LMFDB - Embedded newform 1925.2.b.q.1849.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1925,2,Mod(1849,1925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1925.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,-26,0,-2,0,0,-26,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1442x^{8} + 3659x^{6} + 4315x^{4} + 2225x^{2} + 400 $ x^14 + 27x^12 + 283x^10 + 1442x^8 + 3659x^6 + 4315x^4 + 2225x^2 + 400
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.10
Root			$1.06204i$ of defining polynomial
Character	$\chi$	$=$	1925.1849
Dual form			1925.2.b.q.1849.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.06204i q^{2} -1.17047i q^{3} +0.872076 q^{4} +1.24309 q^{6} -1.00000i q^{7} +3.05025i q^{8} +1.63000 q^{9} -1.00000 q^{11} -1.02074i q^{12} -6.82997i q^{13} +1.06204 q^{14} -1.49533 q^{16} -2.26325i q^{17} +1.73112i q^{18} -6.41641 q^{19} -1.17047 q^{21} -1.06204i q^{22} -9.35726i q^{23} +3.57024 q^{24} +7.25368 q^{26} -5.41928i q^{27} -0.872076i q^{28} -4.96420 q^{29} -0.856196 q^{31} +4.51241i q^{32} +1.17047i q^{33} +2.40366 q^{34} +1.42148 q^{36} +6.61436i q^{37} -6.81447i q^{38} -7.99428 q^{39} +0.527296 q^{41} -1.24309i q^{42} -4.95746i q^{43} -0.872076 q^{44} +9.93777 q^{46} +1.24309i q^{47} +1.75024i q^{48} -1.00000 q^{49} -2.64907 q^{51} -5.95625i q^{52} +3.97763i q^{53} +5.75548 q^{54} +3.05025 q^{56} +7.51023i q^{57} -5.27217i q^{58} +11.6526 q^{59} +3.40640 q^{61} -0.909312i q^{62} -1.63000i q^{63} -7.78301 q^{64} -1.24309 q^{66} +5.32284i q^{67} -1.97373i q^{68} -10.9524 q^{69} +4.43143 q^{71} +4.97190i q^{72} -1.06974i q^{73} -7.02470 q^{74} -5.59560 q^{76} +1.00000i q^{77} -8.49023i q^{78} +0.966826 q^{79} -1.45313 q^{81} +0.560008i q^{82} +6.07331i q^{83} -1.02074 q^{84} +5.26501 q^{86} +5.81046i q^{87} -3.05025i q^{88} +3.55617 q^{89} -6.82997 q^{91} -8.16024i q^{92} +1.00215i q^{93} -1.32020 q^{94} +5.28165 q^{96} -16.8926i q^{97} -1.06204i q^{98} -1.63000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 26 q^{4} - 2 q^{6} - 26 q^{9} - 14 q^{11} - 2 q^{14} + 58 q^{16} - 36 q^{19} + 44 q^{24} + 26 q^{26} + 4 q^{29} + 48 q^{31} - 66 q^{34} + 88 q^{36} - 20 q^{39} - 20 q^{41} + 26 q^{44} + 6 q^{46}+ \cdots + 26 q^{99}+O(q^{100}) $ 14 * q - 26 * q^4 - 2 * q^6 - 26 * q^9 - 14 * q^11 - 2 * q^14 + 58 * q^16 - 36 * q^19 + 44 * q^24 + 26 * q^26 + 4 * q^29 + 48 * q^31 - 66 * q^34 + 88 * q^36 - 20 * q^39 - 20 * q^41 + 26 * q^44 + 6 * q^46 - 14 * q^49 + 58 * q^51 + 94 * q^54 + 12 * q^56 - 2 * q^59 + 56 * q^61 - 96 * q^64 + 2 * q^66 - 66 * q^69 - 20 * q^71 + 12 * q^74 + 40 * q^76 - 38 * q^79 + 54 * q^81 + 42 * q^84 + 110 * q^86 + 4 * q^89 - 6 * q^91 - 42 * q^94 - 46 * q^96 + 26 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$.

$n$	$276$	$1002$	$1751$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.06204i	0.750974i	0.926828	+	0.375487i	$0.122525\pi$
$2$	1.06204i	0.750974i	−0.926828	+	0.375487i	$0.877475\pi$
$3$	− 1.17047i	− 0.675772i	−0.941187	−	0.337886i	$-0.890288\pi$
$3$	− 1.17047i	− 0.675772i	0.941187	−	0.337886i	$-0.109712\pi$
$4$	0.872076	0.436038
$5$	0	0
$5$	0	0
$6$	1.24309	0.507487
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	3.05025i	1.07843i
$9$	1.63000	0.543332
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 1.02074i	− 0.294662i
$13$	− 6.82997i	− 1.89429i	−0.320803	−	0.947146i	$-0.603953\pi$
$13$	− 6.82997i	− 1.89429i	0.320803	−	0.947146i	$-0.396047\pi$
$14$	1.06204	0.283841
$15$	0	0
$16$	−1.49533	−0.373833
$17$	− 2.26325i	− 0.548919i	−0.961599	−	0.274460i	$-0.911501\pi$
$17$	− 2.26325i	− 0.548919i	0.961599	−	0.274460i	$-0.0884990\pi$
$18$	1.73112i	0.408028i
$19$	−6.41641	−1.47203	−0.736013	−	0.676968i	$-0.763294\pi$
$19$	−6.41641	−1.47203	−0.736013	+	0.676968i	$0.763294\pi$
$20$	0	0
$21$	−1.17047	−0.255418
$22$	− 1.06204i	− 0.226427i
$23$	− 9.35726i	− 1.95112i	−0.219724	−	0.975562i	$-0.570516\pi$
$23$	− 9.35726i	− 1.95112i	0.219724	−	0.975562i	$-0.429484\pi$
$24$	3.57024	0.728771
$25$	0	0
$26$	7.25368	1.42256
$27$	− 5.41928i	− 1.04294i
$28$	− 0.872076i	− 0.164807i
$29$	−4.96420	−0.921829	−0.460914	−	0.887445i	$-0.652478\pi$
$29$	−4.96420	−0.921829	−0.460914	+	0.887445i	$0.652478\pi$
$30$	0	0
$31$	−0.856196	−0.153777	−0.0768887	−	0.997040i	$-0.524499\pi$
$31$	−0.856196	−0.153777	−0.0768887	+	0.997040i	$0.524499\pi$
$32$	4.51241i	0.797689i
$33$	1.17047i	0.203753i
$34$	2.40366	0.412224
$35$	0	0
$36$	1.42148	0.236913
$37$	6.61436i	1.08739i	0.839281	+	0.543697i	$0.182976\pi$
$37$	6.61436i	1.08739i	−0.839281	+	0.543697i	$0.817024\pi$
$38$	− 6.81447i	− 1.10545i
$39$	−7.99428	−1.28011
$40$	0	0
$41$	0.527296	0.0823498	0.0411749	−	0.999152i	$-0.486890\pi$
$41$	0.527296	0.0823498	0.0411749	+	0.999152i	$0.486890\pi$
$42$	− 1.24309i	− 0.191812i
$43$	− 4.95746i	− 0.756005i	−0.925804	−	0.378003i	$-0.876611\pi$
$43$	− 4.95746i	− 0.756005i	0.925804	−	0.378003i	$-0.123389\pi$
$44$	−0.872076	−0.131470
$45$	0	0
$46$	9.93777	1.46524
$47$	1.24309i	0.181323i	0.995882	+	0.0906613i	$0.0288981\pi$
$47$	1.24309i	0.181323i	−0.995882	+	0.0906613i	$0.971102\pi$
$48$	1.75024i	0.252626i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−2.64907	−0.370945
$52$	− 5.95625i	− 0.825983i
$53$	3.97763i	0.546369i	0.961962	+	0.273185i	$0.0880770\pi$
$53$	3.97763i	0.546369i	−0.961962	+	0.273185i	$0.911923\pi$
$54$	5.75548	0.783222
$55$	0	0
$56$	3.05025	0.407607
$57$	7.51023i	0.994754i
$58$	− 5.27217i	− 0.692269i
$59$	11.6526	1.51704	0.758518	−	0.651652i	$-0.225924\pi$
$59$	11.6526	1.51704	0.758518	+	0.651652i	$0.225924\pi$
$60$	0	0
$61$	3.40640	0.436145	0.218072	−	0.975933i	$-0.430023\pi$
$61$	3.40640	0.436145	0.218072	+	0.975933i	$0.430023\pi$
$62$	− 0.909312i	− 0.115483i
$63$	− 1.63000i	− 0.205360i
$64$	−7.78301	−0.972876
$65$	0	0
$66$	−1.24309	−0.153013
$67$	5.32284i	0.650288i	0.945665	+	0.325144i	$0.105413\pi$
$67$	5.32284i	0.650288i	−0.945665	+	0.325144i	$0.894587\pi$
$68$	− 1.97373i	− 0.239350i
$69$	−10.9524	−1.31852
$70$	0	0
$71$	4.43143	0.525914	0.262957	−	0.964808i	$-0.415302\pi$
$71$	4.43143	0.525914	0.262957	+	0.964808i	$0.415302\pi$
$72$	4.97190i	0.585944i
$73$	− 1.06974i	− 0.125204i	−0.998039	−	0.0626019i	$-0.980060\pi$
$73$	− 1.06974i	− 0.125204i	0.998039	−	0.0626019i	$-0.0199399\pi$
$74$	−7.02470	−0.816605
$75$	0	0
$76$	−5.59560	−0.641859
$77$	1.00000i	0.113961i
$78$	− 8.49023i	− 0.961329i
$79$	0.966826	0.108776	0.0543882	−	0.998520i	$-0.482679\pi$
$79$	0.966826	0.108776	0.0543882	+	0.998520i	$0.482679\pi$
$80$	0	0
$81$	−1.45313	−0.161459
$82$	0.560008i	0.0618426i
$83$	6.07331i	0.666633i	0.942815	+	0.333316i	$0.108168\pi$
$83$	6.07331i	0.666633i	−0.942815	+	0.333316i	$0.891832\pi$
$84$	−1.02074	−0.111372
$85$	0	0
$86$	5.26501	0.567740
$87$	5.81046i	0.622946i
$88$	− 3.05025i	− 0.325158i
$89$	3.55617	0.376954	0.188477	−	0.982078i	$-0.439645\pi$
$89$	3.55617	0.376954	0.188477	+	0.982078i	$0.439645\pi$
$90$	0	0
$91$	−6.82997	−0.715975
$92$	− 8.16024i	− 0.850764i
$93$	1.00215i	0.103918i
$94$	−1.32020	−0.136169
$95$	0	0
$96$	5.28165	0.539056
$97$	− 16.8926i	− 1.71518i	−0.514331	−	0.857592i	$-0.671960\pi$
$97$	− 16.8926i	− 1.71518i	0.514331	−	0.857592i	$-0.328040\pi$
$98$	− 1.06204i	− 0.107282i
$99$	−1.63000	−0.163821
$100$	0	0
$101$	8.23058	0.818974	0.409487	−	0.912316i	$-0.365708\pi$
$101$	8.23058	0.818974	0.409487	+	0.912316i	$0.365708\pi$
$102$	− 2.81342i	− 0.278570i
$103$	− 17.1467i	− 1.68951i	−0.535150	−	0.844757i	$-0.679745\pi$
$103$	− 17.1467i	− 1.68951i	0.535150	−	0.844757i	$-0.320255\pi$
$104$	20.8331	2.04286
$105$	0	0
$106$	−4.22439	−0.410309
$107$	− 17.4535i	− 1.68729i	−0.536902	−	0.843645i	$-0.680406\pi$
$107$	− 17.4535i	− 1.68729i	0.536902	−	0.843645i	$-0.319594\pi$
$108$	− 4.72602i	− 0.454762i
$109$	−9.79014	−0.937725	−0.468863	−	0.883271i	$-0.655336\pi$
$109$	−9.79014	−0.937725	−0.468863	+	0.883271i	$0.655336\pi$
$110$	0	0
$111$	7.74193	0.734831
$112$	1.49533i	0.141296i
$113$	16.6228i	1.56374i	0.623439	+	0.781872i	$0.285735\pi$
$113$	16.6228i	1.56374i	−0.623439	+	0.781872i	$0.714265\pi$
$114$	−7.97614	−0.747034
$115$	0	0
$116$	−4.32916	−0.401952
$117$	− 11.1328i	− 1.02923i
$118$	12.3755i	1.13925i
$119$	−2.26325	−0.207472
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.61772i	0.327533i
$123$	− 0.617186i	− 0.0556497i
$124$	−0.746668	−0.0670528
$125$	0	0
$126$	1.73112	0.154220
$127$	− 16.1524i	− 1.43330i	−0.697435	−	0.716648i	$-0.745675\pi$
$127$	− 16.1524i	− 1.43330i	0.697435	−	0.716648i	$-0.254325\pi$
$128$	0.758967i	0.0670838i
$129$	−5.80257	−0.510888
$130$	0	0
$131$	9.91338	0.866136	0.433068	−	0.901361i	$-0.357431\pi$
$131$	9.91338	0.866136	0.433068	+	0.901361i	$0.357431\pi$
$132$	1.02074i	0.0888441i
$133$	6.41641i	0.556373i
$134$	−5.65305	−0.488349
$135$	0	0
$136$	6.90349	0.591970
$137$	4.06402i	0.347213i	0.984815	+	0.173606i	$0.0555420\pi$
$137$	4.06402i	0.347213i	−0.984815	+	0.173606i	$0.944458\pi$
$138$	− 11.6319i	− 0.990171i
$139$	−14.0507	−1.19176	−0.595881	−	0.803073i	$-0.703197\pi$
$139$	−14.0507	−1.19176	−0.595881	+	0.803073i	$0.703197\pi$
$140$	0	0
$141$	1.45500	0.122533
$142$	4.70634i	0.394947i
$143$	6.82997i	0.571150i
$144$	−2.43738	−0.203115
$145$	0	0
$146$	1.13611	0.0940249
$147$	1.17047i	0.0965389i
$148$	5.76823i	0.474145i
$149$	4.68802	0.384058	0.192029	−	0.981389i	$-0.438493\pi$
$149$	4.68802	0.384058	0.192029	+	0.981389i	$0.438493\pi$
$150$	0	0
$151$	4.15871	0.338431	0.169215	−	0.985579i	$-0.445877\pi$
$151$	4.15871	0.338431	0.169215	+	0.985579i	$0.445877\pi$
$152$	− 19.5717i	− 1.58747i
$153$	− 3.68909i	− 0.298245i
$154$	−1.06204	−0.0855814
$155$	0	0
$156$	−6.97162	−0.558177
$157$	9.50165i	0.758314i	0.925332	+	0.379157i	$0.123786\pi$
$157$	9.50165i	0.758314i	−0.925332	+	0.379157i	$0.876214\pi$
$158$	1.02681i	0.0816883i
$159$	4.65570	0.369221
$160$	0	0
$161$	−9.35726	−0.737456
$162$	− 1.54328i	− 0.121251i
$163$	23.3193i	1.82651i	0.407386	+	0.913256i	$0.366440\pi$
$163$	23.3193i	1.82651i	−0.407386	+	0.913256i	$0.633560\pi$
$164$	0.459842	0.0359077
$165$	0	0
$166$	−6.45009	−0.500624
$167$	− 12.6057i	− 0.975458i	−0.872995	−	0.487729i	$-0.837825\pi$
$167$	− 12.6057i	− 0.975458i	0.872995	−	0.487729i	$-0.162175\pi$
$168$	− 3.57024i	− 0.275450i
$169$	−33.6484	−2.58834
$170$	0	0
$171$	−10.4587	−0.799798
$172$	− 4.32328i	− 0.329647i
$173$	− 9.26682i	− 0.704543i	−0.935898	−	0.352272i	$-0.885409\pi$
$173$	− 9.26682i	− 0.704543i	0.935898	−	0.352272i	$-0.114591\pi$
$174$	−6.17092	−0.467817
$175$	0	0
$176$	1.49533	0.112715
$177$	− 13.6390i	− 1.02517i
$178$	3.77679i	0.283082i
$179$	15.5397	1.16149	0.580747	−	0.814084i	$-0.302761\pi$
$179$	15.5397	1.16149	0.580747	+	0.814084i	$0.302761\pi$
$180$	0	0
$181$	−3.33690	−0.248030	−0.124015	−	0.992280i	$-0.539577\pi$
$181$	−3.33690	−0.248030	−0.124015	+	0.992280i	$0.539577\pi$
$182$	− 7.25368i	− 0.537679i
$183$	− 3.98710i	− 0.294735i
$184$	28.5420	2.10415
$185$	0	0
$186$	−1.06432	−0.0780401
$187$	2.26325i	0.165505i
$188$	1.08406i	0.0790636i
$189$	−5.41928	−0.394195
$190$	0	0
$191$	20.9541	1.51618	0.758092	−	0.652147i	$-0.226132\pi$
$191$	20.9541	1.51618	0.758092	+	0.652147i	$0.226132\pi$
$192$	9.10979i	0.657443i
$193$	13.7901i	0.992636i	0.868141	+	0.496318i	$0.165315\pi$
$193$	13.7901i	0.992636i	−0.868141	+	0.496318i	$0.834685\pi$
$194$	17.9406	1.28806
$195$	0	0
$196$	−0.872076	−0.0622911
$197$	11.4424i	0.815234i	0.913153	+	0.407617i	$0.133640\pi$
$197$	11.4424i	0.815234i	−0.913153	+	0.407617i	$0.866360\pi$
$198$	− 1.73112i	− 0.123025i
$199$	−21.8077	−1.54591	−0.772955	−	0.634461i	$-0.781222\pi$
$199$	−21.8077	−1.54591	−0.772955	+	0.634461i	$0.781222\pi$
$200$	0	0
$201$	6.23023	0.439447
$202$	8.74119i	0.615028i
$203$	4.96420i	0.348418i
$204$	−2.31019	−0.161746
$205$	0	0
$206$	18.2104	1.26878
$207$	− 15.2523i	− 1.06011i
$208$	10.2131i	0.708148i
$209$	6.41641	0.443832
$210$	0	0
$211$	23.1975	1.59698	0.798490	−	0.602008i	$-0.205632\pi$
$211$	23.1975	1.59698	0.798490	+	0.602008i	$0.205632\pi$
$212$	3.46879i	0.238238i
$213$	− 5.18686i	− 0.355398i
$214$	18.5362	1.26711
$215$	0	0
$216$	16.5302	1.12474
$217$	0.856196i	0.0581224i
$218$	− 10.3975i	− 0.704207i
$219$	−1.25210	−0.0846093
$220$	0	0
$221$	−15.4579	−1.03981
$222$	8.22222i	0.551839i
$223$	− 2.36026i	− 0.158055i	−0.996872	−	0.0790274i	$-0.974819\pi$
$223$	− 2.36026i	− 0.158055i	0.996872	−	0.0790274i	$-0.0251815\pi$
$224$	4.51241	0.301498
$225$	0	0
$226$	−17.6541	−1.17433
$227$	28.4393i	1.88758i	0.330545	+	0.943790i	$0.392768\pi$
$227$	28.4393i	1.88758i	−0.330545	+	0.943790i	$0.607232\pi$
$228$	6.54949i	0.433751i
$229$	18.9280	1.25080	0.625400	−	0.780305i	$-0.284936\pi$
$229$	18.9280	1.25080	0.625400	+	0.780305i	$0.284936\pi$
$230$	0	0
$231$	1.17047	0.0770114
$232$	− 15.1421i	− 0.994125i
$233$	− 17.8126i	− 1.16694i	−0.812135	−	0.583470i	$-0.801695\pi$
$233$	− 17.8126i	− 1.16694i	0.812135	−	0.583470i	$-0.198305\pi$
$234$	11.8235	0.772924
$235$	0	0
$236$	10.1619	0.661485
$237$	− 1.13164i	− 0.0735081i
$238$	− 2.40366i	− 0.155806i
$239$	13.3135	0.861179	0.430589	−	0.902548i	$-0.358306\pi$
$239$	13.3135	0.861179	0.430589	+	0.902548i	$0.358306\pi$
$240$	0	0
$241$	8.43397	0.543280	0.271640	−	0.962399i	$-0.412434\pi$
$241$	8.43397	0.543280	0.271640	+	0.962399i	$0.412434\pi$
$242$	1.06204i	0.0682704i
$243$	− 14.5570i	− 0.933831i
$244$	2.97064	0.190176
$245$	0	0
$246$	0.655474	0.0417915
$247$	43.8239i	2.78845i
$248$	− 2.61161i	− 0.165838i
$249$	7.10864	0.450492
$250$	0	0
$251$	13.9949	0.883350	0.441675	−	0.897175i	$-0.354384\pi$
$251$	13.9949	0.883350	0.441675	+	0.897175i	$0.354384\pi$
$252$	− 1.42148i	− 0.0895448i
$253$	9.35726i	0.588286i
$254$	17.1545	1.07637
$255$	0	0
$256$	−16.3721	−1.02325
$257$	7.16272i	0.446798i	0.974727	+	0.223399i	$0.0717153\pi$
$257$	7.16272i	0.446798i	−0.974727	+	0.223399i	$0.928285\pi$
$258$	− 6.16255i	− 0.383663i
$259$	6.61436	0.410996
$260$	0	0
$261$	−8.09162	−0.500859
$262$	10.5284i	0.650446i
$263$	− 7.01651i	− 0.432657i	−0.976321	−	0.216328i	$-0.930592\pi$
$263$	− 7.01651i	− 0.432657i	0.976321	−	0.216328i	$-0.0694081\pi$
$264$	−3.57024	−0.219733
$265$	0	0
$266$	−6.81447	−0.417822
$267$	− 4.16240i	− 0.254735i
$268$	4.64192i	0.283550i
$269$	4.13429	0.252072	0.126036	−	0.992026i	$-0.459774\pi$
$269$	4.13429	0.252072	0.126036	+	0.992026i	$0.459774\pi$
$270$	0	0
$271$	−12.8230	−0.778939	−0.389469	−	0.921039i	$-0.627342\pi$
$271$	−12.8230	−0.778939	−0.389469	+	0.921039i	$0.627342\pi$
$272$	3.38431i	0.205204i
$273$	7.99428i	0.483836i
$274$	−4.31614	−0.260748
$275$	0	0
$276$	−9.55134	−0.574923
$277$	− 10.8885i	− 0.654227i	−0.944985	−	0.327114i	$-0.893924\pi$
$277$	− 10.8885i	− 0.654227i	0.944985	−	0.327114i	$-0.106076\pi$
$278$	− 14.9223i	− 0.894983i
$279$	−1.39560	−0.0835521
$280$	0	0
$281$	−3.16841	−0.189011	−0.0945056	−	0.995524i	$-0.530127\pi$
$281$	−3.16841	−0.189011	−0.0945056	+	0.995524i	$0.530127\pi$
$282$	1.54526i	0.0920190i
$283$	33.0957i	1.96733i	0.179997	+	0.983667i	$0.442391\pi$
$283$	33.0957i	1.96733i	−0.179997	+	0.983667i	$0.557609\pi$
$284$	3.86454	0.229318
$285$	0	0
$286$	−7.25368	−0.428919
$287$	− 0.527296i	− 0.0311253i
$288$	7.35520i	0.433409i
$289$	11.8777	0.698687
$290$	0	0
$291$	−19.7723	−1.15907
$292$	− 0.932897i	− 0.0545937i
$293$	17.3630i	1.01436i	0.861841	+	0.507178i	$0.169311\pi$
$293$	17.3630i	1.01436i	−0.861841	+	0.507178i	$0.830689\pi$
$294$	−1.24309	−0.0724982
$295$	0	0
$296$	−20.1755	−1.17268
$297$	5.41928i	0.314459i
$298$	4.97886i	0.288417i
$299$	−63.9098	−3.69600
$300$	0	0
$301$	−4.95746	−0.285743
$302$	4.41670i	0.254153i
$303$	− 9.63367i	− 0.553440i
$304$	9.59466	0.550291
$305$	0	0
$306$	3.91795	0.223974
$307$	− 11.7256i	− 0.669217i	−0.942357	−	0.334608i	$-0.891396\pi$
$307$	− 11.7256i	− 0.669217i	0.942357	−	0.334608i	$-0.108604\pi$
$308$	0.872076i	0.0496911i
$309$	−20.0697	−1.14173
$310$	0	0
$311$	10.4874	0.594686	0.297343	−	0.954771i	$-0.403900\pi$
$311$	10.4874	0.594686	0.297343	+	0.954771i	$0.403900\pi$
$312$	− 24.3846i	− 1.38051i
$313$	22.0342i	1.24545i	0.782443	+	0.622723i	$0.213973\pi$
$313$	22.0342i	1.24545i	−0.782443	+	0.622723i	$0.786027\pi$
$314$	−10.0911	−0.569474
$315$	0	0
$316$	0.843146	0.0474307
$317$	14.8993i	0.836827i	0.908257	+	0.418414i	$0.137414\pi$
$317$	14.8993i	0.836827i	−0.908257	+	0.418414i	$0.862586\pi$
$318$	4.94453i	0.277275i
$319$	4.96420	0.277942
$320$	0	0
$321$	−20.4288	−1.14022
$322$	− 9.93777i	− 0.553810i
$323$	14.5220i	0.808023i
$324$	−1.26724	−0.0704022
$325$	0	0
$326$	−24.7660	−1.37166
$327$	11.4591i	0.633689i
$328$	1.60839i	0.0888083i
$329$	1.24309	0.0685335
$330$	0	0
$331$	13.5775	0.746290	0.373145	−	0.927773i	$-0.378279\pi$
$331$	13.5775	0.746290	0.373145	+	0.927773i	$0.378279\pi$
$332$	5.29639i	0.290677i
$333$	10.7814i	0.590816i
$334$	13.3877	0.732543
$335$	0	0
$336$	1.75024	0.0954836
$337$	3.68755i	0.200874i	0.994943	+	0.100437i	$0.0320241\pi$
$337$	3.68755i	0.200874i	−0.994943	+	0.100437i	$0.967976\pi$
$338$	− 35.7359i	− 1.94378i
$339$	19.4566	1.05674
$340$	0	0
$341$	0.856196	0.0463656
$342$	− 11.1075i	− 0.600628i
$343$	1.00000i	0.0539949i
$344$	15.1215	0.815297
$345$	0	0
$346$	9.84171	0.529094
$347$	− 19.0044i	− 1.02021i	−0.860112	−	0.510106i	$-0.829606\pi$
$347$	− 19.0044i	− 1.02021i	0.860112	−	0.510106i	$-0.170394\pi$
$348$	5.06716i	0.271628i
$349$	5.67289	0.303663	0.151831	−	0.988406i	$-0.451483\pi$
$349$	5.67289	0.303663	0.151831	+	0.988406i	$0.451483\pi$
$350$	0	0
$351$	−37.0135	−1.97563
$352$	− 4.51241i	− 0.240512i
$353$	− 2.84951i	− 0.151664i	−0.997121	−	0.0758320i	$-0.975839\pi$
$353$	− 2.84951i	− 0.151664i	0.997121	−	0.0758320i	$-0.0241613\pi$
$354$	14.4851	0.769876
$355$	0	0
$356$	3.10125	0.164366
$357$	2.64907i	0.140204i
$358$	16.5038i	0.872251i
$359$	−20.9541	−1.10592	−0.552958	−	0.833209i	$-0.686501\pi$
$359$	−20.9541	−1.10592	−0.552958	+	0.833209i	$0.686501\pi$
$360$	0	0
$361$	22.1703	1.16686
$362$	− 3.54392i	− 0.186264i
$363$	− 1.17047i	− 0.0614339i
$364$	−5.95625	−0.312192
$365$	0	0
$366$	4.23445	0.221338
$367$	3.69812i	0.193040i	0.995331	+	0.0965201i	$0.0307712\pi$
$367$	3.69812i	0.193040i	−0.995331	+	0.0965201i	$0.969229\pi$
$368$	13.9922i	0.729394i
$369$	0.859490	0.0447433
$370$	0	0
$371$	3.97763	0.206508
$372$	0.873954i	0.0453124i
$373$	30.7541i	1.59238i	0.605043	+	0.796192i	$0.293156\pi$
$373$	30.7541i	1.59238i	−0.605043	+	0.796192i	$0.706844\pi$
$374$	−2.40366	−0.124290
$375$	0	0
$376$	−3.79172	−0.195543
$377$	33.9053i	1.74621i
$378$	− 5.75548i	− 0.296030i
$379$	−18.6017	−0.955506	−0.477753	−	0.878494i	$-0.658549\pi$
$379$	−18.6017	−0.955506	−0.477753	+	0.878494i	$0.658549\pi$
$380$	0	0
$381$	−18.9060	−0.968582
$382$	22.2540i	1.13862i
$383$	− 15.1169i	− 0.772440i	−0.922407	−	0.386220i	$-0.873781\pi$
$383$	− 15.1169i	− 0.772440i	0.922407	−	0.386220i	$-0.126219\pi$
$384$	0.888349	0.0453334
$385$	0	0
$386$	−14.6456	−0.745444
$387$	− 8.08063i	− 0.410762i
$388$	− 14.7316i	− 0.747885i
$389$	13.9240	0.705973	0.352986	−	0.935628i	$-0.385166\pi$
$389$	13.9240	0.705973	0.352986	+	0.935628i	$0.385166\pi$
$390$	0	0
$391$	−21.1778	−1.07101
$392$	− 3.05025i	− 0.154061i
$393$	− 11.6033i	− 0.585311i
$394$	−12.1522	−0.612220
$395$	0	0
$396$	−1.42148	−0.0714320
$397$	− 1.02403i	− 0.0513946i	−0.999670	−	0.0256973i	$-0.991819\pi$
$397$	− 1.02403i	− 0.0513946i	0.999670	−	0.0256973i	$-0.00818061\pi$
$398$	− 23.1606i	− 1.16094i
$399$	7.51023	0.375982
$400$	0	0
$401$	3.00889	0.150257	0.0751284	−	0.997174i	$-0.476063\pi$
$401$	3.00889	0.150257	0.0751284	+	0.997174i	$0.476063\pi$
$402$	6.61674i	0.330013i
$403$	5.84779i	0.291299i
$404$	7.17769	0.357104
$405$	0	0
$406$	−5.27217	−0.261653
$407$	− 6.61436i	− 0.327862i
$408$	− 8.08035i	− 0.400037i
$409$	31.1886	1.54218	0.771090	−	0.636727i	$-0.219712\pi$
$409$	31.1886	1.54218	0.771090	+	0.636727i	$0.219712\pi$
$410$	0	0
$411$	4.75682	0.234637
$412$	− 14.9532i	− 0.736692i
$413$	− 11.6526i	− 0.573385i
$414$	16.1985	0.796113
$415$	0	0
$416$	30.8196	1.51105
$417$	16.4459i	0.805360i
$418$	6.81447i	0.333307i
$419$	31.4805	1.53792	0.768962	−	0.639294i	$-0.220773\pi$
$419$	31.4805	1.53792	0.768962	+	0.639294i	$0.220773\pi$
$420$	0	0
$421$	16.6545	0.811689	0.405845	−	0.913942i	$-0.366977\pi$
$421$	16.6545	0.811689	0.405845	+	0.913942i	$0.366977\pi$
$422$	24.6366i	1.19929i
$423$	2.02622i	0.0985183i
$424$	−12.1328	−0.589219
$425$	0	0
$426$	5.50864	0.266895
$427$	− 3.40640i	− 0.164847i
$428$	− 15.2207i	− 0.735722i
$429$	7.99428	0.385968
$430$	0	0
$431$	−31.0206	−1.49421	−0.747104	−	0.664707i	$-0.768556\pi$
$431$	−31.0206	−1.49421	−0.747104	+	0.664707i	$0.768556\pi$
$432$	8.10362i	0.389886i
$433$	− 14.6496i	− 0.704016i	−0.935997	−	0.352008i	$-0.885499\pi$
$433$	− 14.6496i	− 0.704016i	0.935997	−	0.352008i	$-0.114501\pi$
$434$	−0.909312	−0.0436484
$435$	0	0
$436$	−8.53775	−0.408884
$437$	60.0400i	2.87210i
$438$	− 1.32978i	− 0.0635394i
$439$	−16.8201	−0.802778	−0.401389	−	0.915908i	$-0.631473\pi$
$439$	−16.8201	−0.802778	−0.401389	+	0.915908i	$0.631473\pi$
$440$	0	0
$441$	−1.63000	−0.0776188
$442$	− 16.4169i	− 0.780873i
$443$	18.3489i	0.871784i	0.899999	+	0.435892i	$0.143567\pi$
$443$	18.3489i	0.871784i	−0.899999	+	0.435892i	$0.856433\pi$
$444$	6.75155	0.320414
$445$	0	0
$446$	2.50669	0.118695
$447$	− 5.48720i	− 0.259536i
$448$	7.78301i	0.367713i
$449$	−11.1951	−0.528329	−0.264164	−	0.964478i	$-0.585096\pi$
$449$	−11.1951	−0.528329	−0.264164	+	0.964478i	$0.585096\pi$
$450$	0	0
$451$	−0.527296	−0.0248294
$452$	14.4964i	0.681852i
$453$	− 4.86765i	− 0.228702i
$454$	−30.2036	−1.41752
$455$	0	0
$456$	−22.9081	−1.07277
$457$	12.0566i	0.563984i	0.959417	+	0.281992i	$0.0909951\pi$
$457$	12.0566i	0.563984i	−0.959417	+	0.281992i	$0.909005\pi$
$458$	20.1023i	0.939318i
$459$	−12.2652	−0.572490
$460$	0	0
$461$	−41.1825	−1.91806	−0.959030	−	0.283303i	$-0.908570\pi$
$461$	−41.1825	−1.91806	−0.959030	+	0.283303i	$0.908570\pi$
$462$	1.24309i	0.0578336i
$463$	− 33.1629i	− 1.54121i	−0.637311	−	0.770606i	$-0.719953\pi$
$463$	− 33.1629i	− 1.54121i	0.637311	−	0.770606i	$-0.280047\pi$
$464$	7.42312	0.344610
$465$	0	0
$466$	18.9176	0.876341
$467$	− 20.8688i	− 0.965695i	−0.875705	−	0.482847i	$-0.839603\pi$
$467$	− 20.8688i	− 0.965695i	0.875705	−	0.482847i	$-0.160397\pi$
$468$	− 9.70866i	− 0.448783i
$469$	5.32284	0.245786
$470$	0	0
$471$	11.1214	0.512448
$472$	35.5433i	1.63601i
$473$	4.95746i	0.227944i
$474$	1.20185	0.0552027
$475$	0	0
$476$	−1.97373	−0.0904657
$477$	6.48351i	0.296860i
$478$	14.1394i	0.646723i
$479$	−4.71764	−0.215555	−0.107777	−	0.994175i	$-0.534373\pi$
$479$	−4.71764	−0.215555	−0.107777	+	0.994175i	$0.534373\pi$
$480$	0	0
$481$	45.1759	2.05984
$482$	8.95720i	0.407989i
$483$	10.9524i	0.498352i
$484$	0.872076	0.0396398
$485$	0	0
$486$	15.4601	0.701283
$487$	− 5.80132i	− 0.262883i	−0.991324	−	0.131441i	$-0.958039\pi$
$487$	− 5.80132i	− 0.262883i	0.991324	−	0.131441i	$-0.0419605\pi$
$488$	10.3904i	0.470350i
$489$	27.2946	1.23431
$490$	0	0
$491$	−30.4064	−1.37222	−0.686111	−	0.727497i	$-0.740683\pi$
$491$	−30.4064	−1.37222	−0.686111	+	0.727497i	$0.740683\pi$
$492$	− 0.538233i	− 0.0242654i
$493$	11.2352i	0.506010i
$494$	−46.5426	−2.09405
$495$	0	0
$496$	1.28030	0.0574870
$497$	− 4.43143i	− 0.198777i
$498$	7.54964i	0.338308i
$499$	−40.7188	−1.82282	−0.911411	−	0.411497i	$-0.865006\pi$
$499$	−40.7188	−1.82282	−0.911411	+	0.411497i	$0.865006\pi$
$500$	0	0
$501$	−14.7546	−0.659187
$502$	14.8631i	0.663373i
$503$	16.6048i	0.740370i	0.928958	+	0.370185i	$0.120706\pi$
$503$	16.6048i	0.740370i	−0.928958	+	0.370185i	$0.879294\pi$
$504$	4.97190	0.221466
$505$	0	0
$506$	−9.93777	−0.441788
$507$	39.3846i	1.74913i
$508$	− 14.0861i	− 0.624972i
$509$	8.91546	0.395171	0.197585	−	0.980286i	$-0.436690\pi$
$509$	8.91546	0.395171	0.197585	+	0.980286i	$0.436690\pi$
$510$	0	0
$511$	−1.06974	−0.0473226
$512$	− 15.8698i	− 0.701354i
$513$	34.7723i	1.53524i
$514$	−7.60708	−0.335534
$515$	0	0
$516$	−5.06028	−0.222766
$517$	− 1.24309i	− 0.0546708i
$518$	7.02470i	0.308648i
$519$	−10.8466	−0.476111
$520$	0	0
$521$	−12.0474	−0.527806	−0.263903	−	0.964549i	$-0.585010\pi$
$521$	−12.0474	−0.527806	−0.263903	+	0.964549i	$0.585010\pi$
$522$	− 8.59361i	− 0.376132i
$523$	− 6.05630i	− 0.264823i	−0.991195	−	0.132412i	$-0.957728\pi$
$523$	− 6.05630i	− 0.264823i	0.991195	−	0.132412i	$-0.0422721\pi$
$524$	8.64522	0.377668
$525$	0	0
$526$	7.45180	0.324914
$527$	1.93779i	0.0844114i
$528$	− 1.75024i	− 0.0761696i
$529$	−64.5584	−2.80689
$530$	0	0
$531$	18.9936	0.824253
$532$	5.59560i	0.242600i
$533$	− 3.60142i	− 0.155995i
$534$	4.42063	0.191299
$535$	0	0
$536$	−16.2360	−0.701288
$537$	− 18.1888i	− 0.784905i
$538$	4.39077i	0.189300i
$539$	1.00000	0.0430730
$540$	0	0
$541$	−2.81266	−0.120926	−0.0604629	−	0.998170i	$-0.519258\pi$
$541$	−2.81266	−0.120926	−0.0604629	+	0.998170i	$0.519258\pi$
$542$	− 13.6185i	− 0.584963i
$543$	3.90575i	0.167612i
$544$	10.2127	0.437867
$545$	0	0
$546$	−8.49023	−0.363348
$547$	− 14.1171i	− 0.603603i	−0.953371	−	0.301802i	$-0.902412\pi$
$547$	− 14.1171i	− 0.603603i	0.953371	−	0.301802i	$-0.0975880\pi$
$548$	3.54413i	0.151398i
$549$	5.55241	0.236971
$550$	0	0
$551$	31.8523	1.35696
$552$	− 33.4076i	− 1.42192i
$553$	− 0.966826i	− 0.0411136i
$554$	11.5640	0.491308
$555$	0	0
$556$	−12.2533	−0.519654
$557$	1.95691i	0.0829168i	0.999140	+	0.0414584i	$0.0132004\pi$
$557$	1.95691i	0.0829168i	−0.999140	+	0.0414584i	$0.986800\pi$
$558$	− 1.48217i	− 0.0627455i
$559$	−33.8593	−1.43209
$560$	0	0
$561$	2.64907	0.111844
$562$	− 3.36497i	− 0.141943i
$563$	− 15.5164i	− 0.653939i	−0.945035	−	0.326970i	$-0.893973\pi$
$563$	− 15.5164i	− 0.653939i	0.945035	−	0.326970i	$-0.106027\pi$
$564$	1.26887	0.0534290
$565$	0	0
$566$	−35.1489	−1.47742
$567$	1.45313i	0.0610257i
$568$	13.5170i	0.567160i
$569$	37.7170	1.58118	0.790589	−	0.612347i	$-0.209774\pi$
$569$	37.7170	1.58118	0.790589	+	0.612347i	$0.209774\pi$
$570$	0	0
$571$	−12.4679	−0.521767	−0.260883	−	0.965370i	$-0.584014\pi$
$571$	−12.4679	−0.521767	−0.260883	+	0.965370i	$0.584014\pi$
$572$	5.95625i	0.249043i
$573$	− 24.5262i	− 1.02460i
$574$	0.560008	0.0233743
$575$	0	0
$576$	−12.6863	−0.528594
$577$	7.71735i	0.321278i	0.987013	+	0.160639i	$0.0513554\pi$
$577$	7.71735i	0.321278i	−0.987013	+	0.160639i	$0.948645\pi$
$578$	12.6146i	0.524696i
$579$	16.1410	0.670796
$580$	0	0
$581$	6.07331	0.251963
$582$	− 20.9989i	− 0.870434i
$583$	− 3.97763i	− 0.164736i
$584$	3.26298	0.135023
$585$	0	0
$586$	−18.4401	−0.761755
$587$	− 30.6554i	− 1.26528i	−0.774445	−	0.632641i	$-0.781971\pi$
$587$	− 30.6554i	− 1.26528i	0.774445	−	0.632641i	$-0.218029\pi$
$588$	1.02074i	0.0420946i
$589$	5.49370	0.226364
$590$	0	0
$591$	13.3930	0.550913
$592$	− 9.89066i	− 0.406504i
$593$	− 42.4286i	− 1.74233i	−0.490986	−	0.871167i	$-0.663364\pi$
$593$	− 42.4286i	− 1.74233i	0.490986	−	0.871167i	$-0.336636\pi$
$594$	−5.75548	−0.236150
$595$	0	0
$596$	4.08831	0.167464
$597$	25.5253i	1.04468i
$598$	− 67.8746i	− 2.77560i
$599$	−8.49535	−0.347111	−0.173555	−	0.984824i	$-0.555526\pi$
$599$	−8.49535	−0.347111	−0.173555	+	0.984824i	$0.555526\pi$
$600$	0	0
$601$	−23.1651	−0.944925	−0.472463	−	0.881351i	$-0.656635\pi$
$601$	−23.1651	−0.944925	−0.472463	+	0.881351i	$0.656635\pi$
$602$	− 5.26501i	− 0.214586i
$603$	8.67620i	0.353322i
$604$	3.62671	0.147569
$605$	0	0
$606$	10.2313	0.415619
$607$	2.87209i	0.116575i	0.998300	+	0.0582873i	$0.0185640\pi$
$607$	2.87209i	0.116575i	−0.998300	+	0.0582873i	$0.981436\pi$
$608$	− 28.9535i	− 1.17422i
$609$	5.81046	0.235452
$610$	0	0
$611$	8.49023	0.343478
$612$	− 3.21717i	− 0.130046i
$613$	33.8112i	1.36562i	0.730595	+	0.682811i	$0.239243\pi$
$613$	33.8112i	1.36562i	−0.730595	+	0.682811i	$0.760757\pi$
$614$	12.4531	0.502564
$615$	0	0
$616$	−3.05025	−0.122898
$617$	− 26.0998i	− 1.05074i	−0.850875	−	0.525369i	$-0.823927\pi$
$617$	− 26.0998i	− 1.05074i	0.850875	−	0.525369i	$-0.176073\pi$
$618$	− 21.3148i	− 0.857407i
$619$	27.1306	1.09047	0.545235	−	0.838283i	$-0.316440\pi$
$619$	27.1306	1.09047	0.545235	+	0.838283i	$0.316440\pi$
$620$	0	0
$621$	−50.7096	−2.03491
$622$	11.1380i	0.446594i
$623$	− 3.55617i	− 0.142475i
$624$	11.9541	0.478547
$625$	0	0
$626$	−23.4011	−0.935297
$627$	− 7.51023i	− 0.299930i
$628$	8.28616i	0.330654i
$629$	14.9700	0.596892
$630$	0	0
$631$	−24.0346	−0.956802	−0.478401	−	0.878141i	$-0.658784\pi$
$631$	−24.0346	−0.956802	−0.478401	+	0.878141i	$0.658784\pi$
$632$	2.94906i	0.117307i
$633$	− 27.1520i	− 1.07920i
$634$	−15.8236	−0.628436
$635$	0	0
$636$	4.06013	0.160994
$637$	6.82997i	0.270613i
$638$	5.27217i	0.208727i
$639$	7.22320	0.285746
$640$	0	0
$641$	9.48821	0.374762	0.187381	−	0.982287i	$-0.440000\pi$
$641$	9.48821	0.374762	0.187381	+	0.982287i	$0.440000\pi$
$642$	− 21.6961i	− 0.856278i
$643$	34.1937i	1.34847i	0.738517	+	0.674235i	$0.235526\pi$
$643$	34.1937i	1.34847i	−0.738517	+	0.674235i	$0.764474\pi$
$644$	−8.16024	−0.321559
$645$	0	0
$646$	−15.4229	−0.606804
$647$	− 25.8239i	− 1.01524i	−0.861581	−	0.507620i	$-0.830525\pi$
$647$	− 25.8239i	− 1.01524i	0.861581	−	0.507620i	$-0.169475\pi$
$648$	− 4.43241i	− 0.174122i
$649$	−11.6526	−0.457403
$650$	0	0
$651$	1.00215	0.0392775
$652$	20.3362i	0.796429i
$653$	21.1459i	0.827502i	0.910390	+	0.413751i	$0.135782\pi$
$653$	21.1459i	0.827502i	−0.910390	+	0.413751i	$0.864218\pi$
$654$	−12.1700	−0.475884
$655$	0	0
$656$	−0.788483	−0.0307851
$657$	− 1.74367i	− 0.0680272i
$658$	1.32020i	0.0514669i
$659$	12.9835	0.505764	0.252882	−	0.967497i	$-0.418621\pi$
$659$	12.9835	0.505764	0.252882	+	0.967497i	$0.418621\pi$
$660$	0	0
$661$	32.6438	1.26970	0.634849	−	0.772636i	$-0.281062\pi$
$661$	32.6438	1.26970	0.634849	+	0.772636i	$0.281062\pi$
$662$	14.4199i	0.560444i
$663$	18.0931i	0.702677i
$664$	−18.5251	−0.718915
$665$	0	0
$666$	−11.4502	−0.443687
$667$	46.4513i	1.79860i
$668$	− 10.9931i	− 0.425337i
$669$	−2.76262	−0.106809
$670$	0	0
$671$	−3.40640	−0.131503
$672$	− 5.28165i	− 0.203744i
$673$	19.5798i	0.754745i	0.926062	+	0.377372i	$0.123172\pi$
$673$	19.5798i	0.754745i	−0.926062	+	0.377372i	$0.876828\pi$
$674$	−3.91632	−0.150851
$675$	0	0
$676$	−29.3440	−1.12862
$677$	40.2314i	1.54622i	0.634273	+	0.773109i	$0.281299\pi$
$677$	40.2314i	1.54622i	−0.634273	+	0.773109i	$0.718701\pi$
$678$	20.6636i	0.793581i
$679$	−16.8926	−0.648278
$680$	0	0
$681$	33.2874	1.27557
$682$	0.909312i	0.0348194i
$683$	37.6157i	1.43933i	0.694323	+	0.719663i	$0.255704\pi$
$683$	37.6157i	1.43933i	−0.694323	+	0.719663i	$0.744296\pi$
$684$	−9.12080	−0.348742
$685$	0	0
$686$	−1.06204	−0.0405488
$687$	− 22.1547i	− 0.845256i
$688$	7.41304i	0.282620i
$689$	27.1671	1.03498
$690$	0	0
$691$	14.7643	0.561659	0.280829	−	0.959758i	$-0.409391\pi$
$691$	14.7643	0.561659	0.280829	+	0.959758i	$0.409391\pi$
$692$	− 8.08137i	− 0.307208i
$693$	1.63000i	0.0619184i
$694$	20.1834	0.766152
$695$	0	0
$696$	−17.7234	−0.671802
$697$	− 1.19340i	− 0.0452034i
$698$	6.02482i	0.228043i
$699$	−20.8491	−0.788585
$700$	0	0
$701$	16.1271	0.609111	0.304555	−	0.952495i	$-0.401492\pi$
$701$	16.1271	0.609111	0.304555	+	0.952495i	$0.401492\pi$
$702$	− 39.3097i	− 1.48365i
$703$	− 42.4405i	− 1.60067i
$704$	7.78301	0.293333
$705$	0	0
$706$	3.02629	0.113896
$707$	− 8.23058i	− 0.309543i
$708$	− 11.8943i	− 0.447013i
$709$	36.9892	1.38916	0.694579	−	0.719416i	$-0.255591\pi$
$709$	36.9892	1.38916	0.694579	+	0.719416i	$0.255591\pi$
$710$	0	0
$711$	1.57592	0.0591017
$712$	10.8472i	0.406517i
$713$	8.01165i	0.300039i
$714$	−2.81342	−0.105289
$715$	0	0
$716$	13.5518	0.506455
$717$	− 15.5831i	− 0.581961i
$718$	− 22.2541i	− 0.830515i
$719$	−6.71788	−0.250535	−0.125267	−	0.992123i	$-0.539979\pi$
$719$	−6.71788	−0.250535	−0.125267	+	0.992123i	$0.539979\pi$
$720$	0	0
$721$	−17.1467	−0.638576
$722$	23.5457i	0.876280i
$723$	− 9.87173i	− 0.367134i
$724$	−2.91003	−0.108151
$725$	0	0
$726$	1.24309	0.0461352
$727$	9.41666i	0.349245i	0.984636	+	0.174622i	$0.0558704\pi$
$727$	9.41666i	0.349245i	−0.984636	+	0.174622i	$0.944130\pi$
$728$	− 20.8331i	− 0.772127i
$729$	−21.3979	−0.792516
$730$	0	0
$731$	−11.2200	−0.414986
$732$	− 3.47705i	− 0.128515i
$733$	7.09353i	0.262006i	0.991382	+	0.131003i	$0.0418197\pi$
$733$	7.09353i	0.262006i	−0.991382	+	0.131003i	$0.958180\pi$
$734$	−3.92754	−0.144968
$735$	0	0
$736$	42.2238	1.55639
$737$	− 5.32284i	− 0.196069i
$738$	0.912811i	0.0336010i
$739$	−28.4731	−1.04740	−0.523701	−	0.851902i	$-0.675449\pi$
$739$	−28.4731	−1.04740	−0.523701	+	0.851902i	$0.675449\pi$
$740$	0	0
$741$	51.2946	1.88435
$742$	4.22439i	0.155082i
$743$	− 1.20393i	− 0.0441681i	−0.999756	−	0.0220840i	$-0.992970\pi$
$743$	− 1.20393i	− 0.0441681i	0.999756	−	0.0220840i	$-0.00703014\pi$
$744$	−3.05682	−0.112069
$745$	0	0
$746$	−32.6620	−1.19584
$747$	9.89947i	0.362203i
$748$	1.97373i	0.0721667i
$749$	−17.4535	−0.637735
$750$	0	0
$751$	13.1877	0.481227	0.240614	−	0.970621i	$-0.422651\pi$
$751$	13.1877	0.481227	0.240614	+	0.970621i	$0.422651\pi$
$752$	− 1.85882i	− 0.0677844i
$753$	− 16.3806i	− 0.596943i
$754$	−36.0087	−1.31136
$755$	0	0
$756$	−4.72602	−0.171884
$757$	− 28.5746i	− 1.03856i	−0.854603	−	0.519282i	$-0.826200\pi$
$757$	− 28.5746i	− 1.03856i	0.854603	−	0.519282i	$-0.173800\pi$
$758$	− 19.7557i	− 0.717560i
$759$	10.9524	0.397547
$760$	0	0
$761$	22.9475	0.831847	0.415924	−	0.909400i	$-0.363458\pi$
$761$	22.9475	0.831847	0.415924	+	0.909400i	$0.363458\pi$
$762$	− 20.0788i	− 0.727380i
$763$	9.79014i	0.354427i
$764$	18.2736	0.661114
$765$	0	0
$766$	16.0548	0.580082
$767$	− 79.5867i	− 2.87371i
$768$	19.1630i	0.691487i
$769$	5.02937	0.181364	0.0906818	−	0.995880i	$-0.471095\pi$
$769$	5.02937	0.181364	0.0906818	+	0.995880i	$0.471095\pi$
$770$	0	0
$771$	8.38376	0.301934
$772$	12.0260i	0.432827i
$773$	3.44069i	0.123753i	0.998084	+	0.0618765i	$0.0197085\pi$
$773$	3.44069i	0.123753i	−0.998084	+	0.0618765i	$0.980292\pi$
$774$	8.58194	0.308471
$775$	0	0
$776$	51.5267	1.84970
$777$	− 7.74193i	− 0.277740i
$778$	14.7878i	0.530167i
$779$	−3.38335	−0.121221
$780$	0	0
$781$	−4.43143	−0.158569
$782$	− 22.4917i	− 0.804301i
$783$	26.9024i	0.961413i
$784$	1.49533	0.0534047
$785$	0	0
$786$	12.3232	0.439553
$787$	− 16.1901i	− 0.577116i	−0.957462	−	0.288558i	$-0.906824\pi$
$787$	− 16.1901i	− 0.577116i	0.957462	−	0.288558i	$-0.0931759\pi$
$788$	9.97861i	0.355473i
$789$	−8.21263	−0.292377
$790$	0	0
$791$	16.6228	0.591040
$792$	− 4.97190i	− 0.176669i
$793$	− 23.2656i	− 0.826185i
$794$	1.08756	0.0385960
$795$	0	0
$796$	−19.0180	−0.674075
$797$	20.1288i	0.712999i	0.934296	+	0.356499i	$0.116030\pi$
$797$	20.1288i	0.712999i	−0.934296	+	0.356499i	$0.883970\pi$
$798$	7.97614i	0.282352i
$799$	2.81342	0.0995315
$800$	0	0
$801$	5.79654	0.204811
$802$	3.19555i	0.112839i
$803$	1.06974i	0.0377504i
$804$	5.43324	0.191615
$805$	0	0
$806$	−6.21057	−0.218758
$807$	− 4.83907i	− 0.170343i
$808$	25.1054i	0.883203i
$809$	37.2899	1.31104	0.655522	−	0.755176i	$-0.272449\pi$
$809$	37.2899	1.31104	0.655522	+	0.755176i	$0.272449\pi$
$810$	0	0
$811$	−14.5835	−0.512095	−0.256047	−	0.966664i	$-0.582420\pi$
$811$	−14.5835	−0.512095	−0.256047	+	0.966664i	$0.582420\pi$
$812$	4.32916i	0.151924i
$813$	15.0089i	0.526385i
$814$	7.02470	0.246216
$815$	0	0
$816$	3.96124	0.138671
$817$	31.8091i	1.11286i
$818$	33.1235i	1.15814i
$819$	−11.1328	−0.389012
$820$	0	0
$821$	−30.9951	−1.08174	−0.540869	−	0.841107i	$-0.681905\pi$
$821$	−30.9951	−1.08174	−0.540869	+	0.841107i	$0.681905\pi$
$822$	5.05192i	0.176206i
$823$	− 55.0098i	− 1.91752i	−0.284215	−	0.958761i	$-0.591733\pi$
$823$	− 55.0098i	− 1.91752i	0.284215	−	0.958761i	$-0.408267\pi$
$824$	52.3017	1.82202
$825$	0	0
$826$	12.3755	0.430598
$827$	− 44.3005i	− 1.54048i	−0.637755	−	0.770239i	$-0.720137\pi$
$827$	− 44.3005i	− 1.54048i	0.637755	−	0.770239i	$-0.279863\pi$
$828$	− 13.3012i	− 0.462247i
$829$	9.84740	0.342014	0.171007	−	0.985270i	$-0.445298\pi$
$829$	9.84740	0.342014	0.171007	+	0.985270i	$0.445298\pi$
$830$	0	0
$831$	−12.7447	−0.442109
$832$	53.1577i	1.84291i
$833$	2.26325i	0.0784171i
$834$	−17.4662	−0.604804
$835$	0	0
$836$	5.59560	0.193528
$837$	4.63997i	0.160381i
$838$	33.4335i	1.15494i
$839$	−11.3177	−0.390730	−0.195365	−	0.980731i	$-0.562589\pi$
$839$	−11.3177	−0.390730	−0.195365	+	0.980731i	$0.562589\pi$
$840$	0	0
$841$	−4.35673	−0.150232
$842$	17.6877i	0.609558i
$843$	3.70853i	0.127729i
$844$	20.2300	0.696344
$845$	0	0
$846$	−2.15193	−0.0739847
$847$	− 1.00000i	− 0.0343604i
$848$	− 5.94787i	− 0.204251i
$849$	38.7376	1.32947
$850$	0	0
$851$	61.8923	2.12164
$852$	− 4.52334i	− 0.154967i
$853$	− 26.3172i	− 0.901083i	−0.892755	−	0.450541i	$-0.851231\pi$
$853$	− 26.3172i	− 0.901083i	0.892755	−	0.450541i	$-0.148769\pi$
$854$	3.61772	0.123796
$855$	0	0
$856$	53.2375	1.81962
$857$	39.8259i	1.36043i	0.733014	+	0.680213i	$0.238113\pi$
$857$	39.8259i	1.36043i	−0.733014	+	0.680213i	$0.761887\pi$
$858$	8.49023i	0.289852i
$859$	−16.5702	−0.565367	−0.282684	−	0.959213i	$-0.591225\pi$
$859$	−16.5702	−0.565367	−0.282684	+	0.959213i	$0.591225\pi$
$860$	0	0
$861$	−0.617186	−0.0210336
$862$	− 32.9450i	− 1.12211i
$863$	23.2437i	0.791225i	0.918418	+	0.395613i	$0.129468\pi$
$863$	23.2437i	0.791225i	−0.918418	+	0.395613i	$0.870532\pi$
$864$	24.4540	0.831942
$865$	0	0
$866$	15.5585	0.528698
$867$	− 13.9025i	− 0.472154i
$868$	0.746668i	0.0253436i
$869$	−0.966826	−0.0327973
$870$	0	0
$871$	36.3548	1.23184
$872$	− 29.8624i	− 1.01127i
$873$	− 27.5349i	− 0.931914i
$874$	−63.7648	−2.15688
$875$	0	0
$876$	−1.09193	−0.0368929
$877$	− 56.2380i	− 1.89902i	−0.313735	−	0.949511i	$-0.601580\pi$
$877$	− 56.2380i	− 1.89902i	0.313735	−	0.949511i	$-0.398420\pi$
$878$	− 17.8636i	− 0.602866i
$879$	20.3229	0.685474
$880$	0	0
$881$	−50.4473	−1.69961	−0.849807	−	0.527094i	$-0.823282\pi$
$881$	−50.4473	−1.69961	−0.849807	+	0.527094i	$0.823282\pi$
$882$	− 1.73112i	− 0.0582897i
$883$	10.2912i	0.346328i	0.984893	+	0.173164i	$0.0553990\pi$
$883$	10.2912i	0.346328i	−0.984893	+	0.173164i	$0.944601\pi$
$884$	−13.4805	−0.453398
$885$	0	0
$886$	−19.4872	−0.654687
$887$	40.8093i	1.37024i	0.728430	+	0.685120i	$0.240250\pi$
$887$	40.8093i	1.37024i	−0.728430	+	0.685120i	$0.759750\pi$
$888$	23.6148i	0.792462i
$889$	−16.1524	−0.541735
$890$	0	0
$891$	1.45313	0.0486817
$892$	− 2.05833i	− 0.0689179i
$893$	− 7.97614i	− 0.266911i
$894$	5.82761	0.194904
$895$	0	0
$896$	0.758967	0.0253553
$897$	74.8046i	2.49765i
$898$	− 11.8896i	− 0.396761i
$899$	4.25033	0.141756
$900$	0	0
$901$	9.00237	0.299913
$902$	− 0.560008i	− 0.0186462i
$903$	5.80257i	0.193097i
$904$	−50.7038	−1.68638
$905$	0	0
$906$	5.16963	0.171749
$907$	21.3652i	0.709420i	0.934976	+	0.354710i	$0.115420\pi$
$907$	21.3652i	0.709420i	−0.934976	+	0.354710i	$0.884580\pi$
$908$	24.8012i	0.823057i
$909$	13.4158	0.444974
$910$	0	0
$911$	17.4802	0.579144	0.289572	−	0.957156i	$-0.406487\pi$
$911$	17.4802	0.579144	0.289572	+	0.957156i	$0.406487\pi$
$912$	− 11.2303i	− 0.371872i
$913$	− 6.07331i	− 0.200997i
$914$	−12.8046	−0.423537
$915$	0	0
$916$	16.5067	0.545396
$917$	− 9.91338i	− 0.327369i
$918$	− 13.0261i	− 0.429925i
$919$	22.1767	0.731541	0.365771	−	0.930705i	$-0.380805\pi$
$919$	22.1767	0.731541	0.365771	+	0.930705i	$0.380805\pi$
$920$	0	0
$921$	−13.7245	−0.452238
$922$	− 43.7374i	− 1.44041i
$923$	− 30.2665i	− 0.996234i
$924$	1.02074	0.0335799
$925$	0	0
$926$	35.2203	1.15741
$927$	− 27.9490i	− 0.917966i
$928$	− 22.4005i	− 0.735332i
$929$	8.89242	0.291751	0.145875	−	0.989303i	$-0.453400\pi$
$929$	8.89242	0.291751	0.145875	+	0.989303i	$0.453400\pi$
$930$	0	0
$931$	6.41641	0.210289
$932$	− 15.5339i	− 0.508830i
$933$	− 12.2752i	− 0.401872i
$934$	22.1635	0.725212
$935$	0	0
$936$	33.9579	1.10995
$937$	57.8417i	1.88960i	0.327641	+	0.944802i	$0.393746\pi$
$937$	57.8417i	1.88960i	−0.327641	+	0.944802i	$0.606254\pi$
$938$	5.65305i	0.184579i
$939$	25.7904	0.841637
$940$	0	0
$941$	35.3237	1.15152	0.575760	−	0.817619i	$-0.304706\pi$
$941$	35.3237	1.15152	0.575760	+	0.817619i	$0.304706\pi$
$942$	11.8114i	0.384835i
$943$	− 4.93405i	− 0.160675i
$944$	−17.4245	−0.567118
$945$	0	0
$946$	−5.26501	−0.171180
$947$	28.9507i	0.940773i	0.882461	+	0.470386i	$0.155885\pi$
$947$	28.9507i	0.940773i	−0.882461	+	0.470386i	$0.844115\pi$
$948$	− 0.986879i	− 0.0320523i
$949$	−7.30630	−0.237173
$950$	0	0
$951$	17.4392	0.565505
$952$	− 6.90349i	− 0.223743i
$953$	6.75259i	0.218738i	0.994001	+	0.109369i	$0.0348830\pi$
$953$	6.75259i	0.218738i	−0.994001	+	0.109369i	$0.965117\pi$
$954$	−6.88573	−0.222934
$955$	0	0
$956$	11.6104	0.375507
$957$	− 5.81046i	− 0.187825i
$958$	− 5.01031i	− 0.161876i
$959$	4.06402	0.131234
$960$	0	0
$961$	−30.2669	−0.976353
$962$	47.9785i	1.54689i
$963$	− 28.4490i	− 0.916758i
$964$	7.35506	0.236891
$965$	0	0
$966$	−11.6319	−0.374249
$967$	2.17445i	0.0699257i	0.999389	+	0.0349628i	$0.0111313\pi$
$967$	2.17445i	0.0699257i	−0.999389	+	0.0349628i	$0.988869\pi$
$968$	3.05025i	0.0980388i
$969$	16.9975	0.546040
$970$	0	0
$971$	55.7879	1.79032	0.895159	−	0.445746i	$-0.147062\pi$
$971$	55.7879	1.79032	0.895159	+	0.445746i	$0.147062\pi$
$972$	− 12.6948i	− 0.407186i
$973$	14.0507i	0.450444i
$974$	6.16122	0.197418
$975$	0	0
$976$	−5.09370	−0.163045
$977$	− 45.5825i	− 1.45831i	−0.684347	−	0.729156i	$-0.739913\pi$
$977$	− 45.5825i	− 1.45831i	0.684347	−	0.729156i	$-0.260087\pi$
$978$	28.9879i	0.926932i
$979$	−3.55617	−0.113656
$980$	0	0
$981$	−15.9579	−0.509496
$982$	− 32.2927i	− 1.03050i
$983$	− 1.30557i	− 0.0416413i	−0.999783	−	0.0208206i	$-0.993372\pi$
$983$	− 1.30557i	− 0.0416413i	0.999783	−	0.0208206i	$-0.00662790\pi$
$984$	1.88257	0.0600142
$985$	0	0
$986$	−11.9322	−0.380000
$987$	− 1.45500i	− 0.0463131i
$988$	38.2177i	1.21587i
$989$	−46.3882	−1.47506
$990$	0	0
$991$	34.4682	1.09492	0.547459	−	0.836833i	$-0.315595\pi$
$991$	34.4682	1.09492	0.547459	+	0.836833i	$0.315595\pi$
$992$	− 3.86351i	− 0.122666i
$993$	− 15.8921i	− 0.504322i
$994$	4.70634	0.149276
$995$	0	0
$996$	6.19928	0.196432
$997$	− 40.4670i	− 1.28160i	−0.767707	−	0.640801i	$-0.778602\pi$
$997$	− 40.4670i	− 1.28160i	0.767707	−	0.640801i	$-0.221398\pi$
$998$	− 43.2449i	− 1.36889i
$999$	35.8451	1.13409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.b.q.1849.10		14
5.2	odd	4		1925.2.a.bc.1.3	yes	7
5.3	odd	4		1925.2.a.ba.1.5	✓	7
5.4	even	2	inner	1925.2.b.q.1849.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.ba.1.5	✓	7	5.3	odd	4
1925.2.a.bc.1.3	yes	7	5.2	odd	4
1925.2.b.q.1849.5		14	5.4	even	2	inner
1925.2.b.q.1849.10		14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.06204i q^{2} -1.17047i q^{3} +0.872076 q^{4} +1.24309 q^{6} -1.00000i q^{7} +3.05025i q^{8} +1.63000 q^{9} -1.00000 q^{11} -1.02074i q^{12} -6.82997i q^{13} +1.06204 q^{14} -1.49533 q^{16} -2.26325i q^{17} +1.73112i q^{18} -6.41641 q^{19} -1.17047 q^{21} -1.06204i q^{22} -9.35726i q^{23} +3.57024 q^{24} +7.25368 q^{26} -5.41928i q^{27} -0.872076i q^{28} -4.96420 q^{29} -0.856196 q^{31} +4.51241i q^{32} +1.17047i q^{33} +2.40366 q^{34} +1.42148 q^{36} +6.61436i q^{37} -6.81447i q^{38} -7.99428 q^{39} +0.527296 q^{41} -1.24309i q^{42} -4.95746i q^{43} -0.872076 q^{44} +9.93777 q^{46} +1.24309i q^{47} +1.75024i q^{48} -1.00000 q^{49} -2.64907 q^{51} -5.95625i q^{52} +3.97763i q^{53} +5.75548 q^{54} +3.05025 q^{56} +7.51023i q^{57} -5.27217i q^{58} +11.6526 q^{59} +3.40640 q^{61} -0.909312i q^{62} -1.63000i q^{63} -7.78301 q^{64} -1.24309 q^{66} +5.32284i q^{67} -1.97373i q^{68} -10.9524 q^{69} +4.43143 q^{71} +4.97190i q^{72} -1.06974i q^{73} -7.02470 q^{74} -5.59560 q^{76} +1.00000i q^{77} -8.49023i q^{78} +0.966826 q^{79} -1.45313 q^{81} +0.560008i q^{82} +6.07331i q^{83} -1.02074 q^{84} +5.26501 q^{86} +5.81046i q^{87} -3.05025i q^{88} +3.55617 q^{89} -6.82997 q^{91} -8.16024i q^{92} +1.00215i q^{93} -1.32020 q^{94} +5.28165 q^{96} -16.8926i q^{97} -1.06204i q^{98} -1.63000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 26 q^{4} - 2 q^{6} - 26 q^{9} - 14 q^{11} - 2 q^{14} + 58 q^{16} - 36 q^{19} + 44 q^{24} + 26 q^{26} + 4 q^{29} + 48 q^{31} - 66 q^{34} + 88 q^{36} - 20 q^{39} - 20 q^{41} + 26 q^{44} + 6 q^{46}+ \cdots + 26 q^{99}+O(q^{100}) \) 14 * q - 26 * q^4 - 2 * q^6 - 26 * q^9 - 14 * q^11 - 2 * q^14 + 58 * q^16 - 36 * q^19 + 44 * q^24 + 26 * q^26 + 4 * q^29 + 48 * q^31 - 66 * q^34 + 88 * q^36 - 20 * q^39 - 20 * q^41 + 26 * q^44 + 6 * q^46 - 14 * q^49 + 58 * q^51 + 94 * q^54 + 12 * q^56 - 2 * q^59 + 56 * q^61 - 96 * q^64 + 2 * q^66 - 66 * q^69 - 20 * q^71 + 12 * q^74 + 40 * q^76 - 38 * q^79 + 54 * q^81 + 42 * q^84 + 110 * q^86 + 4 * q^89 - 6 * q^91 - 42 * q^94 - 46 * q^96 + 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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