LMFDB - Embedded newform 253.3.k.b.186.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [253,3,Mod(32,253)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(253, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 10]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("253.32");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 253 = 11 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	253.k (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:	$6.89375068832$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			186.1
Root			$-0.841254 - 0.540641i$ of defining polynomial
Character	$\chi$	$=$	253.186
Dual form			253.3.k.b.219.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.109264 - 0.759951i) q^{3} +(1.66166 + 3.63853i) q^{4} +(5.96859 + 1.75253i) q^{5} +(8.06985 - 2.36952i) q^{9} +(-9.25379 + 5.94705i) q^{11} +(2.58354 - 1.66034i) q^{12} +(0.679686 - 4.72732i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(3.54111 + 24.6290i) q^{20} +(22.7909 + 3.09434i) q^{23} +(11.5213 + 7.40429i) q^{25} +(-5.55294 - 12.1592i) q^{27} +(-3.05767 + 21.2665i) q^{31} +(5.53058 + 6.38263i) q^{33} +(22.0309 + 25.4250i) q^{36} +(70.7534 - 20.7751i) q^{37} +(-37.0152 - 23.7882i) q^{44} +52.3183 q^{45} -25.5490 q^{47} +(10.3342 + 6.64137i) q^{48} +(-6.97343 - 48.5013i) q^{49} +(-28.3730 + 32.7442i) q^{53} +(-65.6544 + 19.2779i) q^{55} +(0.526648 + 0.607784i) q^{59} +(18.3299 - 5.38215i) q^{60} +(-61.4076 - 18.0309i) q^{64} +(-101.518 - 65.2419i) q^{67} +(-0.138691 - 17.6581i) q^{69} +(-71.4983 - 45.9492i) q^{71} +(4.36803 - 9.56465i) q^{75} +(-83.7291 + 53.8094i) q^{80} +(55.0448 - 35.3752i) q^{81} +(-25.0923 - 174.521i) q^{89} +(26.6119 + 88.0671i) q^{92} +16.4956 q^{93} +(133.184 + 39.1063i) q^{97} +(-60.5850 + 69.9188i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 5 q^{3} - 4 q^{4} + 34 q^{5} - 16 q^{9} + 11 q^{11} - 24 q^{12} + 50 q^{15} - 16 q^{16} - 128 q^{20} + 32 q^{23} + 57 q^{25} + 134 q^{27} + 202 q^{31} + 187 q^{33} + 156 q^{36} + 256 q^{37} + 44 q^{44}+ \cdots - 429 q^{99}+O(q^{100}) $ 10 * q + 5 * q^3 - 4 * q^4 + 34 * q^5 - 16 * q^9 + 11 * q^11 - 24 * q^12 + 50 * q^15 - 16 * q^16 - 128 * q^20 + 32 * q^23 + 57 * q^25 + 134 * q^27 + 202 * q^31 + 187 * q^33 + 156 * q^36 + 256 * q^37 + 44 * q^44 - 512 * q^45 + 214 * q^47 - 96 * q^48 - 49 * q^49 + 70 * q^53 - 374 * q^55 - 107 * q^59 - 504 * q^60 - 64 * q^64 - 464 * q^67 - 446 * q^69 - 681 * q^71 + 914 * q^75 + 192 * q^80 - 31 * q^81 + 97 * q^89 - 268 * q^92 + 640 * q^93 + 708 * q^97 - 429 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/253\mathbb{Z}\right)^\times$.

$n$	$24$	$166$
$\chi(n)$	$-1$	$e\left(\frac{1}{11}\right)$

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		−0.841254	−	0.540641i	$-0.818182\pi$
$2$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$3$	−0.109264	−	0.759951i	−0.0364215	−	0.253317i	0.963474	−	0.267803i	$-0.0862978\pi$
$3$	−0.109264	−	0.759951i	−0.0364215	−	0.253317i	−0.999895	+	0.0144864i	$0.995389\pi$
$4$	1.66166	+	3.63853i	0.415415	+	0.909632i
$5$	5.96859	+	1.75253i	1.19372	+	0.350507i	0.817447	−	0.576003i	$-0.195388\pi$
$5$	5.96859	+	1.75253i	1.19372	+	0.350507i	0.376270	+	0.926510i	$0.377207\pi$
$6$		0			0
$7$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$7$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$8$		0			0
$9$	8.06985	−	2.36952i	0.896650	−	0.263280i
$10$		0			0
$11$	−9.25379	+	5.94705i	−0.841254	+	0.540641i
$11$	−9.25379	+	5.94705i	−0.841254	+	0.540641i
$12$	2.58354	−	1.66034i	0.215295	−	0.138362i
$13$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$13$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$14$		0			0
$15$	0.679686	−	4.72732i	0.0453124	−	0.315155i
$16$	−10.4778	+	12.0920i	−0.654861	+	0.755750i
$17$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$17$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$18$		0			0
$19$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$19$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$20$	3.54111	+	24.6290i	0.177056	+	1.23145i
$21$		0			0
$22$		0			0
$23$	22.7909	+	3.09434i	0.990909	+	0.134536i
$23$	22.7909	+	3.09434i	0.990909	+	0.134536i
$24$		0			0
$25$	11.5213	+	7.40429i	0.460852	+	0.296171i
$26$		0			0
$27$	−5.55294	−	12.1592i	−0.205665	−	0.450342i
$28$		0			0
$29$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$29$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$30$		0			0
$31$	−3.05767	+	21.2665i	−0.0986344	+	0.686017i	0.879172	+	0.476505i	$0.158097\pi$
$31$	−3.05767	+	21.2665i	−0.0986344	+	0.686017i	−0.977806	+	0.209512i	$0.932812\pi$
$32$		0			0
$33$	5.53058	+	6.38263i	0.167593	+	0.193413i
$34$		0			0
$35$		0			0
$36$	22.0309	+	25.4250i	0.611970	+	0.706251i
$37$	70.7534	−	20.7751i	1.91226	−	0.561489i	0.932552	−	0.361037i	$-0.117577\pi$
$37$	70.7534	−	20.7751i	1.91226	−	0.561489i	0.979704	−	0.200452i	$-0.0642410\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$41$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$42$		0			0
$43$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$43$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$44$	−37.0152	−	23.7882i	−0.841254	−	0.540641i
$45$	52.3183			1.16263
$46$		0			0
$47$	−25.5490			−0.543596			−0.271798	−	0.962354i	$-0.587618\pi$
$47$	−25.5490			−0.543596			−0.271798	+	0.962354i	$0.587618\pi$
$48$	10.3342	+	6.64137i	0.215295	+	0.138362i
$49$	−6.97343	−	48.5013i	−0.142315	−	0.989821i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−28.3730	+	32.7442i	−0.535339	+	0.617814i	−0.957404	−	0.288751i	$-0.906760\pi$
$53$	−28.3730	+	32.7442i	−0.535339	+	0.617814i	0.422065	+	0.906566i	$0.361305\pi$
$54$		0			0
$55$	−65.6544	+	19.2779i	−1.19372	+	0.350507i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.526648	+	0.607784i	0.00892624	+	0.0103014i	0.760195	−	0.649695i	$-0.225103\pi$
$59$	0.526648	+	0.607784i	0.00892624	+	0.0103014i	−0.751269	+	0.659996i	$0.770558\pi$
$60$	18.3299	−	5.38215i	0.305498	−	0.0897024i
$61$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$61$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$62$		0			0
$63$		0			0
$64$	−61.4076	−	18.0309i	−0.959493	−	0.281733i
$65$		0			0
$66$		0			0
$67$	−101.518	−	65.2419i	−1.51520	−	0.973759i	−0.992633	−	0.121161i	$-0.961338\pi$
$67$	−101.518	−	65.2419i	−1.51520	−	0.973759i	−0.522566	−	0.852599i	$-0.675025\pi$
$68$		0			0
$69$	−0.138691	−	17.6581i	−0.00201002	−	0.255914i
$70$		0			0
$71$	−71.4983	−	45.9492i	−1.00702	−	0.647171i	−0.0703984	−	0.997519i	$-0.522427\pi$
$71$	−71.4983	−	45.9492i	−1.00702	−	0.647171i	−0.936620	+	0.350348i	$0.886063\pi$
$72$		0			0
$73$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$73$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$74$		0			0
$75$	4.36803	−	9.56465i	0.0582404	−	0.127529i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$79$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$80$	−83.7291	+	53.8094i	−1.04661	+	0.672618i
$81$	55.0448	−	35.3752i	0.679566	−	0.436731i
$82$		0			0
$83$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$83$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−25.0923	−	174.521i	−0.281936	−	1.96091i	−0.276813	−	0.960924i	$-0.589278\pi$
$89$	−25.0923	−	174.521i	−0.281936	−	1.96091i	−0.00512342	−	0.999987i	$-0.501631\pi$
$90$		0			0
$91$		0			0
$92$	26.6119	+	88.0671i	0.289260	+	0.957251i
$93$	16.4956			0.177372
$94$		0			0
$95$		0			0
$96$		0			0
$97$	133.184	+	39.1063i	1.37303	+	0.403157i	0.883337	−	0.468739i	$-0.155292\pi$
$97$	133.184	+	39.1063i	1.37303	+	0.403157i	0.489691	+	0.871896i	$0.337110\pi$
$98$		0			0
$99$	−60.5850	+	69.9188i	−0.611970	+	0.706251i
$100$	−7.79623	+	54.2240i	−0.0779623	+	0.542240i
$101$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$101$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$102$		0			0
$103$	89.0287	−	57.2153i	0.864356	−	0.555488i	−0.0316653	−	0.999499i	$-0.510081\pi$
$103$	89.0287	−	57.2153i	0.864356	−	0.555488i	0.896022	+	0.444010i	$0.146445\pi$
$104$		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	$0.545455\pi$
$108$	35.0147	−	40.4091i	0.324210	−	0.374158i
$109$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$109$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$110$		0			0
$111$	−23.5189	−	51.4992i	−0.211882	−	0.463957i
$112$		0			0
$113$	−190.050	−	122.138i	−1.68186	−	1.08087i	−0.855895	−	0.517149i	$-0.826993\pi$
$113$	−190.050	−	122.138i	−1.68186	−	1.08087i	−0.825967	−	0.563718i	$-0.809370\pi$
$114$		0			0
$115$	130.606	+	58.4107i	1.13571	+	0.507919i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	50.2652	−	110.065i	0.415415	−	0.909632i
$122$		0			0
$123$		0			0
$124$	−82.4597	+	24.2123i	−0.664997	+	0.195261i
$125$	−46.0505	−	53.1451i	−0.368404	−	0.425161i
$126$		0			0
$127$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$127$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$131$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$132$	−14.0334	+	30.7289i	−0.106314	+	0.232795i
$133$		0			0
$134$		0			0
$135$	−11.8337	−	82.3052i	−0.0876571	−	0.609668i
$136$		0			0
$137$	−31.2268			−0.227933			−0.113966	−	0.993485i	$-0.536356\pi$
$137$	−31.2268			−0.227933			−0.113966	+	0.993485i	$0.536356\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	2.79160	+	19.4160i	0.0197986	+	0.137702i
$142$		0			0
$143$		0			0
$144$	−55.9018	+	122.408i	−0.388207	+	0.850055i
$145$		0			0
$146$		0			0
$147$	−36.0966	+	10.5989i	−0.245555	+	0.0721015i
$148$	193.159	+	222.917i	1.30513	+	1.50620i
$149$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$149$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$150$		0			0
$151$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$151$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−55.5203	+	121.572i	−0.358195	+	0.784338i
$156$		0			0
$157$	89.3142	+	195.571i	0.568880	+	1.24567i	0.947391	+	0.320078i	$0.103709\pi$
$157$	89.3142	+	195.571i	0.568880	+	1.24567i	−0.378511	+	0.925597i	$0.623564\pi$
$158$		0			0
$159$	27.9841	+	17.9843i	0.176001	+	0.113109i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	273.506	+	175.772i	1.67795	+	1.07835i	0.878736	+	0.477308i	$0.158388\pi$
$163$	273.506	+	175.772i	1.67795	+	1.07835i	0.799215	+	0.601045i	$0.205249\pi$
$164$		0			0
$165$	21.8239	+	47.7878i	0.132266	+	0.289623i
$166$		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
$169$	−24.0512	+	167.280i	−0.142315	+	0.989821i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$173$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$174$		0			0
$175$		0			0
$176$	25.0474	−	174.209i	0.142315	−	0.989821i
$177$	0.404343	−	0.466636i	0.00228442	−	0.00263636i
$178$		0			0
$179$	−319.404	−	93.7856i	−1.78438	−	0.523942i	−0.788536	−	0.614989i	$-0.789161\pi$
$179$	−319.404	−	93.7856i	−1.78438	−	0.523942i	−0.995847	+	0.0910465i	$0.970979\pi$
$180$	86.9352	+	190.361i	0.482973	+	1.05756i
$181$	−29.7134	−	206.661i	−0.164162	−	1.14177i	−0.890682	−	0.454628i	$-0.849772\pi$
$181$	−29.7134	−	206.661i	−0.164162	−	1.14177i	0.726519	−	0.687146i	$-0.241137\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	458.707			2.47950
$186$		0			0
$187$		0			0
$188$	−42.4538	−	92.9608i	−0.225818	−	0.494472i
$189$		0			0
$190$		0			0
$191$	105.022	−	121.202i	0.549854	−	0.634566i	−0.410995	−	0.911638i	$-0.634819\pi$
$191$	105.022	−	121.202i	0.549854	−	0.634566i	0.960849	+	0.277072i	$0.0893640\pi$
$192$	−6.99293	+	48.6369i	−0.0364215	+	0.253317i
$193$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$193$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$194$		0			0
$195$		0			0
$196$	164.886	−	105.966i	0.841254	−	0.540641i
$197$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$197$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$198$		0			0
$199$	−42.6197	+	296.427i	−0.214169	+	1.48958i	0.544861	+	0.838526i	$0.316582\pi$
$199$	−42.6197	+	296.427i	−0.214169	+	1.48958i	−0.759031	+	0.651055i	$0.774327\pi$
$200$		0			0
$201$	−38.4883	+	84.2776i	−0.191484	+	0.419292i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	191.251	−	29.0327i	0.923919	−	0.140255i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$211$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$212$	−166.287	−	48.8262i	−0.784372	−	0.230312i
$213$	−27.1069	+	59.3558i	−0.127262	+	0.278666i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−179.238	−	206.852i	−0.814720	−	0.940237i
$221$		0			0
$222$		0			0
$223$	−60.8208	+	70.1909i	−0.272739	+	0.314757i	−0.875551	−	0.483126i	$-0.839501\pi$
$223$	−60.8208	+	70.1909i	−0.272739	+	0.314757i	0.602812	+	0.797883i	$0.294047\pi$
$224$		0			0
$225$	110.520	+	32.4515i	0.491199	+	0.144229i
$226$		0			0
$227$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$227$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$228$		0			0
$229$	373.412			1.63062			0.815311	−	0.579023i	$-0.196566\pi$
$229$	373.412			1.63062			0.815311	+	0.579023i	$0.196566\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$233$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$234$		0			0
$235$	−152.492	−	44.7755i	−0.648900	−	0.190534i
$236$	−1.33633	+	2.92616i	−0.00566242	+	0.0123990i
$237$		0			0
$238$		0			0
$239$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$239$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$240$	50.0412	+	57.7506i	0.208505	+	0.240627i
$241$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$241$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$242$		0			0
$243$	−111.681	−	128.887i	−0.459592	−	0.530397i
$244$		0			0
$245$	43.3786	−	301.705i	0.177056	−	1.23145i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−263.298	−	169.212i	−1.04900	−	0.674150i	−0.101801	−	0.994805i	$-0.532461\pi$
$251$	−263.298	−	169.212i	−1.04900	−	0.674150i	−0.947196	+	0.320655i	$0.896097\pi$
$252$		0			0
$253$	−229.304	+	106.904i	−0.906341	+	0.422546i
$254$		0			0
$255$		0			0
$256$	−36.4326	−	253.394i	−0.142315	−	0.989821i
$257$	147.682	+	323.380i	0.574640	+	1.25829i	0.944290	+	0.329115i	$0.106750\pi$
$257$	147.682	+	323.380i	0.574640	+	1.25829i	−0.369650	+	0.929171i	$0.620522\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$263$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$264$		0			0
$265$	−226.732	+	145.712i	−0.855592	+	0.549856i
$266$		0			0
$267$	−129.886	+	38.1379i	−0.486464	+	0.142839i
$268$	68.6954	−	477.787i	0.256326	−	1.78279i
$269$	−340.335	+	392.768i	−1.26519	+	1.46010i	−0.437190	+	0.899369i	$0.644026\pi$
$269$	−340.335	+	392.768i	−1.26519	+	1.46010i	−0.827996	+	0.560734i	$0.810519\pi$
$270$		0			0
$271$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$271$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−150.649			−0.547816
$276$	64.0189	−	29.8463i	0.231953	−	0.108139i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	25.7166	+	178.863i	0.0921742	+	0.641086i
$280$		0			0
$281$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$281$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$282$		0			0
$283$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$283$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$284$	48.3815	−	336.500i	0.170357	−	1.18486i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−189.255	−	218.412i	−0.654861	−	0.755750i
$290$		0			0
$291$	15.1666	−	105.486i	0.0521189	−	0.362495i
$292$		0			0
$293$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$293$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$294$		0			0
$295$	2.07818	+	4.55058i	0.00704468	+	0.0154257i
$296$		0			0
$297$	123.697	+	79.4955i	0.416490	+	0.267662i
$298$		0			0
$299$		0			0
$300$	42.0594			0.140198
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$307$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$308$		0			0
$309$	−53.2085	−	61.4059i	−0.172196	−	0.198724i
$310$		0			0
$311$	−471.604	+	303.082i	−1.51641	+	0.974540i	−0.523983	+	0.851729i	$0.675554\pi$
$311$	−471.604	+	303.082i	−1.51641	+	0.974540i	−0.992430	+	0.122811i	$0.960809\pi$
$312$		0			0
$313$	−80.2984	+	23.5777i	−0.256544	+	0.0753282i	−0.407476	−	0.913216i	$-0.633591\pi$
$313$	−80.2984	+	23.5777i	−0.256544	+	0.0753282i	0.150931	+	0.988544i	$0.451773\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	256.908	+	75.4349i	0.810435	+	0.237965i	0.660592	−	0.750745i	$-0.270305\pi$
$317$	256.908	+	75.4349i	0.810435	+	0.237965i	0.149842	+	0.988710i	$0.452123\pi$
$318$		0			0
$319$		0			0
$320$	−334.916	−	215.238i	−1.04661	−	0.672618i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	220.179	+	141.501i	0.679566	+	0.436731i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−273.791	+	80.3924i	−0.827164	+	0.242877i	−0.667798	−	0.744342i	$-0.732763\pi$
$331$	−273.791	+	80.3924i	−0.827164	+	0.242877i	−0.159366	+	0.987220i	$0.550945\pi$
$332$		0			0
$333$	521.743	−	335.304i	1.56679	−	1.00692i
$334$		0			0
$335$	−491.582	−	567.316i	−1.46741	−	1.69348i
$336$		0			0
$337$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$337$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$338$		0			0
$339$	−72.0532	+	157.774i	−0.212546	+	0.465411i
$340$		0			0
$341$	−98.1781	−	214.980i	−0.287912	−	0.630440i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	30.1186	−	105.637i	0.0873003	−	0.306194i
$346$		0			0
$347$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$347$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$348$		0			0
$349$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$349$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	33.5944	−	233.654i	0.0951683	−	0.661910i	−0.885269	−	0.465079i	$-0.846026\pi$
$353$	33.5944	−	233.654i	0.0951683	−	0.661910i	0.980437	−	0.196831i	$-0.0630650\pi$
$354$		0			0
$355$	−346.216	−	399.555i	−0.975257	−	1.12551i
$356$	593.305	−	381.294i	1.66659	−	1.07105i
$357$		0			0
$358$		0			0
$359$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$359$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$360$		0			0
$361$	−236.405	+	272.826i	−0.654861	+	0.755750i
$362$		0			0
$363$	−89.1366	−	26.1729i	−0.245555	−	0.0721015i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	733.230			1.99790			0.998951	−	0.0457811i	$-0.0145777\pi$
$367$	733.230			1.99790			0.998951	+	0.0457811i	$0.0145777\pi$
$368$	−276.215	+	243.166i	−0.750583	+	0.660776i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	27.4101	+	60.0198i	0.0736831	+	0.161343i
$373$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$373$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$374$		0			0
$375$	−35.3560	+	40.8030i	−0.0942826	+	0.108808i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	524.567	−	337.119i	1.38408	−	0.889496i	0.384646	−	0.923064i	$-0.374324\pi$
$379$	524.567	−	337.119i	1.38408	−	0.889496i	0.999436	+	0.0335688i	$0.0106873\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2.92942	+	20.3746i	−0.00764862	+	0.0531973i	−0.993288	−	0.115665i	$-0.963100\pi$
$383$	−2.92942	+	20.3746i	−0.00764862	+	0.0531973i	0.985640	+	0.168862i	$0.0540093\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	79.0168	+	549.574i	0.203651	+	1.41643i
$389$	521.891	+	335.399i	1.34162	+	0.862208i	0.997065	−	0.0765572i	$-0.0243928\pi$
$389$	521.891	+	335.399i	1.34162	+	0.862208i	0.344557	+	0.938765i	$0.388029\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−355.073	−	104.259i	−0.896650	−	0.263280i
$397$	−293.958	+	643.678i	−0.740448	+	1.62136i	0.0423677	+	0.999102i	$0.486510\pi$
$397$	−293.958	+	643.678i	−0.740448	+	1.62136i	−0.782816	+	0.622253i	$0.786217\pi$
$398$		0			0
$399$		0			0
$400$	−210.250	+	61.7350i	−0.525625	+	0.154337i
$401$	208.433	+	240.545i	0.519784	+	0.599863i	0.953577	−	0.301150i	$-0.0973705\pi$
$401$	208.433	+	240.545i	0.519784	+	0.599863i	−0.433793	+	0.901013i	$0.642825\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	390.536	−	114.672i	0.964287	−	0.283140i
$406$		0			0
$407$	−531.187	+	613.022i	−1.30513	+	1.50620i
$408$		0			0
$409$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$409$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$410$		0			0
$411$	3.41198	+	23.7308i	0.00830164	+	0.0577392i
$412$	356.115	+	228.861i	0.864356	+	0.555488i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	720.197	+	211.469i	1.71885	+	0.504699i	0.984695	−	0.174286i	$-0.0557619\pi$
$419$	720.197	+	211.469i	1.71885	+	0.504699i	0.734152	+	0.678985i	$0.237580\pi$
$420$		0			0
$421$	267.863	−	309.130i	0.636254	−	0.734276i	−0.342454	−	0.939535i	$-0.611258\pi$
$421$	267.863	−	309.130i	0.636254	−	0.734276i	0.978708	+	0.205259i	$0.0658036\pi$
$422$		0			0
$423$	−206.177	+	60.5390i	−0.487415	+	0.143118i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$431$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$432$	205.212	+	60.2557i	0.475028	+	0.139481i
$433$	357.417	+	782.635i	0.825444	+	1.80747i	0.516130	+	0.856510i	$0.327372\pi$
$433$	357.417	+	782.635i	0.825444	+	1.80747i	0.309314	+	0.950960i	$0.399901\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$439$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$440$		0			0
$441$	−171.199	−	374.874i	−0.388207	−	0.850055i
$442$		0			0
$443$	−327.582	+	717.304i	−0.739462	+	1.61920i	0.0449762	+	0.998988i	$0.485679\pi$
$443$	−327.582	+	717.304i	−0.739462	+	1.61920i	−0.784438	+	0.620207i	$0.787048\pi$
$444$	148.301	−	171.148i	0.334011	−	0.385469i
$445$	156.088	−	1085.62i	0.350761	−	2.43959i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−656.154	+	421.685i	−1.46137	+	0.939165i	−0.462757	+	0.886485i	$0.653140\pi$
$449$	−656.154	+	421.685i	−1.46137	+	0.939165i	−0.998611	+	0.0526795i	$0.983224\pi$
$450$		0			0
$451$		0			0
$452$	128.603	−	894.456i	0.284521	−	1.97888i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$457$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$458$		0			0
$459$		0			0
$460$	4.49480	+	572.274i	0.00977131	+	1.24407i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−32.3219	−	224.804i	−0.0698098	−	0.485538i	−0.994493	−	0.104801i	$-0.966579\pi$
$463$	−32.3219	−	224.804i	−0.0698098	−	0.485538i	0.924683	−	0.380737i	$-0.124330\pi$
$464$		0			0
$465$	98.4555	+	28.9091i	0.211732	+	0.0621702i
$466$		0			0
$467$	−605.073	+	698.292i	−1.29566	+	1.49527i	−0.537386	+	0.843337i	$0.680588\pi$
$467$	−605.073	+	698.292i	−1.29566	+	1.49527i	−0.758275	+	0.651935i	$0.773957\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	138.865	−	89.2434i	0.294831	−	0.189476i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−151.378	+	331.471i	−0.317354	+	0.694908i
$478$		0			0
$479$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$479$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	484.000			1.00000
$485$	726.383	+	466.818i	1.49770	+	0.962511i
$486$		0			0
$487$	−103.712	−	227.099i	−0.212962	−	0.466321i	0.772761	−	0.634697i	$-0.218875\pi$
$487$	−103.712	−	227.099i	−0.212962	−	0.466321i	−0.985723	+	0.168376i	$0.946148\pi$
$488$		0			0
$489$	103.693	−	227.057i	0.212052	−	0.464329i
$490$		0			0
$491$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$491$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−484.142	+	311.139i	−0.978065	+	0.628564i
$496$	−225.117	−	259.799i	−0.453865	−	0.523789i
$497$		0			0
$498$		0			0
$499$	−135.781	+	156.699i	−0.272106	+	0.314027i	−0.875312	−	0.483558i	$-0.839344\pi$
$499$	−135.781	+	156.699i	−0.272106	+	0.314027i	0.603206	+	0.797585i	$0.293890\pi$
$500$	116.850	−	255.865i	0.233699	−	0.511730i
$501$		0			0
$502$		0			0
$503$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$503$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	129.752			0.255922
$508$		0			0
$509$	77.7965	+	541.086i	0.152842	+	1.06304i	0.911426	+	0.411463i	$0.134982\pi$
$509$	77.7965	+	541.086i	0.152842	+	1.06304i	−0.758584	+	0.651575i	$0.774109\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	631.647	−	185.468i	1.22650	−	0.360133i
$516$		0			0
$517$	236.425	−	151.941i	0.457302	−	0.293890i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	132.024	−	918.249i	0.253405	−	1.76247i	−0.324039	−	0.946044i	$-0.605041\pi$
$521$	132.024	−	918.249i	0.253405	−	1.76247i	0.577445	−	0.816430i	$-0.304050\pi$
$522$		0			0
$523$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$523$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−135.127			−0.255922
$529$	509.850	+	141.045i	0.963800	+	0.266626i
$530$		0			0
$531$	5.69013	+	3.65682i	0.0107159	+	0.00688667i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−36.3729	+	252.979i	−0.0677336	+	0.471097i
$538$		0			0
$539$	352.970	+	407.349i	0.654861	+	0.755750i
$540$	279.806	−	179.821i	0.518160	−	0.333001i
$541$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$541$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$542$		0			0
$543$	−153.806	+	45.1614i	−0.283252	+	0.0831702i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$547$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$548$	−51.8883	−	113.619i	−0.0946866	−	0.207335i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−50.1204	−	348.595i	−0.0903070	−	0.628099i
$556$		0			0
$557$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$557$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$563$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$564$	−66.0070	+	42.4201i	−0.117034	+	0.0752130i
$565$	−920.281	−	1062.06i	−1.62882	−	1.87975i
$566$		0			0
$567$		0			0
$568$		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
$571$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$571$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$572$		0			0
$573$	−103.583	−	66.5687i	−0.180773	−	0.116176i
$574$		0			0
$575$	239.669	+	204.401i	0.416816	+	0.355480i
$576$	−538.274			−0.934504
$577$	726.546	+	466.923i	1.25918	+	0.809225i	0.988172	−	0.153353i	$-0.0490070\pi$
$577$	726.546	+	466.923i	1.25918	+	0.809225i	0.271007	+	0.962577i	$0.412643\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	67.8264	−	471.743i	0.116340	−	0.809165i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−824.950	+	530.163i	−1.40537	+	0.903174i	−0.999940	−	0.0109532i	$-0.996513\pi$
$587$	−824.950	+	530.163i	−1.40537	+	0.903174i	−0.405427	+	0.914128i	$0.632877\pi$
$588$	−98.5448	−	113.727i	−0.167593	−	0.193413i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−490.126	+	1073.23i	−0.827916	+	1.81288i
$593$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$593$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	229.927			0.385137
$598$		0			0
$599$	838.177			1.39929			0.699647	−	0.714489i	$-0.253341\pi$
$599$	838.177			1.39929			0.699647	+	0.714489i	$0.253341\pi$
$600$		0			0
$601$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$601$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$602$		0			0
$603$	−973.830	−	285.942i	−1.61497	−	0.474199i
$604$		0			0
$605$	492.906	−	568.844i	0.814720	−	0.940237i
$606$		0			0
$607$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$607$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$613$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	205.680	+	450.377i	0.333356	+	0.729947i	0.999879	−	0.0155367i	$-0.00494569\pi$
$617$	205.680	+	450.377i	0.333356	+	0.729947i	−0.666524	+	0.745484i	$0.732218\pi$
$618$		0			0
$619$	−852.807	−	548.066i	−1.37772	−	0.885406i	−0.378526	−	0.925591i	$-0.623569\pi$
$619$	−852.807	−	548.066i	−1.37772	−	0.885406i	−0.999192	+	0.0401853i	$0.987205\pi$
$620$	−534.601			−0.862259
$621$	−88.9317	−	294.303i	−0.143207	−	0.473918i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−323.949	−	709.350i	−0.518319	−	1.13496i
$626$		0			0
$627$		0			0
$628$	−563.180	+	649.945i	−0.896784	+	1.03494i
$629$		0			0
$630$		0			0
$631$	−826.419	−	953.739i	−1.30970	−	1.51147i	−0.659402	−	0.751791i	$-0.729190\pi$
$631$	−826.419	−	953.739i	−1.30970	−	1.51147i	−0.650296	−	0.759681i	$-0.725355\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−18.9363	+	131.705i	−0.0297740	+	0.207083i
$637$		0			0
$638$		0			0
$639$	−685.858	−	201.386i	−1.07333	−	0.315158i
$640$		0			0
$641$	−179.171	−	1246.16i	−0.279519	−	1.94409i	−0.326495	−	0.945199i	$-0.605868\pi$
$641$	−179.171	−	1246.16i	−0.279519	−	1.94409i	0.0469766	−	0.998896i	$-0.485041\pi$
$642$		0			0
$643$	−34.2057			−0.0531970			−0.0265985	−	0.999646i	$-0.508468\pi$
$643$	−34.2057			−0.0531970			−0.0265985	+	0.999646i	$0.508468\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−175.431	−	1220.15i	−0.271145	−	1.88585i	−0.436631	−	0.899641i	$-0.643828\pi$
$647$	−175.431	−	1220.15i	−0.271145	−	1.88585i	0.165486	−	0.986212i	$-0.447081\pi$
$648$		0			0
$649$	−8.48801	−	2.49231i	−0.0130786	−	0.00384022i
$650$		0			0
$651$		0			0
$652$	−185.076	+	1287.23i	−0.283859	+	1.97428i
$653$	337.477	−	99.0921i	0.516810	−	0.151749i	−0.0129185	−	0.999917i	$-0.504112\pi$
$653$	337.477	−	99.0921i	0.516810	−	0.151749i	0.529728	+	0.848168i	$0.322294\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$659$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$660$	−137.613	+	158.814i	−0.208505	+	0.240627i
$661$	−197.767	+	433.048i	−0.299193	+	0.655141i	−0.998200	−	0.0599735i	$-0.980898\pi$
$661$	−197.767	+	433.048i	−0.299193	+	0.655141i	0.699007	+	0.715115i	$0.253626\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	59.9872	+	38.5514i	0.0896670	+	0.0576255i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$673$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$674$		0			0
$675$	26.0535	−	181.206i	0.0385977	−	0.268453i
$676$	−648.617	+	190.451i	−0.959493	+	0.281733i
$677$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$677$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	648.714	−	748.656i	0.949802	−	1.09613i	−0.0454669	−	0.998966i	$-0.514478\pi$
$683$	648.714	−	748.656i	0.949802	−	1.09613i	0.995268	−	0.0971636i	$-0.0309770\pi$
$684$		0			0
$685$	−186.380	−	54.7260i	−0.272087	−	0.0798919i
$686$		0			0
$687$	−40.8007	−	283.775i	−0.0593897	−	0.413064i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	992.681			1.43659			0.718293	−	0.695741i	$-0.244924\pi$
$691$	992.681			1.43659			0.718293	+	0.695741i	$0.244924\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$701$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$702$		0			0
$703$		0			0
$704$	675.483	−	198.340i	0.959493	−	0.281733i
$705$	−17.3653	+	120.778i	−0.0246317	+	0.171317i
$706$		0			0
$707$		0			0
$708$	2.36975	+	0.695821i	0.00334710	+	0.000982798i
$709$	−9.21202	−	20.1715i	−0.0129930	−	0.0284507i	0.903024	−	0.429590i	$-0.141342\pi$
$709$	−9.21202	−	20.1715i	−0.0129930	−	0.0284507i	−0.916017	+	0.401139i	$0.868614\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−135.493	+	475.222i	−0.190032	+	0.666510i
$714$		0			0
$715$		0			0
$716$	−189.500	−	1318.00i	−0.264665	−	1.84078i
$717$		0			0
$718$		0			0
$719$	−581.901	+	1274.19i	−0.809320	+	1.77216i	−0.199015	+	0.979996i	$0.563774\pi$
$719$	−581.901	+	1274.19i	−0.809320	+	1.77216i	−0.610305	+	0.792167i	$0.708953\pi$
$720$	−548.179	+	632.632i	−0.761359	+	0.878656i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	702.568	−	451.513i	0.970398	−	0.623637i
$725$		0			0
$726$		0			0
$727$	−79.8288	+	23.4399i	−0.109806	+	0.0322419i	−0.336174	−	0.941800i	$-0.609133\pi$
$727$	−79.8288	+	23.4399i	−0.109806	+	0.0322419i	0.226368	+	0.974042i	$0.427315\pi$
$728$		0			0
$729$	299.894	−	346.097i	0.411378	−	0.474755i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$733$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$734$		0			0
$735$	−234.021			−0.318396
$736$		0			0
$737$	1327.43			1.80112
$738$		0			0
$739$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$739$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$740$	762.215	+	1669.02i	1.03002	+	2.25543i
$741$		0			0
$742$		0			0
$743$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$743$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	138.472	−	963.096i	0.184384	−	1.28242i	−0.661862	−	0.749626i	$-0.730233\pi$
$751$	138.472	−	963.096i	0.184384	−	1.28242i	0.846246	−	0.532793i	$-0.178857\pi$
$752$	267.697	−	308.939i	0.355980	−	0.410823i
$753$	−99.8234	+	218.583i	−0.132568	+	0.290282i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1250.97	−	803.946i	−1.65253	−	1.06202i	−0.927866	−	0.372914i	$-0.878359\pi$
$757$	−1250.97	−	803.946i	−1.65253	−	1.06202i	−0.724664	−	0.689103i	$-0.758005\pi$
$758$		0			0
$759$	106.297	+	162.579i	0.140049	+	0.214202i
$760$		0			0
$761$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$761$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$762$		0			0
$763$		0			0
$764$	615.508	+	180.730i	0.805639	+	0.236557i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−188.586	+	55.3740i	−0.245555	+	0.0721015i
$769$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$769$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$770$		0			0
$771$	229.616	−	147.565i	0.297816	−	0.191395i
$772$		0			0
$773$	1067.60	−	313.477i	1.38112	−	0.405533i	0.494959	−	0.868916i	$-0.335183\pi$
$773$	1067.60	−	313.477i	1.38112	−	0.405533i	0.886158	+	0.463384i	$0.153365\pi$
$774$		0			0
$775$	−192.692	+	222.378i	−0.248635	+	0.286940i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	934.892			1.19704
$782$		0			0
$783$		0			0
$784$	659.543	+	423.862i	0.841254	+	0.540641i
$785$	190.335	+	1323.81i	0.242465	+	1.68638i
$786$		0			0
$787$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$787$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	135.508	+	156.384i	0.170450	+	0.196709i
$796$	−1149.38	+	337.487i	−1.44394	+	0.423979i
$797$	−165.909	+	1153.92i	−0.208167	+	1.44783i	0.570967	+	0.820973i	$0.306569\pi$
$797$	−165.909	+	1153.92i	−0.208167	+	1.44783i	−0.779133	+	0.626858i	$0.784341\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−616.023	−	1348.90i	−0.769067	−	1.68402i
$802$		0			0
$803$		0			0
$804$	−370.601			−0.460946
$805$		0			0
$806$		0			0
$807$	335.671	+	215.723i	0.415949	+	0.267314i
$808$		0			0
$809$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$809$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$810$		0			0
$811$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$811$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1324.40	+	1528.44i	1.62503	+	1.87538i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		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
$823$	−1577.50	−	463.195i	−1.91676	−	0.562813i	−0.971934	−	0.235255i	$-0.924408\pi$
$823$	−1577.50	−	463.195i	−1.91676	−	0.562813i	−0.944831	−	0.327558i	$-0.893774\pi$
$824$		0			0
$825$	16.4606	+	114.486i	0.0199523	+	0.138771i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	423.431	+	647.630i	0.511390	+	0.782162i
$829$	−1311.78			−1.58236			−0.791180	−	0.611584i	$-0.790533\pi$
$829$	−1311.78			−1.58236			−0.791180	+	0.611584i	$0.790533\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	275.564	−	80.9129i	0.329228	−	0.0966701i
$838$		0			0
$839$	109.013	−	70.0587i	0.129933	−	0.0835026i	−0.474060	−	0.880493i	$-0.657212\pi$
$839$	109.013	−	70.0587i	0.129933	−	0.0835026i	0.603993	+	0.796990i	$0.293576\pi$
$840$		0			0
$841$	−550.738	−	635.585i	−0.654861	−	0.755750i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−436.715	+	956.273i	−0.516823	+	1.13168i
$846$		0			0
$847$		0			0
$848$	−98.6566	−	686.172i	−0.116340	−	0.809165i
$849$		0			0
$850$		0			0
$851$	1676.82	−	254.548i	1.97041	−	0.299116i
$852$	−261.010			−0.306350
$853$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$853$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$857$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$858$		0			0
$859$	−236.423	+	1644.36i	−0.275231	+	1.91427i	0.114647	+	0.993406i	$0.463426\pi$
$859$	−236.423	+	1644.36i	−0.275231	+	1.91427i	−0.389878	+	0.920866i	$0.627483\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	455.947	−	293.019i	0.528327	−	0.339535i	−0.249131	−	0.968470i	$-0.580145\pi$
$863$	455.947	−	293.019i	0.528327	−	0.339535i	0.777458	+	0.628934i	$0.216509\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−145.303	+	167.689i	−0.167593	+	0.193413i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1167.44			1.33727
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$877$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$878$		0			0
$879$		0			0
$880$	454.804	−	995.882i	0.516823	−	1.13168i
$881$	−474.116	+	547.159i	−0.538156	+	0.621066i	−0.958082	−	0.286493i	$-0.907511\pi$
$881$	−474.116	+	547.159i	−0.538156	+	0.621066i	0.419926	+	0.907558i	$0.362056\pi$
$882$		0			0
$883$	1562.50	−	458.791i	1.76954	−	0.519583i	0.775764	−	0.631023i	$-0.217364\pi$
$883$	1562.50	−	458.791i	1.76954	−	0.519583i	0.993771	+	0.111440i	$0.0355463\pi$
$884$		0			0
$885$	3.23115	−	2.07653i	0.00365102	−	0.00234637i
$886$		0			0
$887$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$887$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−298.995	+	654.709i	−0.335573	+	0.734802i
$892$	−356.455	−	104.665i	−0.399613	−	0.117337i
$893$		0			0
$894$		0			0
$895$	−1742.03	−	1119.53i	−1.94640	−	1.25088i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	65.5705	+	456.053i	0.0728561	+	0.506725i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	184.834	−	1285.55i	0.204236	−	1.42049i
$906$		0			0
$907$	1133.17	+	1307.75i	1.24936	+	1.44184i	0.851466	+	0.524410i	$0.175714\pi$
$907$	1133.17	+	1307.75i	1.24936	+	1.44184i	0.397896	+	0.917430i	$0.369740\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	620.773	−	182.275i	0.681419	−	0.200083i	0.0773371	−	0.997005i	$-0.475358\pi$
$911$	620.773	−	182.275i	0.681419	−	0.200083i	0.604082	+	0.796922i	$0.293540\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	620.484	+	1358.67i	0.677385	+	1.48327i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	968.996	+	284.523i	1.04756	+	0.307592i
$926$		0			0
$927$	582.875	−	672.674i	0.628776	−	0.725646i
$928$		0			0
$929$	−1756.35	+	515.710i	−1.89058	+	0.555124i	−0.896991	+	0.442048i	$0.854252\pi$
$929$	−1756.35	+	515.710i	−1.89058	+	0.555124i	−0.993586	+	0.113076i	$0.963930\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	281.857	+	325.280i	0.302098	+	0.348639i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$937$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$938$		0			0
$939$	26.6917	+	58.4467i	0.0284257	+	0.0622435i
$940$	−90.4720	−	629.246i	−0.0962468	−	0.669411i
$941$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$941$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$942$		0			0
$943$		0			0
$944$	−12.8674			−0.0136307
$945$		0			0
$946$		0			0
$947$	464.030	+	1016.08i	0.490000	+	1.07295i	0.979592	+	0.200998i	$0.0644185\pi$
$947$	464.030	+	1016.08i	0.490000	+	1.07295i	−0.489592	+	0.871952i	$0.662854\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	29.2560	−	203.480i	0.0307634	−	0.213964i
$952$		0			0
$953$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$953$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$954$		0			0
$955$	839.245	−	539.350i	0.878790	−	0.564764i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−126.976	+	278.038i	−0.132266	+	0.289623i
$961$	479.157	+	140.693i	0.498602	+	0.146403i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1854.02	+	544.389i	1.90939	+	0.560647i	0.982835	+	0.184489i	$0.0590631\pi$
$971$	1854.02	+	544.389i	1.90939	+	0.560647i	0.926555	+	0.376158i	$0.122755\pi$
$972$	283.382	−	620.519i	0.291545	−	0.638394i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1120.97	+	720.404i	−1.14736	+	0.737363i	−0.969112	−	0.246622i	$-0.920679\pi$
$977$	−1120.97	+	720.404i	−1.14736	+	0.737363i	−0.178248	+	0.983986i	$0.557043\pi$
$978$		0			0
$979$	1270.08	+	1465.76i	1.29733	+	1.49720i
$980$	1169.84	−	343.497i	1.19372	−	0.350507i
$981$		0			0
$982$		0			0
$983$	812.666	−	1779.49i	0.826720	−	1.81026i	0.322991	−	0.946402i	$-0.395312\pi$
$983$	812.666	−	1779.49i	0.826720	−	1.81026i	0.503729	−	0.863862i	$-0.331961\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−576.550	−	370.526i	−0.581786	−	0.373891i	0.216398	−	0.976305i	$-0.430569\pi$
$991$	−576.550	−	370.526i	−0.581786	−	0.373891i	−0.798184	+	0.602414i	$0.794206\pi$
$992$		0			0
$993$	91.0099	+	199.284i	0.0916515	+	0.200689i
$994$		0			0
$995$	−773.877	+	1694.55i	−0.777766	+	1.70307i
$996$		0			0
$997$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$997$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$998$		0			0
$999$	−645.499	−	744.946i	−0.646145	−	0.745691i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	253.3.k.b.186.1	✓	10
11.10	odd	2	CM	253.3.k.b.186.1	✓	10
23.12	even	11	inner	253.3.k.b.219.1	yes	10
253.219	odd	22	inner	253.3.k.b.219.1	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.3.k.b.186.1	✓	10	1.1	even	1	trivial
253.3.k.b.186.1	✓	10	11.10	odd	2	CM
253.3.k.b.219.1	yes	10	23.12	even	11	inner
253.3.k.b.219.1	yes	10	253.219	odd	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:	\( N \)	\(=\)	\( 253 = 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	253.k (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.109264 - 0.759951i) q^{3} +(1.66166 + 3.63853i) q^{4} +(5.96859 + 1.75253i) q^{5} +(8.06985 - 2.36952i) q^{9} +(-9.25379 + 5.94705i) q^{11} +(2.58354 - 1.66034i) q^{12} +(0.679686 - 4.72732i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(3.54111 + 24.6290i) q^{20} +(22.7909 + 3.09434i) q^{23} +(11.5213 + 7.40429i) q^{25} +(-5.55294 - 12.1592i) q^{27} +(-3.05767 + 21.2665i) q^{31} +(5.53058 + 6.38263i) q^{33} +(22.0309 + 25.4250i) q^{36} +(70.7534 - 20.7751i) q^{37} +(-37.0152 - 23.7882i) q^{44} +52.3183 q^{45} -25.5490 q^{47} +(10.3342 + 6.64137i) q^{48} +(-6.97343 - 48.5013i) q^{49} +(-28.3730 + 32.7442i) q^{53} +(-65.6544 + 19.2779i) q^{55} +(0.526648 + 0.607784i) q^{59} +(18.3299 - 5.38215i) q^{60} +(-61.4076 - 18.0309i) q^{64} +(-101.518 - 65.2419i) q^{67} +(-0.138691 - 17.6581i) q^{69} +(-71.4983 - 45.9492i) q^{71} +(4.36803 - 9.56465i) q^{75} +(-83.7291 + 53.8094i) q^{80} +(55.0448 - 35.3752i) q^{81} +(-25.0923 - 174.521i) q^{89} +(26.6119 + 88.0671i) q^{92} +16.4956 q^{93} +(133.184 + 39.1063i) q^{97} +(-60.5850 + 69.9188i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 5 q^{3} - 4 q^{4} + 34 q^{5} - 16 q^{9} + 11 q^{11} - 24 q^{12} + 50 q^{15} - 16 q^{16} - 128 q^{20} + 32 q^{23} + 57 q^{25} + 134 q^{27} + 202 q^{31} + 187 q^{33} + 156 q^{36} + 256 q^{37} + 44 q^{44}+ \cdots - 429 q^{99}+O(q^{100}) \) 10 * q + 5 * q^3 - 4 * q^4 + 34 * q^5 - 16 * q^9 + 11 * q^11 - 24 * q^12 + 50 * q^15 - 16 * q^16 - 128 * q^20 + 32 * q^23 + 57 * q^25 + 134 * q^27 + 202 * q^31 + 187 * q^33 + 156 * q^36 + 256 * q^37 + 44 * q^44 - 512 * q^45 + 214 * q^47 - 96 * q^48 - 49 * q^49 + 70 * q^53 - 374 * q^55 - 107 * q^59 - 504 * q^60 - 64 * q^64 - 464 * q^67 - 446 * q^69 - 681 * q^71 + 914 * q^75 + 192 * q^80 - 31 * q^81 + 97 * q^89 - 268 * q^92 + 640 * q^93 + 708 * q^97 - 429 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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