LMFDB - Embedded newform 2880.2.b.b.2591.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(2591,2880)]

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([1, 1, 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("2880.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.b (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:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.7
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2880.2591
Dual form			2880.2.b.b.2591.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +4.44949i q^{7} +O(q^{10})$	$q-1.00000 q^{5} +4.44949i q^{7} -4.44949i q^{11} -1.41421i q^{13} +2.82843i q^{17} -2.82843 q^{19} -3.46410 q^{23} +1.00000 q^{25} +2.00000 q^{29} +4.89898i q^{31} -4.44949i q^{35} +8.34242i q^{37} -8.34242i q^{41} -9.75663 q^{47} -12.7980 q^{49} +4.00000 q^{53} +4.44949i q^{55} +9.34847i q^{59} -15.4135i q^{61} +1.41421i q^{65} -11.3137 q^{67} +5.65685 q^{71} -11.7980 q^{73} +19.7980 q^{77} -3.10102i q^{79} +0.898979i q^{83} -2.82843i q^{85} -13.9993i q^{89} +6.29253 q^{91} +2.82843 q^{95} -11.7980 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 8 q^{25} + 16 q^{29} - 24 q^{49} + 32 q^{53} - 16 q^{73} + 80 q^{77} - 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^25 + 16 * q^29 - 24 * q^49 + 32 * q^53 - 16 * q^73 + 80 * q^77 - 16 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.44949i	1.68175i	0.541230	+	0.840875i	$0.317959\pi$
$7$	4.44949i	1.68175i	−0.541230	+	0.840875i	$0.682041\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.44949i	− 1.34157i	−0.741651	−	0.670786i	$-0.765957\pi$
$11$	− 4.44949i	− 1.34157i	0.741651	−	0.670786i	$-0.234043\pi$
$12$	0	0
$13$	− 1.41421i	− 0.392232i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	− 1.41421i	− 0.392232i	0.980581	−	0.196116i	$-0.0628330\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843i	0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$	2.82843i	0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.89898i	0.879883i	0.898027	+	0.439941i	$0.145001\pi$
$31$	4.89898i	0.879883i	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.44949i	− 0.752101i
$36$	0	0
$37$	8.34242i	1.37148i	0.727844	+	0.685742i	$0.240522\pi$
$37$	8.34242i	1.37148i	−0.727844	+	0.685742i	$0.759478\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 8.34242i	− 1.30287i	−0.758706	−	0.651433i	$-0.774168\pi$
$41$	− 8.34242i	− 1.30287i	0.758706	−	0.651433i	$-0.225832\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.75663	−1.42315	−0.711575	−	0.702610i	$-0.752018\pi$
$47$	−9.75663	−1.42315	−0.711575	+	0.702610i	$0.752018\pi$
$48$	0	0
$49$	−12.7980	−1.82828
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	4.44949i	0.599969i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.34847i	1.21707i	0.793528	+	0.608534i	$0.208242\pi$
$59$	9.34847i	1.21707i	−0.793528	+	0.608534i	$0.791758\pi$
$60$	0	0
$61$	− 15.4135i	− 1.97349i	−0.162265	−	0.986747i	$-0.551880\pi$
$61$	− 15.4135i	− 1.97349i	0.162265	−	0.986747i	$-0.448120\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.41421i	0.175412i
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	−11.7980	−1.38085	−0.690423	−	0.723406i	$-0.742576\pi$
$73$	−11.7980	−1.38085	−0.690423	+	0.723406i	$0.742576\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.7980	2.25619
$78$	0	0
$79$	− 3.10102i	− 0.348892i	−0.984667	−	0.174446i	$-0.944187\pi$
$79$	− 3.10102i	− 0.348892i	0.984667	−	0.174446i	$-0.0558135\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.898979i	0.0986758i	0.998782	+	0.0493379i	$0.0157111\pi$
$83$	0.898979i	0.0986758i	−0.998782	+	0.0493379i	$0.984289\pi$
$84$	0	0
$85$	− 2.82843i	− 0.306786i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.9993i	− 1.48392i	−0.670444	−	0.741960i	$-0.733896\pi$
$89$	− 13.9993i	− 1.48392i	0.670444	−	0.741960i	$-0.266104\pi$
$90$	0	0
$91$	6.29253	0.659636
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	−11.7980	−1.19790	−0.598951	−	0.800786i	$-0.704415\pi$
$97$	−11.7980	−1.19790	−0.598951	+	0.800786i	$0.704415\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.7980	−1.57196	−0.785978	−	0.618255i	$-0.787840\pi$
$101$	−15.7980	−1.57196	−0.785978	+	0.618255i	$0.787840\pi$
$102$	0	0
$103$	1.34847i	0.132869i	0.997791	+	0.0664343i	$0.0211623\pi$
$103$	1.34847i	0.132869i	−0.997791	+	0.0664343i	$0.978838\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.898979i	− 0.0869076i	−0.999055	−	0.0434538i	$-0.986164\pi$
$107$	− 0.898979i	− 0.0869076i	0.999055	−	0.0434538i	$-0.0138361\pi$
$108$	0	0
$109$	− 15.4135i	− 1.47634i	−0.674613	−	0.738172i	$-0.735689\pi$
$109$	− 15.4135i	− 1.47634i	0.674613	−	0.738172i	$-0.264311\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 19.5133i	− 1.83565i	−0.396982	−	0.917827i	$-0.629942\pi$
$113$	− 19.5133i	− 1.83565i	0.396982	−	0.917827i	$-0.370058\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.5851	−1.15367
$120$	0	0
$121$	−8.79796	−0.799814
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.34847i	0.829543i	0.909926	+	0.414771i	$0.136138\pi$
$127$	9.34847i	0.829543i	−0.909926	+	0.414771i	$0.863862\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20.4495i	− 1.78668i	−0.449381	−	0.893340i	$-0.648355\pi$
$131$	− 20.4495i	− 1.78668i	0.449381	−	0.893340i	$-0.351645\pi$
$132$	0	0
$133$	− 12.5851i	− 1.09126i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.3137i	0.966595i	0.875456	+	0.483298i	$0.160561\pi$
$137$	11.3137i	0.966595i	−0.875456	+	0.483298i	$0.839439\pi$
$138$	0	0
$139$	−10.3923	−0.881464	−0.440732	−	0.897639i	$-0.645281\pi$
$139$	−10.3923	−0.881464	−0.440732	+	0.897639i	$0.645281\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.29253	−0.526208
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0.202041	0.0165518	0.00827592	−	0.999966i	$-0.497366\pi$
$149$	0.202041	0.0165518	0.00827592	+	0.999966i	$0.497366\pi$
$150$	0	0
$151$	13.7980i	1.12286i	0.827524	+	0.561431i	$0.189749\pi$
$151$	13.7980i	1.12286i	−0.827524	+	0.561431i	$0.810251\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.89898i	− 0.393496i
$156$	0	0
$157$	5.51399i	0.440064i	0.975493	+	0.220032i	$0.0706162\pi$
$157$	5.51399i	0.440064i	−0.975493	+	0.220032i	$0.929384\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 15.4135i	− 1.21475i
$162$	0	0
$163$	−23.8988	−1.87190	−0.935948	−	0.352138i	$-0.885455\pi$
$163$	−23.8988	−1.87190	−0.935948	+	0.352138i	$0.885455\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.921404	0.0713004	0.0356502	−	0.999364i	$-0.488650\pi$
$167$	0.921404	0.0713004	0.0356502	+	0.999364i	$0.488650\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.79796	−0.744925	−0.372463	−	0.928047i	$-0.621486\pi$
$173$	−9.79796	−0.744925	−0.372463	+	0.928047i	$0.621486\pi$
$174$	0	0
$175$	4.44949i	0.336350i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.24745i	0.466956i	0.972362	+	0.233478i	$0.0750107\pi$
$179$	6.24745i	0.466956i	−0.972362	+	0.233478i	$0.924989\pi$
$180$	0	0
$181$	4.09978i	0.304734i	0.988324	+	0.152367i	$0.0486896\pi$
$181$	4.09978i	0.304734i	−0.988324	+	0.152367i	$0.951310\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 8.34242i	− 0.613347i
$186$	0	0
$187$	12.5851	0.920311
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.2419	−1.31994	−0.659969	−	0.751293i	$-0.729431\pi$
$191$	−18.2419	−1.31994	−0.659969	+	0.751293i	$0.729431\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.5959	1.25366	0.626829	−	0.779157i	$-0.284353\pi$
$197$	17.5959	1.25366	0.626829	+	0.779157i	$0.284353\pi$
$198$	0	0
$199$	13.7980i	0.978111i	0.872253	+	0.489056i	$0.162659\pi$
$199$	13.7980i	0.978111i	−0.872253	+	0.489056i	$0.837341\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	8.89898i	0.624586i
$204$	0	0
$205$	8.34242i	0.582660i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.5851i	0.870527i
$210$	0	0
$211$	−7.84961	−0.540389	−0.270195	−	0.962806i	$-0.587088\pi$
$211$	−7.84961	−0.540389	−0.270195	+	0.962806i	$0.587088\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−21.7980	−1.47974
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	22.2474i	1.48980i	0.667176	+	0.744900i	$0.267503\pi$
$223$	22.2474i	1.48980i	−0.667176	+	0.744900i	$0.732497\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 27.5959i	− 1.83161i	−0.401628	−	0.915803i	$-0.631556\pi$
$227$	− 27.5959i	− 1.83161i	0.401628	−	0.915803i	$-0.368444\pi$
$228$	0	0
$229$	− 1.27135i	− 0.0840131i	−0.999117	−	0.0420066i	$-0.986625\pi$
$229$	− 1.27135i	− 0.0840131i	0.999117	−	0.0420066i	$-0.0133750\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	22.3417i	1.46365i	0.681491	+	0.731826i	$0.261332\pi$
$233$	22.3417i	1.46365i	−0.681491	+	0.731826i	$0.738668\pi$
$234$	0	0
$235$	9.75663	0.636452
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.1701	−1.62812	−0.814060	−	0.580781i	$-0.802747\pi$
$239$	−25.1701	−1.62812	−0.814060	+	0.580781i	$0.802747\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.7980	0.817632
$246$	0	0
$247$	4.00000i	0.254514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.3485i	1.59998i	0.600013	+	0.799991i	$0.295162\pi$
$251$	25.3485i	1.59998i	−0.600013	+	0.799991i	$0.704838\pi$
$252$	0	0
$253$	15.4135i	0.969037i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 8.19955i	− 0.511474i	−0.966746	−	0.255737i	$-0.917682\pi$
$257$	− 8.19955i	− 0.511474i	0.966746	−	0.255737i	$-0.0823181\pi$
$258$	0	0
$259$	−37.1195	−2.30649
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.84961	−0.484027	−0.242014	−	0.970273i	$-0.577808\pi$
$263$	−7.84961	−0.484027	−0.242014	+	0.970273i	$0.577808\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−25.5959	−1.56061	−0.780305	−	0.625399i	$-0.784936\pi$
$269$	−25.5959	−1.56061	−0.780305	+	0.625399i	$0.784936\pi$
$270$	0	0
$271$	− 4.00000i	− 0.242983i	−0.992592	−	0.121491i	$-0.961232\pi$
$271$	− 4.00000i	− 0.242983i	0.992592	−	0.121491i	$-0.0387677\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 4.44949i	− 0.268314i
$276$	0	0
$277$	− 5.79972i	− 0.348471i	−0.984704	−	0.174236i	$-0.944255\pi$
$277$	− 5.79972i	− 0.348471i	0.984704	−	0.174236i	$-0.0557455\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	23.7559i	1.41716i	0.705631	+	0.708579i	$0.250663\pi$
$281$	23.7559i	1.41716i	−0.705631	+	0.708579i	$0.749337\pi$
$282$	0	0
$283$	10.0424	0.596956	0.298478	−	0.954416i	$-0.403521\pi$
$283$	10.0424	0.596956	0.298478	+	0.954416i	$0.403521\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	37.1195	2.19109
$288$	0	0
$289$	9.00000	0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	− 9.34847i	− 0.544289i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.89898i	0.283315i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.4135i	0.882574i
$306$	0	0
$307$	22.0560	1.25880	0.629400	−	0.777081i	$-0.283301\pi$
$307$	22.0560	1.25880	0.629400	+	0.777081i	$0.283301\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.27135	−0.0720916	−0.0360458	−	0.999350i	$-0.511476\pi$
$311$	−1.27135	−0.0720916	−0.0360458	+	0.999350i	$0.511476\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.5959	0.763623	0.381811	−	0.924240i	$-0.375300\pi$
$317$	13.5959	0.763623	0.381811	+	0.924240i	$0.375300\pi$
$318$	0	0
$319$	− 8.89898i	− 0.498247i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	− 1.41421i	− 0.0784465i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 43.4120i	− 2.39338i
$330$	0	0
$331$	−5.37113	−0.295224	−0.147612	−	0.989045i	$-0.547159\pi$
$331$	−5.37113	−0.295224	−0.147612	+	0.989045i	$0.547159\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	11.3137	0.618134
$336$	0	0
$337$	−4.20204	−0.228900	−0.114450	−	0.993429i	$-0.536511\pi$
$337$	−4.20204	−0.228900	−0.114450	+	0.993429i	$0.536511\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.7980	1.18043
$342$	0	0
$343$	− 25.7980i	− 1.39296i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 29.3939i	− 1.57795i	−0.614428	−	0.788973i	$-0.710613\pi$
$347$	− 29.3939i	− 1.57795i	0.614428	−	0.788973i	$-0.289387\pi$
$348$	0	0
$349$	7.21393i	0.386153i	0.981184	+	0.193076i	$0.0618465\pi$
$349$	7.21393i	0.386153i	−0.981184	+	0.193076i	$0.938153\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 19.7990i	− 1.05379i	−0.849929	−	0.526897i	$-0.823355\pi$
$353$	− 19.7990i	− 1.05379i	0.849929	−	0.526897i	$-0.176645\pi$
$354$	0	0
$355$	−5.65685	−0.300235
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.54270	0.134198	0.0670992	−	0.997746i	$-0.478626\pi$
$359$	2.54270	0.134198	0.0670992	+	0.997746i	$0.478626\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11.7980	0.617533
$366$	0	0
$367$	− 10.2474i	− 0.534912i	−0.963570	−	0.267456i	$-0.913817\pi$
$367$	− 10.2474i	− 0.534912i	0.963570	−	0.267456i	$-0.0861831\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	17.7980i	0.924024i
$372$	0	0
$373$	− 36.3410i	− 1.88166i	−0.338874	−	0.940832i	$-0.610046\pi$
$373$	− 36.3410i	− 1.88166i	0.338874	−	0.940832i	$-0.389954\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.82843i	− 0.145671i
$378$	0	0
$379$	37.4052	1.92138	0.960689	−	0.277628i	$-0.0895482\pi$
$379$	37.4052	1.92138	0.960689	+	0.277628i	$0.0895482\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.21393	0.368615	0.184307	−	0.982869i	$-0.440996\pi$
$383$	7.21393	0.368615	0.184307	+	0.982869i	$0.440996\pi$
$384$	0	0
$385$	−19.7980	−1.00900
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−37.5959	−1.90619	−0.953094	−	0.302673i	$-0.902121\pi$
$389$	−37.5959	−1.90619	−0.953094	+	0.302673i	$0.902121\pi$
$390$	0	0
$391$	− 9.79796i	− 0.495504i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.10102i	0.156029i
$396$	0	0
$397$	9.61377i	0.482501i	0.970463	+	0.241251i	$0.0775576\pi$
$397$	9.61377i	0.482501i	−0.970463	+	0.241251i	$0.922442\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.142865i	0.00713432i	0.999994	+	0.00356716i	$0.00113546\pi$
$401$	0.142865i	0.00713432i	−0.999994	+	0.00356716i	$0.998865\pi$
$402$	0	0
$403$	6.92820	0.345118
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.1195	1.83995
$408$	0	0
$409$	33.5959	1.66121	0.830606	−	0.556861i	$-0.187994\pi$
$409$	33.5959	1.66121	0.830606	+	0.556861i	$0.187994\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−41.5959	−2.04680
$414$	0	0
$415$	− 0.898979i	− 0.0441292i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.04541i	0.197631i	0.995106	+	0.0988155i	$0.0315054\pi$
$419$	4.04541i	0.197631i	−0.995106	+	0.0988155i	$0.968495\pi$
$420$	0	0
$421$	15.6992i	0.765133i	0.923928	+	0.382566i	$0.124960\pi$
$421$	15.6992i	0.765133i	−0.923928	+	0.382566i	$0.875040\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.82843i	0.137199i
$426$	0	0
$427$	68.5821	3.31892
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15.1278	0.728678	0.364339	−	0.931266i	$-0.381295\pi$
$431$	15.1278	0.728678	0.364339	+	0.931266i	$0.381295\pi$
$432$	0	0
$433$	15.7980	0.759201	0.379601	−	0.925150i	$-0.376061\pi$
$433$	15.7980	0.759201	0.379601	+	0.925150i	$0.376061\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.79796	0.468700
$438$	0	0
$439$	− 21.7980i	− 1.04036i	−0.854057	−	0.520180i	$-0.825865\pi$
$439$	− 21.7980i	− 1.04036i	0.854057	−	0.520180i	$-0.174135\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.5959i	1.69121i	0.533807	+	0.845607i	$0.320761\pi$
$443$	35.5959i	1.69121i	−0.533807	+	0.845607i	$0.679239\pi$
$444$	0	0
$445$	13.9993i	0.663629i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.6417i	0.974144i	0.873362	+	0.487072i	$0.161935\pi$
$449$	20.6417i	0.974144i	−0.873362	+	0.487072i	$0.838065\pi$
$450$	0	0
$451$	−37.1195	−1.74789
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.29253	−0.294998
$456$	0	0
$457$	3.79796	0.177661	0.0888305	−	0.996047i	$-0.471687\pi$
$457$	3.79796	0.177661	0.0888305	+	0.996047i	$0.471687\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.7980	0.922083	0.461041	−	0.887379i	$-0.347476\pi$
$461$	19.7980	0.922083	0.461041	+	0.887379i	$0.347476\pi$
$462$	0	0
$463$	− 11.1464i	− 0.518018i	−0.965875	−	0.259009i	$-0.916604\pi$
$463$	− 11.1464i	− 0.518018i	0.965875	−	0.259009i	$-0.0833960\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.4949i	− 1.31859i	−0.751886	−	0.659293i	$-0.770856\pi$
$467$	− 28.4949i	− 1.31859i	0.751886	−	0.659293i	$-0.229144\pi$
$468$	0	0
$469$	− 50.3402i	− 2.32450i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.0560	1.00776	0.503881	−	0.863773i	$-0.331905\pi$
$479$	22.0560	1.00776	0.503881	+	0.863773i	$0.331905\pi$
$480$	0	0
$481$	11.7980	0.537941
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.7980	0.535718
$486$	0	0
$487$	23.5505i	1.06718i	0.845745	+	0.533588i	$0.179157\pi$
$487$	23.5505i	1.06718i	−0.845745	+	0.533588i	$0.820843\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	25.3485i	1.14396i	0.820267	+	0.571980i	$0.193825\pi$
$491$	25.3485i	1.14396i	−0.820267	+	0.571980i	$0.806175\pi$
$492$	0	0
$493$	5.65685i	0.254772i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.1701i	1.12903i
$498$	0	0
$499$	−22.3417	−1.00015	−0.500076	−	0.865982i	$-0.666694\pi$
$499$	−22.3417	−1.00015	−0.500076	+	0.865982i	$0.666694\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−29.8413	−1.33056	−0.665280	−	0.746594i	$-0.731688\pi$
$503$	−29.8413	−1.33056	−0.665280	+	0.746594i	$0.731688\pi$
$504$	0	0
$505$	15.7980	0.703000
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.3939	−1.21421	−0.607106	−	0.794621i	$-0.707670\pi$
$509$	−27.3939	−1.21421	−0.607106	+	0.794621i	$0.707670\pi$
$510$	0	0
$511$	− 52.4949i	− 2.32224i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.34847i	− 0.0594207i
$516$	0	0
$517$	43.4120i	1.90926i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	23.4702i	1.02825i	0.857716	+	0.514123i	$0.171883\pi$
$521$	23.4702i	1.02825i	−0.857716	+	0.514123i	$0.828117\pi$
$522$	0	0
$523$	−9.47090	−0.414134	−0.207067	−	0.978327i	$-0.566392\pi$
$523$	−9.47090	−0.414134	−0.207067	+	0.978327i	$0.566392\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.8564	−0.603595
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.7980	−0.511026
$534$	0	0
$535$	0.898979i	0.0388663i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	56.9444i	2.45277i
$540$	0	0
$541$	32.3840i	1.39230i	0.717897	+	0.696149i	$0.245105\pi$
$541$	32.3840i	1.39230i	−0.717897	+	0.696149i	$0.754895\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.4135i	0.660241i
$546$	0	0
$547$	1.27135	0.0543590	0.0271795	−	0.999631i	$-0.491347\pi$
$547$	1.27135	0.0543590	0.0271795	+	0.999631i	$0.491347\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.65685	−0.240990
$552$	0	0
$553$	13.7980	0.586749
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.2020	−0.771245	−0.385623	−	0.922657i	$-0.626013\pi$
$557$	−18.2020	−0.771245	−0.385623	+	0.922657i	$0.626013\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 14.2020i	− 0.598545i	−0.954168	−	0.299272i	$-0.903256\pi$
$563$	− 14.2020i	− 0.598545i	0.954168	−	0.299272i	$-0.0967439\pi$
$564$	0	0
$565$	19.5133i	0.820929i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 14.2850i	− 0.598858i	−0.954119	−	0.299429i	$-0.903204\pi$
$569$	− 14.2850i	− 0.598858i	0.954119	−	0.299429i	$-0.0967962\pi$
$570$	0	0
$571$	19.7990	0.828562	0.414281	−	0.910149i	$-0.364033\pi$
$571$	19.7990	0.828562	0.414281	+	0.910149i	$0.364033\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	31.3939	1.30694	0.653472	−	0.756951i	$-0.273312\pi$
$577$	31.3939	1.30694	0.653472	+	0.756951i	$0.273312\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	− 17.7980i	− 0.737116i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.4949i	− 0.515720i	−0.966182	−	0.257860i	$-0.916983\pi$
$587$	− 12.4949i	− 0.515720i	0.966182	−	0.257860i	$-0.0830173\pi$
$588$	0	0
$589$	− 13.8564i	− 0.570943i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.8564i	0.569014i	0.958674	+	0.284507i	$0.0918300\pi$
$593$	13.8564i	0.569014i	−0.958674	+	0.284507i	$0.908170\pi$
$594$	0	0
$595$	12.5851	0.515937
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.5851	−0.514212	−0.257106	−	0.966383i	$-0.582769\pi$
$599$	−12.5851	−0.514212	−0.257106	+	0.966383i	$0.582769\pi$
$600$	0	0
$601$	35.5959	1.45199	0.725994	−	0.687701i	$-0.241380\pi$
$601$	35.5959	1.45199	0.725994	+	0.687701i	$0.241380\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.79796	0.357688
$606$	0	0
$607$	28.4495i	1.15473i	0.816486	+	0.577365i	$0.195919\pi$
$607$	28.4495i	1.15473i	−0.816486	+	0.577365i	$0.804081\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.7980i	0.558206i
$612$	0	0
$613$	38.1838i	1.54223i	0.636697	+	0.771114i	$0.280300\pi$
$613$	38.1838i	1.54223i	−0.636697	+	0.771114i	$0.719700\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.7696i	1.48029i	0.672449	+	0.740143i	$0.265242\pi$
$617$	36.7696i	1.48029i	−0.672449	+	0.740143i	$0.734758\pi$
$618$	0	0
$619$	−19.8632	−0.798370	−0.399185	−	0.916870i	$-0.630707\pi$
$619$	−19.8632	−0.798370	−0.399185	+	0.916870i	$0.630707\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	62.2896	2.49558
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−23.5959	−0.940831
$630$	0	0
$631$	− 8.49490i	− 0.338177i	−0.985601	−	0.169088i	$-0.945918\pi$
$631$	− 8.49490i	− 0.338177i	0.985601	−	0.169088i	$-0.0540823\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 9.34847i	− 0.370983i
$636$	0	0
$637$	18.0990i	0.717110i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.78534i	0.268005i	0.990981	+	0.134002i	$0.0427830\pi$
$641$	6.78534i	0.268005i	−0.990981	+	0.134002i	$0.957217\pi$
$642$	0	0
$643$	−25.1701	−0.992612	−0.496306	−	0.868148i	$-0.665311\pi$
$643$	−25.1701	−0.992612	−0.496306	+	0.868148i	$0.665311\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.5836	1.59551	0.797753	−	0.602984i	$-0.206022\pi$
$647$	40.5836	1.59551	0.797753	+	0.602984i	$0.206022\pi$
$648$	0	0
$649$	41.5959	1.63278
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	20.4495i	0.799028i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.2474i	0.710820i	0.934711	+	0.355410i	$0.115659\pi$
$659$	18.2474i	0.710820i	−0.934711	+	0.355410i	$0.884341\pi$
$660$	0	0
$661$	− 7.21393i	− 0.280589i	−0.990110	−	0.140295i	$-0.955195\pi$
$661$	− 7.21393i	− 0.280589i	0.990110	−	0.140295i	$-0.0448050\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.5851i	0.488028i
$666$	0	0
$667$	−6.92820	−0.268261
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−68.5821	−2.64758
$672$	0	0
$673$	35.3939	1.36433	0.682167	−	0.731197i	$-0.261038\pi$
$673$	35.3939	1.36433	0.682167	+	0.731197i	$0.261038\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.59592	0.0613361	0.0306681	−	0.999530i	$-0.490237\pi$
$677$	1.59592	0.0613361	0.0306681	+	0.999530i	$0.490237\pi$
$678$	0	0
$679$	− 52.4949i	− 2.01457i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.1010i	0.577824i	0.957356	+	0.288912i	$0.0932936\pi$
$683$	15.1010i	0.577824i	−0.957356	+	0.288912i	$0.906706\pi$
$684$	0	0
$685$	− 11.3137i	− 0.432275i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 5.65685i	− 0.215509i
$690$	0	0
$691$	−2.82843	−0.107598	−0.0537992	−	0.998552i	$-0.517133\pi$
$691$	−2.82843	−0.107598	−0.0537992	+	0.998552i	$0.517133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.3923	0.394203
$696$	0	0
$697$	23.5959	0.893759
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−27.7980	−1.04991	−0.524957	−	0.851129i	$-0.675919\pi$
$701$	−27.7980	−1.04991	−0.524957	+	0.851129i	$0.675919\pi$
$702$	0	0
$703$	− 23.5959i	− 0.889937i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 70.2929i	− 2.64363i
$708$	0	0
$709$	21.3561i	0.802044i	0.916068	+	0.401022i	$0.131345\pi$
$709$	21.3561i	0.802044i	−0.916068	+	0.401022i	$0.868655\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 16.9706i	− 0.635553i
$714$	0	0
$715$	6.29253	0.235327
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.11416	−0.116138	−0.0580692	−	0.998313i	$-0.518494\pi$
$719$	−3.11416	−0.116138	−0.0580692	+	0.998313i	$0.518494\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	6.65153i	0.246692i	0.992364	+	0.123346i	$0.0393624\pi$
$727$	6.65153i	0.246692i	−0.992364	+	0.123346i	$0.960638\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 30.3984i	− 1.12279i	−0.827548	−	0.561395i	$-0.810265\pi$
$733$	− 30.3984i	− 1.12279i	0.827548	−	0.561395i	$-0.189735\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	50.3402i	1.85431i
$738$	0	0
$739$	33.6554	1.23803	0.619017	−	0.785378i	$-0.287531\pi$
$739$	33.6554	1.23803	0.619017	+	0.785378i	$0.287531\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.0409	−1.39559	−0.697793	−	0.716300i	$-0.745834\pi$
$743$	−38.0409	−1.39559	−0.697793	+	0.716300i	$0.745834\pi$
$744$	0	0
$745$	−0.202041	−0.00740221
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	22.6969i	0.828223i	0.910226	+	0.414112i	$0.135908\pi$
$751$	22.6969i	0.828223i	−0.910226	+	0.414112i	$0.864092\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 13.7980i	− 0.502159i
$756$	0	0
$757$	19.3704i	0.704029i	0.935995	+	0.352015i	$0.114503\pi$
$757$	19.3704i	0.704029i	−0.935995	+	0.352015i	$0.885497\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 9.61377i	− 0.348499i	−0.984702	−	0.174249i	$-0.944250\pi$
$761$	− 9.61377i	− 0.348499i	0.984702	−	0.174249i	$-0.0557499\pi$
$762$	0	0
$763$	68.5821	2.48284
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.2207	0.477373
$768$	0	0
$769$	−9.79796	−0.353323	−0.176662	−	0.984272i	$-0.556530\pi$
$769$	−9.79796	−0.353323	−0.176662	+	0.984272i	$0.556530\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−53.5959	−1.92771	−0.963856	−	0.266425i	$-0.914157\pi$
$773$	−53.5959	−1.92771	−0.963856	+	0.266425i	$0.914157\pi$
$774$	0	0
$775$	4.89898i	0.175977i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.5959i	0.845411i
$780$	0	0
$781$	− 25.1701i	− 0.900658i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 5.51399i	− 0.196803i
$786$	0	0
$787$	8.19955	0.292282	0.146141	−	0.989264i	$-0.453315\pi$
$787$	8.19955	0.292282	0.146141	+	0.989264i	$0.453315\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	86.8241	3.08711
$792$	0	0
$793$	−21.7980	−0.774068
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.1918	1.67162	0.835810	−	0.549018i	$-0.184998\pi$
$797$	47.1918	1.67162	0.835810	+	0.549018i	$0.184998\pi$
$798$	0	0
$799$	− 27.5959i	− 0.976273i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	52.4949i	1.85250i
$804$	0	0
$805$	15.4135i	0.543254i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1.69994i	− 0.0597668i	−0.999553	−	0.0298834i	$-0.990486\pi$
$809$	− 1.69994i	− 0.0597668i	0.999553	−	0.0298834i	$-0.00951360\pi$
$810$	0	0
$811$	39.9479	1.40276	0.701381	−	0.712787i	$-0.252567\pi$
$811$	39.9479	1.40276	0.701381	+	0.712787i	$0.252567\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	23.8988	0.837137
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−29.5959	−1.03290	−0.516452	−	0.856316i	$-0.672748\pi$
$821$	−29.5959	−1.03290	−0.516452	+	0.856316i	$0.672748\pi$
$822$	0	0
$823$	− 19.5505i	− 0.681488i	−0.940156	−	0.340744i	$-0.889321\pi$
$823$	− 19.5505i	− 0.681488i	0.940156	−	0.340744i	$-0.110679\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.39388i	− 0.187563i	−0.995593	−	0.0937817i	$-0.970104\pi$
$827$	− 5.39388i	− 0.187563i	0.995593	−	0.0937817i	$-0.0298956\pi$
$828$	0	0
$829$	− 37.4694i	− 1.30137i	−0.759349	−	0.650684i	$-0.774482\pi$
$829$	− 37.4694i	− 1.30137i	0.759349	−	0.650684i	$-0.225518\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 36.1981i	− 1.25419i
$834$	0	0
$835$	−0.921404	−0.0318865
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25.1701	0.868969	0.434484	−	0.900679i	$-0.356931\pi$
$839$	25.1701	0.868969	0.434484	+	0.900679i	$0.356931\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.0000	−0.378412
$846$	0	0
$847$	− 39.1464i	− 1.34509i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 28.8990i	− 0.990644i
$852$	0	0
$853$	3.67118i	0.125699i	0.998023	+	0.0628494i	$0.0200188\pi$
$853$	3.67118i	0.125699i	−0.998023	+	0.0628494i	$0.979981\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.0280i	0.376709i	0.982101	+	0.188354i	$0.0603153\pi$
$857$	11.0280i	0.376709i	−0.982101	+	0.188354i	$0.939685\pi$
$858$	0	0
$859$	−10.3923	−0.354581	−0.177290	−	0.984159i	$-0.556733\pi$
$859$	−10.3923	−0.354581	−0.177290	+	0.984159i	$0.556733\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	33.0197	1.12400	0.562002	−	0.827136i	$-0.310031\pi$
$863$	33.0197	1.12400	0.562002	+	0.827136i	$0.310031\pi$
$864$	0	0
$865$	9.79796	0.333141
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.7980	−0.468064
$870$	0	0
$871$	16.0000i	0.542139i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 4.44949i	− 0.150420i
$876$	0	0
$877$	16.5420i	0.558583i	0.960206	+	0.279291i	$0.0900995\pi$
$877$	16.5420i	0.558583i	−0.960206	+	0.279291i	$0.909900\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 2.97129i	− 0.100105i	−0.998747	−	0.0500527i	$-0.984061\pi$
$881$	− 2.97129i	− 0.100105i	0.998747	−	0.0500527i	$-0.0159389\pi$
$882$	0	0
$883$	−51.6116	−1.73687	−0.868434	−	0.495805i	$-0.834873\pi$
$883$	−51.6116	−1.73687	−0.868434	+	0.495805i	$0.834873\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.7914	−0.899567	−0.449784	−	0.893137i	$-0.648499\pi$
$887$	−26.7914	−0.899567	−0.449784	+	0.893137i	$0.648499\pi$
$888$	0	0
$889$	−41.5959	−1.39508
$890$	0	0
$891$	0	0
$892$	0	0
$893$	27.5959	0.923462
$894$	0	0
$895$	− 6.24745i	− 0.208829i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.79796i	0.326780i
$900$	0	0
$901$	11.3137i	0.376914i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 4.09978i	− 0.136281i
$906$	0	0
$907$	32.0983	1.06581	0.532904	−	0.846176i	$-0.321101\pi$
$907$	32.0983	1.06581	0.532904	+	0.846176i	$0.321101\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	47.2261	1.56467	0.782335	−	0.622858i	$-0.214029\pi$
$911$	47.2261	1.56467	0.782335	+	0.622858i	$0.214029\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	90.9898	3.00475
$918$	0	0
$919$	− 3.10102i	− 0.102293i	−0.998691	−	0.0511466i	$-0.983712\pi$
$919$	− 3.10102i	− 0.102293i	0.998691	−	0.0511466i	$-0.0162876\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	8.34242i	0.274297i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 32.2412i	− 1.05780i	−0.848684	−	0.528899i	$-0.822605\pi$
$929$	− 32.2412i	− 1.05780i	0.848684	−	0.528899i	$-0.177395\pi$
$930$	0	0
$931$	36.1981	1.18634
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.5851	−0.411575
$936$	0	0
$937$	−45.5959	−1.48955	−0.744777	−	0.667314i	$-0.767444\pi$
$937$	−45.5959	−1.48955	−0.744777	+	0.667314i	$0.767444\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.5959	−0.443214	−0.221607	−	0.975136i	$-0.571130\pi$
$941$	−13.5959	−0.443214	−0.221607	+	0.975136i	$0.571130\pi$
$942$	0	0
$943$	28.8990i	0.941080i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0000i	1.03986i	0.854209	+	0.519930i	$0.174042\pi$
$947$	32.0000i	1.03986i	−0.854209	+	0.519930i	$0.825958\pi$
$948$	0	0
$949$	16.6848i	0.541613i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52.8829i	1.71305i	0.516109	+	0.856523i	$0.327380\pi$
$953$	52.8829i	1.71305i	−0.516109	+	0.856523i	$0.672620\pi$
$954$	0	0
$955$	18.2419	0.590294
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−50.3402	−1.62557
$960$	0	0
$961$	7.00000	0.225806
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.0000	−0.321911
$966$	0	0
$967$	− 6.24745i	− 0.200904i	−0.994942	−	0.100452i	$-0.967971\pi$
$967$	− 6.24745i	− 0.200904i	0.994942	−	0.100452i	$-0.0320289\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 17.3485i	− 0.556739i	−0.960474	−	0.278369i	$-0.910206\pi$
$971$	− 17.3485i	− 0.556739i	0.960474	−	0.278369i	$-0.0897940\pi$
$972$	0	0
$973$	− 46.2405i	− 1.48240i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 24.8844i	− 0.796122i	−0.917359	−	0.398061i	$-0.869683\pi$
$977$	− 24.8844i	− 0.796122i	0.917359	−	0.398061i	$-0.130317\pi$
$978$	0	0
$979$	−62.2896	−1.99078
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.5276	−0.590940	−0.295470	−	0.955352i	$-0.595476\pi$
$983$	−18.5276	−0.590940	−0.295470	+	0.955352i	$0.595476\pi$
$984$	0	0
$985$	−17.5959	−0.560653
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 45.7980i	− 1.45482i	−0.686203	−	0.727410i	$-0.740724\pi$
$991$	− 45.7980i	− 1.45482i	0.686203	−	0.727410i	$-0.259276\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 13.7980i	− 0.437425i
$996$	0	0
$997$	29.6985i	0.940560i	0.882517	+	0.470280i	$0.155847\pi$
$997$	29.6985i	0.940560i	−0.882517	+	0.470280i	$0.844153\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.b.b.2591.7	yes	8
3.2	odd	2		2880.2.b.c.2591.7	yes	8
4.3	odd	2	inner	2880.2.b.b.2591.1	✓	8
8.3	odd	2		2880.2.b.c.2591.2	yes	8
8.5	even	2		2880.2.b.c.2591.8	yes	8
12.11	even	2		2880.2.b.c.2591.1	yes	8
24.5	odd	2	inner	2880.2.b.b.2591.8	yes	8
24.11	even	2	inner	2880.2.b.b.2591.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.b.b.2591.1	✓	8	4.3	odd	2	inner
2880.2.b.b.2591.2	yes	8	24.11	even	2	inner
2880.2.b.b.2591.7	yes	8	1.1	even	1	trivial
2880.2.b.b.2591.8	yes	8	24.5	odd	2	inner
2880.2.b.c.2591.1	yes	8	12.11	even	2
2880.2.b.c.2591.2	yes	8	8.3	odd	2
2880.2.b.c.2591.7	yes	8	3.2	odd	2
2880.2.b.c.2591.8	yes	8	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.44949i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +4.44949i q^{7} -4.44949i q^{11} -1.41421i q^{13} +2.82843i q^{17} -2.82843 q^{19} -3.46410 q^{23} +1.00000 q^{25} +2.00000 q^{29} +4.89898i q^{31} -4.44949i q^{35} +8.34242i q^{37} -8.34242i q^{41} -9.75663 q^{47} -12.7980 q^{49} +4.00000 q^{53} +4.44949i q^{55} +9.34847i q^{59} -15.4135i q^{61} +1.41421i q^{65} -11.3137 q^{67} +5.65685 q^{71} -11.7980 q^{73} +19.7980 q^{77} -3.10102i q^{79} +0.898979i q^{83} -2.82843i q^{85} -13.9993i q^{89} +6.29253 q^{91} +2.82843 q^{95} -11.7980 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 8 q^{25} + 16 q^{29} - 24 q^{49} + 32 q^{53} - 16 q^{73} + 80 q^{77} - 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^25 + 16 * q^29 - 24 * q^49 + 32 * q^53 - 16 * q^73 + 80 * q^77 - 16 * q^97

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