LMFDB - Embedded newform 201.4.j.a.110.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("201.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			110.1
Root			$-0.327068 - 0.945001i$ of defining polynomial
Character	$\chi$	$=$	201.110
Dual form			201.4.j.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.80925 - 4.37128i) q^{3} +(5.23889 + 6.04600i) q^{4} +(4.82246 + 2.20234i) q^{7} +(-11.2162 + 24.5601i) q^{9} +O(q^{10})$	$q+(-2.80925 - 4.37128i) q^{3} +(5.23889 + 6.04600i) q^{4} +(4.82246 + 2.20234i) q^{7} +(-11.2162 + 24.5601i) q^{9} +(11.7114 - 39.8854i) q^{12} +(-10.6415 + 36.2415i) q^{13} +(-9.10815 + 63.3486i) q^{16} +(15.4399 + 33.8086i) q^{19} +(-3.92044 - 27.2672i) q^{21} +(119.937 + 35.2166i) q^{25} +(138.868 - 19.9662i) q^{27} +(11.9489 + 40.6944i) q^{28} +(92.1405 + 313.802i) q^{31} +(-207.250 + 60.8542i) q^{36} +228.514 q^{37} +(188.316 - 55.2947i) q^{39} +(268.971 + 233.064i) q^{43} +(302.502 - 138.148i) q^{48} +(-206.211 - 237.981i) q^{49} +(-274.865 + 125.527i) q^{52} +(104.412 - 162.469i) q^{57} +(-681.427 + 97.9744i) q^{61} +(-108.179 + 93.7379i) q^{63} +(-430.722 + 276.808i) q^{64} +(-514.404 + 190.135i) q^{67} +(-76.8030 - 534.177i) q^{73} +(-182.991 - 623.209i) q^{75} +(-123.519 + 270.469i) q^{76} +(224.806 - 765.617i) q^{79} +(-477.393 - 550.941i) q^{81} +(144.319 - 166.553i) q^{84} +(-131.134 + 151.337i) q^{91} +(1112.87 - 1284.32i) q^{93} -434.966i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) $ 20 * q + 16 * q^4 + 54 * q^9	$ 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) $ 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{3}{22}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$2$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$3$	−2.80925	−	4.37128i	−0.540641	−	0.841254i
$3$	−2.80925	−	4.37128i	−0.540641	−	0.841254i
$4$	5.23889	+	6.04600i	0.654861	+	0.755750i
$5$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$5$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$6$		0			0
$7$	4.82246	+	2.20234i	0.260388	+	0.118915i	0.541332	−	0.840809i	$-0.317920\pi$
$7$	4.82246	+	2.20234i	0.260388	+	0.118915i	−0.280944	+	0.959724i	$0.590647\pi$
$8$		0			0
$9$	−11.2162	+	24.5601i	−0.415415	+	0.909632i
$10$		0			0
$11$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$11$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$12$	11.7114	−	39.8854i	0.281733	−	0.959493i
$13$	−10.6415	+	36.2415i	−0.227032	+	0.773199i	0.764646	+	0.644451i	$0.222914\pi$
$13$	−10.6415	+	36.2415i	−0.227032	+	0.773199i	−0.991677	+	0.128748i	$0.958904\pi$
$14$		0			0
$15$		0			0
$16$	−9.10815	+	63.3486i	−0.142315	+	0.989821i
$17$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$17$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$18$		0			0
$19$	15.4399	+	33.8086i	0.186429	+	0.408223i	0.979651	−	0.200710i	$-0.0643250\pi$
$19$	15.4399	+	33.8086i	0.186429	+	0.408223i	−0.793222	+	0.608933i	$0.791598\pi$
$20$		0			0
$21$	−3.92044	−	27.2672i	−0.0407386	−	0.283343i
$22$		0			0
$23$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$23$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$24$		0			0
$25$	119.937	+	35.2166i	0.959493	+	0.281733i
$26$		0			0
$27$	138.868	−	19.9662i	0.989821	−	0.142315i
$28$	11.9489	+	40.6944i	0.0806478	+	0.274661i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	92.1405	+	313.802i	0.533836	+	1.81808i	0.573957	+	0.818885i	$0.305408\pi$
$31$	92.1405	+	313.802i	0.533836	+	1.81808i	−0.0401208	+	0.999195i	$0.512774\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−207.250	+	60.8542i	−0.959493	+	0.281733i
$37$	228.514			1.01534			0.507668	−	0.861553i	$-0.330508\pi$
$37$	228.514			1.01534			0.507668	+	0.861553i	$0.330508\pi$
$38$		0			0
$39$	188.316	−	55.2947i	0.773199	−	0.227032i
$40$		0			0
$41$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$41$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$42$		0			0
$43$	268.971	+	233.064i	0.953899	+	0.826558i	0.984929	−	0.172961i	$-0.0553334\pi$
$43$	268.971	+	233.064i	0.953899	+	0.826558i	−0.0310302	+	0.999518i	$0.509879\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$47$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$48$	302.502	−	138.148i	0.909632	−	0.415415i
$49$	−206.211	−	237.981i	−0.601200	−	0.693821i
$50$		0			0
$51$		0			0
$52$	−274.865	+	125.527i	−0.733019	+	0.334759i
$53$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$53$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	104.412	−	162.469i	0.242628	−	0.377536i
$58$		0			0
$59$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$59$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$60$		0			0
$61$	−681.427	+	97.9744i	−1.43029	+	0.205645i	−0.813538	−	0.581512i	$-0.802461\pi$
$61$	−681.427	+	97.9744i	−1.43029	+	0.205645i	−0.616753	+	0.787157i	$0.711552\pi$
$62$		0			0
$63$	−108.179	+	93.7379i	−0.216338	+	0.187458i
$64$	−430.722	+	276.808i	−0.841254	+	0.540641i
$65$		0			0
$66$		0			0
$67$	−514.404	+	190.135i	−0.937977	+	0.346697i
$67$	−514.404	+	190.135i	−0.937977	+	0.346697i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$71$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$72$		0			0
$73$	−76.8030	−	534.177i	−0.123138	−	0.856447i	−0.953966	−	0.299916i	$-0.903041\pi$
$73$	−76.8030	−	534.177i	−0.123138	−	0.856447i	0.830827	−	0.556531i	$-0.187868\pi$
$74$		0			0
$75$	−182.991	−	623.209i	−0.281733	−	0.959493i
$76$	−123.519	+	270.469i	−0.186429	+	0.408223i
$77$		0			0
$78$		0			0
$79$	224.806	−	765.617i	0.320159	−	1.09036i	−0.629480	−	0.777017i	$-0.716732\pi$
$79$	224.806	−	765.617i	0.320159	−	1.09036i	0.949639	−	0.313346i	$-0.101450\pi$
$80$		0			0
$81$	−477.393	−	550.941i	−0.654861	−	0.755750i
$82$		0			0
$83$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$83$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$84$	144.319	−	166.553i	0.187458	−	0.216338i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$89$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$90$		0			0
$91$	−131.134	+	151.337i	−0.151062	+	0.174334i
$92$		0			0
$93$	1112.87	−	1284.32i	1.24085	−	1.43202i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	434.966i		−	0.455300i	−0.973743	−	0.227650i	$-0.926896\pi$
$97$		−	434.966i		−	0.455300i	0.973743	−	0.227650i	$-0.0731042\pi$
$98$		0			0
$99$		0			0
$100$	415.415	+	909.632i	0.415415	+	0.909632i
$101$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$101$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$102$		0			0
$103$	−172.525	+	50.6580i	−0.165043	+	0.0484610i	−0.363210	−	0.931707i	$-0.618319\pi$
$103$	−172.525	+	50.6580i	−0.165043	+	0.0484610i	0.198167	+	0.980168i	$0.436501\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$107$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$108$	848.230	+	734.995i	0.755750	+	0.654861i
$109$	92.6447	−	315.519i	0.0814105	−	0.277259i	−0.908720	−	0.417406i	$-0.862939\pi$
$109$	92.6447	−	315.519i	0.0814105	−	0.277259i	0.990131	+	0.140147i	$0.0447575\pi$
$110$		0			0
$111$	−641.953	−	998.899i	−0.548933	−	0.854156i
$112$	−183.439	+	285.436i	−0.154762	+	0.240814i
$113$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$113$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−770.737	−	667.847i	−0.609014	−	0.527714i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1277.09	+	374.986i	0.959493	+	0.281733i
$122$		0			0
$123$		0			0
$124$	−1414.53	+	2201.05i	−1.02442	+	1.59404i
$125$		0			0
$126$		0			0
$127$	483.506	−	1058.73i	0.337828	−	0.739741i	−0.662125	−	0.749393i	$-0.730345\pi$
$127$	483.506	−	1058.73i	0.337828	−	0.739741i	0.999953	+	0.00965233i	$0.00307248\pi$
$128$		0			0
$129$	263.184	−	1830.48i	0.179628	−	1.24934i
$130$		0			0
$131$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$131$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$132$		0			0
$133$			197.044i			0.128466i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$137$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$138$		0			0
$139$	−3065.96	−	440.819i	−1.87087	−	0.268991i	−0.888914	−	0.458075i	$-0.848539\pi$
$139$	−3065.96	−	440.819i	−1.87087	−	0.268991i	−0.981961	+	0.189084i	$0.939448\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1453.69	−	934.227i	−0.841254	−	0.540641i
$145$		0			0
$146$		0			0
$147$	−460.981	+	1569.96i	−0.258647	+	0.880869i
$148$	1197.16	+	1381.59i	0.664904	+	0.767340i
$149$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$149$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$150$		0			0
$151$	−1959.01	+	2260.82i	−1.05577	+	1.21843i	−0.0806560	+	0.996742i	$0.525702\pi$
$151$	−1959.01	+	2260.82i	−1.05577	+	1.21843i	−0.975118	+	0.221686i	$0.928844\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1320.88	+	848.878i	0.677917	+	0.435671i
$157$	−725.161	+	466.033i	−0.368625	+	0.236901i	−0.711818	−	0.702364i	$-0.752128\pi$
$157$	−725.161	+	466.033i	−0.368625	+	0.236901i	0.343193	+	0.939265i	$0.388492\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4040.80			1.94172			0.970858	−	0.239657i	$-0.0770351\pi$
$163$	4040.80			1.94172			0.970858	+	0.239657i	$0.0770351\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$167$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$168$		0			0
$169$	648.028	+	416.462i	0.294960	+	0.189559i
$170$		0			0
$171$	−1003.52			−0.448778
$172$			2847.19i			1.26219i
$173$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$173$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$174$		0			0
$175$	500.830	+	433.972i	0.216338	+	0.187458i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$179$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$180$		0			0
$181$	4007.03	−	2575.16i	1.64552	−	1.05751i	0.710031	−	0.704171i	$-0.248681\pi$
$181$	4007.03	−	2575.16i	1.64552	−	1.05751i	0.935494	−	0.353344i	$-0.114955\pi$
$182$		0			0
$183$	2342.57	+	2703.47i	0.946273	+	1.09206i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	713.658	+	209.549i	0.274661	+	0.0806478i
$190$		0			0
$191$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$191$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$192$	2420.01	+	1105.18i	0.909632	+	0.415415i
$193$	5029.78	−	1476.88i	1.87592	−	0.550819i	0.878603	−	0.477552i	$-0.158476\pi$
$193$	5029.78	−	1476.88i	1.87592	−	0.550819i	0.997312	−	0.0732663i	$-0.0233423\pi$
$194$		0			0
$195$		0			0
$196$	358.512	−	2493.51i	0.130653	−	0.908713i
$197$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$197$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$198$		0			0
$199$	−788.177	+	1725.87i	−0.280766	+	0.614791i	−0.996501	−	0.0835804i	$-0.973364\pi$
$199$	−788.177	+	1725.87i	−0.280766	+	0.614791i	0.715735	+	0.698372i	$0.246092\pi$
$200$		0			0
$201$	2276.22	+	1714.47i	0.798769	+	0.601638i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−2198.92	−	1004.21i	−0.733019	−	0.334759i
$209$		0			0
$210$		0			0
$211$	−186.457	−	119.829i	−0.0608353	−	0.0390965i	0.509870	−	0.860252i	$-0.329694\pi$
$211$	−186.457	−	119.829i	−0.0608353	−	0.0390965i	−0.570705	+	0.821155i	$0.693330\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−246.755	+	1716.22i	−0.0771928	+	0.536888i
$218$		0			0
$219$	−2119.28	+	1836.36i	−0.653915	+	0.566621i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	372.797	+	239.582i	0.111948	+	0.0719444i	0.595417	−	0.803417i	$-0.296987\pi$
$223$	372.797	+	239.582i	0.111948	+	0.0719444i	−0.483469	+	0.875362i	$0.660623\pi$
$224$		0			0
$225$	−2210.15	+	2550.65i	−0.654861	+	0.755750i
$226$		0			0
$227$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$227$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$228$	1529.29	−	219.879i	0.444210	−	0.0638677i
$229$	−1078.24	−	3672.14i	−0.311144	−	1.05966i	−0.955514	−	0.294945i	$-0.904699\pi$
$229$	−1078.24	−	3672.14i	−0.311144	−	1.05966i	0.644370	−	0.764714i	$-0.277120\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$233$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−3978.26	+	1168.12i	−1.09036	+	0.320159i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−1057.47	−	7354.85i	−0.282645	−	1.96584i	−0.258004	−	0.966144i	$-0.583065\pi$
$241$	−1057.47	−	7354.85i	−0.282645	−	1.96584i	−0.0246416	−	0.999696i	$-0.507844\pi$
$242$		0			0
$243$	−1067.20	+	3634.55i	−0.281733	+	0.959493i
$244$	−4162.27	−	3606.63i	−1.09206	−	0.946273i
$245$		0			0
$246$		0			0
$247$	−1389.58	+	199.791i	−0.357963	+	0.0514673i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$251$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$252$	−1133.48	−	162.970i	−0.283343	−	0.0407386i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−3930.08	−	1153.98i	−0.959493	−	0.281733i
$257$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$257$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$258$		0			0
$259$	1102.00	+	503.266i	0.264382	+	0.120739i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$263$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3844.46	−	2113.99i	−0.876260	−	0.481838i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−2394.91	−	3726.55i	−0.536828	−	0.835320i	0.461838	−	0.886964i	$-0.347190\pi$
$271$	−2394.91	−	3726.55i	−0.536828	−	0.835320i	−0.998666	+	0.0516442i	$0.983554\pi$
$272$		0			0
$273$	1029.93	+	148.081i	0.228329	+	0.0328288i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1507.29	−	3300.50i	0.326946	−	0.715912i	−0.672767	−	0.739854i	$-0.734894\pi$
$277$	1507.29	−	3300.50i	0.326946	−	0.715912i	0.999713	+	0.0239421i	$0.00762173\pi$
$278$		0			0
$279$	−8740.46	−	1256.69i	−1.87555	−	0.269663i
$280$		0			0
$281$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$281$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$282$		0			0
$283$	2835.35	+	6208.56i	0.595563	+	1.30410i	0.932022	+	0.362403i	$0.118043\pi$
$283$	2835.35	+	6208.56i	0.595563	+	1.30410i	−0.336459	+	0.941698i	$0.609229\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−699.193	−	4862.99i	−0.142315	−	0.989821i
$290$		0			0
$291$	−1901.36	+	1221.93i	−0.383023	+	0.246154i
$292$	2827.27	−	3262.84i	0.566621	−	0.653915i
$293$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$293$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	2809.25	−	4371.28i	0.540641	−	0.841254i
$301$	783.812	+	1716.31i	0.150094	+	0.328659i
$302$		0			0
$303$		0			0
$304$	−2282.36	+	670.160i	−0.430599	+	0.126435i
$305$		0			0
$306$		0			0
$307$	7838.46	−	2301.58i	1.45721	−	0.427876i	0.545294	−	0.838245i	$-0.316418\pi$
$307$	7838.46	−	2301.58i	1.45721	−	0.427876i	0.911920	+	0.410369i	$0.134600\pi$
$308$		0			0
$309$	706.107	+	611.846i	0.129997	+	0.112643i
$310$		0			0
$311$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$311$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$312$		0			0
$313$	3858.45	−	6003.87i	0.696781	−	1.08421i	−0.294904	−	0.955527i	$-0.595288\pi$
$313$	3858.45	−	6003.87i	0.696781	−	1.08421i	0.991685	−	0.128686i	$-0.0410759\pi$
$314$		0			0
$315$		0			0
$316$	5806.65	−	2651.81i	1.03370	−	0.472076i
$317$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$317$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	829.980	−	5772.64i	0.142315	−	0.989821i
$325$	−2552.60	+	3971.93i	−0.435671	+	0.677917i
$326$		0			0
$327$	−1639.48	+	481.396i	−0.277259	+	0.0814105i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8036.62	+	6963.77i	−1.33454	+	1.15639i	−0.359803	+	0.933028i	$0.617156\pi$
$331$	−8036.62	+	6963.77i	−1.33454	+	1.15639i	−0.974737	+	0.223357i	$0.928298\pi$
$332$		0			0
$333$	−2563.06	+	5612.32i	−0.421786	+	0.923583i
$334$		0			0
$335$		0			0
$336$	1763.05			0.286257
$337$	11168.1	+	5100.32i	1.80524	+	0.824428i	0.954095	+	0.299505i	$0.0968215\pi$
$337$	11168.1	+	5100.32i	1.80524	+	0.824428i	0.851150	+	0.524923i	$0.175906\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−982.642	−	3346.57i	−0.154687	−	0.526816i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$347$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$348$		0			0
$349$	−6398.23	−	7383.96i	−0.981346	−	1.13253i	−0.991172	−	0.132581i	$-0.957673\pi$
$349$	−6398.23	−	7383.96i	−0.981346	−	1.13253i	0.00982661	−	0.999952i	$-0.496872\pi$
$350$		0			0
$351$	−754.154	+	5245.26i	−0.114683	+	0.797639i
$352$		0			0
$353$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$353$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$359$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$360$		0			0
$361$	3587.06	−	4139.68i	0.522971	−	0.603540i
$362$		0			0
$363$	−1948.48	−	6635.93i	−0.281733	−	0.959493i
$364$	−1601.98			−0.230677
$365$		0			0
$366$		0			0
$367$	−1293.42	+	2012.60i	−0.183967	+	0.286258i	−0.920972	−	0.389629i	$-0.872603\pi$
$367$	−1293.42	+	2012.60i	−0.183967	+	0.286258i	0.737005	+	0.675887i	$0.236239\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	13595.2			1.89483
$373$			8151.53i			1.13156i	0.824558	+	0.565778i	$0.191424\pi$
$373$			8151.53i			1.13156i	−0.824558	+	0.565778i	$0.808576\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1525.69	−	2374.02i	−0.206780	−	0.321756i	0.722338	−	0.691541i	$-0.243068\pi$
$379$	−1525.69	−	2374.02i	−0.206780	−	0.321756i	−0.929117	+	0.369785i	$0.879431\pi$
$380$		0			0
$381$	−5986.30	+	860.700i	−0.804954	+	0.115735i
$382$		0			0
$383$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$383$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8740.91	+	3991.84i	−1.14813	+	0.524332i
$388$	2629.80	−	2278.74i	0.344093	−	0.298158i
$389$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$389$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1807.63	+	12572.3i	−0.228520	+	1.58939i	0.475830	+	0.879537i	$0.342148\pi$
$397$	−1807.63	+	12572.3i	−0.228520	+	1.58939i	−0.704350	+	0.709853i	$0.748761\pi$
$398$		0			0
$399$	861.337	−	553.548i	0.108072	−	0.0694537i
$400$	−3323.32	+	7277.06i	−0.415415	+	0.909632i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−12353.2			−1.52694
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	5567.93	+	2542.79i	0.673145	+	0.307415i	0.722505	−	0.691366i	$-0.242991\pi$
$409$	5567.93	+	2542.79i	0.673145	+	0.307415i	−0.0493597	+	0.998781i	$0.515718\pi$
$410$		0			0
$411$		0			0
$412$	−1210.12	−	777.696i	−0.144704	−	0.0929959i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	6686.12	+	14640.6i	0.785181	+	1.71931i
$418$		0			0
$419$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$419$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$420$		0			0
$421$	6575.58	+	14398.5i	0.761221	+	1.66684i	0.745082	+	0.666972i	$0.232410\pi$
$421$	6575.58	+	14398.5i	0.761221	+	1.66684i	0.0161389	+	0.999870i	$0.494863\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3501.92	−	1028.26i	−0.396885	−	0.116536i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			8978.95i			1.00000i
$433$	−4615.10	−	15717.6i	−0.512211	−	1.74443i	−0.655938	−	0.754815i	$-0.727727\pi$
$433$	−4615.10	−	15717.6i	−0.512211	−	1.74443i	0.143727	−	0.989617i	$-0.454091\pi$
$434$		0			0
$435$		0			0
$436$	2392.98	−	1092.84i	0.262851	−	0.120040i
$437$		0			0
$438$		0			0
$439$	17902.0			1.94628			0.973139	−	0.230217i	$-0.0739435\pi$
$439$	17902.0			1.94628			0.973139	+	0.230217i	$0.0739435\pi$
$440$		0			0
$441$	8157.73	−	2395.33i	0.880869	−	0.258647i
$442$		0			0
$443$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$443$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$444$	2676.22	−	9114.36i	0.286053	−	0.974209i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−2686.76	+	386.298i	−0.283343	+	0.0407386i
$449$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$449$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	15386.0	+	2212.18i	1.59580	+	0.229442i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7886.27	+	2315.62i	0.807230	+	0.237024i	0.659209	−	0.751960i	$-0.270891\pi$
$457$	7886.27	+	2315.62i	0.807230	+	0.237024i	0.148021	+	0.988984i	$0.452710\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$461$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$462$		0			0
$463$	16822.3	−	2418.68i	1.68855	−	0.242777i	0.769988	−	0.638058i	$-0.220262\pi$
$463$	16822.3	−	2418.68i	1.68855	−	0.242777i	0.918560	+	0.395282i	$0.129353\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$467$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$468$		−	8158.65i		−	0.805841i
$469$	−2899.43	−	215.976i	−0.285466	−	0.0212641i
$470$		0			0
$471$	4074.32	+	1860.68i	0.398587	+	0.182029i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	661.183	+	4598.63i	0.0638677	+	0.444210i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$479$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$480$		0			0
$481$	−2431.72	+	8281.69i	−0.230514	+	0.785057i
$482$		0			0
$483$		0			0
$484$	4423.34	+	9685.76i	0.415415	+	0.909632i
$485$		0			0
$486$		0			0
$487$	−14421.4	+	12496.2i	−1.34188	+	1.16275i	−0.369529	+	0.929219i	$0.620481\pi$
$487$	−14421.4	+	12496.2i	−1.34188	+	1.16275i	−0.972351	+	0.233526i	$0.924974\pi$
$488$		0			0
$489$	−11351.6	−	17663.5i	−1.04977	−	1.63347i
$490$		0			0
$491$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$491$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−20718.1	+	2978.82i	−1.87555	+	0.269663i
$497$		0			0
$498$		0			0
$499$			11440.6i			1.02635i	0.858283	+	0.513176i	$0.171531\pi$
$499$			11440.6i			1.02635i	−0.858283	+	0.513176i	$0.828469\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$503$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	4002.66i		−	0.350620i
$508$	8934.11	−	2623.29i	0.780290	−	0.229114i
$509$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$509$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$510$		0			0
$511$	806.060	−	2745.19i	0.0697808	−	0.237652i
$512$		0			0
$513$	2819.14	+	4386.66i	0.242628	+	0.377536i
$514$		0			0
$515$		0			0
$516$	12445.9	−	7998.48i	1.06182	−	0.682391i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$521$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$522$		0			0
$523$	−22839.7	−	6706.35i	−1.90958	−	0.560704i	−0.982648	−	0.185482i	$-0.940615\pi$
$523$	−22839.7	−	6706.35i	−1.90958	−	0.560704i	−0.926935	−	0.375222i	$-0.877566\pi$
$524$		0			0
$525$	490.055	−	3408.41i	0.0407386	−	0.283343i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	5054.35	−	11067.5i	0.415415	−	0.909632i
$530$		0			0
$531$		0			0
$532$	−1191.33	+	1032.29i	−0.0970878	+	0.0841271i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9891.39	+	1422.17i	0.786070	+	0.113020i	0.523649	−	0.851934i	$-0.324570\pi$
$541$	9891.39	+	1422.17i	0.786070	+	0.113020i	0.262421	+	0.964954i	$0.415479\pi$
$542$		0			0
$543$	−22513.5	−	10281.6i	−1.77928	−	0.812568i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−15324.1	−	2203.28i	−1.19783	−	0.172222i	−0.485608	−	0.874177i	$-0.661402\pi$
$547$	−15324.1	−	2203.28i	−1.19783	−	0.172222i	−0.712221	+	0.701955i	$0.752311\pi$
$548$		0			0
$549$	5236.77	−	17834.8i	0.407103	−	1.38647i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2770.27	−	3197.06i	0.213027	−	0.245846i
$554$		0			0
$555$		0			0
$556$	−13397.0	−	20846.2i	−1.02187	−	1.59006i
$557$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$557$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$558$		0			0
$559$	−11308.8	+	7267.75i	−0.855659	+	0.549899i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$563$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1088.85	−	3708.27i	−0.0806478	−	0.274661i
$568$		0			0
$569$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$569$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$570$		0			0
$571$	22949.9	+	14749.0i	1.68200	+	1.08096i	0.854270	+	0.519829i	$0.174004\pi$
$571$	22949.9	+	14749.0i	1.68200	+	1.08096i	0.827730	+	0.561127i	$0.189632\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1967.36	−	13683.3i	−0.142315	−	0.989821i
$577$	10954.5	+	9492.12i	0.790367	+	0.684857i	0.953382	−	0.301767i	$-0.0975765\pi$
$577$	10954.5	+	9492.12i	0.790367	+	0.684857i	−0.163015	+	0.986624i	$0.552122\pi$
$578$		0			0
$579$	−20585.8	−	17837.7i	−1.47757	−	1.28033i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$587$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$588$	−11907.0	+	5437.73i	−0.835094	+	0.381375i
$589$	−9186.57	+	7960.21i	−0.642659	+	0.556867i
$590$		0			0
$591$		0			0
$592$	−2081.34	+	14476.0i	−0.144497	+	1.00500i
$593$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$593$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	9758.44	−	1403.05i	0.668989	−	0.0961861i
$598$		0			0
$599$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$599$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$600$		0			0
$601$	−3014.28	+	6600.35i	−0.204584	+	0.447977i	−0.983915	−	0.178636i	$-0.942832\pi$
$601$	−3014.28	+	6600.35i	−0.204584	+	0.447977i	0.779331	+	0.626612i	$0.215559\pi$
$602$		0			0
$603$	1099.94	−	14766.4i	0.0742833	−	0.997237i
$604$	−23931.9			−1.61221
$605$		0			0
$606$		0			0
$607$	13284.2	+	15330.8i	0.888288	+	1.02514i	0.999509	+	0.0313428i	$0.00997836\pi$
$607$	13284.2	+	15330.8i	0.888288	+	1.02514i	−0.111221	+	0.993796i	$0.535476\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−25397.8	−	16322.2i	−1.67342	−	1.07544i	−0.892815	−	0.450424i	$-0.851273\pi$
$613$	−25397.8	−	16322.2i	−1.67342	−	1.07544i	−0.780609	−	0.625020i	$-0.785091\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$617$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$618$		0			0
$619$	1634.05	−	11365.1i	0.106104	−	0.737966i	−0.865424	−	0.501040i	$-0.832951\pi$
$619$	1634.05	−	11365.1i	0.106104	−	0.737966i	0.971528	−	0.236926i	$-0.0761399\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1787.63	+	12433.2i	0.114683	+	0.797639i
$625$	13144.6	+	8447.51i	0.841254	+	0.540641i
$626$		0			0
$627$		0			0
$628$	−6616.67	−	1942.83i	−0.420436	−	0.123451i
$629$		0			0
$630$		0			0
$631$	−7211.85	−	24561.3i	−0.454991	−	1.54956i	−0.793486	−	0.608588i	$-0.791736\pi$
$631$	−7211.85	−	24561.3i	−0.454991	−	1.54956i	0.338495	−	0.940968i	$-0.390082\pi$
$632$		0			0
$633$			1151.69i			0.0723150i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	10819.2	−	4940.95i	0.672953	−	0.307327i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−4435.49	−	30849.5i	−0.272035	−	1.89205i	−0.427199	−	0.904158i	$-0.640500\pi$
$643$	−4435.49	−	30849.5i	−0.272035	−	1.89205i	0.155163	−	0.987889i	$-0.450410\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$647$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	8195.28	−	3742.66i	0.493392	−	0.225325i
$652$	21169.3	+	24430.6i	1.27155	+	1.46745i
$653$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$653$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	13980.8	+	4105.15i	0.830205	+	0.243770i
$658$		0			0
$659$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$659$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$660$		0			0
$661$	−2076.80	−	948.444i	−0.122206	−	0.0558097i	0.353374	−	0.935482i	$-0.385034\pi$
$661$	−2076.80	−	948.444i	−0.122206	−	0.0558097i	−0.475580	+	0.879673i	$0.657762\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	2302.65i		−	0.133072i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14817.9	−	23057.1i	−0.848718	−	1.32063i	−0.945599	−	0.325335i	$-0.894523\pi$
$673$	−14817.9	−	23057.1i	−0.848718	−	1.32063i	0.0968804	−	0.995296i	$-0.469114\pi$
$674$		0			0
$675$	17358.5	+	2495.78i	0.989821	+	0.142315i
$676$	877.015	+	6099.77i	0.0498984	+	0.347051i
$677$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$677$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$678$		0			0
$679$	957.944	−	2097.60i	0.0541421	−	0.118555i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$683$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$684$	−5257.32	−	6067.27i	−0.293887	−	0.339164i
$685$		0			0
$686$		0			0
$687$	−13022.9	+	15029.2i	−0.723225	+	0.834646i
$688$	−17214.1	+	14916.1i	−0.953899	+	0.826558i
$689$		0			0
$690$		0			0
$691$	888.352	+	6178.62i	0.0489067	+	0.340153i	0.999554	+	0.0298784i	$0.00951200\pi$
$691$	888.352	+	6178.62i	0.0489067	+	0.340153i	−0.950647	+	0.310275i	$0.899579\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$			5301.55i			0.286257i
$701$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$701$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$702$		0			0
$703$	3528.23	+	7725.74i	0.189288	+	0.414483i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	34836.7	−	10229.0i	1.84530	−	0.541829i	0.845330	−	0.534244i	$-0.179404\pi$
$709$	34836.7	−	10229.0i	1.84530	−	0.541829i	0.999971	−	0.00758547i	$-0.00241455\pi$
$710$		0			0
$711$	16282.1	+	14108.6i	0.858830	+	0.744180i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$719$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$720$		0			0
$721$	−943.562	−	135.664i	−0.0487380	−	0.00700746i
$722$		0			0
$723$	−29179.4	+	25284.1i	−1.50096	+	1.30059i
$724$	36561.8	+	10735.5i	1.87681	+	0.551080i
$725$		0			0
$726$		0			0
$727$	8940.37	−	13911.5i	0.456093	−	0.709695i	−0.534705	−	0.845039i	$-0.679577\pi$
$727$	8940.37	−	13911.5i	0.456093	−	0.709695i	0.990799	+	0.135343i	$0.0432138\pi$
$728$		0			0
$729$	18885.7	−	5545.34i	0.959493	−	0.281733i
$730$		0			0
$731$		0			0
$732$	−4072.72	+	28326.4i	−0.205645	+	1.43029i
$733$	−21876.4	+	18956.0i	−1.10235	+	0.955194i	−0.999221	−	0.0394740i	$-0.987432\pi$
$733$	−21876.4	+	18956.0i	−1.10235	+	0.955194i	−0.103132	+	0.994668i	$0.532886\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6619.47	−	3023.01i	−0.329501	−	0.150478i	0.243794	−	0.969827i	$-0.421608\pi$
$739$	−6619.47	−	3023.01i	−0.329501	−	0.150478i	−0.573295	+	0.819349i	$0.694335\pi$
$740$		0			0
$741$	4777.02	+	5512.98i	0.236826	+	0.273312i
$742$		0			0
$743$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$743$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	6684.11	+	7713.87i	0.324776	+	0.374811i	0.894533	−	0.447002i	$-0.147508\pi$
$751$	6684.11	+	7713.87i	0.324776	+	0.374811i	−0.569757	+	0.821813i	$0.692963\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	2471.84	+	5412.57i	0.118915	+	0.260388i
$757$	−21889.9	−	34061.4i	−1.05099	−	1.63538i	−0.723212	−	0.690626i	$-0.757335\pi$
$757$	−21889.9	−	34061.4i	−1.05099	−	1.63538i	−0.327782	−	0.944753i	$-0.606301\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$761$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$762$		0			0
$763$	1141.65	−	1317.54i	0.0541686	−	0.0625140i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	5996.24	+	20421.3i	0.281733	+	0.959493i
$769$	22696.5	−	35316.4i	1.06431	−	1.65610i	0.383705	−	0.923456i	$-0.374648\pi$
$769$	22696.5	−	35316.4i	1.06431	−	1.65610i	0.680608	−	0.732648i	$-0.261716\pi$
$770$		0			0
$771$		0			0
$772$	35279.7	+	22672.9i	1.64474	+	1.05701i
$773$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$773$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$774$		0			0
$775$			40881.2i			1.89483i
$776$		0			0
$777$	−895.875	−	6230.95i	−0.0413634	−	0.287688i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	16953.9	−	10895.6i	0.772319	−	0.496339i
$785$		0			0
$786$		0			0
$787$	3340.13	+	2894.24i	0.151287	+	0.131091i	0.727216	−	0.686409i	$-0.240814\pi$
$787$	3340.13	+	2894.24i	0.151287	+	0.131091i	−0.575929	+	0.817500i	$0.695359\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	3700.64	−	25738.5i	0.165717	−	1.15259i
$794$		0			0
$795$		0			0
$796$	−14563.8	+	4276.31i	−0.648491	+	0.190414i
$797$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$797$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1559.21	+	22744.0i	0.0683944	+	0.997658i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$809$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$810$		0			0
$811$	−10999.4	−	5023.25i	−0.476252	−	0.217497i	0.162802	−	0.986659i	$-0.447947\pi$
$811$	−10999.4	−	5023.25i	−0.476252	−	0.217497i	−0.639054	+	0.769162i	$0.720674\pi$
$812$		0			0
$813$	−9561.90	+	20937.6i	−0.412485	+	0.903216i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3726.71	+	12692.0i	−0.159585	+	0.543497i
$818$		0			0
$819$	−2246.02	−	4918.09i	−0.0958268	−	0.209831i
$820$		0			0
$821$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$821$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$822$		0			0
$823$	19344.4	+	42358.2i	0.819322	+	1.79406i	0.560536	+	0.828130i	$0.310595\pi$
$823$	19344.4	+	42358.2i	0.819322	+	1.79406i	0.258786	+	0.965935i	$0.416677\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$827$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$828$		0			0
$829$	−43051.9	−	12641.2i	−1.80368	−	0.529610i	−0.805657	−	0.592383i	$-0.798187\pi$
$829$	−43051.9	−	12641.2i	−1.80368	−	0.529610i	−0.998028	+	0.0627735i	$0.980005\pi$
$830$		0			0
$831$	−18661.8	+	2683.15i	−0.779024	+	0.112007i
$832$	−5448.43	−	18555.7i	−0.227032	−	0.773199i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	19060.8	+	41737.4i	0.787143	+	1.72360i
$838$		0			0
$839$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$839$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	−252.344	−	1755.09i	−0.0102915	−	0.0715790i
$845$		0			0
$846$		0			0
$847$	5332.84	+	4620.93i	0.216338	+	0.187458i
$848$		0			0
$849$	19174.1	−	29835.5i	0.775094	−	1.20607i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−32546.5	−	37560.7i	−1.30642	−	1.50768i	−0.706871	−	0.707343i	$-0.749894\pi$
$853$	−32546.5	−	37560.7i	−1.30642	−	1.50768i	−0.599545	−	0.800341i	$-0.704652\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$857$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$858$		0			0
$859$	18158.2	+	5331.73i	0.721245	+	0.211777i	0.621699	−	0.783256i	$-0.286443\pi$
$859$	18158.2	+	5331.73i	0.721245	+	0.211777i	0.0995460	+	0.995033i	$0.468261\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$863$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−19293.3	+	16717.7i	−0.755750	+	0.654861i
$868$	−11669.0	+	7499.20i	−0.456303	+	0.293248i
$869$		0			0
$870$		0			0
$871$	−1416.76	−	20666.1i	−0.0551150	−	0.803954i
$872$		0			0
$873$	10682.8	+	4878.67i	0.414156	+	0.189139i
$874$		0			0
$875$		0			0
$876$	−22205.3	−	3192.64i	−0.856447	−	0.123138i
$877$	4716.47	+	32803.8i	0.181601	+	1.26306i	0.852979	+	0.521946i	$0.174794\pi$
$877$	4716.47	+	32803.8i	0.181601	+	1.26306i	−0.671378	+	0.741115i	$0.734297\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$881$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$882$		0			0
$883$	14581.0	−	49658.3i	0.555707	−	1.89256i	0.119179	−	0.992873i	$-0.461974\pi$
$883$	14581.0	−	49658.3i	0.555707	−	1.89256i	0.436528	−	0.899691i	$-0.356208\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$887$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$888$		0			0
$889$	4663.37	−	4040.83i	0.175933	−	0.152447i
$890$		0			0
$891$		0			0
$892$	504.528	+	3509.07i	0.0189382	+	0.131718i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−27000.0			−1.00000
$901$		0			0
$902$		0			0
$903$	5300.54	−	8247.80i	0.195339	−	0.303953i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−23516.3	+	6904.99i	−0.860909	+	0.252786i	−0.682244	−	0.731124i	$-0.738996\pi$
$907$	−23516.3	+	6904.99i	−0.860909	+	0.252786i	−0.178665	+	0.983910i	$0.557178\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$911$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$912$	9341.18	+	8094.17i	0.339164	+	0.293887i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	16553.0	−	25756.9i	0.597081	−	0.929076i
$917$		0			0
$918$		0			0
$919$	9685.65	−	4423.29i	0.347661	−	0.158771i	−0.233929	−	0.972254i	$-0.575158\pi$
$919$	9685.65	−	4423.29i	0.347661	−	0.158771i	0.581590	+	0.813482i	$0.302431\pi$
$920$		0			0
$921$	−32081.1	−	27798.4i	−1.14778	−	0.994558i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	27407.2	+	8047.48i	0.974209	+	0.286053i
$926$		0			0
$927$	690.915	−	4805.42i	0.0244797	−	0.170260i
$928$		0			0
$929$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$929$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$930$		0			0
$931$	4861.92	−	10646.1i	0.171153	−	0.374772i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	32478.4i		−	1.13236i	−0.824281	−	0.566181i	$-0.808420\pi$
$937$		−	32478.4i		−	1.13236i	0.824281	−	0.566181i	$-0.191580\pi$
$938$		0			0
$939$	−37084.0			−1.28881
$940$		0			0
$941$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$941$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$947$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$948$	−27904.1	−	17932.9i	−0.955996	−	0.614382i
$949$	20176.7	+	2900.97i	0.690160	+	0.0992301i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$953$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−64920.0	+	41721.5i	−2.17918	+	1.40047i
$962$		0			0
$963$		0			0
$964$	38927.4	−	44924.7i	1.30059	−	1.50096i
$965$		0			0
$966$		0			0
$967$	60131.8			1.99970			0.999848	−	0.0174092i	$-0.00554179\pi$
$967$	60131.8			1.99970			0.999848	+	0.0174092i	$0.00554179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$971$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$972$	−27565.5	+	12588.7i	−0.909632	+	0.415415i
$973$	−13814.6	−	8878.13i	−0.455166	−	0.292518i
$974$		0			0
$975$	24533.3			0.805841
$976$		−	44059.8i		−	1.44500i
$977$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$977$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	6710.04	+	5814.28i	0.218384	+	0.189231i
$982$		0			0
$983$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$983$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−8487.78	−	7354.70i	−0.273312	−	0.236826i
$989$		0			0
$990$		0			0
$991$	−29555.2	+	25609.7i	−0.947379	+	0.820908i	−0.983954	−	0.178423i	$-0.942901\pi$
$991$	−29555.2	+	25609.7i	−0.947379	+	0.820908i	0.0365752	+	0.999331i	$0.488355\pi$
$992$		0			0
$993$	53017.5	+	15567.3i	1.69432	+	0.497497i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5678.42	+	1667.34i	−0.180379	+	0.0529639i	−0.370674	−	0.928763i	$-0.620873\pi$
$997$	−5678.42	+	1667.34i	−0.180379	+	0.0529639i	0.190296	+	0.981727i	$0.439055\pi$
$998$		0			0
$999$	31733.3	−	4562.56i	1.00500	−	0.144497i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.a.110.1	yes	20
3.2	odd	2	CM	201.4.j.a.110.1	yes	20
67.53	odd	22	inner	201.4.j.a.53.1	✓	20
201.53	even	22	inner	201.4.j.a.53.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.a.53.1	✓	20	67.53	odd	22	inner
201.4.j.a.53.1	✓	20	201.53	even	22	inner
201.4.j.a.110.1	yes	20	1.1	even	1	trivial
201.4.j.a.110.1	yes	20	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-2.80925 - 4.37128i) q^{3} +(5.23889 + 6.04600i) q^{4} +(4.82246 + 2.20234i) q^{7} +(-11.2162 + 24.5601i) q^{9} +O(q^{10})\)	\(q+(-2.80925 - 4.37128i) q^{3} +(5.23889 + 6.04600i) q^{4} +(4.82246 + 2.20234i) q^{7} +(-11.2162 + 24.5601i) q^{9} +(11.7114 - 39.8854i) q^{12} +(-10.6415 + 36.2415i) q^{13} +(-9.10815 + 63.3486i) q^{16} +(15.4399 + 33.8086i) q^{19} +(-3.92044 - 27.2672i) q^{21} +(119.937 + 35.2166i) q^{25} +(138.868 - 19.9662i) q^{27} +(11.9489 + 40.6944i) q^{28} +(92.1405 + 313.802i) q^{31} +(-207.250 + 60.8542i) q^{36} +228.514 q^{37} +(188.316 - 55.2947i) q^{39} +(268.971 + 233.064i) q^{43} +(302.502 - 138.148i) q^{48} +(-206.211 - 237.981i) q^{49} +(-274.865 + 125.527i) q^{52} +(104.412 - 162.469i) q^{57} +(-681.427 + 97.9744i) q^{61} +(-108.179 + 93.7379i) q^{63} +(-430.722 + 276.808i) q^{64} +(-514.404 + 190.135i) q^{67} +(-76.8030 - 534.177i) q^{73} +(-182.991 - 623.209i) q^{75} +(-123.519 + 270.469i) q^{76} +(224.806 - 765.617i) q^{79} +(-477.393 - 550.941i) q^{81} +(144.319 - 166.553i) q^{84} +(-131.134 + 151.337i) q^{91} +(1112.87 - 1284.32i) q^{93} -434.966i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) 20 * q + 16 * q^4 + 54 * q^9	\( 20 q + 16 q^{4} + 54 q^{9} - 128 q^{16} + 112 q^{19} + 324 q^{21} + 250 q^{25} - 432 q^{36} + 220 q^{37} - 648 q^{39} + 1258 q^{49} - 6160 q^{52} + 8910 q^{57} + 5940 q^{63} + 1024 q^{64} - 880 q^{67} - 10710 q^{73} - 896 q^{76} - 9724 q^{79} - 1458 q^{81} + 11664 q^{84} - 3888 q^{91} - 1620 q^{93}+O(q^{100}) \) 20 * q + 16 * q^4 + 54 * q^9 - 128 * q^16 + 112 * q^19 + 324 * q^21 + 250 * q^25 - 432 * q^36 + 220 * q^37 - 648 * q^39 + 1258 * q^49 - 6160 * q^52 + 8910 * q^57 + 5940 * q^63 + 1024 * q^64 - 880 * q^67 - 10710 * q^73 - 896 * q^76 - 9724 * q^79 - 1458 * q^81 + 11664 * q^84 - 3888 * q^91 - 1620 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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