LMFDB - Embedded newform 2880.2.k.g.1441.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(1441,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2880.1441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.k (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:	$22.9969157821$
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^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1441.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2880.1441
Dual form			2880.2.k.g.1441.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -3.46410 q^{7} +O(q^{10})$	$q-1.00000i q^{5} -3.46410 q^{7} +3.46410i q^{11} +6.92820 q^{17} -2.00000i q^{19} -6.00000 q^{23} -1.00000 q^{25} -6.00000i q^{29} +6.92820 q^{31} +3.46410i q^{35} +6.92820i q^{37} -6.92820 q^{41} -4.00000i q^{43} -6.00000 q^{47} +5.00000 q^{49} -6.00000i q^{53} +3.46410 q^{55} -10.3923i q^{59} -13.8564i q^{61} +8.00000i q^{67} -12.0000 q^{71} +2.00000 q^{73} -12.0000i q^{77} -6.92820 q^{79} -6.92820i q^{83} -6.92820i q^{85} -6.92820 q^{89} -2.00000 q^{95} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{23} - 4 q^{25} - 24 q^{47} + 20 q^{49} - 48 q^{71} + 8 q^{73} - 8 q^{95} - 56 q^{97}+O(q^{100}) $ 4 * q - 24 * q^23 - 4 * q^25 - 24 * q^47 + 20 * q^49 - 48 * q^71 + 8 * q^73 - 8 * q^95 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.46410i	1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$	3.46410i	1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.92820	1.68034	0.840168	−	0.542326i	$-0.182456\pi$
$17$	6.92820	1.68034	0.840168	+	0.542326i	$0.182456\pi$
$18$	0	0
$19$	− 2.00000i	− 0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$	− 2.00000i	− 0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	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
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.46410i	0.585540i
$36$	0	0
$37$	6.92820i	1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$	6.92820i	1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.3923i	− 1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	− 10.3923i	− 1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$	0	0
$61$	− 13.8564i	− 1.77413i	−0.461644	−	0.887066i	$-0.652740\pi$
$61$	− 13.8564i	− 1.77413i	0.461644	−	0.887066i	$-0.347260\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	−6.92820	−0.779484	−0.389742	−	0.920924i	$-0.627436\pi$
$79$	−6.92820	−0.779484	−0.389742	+	0.920924i	$0.627436\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 6.92820i	− 0.760469i	−0.924890	−	0.380235i	$-0.875843\pi$
$83$	− 6.92820i	− 0.760469i	0.924890	−	0.380235i	$-0.124157\pi$
$84$	0	0
$85$	− 6.92820i	− 0.751469i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.92820	−0.734388	−0.367194	−	0.930144i	$-0.619682\pi$
$89$	−6.92820	−0.734388	−0.367194	+	0.930144i	$0.619682\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.00000	−0.205196
$96$	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
$101$	− 18.0000i	− 1.79107i	−0.444994	−	0.895533i	$-0.646794\pi$
$101$	− 18.0000i	− 1.79107i	0.444994	−	0.895533i	$-0.353206\pi$
$102$	0	0
$103$	−3.46410	−0.341328	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	−3.46410	−0.341328	−0.170664	+	0.985329i	$0.554591\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.92820i	0.669775i	0.942258	+	0.334887i	$0.108698\pi$
$107$	6.92820i	0.669775i	−0.942258	+	0.334887i	$0.891302\pi$
$108$	0	0
$109$	13.8564i	1.32720i	0.748086	+	0.663602i	$0.230973\pi$
$109$	13.8564i	1.32720i	−0.748086	+	0.663602i	$0.769027\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.92820	−0.651751	−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820	−0.651751	−0.325875	+	0.945413i	$0.605659\pi$
$114$	0	0
$115$	6.00000i	0.559503i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−24.0000	−2.20008
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−10.3923	−0.922168	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	−10.3923	−0.922168	−0.461084	+	0.887357i	$0.652539\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 10.3923i	− 0.907980i	−0.891007	−	0.453990i	$-0.850000\pi$
$131$	− 10.3923i	− 0.907980i	0.891007	−	0.453990i	$-0.150000\pi$
$132$	0	0
$133$	6.92820i	0.600751i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	0	0
$139$	− 14.0000i	− 1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$	− 14.0000i	− 1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	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
$153$	0	0
$154$	0	0
$155$	− 6.92820i	− 0.556487i
$156$	0	0
$157$	6.92820i	0.552931i	0.961024	+	0.276465i	$0.0891631\pi$
$157$	6.92820i	0.552931i	−0.961024	+	0.276465i	$0.910837\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.7846	1.63806
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	3.46410	0.261861
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.2487i	1.81243i	0.422813	+	0.906217i	$0.361043\pi$
$179$	24.2487i	1.81243i	−0.422813	+	0.906217i	$0.638957\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.92820	0.509372
$186$	0	0
$187$	24.0000i	1.75505i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.7846i	1.45879i
$204$	0	0
$205$	6.92820i	0.483887i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.92820	0.479234
$210$	0	0
$211$	− 10.0000i	− 0.688428i	−0.938891	−	0.344214i	$-0.888145\pi$
$211$	− 10.0000i	− 0.688428i	0.938891	−	0.344214i	$-0.111855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3.46410	0.231973	0.115987	−	0.993251i	$-0.462997\pi$
$223$	3.46410	0.231973	0.115987	+	0.993251i	$0.462997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 27.7128i	− 1.83131i	−0.401960	−	0.915657i	$-0.631671\pi$
$229$	− 27.7128i	− 1.83131i	0.401960	−	0.915657i	$-0.368329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.7846	1.36165	0.680823	−	0.732448i	$-0.261622\pi$
$233$	20.7846	1.36165	0.680823	+	0.732448i	$0.261622\pi$
$234$	0	0
$235$	6.00000i	0.391397i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 5.00000i	− 0.319438i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.46410i	− 0.218652i	−0.994006	−	0.109326i	$-0.965131\pi$
$251$	− 3.46410i	− 0.218652i	0.994006	−	0.109326i	$-0.0348693\pi$
$252$	0	0
$253$	− 20.7846i	− 1.30672i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.7846	−1.29651	−0.648254	−	0.761424i	$-0.724501\pi$
$257$	−20.7846	−1.29651	−0.648254	+	0.761424i	$0.724501\pi$
$258$	0	0
$259$	− 24.0000i	− 1.49129i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.0000	−1.84988	−0.924940	−	0.380114i	$-0.875885\pi$
$263$	−30.0000	−1.84988	−0.924940	+	0.380114i	$0.875885\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	−13.8564	−0.841717	−0.420858	−	0.907126i	$-0.638271\pi$
$271$	−13.8564	−0.841717	−0.420858	+	0.907126i	$0.638271\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.46410i	− 0.208893i
$276$	0	0
$277$	− 6.92820i	− 0.416275i	−0.978100	−	0.208138i	$-0.933260\pi$
$277$	− 6.92820i	− 0.416275i	0.978100	−	0.208138i	$-0.0667402\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	−10.3923	−0.605063
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	13.8564i	0.798670i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−13.8564	−0.793416
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	20.7846	1.16371
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 13.8564i	− 0.770991i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.7846	1.14589
$330$	0	0
$331$	10.0000i	0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$	10.0000i	0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.0000i	1.29967i
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 27.7128i	− 1.48770i	−0.668346	−	0.743851i	$-0.732997\pi$
$347$	− 27.7128i	− 1.48770i	0.668346	−	0.743851i	$-0.267003\pi$
$348$	0	0
$349$	27.7128i	1.48343i	0.670714	+	0.741716i	$0.265988\pi$
$349$	27.7128i	1.48343i	−0.670714	+	0.741716i	$0.734012\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.7846	−1.10625	−0.553127	−	0.833097i	$-0.686565\pi$
$353$	−20.7846	−1.10625	−0.553127	+	0.833097i	$0.686565\pi$
$354$	0	0
$355$	12.0000i	0.636894i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	15.0000	0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.00000i	− 0.104685i
$366$	0	0
$367$	31.1769	1.62742	0.813711	−	0.581270i	$-0.197444\pi$
$367$	31.1769	1.62742	0.813711	+	0.581270i	$0.197444\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	20.7846i	1.07908i
$372$	0	0
$373$	34.6410i	1.79364i	0.442392	+	0.896822i	$0.354130\pi$
$373$	34.6410i	1.79364i	−0.442392	+	0.896822i	$0.645870\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 22.0000i	− 1.13006i	−0.825069	−	0.565032i	$-0.808864\pi$
$379$	− 22.0000i	− 1.13006i	0.825069	−	0.565032i	$-0.191136\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 6.00000i	− 0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$	− 6.00000i	− 0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$	0	0
$391$	−41.5692	−2.10225
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.92820i	0.348596i
$396$	0	0
$397$	− 27.7128i	− 1.39087i	−0.718591	−	0.695433i	$-0.755213\pi$
$397$	− 27.7128i	− 1.39087i	0.718591	−	0.695433i	$-0.244787\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.6410	−1.72989	−0.864945	−	0.501867i	$-0.832647\pi$
$401$	−34.6410	−1.72989	−0.864945	+	0.501867i	$0.832647\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	36.0000i	1.77144i
$414$	0	0
$415$	−6.92820	−0.340092
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 24.2487i	− 1.18463i	−0.805708	−	0.592314i	$-0.798215\pi$
$419$	− 24.2487i	− 1.18463i	0.805708	−	0.592314i	$-0.201785\pi$
$420$	0	0
$421$	27.7128i	1.35064i	0.737525	+	0.675320i	$0.235994\pi$
$421$	27.7128i	1.35064i	−0.737525	+	0.675320i	$0.764006\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.92820	−0.336067
$426$	0	0
$427$	48.0000i	2.32288i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.0000i	0.574038i
$438$	0	0
$439$	−13.8564	−0.661330	−0.330665	−	0.943748i	$-0.607273\pi$
$439$	−13.8564	−0.661330	−0.330665	+	0.943748i	$0.607273\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 13.8564i	− 0.658338i	−0.944271	−	0.329169i	$-0.893231\pi$
$443$	− 13.8564i	− 0.658338i	0.944271	−	0.329169i	$-0.106769\pi$
$444$	0	0
$445$	6.92820i	0.328428i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−13.8564	−0.653924	−0.326962	−	0.945037i	$-0.606025\pi$
$449$	−13.8564	−0.653924	−0.326962	+	0.945037i	$0.606025\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.0000i	− 0.838344i	−0.907907	−	0.419172i	$-0.862320\pi$
$461$	− 18.0000i	− 0.838344i	0.907907	−	0.419172i	$-0.137680\pi$
$462$	0	0
$463$	−17.3205	−0.804952	−0.402476	−	0.915430i	$-0.631850\pi$
$463$	−17.3205	−0.804952	−0.402476	+	0.915430i	$0.631850\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20.7846i	− 0.961797i	−0.876776	−	0.480899i	$-0.840311\pi$
$467$	− 20.7846i	− 0.961797i	0.876776	−	0.480899i	$-0.159689\pi$
$468$	0	0
$469$	− 27.7128i	− 1.27966i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.8564	0.637118
$474$	0	0
$475$	2.00000i	0.0917663i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	14.0000i	0.635707i
$486$	0	0
$487$	31.1769	1.41276	0.706380	−	0.707832i	$-0.250327\pi$
$487$	31.1769	1.41276	0.706380	+	0.707832i	$0.250327\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.3923i	0.468998i	0.972116	+	0.234499i	$0.0753450\pi$
$491$	10.3923i	0.468998i	−0.972116	+	0.234499i	$0.924655\pi$
$492$	0	0
$493$	− 41.5692i	− 1.87218i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	41.5692	1.86463
$498$	0	0
$499$	− 34.0000i	− 1.52205i	−0.648723	−	0.761025i	$-0.724697\pi$
$499$	− 34.0000i	− 1.52205i	0.648723	−	0.761025i	$-0.275303\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	0	0
$507$	0	0
$508$	0	0
$509$	42.0000i	1.86162i	0.365507	+	0.930809i	$0.380896\pi$
$509$	42.0000i	1.86162i	−0.365507	+	0.930809i	$0.619104\pi$
$510$	0	0
$511$	−6.92820	−0.306486
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.46410i	0.152647i
$516$	0	0
$517$	− 20.7846i	− 0.914106i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.8564	−0.607060	−0.303530	−	0.952822i	$-0.598165\pi$
$521$	−13.8564	−0.607060	−0.303530	+	0.952822i	$0.598165\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.92820	0.299532
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17.3205i	0.746047i
$540$	0	0
$541$	13.8564i	0.595733i	0.954607	+	0.297867i	$0.0962751\pi$
$541$	13.8564i	0.595733i	−0.954607	+	0.297867i	$0.903725\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.8564	0.593543
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	24.0000	1.02058
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	41.5692i	1.75193i	0.482371	+	0.875967i	$0.339776\pi$
$563$	41.5692i	1.75193i	−0.482371	+	0.875967i	$0.660224\pi$
$564$	0	0
$565$	6.92820i	0.291472i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.92820	−0.290445	−0.145223	−	0.989399i	$-0.546390\pi$
$569$	−6.92820	−0.290445	−0.145223	+	0.989399i	$0.546390\pi$
$570$	0	0
$571$	34.0000i	1.42286i	0.702759	+	0.711428i	$0.251951\pi$
$571$	34.0000i	1.42286i	−0.702759	+	0.711428i	$0.748049\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24.0000i	0.995688i
$582$	0	0
$583$	20.7846	0.860811
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.6410i	1.42979i	0.699233	+	0.714894i	$0.253525\pi$
$587$	34.6410i	1.42979i	−0.699233	+	0.714894i	$0.746475\pi$
$588$	0	0
$589$	− 13.8564i	− 0.570943i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.92820	0.284507	0.142254	−	0.989830i	$-0.454565\pi$
$593$	6.92820	0.284507	0.142254	+	0.989830i	$0.454565\pi$
$594$	0	0
$595$	24.0000i	0.983904i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.00000i	0.0406558i
$606$	0	0
$607$	−17.3205	−0.703018	−0.351509	−	0.936185i	$-0.614331\pi$
$607$	−17.3205	−0.703018	−0.351509	+	0.936185i	$0.614331\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	13.8564i	0.559655i	0.960050	+	0.279827i	$0.0902773\pi$
$613$	13.8564i	0.559655i	−0.960050	+	0.279827i	$0.909723\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.7846	0.836757	0.418378	−	0.908273i	$-0.362599\pi$
$617$	20.7846	0.836757	0.418378	+	0.908273i	$0.362599\pi$
$618$	0	0
$619$	− 14.0000i	− 0.562708i	−0.959604	−	0.281354i	$-0.909217\pi$
$619$	− 14.0000i	− 0.562708i	0.959604	−	0.281354i	$-0.0907834\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	48.0000i	1.91389i
$630$	0	0
$631$	20.7846	0.827422	0.413711	−	0.910408i	$-0.364232\pi$
$631$	20.7846	0.827422	0.413711	+	0.910408i	$0.364232\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.3923i	0.412406i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.5692	1.64189	0.820943	−	0.571011i	$-0.193448\pi$
$641$	41.5692	1.64189	0.820943	+	0.571011i	$0.193448\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	18.0000	0.707653	0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000	0.707653	0.353827	+	0.935311i	$0.384880\pi$
$648$	0	0
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	−10.3923	−0.406061
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 24.2487i	− 0.944596i	−0.881439	−	0.472298i	$-0.843425\pi$
$659$	− 24.2487i	− 0.944596i	0.881439	−	0.472298i	$-0.156575\pi$
$660$	0	0
$661$	13.8564i	0.538952i	0.963007	+	0.269476i	$0.0868504\pi$
$661$	13.8564i	0.538952i	−0.963007	+	0.269476i	$0.913150\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.92820	0.268664
$666$	0	0
$667$	36.0000i	1.39393i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	48.0000	1.85302
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	48.4974	1.86116
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 48.4974i	− 1.85570i	−0.372951	−	0.927851i	$-0.621654\pi$
$683$	− 48.4974i	− 1.85570i	0.372951	−	0.927851i	$-0.378346\pi$
$684$	0	0
$685$	6.92820i	0.264713i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 38.0000i	− 1.44559i	−0.691063	−	0.722794i	$-0.742858\pi$
$691$	− 38.0000i	− 1.44559i	0.691063	−	0.722794i	$-0.257142\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.0000	−0.531050
$696$	0	0
$697$	−48.0000	−1.81813
$698$	0	0
$699$	0	0
$700$	0	0
$701$	42.0000i	1.58632i	0.609015	+	0.793159i	$0.291565\pi$
$701$	42.0000i	1.58632i	−0.609015	+	0.793159i	$0.708435\pi$
$702$	0	0
$703$	13.8564	0.522604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	62.3538i	2.34506i
$708$	0	0
$709$	13.8564i	0.520388i	0.965556	+	0.260194i	$0.0837866\pi$
$709$	13.8564i	0.520388i	−0.965556	+	0.260194i	$0.916213\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−41.5692	−1.55678
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	−38.1051	−1.41324	−0.706620	−	0.707593i	$-0.749781\pi$
$727$	−38.1051	−1.41324	−0.706620	+	0.707593i	$0.749781\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 27.7128i	− 1.02500i
$732$	0	0
$733$	20.7846i	0.767697i	0.923396	+	0.383849i	$0.125402\pi$
$733$	20.7846i	0.767697i	−0.923396	+	0.383849i	$0.874598\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−27.7128	−1.02081
$738$	0	0
$739$	− 34.0000i	− 1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$	− 34.0000i	− 1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 24.0000i	− 0.876941i
$750$	0	0
$751$	−6.92820	−0.252814	−0.126407	−	0.991978i	$-0.540344\pi$
$751$	−6.92820	−0.252814	−0.126407	+	0.991978i	$0.540344\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 13.8564i	− 0.504286i
$756$	0	0
$757$	34.6410i	1.25905i	0.776981	+	0.629525i	$0.216750\pi$
$757$	34.6410i	1.25905i	−0.776981	+	0.629525i	$0.783250\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.7128	1.00459	0.502294	−	0.864697i	$-0.332489\pi$
$761$	27.7128	1.00459	0.502294	+	0.864697i	$0.332489\pi$
$762$	0	0
$763$	− 48.0000i	− 1.73772i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	−6.92820	−0.248868
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.8564i	0.496457i
$780$	0	0
$781$	− 41.5692i	− 1.48746i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.92820	0.247278
$786$	0	0
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	−41.5692	−1.47061
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.92820i	0.244491i
$804$	0	0
$805$	− 20.7846i	− 0.732561i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−41.5692	−1.46150	−0.730748	−	0.682647i	$-0.760829\pi$
$809$	−41.5692	−1.46150	−0.730748	+	0.682647i	$0.760829\pi$
$810$	0	0
$811$	− 10.0000i	− 0.351147i	−0.984466	−	0.175574i	$-0.943822\pi$
$811$	− 10.0000i	− 0.351147i	0.984466	−	0.175574i	$-0.0561780\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.00000i	0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$	6.00000i	0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$	0	0
$823$	−51.9615	−1.81126	−0.905632	−	0.424064i	$-0.860603\pi$
$823$	−51.9615	−1.81126	−0.905632	+	0.424064i	$0.860603\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.8564i	0.481834i	0.970546	+	0.240917i	$0.0774482\pi$
$827$	13.8564i	0.481834i	−0.970546	+	0.240917i	$0.922552\pi$
$828$	0	0
$829$	13.8564i	0.481253i	0.970618	+	0.240626i	$0.0773529\pi$
$829$	13.8564i	0.481253i	−0.970618	+	0.240626i	$0.922647\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	34.6410	1.20024
$834$	0	0
$835$	18.0000i	0.622916i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 13.0000i	− 0.447214i
$846$	0	0
$847$	3.46410	0.119028
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 41.5692i	− 1.42497i
$852$	0	0
$853$	− 41.5692i	− 1.42330i	−0.702533	−	0.711651i	$-0.747948\pi$
$853$	− 41.5692i	− 1.42330i	0.702533	−	0.711651i	$-0.252052\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.6410	−1.18331	−0.591657	−	0.806190i	$-0.701526\pi$
$857$	−34.6410	−1.18331	−0.591657	+	0.806190i	$0.701526\pi$
$858$	0	0
$859$	− 22.0000i	− 0.750630i	−0.926897	−	0.375315i	$-0.877534\pi$
$859$	− 22.0000i	− 0.750630i	0.926897	−	0.375315i	$-0.122466\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 24.0000i	− 0.814144i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 3.46410i	− 0.117108i
$876$	0	0
$877$	34.6410i	1.16974i	0.811126	+	0.584872i	$0.198855\pi$
$877$	34.6410i	1.16974i	−0.811126	+	0.584872i	$0.801145\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.7846	−0.700251	−0.350126	−	0.936703i	$-0.613861\pi$
$881$	−20.7846	−0.700251	−0.350126	+	0.936703i	$0.613861\pi$
$882$	0	0
$883$	52.0000i	1.74994i	0.484178	+	0.874970i	$0.339119\pi$
$883$	52.0000i	1.74994i	−0.484178	+	0.874970i	$0.660881\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	6.00000	0.201460	0.100730	−	0.994914i	$-0.467882\pi$
$887$	6.00000	0.201460	0.100730	+	0.994914i	$0.467882\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.0000i	0.401565i
$894$	0	0
$895$	24.2487	0.810545
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 41.5692i	− 1.38641i
$900$	0	0
$901$	− 41.5692i	− 1.38487i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	36.0000i	1.18882i
$918$	0	0
$919$	−34.6410	−1.14270	−0.571351	−	0.820706i	$-0.693580\pi$
$919$	−34.6410	−1.14270	−0.571351	+	0.820706i	$0.693580\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 6.92820i	− 0.227798i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	− 10.0000i	− 0.327737i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.0000	0.784884
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 18.0000i	− 0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$	− 18.0000i	− 0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$	0	0
$943$	41.5692	1.35368
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.5692i	− 1.35082i	−0.737443	−	0.675409i	$-0.763967\pi$
$947$	− 41.5692i	− 1.35082i	0.737443	−	0.675409i	$-0.236033\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48.4974	−1.57099	−0.785493	−	0.618871i	$-0.787590\pi$
$953$	−48.4974	−1.57099	−0.785493	+	0.618871i	$0.787590\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 14.0000i	− 0.450676i
$966$	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
$969$	0	0
$970$	0	0
$971$	− 17.3205i	− 0.555842i	−0.960604	−	0.277921i	$-0.910355\pi$
$971$	− 17.3205i	− 0.555842i	0.960604	−	0.277921i	$-0.0896453\pi$
$972$	0	0
$973$	48.4974i	1.55476i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.92820	−0.221653	−0.110826	−	0.993840i	$-0.535350\pi$
$977$	−6.92820	−0.221653	−0.110826	+	0.993840i	$0.535350\pi$
$978$	0	0
$979$	− 24.0000i	− 0.767043i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.0000	−1.33959	−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000	−1.33959	−0.669796	+	0.742545i	$0.733618\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.0000i	0.763156i
$990$	0	0
$991$	55.4256	1.76065	0.880327	−	0.474368i	$-0.157323\pi$
$991$	55.4256	1.76065	0.880327	+	0.474368i	$0.157323\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.k.g.1441.1	✓	4
3.2	odd	2		2880.2.k.j.1441.3	yes	4
4.3	odd	2		2880.2.k.j.1441.2	yes	4
8.3	odd	2		2880.2.k.j.1441.4	yes	4
8.5	even	2	inner	2880.2.k.g.1441.3	yes	4
12.11	even	2	inner	2880.2.k.g.1441.4	yes	4
24.5	odd	2		2880.2.k.j.1441.1	yes	4
24.11	even	2	inner	2880.2.k.g.1441.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.k.g.1441.1	✓	4	1.1	even	1	trivial
2880.2.k.g.1441.2	yes	4	24.11	even	2	inner
2880.2.k.g.1441.3	yes	4	8.5	even	2	inner
2880.2.k.g.1441.4	yes	4	12.11	even	2	inner
2880.2.k.j.1441.1	yes	4	24.5	odd	2
2880.2.k.j.1441.2	yes	4	4.3	odd	2
2880.2.k.j.1441.3	yes	4	3.2	odd	2
2880.2.k.j.1441.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -3.46410 q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} -3.46410 q^{7} +3.46410i q^{11} +6.92820 q^{17} -2.00000i q^{19} -6.00000 q^{23} -1.00000 q^{25} -6.00000i q^{29} +6.92820 q^{31} +3.46410i q^{35} +6.92820i q^{37} -6.92820 q^{41} -4.00000i q^{43} -6.00000 q^{47} +5.00000 q^{49} -6.00000i q^{53} +3.46410 q^{55} -10.3923i q^{59} -13.8564i q^{61} +8.00000i q^{67} -12.0000 q^{71} +2.00000 q^{73} -12.0000i q^{77} -6.92820 q^{79} -6.92820i q^{83} -6.92820i q^{85} -6.92820 q^{89} -2.00000 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{23} - 4 q^{25} - 24 q^{47} + 20 q^{49} - 48 q^{71} + 8 q^{73} - 8 q^{95} - 56 q^{97}+O(q^{100}) \) 4 * q - 24 * q^23 - 4 * q^25 - 24 * q^47 + 20 * q^49 - 48 * q^71 + 8 * q^73 - 8 * q^95 - 56 * q^97

Introduction

L-functions

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