LMFDB - Embedded newform 1176.2.q.j.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(361,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1176.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.q (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:	$9.39040727770$
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 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1176.961
Dual form			1176.2.q.j.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{5} +(-0.500000 + 0.866025i) q^{9} +2.00000 q^{13} +2.00000 q^{15} +(3.00000 + 5.19615i) q^{17} +(-2.00000 + 3.46410i) q^{19} +(2.00000 - 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +6.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(5.00000 - 8.66025i) q^{37} +(1.00000 + 1.73205i) q^{39} +10.0000 q^{41} +12.0000 q^{43} +(1.00000 + 1.73205i) q^{45} +(-4.00000 + 6.92820i) q^{47} +(-3.00000 + 5.19615i) q^{51} +(-3.00000 - 5.19615i) q^{53} -4.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(-5.00000 + 8.66025i) q^{61} +(2.00000 - 3.46410i) q^{65} +(-6.00000 - 10.3923i) q^{67} +4.00000 q^{69} +4.00000 q^{71} +(1.00000 + 1.73205i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} -4.00000 q^{83} +12.0000 q^{85} +(3.00000 + 5.19615i) q^{87} +(3.00000 - 5.19615i) q^{89} +(4.00000 - 6.92820i) q^{93} +(4.00000 + 6.92820i) q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} - q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 - q^9	$ 2 q + q^{3} + 2 q^{5} - q^{9} + 4 q^{13} + 4 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{31} + 10 q^{37} + 2 q^{39} + 20 q^{41} + 24 q^{43} + 2 q^{45} - 8 q^{47} - 6 q^{51} - 6 q^{53} - 8 q^{57} + 4 q^{59} - 10 q^{61} + 4 q^{65} - 12 q^{67} + 8 q^{69} + 8 q^{71} + 2 q^{73} - q^{75} - 8 q^{79} - q^{81} - 8 q^{83} + 24 q^{85} + 6 q^{87} + 6 q^{89} + 8 q^{93} + 8 q^{95} - 20 q^{97}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 - q^9 + 4 * q^13 + 4 * q^15 + 6 * q^17 - 4 * q^19 + 4 * q^23 + q^25 - 2 * q^27 + 12 * q^29 - 8 * q^31 + 10 * q^37 + 2 * q^39 + 20 * q^41 + 24 * q^43 + 2 * q^45 - 8 * q^47 - 6 * q^51 - 6 * q^53 - 8 * q^57 + 4 * q^59 - 10 * q^61 + 4 * q^65 - 12 * q^67 + 8 * q^69 + 8 * q^71 + 2 * q^73 - q^75 - 8 * q^79 - q^81 - 8 * q^83 + 24 * q^85 + 6 * q^87 + 6 * q^89 + 8 * q^93 + 8 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$4$		0			0
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	2.00000			0.516398
$16$		0			0
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	−0.998899	−	0.0469020i	$-0.985065\pi$
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	0.540068	+	0.841621i	$0.318398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.00000	−	8.66025i	0.821995	−	1.42374i	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	5.00000	−	8.66025i	0.821995	−	1.42374i	0.904194	−	0.427121i	$-0.140472\pi$
$38$		0			0
$39$	1.00000	+	1.73205i	0.160128	+	0.277350i
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$	12.0000			1.82998			0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000			1.82998			0.914991	+	0.403473i	$0.132197\pi$
$44$		0			0
$45$	1.00000	+	1.73205i	0.149071	+	0.258199i
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		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	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.00000			−0.529813
$58$		0			0
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	0.345207	+	0.938527i	$0.387809\pi$
$61$	−5.00000	+	8.66025i	−0.640184	+	1.10883i	−0.985391	+	0.170305i	$0.945525\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	−	3.46410i	0.248069	−	0.429669i
$66$		0			0
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	−0.955588	−	0.294706i	$-0.904778\pi$
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	0.222571	−	0.974916i	$-0.428555\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	0.918594	−	0.395203i	$-0.129326\pi$
$73$	1.00000	+	1.73205i	0.117041	+	0.202721i	−0.801553	+	0.597924i	$0.795992\pi$
$74$		0			0
$75$	−0.500000	+	0.866025i	−0.0577350	+	0.100000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$	3.00000	+	5.19615i	0.321634	+	0.557086i
$88$		0			0
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	−0.662071	−	0.749441i	$-0.730322\pi$
$89$	3.00000	−	5.19615i	0.317999	−	0.550791i	0.980071	+	0.198650i	$0.0636557\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	4.00000	−	6.92820i	0.414781	−	0.718421i
$94$		0			0
$95$	4.00000	+	6.92820i	0.410391	+	0.710819i
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.00000	+	8.66025i	0.497519	+	0.861727i	0.999996	−	0.00286291i	$-0.000911295\pi$
$101$	5.00000	+	8.66025i	0.497519	+	0.861727i	−0.502477	+	0.864590i	$0.667578\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000	+	13.8564i	−0.773389	+	1.33955i	0.162306	+	0.986740i	$0.448107\pi$
$107$	−8.00000	+	13.8564i	−0.773389	+	1.33955i	−0.935695	+	0.352809i	$0.885227\pi$
$108$		0			0
$109$	−7.00000	−	12.1244i	−0.670478	−	1.16130i	−0.977769	−	0.209687i	$-0.932756\pi$
$109$	−7.00000	−	12.1244i	−0.670478	−	1.16130i	0.307290	−	0.951616i	$-0.400578\pi$
$110$		0			0
$111$	10.0000			0.949158
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	−4.00000	−	6.92820i	−0.373002	−	0.646058i
$116$		0			0
$117$	−1.00000	+	1.73205i	−0.0924500	+	0.160128i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$	5.00000	+	8.66025i	0.450835	+	0.780869i
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$	6.00000	+	10.3923i	0.528271	+	0.914991i
$130$		0			0
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.00000	+	1.73205i	−0.0860663	+	0.149071i
$136$		0			0
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	0.993109	+	0.117198i	$0.0373911\pi$
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	−0.395058	+	0.918656i	$0.629276\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000	−	10.3923i	0.498273	−	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	−16.0000			−1.28515
$156$		0			0
$157$	−5.00000	−	8.66025i	−0.399043	−	0.691164i	0.594565	−	0.804048i	$-0.297324\pi$
$157$	−5.00000	−	8.66025i	−0.399043	−	0.691164i	−0.993608	+	0.112884i	$0.963991\pi$
$158$		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	−0.777007	−	0.629492i	$-0.783263\pi$
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	0.933659	+	0.358162i	$0.116597\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−16.0000			−1.23812			−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000			−1.23812			−0.619059	+	0.785345i	$0.712486\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−2.00000	−	3.46410i	−0.152944	−	0.264906i
$172$		0			0
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	−0.825505	−	0.564396i	$-0.809109\pi$
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	0.901533	+	0.432710i	$0.142443\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.00000	+	3.46410i	−0.150329	+	0.260378i
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$	−10.0000	−	17.3205i	−0.735215	−	1.27343i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.00000	−	3.46410i	0.144715	−	0.250654i	−0.784552	−	0.620063i	$-0.787107\pi$
$191$	2.00000	−	3.46410i	0.144715	−	0.250654i	0.929267	+	0.369410i	$0.120440\pi$
$192$		0			0
$193$	−9.00000	−	15.5885i	−0.647834	−	1.12208i	−0.983639	−	0.180150i	$-0.942342\pi$
$193$	−9.00000	−	15.5885i	−0.647834	−	1.12208i	0.335805	−	0.941932i	$-0.390992\pi$
$194$		0			0
$195$	4.00000			0.286446
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$		0			0
$199$	−12.0000	−	20.7846i	−0.850657	−	1.47338i	−0.880616	−	0.473831i	$-0.842871\pi$
$199$	−12.0000	−	20.7846i	−0.850657	−	1.47338i	0.0299585	−	0.999551i	$-0.490462\pi$
$200$		0			0
$201$	6.00000	−	10.3923i	0.423207	−	0.733017i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	10.0000	−	17.3205i	0.698430	−	1.20972i
$206$		0			0
$207$	2.00000	+	3.46410i	0.139010	+	0.240772i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$	2.00000	+	3.46410i	0.137038	+	0.237356i
$214$		0			0
$215$	12.0000	−	20.7846i	0.818393	−	1.41750i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−1.00000	+	1.73205i	−0.0675737	+	0.117041i
$220$		0			0
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	0.993508	−	0.113761i	$-0.0362899\pi$
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	−0.595274	+	0.803523i	$0.702957\pi$
$228$		0			0
$229$	3.00000	−	5.19615i	0.198246	−	0.343371i	−0.749714	−	0.661762i	$-0.769809\pi$
$229$	3.00000	−	5.19615i	0.198246	−	0.343371i	0.947960	+	0.318390i	$0.103142\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	0.404674	+	0.914461i	$0.367385\pi$
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	−0.994283	+	0.106773i	$0.965948\pi$
$234$		0			0
$235$	8.00000	+	13.8564i	0.521862	+	0.903892i
$236$		0			0
$237$	−8.00000			−0.519656
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	−0.965387	−	0.260822i	$-0.916006\pi$
$241$	−11.0000	−	19.0526i	−0.708572	−	1.22728i	0.256814	−	0.966461i	$-0.417327\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.00000	+	6.92820i	−0.254514	+	0.440831i
$248$		0			0
$249$	−2.00000	−	3.46410i	−0.126745	−	0.219529i
$250$		0			0
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	6.00000	+	10.3923i	0.375735	+	0.650791i
$256$		0			0
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	−0.560781	−	0.827964i	$-0.689499\pi$
$257$	7.00000	−	12.1244i	0.436648	−	0.756297i	0.997429	+	0.0716680i	$0.0228322\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.00000	+	5.19615i	−0.185695	+	0.321634i
$262$		0			0
$263$	−14.0000	−	24.2487i	−0.863277	−	1.49524i	−0.868748	−	0.495255i	$-0.835075\pi$
$263$	−14.0000	−	24.2487i	−0.863277	−	1.49524i	0.00547092	−	0.999985i	$-0.498259\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$	6.00000			0.367194
$268$		0			0
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	0.977229	−	0.212187i	$-0.0680585\pi$
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	−0.672374	+	0.740212i	$0.734725\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	−6.00000	−	10.3923i	−0.356663	−	0.617758i	0.630738	−	0.775996i	$-0.282752\pi$
$283$	−6.00000	−	10.3923i	−0.356663	−	0.617758i	−0.987401	+	0.158237i	$0.949419\pi$
$284$		0			0
$285$	−4.00000	+	6.92820i	−0.236940	+	0.410391i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$	−5.00000	−	8.66025i	−0.293105	−	0.507673i
$292$		0			0
$293$	22.0000			1.28525			0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000			1.28525			0.642627	+	0.766179i	$0.277845\pi$
$294$		0			0
$295$	8.00000			0.465778
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.00000	−	6.92820i	0.231326	−	0.400668i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−5.00000	+	8.66025i	−0.287242	+	0.497519i
$304$		0			0
$305$	10.0000	+	17.3205i	0.572598	+	0.991769i
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−3.00000	+	5.19615i	−0.169570	+	0.293704i	−0.938269	−	0.345907i	$-0.887571\pi$
$313$	−3.00000	+	5.19615i	−0.169570	+	0.293704i	0.768699	+	0.639611i	$0.220905\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	−0.836576	−	0.547852i	$-0.815446\pi$
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	0.892741	+	0.450570i	$0.148779\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−16.0000			−0.893033
$322$		0			0
$323$	−24.0000			−1.33540
$324$		0			0
$325$	1.00000	+	1.73205i	0.0554700	+	0.0960769i
$326$		0			0
$327$	7.00000	−	12.1244i	0.387101	−	0.670478i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	−0.448649	−	0.893708i	$-0.648095\pi$
$331$	10.0000	−	17.3205i	0.549650	−	0.952021i	0.998298	−	0.0583130i	$-0.0185721\pi$
$332$		0			0
$333$	5.00000	+	8.66025i	0.273998	+	0.474579i
$334$		0			0
$335$	−24.0000			−1.31126
$336$		0			0
$337$	−30.0000			−1.63420			−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000			−1.63420			−0.817102	+	0.576493i	$0.804421\pi$
$338$		0			0
$339$	−7.00000	−	12.1244i	−0.380188	−	0.658505i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	4.00000	−	6.92820i	0.215353	−	0.373002i
$346$		0			0
$347$	4.00000	+	6.92820i	0.214731	+	0.371925i	0.953189	−	0.302374i	$-0.0977791\pi$
$347$	4.00000	+	6.92820i	0.214731	+	0.371925i	−0.738458	+	0.674299i	$0.764446\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	−2.00000			−0.106752
$352$		0			0
$353$	−5.00000	−	8.66025i	−0.266123	−	0.460939i	0.701734	−	0.712439i	$-0.252409\pi$
$353$	−5.00000	−	8.66025i	−0.266123	−	0.460939i	−0.967857	+	0.251500i	$0.919076\pi$
$354$		0			0
$355$	4.00000	−	6.92820i	0.212298	−	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.00000	−	10.3923i	0.316668	−	0.548485i	−0.663123	−	0.748511i	$-0.730769\pi$
$359$	6.00000	−	10.3923i	0.316668	−	0.548485i	0.979791	+	0.200026i	$0.0641026\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$	11.0000			0.577350
$364$		0			0
$365$	4.00000			0.209370
$366$		0			0
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	0.893873	+	0.448320i	$0.147978\pi$
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	−0.0586798	+	0.998277i	$0.518689\pi$
$368$		0			0
$369$	−5.00000	+	8.66025i	−0.260290	+	0.450835i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	0.427051	+	0.904227i	$0.359552\pi$
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$	6.00000	+	10.3923i	0.309839	+	0.536656i
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$	4.00000	+	6.92820i	0.204926	+	0.354943i
$382$		0			0
$383$	−8.00000	+	13.8564i	−0.408781	+	0.708029i	−0.994753	−	0.102302i	$-0.967379\pi$
$383$	−8.00000	+	13.8564i	−0.408781	+	0.708029i	0.585973	+	0.810331i	$0.300713\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−6.00000	+	10.3923i	−0.304997	+	0.528271i
$388$		0			0
$389$	−7.00000	−	12.1244i	−0.354914	−	0.614729i	0.632189	−	0.774814i	$-0.282157\pi$
$389$	−7.00000	−	12.1244i	−0.354914	−	0.614729i	−0.987103	+	0.160085i	$0.948823\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$	−12.0000			−0.605320
$394$		0			0
$395$	8.00000	+	13.8564i	0.402524	+	0.697191i
$396$		0			0
$397$	−13.0000	+	22.5167i	−0.652451	+	1.13008i	0.330075	+	0.943955i	$0.392926\pi$
$397$	−13.0000	+	22.5167i	−0.652451	+	1.13008i	−0.982526	+	0.186124i	$0.940407\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.00000	+	8.66025i	−0.249688	+	0.432472i	−0.963439	−	0.267927i	$-0.913661\pi$
$401$	−5.00000	+	8.66025i	−0.249688	+	0.432472i	0.713751	+	0.700399i	$0.246995\pi$
$402$		0			0
$403$	−8.00000	−	13.8564i	−0.398508	−	0.690237i
$404$		0			0
$405$	−2.00000			−0.0993808
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$		0			0
$411$	−7.00000	+	12.1244i	−0.345285	+	0.598050i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.00000	+	6.92820i	−0.196352	+	0.340092i
$416$		0			0
$417$	2.00000	+	3.46410i	0.0979404	+	0.169638i
$418$		0			0
$419$	36.0000			1.75872			0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000			1.75872			0.879358	+	0.476162i	$0.157972\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	−4.00000	−	6.92820i	−0.194487	−	0.336861i
$424$		0			0
$425$	−3.00000	+	5.19615i	−0.145521	+	0.252050i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	0.684564	−	0.728953i	$-0.259993\pi$
$431$	−6.00000	−	10.3923i	−0.289010	−	0.500580i	−0.973574	+	0.228373i	$0.926659\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$	12.0000			0.575356
$436$		0			0
$437$	8.00000	+	13.8564i	0.382692	+	0.662842i
$438$		0			0
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	−0.423556	−	0.905870i	$-0.639218\pi$
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	0.996284	−	0.0861252i	$-0.0274485\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.0000	−	34.6410i	0.950229	−	1.64584i	0.205301	−	0.978699i	$-0.434183\pi$
$443$	20.0000	−	34.6410i	0.950229	−	1.64584i	0.744927	−	0.667146i	$-0.232484\pi$
$444$		0			0
$445$	−6.00000	−	10.3923i	−0.284427	−	0.492642i
$446$		0			0
$447$	−6.00000			−0.283790
$448$		0			0
$449$	34.0000			1.60456			0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000			1.60456			0.802280	+	0.596948i	$0.203620\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−4.00000	+	6.92820i	−0.187936	+	0.325515i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$	−3.00000	−	5.19615i	−0.140028	−	0.242536i
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$		0			0
$465$	−8.00000	−	13.8564i	−0.370991	−	0.642575i
$466$		0			0
$467$	−2.00000	+	3.46410i	−0.0925490	+	0.160300i	−0.908583	−	0.417704i	$-0.862835\pi$
$467$	−2.00000	+	3.46410i	−0.0925490	+	0.160300i	0.816034	+	0.578004i	$0.196168\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	5.00000	−	8.66025i	0.230388	−	0.399043i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.00000			−0.183533
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		0			0
$481$	10.0000	−	17.3205i	0.455961	−	0.789747i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.0000	+	17.3205i	−0.454077	+	0.786484i
$486$		0			0
$487$	4.00000	+	6.92820i	0.181257	+	0.313947i	0.942309	−	0.334744i	$-0.108650\pi$
$487$	4.00000	+	6.92820i	0.181257	+	0.313947i	−0.761052	+	0.648691i	$0.775317\pi$
$488$		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
$493$	18.0000	+	31.1769i	0.810679	+	1.40414i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	−0.907314	−	0.420455i	$-0.861871\pi$
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	0.817781	+	0.575529i	$0.195204\pi$
$500$		0			0
$501$	−8.00000	−	13.8564i	−0.357414	−	0.619059i
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	20.0000			0.889988
$506$		0			0
$507$	−4.50000	−	7.79423i	−0.199852	−	0.346154i
$508$		0			0
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	−0.924821	−	0.380402i	$-0.875786\pi$
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	0.791849	+	0.610718i	$0.209119\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	2.00000	−	3.46410i	0.0883022	−	0.152944i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	2.00000			0.0877903
$520$		0			0
$521$	−9.00000	−	15.5885i	−0.394297	−	0.682943i	0.598714	−	0.800963i	$-0.295679\pi$
$521$	−9.00000	−	15.5885i	−0.394297	−	0.682943i	−0.993011	+	0.118020i	$0.962345\pi$
$522$		0			0
$523$	6.00000	−	10.3923i	0.262362	−	0.454424i	−0.704507	−	0.709697i	$-0.748832\pi$
$523$	6.00000	−	10.3923i	0.262362	−	0.454424i	0.966869	+	0.255273i	$0.0821653\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	24.0000	−	41.5692i	1.04546	−	1.81078i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$	20.0000			0.866296
$534$		0			0
$535$	16.0000	+	27.7128i	0.691740	+	1.19813i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.00000	−	1.73205i	0.0429934	−	0.0744667i	−0.843728	−	0.536771i	$-0.819644\pi$
$541$	1.00000	−	1.73205i	0.0429934	−	0.0744667i	0.886721	+	0.462304i	$0.152977\pi$
$542$		0			0
$543$	−3.00000	−	5.19615i	−0.128742	−	0.222988i
$544$		0			0
$545$	−28.0000			−1.19939
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		0			0
$549$	−5.00000	−	8.66025i	−0.213395	−	0.369611i
$550$		0			0
$551$	−12.0000	+	20.7846i	−0.511217	+	0.885454i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	10.0000	−	17.3205i	0.424476	−	0.735215i
$556$		0			0
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	0.991254	−	0.131965i	$-0.0421286\pi$
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	−0.609912	+	0.792469i	$0.708795\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.00000	+	10.3923i	0.252870	+	0.437983i	0.964315	−	0.264758i	$-0.0852922\pi$
$563$	6.00000	+	10.3923i	0.252870	+	0.437983i	−0.711445	+	0.702742i	$0.751959\pi$
$564$		0			0
$565$	−14.0000	+	24.2487i	−0.588984	+	1.02015i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	−0.990668	−	0.136295i	$-0.956481\pi$
$569$	−9.00000	+	15.5885i	−0.377300	+	0.653502i	0.613369	+	0.789797i	$0.289814\pi$
$570$		0			0
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	0.821138	−	0.570730i	$-0.193340\pi$
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	−0.904835	+	0.425762i	$0.860006\pi$
$572$		0			0
$573$	4.00000			0.167102
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	9.00000	+	15.5885i	0.374675	+	0.648956i	0.990278	−	0.139100i	$-0.0444210\pi$
$577$	9.00000	+	15.5885i	0.374675	+	0.648956i	−0.615603	+	0.788056i	$0.711088\pi$
$578$		0			0
$579$	9.00000	−	15.5885i	0.374027	−	0.647834i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	2.00000	+	3.46410i	0.0826898	+	0.143223i
$586$		0			0
$587$	−36.0000			−1.48588			−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000			−1.48588			−0.742940	+	0.669359i	$0.766569\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$		0			0
$591$	11.0000	+	19.0526i	0.452480	+	0.783718i
$592$		0			0
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	−0.797831	−	0.602881i	$-0.794019\pi$
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	0.921026	+	0.389501i	$0.127353\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	12.0000	−	20.7846i	0.491127	−	0.850657i
$598$		0			0
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	−0.954521	−	0.298143i	$-0.903633\pi$
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	0.219061	−	0.975711i	$-0.429701\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$	−11.0000	−	19.0526i	−0.447214	−	0.774597i
$606$		0			0
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	0.333842	+	0.942629i	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.00000	+	13.8564i	−0.323645	+	0.560570i
$612$		0			0
$613$	5.00000	+	8.66025i	0.201948	+	0.349784i	0.949156	−	0.314806i	$-0.101939\pi$
$613$	5.00000	+	8.66025i	0.201948	+	0.349784i	−0.747208	+	0.664590i	$0.768606\pi$
$614$		0			0
$615$	20.0000			0.806478
$616$		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
$619$	6.00000	+	10.3923i	0.241160	+	0.417702i	0.961045	−	0.276392i	$-0.0891387\pi$
$619$	6.00000	+	10.3923i	0.241160	+	0.417702i	−0.719885	+	0.694094i	$0.755805\pi$
$620$		0			0
$621$	−2.00000	+	3.46410i	−0.0802572	+	0.139010i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	60.0000			2.39236
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$		0			0
$633$	−6.00000	−	10.3923i	−0.238479	−	0.413057i
$634$		0			0
$635$	8.00000	−	13.8564i	0.317470	−	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−2.00000	+	3.46410i	−0.0791188	+	0.137038i
$640$		0			0
$641$	−13.0000	−	22.5167i	−0.513469	−	0.889355i	−0.999878	−	0.0156233i	$-0.995027\pi$
$641$	−13.0000	−	22.5167i	−0.513469	−	0.889355i	0.486409	−	0.873731i	$-0.338307\pi$
$642$		0			0
$643$	20.0000			0.788723			0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000			0.788723			0.394362	+	0.918955i	$0.370966\pi$
$644$		0			0
$645$	24.0000			0.944999
$646$		0			0
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	−0.987748	−	0.156059i	$-0.950121\pi$
$647$	−16.0000	−	27.7128i	−0.629025	−	1.08950i	0.358723	−	0.933444i	$-0.383212\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	−0.918736	−	0.394872i	$-0.870789\pi$
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	0.801337	+	0.598213i	$0.204122\pi$
$654$		0			0
$655$	12.0000	+	20.7846i	0.468879	+	0.812122i
$656$		0			0
$657$	−2.00000			−0.0780274
$658$		0			0
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	0.969442	−	0.245319i	$-0.0788928\pi$
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	−0.697174	+	0.716902i	$0.745559\pi$
$662$		0			0
$663$	−6.00000	+	10.3923i	−0.233021	+	0.403604i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	12.0000	−	20.7846i	0.464642	−	0.804783i
$668$		0			0
$669$	−8.00000	−	13.8564i	−0.309298	−	0.535720i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.0192450	−	0.0333333i
$676$		0			0
$677$	−19.0000	+	32.9090i	−0.730229	+	1.26479i	0.226556	+	0.973998i	$0.427253\pi$
$677$	−19.0000	+	32.9090i	−0.730229	+	1.26479i	−0.956785	+	0.290796i	$0.906080\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.00000	+	10.3923i	−0.229920	+	0.398234i
$682$		0			0
$683$	20.0000	+	34.6410i	0.765279	+	1.32550i	0.940099	+	0.340901i	$0.110732\pi$
$683$	20.0000	+	34.6410i	0.765279	+	1.32550i	−0.174820	+	0.984600i	$0.555934\pi$
$684$		0			0
$685$	28.0000			1.06983
$686$		0			0
$687$	6.00000			0.228914
$688$		0			0
$689$	−6.00000	−	10.3923i	−0.228582	−	0.395915i
$690$		0			0
$691$	18.0000	−	31.1769i	0.684752	−	1.18603i	−0.288762	−	0.957401i	$-0.593244\pi$
$691$	18.0000	−	31.1769i	0.684752	−	1.18603i	0.973515	−	0.228625i	$-0.0734229\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.00000	−	6.92820i	0.151729	−	0.262802i
$696$		0			0
$697$	30.0000	+	51.9615i	1.13633	+	1.96818i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$	38.0000			1.43524			0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000			1.43524			0.717620	+	0.696435i	$0.245231\pi$
$702$		0			0
$703$	20.0000	+	34.6410i	0.754314	+	1.30651i
$704$		0			0
$705$	−8.00000	+	13.8564i	−0.301297	+	0.521862i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.00000	+	5.19615i	−0.112667	+	0.195146i	−0.916845	−	0.399244i	$-0.869273\pi$
$709$	−3.00000	+	5.19615i	−0.112667	+	0.195146i	0.804178	+	0.594389i	$0.202606\pi$
$710$		0			0
$711$	−4.00000	−	6.92820i	−0.150012	−	0.259828i
$712$		0			0
$713$	−32.0000			−1.19841
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−6.00000	−	10.3923i	−0.224074	−	0.388108i
$718$		0			0
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	−0.975759	−	0.218850i	$-0.929769\pi$
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	0.677409	+	0.735607i	$0.263103\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	11.0000	−	19.0526i	0.409094	−	0.708572i
$724$		0			0
$725$	3.00000	+	5.19615i	0.111417	+	0.192980i
$726$		0			0
$727$	16.0000			0.593407			0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000			0.593407			0.296704	+	0.954970i	$0.404113\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	36.0000	+	62.3538i	1.33151	+	2.30624i
$732$		0			0
$733$	7.00000	−	12.1244i	0.258551	−	0.447823i	−0.707303	−	0.706910i	$-0.750088\pi$
$733$	7.00000	−	12.1244i	0.258551	−	0.447823i	0.965854	+	0.259087i	$0.0834217\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−14.0000	−	24.2487i	−0.514998	−	0.892003i	−0.999849	−	0.0174060i	$-0.994459\pi$
$739$	−14.0000	−	24.2487i	−0.514998	−	0.892003i	0.484850	−	0.874597i	$-0.338874\pi$
$740$		0			0
$741$	−8.00000			−0.293887
$742$		0			0
$743$	52.0000			1.90769			0.953847	−	0.300291i	$-0.0970839\pi$
$743$	52.0000			1.90769			0.953847	+	0.300291i	$0.0970839\pi$
$744$		0			0
$745$	6.00000	+	10.3923i	0.219823	+	0.380745i
$746$		0			0
$747$	2.00000	−	3.46410i	0.0731762	−	0.126745i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$	−10.0000	−	17.3205i	−0.364420	−	0.631194i
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.00000	−	5.19615i	0.108750	−	0.188360i	−0.806514	−	0.591215i	$-0.798649\pi$
$761$	3.00000	−	5.19615i	0.108750	−	0.188360i	0.915264	+	0.402854i	$0.131982\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−6.00000	+	10.3923i	−0.216930	+	0.375735i
$766$		0			0
$767$	4.00000	+	6.92820i	0.144432	+	0.250163i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	14.0000			0.504198
$772$		0			0
$773$	−19.0000	−	32.9090i	−0.683383	−	1.18365i	−0.973942	−	0.226796i	$-0.927175\pi$
$773$	−19.0000	−	32.9090i	−0.683383	−	1.18365i	0.290560	−	0.956857i	$-0.406159\pi$
$774$		0			0
$775$	4.00000	−	6.92820i	0.143684	−	0.248868i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.0000	+	34.6410i	−0.716574	+	1.24114i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$	−20.0000			−0.713831
$786$		0			0
$787$	10.0000	+	17.3205i	0.356462	+	0.617409i	0.987367	−	0.158450i	$-0.0506498\pi$
$787$	10.0000	+	17.3205i	0.356462	+	0.617409i	−0.630905	+	0.775860i	$0.717316\pi$
$788$		0			0
$789$	14.0000	−	24.2487i	0.498413	−	0.863277i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−10.0000	+	17.3205i	−0.355110	+	0.615069i
$794$		0			0
$795$	−6.00000	−	10.3923i	−0.212798	−	0.368577i
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$	3.00000	+	5.19615i	0.106000	+	0.183597i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−5.00000	+	8.66025i	−0.176008	+	0.304855i
$808$		0			0
$809$	−13.0000	−	22.5167i	−0.457056	−	0.791644i	0.541748	−	0.840541i	$-0.317763\pi$
$809$	−13.0000	−	22.5167i	−0.457056	−	0.791644i	−0.998804	+	0.0488972i	$0.984429\pi$
$810$		0			0
$811$	4.00000			0.140459			0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000			0.140459			0.0702295	+	0.997531i	$0.477627\pi$
$812$		0			0
$813$	−8.00000			−0.280572
$814$		0			0
$815$	−4.00000	−	6.92820i	−0.140114	−	0.242684i
$816$		0			0
$817$	−24.0000	+	41.5692i	−0.839654	+	1.45432i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.0000	−	36.3731i	0.732905	−	1.26943i	−0.222731	−	0.974880i	$-0.571497\pi$
$821$	21.0000	−	36.3731i	0.732905	−	1.26943i	0.955636	−	0.294549i	$-0.0951694\pi$
$822$		0			0
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	0.439961	−	0.898017i	$-0.354992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.0000			0.556375			0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000			0.556375			0.278187	+	0.960527i	$0.410266\pi$
$828$		0			0
$829$	7.00000	+	12.1244i	0.243120	+	0.421096i	0.961601	−	0.274450i	$-0.0884958\pi$
$829$	7.00000	+	12.1244i	0.243120	+	0.421096i	−0.718481	+	0.695546i	$0.755162\pi$
$830$		0			0
$831$	−5.00000	+	8.66025i	−0.173448	+	0.300421i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−16.0000	+	27.7128i	−0.553703	+	0.959041i
$836$		0			0
$837$	4.00000	+	6.92820i	0.138260	+	0.239474i
$838$		0			0
$839$	−48.0000			−1.65714			−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000			−1.65714			−0.828572	+	0.559883i	$0.810846\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	−15.0000	−	25.9808i	−0.516627	−	0.894825i
$844$		0			0
$845$	−9.00000	+	15.5885i	−0.309609	+	0.536259i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	6.00000	−	10.3923i	0.205919	−	0.356663i
$850$		0			0
$851$	−20.0000	−	34.6410i	−0.685591	−	1.18748i
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$	−8.00000			−0.273594
$856$		0			0
$857$	19.0000	+	32.9090i	0.649028	+	1.12415i	0.983355	+	0.181692i	$0.0581574\pi$
$857$	19.0000	+	32.9090i	0.649028	+	1.12415i	−0.334328	+	0.942457i	$0.608509\pi$
$858$		0			0
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	0.196887	+	0.980426i	$0.436917\pi$
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	−0.947518	+	0.319704i	$0.896417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.00000	+	3.46410i	−0.0680808	+	0.117919i	−0.898056	−	0.439880i	$-0.855021\pi$
$863$	−2.00000	+	3.46410i	−0.0680808	+	0.117919i	0.829976	+	0.557800i	$0.188354\pi$
$864$		0			0
$865$	−2.00000	−	3.46410i	−0.0680020	−	0.117783i
$866$		0			0
$867$	−19.0000			−0.645274
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−12.0000	−	20.7846i	−0.406604	−	0.704260i
$872$		0			0
$873$	5.00000	−	8.66025i	0.169224	−	0.293105i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.00000	−	1.73205i	0.0337676	−	0.0584872i	−0.848648	−	0.528958i	$-0.822583\pi$
$877$	1.00000	−	1.73205i	0.0337676	−	0.0584872i	0.882415	+	0.470471i	$0.155916\pi$
$878$		0			0
$879$	11.0000	+	19.0526i	0.371021	+	0.642627i
$880$		0			0
$881$	42.0000			1.41502			0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000			1.41502			0.707508	+	0.706705i	$0.249819\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$	4.00000	+	6.92820i	0.134459	+	0.232889i
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.0000	−	27.7128i	−0.535420	−	0.927374i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	8.00000			0.267112
$898$		0			0
$899$	−24.0000	−	41.5692i	−0.800445	−	1.38641i
$900$		0			0
$901$	18.0000	−	31.1769i	0.599667	−	1.03865i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.00000	+	10.3923i	−0.199447	+	0.345452i
$906$		0			0
$907$	26.0000	+	45.0333i	0.863316	+	1.49531i	0.868710	+	0.495321i	$0.164950\pi$
$907$	26.0000	+	45.0333i	0.863316	+	1.49531i	−0.00539395	+	0.999985i	$0.501717\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		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
$913$		0			0
$914$		0			0
$915$	−10.0000	+	17.3205i	−0.330590	+	0.572598i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−4.00000	+	6.92820i	−0.131948	+	0.228540i	−0.924427	−	0.381358i	$-0.875456\pi$
$919$	−4.00000	+	6.92820i	−0.131948	+	0.228540i	0.792480	+	0.609898i	$0.208790\pi$
$920$		0			0
$921$	2.00000	+	3.46410i	0.0659022	+	0.114146i
$922$		0			0
$923$	8.00000			0.263323
$924$		0			0
$925$	10.0000			0.328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−10.0000			−0.326686			−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000			−0.326686			−0.163343	+	0.986569i	$0.552228\pi$
$938$		0			0
$939$	−6.00000			−0.195803
$940$		0			0
$941$	−11.0000	−	19.0526i	−0.358590	−	0.621096i	0.629136	−	0.777295i	$-0.283409\pi$
$941$	−11.0000	−	19.0526i	−0.358590	−	0.621096i	−0.987725	+	0.156200i	$0.950076\pi$
$942$		0			0
$943$	20.0000	−	34.6410i	0.651290	−	1.12807i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.0000	−	20.7846i	0.389948	−	0.675409i	−0.602494	−	0.798123i	$-0.705826\pi$
$947$	12.0000	−	20.7846i	0.389948	−	0.675409i	0.992442	+	0.122714i	$0.0391598\pi$
$948$		0			0
$949$	2.00000	+	3.46410i	0.0649227	+	0.112449i
$950$		0			0
$951$	2.00000			0.0648544
$952$		0			0
$953$	−38.0000			−1.23094			−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000			−1.23094			−0.615470	+	0.788160i	$0.711034\pi$
$954$		0			0
$955$	−4.00000	−	6.92820i	−0.129437	−	0.224191i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$	−8.00000	−	13.8564i	−0.257796	−	0.446516i
$964$		0			0
$965$	−36.0000			−1.15888
$966$		0			0
$967$	−48.0000			−1.54358			−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000			−1.54358			−0.771788	+	0.635880i	$0.780637\pi$
$968$		0			0
$969$	−12.0000	−	20.7846i	−0.385496	−	0.667698i
$970$		0			0
$971$	14.0000	−	24.2487i	0.449281	−	0.778178i	−0.549058	−	0.835784i	$-0.685013\pi$
$971$	14.0000	−	24.2487i	0.449281	−	0.778178i	0.998339	+	0.0576061i	$0.0183467\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−1.00000	+	1.73205i	−0.0320256	+	0.0554700i
$976$		0			0
$977$	11.0000	+	19.0526i	0.351921	+	0.609545i	0.986586	−	0.163242i	$-0.0521952\pi$
$977$	11.0000	+	19.0526i	0.351921	+	0.609545i	−0.634665	+	0.772787i	$0.718862\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	14.0000			0.446986
$982$		0			0
$983$	−20.0000	−	34.6410i	−0.637901	−	1.10488i	−0.985893	−	0.167379i	$-0.946470\pi$
$983$	−20.0000	−	34.6410i	−0.637901	−	1.10488i	0.347992	−	0.937498i	$-0.386864\pi$
$984$		0			0
$985$	22.0000	−	38.1051i	0.700978	−	1.21413i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000	−	41.5692i	0.763156	−	1.32182i
$990$		0			0
$991$	24.0000	+	41.5692i	0.762385	+	1.32049i	0.941618	+	0.336683i	$0.109305\pi$
$991$	24.0000	+	41.5692i	0.762385	+	1.32049i	−0.179233	+	0.983807i	$0.557362\pi$
$992$		0			0
$993$	20.0000			0.634681
$994$		0			0
$995$	−48.0000			−1.52170
$996$		0			0
$997$	−1.00000	−	1.73205i	−0.0316703	−	0.0548546i	0.849756	−	0.527176i	$-0.176749\pi$
$997$	−1.00000	−	1.73205i	−0.0316703	−	0.0548546i	−0.881426	+	0.472322i	$0.843416\pi$
$998$		0			0
$999$	−5.00000	+	8.66025i	−0.158193	+	0.273998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.q.j.961.1		2
3.2	odd	2		3528.2.s.h.3313.1		2
4.3	odd	2		2352.2.q.j.961.1		2
7.2	even	3		1176.2.a.a.1.1		1
7.3	odd	6		1176.2.q.b.361.1		2
7.4	even	3	inner	1176.2.q.j.361.1		2
7.5	odd	6		168.2.a.b.1.1	✓	1
7.6	odd	2		1176.2.q.b.961.1		2
21.2	odd	6		3528.2.a.w.1.1		1
21.5	even	6		504.2.a.b.1.1		1
21.11	odd	6		3528.2.s.h.361.1		2
21.17	even	6		3528.2.s.v.361.1		2
21.20	even	2		3528.2.s.v.3313.1		2
28.3	even	6		2352.2.q.o.1537.1		2
28.11	odd	6		2352.2.q.j.1537.1		2
28.19	even	6		336.2.a.c.1.1		1
28.23	odd	6		2352.2.a.q.1.1		1
28.27	even	2		2352.2.q.o.961.1		2
35.12	even	12		4200.2.t.m.1849.1		2
35.19	odd	6		4200.2.a.i.1.1		1
35.33	even	12		4200.2.t.m.1849.2		2
56.5	odd	6		1344.2.a.c.1.1		1
56.19	even	6		1344.2.a.n.1.1		1
56.37	even	6		9408.2.a.cy.1.1		1
56.51	odd	6		9408.2.a.bc.1.1		1
84.23	even	6		7056.2.a.br.1.1		1
84.47	odd	6		1008.2.a.e.1.1		1
112.5	odd	12		5376.2.c.bd.2689.1		2
112.19	even	12		5376.2.c.f.2689.1		2
112.61	odd	12		5376.2.c.bd.2689.2		2
112.75	even	12		5376.2.c.f.2689.2		2
140.19	even	6		8400.2.a.bx.1.1		1
168.5	even	6		4032.2.a.be.1.1		1
168.131	odd	6		4032.2.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.a.b.1.1	✓	1	7.5	odd	6
336.2.a.c.1.1		1	28.19	even	6
504.2.a.b.1.1		1	21.5	even	6
1008.2.a.e.1.1		1	84.47	odd	6
1176.2.a.a.1.1		1	7.2	even	3
1176.2.q.b.361.1		2	7.3	odd	6
1176.2.q.b.961.1		2	7.6	odd	2
1176.2.q.j.361.1		2	7.4	even	3	inner
1176.2.q.j.961.1		2	1.1	even	1	trivial
1344.2.a.c.1.1		1	56.5	odd	6
1344.2.a.n.1.1		1	56.19	even	6
2352.2.a.q.1.1		1	28.23	odd	6
2352.2.q.j.961.1		2	4.3	odd	2
2352.2.q.j.1537.1		2	28.11	odd	6
2352.2.q.o.961.1		2	28.27	even	2
2352.2.q.o.1537.1		2	28.3	even	6
3528.2.a.w.1.1		1	21.2	odd	6
3528.2.s.h.361.1		2	21.11	odd	6
3528.2.s.h.3313.1		2	3.2	odd	2
3528.2.s.v.361.1		2	21.17	even	6
3528.2.s.v.3313.1		2	21.20	even	2
4032.2.a.be.1.1		1	168.5	even	6
4032.2.a.bj.1.1		1	168.131	odd	6
4200.2.a.i.1.1		1	35.19	odd	6
4200.2.t.m.1849.1		2	35.12	even	12
4200.2.t.m.1849.2		2	35.33	even	12
5376.2.c.f.2689.1		2	112.19	even	12
5376.2.c.f.2689.2		2	112.75	even	12
5376.2.c.bd.2689.1		2	112.5	odd	12
5376.2.c.bd.2689.2		2	112.61	odd	12
7056.2.a.br.1.1		1	84.23	even	6
8400.2.a.bx.1.1		1	140.19	even	6
9408.2.a.bc.1.1		1	56.51	odd	6
9408.2.a.cy.1.1		1	56.37	even	6

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{3} +(1.00000 - 1.73205i) q^{5} +(-0.500000 + 0.866025i) q^{9} +2.00000 q^{13} +2.00000 q^{15} +(3.00000 + 5.19615i) q^{17} +(-2.00000 + 3.46410i) q^{19} +(2.00000 - 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} -1.00000 q^{27} +6.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(5.00000 - 8.66025i) q^{37} +(1.00000 + 1.73205i) q^{39} +10.0000 q^{41} +12.0000 q^{43} +(1.00000 + 1.73205i) q^{45} +(-4.00000 + 6.92820i) q^{47} +(-3.00000 + 5.19615i) q^{51} +(-3.00000 - 5.19615i) q^{53} -4.00000 q^{57} +(2.00000 + 3.46410i) q^{59} +(-5.00000 + 8.66025i) q^{61} +(2.00000 - 3.46410i) q^{65} +(-6.00000 - 10.3923i) q^{67} +4.00000 q^{69} +4.00000 q^{71} +(1.00000 + 1.73205i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(-4.00000 + 6.92820i) q^{79} +(-0.500000 - 0.866025i) q^{81} -4.00000 q^{83} +12.0000 q^{85} +(3.00000 + 5.19615i) q^{87} +(3.00000 - 5.19615i) q^{89} +(4.00000 - 6.92820i) q^{93} +(4.00000 + 6.92820i) q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} - q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 - q^9	\( 2 q + q^{3} + 2 q^{5} - q^{9} + 4 q^{13} + 4 q^{15} + 6 q^{17} - 4 q^{19} + 4 q^{23} + q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{31} + 10 q^{37} + 2 q^{39} + 20 q^{41} + 24 q^{43} + 2 q^{45} - 8 q^{47} - 6 q^{51} - 6 q^{53} - 8 q^{57} + 4 q^{59} - 10 q^{61} + 4 q^{65} - 12 q^{67} + 8 q^{69} + 8 q^{71} + 2 q^{73} - q^{75} - 8 q^{79} - q^{81} - 8 q^{83} + 24 q^{85} + 6 q^{87} + 6 q^{89} + 8 q^{93} + 8 q^{95} - 20 q^{97}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 - q^9 + 4 * q^13 + 4 * q^15 + 6 * q^17 - 4 * q^19 + 4 * q^23 + q^25 - 2 * q^27 + 12 * q^29 - 8 * q^31 + 10 * q^37 + 2 * q^39 + 20 * q^41 + 24 * q^43 + 2 * q^45 - 8 * q^47 - 6 * q^51 - 6 * q^53 - 8 * q^57 + 4 * q^59 - 10 * q^61 + 4 * q^65 - 12 * q^67 + 8 * q^69 + 8 * q^71 + 2 * q^73 - q^75 - 8 * q^79 - q^81 - 8 * q^83 + 24 * q^85 + 6 * q^87 + 6 * q^89 + 8 * q^93 + 8 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more