LMFDB - Embedded newform 4598.2.a.bs.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.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:	$36.7152148494$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.258228.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 12x^{2} + 6x + 24 $ x^4 - 2x^3 - 12x^2 + 6*x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.63927$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -0.639273 q^{3} +1.00000 q^{4} -1.63927 q^{5} +0.639273 q^{6} -1.54956 q^{7} -1.00000 q^{8} -2.59133 q^{9} +1.63927 q^{10} -0.639273 q^{12} -6.04177 q^{13} +1.54956 q^{14} +1.04794 q^{15} +1.00000 q^{16} +2.54956 q^{17} +2.59133 q^{18} -1.00000 q^{19} -1.63927 q^{20} +0.990593 q^{21} -7.04177 q^{23} +0.639273 q^{24} -2.31278 q^{25} +6.04177 q^{26} +3.57439 q^{27} -1.54956 q^{28} -1.95206 q^{29} -1.04794 q^{30} -1.27855 q^{31} -1.00000 q^{32} -2.54956 q^{34} +2.54015 q^{35} -2.59133 q^{36} +5.73839 q^{37} +1.00000 q^{38} +3.86234 q^{39} +1.63927 q^{40} -9.13148 q^{41} -0.990593 q^{42} -12.8050 q^{43} +4.24790 q^{45} +7.04177 q^{46} +3.59133 q^{47} -0.639273 q^{48} -4.59886 q^{49} +2.31278 q^{50} -1.62987 q^{51} -6.04177 q^{52} -7.95206 q^{53} -3.57439 q^{54} +1.54956 q^{56} +0.639273 q^{57} +1.95206 q^{58} +5.40250 q^{59} +1.04794 q^{60} -13.5434 q^{61} +1.27855 q^{62} +4.01542 q^{63} +1.00000 q^{64} +9.90411 q^{65} +10.2212 q^{67} +2.54956 q^{68} +4.50162 q^{69} -2.54015 q^{70} -12.4100 q^{71} +2.59133 q^{72} +1.45044 q^{73} -5.73839 q^{74} +1.47850 q^{75} -1.00000 q^{76} -3.86234 q^{78} +1.13148 q^{79} -1.63927 q^{80} +5.48898 q^{81} +9.13148 q^{82} +4.36073 q^{83} +0.990593 q^{84} -4.17943 q^{85} +12.8050 q^{86} +1.24790 q^{87} -5.85294 q^{89} -4.24790 q^{90} +9.36209 q^{91} -7.04177 q^{92} +0.817341 q^{93} -3.59133 q^{94} +1.63927 q^{95} +0.639273 q^{96} -10.0000 q^{97} +4.59886 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 4 q^{8} + 16 q^{9} + 2 q^{10} + 2 q^{12} - q^{13} + 3 q^{14} + 26 q^{15} + 4 q^{16} + 7 q^{17} - 16 q^{18} - 4 q^{19} - 2 q^{20} + 9 q^{21}+ \cdots - 23 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 3 * q^7 - 4 * q^8 + 16 * q^9 + 2 * q^10 + 2 * q^12 - q^13 + 3 * q^14 + 26 * q^15 + 4 * q^16 + 7 * q^17 - 16 * q^18 - 4 * q^19 - 2 * q^20 + 9 * q^21 - 5 * q^23 - 2 * q^24 + 8 * q^25 + q^26 + 8 * q^27 - 3 * q^28 + 14 * q^29 - 26 * q^30 + 4 * q^31 - 4 * q^32 - 7 * q^34 + 12 * q^35 + 16 * q^36 + 12 * q^37 + 4 * q^38 - 5 * q^39 + 2 * q^40 - 12 * q^41 - 9 * q^42 - 14 * q^43 - 2 * q^45 + 5 * q^46 - 12 * q^47 + 2 * q^48 + 23 * q^49 - 8 * q^50 - 7 * q^51 - q^52 - 10 * q^53 - 8 * q^54 + 3 * q^56 - 2 * q^57 - 14 * q^58 + 3 * q^59 + 26 * q^60 - 6 * q^61 - 4 * q^62 + 18 * q^63 + 4 * q^64 - 4 * q^65 + 15 * q^67 + 7 * q^68 - 7 * q^69 - 12 * q^70 - 16 * q^71 - 16 * q^72 + 9 * q^73 - 12 * q^74 - 44 * q^75 - 4 * q^76 + 5 * q^78 - 20 * q^79 - 2 * q^80 + 52 * q^81 + 12 * q^82 + 22 * q^83 + 9 * q^84 - 14 * q^85 + 14 * q^86 - 14 * q^87 - 8 * q^89 + 2 * q^90 - 18 * q^91 - 5 * q^92 + 56 * q^93 + 12 * q^94 + 2 * q^95 - 2 * q^96 - 40 * q^97 - 23 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.639273	−0.369085	−0.184542	−	0.982825i	$-0.559080\pi$
$3$	−0.639273	−0.369085	−0.184542	+	0.982825i	$0.559080\pi$
$4$	1.00000	0.500000
$5$	−1.63927	−0.733105	−0.366553	−	0.930397i	$-0.619462\pi$
$5$	−1.63927	−0.733105	−0.366553	+	0.930397i	$0.619462\pi$
$6$	0.639273	0.260982
$7$	−1.54956	−0.585679	−0.292839	−	0.956162i	$-0.594600\pi$
$7$	−1.54956	−0.585679	−0.292839	+	0.956162i	$0.594600\pi$
$8$	−1.00000	−0.353553
$9$	−2.59133	−0.863776
$10$	1.63927	0.518384
$11$	0	0
$11$	0	0
$12$	−0.639273	−0.184542
$13$	−6.04177	−1.67569	−0.837843	−	0.545912i	$-0.816183\pi$
$13$	−6.04177	−1.67569	−0.837843	+	0.545912i	$0.816183\pi$
$14$	1.54956	0.414137
$15$	1.04794	0.270578
$16$	1.00000	0.250000
$17$	2.54956	0.618359	0.309180	−	0.951004i	$-0.399946\pi$
$17$	2.54956	0.618359	0.309180	+	0.951004i	$0.399946\pi$
$18$	2.59133	0.610782
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.63927	−0.366553
$21$	0.990593	0.216165
$22$	0	0
$23$	−7.04177	−1.46831	−0.734155	−	0.678982i	$-0.762422\pi$
$23$	−7.04177	−1.46831	−0.734155	+	0.678982i	$0.762422\pi$
$24$	0.639273	0.130491
$25$	−2.31278	−0.462557
$26$	6.04177	1.18489
$27$	3.57439	0.687891
$28$	−1.54956	−0.292839
$29$	−1.95206	−0.362488	−0.181244	−	0.983438i	$-0.558012\pi$
$29$	−1.95206	−0.362488	−0.181244	+	0.983438i	$0.558012\pi$
$30$	−1.04794	−0.191328
$31$	−1.27855	−0.229634	−0.114817	−	0.993387i	$-0.536628\pi$
$31$	−1.27855	−0.229634	−0.114817	+	0.993387i	$0.536628\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.54956	−0.437246
$35$	2.54015	0.429364
$36$	−2.59133	−0.431888
$37$	5.73839	0.943386	0.471693	−	0.881763i	$-0.343643\pi$
$37$	5.73839	0.943386	0.471693	+	0.881763i	$0.343643\pi$
$38$	1.00000	0.162221
$39$	3.86234	0.618470
$40$	1.63927	0.259192
$41$	−9.13148	−1.42610	−0.713049	−	0.701114i	$-0.752686\pi$
$41$	−9.13148	−1.42610	−0.713049	+	0.701114i	$0.752686\pi$
$42$	−0.990593	−0.152852
$43$	−12.8050	−1.95274	−0.976371	−	0.216099i	$-0.930667\pi$
$43$	−12.8050	−1.95274	−0.976371	+	0.216099i	$0.930667\pi$
$44$	0	0
$45$	4.24790	0.633239
$46$	7.04177	1.03825
$47$	3.59133	0.523849	0.261925	−	0.965088i	$-0.415643\pi$
$47$	3.59133	0.523849	0.261925	+	0.965088i	$0.415643\pi$
$48$	−0.639273	−0.0922712
$49$	−4.59886	−0.656980
$50$	2.31278	0.327077
$51$	−1.62987	−0.228227
$52$	−6.04177	−0.837843
$53$	−7.95206	−1.09230	−0.546149	−	0.837688i	$-0.683907\pi$
$53$	−7.95206	−1.09230	−0.546149	+	0.837688i	$0.683907\pi$
$54$	−3.57439	−0.486413
$55$	0	0
$56$	1.54956	0.207069
$57$	0.639273	0.0846738
$58$	1.95206	0.256318
$59$	5.40250	0.703345	0.351673	−	0.936123i	$-0.385613\pi$
$59$	5.40250	0.703345	0.351673	+	0.936123i	$0.385613\pi$
$60$	1.04794	0.135289
$61$	−13.5434	−1.73405	−0.867026	−	0.498262i	$-0.833972\pi$
$61$	−13.5434	−1.73405	−0.867026	+	0.498262i	$0.833972\pi$
$62$	1.27855	0.162376
$63$	4.01542	0.505896
$64$	1.00000	0.125000
$65$	9.90411	1.22845
$66$	0	0
$67$	10.2212	1.24872	0.624359	−	0.781138i	$-0.285360\pi$
$67$	10.2212	1.24872	0.624359	+	0.781138i	$0.285360\pi$
$68$	2.54956	0.309180
$69$	4.50162	0.541931
$70$	−2.54015	−0.303606
$71$	−12.4100	−1.47280	−0.736400	−	0.676547i	$-0.763476\pi$
$71$	−12.4100	−1.47280	−0.736400	+	0.676547i	$0.763476\pi$
$72$	2.59133	0.305391
$73$	1.45044	0.169761	0.0848806	−	0.996391i	$-0.472949\pi$
$73$	1.45044	0.169761	0.0848806	+	0.996391i	$0.472949\pi$
$74$	−5.73839	−0.667075
$75$	1.47850	0.170723
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−3.86234	−0.437324
$79$	1.13148	0.127302	0.0636509	−	0.997972i	$-0.479726\pi$
$79$	1.13148	0.127302	0.0636509	+	0.997972i	$0.479726\pi$
$80$	−1.63927	−0.183276
$81$	5.48898	0.609886
$82$	9.13148	1.00840
$83$	4.36073	0.478652	0.239326	−	0.970939i	$-0.423074\pi$
$83$	4.36073	0.478652	0.239326	+	0.970939i	$0.423074\pi$
$84$	0.990593	0.108083
$85$	−4.17943	−0.453322
$86$	12.8050	1.38080
$87$	1.24790	0.133789
$88$	0	0
$89$	−5.85294	−0.620410	−0.310205	−	0.950670i	$-0.600398\pi$
$89$	−5.85294	−0.620410	−0.310205	+	0.950670i	$0.600398\pi$
$90$	−4.24790	−0.447768
$91$	9.36209	0.981413
$92$	−7.04177	−0.734155
$93$	0.817341	0.0847543
$94$	−3.59133	−0.370417
$95$	1.63927	0.168186
$96$	0.639273	0.0652456
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	4.59886	0.464555
$99$	0	0
$100$	−2.31278	−0.231278
$101$	8.26484	0.822382	0.411191	−	0.911549i	$-0.365113\pi$
$101$	8.26484	0.822382	0.411191	+	0.911549i	$0.365113\pi$
$102$	1.62987	0.161381
$103$	8.14706	0.802754	0.401377	−	0.915913i	$-0.368532\pi$
$103$	8.14706	0.802754	0.401377	+	0.915913i	$0.368532\pi$
$104$	6.04177	0.592444
$105$	−1.62385	−0.158472
$106$	7.95206	0.772372
$107$	−12.5509	−1.21334	−0.606672	−	0.794953i	$-0.707496\pi$
$107$	−12.5509	−1.21334	−0.606672	+	0.794953i	$0.707496\pi$
$108$	3.57439	0.343946
$109$	3.40250	0.325900	0.162950	−	0.986634i	$-0.447899\pi$
$109$	3.40250	0.325900	0.162950	+	0.986634i	$0.447899\pi$
$110$	0	0
$111$	−3.66840	−0.348189
$112$	−1.54956	−0.146420
$113$	−3.13148	−0.294585	−0.147293	−	0.989093i	$-0.547056\pi$
$113$	−3.13148	−0.294585	−0.147293	+	0.989093i	$0.547056\pi$
$114$	−0.639273	−0.0598734
$115$	11.5434	1.07643
$116$	−1.95206	−0.181244
$117$	15.6562	1.44742
$118$	−5.40250	−0.497340
$119$	−3.95070	−0.362160
$120$	−1.04794	−0.0956638
$121$	0	0
$122$	13.5434	1.22616
$123$	5.83751	0.526351
$124$	−1.27855	−0.114817
$125$	11.9877	1.07221
$126$	−4.01542	−0.357722
$127$	5.82057	0.516492	0.258246	−	0.966079i	$-0.416855\pi$
$127$	5.82057	0.516492	0.258246	+	0.966079i	$0.416855\pi$
$128$	−1.00000	−0.0883883
$129$	8.18589	0.720728
$130$	−9.90411	−0.868648
$131$	−14.9844	−1.30919	−0.654597	−	0.755978i	$-0.727162\pi$
$131$	−14.9844	−1.30919	−0.654597	+	0.755978i	$0.727162\pi$
$132$	0	0
$133$	1.54956	0.134364
$134$	−10.2212	−0.882977
$135$	−5.85940	−0.504297
$136$	−2.54956	−0.218623
$137$	19.3546	1.65357	0.826785	−	0.562517i	$-0.190167\pi$
$137$	19.3546	1.65357	0.826785	+	0.562517i	$0.190167\pi$
$138$	−4.50162	−0.383203
$139$	−15.7398	−1.33503	−0.667514	−	0.744597i	$-0.732642\pi$
$139$	−15.7398	−1.33503	−0.667514	+	0.744597i	$0.732642\pi$
$140$	2.54015	0.214682
$141$	−2.29584	−0.193345
$142$	12.4100	1.04143
$143$	0	0
$144$	−2.59133	−0.215944
$145$	3.19995	0.265742
$146$	−1.45044	−0.120039
$147$	2.93993	0.242481
$148$	5.73839	0.471693
$149$	−19.4456	−1.59305	−0.796524	−	0.604607i	$-0.793330\pi$
$149$	−19.4456	−1.59305	−0.796524	+	0.604607i	$0.793330\pi$
$150$	−1.47850	−0.120719
$151$	12.2306	0.995312	0.497656	−	0.867374i	$-0.334194\pi$
$151$	12.2306	0.995312	0.497656	+	0.867374i	$0.334194\pi$
$152$	1.00000	0.0811107
$153$	−6.60675	−0.534124
$154$	0	0
$155$	2.09589	0.168346
$156$	3.86234	0.309235
$157$	24.6425	1.96669	0.983343	−	0.181759i	$-0.0581792\pi$
$157$	24.6425	1.96669	0.983343	+	0.181759i	$0.0581792\pi$
$158$	−1.13148	−0.0900159
$159$	5.08354	0.403151
$160$	1.63927	0.129596
$161$	10.9116	0.859958
$162$	−5.48898	−0.431255
$163$	−19.0032	−1.48845	−0.744224	−	0.667930i	$-0.767181\pi$
$163$	−19.0032	−1.48845	−0.744224	+	0.667930i	$0.767181\pi$
$164$	−9.13148	−0.713049
$165$	0	0
$166$	−4.36073	−0.338458
$167$	1.82057	0.140880	0.0704401	−	0.997516i	$-0.477560\pi$
$167$	1.82057	0.140880	0.0704401	+	0.997516i	$0.477560\pi$
$168$	−0.990593	−0.0764259
$169$	23.5030	1.80792
$170$	4.17943	0.320547
$171$	2.59133	0.198164
$172$	−12.8050	−0.976371
$173$	−10.3109	−0.783924	−0.391962	−	0.919981i	$-0.628204\pi$
$173$	−10.3109	−0.783924	−0.391962	+	0.919981i	$0.628204\pi$
$174$	−1.24790	−0.0946029
$175$	3.58380	0.270909
$176$	0	0
$177$	−3.45367	−0.259594
$178$	5.85294	0.438696
$179$	5.78770	0.432593	0.216296	−	0.976328i	$-0.430602\pi$
$179$	5.78770	0.432593	0.216296	+	0.976328i	$0.430602\pi$
$180$	4.24790	0.316620
$181$	15.4768	1.15038	0.575190	−	0.818020i	$-0.304928\pi$
$181$	15.4768	1.15038	0.575190	+	0.818020i	$0.304928\pi$
$182$	−9.36209	−0.693964
$183$	8.65793	0.640012
$184$	7.04177	0.519126
$185$	−9.40680	−0.691601
$186$	−0.817341	−0.0599304
$187$	0	0
$188$	3.59133	0.261925
$189$	−5.53873	−0.402883
$190$	−1.63927	−0.118925
$191$	15.4537	1.11819	0.559094	−	0.829104i	$-0.311149\pi$
$191$	15.4537	1.11819	0.559094	+	0.829104i	$0.311149\pi$
$192$	−0.639273	−0.0461356
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	10.0000	0.717958
$195$	−6.33144	−0.453404
$196$	−4.59886	−0.328490
$197$	−5.72469	−0.407867	−0.203933	−	0.978985i	$-0.565373\pi$
$197$	−5.72469	−0.407867	−0.203933	+	0.978985i	$0.565373\pi$
$198$	0	0
$199$	4.18696	0.296806	0.148403	−	0.988927i	$-0.452587\pi$
$199$	4.18696	0.296806	0.148403	+	0.988927i	$0.452587\pi$
$200$	2.31278	0.163538
$201$	−6.53414	−0.460883
$202$	−8.26484	−0.581512
$203$	3.02483	0.212301
$204$	−1.62987	−0.114113
$205$	14.9690	1.04548
$206$	−8.14706	−0.567633
$207$	18.2475	1.26829
$208$	−6.04177	−0.418921
$209$	0	0
$210$	1.62385	0.112056
$211$	−12.9022	−0.888227	−0.444113	−	0.895971i	$-0.646481\pi$
$211$	−12.9022	−0.888227	−0.444113	+	0.895971i	$0.646481\pi$
$212$	−7.95206	−0.546149
$213$	7.93340	0.543588
$214$	12.5509	0.857963
$215$	20.9909	1.43157
$216$	−3.57439	−0.243206
$217$	1.98119	0.134492
$218$	−3.40250	−0.230446
$219$	−0.927228	−0.0626562
$220$	0	0
$221$	−15.4039	−1.03618
$222$	3.66840	0.246207
$223$	21.3109	1.42708	0.713542	−	0.700612i	$-0.247090\pi$
$223$	21.3109	1.42708	0.713542	+	0.700612i	$0.247090\pi$
$224$	1.54956	0.103534
$225$	5.99318	0.399545
$226$	3.13148	0.208303
$227$	20.5030	1.36083	0.680415	−	0.732827i	$-0.261799\pi$
$227$	20.5030	1.36083	0.680415	+	0.732827i	$0.261799\pi$
$228$	0.639273	0.0423369
$229$	16.6406	1.09964	0.549822	−	0.835282i	$-0.314696\pi$
$229$	16.6406	1.09964	0.549822	+	0.835282i	$0.314696\pi$
$230$	−11.5434	−0.761148
$231$	0	0
$232$	1.95206	0.128159
$233$	−6.57575	−0.430792	−0.215396	−	0.976527i	$-0.569104\pi$
$233$	−6.57575	−0.430792	−0.215396	+	0.976527i	$0.569104\pi$
$234$	−15.6562	−1.02348
$235$	−5.88717	−0.384037
$236$	5.40250	0.351673
$237$	−0.723327	−0.0469851
$238$	3.95070	0.256086
$239$	−3.63791	−0.235317	−0.117659	−	0.993054i	$-0.537539\pi$
$239$	−3.63791	−0.235317	−0.117659	+	0.993054i	$0.537539\pi$
$240$	1.04794	0.0676445
$241$	−28.1498	−1.81329	−0.906643	−	0.421898i	$-0.861364\pi$
$241$	−28.1498	−1.81329	−0.906643	+	0.421898i	$0.861364\pi$
$242$	0	0
$243$	−14.2321	−0.912991
$244$	−13.5434	−0.867026
$245$	7.53879	0.481636
$246$	−5.83751	−0.372186
$247$	6.04177	0.384429
$248$	1.27855	0.0811878
$249$	−2.78770	−0.176663
$250$	−11.9877	−0.758166
$251$	−10.8050	−0.682005	−0.341003	−	0.940062i	$-0.610766\pi$
$251$	−10.8050	−0.682005	−0.341003	+	0.940062i	$0.610766\pi$
$252$	4.01542	0.252948
$253$	0	0
$254$	−5.82057	−0.365215
$255$	2.67180	0.167314
$256$	1.00000	0.0625000
$257$	−15.0032	−0.935876	−0.467938	−	0.883761i	$-0.655003\pi$
$257$	−15.0032	−0.935876	−0.467938	+	0.883761i	$0.655003\pi$
$258$	−8.18589	−0.509631
$259$	−8.89199	−0.552521
$260$	9.90411	0.614227
$261$	5.05842	0.313108
$262$	14.9844	0.925740
$263$	3.57575	0.220490	0.110245	−	0.993904i	$-0.464836\pi$
$263$	3.57575	0.220490	0.110245	+	0.993904i	$0.464836\pi$
$264$	0	0
$265$	13.0356	0.800770
$266$	−1.54956	−0.0950096
$267$	3.74163	0.228984
$268$	10.2212	0.624359
$269$	−25.5895	−1.56022	−0.780108	−	0.625644i	$-0.784836\pi$
$269$	−25.5895	−1.56022	−0.780108	+	0.625644i	$0.784836\pi$
$270$	5.85940	0.356592
$271$	−10.8125	−0.656814	−0.328407	−	0.944536i	$-0.606512\pi$
$271$	−10.8125	−0.656814	−0.328407	+	0.944536i	$0.606512\pi$
$272$	2.54956	0.154590
$273$	−5.98493	−0.362225
$274$	−19.3546	−1.16925
$275$	0	0
$276$	4.50162	0.270965
$277$	0.277188	0.0166546	0.00832731	−	0.999965i	$-0.497349\pi$
$277$	0.277188	0.0166546	0.00832731	+	0.999965i	$0.497349\pi$
$278$	15.7398	0.944008
$279$	3.31314	0.198352
$280$	−2.54015	−0.151803
$281$	8.23060	0.490997	0.245498	−	0.969397i	$-0.421048\pi$
$281$	8.23060	0.490997	0.245498	+	0.969397i	$0.421048\pi$
$282$	2.29584	0.136715
$283$	20.4612	1.21629	0.608146	−	0.793825i	$-0.291913\pi$
$283$	20.4612	1.21629	0.608146	+	0.793825i	$0.291913\pi$
$284$	−12.4100	−0.736400
$285$	−1.04794	−0.0620748
$286$	0	0
$287$	14.1498	0.835235
$288$	2.59133	0.152696
$289$	−10.4997	−0.617632
$290$	−3.19995	−0.187908
$291$	6.39273	0.374749
$292$	1.45044	0.0848806
$293$	−11.5402	−0.674183	−0.337091	−	0.941472i	$-0.609443\pi$
$293$	−11.5402	−0.674183	−0.337091	+	0.941472i	$0.609443\pi$
$294$	−2.93993	−0.171460
$295$	−8.85617	−0.515626
$296$	−5.73839	−0.333537
$297$	0	0
$298$	19.4456	1.12645
$299$	42.5447	2.46043
$300$	1.47850	0.0853613
$301$	19.8421	1.14368
$302$	−12.2306	−0.703792
$303$	−5.28349	−0.303529
$304$	−1.00000	−0.0573539
$305$	22.2013	1.27124
$306$	6.60675	0.377683
$307$	1.75382	0.100096	0.0500478	−	0.998747i	$-0.484063\pi$
$307$	1.75382	0.100096	0.0500478	+	0.998747i	$0.484063\pi$
$308$	0	0
$309$	−5.20820	−0.296284
$310$	−2.09589	−0.119038
$311$	−17.7473	−1.00636	−0.503178	−	0.864183i	$-0.667836\pi$
$311$	−17.7473	−1.00636	−0.503178	+	0.864183i	$0.667836\pi$
$312$	−3.86234	−0.218662
$313$	−16.4381	−0.929136	−0.464568	−	0.885538i	$-0.653790\pi$
$313$	−16.4381	−0.929136	−0.464568	+	0.885538i	$0.653790\pi$
$314$	−24.6425	−1.39066
$315$	−6.58237	−0.370875
$316$	1.13148	0.0636509
$317$	6.78204	0.380917	0.190459	−	0.981695i	$-0.439003\pi$
$317$	6.78204	0.380917	0.190459	+	0.981695i	$0.439003\pi$
$318$	−5.08354	−0.285071
$319$	0	0
$320$	−1.63927	−0.0916382
$321$	8.02347	0.447826
$322$	−10.9116	−0.608082
$323$	−2.54956	−0.141861
$324$	5.48898	0.304943
$325$	13.9733	0.775099
$326$	19.0032	1.05249
$327$	−2.17513	−0.120285
$328$	9.13148	0.504202
$329$	−5.56498	−0.306807
$330$	0	0
$331$	−7.61616	−0.418622	−0.209311	−	0.977849i	$-0.567122\pi$
$331$	−7.61616	−0.418622	−0.209311	+	0.977849i	$0.567122\pi$
$332$	4.36073	0.239326
$333$	−14.8701	−0.814875
$334$	−1.82057	−0.0996174
$335$	−16.7553	−0.915442
$336$	0.990593	0.0540413
$337$	20.0200	1.09056	0.545280	−	0.838254i	$-0.316423\pi$
$337$	20.0200	1.09056	0.545280	+	0.838254i	$0.316423\pi$
$338$	−23.5030	−1.27839
$339$	2.00187	0.108727
$340$	−4.17943	−0.226661
$341$	0	0
$342$	−2.59133	−0.140123
$343$	17.9731	0.970458
$344$	12.8050	0.690399
$345$	−7.37938	−0.397292
$346$	10.3109	0.554318
$347$	16.3296	0.876617	0.438308	−	0.898825i	$-0.355578\pi$
$347$	16.3296	0.876617	0.438308	+	0.898825i	$0.355578\pi$
$348$	1.24790	0.0668943
$349$	26.9995	1.44525	0.722625	−	0.691241i	$-0.242936\pi$
$349$	26.9995	1.44525	0.722625	+	0.691241i	$0.242936\pi$
$350$	−3.58380	−0.191562
$351$	−21.5956	−1.15269
$352$	0	0
$353$	14.0525	0.747941	0.373970	−	0.927441i	$-0.377996\pi$
$353$	14.0525	0.747941	0.373970	+	0.927441i	$0.377996\pi$
$354$	3.45367	0.183561
$355$	20.3434	1.07972
$356$	−5.85294	−0.310205
$357$	2.52558	0.133668
$358$	−5.78770	−0.305889
$359$	−25.3546	−1.33816	−0.669081	−	0.743189i	$-0.733312\pi$
$359$	−25.3546	−1.33816	−0.669081	+	0.743189i	$0.733312\pi$
$360$	−4.24790	−0.223884
$361$	1.00000	0.0526316
$362$	−15.4768	−0.813442
$363$	0	0
$364$	9.36209	0.490707
$365$	−2.37767	−0.124453
$366$	−8.65793	−0.452557
$367$	19.3931	1.01231	0.506155	−	0.862442i	$-0.331066\pi$
$367$	19.3931	1.01231	0.506155	+	0.862442i	$0.331066\pi$
$368$	−7.04177	−0.367078
$369$	23.6627	1.23183
$370$	9.40680	0.489036
$371$	12.3222	0.639736
$372$	0.817341	0.0423772
$373$	35.3018	1.82786	0.913929	−	0.405875i	$-0.133033\pi$
$373$	35.3018	1.82786	0.913929	+	0.405875i	$0.133033\pi$
$374$	0	0
$375$	−7.66339	−0.395736
$376$	−3.59133	−0.185209
$377$	11.7939	0.607415
$378$	5.53873	0.284882
$379$	−5.17002	−0.265566	−0.132783	−	0.991145i	$-0.542391\pi$
$379$	−5.17002	−0.265566	−0.132783	+	0.991145i	$0.542391\pi$
$380$	1.63927	0.0840930
$381$	−3.72094	−0.190629
$382$	−15.4537	−0.790679
$383$	−5.16759	−0.264052	−0.132026	−	0.991246i	$-0.542148\pi$
$383$	−5.16759	−0.264052	−0.132026	+	0.991246i	$0.542148\pi$
$384$	0.639273	0.0326228
$385$	0	0
$386$	8.00000	0.407189
$387$	33.1820	1.68673
$388$	−10.0000	−0.507673
$389$	4.91782	0.249343	0.124672	−	0.992198i	$-0.460212\pi$
$389$	4.91782	0.249343	0.124672	+	0.992198i	$0.460212\pi$
$390$	6.33144	0.320605
$391$	−17.9534	−0.907943
$392$	4.59886	0.232278
$393$	9.57914	0.483204
$394$	5.72469	0.288405
$395$	−1.85481	−0.0933256
$396$	0	0
$397$	−7.54151	−0.378498	−0.189249	−	0.981929i	$-0.560605\pi$
$397$	−7.54151	−0.378498	−0.189249	+	0.981929i	$0.560605\pi$
$398$	−4.18696	−0.209873
$399$	−0.990593	−0.0495917
$400$	−2.31278	−0.115639
$401$	4.03236	0.201367	0.100683	−	0.994919i	$-0.467897\pi$
$401$	4.03236	0.201367	0.100683	+	0.994919i	$0.467897\pi$
$402$	6.53414	0.325893
$403$	7.72469	0.384794
$404$	8.26484	0.411191
$405$	−8.99793	−0.447111
$406$	−3.02483	−0.150120
$407$	0	0
$408$	1.62987	0.0806904
$409$	0.358853	0.0177441	0.00887207	−	0.999961i	$-0.497176\pi$
$409$	0.358853	0.0177441	0.00887207	+	0.999961i	$0.497176\pi$
$410$	−14.9690	−0.739266
$411$	−12.3729	−0.610308
$412$	8.14706	0.401377
$413$	−8.37149	−0.411934
$414$	−18.2475	−0.896818
$415$	−7.14842	−0.350902
$416$	6.04177	0.296222
$417$	10.0620	0.492739
$418$	0	0
$419$	31.2643	1.52736	0.763681	−	0.645594i	$-0.223390\pi$
$419$	31.2643	1.52736	0.763681	+	0.645594i	$0.223390\pi$
$420$	−1.62385	−0.0792359
$421$	18.7303	0.912861	0.456431	−	0.889759i	$-0.349128\pi$
$421$	18.7303	0.912861	0.456431	+	0.889759i	$0.349128\pi$
$422$	12.9022	0.628071
$423$	−9.30632	−0.452489
$424$	7.95206	0.386186
$425$	−5.89658	−0.286026
$426$	−7.93340	−0.384375
$427$	20.9863	1.01560
$428$	−12.5509	−0.606672
$429$	0	0
$430$	−20.9909	−1.01227
$431$	7.03559	0.338893	0.169446	−	0.985539i	$-0.445802\pi$
$431$	7.03559	0.338893	0.169446	+	0.985539i	$0.445802\pi$
$432$	3.57439	0.171973
$433$	11.2785	0.542012	0.271006	−	0.962578i	$-0.412644\pi$
$433$	11.2785	0.542012	0.271006	+	0.962578i	$0.412644\pi$
$434$	−1.98119	−0.0950999
$435$	−2.04565	−0.0980812
$436$	3.40250	0.162950
$437$	7.04177	0.336854
$438$	0.927228	0.0443047
$439$	2.47356	0.118056	0.0590282	−	0.998256i	$-0.481200\pi$
$439$	2.47356	0.118056	0.0590282	+	0.998256i	$0.481200\pi$
$440$	0	0
$441$	11.9172	0.567484
$442$	15.4039	0.732687
$443$	23.0680	1.09599	0.547996	−	0.836481i	$-0.315391\pi$
$443$	23.0680	1.09599	0.547996	+	0.836481i	$0.315391\pi$
$444$	−3.66840	−0.174095
$445$	9.59456	0.454826
$446$	−21.3109	−1.00910
$447$	12.4311	0.587969
$448$	−1.54956	−0.0732098
$449$	−22.8050	−1.07623	−0.538117	−	0.842870i	$-0.680864\pi$
$449$	−22.8050	−1.07623	−0.538117	+	0.842870i	$0.680864\pi$
$450$	−5.99318	−0.282521
$451$	0	0
$452$	−3.13148	−0.147293
$453$	−7.81870	−0.367355
$454$	−20.5030	−0.962252
$455$	−15.3470	−0.719479
$456$	−0.639273	−0.0299367
$457$	19.1253	0.894644	0.447322	−	0.894373i	$-0.352378\pi$
$457$	19.1253	0.894644	0.447322	+	0.894373i	$0.352378\pi$
$458$	−16.6406	−0.777566
$459$	9.11312	0.425364
$460$	11.5434	0.538213
$461$	16.2799	0.758231	0.379115	−	0.925349i	$-0.376228\pi$
$461$	16.2799	0.758231	0.379115	+	0.925349i	$0.376228\pi$
$462$	0	0
$463$	−21.0525	−0.978394	−0.489197	−	0.872173i	$-0.662710\pi$
$463$	−21.0525	−0.978394	−0.489197	+	0.872173i	$0.662710\pi$
$464$	−1.95206	−0.0906219
$465$	−1.33985	−0.0621339
$466$	6.57575	0.304616
$467$	−32.9909	−1.52664	−0.763318	−	0.646023i	$-0.776431\pi$
$467$	−32.9909	−1.52664	−0.763318	+	0.646023i	$0.776431\pi$
$468$	15.6562	0.723709
$469$	−15.8384	−0.731347
$470$	5.88717	0.271555
$471$	−15.7533	−0.725874
$472$	−5.40250	−0.248670
$473$	0	0
$474$	0.723327	0.0332235
$475$	2.31278	0.106118
$476$	−3.95070	−0.181080
$477$	20.6064	0.943502
$478$	3.63791	0.166394
$479$	21.2630	0.971530	0.485765	−	0.874090i	$-0.338541\pi$
$479$	21.2630	0.971530	0.485765	+	0.874090i	$0.338541\pi$
$480$	−1.04794	−0.0478319
$481$	−34.6701	−1.58082
$482$	28.1498	1.28219
$483$	−6.97553	−0.317397
$484$	0	0
$485$	16.3927	0.744356
$486$	14.2321	0.645582
$487$	19.4306	0.880483	0.440241	−	0.897880i	$-0.354893\pi$
$487$	19.4306	0.880483	0.440241	+	0.897880i	$0.354893\pi$
$488$	13.5434	0.613080
$489$	12.1483	0.549363
$490$	−7.53879	−0.340568
$491$	−32.2968	−1.45754	−0.728768	−	0.684761i	$-0.759907\pi$
$491$	−32.2968	−1.45754	−0.728768	+	0.684761i	$0.759907\pi$
$492$	5.83751	0.263175
$493$	−4.97688	−0.224148
$494$	−6.04177	−0.271832
$495$	0	0
$496$	−1.27855	−0.0574085
$497$	19.2301	0.862587
$498$	2.78770	0.124920
$499$	4.26484	0.190920	0.0954602	−	0.995433i	$-0.469568\pi$
$499$	4.26484	0.190920	0.0954602	+	0.995433i	$0.469568\pi$
$500$	11.9877	0.536104
$501$	−1.16384	−0.0519967
$502$	10.8050	0.482250
$503$	−22.0108	−0.981412	−0.490706	−	0.871325i	$-0.663261\pi$
$503$	−22.0108	−0.981412	−0.490706	+	0.871325i	$0.663261\pi$
$504$	−4.01542	−0.178861
$505$	−13.5483	−0.602893
$506$	0	0
$507$	−15.0248	−0.667276
$508$	5.82057	0.258246
$509$	−7.31107	−0.324057	−0.162029	−	0.986786i	$-0.551804\pi$
$509$	−7.31107	−0.324057	−0.162029	+	0.986786i	$0.551804\pi$
$510$	−2.67180	−0.118309
$511$	−2.24754	−0.0994255
$512$	−1.00000	−0.0441942
$513$	−3.57439	−0.157813
$514$	15.0032	0.661764
$515$	−13.3553	−0.588503
$516$	8.18589	0.360364
$517$	0	0
$518$	8.89199	0.390691
$519$	6.59149	0.289334
$520$	−9.90411	−0.434324
$521$	−31.4644	−1.37848	−0.689241	−	0.724532i	$-0.742056\pi$
$521$	−31.4644	−1.37848	−0.689241	+	0.724532i	$0.742056\pi$
$522$	−5.05842	−0.221401
$523$	4.01264	0.175460	0.0877302	−	0.996144i	$-0.472039\pi$
$523$	4.01264	0.175460	0.0877302	+	0.996144i	$0.472039\pi$
$524$	−14.9844	−0.654597
$525$	−2.29103	−0.0999886
$526$	−3.57575	−0.155910
$527$	−3.25973	−0.141996
$528$	0	0
$529$	26.5865	1.15594
$530$	−13.0356	−0.566230
$531$	−13.9996	−0.607533
$532$	1.54956	0.0671820
$533$	55.1703	2.38969
$534$	−3.74163	−0.161916
$535$	20.5744	0.889508
$536$	−10.2212	−0.441488
$537$	−3.69992	−0.159663
$538$	25.5895	1.10324
$539$	0	0
$540$	−5.85940	−0.252148
$541$	6.65298	0.286034	0.143017	−	0.989720i	$-0.454320\pi$
$541$	6.65298	0.286034	0.143017	+	0.989720i	$0.454320\pi$
$542$	10.8125	0.464438
$543$	−9.89390	−0.424588
$544$	−2.54956	−0.109311
$545$	−5.57762	−0.238919
$546$	5.98493	0.256131
$547$	−22.5622	−0.964690	−0.482345	−	0.875981i	$-0.660215\pi$
$547$	−22.5622	−0.964690	−0.482345	+	0.875981i	$0.660215\pi$
$548$	19.3546	0.826785
$549$	35.0954	1.49783
$550$	0	0
$551$	1.95206	0.0831604
$552$	−4.50162	−0.191602
$553$	−1.75330	−0.0745579
$554$	−0.277188	−0.0117766
$555$	6.01352	0.255260
$556$	−15.7398	−0.667514
$557$	−25.8270	−1.09433	−0.547163	−	0.837026i	$-0.684292\pi$
$557$	−25.8270	−1.09433	−0.547163	+	0.837026i	$0.684292\pi$
$558$	−3.31314	−0.140256
$559$	77.3648	3.27218
$560$	2.54015	0.107341
$561$	0	0
$562$	−8.23060	−0.347187
$563$	−22.0698	−0.930132	−0.465066	−	0.885276i	$-0.653969\pi$
$563$	−22.0698	−0.930132	−0.465066	+	0.885276i	$0.653969\pi$
$564$	−2.29584	−0.0966724
$565$	5.13336	0.215962
$566$	−20.4612	−0.860049
$567$	−8.50550	−0.357197
$568$	12.4100	0.520713
$569$	0.275315	0.0115418	0.00577089	−	0.999983i	$-0.498163\pi$
$569$	0.275315	0.0115418	0.00577089	+	0.999983i	$0.498163\pi$
$570$	1.04794	0.0438935
$571$	37.4287	1.56634	0.783171	−	0.621807i	$-0.213601\pi$
$571$	37.4287	1.56634	0.783171	+	0.621807i	$0.213601\pi$
$572$	0	0
$573$	−9.87912	−0.412706
$574$	−14.1498	−0.590600
$575$	16.2861	0.679177
$576$	−2.59133	−0.107972
$577$	−14.2893	−0.594872	−0.297436	−	0.954742i	$-0.596131\pi$
$577$	−14.2893	−0.594872	−0.297436	+	0.954742i	$0.596131\pi$
$578$	10.4997	0.436732
$579$	5.11419	0.212538
$580$	3.19995	0.132871
$581$	−6.75721	−0.280336
$582$	−6.39273	−0.264987
$583$	0	0
$584$	−1.45044	−0.0600196
$585$	−25.6648	−1.06111
$586$	11.5402	0.476719
$587$	0.100479	0.00414722	0.00207361	−	0.999998i	$-0.499340\pi$
$587$	0.100479	0.00414722	0.00207361	+	0.999998i	$0.499340\pi$
$588$	2.93993	0.121241
$589$	1.27855	0.0526816
$590$	8.85617	0.364603
$591$	3.65964	0.150537
$592$	5.73839	0.235847
$593$	19.4270	0.797770	0.398885	−	0.917001i	$-0.369397\pi$
$593$	19.4270	0.797770	0.398885	+	0.917001i	$0.369397\pi$
$594$	0	0
$595$	6.47627	0.265501
$596$	−19.4456	−0.796524
$597$	−2.67661	−0.109546
$598$	−42.5447	−1.73978
$599$	−36.8389	−1.50520	−0.752598	−	0.658481i	$-0.771199\pi$
$599$	−36.8389	−1.50520	−0.752598	+	0.658481i	$0.771199\pi$
$600$	−1.47850	−0.0603595
$601$	−2.98170	−0.121626	−0.0608130	−	0.998149i	$-0.519369\pi$
$601$	−2.98170	−0.121626	−0.0608130	+	0.998149i	$0.519369\pi$
$602$	−19.8421	−0.808704
$603$	−26.4865	−1.07861
$604$	12.2306	0.497656
$605$	0	0
$606$	5.28349	0.214627
$607$	−28.2118	−1.14508	−0.572541	−	0.819876i	$-0.694042\pi$
$607$	−28.2118	−1.14508	−0.572541	+	0.819876i	$0.694042\pi$
$608$	1.00000	0.0405554
$609$	−1.93369	−0.0783572
$610$	−22.2013	−0.898905
$611$	−21.6980	−0.877806
$612$	−6.60675	−0.267062
$613$	33.8872	1.36869	0.684345	−	0.729158i	$-0.260088\pi$
$613$	33.8872	1.36869	0.684345	+	0.729158i	$0.260088\pi$
$614$	−1.75382	−0.0707782
$615$	−9.56928	−0.385871
$616$	0	0
$617$	−14.9346	−0.601244	−0.300622	−	0.953743i	$-0.597194\pi$
$617$	−14.9346	−0.601244	−0.300622	+	0.953743i	$0.597194\pi$
$618$	5.20820	0.209505
$619$	−30.9909	−1.24563	−0.622814	−	0.782370i	$-0.714011\pi$
$619$	−30.9909	−1.24563	−0.622814	+	0.782370i	$0.714011\pi$
$620$	2.09589	0.0841729
$621$	−25.1700	−1.01004
$622$	17.7473	0.711601
$623$	9.06948	0.363361
$624$	3.86234	0.154617
$625$	−8.08712	−0.323485
$626$	16.4381	0.656998
$627$	0	0
$628$	24.6425	0.983343
$629$	14.6304	0.583351
$630$	6.58237	0.262248
$631$	−8.54526	−0.340181	−0.170091	−	0.985428i	$-0.554406\pi$
$631$	−8.54526	−0.340181	−0.170091	+	0.985428i	$0.554406\pi$
$632$	−1.13148	−0.0450080
$633$	8.24806	0.327831
$634$	−6.78204	−0.269349
$635$	−9.54151	−0.378643
$636$	5.08354	0.201575
$637$	27.7853	1.10089
$638$	0	0
$639$	32.1585	1.27217
$640$	1.63927	0.0647980
$641$	−41.5776	−1.64222	−0.821109	−	0.570772i	$-0.806644\pi$
$641$	−41.5776	−1.64222	−0.821109	+	0.570772i	$0.806644\pi$
$642$	−8.02347	−0.316661
$643$	−9.38090	−0.369947	−0.184973	−	0.982744i	$-0.559220\pi$
$643$	−9.38090	−0.369947	−0.184973	+	0.982744i	$0.559220\pi$
$644$	10.9116	0.429979
$645$	−13.4189	−0.528369
$646$	2.54956	0.100311
$647$	−44.9256	−1.76621	−0.883105	−	0.469176i	$-0.844551\pi$
$647$	−44.9256	−1.76621	−0.883105	+	0.469176i	$0.844551\pi$
$648$	−5.48898	−0.215627
$649$	0	0
$650$	−13.9733	−0.548078
$651$	−1.26652	−0.0496388
$652$	−19.0032	−0.744224
$653$	4.78618	0.187298	0.0936488	−	0.995605i	$-0.470147\pi$
$653$	4.78618	0.187298	0.0936488	+	0.995605i	$0.470147\pi$
$654$	2.17513	0.0850541
$655$	24.5636	0.959778
$656$	−9.13148	−0.356524
$657$	−3.75857	−0.146636
$658$	5.56498	0.216946
$659$	−32.1934	−1.25408	−0.627039	−	0.778988i	$-0.715733\pi$
$659$	−32.1934	−1.25408	−0.627039	+	0.778988i	$0.715733\pi$
$660$	0	0
$661$	−42.1684	−1.64016	−0.820081	−	0.572247i	$-0.806072\pi$
$661$	−42.1684	−1.64016	−0.820081	+	0.572247i	$0.806072\pi$
$662$	7.61616	0.296010
$663$	9.84728	0.382436
$664$	−4.36073	−0.169229
$665$	−2.54015	−0.0985029
$666$	14.8701	0.576203
$667$	13.7459	0.532244
$668$	1.82057	0.0704401
$669$	−13.6235	−0.526715
$670$	16.7553	0.647315
$671$	0	0
$672$	−0.990593	−0.0382129
$673$	37.4009	1.44170	0.720850	−	0.693092i	$-0.243752\pi$
$673$	37.4009	1.44170	0.720850	+	0.693092i	$0.243752\pi$
$674$	−20.0200	−0.771142
$675$	−8.26678	−0.318189
$676$	23.5030	0.903961
$677$	40.3742	1.55171	0.775853	−	0.630913i	$-0.217320\pi$
$677$	40.3742	1.55171	0.775853	+	0.630913i	$0.217320\pi$
$678$	−2.00187	−0.0768815
$679$	15.4956	0.594667
$680$	4.17943	0.160274
$681$	−13.1070	−0.502262
$682$	0	0
$683$	−8.06472	−0.308588	−0.154294	−	0.988025i	$-0.549310\pi$
$683$	−8.06472	−0.308588	−0.154294	+	0.988025i	$0.549310\pi$
$684$	2.59133	0.0990820
$685$	−31.7274	−1.21224
$686$	−17.9731	−0.686218
$687$	−10.6379	−0.405862
$688$	−12.8050	−0.488186
$689$	48.0445	1.83035
$690$	7.37938	0.280928
$691$	27.8402	1.05909	0.529546	−	0.848281i	$-0.322362\pi$
$691$	27.8402	1.05909	0.529546	+	0.848281i	$0.322362\pi$
$692$	−10.3109	−0.391962
$693$	0	0
$694$	−16.3296	−0.619862
$695$	25.8018	0.978717
$696$	−1.24790	−0.0473014
$697$	−23.2813	−0.881841
$698$	−26.9995	−1.02195
$699$	4.20370	0.158999
$700$	3.58380	0.135455
$701$	36.2010	1.36729	0.683646	−	0.729814i	$-0.260393\pi$
$701$	36.2010	1.36729	0.683646	+	0.729814i	$0.260393\pi$
$702$	21.5956	0.815075
$703$	−5.73839	−0.216428
$704$	0	0
$705$	3.76351	0.141742
$706$	−14.0525	−0.528874
$707$	−12.8069	−0.481652
$708$	−3.45367	−0.129797
$709$	22.7091	0.852858	0.426429	−	0.904521i	$-0.359771\pi$
$709$	22.7091	0.852858	0.426429	+	0.904521i	$0.359771\pi$
$710$	−20.3434	−0.763475
$711$	−2.93204	−0.109960
$712$	5.85294	0.219348
$713$	9.00323	0.337174
$714$	−2.52558	−0.0945173
$715$	0	0
$716$	5.78770	0.216296
$717$	2.32562	0.0868520
$718$	25.3546	0.946223
$719$	22.2859	0.831125	0.415562	−	0.909565i	$-0.363585\pi$
$719$	22.2859	0.831125	0.415562	+	0.909565i	$0.363585\pi$
$720$	4.24790	0.158310
$721$	−12.6244	−0.470156
$722$	−1.00000	−0.0372161
$723$	17.9954	0.669256
$724$	15.4768	0.575190
$725$	4.51468	0.167671
$726$	0	0
$727$	−4.03100	−0.149502	−0.0747508	−	0.997202i	$-0.523816\pi$
$727$	−4.03100	−0.149502	−0.0747508	+	0.997202i	$0.523816\pi$
$728$	−9.36209	−0.346982
$729$	−7.36871	−0.272915
$730$	2.37767	0.0880014
$731$	−32.6471	−1.20750
$732$	8.65793	0.320006
$733$	36.0712	1.33232	0.666160	−	0.745809i	$-0.267937\pi$
$733$	36.0712	1.33232	0.666160	+	0.745809i	$0.267937\pi$
$734$	−19.3931	−0.715812
$735$	−4.81935	−0.177764
$736$	7.04177	0.259563
$737$	0	0
$738$	−23.6627	−0.871035
$739$	−38.6388	−1.42135	−0.710675	−	0.703521i	$-0.751610\pi$
$739$	−38.6388	−1.42135	−0.710675	+	0.703521i	$0.751610\pi$
$740$	−9.40680	−0.345801
$741$	−3.86234	−0.141887
$742$	−12.3222	−0.452362
$743$	−50.8416	−1.86520	−0.932599	−	0.360915i	$-0.882464\pi$
$743$	−50.8416	−1.86520	−0.932599	+	0.360915i	$0.882464\pi$
$744$	−0.817341	−0.0299652
$745$	31.8767	1.16787
$746$	−35.3018	−1.29249
$747$	−11.3001	−0.413448
$748$	0	0
$749$	19.4484	0.710629
$750$	7.66339	0.279827
$751$	39.4521	1.43963	0.719814	−	0.694167i	$-0.244227\pi$
$751$	39.4521	1.43963	0.719814	+	0.694167i	$0.244227\pi$
$752$	3.59133	0.130962
$753$	6.90734	0.251718
$754$	−11.7939	−0.429507
$755$	−20.0493	−0.729669
$756$	−5.53873	−0.201442
$757$	16.5402	0.601162	0.300581	−	0.953756i	$-0.402819\pi$
$757$	16.5402	0.601162	0.300581	+	0.953756i	$0.402819\pi$
$758$	5.17002	0.187784
$759$	0	0
$760$	−1.63927	−0.0594627
$761$	23.5341	0.853112	0.426556	−	0.904461i	$-0.359727\pi$
$761$	23.5341	0.853112	0.426556	+	0.904461i	$0.359727\pi$
$762$	3.72094	0.134795
$763$	−5.27237	−0.190873
$764$	15.4537	0.559094
$765$	10.8303	0.391569
$766$	5.16759	0.186713
$767$	−32.6406	−1.17858
$768$	−0.639273	−0.0230678
$769$	0.0803060	0.00289591	0.00144795	−	0.999999i	$-0.499539\pi$
$769$	0.0803060	0.00289591	0.00144795	+	0.999999i	$0.499539\pi$
$770$	0	0
$771$	9.59117	0.345417
$772$	−8.00000	−0.287926
$773$	16.2541	0.584618	0.292309	−	0.956324i	$-0.405576\pi$
$773$	16.2541	0.584618	0.292309	+	0.956324i	$0.405576\pi$
$774$	−33.1820	−1.19270
$775$	2.95700	0.106219
$776$	10.0000	0.358979
$777$	5.68441	0.203927
$778$	−4.91782	−0.176312
$779$	9.13148	0.327169
$780$	−6.33144	−0.226702
$781$	0	0
$782$	17.9534	0.642013
$783$	−6.97741	−0.249352
$784$	−4.59886	−0.164245
$785$	−40.3958	−1.44179
$786$	−9.57914	−0.341677
$787$	−11.7726	−0.419649	−0.209824	−	0.977739i	$-0.567289\pi$
$787$	−11.7726	−0.419649	−0.209824	+	0.977739i	$0.567289\pi$
$788$	−5.72469	−0.203933
$789$	−2.28588	−0.0813795
$790$	1.85481	0.0659911
$791$	4.85242	0.172532
$792$	0	0
$793$	81.8260	2.90573
$794$	7.54151	0.267638
$795$	−8.33331	−0.295552
$796$	4.18696	0.148403
$797$	12.5003	0.442782	0.221391	−	0.975185i	$-0.428940\pi$
$797$	12.5003	0.442782	0.221391	+	0.975185i	$0.428940\pi$
$798$	0.990593	0.0350666
$799$	9.15631	0.323927
$800$	2.31278	0.0817692
$801$	15.1669	0.535895
$802$	−4.03236	−0.142388
$803$	0	0
$804$	−6.53414	−0.230441
$805$	−17.8872	−0.630440
$806$	−7.72469	−0.272090
$807$	16.3587	0.575852
$808$	−8.26484	−0.290756
$809$	6.86988	0.241532	0.120766	−	0.992681i	$-0.461465\pi$
$809$	6.86988	0.241532	0.120766	+	0.992681i	$0.461465\pi$
$810$	8.99793	0.316155
$811$	34.8255	1.22289	0.611445	−	0.791287i	$-0.290589\pi$
$811$	34.8255	1.22289	0.611445	+	0.791287i	$0.290589\pi$
$812$	3.02483	0.106151
$813$	6.91216	0.242420
$814$	0	0
$815$	31.1515	1.09119
$816$	−1.62987	−0.0570567
$817$	12.8050	0.447990
$818$	−0.358853	−0.0125470
$819$	−24.2602	−0.847722
$820$	14.9690	0.522740
$821$	8.45933	0.295233	0.147616	−	0.989045i	$-0.452840\pi$
$821$	8.45933	0.295233	0.147616	+	0.989045i	$0.452840\pi$
$822$	12.3729	0.431553
$823$	−7.47627	−0.260606	−0.130303	−	0.991474i	$-0.541595\pi$
$823$	−7.47627	−0.260606	−0.130303	+	0.991474i	$0.541595\pi$
$824$	−8.14706	−0.283816
$825$	0	0
$826$	8.37149	0.291281
$827$	−16.4386	−0.571626	−0.285813	−	0.958285i	$-0.592264\pi$
$827$	−16.4386	−0.571626	−0.285813	+	0.958285i	$0.592264\pi$
$828$	18.2475	0.634146
$829$	7.49974	0.260477	0.130238	−	0.991483i	$-0.458426\pi$
$829$	7.49974	0.260477	0.130238	+	0.991483i	$0.458426\pi$
$830$	7.14842	0.248125
$831$	−0.177199	−0.00614697
$832$	−6.04177	−0.209461
$833$	−11.7251	−0.406250
$834$	−10.0620	−0.348419
$835$	−2.98442	−0.103280
$836$	0	0
$837$	−4.57002	−0.157963
$838$	−31.2643	−1.08001
$839$	5.60879	0.193637	0.0968184	−	0.995302i	$-0.469133\pi$
$839$	5.60879	0.193637	0.0968184	+	0.995302i	$0.469133\pi$
$840$	1.62385	0.0560282
$841$	−25.1895	−0.868603
$842$	−18.7303	−0.645490
$843$	−5.26161	−0.181219
$844$	−12.9022	−0.444113
$845$	−38.5278	−1.32540
$846$	9.30632	0.319958
$847$	0	0
$848$	−7.95206	−0.273075
$849$	−13.0803	−0.448915
$850$	5.89658	0.202251
$851$	−40.4084	−1.38518
$852$	7.93340	0.271794
$853$	5.80176	0.198648	0.0993242	−	0.995055i	$-0.468332\pi$
$853$	5.80176	0.198648	0.0993242	+	0.995055i	$0.468332\pi$
$854$	−20.9863	−0.718136
$855$	−4.24790	−0.145275
$856$	12.5509	0.428982
$857$	18.6444	0.636880	0.318440	−	0.947943i	$-0.396841\pi$
$857$	18.6444	0.636880	0.318440	+	0.947943i	$0.396841\pi$
$858$	0	0
$859$	−33.4502	−1.14131	−0.570653	−	0.821191i	$-0.693310\pi$
$859$	−33.4502	−1.14131	−0.570653	+	0.821191i	$0.693310\pi$
$860$	20.9909	0.715783
$861$	−9.04558	−0.308273
$862$	−7.03559	−0.239633
$863$	−29.9256	−1.01868	−0.509340	−	0.860565i	$-0.670111\pi$
$863$	−29.9256	−1.01868	−0.509340	+	0.860565i	$0.670111\pi$
$864$	−3.57439	−0.121603
$865$	16.9024	0.574699
$866$	−11.2785	−0.383261
$867$	6.71221	0.227958
$868$	1.98119	0.0672458
$869$	0	0
$870$	2.04565	0.0693539
$871$	−61.7541	−2.09246
$872$	−3.40250	−0.115223
$873$	25.9133	0.877032
$874$	−7.04177	−0.238191
$875$	−18.5756	−0.627969
$876$	−0.927228	−0.0313281
$877$	15.6024	0.526857	0.263429	−	0.964679i	$-0.415147\pi$
$877$	15.6024	0.526857	0.263429	+	0.964679i	$0.415147\pi$
$878$	−2.47356	−0.0834785
$879$	7.37731	0.248831
$880$	0	0
$881$	−8.93799	−0.301129	−0.150564	−	0.988600i	$-0.548109\pi$
$881$	−8.93799	−0.301129	−0.150564	+	0.988600i	$0.548109\pi$
$882$	−11.9172	−0.401272
$883$	32.7196	1.10110	0.550551	−	0.834802i	$-0.314418\pi$
$883$	32.7196	1.10110	0.550551	+	0.834802i	$0.314418\pi$
$884$	−15.4039	−0.518088
$885$	5.66151	0.190310
$886$	−23.0680	−0.774983
$887$	−56.4812	−1.89645	−0.948227	−	0.317594i	$-0.897125\pi$
$887$	−56.4812	−1.89645	−0.948227	+	0.317594i	$0.897125\pi$
$888$	3.66840	0.123104
$889$	−9.01933	−0.302499
$890$	−9.59456	−0.321610
$891$	0	0
$892$	21.3109	0.713542
$893$	−3.59133	−0.120179
$894$	−12.4311	−0.415757
$895$	−9.48762	−0.317136
$896$	1.54956	0.0517672
$897$	−27.1977	−0.908106
$898$	22.8050	0.761012
$899$	2.49580	0.0832394
$900$	5.99318	0.199773
$901$	−20.2742	−0.675433
$902$	0	0
$903$	−12.6845	−0.422115
$904$	3.13148	0.104152
$905$	−25.3707	−0.843350
$906$	7.81870	0.259759
$907$	−5.76135	−0.191302	−0.0956512	−	0.995415i	$-0.530493\pi$
$907$	−5.76135	−0.191302	−0.0956512	+	0.995415i	$0.530493\pi$
$908$	20.5030	0.680415
$909$	−21.4169	−0.710354
$910$	15.3470	0.508749
$911$	9.78821	0.324298	0.162149	−	0.986766i	$-0.448157\pi$
$911$	9.78821	0.324298	0.162149	+	0.986766i	$0.448157\pi$
$912$	0.639273	0.0211685
$913$	0	0
$914$	−19.1253	−0.632609
$915$	−14.1927	−0.469197
$916$	16.6406	0.549822
$917$	23.2193	0.766767
$918$	−9.11312	−0.300778
$919$	33.1748	1.09433	0.547167	−	0.837023i	$-0.315706\pi$
$919$	33.1748	1.09433	0.547167	+	0.837023i	$0.315706\pi$
$920$	−11.5434	−0.380574
$921$	−1.12117	−0.0369437
$922$	−16.2799	−0.536150
$923$	74.9785	2.46795
$924$	0	0
$925$	−13.2717	−0.436369
$926$	21.0525	0.691829
$927$	−21.1117	−0.693400
$928$	1.95206	0.0640794
$929$	−12.5415	−0.411474	−0.205737	−	0.978607i	$-0.565959\pi$
$929$	−12.5415	−0.411474	−0.205737	+	0.978607i	$0.565959\pi$
$930$	1.33985	0.0439353
$931$	4.59886	0.150722
$932$	−6.57575	−0.215396
$933$	11.3454	0.371431
$934$	32.9909	1.07949
$935$	0	0
$936$	−15.6562	−0.511739
$937$	−52.2470	−1.70684	−0.853418	−	0.521227i	$-0.825474\pi$
$937$	−52.2470	−1.70684	−0.853418	+	0.521227i	$0.825474\pi$
$938$	15.8384	0.517141
$939$	10.5084	0.342930
$940$	−5.88717	−0.192018
$941$	−29.4189	−0.959029	−0.479515	−	0.877534i	$-0.659187\pi$
$941$	−29.4189	−0.959029	−0.479515	+	0.877534i	$0.659187\pi$
$942$	15.7533	0.513270
$943$	64.3018	2.09395
$944$	5.40250	0.175836
$945$	9.07949	0.295356
$946$	0	0
$947$	−47.4370	−1.54150	−0.770748	−	0.637140i	$-0.780117\pi$
$947$	−47.4370	−1.54150	−0.770748	+	0.637140i	$0.780117\pi$
$948$	−0.723327	−0.0234926
$949$	−8.76322	−0.284466
$950$	−2.31278	−0.0750366
$951$	−4.33558	−0.140591
$952$	3.95070	0.128043
$953$	−42.4785	−1.37601	−0.688007	−	0.725704i	$-0.741514\pi$
$953$	−42.4785	−1.37601	−0.688007	+	0.725704i	$0.741514\pi$
$954$	−20.6064	−0.667157
$955$	−25.3328	−0.819750
$956$	−3.63791	−0.117659
$957$	0	0
$958$	−21.2630	−0.686975
$959$	−29.9910	−0.968461
$960$	1.04794	0.0338222
$961$	−29.3653	−0.947268
$962$	34.6701	1.11781
$963$	32.5236	1.04806
$964$	−28.1498	−0.906643
$965$	13.1142	0.422161
$966$	6.97553	0.224434
$967$	−40.6096	−1.30592	−0.652959	−	0.757393i	$-0.726473\pi$
$967$	−40.6096	−1.30592	−0.652959	+	0.757393i	$0.726473\pi$
$968$	0	0
$969$	1.62987	0.0523588
$970$	−16.3927	−0.526339
$971$	35.5000	1.13925	0.569625	−	0.821905i	$-0.307088\pi$
$971$	35.5000	1.13925	0.569625	+	0.821905i	$0.307088\pi$
$972$	−14.2321	−0.456496
$973$	24.3897	0.781898
$974$	−19.4306	−0.622595
$975$	−8.93276	−0.286077
$976$	−13.5434	−0.433513
$977$	22.0986	0.706997	0.353499	−	0.935435i	$-0.384992\pi$
$977$	22.0986	0.706997	0.353499	+	0.935435i	$0.384992\pi$
$978$	−12.1483	−0.388459
$979$	0	0
$980$	7.53879	0.240818
$981$	−8.81699	−0.281505
$982$	32.2968	1.03063
$983$	−26.1310	−0.833448	−0.416724	−	0.909033i	$-0.636822\pi$
$983$	−26.1310	−0.833448	−0.416724	+	0.909033i	$0.636822\pi$
$984$	−5.83751	−0.186093
$985$	9.38433	0.299009
$986$	4.97688	0.158496
$987$	3.55754	0.113238
$988$	6.04177	0.192214
$989$	90.1698	2.86723
$990$	0	0
$991$	−11.0168	−0.349960	−0.174980	−	0.984572i	$-0.555986\pi$
$991$	−11.0168	−0.349960	−0.174980	+	0.984572i	$0.555986\pi$
$992$	1.27855	0.0405939
$993$	4.86881	0.154507
$994$	−19.2301	−0.609941
$995$	−6.86357	−0.217590
$996$	−2.78770	−0.0883316
$997$	11.0945	0.351367	0.175684	−	0.984447i	$-0.443786\pi$
$997$	11.0945	0.351367	0.175684	+	0.984447i	$0.443786\pi$
$998$	−4.26484	−0.135001
$999$	20.5112	0.648947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bs.1.2	✓	4
11.10	odd	2		4598.2.a.bv.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.bs.1.2	✓	4	1.1	even	1	trivial
4598.2.a.bv.1.2	yes	4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.639273 q^{3} +1.00000 q^{4} -1.63927 q^{5} +0.639273 q^{6} -1.54956 q^{7} -1.00000 q^{8} -2.59133 q^{9} +1.63927 q^{10} -0.639273 q^{12} -6.04177 q^{13} +1.54956 q^{14} +1.04794 q^{15} +1.00000 q^{16} +2.54956 q^{17} +2.59133 q^{18} -1.00000 q^{19} -1.63927 q^{20} +0.990593 q^{21} -7.04177 q^{23} +0.639273 q^{24} -2.31278 q^{25} +6.04177 q^{26} +3.57439 q^{27} -1.54956 q^{28} -1.95206 q^{29} -1.04794 q^{30} -1.27855 q^{31} -1.00000 q^{32} -2.54956 q^{34} +2.54015 q^{35} -2.59133 q^{36} +5.73839 q^{37} +1.00000 q^{38} +3.86234 q^{39} +1.63927 q^{40} -9.13148 q^{41} -0.990593 q^{42} -12.8050 q^{43} +4.24790 q^{45} +7.04177 q^{46} +3.59133 q^{47} -0.639273 q^{48} -4.59886 q^{49} +2.31278 q^{50} -1.62987 q^{51} -6.04177 q^{52} -7.95206 q^{53} -3.57439 q^{54} +1.54956 q^{56} +0.639273 q^{57} +1.95206 q^{58} +5.40250 q^{59} +1.04794 q^{60} -13.5434 q^{61} +1.27855 q^{62} +4.01542 q^{63} +1.00000 q^{64} +9.90411 q^{65} +10.2212 q^{67} +2.54956 q^{68} +4.50162 q^{69} -2.54015 q^{70} -12.4100 q^{71} +2.59133 q^{72} +1.45044 q^{73} -5.73839 q^{74} +1.47850 q^{75} -1.00000 q^{76} -3.86234 q^{78} +1.13148 q^{79} -1.63927 q^{80} +5.48898 q^{81} +9.13148 q^{82} +4.36073 q^{83} +0.990593 q^{84} -4.17943 q^{85} +12.8050 q^{86} +1.24790 q^{87} -5.85294 q^{89} -4.24790 q^{90} +9.36209 q^{91} -7.04177 q^{92} +0.817341 q^{93} -3.59133 q^{94} +1.63927 q^{95} +0.639273 q^{96} -10.0000 q^{97} +4.59886 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 4 q^{8} + 16 q^{9} + 2 q^{10} + 2 q^{12} - q^{13} + 3 q^{14} + 26 q^{15} + 4 q^{16} + 7 q^{17} - 16 q^{18} - 4 q^{19} - 2 q^{20} + 9 q^{21}+ \cdots - 23 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 - 3 * q^7 - 4 * q^8 + 16 * q^9 + 2 * q^10 + 2 * q^12 - q^13 + 3 * q^14 + 26 * q^15 + 4 * q^16 + 7 * q^17 - 16 * q^18 - 4 * q^19 - 2 * q^20 + 9 * q^21 - 5 * q^23 - 2 * q^24 + 8 * q^25 + q^26 + 8 * q^27 - 3 * q^28 + 14 * q^29 - 26 * q^30 + 4 * q^31 - 4 * q^32 - 7 * q^34 + 12 * q^35 + 16 * q^36 + 12 * q^37 + 4 * q^38 - 5 * q^39 + 2 * q^40 - 12 * q^41 - 9 * q^42 - 14 * q^43 - 2 * q^45 + 5 * q^46 - 12 * q^47 + 2 * q^48 + 23 * q^49 - 8 * q^50 - 7 * q^51 - q^52 - 10 * q^53 - 8 * q^54 + 3 * q^56 - 2 * q^57 - 14 * q^58 + 3 * q^59 + 26 * q^60 - 6 * q^61 - 4 * q^62 + 18 * q^63 + 4 * q^64 - 4 * q^65 + 15 * q^67 + 7 * q^68 - 7 * q^69 - 12 * q^70 - 16 * q^71 - 16 * q^72 + 9 * q^73 - 12 * q^74 - 44 * q^75 - 4 * q^76 + 5 * q^78 - 20 * q^79 - 2 * q^80 + 52 * q^81 + 12 * q^82 + 22 * q^83 + 9 * q^84 - 14 * q^85 + 14 * q^86 - 14 * q^87 - 8 * q^89 + 2 * q^90 - 18 * q^91 - 5 * q^92 + 56 * q^93 + 12 * q^94 + 2 * q^95 - 2 * q^96 - 40 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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