LMFDB - Embedded newform 201.4.j.a.8.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [201,4,Mod(5,201)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(201, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("201.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 201 = 3 \cdot 67 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	201.j (of order $22$, degree $10$, minimal)

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:	no
Analytic conductor:	$11.8593839112$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			8.2
Root			$0.723734 + 0.690079i$ of defining polynomial
Character	$\chi$	$=$	201.8
Dual form			201.4.j.a.176.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.92699 - 3.40276i) q^{3} +(7.67594 + 2.25386i) q^{4} +(35.1868 + 5.05910i) q^{7} +(3.84250 - 26.7252i) q^{9} +O(q^{10})$	$q+(3.92699 - 3.40276i) q^{3} +(7.67594 + 2.25386i) q^{4} +(35.1868 + 5.05910i) q^{7} +(3.84250 - 26.7252i) q^{9} +(37.8127 - 17.2685i) q^{12} +(-71.0981 + 32.4694i) q^{13} +(53.8402 + 34.6010i) q^{16} +(11.5471 + 80.3116i) q^{19} +(155.393 - 99.8652i) q^{21} +(-51.9269 - 113.704i) q^{25} +(-75.8498 - 118.025i) q^{27} +(258.690 + 118.140i) q^{28} +(-270.757 - 123.651i) q^{31} +(89.7296 - 196.481i) q^{36} +351.338 q^{37} +(-168.716 + 369.437i) q^{39} +(-107.720 - 366.861i) q^{43} +(329.169 - 47.3273i) q^{48} +(883.414 + 259.394i) q^{49} +(-618.926 + 88.9882i) q^{52} +(318.626 + 276.091i) q^{57} +(-256.777 - 399.553i) q^{61} +(270.411 - 920.935i) q^{63} +(335.289 + 386.944i) q^{64} +(-114.992 + 536.227i) q^{67} +(-1012.33 + 650.584i) q^{73} +(-590.823 - 269.820i) q^{75} +(-92.3766 + 642.493i) q^{76} +(-1143.78 + 522.348i) q^{79} +(-699.470 - 205.383i) q^{81} +(1417.87 - 416.325i) q^{84} +(-2665.98 + 782.803i) q^{91} +(-1484.01 + 435.746i) q^{93} +106.819i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) $ 20 * q + 16 * q^4 + 54 * q^9	$ 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) $ 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{1}{22}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$2$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$3$	3.92699	−	3.40276i	0.755750	−	0.654861i
$3$	3.92699	−	3.40276i	0.755750	−	0.654861i
$4$	7.67594	+	2.25386i	0.959493	+	0.281733i
$5$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$5$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$6$		0			0
$7$	35.1868	+	5.05910i	1.89991	+	0.273166i	0.989938	−	0.141501i	$-0.0451929\pi$
$7$	35.1868	+	5.05910i	1.89991	+	0.273166i	0.909973	+	0.414667i	$0.136102\pi$
$8$		0			0
$9$	3.84250	−	26.7252i	0.142315	−	0.989821i
$10$		0			0
$11$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$11$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$12$	37.8127	−	17.2685i	0.909632	−	0.415415i
$13$	−71.0981	+	32.4694i	−1.51685	+	0.692723i	−0.987780	−	0.155855i	$-0.950187\pi$
$13$	−71.0981	+	32.4694i	−1.51685	+	0.692723i	−0.529071	+	0.848578i	$0.677459\pi$
$14$		0			0
$15$		0			0
$16$	53.8402	+	34.6010i	0.841254	+	0.540641i
$17$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$17$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$18$		0			0
$19$	11.5471	+	80.3116i	0.139425	+	0.969724i	0.932647	+	0.360791i	$0.117493\pi$
$19$	11.5471	+	80.3116i	0.139425	+	0.969724i	−0.793222	+	0.608933i	$0.791598\pi$
$20$		0			0
$21$	155.393	−	99.8652i	1.61474	−	1.03773i
$22$		0			0
$23$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$23$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$24$		0			0
$25$	−51.9269	−	113.704i	−0.415415	−	0.909632i
$26$		0			0
$27$	−75.8498	−	118.025i	−0.540641	−	0.841254i
$28$	258.690	+	118.140i	1.74599	+	0.797368i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−270.757	−	123.651i	−1.56869	−	0.716397i	−0.573957	−	0.818885i	$-0.694592\pi$
$31$	−270.757	−	123.651i	−1.56869	−	0.716397i	−0.994734	+	0.102488i	$0.967320\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	89.7296	−	196.481i	0.415415	−	0.909632i
$37$	351.338			1.56107			0.780534	−	0.625113i	$-0.214947\pi$
$37$	351.338			1.56107			0.780534	+	0.625113i	$0.214947\pi$
$38$		0			0
$39$	−168.716	+	369.437i	−0.692723	+	1.51685i
$40$		0			0
$41$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$41$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$42$		0			0
$43$	−107.720	−	366.861i	−0.382027	−	1.30107i	−0.896304	−	0.443441i	$-0.853758\pi$
$43$	−107.720	−	366.861i	−0.382027	−	1.30107i	0.514276	−	0.857625i	$-0.328061\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$47$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$48$	329.169	−	47.3273i	0.989821	−	0.142315i
$49$	883.414	+	259.394i	2.57555	+	0.756250i
$50$		0			0
$51$		0			0
$52$	−618.926	+	88.9882i	−1.65057	+	0.237316i
$53$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$53$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	318.626	+	276.091i	0.740405	+	0.641564i
$58$		0			0
$59$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$59$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$60$		0			0
$61$	−256.777	−	399.553i	−0.538966	−	0.838648i	0.459814	−	0.888015i	$-0.347916\pi$
$61$	−256.777	−	399.553i	−0.538966	−	0.838648i	−0.998781	+	0.0493670i	$0.984280\pi$
$62$		0			0
$63$	270.411	−	920.935i	0.540771	−	1.84170i
$64$	335.289	+	386.944i	0.654861	+	0.755750i
$65$		0			0
$66$		0			0
$67$	−114.992	+	536.227i	−0.209680	+	0.977770i
$67$	−114.992	+	536.227i	−0.209680	+	0.977770i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$71$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$72$		0			0
$73$	−1012.33	+	650.584i	−1.62307	+	1.04308i	−0.669105	+	0.743168i	$0.733322\pi$
$73$	−1012.33	+	650.584i	−1.62307	+	1.04308i	−0.953966	+	0.299916i	$0.903041\pi$
$74$		0			0
$75$	−590.823	−	269.820i	−0.909632	−	0.415415i
$76$	−92.3766	+	642.493i	−0.139425	+	0.969724i
$77$		0			0
$78$		0			0
$79$	−1143.78	+	522.348i	−1.62893	+	0.743908i	−0.999452	−	0.0331090i	$-0.989459\pi$
$79$	−1143.78	+	522.348i	−1.62893	+	0.743908i	−0.629480	+	0.777017i	$0.716732\pi$
$80$		0			0
$81$	−699.470	−	205.383i	−0.959493	−	0.281733i
$82$		0			0
$83$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$83$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$84$	1417.87	−	416.325i	1.84170	−	0.540771i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$89$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$90$		0			0
$91$	−2665.98	+	782.803i	−3.07111	+	0.901759i
$92$		0			0
$93$	−1484.01	+	435.746i	−1.65468	+	0.485857i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			106.819i			0.111813i	0.998436	+	0.0559065i	$0.0178049\pi$
$97$			106.819i			0.111813i	−0.998436	+	0.0559065i	$0.982195\pi$
$98$		0			0
$99$		0			0
$100$	−142.315	−	989.821i	−0.142315	−	0.989821i
$101$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$101$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$102$		0			0
$103$	530.642	−	1161.94i	0.507629	−	1.11155i	−0.466285	−	0.884634i	$-0.654408\pi$
$103$	530.642	−	1161.94i	0.507629	−	1.11155i	0.973914	−	0.226917i	$-0.0728647\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$107$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$108$	−316.208	−	1076.91i	−0.281733	−	0.959493i
$109$	1352.34	−	617.591i	1.18835	−	0.542702i	0.279631	−	0.960108i	$-0.409788\pi$
$109$	1352.34	−	617.591i	1.18835	−	0.542702i	0.908720	+	0.417406i	$0.137061\pi$
$110$		0			0
$111$	1379.70	−	1195.52i	1.17978	−	1.02228i
$112$	1719.42	+	1489.88i	1.45062	+	1.25697i
$113$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$113$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	594.556	+	2024.87i	0.469801	+	1.60000i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−552.917	−	1210.72i	−0.415415	−	0.909632i
$122$		0			0
$123$		0			0
$124$	−1799.63	−	1559.38i	−1.30332	−	1.12933i
$125$		0			0
$126$		0			0
$127$	−344.809	+	2398.20i	−0.240920	+	1.67563i	0.406616	+	0.913599i	$0.366709\pi$
$127$	−344.809	+	2398.20i	−0.240920	+	1.67563i	−0.647535	+	0.762035i	$0.724200\pi$
$128$		0			0
$129$	−1671.36	−	1074.12i	−1.14073	−	0.733105i
$130$		0			0
$131$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$131$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$132$		0			0
$133$			2884.33i			1.88048i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$137$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$138$		0			0
$139$	−1770.00	+	2754.17i	−1.08007	+	1.68061i	−0.500148	+	0.865940i	$0.666721\pi$
$139$	−1770.00	+	2754.17i	−1.08007	+	1.68061i	−0.579918	+	0.814675i	$0.696915\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1131.60	−	1305.94i	0.654861	−	0.755750i
$145$		0			0
$146$		0			0
$147$	4351.81	−	1987.40i	2.44171	−	1.11509i
$148$	2696.85	+	791.866i	1.49783	+	0.439804i
$149$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$149$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$150$		0			0
$151$	287.193	−	84.3275i	0.154778	−	0.0454469i	−0.203426	−	0.979090i	$-0.565208\pi$
$151$	287.193	−	84.3275i	0.154778	−	0.0454469i	0.358204	+	0.933644i	$0.383389\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2127.71	+	2455.51i	−1.09201	+	1.26025i
$157$	−884.233	−	1020.46i	−0.449487	−	0.518736i	0.485106	−	0.874456i	$-0.338781\pi$
$157$	−884.233	−	1020.46i	−0.449487	−	0.518736i	−0.934593	+	0.355720i	$0.884236\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2585.94			1.24262			0.621309	−	0.783566i	$-0.286601\pi$
$163$	2585.94			1.24262			0.621309	+	0.783566i	$0.286601\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$167$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$168$		0			0
$169$	2561.95	−	2956.64i	1.16611	−	1.34576i
$170$		0			0
$171$	2190.71			0.979696
$172$		−	3058.79i		−	1.35599i
$173$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$173$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$174$		0			0
$175$	−1251.90	−	4263.59i	−0.540771	−	1.84170i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$179$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$180$		0			0
$181$	3180.21	+	3670.16i	1.30599	+	1.50719i	0.710031	+	0.704171i	$0.248681\pi$
$181$	3180.21	+	3670.16i	1.30599	+	1.50719i	0.595955	+	0.803018i	$0.296774\pi$
$182$		0			0
$183$	−2367.94	−	695.291i	−0.956521	−	0.280860i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2071.82	−	4536.65i	−0.797368	−	1.74599i
$190$		0			0
$191$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$191$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$192$	2633.35	+	378.619i	0.989821	+	0.142315i
$193$	−2085.69	+	4567.03i	−0.777884	+	1.70333i	−0.0694118	+	0.997588i	$0.522112\pi$
$193$	−2085.69	+	4567.03i	−0.777884	+	1.70333i	−0.708472	+	0.705739i	$0.750615\pi$
$194$		0			0
$195$		0			0
$196$	6196.40	+	3982.18i	2.25816	+	1.45123i
$197$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$197$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$198$		0			0
$199$	796.228	−	5537.89i	0.283634	−	1.97272i	0.0590873	−	0.998253i	$-0.481181\pi$
$199$	796.228	−	5537.89i	0.283634	−	1.97272i	0.224547	−	0.974463i	$-0.427910\pi$
$200$		0			0
$201$	1373.08	+	2497.05i	0.481838	+	0.876260i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4951.41	−	711.906i	−1.65057	−	0.237316i
$209$		0			0
$210$		0			0
$211$	2936.72	−	3389.16i	0.958162	−	1.10578i	−0.0361575	−	0.999346i	$-0.511512\pi$
$211$	2936.72	−	3389.16i	0.958162	−	1.10578i	0.994320	−	0.106432i	$-0.0339427\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−8901.53	−	5720.67i	−2.78468	−	1.78960i
$218$		0			0
$219$	−1761.63	+	5999.55i	−0.543560	+	1.85120i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1504.50	−	1736.28i	0.451787	−	0.521391i	−0.483469	−	0.875362i	$-0.660623\pi$
$223$	1504.50	−	1736.28i	0.451787	−	0.521391i	0.935256	+	0.353971i	$0.115169\pi$
$224$		0			0
$225$	−3238.29	+	950.847i	−0.959493	+	0.281733i
$226$		0			0
$227$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$227$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$228$	1823.49	+	2837.40i	0.529663	+	0.824172i
$229$	5698.07	+	2602.22i	1.64428	+	0.750915i	0.999905	−	0.0137984i	$-0.00439230\pi$
$229$	5698.07	+	2602.22i	1.64428	+	0.750915i	0.644370	+	0.764714i	$0.277120\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$233$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2714.20	+	5943.27i	−0.743908	+	1.62893i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1624.08	+	1043.73i	−0.434093	+	0.278975i	−0.739383	−	0.673285i	$-0.764883\pi$
$241$	−1624.08	+	1043.73i	−0.434093	+	0.278975i	0.305290	+	0.952259i	$0.401246\pi$
$242$		0			0
$243$	−3445.68	+	1573.59i	−0.909632	+	0.415415i
$244$	−1070.47	−	3645.69i	−0.280860	−	0.956521i
$245$		0			0
$246$		0			0
$247$	−3428.65	−	5335.08i	−0.883237	−	1.37434i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$251$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$252$	4151.32	−	6459.58i	1.03773	−	1.61474i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1701.54	+	3725.85i	0.415415	+	0.909632i
$257$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$257$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$258$		0			0
$259$	12362.5	+	1777.45i	2.96589	+	0.426431i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$263$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−2091.26	+	3856.87i	−0.476656	+	0.879090i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−348.244	+	301.755i	−0.0780601	+	0.0676395i	−0.693017	−	0.720921i	$-0.743719\pi$
$271$	−348.244	+	301.755i	−0.0780601	+	0.0676395i	0.614957	+	0.788561i	$0.289173\pi$
$272$		0			0
$273$	−7805.60	+	12145.8i	−1.73046	+	2.69265i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	882.805	−	6140.04i	0.191490	−	1.33184i	−0.636579	−	0.771212i	$-0.719651\pi$
$277$	882.805	−	6140.04i	0.191490	−	1.33184i	0.828068	−	0.560627i	$-0.189440\pi$
$278$		0			0
$279$	−4344.97	+	6760.91i	−0.932354	+	1.45077i
$280$		0			0
$281$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$281$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$282$		0			0
$283$	−77.9470	−	542.133i	−0.0163727	−	0.113874i	0.979996	−	0.199016i	$-0.0637747\pi$
$283$	−77.9470	−	542.133i	−0.0163727	−	0.113874i	−0.996369	+	0.0851419i	$0.972866\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4133.08	−	2656.17i	0.841254	−	0.540641i
$290$		0			0
$291$	363.480	+	419.478i	0.0732219	+	0.0845026i
$292$	−9236.91	+	2712.20i	−1.85120	+	0.543560i
$293$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$293$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−3926.99	−	3402.76i	−0.755750	−	0.654861i
$301$	−1934.34	−	13453.7i	−0.370411	−	2.57627i
$302$		0			0
$303$		0			0
$304$	−2157.17	+	4723.54i	−0.406980	+	0.891163i
$305$		0			0
$306$		0			0
$307$	−4054.88	+	8878.96i	−0.753826	+	1.65065i	0.00554079	+	0.999985i	$0.498236\pi$
$307$	−4054.88	+	8878.96i	−0.753826	+	1.65065i	−0.759366	+	0.650663i	$0.774491\pi$
$308$		0			0
$309$	−1869.99	−	6368.59i	−0.344271	−	1.17248i
$310$		0			0
$311$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$311$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$312$		0			0
$313$	−7997.74	−	6930.08i	−1.44428	−	1.25147i	−0.915260	−	0.402863i	$-0.868015\pi$
$313$	−7997.74	−	6930.08i	−1.44428	−	1.25147i	−0.529018	−	0.848611i	$-0.677440\pi$
$314$		0			0
$315$		0			0
$316$	−9956.91	+	1431.59i	−1.77253	+	0.254852i
$317$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$317$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−4906.19	−	3153.02i	−0.841254	−	0.540641i
$325$	7383.80	+	6398.10i	1.26025	+	1.09201i
$326$		0			0
$327$	3209.10	−	7026.94i	0.542702	−	1.18835i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2425.74	+	8261.30i	−0.402811	+	1.37185i	0.469515	+	0.882925i	$0.344429\pi$
$331$	−2425.74	+	8261.30i	−0.402811	+	1.37185i	−0.872326	+	0.488925i	$0.837389\pi$
$332$		0			0
$333$	1350.02	−	9389.56i	0.222163	−	1.54518i
$334$		0			0
$335$		0			0
$336$	11821.8			1.91945
$337$	−227.470	−	32.7052i	−0.0367687	−	0.00528654i	0.123906	−	0.992294i	$-0.460458\pi$
$337$	−227.470	−	32.7052i	−0.0367687	−	0.00528654i	−0.160675	+	0.987007i	$0.551367\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18680.9	+	8531.29i	2.94074	+	1.34299i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$347$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$348$		0			0
$349$	−3406.77	−	1000.32i	−0.522521	−	0.153426i	0.00982661	−	0.999952i	$-0.496872\pi$
$349$	−3406.77	−	1000.32i	−0.522521	−	0.153426i	−0.532348	+	0.846526i	$0.678690\pi$
$350$		0			0
$351$	9224.97	+	5928.53i	1.40283	+	0.901542i
$352$		0			0
$353$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$353$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$359$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	264.538	−	77.6755i	0.0385681	−	0.0113246i
$362$		0			0
$363$	−6291.09	−	2873.04i	−0.909632	−	0.415415i
$364$	−22228.3			−3.20076
$365$		0			0
$366$		0			0
$367$	6730.17	+	5831.73i	0.957254	+	0.829465i	0.985420	−	0.170138i	$-0.0544213\pi$
$367$	6730.17	+	5831.73i	0.957254	+	0.829465i	−0.0281661	+	0.999603i	$0.508967\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−12373.3			−1.72453
$373$		−	13280.3i		−	1.84350i	−0.387779	−	0.921752i	$-0.626758\pi$
$373$		−	13280.3i		−	1.84350i	0.387779	−	0.921752i	$-0.373242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	7712.31	−	6682.75i	1.04526	−	0.905726i	0.0496011	−	0.998769i	$-0.484205\pi$
$379$	7712.31	−	6682.75i	1.04526	−	0.905726i	0.995662	+	0.0930434i	$0.0296595\pi$
$380$		0			0
$381$	6806.42	+	10591.0i	0.915232	+	1.42413i
$382$		0			0
$383$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$383$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10218.3	+	1469.18i	−1.34219	+	0.192978i
$388$	−240.756	+	819.939i	−0.0315013	+	0.107284i
$389$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$389$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2778.39	−	1785.56i	−0.351243	−	0.225730i	0.353107	−	0.935583i	$-0.385125\pi$
$397$	−2778.39	−	1785.56i	−0.351243	−	0.225730i	−0.704350	+	0.709853i	$0.748761\pi$
$398$		0			0
$399$	9814.68	+	11326.7i	1.23145	+	1.42117i
$400$	1138.52	−	7918.57i	0.142315	−	0.989821i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	23265.2			2.87574
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	7529.19	+	1082.53i	0.910255	+	0.130875i	0.581506	−	0.813542i	$-0.302464\pi$
$409$	7529.19	+	1082.53i	0.910255	+	0.130875i	0.328749	+	0.944417i	$0.393373\pi$
$410$		0			0
$411$		0			0
$412$	6692.04	−	7723.03i	0.800226	−	0.923511i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2421.00	+	16838.5i	0.284310	+	1.97742i
$418$		0			0
$419$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$419$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$420$		0			0
$421$	1295.71	+	9011.88i	0.149998	+	1.04326i	0.916218	+	0.400680i	$0.131226\pi$
$421$	1295.71	+	9011.88i	0.149998	+	1.04326i	−0.766220	+	0.642579i	$0.777865\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7013.80	−	15358.1i	−0.794898	−	1.74058i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	8978.95i		−	1.00000i
$433$	8881.76	+	4056.16i	0.985751	+	0.450177i	0.842024	−	0.539440i	$-0.181364\pi$
$433$	8881.76	+	4056.16i	0.985751	+	0.450177i	0.143727	+	0.989617i	$0.454091\pi$
$434$		0			0
$435$		0			0
$436$	11772.4	−	1692.62i	1.29311	−	0.185921i
$437$		0			0
$438$		0			0
$439$	11289.1			1.22734			0.613669	−	0.789563i	$-0.289693\pi$
$439$	11289.1			1.22734			0.613669	+	0.789563i	$0.289693\pi$
$440$		0			0
$441$	10326.9	−	22612.7i	1.11509	−	2.44171i
$442$		0			0
$443$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$443$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$444$	13285.0	−	6067.07i	1.42000	−	0.648491i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	9840.16	+	15311.6i	1.03773	+	1.61474i
$449$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$449$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	840.859	−	1308.40i	0.0872119	−	0.135704i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6853.51	−	15007.1i	−0.701518	−	1.53611i	−0.838121	−	0.545484i	$-0.816346\pi$
$457$	−6853.51	−	15007.1i	−0.701518	−	1.53611i	0.136604	−	0.990626i	$-0.456381\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$461$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$462$		0			0
$463$	−10769.7	−	16758.0i	−1.08102	−	1.68210i	−0.559036	−	0.829143i	$-0.688829\pi$
$463$	−10769.7	−	16758.0i	−1.08102	−	1.68210i	−0.521983	−	0.852956i	$-0.674808\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$467$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$468$			16882.9i			1.66754i
$469$	−6759.05	+	18286.4i	−0.665467	+	1.80040i
$470$		0			0
$471$	−6944.75	−	998.504i	−0.679399	−	0.0976829i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	8532.15	−	5483.28i	0.824172	−	0.529663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$479$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$480$		0			0
$481$	−24979.4	+	11407.7i	−2.36791	+	1.08139i
$482$		0			0
$483$		0			0
$484$	−1515.37	−	10539.6i	−0.142315	−	0.989821i
$485$		0			0
$486$		0			0
$487$	−302.032	+	1028.63i	−0.0281034	+	0.0957115i	−0.972351	−	0.233526i	$-0.924974\pi$
$487$	−302.032	+	1028.63i	−0.0281034	+	0.0957115i	0.944247	+	0.329238i	$0.106792\pi$
$488$		0			0
$489$	10155.0	−	8799.33i	0.939108	−	0.813741i
$490$		0			0
$491$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$491$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−10299.2	−	16025.9i	−0.932354	−	1.45077i
$497$		0			0
$498$		0			0
$499$			20567.7i			1.84516i	0.385806	+	0.922580i	$0.373923\pi$
$499$			20567.7i			1.84516i	−0.385806	+	0.922580i	$0.626077\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$503$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	20328.4i		−	1.78070i
$508$	−8051.93	+	17631.3i	−0.703242	+	1.53988i
$509$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$509$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$510$		0			0
$511$	−38912.0	+	17770.5i	−3.36863	+	1.53840i
$512$		0			0
$513$	8602.91	−	7454.46i	0.740405	−	0.641564i
$514$		0			0
$515$		0			0
$516$	−10408.3	−	12011.8i	−0.887987	−	1.02479i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$521$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$522$		0			0
$523$	7787.55	+	17052.4i	0.651101	+	1.42571i	0.890587	+	0.454812i	$0.150294\pi$
$523$	7787.55	+	17052.4i	0.651101	+	1.42571i	−0.239487	+	0.970900i	$0.576979\pi$
$524$		0			0
$525$	−19424.2	−	12483.2i	−1.61474	−	1.03773i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1731.54	+	12043.2i	−0.142315	+	0.989821i
$530$		0			0
$531$		0			0
$532$	−6500.88	+	22140.0i	−0.529791	+	1.80430i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9114.69	−	14182.7i	0.724346	−	1.12710i	−0.262421	−	0.964954i	$-0.584521\pi$
$541$	9114.69	−	14182.7i	0.724346	−	1.12710i	0.986766	−	0.162150i	$-0.0518429\pi$
$542$		0			0
$543$	24977.3	+	3591.20i	1.97400	+	0.283818i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13495.3	−	20999.1i	1.05487	−	1.64142i	0.342653	−	0.939462i	$-0.388674\pi$
$547$	13495.3	−	20999.1i	1.05487	−	1.64142i	0.712221	−	0.701955i	$-0.247689\pi$
$548$		0			0
$549$	−11664.8	+	5327.13i	−0.906815	+	0.414128i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−42888.7	+	12593.3i	−3.29804	+	0.968390i
$554$		0			0
$555$		0			0
$556$	−19793.9	+	17151.5i	−1.50980	+	1.30825i
$557$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$557$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$558$		0			0
$559$	19570.5	+	22585.5i	1.48076	+	1.70888i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$563$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−23573.1	−	10765.5i	−1.74599	−	0.797368i
$568$		0			0
$569$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$569$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$570$		0			0
$571$	17017.7	−	19639.5i	1.24723	−	1.43938i	0.392962	−	0.919555i	$-0.371450\pi$
$571$	17017.7	−	19639.5i	1.24723	−	1.43938i	0.854270	−	0.519829i	$-0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	11629.5	−	7473.82i	0.841254	−	0.540641i
$577$	−2356.69	−	8026.15i	−0.170035	−	0.579087i	−0.999781	−	0.0209447i	$-0.993333\pi$
$577$	−2356.69	−	8026.15i	−0.170035	−	0.579087i	0.829745	−	0.558142i	$-0.188486\pi$
$578$		0			0
$579$	7350.00	+	25031.8i	0.527557	+	1.79669i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$587$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$588$	37883.6	−	5446.84i	2.65696	−	0.382013i
$589$	6804.14	−	23172.8i	0.475992	−	1.62108i
$590$		0			0
$591$		0			0
$592$	18916.1	+	12156.6i	1.31325	+	0.843978i
$593$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$593$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−15717.3	−	24456.6i	−1.07750	−	1.67662i
$598$		0			0
$599$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$599$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$600$		0			0
$601$	−3748.00	+	26067.9i	−0.254383	+	1.76927i	0.316843	+	0.948478i	$0.397377\pi$
$601$	−3748.00	+	26067.9i	−0.254383	+	1.76927i	−0.571226	+	0.820793i	$0.693532\pi$
$602$		0			0
$603$	13888.9	+	5133.64i	0.937977	+	0.346697i
$604$	2394.54			0.161312
$605$		0			0
$606$		0			0
$607$	4972.50	+	1460.06i	0.332500	+	0.0976309i	0.443721	−	0.896165i	$-0.353658\pi$
$607$	4972.50	+	1460.06i	0.332500	+	0.0976309i	−0.111221	+	0.993796i	$0.535476\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	6336.63	−	7312.87i	0.417511	−	0.481833i	−0.507566	−	0.861613i	$-0.669455\pi$
$613$	6336.63	−	7312.87i	0.417511	−	0.481833i	0.925077	+	0.379779i	$0.124000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$617$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$618$		0			0
$619$	−11845.7	−	7612.75i	−0.769172	−	0.494317i	0.0962518	−	0.995357i	$-0.469315\pi$
$619$	−11845.7	−	7612.75i	−0.769172	−	0.494317i	−0.865424	+	0.501040i	$0.832951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−21866.6	+	14052.8i	−1.40283	+	0.901542i
$625$	−10232.2	+	11808.6i	−0.654861	+	0.755750i
$626$		0			0
$627$		0			0
$628$	−4487.35	−	9825.93i	−0.285135	−	0.624358i
$629$		0			0
$630$		0			0
$631$	17549.4	+	8014.56i	1.10718	+	0.505633i	0.883217	−	0.468964i	$-0.155372\pi$
$631$	17549.4	+	8014.56i	1.10718	+	0.505633i	0.223965	+	0.974597i	$0.428100\pi$
$632$		0			0
$633$		−	23302.1i		−	1.46315i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−71231.4	+	10241.5i	−4.43060	+	0.637024i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	3550.99	−	2282.08i	0.217787	−	0.139963i	−0.427199	−	0.904158i	$-0.640500\pi$
$643$	3550.99	−	2282.08i	0.217787	−	0.139963i	0.644986	+	0.764194i	$0.276863\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$647$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−54422.3	+	7824.74i	−3.27646	+	0.471084i
$652$	19849.6	+	5828.35i	1.19228	+	0.350086i
$653$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$653$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	13497.1	+	29554.5i	0.801480	+	1.75500i
$658$		0			0
$659$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$659$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$660$		0			0
$661$	−31472.0	−	4524.99i	−1.85192	−	0.266266i	−0.875670	−	0.482910i	$-0.839580\pi$
$661$	−31472.0	−	4524.99i	−1.85192	−	0.266266i	−0.976251	+	0.216644i	$0.930489\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	11937.8i		−	0.689899i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−25921.7	+	22461.3i	−1.48471	+	1.28651i	−0.619599	+	0.784919i	$0.712705\pi$
$673$	−25921.7	+	22461.3i	−1.48471	+	1.28651i	−0.865108	+	0.501586i	$0.832750\pi$
$674$		0			0
$675$	−9481.23	+	14753.1i	−0.540641	+	0.841254i
$676$	26329.2	−	16920.8i	1.49802	−	0.962720i
$677$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$677$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$678$		0			0
$679$	−540.410	+	3758.63i	−0.0305435	+	0.212435i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$683$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$684$	16815.8	+	4937.56i	0.940011	+	0.276012i
$685$		0			0
$686$		0			0
$687$	31231.0	−	9170.24i	1.73441	−	0.509267i
$688$	6894.09	−	23479.1i	0.382027	−	1.30107i
$689$		0			0
$690$		0			0
$691$	30547.8	−	19631.9i	1.68176	−	1.08080i	0.824724	−	0.565535i	$-0.191330\pi$
$691$	30547.8	−	19631.9i	1.68176	−	1.08080i	0.857031	−	0.515264i	$-0.172306\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		−	35548.7i		−	1.91945i
$701$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$701$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$702$		0			0
$703$	4056.92	+	28216.5i	0.217652	+	1.51381i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	13258.9	−	29033.0i	0.702326	−	1.53788i	−0.134802	−	0.990872i	$-0.543040\pi$
$709$	13258.9	−	29033.0i	0.702326	−	1.53788i	0.837128	−	0.547007i	$-0.184233\pi$
$710$		0			0
$711$	9564.86	+	32574.9i	0.504515	+	1.71822i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$719$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$720$		0			0
$721$	24550.0	−	38200.6i	1.26809	−	1.97318i
$722$		0			0
$723$	−2826.18	+	9625.09i	−0.145376	+	0.495105i
$724$	16139.1	+	35339.7i	0.828460	+	1.81408i
$725$		0			0
$726$		0			0
$727$	−29627.8	−	25672.6i	−1.51146	−	1.30969i	−0.760341	−	0.649524i	$-0.774968\pi$
$727$	−29627.8	−	25672.6i	−1.51146	−	1.30969i	−0.751121	−	0.660165i	$-0.770486\pi$
$728$		0			0
$729$	−8176.61	+	17904.3i	−0.415415	+	0.909632i
$730$		0			0
$731$		0			0
$732$	−16609.1	−	10674.0i	−0.838648	−	0.538966i
$733$	2724.38	−	9278.39i	0.137282	−	0.467538i	−0.861939	−	0.507012i	$-0.830750\pi$
$733$	2724.38	−	9278.39i	0.137282	−	0.467538i	0.999221	+	0.0394740i	$0.0125682\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	32585.4	+	4685.07i	1.62202	+	0.233211i	0.892598	−	0.450854i	$-0.148881\pi$
$739$	32585.4	+	4685.07i	1.62202	+	0.233211i	0.729421	+	0.684065i	$0.239790\pi$
$740$		0			0
$741$	−31618.2	−	9283.95i	−1.56751	−	0.460262i
$742$		0			0
$743$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$743$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−30734.7	−	9024.51i	−1.49337	−	0.438494i	−0.569757	−	0.821813i	$-0.692963\pi$
$751$	−30734.7	−	9024.51i	−1.49337	−	0.438494i	−0.923616	+	0.383319i	$0.874781\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−5678.18	−	39492.6i	−0.273166	−	1.89991i
$757$	−14446.7	+	12518.1i	−0.693624	+	0.601028i	−0.928648	−	0.370961i	$-0.879028\pi$
$757$	−14446.7	+	12518.1i	−0.693624	+	0.601028i	0.235025	+	0.971989i	$0.424483\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$761$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$	50708.9	−	14889.5i	2.40601	−	0.706468i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	19360.1	+	8841.46i	0.909632	+	0.415415i
$769$	10145.2	+	8790.90i	0.475744	+	0.412234i	0.859449	−	0.511221i	$-0.170807\pi$
$769$	10145.2	+	8790.90i	0.475744	+	0.412234i	−0.383705	+	0.923456i	$0.625352\pi$
$770$		0			0
$771$		0			0
$772$	−26303.1	+	30355.4i	−1.22626	+	1.41518i
$773$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$773$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$774$		0			0
$775$			37207.0i			1.72453i
$776$		0			0
$777$	54595.5	−	35086.4i	2.52072	−	1.61997i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	38587.9	+	44532.8i	1.75783	+	2.02865i
$785$		0			0
$786$		0			0
$787$	−12074.6	−	41122.3i	−0.546903	−	1.86258i	−0.504375	−	0.863485i	$-0.668277\pi$
$787$	−12074.6	−	41122.3i	−0.546903	−	1.86258i	−0.0425282	−	0.999095i	$-0.513541\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	31229.6	+	20070.1i	1.39848	+	0.898750i
$794$		0			0
$795$		0			0
$796$	18593.4	−	40713.9i	0.827923	−	1.81290i
$797$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$797$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	4911.66	+	22261.9i	0.215449	+	0.976515i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$809$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$810$		0			0
$811$	45381.0	+	6524.81i	1.96491	+	0.282512i	0.999785	+	0.0207287i	$0.00659863\pi$
$811$	45381.0	+	6524.81i	1.96491	+	0.282512i	0.965127	+	0.261783i	$0.0843105\pi$
$812$		0			0
$813$	−340.751	+	2369.98i	−0.0146995	+	0.102237i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	28219.4	−	12887.4i	1.20841	−	0.551862i
$818$		0			0
$819$	10676.5	+	74256.8i	0.455516	+	3.16818i
$820$		0			0
$821$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$821$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$822$		0			0
$823$	−6177.67	−	42966.6i	−0.261653	−	1.81983i	−0.520439	−	0.853899i	$-0.674232\pi$
$823$	−6177.67	−	42966.6i	−0.261653	−	1.81983i	0.258786	−	0.965935i	$-0.416677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$827$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$828$		0			0
$829$	1584.49	+	3469.54i	0.0663830	+	0.145359i	0.939915	−	0.341408i	$-0.110904\pi$
$829$	1584.49	+	3469.54i	0.0663830	+	0.145359i	−0.873532	+	0.486766i	$0.838176\pi$
$830$		0			0
$831$	−17426.3	−	27115.9i	−0.727451	−	1.13194i
$832$	−36402.2	−	16624.3i	−1.51685	−	0.692723i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	5943.06	+	41334.9i	0.245427	+	1.70698i
$838$		0			0
$839$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$839$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	30180.8	−	19396.0i	1.23088	−	0.791041i
$845$		0			0
$846$		0			0
$847$	−13330.3	−	45398.7i	−0.540771	−	1.84170i
$848$		0			0
$849$	−2150.84	−	1863.72i	−0.0869456	−	0.0753388i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	3426.46	+	1006.10i	0.137538	+	0.0403847i	0.349777	−	0.936833i	$-0.386257\pi$
$853$	3426.46	+	1006.10i	0.137538	+	0.0403847i	−0.212239	+	0.977218i	$0.568076\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$857$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$858$		0			0
$859$	20305.0	+	44461.9i	0.806518	+	1.76603i	0.621699	+	0.783256i	$0.286443\pi$
$859$	20305.0	+	44461.9i	0.806518	+	1.76603i	0.184819	+	0.982772i	$0.440830\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$863$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	7192.27	−	24494.6i	0.281733	−	0.959493i
$868$	−55434.1	−	63974.3i	−2.16769	−	2.50165i
$869$		0			0
$870$		0			0
$871$	−9235.25	−	41858.5i	−0.359270	−	1.62838i
$872$		0			0
$873$	2854.76	+	410.453i	0.110675	+	0.0159126i
$874$		0			0
$875$		0			0
$876$	−27044.3	+	42081.7i	−1.04308	+	1.62307i
$877$	−5262.86	+	3382.23i	−0.202639	+	0.130228i	−0.638025	−	0.770016i	$-0.720248\pi$
$877$	−5262.86	+	3382.23i	−0.202639	+	0.130228i	0.435386	+	0.900244i	$0.356612\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$881$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$882$		0			0
$883$	−42946.7	+	19613.1i	−1.63677	+	0.747490i	−0.999728	−	0.0233351i	$-0.992572\pi$
$883$	−42946.7	+	19613.1i	−1.63677	+	0.747490i	−0.637047	+	0.770825i	$0.719844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$887$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$888$		0			0
$889$	−24265.5	+	82640.6i	−0.915453	+	3.11775i
$890$		0			0
$891$		0			0
$892$	15461.8	−	9936.68i	0.580380	−	0.372987i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−27000.0			−1.00000
$901$		0			0
$902$		0			0
$903$	−53375.7	−	46250.3i	−1.96703	−	1.70444i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	8661.17	−	18965.3i	0.317078	−	0.694303i	−0.682244	−	0.731124i	$-0.738996\pi$
$907$	8661.17	−	18965.3i	0.317078	−	0.694303i	0.999322	+	0.0368211i	$0.0117232\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$911$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$912$	7601.87	+	25889.6i	0.276012	+	0.940011i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	37873.0	+	32817.2i	1.36611	+	1.18374i
$917$		0			0
$918$		0			0
$919$	−47814.7	+	6874.72i	−1.71628	+	0.246764i	−0.929066	−	0.369914i	$-0.879387\pi$
$919$	−47814.7	+	6874.72i	−1.71628	+	0.246764i	−0.787216	+	0.616678i	$0.788478\pi$
$920$		0			0
$921$	14289.4	+	48665.4i	0.511241	+	1.74113i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−18243.9	−	39948.5i	−0.648491	−	1.42000i
$926$		0			0
$927$	−29014.2	−	18646.3i	−1.02799	−	0.660652i
$928$		0			0
$929$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$929$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$930$		0			0
$931$	−10631.5	+	73943.6i	−0.374257	+	2.60301i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	29519.1i		−	1.02919i	−0.857435	−	0.514593i	$-0.827943\pi$
$937$		−	29519.1i		−	1.02919i	0.857435	−	0.514593i	$-0.172057\pi$
$938$		0			0
$939$	−54988.4			−1.91105
$940$		0			0
$941$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$941$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$947$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$948$	−34229.3	+	39502.8i	−1.17270	+	1.35337i
$949$	50850.6	−	79125.0i	1.73939	−	2.70654i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$953$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	38511.0	+	44444.1i	1.29271	+	1.49186i
$962$		0			0
$963$		0			0
$964$	−14818.8	+	4351.19i	−0.495105	+	0.145376i
$965$		0			0
$966$		0			0
$967$	−57991.0			−1.92850			−0.964252	−	0.264986i	$-0.914633\pi$
$967$	−57991.0			−1.92850			−0.964252	+	0.264986i	$0.914633\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$971$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$972$	−29995.5	+	4312.70i	−0.989821	+	0.142315i
$973$	−76214.2	+	87955.9i	−2.51112	+	2.89798i
$974$		0			0
$975$	50767.3			1.66754
$976$		−	30396.8i		−	0.996903i
$977$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$977$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−11308.9	−	38514.5i	−0.368058	−	1.25349i
$982$		0			0
$983$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$983$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−14293.6	−	48679.4i	−0.460262	−	1.56751i
$989$		0			0
$990$		0			0
$991$	3136.37	−	10681.5i	0.100535	−	0.342390i	−0.893829	−	0.448407i	$-0.851992\pi$
$991$	3136.37	−	10681.5i	0.100535	−	0.342390i	0.994364	+	0.106017i	$0.0338097\pi$
$992$		0			0
$993$	18585.4	+	40696.2i	0.593946	+	1.30056i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−9695.00	+	21229.1i	−0.307967	+	0.674355i	−0.998816	−	0.0486447i	$-0.984510\pi$
$997$	−9695.00	+	21229.1i	−0.307967	+	0.674355i	0.690849	+	0.722999i	$0.257237\pi$
$998$		0			0
$999$	−26648.9	−	41466.5i	−0.843978	−	1.31325i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.a.8.2	✓	20
3.2	odd	2	CM	201.4.j.a.8.2	✓	20
67.42	odd	22	inner	201.4.j.a.176.2	yes	20
201.176	even	22	inner	201.4.j.a.176.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.a.8.2	✓	20	1.1	even	1	trivial
201.4.j.a.8.2	✓	20	3.2	odd	2	CM
201.4.j.a.176.2	yes	20	67.42	odd	22	inner
201.4.j.a.176.2	yes	20	201.176	even	22	inner

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:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(3.92699 - 3.40276i) q^{3} +(7.67594 + 2.25386i) q^{4} +(35.1868 + 5.05910i) q^{7} +(3.84250 - 26.7252i) q^{9} +O(q^{10})\)	\(q+(3.92699 - 3.40276i) q^{3} +(7.67594 + 2.25386i) q^{4} +(35.1868 + 5.05910i) q^{7} +(3.84250 - 26.7252i) q^{9} +(37.8127 - 17.2685i) q^{12} +(-71.0981 + 32.4694i) q^{13} +(53.8402 + 34.6010i) q^{16} +(11.5471 + 80.3116i) q^{19} +(155.393 - 99.8652i) q^{21} +(-51.9269 - 113.704i) q^{25} +(-75.8498 - 118.025i) q^{27} +(258.690 + 118.140i) q^{28} +(-270.757 - 123.651i) q^{31} +(89.7296 - 196.481i) q^{36} +351.338 q^{37} +(-168.716 + 369.437i) q^{39} +(-107.720 - 366.861i) q^{43} +(329.169 - 47.3273i) q^{48} +(883.414 + 259.394i) q^{49} +(-618.926 + 88.9882i) q^{52} +(318.626 + 276.091i) q^{57} +(-256.777 - 399.553i) q^{61} +(270.411 - 920.935i) q^{63} +(335.289 + 386.944i) q^{64} +(-114.992 + 536.227i) q^{67} +(-1012.33 + 650.584i) q^{73} +(-590.823 - 269.820i) q^{75} +(-92.3766 + 642.493i) q^{76} +(-1143.78 + 522.348i) q^{79} +(-699.470 - 205.383i) q^{81} +(1417.87 - 416.325i) q^{84} +(-2665.98 + 782.803i) q^{91} +(-1484.01 + 435.746i) q^{93} +106.819i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) 20 * q + 16 * q^4 + 54 * q^9	\( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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