LMFDB - Embedded newform 1184.2.g.a.961.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1184,2,Mod(961,1184)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1184.961"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1184 = 2^{5} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1184.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$9.45428759932$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1184.961
Dual form			1184.2.g.a.961.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -3.00000 q^{7} +6.00000 q^{9} +3.00000 q^{11} -6.00000i q^{13} +6.00000i q^{17} +9.00000 q^{21} +4.00000i q^{23} +5.00000 q^{25} -9.00000 q^{27} -6.00000i q^{29} -4.00000i q^{31} -9.00000 q^{33} +(-1.00000 + 6.00000i) q^{37} +18.0000i q^{39} -7.00000 q^{41} -10.0000i q^{43} -3.00000 q^{47} +2.00000 q^{49} -18.0000i q^{51} -11.0000 q^{53} +10.0000i q^{59} +6.00000i q^{61} -18.0000 q^{63} -12.0000 q^{67} -12.0000i q^{69} +3.00000 q^{71} -7.00000 q^{73} -15.0000 q^{75} -9.00000 q^{77} -6.00000i q^{79} +9.00000 q^{81} -9.00000 q^{83} +18.0000i q^{87} +12.0000i q^{89} +18.0000i q^{91} +12.0000i q^{93} +6.00000i q^{97} +18.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 6 q^{7} + 12 q^{9} + 6 q^{11} + 18 q^{21} + 10 q^{25} - 18 q^{27} - 18 q^{33} - 2 q^{37} - 14 q^{41} - 6 q^{47} + 4 q^{49} - 22 q^{53} - 36 q^{63} - 24 q^{67} + 6 q^{71} - 14 q^{73} - 30 q^{75}+ \cdots + 36 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 6 * q^7 + 12 * q^9 + 6 * q^11 + 18 * q^21 + 10 * q^25 - 18 * q^27 - 18 * q^33 - 2 * q^37 - 14 * q^41 - 6 * q^47 + 4 * q^49 - 22 * q^53 - 36 * q^63 - 24 * q^67 + 6 * q^71 - 14 * q^73 - 30 * q^75 - 18 * q^77 + 18 * q^81 - 18 * q^83 + 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1184\mathbb{Z}\right)^\times$.

$n$	$223$	$705$	$741$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−3.00000			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−3.00000			−1.13389			−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000			−1.13389			−0.566947	+	0.823754i	$0.691875\pi$
$8$		0			0
$9$	6.00000			2.00000
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	9.00000			1.96396
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	−9.00000			−1.73205
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		0			0
$33$	−9.00000			−1.56670
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	+	6.00000i	−0.164399	+	0.986394i
$37$	−1.00000	+	6.00000i	−0.164399	+	0.986394i
$38$		0			0
$39$			18.0000i			2.88231i
$40$		0			0
$41$	−7.00000			−1.09322			−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000			−1.09322			−0.546608	+	0.837389i	$0.684081\pi$
$42$		0			0
$43$		−	10.0000i		−	1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$		−	10.0000i		−	1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.00000			−0.437595			−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000			−0.437595			−0.218797	+	0.975770i	$0.570213\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		−	18.0000i		−	2.52050i
$52$		0			0
$53$	−11.0000			−1.51097			−0.755483	−	0.655168i	$-0.772598\pi$
$53$	−11.0000			−1.51097			−0.755483	+	0.655168i	$0.772598\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			10.0000i			1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$			6.00000i			0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$			6.00000i			0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$		0			0
$63$	−18.0000			−2.26779
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$		0			0
$69$		−	12.0000i		−	1.44463i
$70$		0			0
$71$	3.00000			0.356034			0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000			0.356034			0.178017	+	0.984027i	$0.443032\pi$
$72$		0			0
$73$	−7.00000			−0.819288			−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000			−0.819288			−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$	−15.0000			−1.73205
$76$		0			0
$77$	−9.00000			−1.02565
$78$		0			0
$79$		−	6.00000i		−	0.675053i	−0.941316	−	0.337526i	$-0.890410\pi$
$79$		−	6.00000i		−	0.675053i	0.941316	−	0.337526i	$-0.109590\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	−9.00000			−0.987878			−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000			−0.987878			−0.493939	+	0.869496i	$0.664443\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			18.0000i			1.92980i
$88$		0			0
$89$			12.0000i			1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$			12.0000i			1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$		0			0
$91$			18.0000i			1.88691i
$92$		0			0
$93$			12.0000i			1.24434i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$	18.0000			1.80907
$100$		0			0
$101$	−11.0000			−1.09454			−0.547270	−	0.836956i	$-0.684333\pi$
$101$	−11.0000			−1.09454			−0.547270	+	0.836956i	$0.684333\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$		−	12.0000i		−	1.14939i	−0.818367	−	0.574696i	$-0.805120\pi$
$109$		−	12.0000i		−	1.14939i	0.818367	−	0.574696i	$-0.194880\pi$
$110$		0			0
$111$	3.00000	−	18.0000i	0.284747	−	1.70848i
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	36.0000i		−	3.32820i
$118$		0			0
$119$		−	18.0000i		−	1.65006i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$	21.0000			1.89351
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.00000			0.266207			0.133103	−	0.991102i	$-0.457506\pi$
$127$	3.00000			0.266207			0.133103	+	0.991102i	$0.457506\pi$
$128$		0			0
$129$			30.0000i			2.64135i
$130$		0			0
$131$		−	2.00000i		−	0.174741i	−0.996176	−	0.0873704i	$-0.972154\pi$
$131$		−	2.00000i		−	0.174741i	0.996176	−	0.0873704i	$-0.0278464\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	9.00000			0.757937
$142$		0			0
$143$		−	18.0000i		−	1.50524i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−6.00000			−0.494872
$148$		0			0
$149$	7.00000			0.573462			0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000			0.573462			0.286731	+	0.958011i	$0.407431\pi$
$150$		0			0
$151$	−12.0000			−0.976546			−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000			−0.976546			−0.488273	+	0.872691i	$0.662373\pi$
$152$		0			0
$153$			36.0000i			2.91043i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−21.0000			−1.67598			−0.837991	−	0.545684i	$-0.816270\pi$
$157$	−21.0000			−1.67598			−0.837991	+	0.545684i	$0.816270\pi$
$158$		0			0
$159$	33.0000			2.61707
$160$		0			0
$161$		−	12.0000i		−	0.945732i
$162$		0			0
$163$			20.0000i			1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$			20.0000i			1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.00000i		−	0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$		−	4.00000i		−	0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000			−0.0760286			−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000			−0.0760286			−0.0380143	+	0.999277i	$0.512103\pi$
$174$		0			0
$175$	−15.0000			−1.13389
$176$		0			0
$177$		−	30.0000i		−	2.25494i
$178$		0			0
$179$			10.0000i			0.747435i	0.927543	+	0.373718i	$0.121917\pi$
$179$			10.0000i			0.747435i	−0.927543	+	0.373718i	$0.878083\pi$
$180$		0			0
$181$	25.0000			1.85824			0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000			1.85824			0.929118	+	0.369784i	$0.120568\pi$
$182$		0			0
$183$		−	18.0000i		−	1.33060i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			18.0000i			1.31629i
$188$		0			0
$189$	27.0000			1.96396
$190$		0			0
$191$		−	14.0000i		−	1.01300i	−0.862239	−	0.506502i	$-0.830938\pi$
$191$		−	14.0000i		−	1.01300i	0.862239	−	0.506502i	$-0.169062\pi$
$192$		0			0
$193$		−	12.0000i		−	0.863779i	−0.901927	−	0.431889i	$-0.857847\pi$
$193$		−	12.0000i		−	0.863779i	0.901927	−	0.431889i	$-0.142153\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.0000			−0.926212			−0.463106	−	0.886303i	$-0.653265\pi$
$197$	−13.0000			−0.926212			−0.463106	+	0.886303i	$0.653265\pi$
$198$		0			0
$199$			2.00000i			0.141776i	0.997484	+	0.0708881i	$0.0225833\pi$
$199$			2.00000i			0.141776i	−0.997484	+	0.0708881i	$0.977417\pi$
$200$		0			0
$201$	36.0000			2.53924
$202$		0			0
$203$			18.0000i			1.26335i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			24.0000i			1.66812i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	21.0000			1.44570			0.722850	−	0.691005i	$-0.242832\pi$
$211$	21.0000			1.44570			0.722850	+	0.691005i	$0.242832\pi$
$212$		0			0
$213$	−9.00000			−0.616670
$214$		0			0
$215$		0			0
$216$		0			0
$217$			12.0000i			0.814613i
$218$		0			0
$219$	21.0000			1.41905
$220$		0			0
$221$	36.0000			2.42162
$222$		0			0
$223$	−15.0000			−1.00447			−0.502237	−	0.864730i	$-0.667490\pi$
$223$	−15.0000			−1.00447			−0.502237	+	0.864730i	$0.667490\pi$
$224$		0			0
$225$	30.0000			2.00000
$226$		0			0
$227$		−	28.0000i		−	1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$		−	28.0000i		−	1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$		0			0
$229$	−9.00000			−0.594737			−0.297368	−	0.954763i	$-0.596109\pi$
$229$	−9.00000			−0.594737			−0.297368	+	0.954763i	$0.596109\pi$
$230$		0			0
$231$	27.0000			1.77647
$232$		0			0
$233$	2.00000			0.131024			0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000			0.131024			0.0655122	+	0.997852i	$0.479132\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			18.0000i			1.16923i
$238$		0			0
$239$		−	2.00000i		−	0.129369i	−0.997906	−	0.0646846i	$-0.979396\pi$
$239$		−	2.00000i		−	0.129369i	0.997906	−	0.0646846i	$-0.0206041\pi$
$240$		0			0
$241$		−	18.0000i		−	1.15948i	−0.814801	−	0.579741i	$-0.803154\pi$
$241$		−	18.0000i		−	1.15948i	0.814801	−	0.579741i	$-0.196846\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	27.0000			1.71106
$250$		0			0
$251$			20.0000i			1.26239i	0.775625	+	0.631194i	$0.217435\pi$
$251$			20.0000i			1.26239i	−0.775625	+	0.631194i	$0.782565\pi$
$252$		0			0
$253$			12.0000i			0.754434i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		0			0
$259$	3.00000	−	18.0000i	0.186411	−	1.11847i
$260$		0			0
$261$		−	36.0000i		−	2.22834i
$262$		0			0
$263$	27.0000			1.66489			0.832446	−	0.554107i	$-0.186940\pi$
$263$	27.0000			1.66489			0.832446	+	0.554107i	$0.186940\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	36.0000i		−	2.20316i
$268$		0			0
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$	−9.00000			−0.546711			−0.273356	−	0.961913i	$-0.588134\pi$
$271$	−9.00000			−0.546711			−0.273356	+	0.961913i	$0.588134\pi$
$272$		0			0
$273$		−	54.0000i		−	3.26823i
$274$		0			0
$275$	15.0000			0.904534
$276$		0			0
$277$			24.0000i			1.44202i	0.692925	+	0.721010i	$0.256322\pi$
$277$			24.0000i			1.44202i	−0.692925	+	0.721010i	$0.743678\pi$
$278$		0			0
$279$		−	24.0000i		−	1.43684i
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$		0			0
$283$			6.00000i			0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$			6.00000i			0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	21.0000			1.23959
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$		−	18.0000i		−	1.05518i
$292$		0			0
$293$	−2.00000			−0.116841			−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000			−0.116841			−0.0584206	+	0.998292i	$0.518606\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−27.0000			−1.56670
$298$		0			0
$299$	24.0000			1.38796
$300$		0			0
$301$			30.0000i			1.72917i
$302$		0			0
$303$	33.0000			1.89580
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−21.0000			−1.19853			−0.599267	−	0.800549i	$-0.704541\pi$
$307$	−21.0000			−1.19853			−0.599267	+	0.800549i	$0.704541\pi$
$308$		0			0
$309$		−	18.0000i		−	1.02398i
$310$		0			0
$311$			16.0000i			0.907277i	0.891186	+	0.453638i	$0.149874\pi$
$311$			16.0000i			0.907277i	−0.891186	+	0.453638i	$0.850126\pi$
$312$		0			0
$313$			6.00000i			0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$			6.00000i			0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.0000			0.561656			0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000			0.561656			0.280828	+	0.959758i	$0.409391\pi$
$318$		0			0
$319$		−	18.0000i		−	1.00781i
$320$		0			0
$321$	−36.0000			−2.00932
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	30.0000i		−	1.66410i
$326$		0			0
$327$			36.0000i			1.99080i
$328$		0			0
$329$	9.00000			0.496186
$330$		0			0
$331$		−	24.0000i		−	1.31916i	−0.751635	−	0.659580i	$-0.770734\pi$
$331$		−	24.0000i		−	1.31916i	0.751635	−	0.659580i	$-0.229266\pi$
$332$		0			0
$333$	−6.00000	+	36.0000i	−0.328798	+	1.97279i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−15.0000			−0.817102			−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000			−0.817102			−0.408551	+	0.912735i	$0.633966\pi$
$338$		0			0
$339$			18.0000i			0.977626i
$340$		0			0
$341$		−	12.0000i		−	0.649836i
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$			34.0000i			1.82522i	0.408836	+	0.912608i	$0.365935\pi$
$347$			34.0000i			1.82522i	−0.408836	+	0.912608i	$0.634065\pi$
$348$		0			0
$349$	−30.0000			−1.60586			−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000			−1.60586			−0.802932	+	0.596071i	$0.796728\pi$
$350$		0			0
$351$			54.0000i			2.88231i
$352$		0			0
$353$		−	18.0000i		−	0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$		−	18.0000i		−	0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			54.0000i			2.85798i
$358$		0			0
$359$	−33.0000			−1.74167			−0.870837	−	0.491572i	$-0.836422\pi$
$359$	−33.0000			−1.74167			−0.870837	+	0.491572i	$0.836422\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	6.00000			0.314918
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−42.0000			−2.18643
$370$		0			0
$371$	33.0000			1.71327
$372$		0			0
$373$	−33.0000			−1.70868			−0.854338	−	0.519718i	$-0.826037\pi$
$373$	−33.0000			−1.70868			−0.854338	+	0.519718i	$0.826037\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−36.0000			−1.85409
$378$		0			0
$379$	−27.0000			−1.38690			−0.693448	−	0.720506i	$-0.743909\pi$
$379$	−27.0000			−1.38690			−0.693448	+	0.720506i	$0.743909\pi$
$380$		0			0
$381$	−9.00000			−0.461084
$382$		0			0
$383$		−	8.00000i		−	0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$		−	8.00000i		−	0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	60.0000i		−	3.04997i
$388$		0			0
$389$		−	24.0000i		−	1.21685i	−0.793612	−	0.608424i	$-0.791802\pi$
$389$		−	24.0000i		−	1.21685i	0.793612	−	0.608424i	$-0.208198\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$			6.00000i			0.302660i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.00000			−0.150566			−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000			−0.150566			−0.0752828	+	0.997162i	$0.523986\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$	−24.0000			−1.19553
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.00000	+	18.0000i	−0.148704	+	0.892227i
$408$		0			0
$409$		−	18.0000i		−	0.890043i	−0.895520	−	0.445021i	$-0.853196\pi$
$409$		−	18.0000i		−	0.890043i	0.895520	−	0.445021i	$-0.146804\pi$
$410$		0			0
$411$	−6.00000			−0.295958
$412$		0			0
$413$		−	30.0000i		−	1.47620i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	36.0000			1.76293
$418$		0			0
$419$	−3.00000			−0.146560			−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000			−0.146560			−0.0732798	+	0.997311i	$0.523347\pi$
$420$		0			0
$421$		−	12.0000i		−	0.584844i	−0.956289	−	0.292422i	$-0.905539\pi$
$421$		−	12.0000i		−	0.584844i	0.956289	−	0.292422i	$-0.0944612\pi$
$422$		0			0
$423$	−18.0000			−0.875190
$424$		0			0
$425$			30.0000i			1.45521i
$426$		0			0
$427$		−	18.0000i		−	0.871081i
$428$		0			0
$429$			54.0000i			2.60714i
$430$		0			0
$431$			2.00000i			0.0963366i	0.998839	+	0.0481683i	$0.0153384\pi$
$431$			2.00000i			0.0963366i	−0.998839	+	0.0481683i	$0.984662\pi$
$432$		0			0
$433$	−15.0000			−0.720854			−0.360427	−	0.932787i	$-0.617369\pi$
$433$	−15.0000			−0.720854			−0.360427	+	0.932787i	$0.617369\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	10.0000i		−	0.477274i	−0.971109	−	0.238637i	$-0.923299\pi$
$439$		−	10.0000i		−	0.477274i	0.971109	−	0.238637i	$-0.0767006\pi$
$440$		0			0
$441$	12.0000			0.571429
$442$		0			0
$443$	−27.0000			−1.28281			−0.641404	−	0.767203i	$-0.721648\pi$
$443$	−27.0000			−1.28281			−0.641404	+	0.767203i	$0.721648\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−21.0000			−0.993266
$448$		0			0
$449$			36.0000i			1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$			36.0000i			1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$		0			0
$451$	−21.0000			−0.988851
$452$		0			0
$453$	36.0000			1.69143
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		0			0
$459$		−	54.0000i		−	2.52050i
$460$		0			0
$461$		−	30.0000i		−	1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$		−	30.0000i		−	1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$		0			0
$463$			22.0000i			1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$			22.0000i			1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		0			0
$469$	36.0000			1.66233
$470$		0			0
$471$	63.0000			2.90289
$472$		0			0
$473$		−	30.0000i		−	1.37940i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−66.0000			−3.02193
$478$		0			0
$479$			26.0000i			1.18797i	0.804476	+	0.593985i	$0.202446\pi$
$479$			26.0000i			1.18797i	−0.804476	+	0.593985i	$0.797554\pi$
$480$		0			0
$481$	36.0000	+	6.00000i	1.64146	+	0.273576i
$482$		0			0
$483$			36.0000i			1.63806i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	6.00000i		−	0.271886i	−0.990717	−	0.135943i	$-0.956594\pi$
$487$		−	6.00000i		−	0.271886i	0.990717	−	0.135943i	$-0.0434064\pi$
$488$		0			0
$489$		−	60.0000i		−	2.71329i
$490$		0			0
$491$	−24.0000			−1.08310			−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000			−1.08310			−0.541552	+	0.840667i	$0.682163\pi$
$492$		0			0
$493$	36.0000			1.62136
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.00000			−0.403705
$498$		0			0
$499$			32.0000i			1.43252i	0.697835	+	0.716258i	$0.254147\pi$
$499$			32.0000i			1.43252i	−0.697835	+	0.716258i	$0.745853\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		0			0
$503$		−	40.0000i		−	1.78351i	−0.452517	−	0.891756i	$-0.649474\pi$
$503$		−	40.0000i		−	1.78351i	0.452517	−	0.891756i	$-0.350526\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	69.0000			3.06440
$508$		0			0
$509$	11.0000			0.487566			0.243783	−	0.969830i	$-0.421611\pi$
$509$	11.0000			0.487566			0.243783	+	0.969830i	$0.421611\pi$
$510$		0			0
$511$	21.0000			0.928985
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−9.00000			−0.395820
$518$		0			0
$519$	3.00000			0.131685
$520$		0			0
$521$	−7.00000			−0.306676			−0.153338	−	0.988174i	$-0.549002\pi$
$521$	−7.00000			−0.306676			−0.153338	+	0.988174i	$0.549002\pi$
$522$		0			0
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$		0			0
$525$	45.0000			1.96396
$526$		0			0
$527$	24.0000			1.04546
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$			60.0000i			2.60378i
$532$		0			0
$533$			42.0000i			1.81922i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	30.0000i		−	1.29460i
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$			36.0000i			1.54776i	0.633332	+	0.773880i	$0.281687\pi$
$541$			36.0000i			1.54776i	−0.633332	+	0.773880i	$0.718313\pi$
$542$		0			0
$543$	−75.0000			−3.21856
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$		0			0
$549$			36.0000i			1.53644i
$550$		0			0
$551$		0			0
$552$		0			0
$553$			18.0000i			0.765438i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$	−60.0000			−2.53773
$560$		0			0
$561$		−	54.0000i		−	2.27988i
$562$		0			0
$563$		−	14.0000i		−	0.590030i	−0.955493	−	0.295015i	$-0.904675\pi$
$563$		−	14.0000i		−	0.590030i	0.955493	−	0.295015i	$-0.0953246\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−27.0000			−1.13389
$568$		0			0
$569$		−	6.00000i		−	0.251533i	−0.992060	−	0.125767i	$-0.959861\pi$
$569$		−	6.00000i		−	0.251533i	0.992060	−	0.125767i	$-0.0401390\pi$
$570$		0			0
$571$	−15.0000			−0.627730			−0.313865	−	0.949468i	$-0.601624\pi$
$571$	−15.0000			−0.627730			−0.313865	+	0.949468i	$0.601624\pi$
$572$		0			0
$573$			42.0000i			1.75458i
$574$		0			0
$575$			20.0000i			0.834058i
$576$		0			0
$577$		−	36.0000i		−	1.49870i	−0.662174	−	0.749350i	$-0.730366\pi$
$577$		−	36.0000i		−	1.49870i	0.662174	−	0.749350i	$-0.269634\pi$
$578$		0			0
$579$			36.0000i			1.49611i
$580$		0			0
$581$	27.0000			1.12015
$582$		0			0
$583$	−33.0000			−1.36672
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	32.0000i		−	1.32078i	−0.750922	−	0.660391i	$-0.770391\pi$
$587$		−	32.0000i		−	1.32078i	0.750922	−	0.660391i	$-0.229609\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	39.0000			1.60425
$592$		0			0
$593$	47.0000			1.93006			0.965029	−	0.262142i	$-0.0844289\pi$
$593$	47.0000			1.93006			0.965029	+	0.262142i	$0.0844289\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	6.00000i		−	0.245564i
$598$		0			0
$599$	15.0000			0.612883			0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000			0.612883			0.306442	+	0.951889i	$0.400862\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$	−72.0000			−2.93207
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	12.0000i		−	0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$		−	12.0000i		−	0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$		0			0
$609$		−	54.0000i		−	2.18819i
$610$		0			0
$611$			18.0000i			0.728202i
$612$		0			0
$613$	25.0000			1.00974			0.504870	−	0.863195i	$-0.331540\pi$
$613$	25.0000			1.00974			0.504870	+	0.863195i	$0.331540\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	19.0000			0.764911			0.382456	−	0.923974i	$-0.375078\pi$
$617$	19.0000			0.764911			0.382456	+	0.923974i	$0.375078\pi$
$618$		0			0
$619$	−45.0000			−1.80870			−0.904351	−	0.426789i	$-0.859645\pi$
$619$	−45.0000			−1.80870			−0.904351	+	0.426789i	$0.859645\pi$
$620$		0			0
$621$		−	36.0000i		−	1.44463i
$622$		0			0
$623$		−	36.0000i		−	1.44231i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−36.0000	−	6.00000i	−1.43541	−	0.239236i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−63.0000			−2.50403
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	12.0000i		−	0.475457i
$638$		0			0
$639$	18.0000			0.712069
$640$		0			0
$641$	31.0000			1.22443			0.612213	−	0.790693i	$-0.290279\pi$
$641$	31.0000			1.22443			0.612213	+	0.790693i	$0.290279\pi$
$642$		0			0
$643$			42.0000i			1.65632i	0.560493	+	0.828159i	$0.310612\pi$
$643$			42.0000i			1.65632i	−0.560493	+	0.828159i	$0.689388\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.0000i		−	1.02217i	−0.859532	−	0.511083i	$-0.829245\pi$
$647$		−	26.0000i		−	1.02217i	0.859532	−	0.511083i	$-0.170755\pi$
$648$		0			0
$649$			30.0000i			1.17760i
$650$		0			0
$651$		−	36.0000i		−	1.41095i
$652$		0			0
$653$		−	24.0000i		−	0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$		−	24.0000i		−	0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−42.0000			−1.63858
$658$		0			0
$659$	−15.0000			−0.584317			−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000			−0.584317			−0.292159	+	0.956370i	$0.594373\pi$
$660$		0			0
$661$			24.0000i			0.933492i	0.884391	+	0.466746i	$0.154574\pi$
$661$			24.0000i			0.933492i	−0.884391	+	0.466746i	$0.845426\pi$
$662$		0			0
$663$	−108.000			−4.19437
$664$		0			0
$665$		0			0
$666$		0			0
$667$	24.0000			0.929284
$668$		0			0
$669$	45.0000			1.73980
$670$		0			0
$671$			18.0000i			0.694882i
$672$		0			0
$673$	−9.00000			−0.346925			−0.173462	−	0.984841i	$-0.555495\pi$
$673$	−9.00000			−0.346925			−0.173462	+	0.984841i	$0.555495\pi$
$674$		0			0
$675$	−45.0000			−1.73205
$676$		0			0
$677$	19.0000			0.730229			0.365115	−	0.930963i	$-0.381030\pi$
$677$	19.0000			0.730229			0.365115	+	0.930963i	$0.381030\pi$
$678$		0			0
$679$		−	18.0000i		−	0.690777i
$680$		0			0
$681$			84.0000i			3.21889i
$682$		0			0
$683$			2.00000i			0.0765279i	0.999268	+	0.0382639i	$0.0121828\pi$
$683$			2.00000i			0.0765279i	−0.999268	+	0.0382639i	$0.987817\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	27.0000			1.03011
$688$		0			0
$689$			66.0000i			2.51440i
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$	−54.0000			−2.05129
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	42.0000i		−	1.59086i
$698$		0			0
$699$	−6.00000			−0.226941
$700$		0			0
$701$			6.00000i			0.226617i	0.993560	+	0.113308i	$0.0361448\pi$
$701$			6.00000i			0.226617i	−0.993560	+	0.113308i	$0.963855\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	33.0000			1.24109
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$		−	36.0000i		−	1.35011i
$712$		0			0
$713$	16.0000			0.599205
$714$		0			0
$715$		0			0
$716$		0			0
$717$			6.00000i			0.224074i
$718$		0			0
$719$	−51.0000			−1.90198			−0.950990	−	0.309223i	$-0.899931\pi$
$719$	−51.0000			−1.90198			−0.950990	+	0.309223i	$0.899931\pi$
$720$		0			0
$721$		−	18.0000i		−	0.670355i
$722$		0			0
$723$			54.0000i			2.00828i
$724$		0			0
$725$		−	30.0000i		−	1.11417i
$726$		0			0
$727$			46.0000i			1.70605i	0.521874	+	0.853023i	$0.325233\pi$
$727$			46.0000i			1.70605i	−0.521874	+	0.853023i	$0.674767\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	60.0000			2.21918
$732$		0			0
$733$	33.0000			1.21888			0.609441	−	0.792831i	$-0.291394\pi$
$733$	33.0000			1.21888			0.609441	+	0.792831i	$0.291394\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−36.0000			−1.32608
$738$		0			0
$739$	3.00000			0.110357			0.0551784	−	0.998477i	$-0.482427\pi$
$739$	3.00000			0.110357			0.0551784	+	0.998477i	$0.482427\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9.00000			0.330178			0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000			0.330178			0.165089	+	0.986279i	$0.447209\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−54.0000			−1.97576
$748$		0			0
$749$	−36.0000			−1.31541
$750$		0			0
$751$	−15.0000			−0.547358			−0.273679	−	0.961821i	$-0.588241\pi$
$751$	−15.0000			−0.547358			−0.273679	+	0.961821i	$0.588241\pi$
$752$		0			0
$753$		−	60.0000i		−	2.18652i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	18.0000i		−	0.654221i	−0.944986	−	0.327111i	$-0.893925\pi$
$757$		−	18.0000i		−	0.654221i	0.944986	−	0.327111i	$-0.106075\pi$
$758$		0			0
$759$		−	36.0000i		−	1.30672i
$760$		0			0
$761$	−1.00000			−0.0362500			−0.0181250	−	0.999836i	$-0.505770\pi$
$761$	−1.00000			−0.0362500			−0.0181250	+	0.999836i	$0.505770\pi$
$762$		0			0
$763$			36.0000i			1.30329i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	60.0000			2.16647
$768$		0			0
$769$		−	18.0000i		−	0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$		−	18.0000i		−	0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$		0			0
$771$		−	54.0000i		−	1.94476i
$772$		0			0
$773$	−5.00000			−0.179838			−0.0899188	−	0.995949i	$-0.528661\pi$
$773$	−5.00000			−0.179838			−0.0899188	+	0.995949i	$0.528661\pi$
$774$		0			0
$775$		−	20.0000i		−	0.718421i
$776$		0			0
$777$	−9.00000	+	54.0000i	−0.322873	+	1.93724i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	9.00000			0.322045
$782$		0			0
$783$			54.0000i			1.92980i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	9.00000			0.320815			0.160408	−	0.987051i	$-0.448719\pi$
$787$	9.00000			0.320815			0.160408	+	0.987051i	$0.448719\pi$
$788$		0			0
$789$	−81.0000			−2.88368
$790$		0			0
$791$			18.0000i			0.640006i
$792$		0			0
$793$	36.0000			1.27840
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	18.0000i		−	0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$		−	18.0000i		−	0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$		0			0
$799$		−	18.0000i		−	0.636794i
$800$		0			0
$801$			72.0000i			2.54399i
$802$		0			0
$803$	−21.0000			−0.741074
$804$		0			0
$805$		0			0
$806$		0			0
$807$	42.0000			1.47847
$808$		0			0
$809$		−	12.0000i		−	0.421898i	−0.977497	−	0.210949i	$-0.932345\pi$
$809$		−	12.0000i		−	0.421898i	0.977497	−	0.210949i	$-0.0676553\pi$
$810$		0			0
$811$	33.0000			1.15879			0.579393	−	0.815048i	$-0.303290\pi$
$811$	33.0000			1.15879			0.579393	+	0.815048i	$0.303290\pi$
$812$		0			0
$813$	27.0000			0.946931
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$			108.000i			3.77383i
$820$		0			0
$821$	25.0000			0.872506			0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000			0.872506			0.436253	+	0.899824i	$0.356305\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	−45.0000			−1.56670
$826$		0			0
$827$		−	4.00000i		−	0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$		−	4.00000i		−	0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$		0			0
$829$			30.0000i			1.04194i	0.853574	+	0.520972i	$0.174430\pi$
$829$			30.0000i			1.04194i	−0.853574	+	0.520972i	$0.825570\pi$
$830$		0			0
$831$		−	72.0000i		−	2.49765i
$832$		0			0
$833$			12.0000i			0.415775i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			36.0000i			1.24434i
$838$		0			0
$839$	36.0000			1.24286			0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000			1.24286			0.621429	+	0.783470i	$0.286552\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		−	72.0000i		−	2.47981i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	6.00000			0.206162
$848$		0			0
$849$		−	18.0000i		−	0.617758i
$850$		0			0
$851$	−24.0000	−	4.00000i	−0.822709	−	0.137118i
$852$		0			0
$853$		−	12.0000i		−	0.410872i	−0.978671	−	0.205436i	$-0.934139\pi$
$853$		−	12.0000i		−	0.410872i	0.978671	−	0.205436i	$-0.0658613\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	42.0000i		−	1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$		−	42.0000i		−	1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$		0			0
$859$			14.0000i			0.477674i	0.971060	+	0.238837i	$0.0767661\pi$
$859$			14.0000i			0.477674i	−0.971060	+	0.238837i	$0.923234\pi$
$860$		0			0
$861$	−63.0000			−2.14703
$862$		0			0
$863$	−48.0000			−1.63394			−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000			−1.63394			−0.816970	+	0.576681i	$0.804348\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	57.0000			1.93582
$868$		0			0
$869$		−	18.0000i		−	0.610608i
$870$		0			0
$871$			72.0000i			2.43963i
$872$		0			0
$873$			36.0000i			1.21842i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	18.0000			0.607817			0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000			0.607817			0.303908	+	0.952701i	$0.401708\pi$
$878$		0			0
$879$	6.00000			0.202375
$880$		0			0
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$		0			0
$883$		−	6.00000i		−	0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$		−	6.00000i		−	0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21.0000			0.705111			0.352555	−	0.935791i	$-0.385313\pi$
$887$	21.0000			0.705111			0.352555	+	0.935791i	$0.385313\pi$
$888$		0			0
$889$	−9.00000			−0.301850
$890$		0			0
$891$	27.0000			0.904534
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−72.0000			−2.40401
$898$		0			0
$899$	−24.0000			−0.800445
$900$		0			0
$901$		−	66.0000i		−	2.19878i
$902$		0			0
$903$		−	90.0000i		−	2.99501i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	26.0000i		−	0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$		−	26.0000i		−	0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$		0			0
$909$	−66.0000			−2.18908
$910$		0			0
$911$			50.0000i			1.65657i	0.560304	+	0.828287i	$0.310684\pi$
$911$			50.0000i			1.65657i	−0.560304	+	0.828287i	$0.689316\pi$
$912$		0			0
$913$	−27.0000			−0.893570
$914$		0			0
$915$		0			0
$916$		0			0
$917$			6.00000i			0.198137i
$918$		0			0
$919$		−	48.0000i		−	1.58337i	−0.610927	−	0.791687i	$-0.709203\pi$
$919$		−	48.0000i		−	1.58337i	0.610927	−	0.791687i	$-0.290797\pi$
$920$		0			0
$921$	63.0000			2.07592
$922$		0			0
$923$		−	18.0000i		−	0.592477i
$924$		0			0
$925$	−5.00000	+	30.0000i	−0.164399	+	0.986394i
$926$		0			0
$927$			36.0000i			1.18240i
$928$		0			0
$929$	46.0000			1.50921			0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000			1.50921			0.754606	+	0.656179i	$0.227828\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	48.0000i		−	1.57145i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−7.00000			−0.228680			−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000			−0.228680			−0.114340	+	0.993442i	$0.536475\pi$
$938$		0			0
$939$		−	18.0000i		−	0.587408i
$940$		0			0
$941$	−10.0000			−0.325991			−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000			−0.325991			−0.162995	+	0.986627i	$0.552116\pi$
$942$		0			0
$943$		−	28.0000i		−	0.911805i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	16.0000i		−	0.519930i	−0.965618	−	0.259965i	$-0.916289\pi$
$947$		−	16.0000i		−	0.519930i	0.965618	−	0.259965i	$-0.0837111\pi$
$948$		0			0
$949$			42.0000i			1.36338i
$950$		0			0
$951$	−30.0000			−0.972817
$952$		0			0
$953$	43.0000			1.39291			0.696453	−	0.717602i	$-0.254760\pi$
$953$	43.0000			1.39291			0.696453	+	0.717602i	$0.254760\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			54.0000i			1.74557i
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	72.0000			2.32017
$964$		0			0
$965$		0			0
$966$		0			0
$967$			22.0000i			0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$			22.0000i			0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.0000			0.385098			0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000			0.385098			0.192549	+	0.981287i	$0.438325\pi$
$972$		0			0
$973$	36.0000			1.15411
$974$		0			0
$975$			90.0000i			2.88231i
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$			36.0000i			1.15056i
$980$		0			0
$981$		−	72.0000i		−	2.29878i
$982$		0			0
$983$	−15.0000			−0.478426			−0.239213	−	0.970967i	$-0.576889\pi$
$983$	−15.0000			−0.478426			−0.239213	+	0.970967i	$0.576889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−27.0000			−0.859419
$988$		0			0
$989$	40.0000			1.27193
$990$		0			0
$991$		−	56.0000i		−	1.77890i	−0.457034	−	0.889449i	$-0.651088\pi$
$991$		−	56.0000i		−	1.77890i	0.457034	−	0.889449i	$-0.348912\pi$
$992$		0			0
$993$			72.0000i			2.28485i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			42.0000i			1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$			42.0000i			1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$		0			0
$999$	9.00000	−	54.0000i	0.284747	−	1.70848i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1184.2.g.a.961.1	✓	2
4.3	odd	2		1184.2.g.c.961.1	yes	2
8.3	odd	2		2368.2.g.a.961.2		2
8.5	even	2		2368.2.g.g.961.2		2
37.36	even	2	inner	1184.2.g.a.961.2	yes	2
148.147	odd	2		1184.2.g.c.961.2	yes	2
296.147	odd	2		2368.2.g.a.961.1		2
296.221	even	2		2368.2.g.g.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.g.a.961.1	✓	2	1.1	even	1	trivial
1184.2.g.a.961.2	yes	2	37.36	even	2	inner
1184.2.g.c.961.1	yes	2	4.3	odd	2
1184.2.g.c.961.2	yes	2	148.147	odd	2
2368.2.g.a.961.1		2	296.147	odd	2
2368.2.g.a.961.2		2	8.3	odd	2
2368.2.g.g.961.1		2	296.221	even	2
2368.2.g.g.961.2		2	8.5	even	2

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:	\( N \)	\(=\)	\( 1184 = 2^{5} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1184.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -3.00000 q^{7} +6.00000 q^{9} +3.00000 q^{11} -6.00000i q^{13} +6.00000i q^{17} +9.00000 q^{21} +4.00000i q^{23} +5.00000 q^{25} -9.00000 q^{27} -6.00000i q^{29} -4.00000i q^{31} -9.00000 q^{33} +(-1.00000 + 6.00000i) q^{37} +18.0000i q^{39} -7.00000 q^{41} -10.0000i q^{43} -3.00000 q^{47} +2.00000 q^{49} -18.0000i q^{51} -11.0000 q^{53} +10.0000i q^{59} +6.00000i q^{61} -18.0000 q^{63} -12.0000 q^{67} -12.0000i q^{69} +3.00000 q^{71} -7.00000 q^{73} -15.0000 q^{75} -9.00000 q^{77} -6.00000i q^{79} +9.00000 q^{81} -9.00000 q^{83} +18.0000i q^{87} +12.0000i q^{89} +18.0000i q^{91} +12.0000i q^{93} +6.00000i q^{97} +18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 6 q^{7} + 12 q^{9} + 6 q^{11} + 18 q^{21} + 10 q^{25} - 18 q^{27} - 18 q^{33} - 2 q^{37} - 14 q^{41} - 6 q^{47} + 4 q^{49} - 22 q^{53} - 36 q^{63} - 24 q^{67} + 6 q^{71} - 14 q^{73} - 30 q^{75}+ \cdots + 36 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 6 * q^7 + 12 * q^9 + 6 * q^11 + 18 * q^21 + 10 * q^25 - 18 * q^27 - 18 * q^33 - 2 * q^37 - 14 * q^41 - 6 * q^47 + 4 * q^49 - 22 * q^53 - 36 * q^63 - 24 * q^67 + 6 * q^71 - 14 * q^73 - 30 * q^75 - 18 * q^77 + 18 * q^81 - 18 * q^83 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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