LMFDB - Embedded newform 2880.2.b.a.2591.4

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:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.4
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	2880.2591
Dual form			2880.2.b.a.2591.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.16228i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.16228i q^{7} -1.16228i q^{11} +5.88635i q^{13} +1.64371i q^{17} -1.64371 q^{19} +1.00000 q^{25} -8.32456 q^{29} +4.32456i q^{31} +3.16228i q^{35} +2.59893i q^{37} -7.53006i q^{41} +8.48528 q^{43} -10.1290 q^{47} -3.00000 q^{49} +2.32456 q^{53} +1.16228i q^{55} +13.1623i q^{59} +8.48528i q^{61} -5.88635i q^{65} +11.7727 q^{67} +11.7727 q^{71} -2.00000 q^{73} -3.67544 q^{77} -0.324555i q^{79} -2.32456i q^{83} -1.64371i q^{85} +16.0153i q^{89} +18.6143 q^{91} +1.64371 q^{95} -0.324555 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 48 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 + 48 * 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
$7$	− 3.16228i	− 1.19523i	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	− 3.16228i	− 1.19523i	0.801784	−	0.597614i	$-0.203885\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.16228i	− 0.350440i	−0.984529	−	0.175220i	$-0.943936\pi$
$11$	− 1.16228i	− 0.350440i	0.984529	−	0.175220i	$-0.0560637\pi$
$12$	0	0
$13$	5.88635i	1.63258i	0.577643	+	0.816290i	$0.303973\pi$
$13$	5.88635i	1.63258i	−0.577643	+	0.816290i	$0.696027\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.64371i	0.398658i	0.979933	+	0.199329i	$0.0638762\pi$
$17$	1.64371i	0.398658i	−0.979933	+	0.199329i	$0.936124\pi$
$18$	0	0
$19$	−1.64371	−0.377093	−0.188546	−	0.982064i	$-0.560378\pi$
$19$	−1.64371	−0.377093	−0.188546	+	0.982064i	$0.560378\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.32456	−1.54583	−0.772916	−	0.634509i	$-0.781202\pi$
$29$	−8.32456	−1.54583	−0.772916	+	0.634509i	$0.781202\pi$
$30$	0	0
$31$	4.32456i	0.776713i	0.921509	+	0.388357i	$0.126957\pi$
$31$	4.32456i	0.776713i	−0.921509	+	0.388357i	$0.873043\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.16228i	0.534522i
$36$	0	0
$37$	2.59893i	0.427262i	0.976914	+	0.213631i	$0.0685290\pi$
$37$	2.59893i	0.427262i	−0.976914	+	0.213631i	$0.931471\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.53006i	− 1.17600i	−0.808862	−	0.587999i	$-0.799916\pi$
$41$	− 7.53006i	− 1.17600i	0.808862	−	0.587999i	$-0.200084\pi$
$42$	0	0
$43$	8.48528	1.29399	0.646997	−	0.762493i	$-0.276025\pi$
$43$	8.48528	1.29399	0.646997	+	0.762493i	$0.276025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.1290	−1.47747	−0.738733	−	0.673999i	$-0.764575\pi$
$47$	−10.1290	−1.47747	−0.738733	+	0.673999i	$0.764575\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.32456	0.319302	0.159651	−	0.987174i	$-0.448963\pi$
$53$	2.32456	0.319302	0.159651	+	0.987174i	$0.448963\pi$
$54$	0	0
$55$	1.16228i	0.156721i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.1623i	1.71358i	0.515663	+	0.856791i	$0.327546\pi$
$59$	13.1623i	1.71358i	−0.515663	+	0.856791i	$0.672454\pi$
$60$	0	0
$61$	8.48528i	1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$	8.48528i	1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 5.88635i	− 0.730112i
$66$	0	0
$67$	11.7727	1.43826	0.719132	−	0.694873i	$-0.244540\pi$
$67$	11.7727	1.43826	0.719132	+	0.694873i	$0.244540\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.7727	1.39716	0.698581	−	0.715531i	$-0.253815\pi$
$71$	11.7727	1.39716	0.698581	+	0.715531i	$0.253815\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.67544	−0.418856
$78$	0	0
$79$	− 0.324555i	− 0.0365153i	−0.999833	−	0.0182577i	$-0.994188\pi$
$79$	− 0.324555i	− 0.0365153i	0.999833	−	0.0182577i	$-0.00581192\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.32456i	− 0.255153i	−0.991829	−	0.127577i	$-0.959280\pi$
$83$	− 2.32456i	− 0.255153i	0.991829	−	0.127577i	$-0.0407198\pi$
$84$	0	0
$85$	− 1.64371i	− 0.178285i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.0153i	1.69762i	0.528696	+	0.848811i	$0.322681\pi$
$89$	16.0153i	1.69762i	−0.528696	+	0.848811i	$0.677319\pi$
$90$	0	0
$91$	18.6143	1.95131
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.64371	0.168641
$96$	0	0
$97$	−0.324555	−0.0329536	−0.0164768	−	0.999864i	$-0.505245\pi$
$97$	−0.324555	−0.0329536	−0.0164768	+	0.999864i	$0.505245\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	0.837722i	0.0825432i	0.999148	+	0.0412716i	$0.0131409\pi$
$103$	0.837722i	0.0825432i	−0.999148	+	0.0412716i	$0.986859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.3246i	1.38481i	0.721510	+	0.692404i	$0.243448\pi$
$107$	14.3246i	1.38481i	−0.721510	+	0.692404i	$0.756552\pi$
$108$	0	0
$109$	8.48528i	0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$	8.48528i	0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.1290i	0.952855i	0.879214	+	0.476428i	$0.158069\pi$
$113$	10.1290i	0.952855i	−0.879214	+	0.476428i	$0.841931\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.19786	0.476487
$120$	0	0
$121$	9.64911	0.877192
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	− 15.1623i	− 1.34543i	−0.739900	−	0.672717i	$-0.765127\pi$
$127$	− 15.1623i	− 1.34543i	0.739900	−	0.672717i	$-0.234873\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	15.4868i	1.35309i	0.736401	+	0.676545i	$0.236524\pi$
$131$	15.4868i	1.35309i	−0.736401	+	0.676545i	$0.763476\pi$
$132$	0	0
$133$	5.19786i	0.450712i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.84157i	− 0.584515i	−0.956340	−	0.292257i	$-0.905594\pi$
$137$	− 6.84157i	− 0.584515i	0.956340	−	0.292257i	$-0.0944064\pi$
$138$	0	0
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.84157	0.572121
$144$	0	0
$145$	8.32456	0.691317
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.67544	−0.301104	−0.150552	−	0.988602i	$-0.548105\pi$
$149$	−3.67544	−0.301104	−0.150552	+	0.988602i	$0.548105\pi$
$150$	0	0
$151$	10.0000i	0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$	10.0000i	0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 4.32456i	− 0.347357i
$156$	0	0
$157$	14.3716i	1.14698i	0.819212	+	0.573491i	$0.194411\pi$
$157$	14.3716i	1.14698i	−0.819212	+	0.573491i	$0.805589\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.2580	−1.56761	−0.783805	−	0.621007i	$-0.786724\pi$
$167$	−20.2580	−1.56761	−0.783805	+	0.621007i	$0.786724\pi$
$168$	0	0
$169$	−21.6491	−1.66532
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.9737	−1.44254	−0.721271	−	0.692653i	$-0.756442\pi$
$173$	−18.9737	−1.44254	−0.721271	+	0.692653i	$0.756442\pi$
$174$	0	0
$175$	− 3.16228i	− 0.239046i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 15.4868i	− 1.15754i	−0.815491	−	0.578770i	$-0.803533\pi$
$179$	− 15.4868i	− 1.15754i	0.815491	−	0.578770i	$-0.196467\pi$
$180$	0	0
$181$	11.7727i	0.875058i	0.899204	+	0.437529i	$0.144146\pi$
$181$	11.7727i	0.875058i	−0.899204	+	0.437529i	$0.855854\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.59893i	− 0.191077i
$186$	0	0
$187$	1.91045	0.139706
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.28742	0.237869	0.118935	−	0.992902i	$-0.462052\pi$
$191$	3.28742	0.237869	0.118935	+	0.992902i	$0.462052\pi$
$192$	0	0
$193$	−12.3246	−0.887141	−0.443570	−	0.896240i	$-0.646288\pi$
$193$	−12.3246	−0.887141	−0.443570	+	0.896240i	$0.646288\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.6491	−0.758718	−0.379359	−	0.925250i	$-0.623855\pi$
$197$	−10.6491	−0.758718	−0.379359	+	0.925250i	$0.623855\pi$
$198$	0	0
$199$	18.6491i	1.32200i	0.750386	+	0.661000i	$0.229868\pi$
$199$	18.6491i	1.32200i	−0.750386	+	0.661000i	$0.770132\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	26.3246i	1.84762i
$204$	0	0
$205$	7.53006i	0.525922i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.91045i	0.132148i
$210$	0	0
$211$	−23.5454	−1.62093	−0.810466	−	0.585786i	$-0.800786\pi$
$211$	−23.5454	−1.62093	−0.810466	+	0.585786i	$0.800786\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.48528	−0.578691
$216$	0	0
$217$	13.6754	0.928350
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.67544	−0.650841
$222$	0	0
$223$	13.4868i	0.903145i	0.892234	+	0.451573i	$0.149137\pi$
$223$	13.4868i	0.903145i	−0.892234	+	0.451573i	$0.850863\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 16.6491i	− 1.10504i	−0.833500	−	0.552520i	$-0.813666\pi$
$227$	− 16.6491i	− 1.10504i	0.833500	−	0.552520i	$-0.186334\pi$
$228$	0	0
$229$	20.2580i	1.33869i	0.742954	+	0.669343i	$0.233424\pi$
$229$	20.2580i	1.33869i	−0.742954	+	0.669343i	$0.766576\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.93113i	− 0.323049i	−0.986869	−	0.161524i	$-0.948359\pi$
$233$	− 4.93113i	− 0.323049i	0.986869	−	0.161524i	$-0.0516411\pi$
$234$	0	0
$235$	10.1290	0.660742
$236$	0	0
$237$	0	0
$238$	0	0
$239$	20.2580	1.31038	0.655190	−	0.755464i	$-0.272589\pi$
$239$	20.2580	1.31038	0.655190	+	0.755464i	$0.272589\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	− 9.67544i	− 0.615634i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.48683i	− 0.220087i	−0.993927	−	0.110043i	$-0.964901\pi$
$251$	− 3.48683i	− 0.220087i	0.993927	−	0.110043i	$-0.0350990\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 27.0996i	− 1.69042i	−0.534431	−	0.845212i	$-0.679474\pi$
$257$	− 27.0996i	− 1.69042i	0.534431	−	0.845212i	$-0.320526\pi$
$258$	0	0
$259$	8.21854	0.510675
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.28742	0.202711	0.101355	−	0.994850i	$-0.467682\pi$
$263$	3.28742	0.202711	0.101355	+	0.994850i	$0.467682\pi$
$264$	0	0
$265$	−2.32456	−0.142796
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.9737	1.52267	0.761336	−	0.648358i	$-0.224544\pi$
$269$	24.9737	1.52267	0.761336	+	0.648358i	$0.224544\pi$
$270$	0	0
$271$	− 18.6491i	− 1.13285i	−0.824112	−	0.566426i	$-0.808326\pi$
$271$	− 18.6491i	− 1.13285i	0.824112	−	0.566426i	$-0.191674\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.16228i	− 0.0700880i
$276$	0	0
$277$	0.688486i	0.0413671i	0.999786	+	0.0206836i	$0.00658425\pi$
$277$	0.688486i	0.0413671i	−0.999786	+	0.0206836i	$0.993416\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.53006i	0.449206i	0.974450	+	0.224603i	$0.0721085\pi$
$281$	7.53006i	0.449206i	−0.974450	+	0.224603i	$0.927892\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−23.8121	−1.40559
$288$	0	0
$289$	14.2982	0.841072
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	− 13.1623i	− 0.766337i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 26.8328i	− 1.54662i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.48528i	− 0.485866i
$306$	0	0
$307$	−20.2580	−1.15618	−0.578092	−	0.815972i	$-0.696203\pi$
$307$	−20.2580	−1.15618	−0.578092	+	0.815972i	$0.696203\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.57484	−0.372825	−0.186412	−	0.982472i	$-0.559686\pi$
$311$	−6.57484	−0.372825	−0.186412	+	0.982472i	$0.559686\pi$
$312$	0	0
$313$	16.9737	0.959408	0.479704	−	0.877430i	$-0.340744\pi$
$313$	16.9737	0.959408	0.479704	+	0.877430i	$0.340744\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	9.67544i	0.541721i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.70178i	− 0.150331i
$324$	0	0
$325$	5.88635i	0.326516i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	32.0307i	1.76591i
$330$	0	0
$331$	21.9017	1.20383	0.601913	−	0.798562i	$-0.294405\pi$
$331$	21.9017	1.20383	0.601913	+	0.798562i	$0.294405\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.7727	−0.643211
$336$	0	0
$337$	30.6491	1.66956	0.834782	−	0.550581i	$-0.185594\pi$
$337$	30.6491	1.66956	0.834782	+	0.550581i	$0.185594\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.02633	0.272191
$342$	0	0
$343$	− 12.6491i	− 0.682988i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000i	1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$	24.0000i	1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$	0	0
$349$	− 15.0601i	− 0.806150i	−0.915167	−	0.403075i	$-0.867941\pi$
$349$	− 15.0601i	− 0.806150i	0.915167	−	0.403075i	$-0.132059\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.1891i	1.34068i	0.742054	+	0.670340i	$0.233852\pi$
$353$	25.1891i	1.34068i	−0.742054	+	0.670340i	$0.766148\pi$
$354$	0	0
$355$	−11.7727	−0.624830
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.7433	1.51701	0.758506	−	0.651666i	$-0.225930\pi$
$359$	28.7433	1.51701	0.758506	+	0.651666i	$0.225930\pi$
$360$	0	0
$361$	−16.2982	−0.857801
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	− 6.51317i	− 0.339985i	−0.985445	−	0.169992i	$-0.945626\pi$
$367$	− 6.51317i	− 0.339985i	0.985445	−	0.169992i	$-0.0543743\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 7.35089i	− 0.381639i
$372$	0	0
$373$	− 14.3716i	− 0.744135i	−0.928206	−	0.372067i	$-0.878649\pi$
$373$	− 14.3716i	− 0.744135i	0.928206	−	0.372067i	$-0.121351\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 49.0012i	− 2.52369i
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.8121	1.21674	0.608372	−	0.793652i	$-0.291823\pi$
$383$	23.8121	1.21674	0.608372	+	0.793652i	$0.291823\pi$
$384$	0	0
$385$	3.67544	0.187318
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.324555i	0.0163302i
$396$	0	0
$397$	− 26.1443i	− 1.31215i	−0.754697	−	0.656073i	$-0.772216\pi$
$397$	− 26.1443i	− 1.31215i	0.754697	−	0.656073i	$-0.227784\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.7880i	1.38767i	0.720135	+	0.693834i	$0.244080\pi$
$401$	27.7880i	1.38767i	−0.720135	+	0.693834i	$0.755920\pi$
$402$	0	0
$403$	−25.4558	−1.26805
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.02068	0.149730
$408$	0	0
$409$	−30.6491	−1.51550	−0.757750	−	0.652544i	$-0.773702\pi$
$409$	−30.6491	−1.51550	−0.757750	+	0.652544i	$0.773702\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	41.6228	2.04812
$414$	0	0
$415$	2.32456i	0.114108i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.48683i	0.170343i	0.996366	+	0.0851715i	$0.0271438\pi$
$419$	3.48683i	0.170343i	−0.996366	+	0.0851715i	$0.972856\pi$
$420$	0	0
$421$	3.28742i	0.160219i	0.996786	+	0.0801095i	$0.0255270\pi$
$421$	3.28742i	0.160219i	−0.996786	+	0.0801095i	$0.974473\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.64371i	0.0797316i
$426$	0	0
$427$	26.8328	1.29853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−38.6055	−1.85956	−0.929781	−	0.368113i	$-0.880004\pi$
$431$	−38.6055	−1.85956	−0.929781	+	0.368113i	$0.880004\pi$
$432$	0	0
$433$	−35.2982	−1.69632	−0.848162	−	0.529737i	$-0.822291\pi$
$433$	−35.2982	−1.69632	−0.848162	+	0.529737i	$0.822291\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	9.35089i	0.446294i	0.974785	+	0.223147i	$0.0716329\pi$
$439$	9.35089i	0.446294i	−0.974785	+	0.223147i	$0.928367\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 16.6491i	− 0.791023i	−0.918461	−	0.395512i	$-0.870567\pi$
$443$	− 16.6491i	− 0.791023i	0.918461	−	0.395512i	$-0.129433\pi$
$444$	0	0
$445$	− 16.0153i	− 0.759200i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2.33219i	− 0.110063i	−0.998485	−	0.0550315i	$-0.982474\pi$
$449$	− 2.33219i	− 0.110063i	0.998485	−	0.0550315i	$-0.0175259\pi$
$450$	0	0
$451$	−8.75202	−0.412116
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.6143	−0.872651
$456$	0	0
$457$	33.6228	1.57281	0.786404	−	0.617713i	$-0.211941\pi$
$457$	33.6228	1.57281	0.786404	+	0.617713i	$0.211941\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.3246	0.946609	0.473304	−	0.880899i	$-0.343061\pi$
$461$	20.3246	0.946609	0.473304	+	0.880899i	$0.343061\pi$
$462$	0	0
$463$	35.1623i	1.63413i	0.576546	+	0.817065i	$0.304400\pi$
$463$	35.1623i	1.63413i	−0.576546	+	0.817065i	$0.695600\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 38.3246i	− 1.77345i	−0.462298	−	0.886724i	$-0.652975\pi$
$467$	− 38.3246i	− 1.77345i	0.462298	−	0.886724i	$-0.347025\pi$
$468$	0	0
$469$	− 37.2285i	− 1.71905i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 9.86225i	− 0.453467i
$474$	0	0
$475$	−1.64371	−0.0754185
$476$	0	0
$477$	0	0
$478$	0	0
$479$	22.1684	1.01290	0.506451	−	0.862269i	$-0.330957\pi$
$479$	22.1684	1.01290	0.506451	+	0.862269i	$0.330957\pi$
$480$	0	0
$481$	−15.2982	−0.697539
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.324555	0.0147373
$486$	0	0
$487$	− 15.1623i	− 0.687068i	−0.939140	−	0.343534i	$-0.888376\pi$
$487$	− 15.1623i	− 0.687068i	0.939140	−	0.343534i	$-0.111624\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.51317i	0.384194i	0.981376	+	0.192097i	$0.0615288\pi$
$491$	8.51317i	0.384194i	−0.981376	+	0.192097i	$0.938471\pi$
$492$	0	0
$493$	− 13.6831i	− 0.616258i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 37.2285i	− 1.66993i
$498$	0	0
$499$	−38.8723	−1.74016	−0.870080	−	0.492910i	$-0.835933\pi$
$499$	−38.8723	−1.74016	−0.870080	+	0.492910i	$0.835933\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.1290	−0.451629	−0.225815	−	0.974170i	$-0.572504\pi$
$503$	−10.1290	−0.451629	−0.225815	+	0.974170i	$0.572504\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−39.2982	−1.74186	−0.870932	−	0.491404i	$-0.836484\pi$
$509$	−39.2982	−1.74186	−0.870932	+	0.491404i	$0.836484\pi$
$510$	0	0
$511$	6.32456i	0.279782i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 0.837722i	− 0.0369145i
$516$	0	0
$517$	11.7727i	0.517763i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.7279i	0.557620i	0.960346	+	0.278810i	$0.0899400\pi$
$521$	12.7279i	0.557620i	−0.960346	+	0.278810i	$0.910060\pi$
$522$	0	0
$523$	−3.28742	−0.143749	−0.0718744	−	0.997414i	$-0.522898\pi$
$523$	−3.28742	−0.143749	−0.0718744	+	0.997414i	$0.522898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.10831	−0.309643
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	44.3246	1.91991
$534$	0	0
$535$	− 14.3246i	− 0.619305i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.48683i	0.150189i
$540$	0	0
$541$	18.3475i	0.788822i	0.918934	+	0.394411i	$0.129051\pi$
$541$	18.3475i	0.788822i	−0.918934	+	0.394411i	$0.870949\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.48528i	− 0.363470i
$546$	0	0
$547$	−45.7138	−1.95458	−0.977291	−	0.211902i	$-0.932034\pi$
$547$	−45.7138	−1.95458	−0.977291	+	0.211902i	$0.932034\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13.6831	0.582922
$552$	0	0
$553$	−1.02633	−0.0436442
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.6754	0.918418	0.459209	−	0.888328i	$-0.348133\pi$
$557$	21.6754	0.918418	0.459209	+	0.888328i	$0.348133\pi$
$558$	0	0
$559$	49.9473i	2.11255i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 37.9473i	− 1.59929i	−0.600473	−	0.799645i	$-0.705021\pi$
$563$	− 37.9473i	− 1.59929i	0.600473	−	0.799645i	$-0.294979\pi$
$564$	0	0
$565$	− 10.1290i	− 0.426130i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	38.1838i	1.60075i	0.599502	+	0.800373i	$0.295365\pi$
$569$	38.1838i	1.60075i	−0.599502	+	0.800373i	$0.704635\pi$
$570$	0	0
$571$	−25.1891	−1.05413	−0.527066	−	0.849825i	$-0.676708\pi$
$571$	−25.1891	−1.05413	−0.527066	+	0.849825i	$0.676708\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.3246	1.01264	0.506322	−	0.862344i	$-0.331005\pi$
$577$	24.3246	1.01264	0.506322	+	0.862344i	$0.331005\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.35089	−0.304966
$582$	0	0
$583$	− 2.70178i	− 0.111896i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.02633i	− 0.207459i	−0.994606	−	0.103730i	$-0.966922\pi$
$587$	− 5.02633i	− 0.207459i	0.994606	−	0.103730i	$-0.0330776\pi$
$588$	0	0
$589$	− 7.10831i	− 0.292893i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.3870i	1.24784i	0.781487	+	0.623922i	$0.214462\pi$
$593$	30.3870i	1.24784i	−0.781487	+	0.623922i	$0.785538\pi$
$594$	0	0
$595$	−5.19786	−0.213092
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.0307	1.30874	0.654369	−	0.756175i	$-0.272934\pi$
$599$	32.0307	1.30874	0.654369	+	0.756175i	$0.272934\pi$
$600$	0	0
$601$	−11.3509	−0.463012	−0.231506	−	0.972833i	$-0.574365\pi$
$601$	−11.3509	−0.463012	−0.231506	+	0.972833i	$0.574365\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.64911	−0.392292
$606$	0	0
$607$	− 27.1623i	− 1.10248i	−0.834346	−	0.551241i	$-0.814154\pi$
$607$	− 27.1623i	− 1.10248i	0.834346	−	0.551241i	$-0.185846\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 59.6228i	− 2.41208i
$612$	0	0
$613$	9.17377i	0.370525i	0.982689	+	0.185262i	$0.0593135\pi$
$613$	9.17377i	0.370525i	−0.982689	+	0.185262i	$0.940686\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 15.3269i	− 0.617036i	−0.951219	−	0.308518i	$-0.900167\pi$
$617$	− 15.3269i	− 0.617036i	0.951219	−	0.308518i	$-0.0998330\pi$
$618$	0	0
$619$	−30.1202	−1.21063	−0.605317	−	0.795984i	$-0.706954\pi$
$619$	−30.1202	−1.21063	−0.605317	+	0.795984i	$0.706954\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	50.6450	2.02905
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.27189	−0.170331
$630$	0	0
$631$	− 8.97367i	− 0.357236i	−0.983918	−	0.178618i	$-0.942837\pi$
$631$	− 8.97367i	− 0.357236i	0.983918	−	0.178618i	$-0.0571626\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.1623i	0.601697i
$636$	0	0
$637$	− 17.6590i	− 0.699677i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.53006i	0.297419i	0.988881	+	0.148710i	$0.0475120\pi$
$641$	7.53006i	0.297419i	−0.988881	+	0.148710i	$0.952488\pi$
$642$	0	0
$643$	−13.6831	−0.539611	−0.269805	−	0.962915i	$-0.586959\pi$
$643$	−13.6831	−0.539611	−0.269805	+	0.962915i	$0.586959\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.6744	−1.32388	−0.661938	−	0.749558i	$-0.730266\pi$
$647$	−33.6744	−1.32388	−0.661938	+	0.749558i	$0.730266\pi$
$648$	0	0
$649$	15.2982	0.600508
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.2982	−1.06826	−0.534131	−	0.845402i	$-0.679361\pi$
$653$	−27.2982	−1.06826	−0.534131	+	0.845402i	$0.679361\pi$
$654$	0	0
$655$	− 15.4868i	− 0.605121i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 8.51317i	− 0.331626i	−0.986157	−	0.165813i	$-0.946975\pi$
$659$	− 8.51317i	− 0.331626i	0.986157	−	0.165813i	$-0.0530248\pi$
$660$	0	0
$661$	− 45.7138i	− 1.77806i	−0.457847	−	0.889031i	$-0.651379\pi$
$661$	− 45.7138i	− 1.77806i	0.457847	−	0.889031i	$-0.348621\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.19786i	− 0.201565i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.86225	0.380728
$672$	0	0
$673$	26.6491	1.02725	0.513624	−	0.858015i	$-0.328303\pi$
$673$	26.6491	1.02725	0.513624	+	0.858015i	$0.328303\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.9473	1.22784	0.613918	−	0.789370i	$-0.289593\pi$
$677$	31.9473	1.22784	0.613918	+	0.789370i	$0.289593\pi$
$678$	0	0
$679$	1.02633i	0.0393871i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 26.3246i	− 1.00728i	−0.863913	−	0.503641i	$-0.831994\pi$
$683$	− 26.3246i	− 1.00728i	0.863913	−	0.503641i	$-0.168006\pi$
$684$	0	0
$685$	6.84157i	0.261403i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.6831i	0.521286i
$690$	0	0
$691$	−18.6143	−0.708120	−0.354060	−	0.935223i	$-0.615199\pi$
$691$	−18.6143	−0.708120	−0.354060	+	0.935223i	$0.615199\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.9706	−0.643730
$696$	0	0
$697$	12.3772	0.468821
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	− 4.27189i	− 0.161117i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.9737i	0.713578i
$708$	0	0
$709$	− 3.28742i	− 0.123462i	−0.998093	−	0.0617308i	$-0.980338\pi$
$709$	− 3.28742i	− 0.123462i	0.998093	−	0.0617308i	$-0.0196620\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−6.84157	−0.255860
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.4558	0.949343	0.474671	−	0.880163i	$-0.342567\pi$
$719$	25.4558	0.949343	0.474671	+	0.880163i	$0.342567\pi$
$720$	0	0
$721$	2.64911	0.0986580
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.32456	−0.309166
$726$	0	0
$727$	3.81139i	0.141357i	0.997499	+	0.0706783i	$0.0225164\pi$
$727$	3.81139i	0.141357i	−0.997499	+	0.0706783i	$0.977484\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13.9473i	0.515861i
$732$	0	0
$733$	17.6590i	0.652252i	0.945326	+	0.326126i	$0.105743\pi$
$733$	17.6590i	0.652252i	−0.945326	+	0.326126i	$0.894257\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 13.6831i	− 0.504025i
$738$	0	0
$739$	−42.1597	−1.55087	−0.775434	−	0.631428i	$-0.782469\pi$
$739$	−42.1597	−1.55087	−0.775434	+	0.631428i	$0.782469\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.6744	−1.23539	−0.617697	−	0.786416i	$-0.711934\pi$
$743$	−33.6744	−1.23539	−0.617697	+	0.786416i	$0.711934\pi$
$744$	0	0
$745$	3.67544	0.134658
$746$	0	0
$747$	0	0
$748$	0	0
$749$	45.2982	1.65516
$750$	0	0
$751$	− 28.9737i	− 1.05726i	−0.848851	−	0.528632i	$-0.822705\pi$
$751$	− 28.9737i	− 1.05726i	0.848851	−	0.528632i	$-0.177295\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 10.0000i	− 0.363937i
$756$	0	0
$757$	− 46.4023i	− 1.68652i	−0.537505	−	0.843260i	$-0.680633\pi$
$757$	− 46.4023i	− 1.68652i	0.537505	−	0.843260i	$-0.319367\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 12.7279i	− 0.461387i	−0.973026	−	0.230693i	$-0.925901\pi$
$761$	− 12.7279i	− 0.461387i	0.973026	−	0.230693i	$-0.0740994\pi$
$762$	0	0
$763$	26.8328	0.971413
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−77.4778	−2.79756
$768$	0	0
$769$	−20.6491	−0.744626	−0.372313	−	0.928107i	$-0.621435\pi$
$769$	−20.6491	−0.744626	−0.372313	+	0.928107i	$0.621435\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15.2982	−0.550239	−0.275119	−	0.961410i	$-0.588717\pi$
$773$	−15.2982	−0.550239	−0.275119	+	0.961410i	$0.588717\pi$
$774$	0	0
$775$	4.32456i	0.155343i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.3772i	0.443460i
$780$	0	0
$781$	− 13.6831i	− 0.489621i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 14.3716i	− 0.512946i
$786$	0	0
$787$	−10.3957	−0.370568	−0.185284	−	0.982685i	$-0.559320\pi$
$787$	−10.3957	−0.370568	−0.185284	+	0.982685i	$0.559320\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32.0307	1.13888
$792$	0	0
$793$	−49.9473	−1.77368
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−38.3246	−1.35753	−0.678763	−	0.734358i	$-0.737484\pi$
$797$	−38.3246	−1.35753	−0.678763	+	0.734358i	$0.737484\pi$
$798$	0	0
$799$	− 16.6491i	− 0.589003i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.32456i	0.0820318i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	16.0153i	0.563069i	0.959551	+	0.281535i	$0.0908434\pi$
$809$	16.0153i	0.563069i	−0.959551	+	0.281535i	$0.909157\pi$
$810$	0	0
$811$	−6.57484	−0.230874	−0.115437	−	0.993315i	$-0.536827\pi$
$811$	−6.57484	−0.230874	−0.115437	+	0.993315i	$0.536827\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−13.9473	−0.487955
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.6228	1.45264	0.726322	−	0.687354i	$-0.241228\pi$
$821$	41.6228	1.45264	0.726322	+	0.687354i	$0.241228\pi$
$822$	0	0
$823$	22.7851i	0.794237i	0.917767	+	0.397119i	$0.129990\pi$
$823$	22.7851i	0.794237i	−0.917767	+	0.397119i	$0.870010\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 49.9473i	− 1.73684i	−0.495830	−	0.868419i	$-0.665136\pi$
$827$	− 49.9473i	− 1.73684i	0.495830	−	0.868419i	$-0.334864\pi$
$828$	0	0
$829$	− 5.19786i	− 0.180529i	−0.995918	−	0.0902646i	$-0.971229\pi$
$829$	− 5.19786i	− 0.180529i	0.995918	−	0.0902646i	$-0.0287713\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 4.93113i	− 0.170853i
$834$	0	0
$835$	20.2580	0.701056
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.19786	0.179450	0.0897251	−	0.995967i	$-0.471401\pi$
$839$	5.19786	0.179450	0.0897251	+	0.995967i	$0.471401\pi$
$840$	0	0
$841$	40.2982	1.38959
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21.6491	0.744752
$846$	0	0
$847$	− 30.5132i	− 1.04844i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 31.3422i	− 1.07314i	−0.843857	−	0.536568i	$-0.819720\pi$
$853$	− 31.3422i	− 1.07314i	0.843857	−	0.536568i	$-0.180280\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 35.5848i	− 1.21555i	−0.794107	−	0.607777i	$-0.792061\pi$
$857$	− 35.5848i	− 1.21555i	0.794107	−	0.607777i	$-0.207939\pi$
$858$	0	0
$859$	53.6656	1.83105	0.915524	−	0.402264i	$-0.131776\pi$
$859$	53.6656	1.83105	0.915524	+	0.402264i	$0.131776\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	37.2285	1.26727	0.633637	−	0.773630i	$-0.281561\pi$
$863$	37.2285	1.26727	0.633637	+	0.773630i	$0.281561\pi$
$864$	0	0
$865$	18.9737	0.645124
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.377223	−0.0127964
$870$	0	0
$871$	69.2982i	2.34808i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.16228i	0.106904i
$876$	0	0
$877$	9.17377i	0.309776i	0.987932	+	0.154888i	$0.0495017\pi$
$877$	9.17377i	0.309776i	−0.987932	+	0.154888i	$0.950498\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 16.0153i	− 0.539571i	−0.962921	−	0.269785i	$-0.913047\pi$
$881$	− 16.0153i	− 0.539571i	0.962921	−	0.269785i	$-0.0869527\pi$
$882$	0	0
$883$	−16.9706	−0.571105	−0.285552	−	0.958363i	$-0.592177\pi$
$883$	−16.9706	−0.571105	−0.285552	+	0.958363i	$0.592177\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.2580	−0.680196	−0.340098	−	0.940390i	$-0.610460\pi$
$887$	−20.2580	−0.680196	−0.340098	+	0.940390i	$0.610460\pi$
$888$	0	0
$889$	−47.9473	−1.60810
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.6491	0.557141
$894$	0	0
$895$	15.4868i	0.517668i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 36.0000i	− 1.20067i
$900$	0	0
$901$	3.82089i	0.127292i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 11.7727i	− 0.391338i
$906$	0	0
$907$	15.0601	0.500063	0.250031	−	0.968238i	$-0.419559\pi$
$907$	15.0601	0.500063	0.250031	+	0.968238i	$0.419559\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−30.6537	−1.01560	−0.507801	−	0.861474i	$-0.669542\pi$
$911$	−30.6537	−1.01560	−0.507801	+	0.861474i	$0.669542\pi$
$912$	0	0
$913$	−2.70178	−0.0894158
$914$	0	0
$915$	0	0
$916$	0	0
$917$	48.9737	1.61725
$918$	0	0
$919$	40.9737i	1.35160i	0.737087	+	0.675798i	$0.236201\pi$
$919$	40.9737i	1.35160i	−0.737087	+	0.675798i	$0.763799\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	69.2982i	2.28098i
$924$	0	0
$925$	2.59893i	0.0854524i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.7880i	0.911696i	0.890058	+	0.455848i	$0.150664\pi$
$929$	27.7880i	0.911696i	−0.890058	+	0.455848i	$0.849336\pi$
$930$	0	0
$931$	4.93113	0.161611
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.91045	−0.0624783
$936$	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
$941$	−55.9473	−1.82383	−0.911915	−	0.410378i	$-0.865397\pi$
$941$	−55.9473	−1.82383	−0.911915	+	0.410378i	$0.865397\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	54.5964i	1.77415i	0.461629	+	0.887073i	$0.347265\pi$
$947$	54.5964i	1.77415i	−0.461629	+	0.887073i	$0.652735\pi$
$948$	0	0
$949$	− 11.7727i	− 0.382158i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.9618i	1.19731i	0.801007	+	0.598655i	$0.204298\pi$
$953$	36.9618i	1.19731i	−0.801007	+	0.598655i	$0.795702\pi$
$954$	0	0
$955$	−3.28742	−0.106378
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−21.6350	−0.698629
$960$	0	0
$961$	12.2982	0.396717
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.3246	0.396741
$966$	0	0
$967$	− 42.1359i	− 1.35500i	−0.735522	−	0.677500i	$-0.763063\pi$
$967$	− 42.1359i	− 1.35500i	0.735522	−	0.677500i	$-0.236937\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	32.1359i	1.03129i	0.856802	+	0.515646i	$0.172448\pi$
$971$	32.1359i	1.03129i	−0.856802	+	0.515646i	$0.827552\pi$
$972$	0	0
$973$	− 53.6656i	− 1.72044i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15.3269i	− 0.490349i	−0.969479	−	0.245175i	$-0.921155\pi$
$977$	− 15.3269i	− 0.490349i	0.969479	−	0.245175i	$-0.0788453\pi$
$978$	0	0
$979$	18.6143	0.594915
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−30.3870	−0.969194	−0.484597	−	0.874738i	$-0.661034\pi$
$983$	−30.3870	−0.969194	−0.484597	+	0.874738i	$0.661034\pi$
$984$	0	0
$985$	10.6491	0.339309
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 0.701779i	− 0.0222927i	−0.999938	−	0.0111464i	$-0.996452\pi$
$991$	− 0.701779i	− 0.0222927i	0.999938	−	0.0111464i	$-0.00354807\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 18.6491i	− 0.591217i
$996$	0	0
$997$	7.79680i	0.246927i	0.992349	+	0.123463i	$0.0394002\pi$
$997$	7.79680i	0.246927i	−0.992349	+	0.123463i	$0.960600\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.b.a.2591.4	yes	8
3.2	odd	2		2880.2.b.d.2591.2	yes	8
4.3	odd	2	inner	2880.2.b.a.2591.6	yes	8
8.3	odd	2		2880.2.b.d.2591.7	yes	8
8.5	even	2		2880.2.b.d.2591.1	yes	8
12.11	even	2		2880.2.b.d.2591.8	yes	8
24.5	odd	2	inner	2880.2.b.a.2591.3	✓	8
24.11	even	2	inner	2880.2.b.a.2591.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.b.a.2591.3	✓	8	24.5	odd	2	inner
2880.2.b.a.2591.4	yes	8	1.1	even	1	trivial
2880.2.b.a.2591.5	yes	8	24.11	even	2	inner
2880.2.b.a.2591.6	yes	8	4.3	odd	2	inner
2880.2.b.d.2591.1	yes	8	8.5	even	2
2880.2.b.d.2591.2	yes	8	3.2	odd	2
2880.2.b.d.2591.7	yes	8	8.3	odd	2
2880.2.b.d.2591.8	yes	8	12.11	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.16228i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.16228i q^{7} -1.16228i q^{11} +5.88635i q^{13} +1.64371i q^{17} -1.64371 q^{19} +1.00000 q^{25} -8.32456 q^{29} +4.32456i q^{31} +3.16228i q^{35} +2.59893i q^{37} -7.53006i q^{41} +8.48528 q^{43} -10.1290 q^{47} -3.00000 q^{49} +2.32456 q^{53} +1.16228i q^{55} +13.1623i q^{59} +8.48528i q^{61} -5.88635i q^{65} +11.7727 q^{67} +11.7727 q^{71} -2.00000 q^{73} -3.67544 q^{77} -0.324555i q^{79} -2.32456i q^{83} -1.64371i q^{85} +16.0153i q^{89} +18.6143 q^{91} +1.64371 q^{95} -0.324555 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 8 q^{25} - 16 q^{29} - 24 q^{49} - 32 q^{53} - 16 q^{73} - 80 q^{77} + 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^25 - 16 * q^29 - 24 * q^49 - 32 * q^53 - 16 * q^73 - 80 * q^77 + 48 * q^97

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