LMFDB - Embedded newform 1216.2.b.a.607.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,2,Mod(607,1216)] 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("1216.607"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-4,0,0,0,-16] 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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			607.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1216.607
Dual form			1216.2.b.a.607.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} -1.73205i q^{5} +1.00000i q^{7} -1.00000 q^{9} +5.19615 q^{11} -4.00000 q^{13} +3.46410 q^{15} +3.00000 q^{17} +(1.73205 - 4.00000i) q^{19} -2.00000 q^{21} +2.00000 q^{25} +4.00000i q^{27} -6.00000 q^{29} +10.3923 q^{31} +10.3923i q^{33} +1.73205 q^{35} +2.00000 q^{37} -8.00000i q^{39} +3.46410i q^{41} -5.19615 q^{43} +1.73205i q^{45} +9.00000i q^{47} +6.00000 q^{49} +6.00000i q^{51} +6.00000 q^{53} -9.00000i q^{55} +(8.00000 + 3.46410i) q^{57} -1.73205i q^{61} -1.00000i q^{63} +6.92820i q^{65} +2.00000i q^{67} +6.92820 q^{71} -11.0000 q^{73} +4.00000i q^{75} +5.19615i q^{77} +3.46410 q^{79} -11.0000 q^{81} -3.46410 q^{83} -5.19615i q^{85} -12.0000i q^{87} -3.46410i q^{89} -4.00000i q^{91} +20.7846i q^{93} +(-6.92820 - 3.00000i) q^{95} +13.8564i q^{97} -5.19615 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9} - 16 q^{13} + 12 q^{17} - 8 q^{21} + 8 q^{25} - 24 q^{29} + 8 q^{37} + 24 q^{49} + 24 q^{53} + 32 q^{57} - 44 q^{73} - 44 q^{81}+O(q^{100}) $ 4 * q - 4 * q^9 - 16 * q^13 + 12 * q^17 - 8 * q^21 + 8 * q^25 - 24 * q^29 + 8 * q^37 + 24 * q^49 + 24 * q^53 + 32 * q^57 - 44 * q^73 - 44 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			2.00000i			1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$		−	1.73205i		−	0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$		−	1.73205i		−	0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	5.19615			1.56670			0.783349	−	0.621582i	$-0.213510\pi$
$11$	5.19615			1.56670			0.783349	+	0.621582i	$0.213510\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	3.46410			0.894427
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	1.73205	−	4.00000i	0.397360	−	0.917663i
$19$	1.73205	−	4.00000i	0.397360	−	0.917663i
$20$		0			0
$21$	−2.00000			−0.436436
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	2.00000			0.400000
$26$		0			0
$27$			4.00000i			0.769800i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	10.3923			1.86651			0.933257	−	0.359211i	$-0.116954\pi$
$31$	10.3923			1.86651			0.933257	+	0.359211i	$0.116954\pi$
$32$		0			0
$33$			10.3923i			1.80907i
$34$		0			0
$35$	1.73205			0.292770
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		−	8.00000i		−	1.28103i
$40$		0			0
$41$			3.46410i			0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$			3.46410i			0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$		0			0
$43$	−5.19615			−0.792406			−0.396203	−	0.918163i	$-0.629672\pi$
$43$	−5.19615			−0.792406			−0.396203	+	0.918163i	$0.629672\pi$
$44$		0			0
$45$			1.73205i			0.258199i
$46$		0			0
$47$			9.00000i			1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$			9.00000i			1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$			6.00000i			0.840168i
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		−	9.00000i		−	1.21356i
$56$		0			0
$57$	8.00000	+	3.46410i	1.05963	+	0.458831i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		−	1.73205i		−	0.221766i	−0.993833	−	0.110883i	$-0.964632\pi$
$61$		−	1.73205i		−	0.221766i	0.993833	−	0.110883i	$-0.0353679\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$			6.92820i			0.859338i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.92820			0.822226			0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820			0.822226			0.411113	+	0.911584i	$0.365140\pi$
$72$		0			0
$73$	−11.0000			−1.28745			−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000			−1.28745			−0.643726	+	0.765256i	$0.722612\pi$
$74$		0			0
$75$			4.00000i			0.461880i
$76$		0			0
$77$			5.19615i			0.592157i
$78$		0			0
$79$	3.46410			0.389742			0.194871	−	0.980829i	$-0.437571\pi$
$79$	3.46410			0.389742			0.194871	+	0.980829i	$0.437571\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$		0			0
$85$		−	5.19615i		−	0.563602i
$86$		0			0
$87$		−	12.0000i		−	1.28654i
$88$		0			0
$89$		−	3.46410i		−	0.367194i	−0.983002	−	0.183597i	$-0.941226\pi$
$89$		−	3.46410i		−	0.367194i	0.983002	−	0.183597i	$-0.0587741\pi$
$90$		0			0
$91$		−	4.00000i		−	0.419314i
$92$		0			0
$93$			20.7846i			2.15526i
$94$		0			0
$95$	−6.92820	−	3.00000i	−0.710819	−	0.307794i
$96$		0			0
$97$			13.8564i			1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$			13.8564i			1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−5.19615			−0.522233
$100$		0			0
$101$			13.8564i			1.37876i	0.724398	+	0.689382i	$0.242118\pi$
$101$			13.8564i			1.37876i	−0.724398	+	0.689382i	$0.757882\pi$
$102$		0			0
$103$	6.92820			0.682656			0.341328	−	0.939944i	$-0.389123\pi$
$103$	6.92820			0.682656			0.341328	+	0.939944i	$0.389123\pi$
$104$		0			0
$105$			3.46410i			0.338062i
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$			4.00000i			0.379663i
$112$		0			0
$113$		−	17.3205i		−	1.62938i	−0.579899	−	0.814688i	$-0.696908\pi$
$113$		−	17.3205i		−	1.62938i	0.579899	−	0.814688i	$-0.303092\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	4.00000			0.369800
$118$		0			0
$119$			3.00000i			0.275010i
$120$		0			0
$121$	16.0000			1.45455
$122$		0			0
$123$	−6.92820			−0.624695
$124$		0			0
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$	10.3923			0.922168			0.461084	−	0.887357i	$-0.347461\pi$
$127$	10.3923			0.922168			0.461084	+	0.887357i	$0.347461\pi$
$128$		0			0
$129$		−	10.3923i		−	0.914991i
$130$		0			0
$131$	1.73205			0.151330			0.0756650	−	0.997133i	$-0.475892\pi$
$131$	1.73205			0.151330			0.0756650	+	0.997133i	$0.475892\pi$
$132$		0			0
$133$	4.00000	+	1.73205i	0.346844	+	0.150188i
$134$		0			0
$135$	6.92820			0.596285
$136$		0			0
$137$	−9.00000			−0.768922			−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000			−0.768922			−0.384461	+	0.923141i	$0.625613\pi$
$138$		0			0
$139$	1.73205			0.146911			0.0734553	−	0.997299i	$-0.476597\pi$
$139$	1.73205			0.146911			0.0734553	+	0.997299i	$0.476597\pi$
$140$		0			0
$141$	−18.0000			−1.51587
$142$		0			0
$143$	−20.7846			−1.73810
$144$		0			0
$145$			10.3923i			0.863034i
$146$		0			0
$147$			12.0000i			0.989743i
$148$		0			0
$149$		−	19.0526i		−	1.56085i	−0.625252	−	0.780423i	$-0.715004\pi$
$149$		−	19.0526i		−	1.56085i	0.625252	−	0.780423i	$-0.284996\pi$
$150$		0			0
$151$	−13.8564			−1.12762			−0.563809	−	0.825905i	$-0.690665\pi$
$151$	−13.8564			−1.12762			−0.563809	+	0.825905i	$0.690665\pi$
$152$		0			0
$153$	−3.00000			−0.242536
$154$		0			0
$155$		−	18.0000i		−	1.44579i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$			12.0000i			0.951662i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	17.3205			1.35665			0.678323	−	0.734763i	$-0.262707\pi$
$163$	17.3205			1.35665			0.678323	+	0.734763i	$0.262707\pi$
$164$		0			0
$165$	18.0000			1.40130
$166$		0			0
$167$	−17.3205			−1.34030			−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205			−1.34030			−0.670151	+	0.742225i	$0.733770\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−1.73205	+	4.00000i	−0.132453	+	0.305888i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$			2.00000i			0.151186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	6.00000i		−	0.448461i	−0.974536	−	0.224231i	$-0.928013\pi$
$179$		−	6.00000i		−	0.448461i	0.974536	−	0.224231i	$-0.0719869\pi$
$180$		0			0
$181$	20.0000			1.48659			0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000			1.48659			0.743294	+	0.668965i	$0.233262\pi$
$182$		0			0
$183$	3.46410			0.256074
$184$		0			0
$185$		−	3.46410i		−	0.254686i
$186$		0			0
$187$	15.5885			1.13994
$188$		0			0
$189$	−4.00000			−0.290957
$190$		0			0
$191$		−	9.00000i		−	0.651217i	−0.945505	−	0.325609i	$-0.894431\pi$
$191$		−	9.00000i		−	0.651217i	0.945505	−	0.325609i	$-0.105569\pi$
$192$		0			0
$193$		−	17.3205i		−	1.24676i	−0.781920	−	0.623379i	$-0.785760\pi$
$193$		−	17.3205i		−	1.24676i	0.781920	−	0.623379i	$-0.214240\pi$
$194$		0			0
$195$	−13.8564			−0.992278
$196$		0			0
$197$		−	20.7846i		−	1.48084i	−0.672143	−	0.740421i	$-0.734626\pi$
$197$		−	20.7846i		−	1.48084i	0.672143	−	0.740421i	$-0.265374\pi$
$198$		0			0
$199$			19.0000i			1.34687i	0.739244	+	0.673437i	$0.235183\pi$
$199$			19.0000i			1.34687i	−0.739244	+	0.673437i	$0.764817\pi$
$200$		0			0
$201$	−4.00000			−0.282138
$202$		0			0
$203$		−	6.00000i		−	0.421117i
$204$		0			0
$205$	6.00000			0.419058
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.00000	−	20.7846i	0.622543	−	1.43770i
$210$		0			0
$211$		−	4.00000i		−	0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$		−	4.00000i		−	0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$		0			0
$213$			13.8564i			0.949425i
$214$		0			0
$215$			9.00000i			0.613795i
$216$		0			0
$217$			10.3923i			0.705476i
$218$		0			0
$219$		−	22.0000i		−	1.48662i
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$	10.3923			0.695920			0.347960	−	0.937509i	$-0.386874\pi$
$223$	10.3923			0.695920			0.347960	+	0.937509i	$0.386874\pi$
$224$		0			0
$225$	−2.00000			−0.133333
$226$		0			0
$227$		−	24.0000i		−	1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$		−	24.0000i		−	1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$		0			0
$229$		−	25.9808i		−	1.71686i	−0.512933	−	0.858429i	$-0.671441\pi$
$229$		−	25.9808i		−	1.71686i	0.512933	−	0.858429i	$-0.328559\pi$
$230$		0			0
$231$	−10.3923			−0.683763
$232$		0			0
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$		0			0
$235$	15.5885			1.01688
$236$		0			0
$237$			6.92820i			0.450035i
$238$		0			0
$239$			9.00000i			0.582162i	0.956698	+	0.291081i	$0.0940149\pi$
$239$			9.00000i			0.582162i	−0.956698	+	0.291081i	$0.905985\pi$
$240$		0			0
$241$			3.46410i			0.223142i	0.993756	+	0.111571i	$0.0355883\pi$
$241$			3.46410i			0.223142i	−0.993756	+	0.111571i	$0.964412\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$		−	10.3923i		−	0.663940i
$246$		0			0
$247$	−6.92820	+	16.0000i	−0.440831	+	1.01806i
$248$		0			0
$249$		−	6.92820i		−	0.439057i
$250$		0			0
$251$	−12.1244			−0.765283			−0.382641	−	0.923897i	$-0.624985\pi$
$251$	−12.1244			−0.765283			−0.382641	+	0.923897i	$0.624985\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	10.3923			0.650791
$256$		0			0
$257$		−	27.7128i		−	1.72868i	−0.502910	−	0.864339i	$-0.667737\pi$
$257$		−	27.7128i		−	1.72868i	0.502910	−	0.864339i	$-0.332263\pi$
$258$		0			0
$259$			2.00000i			0.124274i
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	3.00000i		−	0.184988i	−0.995713	−	0.0924940i	$-0.970516\pi$
$263$		−	3.00000i		−	0.184988i	0.995713	−	0.0924940i	$-0.0294839\pi$
$264$		0			0
$265$		−	10.3923i		−	0.638394i
$266$		0			0
$267$	6.92820			0.423999
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$		−	28.0000i		−	1.70088i	−0.526073	−	0.850439i	$-0.676336\pi$
$271$		−	28.0000i		−	1.70088i	0.526073	−	0.850439i	$-0.323664\pi$
$272$		0			0
$273$	8.00000			0.484182
$274$		0			0
$275$	10.3923			0.626680
$276$		0			0
$277$			5.19615i			0.312207i	0.987741	+	0.156103i	$0.0498933\pi$
$277$			5.19615i			0.312207i	−0.987741	+	0.156103i	$0.950107\pi$
$278$		0			0
$279$	−10.3923			−0.622171
$280$		0			0
$281$		−	27.7128i		−	1.65321i	−0.562784	−	0.826604i	$-0.690270\pi$
$281$		−	27.7128i		−	1.65321i	0.562784	−	0.826604i	$-0.309730\pi$
$282$		0			0
$283$	−25.9808			−1.54440			−0.772198	−	0.635382i	$-0.780843\pi$
$283$	−25.9808			−1.54440			−0.772198	+	0.635382i	$0.780843\pi$
$284$		0			0
$285$	6.00000	−	13.8564i	0.355409	−	0.820783i
$286$		0			0
$287$	−3.46410			−0.204479
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	−27.7128			−1.62455
$292$		0			0
$293$	24.0000			1.40209			0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			20.7846i			1.20605i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	5.19615i		−	0.299501i
$302$		0			0
$303$	−27.7128			−1.59206
$304$		0			0
$305$	−3.00000			−0.171780
$306$		0			0
$307$			22.0000i			1.25561i	0.778372	+	0.627803i	$0.216046\pi$
$307$			22.0000i			1.25561i	−0.778372	+	0.627803i	$0.783954\pi$
$308$		0			0
$309$			13.8564i			0.788263i
$310$		0			0
$311$			33.0000i			1.87126i	0.352985	+	0.935629i	$0.385167\pi$
$311$			33.0000i			1.87126i	−0.352985	+	0.935629i	$0.614833\pi$
$312$		0			0
$313$	14.0000			0.791327			0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000			0.791327			0.395663	+	0.918396i	$0.370515\pi$
$314$		0			0
$315$	−1.73205			−0.0975900
$316$		0			0
$317$	−24.0000			−1.34797			−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000			−1.34797			−0.673987	+	0.738743i	$0.735420\pi$
$318$		0			0
$319$	−31.1769			−1.74557
$320$		0			0
$321$	−24.0000			−1.33955
$322$		0			0
$323$	5.19615	−	12.0000i	0.289122	−	0.667698i
$324$		0			0
$325$	−8.00000			−0.443760
$326$		0			0
$327$		−	28.0000i		−	1.54840i
$328$		0			0
$329$	−9.00000			−0.496186
$330$		0			0
$331$			26.0000i			1.42909i	0.699590	+	0.714545i	$0.253366\pi$
$331$			26.0000i			1.42909i	−0.699590	+	0.714545i	$0.746634\pi$
$332$		0			0
$333$	−2.00000			−0.109599
$334$		0			0
$335$	3.46410			0.189264
$336$		0			0
$337$			27.7128i			1.50961i	0.655947	+	0.754807i	$0.272269\pi$
$337$			27.7128i			1.50961i	−0.655947	+	0.754807i	$0.727731\pi$
$338$		0			0
$339$	34.6410			1.88144
$340$		0			0
$341$	54.0000			2.92426
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.5885			−0.836832			−0.418416	−	0.908255i	$-0.637415\pi$
$347$	−15.5885			−0.836832			−0.418416	+	0.908255i	$0.637415\pi$
$348$		0			0
$349$			8.66025i			0.463573i	0.972767	+	0.231786i	$0.0744570\pi$
$349$			8.66025i			0.463573i	−0.972767	+	0.231786i	$0.925543\pi$
$350$		0			0
$351$		−	16.0000i		−	0.854017i
$352$		0			0
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		−	12.0000i		−	0.636894i
$356$		0			0
$357$	−6.00000			−0.317554
$358$		0			0
$359$			15.0000i			0.791670i	0.918322	+	0.395835i	$0.129545\pi$
$359$			15.0000i			0.791670i	−0.918322	+	0.395835i	$0.870455\pi$
$360$		0			0
$361$	−13.0000	−	13.8564i	−0.684211	−	0.729285i
$362$		0			0
$363$			32.0000i			1.67956i
$364$		0			0
$365$			19.0526i			0.997257i
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		0			0
$369$		−	3.46410i		−	0.180334i
$370$		0			0
$371$			6.00000i			0.311504i
$372$		0			0
$373$	−16.0000			−0.828449			−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000			−0.828449			−0.414224	+	0.910175i	$0.635947\pi$
$374$		0			0
$375$	24.2487			1.25220
$376$		0			0
$377$	24.0000			1.23606
$378$		0			0
$379$		−	20.0000i		−	1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$		−	20.0000i		−	1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$		0			0
$381$			20.7846i			1.06483i
$382$		0			0
$383$	−31.1769			−1.59307			−0.796533	−	0.604595i	$-0.793335\pi$
$383$	−31.1769			−1.59307			−0.796533	+	0.604595i	$0.793335\pi$
$384$		0			0
$385$	9.00000			0.458682
$386$		0			0
$387$	5.19615			0.264135
$388$		0			0
$389$			5.19615i			0.263455i	0.991286	+	0.131728i	$0.0420524\pi$
$389$			5.19615i			0.263455i	−0.991286	+	0.131728i	$0.957948\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			3.46410i			0.174741i
$394$		0			0
$395$		−	6.00000i		−	0.301893i
$396$		0			0
$397$			25.9808i			1.30394i	0.758246	+	0.651969i	$0.226057\pi$
$397$			25.9808i			1.30394i	−0.758246	+	0.651969i	$0.773943\pi$
$398$		0			0
$399$	−3.46410	+	8.00000i	−0.173422	+	0.400501i
$400$		0			0
$401$		−	10.3923i		−	0.518967i	−0.965748	−	0.259483i	$-0.916448\pi$
$401$		−	10.3923i		−	0.518967i	0.965748	−	0.259483i	$-0.0835523\pi$
$402$		0			0
$403$	−41.5692			−2.07071
$404$		0			0
$405$			19.0526i			0.946729i
$406$		0			0
$407$	10.3923			0.515127
$408$		0			0
$409$		−	24.2487i		−	1.19902i	−0.800367	−	0.599511i	$-0.795362\pi$
$409$		−	24.2487i		−	1.19902i	0.800367	−	0.599511i	$-0.204638\pi$
$410$		0			0
$411$		−	18.0000i		−	0.887875i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			6.00000i			0.294528i
$416$		0			0
$417$			3.46410i			0.169638i
$418$		0			0
$419$	−10.3923			−0.507697			−0.253849	−	0.967244i	$-0.581697\pi$
$419$	−10.3923			−0.507697			−0.253849	+	0.967244i	$0.581697\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		−	9.00000i		−	0.437595i
$424$		0			0
$425$	6.00000			0.291043
$426$		0			0
$427$	1.73205			0.0838198
$428$		0			0
$429$		−	41.5692i		−	2.00698i
$430$		0			0
$431$	−27.7128			−1.33488			−0.667440	−	0.744664i	$-0.732610\pi$
$431$	−27.7128			−1.33488			−0.667440	+	0.744664i	$0.732610\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$		0			0
$435$	−20.7846			−0.996546
$436$		0			0
$437$		0			0
$438$		0			0
$439$	10.3923			0.495998			0.247999	−	0.968760i	$-0.420227\pi$
$439$	10.3923			0.495998			0.247999	+	0.968760i	$0.420227\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$	−1.73205			−0.0822922			−0.0411461	−	0.999153i	$-0.513101\pi$
$443$	−1.73205			−0.0822922			−0.0411461	+	0.999153i	$0.513101\pi$
$444$		0			0
$445$	−6.00000			−0.284427
$446$		0			0
$447$	38.1051			1.80231
$448$		0			0
$449$			6.92820i			0.326962i	0.986546	+	0.163481i	$0.0522723\pi$
$449$			6.92820i			0.326962i	−0.986546	+	0.163481i	$0.947728\pi$
$450$		0			0
$451$			18.0000i			0.847587i
$452$		0			0
$453$		−	27.7128i		−	1.30206i
$454$		0			0
$455$	−6.92820			−0.324799
$456$		0			0
$457$	−29.0000			−1.35656			−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000			−1.35656			−0.678281	+	0.734802i	$0.737275\pi$
$458$		0			0
$459$			12.0000i			0.560112i
$460$		0			0
$461$			22.5167i			1.04871i	0.851501	+	0.524353i	$0.175693\pi$
$461$			22.5167i			1.04871i	−0.851501	+	0.524353i	$0.824307\pi$
$462$		0			0
$463$		−	25.0000i		−	1.16185i	−0.813958	−	0.580924i	$-0.802691\pi$
$463$		−	25.0000i		−	1.16185i	0.813958	−	0.580924i	$-0.197309\pi$
$464$		0			0
$465$	36.0000			1.66946
$466$		0			0
$467$	−15.5885			−0.721348			−0.360674	−	0.932692i	$-0.617453\pi$
$467$	−15.5885			−0.721348			−0.360674	+	0.932692i	$0.617453\pi$
$468$		0			0
$469$	−2.00000			−0.0923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−27.0000			−1.24146
$474$		0			0
$475$	3.46410	−	8.00000i	0.158944	−	0.367065i
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$		0			0
$481$	−8.00000			−0.364769
$482$		0			0
$483$		0			0
$484$		0			0
$485$	24.0000			1.08978
$486$		0			0
$487$	27.7128			1.25579			0.627894	−	0.778299i	$-0.283917\pi$
$487$	27.7128			1.25579			0.627894	+	0.778299i	$0.283917\pi$
$488$		0			0
$489$			34.6410i			1.56652i
$490$		0			0
$491$	24.2487			1.09433			0.547165	−	0.837025i	$-0.315707\pi$
$491$	24.2487			1.09433			0.547165	+	0.837025i	$0.315707\pi$
$492$		0			0
$493$	−18.0000			−0.810679
$494$		0			0
$495$			9.00000i			0.404520i
$496$		0			0
$497$			6.92820i			0.310772i
$498$		0			0
$499$	36.3731			1.62828			0.814141	−	0.580667i	$-0.197208\pi$
$499$	36.3731			1.62828			0.814141	+	0.580667i	$0.197208\pi$
$500$		0			0
$501$		−	34.6410i		−	1.54765i
$502$		0			0
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$		0			0
$505$	24.0000			1.06799
$506$		0			0
$507$			6.00000i			0.266469i
$508$		0			0
$509$	24.0000			1.06378			0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000			1.06378			0.531891	+	0.846813i	$0.321482\pi$
$510$		0			0
$511$		−	11.0000i		−	0.486611i
$512$		0			0
$513$	16.0000	+	6.92820i	0.706417	+	0.305888i
$514$		0			0
$515$		−	12.0000i		−	0.528783i
$516$		0			0
$517$			46.7654i			2.05674i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.3205i			0.758825i	0.925228	+	0.379413i	$0.123874\pi$
$521$			17.3205i			0.758825i	−0.925228	+	0.379413i	$0.876126\pi$
$522$		0			0
$523$			8.00000i			0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$			8.00000i			0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$		0			0
$525$	−4.00000			−0.174574
$526$		0			0
$527$	31.1769			1.35809
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	13.8564i		−	0.600188i
$534$		0			0
$535$	20.7846			0.898597
$536$		0			0
$537$	12.0000			0.517838
$538$		0			0
$539$	31.1769			1.34288
$540$		0			0
$541$		−	8.66025i		−	0.372333i	−0.982518	−	0.186167i	$-0.940394\pi$
$541$		−	8.66025i		−	0.372333i	0.982518	−	0.186167i	$-0.0596065\pi$
$542$		0			0
$543$			40.0000i			1.71656i
$544$		0			0
$545$			24.2487i			1.03870i
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$			1.73205i			0.0739221i
$550$		0			0
$551$	−10.3923	+	24.0000i	−0.442727	+	1.02243i
$552$		0			0
$553$			3.46410i			0.147309i
$554$		0			0
$555$	6.92820			0.294086
$556$		0			0
$557$		−	8.66025i		−	0.366947i	−0.983025	−	0.183473i	$-0.941266\pi$
$557$		−	8.66025i		−	0.366947i	0.983025	−	0.183473i	$-0.0587341\pi$
$558$		0			0
$559$	20.7846			0.879095
$560$		0			0
$561$			31.1769i			1.31629i
$562$		0			0
$563$			30.0000i			1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$			30.0000i			1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$		0			0
$565$	−30.0000			−1.26211
$566$		0			0
$567$		−	11.0000i		−	0.461957i
$568$		0			0
$569$		−	27.7128i		−	1.16178i	−0.813982	−	0.580891i	$-0.802704\pi$
$569$		−	27.7128i		−	1.16178i	0.813982	−	0.580891i	$-0.197296\pi$
$570$		0			0
$571$	−3.46410			−0.144968			−0.0724841	−	0.997370i	$-0.523093\pi$
$571$	−3.46410			−0.144968			−0.0724841	+	0.997370i	$0.523093\pi$
$572$		0			0
$573$	18.0000			0.751961
$574$		0			0
$575$		0			0
$576$		0			0
$577$	29.0000			1.20729			0.603643	−	0.797255i	$-0.293715\pi$
$577$	29.0000			1.20729			0.603643	+	0.797255i	$0.293715\pi$
$578$		0			0
$579$	34.6410			1.43963
$580$		0			0
$581$		−	3.46410i		−	0.143715i
$582$		0			0
$583$	31.1769			1.29122
$584$		0			0
$585$		−	6.92820i		−	0.286446i
$586$		0			0
$587$	−32.9090			−1.35830			−0.679149	−	0.734000i	$-0.737651\pi$
$587$	−32.9090			−1.35830			−0.679149	+	0.734000i	$0.737651\pi$
$588$		0			0
$589$	18.0000	−	41.5692i	0.741677	−	1.71283i
$590$		0			0
$591$	41.5692			1.70993
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$	5.19615			0.213021
$596$		0			0
$597$	−38.0000			−1.55524
$598$		0			0
$599$	−38.1051			−1.55693			−0.778466	−	0.627686i	$-0.784002\pi$
$599$	−38.1051			−1.55693			−0.778466	+	0.627686i	$0.784002\pi$
$600$		0			0
$601$			41.5692i			1.69564i	0.530281	+	0.847822i	$0.322086\pi$
$601$			41.5692i			1.69564i	−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$		−	27.7128i		−	1.12669i
$606$		0			0
$607$	6.92820			0.281207			0.140604	−	0.990066i	$-0.455096\pi$
$607$	6.92820			0.281207			0.140604	+	0.990066i	$0.455096\pi$
$608$		0			0
$609$	12.0000			0.486265
$610$		0			0
$611$		−	36.0000i		−	1.45640i
$612$		0			0
$613$			39.8372i			1.60901i	0.593947	+	0.804504i	$0.297569\pi$
$613$			39.8372i			1.60901i	−0.593947	+	0.804504i	$0.702431\pi$
$614$		0			0
$615$			12.0000i			0.483887i
$616$		0			0
$617$	−3.00000			−0.120775			−0.0603877	−	0.998175i	$-0.519234\pi$
$617$	−3.00000			−0.120775			−0.0603877	+	0.998175i	$0.519234\pi$
$618$		0			0
$619$	24.2487			0.974638			0.487319	−	0.873224i	$-0.337975\pi$
$619$	24.2487			0.974638			0.487319	+	0.873224i	$0.337975\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.46410			0.138786
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	41.5692	+	18.0000i	1.66011	+	0.718851i
$628$		0			0
$629$	6.00000			0.239236
$630$		0			0
$631$			7.00000i			0.278666i	0.990246	+	0.139333i	$0.0444958\pi$
$631$			7.00000i			0.278666i	−0.990246	+	0.139333i	$0.955504\pi$
$632$		0			0
$633$	8.00000			0.317971
$634$		0			0
$635$		−	18.0000i		−	0.714308i
$636$		0			0
$637$	−24.0000			−0.950915
$638$		0			0
$639$	−6.92820			−0.274075
$640$		0			0
$641$		−	34.6410i		−	1.36824i	−0.729370	−	0.684119i	$-0.760187\pi$
$641$		−	34.6410i		−	1.36824i	0.729370	−	0.684119i	$-0.239813\pi$
$642$		0			0
$643$	−5.19615			−0.204916			−0.102458	−	0.994737i	$-0.532671\pi$
$643$	−5.19615			−0.204916			−0.102458	+	0.994737i	$0.532671\pi$
$644$		0			0
$645$	−18.0000			−0.708749
$646$		0			0
$647$		−	9.00000i		−	0.353827i	−0.984226	−	0.176913i	$-0.943389\pi$
$647$		−	9.00000i		−	0.353827i	0.984226	−	0.176913i	$-0.0566112\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−20.7846			−0.814613
$652$		0			0
$653$		−	8.66025i		−	0.338902i	−0.985539	−	0.169451i	$-0.945801\pi$
$653$		−	8.66025i		−	0.338902i	0.985539	−	0.169451i	$-0.0541994\pi$
$654$		0			0
$655$		−	3.00000i		−	0.117220i
$656$		0			0
$657$	11.0000			0.429151
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−40.0000			−1.55582			−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000			−1.55582			−0.777910	+	0.628376i	$0.783720\pi$
$662$		0			0
$663$		−	24.0000i		−	0.932083i
$664$		0			0
$665$	3.00000	−	6.92820i	0.116335	−	0.268664i
$666$		0			0
$667$		0			0
$668$		0			0
$669$			20.7846i			0.803579i
$670$		0			0
$671$		−	9.00000i		−	0.347441i
$672$		0			0
$673$			13.8564i			0.534125i	0.963679	+	0.267063i	$0.0860531\pi$
$673$			13.8564i			0.534125i	−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$			8.00000i			0.307920i
$676$		0			0
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$	−13.8564			−0.531760
$680$		0			0
$681$	48.0000			1.83936
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$			15.5885i			0.595604i
$686$		0			0
$687$	51.9615			1.98246
$688$		0			0
$689$	−24.0000			−0.914327
$690$		0			0
$691$	5.19615			0.197671			0.0988355	−	0.995104i	$-0.468488\pi$
$691$	5.19615			0.197671			0.0988355	+	0.995104i	$0.468488\pi$
$692$		0			0
$693$		−	5.19615i		−	0.197386i
$694$		0			0
$695$		−	3.00000i		−	0.113796i
$696$		0			0
$697$			10.3923i			0.393637i
$698$		0			0
$699$		−	18.0000i		−	0.680823i
$700$		0			0
$701$			27.7128i			1.04670i	0.852118	+	0.523349i	$0.175318\pi$
$701$			27.7128i			1.04670i	−0.852118	+	0.523349i	$0.824682\pi$
$702$		0			0
$703$	3.46410	−	8.00000i	0.130651	−	0.301726i
$704$		0			0
$705$			31.1769i			1.17419i
$706$		0			0
$707$	−13.8564			−0.521124
$708$		0			0
$709$		−	13.8564i		−	0.520388i	−0.965556	−	0.260194i	$-0.916213\pi$
$709$		−	13.8564i		−	0.520388i	0.965556	−	0.260194i	$-0.0837866\pi$
$710$		0			0
$711$	−3.46410			−0.129914
$712$		0			0
$713$		0			0
$714$		0			0
$715$			36.0000i			1.34632i
$716$		0			0
$717$	−18.0000			−0.672222
$718$		0			0
$719$			51.0000i			1.90198i	0.309223	+	0.950990i	$0.399931\pi$
$719$			51.0000i			1.90198i	−0.309223	+	0.950990i	$0.600069\pi$
$720$		0			0
$721$			6.92820i			0.258020i
$722$		0			0
$723$	−6.92820			−0.257663
$724$		0			0
$725$	−12.0000			−0.445669
$726$		0			0
$727$		−	43.0000i		−	1.59478i	−0.603463	−	0.797391i	$-0.706213\pi$
$727$		−	43.0000i		−	1.59478i	0.603463	−	0.797391i	$-0.293787\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	−15.5885			−0.576560
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	20.7846			0.766652
$736$		0			0
$737$			10.3923i			0.382805i
$738$		0			0
$739$	−15.5885			−0.573431			−0.286715	−	0.958016i	$-0.592563\pi$
$739$	−15.5885			−0.573431			−0.286715	+	0.958016i	$0.592563\pi$
$740$		0			0
$741$	−32.0000	−	13.8564i	−1.17555	−	0.509028i
$742$		0			0
$743$	−24.2487			−0.889599			−0.444799	−	0.895630i	$-0.646725\pi$
$743$	−24.2487			−0.889599			−0.444799	+	0.895630i	$0.646725\pi$
$744$		0			0
$745$	−33.0000			−1.20903
$746$		0			0
$747$	3.46410			0.126745
$748$		0			0
$749$	−12.0000			−0.438470
$750$		0			0
$751$	−27.7128			−1.01125			−0.505627	−	0.862752i	$-0.668739\pi$
$751$	−27.7128			−1.01125			−0.505627	+	0.862752i	$0.668739\pi$
$752$		0			0
$753$		−	24.2487i		−	0.883672i
$754$		0			0
$755$			24.0000i			0.873449i
$756$		0			0
$757$		−	12.1244i		−	0.440667i	−0.975425	−	0.220334i	$-0.929285\pi$
$757$		−	12.1244i		−	0.440667i	0.975425	−	0.220334i	$-0.0707146\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	27.0000			0.978749			0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000			0.978749			0.489375	+	0.872074i	$0.337225\pi$
$762$		0			0
$763$		−	14.0000i		−	0.506834i
$764$		0			0
$765$			5.19615i			0.187867i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−23.0000			−0.829401			−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000			−0.829401			−0.414701	+	0.909958i	$0.636114\pi$
$770$		0			0
$771$	55.4256			1.99611
$772$		0			0
$773$	24.0000			0.863220			0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000			0.863220			0.431610	+	0.902060i	$0.357946\pi$
$774$		0			0
$775$	20.7846			0.746605
$776$		0			0
$777$	−4.00000			−0.143499
$778$		0			0
$779$	13.8564	+	6.00000i	0.496457	+	0.214972i
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$		−	24.0000i		−	0.857690i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			4.00000i			0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$			4.00000i			0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$		0			0
$789$	6.00000			0.213606
$790$		0			0
$791$	17.3205			0.615846
$792$		0			0
$793$			6.92820i			0.246028i
$794$		0			0
$795$	20.7846			0.737154
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$			27.0000i			0.955191i
$800$		0			0
$801$			3.46410i			0.122398i
$802$		0			0
$803$	−57.1577			−2.01705
$804$		0			0
$805$		0			0
$806$		0			0
$807$			12.0000i			0.422420i
$808$		0			0
$809$	21.0000			0.738321			0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000			0.738321			0.369160	+	0.929366i	$0.379645\pi$
$810$		0			0
$811$			40.0000i			1.40459i	0.711886	+	0.702295i	$0.247841\pi$
$811$			40.0000i			1.40459i	−0.711886	+	0.702295i	$0.752159\pi$
$812$		0			0
$813$	56.0000			1.96401
$814$		0			0
$815$		−	30.0000i		−	1.05085i
$816$		0			0
$817$	−9.00000	+	20.7846i	−0.314870	+	0.727161i
$818$		0			0
$819$			4.00000i			0.139771i
$820$		0			0
$821$			22.5167i			0.785837i	0.919573	+	0.392918i	$0.128535\pi$
$821$			22.5167i			0.785837i	−0.919573	+	0.392918i	$0.871465\pi$
$822$		0			0
$823$		−	29.0000i		−	1.01088i	−0.862863	−	0.505438i	$-0.831331\pi$
$823$		−	29.0000i		−	1.01088i	0.862863	−	0.505438i	$-0.168669\pi$
$824$		0			0
$825$			20.7846i			0.723627i
$826$		0			0
$827$		−	18.0000i		−	0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$		−	18.0000i		−	0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$	−10.3923			−0.360505
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$			30.0000i			1.03819i
$836$		0			0
$837$			41.5692i			1.43684i
$838$		0			0
$839$	−20.7846			−0.717564			−0.358782	−	0.933421i	$-0.616808\pi$
$839$	−20.7846			−0.717564			−0.358782	+	0.933421i	$0.616808\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	55.4256			1.90896
$844$		0			0
$845$		−	5.19615i		−	0.178753i
$846$		0			0
$847$			16.0000i			0.549767i
$848$		0			0
$849$		−	51.9615i		−	1.78331i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			55.4256i			1.89774i	0.315671	+	0.948869i	$0.397770\pi$
$853$			55.4256i			1.89774i	−0.315671	+	0.948869i	$0.602230\pi$
$854$		0			0
$855$	6.92820	+	3.00000i	0.236940	+	0.102598i
$856$		0			0
$857$			41.5692i			1.41998i	0.704213	+	0.709989i	$0.251300\pi$
$857$			41.5692i			1.41998i	−0.704213	+	0.709989i	$0.748700\pi$
$858$		0			0
$859$	−53.6936			−1.83200			−0.916001	−	0.401177i	$-0.868601\pi$
$859$	−53.6936			−1.83200			−0.916001	+	0.401177i	$0.868601\pi$
$860$		0			0
$861$		−	6.92820i		−	0.236113i
$862$		0			0
$863$	−24.2487			−0.825436			−0.412718	−	0.910859i	$-0.635420\pi$
$863$	−24.2487			−0.825436			−0.412718	+	0.910859i	$0.635420\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	16.0000i		−	0.543388i
$868$		0			0
$869$	18.0000			0.610608
$870$		0			0
$871$		−	8.00000i		−	0.271070i
$872$		0			0
$873$		−	13.8564i		−	0.468968i
$874$		0			0
$875$	12.1244			0.409878
$876$		0			0
$877$	52.0000			1.75592			0.877958	−	0.478738i	$-0.158906\pi$
$877$	52.0000			1.75592			0.877958	+	0.478738i	$0.158906\pi$
$878$		0			0
$879$			48.0000i			1.61900i
$880$		0			0
$881$	−51.0000			−1.71823			−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000			−1.71823			−0.859117	+	0.511780i	$0.828986\pi$
$882$		0			0
$883$	−8.66025			−0.291441			−0.145720	−	0.989326i	$-0.546550\pi$
$883$	−8.66025			−0.291441			−0.145720	+	0.989326i	$0.546550\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.92820			0.232626			0.116313	−	0.993213i	$-0.462892\pi$
$887$	6.92820			0.232626			0.116313	+	0.993213i	$0.462892\pi$
$888$		0			0
$889$			10.3923i			0.348547i
$890$		0			0
$891$	−57.1577			−1.91485
$892$		0			0
$893$	36.0000	+	15.5885i	1.20469	+	0.521648i
$894$		0			0
$895$	−10.3923			−0.347376
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−62.3538			−2.07962
$900$		0			0
$901$	18.0000			0.599667
$902$		0			0
$903$	10.3923			0.345834
$904$		0			0
$905$		−	34.6410i		−	1.15151i
$906$		0			0
$907$		−	10.0000i		−	0.332045i	−0.986122	−	0.166022i	$-0.946908\pi$
$907$		−	10.0000i		−	0.332045i	0.986122	−	0.166022i	$-0.0530924\pi$
$908$		0			0
$909$		−	13.8564i		−	0.459588i
$910$		0			0
$911$	13.8564			0.459083			0.229542	−	0.973299i	$-0.426277\pi$
$911$	13.8564			0.459083			0.229542	+	0.973299i	$0.426277\pi$
$912$		0			0
$913$	−18.0000			−0.595713
$914$		0			0
$915$		−	6.00000i		−	0.198354i
$916$		0			0
$917$			1.73205i			0.0571974i
$918$		0			0
$919$			16.0000i			0.527791i	0.964551	+	0.263896i	$0.0850075\pi$
$919$			16.0000i			0.527791i	−0.964551	+	0.263896i	$0.914993\pi$
$920$		0			0
$921$	−44.0000			−1.44985
$922$		0			0
$923$	−27.7128			−0.912178
$924$		0			0
$925$	4.00000			0.131519
$926$		0			0
$927$	−6.92820			−0.227552
$928$		0			0
$929$	−6.00000			−0.196854			−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000			−0.196854			−0.0984268	+	0.995144i	$0.531381\pi$
$930$		0			0
$931$	10.3923	−	24.0000i	0.340594	−	0.786568i
$932$		0			0
$933$	−66.0000			−2.16074
$934$		0			0
$935$		−	27.0000i		−	0.882994i
$936$		0			0
$937$	−17.0000			−0.555366			−0.277683	−	0.960673i	$-0.589566\pi$
$937$	−17.0000			−0.555366			−0.277683	+	0.960673i	$0.589566\pi$
$938$		0			0
$939$			28.0000i			0.913745i
$940$		0			0
$941$	−36.0000			−1.17357			−0.586783	−	0.809744i	$-0.699606\pi$
$941$	−36.0000			−1.17357			−0.586783	+	0.809744i	$0.699606\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$			6.92820i			0.225374i
$946$		0			0
$947$	24.2487			0.787977			0.393989	−	0.919115i	$-0.371095\pi$
$947$	24.2487			0.787977			0.393989	+	0.919115i	$0.371095\pi$
$948$		0			0
$949$	44.0000			1.42830
$950$		0			0
$951$		−	48.0000i		−	1.55651i
$952$		0			0
$953$		−	10.3923i		−	0.336640i	−0.985732	−	0.168320i	$-0.946166\pi$
$953$		−	10.3923i		−	0.336640i	0.985732	−	0.168320i	$-0.0538342\pi$
$954$		0			0
$955$	−15.5885			−0.504431
$956$		0			0
$957$		−	62.3538i		−	2.01561i
$958$		0			0
$959$		−	9.00000i		−	0.290625i
$960$		0			0
$961$	77.0000			2.48387
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$	−30.0000			−0.965734
$966$		0			0
$967$		−	52.0000i		−	1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$		−	52.0000i		−	1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$		0			0
$969$	24.0000	+	10.3923i	0.770991	+	0.333849i
$970$		0			0
$971$		−	48.0000i		−	1.54039i	−0.637806	−	0.770197i	$-0.720158\pi$
$971$		−	48.0000i		−	1.54039i	0.637806	−	0.770197i	$-0.279842\pi$
$972$		0			0
$973$			1.73205i			0.0555270i
$974$		0			0
$975$		−	16.0000i		−	0.512410i
$976$		0			0
$977$		−	31.1769i		−	0.997438i	−0.866764	−	0.498719i	$-0.833804\pi$
$977$		−	31.1769i		−	0.997438i	0.866764	−	0.498719i	$-0.166196\pi$
$978$		0			0
$979$		−	18.0000i		−	0.575282i
$980$		0			0
$981$	14.0000			0.446986
$982$		0			0
$983$	−34.6410			−1.10488			−0.552438	−	0.833554i	$-0.686303\pi$
$983$	−34.6410			−1.10488			−0.552438	+	0.833554i	$0.686303\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$		0			0
$987$		−	18.0000i		−	0.572946i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−38.1051			−1.21045			−0.605224	−	0.796055i	$-0.706917\pi$
$991$	−38.1051			−1.21045			−0.605224	+	0.796055i	$0.706917\pi$
$992$		0			0
$993$	−52.0000			−1.65017
$994$		0			0
$995$	32.9090			1.04328
$996$		0			0
$997$		−	36.3731i		−	1.15195i	−0.817469	−	0.575973i	$-0.804623\pi$
$997$		−	36.3731i		−	1.15195i	0.817469	−	0.575973i	$-0.195377\pi$
$998$		0			0
$999$			8.00000i			0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.b.a.607.3	yes	4
4.3	odd	2	inner	1216.2.b.a.607.1	✓	4
8.3	odd	2		1216.2.b.b.607.4	yes	4
8.5	even	2		1216.2.b.b.607.2	yes	4
19.18	odd	2		1216.2.b.b.607.1	yes	4
76.75	even	2		1216.2.b.b.607.3	yes	4
152.37	odd	2	inner	1216.2.b.a.607.4	yes	4
152.75	even	2	inner	1216.2.b.a.607.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.b.a.607.1	✓	4	4.3	odd	2	inner
1216.2.b.a.607.2	yes	4	152.75	even	2	inner
1216.2.b.a.607.3	yes	4	1.1	even	1	trivial
1216.2.b.a.607.4	yes	4	152.37	odd	2	inner
1216.2.b.b.607.1	yes	4	19.18	odd	2
1216.2.b.b.607.2	yes	4	8.5	even	2
1216.2.b.b.607.3	yes	4	76.75	even	2
1216.2.b.b.607.4	yes	4	8.3	odd	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} -1.73205i q^{5} +1.00000i q^{7} -1.00000 q^{9} +5.19615 q^{11} -4.00000 q^{13} +3.46410 q^{15} +3.00000 q^{17} +(1.73205 - 4.00000i) q^{19} -2.00000 q^{21} +2.00000 q^{25} +4.00000i q^{27} -6.00000 q^{29} +10.3923 q^{31} +10.3923i q^{33} +1.73205 q^{35} +2.00000 q^{37} -8.00000i q^{39} +3.46410i q^{41} -5.19615 q^{43} +1.73205i q^{45} +9.00000i q^{47} +6.00000 q^{49} +6.00000i q^{51} +6.00000 q^{53} -9.00000i q^{55} +(8.00000 + 3.46410i) q^{57} -1.73205i q^{61} -1.00000i q^{63} +6.92820i q^{65} +2.00000i q^{67} +6.92820 q^{71} -11.0000 q^{73} +4.00000i q^{75} +5.19615i q^{77} +3.46410 q^{79} -11.0000 q^{81} -3.46410 q^{83} -5.19615i q^{85} -12.0000i q^{87} -3.46410i q^{89} -4.00000i q^{91} +20.7846i q^{93} +(-6.92820 - 3.00000i) q^{95} +13.8564i q^{97} -5.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9} - 16 q^{13} + 12 q^{17} - 8 q^{21} + 8 q^{25} - 24 q^{29} + 8 q^{37} + 24 q^{49} + 24 q^{53} + 32 q^{57} - 44 q^{73} - 44 q^{81}+O(q^{100}) \) 4 * q - 4 * q^9 - 16 * q^13 + 12 * q^17 - 8 * q^21 + 8 * q^25 - 24 * q^29 + 8 * q^37 + 24 * q^49 + 24 * q^53 + 32 * q^57 - 44 * q^73 - 44 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more