LMFDB - Embedded newform 1176.3.d.c.785.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(785,1176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1176.785"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1176.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,-14,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1176.785
Dual form			1176.3.d.c.785.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 2.82843i) q^{3} +4.24264i q^{5} +(-7.00000 + 5.65685i) q^{9} +4.24264i q^{11} -4.00000 q^{13} +(-12.0000 + 4.24264i) q^{15} -7.07107i q^{17} +16.0000 q^{19} +41.0122i q^{23} +7.00000 q^{25} +(-23.0000 - 14.1421i) q^{27} +22.6274i q^{29} -50.0000 q^{31} +(-12.0000 + 4.24264i) q^{33} -48.0000 q^{37} +(-4.00000 - 11.3137i) q^{39} -4.24264i q^{41} -44.0000 q^{43} +(-24.0000 - 29.6985i) q^{45} -28.2843i q^{47} +(20.0000 - 7.07107i) q^{51} -16.9706i q^{53} -18.0000 q^{55} +(16.0000 + 45.2548i) q^{57} -33.9411i q^{59} +68.0000 q^{61} -16.9706i q^{65} +44.0000 q^{67} +(-116.000 + 41.0122i) q^{69} -4.24264i q^{71} -128.000 q^{73} +(7.00000 + 19.7990i) q^{75} +80.0000 q^{79} +(17.0000 - 79.1960i) q^{81} -124.451i q^{83} +30.0000 q^{85} +(-64.0000 + 22.6274i) q^{87} +162.635i q^{89} +(-50.0000 - 141.421i) q^{93} +67.8823i q^{95} -136.000 q^{97} +(-24.0000 - 29.6985i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 14 q^{9} - 8 q^{13} - 24 q^{15} + 32 q^{19} + 14 q^{25} - 46 q^{27} - 100 q^{31} - 24 q^{33} - 96 q^{37} - 8 q^{39} - 88 q^{43} - 48 q^{45} + 40 q^{51} - 36 q^{55} + 32 q^{57} + 136 q^{61}+ \cdots - 48 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 14 * q^9 - 8 * q^13 - 24 * q^15 + 32 * q^19 + 14 * q^25 - 46 * q^27 - 100 * q^31 - 24 * q^33 - 96 * q^37 - 8 * q^39 - 88 * q^43 - 48 * q^45 + 40 * q^51 - 36 * q^55 + 32 * q^57 + 136 * q^61 + 88 * q^67 - 232 * q^69 - 256 * q^73 + 14 * q^75 + 160 * q^79 + 34 * q^81 + 60 * q^85 - 128 * q^87 - 100 * q^93 - 272 * q^97 - 48 * q^99

Character values

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

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	1.00000	+	2.82843i	0.333333	+	0.942809i
$3$	1.00000	+	2.82843i	0.333333	+	0.942809i
$4$		0			0
$5$			4.24264i			0.848528i	0.905539	+	0.424264i	$0.139467\pi$
$5$			4.24264i			0.848528i	−0.905539	+	0.424264i	$0.860533\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$10$		0			0
$11$			4.24264i			0.385695i	0.981229	+	0.192847i	$0.0617722\pi$
$11$			4.24264i			0.385695i	−0.981229	+	0.192847i	$0.938228\pi$
$12$		0			0
$13$	−4.00000			−0.307692			−0.153846	−	0.988095i	$-0.549166\pi$
$13$	−4.00000			−0.307692			−0.153846	+	0.988095i	$0.549166\pi$
$14$		0			0
$15$	−12.0000	+	4.24264i	−0.800000	+	0.282843i
$16$		0			0
$17$		−	7.07107i		−	0.415945i	−0.978135	−	0.207973i	$-0.933314\pi$
$17$		−	7.07107i		−	0.415945i	0.978135	−	0.207973i	$-0.0666865\pi$
$18$		0			0
$19$	16.0000			0.842105			0.421053	−	0.907036i	$-0.361661\pi$
$19$	16.0000			0.842105			0.421053	+	0.907036i	$0.361661\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			41.0122i			1.78314i	0.452884	+	0.891569i	$0.350395\pi$
$23$			41.0122i			1.78314i	−0.452884	+	0.891569i	$0.649605\pi$
$24$		0			0
$25$	7.00000			0.280000
$26$		0			0
$27$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$28$		0			0
$29$			22.6274i			0.780256i	0.920761	+	0.390128i	$0.127569\pi$
$29$			22.6274i			0.780256i	−0.920761	+	0.390128i	$0.872431\pi$
$30$		0			0
$31$	−50.0000			−1.61290			−0.806452	−	0.591300i	$-0.798615\pi$
$31$	−50.0000			−1.61290			−0.806452	+	0.591300i	$0.798615\pi$
$32$		0			0
$33$	−12.0000	+	4.24264i	−0.363636	+	0.128565i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−48.0000			−1.29730			−0.648649	−	0.761088i	$-0.724666\pi$
$37$	−48.0000			−1.29730			−0.648649	+	0.761088i	$0.724666\pi$
$38$		0			0
$39$	−4.00000	−	11.3137i	−0.102564	−	0.290095i
$40$		0			0
$41$		−	4.24264i		−	0.103479i	−0.998661	−	0.0517395i	$-0.983523\pi$
$41$		−	4.24264i		−	0.103479i	0.998661	−	0.0517395i	$-0.0164766\pi$
$42$		0			0
$43$	−44.0000			−1.02326			−0.511628	−	0.859207i	$-0.670957\pi$
$43$	−44.0000			−1.02326			−0.511628	+	0.859207i	$0.670957\pi$
$44$		0			0
$45$	−24.0000	−	29.6985i	−0.533333	−	0.659966i
$46$		0			0
$47$		−	28.2843i		−	0.601793i	−0.953657	−	0.300897i	$-0.902714\pi$
$47$		−	28.2843i		−	0.601793i	0.953657	−	0.300897i	$-0.0972859\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	20.0000	−	7.07107i	0.392157	−	0.138648i
$52$		0			0
$53$		−	16.9706i		−	0.320199i	−0.987101	−	0.160100i	$-0.948818\pi$
$53$		−	16.9706i		−	0.320199i	0.987101	−	0.160100i	$-0.0511816\pi$
$54$		0			0
$55$	−18.0000			−0.327273
$56$		0			0
$57$	16.0000	+	45.2548i	0.280702	+	0.793944i
$58$		0			0
$59$		−	33.9411i		−	0.575273i	−0.957740	−	0.287637i	$-0.907130\pi$
$59$		−	33.9411i		−	0.575273i	0.957740	−	0.287637i	$-0.0928695\pi$
$60$		0			0
$61$	68.0000			1.11475			0.557377	−	0.830259i	$-0.311808\pi$
$61$	68.0000			1.11475			0.557377	+	0.830259i	$0.311808\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	16.9706i		−	0.261086i
$66$		0			0
$67$	44.0000			0.656716			0.328358	−	0.944553i	$-0.393505\pi$
$67$	44.0000			0.656716			0.328358	+	0.944553i	$0.393505\pi$
$68$		0			0
$69$	−116.000	+	41.0122i	−1.68116	+	0.594380i
$70$		0			0
$71$		−	4.24264i		−	0.0597555i	−0.999554	−	0.0298778i	$-0.990488\pi$
$71$		−	4.24264i		−	0.0597555i	0.999554	−	0.0298778i	$-0.00951180\pi$
$72$		0			0
$73$	−128.000			−1.75342			−0.876712	−	0.481015i	$-0.840268\pi$
$73$	−128.000			−1.75342			−0.876712	+	0.481015i	$0.840268\pi$
$74$		0			0
$75$	7.00000	+	19.7990i	0.0933333	+	0.263987i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	80.0000			1.01266			0.506329	−	0.862340i	$-0.331002\pi$
$79$	80.0000			1.01266			0.506329	+	0.862340i	$0.331002\pi$
$80$		0			0
$81$	17.0000	−	79.1960i	0.209877	−	0.977728i
$82$		0			0
$83$		−	124.451i		−	1.49941i	−0.661774	−	0.749704i	$-0.730196\pi$
$83$		−	124.451i		−	1.49941i	0.661774	−	0.749704i	$-0.269804\pi$
$84$		0			0
$85$	30.0000			0.352941
$86$		0			0
$87$	−64.0000	+	22.6274i	−0.735632	+	0.260085i
$88$		0			0
$89$			162.635i			1.82735i	0.406440	+	0.913677i	$0.366770\pi$
$89$			162.635i			1.82735i	−0.406440	+	0.913677i	$0.633230\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−50.0000	−	141.421i	−0.537634	−	1.52066i
$94$		0			0
$95$			67.8823i			0.714550i
$96$		0			0
$97$	−136.000			−1.40206			−0.701031	−	0.713131i	$-0.747277\pi$
$97$	−136.000			−1.40206			−0.701031	+	0.713131i	$0.747277\pi$
$98$		0			0
$99$	−24.0000	−	29.6985i	−0.242424	−	0.299985i
$100$		0			0
$101$		−	41.0122i		−	0.406061i	−0.979172	−	0.203031i	$-0.934921\pi$
$101$		−	41.0122i		−	0.406061i	0.979172	−	0.203031i	$-0.0650791\pi$
$102$		0			0
$103$	146.000			1.41748			0.708738	−	0.705472i	$-0.249265\pi$
$103$	146.000			1.41748			0.708738	+	0.705472i	$0.249265\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			18.3848i			0.171820i	0.996303	+	0.0859102i	$0.0273798\pi$
$107$			18.3848i			0.171820i	−0.996303	+	0.0859102i	$0.972620\pi$
$108$		0			0
$109$	−158.000			−1.44954			−0.724771	−	0.688990i	$-0.758054\pi$
$109$	−158.000			−1.44954			−0.724771	+	0.688990i	$0.758054\pi$
$110$		0			0
$111$	−48.0000	−	135.765i	−0.432432	−	1.22310i
$112$		0			0
$113$		−	130.108i		−	1.15140i	−0.817663	−	0.575698i	$-0.804731\pi$
$113$		−	130.108i		−	1.15140i	0.817663	−	0.575698i	$-0.195269\pi$
$114$		0			0
$115$	−174.000			−1.51304
$116$		0			0
$117$	28.0000	−	22.6274i	0.239316	−	0.193397i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	103.000			0.851240
$122$		0			0
$123$	12.0000	−	4.24264i	0.0975610	−	0.0344930i
$124$		0			0
$125$			135.765i			1.08612i
$126$		0			0
$127$	176.000			1.38583			0.692913	−	0.721021i	$-0.256327\pi$
$127$	176.000			1.38583			0.692913	+	0.721021i	$0.256327\pi$
$128$		0			0
$129$	−44.0000	−	124.451i	−0.341085	−	0.964735i
$130$		0			0
$131$		−	192.333i		−	1.46819i	−0.679046	−	0.734096i	$-0.737606\pi$
$131$		−	192.333i		−	1.46819i	0.679046	−	0.734096i	$-0.262394\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	60.0000	−	97.5807i	0.444444	−	0.722820i
$136$		0			0
$137$			181.019i			1.32131i	0.750690	+	0.660655i	$0.229721\pi$
$137$			181.019i			1.32131i	−0.750690	+	0.660655i	$0.770279\pi$
$138$		0			0
$139$	−78.0000			−0.561151			−0.280576	−	0.959832i	$-0.590525\pi$
$139$	−78.0000			−0.561151			−0.280576	+	0.959832i	$0.590525\pi$
$140$		0			0
$141$	80.0000	−	28.2843i	0.567376	−	0.200598i
$142$		0			0
$143$		−	16.9706i		−	0.118675i
$144$		0			0
$145$	−96.0000			−0.662069
$146$		0			0
$147$		0			0
$148$		0			0
$149$			152.735i			1.02507i	0.858667	+	0.512534i	$0.171293\pi$
$149$			152.735i			1.02507i	−0.858667	+	0.512534i	$0.828707\pi$
$150$		0			0
$151$	64.0000			0.423841			0.211921	−	0.977287i	$-0.432028\pi$
$151$	64.0000			0.423841			0.211921	+	0.977287i	$0.432028\pi$
$152$		0			0
$153$	40.0000	+	49.4975i	0.261438	+	0.323513i
$154$		0			0
$155$		−	212.132i		−	1.36859i
$156$		0			0
$157$	−100.000			−0.636943			−0.318471	−	0.947932i	$-0.603169\pi$
$157$	−100.000			−0.636943			−0.318471	+	0.947932i	$0.603169\pi$
$158$		0			0
$159$	48.0000	−	16.9706i	0.301887	−	0.106733i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−180.000			−1.10429			−0.552147	−	0.833747i	$-0.686191\pi$
$163$	−180.000			−1.10429			−0.552147	+	0.833747i	$0.686191\pi$
$164$		0			0
$165$	−18.0000	−	50.9117i	−0.109091	−	0.308556i
$166$		0			0
$167$		−	147.078i		−	0.880708i	−0.897824	−	0.440354i	$-0.854853\pi$
$167$		−	147.078i		−	0.880708i	0.897824	−	0.440354i	$-0.145147\pi$
$168$		0			0
$169$	−153.000			−0.905325
$170$		0			0
$171$	−112.000	+	90.5097i	−0.654971	+	0.529296i
$172$		0			0
$173$			196.576i			1.13628i	0.822933	+	0.568138i	$0.192336\pi$
$173$			196.576i			1.13628i	−0.822933	+	0.568138i	$0.807664\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	96.0000	−	33.9411i	0.542373	−	0.191758i
$178$		0			0
$179$			233.345i			1.30360i	0.758389	+	0.651802i	$0.225987\pi$
$179$			233.345i			1.30360i	−0.758389	+	0.651802i	$0.774013\pi$
$180$		0			0
$181$	−204.000			−1.12707			−0.563536	−	0.826092i	$-0.690559\pi$
$181$	−204.000			−1.12707			−0.563536	+	0.826092i	$0.690559\pi$
$182$		0			0
$183$	68.0000	+	192.333i	0.371585	+	1.05100i
$184$		0			0
$185$		−	203.647i		−	1.10079i
$186$		0			0
$187$	30.0000			0.160428
$188$		0			0
$189$		0			0
$190$		0			0
$191$			241.831i			1.26613i	0.774099	+	0.633064i	$0.218203\pi$
$191$			241.831i			1.26613i	−0.774099	+	0.633064i	$0.781797\pi$
$192$		0			0
$193$	16.0000			0.0829016			0.0414508	−	0.999141i	$-0.486802\pi$
$193$	16.0000			0.0829016			0.0414508	+	0.999141i	$0.486802\pi$
$194$		0			0
$195$	48.0000	−	16.9706i	0.246154	−	0.0870285i
$196$		0			0
$197$		−	33.9411i		−	0.172290i	−0.996283	−	0.0861450i	$-0.972545\pi$
$197$		−	33.9411i		−	0.172290i	0.996283	−	0.0861450i	$-0.0274548\pi$
$198$		0			0
$199$	−16.0000			−0.0804020			−0.0402010	−	0.999192i	$-0.512800\pi$
$199$	−16.0000			−0.0804020			−0.0402010	+	0.999192i	$0.512800\pi$
$200$		0			0
$201$	44.0000	+	124.451i	0.218905	+	0.619158i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	18.0000			0.0878049
$206$		0			0
$207$	−232.000	−	287.085i	−1.12077	−	1.38689i
$208$		0			0
$209$			67.8823i			0.324795i
$210$		0			0
$211$	20.0000			0.0947867			0.0473934	−	0.998876i	$-0.484909\pi$
$211$	20.0000			0.0947867			0.0473934	+	0.998876i	$0.484909\pi$
$212$		0			0
$213$	12.0000	−	4.24264i	0.0563380	−	0.0199185i
$214$		0			0
$215$		−	186.676i		−	0.868261i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−128.000	−	362.039i	−0.584475	−	1.65314i
$220$		0			0
$221$			28.2843i			0.127983i
$222$		0			0
$223$	16.0000			0.0717489			0.0358744	−	0.999356i	$-0.488578\pi$
$223$	16.0000			0.0717489			0.0358744	+	0.999356i	$0.488578\pi$
$224$		0			0
$225$	−49.0000	+	39.5980i	−0.217778	+	0.175991i
$226$		0			0
$227$			175.362i			0.772522i	0.922390	+	0.386261i	$0.126234\pi$
$227$			175.362i			0.772522i	−0.922390	+	0.386261i	$0.873766\pi$
$228$		0			0
$229$	−68.0000			−0.296943			−0.148472	−	0.988917i	$-0.547435\pi$
$229$	−68.0000			−0.296943			−0.148472	+	0.988917i	$0.547435\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			356.382i			1.52954i	0.644306	+	0.764768i	$0.277146\pi$
$233$			356.382i			1.52954i	−0.644306	+	0.764768i	$0.722854\pi$
$234$		0			0
$235$	120.000			0.510638
$236$		0			0
$237$	80.0000	+	226.274i	0.337553	+	0.954743i
$238$		0			0
$239$			210.718i			0.881665i	0.897590	+	0.440832i	$0.145317\pi$
$239$			210.718i			0.881665i	−0.897590	+	0.440832i	$0.854683\pi$
$240$		0			0
$241$	432.000			1.79253			0.896266	−	0.443518i	$-0.146270\pi$
$241$	432.000			1.79253			0.896266	+	0.443518i	$0.146270\pi$
$242$		0			0
$243$	241.000	−	31.1127i	0.991770	−	0.128036i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−64.0000			−0.259109
$248$		0			0
$249$	352.000	−	124.451i	1.41365	−	0.499802i
$250$		0			0
$251$		−	288.500i		−	1.14940i	−0.818364	−	0.574700i	$-0.805119\pi$
$251$		−	288.500i		−	1.14940i	0.818364	−	0.574700i	$-0.194881\pi$
$252$		0			0
$253$	−174.000			−0.687747
$254$		0			0
$255$	30.0000	+	84.8528i	0.117647	+	0.332756i
$256$		0			0
$257$			165.463i			0.643825i	0.946769	+	0.321912i	$0.104326\pi$
$257$			165.463i			0.643825i	−0.946769	+	0.321912i	$0.895674\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−128.000	−	158.392i	−0.490421	−	0.606866i
$262$		0			0
$263$		−	131.522i		−	0.500083i	−0.968235	−	0.250042i	$-0.919556\pi$
$263$		−	131.522i		−	0.500083i	0.968235	−	0.250042i	$-0.0804443\pi$
$264$		0			0
$265$	72.0000			0.271698
$266$		0			0
$267$	−460.000	+	162.635i	−1.72285	+	0.609118i
$268$		0			0
$269$			301.227i			1.11980i	0.828559	+	0.559902i	$0.189161\pi$
$269$			301.227i			1.11980i	−0.828559	+	0.559902i	$0.810839\pi$
$270$		0			0
$271$	−322.000			−1.18819			−0.594096	−	0.804394i	$-0.702490\pi$
$271$	−322.000			−1.18819			−0.594096	+	0.804394i	$0.702490\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			29.6985i			0.107994i
$276$		0			0
$277$	80.0000			0.288809			0.144404	−	0.989519i	$-0.453873\pi$
$277$	80.0000			0.288809			0.144404	+	0.989519i	$0.453873\pi$
$278$		0			0
$279$	350.000	−	282.843i	1.25448	−	1.01377i
$280$		0			0
$281$			164.049i			0.583803i	0.956448	+	0.291902i	$0.0942880\pi$
$281$			164.049i			0.583803i	−0.956448	+	0.291902i	$0.905712\pi$
$282$		0			0
$283$	304.000			1.07420			0.537102	−	0.843517i	$-0.319519\pi$
$283$	304.000			1.07420			0.537102	+	0.843517i	$0.319519\pi$
$284$		0			0
$285$	−192.000	+	67.8823i	−0.673684	+	0.238183i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	239.000			0.826990
$290$		0			0
$291$	−136.000	−	384.666i	−0.467354	−	1.32188i
$292$		0			0
$293$			547.301i			1.86792i	0.357377	+	0.933960i	$0.383671\pi$
$293$			547.301i			1.86792i	−0.357377	+	0.933960i	$0.616329\pi$
$294$		0			0
$295$	144.000			0.488136
$296$		0			0
$297$	60.0000	−	97.5807i	0.202020	−	0.328555i
$298$		0			0
$299$		−	164.049i		−	0.548658i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	116.000	−	41.0122i	0.382838	−	0.135354i
$304$		0			0
$305$			288.500i			0.945900i
$306$		0			0
$307$	432.000			1.40717			0.703583	−	0.710613i	$-0.251582\pi$
$307$	432.000			1.40717			0.703583	+	0.710613i	$0.251582\pi$
$308$		0			0
$309$	146.000	+	412.950i	0.472492	+	1.33641i
$310$		0			0
$311$			96.1665i			0.309217i	0.987976	+	0.154609i	$0.0494116\pi$
$311$			96.1665i			0.309217i	−0.987976	+	0.154609i	$0.950588\pi$
$312$		0			0
$313$	−112.000			−0.357827			−0.178914	−	0.983865i	$-0.557258\pi$
$313$	−112.000			−0.357827			−0.178914	+	0.983865i	$0.557258\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			322.441i			1.01716i	0.861014	+	0.508582i	$0.169830\pi$
$317$			322.441i			1.01716i	−0.861014	+	0.508582i	$0.830170\pi$
$318$		0			0
$319$	−96.0000			−0.300940
$320$		0			0
$321$	−52.0000	+	18.3848i	−0.161994	+	0.0572734i
$322$		0			0
$323$		−	113.137i		−	0.350270i
$324$		0			0
$325$	−28.0000			−0.0861538
$326$		0			0
$327$	−158.000	−	446.891i	−0.483180	−	1.36664i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−476.000			−1.43807			−0.719033	−	0.694976i	$-0.755415\pi$
$331$	−476.000			−1.43807			−0.719033	+	0.694976i	$0.755415\pi$
$332$		0			0
$333$	336.000	−	271.529i	1.00901	−	0.815402i
$334$		0			0
$335$			186.676i			0.557242i
$336$		0			0
$337$	−174.000			−0.516320			−0.258160	−	0.966102i	$-0.583116\pi$
$337$	−174.000			−0.516320			−0.258160	+	0.966102i	$0.583116\pi$
$338$		0			0
$339$	368.000	−	130.108i	1.08555	−	0.383798i
$340$		0			0
$341$		−	212.132i		−	0.622088i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−174.000	−	492.146i	−0.504348	−	1.42651i
$346$		0			0
$347$			63.6396i			0.183399i	0.995787	+	0.0916997i	$0.0292300\pi$
$347$			63.6396i			0.183399i	−0.995787	+	0.0916997i	$0.970770\pi$
$348$		0			0
$349$	132.000			0.378223			0.189112	−	0.981956i	$-0.439439\pi$
$349$	132.000			0.378223			0.189112	+	0.981956i	$0.439439\pi$
$350$		0			0
$351$	92.0000	+	56.5685i	0.262108	+	0.161164i
$352$		0			0
$353$		−	490.732i		−	1.39018i	−0.718925	−	0.695088i	$-0.755365\pi$
$353$		−	490.732i		−	1.39018i	0.718925	−	0.695088i	$-0.244635\pi$
$354$		0			0
$355$	18.0000			0.0507042
$356$		0			0
$357$		0			0
$358$		0			0
$359$			312.541i			0.870588i	0.900288	+	0.435294i	$0.143356\pi$
$359$			312.541i			0.870588i	−0.900288	+	0.435294i	$0.856644\pi$
$360$		0			0
$361$	−105.000			−0.290859
$362$		0			0
$363$	103.000	+	291.328i	0.283747	+	0.802556i
$364$		0			0
$365$		−	543.058i		−	1.48783i
$366$		0			0
$367$	400.000			1.08992			0.544959	−	0.838463i	$-0.316545\pi$
$367$	400.000			1.08992			0.544959	+	0.838463i	$0.316545\pi$
$368$		0			0
$369$	24.0000	+	29.6985i	0.0650407	+	0.0804837i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	50.0000			0.134048			0.0670241	−	0.997751i	$-0.478650\pi$
$373$	50.0000			0.134048			0.0670241	+	0.997751i	$0.478650\pi$
$374$		0			0
$375$	−384.000	+	135.765i	−1.02400	+	0.362039i
$376$		0			0
$377$		−	90.5097i		−	0.240079i
$378$		0			0
$379$	−156.000			−0.411609			−0.205805	−	0.978593i	$-0.565981\pi$
$379$	−156.000			−0.411609			−0.205805	+	0.978593i	$0.565981\pi$
$380$		0			0
$381$	176.000	+	497.803i	0.461942	+	1.30657i
$382$		0			0
$383$			356.382i			0.930501i	0.885179	+	0.465250i	$0.154036\pi$
$383$			356.382i			0.930501i	−0.885179	+	0.465250i	$0.845964\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	308.000	−	248.902i	0.795866	−	0.643157i
$388$		0			0
$389$			599.627i			1.54146i	0.637164	+	0.770728i	$0.280107\pi$
$389$			599.627i			1.54146i	−0.637164	+	0.770728i	$0.719893\pi$
$390$		0			0
$391$	290.000			0.741688
$392$		0			0
$393$	544.000	−	192.333i	1.38422	−	0.489397i
$394$		0			0
$395$			339.411i			0.859269i
$396$		0			0
$397$	−436.000			−1.09824			−0.549118	−	0.835745i	$-0.685036\pi$
$397$	−436.000			−1.09824			−0.549118	+	0.835745i	$0.685036\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			373.352i			0.931053i	0.885034	+	0.465527i	$0.154135\pi$
$401$			373.352i			0.931053i	−0.885034	+	0.465527i	$0.845865\pi$
$402$		0			0
$403$	200.000			0.496278
$404$		0			0
$405$	336.000	+	72.1249i	0.829630	+	0.178086i
$406$		0			0
$407$		−	203.647i		−	0.500361i
$408$		0			0
$409$	640.000			1.56479			0.782396	−	0.622781i	$-0.213997\pi$
$409$	640.000			1.56479			0.782396	+	0.622781i	$0.213997\pi$
$410$		0			0
$411$	−512.000	+	181.019i	−1.24574	+	0.440436i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	528.000			1.27229
$416$		0			0
$417$	−78.0000	−	220.617i	−0.187050	−	0.529058i
$418$		0			0
$419$		−	226.274i		−	0.540034i	−0.962856	−	0.270017i	$-0.912971\pi$
$419$		−	226.274i		−	0.540034i	0.962856	−	0.270017i	$-0.0870293\pi$
$420$		0			0
$421$	−48.0000			−0.114014			−0.0570071	−	0.998374i	$-0.518156\pi$
$421$	−48.0000			−0.114014			−0.0570071	+	0.998374i	$0.518156\pi$
$422$		0			0
$423$	160.000	+	197.990i	0.378251	+	0.468061i
$424$		0			0
$425$		−	49.4975i		−	0.116465i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	48.0000	−	16.9706i	0.111888	−	0.0395584i
$430$		0			0
$431$			513.360i			1.19109i	0.803322	+	0.595545i	$0.203064\pi$
$431$			513.360i			1.19109i	−0.803322	+	0.595545i	$0.796936\pi$
$432$		0			0
$433$	832.000			1.92148			0.960739	−	0.277454i	$-0.0894905\pi$
$433$	832.000			1.92148			0.960739	+	0.277454i	$0.0894905\pi$
$434$		0			0
$435$	−96.0000	−	271.529i	−0.220690	−	0.624205i
$436$		0			0
$437$			656.195i			1.50159i
$438$		0			0
$439$	480.000			1.09339			0.546697	−	0.837330i	$-0.315885\pi$
$439$	480.000			1.09339			0.546697	+	0.837330i	$0.315885\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			253.144i			0.571432i	0.958314	+	0.285716i	$0.0922313\pi$
$443$			253.144i			0.571432i	−0.958314	+	0.285716i	$0.907769\pi$
$444$		0			0
$445$	−690.000			−1.55056
$446$		0			0
$447$	−432.000	+	152.735i	−0.966443	+	0.341689i
$448$		0			0
$449$		−	107.480i		−	0.239377i	−0.992811	−	0.119688i	$-0.961810\pi$
$449$		−	107.480i		−	0.239377i	0.992811	−	0.119688i	$-0.0381896\pi$
$450$		0			0
$451$	18.0000			0.0399113
$452$		0			0
$453$	64.0000	+	181.019i	0.141280	+	0.399601i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−194.000			−0.424508			−0.212254	−	0.977215i	$-0.568080\pi$
$457$	−194.000			−0.424508			−0.212254	+	0.977215i	$0.568080\pi$
$458$		0			0
$459$	−100.000	+	162.635i	−0.217865	+	0.354324i
$460$		0			0
$461$			210.718i			0.457089i	0.973534	+	0.228544i	$0.0733966\pi$
$461$			210.718i			0.457089i	−0.973534	+	0.228544i	$0.926603\pi$
$462$		0			0
$463$	760.000			1.64147			0.820734	−	0.571310i	$-0.193565\pi$
$463$	760.000			1.64147			0.820734	+	0.571310i	$0.193565\pi$
$464$		0			0
$465$	600.000	−	212.132i	1.29032	−	0.456198i
$466$		0			0
$467$		−	231.931i		−	0.496640i	−0.968678	−	0.248320i	$-0.920122\pi$
$467$		−	231.931i		−	0.496640i	0.968678	−	0.248320i	$-0.0798785\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−100.000	−	282.843i	−0.212314	−	0.600515i
$472$		0			0
$473$		−	186.676i		−	0.394664i
$474$		0			0
$475$	112.000			0.235789
$476$		0			0
$477$	96.0000	+	118.794i	0.201258	+	0.249044i
$478$		0			0
$479$			277.186i			0.578676i	0.957227	+	0.289338i	$0.0934352\pi$
$479$			277.186i			0.578676i	−0.957227	+	0.289338i	$0.906565\pi$
$480$		0			0
$481$	192.000			0.399168
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	576.999i		−	1.18969i
$486$		0			0
$487$	776.000			1.59343			0.796715	−	0.604356i	$-0.206569\pi$
$487$	776.000			1.59343			0.796715	+	0.604356i	$0.206569\pi$
$488$		0			0
$489$	−180.000	−	509.117i	−0.368098	−	1.04114i
$490$		0			0
$491$		−	674.580i		−	1.37389i	−0.726710	−	0.686945i	$-0.758951\pi$
$491$		−	674.580i		−	1.37389i	0.726710	−	0.686945i	$-0.241049\pi$
$492$		0			0
$493$	160.000			0.324544
$494$		0			0
$495$	126.000	−	101.823i	0.254545	−	0.205704i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	12.0000			0.0240481			0.0120240	−	0.999928i	$-0.496173\pi$
$499$	12.0000			0.0240481			0.0120240	+	0.999928i	$0.496173\pi$
$500$		0			0
$501$	416.000	−	147.078i	0.830339	−	0.293569i
$502$		0			0
$503$			435.578i			0.865960i	0.901404	+	0.432980i	$0.142538\pi$
$503$			435.578i			0.865960i	−0.901404	+	0.432980i	$0.857462\pi$
$504$		0			0
$505$	174.000			0.344554
$506$		0			0
$507$	−153.000	−	432.749i	−0.301775	−	0.853549i
$508$		0			0
$509$		−	603.869i		−	1.18638i	−0.805061	−	0.593192i	$-0.797868\pi$
$509$		−	603.869i		−	1.18638i	0.805061	−	0.593192i	$-0.202132\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−368.000	−	226.274i	−0.717349	−	0.441080i
$514$		0			0
$515$			619.426i			1.20277i
$516$		0			0
$517$	120.000			0.232108
$518$		0			0
$519$	−556.000	+	196.576i	−1.07129	+	0.378759i
$520$		0			0
$521$		−	683.065i		−	1.31107i	−0.755167	−	0.655533i	$-0.772444\pi$
$521$		−	683.065i		−	1.31107i	0.755167	−	0.655533i	$-0.227556\pi$
$522$		0			0
$523$	−466.000			−0.891013			−0.445507	−	0.895279i	$-0.646976\pi$
$523$	−466.000			−0.891013			−0.445507	+	0.895279i	$0.646976\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			353.553i			0.670879i
$528$		0			0
$529$	−1153.00			−2.17958
$530$		0			0
$531$	192.000	+	237.588i	0.361582	+	0.447435i
$532$		0			0
$533$			16.9706i			0.0318397i
$534$		0			0
$535$	−78.0000			−0.145794
$536$		0			0
$537$	−660.000	+	233.345i	−1.22905	+	0.434535i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	528.000			0.975970			0.487985	−	0.872852i	$-0.337732\pi$
$541$	528.000			0.975970			0.487985	+	0.872852i	$0.337732\pi$
$542$		0			0
$543$	−204.000	−	576.999i	−0.375691	−	1.06261i
$544$		0			0
$545$		−	670.337i		−	1.22998i
$546$		0			0
$547$	668.000			1.22121			0.610603	−	0.791937i	$-0.290927\pi$
$547$	668.000			1.22121			0.610603	+	0.791937i	$0.290927\pi$
$548$		0			0
$549$	−476.000	+	384.666i	−0.867031	+	0.700667i
$550$		0			0
$551$			362.039i			0.657057i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	576.000	−	203.647i	1.03784	−	0.366931i
$556$		0			0
$557$		−	333.754i		−	0.599200i	−0.954065	−	0.299600i	$-0.903147\pi$
$557$		−	333.754i		−	0.599200i	0.954065	−	0.299600i	$-0.0968532\pi$
$558$		0			0
$559$	176.000			0.314848
$560$		0			0
$561$	30.0000	+	84.8528i	0.0534759	+	0.151253i
$562$		0			0
$563$		−	203.647i		−	0.361717i	−0.983509	−	0.180859i	$-0.942112\pi$
$563$		−	203.647i		−	0.361717i	0.983509	−	0.180859i	$-0.0578876\pi$
$564$		0			0
$565$	552.000			0.976991
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	605.283i		−	1.06377i	−0.846818	−	0.531883i	$-0.821484\pi$
$569$		−	605.283i		−	1.06377i	0.846818	−	0.531883i	$-0.178516\pi$
$570$		0			0
$571$	780.000			1.36602			0.683012	−	0.730407i	$-0.260670\pi$
$571$	780.000			1.36602			0.683012	+	0.730407i	$0.260670\pi$
$572$		0			0
$573$	−684.000	+	241.831i	−1.19372	+	0.422043i
$574$		0			0
$575$			287.085i			0.499279i
$576$		0			0
$577$	−728.000			−1.26170			−0.630849	−	0.775905i	$-0.717293\pi$
$577$	−728.000			−1.26170			−0.630849	+	0.775905i	$0.717293\pi$
$578$		0			0
$579$	16.0000	+	45.2548i	0.0276339	+	0.0781603i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	72.0000			0.123499
$584$		0			0
$585$	96.0000	+	118.794i	0.164103	+	0.203067i
$586$		0			0
$587$			322.441i			0.549303i	0.961544	+	0.274651i	$0.0885624\pi$
$587$			322.441i			0.549303i	−0.961544	+	0.274651i	$0.911438\pi$
$588$		0			0
$589$	−800.000			−1.35823
$590$		0			0
$591$	96.0000	−	33.9411i	0.162437	−	0.0574300i
$592$		0			0
$593$			60.8112i			0.102548i	0.998685	+	0.0512742i	$0.0163282\pi$
$593$			60.8112i			0.102548i	−0.998685	+	0.0512742i	$0.983672\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−16.0000	−	45.2548i	−0.0268007	−	0.0758037i
$598$		0			0
$599$		−	275.772i		−	0.460387i	−0.973145	−	0.230193i	$-0.926064\pi$
$599$		−	275.772i		−	0.460387i	0.973145	−	0.230193i	$-0.0739359\pi$
$600$		0			0
$601$	8.00000			0.0133111			0.00665557	−	0.999978i	$-0.497881\pi$
$601$	8.00000			0.0133111			0.00665557	+	0.999978i	$0.497881\pi$
$602$		0			0
$603$	−308.000	+	248.902i	−0.510779	+	0.412772i
$604$		0			0
$605$			436.992i			0.722301i
$606$		0			0
$607$	−432.000			−0.711697			−0.355848	−	0.934544i	$-0.615808\pi$
$607$	−432.000			−0.711697			−0.355848	+	0.934544i	$0.615808\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			113.137i			0.185167i
$612$		0			0
$613$	146.000			0.238173			0.119086	−	0.992884i	$-0.462003\pi$
$613$	146.000			0.238173			0.119086	+	0.992884i	$0.462003\pi$
$614$		0			0
$615$	18.0000	+	50.9117i	0.0292683	+	0.0827832i
$616$		0			0
$617$			181.019i			0.293386i	0.989182	+	0.146693i	$0.0468630\pi$
$617$			181.019i			0.293386i	−0.989182	+	0.146693i	$0.953137\pi$
$618$		0			0
$619$	238.000			0.384491			0.192246	−	0.981347i	$-0.438423\pi$
$619$	238.000			0.384491			0.192246	+	0.981347i	$0.438423\pi$
$620$		0			0
$621$	580.000	−	943.280i	0.933977	−	1.51897i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−401.000			−0.641600
$626$		0			0
$627$	−192.000	+	67.8823i	−0.306220	+	0.108265i
$628$		0			0
$629$			339.411i			0.539605i
$630$		0			0
$631$	−552.000			−0.874802			−0.437401	−	0.899267i	$-0.644101\pi$
$631$	−552.000			−0.874802			−0.437401	+	0.899267i	$0.644101\pi$
$632$		0			0
$633$	20.0000	+	56.5685i	0.0315956	+	0.0893658i
$634$		0			0
$635$			746.705i			1.17591i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	24.0000	+	29.6985i	0.0375587	+	0.0464765i
$640$		0			0
$641$		−	814.587i		−	1.27081i	−0.772180	−	0.635403i	$-0.780834\pi$
$641$		−	814.587i		−	1.27081i	0.772180	−	0.635403i	$-0.219166\pi$
$642$		0			0
$643$	−784.000			−1.21928			−0.609642	−	0.792677i	$-0.708687\pi$
$643$	−784.000			−1.21928			−0.609642	+	0.792677i	$0.708687\pi$
$644$		0			0
$645$	528.000	−	186.676i	0.818605	−	0.289420i
$646$		0			0
$647$		−	1097.43i		−	1.69618i	−0.529851	−	0.848091i	$-0.677752\pi$
$647$		−	1097.43i		−	1.69618i	0.529851	−	0.848091i	$-0.322248\pi$
$648$		0			0
$649$	144.000			0.221880
$650$		0			0
$651$		0			0
$652$		0			0
$653$			718.420i			1.10018i	0.835104	+	0.550092i	$0.185407\pi$
$653$			718.420i			1.10018i	−0.835104	+	0.550092i	$0.814593\pi$
$654$		0			0
$655$	816.000			1.24580
$656$		0			0
$657$	896.000	−	724.077i	1.36377	−	1.10210i
$658$		0			0
$659$			1025.30i			1.55585i	0.628357	+	0.777925i	$0.283728\pi$
$659$			1025.30i			1.55585i	−0.628357	+	0.777925i	$0.716272\pi$
$660$		0			0
$661$	−484.000			−0.732224			−0.366112	−	0.930571i	$-0.619311\pi$
$661$	−484.000			−0.732224			−0.366112	+	0.930571i	$0.619311\pi$
$662$		0			0
$663$	−80.0000	+	28.2843i	−0.120664	+	0.0426610i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−928.000			−1.39130
$668$		0			0
$669$	16.0000	+	45.2548i	0.0239163	+	0.0676455i
$670$		0			0
$671$			288.500i			0.429955i
$672$		0			0
$673$	−736.000			−1.09361			−0.546805	−	0.837260i	$-0.684156\pi$
$673$	−736.000			−1.09361			−0.546805	+	0.837260i	$0.684156\pi$
$674$		0			0
$675$	−161.000	−	98.9949i	−0.238519	−	0.146659i
$676$		0			0
$677$		−	796.202i		−	1.17607i	−0.808834	−	0.588037i	$-0.799901\pi$
$677$		−	796.202i		−	1.17607i	0.808834	−	0.588037i	$-0.200099\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−496.000	+	175.362i	−0.728341	+	0.257507i
$682$		0			0
$683$		−	335.169i		−	0.490730i	−0.969431	−	0.245365i	$-0.921092\pi$
$683$		−	335.169i		−	0.490730i	0.969431	−	0.245365i	$-0.0789078\pi$
$684$		0			0
$685$	−768.000			−1.12117
$686$		0			0
$687$	−68.0000	−	192.333i	−0.0989811	−	0.279961i
$688$		0			0
$689$			67.8823i			0.0985229i
$690$		0			0
$691$	578.000			0.836469			0.418234	−	0.908339i	$-0.362649\pi$
$691$	578.000			0.836469			0.418234	+	0.908339i	$0.362649\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	330.926i		−	0.476152i
$696$		0			0
$697$	−30.0000			−0.0430416
$698$		0			0
$699$	−1008.00	+	356.382i	−1.44206	+	0.509845i
$700$		0			0
$701$			820.244i			1.17011i	0.810995	+	0.585053i	$0.198926\pi$
$701$			820.244i			1.17011i	−0.810995	+	0.585053i	$0.801074\pi$
$702$		0			0
$703$	−768.000			−1.09246
$704$		0			0
$705$	120.000	+	339.411i	0.170213	+	0.481434i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	206.000			0.290550			0.145275	−	0.989391i	$-0.453593\pi$
$709$	206.000			0.290550			0.145275	+	0.989391i	$0.453593\pi$
$710$		0			0
$711$	−560.000	+	452.548i	−0.787623	+	0.636496i
$712$		0			0
$713$		−	2050.61i		−	2.87603i
$714$		0			0
$715$	72.0000			0.100699
$716$		0			0
$717$	−596.000	+	210.718i	−0.831241	+	0.293888i
$718$		0			0
$719$			1159.66i			1.61287i	0.591321	+	0.806436i	$0.298607\pi$
$719$			1159.66i			1.61287i	−0.591321	+	0.806436i	$0.701393\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	432.000	+	1221.88i	0.597510	+	1.69001i
$724$		0			0
$725$			158.392i			0.218472i
$726$		0			0
$727$	−786.000			−1.08116			−0.540578	−	0.841294i	$-0.681794\pi$
$727$	−786.000			−1.08116			−0.540578	+	0.841294i	$0.681794\pi$
$728$		0			0
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$			311.127i			0.425618i
$732$		0			0
$733$	892.000			1.21692			0.608458	−	0.793586i	$-0.291788\pi$
$733$	892.000			1.21692			0.608458	+	0.793586i	$0.291788\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			186.676i			0.253292i
$738$		0			0
$739$	−748.000			−1.01218			−0.506089	−	0.862481i	$-0.668909\pi$
$739$	−748.000			−1.01218			−0.506089	+	0.862481i	$0.668909\pi$
$740$		0			0
$741$	−64.0000	−	181.019i	−0.0863698	−	0.244291i
$742$		0			0
$743$		−	719.835i		−	0.968822i	−0.874841	−	0.484411i	$-0.839034\pi$
$743$		−	719.835i		−	0.968822i	0.874841	−	0.484411i	$-0.160966\pi$
$744$		0			0
$745$	−648.000			−0.869799
$746$		0			0
$747$	704.000	+	871.156i	0.942436	+	1.16621i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−280.000			−0.372836			−0.186418	−	0.982471i	$-0.559688\pi$
$751$	−280.000			−0.372836			−0.186418	+	0.982471i	$0.559688\pi$
$752$		0			0
$753$	816.000	−	288.500i	1.08367	−	0.383134i
$754$		0			0
$755$			271.529i			0.359641i
$756$		0			0
$757$	14.0000			0.0184941			0.00924703	−	0.999957i	$-0.497057\pi$
$757$	14.0000			0.0184941			0.00924703	+	0.999957i	$0.497057\pi$
$758$		0			0
$759$	−174.000	−	492.146i	−0.229249	−	0.648414i
$760$		0			0
$761$			199.404i			0.262029i	0.991380	+	0.131015i	$0.0418235\pi$
$761$			199.404i			0.262029i	−0.991380	+	0.131015i	$0.958177\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−210.000	+	169.706i	−0.274510	+	0.221837i
$766$		0			0
$767$			135.765i			0.177007i
$768$		0			0
$769$	−784.000			−1.01951			−0.509753	−	0.860321i	$-0.670263\pi$
$769$	−784.000			−1.01951			−0.509753	+	0.860321i	$0.670263\pi$
$770$		0			0
$771$	−468.000	+	165.463i	−0.607004	+	0.214608i
$772$		0			0
$773$			1149.76i			1.48739i	0.668517	+	0.743697i	$0.266929\pi$
$773$			1149.76i			1.48739i	−0.668517	+	0.743697i	$0.733071\pi$
$774$		0			0
$775$	−350.000			−0.451613
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	67.8823i		−	0.0871402i
$780$		0			0
$781$	18.0000			0.0230474
$782$		0			0
$783$	320.000	−	520.431i	0.408685	−	0.664662i
$784$		0			0
$785$		−	424.264i		−	0.540464i
$786$		0			0
$787$	−578.000			−0.734435			−0.367217	−	0.930135i	$-0.619689\pi$
$787$	−578.000			−0.734435			−0.367217	+	0.930135i	$0.619689\pi$
$788$		0			0
$789$	372.000	−	131.522i	0.471483	−	0.166694i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−272.000			−0.343001
$794$		0			0
$795$	72.0000	+	203.647i	0.0905660	+	0.256159i
$796$		0			0
$797$		−	15.5563i		−	0.0195186i	−0.999952	−	0.00975932i	$-0.996893\pi$
$797$		−	15.5563i		−	0.0195186i	0.999952	−	0.00975932i	$-0.00310654\pi$
$798$		0			0
$799$	−200.000			−0.250313
$800$		0			0
$801$	−920.000	−	1138.44i	−1.14856	−	1.42128i
$802$		0			0
$803$		−	543.058i		−	0.676286i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−852.000	+	301.227i	−1.05576	+	0.373268i
$808$		0			0
$809$			1419.87i			1.75509i	0.479491	+	0.877547i	$0.340821\pi$
$809$			1419.87i			1.75509i	−0.479491	+	0.877547i	$0.659179\pi$
$810$		0			0
$811$	−754.000			−0.929716			−0.464858	−	0.885385i	$-0.653895\pi$
$811$	−754.000			−0.929716			−0.464858	+	0.885385i	$0.653895\pi$
$812$		0			0
$813$	−322.000	−	910.754i	−0.396064	−	1.12024i
$814$		0			0
$815$		−	763.675i		−	0.937025i
$816$		0			0
$817$	−704.000			−0.861689
$818$		0			0
$819$		0			0
$820$		0			0
$821$			571.342i			0.695910i	0.937511	+	0.347955i	$0.113124\pi$
$821$			571.342i			0.695910i	−0.937511	+	0.347955i	$0.886876\pi$
$822$		0			0
$823$	352.000			0.427704			0.213852	−	0.976866i	$-0.431399\pi$
$823$	352.000			0.427704			0.213852	+	0.976866i	$0.431399\pi$
$824$		0			0
$825$	−84.0000	+	29.6985i	−0.101818	+	0.0359982i
$826$		0			0
$827$			456.791i			0.552347i	0.961108	+	0.276174i	$0.0890664\pi$
$827$			456.791i			0.552347i	−0.961108	+	0.276174i	$0.910934\pi$
$828$		0			0
$829$	−604.000			−0.728589			−0.364294	−	0.931284i	$-0.618690\pi$
$829$	−604.000			−0.728589			−0.364294	+	0.931284i	$0.618690\pi$
$830$		0			0
$831$	80.0000	+	226.274i	0.0962696	+	0.272291i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	624.000			0.747305
$836$		0			0
$837$	1150.00	+	707.107i	1.37395	+	0.844811i
$838$		0			0
$839$			1052.17i			1.25408i	0.778986	+	0.627041i	$0.215734\pi$
$839$			1052.17i			1.25408i	−0.778986	+	0.627041i	$0.784266\pi$
$840$		0			0
$841$	329.000			0.391201
$842$		0			0
$843$	−464.000	+	164.049i	−0.550415	+	0.194601i
$844$		0			0
$845$		−	649.124i		−	0.768194i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	304.000	+	859.842i	0.358068	+	1.01277i
$850$		0			0
$851$		−	1968.59i		−	2.31326i
$852$		0			0
$853$	−644.000			−0.754982			−0.377491	−	0.926013i	$-0.623213\pi$
$853$	−644.000			−0.754982			−0.377491	+	0.926013i	$0.623213\pi$
$854$		0			0
$855$	−384.000	−	475.176i	−0.449123	−	0.555761i
$856$		0			0
$857$		−	1633.42i		−	1.90597i	−0.303017	−	0.952985i	$-0.597994\pi$
$857$		−	1633.42i		−	1.90597i	0.303017	−	0.952985i	$-0.402006\pi$
$858$		0			0
$859$	240.000			0.279395			0.139697	−	0.990194i	$-0.455387\pi$
$859$	240.000			0.279395			0.139697	+	0.990194i	$0.455387\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1033.79i		−	1.19790i	−0.800785	−	0.598951i	$-0.795584\pi$
$863$		−	1033.79i		−	1.19790i	0.800785	−	0.598951i	$-0.204416\pi$
$864$		0			0
$865$	−834.000			−0.964162
$866$		0			0
$867$	239.000	+	675.994i	0.275663	+	0.779693i
$868$		0			0
$869$			339.411i			0.390577i
$870$		0			0
$871$	−176.000			−0.202067
$872$		0			0
$873$	952.000	−	769.332i	1.09049	−	0.881251i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	802.000			0.914481			0.457241	−	0.889343i	$-0.348838\pi$
$877$	802.000			0.914481			0.457241	+	0.889343i	$0.348838\pi$
$878$		0			0
$879$	−1548.00	+	547.301i	−1.76109	+	0.622640i
$880$		0			0
$881$		−	1260.06i		−	1.43027i	−0.698988	−	0.715133i	$-0.746366\pi$
$881$		−	1260.06i		−	1.43027i	0.698988	−	0.715133i	$-0.253634\pi$
$882$		0			0
$883$	1732.00			1.96149			0.980747	−	0.195280i	$-0.0625617\pi$
$883$	1732.00			1.96149			0.980747	+	0.195280i	$0.0625617\pi$
$884$		0			0
$885$	144.000	+	407.294i	0.162712	+	0.460219i
$886$		0			0
$887$		−	520.431i		−	0.586731i	−0.956000	−	0.293366i	$-0.905225\pi$
$887$		−	520.431i		−	0.586731i	0.956000	−	0.293366i	$-0.0947753\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	336.000	+	72.1249i	0.377104	+	0.0809483i
$892$		0			0
$893$		−	452.548i		−	0.506773i
$894$		0			0
$895$	−990.000			−1.10615
$896$		0			0
$897$	464.000	−	164.049i	0.517280	−	0.182886i
$898$		0			0
$899$		−	1131.37i		−	1.25848i
$900$		0			0
$901$	−120.000			−0.133185
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	865.499i		−	0.956352i
$906$		0			0
$907$	−1156.00			−1.27453			−0.637266	−	0.770644i	$-0.719935\pi$
$907$	−1156.00			−1.27453			−0.637266	+	0.770644i	$0.719935\pi$
$908$		0			0
$909$	232.000	+	287.085i	0.255226	+	0.315825i
$910$		0			0
$911$		−	106.066i		−	0.116428i	−0.998304	−	0.0582141i	$-0.981459\pi$
$911$		−	106.066i		−	0.116428i	0.998304	−	0.0582141i	$-0.0185406\pi$
$912$		0			0
$913$	528.000			0.578313
$914$		0			0
$915$	−816.000	+	288.500i	−0.891803	+	0.315300i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−936.000			−1.01850			−0.509249	−	0.860619i	$-0.670077\pi$
$919$	−936.000			−1.01850			−0.509249	+	0.860619i	$0.670077\pi$
$920$		0			0
$921$	432.000	+	1221.88i	0.469055	+	1.32669i
$922$		0			0
$923$			16.9706i			0.0183863i
$924$		0			0
$925$	−336.000			−0.363243
$926$		0			0
$927$	−1022.00	+	825.901i	−1.10248	+	0.890939i
$928$		0			0
$929$			934.795i			1.00624i	0.864217	+	0.503119i	$0.167814\pi$
$929$			934.795i			1.00624i	−0.864217	+	0.503119i	$0.832186\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−272.000	+	96.1665i	−0.291533	+	0.103072i
$934$		0			0
$935$			127.279i			0.136128i
$936$		0			0
$937$	1008.00			1.07577			0.537887	−	0.843017i	$-0.319223\pi$
$937$	1008.00			1.07577			0.537887	+	0.843017i	$0.319223\pi$
$938$		0			0
$939$	−112.000	−	316.784i	−0.119276	−	0.337363i
$940$		0			0
$941$			920.653i			0.978377i	0.872178	+	0.489189i	$0.162707\pi$
$941$			920.653i			0.978377i	−0.872178	+	0.489189i	$0.837293\pi$
$942$		0			0
$943$	174.000			0.184517
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1161.07i		−	1.22605i	−0.790064	−	0.613025i	$-0.789952\pi$
$947$		−	1161.07i		−	1.22605i	0.790064	−	0.613025i	$-0.210048\pi$
$948$		0			0
$949$	512.000			0.539515
$950$		0			0
$951$	−912.000	+	322.441i	−0.958991	+	0.339054i
$952$		0			0
$953$		−	135.765i		−	0.142460i	−0.997460	−	0.0712301i	$-0.977308\pi$
$953$		−	135.765i		−	0.142460i	0.997460	−	0.0712301i	$-0.0226925\pi$
$954$		0			0
$955$	−1026.00			−1.07435
$956$		0			0
$957$	−96.0000	−	271.529i	−0.100313	−	0.283729i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1539.00			1.60146
$962$		0			0
$963$	−104.000	−	128.693i	−0.107996	−	0.133638i
$964$		0			0
$965$			67.8823i			0.0703443i
$966$		0			0
$967$	−728.000			−0.752844			−0.376422	−	0.926448i	$-0.622846\pi$
$967$	−728.000			−0.752844			−0.376422	+	0.926448i	$0.622846\pi$
$968$		0			0
$969$	320.000	−	113.137i	0.330237	−	0.116757i
$970$		0			0
$971$			786.303i			0.809787i	0.914364	+	0.404893i	$0.132691\pi$
$971$			786.303i			0.809787i	−0.914364	+	0.404893i	$0.867309\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−28.0000	−	79.1960i	−0.0287179	−	0.0812266i
$976$		0			0
$977$		−	1555.63i		−	1.59226i	−0.605128	−	0.796128i	$-0.706878\pi$
$977$		−	1555.63i		−	1.59226i	0.605128	−	0.796128i	$-0.293122\pi$
$978$		0			0
$979$	−690.000			−0.704801
$980$		0			0
$981$	1106.00	−	893.783i	1.12742	−	0.911094i
$982$		0			0
$983$			328.098i			0.333772i	0.985976	+	0.166886i	$0.0533711\pi$
$983$			328.098i			0.333772i	−0.985976	+	0.166886i	$0.946629\pi$
$984$		0			0
$985$	144.000			0.146193
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1804.54i		−	1.82461i
$990$		0			0
$991$	952.000			0.960646			0.480323	−	0.877092i	$-0.340519\pi$
$991$	952.000			0.960646			0.480323	+	0.877092i	$0.340519\pi$
$992$		0			0
$993$	−476.000	−	1346.33i	−0.479355	−	1.35582i
$994$		0			0
$995$		−	67.8823i		−	0.0682234i
$996$		0			0
$997$	−1148.00			−1.15145			−0.575727	−	0.817642i	$-0.695281\pi$
$997$	−1148.00			−1.15145			−0.575727	+	0.817642i	$0.695281\pi$
$998$		0			0
$999$	1104.00	+	678.823i	1.10511	+	0.679502i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.d.c.785.2	yes	2
3.2	odd	2	inner	1176.3.d.c.785.1	yes	2
7.6	odd	2		1176.3.d.b.785.1	✓	2
21.20	even	2		1176.3.d.b.785.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.3.d.b.785.1	✓	2	7.6	odd	2
1176.3.d.b.785.2	yes	2	21.20	even	2
1176.3.d.c.785.1	yes	2	3.2	odd	2	inner
1176.3.d.c.785.2	yes	2	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 2.82843i) q^{3} +4.24264i q^{5} +(-7.00000 + 5.65685i) q^{9} +4.24264i q^{11} -4.00000 q^{13} +(-12.0000 + 4.24264i) q^{15} -7.07107i q^{17} +16.0000 q^{19} +41.0122i q^{23} +7.00000 q^{25} +(-23.0000 - 14.1421i) q^{27} +22.6274i q^{29} -50.0000 q^{31} +(-12.0000 + 4.24264i) q^{33} -48.0000 q^{37} +(-4.00000 - 11.3137i) q^{39} -4.24264i q^{41} -44.0000 q^{43} +(-24.0000 - 29.6985i) q^{45} -28.2843i q^{47} +(20.0000 - 7.07107i) q^{51} -16.9706i q^{53} -18.0000 q^{55} +(16.0000 + 45.2548i) q^{57} -33.9411i q^{59} +68.0000 q^{61} -16.9706i q^{65} +44.0000 q^{67} +(-116.000 + 41.0122i) q^{69} -4.24264i q^{71} -128.000 q^{73} +(7.00000 + 19.7990i) q^{75} +80.0000 q^{79} +(17.0000 - 79.1960i) q^{81} -124.451i q^{83} +30.0000 q^{85} +(-64.0000 + 22.6274i) q^{87} +162.635i q^{89} +(-50.0000 - 141.421i) q^{93} +67.8823i q^{95} -136.000 q^{97} +(-24.0000 - 29.6985i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 14 q^{9} - 8 q^{13} - 24 q^{15} + 32 q^{19} + 14 q^{25} - 46 q^{27} - 100 q^{31} - 24 q^{33} - 96 q^{37} - 8 q^{39} - 88 q^{43} - 48 q^{45} + 40 q^{51} - 36 q^{55} + 32 q^{57} + 136 q^{61}+ \cdots - 48 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 14 * q^9 - 8 * q^13 - 24 * q^15 + 32 * q^19 + 14 * q^25 - 46 * q^27 - 100 * q^31 - 24 * q^33 - 96 * q^37 - 8 * q^39 - 88 * q^43 - 48 * q^45 + 40 * q^51 - 36 * q^55 + 32 * q^57 + 136 * q^61 + 88 * q^67 - 232 * q^69 - 256 * q^73 + 14 * q^75 + 160 * q^79 + 34 * q^81 + 60 * q^85 - 128 * q^87 - 100 * q^93 - 272 * q^97 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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