LMFDB - Embedded newform 201.2.p.a.113.1

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			113.1
Root			$-0.995472 + 0.0950560i$ of defining polynomial
Character	$\chi$	$=$	201.113
Dual form			201.2.p.a.185.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.57553 - 0.719520i) q^{3} +(-1.85674 + 0.743325i) q^{4} +(4.20741 + 1.45620i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})$	$q+(-1.57553 - 0.719520i) q^{3} +(-1.85674 + 0.743325i) q^{4} +(4.20741 + 1.45620i) q^{7} +(1.96458 + 2.26725i) q^{9} +(3.46018 + 0.164829i) q^{12} +(1.90439 + 3.69400i) q^{13} +(2.89494 - 2.76032i) q^{16} +(2.11756 + 6.11829i) q^{19} +(-5.58113 - 5.32159i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(-1.46393 - 4.98567i) q^{27} +(-8.89447 + 0.423696i) q^{28} +(2.90393 - 5.63284i) q^{31} +(-5.33301 - 2.74936i) q^{36} +(-6.07910 + 10.5293i) q^{37} +(-0.342513 - 7.19024i) q^{39} +(7.42015 - 1.06686i) q^{43} +(-6.54716 + 2.26599i) q^{48} +(10.0794 + 7.92652i) q^{49} +(-6.28178 - 5.44320i) q^{52} +(1.06595 - 11.1632i) q^{57} +(-7.99569 - 1.93973i) q^{61} +(4.96423 + 12.4001i) q^{63} +(-3.32332 + 7.27706i) q^{64} +(2.11074 - 7.90852i) q^{67} +(3.96060 - 16.3258i) q^{73} +(4.68209 + 7.28547i) q^{75} +(-8.47963 - 9.78602i) q^{76} +(13.5468 + 0.645314i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(14.3184 + 5.73220i) q^{84} +(2.63334 + 18.3153i) q^{91} +(-8.62817 + 6.78526i) q^{93} +(-16.8369 - 9.72078i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{29}{66}\right)$

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		−0.189251	−	0.981929i	$-0.560606\pi$
$2$		0			0		0.189251	+	0.981929i	$0.439394\pi$
$3$	−1.57553	−	0.719520i	−0.909632	−	0.415415i
$3$	−1.57553	−	0.719520i	−0.909632	−	0.415415i
$4$	−1.85674	+	0.743325i	−0.928368	+	0.371662i
$5$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$5$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$6$		0			0
$7$	4.20741	+	1.45620i	1.59025	+	0.550391i	0.971876	−	0.235493i	$-0.0756704\pi$
$7$	4.20741	+	1.45620i	1.59025	+	0.550391i	0.618374	+	0.785884i	$0.287792\pi$
$8$		0			0
$9$	1.96458	+	2.26725i	0.654861	+	0.755750i
$10$		0			0
$11$		0			0		0.971812	−	0.235759i	$-0.0757576\pi$
$11$		0			0		−0.971812	+	0.235759i	$0.924242\pi$
$12$	3.46018	+	0.164829i	0.998867	+	0.0475819i
$13$	1.90439	+	3.69400i	0.528182	+	1.02453i	0.990461	+	0.137795i	$0.0440016\pi$
$13$	1.90439	+	3.69400i	0.528182	+	1.02453i	−0.462279	+	0.886735i	$0.652968\pi$
$14$		0			0
$15$		0			0
$16$	2.89494	−	2.76032i	0.723734	−	0.690079i
$17$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$17$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$18$		0			0
$19$	2.11756	+	6.11829i	0.485802	+	1.40363i	0.875736	+	0.482790i	$0.160376\pi$
$19$	2.11756	+	6.11829i	0.485802	+	1.40363i	−0.389934	+	0.920843i	$0.627502\pi$
$20$		0			0
$21$	−5.58113	−	5.32159i	−1.21790	−	1.16127i
$22$		0			0
$23$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$23$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$24$		0			0
$25$	−4.20627	−	2.70320i	−0.841254	−	0.540641i
$26$		0			0
$27$	−1.46393	−	4.98567i	−0.281733	−	0.959493i
$28$	−8.89447	+	0.423696i	−1.68090	+	0.0800710i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	2.90393	−	5.63284i	0.521561	−	1.01169i	−0.470075	−	0.882626i	$-0.655773\pi$
$31$	2.90393	−	5.63284i	0.521561	−	1.01169i	0.991637	−	0.129061i	$-0.0411964\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.33301	−	2.74936i	−0.888835	−	0.458227i
$37$	−6.07910	+	10.5293i	−0.999398	+	1.73101i	−0.469661	+	0.882847i	$0.655624\pi$
$37$	−6.07910	+	10.5293i	−0.999398	+	1.73101i	−0.529738	+	0.848162i	$0.677710\pi$
$38$		0			0
$39$	−0.342513	−	7.19024i	−0.0548460	−	1.15136i
$40$		0			0
$41$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$41$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$42$		0			0
$43$	7.42015	−	1.06686i	1.13156	−	0.162694i	0.449007	−	0.893528i	$-0.351778\pi$
$43$	7.42015	−	1.06686i	1.13156	−	0.162694i	0.682555	+	0.730834i	$0.260869\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$47$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$48$	−6.54716	+	2.26599i	−0.945001	+	0.327068i
$49$	10.0794	+	7.92652i	1.43991	+	1.13236i
$50$		0			0
$51$		0			0
$52$	−6.28178	−	5.44320i	−0.871127	−	0.754835i
$53$		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
$57$	1.06595	−	11.1632i	0.141189	−	1.47860i
$58$		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
$61$	−7.99569	−	1.93973i	−1.02374	−	0.248357i	−0.311467	−	0.950257i	$-0.600820\pi$
$61$	−7.99569	−	1.93973i	−1.02374	−	0.248357i	−0.712276	+	0.701899i	$0.752336\pi$
$62$		0			0
$63$	4.96423	+	12.4001i	0.625435	+	1.56226i
$64$	−3.32332	+	7.27706i	−0.415415	+	0.909632i
$65$		0			0
$66$		0			0
$67$	2.11074	−	7.90852i	0.257868	−	0.966180i
$67$	2.11074	−	7.90852i	0.257868	−	0.966180i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.928368	−	0.371662i	$-0.121212\pi$
$71$		0			0		−0.928368	+	0.371662i	$0.878788\pi$
$72$		0			0
$73$	3.96060	−	16.3258i	0.463553	−	1.91079i	0.0539089	−	0.998546i	$-0.482832\pi$
$73$	3.96060	−	16.3258i	0.463553	−	1.91079i	0.409644	−	0.912245i	$-0.365653\pi$
$74$		0			0
$75$	4.68209	+	7.28547i	0.540641	+	0.841254i
$76$	−8.47963	−	9.78602i	−0.972680	−	1.12253i
$77$		0			0
$78$		0			0
$79$	13.5468	+	0.645314i	1.52414	+	0.0726035i	0.792829	−	0.609445i	$-0.208608\pi$
$79$	13.5468	+	0.645314i	1.52414	+	0.0726035i	0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$	−1.28083	+	8.90839i	−0.142315	+	0.989821i
$82$		0			0
$83$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$83$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$84$	14.3184	+	5.73220i	1.56226	+	0.625435i
$85$		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
$91$	2.63334	+	18.3153i	0.276049	+	1.91997i
$92$		0			0
$93$	−8.62817	+	6.78526i	−0.894699	+	0.703599i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−16.8369	−	9.72078i	−1.70953	−	0.986996i	−0.935136	−	0.354289i	$-0.884723\pi$
$97$	−16.8369	−	9.72078i	−1.70953	−	0.986996i	−0.774391	−	0.632707i	$-0.781944\pi$
$98$		0			0
$99$		0			0
$100$	9.81929	+	1.89251i	0.981929	+	0.189251i
$101$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$101$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$102$		0			0
$103$	−9.31430	−	4.80186i	−0.917766	−	0.473141i	−0.0664469	−	0.997790i	$-0.521166\pi$
$103$	−9.31430	−	4.80186i	−0.917766	−	0.473141i	−0.851319	+	0.524649i	$0.824197\pi$
$104$		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	$0.909091\pi$
$108$	6.42410	+	8.16890i	0.618159	+	0.786053i
$109$	−8.87878	+	13.8156i	−0.850432	+	1.32330i	0.0943294	+	0.995541i	$0.469929\pi$
$109$	−8.87878	+	13.8156i	−0.850432	+	1.32330i	−0.944762	+	0.327758i	$0.893707\pi$
$110$		0			0
$111$	17.1539	−	12.2152i	1.62817	−	1.15942i
$112$	16.1997	−	7.39817i	1.53073	−	0.699062i
$113$		0			0		0.690079	−	0.723734i	$-0.257576\pi$
$113$		0			0		−0.690079	+	0.723734i	$0.742424\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.63388	+	11.5749i	−0.428403	+	1.07010i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.77719	−	5.04049i	0.888835	−	0.458227i
$122$		0			0
$123$		0			0
$124$	−1.20480	+	12.6173i	−0.108194	+	1.13306i
$125$		0			0
$126$		0			0
$127$	3.70639	−	10.7089i	0.328889	−	0.950262i	−0.652426	−	0.757853i	$-0.726249\pi$
$127$	3.70639	−	10.7089i	0.328889	−	0.950262i	0.981315	−	0.192410i	$-0.0616302\pi$
$128$		0			0
$129$	−12.4583	−	3.65808i	−1.09689	−	0.322076i
$130$		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
$133$			28.8257i			2.49951i
$134$		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
$139$	0.841348	−	2.86537i	0.0713622	−	0.243037i	−0.916086	−	0.400983i	$-0.868669\pi$
$139$	0.841348	−	2.86537i	0.0713622	−	0.243037i	0.987448	+	0.157945i	$0.0504870\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	11.9457	+	1.14067i	0.995472	+	0.0950560i
$145$		0			0
$146$		0			0
$147$	−10.1771	−	19.7408i	−0.839392	−	1.62819i
$148$	3.46059	−	24.0689i	0.284458	−	1.97845i
$149$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$149$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$150$		0			0
$151$	21.0017	+	8.40780i	1.70909	+	0.684217i	0.999786	−	0.0206838i	$-0.00658434\pi$
$151$	21.0017	+	8.40780i	1.70909	+	0.684217i	0.709306	+	0.704901i	$0.249009\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	5.98064	+	13.0958i	0.478835	+	1.04850i
$157$	−14.4295	−	20.2634i	−1.15160	−	1.61720i	−0.674308	−	0.738451i	$-0.735558\pi$
$157$	−14.4295	−	20.2634i	−1.15160	−	1.61720i	−0.477292	−	0.878745i	$-0.658382\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.21964	−	14.2368i	−0.643812	−	1.11511i	−0.984575	−	0.174966i	$-0.944019\pi$
$163$	−8.21964	−	14.2368i	−0.643812	−	1.11511i	0.340763	−	0.940149i	$-0.389315\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$167$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$168$		0			0
$169$	−2.47817	+	3.48011i	−0.190629	+	0.267701i
$170$		0			0
$171$	−9.71157	+	16.8209i	−0.742662	+	1.28633i
$172$	−12.9842	+	7.49645i	−0.990039	+	0.571599i
$173$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$173$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$174$		0			0
$175$	−13.7611	−	17.4986i	−1.04024	−	1.32277i
$176$		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
$181$	−20.6312	+	1.97004i	−1.53350	+	0.146432i	−0.827374	−	0.561651i	$-0.810166\pi$
$181$	−20.6312	+	1.97004i	−1.53350	+	0.146432i	−0.706129	+	0.708083i	$0.749560\pi$
$182$		0			0
$183$	11.2018	+	8.80917i	0.828058	+	0.651192i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.10079	−	23.1085i	0.0800710	−	1.68090i
$190$		0			0
$191$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$191$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$192$	10.4720	−	9.07402i	0.755750	−	0.654861i
$193$	−2.76001	+	1.77375i	−0.198670	+	0.127677i	−0.636193	−	0.771530i	$-0.719492\pi$
$193$	−2.76001	+	1.77375i	−0.198670	+	0.127677i	0.437524	+	0.899207i	$0.355856\pi$
$194$		0			0
$195$		0			0
$196$	−24.6067	−	7.22519i	−1.75762	−	0.516085i
$197$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$197$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$198$		0			0
$199$	−23.7144	+	4.57058i	−1.68107	+	0.324000i	−0.938372	−	0.345627i	$-0.887666\pi$
$199$	−23.7144	+	4.57058i	−1.68107	+	0.324000i	−0.742698	+	0.669626i	$0.766454\pi$
$200$		0			0
$201$	−9.01588	+	10.9414i	−0.635931	+	0.771746i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	15.7097	+	5.43717i	1.08927	+	0.377000i
$209$		0			0
$210$		0			0
$211$	5.13045	+	0.489898i	0.353194	+	0.0337260i	0.270146	−	0.962819i	$-0.412928\pi$
$211$	5.13045	+	0.489898i	0.353194	+	0.0337260i	0.0830489	+	0.996545i	$0.473534\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	20.4205	−	19.4710i	1.38624	−	1.32177i
$218$		0			0
$219$	−17.9868	+	22.8721i	−1.21543	+	1.54555i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.23038	−	7.07355i	−0.216322	−	0.473680i	0.770097	−	0.637927i	$-0.220208\pi$
$223$	−3.23038	−	7.07355i	−0.216322	−	0.473680i	−0.986419	+	0.164247i	$0.947481\pi$
$224$		0			0
$225$	−2.13472	−	14.8473i	−0.142315	−	0.989821i
$226$		0			0
$227$		0			0		0.786053	−	0.618159i	$-0.212121\pi$
$227$		0			0		−0.786053	+	0.618159i	$0.787879\pi$
$228$	6.31867	+	21.5194i	0.418464	+	1.42516i
$229$	28.5227	−	1.35871i	1.88484	−	0.0897858i	0.926649	−	0.375929i	$-0.122676\pi$
$229$	28.5227	−	1.35871i	1.88484	−	0.0897858i	0.958187	+	0.286143i	$0.0923732\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.814576	−	0.580057i	$-0.803030\pi$
$233$		0			0		0.814576	+	0.580057i	$0.196970\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−20.8791	−	10.7639i	−1.35624	−	0.699191i
$238$		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
$241$	24.1240	−	7.08344i	1.55396	−	0.456285i	0.611681	−	0.791104i	$-0.290494\pi$
$241$	24.1240	−	7.08344i	1.55396	−	0.456285i	0.942282	+	0.334820i	$0.108675\pi$
$242$		0			0
$243$	8.42776	−	13.1138i	0.540641	−	0.841254i
$244$	16.2877	−	2.34182i	1.04272	−	0.149920i
$245$		0			0
$246$		0			0
$247$	−18.5683	+	19.4739i	−1.18147	+	1.23909i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$251$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$252$	−18.4345	−	19.3336i	−1.16127	−	1.21790i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.761311	−	15.9819i	0.0475819	−	0.998867i
$257$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$257$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$258$		0			0
$259$	−40.9100	+	35.4487i	−2.54203	+	2.20268i
$260$		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
$268$	1.95951	+	16.2530i	0.119696	+	0.992811i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	20.2907	+	9.26647i	1.23257	+	0.562898i	0.921830	−	0.387595i	$-0.126694\pi$
$271$	20.2907	+	9.26647i	1.23257	+	0.562898i	0.310745	+	0.950493i	$0.399421\pi$
$272$		0			0
$273$	9.02932	−	30.7510i	0.546479	−	1.86114i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	20.8655	+	24.0800i	1.25368	+	1.44683i	0.845542	+	0.533909i	$0.179278\pi$
$277$	20.8655	+	24.0800i	1.25368	+	1.44683i	0.408142	+	0.912919i	$0.366177\pi$
$278$		0			0
$279$	18.4761	−	4.48224i	1.10613	−	0.268345i
$280$		0			0
$281$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$281$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$282$		0			0
$283$	18.1894	−	20.9916i	1.08125	−	1.24782i	0.114135	−	0.993465i	$-0.463590\pi$
$283$	18.1894	−	20.9916i	1.08125	−	1.24782i	0.967110	−	0.254358i	$-0.0818643\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	12.3035	+	11.7313i	0.723734	+	0.690079i
$290$		0			0
$291$	19.5327	+	27.4299i	1.14503	+	1.60797i
$292$	4.78160	+	33.2567i	0.279822	+	1.94620i
$293$		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
$300$	−14.1089	−	10.0469i	−0.814576	−	0.580057i
$301$	32.7731	+	6.31651i	1.88901	+	0.364077i
$302$		0			0
$303$		0			0
$304$	23.0186	+	11.8669i	1.32021	+	0.680615i
$305$		0			0
$306$		0			0
$307$	−0.318582	−	6.68786i	−0.0181824	−	0.381696i	−0.989557	−	0.144143i	$-0.953958\pi$
$307$	−0.318582	−	6.68786i	−0.0181824	−	0.381696i	0.971374	−	0.237553i	$-0.0763455\pi$
$308$		0			0
$309$	11.2199	+	14.2673i	0.638279	+	0.811638i
$310$		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
$313$	−31.6320	+	14.4459i	−1.78795	+	0.816528i	−0.817227	+	0.576316i	$0.804490\pi$
$313$	−31.6320	+	14.4459i	−1.78795	+	0.816528i	−0.970720	+	0.240212i	$0.922783\pi$
$314$		0			0
$315$		0			0
$316$	−25.6325	+	8.87151i	−1.44194	+	0.499061i
$317$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$317$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−4.24366	−	17.4926i	−0.235759	−	0.971812i
$325$	1.97526	−	20.6859i	0.109568	−	1.14745i
$326$		0			0
$327$	23.9294	−	15.3785i	1.32330	−	0.850432i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.27771	−	13.1831i	−0.290089	−	0.724608i	−0.999798	−	0.0201126i	$-0.993598\pi$
$331$	−5.27771	−	13.1831i	−0.290089	−	0.724608i	0.709708	−	0.704496i	$-0.248827\pi$
$332$		0			0
$333$	−35.8155	+	6.90287i	−1.96268	+	0.378275i
$334$		0			0
$335$		0			0
$336$	−30.8463			−1.68280
$337$	6.55203	+	33.9952i	0.356912	+	1.85183i	0.505352	+	0.862913i	$0.331363\pi$
$337$	6.55203	+	33.9952i	0.356912	+	1.85183i	−0.148441	+	0.988921i	$0.547425\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	14.0159	+	21.8092i	0.756789	+	1.17759i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.998867	−	0.0475819i	$-0.984848\pi$
$347$		0			0		0.998867	+	0.0475819i	$0.0151515\pi$
$348$		0			0
$349$	4.00714	−	27.8702i	0.214497	−	1.49186i	−0.543394	−	0.839478i	$-0.682861\pi$
$349$	4.00714	−	27.8702i	0.214497	−	1.49186i	0.757891	−	0.652382i	$-0.226230\pi$
$350$		0			0
$351$	15.6292	−	14.9024i	0.834223	−	0.795430i
$352$		0			0
$353$		0			0		0.618159	−	0.786053i	$-0.287879\pi$
$353$		0			0		−0.618159	+	0.786053i	$0.712121\pi$
$354$		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
$361$	−18.0144	+	14.1667i	−0.948128	+	0.745616i
$362$		0			0
$363$	−19.0310	+	0.906557i	−0.998867	+	0.0475819i
$364$	−18.5036	−	32.0493i	−0.969855	−	1.67984i
$365$		0			0
$366$		0			0
$367$	−28.8642	−	20.5541i	−1.50670	−	1.07292i	−0.973982	−	0.226623i	$-0.927231\pi$
$367$	−28.8642	−	20.5541i	−1.50670	−	1.07292i	−0.532718	−	0.846293i	$-0.678829\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	10.9766	−	19.0120i	0.569109	−	0.985725i
$373$	12.1196	−	6.99727i	0.627531	−	0.362305i	−0.152264	−	0.988340i	$-0.548656\pi$
$373$	12.1196	−	6.99727i	0.627531	−	0.362305i	0.779795	+	0.626035i	$0.215323\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−31.7119	+	22.5819i	−1.62893	+	1.15996i	−0.804871	+	0.593450i	$0.797765\pi$
$379$	−31.7119	+	22.5819i	−1.62893	+	1.15996i	−0.824059	+	0.566505i	$0.808295\pi$
$380$		0			0
$381$	−13.5448	+	14.2054i	−0.693921	+	0.727764i
$382$		0			0
$383$		0			0		0.945001	−	0.327068i	$-0.106061\pi$
$383$		0			0		−0.945001	+	0.327068i	$0.893939\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	16.9963	+	14.7274i	0.863972	+	0.748636i
$388$	38.4874	+	5.53365i	1.95390	+	0.280928i
$389$		0			0		0.888835	−	0.458227i	$-0.151515\pi$
$389$		0			0		−0.888835	+	0.458227i	$0.848485\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	37.9710	+	11.1493i	1.90571	+	0.559567i	0.985938	+	0.167109i	$0.0534431\pi$
$397$	37.9710	+	11.1493i	1.90571	+	0.559567i	0.919770	+	0.392458i	$0.128375\pi$
$398$		0			0
$399$	20.7407	−	45.4158i	1.03833	−	2.27363i
$400$	−19.6386	+	3.78502i	−0.981929	+	0.189251i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	26.3379			1.31198
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−2.44688	−	0.846873i	−0.120990	−	0.0418752i	0.265902	−	0.964000i	$-0.414330\pi$
$409$	−2.44688	−	0.846873i	−0.120990	−	0.0418752i	−0.386892	+	0.922125i	$0.626451\pi$
$410$		0			0
$411$		0			0
$412$	20.8635	+	1.99223i	1.02787	+	0.0981500i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.38726	+	3.90910i	−0.165875	+	0.191430i
$418$		0			0
$419$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$419$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$420$		0			0
$421$	−11.6595	−	33.6879i	−0.568250	−	1.64185i	−0.752216	−	0.658917i	$-0.771015\pi$
$421$	−11.6595	−	33.6879i	−0.568250	−	1.64185i	0.183966	−	0.982933i	$-0.441106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−30.8165	−	19.8046i	−1.49131	−	0.958410i
$428$		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
$433$	−16.5198	+	32.0439i	−0.793889	+	1.53993i	0.0471069	+	0.998890i	$0.485000\pi$
$433$	−16.5198	+	32.0439i	−0.793889	+	1.53993i	−0.840996	+	0.541041i	$0.818030\pi$
$434$		0			0
$435$		0			0
$436$	6.21603	−	32.2518i	0.297694	−	1.54458i
$437$		0			0
$438$		0			0
$439$	−7.20351	+	12.4768i	−0.343805	+	0.595487i	−0.985136	−	0.171777i	$-0.945049\pi$
$439$	−7.20351	+	12.4768i	−0.343805	+	0.595487i	0.641331	+	0.767264i	$0.278382\pi$
$440$		0			0
$441$	1.83040	+	38.4248i	0.0871618	+	1.82975i
$442$		0			0
$443$		0			0		−0.618159	−	0.786053i	$-0.712121\pi$
$443$		0			0		0.618159	+	0.786053i	$0.287879\pi$
$444$	−22.7703	+	35.4313i	−1.08063	+	1.68149i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−24.5794	+	25.7781i	−1.16127	+	1.21790i
$449$		0			0		0.995472	−	0.0950560i	$-0.0303030\pi$
$449$		0			0		−0.995472	+	0.0950560i	$0.969697\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−27.0392	−	28.3579i	−1.27041	−	1.33237i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.02478	−	42.5054i	0.0947154	−	1.98832i	−0.0495022	−	0.998774i	$-0.515763\pi$
$457$	2.02478	−	42.5054i	0.0947154	−	1.98832i	0.144218	−	0.989546i	$-0.453933\pi$
$458$		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
$463$	9.56122	+	2.31953i	0.444348	+	0.107798i	0.451689	−	0.892175i	$-0.350822\pi$
$463$	9.56122	+	2.31953i	0.444348	+	0.107798i	−0.00734134	+	0.999973i	$0.502337\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.981929	−	0.189251i	$-0.0606061\pi$
$467$		0			0		−0.981929	+	0.189251i	$0.939394\pi$
$468$		−	24.9360i		−	1.15267i
$469$	20.3971	−	30.2007i	0.941852	−	1.39454i
$470$		0			0
$471$	8.15418	+	42.3079i	0.375725	+	1.94944i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	7.63196	−	31.4594i	0.350178	−	1.44346i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$479$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$480$		0			0
$481$	−50.4722	−	2.40429i	−2.30133	−	0.109626i
$482$		0			0
$483$		0			0
$484$	−14.4069	+	16.6265i	−0.654861	+	0.755750i
$485$		0			0
$486$		0			0
$487$	−12.0968	+	15.3824i	−0.548159	+	0.697041i	−0.978645	−	0.205556i	$-0.934100\pi$
$487$	−12.0968	+	15.3824i	−0.548159	+	0.697041i	0.430486	+	0.902597i	$0.358342\pi$
$488$		0			0
$489$	2.70659	+	28.3447i	0.122396	+	1.28179i
$490$		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
$496$	−7.14172	−	24.3225i	−0.320673	−	1.09211i
$497$		0			0
$498$		0			0
$499$	−7.73117	−	4.46359i	−0.346095	−	0.199818i	0.316869	−	0.948469i	$-0.397368\pi$
$499$	−7.73117	−	4.46359i	−0.346095	−	0.199818i	−0.662964	+	0.748651i	$0.730702\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.189251	−	0.981929i	$-0.439394\pi$
$503$		0			0		−0.189251	+	0.981929i	$0.560606\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	6.40844	−	3.69992i	0.284609	−	0.164319i
$508$	1.07841	+	22.6387i	0.0478468	+	1.00443i
$509$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$509$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$510$		0			0
$511$	40.4375	−	62.9219i	1.78885	−	2.78350i
$512$		0			0
$513$	27.4038	−	19.5142i	1.20991	−	0.861573i
$514$		0			0
$515$		0			0
$516$	25.8509	−	2.46846i	1.13802	−	0.108668i
$517$		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
$523$	−34.9071	+	17.9958i	−1.52638	+	0.786903i	−0.998067	−	0.0621473i	$-0.980205\pi$
$523$	−34.9071	+	17.9958i	−1.52638	+	0.786903i	−0.528312	+	0.849050i	$0.677175\pi$
$524$		0			0
$525$	9.09036	+	37.4710i	0.396736	+	1.63537i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−7.52256	+	21.7350i	−0.327068	+	0.945001i
$530$		0			0
$531$		0			0
$532$	−21.4269	−	53.5218i	−0.928973	−	2.32046i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.00455	+	10.2326i	−0.129176	+	0.439932i	−0.998527	−	0.0542631i	$-0.982719\pi$
$541$	−3.00455	+	10.2326i	−0.129176	+	0.439932i	0.869351	+	0.494195i	$0.164537\pi$
$542$		0			0
$543$	33.9225	+	11.7407i	1.45575	+	0.503841i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−36.3246	+	8.81225i	−1.55313	+	0.376785i	−0.918305	−	0.395874i	$-0.870442\pi$
$547$	−36.3246	+	8.81225i	−1.55313	+	0.376785i	−0.634822	+	0.772658i	$0.718927\pi$
$548$		0			0
$549$	−11.3103	−	21.9390i	−0.482713	−	0.936333i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	56.0573	+	22.4419i	2.38380	+	0.954328i
$554$		0			0
$555$		0			0
$556$	0.567738	+	5.94562i	0.0240775	+	0.252151i
$557$		0			0		−0.723734	−	0.690079i	$-0.757576\pi$
$557$		0			0		0.723734	+	0.690079i	$0.242424\pi$
$558$		0			0
$559$	18.0718	+	25.3783i	0.764356	+	1.07339i
$560$		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
$567$	−18.3614	+	35.6161i	−0.771105	+	1.49574i
$568$		0			0
$569$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$569$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$570$		0			0
$571$	6.75584	−	9.48725i	0.282723	−	0.397029i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	6.75584	−	9.48725i	0.282723	−	0.397029i	0.931378	+	0.364054i	$0.118607\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−23.0278	+	6.76158i	−0.959493	+	0.281733i
$577$	−25.8217	−	32.8350i	−1.07497	−	1.36694i	−0.925750	−	0.378135i	$-0.876565\pi$
$577$	−25.8217	−	32.8350i	−1.07497	−	1.36694i	−0.149222	−	0.988804i	$-0.547677\pi$
$578$		0			0
$579$	5.62472	−	0.808712i	0.233755	−	0.0336089i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.690079	−	0.723734i	$-0.742424\pi$
$587$		0			0		0.690079	+	0.723734i	$0.257576\pi$
$588$	33.5700	+	29.0885i	1.38440	+	1.19959i
$589$	40.6126	+	5.83921i	1.67341	+	0.240601i
$590$		0			0
$591$		0			0
$592$	11.4656	+	47.2619i	0.471234	+	1.94245i
$593$		0			0		0.0950560	−	0.995472i	$-0.469697\pi$
$593$		0			0		−0.0950560	+	0.995472i	$0.530303\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	40.6514	+	9.86191i	1.66375	+	0.403621i
$598$		0			0
$599$		0			0		−0.371662	−	0.928368i	$-0.621212\pi$
$599$		0			0		0.371662	+	0.928368i	$0.378788\pi$
$600$		0			0
$601$	33.6485	−	6.48521i	1.37255	−	0.264537i	0.550919	−	0.834559i	$-0.314277\pi$
$601$	33.6485	−	6.48521i	1.37255	−	0.264537i	0.821630	+	0.570021i	$0.193065\pi$
$602$		0			0
$603$	22.0773	−	10.7514i	0.899058	−	0.437829i
$604$	−45.2443			−1.84096
$605$		0			0
$606$		0			0
$607$	10.3197	−	4.13139i	0.418864	−	0.167688i	−0.152650	−	0.988280i	$-0.548781\pi$
$607$	10.3197	−	4.13139i	0.418864	−	0.167688i	0.571514	+	0.820593i	$0.306356\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	22.0184	+	2.10251i	0.889316	+	0.0849194i	0.529704	−	0.848183i	$-0.322303\pi$
$613$	22.0184	+	2.10251i	0.889316	+	0.0849194i	0.359612	+	0.933102i	$0.382909\pi$
$614$		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
$619$	32.2606	−	30.7605i	1.29666	−	1.23637i	0.341644	−	0.939829i	$-0.389016\pi$
$619$	32.2606	−	30.7605i	1.29666	−	1.23637i	0.955021	−	0.296538i	$-0.0958321\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−20.8389	−	19.8698i	−0.834223	−	0.795430i
$625$	10.3854	+	22.7408i	0.415415	+	0.909632i
$626$		0			0
$627$		0			0
$628$	41.8541	+	26.8980i	1.67016	+	1.07335i
$629$		0			0
$630$		0			0
$631$	50.0403	−	2.38372i	1.99207	−	0.0948942i	0.992458	−	0.122585i	$-0.0391183\pi$
$631$	50.0403	−	2.38372i	1.99207	−	0.0948942i	0.999617	+	0.0276903i	$0.00881524\pi$
$632$		0			0
$633$	−7.73067	−	4.46331i	−0.307267	−	0.177400i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−10.0855	+	52.3284i	−0.399601	+	2.07333i
$638$		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
$643$	−24.1475	+	7.09036i	−0.952286	+	0.279616i	−0.720738	−	0.693207i	$-0.756197\pi$
$643$	−24.1475	+	7.09036i	−0.952286	+	0.279616i	−0.231548	+	0.972824i	$0.574379\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.814576	−	0.580057i	$-0.196970\pi$
$647$		0			0		−0.814576	+	0.580057i	$0.803030\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−46.1829	+	15.9841i	−1.81005	+	0.626465i
$652$	25.8443	+	20.3242i	1.01214	+	0.795956i
$653$		0			0		0.371662	−	0.928368i	$-0.378788\pi$
$653$		0			0		−0.371662	+	0.928368i	$0.621212\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	44.7956	−	23.0937i	1.74764	−	0.900972i
$658$		0			0
$659$		0			0		−0.235759	−	0.971812i	$-0.575758\pi$
$659$		0			0		0.235759	+	0.971812i	$0.424242\pi$
$660$		0			0
$661$	−34.9782	+	30.3088i	−1.36049	+	1.17887i	−0.394929	+	0.918711i	$0.629231\pi$
$661$	−34.9782	+	30.3088i	−1.36049	+	1.17887i	−0.965565	+	0.260163i	$0.916224\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			13.4689i			0.520738i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−47.1711	−	21.5423i	−1.81831	−	0.830395i	−0.922590	−	0.385782i	$-0.873932\pi$
$673$	−47.1711	−	21.5423i	−1.81831	−	0.830395i	−0.895722	−	0.444614i	$-0.853341\pi$
$674$		0			0
$675$	−7.31963	+	24.9284i	−0.281733	+	0.959493i
$676$	2.01446	−	8.30373i	0.0774793	−	0.319374i
$677$		0			0		−0.945001	−	0.327068i	$-0.893939\pi$
$677$		0			0		0.945001	+	0.327068i	$0.106061\pi$
$678$		0			0
$679$	−56.6843	−	65.4171i	−2.17534	−	2.51048i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.458227	−	0.888835i	$-0.651515\pi$
$683$		0			0		0.458227	+	0.888835i	$0.348485\pi$
$684$	5.52840	−	38.4509i	0.211384	−	1.47021i
$685$		0			0
$686$		0			0
$687$	−45.9160	−	18.3820i	−1.75181	−	0.701317i
$688$	18.5360	−	23.5704i	0.706678	−	0.898615i
$689$		0			0
$690$		0			0
$691$	−25.2023	−	24.0303i	−0.958739	−	0.914156i	0.0376301	−	0.999292i	$-0.488019\pi$
$691$	−25.2023	−	24.0303i	−0.958739	−	0.914156i	−0.996369	+	0.0851357i	$0.972868\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	38.5579	+	22.2614i	1.45735	+	0.841402i
$701$		0			0		0.458227	−	0.888835i	$-0.348485\pi$
$701$		0			0		−0.458227	+	0.888835i	$0.651515\pi$
$702$		0			0
$703$	−77.2943	−	14.8973i	−2.91521	−	0.561861i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.66421	−	34.9361i	−0.0625007	−	1.31205i	−0.784488	−	0.620145i	$-0.787074\pi$
$709$	−1.66421	−	34.9361i	−0.0625007	−	1.31205i	0.721987	−	0.691907i	$-0.243229\pi$
$710$		0			0
$711$	25.1507	+	31.9818i	0.943227	+	1.19941i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.786053	−	0.618159i	$-0.787879\pi$
$719$		0			0		0.786053	+	0.618159i	$0.212121\pi$
$720$		0			0
$721$	−32.1966	−	33.7668i	−1.19906	−	1.25754i
$722$		0			0
$723$	−43.1047	−	6.19752i	−1.60308	−	0.230488i
$724$	36.8423	−	18.9935i	1.36923	−	0.705888i
$725$		0			0
$726$		0			0
$727$	−0.255918	+	2.68009i	−0.00949147	+	0.0993992i	−0.998976	−	0.0452379i	$-0.985595\pi$
$727$	−0.255918	+	2.68009i	−0.00949147	+	0.0993992i	0.989485	+	0.144637i	$0.0462015\pi$
$728$		0			0
$729$	−22.7138	+	14.5973i	−0.841254	+	0.540641i
$730$		0			0
$731$		0			0
$732$	−27.3468	−	8.02974i	−1.01077	−	0.296788i
$733$	−19.1029	−	47.7167i	−0.705581	−	1.76246i	−0.644858	−	0.764303i	$-0.723083\pi$
$733$	−19.1029	−	47.7167i	−0.705581	−	1.76246i	−0.0607239	−	0.998155i	$-0.519341\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	9.60167	+	49.8182i	0.353203	+	1.83259i	0.529568	+	0.848268i	$0.322354\pi$
$739$	9.60167	+	49.8182i	0.353203	+	1.83259i	−0.176365	+	0.984325i	$0.556434\pi$
$740$		0			0
$741$	43.2667	−	17.3214i	1.58944	−	0.636317i
$742$		0			0
$743$		0			0		0.235759	−	0.971812i	$-0.424242\pi$
$743$		0			0		−0.235759	+	0.971812i	$0.575758\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.38669	−	9.64466i	0.0506011	−	0.351939i	−0.948753	−	0.316017i	$-0.897654\pi$
$751$	1.38669	−	9.64466i	0.0506011	−	0.351939i	0.999355	−	0.0359215i	$-0.0114366\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	15.1333	+	43.7247i	0.550391	+	1.59025i
$757$	−1.31495	−	13.7708i	−0.0477926	−	0.500507i	−0.987399	−	0.158250i	$-0.949415\pi$
$757$	−1.31495	−	13.7708i	−0.0477926	−	0.500507i	0.939606	−	0.342257i	$-0.111191\pi$
$758$		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
$763$	−57.4749	+	45.1988i	−2.08073	+	1.63631i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−12.6987	+	24.6321i	−0.458227	+	0.888835i
$769$	3.77555	+	2.68856i	0.136150	+	0.0969519i	0.646103	−	0.763250i	$-0.276398\pi$
$769$	3.77555	+	2.68856i	0.136150	+	0.0969519i	−0.509953	+	0.860202i	$0.670337\pi$
$770$		0			0
$771$		0			0
$772$	3.80613	−	5.34497i	0.136986	−	0.192370i
$773$		0			0		−0.888835	−	0.458227i	$-0.848485\pi$
$773$		0			0		0.888835	+	0.458227i	$0.151515\pi$
$774$		0			0
$775$	−27.4414	+	15.8433i	−0.985725	+	0.569109i
$776$		0			0
$777$	89.9610	−	26.4149i	3.22733	−	0.947630i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	51.0589	−	4.87553i	1.82353	−	0.174126i
$785$		0			0
$786$		0			0
$787$	−12.3355	+	30.8126i	−0.439713	+	1.09835i	0.528658	+	0.848835i	$0.322696\pi$
$787$	−12.3355	+	30.8126i	−0.439713	+	1.09835i	−0.968371	+	0.249515i	$0.919729\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−8.06153	−	33.2301i	−0.286273	−	1.18003i
$794$		0			0
$795$		0			0
$796$	40.6340	−	26.1139i	1.44023	−	0.925582i
$797$		0			0		0.327068	−	0.945001i	$-0.393939\pi$
$797$		0			0		−0.327068	+	0.945001i	$0.606061\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	8.60710	−	27.0170i	0.303549	−	0.952816i
$805$		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
$811$	15.5203	+	5.37165i	0.544993	+	0.188624i	0.585672	−	0.810548i	$-0.300831\pi$
$811$	15.5203	+	5.37165i	0.544993	+	0.188624i	−0.0406786	+	0.999172i	$0.512952\pi$
$812$		0			0
$813$	−25.3012	−	29.1992i	−0.887353	−	1.02406i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	22.2400	+	43.1395i	0.778078	+	1.50926i
$818$		0			0
$819$	−36.3519	+	41.9524i	−1.27024	+	1.46593i
$820$		0			0
$821$		0			0		−0.928368	−	0.371662i	$-0.878788\pi$
$821$		0			0		0.928368	+	0.371662i	$0.121212\pi$
$822$		0			0
$823$	−15.1992	−	43.9153i	−0.529812	−	1.53079i	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−15.1992	−	43.9153i	−0.529812	−	1.53079i	0.289346	−	0.957224i	$-0.406562\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.580057	−	0.814576i	$-0.696970\pi$
$827$		0			0		0.580057	+	0.814576i	$0.303030\pi$
$828$		0			0
$829$	25.0947	+	16.1274i	0.871576	+	0.560128i	0.898234	−	0.439517i	$-0.144850\pi$
$829$	25.0947	+	16.1274i	0.871576	+	0.560128i	−0.0266585	+	0.999645i	$0.508487\pi$
$830$		0			0
$831$	−15.5481	−	52.9519i	−0.539357	−	1.83688i
$832$	−33.2103	+	1.58200i	−1.15136	+	0.0548460i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−32.3346	−	6.23199i	−1.11765	−	0.215409i
$838$		0			0
$839$		0			0		0.580057	−	0.814576i	$-0.303030\pi$
$839$		0			0		−0.580057	+	0.814576i	$0.696970\pi$
$840$		0			0
$841$	−14.5000	+	25.1147i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	−9.89004	+	2.90398i	−0.340429	+	0.0999590i
$845$		0			0
$846$		0			0
$847$	48.4766	−	6.96988i	1.66567	−	0.239488i
$848$		0			0
$849$	−43.7618	+	19.9853i	−1.50190	+	0.685895i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	7.60908	+	5.98385i	0.260530	+	0.204883i	0.739877	−	0.672743i	$-0.234884\pi$
$853$	7.60908	+	5.98385i	0.260530	+	0.204883i	−0.479347	+	0.877626i	$0.659126\pi$
$854$		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
$859$	−0.696423	+	14.6197i	−0.0237617	+	0.498819i	0.955348	+	0.295484i	$0.0954809\pi$
$859$	−0.696423	+	14.6197i	−0.0237617	+	0.498819i	−0.979109	+	0.203335i	$0.934822\pi$
$860$		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
$867$	−10.9436	−	27.3357i	−0.371662	−	0.928368i
$868$	−23.4423	+	51.3315i	−0.795684	+	1.74231i
$869$		0			0
$870$		0			0
$871$	33.2337	−	7.26381i	1.12608	−	0.246125i
$872$		0			0
$873$	−11.0380	−	57.2707i	−0.373580	−	1.93832i
$874$		0			0
$875$		0			0
$876$	16.3953	−	55.8374i	0.553947	−	1.88657i
$877$	−13.4266	+	55.3451i	−0.453383	+	1.86887i	0.0402330	+	0.999190i	$0.487190\pi$
$877$	−13.4266	+	55.3451i	−0.453383	+	1.86887i	−0.493616	+	0.869680i	$0.664325\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.995472	−	0.0950560i	$-0.969697\pi$
$881$		0			0		0.995472	+	0.0950560i	$0.0303030\pi$
$882$		0			0
$883$	17.1672	+	0.817772i	0.577721	+	0.0275202i	0.334409	−	0.942428i	$-0.391463\pi$
$883$	17.1672	+	0.817772i	0.577721	+	0.0275202i	0.243312	+	0.969948i	$0.421766\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.723734	−	0.690079i	$-0.242424\pi$
$887$		0			0		−0.723734	+	0.690079i	$0.757576\pi$
$888$		0			0
$889$	31.1886	−	39.6595i	1.04603	−	1.33014i
$890$		0			0
$891$		0			0
$892$	11.2559	+	10.7325i	0.376876	+	0.359350i
$893$		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
$903$	−47.0902	−	33.5328i	−1.56706	−	1.11590i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−45.3902	−	23.4003i	−1.50716	−	0.776994i	−0.510594	−	0.859822i	$-0.670574\pi$
$907$	−45.3902	−	23.4003i	−1.50716	−	0.776994i	−0.996564	+	0.0828283i	$0.973605\pi$
$908$		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	$0.909091\pi$
$912$	−27.7280	−	35.2590i	−0.918166	−	1.16754i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−51.9492	+	23.7244i	−1.71645	+	0.783877i
$917$		0			0
$918$		0			0
$919$	14.4244	−	4.99234i	0.475818	−	0.164682i	−0.0786243	−	0.996904i	$-0.525053\pi$
$919$	14.4244	−	4.99234i	0.475818	−	0.164682i	0.554442	+	0.832222i	$0.312932\pi$
$920$		0			0
$921$	−4.31011	+	10.7661i	−0.142023	+	0.354756i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	54.0332	−	27.8561i	1.77660	−	0.915902i
$926$		0			0
$927$	−7.41171	−	30.5515i	−0.243433	−	1.00344i
$928$		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
$931$	−27.1530	+	78.4536i	−0.889905	+	2.57121i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			59.6396i			1.94834i	0.225813	+	0.974171i	$0.427496\pi$
$937$			59.6396i			1.94834i	−0.225813	+	0.974171i	$0.572504\pi$
$938$		0			0
$939$	60.2312			1.96557
$940$		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	$0.272727\pi$
$948$	46.7680	+	4.46580i	1.51895	+	0.145043i
$949$	67.8500	−	16.4602i	2.20250	−	0.534322i
$950$		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
$961$	−5.31430	−	7.46289i	−0.171429	−	0.240738i
$962$		0			0
$963$		0			0
$964$	−39.5266	+	31.0841i	−1.27307	+	1.00115i
$965$		0			0
$966$		0			0
$967$	29.7856	+	51.5903i	0.957842	+	1.65903i	0.727727	+	0.685867i	$0.240577\pi$
$967$	29.7856	+	51.5903i	0.957842	+	1.65903i	0.230115	+	0.973163i	$0.426090\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.981929	−	0.189251i	$-0.939394\pi$
$971$		0			0		0.981929	+	0.189251i	$0.0606061\pi$
$972$	−5.90027	+	30.6135i	−0.189251	+	0.981929i
$973$	7.71243	−	10.8306i	0.247249	−	0.347213i
$974$		0			0
$975$	−17.9960	+	31.1700i	−0.576333	+	0.998238i
$976$	−28.5013	+	16.4552i	−0.912304	+	0.526719i
$977$		0			0		−0.0475819	−	0.998867i	$-0.515152\pi$
$977$		0			0		0.0475819	+	0.998867i	$0.484848\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−48.7666	+	7.01158i	−1.55700	+	0.223862i
$982$		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
$988$	20.0010	−	49.9601i	0.636317	−	1.58944i
$989$		0			0
$990$		0			0
$991$	55.0394	+	7.91347i	1.74838	+	0.251380i	0.940942	−	0.338567i	$-0.109942\pi$
$991$	55.0394	+	7.91347i	1.74838	+	0.251380i	0.807442	+	0.589947i	$0.200851\pi$
$992$		0			0
$993$	−1.17031	+	24.5678i	−0.0371386	+	0.779634i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	51.2098	−	32.9105i	1.62183	−	1.04229i	0.667033	−	0.745028i	$-0.267564\pi$
$997$	51.2098	−	32.9105i	1.62183	−	1.04229i	0.954797	−	0.297259i	$-0.0960725\pi$
$998$		0			0
$999$	61.3951	+	14.8943i	1.94245	+	0.471234i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.a.113.1	✓	20
3.2	odd	2	CM	201.2.p.a.113.1	✓	20
67.51	odd	66	inner	201.2.p.a.185.1	yes	20
201.185	even	66	inner	201.2.p.a.185.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.a.113.1	✓	20	1.1	even	1	trivial
201.2.p.a.113.1	✓	20	3.2	odd	2	CM
201.2.p.a.185.1	yes	20	67.51	odd	66	inner
201.2.p.a.185.1	yes	20	201.185	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)

\(f(q)\)	\(=\)	\(q+(-1.57553 - 0.719520i) q^{3} +(-1.85674 + 0.743325i) q^{4} +(4.20741 + 1.45620i) q^{7} +(1.96458 + 2.26725i) q^{9} +O(q^{10})\)	\(q+(-1.57553 - 0.719520i) q^{3} +(-1.85674 + 0.743325i) q^{4} +(4.20741 + 1.45620i) q^{7} +(1.96458 + 2.26725i) q^{9} +(3.46018 + 0.164829i) q^{12} +(1.90439 + 3.69400i) q^{13} +(2.89494 - 2.76032i) q^{16} +(2.11756 + 6.11829i) q^{19} +(-5.58113 - 5.32159i) q^{21} +(-4.20627 - 2.70320i) q^{25} +(-1.46393 - 4.98567i) q^{27} +(-8.89447 + 0.423696i) q^{28} +(2.90393 - 5.63284i) q^{31} +(-5.33301 - 2.74936i) q^{36} +(-6.07910 + 10.5293i) q^{37} +(-0.342513 - 7.19024i) q^{39} +(7.42015 - 1.06686i) q^{43} +(-6.54716 + 2.26599i) q^{48} +(10.0794 + 7.92652i) q^{49} +(-6.28178 - 5.44320i) q^{52} +(1.06595 - 11.1632i) q^{57} +(-7.99569 - 1.93973i) q^{61} +(4.96423 + 12.4001i) q^{63} +(-3.32332 + 7.27706i) q^{64} +(2.11074 - 7.90852i) q^{67} +(3.96060 - 16.3258i) q^{73} +(4.68209 + 7.28547i) q^{75} +(-8.47963 - 9.78602i) q^{76} +(13.5468 + 0.645314i) q^{79} +(-1.28083 + 8.90839i) q^{81} +(14.3184 + 5.73220i) q^{84} +(2.63334 + 18.3153i) q^{91} +(-8.62817 + 6.78526i) q^{93} +(-16.8369 - 9.72078i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{4} - 6 q^{7} + 6 q^{9}+O(q^{10}) \) 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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