LMFDB - Embedded newform 2156.2.i.c.1145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2156,2,Mod(177,2156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2156, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 0]))

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

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

chi := DirichletCharacter("2156.177");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2156 = 2^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2156.i (of order $3$, degree $2$, not 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:	$17.2157466758$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1145.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2156.1145
Dual form			2156.2.i.c.177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{3} +(-1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} +(0.500000 - 0.866025i) q^{11} +4.00000 q^{13} -3.00000 q^{15} +(3.00000 - 5.19615i) q^{17} +(4.00000 + 6.92820i) q^{19} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +5.00000 q^{27} +(2.50000 - 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{33} +(0.500000 + 0.866025i) q^{37} +(2.00000 - 3.46410i) q^{39} -10.0000 q^{43} +(3.00000 - 5.19615i) q^{45} +(-3.00000 - 5.19615i) q^{51} +(3.00000 - 5.19615i) q^{53} -3.00000 q^{55} +8.00000 q^{57} +(1.50000 - 2.59808i) q^{59} +(-2.00000 - 3.46410i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(0.500000 - 0.866025i) q^{67} +3.00000 q^{69} +15.0000 q^{71} +(-2.00000 + 3.46410i) q^{73} +(2.00000 + 3.46410i) q^{75} +(-1.00000 - 1.73205i) q^{79} +(-0.500000 + 0.866025i) q^{81} -6.00000 q^{83} -18.0000 q^{85} +(-4.50000 - 7.79423i) q^{89} +(-2.50000 - 4.33013i) q^{93} +(12.0000 - 20.7846i) q^{95} +7.00000 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 3 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + q^3 - 3 * q^5 + 2 * q^9	$ 2 q + q^{3} - 3 q^{5} + 2 q^{9} + q^{11} + 8 q^{13} - 6 q^{15} + 6 q^{17} + 8 q^{19} + 3 q^{23} - 4 q^{25} + 10 q^{27} + 5 q^{31} - q^{33} + q^{37} + 4 q^{39} - 20 q^{43} + 6 q^{45} - 6 q^{51} + 6 q^{53} - 6 q^{55} + 16 q^{57} + 3 q^{59} - 4 q^{61} - 12 q^{65} + q^{67} + 6 q^{69} + 30 q^{71} - 4 q^{73} + 4 q^{75} - 2 q^{79} - q^{81} - 12 q^{83} - 36 q^{85} - 9 q^{89} - 5 q^{93} + 24 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + q^3 - 3 * q^5 + 2 * q^9 + q^11 + 8 * q^13 - 6 * q^15 + 6 * q^17 + 8 * q^19 + 3 * q^23 - 4 * q^25 + 10 * q^27 + 5 * q^31 - q^33 + q^37 + 4 * q^39 - 20 * q^43 + 6 * q^45 - 6 * q^51 + 6 * q^53 - 6 * q^55 + 16 * q^57 + 3 * q^59 - 4 * q^61 - 12 * q^65 + q^67 + 6 * q^69 + 30 * q^71 - 4 * q^73 + 4 * q^75 - 2 * q^79 - q^81 - 12 * q^83 - 36 * q^85 - 9 * q^89 - 5 * q^93 + 24 * q^95 + 14 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times$.

$n$	$981$	$1079$	$1277$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$2$		0			0
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i	−0.684819	−	0.728714i	$-0.740119\pi$
$3$	0.500000	−	0.866025i	0.288675	−	0.500000i	0.973494	+	0.228714i	$0.0734519\pi$
$4$		0			0
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	0.306851	−	0.951757i	$-0.400725\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$12$		0			0
$13$	4.00000			1.10940			0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000			1.10940			0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$	−3.00000			−0.774597
$16$		0			0
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	3.00000	−	5.19615i	0.727607	−	1.26025i	0.957892	−	0.287129i	$-0.0927008\pi$
$18$		0			0
$19$	4.00000	+	6.92820i	0.917663	+	1.58944i	0.802955	+	0.596040i	$0.203260\pi$
$19$	4.00000	+	6.92820i	0.917663	+	1.58944i	0.114708	+	0.993399i	$0.463407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	$-0.0654092\pi$
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	−0.666190	+	0.745782i	$0.732076\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$	2.50000	−	4.33013i	0.449013	−	0.777714i	0.998322	+	0.0579057i	$0.0184423\pi$
$32$		0			0
$33$	−0.500000	−	0.866025i	−0.0870388	−	0.150756i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	0.904194	−	0.427121i	$-0.140472\pi$
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	2.00000	−	3.46410i	0.320256	−	0.554700i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−10.0000			−1.52499			−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000			−1.52499			−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$	3.00000	−	5.19615i	0.447214	−	0.774597i
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−3.00000	−	5.19615i	−0.420084	−	0.727607i
$52$		0			0
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$	8.00000			1.05963
$58$		0			0
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	−0.751710	−	0.659494i	$-0.770771\pi$
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	0.946993	+	0.321253i	$0.104104\pi$
$60$		0			0
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	0.709113	−	0.705095i	$-0.249096\pi$
$61$	−2.00000	−	3.46410i	−0.256074	−	0.443533i	−0.965187	+	0.261562i	$0.915762\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.00000	−	10.3923i	−0.744208	−	1.28901i
$66$		0			0
$67$	0.500000	−	0.866025i	0.0610847	−	0.105802i	−0.833866	−	0.551967i	$-0.813877\pi$
$67$	0.500000	−	0.866025i	0.0610847	−	0.105802i	0.894951	+	0.446165i	$0.147211\pi$
$68$		0			0
$69$	3.00000			0.361158
$70$		0			0
$71$	15.0000			1.78017			0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000			1.78017			0.890086	+	0.455792i	$0.150644\pi$
$72$		0			0
$73$	−2.00000	+	3.46410i	−0.234082	+	0.405442i	−0.959006	−	0.283387i	$-0.908542\pi$
$73$	−2.00000	+	3.46410i	−0.234082	+	0.405442i	0.724923	+	0.688830i	$0.241875\pi$
$74$		0			0
$75$	2.00000	+	3.46410i	0.230940	+	0.400000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.00000	−	1.73205i	−0.112509	−	0.194871i	0.804272	−	0.594261i	$-0.202555\pi$
$79$	−1.00000	−	1.73205i	−0.112509	−	0.194871i	−0.916781	+	0.399390i	$0.869222\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	−18.0000			−1.95237
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	0.522654	−	0.852545i	$-0.324942\pi$
$89$	−4.50000	−	7.79423i	−0.476999	−	0.826187i	−0.999653	+	0.0263586i	$0.991609\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.50000	−	4.33013i	−0.259238	−	0.449013i
$94$		0			0
$95$	12.0000	−	20.7846i	1.23117	−	2.13246i
$96$		0			0
$97$	7.00000			0.710742			0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000			0.710742			0.355371	+	0.934725i	$0.384354\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	0.683793	−	0.729676i	$-0.260329\pi$
$107$	−3.00000	−	5.19615i	−0.290021	−	0.502331i	−0.973814	+	0.227345i	$0.926996\pi$
$108$		0			0
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−1.00000	+	1.73205i	−0.0957826	+	0.165900i	0.814152	+	0.580651i	$0.197202\pi$
$110$		0			0
$111$	1.00000			0.0949158
$112$		0			0
$113$	−15.0000			−1.41108			−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000			−1.41108			−0.705541	+	0.708669i	$0.749296\pi$
$114$		0			0
$115$	4.50000	−	7.79423i	0.419627	−	0.726816i
$116$		0			0
$117$	4.00000	+	6.92820i	0.369800	+	0.640513i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$		0			0
$129$	−5.00000	+	8.66025i	−0.440225	+	0.762493i
$130$		0			0
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−7.50000	−	12.9904i	−0.645497	−	1.11803i
$136$		0			0
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000	−	3.46410i	0.167248	−	0.289683i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$		0			0
$153$	12.0000			0.970143
$154$		0			0
$155$	−15.0000			−1.20483
$156$		0			0
$157$	2.50000	−	4.33013i	0.199522	−	0.345582i	−0.748852	−	0.662738i	$-0.769394\pi$
$157$	2.50000	−	4.33013i	0.199522	−	0.345582i	0.948373	+	0.317156i	$0.102728\pi$
$158$		0			0
$159$	−3.00000	−	5.19615i	−0.237915	−	0.412082i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	0.933659	−	0.358162i	$-0.116597\pi$
$163$	2.00000	+	3.46410i	0.156652	+	0.271329i	−0.777007	+	0.629492i	$0.783263\pi$
$164$		0			0
$165$	−1.50000	+	2.59808i	−0.116775	+	0.202260i
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−8.00000	+	13.8564i	−0.611775	+	1.05963i
$172$		0			0
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	0.973670	+	0.227964i	$0.0732068\pi$
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	−0.289412	+	0.957205i	$0.593460\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.50000	−	2.59808i	−0.112747	−	0.195283i
$178$		0			0
$179$	4.50000	−	7.79423i	0.336346	−	0.582568i	−0.647397	−	0.762153i	$-0.724142\pi$
$179$	4.50000	−	7.79423i	0.336346	−	0.582568i	0.983742	+	0.179585i	$0.0574756\pi$
$180$		0			0
$181$	13.0000			0.966282			0.483141	−	0.875542i	$-0.339496\pi$
$181$	13.0000			0.966282			0.483141	+	0.875542i	$0.339496\pi$
$182$		0			0
$183$	−4.00000			−0.295689
$184$		0			0
$185$	1.50000	−	2.59808i	0.110282	−	0.191014i
$186$		0			0
$187$	−3.00000	−	5.19615i	−0.219382	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.5000	+	18.1865i	0.759753	+	1.31593i	0.942976	+	0.332860i	$0.108014\pi$
$191$	10.5000	+	18.1865i	0.759753	+	1.31593i	−0.183223	+	0.983071i	$0.558653\pi$
$192$		0			0
$193$	−10.0000	+	17.3205i	−0.719816	+	1.24676i	0.241257	+	0.970461i	$0.422440\pi$
$193$	−10.0000	+	17.3205i	−0.719816	+	1.24676i	−0.961073	+	0.276296i	$0.910893\pi$
$194$		0			0
$195$	−12.0000			−0.859338
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	−0.688705	−	0.725042i	$-0.741820\pi$
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	0.972257	+	0.233915i	$0.0751537\pi$
$200$		0			0
$201$	−0.500000	−	0.866025i	−0.0352673	−	0.0610847i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.00000	+	5.19615i	−0.208514	+	0.361158i
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$	7.50000	−	12.9904i	0.513892	−	0.890086i
$214$		0			0
$215$	15.0000	+	25.9808i	1.02299	+	1.77187i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.00000	+	3.46410i	0.135147	+	0.234082i
$220$		0			0
$221$	12.0000	−	20.7846i	0.807207	−	1.39812i
$222$		0			0
$223$	−17.0000			−1.13840			−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000			−1.13840			−0.569202	+	0.822198i	$0.692748\pi$
$224$		0			0
$225$	−8.00000			−0.533333
$226$		0			0
$227$	3.00000	−	5.19615i	0.199117	−	0.344881i	−0.749125	−	0.662428i	$-0.769526\pi$
$227$	3.00000	−	5.19615i	0.199117	−	0.344881i	0.948242	+	0.317547i	$0.102859\pi$
$228$		0			0
$229$	−6.50000	−	11.2583i	−0.429532	−	0.743971i	0.567300	−	0.823511i	$-0.307988\pi$
$229$	−6.50000	−	11.2583i	−0.429532	−	0.743971i	−0.996832	+	0.0795401i	$0.974655\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	0.928312	+	0.371802i	$0.121260\pi$
$233$	12.0000	+	20.7846i	0.786146	+	1.36165i	−0.142166	+	0.989843i	$0.545407\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.00000			−0.129914
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	4.00000	−	6.92820i	0.257663	−	0.446285i	−0.707953	−	0.706260i	$-0.750381\pi$
$241$	4.00000	−	6.92820i	0.257663	−	0.446285i	0.965615	+	0.259975i	$0.0837143\pi$
$242$		0			0
$243$	8.00000	+	13.8564i	0.513200	+	0.888889i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	16.0000	+	27.7128i	1.01806	+	1.76332i
$248$		0			0
$249$	−3.00000	+	5.19615i	−0.190117	+	0.329293i
$250$		0			0
$251$	9.00000			0.568075			0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000			0.568075			0.284037	+	0.958813i	$0.408326\pi$
$252$		0			0
$253$	3.00000			0.188608
$254$		0			0
$255$	−9.00000	+	15.5885i	−0.563602	+	0.976187i
$256$		0			0
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	−0.997374	−	0.0724199i	$-0.976928\pi$
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	0.435970	−	0.899961i	$-0.356405\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	0.442943	+	0.896550i	$0.353935\pi$
$263$	−9.00000	+	15.5885i	−0.554964	+	0.961225i	−0.997906	+	0.0646755i	$0.979399\pi$
$264$		0			0
$265$	−18.0000			−1.10573
$266$		0			0
$267$	−9.00000			−0.550791
$268$		0			0
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	−0.942871	−	0.333157i	$-0.891886\pi$
$269$	−3.00000	+	5.19615i	−0.182913	+	0.316815i	0.759958	+	0.649972i	$0.225219\pi$
$270$		0			0
$271$	10.0000	+	17.3205i	0.607457	+	1.05215i	0.991658	+	0.128897i	$0.0411435\pi$
$271$	10.0000	+	17.3205i	0.607457	+	1.05215i	−0.384201	+	0.923249i	$0.625523\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	−0.675810	−	0.737075i	$-0.736206\pi$
$277$	5.00000	−	8.66025i	0.300421	−	0.520344i	0.976231	+	0.216731i	$0.0695395\pi$
$278$		0			0
$279$	10.0000			0.598684
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$		0			0
$285$	−12.0000	−	20.7846i	−0.710819	−	1.23117i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$	3.50000	−	6.06218i	0.205174	−	0.355371i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	−9.00000			−0.524000
$296$		0			0
$297$	2.50000	−	4.33013i	0.145065	−	0.251259i
$298$		0			0
$299$	6.00000	+	10.3923i	0.346989	+	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−9.00000	−	15.5885i	−0.517036	−	0.895533i
$304$		0			0
$305$	−6.00000	+	10.3923i	−0.343559	+	0.595062i
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	6.00000	−	10.3923i	0.340229	−	0.589294i	−0.644246	−	0.764818i	$-0.722829\pi$
$311$	6.00000	−	10.3923i	0.340229	−	0.589294i	0.984475	+	0.175525i	$0.0561621\pi$
$312$		0			0
$313$	−0.500000	−	0.866025i	−0.0282617	−	0.0489506i	0.851549	−	0.524276i	$-0.175664\pi$
$313$	−0.500000	−	0.866025i	−0.0282617	−	0.0489506i	−0.879810	+	0.475325i	$0.842331\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.5000	−	28.5788i	−0.926732	−	1.60515i	−0.788751	−	0.614713i	$-0.789272\pi$
$317$	−16.5000	−	28.5788i	−0.926732	−	1.60515i	−0.137981	−	0.990435i	$-0.544061\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−6.00000			−0.334887
$322$		0			0
$323$	48.0000			2.67079
$324$		0			0
$325$	−8.00000	+	13.8564i	−0.443760	+	0.768615i
$326$		0			0
$327$	1.00000	+	1.73205i	0.0553001	+	0.0957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	0.946038	−	0.324057i	$-0.105047\pi$
$331$	3.50000	+	6.06218i	0.192377	+	0.333207i	−0.753660	+	0.657264i	$0.771714\pi$
$332$		0			0
$333$	−1.00000	+	1.73205i	−0.0547997	+	0.0949158i
$334$		0			0
$335$	−3.00000			−0.163908
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$	−7.50000	+	12.9904i	−0.407344	+	0.705541i
$340$		0			0
$341$	−2.50000	−	4.33013i	−0.135383	−	0.234490i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−4.50000	−	7.79423i	−0.242272	−	0.419627i
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	20.0000			1.06752
$352$		0			0
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	0.438733	+	0.898617i	$0.355427\pi$
$353$	−10.5000	+	18.1865i	−0.558859	+	0.967972i	−0.997592	+	0.0693543i	$0.977906\pi$
$354$		0			0
$355$	−22.5000	−	38.9711i	−1.19418	−	2.06837i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.0000	+	31.1769i	0.950004	+	1.64545i	0.745409	+	0.666608i	$0.232254\pi$
$359$	18.0000	+	31.1769i	0.950004	+	1.64545i	0.204595	+	0.978847i	$0.434412\pi$
$360$		0			0
$361$	−22.5000	+	38.9711i	−1.18421	+	2.05111i
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	12.0000			0.628109
$366$		0			0
$367$	−9.50000	+	16.4545i	−0.495896	+	0.858917i	−0.999989	−	0.00473247i	$-0.998494\pi$
$367$	−9.50000	+	16.4545i	−0.495896	+	0.858917i	0.504093	+	0.863649i	$0.331827\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$	−1.50000	+	2.59808i	−0.0774597	+	0.134164i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	29.0000			1.48963			0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000			1.48963			0.744815	+	0.667271i	$0.232538\pi$
$380$		0			0
$381$	−8.00000	+	13.8564i	−0.409852	+	0.709885i
$382$		0			0
$383$	−13.5000	−	23.3827i	−0.689818	−	1.19480i	−0.971897	−	0.235408i	$-0.924357\pi$
$383$	−13.5000	−	23.3827i	−0.689818	−	1.19480i	0.282079	−	0.959391i	$-0.408976\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10.0000	−	17.3205i	−0.508329	−	0.880451i
$388$		0			0
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	−0.289124	−	0.957292i	$-0.593364\pi$
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	0.973601	−	0.228257i	$-0.0733028\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$	−6.00000			−0.302660
$394$		0			0
$395$	−3.00000	+	5.19615i	−0.150946	+	0.261447i
$396$		0			0
$397$	−17.0000	−	29.4449i	−0.853206	−	1.47780i	−0.878300	−	0.478110i	$-0.841322\pi$
$397$	−17.0000	−	29.4449i	−0.853206	−	1.47780i	0.0250943	−	0.999685i	$-0.492011\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	10.0000	−	17.3205i	0.498135	−	0.862796i
$404$		0			0
$405$	3.00000			0.149071
$406$		0			0
$407$	1.00000			0.0495682
$408$		0			0
$409$	1.00000	−	1.73205i	0.0494468	−	0.0856444i	−0.840243	−	0.542211i	$-0.817588\pi$
$409$	1.00000	−	1.73205i	0.0494468	−	0.0856444i	0.889689	+	0.456566i	$0.150921\pi$
$410$		0			0
$411$	4.50000	+	7.79423i	0.221969	+	0.384461i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	9.00000	+	15.5885i	0.441793	+	0.765207i
$416$		0			0
$417$	−7.00000	+	12.1244i	−0.342791	+	0.593732i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.0000	+	20.7846i	0.582086	+	1.00820i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−2.00000	−	3.46410i	−0.0965609	−	0.167248i
$430$		0			0
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	−0.997173	−	0.0751385i	$-0.976060\pi$
$431$	−9.00000	+	15.5885i	−0.433515	+	0.750870i	0.563658	+	0.826008i	$0.309393\pi$
$432$		0			0
$433$	−29.0000			−1.39365			−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000			−1.39365			−0.696826	+	0.717241i	$0.745405\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.0000	+	20.7846i	−0.574038	+	0.994263i
$438$		0			0
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	0.945552	−	0.325471i	$-0.105523\pi$
$439$	4.00000	+	6.92820i	0.190910	+	0.330665i	−0.754642	+	0.656136i	$0.772190\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.5000	+	18.1865i	0.498870	+	0.864068i	0.999999	−	0.00130426i	$-0.000415158\pi$
$443$	10.5000	+	18.1865i	0.498870	+	0.864068i	−0.501129	+	0.865373i	$0.667082\pi$
$444$		0			0
$445$	−13.5000	+	23.3827i	−0.639961	+	1.10845i
$446$		0			0
$447$	−6.00000			−0.283790
$448$		0			0
$449$	3.00000			0.141579			0.0707894	−	0.997491i	$-0.477448\pi$
$449$	3.00000			0.141579			0.0707894	+	0.997491i	$0.477448\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−5.00000	−	8.66025i	−0.234920	−	0.406894i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	0.981921	+	0.189292i	$0.0606194\pi$
$457$	14.0000	+	24.2487i	0.654892	+	1.13431i	−0.327028	+	0.945015i	$0.606047\pi$
$458$		0			0
$459$	15.0000	−	25.9808i	0.700140	−	1.21268i
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	23.0000			1.06890			0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000			1.06890			0.534450	+	0.845200i	$0.320519\pi$
$464$		0			0
$465$	−7.50000	+	12.9904i	−0.347804	+	0.602414i
$466$		0			0
$467$	1.50000	+	2.59808i	0.0694117	+	0.120225i	0.898642	−	0.438682i	$-0.144554\pi$
$467$	1.50000	+	2.59808i	0.0694117	+	0.120225i	−0.829231	+	0.558906i	$0.811221\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−2.50000	−	4.33013i	−0.115194	−	0.199522i
$472$		0			0
$473$	−5.00000	+	8.66025i	−0.229900	+	0.398199i
$474$		0			0
$475$	−32.0000			−1.46826
$476$		0			0
$477$	12.0000			0.549442
$478$		0			0
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	−0.969920	−	0.243426i	$-0.921729\pi$
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	0.695773	+	0.718262i	$0.255062\pi$
$480$		0			0
$481$	2.00000	+	3.46410i	0.0911922	+	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.5000	−	18.1865i	−0.476780	−	0.825808i
$486$		0			0
$487$	−14.5000	+	25.1147i	−0.657058	+	1.13806i	0.324316	+	0.945949i	$0.394866\pi$
$487$	−14.5000	+	25.1147i	−0.657058	+	1.13806i	−0.981374	+	0.192109i	$0.938467\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$	24.0000			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−3.00000	−	5.19615i	−0.134840	−	0.233550i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	0.907314	−	0.420455i	$-0.138129\pi$
$499$	2.00000	+	3.46410i	0.0895323	+	0.155074i	−0.817781	+	0.575529i	$0.804796\pi$
$500$		0			0
$501$	6.00000	−	10.3923i	0.268060	−	0.464294i
$502$		0			0
$503$	30.0000			1.33763			0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000			1.33763			0.668817	+	0.743427i	$0.266801\pi$
$504$		0			0
$505$	−54.0000			−2.40297
$506$		0			0
$507$	1.50000	−	2.59808i	0.0666173	−	0.115385i
$508$		0			0
$509$	−10.5000	−	18.1865i	−0.465404	−	0.806104i	0.533815	−	0.845601i	$-0.320758\pi$
$509$	−10.5000	−	18.1865i	−0.465404	−	0.806104i	−0.999220	+	0.0394971i	$0.987424\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	20.0000	+	34.6410i	0.883022	+	1.52944i
$514$		0			0
$515$	12.0000	−	20.7846i	0.528783	−	0.915879i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	18.0000			0.790112
$520$		0			0
$521$	−13.5000	+	23.3827i	−0.591446	+	1.02441i	0.402592	+	0.915379i	$0.368109\pi$
$521$	−13.5000	+	23.3827i	−0.591446	+	1.02441i	−0.994038	+	0.109035i	$0.965224\pi$
$522$		0			0
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	0.940129	−	0.340818i	$-0.110704\pi$
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.0000	−	25.9808i	−0.653410	−	1.13174i
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$	6.00000			0.260378
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−9.00000	+	15.5885i	−0.389104	+	0.673948i
$536$		0			0
$537$	−4.50000	−	7.79423i	−0.194189	−	0.336346i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	8.00000	+	13.8564i	0.343947	+	0.595733i	0.985162	−	0.171628i	$-0.0549027\pi$
$541$	8.00000	+	13.8564i	0.343947	+	0.595733i	−0.641215	+	0.767361i	$0.721569\pi$
$542$		0			0
$543$	6.50000	−	11.2583i	0.278942	−	0.483141i
$544$		0			0
$545$	6.00000			0.257012
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$	4.00000	−	6.92820i	0.170716	−	0.295689i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.50000	−	2.59808i	−0.0636715	−	0.110282i
$556$		0			0
$557$	9.00000	−	15.5885i	0.381342	−	0.660504i	−0.609912	−	0.792469i	$-0.708795\pi$
$557$	9.00000	−	15.5885i	0.381342	−	0.660504i	0.991254	+	0.131965i	$0.0421286\pi$
$558$		0			0
$559$	−40.0000			−1.69182
$560$		0			0
$561$	−6.00000			−0.253320
$562$		0			0
$563$	−18.0000	+	31.1769i	−0.758610	+	1.31395i	0.184950	+	0.982748i	$0.440788\pi$
$563$	−18.0000	+	31.1769i	−0.758610	+	1.31395i	−0.943560	+	0.331202i	$0.892546\pi$
$564$		0			0
$565$	22.5000	+	38.9711i	0.946582	+	1.63953i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	−22.0000	+	38.1051i	−0.920671	+	1.59465i	−0.122292	+	0.992494i	$0.539025\pi$
$571$	−22.0000	+	38.1051i	−0.920671	+	1.59465i	−0.798379	+	0.602155i	$0.794309\pi$
$572$		0			0
$573$	21.0000			0.877288
$574$		0			0
$575$	−12.0000			−0.500435
$576$		0			0
$577$	8.50000	−	14.7224i	0.353860	−	0.612903i	−0.633062	−	0.774101i	$-0.718202\pi$
$577$	8.50000	−	14.7224i	0.353860	−	0.612903i	0.986922	+	0.161198i	$0.0515357\pi$
$578$		0			0
$579$	10.0000	+	17.3205i	0.415586	+	0.719816i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−3.00000	−	5.19615i	−0.124247	−	0.215203i
$584$		0			0
$585$	12.0000	−	20.7846i	0.496139	−	0.859338i
$586$		0			0
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	40.0000			1.64817
$590$		0			0
$591$	3.00000	−	5.19615i	0.123404	−	0.213741i
$592$		0			0
$593$	−18.0000	−	31.1769i	−0.739171	−	1.28028i	−0.952869	−	0.303383i	$-0.901884\pi$
$593$	−18.0000	−	31.1769i	−0.739171	−	1.28028i	0.213697	−	0.976900i	$-0.431449\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.00000	−	6.92820i	−0.163709	−	0.283552i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	2.00000			0.0814463
$604$		0			0
$605$	−1.50000	+	2.59808i	−0.0609837	+	0.105627i
$606$		0			0
$607$	7.00000	+	12.1244i	0.284121	+	0.492112i	0.972396	−	0.233338i	$-0.0749648\pi$
$607$	7.00000	+	12.1244i	0.284121	+	0.492112i	−0.688274	+	0.725450i	$0.741632\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−16.0000	+	27.7128i	−0.646234	+	1.11931i	0.337781	+	0.941225i	$0.390324\pi$
$613$	−16.0000	+	27.7128i	−0.646234	+	1.11931i	−0.984015	+	0.178085i	$0.943010\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		0			0
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	8.50000	−	14.7224i	0.341644	−	0.591744i	0.984738	+	0.174042i	$0.0556830\pi$
$620$		0			0
$621$	7.50000	+	12.9904i	0.300965	+	0.521286i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$		0			0
$627$	4.00000	−	6.92820i	0.159745	−	0.276686i
$628$		0			0
$629$	6.00000			0.239236
$630$		0			0
$631$	−43.0000			−1.71180			−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000			−1.71180			−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$	10.0000	−	17.3205i	0.397464	−	0.688428i
$634$		0			0
$635$	24.0000	+	41.5692i	0.952411	+	1.64962i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	15.0000	+	25.9808i	0.593391	+	1.02778i
$640$		0			0
$641$	−19.5000	+	33.7750i	−0.770204	+	1.33403i	0.167247	+	0.985915i	$0.446512\pi$
$641$	−19.5000	+	33.7750i	−0.770204	+	1.33403i	−0.937451	+	0.348117i	$0.886821\pi$
$642$		0			0
$643$	13.0000			0.512670			0.256335	−	0.966588i	$-0.417485\pi$
$643$	13.0000			0.512670			0.256335	+	0.966588i	$0.417485\pi$
$644$		0			0
$645$	30.0000			1.18125
$646$		0			0
$647$	1.50000	−	2.59808i	0.0589711	−	0.102141i	−0.835033	−	0.550200i	$-0.814551\pi$
$647$	1.50000	−	2.59808i	0.0589711	−	0.102141i	0.894004	+	0.448059i	$0.147885\pi$
$648$		0			0
$649$	−1.50000	−	2.59808i	−0.0588802	−	0.101983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.50000	−	2.59808i	−0.0586995	−	0.101671i	0.835182	−	0.549973i	$-0.185362\pi$
$653$	−1.50000	−	2.59808i	−0.0586995	−	0.101671i	−0.893882	+	0.448303i	$0.852029\pi$
$654$		0			0
$655$	−9.00000	+	15.5885i	−0.351659	+	0.609091i
$656$		0			0
$657$	−8.00000			−0.312110
$658$		0			0
$659$	−42.0000			−1.63609			−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000			−1.63609			−0.818044	+	0.575156i	$0.804941\pi$
$660$		0			0
$661$	8.50000	−	14.7224i	0.330612	−	0.572636i	−0.652020	−	0.758202i	$-0.726078\pi$
$661$	8.50000	−	14.7224i	0.330612	−	0.572636i	0.982632	+	0.185565i	$0.0594116\pi$
$662$		0			0
$663$	−12.0000	−	20.7846i	−0.466041	−	0.807207i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−8.50000	+	14.7224i	−0.328629	+	0.569202i
$670$		0			0
$671$	−4.00000			−0.154418
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$		0			0
$675$	−10.0000	+	17.3205i	−0.384900	+	0.666667i
$676$		0			0
$677$	−21.0000	−	36.3731i	−0.807096	−	1.39793i	−0.914867	−	0.403755i	$-0.867705\pi$
$677$	−21.0000	−	36.3731i	−0.807096	−	1.39793i	0.107772	−	0.994176i	$-0.465628\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−3.00000	−	5.19615i	−0.114960	−	0.199117i
$682$		0			0
$683$	−24.0000	+	41.5692i	−0.918334	+	1.59060i	−0.116390	+	0.993204i	$0.537132\pi$
$683$	−24.0000	+	41.5692i	−0.918334	+	1.59060i	−0.801945	+	0.597398i	$0.796201\pi$
$684$		0			0
$685$	27.0000			1.03162
$686$		0			0
$687$	−13.0000			−0.495981
$688$		0			0
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	−0.500000	−	0.866025i	−0.0190209	−	0.0329452i	0.856358	−	0.516382i	$-0.172722\pi$
$691$	−0.500000	−	0.866025i	−0.0190209	−	0.0329452i	−0.875379	+	0.483437i	$0.839388\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	21.0000	+	36.3731i	0.796575	+	1.37971i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	24.0000			0.907763
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	−4.00000	+	6.92820i	−0.150863	+	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	18.5000	+	32.0429i	0.694782	+	1.20340i	0.970254	+	0.242089i	$0.0778325\pi$
$709$	18.5000	+	32.0429i	0.694782	+	1.20340i	−0.275472	+	0.961309i	$0.588834\pi$
$710$		0			0
$711$	2.00000	−	3.46410i	0.0750059	−	0.129914i
$712$		0			0
$713$	15.0000			0.561754
$714$		0			0
$715$	−12.0000			−0.448775
$716$		0			0
$717$	3.00000	−	5.19615i	0.112037	−	0.194054i
$718$		0			0
$719$	22.5000	+	38.9711i	0.839108	+	1.45338i	0.890641	+	0.454707i	$0.150256\pi$
$719$	22.5000	+	38.9711i	0.839108	+	1.45338i	−0.0515326	+	0.998671i	$0.516411\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−4.00000	−	6.92820i	−0.148762	−	0.257663i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−17.0000			−0.630495			−0.315248	−	0.949009i	$-0.602088\pi$
$727$	−17.0000			−0.630495			−0.315248	+	0.949009i	$0.602088\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−30.0000	+	51.9615i	−1.10959	+	1.92187i
$732$		0			0
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	0.826723	−	0.562609i	$-0.190202\pi$
$733$	−2.00000	−	3.46410i	−0.0738717	−	0.127950i	−0.900595	+	0.434659i	$0.856869\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.500000	−	0.866025i	−0.0184177	−	0.0319005i
$738$		0			0
$739$	17.0000	−	29.4449i	0.625355	−	1.08315i	−0.363117	−	0.931744i	$-0.618287\pi$
$739$	17.0000	−	29.4449i	0.625355	−	1.08315i	0.988472	−	0.151403i	$-0.0483792\pi$
$740$		0			0
$741$	32.0000			1.17555
$742$		0			0
$743$	12.0000			0.440237			0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000			0.440237			0.220119	+	0.975473i	$0.429356\pi$
$744$		0			0
$745$	−9.00000	+	15.5885i	−0.329734	+	0.571117i
$746$		0			0
$747$	−6.00000	−	10.3923i	−0.219529	−	0.380235i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−17.5000	−	30.3109i	−0.638584	−	1.10606i	−0.985744	−	0.168254i	$-0.946187\pi$
$751$	−17.5000	−	30.3109i	−0.638584	−	1.10606i	0.347160	−	0.937806i	$-0.387146\pi$
$752$		0			0
$753$	4.50000	−	7.79423i	0.163989	−	0.284037i
$754$		0			0
$755$	−30.0000			−1.09181
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$	1.50000	−	2.59808i	0.0544466	−	0.0943042i
$760$		0			0
$761$	18.0000	+	31.1769i	0.652499	+	1.13016i	0.982514	+	0.186187i	$0.0596129\pi$
$761$	18.0000	+	31.1769i	0.652499	+	1.13016i	−0.330015	+	0.943976i	$0.607054\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−18.0000	−	31.1769i	−0.650791	−	1.12720i
$766$		0			0
$767$	6.00000	−	10.3923i	0.216647	−	0.375244i
$768$		0			0
$769$	−44.0000			−1.58668			−0.793340	−	0.608778i	$-0.791660\pi$
$769$	−44.0000			−1.58668			−0.793340	+	0.608778i	$0.791660\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	21.0000	−	36.3731i	0.755318	−	1.30825i	−0.189899	−	0.981804i	$-0.560816\pi$
$773$	21.0000	−	36.3731i	0.755318	−	1.30825i	0.945216	−	0.326445i	$-0.105851\pi$
$774$		0			0
$775$	10.0000	+	17.3205i	0.359211	+	0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	7.50000	−	12.9904i	0.268371	−	0.464832i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.0000			−0.535373
$786$		0			0
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	−0.426193	−	0.904632i	$-0.640145\pi$
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	0.996531	−	0.0832226i	$-0.0265213\pi$
$788$		0			0
$789$	9.00000	+	15.5885i	0.320408	+	0.554964i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−8.00000	−	13.8564i	−0.284088	−	0.492055i
$794$		0			0
$795$	−9.00000	+	15.5885i	−0.319197	+	0.552866i
$796$		0			0
$797$	−9.00000			−0.318796			−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000			−0.318796			−0.159398	+	0.987214i	$0.550955\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	9.00000	−	15.5885i	0.317999	−	0.550791i
$802$		0			0
$803$	2.00000	+	3.46410i	0.0705785	+	0.122245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	3.00000	+	5.19615i	0.105605	+	0.182913i
$808$		0			0
$809$	12.0000	−	20.7846i	0.421898	−	0.730748i	−0.574228	−	0.818696i	$-0.694698\pi$
$809$	12.0000	−	20.7846i	0.421898	−	0.730748i	0.996125	+	0.0879478i	$0.0280309\pi$
$810$		0			0
$811$	−38.0000			−1.33436			−0.667180	−	0.744896i	$-0.732499\pi$
$811$	−38.0000			−1.33436			−0.667180	+	0.744896i	$0.732499\pi$
$812$		0			0
$813$	20.0000			0.701431
$814$		0			0
$815$	6.00000	−	10.3923i	0.210171	−	0.364027i
$816$		0			0
$817$	−40.0000	−	69.2820i	−1.39942	−	2.42387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.0000	−	25.9808i	−0.523504	−	0.906735i	−0.999626	−	0.0273557i	$-0.991291\pi$
$821$	−15.0000	−	25.9808i	−0.523504	−	0.906735i	0.476122	−	0.879379i	$-0.342042\pi$
$822$		0			0
$823$	21.5000	−	37.2391i	0.749443	−	1.29807i	−0.198647	−	0.980071i	$-0.563655\pi$
$823$	21.5000	−	37.2391i	0.749443	−	1.29807i	0.948090	−	0.318002i	$-0.103012\pi$
$824$		0			0
$825$	4.00000			0.139262
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	−9.50000	+	16.4545i	−0.329949	+	0.571488i	−0.982501	−	0.186256i	$-0.940365\pi$
$829$	−9.50000	+	16.4545i	−0.329949	+	0.571488i	0.652553	+	0.757743i	$0.273698\pi$
$830$		0			0
$831$	−5.00000	−	8.66025i	−0.173448	−	0.300421i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−18.0000	−	31.1769i	−0.622916	−	1.07892i
$836$		0			0
$837$	12.5000	−	21.6506i	0.432063	−	0.748355i
$838$		0			0
$839$	39.0000			1.34643			0.673215	−	0.739447i	$-0.264913\pi$
$839$	39.0000			1.34643			0.673215	+	0.739447i	$0.264913\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−9.00000	+	15.5885i	−0.309976	+	0.536895i
$844$		0			0
$845$	−4.50000	−	7.79423i	−0.154805	−	0.268130i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	2.00000	+	3.46410i	0.0686398	+	0.118888i
$850$		0			0
$851$	−1.50000	+	2.59808i	−0.0514193	+	0.0890609i
$852$		0			0
$853$	−38.0000			−1.30110			−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000			−1.30110			−0.650548	+	0.759465i	$0.725461\pi$
$854$		0			0
$855$	48.0000			1.64157
$856$		0			0
$857$	12.0000	−	20.7846i	0.409912	−	0.709989i	−0.584967	−	0.811057i	$-0.698893\pi$
$857$	12.0000	−	20.7846i	0.409912	−	0.709989i	0.994880	+	0.101068i	$0.0322260\pi$
$858$		0			0
$859$	−12.5000	−	21.6506i	−0.426494	−	0.738710i	0.570064	−	0.821600i	$-0.306918\pi$
$859$	−12.5000	−	21.6506i	−0.426494	−	0.738710i	−0.996559	+	0.0828900i	$0.973585\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.0000	−	41.5692i	−0.816970	−	1.41503i	−0.907905	−	0.419176i	$-0.862319\pi$
$863$	−24.0000	−	41.5692i	−0.816970	−	1.41503i	0.0909355	−	0.995857i	$-0.471014\pi$
$864$		0			0
$865$	27.0000	−	46.7654i	0.918028	−	1.59007i
$866$		0			0
$867$	−19.0000			−0.645274
$868$		0			0
$869$	−2.00000			−0.0678454
$870$		0			0
$871$	2.00000	−	3.46410i	0.0677674	−	0.117377i
$872$		0			0
$873$	7.00000	+	12.1244i	0.236914	+	0.410347i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	26.0000	+	45.0333i	0.877958	+	1.52067i	0.853578	+	0.520964i	$0.174428\pi$
$877$	26.0000	+	45.0333i	0.877958	+	1.52067i	0.0243792	+	0.999703i	$0.492239\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	27.0000			0.909653			0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000			0.909653			0.454827	+	0.890580i	$0.349701\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$		0			0
$885$	−4.50000	+	7.79423i	−0.151266	+	0.262000i
$886$		0			0
$887$	15.0000	+	25.9808i	0.503651	+	0.872349i	0.999991	+	0.00422062i	$0.00134347\pi$
$887$	15.0000	+	25.9808i	0.503651	+	0.872349i	−0.496340	+	0.868128i	$0.665323\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.500000	+	0.866025i	0.0167506	+	0.0290129i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−27.0000			−0.902510
$896$		0			0
$897$	12.0000			0.400668
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−18.0000	−	31.1769i	−0.599667	−	1.03865i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19.5000	−	33.7750i	−0.648202	−	1.12272i
$906$		0			0
$907$	2.00000	−	3.46410i	0.0664089	−	0.115024i	−0.830909	−	0.556408i	$-0.812179\pi$
$907$	2.00000	−	3.46410i	0.0664089	−	0.115024i	0.897318	+	0.441384i	$0.145512\pi$
$908$		0			0
$909$	36.0000			1.19404
$910$		0			0
$911$	12.0000			0.397578			0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000			0.397578			0.198789	+	0.980042i	$0.436299\pi$
$912$		0			0
$913$	−3.00000	+	5.19615i	−0.0992855	+	0.171968i
$914$		0			0
$915$	6.00000	+	10.3923i	0.198354	+	0.343559i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	17.0000	+	29.4449i	0.560778	+	0.971296i	0.997429	+	0.0716652i	$0.0228313\pi$
$919$	17.0000	+	29.4449i	0.560778	+	0.971296i	−0.436650	+	0.899631i	$0.643835\pi$
$920$		0			0
$921$	8.00000	−	13.8564i	0.263609	−	0.456584i
$922$		0			0
$923$	60.0000			1.97492
$924$		0			0
$925$	−4.00000			−0.131519
$926$		0			0
$927$	−8.00000	+	13.8564i	−0.262754	+	0.455104i
$928$		0			0
$929$	9.00000	+	15.5885i	0.295280	+	0.511441i	0.975050	−	0.221985i	$-0.0712536\pi$
$929$	9.00000	+	15.5885i	0.295280	+	0.511441i	−0.679770	+	0.733426i	$0.737920\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−6.00000	−	10.3923i	−0.196431	−	0.340229i
$934$		0			0
$935$	−9.00000	+	15.5885i	−0.294331	+	0.509797i
$936$		0			0
$937$	−32.0000			−1.04539			−0.522697	−	0.852518i	$-0.675074\pi$
$937$	−32.0000			−1.04539			−0.522697	+	0.852518i	$0.675074\pi$
$938$		0			0
$939$	−1.00000			−0.0326338
$940$		0			0
$941$	−15.0000	+	25.9808i	−0.488986	+	0.846949i	−0.999920	−	0.0126715i	$-0.995966\pi$
$941$	−15.0000	+	25.9808i	−0.488986	+	0.846949i	0.510934	+	0.859620i	$0.329300\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.5000	−	23.3827i	−0.438691	−	0.759835i	0.558898	−	0.829237i	$-0.311224\pi$
$947$	−13.5000	−	23.3827i	−0.438691	−	0.759835i	−0.997589	+	0.0694014i	$0.977891\pi$
$948$		0			0
$949$	−8.00000	+	13.8564i	−0.259691	+	0.449798i
$950$		0			0
$951$	−33.0000			−1.07010
$952$		0			0
$953$	42.0000			1.36051			0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000			1.36051			0.680257	+	0.732974i	$0.261868\pi$
$954$		0			0
$955$	31.5000	−	54.5596i	1.01932	−	1.76551i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$	6.00000	−	10.3923i	0.193347	−	0.334887i
$964$		0			0
$965$	60.0000			1.93147
$966$		0			0
$967$	−16.0000			−0.514525			−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000			−0.514525			−0.257263	+	0.966342i	$0.582821\pi$
$968$		0			0
$969$	24.0000	−	41.5692i	0.770991	−	1.33540i
$970$		0			0
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	0.720224	−	0.693742i	$-0.244039\pi$
$971$	−7.50000	−	12.9904i	−0.240686	−	0.416881i	−0.960910	+	0.276861i	$0.910706\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	8.00000	+	13.8564i	0.256205	+	0.443760i
$976$		0			0
$977$	−22.5000	+	38.9711i	−0.719839	+	1.24680i	0.241225	+	0.970469i	$0.422451\pi$
$977$	−22.5000	+	38.9711i	−0.719839	+	1.24680i	−0.961063	+	0.276328i	$0.910882\pi$
$978$		0			0
$979$	−9.00000			−0.287641
$980$		0			0
$981$	−4.00000			−0.127710
$982$		0			0
$983$	22.5000	−	38.9711i	0.717639	−	1.24299i	−0.244294	−	0.969701i	$-0.578556\pi$
$983$	22.5000	−	38.9711i	0.717639	−	1.24299i	0.961933	−	0.273285i	$-0.0881103\pi$
$984$		0			0
$985$	−9.00000	−	15.5885i	−0.286764	−	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15.0000	−	25.9808i	−0.476972	−	0.826140i
$990$		0			0
$991$	8.00000	−	13.8564i	0.254128	−	0.440163i	−0.710530	−	0.703667i	$-0.751545\pi$
$991$	8.00000	−	13.8564i	0.254128	−	0.440163i	0.964658	+	0.263504i	$0.0848781\pi$
$992$		0			0
$993$	7.00000			0.222138
$994$		0			0
$995$	−24.0000			−0.760851
$996$		0			0
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	−0.934274	−	0.356555i	$-0.883951\pi$
$997$	−5.00000	+	8.66025i	−0.158352	+	0.274273i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$	2.50000	+	4.33013i	0.0790965	+	0.136999i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2156.2.i.c.1145.1		2
7.2	even	3	inner	2156.2.i.c.177.1		2
7.3	odd	6		44.2.a.a.1.1	✓	1
7.4	even	3		2156.2.a.a.1.1		1
7.5	odd	6		2156.2.i.b.177.1		2
7.6	odd	2		2156.2.i.b.1145.1		2
21.17	even	6		396.2.a.c.1.1		1
28.3	even	6		176.2.a.a.1.1		1
28.11	odd	6		8624.2.a.w.1.1		1
35.3	even	12		1100.2.b.c.749.2		2
35.17	even	12		1100.2.b.c.749.1		2
35.24	odd	6		1100.2.a.b.1.1		1
56.3	even	6		704.2.a.i.1.1		1
56.45	odd	6		704.2.a.f.1.1		1
63.31	odd	6		3564.2.i.j.1189.1		2
63.38	even	6		3564.2.i.a.2377.1		2
63.52	odd	6		3564.2.i.j.2377.1		2
63.59	even	6		3564.2.i.a.1189.1		2
77.3	odd	30		484.2.e.a.9.1		4
77.10	even	6		484.2.a.a.1.1		1
77.17	even	30		484.2.e.b.245.1		4
77.24	even	30		484.2.e.b.81.1		4
77.31	odd	30		484.2.e.a.81.1		4
77.38	odd	30		484.2.e.a.245.1		4
77.52	even	30		484.2.e.b.9.1		4
77.59	odd	30		484.2.e.a.269.1		4
77.73	even	30		484.2.e.b.269.1		4
84.59	odd	6		1584.2.a.p.1.1		1
91.38	odd	6		7436.2.a.d.1.1		1
105.17	odd	12		9900.2.c.g.5149.2		2
105.38	odd	12		9900.2.c.g.5149.1		2
105.59	even	6		9900.2.a.h.1.1		1
112.3	even	12		2816.2.c.k.1409.1		2
112.45	odd	12		2816.2.c.e.1409.2		2
112.59	even	12		2816.2.c.k.1409.2		2
112.101	odd	12		2816.2.c.e.1409.1		2
140.3	odd	12		4400.2.b.k.4049.1		2
140.59	even	6		4400.2.a.v.1.1		1
140.87	odd	12		4400.2.b.k.4049.2		2
168.59	odd	6		6336.2.a.i.1.1		1
168.101	even	6		6336.2.a.j.1.1		1
231.164	odd	6		4356.2.a.j.1.1		1
308.87	odd	6		1936.2.a.c.1.1		1
616.395	odd	6		7744.2.a.bc.1.1		1
616.549	even	6		7744.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.a.a.1.1	✓	1	7.3	odd	6
176.2.a.a.1.1		1	28.3	even	6
396.2.a.c.1.1		1	21.17	even	6
484.2.a.a.1.1		1	77.10	even	6
484.2.e.a.9.1		4	77.3	odd	30
484.2.e.a.81.1		4	77.31	odd	30
484.2.e.a.245.1		4	77.38	odd	30
484.2.e.a.269.1		4	77.59	odd	30
484.2.e.b.9.1		4	77.52	even	30
484.2.e.b.81.1		4	77.24	even	30
484.2.e.b.245.1		4	77.17	even	30
484.2.e.b.269.1		4	77.73	even	30
704.2.a.f.1.1		1	56.45	odd	6
704.2.a.i.1.1		1	56.3	even	6
1100.2.a.b.1.1		1	35.24	odd	6
1100.2.b.c.749.1		2	35.17	even	12
1100.2.b.c.749.2		2	35.3	even	12
1584.2.a.p.1.1		1	84.59	odd	6
1936.2.a.c.1.1		1	308.87	odd	6
2156.2.a.a.1.1		1	7.4	even	3
2156.2.i.b.177.1		2	7.5	odd	6
2156.2.i.b.1145.1		2	7.6	odd	2
2156.2.i.c.177.1		2	7.2	even	3	inner
2156.2.i.c.1145.1		2	1.1	even	1	trivial
2816.2.c.e.1409.1		2	112.101	odd	12
2816.2.c.e.1409.2		2	112.45	odd	12
2816.2.c.k.1409.1		2	112.3	even	12
2816.2.c.k.1409.2		2	112.59	even	12
3564.2.i.a.1189.1		2	63.59	even	6
3564.2.i.a.2377.1		2	63.38	even	6
3564.2.i.j.1189.1		2	63.31	odd	6
3564.2.i.j.2377.1		2	63.52	odd	6
4356.2.a.j.1.1		1	231.164	odd	6
4400.2.a.v.1.1		1	140.59	even	6
4400.2.b.k.4049.1		2	140.3	odd	12
4400.2.b.k.4049.2		2	140.87	odd	12
6336.2.a.i.1.1		1	168.59	odd	6
6336.2.a.j.1.1		1	168.101	even	6
7436.2.a.d.1.1		1	91.38	odd	6
7744.2.a.m.1.1		1	616.549	even	6
7744.2.a.bc.1.1		1	616.395	odd	6
8624.2.a.w.1.1		1	28.11	odd	6
9900.2.a.h.1.1		1	105.59	even	6
9900.2.c.g.5149.1		2	105.38	odd	12
9900.2.c.g.5149.2		2	105.17	odd	12

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:	\( N \)	\(=\)	\( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2156.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{3} +(-1.50000 - 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{9} +(0.500000 - 0.866025i) q^{11} +4.00000 q^{13} -3.00000 q^{15} +(3.00000 - 5.19615i) q^{17} +(4.00000 + 6.92820i) q^{19} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +5.00000 q^{27} +(2.50000 - 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{33} +(0.500000 + 0.866025i) q^{37} +(2.00000 - 3.46410i) q^{39} -10.0000 q^{43} +(3.00000 - 5.19615i) q^{45} +(-3.00000 - 5.19615i) q^{51} +(3.00000 - 5.19615i) q^{53} -3.00000 q^{55} +8.00000 q^{57} +(1.50000 - 2.59808i) q^{59} +(-2.00000 - 3.46410i) q^{61} +(-6.00000 - 10.3923i) q^{65} +(0.500000 - 0.866025i) q^{67} +3.00000 q^{69} +15.0000 q^{71} +(-2.00000 + 3.46410i) q^{73} +(2.00000 + 3.46410i) q^{75} +(-1.00000 - 1.73205i) q^{79} +(-0.500000 + 0.866025i) q^{81} -6.00000 q^{83} -18.0000 q^{85} +(-4.50000 - 7.79423i) q^{89} +(-2.50000 - 4.33013i) q^{93} +(12.0000 - 20.7846i) q^{95} +7.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + q^3 - 3 * q^5 + 2 * q^9	\( 2 q + q^{3} - 3 q^{5} + 2 q^{9} + q^{11} + 8 q^{13} - 6 q^{15} + 6 q^{17} + 8 q^{19} + 3 q^{23} - 4 q^{25} + 10 q^{27} + 5 q^{31} - q^{33} + q^{37} + 4 q^{39} - 20 q^{43} + 6 q^{45} - 6 q^{51} + 6 q^{53} - 6 q^{55} + 16 q^{57} + 3 q^{59} - 4 q^{61} - 12 q^{65} + q^{67} + 6 q^{69} + 30 q^{71} - 4 q^{73} + 4 q^{75} - 2 q^{79} - q^{81} - 12 q^{83} - 36 q^{85} - 9 q^{89} - 5 q^{93} + 24 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + q^3 - 3 * q^5 + 2 * q^9 + q^11 + 8 * q^13 - 6 * q^15 + 6 * q^17 + 8 * q^19 + 3 * q^23 - 4 * q^25 + 10 * q^27 + 5 * q^31 - q^33 + q^37 + 4 * q^39 - 20 * q^43 + 6 * q^45 - 6 * q^51 + 6 * q^53 - 6 * q^55 + 16 * q^57 + 3 * q^59 - 4 * q^61 - 12 * q^65 + q^67 + 6 * q^69 + 30 * q^71 - 4 * q^73 + 4 * q^75 - 2 * q^79 - q^81 - 12 * q^83 - 36 * q^85 - 9 * q^89 - 5 * q^93 + 24 * q^95 + 14 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more