LMFDB - Embedded newform 1925.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.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:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 6 $ x^7 - x^6 - 13x^5 + 12x^4 + 47x^3 - 37x^2 - 35*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.151896$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.151896 q^{2} -2.21537 q^{3} -1.97693 q^{4} +0.336506 q^{6} -1.00000 q^{7} +0.604080 q^{8} +1.90787 q^{9} +O(q^{10})$	\(q-0.151896 q^{2} -2.21537 q^{3} -1.97693 q^{4} +0.336506 q^{6} -1.00000 q^{7} +0.604080 q^{8} +1.90787 q^{9} +1.00000 q^{11} +4.37963 q^{12} -1.09806 q^{13} +0.151896 q^{14} +3.86210 q^{16} -5.71231 q^{17} -0.289798 q^{18} -0.976928 q^{19} +2.21537 q^{21} -0.151896 q^{22} -6.89879 q^{23} -1.33826 q^{24} +0.166791 q^{26} +2.41947 q^{27} +1.97693 q^{28} -9.78172 q^{29} -4.56229 q^{31} -1.79480 q^{32} -2.21537 q^{33} +0.867678 q^{34} -3.77172 q^{36} +1.12882 q^{37} +0.148391 q^{38} +2.43262 q^{39} +7.19315 q^{41} -0.336506 q^{42} -5.03482 q^{43} -1.97693 q^{44} +1.04790 q^{46} -1.17545 q^{47} -8.55598 q^{48} +1.00000 q^{49} +12.6549 q^{51} +2.17079 q^{52} -10.7836 q^{53} -0.367508 q^{54} -0.604080 q^{56} +2.16426 q^{57} +1.48581 q^{58} +4.29833 q^{59} +9.30798 q^{61} +0.692994 q^{62} -1.90787 q^{63} -7.45157 q^{64} +0.336506 q^{66} -4.00012 q^{67} +11.2928 q^{68} +15.2834 q^{69} +9.90832 q^{71} +1.15251 q^{72} +3.24173 q^{73} -0.171464 q^{74} +1.93132 q^{76} -1.00000 q^{77} -0.369505 q^{78} +13.8523 q^{79} -11.0836 q^{81} -1.09261 q^{82} -1.37098 q^{83} -4.37963 q^{84} +0.764770 q^{86} +21.6701 q^{87} +0.604080 q^{88} +4.13811 q^{89} +1.09806 q^{91} +13.6384 q^{92} +10.1072 q^{93} +0.178546 q^{94} +3.97614 q^{96} -16.2236 q^{97} -0.151896 q^{98} +1.90787 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9}+O(q^{10}) $ 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9	$ 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9} + 7 q^{11} + 9 q^{12} + 3 q^{13} + q^{14} + 21 q^{16} + 2 q^{17} - 8 q^{18} + 20 q^{19} - q^{22} - 11 q^{23} + 18 q^{24} - 13 q^{26} + 12 q^{27} - 13 q^{28} + 4 q^{29} + 6 q^{31} - q^{32} - 7 q^{34} + 12 q^{36} - 19 q^{37} - 3 q^{38} - 10 q^{39} + 24 q^{41} - 3 q^{42} + 13 q^{44} + 33 q^{46} + q^{47} + 15 q^{48} + 7 q^{49} + 19 q^{51} + 29 q^{52} - 7 q^{53} + 9 q^{54} + 9 q^{57} - 37 q^{58} + 9 q^{59} + 18 q^{61} + 40 q^{62} - 9 q^{63} + 8 q^{64} + 3 q^{66} - 18 q^{68} - 15 q^{69} + 18 q^{71} + 64 q^{72} - 5 q^{73} - 24 q^{74} + 88 q^{76} - 7 q^{77} - 79 q^{78} + 25 q^{79} - q^{81} + 60 q^{82} + 17 q^{83} - 9 q^{84} - 41 q^{86} + 24 q^{87} - 16 q^{89} - 3 q^{91} - 28 q^{92} - 26 q^{93} - 31 q^{94} + 17 q^{96} + 4 q^{97} - q^{98} + 9 q^{99}+O(q^{100}) $ 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9 + 7 * q^11 + 9 * q^12 + 3 * q^13 + q^14 + 21 * q^16 + 2 * q^17 - 8 * q^18 + 20 * q^19 - q^22 - 11 * q^23 + 18 * q^24 - 13 * q^26 + 12 * q^27 - 13 * q^28 + 4 * q^29 + 6 * q^31 - q^32 - 7 * q^34 + 12 * q^36 - 19 * q^37 - 3 * q^38 - 10 * q^39 + 24 * q^41 - 3 * q^42 + 13 * q^44 + 33 * q^46 + q^47 + 15 * q^48 + 7 * q^49 + 19 * q^51 + 29 * q^52 - 7 * q^53 + 9 * q^54 + 9 * q^57 - 37 * q^58 + 9 * q^59 + 18 * q^61 + 40 * q^62 - 9 * q^63 + 8 * q^64 + 3 * q^66 - 18 * q^68 - 15 * q^69 + 18 * q^71 + 64 * q^72 - 5 * q^73 - 24 * q^74 + 88 * q^76 - 7 * q^77 - 79 * q^78 + 25 * q^79 - q^81 + 60 * q^82 + 17 * q^83 - 9 * q^84 - 41 * q^86 + 24 * q^87 - 16 * q^89 - 3 * q^91 - 28 * q^92 - 26 * q^93 - 31 * q^94 + 17 * q^96 + 4 * q^97 - q^98 + 9 * 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 $

$2$	−0.151896	−0.107407	−0.0537034	−	0.998557i	$-0.517103\pi$
$2$	−0.151896	−0.107407	−0.0537034	+	0.998557i	$0.517103\pi$
$3$	−2.21537	−1.27905	−0.639523	−	0.768772i	$-0.720868\pi$
$3$	−2.21537	−1.27905	−0.639523	+	0.768772i	$0.720868\pi$
$4$	−1.97693	−0.988464
$5$	0	0
$5$	0	0
$6$	0.336506	0.137378
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.604080	0.213574
$9$	1.90787	0.635957
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	4.37963	1.26429
$13$	−1.09806	−0.304548	−0.152274	−	0.988338i	$-0.548660\pi$
$13$	−1.09806	−0.304548	−0.152274	+	0.988338i	$0.548660\pi$
$14$	0.151896	0.0405959
$15$	0	0
$16$	3.86210	0.965524
$17$	−5.71231	−1.38544	−0.692719	−	0.721207i	$-0.743588\pi$
$17$	−5.71231	−1.38544	−0.692719	+	0.721207i	$0.743588\pi$
$18$	−0.289798	−0.0683061
$19$	−0.976928	−0.224123	−0.112061	−	0.993701i	$-0.535745\pi$
$19$	−0.976928	−0.224123	−0.112061	+	0.993701i	$0.535745\pi$
$20$	0	0
$21$	2.21537	0.483434
$22$	−0.151896	−0.0323844
$23$	−6.89879	−1.43850	−0.719249	−	0.694752i	$-0.755514\pi$
$23$	−6.89879	−1.43850	−0.719249	+	0.694752i	$0.755514\pi$
$24$	−1.33826	−0.273171
$25$	0	0
$26$	0.166791	0.0327105
$27$	2.41947	0.465627
$28$	1.97693	0.373604
$29$	−9.78172	−1.81642	−0.908210	−	0.418515i	$-0.862551\pi$
$29$	−9.78172	−1.81642	−0.908210	+	0.418515i	$0.862551\pi$
$30$	0	0
$31$	−4.56229	−0.819411	−0.409705	−	0.912218i	$-0.634369\pi$
$31$	−4.56229	−0.819411	−0.409705	+	0.912218i	$0.634369\pi$
$32$	−1.79480	−0.317278
$33$	−2.21537	−0.385647
$34$	0.867678	0.148806
$35$	0	0
$36$	−3.77172	−0.628621
$37$	1.12882	0.185577	0.0927887	−	0.995686i	$-0.470422\pi$
$37$	1.12882	0.185577	0.0927887	+	0.995686i	$0.470422\pi$
$38$	0.148391	0.0240723
$39$	2.43262	0.389530
$40$	0	0
$41$	7.19315	1.12338	0.561690	−	0.827348i	$-0.310151\pi$
$41$	7.19315	1.12338	0.561690	+	0.827348i	$0.310151\pi$
$42$	−0.336506	−0.0519241
$43$	−5.03482	−0.767803	−0.383902	−	0.923374i	$-0.625420\pi$
$43$	−5.03482	−0.767803	−0.383902	+	0.923374i	$0.625420\pi$
$44$	−1.97693	−0.298033
$45$	0	0
$46$	1.04790	0.154504
$47$	−1.17545	−0.171456	−0.0857282	−	0.996319i	$-0.527322\pi$
$47$	−1.17545	−0.171456	−0.0857282	+	0.996319i	$0.527322\pi$
$48$	−8.55598	−1.23495
$49$	1.00000	0.142857
$50$	0	0
$51$	12.6549	1.77204
$52$	2.17079	0.301034
$53$	−10.7836	−1.48124	−0.740620	−	0.671924i	$-0.765468\pi$
$53$	−10.7836	−1.48124	−0.740620	+	0.671924i	$0.765468\pi$
$54$	−0.367508	−0.0500115
$55$	0	0
$56$	−0.604080	−0.0807236
$57$	2.16426	0.286663
$58$	1.48581	0.195096
$59$	4.29833	0.559596	0.279798	−	0.960059i	$-0.409733\pi$
$59$	4.29833	0.559596	0.279798	+	0.960059i	$0.409733\pi$
$60$	0	0
$61$	9.30798	1.19176	0.595882	−	0.803072i	$-0.296803\pi$
$61$	9.30798	1.19176	0.595882	+	0.803072i	$0.296803\pi$
$62$	0.692994	0.0880103
$63$	−1.90787	−0.240369
$64$	−7.45157	−0.931447
$65$	0	0
$66$	0.336506	0.0414211
$67$	−4.00012	−0.488693	−0.244346	−	0.969688i	$-0.578573\pi$
$67$	−4.00012	−0.488693	−0.244346	+	0.969688i	$0.578573\pi$
$68$	11.2928	1.36946
$69$	15.2834	1.83990
$70$	0	0
$71$	9.90832	1.17590	0.587951	−	0.808897i	$-0.299935\pi$
$71$	9.90832	1.17590	0.587951	+	0.808897i	$0.299935\pi$
$72$	1.15251	0.135824
$73$	3.24173	0.379415	0.189708	−	0.981841i	$-0.439246\pi$
$73$	3.24173	0.379415	0.189708	+	0.981841i	$0.439246\pi$
$74$	−0.171464	−0.0199323
$75$	0	0
$76$	1.93132	0.221537
$77$	−1.00000	−0.113961
$78$	−0.369505	−0.0418382
$79$	13.8523	1.55851	0.779254	−	0.626708i	$-0.215598\pi$
$79$	13.8523	1.55851	0.779254	+	0.626708i	$0.215598\pi$
$80$	0	0
$81$	−11.0836	−1.23152
$82$	−1.09261	−0.120659
$83$	−1.37098	−0.150484	−0.0752421	−	0.997165i	$-0.523973\pi$
$83$	−1.37098	−0.150484	−0.0752421	+	0.997165i	$0.523973\pi$
$84$	−4.37963	−0.477857
$85$	0	0
$86$	0.764770	0.0824673
$87$	21.6701	2.32328
$88$	0.604080	0.0643951
$89$	4.13811	0.438639	0.219319	−	0.975653i	$-0.429616\pi$
$89$	4.13811	0.438639	0.219319	+	0.975653i	$0.429616\pi$
$90$	0	0
$91$	1.09806	0.115108
$92$	13.6384	1.42190
$93$	10.1072	1.04806
$94$	0.178546	0.0184156
$95$	0	0
$96$	3.97614	0.405813
$97$	−16.2236	−1.64726	−0.823630	−	0.567128i	$-0.808055\pi$
$97$	−16.2236	−1.64726	−0.823630	+	0.567128i	$0.808055\pi$
$98$	−0.151896	−0.0153438
$99$	1.90787	0.191748
$100$	0	0
$101$	−0.178676	−0.0177789	−0.00888947	−	0.999960i	$-0.502830\pi$
$101$	−0.178676	−0.0177789	−0.00888947	+	0.999960i	$0.502830\pi$
$102$	−1.92223	−0.190329
$103$	17.2395	1.69866	0.849330	−	0.527862i	$-0.177006\pi$
$103$	17.2395	1.69866	0.849330	+	0.527862i	$0.177006\pi$
$104$	−0.663317	−0.0650436
$105$	0	0
$106$	1.63799	0.159095
$107$	14.5505	1.40665	0.703324	−	0.710869i	$-0.251698\pi$
$107$	14.5505	1.40665	0.703324	+	0.710869i	$0.251698\pi$
$108$	−4.78312	−0.460256
$109$	−6.35273	−0.608481	−0.304241	−	0.952595i	$-0.598403\pi$
$109$	−6.35273	−0.608481	−0.304241	+	0.952595i	$0.598403\pi$
$110$	0	0
$111$	−2.50076	−0.237362
$112$	−3.86210	−0.364934
$113$	0.795037	0.0747908	0.0373954	−	0.999301i	$-0.488094\pi$
$113$	0.795037	0.0747908	0.0373954	+	0.999301i	$0.488094\pi$
$114$	−0.328742	−0.0307895
$115$	0	0
$116$	19.3378	1.79547
$117$	−2.09496	−0.193679
$118$	−0.652900	−0.0601043
$119$	5.71231	0.523647
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.41385	−0.128004
$123$	−15.9355	−1.43686
$124$	9.01931	0.809958
$125$	0	0
$126$	0.289798	0.0258173
$127$	−4.49828	−0.399158	−0.199579	−	0.979882i	$-0.563957\pi$
$127$	−4.49828	−0.399158	−0.199579	+	0.979882i	$0.563957\pi$
$128$	4.72146	0.417322
$129$	11.1540	0.982055
$130$	0	0
$131$	−4.91258	−0.429214	−0.214607	−	0.976700i	$-0.568847\pi$
$131$	−4.91258	−0.429214	−0.214607	+	0.976700i	$0.568847\pi$
$132$	4.37963	0.381198
$133$	0.976928	0.0847104
$134$	0.607603	0.0524889
$135$	0	0
$136$	−3.45069	−0.295894
$137$	16.5865	1.41708	0.708541	−	0.705670i	$-0.249354\pi$
$137$	16.5865	1.41708	0.708541	+	0.705670i	$0.249354\pi$
$138$	−2.32149	−0.197618
$139$	19.9372	1.69105	0.845527	−	0.533932i	$-0.179286\pi$
$139$	19.9372	1.69105	0.845527	+	0.533932i	$0.179286\pi$
$140$	0	0
$141$	2.60405	0.219300
$142$	−1.50504	−0.126300
$143$	−1.09806	−0.0918246
$144$	7.36839	0.614032
$145$	0	0
$146$	−0.492406	−0.0407518
$147$	−2.21537	−0.182721
$148$	−2.23160	−0.183437
$149$	2.26555	0.185601	0.0928006	−	0.995685i	$-0.470418\pi$
$149$	2.26555	0.185601	0.0928006	+	0.995685i	$0.470418\pi$
$150$	0	0
$151$	−18.1130	−1.47402	−0.737008	−	0.675884i	$-0.763762\pi$
$151$	−18.1130	−1.47402	−0.737008	+	0.675884i	$0.763762\pi$
$152$	−0.590142	−0.0478669
$153$	−10.8984	−0.881080
$154$	0.151896	0.0122401
$155$	0	0
$156$	−4.80911	−0.385037
$157$	14.3063	1.14177	0.570885	−	0.821030i	$-0.306600\pi$
$157$	14.3063	1.14177	0.570885	+	0.821030i	$0.306600\pi$
$158$	−2.10411	−0.167394
$159$	23.8897	1.89457
$160$	0	0
$161$	6.89879	0.543701
$162$	1.68356	0.132273
$163$	7.91728	0.620129	0.310065	−	0.950715i	$-0.399649\pi$
$163$	7.91728	0.620129	0.310065	+	0.950715i	$0.399649\pi$
$164$	−14.2203	−1.11042
$165$	0	0
$166$	0.208246	0.0161630
$167$	−7.74877	−0.599618	−0.299809	−	0.953999i	$-0.596923\pi$
$167$	−7.74877	−0.599618	−0.299809	+	0.953999i	$0.596923\pi$
$168$	1.33826	0.103249
$169$	−11.7943	−0.907251
$170$	0	0
$171$	−1.86385	−0.142532
$172$	9.95348	0.758946
$173$	−1.67390	−0.127264	−0.0636322	−	0.997973i	$-0.520268\pi$
$173$	−1.67390	−0.127264	−0.0636322	+	0.997973i	$0.520268\pi$
$174$	−3.29161	−0.249536
$175$	0	0
$176$	3.86210	0.291117
$177$	−9.52241	−0.715748
$178$	−0.628563	−0.0471128
$179$	2.44178	0.182507	0.0912536	−	0.995828i	$-0.470913\pi$
$179$	2.44178	0.182507	0.0912536	+	0.995828i	$0.470913\pi$
$180$	0	0
$181$	−13.9875	−1.03968	−0.519841	−	0.854263i	$-0.674009\pi$
$181$	−13.9875	−1.03968	−0.519841	+	0.854263i	$0.674009\pi$
$182$	−0.166791	−0.0123634
$183$	−20.6206	−1.52432
$184$	−4.16742	−0.307226
$185$	0	0
$186$	−1.53524	−0.112569
$187$	−5.71231	−0.417726
$188$	2.32377	0.169478
$189$	−2.41947	−0.175991
$190$	0	0
$191$	−13.6741	−0.989420	−0.494710	−	0.869058i	$-0.664726\pi$
$191$	−13.6741	−0.989420	−0.494710	+	0.869058i	$0.664726\pi$
$192$	16.5080	1.19136
$193$	2.16256	0.155665	0.0778323	−	0.996966i	$-0.475200\pi$
$193$	2.16256	0.155665	0.0778323	+	0.996966i	$0.475200\pi$
$194$	2.46431	0.176927
$195$	0	0
$196$	−1.97693	−0.141209
$197$	15.0076	1.06925	0.534623	−	0.845091i	$-0.320454\pi$
$197$	15.0076	1.06925	0.534623	+	0.845091i	$0.320454\pi$
$198$	−0.289798	−0.0205951
$199$	8.06287	0.571562	0.285781	−	0.958295i	$-0.407747\pi$
$199$	8.06287	0.571562	0.285781	+	0.958295i	$0.407747\pi$
$200$	0	0
$201$	8.86176	0.625060
$202$	0.0271402	0.00190958
$203$	9.78172	0.686542
$204$	−25.0178	−1.75160
$205$	0	0
$206$	−2.61862	−0.182448
$207$	−13.1620	−0.914823
$208$	−4.24082	−0.294048
$209$	−0.976928	−0.0675755
$210$	0	0
$211$	10.8080	0.744053	0.372026	−	0.928222i	$-0.378663\pi$
$211$	10.8080	0.744053	0.372026	+	0.928222i	$0.378663\pi$
$212$	21.3184	1.46415
$213$	−21.9506	−1.50403
$214$	−2.21016	−0.151084
$215$	0	0
$216$	1.46155	0.0994461
$217$	4.56229	0.309708
$218$	0.964955	0.0653550
$219$	−7.18163	−0.485290
$220$	0	0
$221$	6.27247	0.421932
$222$	0.379856	0.0254943
$223$	23.2388	1.55618	0.778091	−	0.628151i	$-0.216188\pi$
$223$	23.2388	1.55618	0.778091	+	0.628151i	$0.216188\pi$
$224$	1.79480	0.119920
$225$	0	0
$226$	−0.120763	−0.00803304
$227$	10.2621	0.681122	0.340561	−	0.940222i	$-0.389383\pi$
$227$	10.2621	0.681122	0.340561	+	0.940222i	$0.389383\pi$
$228$	−4.27858	−0.283356
$229$	−24.8564	−1.64256	−0.821279	−	0.570527i	$-0.806739\pi$
$229$	−24.8564	−1.64256	−0.821279	+	0.570527i	$0.806739\pi$
$230$	0	0
$231$	2.21537	0.145761
$232$	−5.90894	−0.387941
$233$	21.8456	1.43115	0.715575	−	0.698536i	$-0.246165\pi$
$233$	21.8456	1.43115	0.715575	+	0.698536i	$0.246165\pi$
$234$	0.318217	0.0208025
$235$	0	0
$236$	−8.49750	−0.553140
$237$	−30.6880	−1.99340
$238$	−0.867678	−0.0562432
$239$	−24.7516	−1.60105	−0.800523	−	0.599302i	$-0.795445\pi$
$239$	−24.7516	−1.60105	−0.800523	+	0.599302i	$0.795445\pi$
$240$	0	0
$241$	11.0034	0.708793	0.354397	−	0.935095i	$-0.384686\pi$
$241$	11.0034	0.708793	0.354397	+	0.935095i	$0.384686\pi$
$242$	−0.151896	−0.00976425
$243$	17.2960	1.10954
$244$	−18.4012	−1.17802
$245$	0	0
$246$	2.42054	0.154328
$247$	1.07273	0.0682560
$248$	−2.75599	−0.175005
$249$	3.03722	0.192476
$250$	0	0
$251$	14.6902	0.927237	0.463619	−	0.886035i	$-0.346551\pi$
$251$	14.6902	0.927237	0.463619	+	0.886035i	$0.346551\pi$
$252$	3.77172	0.237596
$253$	−6.89879	−0.433723
$254$	0.683272	0.0428723
$255$	0	0
$256$	14.1860	0.886623
$257$	−0.128375	−0.00800778	−0.00400389	−	0.999992i	$-0.501274\pi$
$257$	−0.128375	−0.00800778	−0.00400389	+	0.999992i	$0.501274\pi$
$258$	−1.69425	−0.105479
$259$	−1.12882	−0.0701417
$260$	0	0
$261$	−18.6623	−1.15517
$262$	0.746202	0.0461005
$263$	13.2770	0.818694	0.409347	−	0.912379i	$-0.365757\pi$
$263$	13.2770	0.818694	0.409347	+	0.912379i	$0.365757\pi$
$264$	−1.33826	−0.0823643
$265$	0	0
$266$	−0.148391	−0.00909847
$267$	−9.16745	−0.561039
$268$	7.90795	0.483055
$269$	17.1671	1.04670	0.523349	−	0.852118i	$-0.324682\pi$
$269$	17.1671	1.04670	0.523349	+	0.852118i	$0.324682\pi$
$270$	0	0
$271$	5.06309	0.307561	0.153780	−	0.988105i	$-0.450855\pi$
$271$	5.06309	0.307561	0.153780	+	0.988105i	$0.450855\pi$
$272$	−22.0615	−1.33768
$273$	−2.43262	−0.147229
$274$	−2.51943	−0.152204
$275$	0	0
$276$	−30.2142	−1.81868
$277$	−18.2278	−1.09520	−0.547600	−	0.836740i	$-0.684458\pi$
$277$	−18.2278	−1.09520	−0.547600	+	0.836740i	$0.684458\pi$
$278$	−3.02839	−0.181631
$279$	−8.70426	−0.521110
$280$	0	0
$281$	−7.15814	−0.427019	−0.213509	−	0.976941i	$-0.568489\pi$
$281$	−7.15814	−0.427019	−0.213509	+	0.976941i	$0.568489\pi$
$282$	−0.395545	−0.0235544
$283$	11.1596	0.663372	0.331686	−	0.943390i	$-0.392383\pi$
$283$	11.1596	0.663372	0.331686	+	0.943390i	$0.392383\pi$
$284$	−19.5880	−1.16234
$285$	0	0
$286$	0.166791	0.00986258
$287$	−7.19315	−0.424598
$288$	−3.42424	−0.201775
$289$	15.6305	0.919441
$290$	0	0
$291$	35.9414	2.10692
$292$	−6.40866	−0.375038
$293$	−27.7716	−1.62244	−0.811218	−	0.584743i	$-0.801195\pi$
$293$	−27.7716	−1.62244	−0.811218	+	0.584743i	$0.801195\pi$
$294$	0.336506	0.0196254
$295$	0	0
$296$	0.681900	0.0396346
$297$	2.41947	0.140392
$298$	−0.344128	−0.0199348
$299$	7.57530	0.438091
$300$	0	0
$301$	5.03482	0.290202
$302$	2.75130	0.158319
$303$	0.395834	0.0227401
$304$	−3.77299	−0.216396
$305$	0	0
$306$	1.65542	0.0946340
$307$	13.5754	0.774790	0.387395	−	0.921914i	$-0.373375\pi$
$307$	13.5754	0.774790	0.387395	+	0.921914i	$0.373375\pi$
$308$	1.97693	0.112646
$309$	−38.1919	−2.17266
$310$	0	0
$311$	−32.3570	−1.83480	−0.917398	−	0.397971i	$-0.869715\pi$
$311$	−32.3570	−1.83480	−0.917398	+	0.397971i	$0.869715\pi$
$312$	1.46949	0.0831937
$313$	−2.12074	−0.119871	−0.0599357	−	0.998202i	$-0.519090\pi$
$313$	−2.12074	−0.119871	−0.0599357	+	0.998202i	$0.519090\pi$
$314$	−2.17308	−0.122634
$315$	0	0
$316$	−27.3850	−1.54053
$317$	−19.4563	−1.09277	−0.546387	−	0.837533i	$-0.683997\pi$
$317$	−19.4563	−1.09277	−0.546387	+	0.837533i	$0.683997\pi$
$318$	−3.62875	−0.203490
$319$	−9.78172	−0.547671
$320$	0	0
$321$	−32.2347	−1.79917
$322$	−1.04790	−0.0583972
$323$	5.58051	0.310508
$324$	21.9116	1.21731
$325$	0	0
$326$	−1.20260	−0.0666061
$327$	14.0737	0.778275
$328$	4.34523	0.239925
$329$	1.17545	0.0648044
$330$	0	0
$331$	7.67314	0.421754	0.210877	−	0.977513i	$-0.432368\pi$
$331$	7.67314	0.421754	0.210877	+	0.977513i	$0.432368\pi$
$332$	2.71032	0.148748
$333$	2.15365	0.118019
$334$	1.17701	0.0644030
$335$	0	0
$336$	8.55598	0.466767
$337$	−24.0394	−1.30951	−0.654754	−	0.755842i	$-0.727228\pi$
$337$	−24.0394	−1.30951	−0.654754	+	0.755842i	$0.727228\pi$
$338$	1.79150	0.0974449
$339$	−1.76130	−0.0956609
$340$	0	0
$341$	−4.56229	−0.247062
$342$	0.283112	0.0153089
$343$	−1.00000	−0.0539949
$344$	−3.04143	−0.163983
$345$	0	0
$346$	0.254259	0.0136691
$347$	−34.5623	−1.85540	−0.927701	−	0.373323i	$-0.878218\pi$
$347$	−34.5623	−1.85540	−0.927701	+	0.373323i	$0.878218\pi$
$348$	−42.8403	−2.29648
$349$	33.5588	1.79636	0.898181	−	0.439626i	$-0.144889\pi$
$349$	33.5588	1.79636	0.898181	+	0.439626i	$0.144889\pi$
$350$	0	0
$351$	−2.65673	−0.141806
$352$	−1.79480	−0.0956630
$353$	12.6905	0.675450	0.337725	−	0.941245i	$-0.390343\pi$
$353$	12.6905	0.675450	0.337725	+	0.941245i	$0.390343\pi$
$354$	1.44642	0.0768762
$355$	0	0
$356$	−8.18075	−0.433579
$357$	−12.6549	−0.669768
$358$	−0.370897	−0.0196025
$359$	−9.98748	−0.527119	−0.263560	−	0.964643i	$-0.584897\pi$
$359$	−9.98748	−0.527119	−0.263560	+	0.964643i	$0.584897\pi$
$360$	0	0
$361$	−18.0456	−0.949769
$362$	2.12465	0.111669
$363$	−2.21537	−0.116277
$364$	−2.17079	−0.113780
$365$	0	0
$366$	3.13219	0.163722
$367$	24.3134	1.26915	0.634574	−	0.772862i	$-0.281176\pi$
$367$	24.3134	1.26915	0.634574	+	0.772862i	$0.281176\pi$
$368$	−26.6438	−1.38890
$369$	13.7236	0.714422
$370$	0	0
$371$	10.7836	0.559856
$372$	−19.9811	−1.03597
$373$	13.9119	0.720333	0.360166	−	0.932888i	$-0.382720\pi$
$373$	13.9119	0.720333	0.360166	+	0.932888i	$0.382720\pi$
$374$	0.867678	0.0448665
$375$	0	0
$376$	−0.710063	−0.0366187
$377$	10.7409	0.553186
$378$	0.367508	0.0189026
$379$	31.0288	1.59384	0.796921	−	0.604084i	$-0.206461\pi$
$379$	31.0288	1.59384	0.796921	+	0.604084i	$0.206461\pi$
$380$	0	0
$381$	9.96537	0.510541
$382$	2.07704	0.106270
$383$	24.4087	1.24723	0.623614	−	0.781733i	$-0.285664\pi$
$383$	24.4087	1.24723	0.623614	+	0.781733i	$0.285664\pi$
$384$	−10.4598	−0.533774
$385$	0	0
$386$	−0.328485	−0.0167194
$387$	−9.60580	−0.488290
$388$	32.0729	1.62826
$389$	−21.4397	−1.08704	−0.543519	−	0.839397i	$-0.682908\pi$
$389$	−21.4397	−1.08704	−0.543519	+	0.839397i	$0.682908\pi$
$390$	0	0
$391$	39.4081	1.99295
$392$	0.604080	0.0305106
$393$	10.8832	0.548985
$394$	−2.27959	−0.114844
$395$	0	0
$396$	−3.77172	−0.189536
$397$	−25.1326	−1.26137	−0.630685	−	0.776039i	$-0.717226\pi$
$397$	−25.1326	−1.26137	−0.630685	+	0.776039i	$0.717226\pi$
$398$	−1.22472	−0.0613896
$399$	−2.16426	−0.108348
$400$	0	0
$401$	2.98499	0.149063	0.0745316	−	0.997219i	$-0.476254\pi$
$401$	2.98499	0.149063	0.0745316	+	0.997219i	$0.476254\pi$
$402$	−1.34607	−0.0671357
$403$	5.00967	0.249550
$404$	0.353230	0.0175738
$405$	0	0
$406$	−1.48581	−0.0737393
$407$	1.12882	0.0559537
$408$	7.64456	0.378462
$409$	−20.4362	−1.01050	−0.505252	−	0.862972i	$-0.668600\pi$
$409$	−20.4362	−1.01050	−0.505252	+	0.862972i	$0.668600\pi$
$410$	0	0
$411$	−36.7453	−1.81251
$412$	−34.0813	−1.67906
$413$	−4.29833	−0.211507
$414$	1.99926	0.0982582
$415$	0	0
$416$	1.97080	0.0966264
$417$	−44.1684	−2.16294
$418$	0.148391	0.00725806
$419$	−5.00321	−0.244423	−0.122212	−	0.992504i	$-0.538999\pi$
$419$	−5.00321	−0.244423	−0.122212	+	0.992504i	$0.538999\pi$
$420$	0	0
$421$	−27.5553	−1.34296	−0.671481	−	0.741022i	$-0.734342\pi$
$421$	−27.5553	−1.34296	−0.671481	+	0.741022i	$0.734342\pi$
$422$	−1.64169	−0.0799163
$423$	−2.24260	−0.109039
$424$	−6.51415	−0.316355
$425$	0	0
$426$	3.33421	0.161543
$427$	−9.30798	−0.450445
$428$	−28.7653	−1.39042
$429$	2.43262	0.117448
$430$	0	0
$431$	−17.3016	−0.833387	−0.416693	−	0.909047i	$-0.636811\pi$
$431$	−17.3016	−0.833387	−0.416693	+	0.909047i	$0.636811\pi$
$432$	9.34423	0.449574
$433$	18.3543	0.882049	0.441025	−	0.897495i	$-0.354615\pi$
$433$	18.3543	0.882049	0.441025	+	0.897495i	$0.354615\pi$
$434$	−0.692994	−0.0332648
$435$	0	0
$436$	12.5589	0.601462
$437$	6.73962	0.322400
$438$	1.09086	0.0521234
$439$	17.4946	0.834973	0.417487	−	0.908683i	$-0.362911\pi$
$439$	17.4946	0.834973	0.417487	+	0.908683i	$0.362911\pi$
$440$	0	0
$441$	1.90787	0.0908510
$442$	−0.952764	−0.0453184
$443$	−23.0816	−1.09664	−0.548321	−	0.836268i	$-0.684733\pi$
$443$	−23.0816	−1.09664	−0.548321	+	0.836268i	$0.684733\pi$
$444$	4.94383	0.234624
$445$	0	0
$446$	−3.52988	−0.167145
$447$	−5.01904	−0.237392
$448$	7.45157	0.352054
$449$	8.03017	0.378967	0.189483	−	0.981884i	$-0.439319\pi$
$449$	8.03017	0.378967	0.189483	+	0.981884i	$0.439319\pi$
$450$	0	0
$451$	7.19315	0.338712
$452$	−1.57173	−0.0739280
$453$	40.1271	1.88533
$454$	−1.55878	−0.0731571
$455$	0	0
$456$	1.30738	0.0612239
$457$	3.11415	0.145674	0.0728368	−	0.997344i	$-0.476795\pi$
$457$	3.11415	0.145674	0.0728368	+	0.997344i	$0.476795\pi$
$458$	3.77559	0.176422
$459$	−13.8208	−0.645098
$460$	0	0
$461$	26.0811	1.21472	0.607360	−	0.794427i	$-0.292229\pi$
$461$	26.0811	1.21472	0.607360	+	0.794427i	$0.292229\pi$
$462$	−0.336506	−0.0156557
$463$	22.6226	1.05136	0.525681	−	0.850682i	$-0.323810\pi$
$463$	22.6226	1.05136	0.525681	+	0.850682i	$0.323810\pi$
$464$	−37.7780	−1.75380
$465$	0	0
$466$	−3.31826	−0.153715
$467$	25.8242	1.19500	0.597500	−	0.801869i	$-0.296161\pi$
$467$	25.8242	1.19500	0.597500	+	0.801869i	$0.296161\pi$
$468$	4.14159	0.191445
$469$	4.00012	0.184709
$470$	0	0
$471$	−31.6939	−1.46038
$472$	2.59654	0.119515
$473$	−5.03482	−0.231501
$474$	4.66139	0.214105
$475$	0	0
$476$	−11.2928	−0.517606
$477$	−20.5737	−0.942006
$478$	3.75967	0.171963
$479$	5.74653	0.262566	0.131283	−	0.991345i	$-0.458090\pi$
$479$	5.74653	0.262566	0.131283	+	0.991345i	$0.458090\pi$
$480$	0	0
$481$	−1.23952	−0.0565172
$482$	−1.67138	−0.0761292
$483$	−15.2834	−0.695418
$484$	−1.97693	−0.0898603
$485$	0	0
$486$	−2.62719	−0.119172
$487$	29.4527	1.33463	0.667315	−	0.744776i	$-0.267444\pi$
$487$	29.4527	1.33463	0.667315	+	0.744776i	$0.267444\pi$
$488$	5.62276	0.254530
$489$	−17.5397	−0.793174
$490$	0	0
$491$	12.7515	0.575467	0.287733	−	0.957711i	$-0.407098\pi$
$491$	12.7515	0.575467	0.287733	+	0.957711i	$0.407098\pi$
$492$	31.5033	1.42028
$493$	55.8762	2.51654
$494$	−0.162943	−0.00733116
$495$	0	0
$496$	−17.6200	−0.791161
$497$	−9.90832	−0.444449
$498$	−0.461342	−0.0206732
$499$	23.5914	1.05609	0.528047	−	0.849215i	$-0.322924\pi$
$499$	23.5914	1.05609	0.528047	+	0.849215i	$0.322924\pi$
$500$	0	0
$501$	17.1664	0.766938
$502$	−2.23138	−0.0995915
$503$	−6.79108	−0.302799	−0.151400	−	0.988473i	$-0.548378\pi$
$503$	−6.79108	−0.302799	−0.151400	+	0.988473i	$0.548378\pi$
$504$	−1.15251	−0.0513367
$505$	0	0
$506$	1.04790	0.0465848
$507$	26.1287	1.16041
$508$	8.89278	0.394553
$509$	−9.77652	−0.433337	−0.216668	−	0.976245i	$-0.569519\pi$
$509$	−9.77652	−0.433337	−0.216668	+	0.976245i	$0.569519\pi$
$510$	0	0
$511$	−3.24173	−0.143406
$512$	−11.5977	−0.512551
$513$	−2.36365	−0.104358
$514$	0.0194996	0.000860090 0
$515$	0	0
$516$	−22.0507	−0.970726
$517$	−1.17545	−0.0516960
$518$	0.171464	0.00753369
$519$	3.70832	0.162777
$520$	0	0
$521$	−31.0849	−1.36185	−0.680927	−	0.732352i	$-0.738423\pi$
$521$	−31.0849	−1.36185	−0.680927	+	0.732352i	$0.738423\pi$
$522$	2.83473	0.124073
$523$	−13.2626	−0.579931	−0.289966	−	0.957037i	$-0.593644\pi$
$523$	−13.2626	−0.579931	−0.289966	+	0.957037i	$0.593644\pi$
$524$	9.71182	0.424263
$525$	0	0
$526$	−2.01672	−0.0879333
$527$	26.0612	1.13524
$528$	−8.55598	−0.372351
$529$	24.5934	1.06928
$530$	0	0
$531$	8.20067	0.355879
$532$	−1.93132	−0.0837331
$533$	−7.89852	−0.342123
$534$	1.39250	0.0602594
$535$	0	0
$536$	−2.41639	−0.104372
$537$	−5.40945	−0.233435
$538$	−2.60762	−0.112423
$539$	1.00000	0.0430730
$540$	0	0
$541$	8.23861	0.354206	0.177103	−	0.984192i	$-0.443328\pi$
$541$	8.23861	0.354206	0.177103	+	0.984192i	$0.443328\pi$
$542$	−0.769063	−0.0330341
$543$	30.9875	1.32980
$544$	10.2524	0.439570
$545$	0	0
$546$	0.369505	0.0158133
$547$	−16.1174	−0.689129	−0.344564	−	0.938763i	$-0.611973\pi$
$547$	−16.1174	−0.689129	−0.344564	+	0.938763i	$0.611973\pi$
$548$	−32.7903	−1.40073
$549$	17.7584	0.757911
$550$	0	0
$551$	9.55603	0.407101
$552$	9.23239	0.392957
$553$	−13.8523	−0.589061
$554$	2.76873	0.117632
$555$	0	0
$556$	−39.4145	−1.67155
$557$	−15.7911	−0.669091	−0.334545	−	0.942380i	$-0.608583\pi$
$557$	−15.7911	−0.669091	−0.334545	+	0.942380i	$0.608583\pi$
$558$	1.32214	0.0559708
$559$	5.52855	0.233833
$560$	0	0
$561$	12.6549	0.534290
$562$	1.08729	0.0458647
$563$	4.28230	0.180477	0.0902387	−	0.995920i	$-0.471237\pi$
$563$	4.28230	0.180477	0.0902387	+	0.995920i	$0.471237\pi$
$564$	−5.14802	−0.216771
$565$	0	0
$566$	−1.69511	−0.0712506
$567$	11.0836	0.465469
$568$	5.98542	0.251142
$569$	3.63395	0.152343	0.0761716	−	0.997095i	$-0.475730\pi$
$569$	3.63395	0.152343	0.0761716	+	0.997095i	$0.475730\pi$
$570$	0	0
$571$	−0.426150	−0.0178338	−0.00891691	−	0.999960i	$-0.502838\pi$
$571$	−0.426150	−0.0178338	−0.00891691	+	0.999960i	$0.502838\pi$
$572$	2.17079	0.0907653
$573$	30.2931	1.26551
$574$	1.09261	0.0456047
$575$	0	0
$576$	−14.2166	−0.592360
$577$	1.77547	0.0739137	0.0369568	−	0.999317i	$-0.488234\pi$
$577$	1.77547	0.0739137	0.0369568	+	0.999317i	$0.488234\pi$
$578$	−2.37421	−0.0987542
$579$	−4.79088	−0.199102
$580$	0	0
$581$	1.37098	0.0568776
$582$	−5.45935	−0.226297
$583$	−10.7836	−0.446611
$584$	1.95826	0.0810335
$585$	0	0
$586$	4.21840	0.174261
$587$	27.4767	1.13409	0.567043	−	0.823688i	$-0.308087\pi$
$587$	27.4767	1.13409	0.567043	+	0.823688i	$0.308087\pi$
$588$	4.37963	0.180613
$589$	4.45702	0.183648
$590$	0	0
$591$	−33.2474	−1.36761
$592$	4.35963	0.179180
$593$	17.3758	0.713539	0.356769	−	0.934192i	$-0.383878\pi$
$593$	17.3758	0.713539	0.356769	+	0.934192i	$0.383878\pi$
$594$	−0.367508	−0.0150790
$595$	0	0
$596$	−4.47883	−0.183460
$597$	−17.8623	−0.731053
$598$	−1.15066	−0.0470540
$599$	−12.7427	−0.520654	−0.260327	−	0.965521i	$-0.583830\pi$
$599$	−12.7427	−0.520654	−0.260327	+	0.965521i	$0.583830\pi$
$600$	0	0
$601$	39.4325	1.60849	0.804243	−	0.594300i	$-0.202571\pi$
$601$	39.4325	1.60849	0.804243	+	0.594300i	$0.202571\pi$
$602$	−0.764770	−0.0311697
$603$	−7.63172	−0.310788
$604$	35.8081	1.45701
$605$	0	0
$606$	−0.0601256	−0.00244244
$607$	−17.2169	−0.698811	−0.349406	−	0.936972i	$-0.613616\pi$
$607$	−17.2169	−0.698811	−0.349406	+	0.936972i	$0.613616\pi$
$608$	1.75339	0.0711092
$609$	−21.6701	−0.878119
$610$	0	0
$611$	1.29071	0.0522166
$612$	21.5453	0.870916
$613$	0.189342	0.00764745	0.00382372	−	0.999993i	$-0.498783\pi$
$613$	0.189342	0.00764745	0.00382372	+	0.999993i	$0.498783\pi$
$614$	−2.06205	−0.0832177
$615$	0	0
$616$	−0.604080	−0.0243391
$617$	−20.0172	−0.805862	−0.402931	−	0.915230i	$-0.632009\pi$
$617$	−20.0172	−0.805862	−0.402931	+	0.915230i	$0.632009\pi$
$618$	5.80121	0.233359
$619$	40.6163	1.63251	0.816254	−	0.577694i	$-0.196047\pi$
$619$	40.6163	1.63251	0.816254	+	0.577694i	$0.196047\pi$
$620$	0	0
$621$	−16.6914	−0.669804
$622$	4.91490	0.197070
$623$	−4.13811	−0.165790
$624$	9.39500	0.376101
$625$	0	0
$626$	0.322133	0.0128750
$627$	2.16426	0.0864321
$628$	−28.2826	−1.12860
$629$	−6.44819	−0.257106
$630$	0	0
$631$	43.4349	1.72912	0.864558	−	0.502532i	$-0.167598\pi$
$631$	43.4349	1.72912	0.864558	+	0.502532i	$0.167598\pi$
$632$	8.36791	0.332858
$633$	−23.9437	−0.951677
$634$	2.95533	0.117371
$635$	0	0
$636$	−47.2281	−1.87272
$637$	−1.09806	−0.0435068
$638$	1.48581	0.0588236
$639$	18.9038	0.747823
$640$	0	0
$641$	−23.6404	−0.933740	−0.466870	−	0.884326i	$-0.654618\pi$
$641$	−23.6404	−0.933740	−0.466870	+	0.884326i	$0.654618\pi$
$642$	4.89633	0.193243
$643$	2.85977	0.112778	0.0563892	−	0.998409i	$-0.482041\pi$
$643$	2.85977	0.112778	0.0563892	+	0.998409i	$0.482041\pi$
$644$	−13.6384	−0.537429
$645$	0	0
$646$	−0.847658	−0.0333507
$647$	−19.2211	−0.755660	−0.377830	−	0.925875i	$-0.623330\pi$
$647$	−19.2211	−0.755660	−0.377830	+	0.925875i	$0.623330\pi$
$648$	−6.69540	−0.263020
$649$	4.29833	0.168724
$650$	0	0
$651$	−10.1072	−0.396131
$652$	−15.6519	−0.612975
$653$	9.59905	0.375640	0.187820	−	0.982203i	$-0.439858\pi$
$653$	9.59905	0.375640	0.187820	+	0.982203i	$0.439858\pi$
$654$	−2.13773	−0.0835920
$655$	0	0
$656$	27.7806	1.08465
$657$	6.18480	0.241292
$658$	−0.178546	−0.00696043
$659$	−14.2403	−0.554723	−0.277361	−	0.960766i	$-0.589460\pi$
$659$	−14.2403	−0.554723	−0.277361	+	0.960766i	$0.589460\pi$
$660$	0	0
$661$	−13.3679	−0.519949	−0.259975	−	0.965615i	$-0.583714\pi$
$661$	−13.3679	−0.519949	−0.259975	+	0.965615i	$0.583714\pi$
$662$	−1.16552	−0.0452992
$663$	−13.8959	−0.539670
$664$	−0.828179	−0.0321396
$665$	0	0
$666$	−0.327131	−0.0126761
$667$	67.4821	2.61292
$668$	15.3188	0.592700
$669$	−51.4825	−1.99043
$670$	0	0
$671$	9.30798	0.359330
$672$	−3.97614	−0.153383
$673$	−34.8230	−1.34233	−0.671164	−	0.741309i	$-0.734205\pi$
$673$	−34.8230	−1.34233	−0.671164	+	0.741309i	$0.734205\pi$
$674$	3.65149	0.140650
$675$	0	0
$676$	23.3164	0.896784
$677$	4.64741	0.178614	0.0893072	−	0.996004i	$-0.471535\pi$
$677$	4.64741	0.178614	0.0893072	+	0.996004i	$0.471535\pi$
$678$	0.267535	0.0102746
$679$	16.2236	0.622606
$680$	0	0
$681$	−22.7344	−0.871186
$682$	0.692994	0.0265361
$683$	36.8978	1.41186	0.705928	−	0.708283i	$-0.250530\pi$
$683$	36.8978	1.41186	0.705928	+	0.708283i	$0.250530\pi$
$684$	3.68470	0.140888
$685$	0	0
$686$	0.151896	0.00579942
$687$	55.0662	2.10091
$688$	−19.4450	−0.741333
$689$	11.8411	0.451108
$690$	0	0
$691$	−37.9569	−1.44395	−0.721974	−	0.691920i	$-0.756765\pi$
$691$	−37.9569	−1.44395	−0.721974	+	0.691920i	$0.756765\pi$
$692$	3.30918	0.125796
$693$	−1.90787	−0.0724741
$694$	5.24988	0.199283
$695$	0	0
$696$	13.0905	0.496194
$697$	−41.0895	−1.55638
$698$	−5.09745	−0.192941
$699$	−48.3960	−1.83051
$700$	0	0
$701$	27.0379	1.02121	0.510604	−	0.859816i	$-0.329422\pi$
$701$	27.0379	1.02121	0.510604	+	0.859816i	$0.329422\pi$
$702$	0.403547	0.0152309
$703$	−1.10278	−0.0415921
$704$	−7.45157	−0.280842
$705$	0	0
$706$	−1.92764	−0.0725479
$707$	0.178676	0.00671981
$708$	18.8251	0.707491
$709$	−9.81423	−0.368581	−0.184291	−	0.982872i	$-0.558999\pi$
$709$	−9.81423	−0.368581	−0.184291	+	0.982872i	$0.558999\pi$
$710$	0	0
$711$	26.4285	0.991145
$712$	2.49975	0.0936821
$713$	31.4743	1.17872
$714$	1.92223	0.0719376
$715$	0	0
$716$	−4.82722	−0.180402
$717$	54.8339	2.04781
$718$	1.51706	0.0566162
$719$	−31.0336	−1.15736	−0.578678	−	0.815556i	$-0.696431\pi$
$719$	−31.0336	−1.15736	−0.578678	+	0.815556i	$0.696431\pi$
$720$	0	0
$721$	−17.2395	−0.642033
$722$	2.74106	0.102012
$723$	−24.3767	−0.906579
$724$	27.6523	1.02769
$725$	0	0
$726$	0.336506	0.0124889
$727$	24.5271	0.909659	0.454829	−	0.890579i	$-0.349700\pi$
$727$	24.5271	0.909659	0.454829	+	0.890579i	$0.349700\pi$
$728$	0.663317	0.0245842
$729$	−5.06609	−0.187633
$730$	0	0
$731$	28.7605	1.06374
$732$	40.7655	1.50674
$733$	50.6246	1.86986	0.934931	−	0.354828i	$-0.115461\pi$
$733$	50.6246	1.86986	0.934931	+	0.354828i	$0.115461\pi$
$734$	−3.69311	−0.136315
$735$	0	0
$736$	12.3819	0.456404
$737$	−4.00012	−0.147346
$738$	−2.08456	−0.0767338
$739$	23.6440	0.869757	0.434879	−	0.900489i	$-0.356791\pi$
$739$	23.6440	0.869757	0.434879	+	0.900489i	$0.356791\pi$
$740$	0	0
$741$	−2.37649	−0.0873025
$742$	−1.63799	−0.0601323
$743$	34.1982	1.25461	0.627305	−	0.778774i	$-0.284158\pi$
$743$	34.1982	1.25461	0.627305	+	0.778774i	$0.284158\pi$
$744$	6.10553	0.223840
$745$	0	0
$746$	−2.11317	−0.0773686
$747$	−2.61565	−0.0957015
$748$	11.2928	0.412907
$749$	−14.5505	−0.531663
$750$	0	0
$751$	−31.5306	−1.15057	−0.575283	−	0.817954i	$-0.695108\pi$
$751$	−31.5306	−1.15057	−0.575283	+	0.817954i	$0.695108\pi$
$752$	−4.53969	−0.165545
$753$	−32.5443	−1.18598
$754$	−1.63151	−0.0594160
$755$	0	0
$756$	4.78312	0.173960
$757$	−43.4816	−1.58036	−0.790182	−	0.612872i	$-0.790014\pi$
$757$	−43.4816	−1.58036	−0.790182	+	0.612872i	$0.790014\pi$
$758$	−4.71315	−0.171189
$759$	15.2834	0.554752
$760$	0	0
$761$	−44.3367	−1.60720	−0.803602	−	0.595167i	$-0.797086\pi$
$761$	−44.3367	−1.60720	−0.803602	+	0.595167i	$0.797086\pi$
$762$	−1.51370	−0.0548356
$763$	6.35273	0.229984
$764$	27.0326	0.978006
$765$	0	0
$766$	−3.70759	−0.133961
$767$	−4.71984	−0.170424
$768$	−31.4272	−1.13403
$769$	−47.4471	−1.71099	−0.855493	−	0.517814i	$-0.826746\pi$
$769$	−47.4471	−1.71099	−0.855493	+	0.517814i	$0.826746\pi$
$770$	0	0
$771$	0.284397	0.0102423
$772$	−4.27523	−0.153869
$773$	10.1254	0.364184	0.182092	−	0.983281i	$-0.441713\pi$
$773$	10.1254	0.364184	0.182092	+	0.983281i	$0.441713\pi$
$774$	1.45908	0.0524457
$775$	0	0
$776$	−9.80036	−0.351813
$777$	2.50076	0.0897144
$778$	3.25661	0.116755
$779$	−7.02718	−0.251775
$780$	0	0
$781$	9.90832	0.354548
$782$	−5.98593	−0.214056
$783$	−23.6666	−0.845774
$784$	3.86210	0.137932
$785$	0	0
$786$	−1.65312	−0.0589647
$787$	9.72145	0.346532	0.173266	−	0.984875i	$-0.444568\pi$
$787$	9.72145	0.346532	0.173266	+	0.984875i	$0.444568\pi$
$788$	−29.6689	−1.05691
$789$	−29.4135	−1.04715
$790$	0	0
$791$	−0.795037	−0.0282683
$792$	1.15251	0.0409525
$793$	−10.2207	−0.362949
$794$	3.81755	0.135480
$795$	0	0
$796$	−15.9397	−0.564968
$797$	19.9546	0.706827	0.353414	−	0.935467i	$-0.385021\pi$
$797$	19.9546	0.706827	0.353414	+	0.935467i	$0.385021\pi$
$798$	0.328742	0.0116374
$799$	6.71451	0.237542
$800$	0	0
$801$	7.89499	0.278956
$802$	−0.453408	−0.0160104
$803$	3.24173	0.114398
$804$	−17.5191	−0.617849
$805$	0	0
$806$	−0.760950	−0.0268033
$807$	−38.0316	−1.33878
$808$	−0.107935	−0.00379713
$809$	16.0119	0.562948	0.281474	−	0.959569i	$-0.409177\pi$
$809$	16.0119	0.562948	0.281474	+	0.959569i	$0.409177\pi$
$810$	0	0
$811$	36.5355	1.28293	0.641467	−	0.767150i	$-0.278326\pi$
$811$	36.5355	1.28293	0.641467	+	0.767150i	$0.278326\pi$
$812$	−19.3378	−0.678622
$813$	−11.2166	−0.393384
$814$	−0.171464	−0.00600981
$815$	0	0
$816$	48.8744	1.71095
$817$	4.91866	0.172082
$818$	3.10418	0.108535
$819$	2.09496	0.0732039
$820$	0	0
$821$	15.2923	0.533705	0.266853	−	0.963737i	$-0.414016\pi$
$821$	15.2923	0.533705	0.266853	+	0.963737i	$0.414016\pi$
$822$	5.58147	0.194676
$823$	−36.4539	−1.27070	−0.635351	−	0.772223i	$-0.719145\pi$
$823$	−36.4539	−1.27070	−0.635351	+	0.772223i	$0.719145\pi$
$824$	10.4140	0.362790
$825$	0	0
$826$	0.652900	0.0227173
$827$	−16.1731	−0.562394	−0.281197	−	0.959650i	$-0.590731\pi$
$827$	−16.1731	−0.562394	−0.281197	+	0.959650i	$0.590731\pi$
$828$	26.0203	0.904270
$829$	31.3240	1.08793	0.543963	−	0.839109i	$-0.316923\pi$
$829$	31.3240	1.08793	0.543963	+	0.839109i	$0.316923\pi$
$830$	0	0
$831$	40.3813	1.40081
$832$	8.18229	0.283670
$833$	−5.71231	−0.197920
$834$	6.70901	0.232314
$835$	0	0
$836$	1.93132	0.0667959
$837$	−11.0383	−0.381540
$838$	0.759969	0.0262527
$839$	−42.2772	−1.45957	−0.729785	−	0.683676i	$-0.760380\pi$
$839$	−42.2772	−1.45957	−0.729785	+	0.683676i	$0.760380\pi$
$840$	0	0
$841$	66.6821	2.29938
$842$	4.18554	0.144243
$843$	15.8579	0.546176
$844$	−21.3666	−0.735469
$845$	0	0
$846$	0.340642	0.0117115
$847$	−1.00000	−0.0343604
$848$	−41.6473	−1.43017
$849$	−24.7227	−0.848482
$850$	0	0
$851$	−7.78752	−0.266953
$852$	43.3948	1.48668
$853$	−15.3640	−0.526054	−0.263027	−	0.964788i	$-0.584721\pi$
$853$	−15.3640	−0.526054	−0.263027	+	0.964788i	$0.584721\pi$
$854$	1.41385	0.0483808
$855$	0	0
$856$	8.78966	0.300424
$857$	26.5700	0.907612	0.453806	−	0.891100i	$-0.350066\pi$
$857$	26.5700	0.907612	0.453806	+	0.891100i	$0.350066\pi$
$858$	−0.369505	−0.0126147
$859$	−30.5761	−1.04324	−0.521621	−	0.853177i	$-0.674673\pi$
$859$	−30.5761	−1.04324	−0.521621	+	0.853177i	$0.674673\pi$
$860$	0	0
$861$	15.9355	0.543080
$862$	2.62804	0.0895114
$863$	−49.1667	−1.67365	−0.836827	−	0.547468i	$-0.815592\pi$
$863$	−49.1667	−1.67365	−0.836827	+	0.547468i	$0.815592\pi$
$864$	−4.34246	−0.147733
$865$	0	0
$866$	−2.78794	−0.0947380
$867$	−34.6274	−1.17601
$868$	−9.01931	−0.306135
$869$	13.8523	0.469908
$870$	0	0
$871$	4.39238	0.148830
$872$	−3.83756	−0.129956
$873$	−30.9526	−1.04759
$874$	−1.02372	−0.0346279
$875$	0	0
$876$	14.1976	0.479691
$877$	3.59305	0.121329	0.0606644	−	0.998158i	$-0.480678\pi$
$877$	3.59305	0.121329	0.0606644	+	0.998158i	$0.480678\pi$
$878$	−2.65737	−0.0896818
$879$	61.5245	2.07517
$880$	0	0
$881$	−19.3117	−0.650628	−0.325314	−	0.945606i	$-0.605470\pi$
$881$	−19.3117	−0.650628	−0.325314	+	0.945606i	$0.605470\pi$
$882$	−0.289798	−0.00975802
$883$	−33.0224	−1.11129	−0.555645	−	0.831419i	$-0.687529\pi$
$883$	−33.0224	−1.11129	−0.555645	+	0.831419i	$0.687529\pi$
$884$	−12.4002	−0.417065
$885$	0	0
$886$	3.50601	0.117787
$887$	39.5514	1.32800	0.664002	−	0.747730i	$-0.268856\pi$
$887$	39.5514	1.32800	0.664002	+	0.747730i	$0.268856\pi$
$888$	−1.51066	−0.0506945
$889$	4.49828	0.150868
$890$	0	0
$891$	−11.0836	−0.371316
$892$	−45.9414	−1.53823
$893$	1.14833	0.0384272
$894$	0.762372	0.0254975
$895$	0	0
$896$	−4.72146	−0.157733
$897$	−16.7821	−0.560338
$898$	−1.21975	−0.0407036
$899$	44.6270	1.48839
$900$	0	0
$901$	61.5992	2.05217
$902$	−1.09261	−0.0363800
$903$	−11.1540	−0.371182
$904$	0.480266	0.0159734
$905$	0	0
$906$	−6.09514	−0.202498
$907$	−19.8019	−0.657511	−0.328755	−	0.944415i	$-0.606629\pi$
$907$	−19.8019	−0.657511	−0.328755	+	0.944415i	$0.606629\pi$
$908$	−20.2875	−0.673264
$909$	−0.340891	−0.0113066
$910$	0	0
$911$	−23.5082	−0.778861	−0.389430	−	0.921056i	$-0.627328\pi$
$911$	−23.5082	−0.778861	−0.389430	+	0.921056i	$0.627328\pi$
$912$	8.35858	0.276780
$913$	−1.37098	−0.0453727
$914$	−0.473027	−0.0156463
$915$	0	0
$916$	49.1393	1.62361
$917$	4.91258	0.162228
$918$	2.09932	0.0692879
$919$	52.5940	1.73492	0.867458	−	0.497510i	$-0.165752\pi$
$919$	52.5940	1.73492	0.867458	+	0.497510i	$0.165752\pi$
$920$	0	0
$921$	−30.0746	−0.990991
$922$	−3.96162	−0.130469
$923$	−10.8800	−0.358118
$924$	−4.37963	−0.144079
$925$	0	0
$926$	−3.43629	−0.112923
$927$	32.8908	1.08028
$928$	17.5562	0.576311
$929$	−55.0184	−1.80510	−0.902548	−	0.430589i	$-0.858306\pi$
$929$	−55.0184	−1.80510	−0.902548	+	0.430589i	$0.858306\pi$
$930$	0	0
$931$	−0.976928	−0.0320175
$932$	−43.1871	−1.41464
$933$	71.6828	2.34679
$934$	−3.92259	−0.128351
$935$	0	0
$936$	−1.26552	−0.0413650
$937$	−26.5714	−0.868048	−0.434024	−	0.900901i	$-0.642907\pi$
$937$	−26.5714	−0.868048	−0.434024	+	0.900901i	$0.642907\pi$
$938$	−0.607603	−0.0198389
$939$	4.69823	0.153321
$940$	0	0
$941$	−22.8193	−0.743887	−0.371943	−	0.928255i	$-0.621308\pi$
$941$	−22.8193	−0.743887	−0.371943	+	0.928255i	$0.621308\pi$
$942$	4.81417	0.156854
$943$	−49.6240	−1.61598
$944$	16.6006	0.540303
$945$	0	0
$946$	0.764770	0.0248648
$947$	2.82112	0.0916741	0.0458371	−	0.998949i	$-0.485404\pi$
$947$	2.82112	0.0916741	0.0458371	+	0.998949i	$0.485404\pi$
$948$	60.6680	1.97041
$949$	−3.55962	−0.115550
$950$	0	0
$951$	43.1029	1.39771
$952$	3.45069	0.111838
$953$	12.3476	0.399977	0.199989	−	0.979798i	$-0.435909\pi$
$953$	12.3476	0.399977	0.199989	+	0.979798i	$0.435909\pi$
$954$	3.12507	0.101178
$955$	0	0
$956$	48.9321	1.58258
$957$	21.6701	0.700496
$958$	−0.872875	−0.0282013
$959$	−16.5865	−0.535606
$960$	0	0
$961$	−10.1855	−0.328566
$962$	0.188278	0.00607033
$963$	27.7605	0.894569
$964$	−21.7530	−0.700617
$965$	0	0
$966$	2.32149	0.0746926
$967$	−8.34422	−0.268332	−0.134166	−	0.990959i	$-0.542836\pi$
$967$	−8.34422	−0.268332	−0.134166	+	0.990959i	$0.542836\pi$
$968$	0.604080	0.0194159
$969$	−12.3629	−0.397154
$970$	0	0
$971$	−8.13737	−0.261141	−0.130570	−	0.991439i	$-0.541681\pi$
$971$	−8.13737	−0.261141	−0.130570	+	0.991439i	$0.541681\pi$
$972$	−34.1929	−1.09674
$973$	−19.9372	−0.639159
$974$	−4.47375	−0.143348
$975$	0	0
$976$	35.9483	1.15068
$977$	27.5701	0.882045	0.441023	−	0.897496i	$-0.354616\pi$
$977$	27.5701	0.882045	0.441023	+	0.897496i	$0.354616\pi$
$978$	2.66422	0.0851922
$979$	4.13811	0.132255
$980$	0	0
$981$	−12.1202	−0.386968
$982$	−1.93690	−0.0618090
$983$	−46.4748	−1.48232	−0.741159	−	0.671330i	$-0.765723\pi$
$983$	−46.4748	−1.48232	−0.741159	+	0.671330i	$0.765723\pi$
$984$	−9.62631	−0.306876
$985$	0	0
$986$	−8.48738	−0.270293
$987$	−2.60405	−0.0828878
$988$	−2.12070	−0.0674686
$989$	34.7342	1.10448
$990$	0	0
$991$	25.4209	0.807521	0.403760	−	0.914865i	$-0.367703\pi$
$991$	25.4209	0.807521	0.403760	+	0.914865i	$0.367703\pi$
$992$	8.18838	0.259981
$993$	−16.9989	−0.539443
$994$	1.50504	0.0477368
$995$	0	0
$996$	−6.00436	−0.190256
$997$	41.8987	1.32695	0.663473	−	0.748200i	$-0.269082\pi$
$997$	41.8987	1.32695	0.663473	+	0.748200i	$0.269082\pi$
$998$	−3.58344	−0.113432
$999$	2.73115	0.0864099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.bb.1.4	✓	7
5.2	odd	4		1925.2.b.r.1849.7		14
5.3	odd	4		1925.2.b.r.1849.8		14
5.4	even	2		1925.2.a.bd.1.4	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.bb.1.4	✓	7	1.1	even	1	trivial
1925.2.a.bd.1.4	yes	7	5.4	even	2
1925.2.b.r.1849.7		14	5.2	odd	4
1925.2.b.r.1849.8		14	5.3	odd	4

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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.151896 q^{2} -2.21537 q^{3} -1.97693 q^{4} +0.336506 q^{6} -1.00000 q^{7} +0.604080 q^{8} +1.90787 q^{9} +O(q^{10})\)	\(q-0.151896 q^{2} -2.21537 q^{3} -1.97693 q^{4} +0.336506 q^{6} -1.00000 q^{7} +0.604080 q^{8} +1.90787 q^{9} +1.00000 q^{11} +4.37963 q^{12} -1.09806 q^{13} +0.151896 q^{14} +3.86210 q^{16} -5.71231 q^{17} -0.289798 q^{18} -0.976928 q^{19} +2.21537 q^{21} -0.151896 q^{22} -6.89879 q^{23} -1.33826 q^{24} +0.166791 q^{26} +2.41947 q^{27} +1.97693 q^{28} -9.78172 q^{29} -4.56229 q^{31} -1.79480 q^{32} -2.21537 q^{33} +0.867678 q^{34} -3.77172 q^{36} +1.12882 q^{37} +0.148391 q^{38} +2.43262 q^{39} +7.19315 q^{41} -0.336506 q^{42} -5.03482 q^{43} -1.97693 q^{44} +1.04790 q^{46} -1.17545 q^{47} -8.55598 q^{48} +1.00000 q^{49} +12.6549 q^{51} +2.17079 q^{52} -10.7836 q^{53} -0.367508 q^{54} -0.604080 q^{56} +2.16426 q^{57} +1.48581 q^{58} +4.29833 q^{59} +9.30798 q^{61} +0.692994 q^{62} -1.90787 q^{63} -7.45157 q^{64} +0.336506 q^{66} -4.00012 q^{67} +11.2928 q^{68} +15.2834 q^{69} +9.90832 q^{71} +1.15251 q^{72} +3.24173 q^{73} -0.171464 q^{74} +1.93132 q^{76} -1.00000 q^{77} -0.369505 q^{78} +13.8523 q^{79} -11.0836 q^{81} -1.09261 q^{82} -1.37098 q^{83} -4.37963 q^{84} +0.764770 q^{86} +21.6701 q^{87} +0.604080 q^{88} +4.13811 q^{89} +1.09806 q^{91} +13.6384 q^{92} +10.1072 q^{93} +0.178546 q^{94} +3.97614 q^{96} -16.2236 q^{97} -0.151896 q^{98} +1.90787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9}+O(q^{10}) \) 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9	\( 7 q - q^{2} + 13 q^{4} + 3 q^{6} - 7 q^{7} + 9 q^{9} + 7 q^{11} + 9 q^{12} + 3 q^{13} + q^{14} + 21 q^{16} + 2 q^{17} - 8 q^{18} + 20 q^{19} - q^{22} - 11 q^{23} + 18 q^{24} - 13 q^{26} + 12 q^{27} - 13 q^{28} + 4 q^{29} + 6 q^{31} - q^{32} - 7 q^{34} + 12 q^{36} - 19 q^{37} - 3 q^{38} - 10 q^{39} + 24 q^{41} - 3 q^{42} + 13 q^{44} + 33 q^{46} + q^{47} + 15 q^{48} + 7 q^{49} + 19 q^{51} + 29 q^{52} - 7 q^{53} + 9 q^{54} + 9 q^{57} - 37 q^{58} + 9 q^{59} + 18 q^{61} + 40 q^{62} - 9 q^{63} + 8 q^{64} + 3 q^{66} - 18 q^{68} - 15 q^{69} + 18 q^{71} + 64 q^{72} - 5 q^{73} - 24 q^{74} + 88 q^{76} - 7 q^{77} - 79 q^{78} + 25 q^{79} - q^{81} + 60 q^{82} + 17 q^{83} - 9 q^{84} - 41 q^{86} + 24 q^{87} - 16 q^{89} - 3 q^{91} - 28 q^{92} - 26 q^{93} - 31 q^{94} + 17 q^{96} + 4 q^{97} - q^{98} + 9 q^{99}+O(q^{100}) \) 7 * q - q^2 + 13 * q^4 + 3 * q^6 - 7 * q^7 + 9 * q^9 + 7 * q^11 + 9 * q^12 + 3 * q^13 + q^14 + 21 * q^16 + 2 * q^17 - 8 * q^18 + 20 * q^19 - q^22 - 11 * q^23 + 18 * q^24 - 13 * q^26 + 12 * q^27 - 13 * q^28 + 4 * q^29 + 6 * q^31 - q^32 - 7 * q^34 + 12 * q^36 - 19 * q^37 - 3 * q^38 - 10 * q^39 + 24 * q^41 - 3 * q^42 + 13 * q^44 + 33 * q^46 + q^47 + 15 * q^48 + 7 * q^49 + 19 * q^51 + 29 * q^52 - 7 * q^53 + 9 * q^54 + 9 * q^57 - 37 * q^58 + 9 * q^59 + 18 * q^61 + 40 * q^62 - 9 * q^63 + 8 * q^64 + 3 * q^66 - 18 * q^68 - 15 * q^69 + 18 * q^71 + 64 * q^72 - 5 * q^73 - 24 * q^74 + 88 * q^76 - 7 * q^77 - 79 * q^78 + 25 * q^79 - q^81 + 60 * q^82 + 17 * q^83 - 9 * q^84 - 41 * q^86 + 24 * q^87 - 16 * q^89 - 3 * q^91 - 28 * q^92 - 26 * q^93 - 31 * q^94 + 17 * q^96 + 4 * q^97 - q^98 + 9 * q^99

Introduction

L-functions

Modular forms

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Database

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