LMFDB - Embedded newform 1216.2.c.g.609.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(609,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1216.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$9.70980888579$
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			609.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.g.609.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -3.46410i q^{5} -1.73205 q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -3.46410i q^{5} -1.73205 q^{7} +2.00000 q^{9} -5.19615i q^{13} -3.46410 q^{15} -3.00000 q^{17} +1.00000i q^{19} +1.73205i q^{21} +1.73205 q^{23} -7.00000 q^{25} -5.00000i q^{27} -1.73205i q^{29} +3.46410 q^{31} +6.00000i q^{35} -5.19615 q^{39} +10.0000i q^{43} -6.92820i q^{45} -10.3923 q^{47} -4.00000 q^{49} +3.00000i q^{51} +5.19615i q^{53} +1.00000 q^{57} -9.00000i q^{59} -10.3923i q^{61} -3.46410 q^{63} -18.0000 q^{65} +13.0000i q^{67} -1.73205i q^{69} +3.46410 q^{71} -1.00000 q^{73} +7.00000i q^{75} -10.3923 q^{79} +1.00000 q^{81} +10.3923i q^{85} -1.73205 q^{87} +18.0000 q^{89} +9.00000i q^{91} -3.46410i q^{93} +3.46410 q^{95} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{9}+O(q^{10}) $ 4 * q + 8 * q^9	$ 4 q + 8 q^{9} - 12 q^{17} - 28 q^{25} - 16 q^{49} + 4 q^{57} - 72 q^{65} - 4 q^{73} + 4 q^{81} + 72 q^{89} - 56 q^{97}+O(q^{100}) $ 4 * q + 8 * q^9 - 12 * q^17 - 28 * q^25 - 16 * q^49 + 4 * q^57 - 72 * q^65 - 4 * q^73 + 4 * q^81 + 72 * q^89 - 56 * q^97

Display 100 coefficients 10 coefficients

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	− 3.46410i	− 1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	− 3.46410i	− 1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$	0	0
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 5.19615i	− 1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	− 5.19615i	− 1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$	0	0
$15$	−3.46410	−0.894427
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	1.73205i	0.377964i
$22$	0	0
$23$	1.73205	0.361158	0.180579	−	0.983561i	$-0.442203\pi$
$23$	1.73205	0.361158	0.180579	+	0.983561i	$0.442203\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	− 1.73205i	− 0.321634i	−0.986984	−	0.160817i	$-0.948587\pi$
$29$	− 1.73205i	− 0.321634i	0.986984	−	0.160817i	$-0.0514129\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.00000i	1.01419i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−5.19615	−0.832050
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	− 6.92820i	− 1.03280i
$46$	0	0
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	3.00000i	0.420084i
$52$	0	0
$53$	5.19615i	0.713746i	0.934153	+	0.356873i	$0.116157\pi$
$53$	5.19615i	0.713746i	−0.934153	+	0.356873i	$0.883843\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	− 9.00000i	− 1.17170i	−0.810419	−	0.585850i	$-0.800761\pi$
$59$	− 9.00000i	− 1.17170i	0.810419	−	0.585850i	$-0.199239\pi$
$60$	0	0
$61$	− 10.3923i	− 1.33060i	−0.746577	−	0.665299i	$-0.768304\pi$
$61$	− 10.3923i	− 1.33060i	0.746577	−	0.665299i	$-0.231696\pi$
$62$	0	0
$63$	−3.46410	−0.436436
$64$	0	0
$65$	−18.0000	−2.23263
$66$	0	0
$67$	13.0000i	1.58820i	0.607785	+	0.794101i	$0.292058\pi$
$67$	13.0000i	1.58820i	−0.607785	+	0.794101i	$0.707942\pi$
$68$	0	0
$69$	− 1.73205i	− 0.208514i
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	−1.00000	−0.117041	−0.0585206	−	0.998286i	$-0.518638\pi$
$73$	−1.00000	−0.117041	−0.0585206	+	0.998286i	$0.518638\pi$
$74$	0	0
$75$	7.00000i	0.808290i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.3923	−1.16923	−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923	−1.16923	−0.584613	+	0.811312i	$0.698754\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	10.3923i	1.12720i
$86$	0	0
$87$	−1.73205	−0.185695
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	9.00000i	0.943456i
$92$	0	0
$93$	− 3.46410i	− 0.359211i
$94$	0	0
$95$	3.46410	0.355409
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−13.8564	−1.36531	−0.682656	−	0.730740i	$-0.739175\pi$
$103$	−13.8564	−1.36531	−0.682656	+	0.730740i	$0.739175\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	0	0
$109$	− 5.19615i	− 0.497701i	−0.968542	−	0.248851i	$-0.919947\pi$
$109$	− 5.19615i	− 0.497701i	0.968542	−	0.248851i	$-0.0800528\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	− 6.00000i	− 0.559503i
$116$	0	0
$117$	− 10.3923i	− 0.960769i
$118$	0	0
$119$	5.19615	0.476331
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	6.92820i	0.619677i
$126$	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
$129$	10.0000	0.880451
$130$	0	0
$131$	− 18.0000i	− 1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$	− 18.0000i	− 1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$	0	0
$133$	− 1.73205i	− 0.150188i
$134$	0	0
$135$	−17.3205	−1.49071
$136$	0	0
$137$	21.0000	1.79415	0.897076	−	0.441877i	$-0.145687\pi$
$137$	21.0000	1.79415	0.897076	+	0.441877i	$0.145687\pi$
$138$	0	0
$139$	− 22.0000i	− 1.86602i	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	− 22.0000i	− 1.86602i	0.359856	−	0.933008i	$-0.382826\pi$
$140$	0	0
$141$	10.3923i	0.875190i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	4.00000i	0.329914i
$148$	0	0
$149$	− 20.7846i	− 1.70274i	−0.524564	−	0.851371i	$-0.675772\pi$
$149$	− 20.7846i	− 1.70274i	0.524564	−	0.851371i	$-0.324228\pi$
$150$	0	0
$151$	13.8564	1.12762	0.563809	−	0.825905i	$-0.309335\pi$
$151$	13.8564	1.12762	0.563809	+	0.825905i	$0.309335\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	− 12.0000i	− 0.963863i
$156$	0	0
$157$	3.46410i	0.276465i	0.990400	+	0.138233i	$0.0441422\pi$
$157$	3.46410i	0.276465i	−0.990400	+	0.138233i	$0.955858\pi$
$158$	0	0
$159$	5.19615	0.412082
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	14.0000i	1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$	14.0000i	1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	−14.0000	−1.07692
$170$	0	0
$171$	2.00000i	0.152944i
$172$	0	0
$173$	20.7846i	1.58022i	0.612962	+	0.790112i	$0.289978\pi$
$173$	20.7846i	1.58022i	−0.612962	+	0.790112i	$0.710022\pi$
$174$	0	0
$175$	12.1244	0.916515
$176$	0	0
$177$	−9.00000	−0.676481
$178$	0	0
$179$	− 12.0000i	− 0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$	− 12.0000i	− 0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$	0	0
$181$	13.8564i	1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$	13.8564i	1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$	0	0
$183$	−10.3923	−0.768221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	8.66025i	0.629941i
$190$	0	0
$191$	1.73205	0.125327	0.0626634	−	0.998035i	$-0.480041\pi$
$191$	1.73205	0.125327	0.0626634	+	0.998035i	$0.480041\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	18.0000i	1.28901i
$196$	0	0
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	22.5167	1.59616	0.798082	−	0.602549i	$-0.205848\pi$
$199$	22.5167	1.59616	0.798082	+	0.602549i	$0.205848\pi$
$200$	0	0
$201$	13.0000	0.916949
$202$	0	0
$203$	3.00000i	0.210559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.46410	0.240772
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 5.00000i	− 0.344214i	−0.985078	−	0.172107i	$-0.944942\pi$
$211$	− 5.00000i	− 0.344214i	0.985078	−	0.172107i	$-0.0550575\pi$
$212$	0	0
$213$	− 3.46410i	− 0.237356i
$214$	0	0
$215$	34.6410	2.36250
$216$	0	0
$217$	−6.00000	−0.407307
$218$	0	0
$219$	1.00000i	0.0675737i
$220$	0	0
$221$	15.5885i	1.04859i
$222$	0	0
$223$	6.92820	0.463947	0.231973	−	0.972722i	$-0.425482\pi$
$223$	6.92820	0.463947	0.231973	+	0.972722i	$0.425482\pi$
$224$	0	0
$225$	−14.0000	−0.933333
$226$	0	0
$227$	− 15.0000i	− 0.995585i	−0.867296	−	0.497792i	$-0.834144\pi$
$227$	− 15.0000i	− 0.995585i	0.867296	−	0.497792i	$-0.165856\pi$
$228$	0	0
$229$	− 3.46410i	− 0.228914i	−0.993428	−	0.114457i	$-0.963487\pi$
$229$	− 3.46410i	− 0.228914i	0.993428	−	0.114457i	$-0.0365129\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	36.0000i	2.34838i
$236$	0	0
$237$	10.3923i	0.675053i
$238$	0	0
$239$	19.0526	1.23241	0.616204	−	0.787587i	$-0.288670\pi$
$239$	19.0526	1.23241	0.616204	+	0.787587i	$0.288670\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	13.8564i	0.885253i
$246$	0	0
$247$	5.19615	0.330623
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.0000i	1.13615i	0.822977	+	0.568075i	$0.192312\pi$
$251$	18.0000i	1.13615i	−0.822977	+	0.568075i	$0.807688\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	10.3923	0.650791
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 3.46410i	− 0.214423i
$262$	0	0
$263$	24.2487	1.49524	0.747620	−	0.664127i	$-0.231197\pi$
$263$	24.2487	1.49524	0.747620	+	0.664127i	$0.231197\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	− 18.0000i	− 1.10158i
$268$	0	0
$269$	27.7128i	1.68968i	0.535019	+	0.844840i	$0.320304\pi$
$269$	27.7128i	1.68968i	−0.535019	+	0.844840i	$0.679696\pi$
$270$	0	0
$271$	−22.5167	−1.36779	−0.683895	−	0.729581i	$-0.739715\pi$
$271$	−22.5167	−1.36779	−0.683895	+	0.729581i	$0.739715\pi$
$272$	0	0
$273$	9.00000	0.544705
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 10.3923i	− 0.624413i	−0.950014	−	0.312207i	$-0.898932\pi$
$277$	− 10.3923i	− 0.624413i	0.950014	−	0.312207i	$-0.101068\pi$
$278$	0	0
$279$	6.92820	0.414781
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	− 28.0000i	− 1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$	− 28.0000i	− 1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$	0	0
$285$	− 3.46410i	− 0.205196i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	14.0000i	0.820695i
$292$	0	0
$293$	− 1.73205i	− 0.101187i	−0.998719	−	0.0505937i	$-0.983889\pi$
$293$	− 1.73205i	− 0.101187i	0.998719	−	0.0505937i	$-0.0161114\pi$
$294$	0	0
$295$	−31.1769	−1.81519
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.00000i	− 0.520483i
$300$	0	0
$301$	− 17.3205i	− 0.998337i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−36.0000	−2.06135
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	0	0
$309$	13.8564i	0.788263i
$310$	0	0
$311$	1.73205	0.0982156	0.0491078	−	0.998793i	$-0.484362\pi$
$311$	1.73205	0.0982156	0.0491078	+	0.998793i	$0.484362\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	0	0
$315$	12.0000i	0.676123i
$316$	0	0
$317$	8.66025i	0.486408i	0.969975	+	0.243204i	$0.0781985\pi$
$317$	8.66025i	0.486408i	−0.969975	+	0.243204i	$0.921801\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.00000	−0.167444
$322$	0	0
$323$	− 3.00000i	− 0.166924i
$324$	0	0
$325$	36.3731i	2.01761i
$326$	0	0
$327$	−5.19615	−0.287348
$328$	0	0
$329$	18.0000	0.992372
$330$	0	0
$331$	− 7.00000i	− 0.384755i	−0.981321	−	0.192377i	$-0.938380\pi$
$331$	− 7.00000i	− 0.384755i	0.981321	−	0.192377i	$-0.0616198\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	45.0333	2.46043
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.0526	1.02874
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	0	0
$349$	− 10.3923i	− 0.556287i	−0.960539	−	0.278144i	$-0.910281\pi$
$349$	− 10.3923i	− 0.556287i	0.960539	−	0.278144i	$-0.0897191\pi$
$350$	0	0
$351$	−25.9808	−1.38675
$352$	0	0
$353$	27.0000	1.43706	0.718532	−	0.695493i	$-0.244814\pi$
$353$	27.0000	1.43706	0.718532	+	0.695493i	$0.244814\pi$
$354$	0	0
$355$	− 12.0000i	− 0.636894i
$356$	0	0
$357$	− 5.19615i	− 0.275010i
$358$	0	0
$359$	8.66025	0.457071	0.228535	−	0.973536i	$-0.426606\pi$
$359$	8.66025	0.457071	0.228535	+	0.973536i	$0.426606\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 11.0000i	− 0.577350i
$364$	0	0
$365$	3.46410i	0.181319i
$366$	0	0
$367$	−31.1769	−1.62742	−0.813711	−	0.581270i	$-0.802556\pi$
$367$	−31.1769	−1.62742	−0.813711	+	0.581270i	$0.802556\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 9.00000i	− 0.467257i
$372$	0	0
$373$	12.1244i	0.627775i	0.949460	+	0.313888i	$0.101632\pi$
$373$	12.1244i	0.627775i	−0.949460	+	0.313888i	$0.898368\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	1.00000i	0.0513665i	0.999670	+	0.0256833i	$0.00817614\pi$
$379$	1.00000i	0.0513665i	−0.999670	+	0.0256833i	$0.991824\pi$
$380$	0	0
$381$	− 10.3923i	− 0.532414i
$382$	0	0
$383$	−34.6410	−1.77007	−0.885037	−	0.465521i	$-0.845867\pi$
$383$	−34.6410	−1.77007	−0.885037	+	0.465521i	$0.845867\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.0000i	1.01666i
$388$	0	0
$389$	27.7128i	1.40510i	0.711637	+	0.702548i	$0.247954\pi$
$389$	27.7128i	1.40510i	−0.711637	+	0.702548i	$0.752046\pi$
$390$	0	0
$391$	−5.19615	−0.262781
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	36.0000i	1.81136i
$396$	0	0
$397$	− 17.3205i	− 0.869291i	−0.900602	−	0.434646i	$-0.856874\pi$
$397$	− 17.3205i	− 0.869291i	0.900602	−	0.434646i	$-0.143126\pi$
$398$	0	0
$399$	−1.73205	−0.0867110
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	− 18.0000i	− 0.896644i
$404$	0	0
$405$	− 3.46410i	− 0.172133i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	− 21.0000i	− 1.03585i
$412$	0	0
$413$	15.5885i	0.767058i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	− 6.00000i	− 0.293119i	−0.989202	−	0.146560i	$-0.953180\pi$
$419$	− 6.00000i	− 0.293119i	0.989202	−	0.146560i	$-0.0468200\pi$
$420$	0	0
$421$	− 8.66025i	− 0.422075i	−0.977478	−	0.211037i	$-0.932316\pi$
$421$	− 8.66025i	− 0.422075i	0.977478	−	0.211037i	$-0.0676842\pi$
$422$	0	0
$423$	−20.7846	−1.01058
$424$	0	0
$425$	21.0000	1.01865
$426$	0	0
$427$	18.0000i	0.871081i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.2487	1.16802	0.584010	−	0.811747i	$-0.301483\pi$
$431$	24.2487	1.16802	0.584010	+	0.811747i	$0.301483\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	6.00000i	0.287678i
$436$	0	0
$437$	1.73205i	0.0828552i
$438$	0	0
$439$	−34.6410	−1.65333	−0.826663	−	0.562698i	$-0.809764\pi$
$439$	−34.6410	−1.65333	−0.826663	+	0.562698i	$0.809764\pi$
$440$	0	0
$441$	−8.00000	−0.380952
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	− 62.3538i	− 2.95585i
$446$	0	0
$447$	−20.7846	−0.983078
$448$	0	0
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 13.8564i	− 0.651031i
$454$	0	0
$455$	31.1769	1.46160
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	0	0
$459$	15.0000i	0.700140i
$460$	0	0
$461$	− 13.8564i	− 0.645357i	−0.946509	−	0.322679i	$-0.895417\pi$
$461$	− 13.8564i	− 0.645357i	0.946509	−	0.322679i	$-0.104583\pi$
$462$	0	0
$463$	−10.3923	−0.482971	−0.241486	−	0.970404i	$-0.577635\pi$
$463$	−10.3923	−0.482971	−0.241486	+	0.970404i	$0.577635\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	− 22.5167i	− 1.03972i
$470$	0	0
$471$	3.46410	0.159617
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 7.00000i	− 0.321182i
$476$	0	0
$477$	10.3923i	0.475831i
$478$	0	0
$479$	31.1769	1.42451	0.712255	−	0.701921i	$-0.247674\pi$
$479$	31.1769	1.42451	0.712255	+	0.701921i	$0.247674\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	48.4974i	2.20215i
$486$	0	0
$487$	−34.6410	−1.56973	−0.784867	−	0.619664i	$-0.787269\pi$
$487$	−34.6410	−1.56973	−0.784867	+	0.619664i	$0.787269\pi$
$488$	0	0
$489$	14.0000	0.633102
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	5.19615i	0.234023i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	− 2.00000i	− 0.0895323i	−0.998997	−	0.0447661i	$-0.985746\pi$
$499$	− 2.00000i	− 0.0895323i	0.998997	−	0.0447661i	$-0.0142543\pi$
$500$	0	0
$501$	13.8564i	0.619059i
$502$	0	0
$503$	5.19615	0.231685	0.115842	−	0.993268i	$-0.463043\pi$
$503$	5.19615	0.231685	0.115842	+	0.993268i	$0.463043\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.0000i	0.621762i
$508$	0	0
$509$	27.7128i	1.22835i	0.789170	+	0.614174i	$0.210511\pi$
$509$	27.7128i	1.22835i	−0.789170	+	0.614174i	$0.789489\pi$
$510$	0	0
$511$	1.73205	0.0766214
$512$	0	0
$513$	5.00000	0.220755
$514$	0	0
$515$	48.0000i	2.11513i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	20.7846	0.912343
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	0	0
$523$	7.00000i	0.306089i	0.988219	+	0.153044i	$0.0489077\pi$
$523$	7.00000i	0.306089i	−0.988219	+	0.153044i	$0.951092\pi$
$524$	0	0
$525$	− 12.1244i	− 0.529150i
$526$	0	0
$527$	−10.3923	−0.452696
$528$	0	0
$529$	−20.0000	−0.869565
$530$	0	0
$531$	− 18.0000i	− 0.781133i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−10.3923	−0.449299
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	13.8564	0.594635
$544$	0	0
$545$	−18.0000	−0.771035
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	− 20.7846i	− 0.887066i
$550$	0	0
$551$	1.73205	0.0737878
$552$	0	0
$553$	18.0000	0.765438
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.2487i	− 1.02745i	−0.857955	−	0.513725i	$-0.828265\pi$
$557$	− 24.2487i	− 1.02745i	0.857955	−	0.513725i	$-0.171735\pi$
$558$	0	0
$559$	51.9615	2.19774
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	− 20.7846i	− 0.874415i
$566$	0	0
$567$	−1.73205	−0.0727393
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	8.00000i	0.334790i	0.985890	+	0.167395i	$0.0535355\pi$
$571$	8.00000i	0.334790i	−0.985890	+	0.167395i	$0.946465\pi$
$572$	0	0
$573$	− 1.73205i	− 0.0723575i
$574$	0	0
$575$	−12.1244	−0.505621
$576$	0	0
$577$	−35.0000	−1.45707	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000	−1.45707	−0.728535	+	0.685009i	$0.759798\pi$
$578$	0	0
$579$	16.0000i	0.664937i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−36.0000	−1.48842
$586$	0	0
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	0	0
$589$	3.46410i	0.142736i
$590$	0	0
$591$	−13.8564	−0.569976
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	− 18.0000i	− 0.737928i
$596$	0	0
$597$	− 22.5167i	− 0.921546i
$598$	0	0
$599$	−27.7128	−1.13231	−0.566157	−	0.824297i	$-0.691571\pi$
$599$	−27.7128	−1.13231	−0.566157	+	0.824297i	$0.691571\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	26.0000i	1.05880i
$604$	0	0
$605$	− 38.1051i	− 1.54919i
$606$	0	0
$607$	3.46410	0.140604	0.0703018	−	0.997526i	$-0.477604\pi$
$607$	3.46410	0.140604	0.0703018	+	0.997526i	$0.477604\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	54.0000i	2.18461i
$612$	0	0
$613$	− 6.92820i	− 0.279827i	−0.990164	−	0.139914i	$-0.955317\pi$
$613$	− 6.92820i	− 0.279827i	0.990164	−	0.139914i	$-0.0446825\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	2.00000i	0.0803868i	0.999192	+	0.0401934i	$0.0127974\pi$
$619$	2.00000i	0.0803868i	−0.999192	+	0.0401934i	$0.987203\pi$
$620$	0	0
$621$	− 8.66025i	− 0.347524i
$622$	0	0
$623$	−31.1769	−1.24908
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.1769	−1.24113	−0.620567	−	0.784154i	$-0.713097\pi$
$631$	−31.1769	−1.24113	−0.620567	+	0.784154i	$0.713097\pi$
$632$	0	0
$633$	−5.00000	−0.198732
$634$	0	0
$635$	− 36.0000i	− 1.42862i
$636$	0	0
$637$	20.7846i	0.823516i
$638$	0	0
$639$	6.92820	0.274075
$640$	0	0
$641$	−48.0000	−1.89589	−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000	−1.89589	−0.947943	+	0.318440i	$0.896841\pi$
$642$	0	0
$643$	− 22.0000i	− 0.867595i	−0.901010	−	0.433798i	$-0.857173\pi$
$643$	− 22.0000i	− 0.867595i	0.901010	−	0.433798i	$-0.142827\pi$
$644$	0	0
$645$	− 34.6410i	− 1.36399i
$646$	0	0
$647$	22.5167	0.885221	0.442611	−	0.896714i	$-0.354052\pi$
$647$	22.5167	0.885221	0.442611	+	0.896714i	$0.354052\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	6.00000i	0.235159i
$652$	0	0
$653$	3.46410i	0.135561i	0.997700	+	0.0677804i	$0.0215917\pi$
$653$	3.46410i	0.135561i	−0.997700	+	0.0677804i	$0.978408\pi$
$654$	0	0
$655$	−62.3538	−2.43637
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	− 51.0000i	− 1.98668i	−0.115229	−	0.993339i	$-0.536760\pi$
$659$	− 51.0000i	− 1.98668i	0.115229	−	0.993339i	$-0.463240\pi$
$660$	0	0
$661$	19.0526i	0.741059i	0.928821	+	0.370529i	$0.120824\pi$
$661$	19.0526i	0.741059i	−0.928821	+	0.370529i	$0.879176\pi$
$662$	0	0
$663$	15.5885	0.605406
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	− 3.00000i	− 0.116160i
$668$	0	0
$669$	− 6.92820i	− 0.267860i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	0	0
$675$	35.0000i	1.34715i
$676$	0	0
$677$	− 36.3731i	− 1.39793i	−0.715156	−	0.698965i	$-0.753644\pi$
$677$	− 36.3731i	− 1.39793i	0.715156	−	0.698965i	$-0.246356\pi$
$678$	0	0
$679$	24.2487	0.930580
$680$	0	0
$681$	−15.0000	−0.574801
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	− 72.7461i	− 2.77949i
$686$	0	0
$687$	−3.46410	−0.132164
$688$	0	0
$689$	27.0000	1.02862
$690$	0	0
$691$	− 34.0000i	− 1.29342i	−0.762736	−	0.646710i	$-0.776144\pi$
$691$	− 34.0000i	− 1.29342i	0.762736	−	0.646710i	$-0.223856\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−76.2102	−2.89082
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 6.00000i	− 0.226941i
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	36.0000	1.35584
$706$	0	0
$707$	0	0
$708$	0	0
$709$	24.2487i	0.910679i	0.890318	+	0.455340i	$0.150482\pi$
$709$	24.2487i	0.910679i	−0.890318	+	0.455340i	$0.849518\pi$
$710$	0	0
$711$	−20.7846	−0.779484
$712$	0	0
$713$	6.00000	0.224702
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 19.0526i	− 0.711531i
$718$	0	0
$719$	8.66025	0.322973	0.161486	−	0.986875i	$-0.448371\pi$
$719$	8.66025	0.322973	0.161486	+	0.986875i	$0.448371\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	0	0
$723$	− 8.00000i	− 0.297523i
$724$	0	0
$725$	12.1244i	0.450287i
$726$	0	0
$727$	−19.0526	−0.706620	−0.353310	−	0.935506i	$-0.614944\pi$
$727$	−19.0526	−0.706620	−0.353310	+	0.935506i	$0.614944\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 30.0000i	− 1.10959i
$732$	0	0
$733$	31.1769i	1.15155i	0.817610	+	0.575773i	$0.195299\pi$
$733$	31.1769i	1.15155i	−0.817610	+	0.575773i	$0.804701\pi$
$734$	0	0
$735$	13.8564	0.511101
$736$	0	0
$737$	0	0
$738$	0	0
$739$	28.0000i	1.03000i	0.857191	+	0.514998i	$0.172207\pi$
$739$	28.0000i	1.03000i	−0.857191	+	0.514998i	$0.827793\pi$
$740$	0	0
$741$	− 5.19615i	− 0.190885i
$742$	0	0
$743$	17.3205	0.635428	0.317714	−	0.948187i	$-0.397085\pi$
$743$	17.3205	0.635428	0.317714	+	0.948187i	$0.397085\pi$
$744$	0	0
$745$	−72.0000	−2.63788
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.19615i	0.189863i
$750$	0	0
$751$	−20.7846	−0.758441	−0.379221	−	0.925306i	$-0.623808\pi$
$751$	−20.7846	−0.758441	−0.379221	+	0.925306i	$0.623808\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	− 48.0000i	− 1.74690i
$756$	0	0
$757$	− 51.9615i	− 1.88857i	−0.329124	−	0.944287i	$-0.606753\pi$
$757$	− 51.9615i	− 1.88857i	0.329124	−	0.944287i	$-0.393247\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	9.00000i	0.325822i
$764$	0	0
$765$	20.7846i	0.751469i
$766$	0	0
$767$	−46.7654	−1.68860
$768$	0	0
$769$	41.0000	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$769$	41.0000	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$770$	0	0
$771$	− 6.00000i	− 0.216085i
$772$	0	0
$773$	32.9090i	1.18365i	0.806065	+	0.591827i	$0.201593\pi$
$773$	32.9090i	1.18365i	−0.806065	+	0.591827i	$0.798407\pi$
$774$	0	0
$775$	−24.2487	−0.871039
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−8.66025	−0.309492
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	53.0000i	1.88925i	0.328158	+	0.944623i	$0.393572\pi$
$787$	53.0000i	1.88925i	−0.328158	+	0.944623i	$0.606428\pi$
$788$	0	0
$789$	− 24.2487i	− 0.863277i
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	−54.0000	−1.91760
$794$	0	0
$795$	− 18.0000i	− 0.638394i
$796$	0	0
$797$	− 29.4449i	− 1.04299i	−0.853254	−	0.521495i	$-0.825374\pi$
$797$	− 29.4449i	− 1.04299i	0.853254	−	0.521495i	$-0.174626\pi$
$798$	0	0
$799$	31.1769	1.10296
$800$	0	0
$801$	36.0000	1.27200
$802$	0	0
$803$	0	0
$804$	0	0
$805$	10.3923i	0.366281i
$806$	0	0
$807$	27.7128	0.975537
$808$	0	0
$809$	−27.0000	−0.949269	−0.474635	−	0.880183i	$-0.657420\pi$
$809$	−27.0000	−0.949269	−0.474635	+	0.880183i	$0.657420\pi$
$810$	0	0
$811$	31.0000i	1.08856i	0.838905	+	0.544279i	$0.183197\pi$
$811$	31.0000i	1.08856i	−0.838905	+	0.544279i	$0.816803\pi$
$812$	0	0
$813$	22.5167i	0.789694i
$814$	0	0
$815$	48.4974	1.69879
$816$	0	0
$817$	−10.0000	−0.349856
$818$	0	0
$819$	18.0000i	0.628971i
$820$	0	0
$821$	− 10.3923i	− 0.362694i	−0.983419	−	0.181347i	$-0.941954\pi$
$821$	− 10.3923i	− 0.362694i	0.983419	−	0.181347i	$-0.0580457\pi$
$822$	0	0
$823$	8.66025	0.301877	0.150939	−	0.988543i	$-0.451770\pi$
$823$	8.66025	0.301877	0.150939	+	0.988543i	$0.451770\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.0000i	1.56480i	0.622774	+	0.782402i	$0.286006\pi$
$827$	45.0000i	1.56480i	−0.622774	+	0.782402i	$0.713994\pi$
$828$	0	0
$829$	22.5167i	0.782036i	0.920383	+	0.391018i	$0.127877\pi$
$829$	22.5167i	0.782036i	−0.920383	+	0.391018i	$0.872123\pi$
$830$	0	0
$831$	−10.3923	−0.360505
$832$	0	0
$833$	12.0000	0.415775
$834$	0	0
$835$	48.0000i	1.66111i
$836$	0	0
$837$	− 17.3205i	− 0.598684i
$838$	0	0
$839$	38.1051	1.31553	0.657767	−	0.753221i	$-0.271501\pi$
$839$	38.1051	1.31553	0.657767	+	0.753221i	$0.271501\pi$
$840$	0	0
$841$	26.0000	0.896552
$842$	0	0
$843$	− 24.0000i	− 0.826604i
$844$	0	0
$845$	48.4974i	1.66836i
$846$	0	0
$847$	−19.0526	−0.654654
$848$	0	0
$849$	−28.0000	−0.960958
$850$	0	0
$851$	0	0
$852$	0	0
$853$	13.8564i	0.474434i	0.971457	+	0.237217i	$0.0762353\pi$
$853$	13.8564i	0.474434i	−0.971457	+	0.237217i	$0.923765\pi$
$854$	0	0
$855$	6.92820	0.236940
$856$	0	0
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	0	0
$859$	− 8.00000i	− 0.272956i	−0.990643	−	0.136478i	$-0.956422\pi$
$859$	− 8.00000i	− 0.272956i	0.990643	−	0.136478i	$-0.0435784\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.0333	1.53295	0.766476	−	0.642273i	$-0.222008\pi$
$863$	45.0333	1.53295	0.766476	+	0.642273i	$0.222008\pi$
$864$	0	0
$865$	72.0000	2.44807
$866$	0	0
$867$	8.00000i	0.271694i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	67.5500	2.28884
$872$	0	0
$873$	−28.0000	−0.947656
$874$	0	0
$875$	− 12.0000i	− 0.405674i
$876$	0	0
$877$	5.19615i	0.175462i	0.996144	+	0.0877308i	$0.0279615\pi$
$877$	5.19615i	0.175462i	−0.996144	+	0.0877308i	$0.972038\pi$
$878$	0	0
$879$	−1.73205	−0.0584206
$880$	0	0
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	0	0
$883$	14.0000i	0.471138i	0.971858	+	0.235569i	$0.0756953\pi$
$883$	14.0000i	0.471138i	−0.971858	+	0.235569i	$0.924305\pi$
$884$	0	0
$885$	31.1769i	1.04800i
$886$	0	0
$887$	−51.9615	−1.74470	−0.872349	−	0.488884i	$-0.837404\pi$
$887$	−51.9615	−1.74470	−0.872349	+	0.488884i	$0.837404\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10.3923i	− 0.347765i
$894$	0	0
$895$	−41.5692	−1.38951
$896$	0	0
$897$	−9.00000	−0.300501
$898$	0	0
$899$	− 6.00000i	− 0.200111i
$900$	0	0
$901$	− 15.5885i	− 0.519327i
$902$	0	0
$903$	−17.3205	−0.576390
$904$	0	0
$905$	48.0000	1.59557
$906$	0	0
$907$	19.0000i	0.630885i	0.948945	+	0.315442i	$0.102153\pi$
$907$	19.0000i	0.630885i	−0.948945	+	0.315442i	$0.897847\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.1769	1.03294	0.516469	−	0.856306i	$-0.327246\pi$
$911$	31.1769	1.03294	0.516469	+	0.856306i	$0.327246\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	36.0000i	1.19012i
$916$	0	0
$917$	31.1769i	1.02955i
$918$	0	0
$919$	−32.9090	−1.08557	−0.542783	−	0.839873i	$-0.682630\pi$
$919$	−32.9090	−1.08557	−0.542783	+	0.839873i	$0.682630\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	− 18.0000i	− 0.592477i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−27.7128	−0.910208
$928$	0	0
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	− 1.73205i	− 0.0567048i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−59.0000	−1.92745	−0.963723	−	0.266904i	$-0.913999\pi$
$937$	−59.0000	−1.92745	−0.963723	+	0.266904i	$0.913999\pi$
$938$	0	0
$939$	− 19.0000i	− 0.620042i
$940$	0	0
$941$	43.3013i	1.41158i	0.708421	+	0.705791i	$0.249408\pi$
$941$	43.3013i	1.41158i	−0.708421	+	0.705791i	$0.750592\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	30.0000	0.975900
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	5.19615i	0.168674i
$950$	0	0
$951$	8.66025	0.280828
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	0	0
$955$	− 6.00000i	− 0.194155i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−36.3731	−1.17455
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	− 6.00000i	− 0.193347i
$964$	0	0
$965$	55.4256i	1.78421i
$966$	0	0
$967$	24.2487	0.779786	0.389893	−	0.920860i	$-0.372512\pi$
$967$	24.2487	0.779786	0.389893	+	0.920860i	$0.372512\pi$
$968$	0	0
$969$	−3.00000	−0.0963739
$970$	0	0
$971$	60.0000i	1.92549i	0.270408	+	0.962746i	$0.412841\pi$
$971$	60.0000i	1.92549i	−0.270408	+	0.962746i	$0.587159\pi$
$972$	0	0
$973$	38.1051i	1.22159i
$974$	0	0
$975$	36.3731	1.16487
$976$	0	0
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 10.3923i	− 0.331801i
$982$	0	0
$983$	45.0333	1.43634	0.718170	−	0.695868i	$-0.244980\pi$
$983$	45.0333	1.43634	0.718170	+	0.695868i	$0.244980\pi$
$984$	0	0
$985$	−48.0000	−1.52941
$986$	0	0
$987$	− 18.0000i	− 0.572946i
$988$	0	0
$989$	17.3205i	0.550760i
$990$	0	0
$991$	31.1769	0.990367	0.495184	−	0.868788i	$-0.335101\pi$
$991$	31.1769	0.990367	0.495184	+	0.868788i	$0.335101\pi$
$992$	0	0
$993$	−7.00000	−0.222138
$994$	0	0
$995$	− 78.0000i	− 2.47277i
$996$	0	0
$997$	− 38.1051i	− 1.20680i	−0.797438	−	0.603401i	$-0.793812\pi$
$997$	− 38.1051i	− 1.20680i	0.797438	−	0.603401i	$-0.206188\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.g.609.1	✓	4
4.3	odd	2	inner	1216.2.c.g.609.3	yes	4
8.3	odd	2	inner	1216.2.c.g.609.2	yes	4
8.5	even	2	inner	1216.2.c.g.609.4	yes	4
16.3	odd	4		4864.2.a.t.1.1		2
16.5	even	4		4864.2.a.t.1.2		2
16.11	odd	4		4864.2.a.w.1.2		2
16.13	even	4		4864.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.g.609.1	✓	4	1.1	even	1	trivial
1216.2.c.g.609.2	yes	4	8.3	odd	2	inner
1216.2.c.g.609.3	yes	4	4.3	odd	2	inner
1216.2.c.g.609.4	yes	4	8.5	even	2	inner
4864.2.a.t.1.1		2	16.3	odd	4
4864.2.a.t.1.2		2	16.5	even	4
4864.2.a.w.1.1		2	16.13	even	4
4864.2.a.w.1.2		2	16.11	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -3.46410i q^{5} -1.73205 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -3.46410i q^{5} -1.73205 q^{7} +2.00000 q^{9} -5.19615i q^{13} -3.46410 q^{15} -3.00000 q^{17} +1.00000i q^{19} +1.73205i q^{21} +1.73205 q^{23} -7.00000 q^{25} -5.00000i q^{27} -1.73205i q^{29} +3.46410 q^{31} +6.00000i q^{35} -5.19615 q^{39} +10.0000i q^{43} -6.92820i q^{45} -10.3923 q^{47} -4.00000 q^{49} +3.00000i q^{51} +5.19615i q^{53} +1.00000 q^{57} -9.00000i q^{59} -10.3923i q^{61} -3.46410 q^{63} -18.0000 q^{65} +13.0000i q^{67} -1.73205i q^{69} +3.46410 q^{71} -1.00000 q^{73} +7.00000i q^{75} -10.3923 q^{79} +1.00000 q^{81} +10.3923i q^{85} -1.73205 q^{87} +18.0000 q^{89} +9.00000i q^{91} -3.46410i q^{93} +3.46410 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{9}+O(q^{10}) \) 4 * q + 8 * q^9	\( 4 q + 8 q^{9} - 12 q^{17} - 28 q^{25} - 16 q^{49} + 4 q^{57} - 72 q^{65} - 4 q^{73} + 4 q^{81} + 72 q^{89} - 56 q^{97}+O(q^{100}) \) 4 * q + 8 * q^9 - 12 * q^17 - 28 * q^25 - 16 * q^49 + 4 * q^57 - 72 * q^65 - 4 * q^73 + 4 * q^81 + 72 * q^89 - 56 * q^97

Introduction

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