LMFDB - Embedded newform 6025.2.a.n.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:	$40$
Twist minimal:	yes
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)

$f(q)$	$=$	$q-0.409457 q^{2} -1.91892 q^{3} -1.83235 q^{4} +0.785714 q^{6} -3.66070 q^{7} +1.56918 q^{8} +0.682248 q^{9} +O(q^{10})$	\(q-0.409457 q^{2} -1.91892 q^{3} -1.83235 q^{4} +0.785714 q^{6} -3.66070 q^{7} +1.56918 q^{8} +0.682248 q^{9} +5.60163 q^{11} +3.51612 q^{12} +4.56597 q^{13} +1.49890 q^{14} +3.02218 q^{16} +6.06372 q^{17} -0.279351 q^{18} -3.67460 q^{19} +7.02459 q^{21} -2.29363 q^{22} +7.36362 q^{23} -3.01113 q^{24} -1.86957 q^{26} +4.44758 q^{27} +6.70767 q^{28} -1.37829 q^{29} +2.62718 q^{31} -4.37581 q^{32} -10.7491 q^{33} -2.48283 q^{34} -1.25011 q^{36} +3.20350 q^{37} +1.50459 q^{38} -8.76172 q^{39} -9.05721 q^{41} -2.87626 q^{42} +8.36887 q^{43} -10.2641 q^{44} -3.01508 q^{46} +8.91682 q^{47} -5.79932 q^{48} +6.40073 q^{49} -11.6358 q^{51} -8.36643 q^{52} +8.02895 q^{53} -1.82109 q^{54} -5.74429 q^{56} +7.05126 q^{57} +0.564351 q^{58} +12.3585 q^{59} +11.2806 q^{61} -1.07571 q^{62} -2.49750 q^{63} -4.25266 q^{64} +4.40128 q^{66} -6.04317 q^{67} -11.1108 q^{68} -14.1302 q^{69} -11.4456 q^{71} +1.07057 q^{72} -16.3574 q^{73} -1.31169 q^{74} +6.73313 q^{76} -20.5059 q^{77} +3.58754 q^{78} -4.63741 q^{79} -10.5813 q^{81} +3.70854 q^{82} +15.9453 q^{83} -12.8715 q^{84} -3.42669 q^{86} +2.64483 q^{87} +8.78997 q^{88} -0.0114651 q^{89} -16.7146 q^{91} -13.4927 q^{92} -5.04134 q^{93} -3.65105 q^{94} +8.39682 q^{96} +11.6532 q^{97} -2.62082 q^{98} +3.82170 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	$ 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) $ 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.409457	−0.289529	−0.144765	−	0.989466i	$-0.546243\pi$
$2$	−0.409457	−0.289529	−0.144765	+	0.989466i	$0.546243\pi$
$3$	−1.91892	−1.10789	−0.553944	−	0.832554i	$-0.686878\pi$
$3$	−1.91892	−1.10789	−0.553944	+	0.832554i	$0.686878\pi$
$4$	−1.83235	−0.916173
$5$	0	0
$5$	0	0
$6$	0.785714	0.320766
$7$	−3.66070	−1.38361	−0.691807	−	0.722082i	$-0.743185\pi$
$7$	−3.66070	−1.38361	−0.691807	+	0.722082i	$0.743185\pi$
$8$	1.56918	0.554788
$9$	0.682248	0.227416
$10$	0	0
$11$	5.60163	1.68896	0.844478	−	0.535590i	$-0.179911\pi$
$11$	5.60163	1.68896	0.844478	+	0.535590i	$0.179911\pi$
$12$	3.51612	1.01502
$13$	4.56597	1.26637	0.633186	−	0.774000i	$-0.281747\pi$
$13$	4.56597	1.26637	0.633186	+	0.774000i	$0.281747\pi$
$14$	1.49890	0.400597
$15$	0	0
$16$	3.02218	0.755545
$17$	6.06372	1.47067	0.735334	−	0.677705i	$-0.237025\pi$
$17$	6.06372	1.47067	0.735334	+	0.677705i	$0.237025\pi$
$18$	−0.279351	−0.0658436
$19$	−3.67460	−0.843011	−0.421505	−	0.906826i	$-0.638498\pi$
$19$	−3.67460	−0.843011	−0.421505	+	0.906826i	$0.638498\pi$
$20$	0	0
$21$	7.02459	1.53289
$22$	−2.29363	−0.489003
$23$	7.36362	1.53542	0.767710	−	0.640797i	$-0.221396\pi$
$23$	7.36362	1.53542	0.767710	+	0.640797i	$0.221396\pi$
$24$	−3.01113	−0.614644
$25$	0	0
$26$	−1.86957	−0.366652
$27$	4.44758	0.855937
$28$	6.70767	1.26763
$29$	−1.37829	−0.255942	−0.127971	−	0.991778i	$-0.540847\pi$
$29$	−1.37829	−0.255942	−0.127971	+	0.991778i	$0.540847\pi$
$30$	0	0
$31$	2.62718	0.471855	0.235927	−	0.971771i	$-0.424187\pi$
$31$	2.62718	0.471855	0.235927	+	0.971771i	$0.424187\pi$
$32$	−4.37581	−0.773541
$33$	−10.7491	−1.87117
$34$	−2.48283	−0.425802
$35$	0	0
$36$	−1.25011	−0.208352
$37$	3.20350	0.526651	0.263326	−	0.964707i	$-0.415181\pi$
$37$	3.20350	0.526651	0.263326	+	0.964707i	$0.415181\pi$
$38$	1.50459	0.244076
$39$	−8.76172	−1.40300
$40$	0	0
$41$	−9.05721	−1.41450	−0.707250	−	0.706964i	$-0.750064\pi$
$41$	−9.05721	−1.41450	−0.707250	+	0.706964i	$0.750064\pi$
$42$	−2.87626	−0.443817
$43$	8.36887	1.27624	0.638120	−	0.769937i	$-0.279712\pi$
$43$	8.36887	1.27624	0.638120	+	0.769937i	$0.279712\pi$
$44$	−10.2641	−1.54738
$45$	0	0
$46$	−3.01508	−0.444549
$47$	8.91682	1.30065	0.650326	−	0.759656i	$-0.274632\pi$
$47$	8.91682	1.30065	0.650326	+	0.759656i	$0.274632\pi$
$48$	−5.79932	−0.837059
$49$	6.40073	0.914390
$50$	0	0
$51$	−11.6358	−1.62934
$52$	−8.36643	−1.16022
$53$	8.02895	1.10286	0.551431	−	0.834221i	$-0.314082\pi$
$53$	8.02895	1.10286	0.551431	+	0.834221i	$0.314082\pi$
$54$	−1.82109	−0.247819
$55$	0	0
$56$	−5.74429	−0.767614
$57$	7.05126	0.933962
$58$	0.564351	0.0741029
$59$	12.3585	1.60895	0.804473	−	0.593989i	$-0.202448\pi$
$59$	12.3585	1.60895	0.804473	+	0.593989i	$0.202448\pi$
$60$	0	0
$61$	11.2806	1.44433	0.722166	−	0.691720i	$-0.243147\pi$
$61$	11.2806	1.44433	0.722166	+	0.691720i	$0.243147\pi$
$62$	−1.07571	−0.136616
$63$	−2.49750	−0.314656
$64$	−4.25266	−0.531582
$65$	0	0
$66$	4.40128	0.541760
$67$	−6.04317	−0.738291	−0.369146	−	0.929372i	$-0.620350\pi$
$67$	−6.04317	−0.738291	−0.369146	+	0.929372i	$0.620350\pi$
$68$	−11.1108	−1.34739
$69$	−14.1302	−1.70107
$70$	0	0
$71$	−11.4456	−1.35834	−0.679169	−	0.733982i	$-0.737660\pi$
$71$	−11.4456	−1.35834	−0.679169	+	0.733982i	$0.737660\pi$
$72$	1.07057	0.126168
$73$	−16.3574	−1.91449	−0.957247	−	0.289272i	$-0.906587\pi$
$73$	−16.3574	−1.91449	−0.957247	+	0.289272i	$0.906587\pi$
$74$	−1.31169	−0.152481
$75$	0	0
$76$	6.73313	0.772343
$77$	−20.5059	−2.33686
$78$	3.58754	0.406209
$79$	−4.63741	−0.521749	−0.260875	−	0.965373i	$-0.584011\pi$
$79$	−4.63741	−0.521749	−0.260875	+	0.965373i	$0.584011\pi$
$80$	0	0
$81$	−10.5813	−1.17570
$82$	3.70854	0.409539
$83$	15.9453	1.75022	0.875111	−	0.483922i	$-0.160788\pi$
$83$	15.9453	1.75022	0.875111	+	0.483922i	$0.160788\pi$
$84$	−12.8715	−1.40439
$85$	0	0
$86$	−3.42669	−0.369509
$87$	2.64483	0.283556
$88$	8.78997	0.937014
$89$	−0.0114651	−0.00121530	−0.000607650	−	1.00000i	$-0.500193\pi$
$89$	−0.0114651	−0.00121530	−0.000607650		1.00000i	$0.500193\pi$
$90$	0	0
$91$	−16.7146	−1.75217
$92$	−13.4927	−1.40671
$93$	−5.04134	−0.522762
$94$	−3.65105	−0.376577
$95$	0	0
$96$	8.39682	0.856997
$97$	11.6532	1.18320	0.591602	−	0.806230i	$-0.298496\pi$
$97$	11.6532	1.18320	0.591602	+	0.806230i	$0.298496\pi$
$98$	−2.62082	−0.264743
$99$	3.82170	0.384095
$100$	0	0
$101$	−6.62572	−0.659284	−0.329642	−	0.944106i	$-0.606928\pi$
$101$	−6.62572	−0.659284	−0.329642	+	0.944106i	$0.606928\pi$
$102$	4.76435	0.471741
$103$	−12.3000	−1.21196	−0.605980	−	0.795480i	$-0.707219\pi$
$103$	−12.3000	−1.21196	−0.605980	+	0.795480i	$0.707219\pi$
$104$	7.16482	0.702568
$105$	0	0
$106$	−3.28751	−0.319311
$107$	−18.9253	−1.82957	−0.914787	−	0.403937i	$-0.867642\pi$
$107$	−18.9253	−1.82957	−0.914787	+	0.403937i	$0.867642\pi$
$108$	−8.14950	−0.784186
$109$	−4.64494	−0.444904	−0.222452	−	0.974944i	$-0.571406\pi$
$109$	−4.64494	−0.444904	−0.222452	+	0.974944i	$0.571406\pi$
$110$	0	0
$111$	−6.14725	−0.583471
$112$	−11.0633	−1.04538
$113$	−5.23432	−0.492403	−0.246201	−	0.969219i	$-0.579182\pi$
$113$	−5.23432	−0.492403	−0.246201	+	0.969219i	$0.579182\pi$
$114$	−2.88718	−0.270409
$115$	0	0
$116$	2.52551	0.234487
$117$	3.11512	0.287993
$118$	−5.06029	−0.465837
$119$	−22.1975	−2.03484
$120$	0	0
$121$	20.3783	1.85257
$122$	−4.61891	−0.418177
$123$	17.3801	1.56711
$124$	−4.81390	−0.432301
$125$	0	0
$126$	1.02262	0.0911022
$127$	5.13608	0.455753	0.227876	−	0.973690i	$-0.426822\pi$
$127$	5.13608	0.455753	0.227876	+	0.973690i	$0.426822\pi$
$128$	10.4929	0.927450
$129$	−16.0592	−1.41393
$130$	0	0
$131$	9.15338	0.799735	0.399867	−	0.916573i	$-0.369056\pi$
$131$	9.15338	0.799735	0.399867	+	0.916573i	$0.369056\pi$
$132$	19.6960	1.71432
$133$	13.4516	1.16640
$134$	2.47442	0.213757
$135$	0	0
$136$	9.51506	0.815910
$137$	−0.726755	−0.0620909	−0.0310454	−	0.999518i	$-0.509884\pi$
$137$	−0.726755	−0.0620909	−0.0310454	+	0.999518i	$0.509884\pi$
$138$	5.78569	0.492511
$139$	5.06719	0.429794	0.214897	−	0.976637i	$-0.431058\pi$
$139$	5.06719	0.429794	0.214897	+	0.976637i	$0.431058\pi$
$140$	0	0
$141$	−17.1106	−1.44098
$142$	4.68646	0.393279
$143$	25.5769	2.13885
$144$	2.06187	0.171823
$145$	0	0
$146$	6.69766	0.554302
$147$	−12.2825	−1.01304
$148$	−5.86991	−0.482504
$149$	8.25572	0.676335	0.338167	−	0.941086i	$-0.390193\pi$
$149$	8.25572	0.676335	0.338167	+	0.941086i	$0.390193\pi$
$150$	0	0
$151$	6.77980	0.551732	0.275866	−	0.961196i	$-0.411035\pi$
$151$	6.77980	0.551732	0.275866	+	0.961196i	$0.411035\pi$
$152$	−5.76610	−0.467693
$153$	4.13696	0.334453
$154$	8.39628	0.676591
$155$	0	0
$156$	16.0545	1.28539
$157$	7.76989	0.620105	0.310053	−	0.950719i	$-0.399653\pi$
$157$	7.76989	0.620105	0.310053	+	0.950719i	$0.399653\pi$
$158$	1.89882	0.151062
$159$	−15.4069	−1.22185
$160$	0	0
$161$	−26.9560	−2.12443
$162$	4.33257	0.340399
$163$	8.40198	0.658093	0.329047	−	0.944314i	$-0.393273\pi$
$163$	8.40198	0.658093	0.329047	+	0.944314i	$0.393273\pi$
$164$	16.5959	1.29593
$165$	0	0
$166$	−6.52890	−0.506741
$167$	0.230648	0.0178481	0.00892404	−	0.999960i	$-0.497159\pi$
$167$	0.230648	0.0178481	0.00892404	+	0.999960i	$0.497159\pi$
$168$	11.0228	0.850430
$169$	7.84806	0.603697
$170$	0	0
$171$	−2.50699	−0.191714
$172$	−15.3347	−1.16926
$173$	7.82620	0.595015	0.297507	−	0.954719i	$-0.403845\pi$
$173$	7.82620	0.595015	0.297507	+	0.954719i	$0.403845\pi$
$174$	−1.08294	−0.0820977
$175$	0	0
$176$	16.9291	1.27608
$177$	−23.7150	−1.78253
$178$	0.00469447	0.000351865 0
$179$	8.64504	0.646161	0.323080	−	0.946372i	$-0.395282\pi$
$179$	8.64504	0.646161	0.323080	+	0.946372i	$0.395282\pi$
$180$	0	0
$181$	3.48479	0.259022	0.129511	−	0.991578i	$-0.458659\pi$
$181$	3.48479	0.259022	0.129511	+	0.991578i	$0.458659\pi$
$182$	6.84392	0.507305
$183$	−21.6465	−1.60016
$184$	11.5548	0.851834
$185$	0	0
$186$	2.06421	0.151355
$187$	33.9668	2.48390
$188$	−16.3387	−1.19162
$189$	−16.2813	−1.18429
$190$	0	0
$191$	−6.43507	−0.465625	−0.232813	−	0.972522i	$-0.574793\pi$
$191$	−6.43507	−0.465625	−0.232813	+	0.972522i	$0.574793\pi$
$192$	8.16050	0.588933
$193$	14.3687	1.03428	0.517140	−	0.855901i	$-0.326997\pi$
$193$	14.3687	1.03428	0.517140	+	0.855901i	$0.326997\pi$
$194$	−4.77148	−0.342572
$195$	0	0
$196$	−11.7283	−0.837739
$197$	2.34264	0.166906	0.0834530	−	0.996512i	$-0.473405\pi$
$197$	2.34264	0.166906	0.0834530	+	0.996512i	$0.473405\pi$
$198$	−1.56482	−0.111207
$199$	19.4188	1.37656	0.688281	−	0.725444i	$-0.258366\pi$
$199$	19.4188	1.37656	0.688281	+	0.725444i	$0.258366\pi$
$200$	0	0
$201$	11.5964	0.817944
$202$	2.71295	0.190882
$203$	5.04552	0.354126
$204$	21.3208	1.49275
$205$	0	0
$206$	5.03633	0.350898
$207$	5.02381	0.349179
$208$	13.7992	0.956801
$209$	−20.5838	−1.42381
$210$	0	0
$211$	9.53448	0.656381	0.328190	−	0.944612i	$-0.393561\pi$
$211$	9.53448	0.656381	0.328190	+	0.944612i	$0.393561\pi$
$212$	−14.7118	−1.01041
$213$	21.9631	1.50489
$214$	7.74907	0.529716
$215$	0	0
$216$	6.97904	0.474864
$217$	−9.61731	−0.652865
$218$	1.90190	0.128813
$219$	31.3886	2.12104
$220$	0	0
$221$	27.6868	1.86241
$222$	2.51703	0.168932
$223$	−4.03312	−0.270078	−0.135039	−	0.990840i	$-0.543116\pi$
$223$	−4.03312	−0.270078	−0.135039	+	0.990840i	$0.543116\pi$
$224$	16.0185	1.07028
$225$	0	0
$226$	2.14322	0.142565
$227$	−9.72480	−0.645457	−0.322729	−	0.946492i	$-0.604600\pi$
$227$	−9.72480	−0.645457	−0.322729	+	0.946492i	$0.604600\pi$
$228$	−12.9203	−0.855670
$229$	−2.98757	−0.197424	−0.0987122	−	0.995116i	$-0.531472\pi$
$229$	−2.98757	−0.197424	−0.0987122	+	0.995116i	$0.531472\pi$
$230$	0	0
$231$	39.3492	2.58898
$232$	−2.16279	−0.141994
$233$	−16.9056	−1.10752	−0.553762	−	0.832675i	$-0.686808\pi$
$233$	−16.9056	−1.10752	−0.553762	+	0.832675i	$0.686808\pi$
$234$	−1.27551	−0.0833825
$235$	0	0
$236$	−22.6451	−1.47407
$237$	8.89881	0.578040
$238$	9.08890	0.589146
$239$	−19.8394	−1.28330	−0.641652	−	0.766996i	$-0.721751\pi$
$239$	−19.8394	−1.28330	−0.641652	+	0.766996i	$0.721751\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−8.34403	−0.536375
$243$	6.96188	0.446605
$244$	−20.6699	−1.32326
$245$	0	0
$246$	−7.11638	−0.453724
$247$	−16.7781	−1.06757
$248$	4.12251	0.261780
$249$	−30.5977	−1.93905
$250$	0	0
$251$	−24.4229	−1.54156	−0.770780	−	0.637101i	$-0.780133\pi$
$251$	−24.4229	−1.54156	−0.770780	+	0.637101i	$0.780133\pi$
$252$	4.57629	0.288279
$253$	41.2483	2.59326
$254$	−2.10300	−0.131954
$255$	0	0
$256$	4.20893	0.263058
$257$	−7.72278	−0.481734	−0.240867	−	0.970558i	$-0.577432\pi$
$257$	−7.72278	−0.481734	−0.240867	+	0.970558i	$0.577432\pi$
$258$	6.57553	0.409375
$259$	−11.7270	−0.728683
$260$	0	0
$261$	−0.940337	−0.0582054
$262$	−3.74791	−0.231547
$263$	−16.3122	−1.00585	−0.502927	−	0.864329i	$-0.667743\pi$
$263$	−16.3122	−1.00585	−0.502927	+	0.864329i	$0.667743\pi$
$264$	−16.8672	−1.03811
$265$	0	0
$266$	−5.50785	−0.337708
$267$	0.0220006	0.00134642
$268$	11.0732	0.676402
$269$	−16.3838	−0.998939	−0.499470	−	0.866331i	$-0.666472\pi$
$269$	−16.3838	−0.998939	−0.499470	+	0.866331i	$0.666472\pi$
$270$	0	0
$271$	−3.92502	−0.238428	−0.119214	−	0.992869i	$-0.538037\pi$
$271$	−3.92502	−0.238428	−0.119214	+	0.992869i	$0.538037\pi$
$272$	18.3257	1.11116
$273$	32.0740	1.94121
$274$	0.297575	0.0179771
$275$	0	0
$276$	25.8914	1.55848
$277$	6.33888	0.380867	0.190433	−	0.981700i	$-0.439011\pi$
$277$	6.33888	0.380867	0.190433	+	0.981700i	$0.439011\pi$
$278$	−2.07479	−0.124438
$279$	1.79238	0.107307
$280$	0	0
$281$	−21.6476	−1.29139	−0.645693	−	0.763597i	$-0.723431\pi$
$281$	−21.6476	−1.29139	−0.645693	+	0.763597i	$0.723431\pi$
$282$	7.00606	0.417205
$283$	24.5424	1.45889	0.729446	−	0.684038i	$-0.239778\pi$
$283$	24.5424	1.45889	0.729446	+	0.684038i	$0.239778\pi$
$284$	20.9722	1.24447
$285$	0	0
$286$	−10.4726	−0.619259
$287$	33.1558	1.95712
$288$	−2.98538	−0.175915
$289$	19.7687	1.16287
$290$	0	0
$291$	−22.3615	−1.31086
$292$	29.9725	1.75401
$293$	−6.67466	−0.389938	−0.194969	−	0.980809i	$-0.562461\pi$
$293$	−6.67466	−0.389938	−0.194969	+	0.980809i	$0.562461\pi$
$294$	5.02914	0.293305
$295$	0	0
$296$	5.02686	0.292180
$297$	24.9137	1.44564
$298$	−3.38036	−0.195819
$299$	33.6220	1.94441
$300$	0	0
$301$	−30.6359	−1.76582
$302$	−2.77603	−0.159743
$303$	12.7142	0.730413
$304$	−11.1053	−0.636933
$305$	0	0
$306$	−1.69390	−0.0968341
$307$	19.2203	1.09696	0.548479	−	0.836164i	$-0.315207\pi$
$307$	19.2203	1.09696	0.548479	+	0.836164i	$0.315207\pi$
$308$	37.5739	2.14097
$309$	23.6028	1.34271
$310$	0	0
$311$	29.5985	1.67838	0.839188	−	0.543842i	$-0.183031\pi$
$311$	29.5985	1.67838	0.839188	+	0.543842i	$0.183031\pi$
$312$	−13.7487	−0.778367
$313$	−23.2674	−1.31515	−0.657575	−	0.753389i	$-0.728418\pi$
$313$	−23.2674	−1.31515	−0.657575	+	0.753389i	$0.728418\pi$
$314$	−3.18143	−0.179539
$315$	0	0
$316$	8.49734	0.478012
$317$	17.6210	0.989692	0.494846	−	0.868981i	$-0.335224\pi$
$317$	17.6210	0.989692	0.494846	+	0.868981i	$0.335224\pi$
$318$	6.30846	0.353761
$319$	−7.72069	−0.432276
$320$	0	0
$321$	36.3160	2.02696
$322$	11.0373	0.615085
$323$	−22.2817	−1.23979
$324$	19.3886	1.07714
$325$	0	0
$326$	−3.44024	−0.190537
$327$	8.91325	0.492904
$328$	−14.2124	−0.784748
$329$	−32.6418	−1.79960
$330$	0	0
$331$	23.8576	1.31133	0.655665	−	0.755052i	$-0.272388\pi$
$331$	23.8576	1.31133	0.655665	+	0.755052i	$0.272388\pi$
$332$	−29.2173	−1.60351
$333$	2.18558	0.119769
$334$	−0.0944404	−0.00516755
$335$	0	0
$336$	21.2296	1.15817
$337$	26.2782	1.43146	0.715732	−	0.698375i	$-0.246093\pi$
$337$	26.2782	1.43146	0.715732	+	0.698375i	$0.246093\pi$
$338$	−3.21344	−0.174788
$339$	10.0442	0.545527
$340$	0	0
$341$	14.7165	0.796942
$342$	1.02650	0.0555069
$343$	2.19375	0.118451
$344$	13.1322	0.708043
$345$	0	0
$346$	−3.20449	−0.172274
$347$	11.2718	0.605103	0.302551	−	0.953133i	$-0.402162\pi$
$347$	11.2718	0.605103	0.302551	+	0.953133i	$0.402162\pi$
$348$	−4.84624	−0.259786
$349$	−20.9940	−1.12378	−0.561892	−	0.827211i	$-0.689926\pi$
$349$	−20.9940	−1.12378	−0.561892	+	0.827211i	$0.689926\pi$
$350$	0	0
$351$	20.3075	1.08393
$352$	−24.5117	−1.30648
$353$	−27.6950	−1.47406	−0.737029	−	0.675861i	$-0.763772\pi$
$353$	−27.6950	−1.47406	−0.737029	+	0.675861i	$0.763772\pi$
$354$	9.71028	0.516096
$355$	0	0
$356$	0.0210081	0.00111343
$357$	42.5951	2.25437
$358$	−3.53977	−0.187083
$359$	22.9268	1.21003	0.605016	−	0.796213i	$-0.293167\pi$
$359$	22.9268	1.21003	0.605016	+	0.796213i	$0.293167\pi$
$360$	0	0
$361$	−5.49732	−0.289333
$362$	−1.42687	−0.0749945
$363$	−39.1043	−2.05244
$364$	30.6270	1.60529
$365$	0	0
$366$	8.86331	0.463293
$367$	31.2253	1.62994	0.814972	−	0.579500i	$-0.196752\pi$
$367$	31.2253	1.62994	0.814972	+	0.579500i	$0.196752\pi$
$368$	22.2542	1.16008
$369$	−6.17926	−0.321680
$370$	0	0
$371$	−29.3916	−1.52594
$372$	9.23747	0.478941
$373$	3.01321	0.156018	0.0780090	−	0.996953i	$-0.475144\pi$
$373$	3.01321	0.156018	0.0780090	+	0.996953i	$0.475144\pi$
$374$	−13.9079	−0.719161
$375$	0	0
$376$	13.9921	0.721586
$377$	−6.29324	−0.324118
$378$	6.66646	0.342886
$379$	−14.6165	−0.750801	−0.375401	−	0.926863i	$-0.622495\pi$
$379$	−14.6165	−0.750801	−0.375401	+	0.926863i	$0.622495\pi$
$380$	0	0
$381$	−9.85571	−0.504923
$382$	2.63488	0.134812
$383$	0.575164	0.0293895	0.0146948	−	0.999892i	$-0.495322\pi$
$383$	0.575164	0.0293895	0.0146948	+	0.999892i	$0.495322\pi$
$384$	−20.1350	−1.02751
$385$	0	0
$386$	−5.88334	−0.299454
$387$	5.70964	0.290237
$388$	−21.3527	−1.08402
$389$	−33.6574	−1.70650	−0.853248	−	0.521505i	$-0.825371\pi$
$389$	−33.6574	−1.70650	−0.853248	+	0.521505i	$0.825371\pi$
$390$	0	0
$391$	44.6509	2.25809
$392$	10.0439	0.507293
$393$	−17.5646	−0.886017
$394$	−0.959208	−0.0483242
$395$	0	0
$396$	−7.00268	−0.351898
$397$	−20.5936	−1.03356	−0.516782	−	0.856117i	$-0.672870\pi$
$397$	−20.5936	−1.03356	−0.516782	+	0.856117i	$0.672870\pi$
$398$	−7.95115	−0.398555
$399$	−25.8125	−1.29224
$400$	0	0
$401$	21.7864	1.08796	0.543981	−	0.839097i	$-0.316916\pi$
$401$	21.7864	1.08796	0.543981	+	0.839097i	$0.316916\pi$
$402$	−4.74820	−0.236819
$403$	11.9956	0.597544
$404$	12.1406	0.604018
$405$	0	0
$406$	−2.06592	−0.102530
$407$	17.9448	0.889491
$408$	−18.2586	−0.903937
$409$	−25.4002	−1.25596	−0.627979	−	0.778230i	$-0.716118\pi$
$409$	−25.4002	−1.25596	−0.627979	+	0.778230i	$0.716118\pi$
$410$	0	0
$411$	1.39458	0.0687897
$412$	22.5379	1.11036
$413$	−45.2410	−2.22616
$414$	−2.05703	−0.101098
$415$	0	0
$416$	−19.9798	−0.979590
$417$	−9.72353	−0.476163
$418$	8.42815	0.412235
$419$	8.48703	0.414619	0.207309	−	0.978275i	$-0.433529\pi$
$419$	8.48703	0.414619	0.207309	+	0.978275i	$0.433529\pi$
$420$	0	0
$421$	−6.07594	−0.296123	−0.148062	−	0.988978i	$-0.547303\pi$
$421$	−6.07594	−0.296123	−0.148062	+	0.988978i	$0.547303\pi$
$422$	−3.90396	−0.190042
$423$	6.08348	0.295789
$424$	12.5989	0.611855
$425$	0	0
$426$	−8.99293	−0.435709
$427$	−41.2949	−1.99840
$428$	34.6776	1.67621
$429$	−49.0800	−2.36960
$430$	0	0
$431$	−9.49129	−0.457179	−0.228590	−	0.973523i	$-0.573411\pi$
$431$	−9.49129	−0.457179	−0.228590	+	0.973523i	$0.573411\pi$
$432$	13.4414	0.646699
$433$	27.9531	1.34334	0.671671	−	0.740850i	$-0.265577\pi$
$433$	27.9531	1.34334	0.671671	+	0.740850i	$0.265577\pi$
$434$	3.93787	0.189024
$435$	0	0
$436$	8.51113	0.407609
$437$	−27.0583	−1.29438
$438$	−12.8523	−0.614105
$439$	−16.3891	−0.782207	−0.391103	−	0.920347i	$-0.627907\pi$
$439$	−16.3891	−0.782207	−0.391103	+	0.920347i	$0.627907\pi$
$440$	0	0
$441$	4.36688	0.207947
$442$	−11.3365	−0.539223
$443$	−1.42690	−0.0677940	−0.0338970	−	0.999425i	$-0.510792\pi$
$443$	−1.42690	−0.0677940	−0.0338970	+	0.999425i	$0.510792\pi$
$444$	11.2639	0.534560
$445$	0	0
$446$	1.65139	0.0781955
$447$	−15.8420	−0.749303
$448$	15.5677	0.735505
$449$	−5.92269	−0.279509	−0.139755	−	0.990186i	$-0.544631\pi$
$449$	−5.92269	−0.279509	−0.139755	+	0.990186i	$0.544631\pi$
$450$	0	0
$451$	−50.7352	−2.38903
$452$	9.59107	0.451126
$453$	−13.0099	−0.611258
$454$	3.98188	0.186879
$455$	0	0
$456$	11.0647	0.518151
$457$	6.96191	0.325664	0.162832	−	0.986654i	$-0.447937\pi$
$457$	6.96191	0.325664	0.162832	+	0.986654i	$0.447937\pi$
$458$	1.22328	0.0571602
$459$	26.9689	1.25880
$460$	0	0
$461$	0.947733	0.0441403	0.0220702	−	0.999756i	$-0.492974\pi$
$461$	0.947733	0.0441403	0.0220702	+	0.999756i	$0.492974\pi$
$462$	−16.1118	−0.749587
$463$	−39.8964	−1.85414	−0.927072	−	0.374882i	$-0.877683\pi$
$463$	−39.8964	−1.85414	−0.927072	+	0.374882i	$0.877683\pi$
$464$	−4.16545	−0.193376
$465$	0	0
$466$	6.92212	0.320661
$467$	32.3690	1.49786	0.748929	−	0.662650i	$-0.230568\pi$
$467$	32.3690	1.49786	0.748929	+	0.662650i	$0.230568\pi$
$468$	−5.70798	−0.263851
$469$	22.1222	1.02151
$470$	0	0
$471$	−14.9098	−0.687007
$472$	19.3928	0.892625
$473$	46.8793	2.15551
$474$	−3.64368	−0.167360
$475$	0	0
$476$	40.6734	1.86426
$477$	5.47773	0.250808
$478$	8.12337	0.371554
$479$	5.49279	0.250972	0.125486	−	0.992095i	$-0.459951\pi$
$479$	5.49279	0.250972	0.125486	+	0.992095i	$0.459951\pi$
$480$	0	0
$481$	14.6271	0.666936
$482$	−0.409457	−0.0186502
$483$	51.7264	2.35363
$484$	−37.3401	−1.69728
$485$	0	0
$486$	−2.85059	−0.129305
$487$	−4.75666	−0.215545	−0.107772	−	0.994176i	$-0.534372\pi$
$487$	−4.75666	−0.215545	−0.107772	+	0.994176i	$0.534372\pi$
$488$	17.7013	0.801299
$489$	−16.1227	−0.729094
$490$	0	0
$491$	4.80239	0.216729	0.108365	−	0.994111i	$-0.465439\pi$
$491$	4.80239	0.216729	0.108365	+	0.994111i	$0.465439\pi$
$492$	−31.8463	−1.43574
$493$	−8.35758	−0.376407
$494$	6.86990	0.309092
$495$	0	0
$496$	7.93980	0.356508
$497$	41.8988	1.87942
$498$	12.5284	0.561412
$499$	19.4936	0.872654	0.436327	−	0.899788i	$-0.356279\pi$
$499$	19.4936	0.872654	0.436327	+	0.899788i	$0.356279\pi$
$500$	0	0
$501$	−0.442595	−0.0197737
$502$	10.0001	0.446327
$503$	−16.5008	−0.735735	−0.367868	−	0.929878i	$-0.619912\pi$
$503$	−16.5008	−0.735735	−0.367868	+	0.929878i	$0.619912\pi$
$504$	−3.91903	−0.174567
$505$	0	0
$506$	−16.8894	−0.750825
$507$	−15.0598	−0.668829
$508$	−9.41106	−0.417548
$509$	−38.6586	−1.71351	−0.856755	−	0.515723i	$-0.827523\pi$
$509$	−38.6586	−1.71351	−0.856755	+	0.515723i	$0.827523\pi$
$510$	0	0
$511$	59.8797	2.64892
$512$	−22.7092	−1.00361
$513$	−16.3431	−0.721564
$514$	3.16214	0.139476
$515$	0	0
$516$	29.4259	1.29540
$517$	49.9487	2.19674
$518$	4.80171	0.210975
$519$	−15.0178	−0.659210
$520$	0	0
$521$	−36.7859	−1.61162	−0.805810	−	0.592175i	$-0.798270\pi$
$521$	−36.7859	−1.61162	−0.805810	+	0.592175i	$0.798270\pi$
$522$	0.385027	0.0168522
$523$	−11.7178	−0.512382	−0.256191	−	0.966626i	$-0.582468\pi$
$523$	−11.7178	−0.512382	−0.256191	+	0.966626i	$0.582468\pi$
$524$	−16.7722	−0.732695
$525$	0	0
$526$	6.67914	0.291224
$527$	15.9305	0.693942
$528$	−32.4857	−1.41376
$529$	31.2229	1.35752
$530$	0	0
$531$	8.43159	0.365900
$532$	−24.6480	−1.06863
$533$	−41.3550	−1.79128
$534$	−0.00900831	−0.000389827 0
$535$	0	0
$536$	−9.48282	−0.409595
$537$	−16.5891	−0.715874
$538$	6.70846	0.289222
$539$	35.8545	1.54436
$540$	0	0
$541$	−2.67948	−0.115200	−0.0576000	−	0.998340i	$-0.518345\pi$
$541$	−2.67948	−0.115200	−0.0576000	+	0.998340i	$0.518345\pi$
$542$	1.60713	0.0690320
$543$	−6.68702	−0.286967
$544$	−26.5337	−1.13762
$545$	0	0
$546$	−13.1329	−0.562037
$547$	−22.6037	−0.966464	−0.483232	−	0.875492i	$-0.660537\pi$
$547$	−22.6037	−0.966464	−0.483232	+	0.875492i	$0.660537\pi$
$548$	1.33167	0.0568860
$549$	7.69616	0.328464
$550$	0	0
$551$	5.06467	0.215762
$552$	−22.1728	−0.943736
$553$	16.9762	0.721900
$554$	−2.59550	−0.110272
$555$	0	0
$556$	−9.28485	−0.393765
$557$	19.2117	0.814026	0.407013	−	0.913422i	$-0.366570\pi$
$557$	19.2117	0.814026	0.407013	+	0.913422i	$0.366570\pi$
$558$	−0.733904	−0.0310686
$559$	38.2120	1.61619
$560$	0	0
$561$	−65.1794	−2.75188
$562$	8.86374	0.373894
$563$	5.59755	0.235909	0.117954	−	0.993019i	$-0.462366\pi$
$563$	5.59755	0.235909	0.117954	+	0.993019i	$0.462366\pi$
$564$	31.3526	1.32018
$565$	0	0
$566$	−10.0490	−0.422392
$567$	38.7349	1.62671
$568$	−17.9601	−0.753590
$569$	35.2738	1.47876	0.739378	−	0.673291i	$-0.235120\pi$
$569$	35.2738	1.47876	0.739378	+	0.673291i	$0.235120\pi$
$570$	0	0
$571$	27.2224	1.13922	0.569612	−	0.821914i	$-0.307094\pi$
$571$	27.2224	1.13922	0.569612	+	0.821914i	$0.307094\pi$
$572$	−46.8657	−1.95955
$573$	12.3484	0.515861
$574$	−13.5758	−0.566645
$575$	0	0
$576$	−2.90136	−0.120890
$577$	2.89393	0.120476	0.0602379	−	0.998184i	$-0.480814\pi$
$577$	2.89393	0.120476	0.0602379	+	0.998184i	$0.480814\pi$
$578$	−8.09443	−0.336684
$579$	−27.5723	−1.14587
$580$	0	0
$581$	−58.3709	−2.42163
$582$	9.15608	0.379532
$583$	44.9752	1.86268
$584$	−25.6678	−1.06214
$585$	0	0
$586$	2.73298	0.112899
$587$	10.4088	0.429619	0.214809	−	0.976656i	$-0.431087\pi$
$587$	10.4088	0.429619	0.214809	+	0.976656i	$0.431087\pi$
$588$	22.5057	0.928121
$589$	−9.65382	−0.397779
$590$	0	0
$591$	−4.49533	−0.184913
$592$	9.68154	0.397909
$593$	35.0305	1.43853	0.719265	−	0.694735i	$-0.244479\pi$
$593$	35.0305	1.43853	0.719265	+	0.694735i	$0.244479\pi$
$594$	−10.2011	−0.418555
$595$	0	0
$596$	−15.1273	−0.619639
$597$	−37.2631	−1.52508
$598$	−13.7668	−0.562965
$599$	28.6866	1.17210	0.586051	−	0.810274i	$-0.300682\pi$
$599$	28.6866	1.17210	0.586051	+	0.810274i	$0.300682\pi$
$600$	0	0
$601$	−36.5265	−1.48995	−0.744973	−	0.667094i	$-0.767538\pi$
$601$	−36.5265	−1.48995	−0.744973	+	0.667094i	$0.767538\pi$
$602$	12.5441	0.511258
$603$	−4.12294	−0.167899
$604$	−12.4229	−0.505482
$605$	0	0
$606$	−5.20592	−0.211476
$607$	−35.4404	−1.43848	−0.719241	−	0.694760i	$-0.755510\pi$
$607$	−35.4404	−1.43848	−0.719241	+	0.694760i	$0.755510\pi$
$608$	16.0793	0.652103
$609$	−9.68193	−0.392332
$610$	0	0
$611$	40.7139	1.64711
$612$	−7.58034	−0.306417
$613$	−24.9873	−1.00923	−0.504614	−	0.863345i	$-0.668365\pi$
$613$	−24.9873	−1.00923	−0.504614	+	0.863345i	$0.668365\pi$
$614$	−7.86986	−0.317602
$615$	0	0
$616$	−32.1774	−1.29647
$617$	15.9569	0.642400	0.321200	−	0.947011i	$-0.395914\pi$
$617$	15.9569	0.642400	0.321200	+	0.947011i	$0.395914\pi$
$618$	−9.66431	−0.388756
$619$	23.5518	0.946627	0.473313	−	0.880894i	$-0.343058\pi$
$619$	23.5518	0.946627	0.473313	+	0.880894i	$0.343058\pi$
$620$	0	0
$621$	32.7503	1.31422
$622$	−12.1193	−0.485939
$623$	0.0419704	0.00168151
$624$	−26.4795	−1.06003
$625$	0	0
$626$	9.52698	0.380775
$627$	39.4986	1.57742
$628$	−14.2371	−0.568123
$629$	19.4251	0.774530
$630$	0	0
$631$	24.0264	0.956476	0.478238	−	0.878230i	$-0.341276\pi$
$631$	24.0264	0.956476	0.478238	+	0.878230i	$0.341276\pi$
$632$	−7.27692	−0.289461
$633$	−18.2959	−0.727197
$634$	−7.21502	−0.286545
$635$	0	0
$636$	28.2308	1.11942
$637$	29.2255	1.15796
$638$	3.16129	0.125157
$639$	−7.80870	−0.308908
$640$	0	0
$641$	−9.84658	−0.388917	−0.194458	−	0.980911i	$-0.562295\pi$
$641$	−9.84658	−0.388917	−0.194458	+	0.980911i	$0.562295\pi$
$642$	−14.8698	−0.586865
$643$	−2.33784	−0.0921956	−0.0460978	−	0.998937i	$-0.514679\pi$
$643$	−2.33784	−0.0921956	−0.0460978	+	0.998937i	$0.514679\pi$
$644$	49.3927	1.94635
$645$	0	0
$646$	9.12341	0.358956
$647$	7.30145	0.287050	0.143525	−	0.989647i	$-0.454156\pi$
$647$	7.30145	0.287050	0.143525	+	0.989647i	$0.454156\pi$
$648$	−16.6039	−0.652264
$649$	69.2281	2.71744
$650$	0	0
$651$	18.4548	0.723302
$652$	−15.3953	−0.602927
$653$	36.0105	1.40920	0.704601	−	0.709604i	$-0.251126\pi$
$653$	36.0105	1.40920	0.704601	+	0.709604i	$0.251126\pi$
$654$	−3.64959	−0.142710
$655$	0	0
$656$	−27.3725	−1.06872
$657$	−11.1598	−0.435386
$658$	13.3654	0.521037
$659$	10.2605	0.399694	0.199847	−	0.979827i	$-0.435955\pi$
$659$	10.2605	0.399694	0.199847	+	0.979827i	$0.435955\pi$
$660$	0	0
$661$	−1.07417	−0.0417805	−0.0208903	−	0.999782i	$-0.506650\pi$
$661$	−1.07417	−0.0417805	−0.0208903	+	0.999782i	$0.506650\pi$
$662$	−9.76863	−0.379669
$663$	−53.1286	−2.06335
$664$	25.0210	0.971003
$665$	0	0
$666$	−0.894899	−0.0346766
$667$	−10.1492	−0.392979
$668$	−0.422627	−0.0163519
$669$	7.73923	0.299216
$670$	0	0
$671$	63.1897	2.43941
$672$	−30.7382	−1.18575
$673$	−38.2772	−1.47548	−0.737738	−	0.675087i	$-0.764106\pi$
$673$	−38.2772	−1.47548	−0.737738	+	0.675087i	$0.764106\pi$
$674$	−10.7598	−0.414451
$675$	0	0
$676$	−14.3804	−0.553091
$677$	−44.5908	−1.71376	−0.856882	−	0.515512i	$-0.827602\pi$
$677$	−44.5908	−1.71376	−0.856882	+	0.515512i	$0.827602\pi$
$678$	−4.11267	−0.157946
$679$	−42.6589	−1.63710
$680$	0	0
$681$	18.6611	0.715094
$682$	−6.02576	−0.230738
$683$	37.5648	1.43738	0.718689	−	0.695332i	$-0.244743\pi$
$683$	37.5648	1.43738	0.718689	+	0.695332i	$0.244743\pi$
$684$	4.59366	0.175643
$685$	0	0
$686$	−0.898246	−0.0342952
$687$	5.73291	0.218724
$688$	25.2922	0.964257
$689$	36.6599	1.39663
$690$	0	0
$691$	37.1653	1.41383	0.706917	−	0.707297i	$-0.250086\pi$
$691$	37.1653	1.41383	0.706917	+	0.707297i	$0.250086\pi$
$692$	−14.3403	−0.545136
$693$	−13.9901	−0.531440
$694$	−4.61532	−0.175195
$695$	0	0
$696$	4.15021	0.157313
$697$	−54.9204	−2.08026
$698$	8.59614	0.325369
$699$	32.4405	1.22701
$700$	0	0
$701$	7.25521	0.274026	0.137013	−	0.990569i	$-0.456250\pi$
$701$	7.25521	0.274026	0.137013	+	0.990569i	$0.456250\pi$
$702$	−8.31504	−0.313831
$703$	−11.7716	−0.443973
$704$	−23.8218	−0.897819
$705$	0	0
$706$	11.3399	0.426783
$707$	24.2548	0.912195
$708$	43.4542	1.63311
$709$	−7.32664	−0.275158	−0.137579	−	0.990491i	$-0.543932\pi$
$709$	−7.32664	−0.275158	−0.137579	+	0.990491i	$0.543932\pi$
$710$	0	0
$711$	−3.16386	−0.118654
$712$	−0.0179908	−0.000674235 0
$713$	19.3455	0.724496
$714$	−17.4409	−0.652708
$715$	0	0
$716$	−15.8407	−0.591995
$717$	38.0702	1.42176
$718$	−9.38754	−0.350340
$719$	0.930727	0.0347103	0.0173551	−	0.999849i	$-0.494475\pi$
$719$	0.930727	0.0347103	0.0173551	+	0.999849i	$0.494475\pi$
$720$	0	0
$721$	45.0268	1.67688
$722$	2.25091	0.0837704
$723$	−1.91892	−0.0713653
$724$	−6.38533	−0.237309
$725$	0	0
$726$	16.0115	0.594243
$727$	−48.4086	−1.79537	−0.897687	−	0.440634i	$-0.854754\pi$
$727$	−48.4086	−1.79537	−0.897687	+	0.440634i	$0.854754\pi$
$728$	−26.2283	−0.972084
$729$	18.3846	0.680910
$730$	0	0
$731$	50.7465	1.87693
$732$	39.6639	1.46602
$733$	−6.34004	−0.234175	−0.117087	−	0.993122i	$-0.537356\pi$
$733$	−6.34004	−0.234175	−0.117087	+	0.993122i	$0.537356\pi$
$734$	−12.7854	−0.471917
$735$	0	0
$736$	−32.2218	−1.18771
$737$	−33.8516	−1.24694
$738$	2.53014	0.0931357
$739$	−17.2160	−0.633299	−0.316650	−	0.948543i	$-0.602558\pi$
$739$	−17.2160	−0.633299	−0.316650	+	0.948543i	$0.602558\pi$
$740$	0	0
$741$	32.1958	1.18274
$742$	12.0346	0.441803
$743$	−14.2958	−0.524460	−0.262230	−	0.965005i	$-0.584458\pi$
$743$	−14.2958	−0.524460	−0.262230	+	0.965005i	$0.584458\pi$
$744$	−7.91076	−0.290023
$745$	0	0
$746$	−1.23378	−0.0451718
$747$	10.8786	0.398028
$748$	−62.2388	−2.27568
$749$	69.2797	2.53143
$750$	0	0
$751$	−40.3232	−1.47141	−0.735707	−	0.677300i	$-0.763150\pi$
$751$	−40.3232	−1.47141	−0.735707	+	0.677300i	$0.763150\pi$
$752$	26.9482	0.982701
$753$	46.8656	1.70788
$754$	2.57681	0.0938418
$755$	0	0
$756$	29.8329	1.08501
$757$	45.1835	1.64222	0.821110	−	0.570769i	$-0.193355\pi$
$757$	45.1835	1.64222	0.821110	+	0.570769i	$0.193355\pi$
$758$	5.98484	0.217379
$759$	−79.1521	−2.87304
$760$	0	0
$761$	−6.47748	−0.234809	−0.117404	−	0.993084i	$-0.537457\pi$
$761$	−6.47748	−0.234809	−0.117404	+	0.993084i	$0.537457\pi$
$762$	4.03548	0.146190
$763$	17.0037	0.615576
$764$	11.7913	0.426593
$765$	0	0
$766$	−0.235505	−0.00850913
$767$	56.4287	2.03752
$768$	−8.07659	−0.291439
$769$	−19.5543	−0.705144	−0.352572	−	0.935785i	$-0.614693\pi$
$769$	−19.5543	−0.705144	−0.352572	+	0.935785i	$0.614693\pi$
$770$	0	0
$771$	14.8194	0.533707
$772$	−26.3284	−0.947578
$773$	−48.2155	−1.73419	−0.867095	−	0.498143i	$-0.834015\pi$
$773$	−48.2155	−1.73419	−0.867095	+	0.498143i	$0.834015\pi$
$774$	−2.33785	−0.0840322
$775$	0	0
$776$	18.2860	0.656427
$777$	22.5032	0.807299
$778$	13.7812	0.494081
$779$	33.2816	1.19244
$780$	0	0
$781$	−64.1138	−2.29417
$782$	−18.2826	−0.653785
$783$	−6.13006	−0.219071
$784$	19.3442	0.690863
$785$	0	0
$786$	7.19194	0.256528
$787$	−51.8867	−1.84956	−0.924781	−	0.380500i	$-0.875752\pi$
$787$	−51.8867	−1.84956	−0.924781	+	0.380500i	$0.875752\pi$
$788$	−4.29252	−0.152915
$789$	31.3018	1.11437
$790$	0	0
$791$	19.1613	0.681296
$792$	5.99693	0.213092
$793$	51.5068	1.82906
$794$	8.43219	0.299247
$795$	0	0
$796$	−35.5819	−1.26117
$797$	−10.2561	−0.363289	−0.181644	−	0.983364i	$-0.558142\pi$
$797$	−10.2561	−0.363289	−0.181644	+	0.983364i	$0.558142\pi$
$798$	10.5691	0.374142
$799$	54.0691	1.91283
$800$	0	0
$801$	−0.00782205	−0.000276379 0
$802$	−8.92059	−0.314997
$803$	−91.6284	−3.23350
$804$	−21.2485	−0.749378
$805$	0	0
$806$	−4.91168	−0.173007
$807$	31.4392	1.10671
$808$	−10.3969	−0.365763
$809$	−30.7169	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$809$	−30.7169	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$810$	0	0
$811$	31.6810	1.11247	0.556235	−	0.831025i	$-0.312245\pi$
$811$	31.6810	1.11247	0.556235	+	0.831025i	$0.312245\pi$
$812$	−9.24513	−0.324440
$813$	7.53180	0.264152
$814$	−7.34762	−0.257534
$815$	0	0
$816$	−35.1654	−1.23104
$817$	−30.7522	−1.07588
$818$	10.4003	0.363637
$819$	−11.4035	−0.398471
$820$	0	0
$821$	25.3540	0.884861	0.442430	−	0.896803i	$-0.354116\pi$
$821$	25.3540	0.884861	0.442430	+	0.896803i	$0.354116\pi$
$822$	−0.571021	−0.0199167
$823$	−37.7181	−1.31477	−0.657384	−	0.753555i	$-0.728337\pi$
$823$	−37.7181	−1.31477	−0.657384	+	0.753555i	$0.728337\pi$
$824$	−19.3010	−0.672381
$825$	0	0
$826$	18.5242	0.644539
$827$	14.4868	0.503756	0.251878	−	0.967759i	$-0.418952\pi$
$827$	14.4868	0.503756	0.251878	+	0.967759i	$0.418952\pi$
$828$	−9.20535	−0.319908
$829$	32.9697	1.14509	0.572543	−	0.819875i	$-0.305957\pi$
$829$	32.9697	1.14509	0.572543	+	0.819875i	$0.305957\pi$
$830$	0	0
$831$	−12.1638	−0.421958
$832$	−19.4175	−0.673181
$833$	38.8122	1.34476
$834$	3.98136	0.137863
$835$	0	0
$836$	37.7166	1.30445
$837$	11.6846	0.403878
$838$	−3.47507	−0.120044
$839$	3.56548	0.123094	0.0615470	−	0.998104i	$-0.480397\pi$
$839$	3.56548	0.123094	0.0615470	+	0.998104i	$0.480397\pi$
$840$	0	0
$841$	−27.1003	−0.934493
$842$	2.48783	0.0857364
$843$	41.5399	1.43071
$844$	−17.4705	−0.601358
$845$	0	0
$846$	−2.49092	−0.0856395
$847$	−74.5989	−2.56325
$848$	24.2649	0.833261
$849$	−47.0948	−1.61629
$850$	0	0
$851$	23.5893	0.808631
$852$	−40.2440	−1.37874
$853$	45.4884	1.55749	0.778747	−	0.627339i	$-0.215856\pi$
$853$	45.4884	1.55749	0.778747	+	0.627339i	$0.215856\pi$
$854$	16.9085	0.578595
$855$	0	0
$856$	−29.6971	−1.01503
$857$	−10.1348	−0.346200	−0.173100	−	0.984904i	$-0.555378\pi$
$857$	−10.1348	−0.346200	−0.173100	+	0.984904i	$0.555378\pi$
$858$	20.0961	0.686070
$859$	39.2120	1.33790	0.668949	−	0.743309i	$-0.266745\pi$
$859$	39.2120	1.33790	0.668949	+	0.743309i	$0.266745\pi$
$860$	0	0
$861$	−63.6232	−2.16827
$862$	3.88627	0.132367
$863$	7.50783	0.255569	0.127785	−	0.991802i	$-0.459213\pi$
$863$	7.50783	0.255569	0.127785	+	0.991802i	$0.459213\pi$
$864$	−19.4618	−0.662102
$865$	0	0
$866$	−11.4456	−0.388937
$867$	−37.9346	−1.28833
$868$	17.6222	0.598137
$869$	−25.9771	−0.881212
$870$	0	0
$871$	−27.5929	−0.934951
$872$	−7.28873	−0.246828
$873$	7.95037	0.269079
$874$	11.0792	0.374760
$875$	0	0
$876$	−57.5148	−1.94324
$877$	36.4998	1.23251	0.616255	−	0.787547i	$-0.288649\pi$
$877$	36.4998	1.23251	0.616255	+	0.787547i	$0.288649\pi$
$878$	6.71061	0.226472
$879$	12.8081	0.432007
$880$	0	0
$881$	19.5062	0.657182	0.328591	−	0.944472i	$-0.393426\pi$
$881$	19.5062	0.657182	0.328591	+	0.944472i	$0.393426\pi$
$882$	−1.78805	−0.0602067
$883$	42.6299	1.43461	0.717305	−	0.696759i	$-0.245375\pi$
$883$	42.6299	1.43461	0.717305	+	0.696759i	$0.245375\pi$
$884$	−50.7317	−1.70629
$885$	0	0
$886$	0.584253	0.0196284
$887$	−51.4853	−1.72871	−0.864353	−	0.502885i	$-0.832272\pi$
$887$	−51.4853	−1.72871	−0.864353	+	0.502885i	$0.832272\pi$
$888$	−9.64613	−0.323703
$889$	−18.8016	−0.630587
$890$	0	0
$891$	−59.2725	−1.98570
$892$	7.39007	0.247438
$893$	−32.7657	−1.09646
$894$	6.48663	0.216945
$895$	0	0
$896$	−38.4114	−1.28323
$897$	−64.5180	−2.15419
$898$	2.42508	0.0809261
$899$	−3.62102	−0.120768
$900$	0	0
$901$	48.6853	1.62194
$902$	20.7739	0.691694
$903$	58.7878	1.95634
$904$	−8.21358	−0.273180
$905$	0	0
$906$	5.32698	0.176977
$907$	26.9411	0.894564	0.447282	−	0.894393i	$-0.352392\pi$
$907$	26.9411	0.894564	0.447282	+	0.894393i	$0.352392\pi$
$908$	17.8192	0.591350
$909$	−4.52038	−0.149932
$910$	0	0
$911$	3.36348	0.111437	0.0557186	−	0.998447i	$-0.482255\pi$
$911$	3.36348	0.111437	0.0557186	+	0.998447i	$0.482255\pi$
$912$	21.3102	0.705650
$913$	89.3197	2.95605
$914$	−2.85060	−0.0942895
$915$	0	0
$916$	5.47427	0.180875
$917$	−33.5078	−1.10652
$918$	−11.0426	−0.364460
$919$	−31.8917	−1.05201	−0.526004	−	0.850482i	$-0.676310\pi$
$919$	−31.8917	−1.05201	−0.526004	+	0.850482i	$0.676310\pi$
$920$	0	0
$921$	−36.8821	−1.21531
$922$	−0.388055	−0.0127799
$923$	−52.2600	−1.72016
$924$	−72.1013	−2.37196
$925$	0	0
$926$	16.3359	0.536830
$927$	−8.39167	−0.275619
$928$	6.03114	0.197982
$929$	−4.66746	−0.153134	−0.0765672	−	0.997064i	$-0.524396\pi$
$929$	−4.66746	−0.153134	−0.0765672	+	0.997064i	$0.524396\pi$
$930$	0	0
$931$	−23.5201	−0.770841
$932$	30.9770	1.01468
$933$	−56.7971	−1.85945
$934$	−13.2537	−0.433674
$935$	0	0
$936$	4.88818	0.159775
$937$	−57.7244	−1.88577	−0.942887	−	0.333112i	$-0.891901\pi$
$937$	−57.7244	−1.88577	−0.942887	+	0.333112i	$0.891901\pi$
$938$	−9.05810	−0.295757
$939$	44.6482	1.45704
$940$	0	0
$941$	−10.1883	−0.332130	−0.166065	−	0.986115i	$-0.553106\pi$
$941$	−10.1883	−0.332130	−0.166065	+	0.986115i	$0.553106\pi$
$942$	6.10491	0.198909
$943$	−66.6939	−2.17185
$944$	37.3498	1.21563
$945$	0	0
$946$	−19.1950	−0.624085
$947$	13.0419	0.423804	0.211902	−	0.977291i	$-0.432034\pi$
$947$	13.0419	0.423804	0.211902	+	0.977291i	$0.432034\pi$
$948$	−16.3057	−0.529584
$949$	−74.6876	−2.42446
$950$	0	0
$951$	−33.8132	−1.09647
$952$	−34.8318	−1.12891
$953$	−37.6240	−1.21876	−0.609380	−	0.792878i	$-0.708582\pi$
$953$	−37.6240	−1.21876	−0.609380	+	0.792878i	$0.708582\pi$
$954$	−2.24289	−0.0726163
$955$	0	0
$956$	36.3526	1.17573
$957$	14.8154	0.478913
$958$	−2.24906	−0.0726638
$959$	2.66043	0.0859099
$960$	0	0
$961$	−24.0979	−0.777353
$962$	−5.98914	−0.193098
$963$	−12.9117	−0.416074
$964$	−1.83235	−0.0590159
$965$	0	0
$966$	−21.1797	−0.681446
$967$	24.0166	0.772322	0.386161	−	0.922431i	$-0.373801\pi$
$967$	24.0166	0.772322	0.386161	+	0.922431i	$0.373801\pi$
$968$	31.9772	1.02779
$969$	42.7569	1.37355
$970$	0	0
$971$	−4.90623	−0.157448	−0.0787241	−	0.996896i	$-0.525085\pi$
$971$	−4.90623	−0.157448	−0.0787241	+	0.996896i	$0.525085\pi$
$972$	−12.7566	−0.409167
$973$	−18.5495	−0.594669
$974$	1.94765	0.0624066
$975$	0	0
$976$	34.0920	1.09126
$977$	−21.1828	−0.677697	−0.338849	−	0.940841i	$-0.610037\pi$
$977$	−21.1828	−0.677697	−0.338849	+	0.940841i	$0.610037\pi$
$978$	6.60155	0.211094
$979$	−0.0642234	−0.00205259
$980$	0	0
$981$	−3.16900	−0.101178
$982$	−1.96637	−0.0627494
$983$	29.2440	0.932738	0.466369	−	0.884590i	$-0.345562\pi$
$983$	29.2440	0.932738	0.466369	+	0.884590i	$0.345562\pi$
$984$	27.2724	0.869413
$985$	0	0
$986$	3.42207	0.108981
$987$	62.6369	1.99376
$988$	30.7433	0.978074
$989$	61.6251	1.95956
$990$	0	0
$991$	34.4279	1.09364	0.546819	−	0.837251i	$-0.315839\pi$
$991$	34.4279	1.09364	0.546819	+	0.837251i	$0.315839\pi$
$992$	−11.4960	−0.364999
$993$	−45.7807	−1.45281
$994$	−17.1557	−0.544146
$995$	0	0
$996$	56.0656	1.77650
$997$	−25.8804	−0.819639	−0.409820	−	0.912167i	$-0.634408\pi$
$997$	−25.8804	−0.819639	−0.409820	+	0.912167i	$0.634408\pi$
$998$	−7.98179	−0.252659
$999$	14.2478	0.450780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.n.1.16	yes	40
5.4	even	2		6025.2.a.m.1.25	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6025.2.a.m.1.25	✓	40	5.4	even	2
6025.2.a.n.1.16	yes	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.409457 q^{2} -1.91892 q^{3} -1.83235 q^{4} +0.785714 q^{6} -3.66070 q^{7} +1.56918 q^{8} +0.682248 q^{9} +O(q^{10})\)	\(q-0.409457 q^{2} -1.91892 q^{3} -1.83235 q^{4} +0.785714 q^{6} -3.66070 q^{7} +1.56918 q^{8} +0.682248 q^{9} +5.60163 q^{11} +3.51612 q^{12} +4.56597 q^{13} +1.49890 q^{14} +3.02218 q^{16} +6.06372 q^{17} -0.279351 q^{18} -3.67460 q^{19} +7.02459 q^{21} -2.29363 q^{22} +7.36362 q^{23} -3.01113 q^{24} -1.86957 q^{26} +4.44758 q^{27} +6.70767 q^{28} -1.37829 q^{29} +2.62718 q^{31} -4.37581 q^{32} -10.7491 q^{33} -2.48283 q^{34} -1.25011 q^{36} +3.20350 q^{37} +1.50459 q^{38} -8.76172 q^{39} -9.05721 q^{41} -2.87626 q^{42} +8.36887 q^{43} -10.2641 q^{44} -3.01508 q^{46} +8.91682 q^{47} -5.79932 q^{48} +6.40073 q^{49} -11.6358 q^{51} -8.36643 q^{52} +8.02895 q^{53} -1.82109 q^{54} -5.74429 q^{56} +7.05126 q^{57} +0.564351 q^{58} +12.3585 q^{59} +11.2806 q^{61} -1.07571 q^{62} -2.49750 q^{63} -4.25266 q^{64} +4.40128 q^{66} -6.04317 q^{67} -11.1108 q^{68} -14.1302 q^{69} -11.4456 q^{71} +1.07057 q^{72} -16.3574 q^{73} -1.31169 q^{74} +6.73313 q^{76} -20.5059 q^{77} +3.58754 q^{78} -4.63741 q^{79} -10.5813 q^{81} +3.70854 q^{82} +15.9453 q^{83} -12.8715 q^{84} -3.42669 q^{86} +2.64483 q^{87} +8.78997 q^{88} -0.0114651 q^{89} -16.7146 q^{91} -13.4927 q^{92} -5.04134 q^{93} -3.65105 q^{94} +8.39682 q^{96} +11.6532 q^{97} -2.62082 q^{98} +3.82170 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9}+O(q^{10}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9	\( 40 q + 9 q^{2} + 8 q^{3} + 41 q^{4} + 3 q^{6} + 20 q^{7} + 27 q^{8} + 38 q^{9} + q^{11} + 26 q^{12} + 11 q^{13} - q^{14} + 43 q^{16} + 20 q^{17} + 18 q^{18} + 2 q^{21} + 23 q^{22} + 79 q^{23} - 2 q^{24} + 2 q^{26} + 26 q^{27} + 30 q^{28} + 2 q^{29} + q^{31} + 68 q^{32} + 20 q^{33} + 5 q^{34} + 32 q^{36} + 16 q^{37} + 45 q^{38} - 2 q^{39} - 2 q^{41} + 19 q^{42} + 25 q^{43} + 3 q^{44} + 14 q^{46} + 88 q^{47} + 75 q^{48} + 40 q^{49} - 10 q^{51} + 18 q^{52} + 34 q^{53} + 4 q^{54} - 15 q^{56} + 51 q^{57} + 53 q^{58} + q^{59} + 9 q^{61} + 39 q^{62} + 110 q^{63} + 17 q^{64} + 26 q^{66} + 30 q^{67} + 44 q^{68} - 7 q^{69} + 5 q^{71} + 18 q^{72} + 23 q^{73} - 18 q^{74} + 43 q^{76} + 30 q^{77} + 46 q^{78} + 5 q^{79} + 44 q^{81} + 5 q^{82} + 65 q^{83} - 65 q^{84} + 40 q^{86} + 33 q^{87} + 71 q^{88} - 9 q^{89} + q^{91} + 117 q^{92} + 68 q^{93} - 72 q^{94} + 83 q^{96} - 8 q^{97} + 76 q^{98} - 40 q^{99}+O(q^{100}) \) 40 * q + 9 * q^2 + 8 * q^3 + 41 * q^4 + 3 * q^6 + 20 * q^7 + 27 * q^8 + 38 * q^9 + q^11 + 26 * q^12 + 11 * q^13 - q^14 + 43 * q^16 + 20 * q^17 + 18 * q^18 + 2 * q^21 + 23 * q^22 + 79 * q^23 - 2 * q^24 + 2 * q^26 + 26 * q^27 + 30 * q^28 + 2 * q^29 + q^31 + 68 * q^32 + 20 * q^33 + 5 * q^34 + 32 * q^36 + 16 * q^37 + 45 * q^38 - 2 * q^39 - 2 * q^41 + 19 * q^42 + 25 * q^43 + 3 * q^44 + 14 * q^46 + 88 * q^47 + 75 * q^48 + 40 * q^49 - 10 * q^51 + 18 * q^52 + 34 * q^53 + 4 * q^54 - 15 * q^56 + 51 * q^57 + 53 * q^58 + q^59 + 9 * q^61 + 39 * q^62 + 110 * q^63 + 17 * q^64 + 26 * q^66 + 30 * q^67 + 44 * q^68 - 7 * q^69 + 5 * q^71 + 18 * q^72 + 23 * q^73 - 18 * q^74 + 43 * q^76 + 30 * q^77 + 46 * q^78 + 5 * q^79 + 44 * q^81 + 5 * q^82 + 65 * q^83 - 65 * q^84 + 40 * q^86 + 33 * q^87 + 71 * q^88 - 9 * q^89 + q^91 + 117 * q^92 + 68 * q^93 - 72 * q^94 + 83 * q^96 - 8 * q^97 + 76 * q^98 - 40 * q^99

Introduction

L-functions

Modular forms

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Database

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