LMFDB - Embedded newform 201.4.j.a.53.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("201.5");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			53.2
Root			$0.981929 - 0.189251i$ of defining polynomial
Character	$\chi$	$=$	201.53
Dual form			201.4.j.a.110.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.80925 - 4.37128i) q^{3} +(5.23889 - 6.04600i) q^{4} +(-32.3206 + 14.7603i) q^{7} +(-11.2162 - 24.5601i) q^{9} +O(q^{10})$	$q+(2.80925 - 4.37128i) q^{3} +(5.23889 - 6.04600i) q^{4} +(-32.3206 + 14.7603i) q^{7} +(-11.2162 - 24.5601i) q^{9} +(-11.7114 - 39.8854i) q^{12} +(-25.2367 - 85.9485i) q^{13} +(-9.10815 - 63.3486i) q^{16} +(-54.5804 + 119.515i) q^{19} +(-26.2752 + 182.748i) q^{21} +(119.937 - 35.2166i) q^{25} +(-138.868 - 19.9662i) q^{27} +(-80.0831 + 272.738i) q^{28} +(79.6402 - 271.230i) q^{31} +(-207.250 - 60.8542i) q^{36} -17.4255 q^{37} +(-446.601 - 131.134i) q^{39} +(-47.5346 + 41.1890i) q^{43} +(-302.502 - 138.148i) q^{48} +(602.138 - 694.904i) q^{49} +(-651.857 - 297.693i) q^{52} +(369.101 + 574.333i) q^{57} +(876.217 + 125.981i) q^{61} +(725.029 + 628.241i) q^{63} +(-430.722 - 276.808i) q^{64} +(-329.950 + 438.060i) q^{67} +(28.5996 - 198.915i) q^{73} +(182.991 - 623.209i) q^{75} +(436.644 + 956.116i) q^{76} +(-365.137 - 1243.54i) q^{79} +(-477.393 + 550.941i) q^{81} +(967.241 + 1116.26i) q^{84} +(2084.29 + 2405.40i) q^{91} +(-961.892 - 1110.08i) q^{93} -1873.07i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/201\mathbb{Z}\right)^\times$.

$n$	$68$	$136$
$\chi(n)$	$-1$	$e\left(\frac{19}{22}\right)$

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	$-0.136364\pi$
$2$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$3$	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	$-0.0454545\pi$
$5$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$6$		0			0
$7$	−32.3206	+	14.7603i	−1.74515	+	0.796983i	−0.755211	+	0.655482i	$0.772466\pi$
$7$	−32.3206	+	14.7603i	−1.74515	+	0.796983i	−0.989938	+	0.141501i	$0.954807\pi$
$8$		0			0
$9$	−11.2162	−	24.5601i	−0.415415	−	0.909632i
$10$		0			0
$11$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$11$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$12$	−11.7114	−	39.8854i	−0.281733	−	0.959493i
$13$	−25.2367	−	85.9485i	−0.538416	−	1.83368i	−0.552109	−	0.833772i	$-0.686177\pi$
$13$	−25.2367	−	85.9485i	−0.538416	−	1.83368i	0.0136929	−	0.999906i	$-0.495641\pi$
$14$		0			0
$15$		0			0
$16$	−9.10815	−	63.3486i	−0.142315	−	0.989821i
$17$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$17$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$18$		0			0
$19$	−54.5804	+	119.515i	−0.659032	+	1.44308i	0.224389	+	0.974500i	$0.427961\pi$
$19$	−54.5804	+	119.515i	−0.659032	+	1.44308i	−0.883421	+	0.468580i	$0.844766\pi$
$20$		0			0
$21$	−26.2752	+	182.748i	−0.273034	+	1.89899i
$22$		0			0
$23$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$23$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$24$		0			0
$25$	119.937	−	35.2166i	0.959493	−	0.281733i
$26$		0			0
$27$	−138.868	−	19.9662i	−0.989821	−	0.142315i
$28$	−80.0831	+	272.738i	−0.540510	+	1.84081i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	79.6402	−	271.230i	0.461413	−	1.57143i	−0.320001	−	0.947417i	$-0.603683\pi$
$31$	79.6402	−	271.230i	0.461413	−	1.57143i	0.781415	−	0.624012i	$-0.214498\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−207.250	−	60.8542i	−0.959493	−	0.281733i
$37$	−17.4255			−0.0774253			−0.0387126	−	0.999250i	$-0.512326\pi$
$37$	−17.4255			−0.0774253			−0.0387126	+	0.999250i	$0.512326\pi$
$38$		0			0
$39$	−446.601	−	131.134i	−1.83368	−	0.538416i
$40$		0			0
$41$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$41$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$42$		0			0
$43$	−47.5346	+	41.1890i	−0.168580	+	0.146076i	−0.735065	−	0.677996i	$-0.762848\pi$
$43$	−47.5346	+	41.1890i	−0.168580	+	0.146076i	0.566485	+	0.824072i	$0.308303\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$47$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$48$	−302.502	−	138.148i	−0.909632	−	0.415415i
$49$	602.138	−	694.904i	1.75550	−	2.02596i
$50$		0			0
$51$		0			0
$52$	−651.857	−	297.693i	−1.73839	−	0.793896i
$53$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$53$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	369.101	+	574.333i	0.857696	+	1.33460i
$58$		0			0
$59$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$59$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$60$		0			0
$61$	876.217	+	125.981i	1.83915	+	0.264430i	0.972231	−	0.234022i	$-0.0751887\pi$
$61$	876.217	+	125.981i	1.83915	+	0.264430i	0.866918	+	0.498451i	$0.166098\pi$
$62$		0			0
$63$	725.029	+	628.241i	1.44992	+	1.25636i
$64$	−430.722	−	276.808i	−0.841254	−	0.540641i
$65$		0			0
$66$		0			0
$67$	−329.950	+	438.060i	−0.601638	+	0.798769i
$67$	−329.950	+	438.060i	−0.601638	+	0.798769i
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$71$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$72$		0			0
$73$	28.5996	−	198.915i	0.0458539	−	0.318921i	−0.953966	−	0.299916i	$-0.903041\pi$
$73$	28.5996	−	198.915i	0.0458539	−	0.318921i	0.999819	−	0.0190045i	$-0.00604967\pi$
$74$		0			0
$75$	182.991	−	623.209i	0.281733	−	0.959493i
$76$	436.644	+	956.116i	0.659032	+	1.44308i
$77$		0			0
$78$		0			0
$79$	−365.137	−	1243.54i	−0.520015	−	1.77101i	−0.629480	−	0.777017i	$-0.716732\pi$
$79$	−365.137	−	1243.54i	−0.520015	−	1.77101i	0.109465	−	0.993991i	$-0.465086\pi$
$80$		0			0
$81$	−477.393	+	550.941i	−0.654861	+	0.755750i
$82$		0			0
$83$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$83$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$84$	967.241	+	1116.26i	1.25636	+	1.44992i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$89$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$90$		0			0
$91$	2084.29	+	2405.40i	2.40103	+	2.77093i
$92$		0			0
$93$	−961.892	−	1110.08i	−1.07251	−	1.23774i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1873.07i		−	1.96063i	−0.197430	−	0.980317i	$-0.563259\pi$
$97$		−	1873.07i		−	1.96063i	0.197430	−	0.980317i	$-0.436741\pi$
$98$		0			0
$99$		0			0
$100$	415.415	−	909.632i	0.415415	−	0.909632i
$101$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$101$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$102$		0			0
$103$	1623.38	+	476.669i	1.55298	+	0.455996i	0.941989	−	0.335643i	$-0.108953\pi$
$103$	1623.38	+	476.669i	1.55298	+	0.455996i	0.610990	+	0.791638i	$0.290772\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$107$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$108$	−848.230	+	734.995i	−0.755750	+	0.654861i
$109$	267.649	+	911.528i	0.235194	+	0.800996i	0.989507	+	0.144482i	$0.0461516\pi$
$109$	267.649	+	911.528i	0.235194	+	0.800996i	−0.754314	+	0.656514i	$0.772030\pi$
$110$		0			0
$111$	−48.9526	+	76.1718i	−0.0418593	+	0.0651343i
$112$	1229.43	+	1913.03i	1.03723	+	1.61396i
$113$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$113$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1827.84	+	1583.83i	−1.44431	+	1.25150i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1277.09	−	374.986i	0.959493	−	0.281733i
$122$		0			0
$123$		0			0
$124$	−1222.63	−	1902.45i	−0.885446	−	1.37778i
$125$		0			0
$126$		0			0
$127$	−180.579	−	395.413i	−0.126172	−	0.276278i	0.835996	−	0.548736i	$-0.184891\pi$
$127$	−180.579	−	395.413i	−0.126172	−	0.276278i	−0.962168	+	0.272458i	$0.912163\pi$
$128$		0			0
$129$	46.5119	+	323.497i	0.0317453	+	0.220793i
$130$		0			0
$131$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$131$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$132$		0			0
$133$		−	4668.41i		−	3.04363i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$137$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$138$		0			0
$139$	308.932	−	44.4178i	0.188513	−	0.0271041i	−0.0474117	−	0.998875i	$-0.515097\pi$
$139$	308.932	−	44.4178i	0.188513	−	0.0271041i	0.235925	+	0.971771i	$0.424188\pi$
$140$		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
$147$	−1346.06	−	4584.27i	−0.755248	−	2.57214i
$148$	−91.2902	+	105.355i	−0.0507028	+	0.0585141i
$149$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$149$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$150$		0			0
$151$	2284.82	+	2636.83i	1.23137	+	1.42107i	0.873163	+	0.487429i	$0.162065\pi$
$151$	2284.82	+	2636.83i	1.23137	+	1.42107i	0.358204	+	0.933644i	$0.383389\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−3132.53	+	2013.15i	−1.60771	+	1.03321i
$157$	−1965.75	−	1263.31i	−0.999262	−	0.642187i	−0.0646697	−	0.997907i	$-0.520599\pi$
$157$	−1965.75	−	1263.31i	−0.999262	−	0.642187i	−0.934593	+	0.355720i	$0.884236\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	412.257			0.198101			0.0990505	−	0.995082i	$-0.468419\pi$
$163$	412.257			0.198101			0.0990505	+	0.995082i	$0.468419\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$167$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$168$		0			0
$169$	−4902.01	+	3150.33i	−2.23123	+	1.43392i
$170$		0			0
$171$	3547.47			1.58644
$172$			503.178i			0.223064i
$173$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$173$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$174$		0			0
$175$	−3356.62	+	2908.53i	−1.44992	+	1.25636i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$179$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$180$		0			0
$181$	887.479	+	570.348i	0.364452	+	0.234219i	0.710031	−	0.704171i	$-0.248681\pi$
$181$	887.479	+	570.348i	0.364452	+	0.234219i	−0.345579	+	0.938390i	$0.612317\pi$
$182$		0			0
$183$	3012.21	−	3476.28i	1.21677	−	1.40423i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4783.01	−	1404.42i	1.84081	−	0.540510i
$190$		0			0
$191$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$191$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$192$	−2420.01	+	1105.18i	−0.909632	+	0.415415i
$193$	−4113.03	−	1207.70i	−1.53400	−	0.450424i	−0.597730	−	0.801698i	$-0.703930\pi$
$193$	−4113.03	−	1207.70i	−1.53400	−	0.450424i	−0.936273	+	0.351274i	$0.885749\pi$
$194$		0			0
$195$		0			0
$196$	−1046.86	−	7281.04i	−0.381507	−	2.65344i
$197$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$197$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$198$		0			0
$199$	−2060.62	−	4512.12i	−0.734036	−	1.60731i	−0.793123	−	0.609061i	$-0.791546\pi$
$199$	−2060.62	−	4512.12i	−0.734036	−	1.60731i	0.0590873	−	0.998253i	$-0.481181\pi$
$200$		0			0
$201$	987.970	+	2672.92i	0.346697	+	0.937977i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−5214.85	+	2381.54i	−1.73839	+	0.793896i
$209$		0			0
$210$		0			0
$211$	1630.79	−	1048.05i	0.532078	−	0.341946i	−0.246855	−	0.969052i	$-0.579397\pi$
$211$	1630.79	−	1048.05i	0.532078	−	0.341946i	0.778934	+	0.627106i	$0.215761\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1429.42	+	9941.83i	0.447167	+	3.11012i
$218$		0			0
$219$	−789.169	−	683.819i	−0.243503	−	0.210996i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4930.43	+	3168.60i	−1.48057	+	0.951502i	−0.483469	+	0.875362i	$0.660623\pi$
$223$	−4930.43	+	3168.60i	−1.48057	+	0.951502i	−0.997097	+	0.0761405i	$0.975740\pi$
$224$		0			0
$225$	−2210.15	−	2550.65i	−0.654861	−	0.755750i
$226$		0			0
$227$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$227$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$228$	5406.09	+	777.279i	1.57030	+	0.225774i
$229$	1787.20	−	6086.64i	0.515727	−	1.75640i	−0.128643	−	0.991691i	$-0.541062\pi$
$229$	1787.20	−	6086.64i	0.515727	−	1.75640i	0.644370	−	0.764714i	$-0.277120\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$233$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−6461.64	−	1897.31i	−1.77101	−	0.520015i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−957.465	+	6659.32i	−0.255916	+	1.77994i	0.305290	+	0.952259i	$0.401246\pi$
$241$	−957.465	+	6659.32i	−0.255916	+	1.77994i	−0.561206	+	0.827676i	$0.689663\pi$
$242$		0			0
$243$	1067.20	+	3634.55i	0.281733	+	0.959493i
$244$	5352.08	−	4637.60i	1.40423	−	1.21677i
$245$		0			0
$246$		0			0
$247$	11649.5	+	1674.95i	3.00098	+	0.431475i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$251$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$252$	7596.69	−	1092.24i	1.89899	−	0.273034i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−3930.08	+	1153.98i	−0.959493	+	0.281733i
$257$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$257$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$258$		0			0
$259$	563.203	−	257.206i	0.135119	−	0.0617066i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$263$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	919.939	+	4289.82i	0.209680	+	0.977770i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	1596.25	−	2483.81i	0.357806	−	0.556756i	−0.614957	−	0.788561i	$-0.710827\pi$
$271$	1596.25	−	2483.81i	0.357806	−	0.556756i	0.972762	+	0.231804i	$0.0744629\pi$
$272$		0			0
$273$	16370.0	−	2353.65i	3.62915	−	0.521793i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3699.92	−	8101.70i	−0.802551	−	1.75734i	−0.636579	−	0.771212i	$-0.719651\pi$
$277$	−3699.92	−	8101.70i	−0.802551	−	1.75734i	−0.165972	−	0.986130i	$-0.553076\pi$
$278$		0			0
$279$	−7554.68	+	1086.20i	−1.62110	+	0.233079i
$280$		0			0
$281$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$281$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$282$		0			0
$283$	−3497.49	+	7658.45i	−0.734645	+	1.60865i	0.0575227	+	0.998344i	$0.481680\pi$
$283$	−3497.49	+	7658.45i	−0.734645	+	1.60865i	−0.792168	+	0.610304i	$0.791047\pi$
$284$		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
$291$	−8187.72	−	5261.93i	−1.64939	−	1.06000i
$292$	−1052.81	−	1215.01i	−0.210996	−	0.243503i
$293$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$293$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$294$		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
$301$	928.385	−	2032.88i	0.177778	−	0.389280i
$302$		0			0
$303$		0			0
$304$	8068.20	+	2369.04i	1.52218	+	0.446953i
$305$		0			0
$306$		0			0
$307$	5532.49	+	1624.49i	1.02852	+	0.302001i	0.752110	−	0.659038i	$-0.229036\pi$
$307$	5532.49	+	1624.49i	1.02852	+	0.302001i	0.276412	+	0.961039i	$0.410855\pi$
$308$		0			0
$309$	6644.15	−	5757.19i	1.22321	−	1.05992i
$310$		0			0
$311$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$311$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$312$		0			0
$313$	5987.07	+	9316.06i	1.08118	+	1.68235i	0.552161	+	0.833737i	$0.313803\pi$
$313$	5987.07	+	9316.06i	1.08118	+	1.68235i	0.529018	+	0.848611i	$0.322560\pi$
$314$		0			0
$315$		0			0
$316$	−9431.37	−	4307.16i	−1.67898	−	0.766762i
$317$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$317$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$318$		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
$325$	−6053.62	−	9419.62i	−1.03321	−	1.60771i
$326$		0			0
$327$	4736.44	+	1390.74i	0.800996	+	0.235194i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7225.98	+	6261.35i	1.19993	+	1.03974i	0.998180	+	0.0603052i	$0.0192074\pi$
$331$	7225.98	+	6261.35i	1.19993	+	1.03974i	0.201747	+	0.979438i	$0.435338\pi$
$332$		0			0
$333$	195.448	+	427.971i	0.0321636	+	0.0704285i
$334$		0			0
$335$		0			0
$336$	11816.1			1.91852
$337$	−8641.29	+	3946.35i	−1.39680	+	0.637897i	−0.964560	−	0.263863i	$-0.915003\pi$
$337$	−8641.29	+	3946.35i	−1.39680	+	0.637897i	−0.432238	+	0.901759i	$0.642276\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−5770.88	+	19653.8i	−0.908450	+	3.09390i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$347$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$348$		0			0
$349$	−83.9114	+	96.8389i	−0.0128701	+	0.0148529i	−0.762148	−	0.647403i	$-0.775855\pi$
$349$	−83.9114	+	96.8389i	−0.0128701	+	0.0148529i	0.749278	+	0.662256i	$0.230401\pi$
$350$		0			0
$351$	1788.51	+	12439.4i	0.271977	+	1.89164i
$352$		0			0
$353$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$353$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$359$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$360$		0			0
$361$	−6813.00	−	7862.62i	−0.993294	−	1.14632i
$362$		0			0
$363$	1948.48	−	6635.93i	0.281733	−	0.959493i
$364$	25462.4			3.66647
$365$		0			0
$366$		0			0
$367$	−3351.58	−	5215.16i	−0.476706	−	0.741769i	0.516731	−	0.856148i	$-0.327149\pi$
$367$	−3351.58	−	5215.16i	−0.476706	−	0.741769i	−0.993437	+	0.114379i	$0.963512\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−11750.8			−1.63777
$373$		−	14316.4i		−	1.98733i	−0.112384	−	0.993665i	$-0.535849\pi$
$373$		−	14316.4i		−	1.98733i	0.112384	−	0.993665i	$-0.464151\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6917.26	+	10763.5i	−0.937508	+	1.45879i	−0.0496011	+	0.998769i	$0.515795\pi$
$379$	−6917.26	+	10763.5i	−0.937508	+	1.45879i	−0.887907	+	0.460022i	$0.847841\pi$
$380$		0			0
$381$	−2235.76	−	321.453i	−0.300633	−	0.0432245i
$382$		0			0
$383$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$383$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1544.76	+	705.469i	0.202906	+	0.0926641i
$388$	−11324.6	−	9812.80i	−1.48175	−	1.28394i
$389$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$389$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1585.82	+	11029.6i	0.200479	+	1.39436i	0.802867	+	0.596158i	$0.203307\pi$
$397$	1585.82	+	11029.6i	0.200479	+	1.39436i	−0.602388	+	0.798203i	$0.705784\pi$
$398$		0			0
$399$	−20406.9	−	13114.7i	−2.56046	−	1.64551i
$400$	−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
$403$	−25321.6			−3.12993
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	9281.03	−	4238.50i	1.12205	−	0.512422i	0.234029	−	0.972230i	$-0.424809\pi$
$409$	9281.03	−	4238.50i	1.12205	−	0.512422i	0.888018	+	0.459808i	$0.152082\pi$
$410$		0			0
$411$		0			0
$412$	11386.7	−	7317.77i	1.36160	−	0.875050i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	673.706	−	1475.21i	0.0791164	−	0.173241i
$418$		0			0
$419$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$419$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$420$		0			0
$421$	7086.41	−	15517.1i	0.820358	−	1.79633i	0.266211	−	0.963915i	$-0.414228\pi$
$421$	7086.41	−	15517.1i	0.820358	−	1.79633i	0.554147	−	0.832419i	$-0.313044\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−30179.4	+	8861.47i	−3.42034	+	1.00430i
$428$		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
$433$	5026.25	−	17117.8i	0.557844	−	1.89984i	0.143727	−	0.989617i	$-0.454091\pi$
$433$	5026.25	−	17117.8i	0.557844	−	1.89984i	0.414117	−	0.910224i	$-0.364090\pi$
$434$		0			0
$435$		0			0
$436$	6913.28	+	3157.19i	0.759371	+	0.346793i
$437$		0			0
$438$		0			0
$439$	1644.27			0.178763			0.0893814	−	0.995997i	$-0.471511\pi$
$439$	1644.27			0.178763			0.0893814	+	0.995997i	$0.471511\pi$
$440$		0			0
$441$	−23820.6	−	6994.35i	−2.57214	−	0.755248i
$442$		0			0
$443$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$443$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$444$	204.077	+	695.023i	0.0218132	+	0.0742890i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	18007.0	+	2589.01i	1.89899	+	0.273034i
$449$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$449$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	17945.0	−	2580.10i	1.86121	−	0.267601i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−18018.4	+	5290.67i	−1.84434	+	0.541547i	−0.844358	+	0.535780i	$0.820018\pi$
$457$	−18018.4	+	5290.67i	−1.84434	+	0.541547i	−0.999983	+	0.00576765i	$0.998164\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$461$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$462$		0			0
$463$	18796.5	+	2702.53i	1.88671	+	0.271268i	0.986447	−	0.164079i	$-0.0524651\pi$
$463$	18796.5	+	2702.53i	1.88671	+	0.271268i	0.900263	+	0.435347i	$0.143374\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$467$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$468$			19348.6i			1.91109i
$469$	4198.27	−	19028.5i	0.413343	−	1.87347i
$470$		0			0
$471$	−11044.6	+	5043.90i	−1.08048	+	0.493441i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2337.30	+	16256.3i	−0.225774	+	1.57030i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$479$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$480$		0			0
$481$	439.763	+	1497.69i	0.0416870	+	0.141973i
$482$		0			0
$483$		0			0
$484$	4423.34	−	9685.76i	0.415415	−	0.909632i
$485$		0			0
$486$		0			0
$487$	−9452.99	−	8191.07i	−0.879581	−	0.762161i	0.0927693	−	0.995688i	$-0.470428\pi$
$487$	−9452.99	−	8191.07i	−0.879581	−	0.762161i	−0.972351	+	0.233526i	$0.924974\pi$
$488$		0			0
$489$	1158.13	−	1802.09i	0.107102	−	0.166653i
$490$		0			0
$491$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$491$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−17907.4	−	2574.69i	−1.62110	−	0.233079i
$497$		0			0
$498$		0			0
$499$			19969.2i			1.79147i	0.444590	+	0.895734i	$0.353349\pi$
$499$			19969.2i			1.79147i	−0.444590	+	0.895734i	$0.646651\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$503$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			30278.1i			2.65227i
$508$	−3336.70	−	979.744i	−0.291422	−	0.0855691i
$509$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$509$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$510$		0			0
$511$	2011.69	+	6851.19i	0.174153	+	0.593109i
$512$		0			0
$513$	9965.74	−	15507.0i	0.857696	−	1.33460i
$514$		0			0
$515$		0			0
$516$	2199.53	+	1413.55i	0.187653	+	0.120597i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$521$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$522$		0			0
$523$	13241.7	−	3888.12i	1.10711	−	0.325078i	0.323439	−	0.946249i	$-0.395161\pi$
$523$	13241.7	−	3888.12i	1.10711	−	0.325078i	0.783675	+	0.621171i	$0.213343\pi$
$524$		0			0
$525$	3284.40	+	22843.5i	0.273034	+	1.89899i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	5054.35	+	11067.5i	0.415415	+	0.909632i
$530$		0			0
$531$		0			0
$532$	−28225.2	−	24457.3i	−2.30022	−	1.99315i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	24037.5	−	3456.07i	1.91026	−	0.274654i	0.917785	−	0.397077i	$-0.129975\pi$
$541$	24037.5	−	3456.07i	1.91026	−	0.274654i	0.992478	+	0.122422i	$0.0390662\pi$
$542$		0			0
$543$	4986.30	−	2277.17i	0.394075	−	0.179968i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	17777.8	−	2556.06i	1.38962	−	0.199797i	0.593450	−	0.804871i	$-0.297765\pi$
$547$	17777.8	−	2556.06i	1.38962	−	0.199797i	0.796170	+	0.605073i	$0.206856\pi$
$548$		0			0
$549$	−6733.73	−	22933.0i	−0.523476	−	1.78280i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	30156.6	+	34802.6i	2.31897	+	2.67623i
$554$		0			0
$555$		0			0
$556$	1349.91	−	2100.50i	0.102966	−	0.160218i
$557$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$557$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$558$		0			0
$559$	4739.75	+	3046.05i	0.358622	+	0.230473i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$563$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7297.57	−	24853.3i	0.540510	−	1.84081i
$568$		0			0
$569$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$569$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$570$		0			0
$571$	10046.3	−	6456.36i	0.736294	−	0.473188i	−0.117976	−	0.993016i	$-0.537641\pi$
$571$	10046.3	−	6456.36i	0.736294	−	0.473188i	0.854270	+	0.519829i	$0.174004\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1967.36	+	13683.3i	−0.142315	+	0.989821i
$577$	15541.7	−	13467.0i	1.12133	−	0.971640i	0.121552	−	0.992585i	$-0.461213\pi$
$577$	15541.7	−	13467.0i	1.12133	−	0.971640i	0.999781	+	0.0209447i	$0.00666738\pi$
$578$		0			0
$579$	−16833.7	+	14586.5i	−1.20826	+	1.04697i
$580$		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	$-0.0454545\pi$
$587$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$588$	−34768.4	−	15878.2i	−2.43848	−	1.11361i
$589$	28069.1	+	24322.0i	1.96361	+	1.70148i
$590$		0			0
$591$		0			0
$592$	158.714	+	1103.88i	0.0110188	+	0.0766372i
$593$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$593$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−25512.5	−	3668.15i	−1.74901	−	0.251469i
$598$		0			0
$599$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$599$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$600$		0			0
$601$	−450.373	−	986.179i	−0.0305675	−	0.0669336i	0.893732	−	0.448601i	$-0.148077\pi$
$601$	−450.373	−	986.179i	−0.0305675	−	0.0669336i	−0.924300	+	0.381667i	$0.875350\pi$
$602$		0			0
$603$	14459.6	+	3190.22i	0.976515	+	0.215449i
$604$	27912.2			1.88035
$605$		0			0
$606$		0			0
$607$	2178.46	−	2514.08i	0.145669	−	0.168111i	−0.678226	−	0.734853i	$-0.737251\pi$
$607$	2178.46	−	2514.08i	0.145669	−	0.168111i	0.823895	+	0.566742i	$0.191797\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	6237.39	−	4008.53i	0.410972	−	0.264116i	−0.318778	−	0.947829i	$-0.603273\pi$
$613$	6237.39	−	4008.53i	0.410972	−	0.264116i	0.729750	+	0.683714i	$0.239636\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$617$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$618$		0			0
$619$	−2713.81	−	18874.9i	−0.176215	−	1.22560i	−0.865424	−	0.501040i	$-0.832951\pi$
$619$	−2713.81	−	18874.9i	−0.176215	−	1.22560i	0.689209	−	0.724562i	$-0.257958\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−4239.44	+	29485.9i	−0.271977	+	1.89164i
$625$	13144.6	−	8447.51i	0.841254	−	0.540641i
$626$		0			0
$627$		0			0
$628$	−17936.3	+	5266.59i	−1.13971	+	0.334649i
$629$		0			0
$630$		0			0
$631$	7788.21	−	26524.2i	0.491353	−	1.67340i	−0.223965	−	0.974597i	$-0.571900\pi$
$631$	7788.21	−	26524.2i	0.491353	−	1.67340i	0.715319	−	0.698798i	$-0.246282\pi$
$632$		0			0
$633$		−	10072.9i		−	0.632482i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−74921.9	−	34215.7i	−4.66015	−	2.12822i
$638$		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
$643$	1982.56	−	13789.0i	0.121593	−	0.845701i	−0.834158	−	0.551525i	$-0.814046\pi$
$643$	1982.56	−	13789.0i	0.121593	−	0.845701i	0.955751	−	0.294175i	$-0.0950450\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$647$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	47474.1	+	21680.7i	2.85815	+	1.30528i
$652$	2159.77	−	2492.51i	0.129729	−	0.149715i
$653$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$653$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−5206.14	+	1528.66i	−0.309149	+	0.0907743i
$658$		0			0
$659$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$659$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$660$		0			0
$661$	30237.7	−	13809.1i	1.77929	−	0.812573i	0.803037	−	0.595930i	$-0.203216\pi$
$661$	30237.7	−	13809.1i	1.77929	−	0.812573i	0.976251	−	0.216644i	$-0.0695110\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			30453.7i			1.75995i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−10922.2	+	16995.4i	−0.625590	+	0.973436i	0.373363	+	0.927685i	$0.378204\pi$
$673$	−10922.2	+	16995.4i	−0.625590	+	0.973436i	−0.998953	+	0.0457511i	$0.985432\pi$
$674$		0			0
$675$	−17358.5	+	2495.78i	−0.989821	+	0.142315i
$676$	−6634.18	+	46141.8i	−0.377457	+	2.62527i
$677$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$677$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$678$		0			0
$679$	27647.1	+	60538.8i	1.56259	+	3.42160i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$683$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$684$	18584.8	−	21448.0i	1.03890	−	1.19895i
$685$		0			0
$686$		0			0
$687$	−21585.7	−	24911.3i	−1.19876	−	1.38344i
$688$	3042.21	+	2636.09i	0.168580	+	0.146076i
$689$		0			0
$690$		0			0
$691$	3500.91	−	24349.4i	0.192737	−	1.34051i	−0.631988	−	0.774978i	$-0.717761\pi$
$691$	3500.91	−	24349.4i	0.192737	−	1.34051i	0.824724	−	0.565535i	$-0.191330\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$			35531.5i			1.91852i
$701$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$701$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$702$		0			0
$703$	951.092	−	2082.60i	0.0510257	−	0.111731i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	23515.8	+	6904.85i	1.24563	+	0.365750i	0.837128	−	0.547007i	$-0.184233\pi$
$709$	23515.8	+	6904.85i	1.24563	+	0.365750i	0.408503	+	0.912757i	$0.366051\pi$
$710$		0			0
$711$	−26446.1	+	22915.6i	−1.39494	+	1.20873i
$712$		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	$-0.272727\pi$
$719$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$720$		0			0
$721$	−59504.6	+	8555.47i	−3.07360	+	0.441917i
$722$		0			0
$723$	26420.0	+	22893.1i	1.35902	+	1.17760i
$724$	8097.72	−	2377.71i	0.415676	−	0.122054i
$725$		0			0
$726$		0			0
$727$	−17749.5	−	27618.8i	−0.905494	−	1.40898i	−0.912534	−	0.409001i	$-0.865877\pi$
$727$	−17749.5	−	27618.8i	−0.905494	−	1.40898i	0.00703957	−	0.999975i	$-0.497759\pi$
$728$		0			0
$729$	18885.7	+	5545.34i	0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$	−5236.93	−	36423.7i	−0.264430	−	1.83915i
$733$	1184.06	+	1026.00i	0.0596648	+	0.0516999i	0.684183	−	0.729311i	$-0.260159\pi$
$733$	1184.06	+	1026.00i	0.0596648	+	0.0516999i	−0.624518	+	0.781010i	$0.714705\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−36519.6	+	16678.0i	−1.81786	+	0.830188i	−0.892598	+	0.450854i	$0.851119\pi$
$739$	−36519.6	+	16678.0i	−1.81786	+	0.830188i	−0.925260	+	0.379333i	$0.876153\pi$
$740$		0			0
$741$	40048.1	−	46218.0i	1.98543	−	2.29131i
$742$		0			0
$743$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$743$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26798.5	+	30927.2i	−1.30212	+	1.50273i	−0.569757	+	0.821813i	$0.692963\pi$
$751$	−26798.5	+	30927.2i	−1.30212	+	1.50273i	−0.732363	+	0.680914i	$0.761583\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	16566.5	−	36275.7i	0.796983	−	1.74515i
$757$	−22494.4	+	35002.0i	−1.08002	+	1.68054i	−0.499346	+	0.866403i	$0.666426\pi$
$757$	−22494.4	+	35002.0i	−1.08002	+	1.68054i	−0.580672	+	0.814137i	$0.697210\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$761$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$762$		0			0
$763$	−22105.0	−	25510.6i	−1.04883	−	1.21041i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−5996.24	+	20421.3i	−0.281733	+	0.959493i
$769$	−21293.4	−	33133.1i	−0.998516	−	1.55372i	−0.822051	−	0.569413i	$-0.807170\pi$
$769$	−21293.4	−	33133.1i	−0.998516	−	1.55372i	−0.176464	−	0.984307i	$-0.556466\pi$
$770$		0			0
$771$		0			0
$772$	−28849.4	+	18540.4i	−1.34497	+	0.864357i
$773$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$773$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$774$		0			0
$775$		−	35335.0i		−	1.63777i
$776$		0			0
$777$	457.858	−	3184.48i	0.0211397	−	0.147030i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−49505.5	−	31815.3i	−2.25517	−	1.44931i
$785$		0			0
$786$		0			0
$787$	15141.3	−	13120.0i	0.685804	−	0.594252i	−0.240672	−	0.970606i	$-0.577368\pi$
$787$	15141.3	−	13120.0i	0.685804	−	0.594252i	0.926476	+	0.376354i	$0.122822\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−11285.0	−	78488.8i	−0.505349	−	3.51478i
$794$		0			0
$795$		0			0
$796$	−38075.6	−	11180.0i	−1.69542	−	0.497820i
$797$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$797$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	21336.3	+	8029.87i	0.935914	+	0.352228i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$809$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$810$		0			0
$811$	31177.3	−	14238.2i	1.34992	−	0.616486i	0.396470	−	0.918048i	$-0.370235\pi$
$811$	31177.3	−	14238.2i	1.34992	−	0.616486i	0.953447	+	0.301561i	$0.0975077\pi$
$812$		0			0
$813$	−6373.18	−	13955.3i	−0.274929	−	0.602010i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2328.22	−	7929.19i	−0.0996990	−	0.339544i
$818$		0			0
$819$	35699.0	−	78169.9i	1.52311	−	3.33514i
$820$		0			0
$821$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$821$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$822$		0			0
$823$	−15126.8	+	33123.0i	−0.640687	+	1.40291i	0.258786	+	0.965935i	$0.416677\pi$
$823$	−15126.8	+	33123.0i	−0.640687	+	1.40291i	−0.899474	+	0.436975i	$0.856050\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$827$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$828$		0			0
$829$	21605.6	−	6343.98i	0.905179	−	0.265785i	0.204169	−	0.978936i	$-0.434551\pi$
$829$	21605.6	−	6343.98i	0.905179	−	0.265785i	0.701010	+	0.713151i	$0.252733\pi$
$830$		0			0
$831$	−45808.8	−	6586.31i	−1.91226	−	0.274942i
$832$	−12921.2	+	44005.6i	−0.538416	+	1.83368i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−16474.9	+	36075.1i	−0.680354	+	1.48977i
$838$		0			0
$839$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$839$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$840$		0			0
$841$	24389.0			1.00000
$842$		0			0
$843$		0			0
$844$	2207.05	−	15350.4i	0.0900116	−	0.626045i
$845$		0			0
$846$		0			0
$847$	−35741.3	+	30970.0i	−1.44992	+	1.25636i
$848$		0			0
$849$	23651.9	+	36803.0i	0.956102	+	1.48772i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−26127.3	+	30152.5i	−1.04875	+	1.21032i	−0.0716721	+	0.997428i	$0.522834\pi$
$853$	−26127.3	+	30152.5i	−1.04875	+	1.21032i	−0.977076	+	0.212891i	$0.931712\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$857$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$858$		0			0
$859$	39480.4	−	11592.5i	1.56817	−	0.460455i	0.621699	−	0.783256i	$-0.286443\pi$
$859$	39480.4	−	11592.5i	1.56817	−	0.460455i	0.946467	+	0.322801i	$0.104625\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$863$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	19293.3	+	16717.7i	0.755750	+	0.654861i
$868$	67596.8	+	43441.8i	2.64330	+	1.69875i
$869$		0			0
$870$		0			0
$871$	45977.4	+	17303.5i	1.78862	+	0.673141i
$872$		0			0
$873$	−46002.7	+	21008.7i	−1.78346	+	0.814477i
$874$		0			0
$875$		0			0
$876$	−8268.73	+	1188.86i	−0.318921	+	0.0458539i
$877$	1213.24	−	8438.26i	0.0467140	−	0.324903i	−0.953043	−	0.302836i	$-0.902066\pi$
$877$	1213.24	−	8438.26i	0.0467140	−	0.324903i	0.999757	−	0.0220663i	$-0.00702451\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$881$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$882$		0			0
$883$	−8281.19	−	28203.1i	−0.315611	−	1.07487i	−0.952657	−	0.304046i	$-0.901662\pi$
$883$	−8281.19	−	28203.1i	−0.315611	−	1.07487i	0.637047	−	0.770825i	$-0.280156\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$887$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$888$		0			0
$889$	11672.9	+	10114.6i	0.440377	+	0.381589i
$890$		0			0
$891$		0			0
$892$	−6672.65	+	46409.3i	−0.250467	+	1.74204i
$893$		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
$903$	−6278.22	−	9769.10i	−0.231369	−	0.360017i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	35762.1	+	10500.7i	1.30922	+	0.384421i	0.860590	−	0.509298i	$-0.170095\pi$
$907$	35762.1	+	10500.7i	1.30922	+	0.384421i	0.448627	+	0.893719i	$0.351913\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$911$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$912$	33021.3	−	28613.2i	1.19895	−	1.03890i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−27436.9	−	42692.6i	−0.989673	−	1.53996i
$917$		0			0
$918$		0			0
$919$	−4722.80	−	2156.83i	−0.169522	−	0.0774181i	0.328847	−	0.944383i	$-0.393340\pi$
$919$	−4722.80	−	2156.83i	−0.169522	−	0.0774181i	−0.498369	+	0.866965i	$0.666067\pi$
$920$		0			0
$921$	22643.3	−	19620.5i	0.810120	−	0.701973i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2089.96	+	613.666i	−0.0742890	+	0.0218132i
$926$		0			0
$927$	−6501.20	−	45216.9i	−0.230342	−	1.60207i
$928$		0			0
$929$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$929$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$930$		0			0
$931$	50186.2	+	109892.i	1.76669	+	3.86850i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			51424.9i			1.79293i	0.443110	+	0.896467i	$0.353875\pi$
$937$			51424.9i			1.79293i	−0.443110	+	0.896467i	$0.646125\pi$
$938$		0			0
$939$	57542.3			1.99981
$940$		0			0
$941$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$941$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$947$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$948$	−45322.9	+	29127.3i	−1.55276	+	0.997901i
$949$	−17818.2	+	2561.87i	−0.609486	+	0.0876309i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$953$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−42161.2	−	27095.4i	−1.41523	−	0.909516i
$962$		0			0
$963$		0			0
$964$	35246.2	+	40676.2i	1.17760	+	1.35902i
$965$		0			0
$966$		0			0
$967$	24027.2			0.799032			0.399516	−	0.916726i	$-0.369178\pi$
$967$	24027.2			0.799032			0.399516	+	0.916726i	$0.369178\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$971$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$972$	27565.5	+	12588.7i	0.909632	+	0.415415i
$973$	−9329.26	+	5995.55i	−0.307382	+	0.197542i
$974$		0			0
$975$	−58181.9			−1.91109
$976$		−	56654.5i		−	1.85806i
$977$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$977$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	19385.2	−	16797.4i	0.630909	−	0.546685i
$982$		0			0
$983$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$983$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	71157.2	−	61658.1i	2.29131	−	1.98543i
$989$		0			0
$990$		0			0
$991$	−38709.5	−	33542.0i	−1.24081	−	1.07517i	−0.994364	−	0.106017i	$-0.966190\pi$
$991$	−38709.5	−	33542.0i	−1.24081	−	1.07517i	−0.246450	−	0.969155i	$-0.579264\pi$
$992$		0			0
$993$	47669.7	−	13997.1i	1.52342	−	0.447316i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	52349.4	+	15371.2i	1.66291	+	0.488275i	0.972062	−	0.234725i	$-0.0754189\pi$
$997$	52349.4	+	15371.2i	1.66291	+	0.488275i	0.690849	+	0.722999i	$0.257237\pi$
$998$		0			0
$999$	2419.85	+	347.921i	0.0766372	+	0.0110188i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.a.53.2	✓	20
3.2	odd	2	CM	201.4.j.a.53.2	✓	20
67.43	odd	22	inner	201.4.j.a.110.2	yes	20
201.110	even	22	inner	201.4.j.a.110.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.a.53.2	✓	20	1.1	even	1	trivial
201.4.j.a.53.2	✓	20	3.2	odd	2	CM
201.4.j.a.110.2	yes	20	67.43	odd	22	inner
201.4.j.a.110.2	yes	20	201.110	even	22	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(2.80925 - 4.37128i) q^{3} +(5.23889 - 6.04600i) q^{4} +(-32.3206 + 14.7603i) q^{7} +(-11.2162 - 24.5601i) q^{9} +O(q^{10})\)	\(q+(2.80925 - 4.37128i) q^{3} +(5.23889 - 6.04600i) q^{4} +(-32.3206 + 14.7603i) q^{7} +(-11.2162 - 24.5601i) q^{9} +(-11.7114 - 39.8854i) q^{12} +(-25.2367 - 85.9485i) q^{13} +(-9.10815 - 63.3486i) q^{16} +(-54.5804 + 119.515i) q^{19} +(-26.2752 + 182.748i) q^{21} +(119.937 - 35.2166i) q^{25} +(-138.868 - 19.9662i) q^{27} +(-80.0831 + 272.738i) q^{28} +(79.6402 - 271.230i) q^{31} +(-207.250 - 60.8542i) q^{36} -17.4255 q^{37} +(-446.601 - 131.134i) q^{39} +(-47.5346 + 41.1890i) q^{43} +(-302.502 - 138.148i) q^{48} +(602.138 - 694.904i) q^{49} +(-651.857 - 297.693i) q^{52} +(369.101 + 574.333i) q^{57} +(876.217 + 125.981i) q^{61} +(725.029 + 628.241i) q^{63} +(-430.722 - 276.808i) q^{64} +(-329.950 + 438.060i) q^{67} +(28.5996 - 198.915i) q^{73} +(182.991 - 623.209i) q^{75} +(436.644 + 956.116i) q^{76} +(-365.137 - 1243.54i) q^{79} +(-477.393 + 550.941i) q^{81} +(967.241 + 1116.26i) q^{84} +(2084.29 + 2405.40i) q^{91} +(-961.892 - 1110.08i) q^{93} -1873.07i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{4} + 54 q^{9}+O(q^{10}) \) 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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