LMFDB - Embedded newform 201.2.p.a.32.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [201,2,Mod(2,201)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(201, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([33, 1]))

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

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

chi := DirichletCharacter("201.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 201 = 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	201.p (of order $66$, degree $20$, 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:	$1.60499308063$
Analytic rank:	$0$
Dimension:	$20$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{66}]$

Embedding invariants

Embedding label			32.1
Root			$-0.786053 + 0.618159i$ of defining polynomial
Character	$\chi$	$=$	201.32
Dual form			201.2.p.a.44.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.71442 + 0.246497i) q^{3} +(1.77767 + 0.916453i) q^{4} +(-3.50336 + 3.67422i) q^{7} +(2.87848 - 0.845198i) q^{9} +O(q^{10})$	$q+(-1.71442 + 0.246497i) q^{3} +(1.77767 + 0.916453i) q^{4} +(-3.50336 + 3.67422i) q^{7} +(2.87848 - 0.845198i) q^{9} +(-3.27358 - 1.13300i) q^{12} +(0.687743 + 3.56835i) q^{13} +(2.32023 + 3.25830i) q^{16} +(-1.65758 + 1.58050i) q^{19} +(5.10055 - 7.16272i) q^{21} +(3.27430 + 3.77875i) q^{25} +(-4.72659 + 2.15856i) q^{27} +(-9.59507 + 3.32089i) q^{28} +(1.88589 - 9.78494i) q^{31} +(5.89157 + 1.13551i) q^{36} +(5.89230 - 10.2058i) q^{37} +(-2.05867 - 5.94812i) q^{39} +(2.24962 - 3.50047i) q^{43} +(-4.78101 - 5.01418i) q^{48} +(-0.893274 - 18.7521i) q^{49} +(-2.04764 + 6.97363i) q^{52} +(2.45221 - 3.11824i) q^{57} +(-1.34031 + 14.0363i) q^{61} +(-6.97891 + 13.5372i) q^{63} +(1.13852 + 7.91857i) q^{64} +(-4.25333 + 6.99351i) q^{67} +(5.46168 + 0.521527i) q^{73} +(-6.54498 - 5.67126i) q^{75} +(-4.39510 + 1.29052i) q^{76} +(15.3567 + 5.31502i) q^{79} +(7.57128 - 4.86577i) q^{81} +(15.6314 - 8.05855i) q^{84} +(-15.5203 - 9.97429i) q^{91} +(-0.821260 + 17.2404i) q^{93} +(-15.6466 - 9.03356i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{4} - 6 q^{7} + 6 q^{9}+O(q^{10}) $ 20 * q - 2 * q^4 - 6 * q^7 + 6 * q^9	$ 20 q - 2 q^{4} - 6 q^{7} + 6 q^{9} - 6 q^{12} - 3 q^{13} + 4 q^{16} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 12 q^{28} + 15 q^{31} + 6 q^{36} + 10 q^{37} - 3 q^{39} - 12 q^{48} - 5 q^{49} - 154 q^{52} - 75 q^{57} + 15 q^{61} - 15 q^{63} + 16 q^{64} + 5 q^{67} + 60 q^{73} - 32 q^{76} + 134 q^{79} - 18 q^{81} + 186 q^{84} - 12 q^{91} - 15 q^{93} + 9 q^{97}+O(q^{100}) $ 20 * q - 2 * q^4 - 6 * q^7 + 6 * q^9 - 6 * q^12 - 3 * q^13 + 4 * q^16 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 12 * q^28 + 15 * q^31 + 6 * q^36 + 10 * q^37 - 3 * q^39 - 12 * q^48 - 5 * q^49 - 154 * q^52 - 75 * q^57 + 15 * q^61 - 15 * q^63 + 16 * q^64 + 5 * q^67 + 60 * q^73 - 32 * q^76 + 134 * q^79 - 18 * q^81 + 186 * q^84 - 12 * q^91 - 15 * q^93 + 9 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{5}{66}\right)$

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		−0.971812	−	0.235759i	$-0.924242\pi$
$2$		0			0		0.971812	+	0.235759i	$0.0757576\pi$
$3$	−1.71442	+	0.246497i	−0.989821	+	0.142315i
$3$	−1.71442	+	0.246497i	−0.989821	+	0.142315i
$4$	1.77767	+	0.916453i	0.888835	+	0.458227i
$5$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$5$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$6$		0			0
$7$	−3.50336	+	3.67422i	−1.32415	+	1.38872i	−0.457983	+	0.888961i	$0.651428\pi$
$7$	−3.50336	+	3.67422i	−1.32415	+	1.38872i	−0.866162	+	0.499763i	$0.833421\pi$
$8$		0			0
$9$	2.87848	−	0.845198i	0.959493	−	0.281733i
$10$		0			0
$11$		0			0		−0.0950560	−	0.995472i	$-0.530303\pi$
$11$		0			0		0.0950560	+	0.995472i	$0.469697\pi$
$12$	−3.27358	−	1.13300i	−0.945001	−	0.327068i
$13$	0.687743	+	3.56835i	0.190745	+	0.989681i	0.943498	+	0.331378i	$0.107514\pi$
$13$	0.687743	+	3.56835i	0.190745	+	0.989681i	−0.752753	+	0.658303i	$0.771274\pi$
$14$		0			0
$15$		0			0
$16$	2.32023	+	3.25830i	0.580057	+	0.814576i
$17$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$17$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$18$		0			0
$19$	−1.65758	+	1.58050i	−0.380276	+	0.362593i	−0.855976	−	0.517015i	$-0.827043\pi$
$19$	−1.65758	+	1.58050i	−0.380276	+	0.362593i	0.475700	+	0.879607i	$0.342195\pi$
$20$		0			0
$21$	5.10055	−	7.16272i	1.11303	−	1.56303i
$22$		0			0
$23$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$23$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$24$		0			0
$25$	3.27430	+	3.77875i	0.654861	+	0.755750i
$26$		0			0
$27$	−4.72659	+	2.15856i	−0.909632	+	0.415415i
$28$	−9.59507	+	3.32089i	−1.81330	+	0.627588i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	1.88589	−	9.78494i	0.338716	−	1.75743i	−0.268872	−	0.963176i	$-0.586651\pi$
$31$	1.88589	−	9.78494i	0.338716	−	1.75743i	0.607589	−	0.794252i	$-0.292137\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	5.89157	+	1.13551i	0.981929	+	0.189251i
$37$	5.89230	−	10.2058i	0.968688	−	1.67782i	0.269325	−	0.963049i	$-0.413199\pi$
$37$	5.89230	−	10.2058i	0.968688	−	1.67782i	0.699363	−	0.714767i	$-0.253467\pi$
$38$		0			0
$39$	−2.05867	−	5.94812i	−0.329650	−	0.952462i
$40$		0			0
$41$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$41$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$42$		0			0
$43$	2.24962	−	3.50047i	0.343063	−	0.533817i	−0.626258	−	0.779616i	$-0.715414\pi$
$43$	2.24962	−	3.50047i	0.343063	−	0.533817i	0.969321	+	0.245800i	$0.0790505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$47$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$48$	−4.78101	−	5.01418i	−0.690079	−	0.723734i
$49$	−0.893274	−	18.7521i	−0.127611	−	2.67888i
$50$		0			0
$51$		0			0
$52$	−2.04764	+	6.97363i	−0.283957	+	0.967069i
$53$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$53$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.45221	−	3.11824i	0.324803	−	0.413021i
$58$		0			0
$59$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$59$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$60$		0			0
$61$	−1.34031	+	14.0363i	−0.171609	+	1.79717i	0.342580	+	0.939489i	$0.388699\pi$
$61$	−1.34031	+	14.0363i	−0.171609	+	1.79717i	−0.514188	+	0.857677i	$0.671907\pi$
$62$		0			0
$63$	−6.97891	+	13.5372i	−0.879259	+	1.70553i
$64$	1.13852	+	7.91857i	0.142315	+	0.989821i
$65$		0			0
$66$		0			0
$67$	−4.25333	+	6.99351i	−0.519627	+	0.854393i
$67$	−4.25333	+	6.99351i	−0.519627	+	0.854393i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$71$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$72$		0			0
$73$	5.46168	+	0.521527i	0.639242	+	0.0610402i	0.409644	−	0.912245i	$-0.365653\pi$
$73$	5.46168	+	0.521527i	0.639242	+	0.0610402i	0.229598	+	0.973286i	$0.426259\pi$
$74$		0			0
$75$	−6.54498	−	5.67126i	−0.755750	−	0.654861i
$76$	−4.39510	+	1.29052i	−0.504152	+	0.148032i
$77$		0			0
$78$		0			0
$79$	15.3567	+	5.31502i	1.72777	+	0.597986i	0.996461	−	0.0840621i	$-0.0267894\pi$
$79$	15.3567	+	5.31502i	1.72777	+	0.597986i	0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$	7.57128	−	4.86577i	0.841254	−	0.540641i
$82$		0			0
$83$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$83$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$84$	15.6314	−	8.05855i	1.70553	−	0.879259i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$89$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$90$		0			0
$91$	−15.5203	−	9.97429i	−1.62697	−	1.04559i
$92$		0			0
$93$	−0.821260	+	17.2404i	−0.0851607	+	1.78774i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−15.6466	−	9.03356i	−1.58867	−	0.917219i	−0.993527	−	0.113599i	$-0.963762\pi$
$97$	−15.6466	−	9.03356i	−1.58867	−	0.917219i	−0.595143	−	0.803620i	$-0.702905\pi$
$98$		0			0
$99$		0			0
$100$	2.35759	+	9.71812i	0.235759	+	0.971812i
$101$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$101$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$102$		0			0
$103$	19.8014	+	3.81640i	1.95109	+	0.376042i	0.997090	+	0.0762298i	$0.0242882\pi$
$103$	19.8014	+	3.81640i	1.95109	+	0.376042i	0.953998	+	0.299812i	$0.0969239\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$107$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$108$	−10.3805	−	0.494486i	−0.998867	−	0.0475819i
$109$	7.88028	−	6.82830i	0.754794	−	0.654033i	−0.189968	−	0.981790i	$-0.560838\pi$
$109$	7.88028	−	6.82830i	0.754794	−	0.654033i	0.944762	+	0.327758i	$0.106293\pi$
$110$		0			0
$111$	−7.58619	+	18.9494i	−0.720050	+	1.79860i
$112$	−20.1003	−	2.88999i	−1.89930	−	0.273078i
$113$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$113$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.99561	+	9.69013i	0.461844	+	0.895853i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−10.8012	+	2.08176i	−0.981929	+	0.189251i
$122$		0			0
$123$		0			0
$124$	12.3199	−	15.6661i	1.10636	−	1.40686i
$125$		0			0
$126$		0			0
$127$	11.8418	+	11.2911i	1.05079	+	1.00193i	0.999995	+	0.00321749i	$0.00102416\pi$
$127$	11.8418	+	11.2911i	1.05079	+	1.00193i	0.0507955	+	0.998709i	$0.483824\pi$
$128$		0			0
$129$	−2.99393	+	6.55580i	−0.263601	+	0.577206i
$130$		0			0
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$		−	11.6274i		−	1.00822i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$137$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$138$		0			0
$139$	−9.21749	−	4.20949i	−0.781817	−	0.357044i	−0.0158092	−	0.999875i	$-0.505032\pi$
$139$	−9.21749	−	4.20949i	−0.781817	−	0.357044i	−0.766008	+	0.642831i	$0.777760\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	9.43264	+	7.41791i	0.786053	+	0.618159i
$145$		0			0
$146$		0			0
$147$	6.15379	+	31.9289i	0.507556	+	2.63345i
$148$	19.8277	−	12.7425i	1.62982	−	1.04742i
$149$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$149$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$150$		0			0
$151$	−5.58800	+	2.88081i	−0.454745	+	0.234437i	−0.670352	−	0.742043i	$-0.733857\pi$
$151$	−5.58800	+	2.88081i	−0.454745	+	0.234437i	0.215608	+	0.976480i	$0.430827\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.79155	−	12.4605i	0.143438	−	0.997636i
$157$	−7.03443	−	2.81616i	−0.561409	−	0.224754i	0.0735617	−	0.997291i	$-0.476563\pi$
$157$	−7.03443	−	2.81616i	−0.561409	−	0.224754i	−0.634970	+	0.772537i	$0.718988\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.00036	−	3.46472i	−0.156680	−	0.271378i	0.776989	−	0.629514i	$-0.216746\pi$
$163$	−2.00036	−	3.46472i	−0.156680	−	0.271378i	−0.933670	+	0.358136i	$0.883413\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$167$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$168$		0			0
$169$	−0.191328	+	0.0765960i	−0.0147175	+	0.00589200i
$170$		0			0
$171$	−3.43549	+	5.95043i	−0.262718	+	0.455041i
$172$	7.20709	−	4.16102i	0.549536	−	0.317275i
$173$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$173$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$174$		0			0
$175$	−25.3550	−	1.20781i	−1.91666	−	0.0913017i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$179$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$180$		0			0
$181$	−20.9975	+	16.5126i	−1.56073	+	1.22737i	−0.706129	+	0.708083i	$0.749560\pi$
$181$	−20.9975	+	16.5126i	−1.56073	+	1.22737i	−0.854599	+	0.519288i	$0.826197\pi$
$182$		0			0
$183$	−1.16205	−	24.3945i	−0.0859016	−	1.80330i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.62791	−	24.9287i	0.627588	−	1.81330i
$190$		0			0
$191$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$191$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$192$	−3.90380	−	13.2951i	−0.281733	−	0.959493i
$193$	17.3277	−	19.9972i	1.24727	−	1.43943i	0.393058	−	0.919514i	$-0.371417\pi$
$193$	17.3277	−	19.9972i	1.24727	−	1.43943i	0.854216	−	0.519918i	$-0.174037\pi$
$194$		0			0
$195$		0			0
$196$	15.5975	−	34.1538i	1.11411	−	2.43956i
$197$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$197$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$198$		0			0
$199$	1.12987	−	4.65738i	0.0800942	−	0.330153i	−0.917642	−	0.397409i	$-0.869909\pi$
$199$	1.12987	−	4.65738i	0.0800942	−	0.330153i	0.997736	+	0.0672563i	$0.0214245\pi$
$200$		0			0
$201$	5.56813	−	13.0382i	0.392745	−	0.919647i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−10.0310	+	10.5203i	−0.695527	+	0.729448i
$209$		0			0
$210$		0			0
$211$	−22.8217	−	17.9472i	−1.57111	−	1.23554i	−0.807524	−	0.589834i	$-0.799193\pi$
$211$	−22.8217	−	17.9472i	−1.57111	−	1.23554i	−0.763589	−	0.645703i	$-0.776565\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	29.3450	+	41.2094i	1.99207	+	2.79747i
$218$		0			0
$219$	−9.49218	+	0.452168i	−0.641422	+	0.0305547i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	3.14969	−	21.9066i	0.210919	−	1.46697i	−0.559178	−	0.829048i	$-0.688883\pi$
$223$	3.14969	−	21.9066i	0.210919	−	1.46697i	0.770097	−	0.637927i	$-0.220208\pi$
$224$		0			0
$225$	12.6188	+	8.10961i	0.841254	+	0.540641i
$226$		0			0
$227$		0			0		0.0475819	−	0.998867i	$-0.484848\pi$
$227$		0			0		−0.0475819	+	0.998867i	$0.515152\pi$
$228$	7.21694	−	3.29587i	0.477954	−	0.218274i
$229$	23.2211	−	8.03689i	1.53449	−	0.531093i	0.576304	−	0.817235i	$-0.304494\pi$
$229$	23.2211	−	8.03689i	1.53449	−	0.531093i	0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$233$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−27.6380	−	5.32680i	−1.79528	−	0.346013i
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	7.88943	+	17.2754i	0.508203	+	1.11281i	0.973715	+	0.227768i	$0.0731428\pi$
$241$	7.88943	+	17.2754i	0.508203	+	1.11281i	−0.465512	+	0.885041i	$0.654130\pi$
$242$		0			0
$243$	−11.7810	+	10.2083i	−0.755750	+	0.654861i
$244$	−15.2462	+	23.7236i	−0.976041	+	1.51875i
$245$		0			0
$246$		0			0
$247$	−6.77978	−	4.82786i	−0.431387	−	0.307189i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$251$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$252$	−24.8124	+	17.6688i	−1.56303	+	1.11303i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−5.23309	+	15.1200i	−0.327068	+	0.945001i
$257$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$257$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$258$		0			0
$259$	16.8553	+	57.4040i	1.04734	+	3.56691i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$263$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−13.9702	+	8.53418i	−0.853369	+	0.521308i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	30.9757	−	4.45363i	1.88164	−	0.270539i	0.896595	−	0.442851i	$-0.146033\pi$
$271$	30.9757	−	4.45363i	1.88164	−	0.270539i	0.985042	+	0.172313i	$0.0551240\pi$
$272$		0			0
$273$	29.0669	+	13.2744i	1.75921	+	0.803405i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.0353	+	3.82752i	−0.783218	+	0.229974i	−0.648808	−	0.760952i	$-0.724732\pi$
$277$	−13.0353	+	3.82752i	−0.783218	+	0.229974i	−0.134410	+	0.990926i	$0.542914\pi$
$278$		0			0
$279$	−2.84171	−	29.7597i	−0.170129	−	1.78167i
$280$		0			0
$281$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$281$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$282$		0			0
$283$	−21.8251	−	6.40842i	−1.29737	−	0.380941i	−0.441091	−	0.897462i	$-0.645408\pi$
$283$	−21.8251	−	6.40842i	−1.29737	−	0.380941i	−0.856274	+	0.516522i	$0.827227\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	9.86097	−	13.8478i	0.580057	−	0.814576i
$290$		0			0
$291$	29.0516	+	11.6305i	1.70303	+	0.681792i
$292$	9.23112	+	5.93248i	0.540210	+	0.347172i
$293$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$293$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−6.43738	−	16.0798i	−0.371662	−	0.928368i
$301$	4.98028	+	20.5290i	0.287058	+	1.18327i
$302$		0			0
$303$		0			0
$304$	−8.99574	−	1.73379i	−0.515941	−	0.0994395i
$305$		0			0
$306$		0			0
$307$	−9.93642	−	28.7094i	−0.567101	−	1.63853i	−0.754539	−	0.656255i	$-0.772139\pi$
$307$	−9.93642	−	28.7094i	−0.567101	−	1.63853i	0.187437	−	0.982277i	$-0.439982\pi$
$308$		0			0
$309$	−34.8886	−	1.66195i	−1.98475	−	0.0945451i
$310$		0			0
$311$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$311$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$312$		0			0
$313$	−20.1846	−	2.90210i	−1.14090	−	0.164037i	−0.454146	−	0.890927i	$-0.650056\pi$
$313$	−20.1846	−	2.90210i	−1.14090	−	0.164037i	−0.686753	+	0.726891i	$0.740965\pi$
$314$		0			0
$315$		0			0
$316$	22.4283	+	23.5221i	1.26169	+	1.32322i
$317$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$317$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	17.9185	−	1.71101i	0.995472	−	0.0950560i
$325$	−11.2320	+	14.2827i	−0.623040	+	0.792259i
$326$		0			0
$327$	−11.8270	+	13.6490i	−0.654033	+	0.754794i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.82554	−	5.48078i	0.155306	−	0.301251i	−0.798327	−	0.602224i	$-0.794281\pi$
$331$	2.82554	−	5.48078i	0.155306	−	0.301251i	0.953633	+	0.300973i	$0.0973116\pi$
$332$		0			0
$333$	8.33497	−	34.3572i	0.456754	−	1.88276i
$334$		0			0
$335$		0			0
$336$	35.1728			1.91883
$337$	−35.6286	−	8.64340i	−1.94081	−	0.470836i	−0.983082	−	0.183168i	$-0.941365\pi$
$337$	−35.6286	−	8.64340i	−1.94081	−	0.470836i	−0.957730	−	0.287668i	$-0.907120\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	45.1716	+	39.1414i	2.43904	+	2.11344i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$347$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$348$		0			0
$349$	−28.6340	+	18.4020i	−1.53274	+	0.985034i	−0.543394	+	0.839478i	$0.682861\pi$
$349$	−28.6340	+	18.4020i	−1.53274	+	0.985034i	−0.989350	+	0.145556i	$0.953503\pi$
$350$		0			0
$351$	−10.9532	−	15.3816i	−0.584637	−	0.821007i
$352$		0			0
$353$		0			0		0.998867	−	0.0475819i	$-0.0151515\pi$
$353$		0			0		−0.998867	+	0.0475819i	$0.984848\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$359$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$360$		0			0
$361$	−0.654462	+	13.7388i	−0.0344453	+	0.723097i
$362$		0			0
$363$	18.0047	−	6.23148i	0.945001	−	0.327068i
$364$	−18.4490	−	31.9546i	−0.966991	−	1.67488i
$365$		0			0
$366$		0			0
$367$	7.23551	+	18.0734i	0.377691	+	0.943426i	0.988441	+	0.151603i	$0.0484435\pi$
$367$	7.23551	+	18.0734i	0.377691	+	0.943426i	−0.610751	+	0.791823i	$0.709132\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−17.2599	+	29.8951i	−0.894886	+	1.54999i
$373$	33.3959	−	19.2811i	1.72917	−	0.998338i	0.835773	−	0.549076i	$-0.185020\pi$
$373$	33.3959	−	19.2811i	1.72917	−	0.998338i	0.893400	−	0.449262i	$-0.148313\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6.22836	+	15.5577i	−0.319929	+	0.799145i	0.678014	+	0.735049i	$0.262841\pi$
$379$	−6.22836	+	15.5577i	−0.319929	+	0.799145i	−0.997944	+	0.0640964i	$0.979583\pi$
$380$		0			0
$381$	−23.0851	−	16.4388i	−1.18268	−	0.842185i
$382$		0			0
$383$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$383$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.51688	−	11.9774i	0.178773	−	0.608845i
$388$	−19.5356	−	30.3981i	−0.991772	−	1.54323i
$389$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$389$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.29651	−	20.3565i	0.466579	−	1.02166i	−0.519360	−	0.854556i	$-0.673830\pi$
$397$	9.29651	−	20.3565i	0.466579	−	1.02166i	0.985938	−	0.167109i	$-0.0534431\pi$
$398$		0			0
$399$	2.86611	+	19.9343i	0.143485	+	0.997961i
$400$	−4.71518	+	19.4362i	−0.235759	+	0.971812i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	36.2131			1.80390
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	27.8255	−	29.1825i	1.37588	−	1.44298i	0.631013	−	0.775772i	$-0.282639\pi$
$409$	27.8255	−	29.1825i	1.37588	−	1.44298i	0.744869	−	0.667211i	$-0.232512\pi$
$410$		0			0
$411$		0			0
$412$	31.7028	+	24.9314i	1.56188	+	1.22828i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	16.8403	+	4.94475i	0.824672	+	0.242146i
$418$		0			0
$419$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$419$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$420$		0			0
$421$	−28.8996	+	27.5557i	−1.40848	+	1.34298i	−0.545159	+	0.838333i	$0.683531\pi$
$421$	−28.8996	+	27.5557i	−1.40848	+	1.34298i	−0.863323	+	0.504652i	$0.831621\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−46.8769	−	54.0988i	−2.26853	−	2.61803i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−18.0000	−	10.3923i	−0.866025	−	0.500000i
$433$	−5.43786	+	28.2143i	−0.261327	+	1.35589i	0.579669	+	0.814852i	$0.303182\pi$
$433$	−5.43786	+	28.2143i	−0.261327	+	1.35589i	−0.840996	+	0.541041i	$0.818030\pi$
$434$		0			0
$435$		0			0
$436$	20.2664	−	4.91657i	0.970583	−	0.235461i
$437$		0			0
$438$		0			0
$439$	19.5867	−	33.9252i	0.934824	−	1.61916i	0.159877	−	0.987137i	$-0.448890\pi$
$439$	19.5867	−	33.9252i	0.934824	−	1.61916i	0.774947	−	0.632026i	$-0.217776\pi$
$440$		0			0
$441$	−18.4205	−	53.2226i	−0.877168	−	2.53441i
$442$		0			0
$443$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$443$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$444$	−30.8520	+	26.7334i	−1.46417	+	1.26871i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−33.0832	−	23.5584i	−1.56303	−	1.11303i
$449$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$449$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	8.87007	−	6.31635i	0.416752	−	0.296768i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.01518	+	20.2690i	−0.328156	+	0.948146i	0.653407	+	0.757007i	$0.273339\pi$
$457$	−7.01518	+	20.2690i	−0.328156	+	0.948146i	−0.981563	+	0.191139i	$0.938782\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$461$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$	4.01101	−	42.0052i	0.186407	−	1.95215i	−0.102362	−	0.994747i	$-0.532640\pi$
$463$	4.01101	−	42.0052i	0.186407	−	1.95215i	0.288769	−	0.957399i	$-0.406754\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$467$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$468$			21.8041i			1.00790i
$469$	−10.7947	−	40.1284i	−0.498454	−	1.85296i
$470$		0			0
$471$	12.7541	+	3.09412i	0.587680	+	0.142570i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−11.3998	−	1.08855i	−0.523057	−	0.0499459i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$479$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$480$		0			0
$481$	40.4701	+	14.0068i	1.84528	+	0.638656i
$482$		0			0
$483$		0			0
$484$	−21.1088	−	6.19812i	−0.959493	−	0.281733i
$485$		0			0
$486$		0			0
$487$	17.0636	−	0.812840i	0.773226	−	0.0368333i	0.342739	−	0.939430i	$-0.388645\pi$
$487$	17.0636	−	0.812840i	0.773226	−	0.0368333i	0.430486	+	0.902597i	$0.358342\pi$
$488$		0			0
$489$	4.28350	+	5.44691i	0.193707	+	0.246318i
$490$		0			0
$491$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$491$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	36.2580	−	16.5585i	1.62803	−	0.743498i
$497$		0			0
$498$		0			0
$499$	−3.26271	−	1.88372i	−0.146059	−	0.0843271i	0.425190	−	0.905104i	$-0.360207\pi$
$499$	−3.26271	−	1.88372i	−0.146059	−	0.0843271i	−0.571248	+	0.820777i	$0.693541\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$503$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.309135	−	0.178479i	0.0137292	−	0.00792655i
$508$	10.7030	+	30.9244i	0.474870	+	1.37205i
$509$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$509$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$	−21.0504	+	18.2403i	−0.931217	+	0.806904i
$512$		0			0
$513$	4.42311	−	11.0484i	0.195285	−	0.487798i
$514$		0			0
$515$		0			0
$516$	−11.3303	+	8.91026i	−0.498789	+	0.392252i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$521$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$522$		0			0
$523$	−4.92085	+	0.948415i	−0.215174	+	0.0414713i	−0.295699	−	0.955281i	$-0.595552\pi$
$523$	−4.92085	+	0.948415i	−0.215174	+	0.0414713i	0.0805251	+	0.996753i	$0.474340\pi$
$524$		0			0
$525$	43.7669	−	4.17923i	1.91014	−	0.182397i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	16.6459	+	15.8718i	0.723734	+	0.690079i
$530$		0			0
$531$		0			0
$532$	10.6560	−	20.6697i	0.461995	−	0.896145i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−24.7751	−	11.3144i	−1.06517	−	0.486445i	−0.195816	−	0.980641i	$-0.562736\pi$
$541$	−24.7751	−	11.3144i	−1.06517	−	0.486445i	−0.869351	+	0.494195i	$0.835463\pi$
$542$		0			0
$543$	31.9282	−	33.4853i	1.37017	−	1.43699i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.306466	+	3.20945i	0.0131035	+	0.137226i	0.999656	−	0.0262168i	$-0.00834602\pi$
$547$	0.306466	+	3.20945i	0.0131035	+	0.137226i	−0.986553	+	0.163443i	$0.947740\pi$
$548$		0			0
$549$	8.00542	+	41.5361i	0.341663	+	1.77272i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−73.3287	+	37.8036i	−3.11825	+	1.60757i
$554$		0			0
$555$		0			0
$556$	−12.5279	−	15.9305i	−0.531300	−	0.675603i
$557$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$557$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$558$		0			0
$559$	14.0380	+	5.61999i	0.593746	+	0.237700i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$563$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−8.64703	+	44.8651i	−0.363141	+	1.88416i
$568$		0			0
$569$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$569$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$570$		0			0
$571$	−14.1677	+	5.67190i	−0.592901	+	0.237362i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−14.1677	+	5.67190i	−0.592901	+	0.237362i	0.0557537	+	0.998445i	$0.482244\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	9.96996	+	21.8312i	0.415415	+	0.909632i
$577$	−29.3997	−	1.40048i	−1.22392	−	0.0583027i	−0.574355	−	0.818606i	$-0.694747\pi$
$577$	−29.3997	−	1.40048i	−1.22392	−	0.0583027i	−0.649568	+	0.760303i	$0.725050\pi$
$578$		0			0
$579$	−24.7777	+	38.5549i	−1.02973	+	1.60229i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$587$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$588$	−18.3219	+	62.3987i	−0.755583	+	2.57328i
$589$	12.3391	+	19.2000i	0.508424	+	0.791124i
$590$		0			0
$591$		0			0
$592$	46.9249	−	4.48079i	1.92860	−	0.184159i
$593$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$593$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.789042	+	8.26322i	−0.0322933	+	0.338191i
$598$		0			0
$599$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$599$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$600$		0			0
$601$	7.03356	−	28.9927i	0.286905	−	1.18264i	−0.627754	−	0.778411i	$-0.716026\pi$
$601$	7.03356	−	28.9927i	0.286905	−	1.18264i	0.914659	−	0.404226i	$-0.132459\pi$
$602$		0			0
$603$	−6.33223	+	23.7256i	−0.257868	+	0.966180i
$604$	−12.5737			−0.511619
$605$		0			0
$606$		0			0
$607$	34.7079	+	17.8932i	1.40875	+	0.726261i	0.983853	−	0.178981i	$-0.0572799\pi$
$607$	34.7079	+	17.8932i	1.40875	+	0.726261i	0.424897	+	0.905242i	$0.360310\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−37.7044	−	29.6510i	−1.52286	−	1.19759i	−0.914934	−	0.403603i	$-0.867758\pi$
$613$	−37.7044	−	29.6510i	−1.52286	−	1.19759i	−0.607930	−	0.793990i	$-0.708000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$617$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$618$		0			0
$619$	−16.3767	−	22.9979i	−0.658237	−	0.924364i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	−16.3767	−	22.9979i	−0.658237	−	0.924364i	−0.999880	+	0.0154656i	$0.995077\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	14.6042	−	20.5088i	0.584637	−	0.821007i
$625$	−3.55787	+	24.7455i	−0.142315	+	0.989821i
$626$		0			0
$627$		0			0
$628$	−9.92402	−	11.4529i	−0.396012	−	0.457022i
$629$		0			0
$630$		0			0
$631$	−37.8957	+	13.1158i	−1.50860	+	0.522132i	−0.951324	−	0.308193i	$-0.900276\pi$
$631$	−37.8957	+	13.1158i	−1.50860	+	0.522132i	−0.557279	+	0.830326i	$0.688154\pi$
$632$		0			0
$633$	43.5500	+	25.1436i	1.73096	+	0.999368i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	66.2998	−	16.0842i	2.62689	−	0.637277i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	6.97674	+	15.2769i	0.275136	+	0.602463i	0.995874	−	0.0907437i	$-0.0289244\pi$
$643$	6.97674	+	15.2769i	0.275136	+	0.602463i	−0.720738	+	0.693207i	$0.756197\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$647$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−60.4677	−	63.4167i	−2.36992	−	2.48550i
$652$	−0.380724	−	7.99238i	−0.0149103	−	0.313006i
$653$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$653$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	16.1621	−	3.11500i	0.630545	−	0.121528i
$658$		0			0
$659$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$659$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$660$		0			0
$661$	−3.76889	−	12.8357i	−0.146593	−	0.499250i	0.853156	−	0.521656i	$-0.174685\pi$
$661$	−3.76889	−	12.8357i	−0.146593	−	0.499250i	−0.999749	+	0.0224061i	$0.992867\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			38.3335i			1.48206i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.94884	−	1.28665i	0.344952	−	0.0495967i	0.0323386	−	0.999477i	$-0.489705\pi$
$673$	8.94884	−	1.28665i	0.344952	−	0.0495967i	0.312614	+	0.949880i	$0.398795\pi$
$674$		0			0
$675$	−23.6329	−	10.7928i	−0.909632	−	0.415415i
$676$	−0.410314	−	0.0391803i	−0.0157813	−	0.00150693i
$677$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$677$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$678$		0			0
$679$	88.0069	−	25.8411i	3.37739	−	0.991692i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.189251	−	0.981929i	$-0.560606\pi$
$683$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$684$	−11.5605	+	7.42945i	−0.442025	+	0.284072i
$685$		0			0
$686$		0			0
$687$	−37.8296	+	19.5025i	−1.44329	+	0.744068i
$688$	16.6252	−	0.791957i	0.633830	−	0.0301931i
$689$		0			0
$690$		0			0
$691$	25.8176	−	36.2558i	0.982149	−	1.37924i	0.0575291	−	0.998344i	$-0.481678\pi$
$691$	25.8176	−	36.2558i	0.982149	−	1.37924i	0.924620	−	0.380891i	$-0.124383\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−43.9660	−	25.3838i	−1.66176	−	0.959416i
$701$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$701$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$702$		0			0
$703$	6.36326	+	26.2297i	0.239995	+	0.989272i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	16.7245	+	48.3223i	0.628103	+	1.81478i	0.577995	+	0.816040i	$0.303835\pi$
$709$	16.7245	+	48.3223i	0.628103	+	1.81478i	0.0501074	+	0.998744i	$0.484044\pi$
$710$		0			0
$711$	48.6963	+	2.31969i	1.82625	+	0.0869952i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$719$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$720$		0			0
$721$	−83.3937	+	59.3844i	−3.10574	+	2.21159i
$722$		0			0
$723$	−17.7841	−	27.6727i	−0.661399	−	1.02916i
$724$	−52.4596	+	10.1108i	−1.94964	+	0.375763i
$725$		0			0
$726$		0			0
$727$	26.2512	−	33.3811i	0.973602	−	1.23804i	0.00234654	−	0.999997i	$-0.499253\pi$
$727$	26.2512	−	33.3811i	0.973602	−	1.23804i	0.971256	−	0.238039i	$-0.0765045\pi$
$728$		0			0
$729$	17.6812	−	20.4052i	0.654861	−	0.755750i
$730$		0			0
$731$		0			0
$732$	20.2907	−	44.4304i	0.749966	−	1.64220i
$733$	−2.68340	+	5.20507i	−0.0991137	+	0.192254i	−0.933179	−	0.359412i	$-0.882977\pi$
$733$	−2.68340	+	5.20507i	−0.0991137	+	0.192254i	0.834065	+	0.551666i	$0.186008\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−17.9341	−	4.35077i	−0.659717	−	0.160046i	−0.108096	−	0.994140i	$-0.534475\pi$
$739$	−17.9341	−	4.35077i	−0.659717	−	0.160046i	−0.551621	+	0.834095i	$0.685990\pi$
$740$		0			0
$741$	12.8134	+	6.60579i	0.470714	+	0.242670i
$742$		0			0
$743$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$743$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−28.9231	+	18.5878i	−1.05542	+	0.678278i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	−28.9231	+	18.5878i	−1.05542	+	0.678278i	−0.106667	+	0.994295i	$0.534018\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	38.1836	−	36.4080i	1.38872	−	1.32415i
$757$	−31.3711	−	39.8916i	−1.14020	−	1.44989i	−0.873972	−	0.485977i	$-0.838464\pi$
$757$	−31.3711	−	39.8916i	−1.14020	−	1.44989i	−0.266231	−	0.963909i	$-0.585778\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$761$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$	−2.51879	+	52.8759i	−0.0911862	+	1.91423i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	5.24468	−	27.2120i	0.189251	−	0.981929i
$769$	−7.43992	−	18.5840i	−0.268290	−	0.670157i	0.731643	−	0.681688i	$-0.238754\pi$
$769$	−7.43992	−	18.5840i	−0.268290	−	0.670157i	−0.999934	+	0.0115309i	$0.996330\pi$
$770$		0			0
$771$		0			0
$772$	49.1294	−	19.6685i	1.76821	−	0.707883i
$773$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$773$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$774$		0			0
$775$	43.1498	−	24.9126i	1.54999	−	0.894886i
$776$		0			0
$777$	−43.0470	−	94.2599i	−1.54430	−	3.38155i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	59.0276	−	46.4198i	2.10813	−	1.65785i
$785$		0			0
$786$		0			0
$787$	20.4325	+	39.6336i	0.728341	+	1.41279i	0.903650	+	0.428272i	$0.140877\pi$
$787$	20.4325	+	39.6336i	0.728341	+	1.41279i	−0.175309	+	0.984514i	$0.556092\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−51.0082	+	4.87070i	−1.81136	+	0.172963i
$794$		0			0
$795$		0			0
$796$	6.27680	−	7.24382i	0.222475	−	0.256750i
$797$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$797$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	21.8472	−	18.0748i	0.770493	−	0.637449i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$809$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$810$		0			0
$811$	−0.271593	+	0.284838i	−0.00953690	+	0.0100020i	−0.728485	−	0.685062i	$-0.759775\pi$
$811$	−0.271593	+	0.284838i	−0.00953690	+	0.0100020i	0.718948	+	0.695064i	$0.244624\pi$
$812$		0			0
$813$	−52.0075	+	15.2708i	−1.82398	+	0.535570i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.80358	+	9.35785i	0.0630992	+	0.327390i
$818$		0			0
$819$	−53.1051	−	15.5931i	−1.85564	−	0.544866i
$820$		0			0
$821$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$821$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$822$		0			0
$823$	−8.18233	+	7.80184i	−0.285218	+	0.271955i	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−8.18233	+	7.80184i	−0.285218	+	0.271955i	0.533940	+	0.845522i	$0.320711\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$827$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$828$		0			0
$829$	37.1698	+	42.8962i	1.29096	+	1.48985i	0.772939	+	0.634481i	$0.218786\pi$
$829$	37.1698	+	42.8962i	1.29096	+	1.48985i	0.518022	+	0.855367i	$0.326668\pi$
$830$		0			0
$831$	21.4046	−	9.77515i	0.742517	−	0.339096i
$832$	−27.4732	+	9.50857i	−0.952462	+	0.329650i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	12.2075	+	50.3202i	0.421955	+	1.73932i
$838$		0			0
$839$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$839$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	−24.1218	−	52.8193i	−0.830305	−	1.81811i
$845$		0			0
$846$		0			0
$847$	30.1917	−	46.9792i	1.03740	−	1.61422i
$848$		0			0
$849$	38.9970	+	5.60692i	1.33837	+	0.192429i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.21415	+	25.4882i	0.0415718	+	0.872699i	0.919304	+	0.393548i	$0.128752\pi$
$853$	1.21415	+	25.4882i	0.0415718	+	0.872699i	−0.877732	+	0.479151i	$0.840945\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$857$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$858$		0			0
$859$	0.637099	−	1.84078i	0.0217376	−	0.0628065i	−0.933610	−	0.358291i	$-0.883360\pi$
$859$	0.637099	−	1.84078i	0.0217376	−	0.0628065i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$863$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−13.4924	+	26.1716i	−0.458227	+	0.888835i
$868$	14.3994	+	100.150i	0.488747	+	3.39931i
$869$		0			0
$870$		0			0
$871$	−27.8805	−	10.3676i	−0.944693	−	0.351294i
$872$		0			0
$873$	−52.6735	−	12.7785i	−1.78273	−	0.432485i
$874$		0			0
$875$		0			0
$876$	−17.2884	−	7.89533i	−0.584120	−	0.266758i
$877$	8.81998	+	0.842206i	0.297829	+	0.0284393i	0.242901	−	0.970051i	$-0.421901\pi$
$877$	8.81998	+	0.842206i	0.297829	+	0.0284393i	0.0549283	+	0.998490i	$0.482507\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$881$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$882$		0			0
$883$	15.4030	+	5.33103i	0.518352	+	0.179403i	0.573703	−	0.819064i	$-0.305506\pi$
$883$	15.4030	+	5.33103i	0.518352	+	0.179403i	−0.0553507	+	0.998467i	$0.517628\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$887$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$888$		0			0
$889$	−82.9722	+	3.95245i	−2.78280	+	0.132561i
$890$		0			0
$891$		0			0
$892$	25.6755	−	36.0562i	0.859679	−	1.20725i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	15.0000	+	25.9808i	0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−13.5986	−	33.9677i	−0.452534	−	1.13037i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	23.9072	+	4.60773i	0.793824	+	0.152997i	0.570013	−	0.821635i	$-0.306938\pi$
$907$	23.9072	+	4.60773i	0.793824	+	0.152997i	0.223811	+	0.974633i	$0.428150\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$911$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$912$	15.8499	+	0.755022i	0.524841	+	0.0250013i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	48.6448	+	6.99407i	1.60727	+	0.231091i
$917$		0			0
$918$		0			0
$919$	−41.5788	−	43.6066i	−1.37156	−	1.43845i	−0.766448	−	0.642307i	$-0.777978\pi$
$919$	−41.5788	−	43.6066i	−1.37156	−	1.43845i	−0.605110	−	0.796142i	$-0.706871\pi$
$920$		0			0
$921$	24.1120	+	46.7707i	0.794517	+	1.54115i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	57.8582	−	11.1512i	1.90236	−	0.366651i
$926$		0			0
$927$	60.2235	−	5.75065i	1.97800	−	0.188876i
$928$		0			0
$929$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$929$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$930$		0			0
$931$	31.1185	+	29.6714i	1.01987	+	0.972442i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	28.6078i		−	0.934576i	−0.884105	−	0.467288i	$-0.845231\pi$
$937$		−	28.6078i		−	0.934576i	0.884105	−	0.467288i	$-0.154769\pi$
$938$		0			0
$939$	35.3202			1.15263
$940$		0			0
$941$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$941$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$947$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$948$	−44.2496	−	34.7983i	−1.43716	−	1.13019i
$949$	1.89524	+	19.8479i	0.0615221	+	0.644289i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$953$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−63.4091	−	25.3852i	−2.04545	−	0.818877i
$962$		0			0
$963$		0			0
$964$	−1.80732	+	37.9403i	−0.0582099	+	1.22198i
$965$		0			0
$966$		0			0
$967$	20.2272	+	35.0346i	0.650464	+	1.12664i	0.983010	+	0.183550i	$0.0587588\pi$
$967$	20.2272	+	35.0346i	0.650464	+	1.12664i	−0.332547	+	0.943087i	$0.607908\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$971$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$972$	−30.2981	+	7.35024i	−0.971812	+	0.235759i
$973$	47.7587	−	19.1197i	1.53107	−	0.612950i
$974$		0			0
$975$	15.7358	−	27.2551i	0.503948	−	0.872863i
$976$	−48.8444	+	28.2003i	−1.56347	+	0.902670i
$977$		0			0		−0.327068	−	0.945001i	$-0.606061\pi$
$977$		0			0		0.327068	+	0.945001i	$0.393939\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	16.9120	−	26.3155i	0.539957	−	0.840190i
$982$		0			0
$983$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$983$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−7.62771	−	14.7957i	−0.242670	−	0.470714i
$989$		0			0
$990$		0			0
$991$	−11.5245	−	17.9324i	−0.366086	−	0.569642i	0.608528	−	0.793533i	$-0.291760\pi$
$991$	−11.5245	−	17.9324i	−0.366086	−	0.569642i	−0.974614	+	0.223891i	$0.928124\pi$
$992$		0			0
$993$	−3.49317	+	10.0929i	−0.110852	+	0.320287i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−39.4855	+	45.5687i	−1.25052	+	1.44317i	−0.400606	+	0.916251i	$0.631200\pi$
$997$	−39.4855	+	45.5687i	−1.25052	+	1.44317i	−0.849912	+	0.526924i	$0.823345\pi$
$998$		0			0
$999$	−5.82071	+	60.9573i	−0.184159	+	1.92860i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.a.32.1	✓	20
3.2	odd	2	CM	201.2.p.a.32.1	✓	20
67.44	odd	66	inner	201.2.p.a.44.1	yes	20
201.44	even	66	inner	201.2.p.a.44.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.a.32.1	✓	20	1.1	even	1	trivial
201.2.p.a.32.1	✓	20	3.2	odd	2	CM
201.2.p.a.44.1	yes	20	67.44	odd	66	inner
201.2.p.a.44.1	yes	20	201.44	even	66	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.p (of order \(66\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.71442 + 0.246497i) q^{3} +(1.77767 + 0.916453i) q^{4} +(-3.50336 + 3.67422i) q^{7} +(2.87848 - 0.845198i) q^{9} +O(q^{10})\)	\(q+(-1.71442 + 0.246497i) q^{3} +(1.77767 + 0.916453i) q^{4} +(-3.50336 + 3.67422i) q^{7} +(2.87848 - 0.845198i) q^{9} +(-3.27358 - 1.13300i) q^{12} +(0.687743 + 3.56835i) q^{13} +(2.32023 + 3.25830i) q^{16} +(-1.65758 + 1.58050i) q^{19} +(5.10055 - 7.16272i) q^{21} +(3.27430 + 3.77875i) q^{25} +(-4.72659 + 2.15856i) q^{27} +(-9.59507 + 3.32089i) q^{28} +(1.88589 - 9.78494i) q^{31} +(5.89157 + 1.13551i) q^{36} +(5.89230 - 10.2058i) q^{37} +(-2.05867 - 5.94812i) q^{39} +(2.24962 - 3.50047i) q^{43} +(-4.78101 - 5.01418i) q^{48} +(-0.893274 - 18.7521i) q^{49} +(-2.04764 + 6.97363i) q^{52} +(2.45221 - 3.11824i) q^{57} +(-1.34031 + 14.0363i) q^{61} +(-6.97891 + 13.5372i) q^{63} +(1.13852 + 7.91857i) q^{64} +(-4.25333 + 6.99351i) q^{67} +(5.46168 + 0.521527i) q^{73} +(-6.54498 - 5.67126i) q^{75} +(-4.39510 + 1.29052i) q^{76} +(15.3567 + 5.31502i) q^{79} +(7.57128 - 4.86577i) q^{81} +(15.6314 - 8.05855i) q^{84} +(-15.5203 - 9.97429i) q^{91} +(-0.821260 + 17.2404i) q^{93} +(-15.6466 - 9.03356i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{4} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 20 * q - 2 * q^4 - 6 * q^7 + 6 * q^9	\( 20 q - 2 q^{4} - 6 q^{7} + 6 q^{9} - 6 q^{12} - 3 q^{13} + 4 q^{16} - 8 q^{19} + 6 q^{21} + 10 q^{25} - 12 q^{28} + 15 q^{31} + 6 q^{36} + 10 q^{37} - 3 q^{39} - 12 q^{48} - 5 q^{49} - 154 q^{52} - 75 q^{57} + 15 q^{61} - 15 q^{63} + 16 q^{64} + 5 q^{67} + 60 q^{73} - 32 q^{76} + 134 q^{79} - 18 q^{81} + 186 q^{84} - 12 q^{91} - 15 q^{93} + 9 q^{97}+O(q^{100}) \) 20 * q - 2 * q^4 - 6 * q^7 + 6 * q^9 - 6 * q^12 - 3 * q^13 + 4 * q^16 - 8 * q^19 + 6 * q^21 + 10 * q^25 - 12 * q^28 + 15 * q^31 + 6 * q^36 + 10 * q^37 - 3 * q^39 - 12 * q^48 - 5 * q^49 - 154 * q^52 - 75 * q^57 + 15 * q^61 - 15 * q^63 + 16 * q^64 + 5 * q^67 + 60 * q^73 - 32 * q^76 + 134 * q^79 - 18 * q^81 + 186 * q^84 - 12 * q^91 - 15 * q^93 + 9 * q^97

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