	−0.631151
$842$	−28.5110	−0.982556
$843$	80.5726	2.77507
$844$	−7.66006	−0.263670
$845$	0	0
$846$	46.4098	1.59560
$847$	21.6182	0.742809
$848$	−1.04557	−0.0359051
$849$	−46.1540	−1.58400
$850$	0	0
$851$	24.5527	0.841656
$852$	−10.0745	−0.345146
$853$	−5.24939	−0.179736	−0.0898679	−	0.995954i	$-0.528644\pi$
$853$	−5.24939	−0.179736	−0.0898679	+	0.995954i	$0.528644\pi$
$854$	6.95691	0.238060
$855$	0	0
$856$	52.9381	1.80939
$857$	−48.0516	−1.64141	−0.820706	−	0.571351i	$-0.806420\pi$
$857$	−48.0516	−1.64141	−0.820706	+	0.571351i	$0.806420\pi$
$858$	27.6969	0.945558
$859$	26.3981	0.900691	0.450345	−	0.892854i	$-0.351301\pi$
$859$	26.3981	0.900691	0.450345	+	0.892854i	$0.351301\pi$
$860$	0	0
$861$	−19.3664	−0.660004
$862$	5.26249	0.179241
$863$	17.6126	0.599540	0.299770	−	0.954012i	$-0.403090\pi$
$863$	17.6126	0.599540	0.299770	+	0.954012i	$0.403090\pi$
$864$	3.15741	0.107417
$865$	0	0
$866$	28.4210	0.965786
$867$	14.7637	0.501401
$868$	0.483239	0.0164022
$869$	−20.2849	−0.688117
$870$	0	0
$871$	23.2939	0.789284
$872$	−31.7035	−1.07362
$873$	19.6044	0.663508
$874$	17.6277	0.596265
$875$	0	0
$876$	6.07808	0.205359
$877$	−36.2317	−1.22346	−0.611729	−	0.791067i	$-0.709526\pi$
$877$	−36.2317	−1.22346	−0.611729	+	0.791067i	$0.709526\pi$
$878$	29.6867	1.00188
$879$	−2.96117	−0.0998777
$880$	0	0
$881$	25.7447	0.867362	0.433681	−	0.901067i	$-0.357215\pi$
$881$	25.7447	0.867362	0.433681	+	0.901067i	$0.357215\pi$
$882$	57.0577	1.92123
$883$	−4.04669	−0.136182	−0.0680910	−	0.997679i	$-0.521691\pi$
$883$	−4.04669	−0.136182	−0.0680910	+	0.997679i	$0.521691\pi$
$884$	16.0845	0.540980
$885$	0	0
$886$	−22.0715	−0.741506
$887$	−4.60254	−0.154538	−0.0772691	−	0.997010i	$-0.524620\pi$
$887$	−4.60254	−0.154538	−0.0772691	+	0.997010i	$0.524620\pi$
$888$	−39.6314	−1.32994
$889$	16.4822	0.552796
$890$	0	0
$891$	20.5295	0.687764
$892$	16.3051	0.545934
$893$	45.0055	1.50605
$894$	−13.4631	−0.450273
$895$	0	0
$896$	0.714366	0.0238653
$897$	−47.8773	−1.59858
$898$	−23.6649	−0.789708
$899$	0.392962	0.0131060
$900$	0	0
$901$	3.07742	0.102524
$902$	−4.41928	−0.147146
$903$	6.26251	0.208403
$904$	61.9345	2.05991
$905$	0	0
$906$	−43.4207	−1.44256
$907$	−21.6827	−0.719963	−0.359982	−	0.932959i	$-0.617217\pi$
$907$	−21.6827	−0.719963	−0.359982	+	0.932959i	$0.617217\pi$
$908$	12.6648	0.420296
$909$	−19.0868	−0.633068
$910$	0	0
$911$	−25.7102	−0.851816	−0.425908	−	0.904767i	$-0.640045\pi$
$911$	−25.7102	−0.851816	−0.425908	+	0.904767i	$0.640045\pi$
$912$	−14.0172	−0.464156
$913$	30.9318	1.02369
$914$	−4.15510	−0.137439
$915$	0	0
$916$	−3.46381	−0.114447
$917$	−15.4260	−0.509411
$918$	3.73101	0.123142
$919$	−0.626457	−0.0206649	−0.0103325	−	0.999947i	$-0.503289\pi$
$919$	−0.626457	−0.0206649	−0.0103325	+	0.999947i	$0.503289\pi$
$920$	0	0
$921$	13.8228	0.455478
$922$	−1.02602	−0.0337901
$923$	−19.2800	−0.634610
$924$	−25.7156	−0.845980
$925$	0	0
$926$	37.7950	1.24202
$927$	43.0070	1.41254
$928$	14.2796	0.468752
$929$	−32.7008	−1.07288	−0.536439	−	0.843939i	$-0.680231\pi$
$929$	−32.7008	−1.07288	−0.536439	+	0.843939i	$0.680231\pi$
$930$	0	0
$931$	55.3312	1.81341
$932$	−7.27664	−0.238354
$933$	42.9033	1.40459
$934$	−28.1152	−0.919956
$935$	0	0
$936$	40.4123	1.32092
$937$	55.9027	1.82626	0.913130	−	0.407668i	$-0.133658\pi$
$937$	55.9027	1.82626	0.913130	+	0.407668i	$0.133658\pi$
$938$	30.0509	0.981196
$939$	36.0855	1.17760
$940$	0	0
$941$	55.1315	1.79723	0.898617	−	0.438734i	$-0.144573\pi$
$941$	55.1315	1.79723	0.898617	+	0.438734i	$0.144573\pi$
$942$	40.0089	1.30356
$943$	7.63923	0.248767
$944$	−18.7525	−0.610342
$945$	0	0
$946$	1.42907	0.0464630
$947$	−4.24224	−0.137854	−0.0689272	−	0.997622i	$-0.521958\pi$
$947$	−4.24224	−0.137854	−0.0689272	+	0.997622i	$0.521958\pi$
$948$	−16.6985	−0.542342
$949$	11.6319	0.377588
$950$	0	0
$951$	81.7741	2.65171
$952$	70.3323	2.27948
$953$	−36.4505	−1.18075	−0.590374	−	0.807130i	$-0.701020\pi$
$953$	−36.4505	−1.18075	−0.590374	+	0.807130i	$0.701020\pi$
$954$	2.28118	0.0738561
$955$	0	0
$956$	−8.49705	−0.274814
$957$	−20.9115	−0.675972
$958$	30.9109	0.998686
$959$	15.4004	0.497305
$960$	0	0
$961$	−30.9856	−0.999534
$962$	−22.3765	−0.721447
$963$	−56.8986	−1.83353
$964$	−0.837004	−0.0269581
$965$	0	0
$966$	−61.7653	−1.98727
$967$	−23.7682	−0.764334	−0.382167	−	0.924093i	$-0.624822\pi$
$967$	−23.7682	−0.764334	−0.382167	+	0.924093i	$0.624822\pi$
$968$	13.7645	0.442408
$969$	41.2567	1.32536
$970$	0	0
$971$	48.7980	1.56600	0.783002	−	0.622020i	$-0.213688\pi$
$971$	48.7980	1.56600	0.783002	+	0.622020i	$0.213688\pi$
$972$	18.7158	0.600309
$973$	45.4290	1.45639
$974$	21.5694	0.691130
$975$	0	0
$976$	2.18215	0.0698488
$977$	−52.4057	−1.67661	−0.838303	−	0.545205i	$-0.816452\pi$
$977$	−52.4057	−1.67661	−0.838303	+	0.545205i	$0.816452\pi$
$978$	−1.56764	−0.0501275
$979$	−9.41697	−0.300968
$980$	0	0
$981$	34.0753	1.08794
$982$	−22.7323	−0.725416
$983$	46.5320	1.48414	0.742071	−	0.670322i	$-0.233844\pi$
$983$	46.5320	1.48414	0.742071	+	0.670322i	$0.233844\pi$
$984$	−12.3307	−0.393090
$985$	0	0
$986$	16.8737	0.537369
$987$	−157.694	−5.01945
$988$	11.5621	0.367839
$989$	−2.47030	−0.0785511
$990$	0	0
$991$	−1.65656	−0.0526225	−0.0263113	−	0.999654i	$-0.508376\pi$
$991$	−1.65656	−0.0526225	−0.0263113	+	0.999654i	$0.508376\pi$
$992$	0.524591	0.0166558
$993$	83.6383	2.65418
$994$	−24.8727	−0.788914
$995$	0	0
$996$	25.4630	0.806827
$997$	−30.9648	−0.980664	−0.490332	−	0.871536i	$-0.663124\pi$
$997$	−30.9648	−0.980664	−0.490332	+	0.871536i	$0.663124\pi$
$998$	16.1892	0.512461
$999$	3.73559	0.118189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.k.1.16		25
5.4	even	2		1205.2.a.d.1.10	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.a.d.1.10	✓	25	5.4	even	2
6025.2.a.k.1.16		25	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.07842 q^{2} +2.50766 q^{3} -0.837004 q^{4} +2.70432 q^{6} -4.80516 q^{7} -3.05949 q^{8} +3.28838 q^{9} +O(q^{10})\)	\(q+1.07842 q^{2} +2.50766 q^{3} -0.837004 q^{4} +2.70432 q^{6} -4.80516 q^{7} -3.05949 q^{8} +3.28838 q^{9} -2.54972 q^{11} -2.09893 q^{12} -4.01681 q^{13} -5.18199 q^{14} -1.62542 q^{16} +4.78408 q^{17} +3.54627 q^{18} +3.43896 q^{19} -12.0497 q^{21} -2.74967 q^{22} +4.75312 q^{23} -7.67218 q^{24} -4.33182 q^{26} +0.723167 q^{27} +4.02193 q^{28} +3.27057 q^{29} +0.120151 q^{31} +4.36609 q^{32} -6.39383 q^{33} +5.15926 q^{34} -2.75239 q^{36} +5.16560 q^{37} +3.70865 q^{38} -10.0728 q^{39} +1.60720 q^{41} -12.9947 q^{42} -0.519723 q^{43} +2.13412 q^{44} +5.12587 q^{46} +13.0869 q^{47} -4.07600 q^{48} +16.0895 q^{49} +11.9969 q^{51} +3.36209 q^{52} +0.643264 q^{53} +0.779879 q^{54} +14.7013 q^{56} +8.62376 q^{57} +3.52706 q^{58} +11.5370 q^{59} -1.34252 q^{61} +0.129574 q^{62} -15.8012 q^{63} +7.95933 q^{64} -6.89525 q^{66} -5.79910 q^{67} -4.00429 q^{68} +11.9192 q^{69} +4.79983 q^{71} -10.0608 q^{72} -2.89581 q^{73} +5.57070 q^{74} -2.87842 q^{76} +12.2518 q^{77} -10.8628 q^{78} +7.95574 q^{79} -8.05169 q^{81} +1.73325 q^{82} -12.1315 q^{83} +10.0857 q^{84} -0.560481 q^{86} +8.20150 q^{87} +7.80083 q^{88} +3.69334 q^{89} +19.3014 q^{91} -3.97838 q^{92} +0.301298 q^{93} +14.1133 q^{94} +10.9487 q^{96} +5.96171 q^{97} +17.3513 q^{98} -8.38444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9}+O(q^{10}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9	\( 25 q + 4 q^{2} - 9 q^{3} + 36 q^{4} + 7 q^{6} - 7 q^{7} + 15 q^{8} + 36 q^{9} + 10 q^{11} - 22 q^{12} - 10 q^{13} + 13 q^{14} + 54 q^{16} - q^{17} + 13 q^{18} + 50 q^{19} + 9 q^{21} - 11 q^{22} + 31 q^{23} + 22 q^{24} + 8 q^{26} - 42 q^{27} - 14 q^{28} + 4 q^{29} + 34 q^{31} + 44 q^{32} - 28 q^{33} + 33 q^{34} + 83 q^{36} - 14 q^{37} + 10 q^{38} + 23 q^{39} + 11 q^{41} - 23 q^{42} - 49 q^{43} + 20 q^{44} + 27 q^{46} + 28 q^{47} - 30 q^{48} + 66 q^{49} + 49 q^{51} - 39 q^{52} + 16 q^{53} + 5 q^{54} + 51 q^{56} - 10 q^{57} + 8 q^{58} + 30 q^{59} + 35 q^{61} + 18 q^{62} + 73 q^{64} - 13 q^{66} - 37 q^{67} - 11 q^{68} - 4 q^{69} + 12 q^{71} + 90 q^{72} - 36 q^{73} - 12 q^{74} + 57 q^{76} + 31 q^{77} + 9 q^{78} + 16 q^{79} + 65 q^{81} + 11 q^{82} - 43 q^{83} - 62 q^{84} - 9 q^{86} + 22 q^{87} - 20 q^{88} + 38 q^{89} + 86 q^{91} + 119 q^{92} - 10 q^{93} - 18 q^{94} - 34 q^{96} - 17 q^{97} + 32 q^{98} + 26 q^{99}+O(q^{100}) \) 25 * q + 4 * q^2 - 9 * q^3 + 36 * q^4 + 7 * q^6 - 7 * q^7 + 15 * q^8 + 36 * q^9 + 10 * q^11 - 22 * q^12 - 10 * q^13 + 13 * q^14 + 54 * q^16 - q^17 + 13 * q^18 + 50 * q^19 + 9 * q^21 - 11 * q^22 + 31 * q^23 + 22 * q^24 + 8 * q^26 - 42 * q^27 - 14 * q^28 + 4 * q^29 + 34 * q^31 + 44 * q^32 - 28 * q^33 + 33 * q^34 + 83 * q^36 - 14 * q^37 + 10 * q^38 + 23 * q^39 + 11 * q^41 - 23 * q^42 - 49 * q^43 + 20 * q^44 + 27 * q^46 + 28 * q^47 - 30 * q^48 + 66 * q^49 + 49 * q^51 - 39 * q^52 + 16 * q^53 + 5 * q^54 + 51 * q^56 - 10 * q^57 + 8 * q^58 + 30 * q^59 + 35 * q^61 + 18 * q^62 + 73 * q^64 - 13 * q^66 - 37 * q^67 - 11 * q^68 - 4 * q^69 + 12 * q^71 + 90 * q^72 - 36 * q^73 - 12 * q^74 + 57 * q^76 + 31 * q^77 + 9 * q^78 + 16 * q^79 + 65 * q^81 + 11 * q^82 - 43 * q^83 - 62 * q^84 - 9 * q^86 + 22 * q^87 - 20 * q^88 + 38 * q^89 + 86 * q^91 + 119 * q^92 - 10 * q^93 - 18 * q^94 - 34 * q^96 - 17 * q^97 + 32 * q^98 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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