LMFDB - Embedded newform 201.4.j.a.161.1

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			161.1
Root			$0.928368 + 0.371662i$ of defining polynomial
Character	$\chi$	$=$	201.161
Dual form			201.4.j.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.72659 - 2.15856i) q^{3} +(1.13852 - 7.91857i) q^{4} +(18.3144 - 15.8695i) q^{7} +(17.6812 + 20.4052i) q^{9} +O(q^{10})$	$q+(-4.72659 - 2.15856i) q^{3} +(1.13852 - 7.91857i) q^{4} +(18.3144 - 15.8695i) q^{7} +(17.6812 + 20.4052i) q^{9} +(-22.4740 + 34.9703i) q^{12} +(50.6773 - 78.8554i) q^{13} +(-61.4076 - 18.0309i) q^{16} +(-77.6235 + 89.5823i) q^{19} +(-120.820 + 35.4758i) q^{21} +(-105.157 - 67.5801i) q^{25} +(-39.5260 - 134.613i) q^{27} +(-104.812 - 163.091i) q^{28} +(-33.9504 - 52.8278i) q^{31} +(181.711 - 116.778i) q^{36} -328.515 q^{37} +(-409.744 + 263.327i) q^{39} +(459.997 - 66.1376i) q^{43} +(251.327 + 217.776i) q^{48} +(34.7613 - 241.770i) q^{49} +(-566.725 - 491.070i) q^{52} +(560.263 - 255.863i) q^{57} +(62.8229 + 213.955i) q^{61} +(647.641 + 93.1167i) q^{63} +(-212.692 + 465.732i) q^{64} +(547.134 - 37.5088i) q^{67} +(-994.413 + 291.986i) q^{73} +(351.157 + 546.410i) q^{75} +(620.988 + 716.658i) q^{76} +(237.905 - 370.187i) q^{79} +(-103.748 + 721.580i) q^{81} +(143.363 + 997.109i) q^{84} +(-323.272 - 2248.41i) q^{91} +(46.4374 + 322.979i) q^{93} +639.951i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{17}{22}\right)$

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		0.755750	−	0.654861i	$-0.227273\pi$
$2$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$3$	−4.72659	−	2.15856i	−0.909632	−	0.415415i
$3$	−4.72659	−	2.15856i	−0.909632	−	0.415415i
$4$	1.13852	−	7.91857i	0.142315	−	0.989821i
$5$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$5$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$6$		0			0
$7$	18.3144	−	15.8695i	0.988882	−	0.856871i	−0.000821854	−	1.00000i	$-0.500262\pi$
$7$	18.3144	−	15.8695i	0.988882	−	0.856871i	0.989704	+	0.143128i	$0.0457161\pi$
$8$		0			0
$9$	17.6812	+	20.4052i	0.654861	+	0.755750i
$10$		0			0
$11$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$11$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$12$	−22.4740	+	34.9703i	−0.540641	+	0.841254i
$13$	50.6773	−	78.8554i	1.08118	−	1.68235i	0.529071	−	0.848578i	$-0.322541\pi$
$13$	50.6773	−	78.8554i	1.08118	−	1.68235i	0.552109	−	0.833772i	$-0.313823\pi$
$14$		0			0
$15$		0			0
$16$	−61.4076	−	18.0309i	−0.959493	−	0.281733i
$17$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$17$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$18$		0			0
$19$	−77.6235	+	89.5823i	−0.937266	+	1.08166i	0.0592487	+	0.998243i	$0.481129\pi$
$19$	−77.6235	+	89.5823i	−0.937266	+	1.08166i	−0.996515	+	0.0834192i	$0.973416\pi$
$20$		0			0
$21$	−120.820	+	35.4758i	−1.25548	+	0.368641i
$22$		0			0
$23$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$23$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$24$		0			0
$25$	−105.157	−	67.5801i	−0.841254	−	0.540641i
$26$		0			0
$27$	−39.5260	−	134.613i	−0.281733	−	0.959493i
$28$	−104.812	−	163.091i	−0.707417	−	1.10076i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−33.9504	−	52.8278i	−0.196699	−	0.306069i	0.728868	−	0.684655i	$-0.240047\pi$
$31$	−33.9504	−	52.8278i	−0.196699	−	0.306069i	−0.925566	+	0.378585i	$0.876411\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	181.711	−	116.778i	0.841254	−	0.540641i
$37$	−328.515			−1.45966			−0.729832	−	0.683627i	$-0.760402\pi$
$37$	−328.515			−1.45966			−0.729832	+	0.683627i	$0.760402\pi$
$38$		0			0
$39$	−409.744	+	263.327i	−1.68235	+	1.08118i
$40$		0			0
$41$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$41$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$42$		0			0
$43$	459.997	−	66.1376i	1.63137	−	0.234555i	0.735065	−	0.677996i	$-0.237152\pi$
$43$	459.997	−	66.1376i	1.63137	−	0.234555i	0.896304	+	0.443441i	$0.146242\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$47$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$48$	251.327	+	217.776i	0.755750	+	0.654861i
$49$	34.7613	−	241.770i	0.101345	−	0.704868i
$50$		0			0
$51$		0			0
$52$	−566.725	−	491.070i	−1.51136	−	1.30960i
$53$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$53$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	560.263	−	255.863i	1.30191	−	0.594560i
$58$		0			0
$59$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$59$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$60$		0			0
$61$	62.8229	+	213.955i	0.131863	+	0.449084i	0.998781	−	0.0493670i	$-0.0157204\pi$
$61$	62.8229	+	213.955i	0.131863	+	0.449084i	−0.866918	+	0.498451i	$0.833902\pi$
$62$		0			0
$63$	647.641	+	93.1167i	1.29516	+	0.186216i
$64$	−212.692	+	465.732i	−0.415415	+	0.909632i
$65$		0			0
$66$		0			0
$67$	547.134	−	37.5088i	0.997658	−	0.0683944i
$67$	547.134	−	37.5088i	0.997658	−	0.0683944i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$71$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$72$		0			0
$73$	−994.413	+	291.986i	−1.59435	+	0.468142i	−0.953966	−	0.299916i	$-0.903041\pi$
$73$	−994.413	+	291.986i	−1.59435	+	0.468142i	−0.640380	+	0.768058i	$0.721223\pi$
$74$		0			0
$75$	351.157	+	546.410i	0.540641	+	0.841254i
$76$	620.988	+	716.658i	0.937266	+	1.08166i
$77$		0			0
$78$		0			0
$79$	237.905	−	370.187i	0.338815	−	0.527207i	−0.629480	−	0.777017i	$-0.716732\pi$
$79$	237.905	−	370.187i	0.338815	−	0.527207i	0.968295	+	0.249810i	$0.0803682\pi$
$80$		0			0
$81$	−103.748	+	721.580i	−0.142315	+	0.989821i
$82$		0			0
$83$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$83$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$84$	143.363	+	997.109i	0.186216	+	1.29516i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$89$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$90$		0			0
$91$	−323.272	−	2248.41i	−0.372397	−	2.59008i
$92$		0			0
$93$	46.4374	+	322.979i	0.0517778	+	0.360122i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			639.951i			0.669868i	0.942242	+	0.334934i	$0.108714\pi$
$97$			639.951i			0.669868i	−0.942242	+	0.334934i	$0.891286\pi$
$98$		0			0
$99$		0			0
$100$	−654.861	+	755.750i	−0.654861	+	0.755750i
$101$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$101$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$102$		0			0
$103$	1656.76	−	1064.73i	1.58490	−	1.01856i	0.610990	−	0.791638i	$-0.290772\pi$
$103$	1656.76	−	1064.73i	1.58490	−	1.01856i	0.973914	−	0.226917i	$-0.0728647\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$107$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$108$	−1110.94	+	159.730i	−0.989821	+	0.142315i
$109$	1230.49	−	1914.68i	1.08128	−	1.68250i	0.538797	−	0.842435i	$-0.318879\pi$
$109$	1230.49	−	1914.68i	1.08128	−	1.68250i	0.542482	−	0.840068i	$-0.317485\pi$
$110$		0			0
$111$	1552.75	+	709.119i	1.32776	+	0.606366i
$112$	−1410.78	+	644.282i	−1.19023	+	0.543562i
$113$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$113$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2505.10	−	360.179i	1.97946	−	0.284603i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1119.71	−	719.593i	−0.841254	−	0.540641i
$122$		0			0
$123$		0			0
$124$	−456.974	+	208.693i	−0.330947	+	0.151139i
$125$		0			0
$126$		0			0
$127$	1213.80	+	1400.80i	0.848091	+	0.978749i	0.999953	−	0.00965233i	$-0.00307248\pi$
$127$	1213.80	+	1400.80i	0.848091	+	0.978749i	−0.151862	+	0.988402i	$0.548527\pi$
$128$		0			0
$129$	−2316.98	−	680.326i	−1.58138	−	0.464336i
$130$		0			0
$131$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$131$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$132$		0			0
$133$			2872.49i			1.87275i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$137$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$138$		0			0
$139$	897.332	−	3056.03i	0.547559	−	1.86482i	0.0474117	−	0.998875i	$-0.484903\pi$
$139$	897.332	−	3056.03i	0.547559	−	1.86482i	0.500148	−	0.865940i	$-0.333279\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−717.837	−	1571.84i	−0.415415	−	0.909632i
$145$		0			0
$146$		0			0
$147$	−686.177	+	1067.71i	−0.384999	+	0.599071i
$148$	−374.021	+	2601.37i	−0.207732	+	1.44481i
$149$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$149$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$150$		0			0
$151$	369.943	+	2573.01i	0.199374	+	1.38668i	0.806106	+	0.591771i	$0.201571\pi$
$151$	369.943	+	2573.01i	0.199374	+	1.38668i	−0.606732	+	0.794907i	$0.707520\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1618.67	+	3544.39i	0.830752	+	1.81909i
$157$	1301.84	−	2850.63i	0.661770	−	1.44908i	−0.219093	−	0.975704i	$-0.570310\pi$
$157$	1301.84	−	2850.63i	0.661770	−	1.44908i	0.880863	−	0.473371i	$-0.156963\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1892.32			−0.909311			−0.454656	−	0.890667i	$-0.650238\pi$
$163$	−1892.32			−0.909311			−0.454656	+	0.890667i	$0.650238\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$167$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$168$		0			0
$169$	−2737.32	−	5993.89i	−1.24593	−	2.72821i
$170$		0			0
$171$	−3200.43			−1.43124
$172$		−	3717.82i		−	1.64814i
$173$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$173$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$174$		0			0
$175$	−2998.34	+	431.096i	−1.29516	+	0.186216i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$179$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$180$		0			0
$181$	1892.65	−	4144.33i	0.777236	−	1.70191i	0.0672055	−	0.997739i	$-0.478592\pi$
$181$	1892.65	−	4144.33i	0.777236	−	1.70191i	0.710031	−	0.704171i	$-0.248681\pi$
$182$		0			0
$183$	164.897	−	1146.88i	0.0666095	−	0.463279i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2860.13	−	1838.10i	−1.10076	−	0.707417i
$190$		0			0
$191$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$191$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$192$	2010.62	−	1742.21i	0.755750	−	0.654861i
$193$	−2696.48	+	1732.92i	−1.00568	+	0.646314i	−0.936273	−	0.351274i	$-0.885749\pi$
$193$	−2696.48	+	1732.92i	−1.00568	+	0.646314i	−0.0694118	+	0.997588i	$0.522112\pi$
$194$		0			0
$195$		0			0
$196$	−1874.90	−	550.519i	−0.683271	−	0.200626i
$197$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$197$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$198$		0			0
$199$	1801.54	+	2079.09i	0.641749	+	0.740618i	0.979683	−	0.200552i	$-0.0642736\pi$
$199$	1801.54	+	2079.09i	0.641749	+	0.740618i	−0.337934	+	0.941170i	$0.609728\pi$
$200$		0			0
$201$	−2667.04	−	1003.73i	−0.935914	−	0.352228i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4533.80	+	3928.56i	−1.51136	+	1.30960i
$209$		0			0
$210$		0			0
$211$	2531.99	+	5544.29i	0.826111	+	1.80893i	0.509870	+	0.860252i	$0.329694\pi$
$211$	2531.99	+	5544.29i	0.826111	+	1.80893i	0.316241	+	0.948679i	$0.397579\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1460.13	−	428.732i	−0.456774	−	0.134121i
$218$		0			0
$219$	5330.45	+	766.403i	1.64474	+	0.236478i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1647.37	+	3607.23i	0.494690	+	1.08322i	0.978159	+	0.207858i	$0.0666493\pi$
$223$	1647.37	+	3607.23i	0.494690	+	1.08322i	−0.483469	+	0.875362i	$0.660623\pi$
$224$		0			0
$225$	−480.313	−	3340.65i	−0.142315	−	0.989821i
$226$		0			0
$227$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$227$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$228$	−1388.20	−	4727.79i	−0.403228	−	1.37327i
$229$	−1105.18	−	1719.69i	−0.318919	−	0.496247i	0.644370	−	0.764714i	$-0.277120\pi$
$229$	−1105.18	−	1719.69i	−0.318919	−	0.496247i	−0.963289	+	0.268466i	$0.913483\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$233$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1923.55	+	1236.19i	−0.527207	+	0.338815i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	6602.26	−	1938.60i	1.76468	−	0.518158i	0.771657	−	0.636039i	$-0.219428\pi$
$241$	6602.26	−	1938.60i	1.76468	−	0.518158i	0.993028	+	0.117881i	$0.0376101\pi$
$242$		0			0
$243$	2047.94	−	3186.66i	0.540641	−	0.841254i
$244$	1765.74	−	253.876i	0.463279	−	0.0666095i
$245$		0			0
$246$		0			0
$247$	3130.30	+	10660.8i	0.806381	+	2.74628i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$251$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$252$	1474.70	−	5022.38i	0.368641	−	1.25548i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	3445.77	+	2214.46i	0.841254	+	0.540641i
$257$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$257$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$258$		0			0
$259$	−6016.54	+	5213.36i	−1.44344	+	1.25074i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$263$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	325.907	−	4375.23i	0.0742833	−	0.997237i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	4734.67	+	2162.25i	1.06130	+	0.484677i	0.868052	−	0.496474i	$-0.165372\pi$
$271$	4734.67	+	2162.25i	1.06130	+	0.484677i	0.193243	+	0.981151i	$0.438099\pi$
$272$		0			0
$273$	−3325.35	+	11325.1i	−0.737213	+	2.51072i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5751.10	+	6637.13i	1.24747	+	1.43966i	0.853956	+	0.520344i	$0.174196\pi$
$277$	5751.10	+	6637.13i	1.24747	+	1.43966i	0.393517	+	0.919317i	$0.371258\pi$
$278$		0			0
$279$	477.679	−	1626.83i	0.102501	−	0.349088i
$280$		0			0
$281$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$281$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$282$		0			0
$283$	4939.42	−	5700.39i	1.03752	−	1.19736i	0.0575227	−	0.998344i	$-0.481680\pi$
$283$	4939.42	−	5700.39i	1.03752	−	1.19736i	0.979996	−	0.199016i	$-0.0637747\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4713.99	+	1384.15i	−0.959493	+	0.281733i
$290$		0			0
$291$	1381.37	−	3024.78i	0.278273	−	0.609333i
$292$	1179.95	+	8206.76i	0.236478	+	1.64474i
$293$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$293$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	4726.59	−	2158.56i	0.909632	−	0.415415i
$301$	7374.98	−	8511.18i	1.41225	−	1.62982i
$302$		0			0
$303$		0			0
$304$	6381.92	−	4101.41i	1.20404	−	0.773789i
$305$		0			0
$306$		0			0
$307$	50.1460	−	32.2269i	0.00932242	−	0.00599115i	−0.535971	−	0.844236i	$-0.680054\pi$
$307$	50.1460	−	32.2269i	0.00932242	−	0.00599115i	0.545294	+	0.838245i	$0.316418\pi$
$308$		0			0
$309$	−10129.1	+	1456.34i	−1.86480	+	0.268118i
$310$		0			0
$311$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$311$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$312$		0			0
$313$	9704.29	−	4431.80i	1.75246	−	0.800320i	0.764686	−	0.644403i	$-0.222894\pi$
$313$	9704.29	−	4431.80i	1.75246	−	0.800320i	0.987770	−	0.155917i	$-0.0498332\pi$
$314$		0			0
$315$		0			0
$316$	−2660.50	−	2305.33i	−0.473622	−	0.410396i
$317$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$317$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	5595.76	+	1643.06i	0.959493	+	0.281733i
$325$	−10658.1	+	4867.39i	−1.81909	+	0.830752i
$326$		0			0
$327$	−9948.96	+	6393.81i	−1.68250	+	1.08128i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	5828.69	+	838.039i	0.967897	+	0.139163i	0.608093	−	0.793866i	$-0.291935\pi$
$331$	5828.69	+	838.039i	0.967897	+	0.139163i	0.359803	+	0.933028i	$0.382844\pi$
$332$		0			0
$333$	−5808.55	−	6703.43i	−0.955876	−	1.10314i
$334$		0			0
$335$		0			0
$336$	8058.90			1.30848
$337$	−9229.39	+	7997.32i	−1.49186	+	1.29270i	−0.640711	+	0.767782i	$0.721360\pi$
$337$	−9229.39	+	7997.32i	−1.49186	+	1.29270i	−0.851150	+	0.524923i	$0.824094\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1293.70	+	2013.03i	0.203654	+	0.316891i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$347$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$348$		0			0
$349$	540.296	−	3757.84i	0.0828692	−	0.576368i	−0.905506	−	0.424334i	$-0.860508\pi$
$349$	540.296	−	3757.84i	0.0828692	−	0.576368i	0.988375	−	0.152035i	$-0.0485825\pi$
$350$		0			0
$351$	−12618.0	−	3704.99i	−1.91881	−	0.563412i
$352$		0			0
$353$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$353$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$359$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$360$		0			0
$361$	−1023.44	−	7118.20i	−0.149212	−	1.03779i
$362$		0			0
$363$	3739.11	+	5818.18i	0.540641	+	0.841254i
$364$	−18172.2			−2.61671
$365$		0			0
$366$		0			0
$367$	12785.6	−	5839.01i	1.81854	−	0.830500i	0.897570	−	0.440871i	$-0.145330\pi$
$367$	12785.6	−	5839.01i	1.81854	−	0.830500i	0.920972	−	0.389629i	$-0.127397\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	2610.40			0.363825
$373$			14192.6i			1.97015i	0.172116	+	0.985077i	$0.444940\pi$
$373$			14192.6i			1.97015i	−0.172116	+	0.985077i	$0.555060\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−13051.1	−	5960.26i	−1.76885	−	0.807805i	−0.981544	−	0.191236i	$-0.938750\pi$
$379$	−13051.1	−	5960.26i	−1.76885	−	0.807805i	−0.787301	−	0.616569i	$-0.788522\pi$
$380$		0			0
$381$	−2713.43	−	9241.08i	−0.364864	−	1.24261i
$382$		0			0
$383$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$383$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9482.87	+	8216.95i	1.24558	+	1.07931i
$388$	5067.50	+	728.596i	0.663049	+	0.0953321i
$389$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$389$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	14819.9	+	4351.53i	1.87353	+	0.550118i	0.997722	+	0.0674577i	$0.0214888\pi$
$397$	14819.9	+	4351.53i	1.87353	+	0.550118i	0.875808	+	0.482660i	$0.160329\pi$
$398$		0			0
$399$	6200.43	−	13577.1i	0.777970	−	1.70352i
$400$	5238.89	+	6046.00i	0.654861	+	0.755750i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−5886.27			−0.727583
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12155.2	+	10532.5i	−1.46952	+	1.27335i	−0.581506	+	0.813542i	$0.697536\pi$
$409$	−12155.2	+	10532.5i	−1.46952	+	1.27335i	−0.888018	+	0.459808i	$0.847918\pi$
$410$		0			0
$411$		0			0
$412$	−6544.92	−	14331.4i	−0.782633	−	1.71373i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−10837.9	+	12507.7i	−1.27275	+	1.46883i
$418$		0			0
$419$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$419$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$420$		0			0
$421$	8668.73	−	10004.2i	1.00353	−	1.15814i	0.0161389	−	0.999870i	$-0.494863\pi$
$421$	8668.73	−	10004.2i	1.00353	−	1.15814i	0.987396	−	0.158271i	$-0.0505919\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4545.92	+	2921.48i	0.515205	+	0.331102i
$428$		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
$433$	−7353.78	−	11442.7i	−0.816167	−	1.26998i	−0.959894	−	0.280363i	$-0.909545\pi$
$433$	−7353.78	−	11442.7i	−0.816167	−	1.26998i	0.143727	−	0.989617i	$-0.454091\pi$
$434$		0			0
$435$		0			0
$436$	−13760.6	−	11923.6i	−1.51150	−	1.30972i
$437$		0			0
$438$		0			0
$439$	−8522.65			−0.926569			−0.463285	−	0.886210i	$-0.653329\pi$
$439$	−8522.65			−0.926569			−0.463285	+	0.886210i	$0.653329\pi$
$440$		0			0
$441$	5547.99	−	3565.48i	0.599071	−	0.384999i
$442$		0			0
$443$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$443$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$444$	7383.05	−	11488.3i	0.789154	−	1.22795i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	3495.59	+	11904.9i	0.368641	+	1.25548i
$449$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$449$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	3805.43	−	12960.1i	0.394690	−	1.34419i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−16437.1	−	10563.5i	−1.68248	−	1.08126i	−0.844358	−	0.535780i	$-0.820018\pi$
$457$	−16437.1	−	10563.5i	−1.68248	−	1.08126i	−0.838121	−	0.545484i	$-0.816346\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$461$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$462$		0			0
$463$	−4654.45	−	15851.6i	−0.467193	−	1.59111i	−0.769988	−	0.638058i	$-0.779738\pi$
$463$	−4654.45	−	15851.6i	−0.467193	−	1.59111i	0.302795	−	0.953056i	$-0.402080\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$467$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$468$		−	20246.9i		−	1.99981i
$469$	9425.17	−	9369.69i	0.927961	−	0.922499i
$470$		0			0
$471$	−12306.5	+	10663.6i	−1.20394	+	1.04322i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	14216.6	−	4174.37i	1.37327	−	0.403228i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$479$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$480$		0			0
$481$	−16648.2	+	25905.2i	−1.57816	+	2.45566i
$482$		0			0
$483$		0			0
$484$	−6972.96	+	8047.22i	−0.654861	+	0.755750i
$485$		0			0
$486$		0			0
$487$	−19769.6	−	2842.44i	−1.83952	−	0.264483i	−0.867172	−	0.498010i	$-0.834064\pi$
$487$	−19769.6	−	2842.44i	−1.83952	−	0.264483i	−0.972351	+	0.233526i	$0.924974\pi$
$488$		0			0
$489$	8944.20	+	4084.68i	0.827138	+	0.377741i
$490$		0			0
$491$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$491$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1132.28	+	3856.18i	0.102501	+	0.349088i
$497$		0			0
$498$		0			0
$499$		−	5586.40i		−	0.501165i	−0.968095	−	0.250583i	$-0.919378\pi$
$499$		−	5586.40i		−	0.501165i	0.968095	−	0.250583i	$-0.0806222\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$503$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			34239.3i			2.99925i
$508$	12474.3	−	8016.74i	1.08948	−	0.700168i
$509$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$509$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$510$		0			0
$511$	−13578.4	+	21128.3i	−1.17548	+	1.82909i
$512$		0			0
$513$	15127.1	+	6908.31i	1.30191	+	0.594560i
$514$		0			0
$515$		0			0
$516$	−8025.13	+	17572.6i	−0.684664	+	1.49920i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$521$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$522$		0			0
$523$	6508.81	+	4182.96i	0.544189	+	0.349729i	0.783675	−	0.621171i	$-0.213343\pi$
$523$	6508.81	+	4182.96i	0.544189	+	0.349729i	−0.239487	+	0.970900i	$0.576979\pi$
$524$		0			0
$525$	15102.5	+	4434.48i	1.25548	+	0.368641i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7967.69	−	9195.21i	−0.654861	−	0.755750i
$530$		0			0
$531$		0			0
$532$	22746.0	+	3270.38i	1.85369	+	0.266521i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	6761.61	−	23027.9i	0.537346	−	1.83003i	−0.0200682	−	0.999799i	$-0.506388\pi$
$541$	6761.61	−	23027.9i	0.537346	−	1.83003i	0.557414	−	0.830234i	$-0.311793\pi$
$542$		0			0
$543$	−17891.6	+	15503.1i	−1.41400	+	1.22524i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1481.11	−	5044.21i	0.115773	−	0.394287i	−0.881135	−	0.472865i	$-0.843220\pi$
$547$	1481.11	−	5044.21i	0.115773	−	0.394287i	0.996908	+	0.0785783i	$0.0250381\pi$
$548$		0			0
$549$	−3255.02	+	5064.91i	−0.253043	+	0.393743i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1517.60	−	10555.2i	−0.116700	−	0.811667i
$554$		0			0
$555$		0			0
$556$	−23177.8	−	10584.9i	−1.76791	−	0.807377i
$557$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$557$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$558$		0			0
$559$	18096.1	−	39624.9i	1.36920	−	2.99813i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$563$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	9551.03	+	14861.7i	0.707417	+	1.10076i
$568$		0			0
$569$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$569$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$570$		0			0
$571$	9383.28	+	20546.5i	0.687703	+	1.50586i	0.854270	+	0.519829i	$0.174004\pi$
$571$	9383.28	+	20546.5i	0.687703	+	1.50586i	−0.166568	+	0.986030i	$0.553268\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−13264.0	+	3894.67i	−0.959493	+	0.281733i
$577$	25191.6	−	3622.01i	1.81758	−	0.261328i	0.852393	−	0.522902i	$-0.175151\pi$
$577$	25191.6	−	3622.01i	1.81758	−	0.261328i	0.965183	+	0.261575i	$0.0842417\pi$
$578$		0			0
$579$	16485.8	−	2370.30i	1.18329	−	0.170132i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$587$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$588$	7673.53	+	6649.15i	0.538182	+	0.466337i
$589$	7367.78	+	1059.33i	0.515423	+	0.0741066i
$590$		0			0
$591$		0			0
$592$	20173.3	+	5923.42i	1.40054	+	0.411235i
$593$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$593$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4027.31	−	13715.7i	−0.276092	−	0.940282i
$598$		0			0
$599$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$599$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$600$		0			0
$601$	17836.3	+	20584.1i	1.21058	+	1.39708i	0.893732	+	0.448601i	$0.148077\pi$
$601$	17836.3	+	20584.1i	1.21058	+	1.39708i	0.316843	+	0.948478i	$0.397377\pi$
$602$		0			0
$603$	10439.4	+	10501.2i	0.705016	+	0.709191i
$604$	20795.7			1.40094
$605$		0			0
$606$		0			0
$607$	−1646.03	+	11448.4i	−0.110066	+	0.765529i	0.857785	+	0.514008i	$0.171840\pi$
$607$	−1646.03	+	11448.4i	−0.110066	+	0.765529i	−0.967852	+	0.251521i	$0.919069\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	9201.89	+	20149.3i	0.606299	+	1.32761i	0.925077	+	0.379779i	$0.124000\pi$
$613$	9201.89	+	20149.3i	0.606299	+	1.32761i	−0.318778	+	0.947829i	$0.603273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$617$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$618$		0			0
$619$	−28712.0	−	8430.59i	−1.86435	−	0.547422i	−0.998924	−	0.0463820i	$-0.985231\pi$
$619$	−28712.0	−	8430.59i	−1.86435	−	0.547422i	−0.865424	−	0.501040i	$-0.832951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	29909.4	−	8782.20i	1.91881	−	0.563412i
$625$	6490.86	+	14213.0i	0.415415	+	0.909632i
$626$		0			0
$627$		0			0
$628$	−21090.7	−	13554.2i	−1.34015	−	0.861259i
$629$		0			0
$630$		0			0
$631$	6176.61	+	9610.99i	0.389678	+	0.606351i	0.979563	−	0.201136i	$-0.0644634\pi$
$631$	6176.61	+	9610.99i	0.389678	+	0.606351i	−0.589885	+	0.807487i	$0.700827\pi$
$632$		0			0
$633$		−	31671.0i		−	1.98864i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−17303.2	−	14993.3i	−1.07626	−	0.932587i
$638$		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
$643$	26539.3	−	7792.65i	1.62770	−	0.477935i	0.664625	−	0.747177i	$-0.268591\pi$
$643$	26539.3	−	7792.65i	1.62770	−	0.477935i	0.963072	+	0.269242i	$0.0867732\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$647$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	5975.98	+	5178.22i	0.359781	+	0.311752i
$652$	−2154.44	+	14984.4i	−0.129408	+	0.900056i
$653$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$653$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−23540.5	−	15128.6i	−1.39787	−	0.898358i
$658$		0			0
$659$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$659$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$660$		0			0
$661$	−18239.0	+	15804.2i	−1.07325	+	0.929972i	−0.997741	−	0.0671734i	$-0.978602\pi$
$661$	−18239.0	+	15804.2i	−1.07325	+	0.929972i	−0.0755037	+	0.997146i	$0.524056\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	20605.8i		−	1.19083i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−15931.8	−	7275.80i	−0.912518	−	0.416733i	−0.0968804	−	0.995296i	$-0.530886\pi$
$673$	−15931.8	−	7275.80i	−0.912518	−	0.416733i	−0.815638	+	0.578563i	$0.803614\pi$
$674$		0			0
$675$	−4940.75	+	16826.6i	−0.281733	+	0.959493i
$676$	−50579.5	+	14851.5i	−2.87776	+	0.844986i
$677$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$677$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$678$		0			0
$679$	10155.7	+	11720.3i	0.573990	+	0.662420i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$683$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$684$	−3643.75	+	25342.8i	−0.203687	+	1.41668i
$685$		0			0
$686$		0			0
$687$	1511.67	+	10513.9i	0.0839501	+	0.583886i
$688$	−29439.8	−	4232.80i	−1.63137	−	0.234555i
$689$		0			0
$690$		0			0
$691$	3927.57	−	1153.24i	0.216226	−	0.0634896i	−0.171826	−	0.985127i	$-0.554967\pi$
$691$	3927.57	−	1153.24i	0.216226	−	0.0634896i	0.388051	+	0.921638i	$0.373148\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$			24233.4i			1.30848i
$701$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$701$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$702$		0			0
$703$	25500.5	−	29429.1i	1.36809	−	1.57886i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−20982.1	+	13484.4i	−1.11142	+	0.714267i	−0.961602	−	0.274446i	$-0.911505\pi$
$709$	−20982.1	+	13484.4i	−1.11142	+	0.714267i	−0.149819	+	0.988713i	$0.547869\pi$
$710$		0			0
$711$	11760.2	−	1690.86i	0.620313	−	0.0891876i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$719$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$720$		0			0
$721$	13445.7	−	45791.8i	0.694512	−	2.36529i
$722$		0			0
$723$	−35390.7	−	5088.42i	−1.82046	−	0.261743i
$724$	−30662.4	−	19705.5i	−1.57397	−	1.01153i
$725$		0			0
$726$		0			0
$727$	14585.5	−	6660.99i	0.744081	−	0.339811i	−0.00703957	−	0.999975i	$-0.502241\pi$
$727$	14585.5	−	6660.99i	0.744081	−	0.339811i	0.751121	+	0.660165i	$0.229514\pi$
$728$		0			0
$729$	−16558.4	+	10641.4i	−0.841254	+	0.540641i
$730$		0			0
$731$		0			0
$732$	−8893.95	−	2611.50i	−0.449084	−	0.131863i
$733$	−12547.6	−	1804.08i	−0.632274	−	0.0909073i	−0.181276	−	0.983432i	$-0.558023\pi$
$733$	−12547.6	−	1804.08i	−0.632274	−	0.0909073i	−0.450998	+	0.892525i	$0.648932\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	26170.4	−	22676.8i	1.30270	−	1.12879i	0.319235	−	0.947675i	$-0.396574\pi$
$739$	26170.4	−	22676.8i	1.30270	−	1.12879i	0.983461	−	0.181118i	$-0.0579715\pi$
$740$		0			0
$741$	8216.38	−	57146.2i	0.407336	−	2.83309i
$742$		0			0
$743$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$743$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−5240.06	+	36445.4i	−0.254611	+	1.77086i	0.315147	+	0.949043i	$0.397946\pi$
$751$	−5240.06	+	36445.4i	−0.254611	+	1.77086i	−0.569757	+	0.821813i	$0.692963\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−17811.4	+	20555.5i	−0.856871	+	0.988882i
$757$	14056.2	+	6419.25i	0.674876	+	0.308206i	0.723212	−	0.690626i	$-0.242665\pi$
$757$	14056.2	+	6419.25i	0.674876	+	0.308206i	−0.0483365	+	0.998831i	$0.515392\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$761$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$762$		0			0
$763$	−7849.33	−	54593.3i	−0.372431	−	2.59031i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−11506.7	−	17904.8i	−0.540641	−	0.841254i
$769$	−34711.4	+	15852.2i	−1.62773	+	0.743360i	−0.999402	−	0.0345865i	$-0.988989\pi$
$769$	−34711.4	+	15852.2i	−1.62773	+	0.743360i	−0.628330	+	0.777947i	$0.716261\pi$
$770$		0			0
$771$		0			0
$772$	10652.3	+	23325.3i	0.496612	+	1.08743i
$773$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$773$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$774$		0			0
$775$			7849.57i			0.363825i
$776$		0			0
$777$	39691.1	−	11654.3i	1.83257	−	0.538092i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−6493.93	+	14219.7i	−0.295824	+	0.647764i
$785$		0			0
$786$		0			0
$787$	16449.2	−	2365.04i	0.745047	−	0.107122i	0.240672	−	0.970606i	$-0.422632\pi$
$787$	16449.2	−	2365.04i	0.745047	−	0.107122i	0.504375	+	0.863485i	$0.331723\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	20055.2	+	5888.74i	0.898085	+	0.263701i
$794$		0			0
$795$		0			0
$796$	18514.5	−	11898.6i	0.824410	−	0.529816i
$797$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$797$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−10984.6	+	19976.4i	−0.481838	+	0.876260i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$809$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$810$		0			0
$811$	−723.621	+	627.022i	−0.0313314	+	0.0271488i	−0.670386	−	0.742013i	$-0.733871\pi$
$811$	−723.621	+	627.022i	−0.0313314	+	0.0271488i	0.639054	+	0.769162i	$0.279326\pi$
$812$		0			0
$813$	−17711.5	−	20440.2i	−0.764046	−	0.881756i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−29781.8	+	46341.4i	−1.27532	+	1.98443i
$818$		0			0
$819$	40163.4	−	46351.1i	1.71358	−	1.97758i
$820$		0			0
$821$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$821$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$822$		0			0
$823$	−17333.3	+	20003.7i	−0.734145	+	0.847249i	−0.992932	−	0.118686i	$-0.962132\pi$
$823$	−17333.3	+	20003.7i	−0.734145	+	0.847249i	0.258786	+	0.965935i	$0.416677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$827$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$828$		0			0
$829$	−35080.7	−	22545.0i	−1.46972	−	0.944534i	−0.998028	−	0.0627735i	$-0.980005\pi$
$829$	−35080.7	−	22545.0i	−1.46972	−	0.944534i	−0.471696	−	0.881761i	$-0.656358\pi$
$830$		0			0
$831$	−12856.5	−	43785.1i	−0.536685	−	1.82778i
$832$	25946.8	+	40373.9i	1.08118	+	1.68235i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5769.39	+	6658.24i	−0.238255	+	0.274961i
$838$		0			0
$839$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$839$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	46785.5	−	13737.5i	1.90809	−	0.560265i
$845$		0			0
$846$		0			0
$847$	−31926.3	+	4590.31i	−1.29516	+	0.186216i
$848$		0			0
$849$	−35651.2	+	16281.4i	−1.44116	+	0.658157i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	6928.38	−	48188.0i	0.278105	−	1.93426i	−0.0716721	−	0.997428i	$-0.522834\pi$
$853$	6928.38	−	48188.0i	0.278105	−	1.93426i	0.349777	−	0.936833i	$-0.386257\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$857$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$858$		0			0
$859$	4216.68	+	2709.90i	0.167487	+	0.107637i	0.621699	−	0.783256i	$-0.286443\pi$
$859$	4216.68	+	2709.90i	0.167487	+	0.107637i	−0.454212	+	0.890894i	$0.650079\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$863$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25268.9	+	3633.11i	0.989821	+	0.142315i
$868$	−5057.33	+	11074.0i	−0.197762	+	0.433037i
$869$		0			0
$870$		0			0
$871$	24769.5	−	45045.3i	0.963585	−	1.75236i
$872$		0			0
$873$	−13058.3	+	11315.1i	−0.506252	+	0.438670i
$874$		0			0
$875$		0			0
$876$	12137.6	−	41337.0i	0.468142	−	1.59435i
$877$	−21699.3	+	6371.49i	−0.835500	+	0.245325i	−0.671378	−	0.741115i	$-0.734297\pi$
$877$	−21699.3	+	6371.49i	−0.835500	+	0.245325i	−0.164122	+	0.986440i	$0.552479\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$881$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$882$		0			0
$883$	−21002.2	+	32680.1i	−0.800432	+	1.24550i	0.165374	+	0.986231i	$0.447117\pi$
$883$	−21002.2	+	32680.1i	−0.800432	+	1.24550i	−0.965806	+	0.259266i	$0.916519\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$887$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$888$		0			0
$889$	44460.0	+	6392.39i	1.67732	+	0.241163i
$890$		0			0
$891$		0			0
$892$	30439.7	−	8937.89i	1.14260	−	0.335496i
$893$		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
$903$	−53230.3	+	24309.5i	−1.96168	+	0.895868i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−43590.4	+	28013.8i	−1.59581	+	1.02556i	−0.626589	+	0.779350i	$0.715550\pi$
$907$	−43590.4	+	28013.8i	−1.59581	+	1.02556i	−0.969216	+	0.246212i	$0.920814\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$911$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$912$	−39017.8	+	5609.91i	−1.41668	+	0.203687i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−14875.8	+	6793.55i	−0.536583	+	0.245049i
$917$		0			0
$918$		0			0
$919$	39767.6	+	34458.8i	1.42743	+	1.23688i	0.929066	+	0.369914i	$0.120613\pi$
$919$	39767.6	+	34458.8i	1.42743	+	1.23688i	0.498369	+	0.866965i	$0.333933\pi$
$920$		0			0
$921$	−306.583	+	44.0800i	−0.0109688	+	0.00157707i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	34545.6	+	22201.1i	1.22795	+	0.789154i
$926$		0			0
$927$	51019.6	+	14980.7i	1.80766	+	0.530778i
$928$		0			0
$929$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$929$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$930$		0			0
$931$	18960.0	+	21881.0i	0.667443	+	0.770270i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			57003.7i			1.98744i	0.111899	+	0.993720i	$0.464307\pi$
$937$			57003.7i			1.98744i	−0.111899	+	0.993720i	$0.535693\pi$
$938$		0			0
$939$	−55434.5			−1.92656
$940$		0			0
$941$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$941$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$947$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$948$	7598.87	+	16639.2i	0.260337	+	0.570059i
$949$	−27369.5	+	93211.8i	−0.936196	+	3.18839i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$953$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	10737.5	−	23511.8i	0.360427	−	0.789225i
$962$		0			0
$963$		0			0
$964$	−7834.13	−	54487.6i	−0.261743	−	1.82046i
$965$		0			0
$966$		0			0
$967$	51152.1			1.70108			0.850538	−	0.525913i	$-0.176276\pi$
$967$	51152.1			1.70108			0.850538	+	0.525913i	$0.176276\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$971$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$972$	−22902.2	−	19844.9i	−0.755750	−	0.654861i
$973$	−32063.6	−	70209.5i	−1.05644	−	2.31327i
$974$		0			0
$975$	60883.0			1.99981
$976$		−	14271.2i		−	0.468043i
$977$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$977$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	60826.0	−	8745.46i	1.97964	−	0.284629i
$982$		0			0
$983$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$983$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	87982.3	−	12649.9i	2.83309	−	0.407336i
$989$		0			0
$990$		0			0
$991$	−53141.1	−	7640.53i	−1.70341	−	0.244914i	−0.779196	−	0.626781i	$-0.784372\pi$
$991$	−53141.1	−	7640.53i	−1.70341	−	0.244914i	−0.924217	+	0.381867i	$0.875281\pi$
$992$		0			0
$993$	−25740.9	−	16542.6i	−0.822620	−	0.528665i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32697.3	−	21013.3i	1.03865	−	0.667499i	0.0939968	−	0.995572i	$-0.470036\pi$
$997$	32697.3	−	21013.3i	1.03865	−	0.667499i	0.944652	+	0.328073i	$0.106399\pi$
$998$		0			0
$999$	12984.9	+	44222.4i	0.411235	+	1.40054i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.a.161.1	yes	20
3.2	odd	2	CM	201.4.j.a.161.1	yes	20
67.5	odd	22	inner	201.4.j.a.5.1	✓	20
201.5	even	22	inner	201.4.j.a.5.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.a.5.1	✓	20	67.5	odd	22	inner
201.4.j.a.5.1	✓	20	201.5	even	22	inner
201.4.j.a.161.1	yes	20	1.1	even	1	trivial
201.4.j.a.161.1	yes	20	3.2	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(-4.72659 - 2.15856i) q^{3} +(1.13852 - 7.91857i) q^{4} +(18.3144 - 15.8695i) q^{7} +(17.6812 + 20.4052i) q^{9} +O(q^{10})\)	\(q+(-4.72659 - 2.15856i) q^{3} +(1.13852 - 7.91857i) q^{4} +(18.3144 - 15.8695i) q^{7} +(17.6812 + 20.4052i) q^{9} +(-22.4740 + 34.9703i) q^{12} +(50.6773 - 78.8554i) q^{13} +(-61.4076 - 18.0309i) q^{16} +(-77.6235 + 89.5823i) q^{19} +(-120.820 + 35.4758i) q^{21} +(-105.157 - 67.5801i) q^{25} +(-39.5260 - 134.613i) q^{27} +(-104.812 - 163.091i) q^{28} +(-33.9504 - 52.8278i) q^{31} +(181.711 - 116.778i) q^{36} -328.515 q^{37} +(-409.744 + 263.327i) q^{39} +(459.997 - 66.1376i) q^{43} +(251.327 + 217.776i) q^{48} +(34.7613 - 241.770i) q^{49} +(-566.725 - 491.070i) q^{52} +(560.263 - 255.863i) q^{57} +(62.8229 + 213.955i) q^{61} +(647.641 + 93.1167i) q^{63} +(-212.692 + 465.732i) q^{64} +(547.134 - 37.5088i) q^{67} +(-994.413 + 291.986i) q^{73} +(351.157 + 546.410i) q^{75} +(620.988 + 716.658i) q^{76} +(237.905 - 370.187i) q^{79} +(-103.748 + 721.580i) q^{81} +(143.363 + 997.109i) q^{84} +(-323.272 - 2248.41i) q^{91} +(46.4374 + 322.979i) q^{93} +639.951i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) 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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