LMFDB - Embedded newform 2304.2.f.g.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,2,Mod(1151,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2304.1151");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2304.f (of order $2$, degree $1$, not 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:	$18.3975326257$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2304.1151
Dual form			2304.2.f.g.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{5} -0.828427i q^{7} +O(q^{10})$	$q+0.585786 q^{5} -0.828427i q^{7} -2.82843i q^{11} -2.82843i q^{13} -2.58579i q^{17} -5.65685 q^{19} -6.82843 q^{23} -4.65685 q^{25} -3.41421 q^{29} +8.82843i q^{31} -0.485281i q^{35} +7.65685i q^{37} +5.41421i q^{41} +1.65685 q^{43} -4.48528 q^{47} +6.31371 q^{49} +9.07107 q^{53} -1.65685i q^{55} -13.6569i q^{59} -3.65685i q^{61} -1.65685i q^{65} -12.0000 q^{67} -12.4853 q^{71} -4.00000 q^{73} -2.34315 q^{77} +10.4853i q^{79} -10.8284i q^{83} -1.51472i q^{85} +3.75736i q^{89} -2.34315 q^{91} -3.31371 q^{95} +2.34315 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5}+O(q^{10}) $ 4 * q + 8 * q^5	$ 4 q + 8 q^{5} - 16 q^{23} + 4 q^{25} - 8 q^{29} - 16 q^{43} + 16 q^{47} - 20 q^{49} + 8 q^{53} - 48 q^{67} - 16 q^{71} - 16 q^{73} - 32 q^{77} - 32 q^{91} + 32 q^{95} + 32 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 - 16 * q^23 + 4 * q^25 - 8 * q^29 - 16 * q^43 + 16 * q^47 - 20 * q^49 + 8 * q^53 - 48 * q^67 - 16 * q^71 - 16 * q^73 - 32 * q^77 - 32 * q^91 + 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	− 0.828427i	− 0.313116i	−0.987669	−	0.156558i	$-0.949960\pi$
$7$	− 0.828427i	− 0.313116i	0.987669	−	0.156558i	$-0.0500398\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.82843i	− 0.852803i	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	− 2.82843i	− 0.852803i	0.904534	−	0.426401i	$-0.140219\pi$
$12$	0	0
$13$	− 2.82843i	− 0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$	− 2.82843i	− 0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.58579i	− 0.627145i	−0.949564	−	0.313573i	$-0.898474\pi$
$17$	− 2.58579i	− 0.627145i	0.949564	−	0.313573i	$-0.101526\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.82843	−1.42383	−0.711913	−	0.702268i	$-0.752171\pi$
$23$	−6.82843	−1.42383	−0.711913	+	0.702268i	$0.752171\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.41421	−0.634004	−0.317002	−	0.948425i	$-0.602676\pi$
$29$	−3.41421	−0.634004	−0.317002	+	0.948425i	$0.602676\pi$
$30$	0	0
$31$	8.82843i	1.58563i	0.609461	+	0.792816i	$0.291386\pi$
$31$	8.82843i	1.58563i	−0.609461	+	0.792816i	$0.708614\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.485281i	− 0.0820275i
$36$	0	0
$37$	7.65685i	1.25878i	0.777090	+	0.629390i	$0.216695\pi$
$37$	7.65685i	1.25878i	−0.777090	+	0.629390i	$0.783305\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.41421i	0.845558i	0.906233	+	0.422779i	$0.138945\pi$
$41$	5.41421i	0.845558i	−0.906233	+	0.422779i	$0.861055\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.48528	−0.654246	−0.327123	−	0.944982i	$-0.606079\pi$
$47$	−4.48528	−0.654246	−0.327123	+	0.944982i	$0.606079\pi$
$48$	0	0
$49$	6.31371	0.901958
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.07107	1.24601	0.623003	−	0.782219i	$-0.285912\pi$
$53$	9.07107	1.24601	0.623003	+	0.782219i	$0.285912\pi$
$54$	0	0
$55$	− 1.65685i	− 0.223410i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 13.6569i	− 1.77797i	−0.457935	−	0.888985i	$-0.651411\pi$
$59$	− 13.6569i	− 1.77797i	0.457935	−	0.888985i	$-0.348589\pi$
$60$	0	0
$61$	− 3.65685i	− 0.468212i	−0.972211	−	0.234106i	$-0.924784\pi$
$61$	− 3.65685i	− 0.468212i	0.972211	−	0.234106i	$-0.0752163\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.65685i	− 0.205507i
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.4853	−1.48173	−0.740865	−	0.671654i	$-0.765584\pi$
$71$	−12.4853	−1.48173	−0.740865	+	0.671654i	$0.765584\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.34315	−0.267026
$78$	0	0
$79$	10.4853i	1.17969i	0.807518	+	0.589843i	$0.200810\pi$
$79$	10.4853i	1.17969i	−0.807518	+	0.589843i	$0.799190\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 10.8284i	− 1.18857i	−0.804253	−	0.594287i	$-0.797434\pi$
$83$	− 10.8284i	− 1.18857i	0.804253	−	0.594287i	$-0.202566\pi$
$84$	0	0
$85$	− 1.51472i	− 0.164294i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.75736i	0.398279i	0.979971	+	0.199140i	$0.0638147\pi$
$89$	3.75736i	0.398279i	−0.979971	+	0.199140i	$0.936185\pi$
$90$	0	0
$91$	−2.34315	−0.245628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.31371	−0.339979
$96$	0	0
$97$	2.34315	0.237910	0.118955	−	0.992900i	$-0.462046\pi$
$97$	2.34315	0.237910	0.118955	+	0.992900i	$0.462046\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.7279	−1.46548	−0.732742	−	0.680507i	$-0.761760\pi$
$101$	−14.7279	−1.46548	−0.732742	+	0.680507i	$0.761760\pi$
$102$	0	0
$103$	8.82843i	0.869891i	0.900457	+	0.434945i	$0.143232\pi$
$103$	8.82843i	0.869891i	−0.900457	+	0.434945i	$0.856768\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.3137i	− 1.09374i	−0.837218	−	0.546869i	$-0.815820\pi$
$107$	− 11.3137i	− 1.09374i	0.837218	−	0.546869i	$-0.184180\pi$
$108$	0	0
$109$	− 5.17157i	− 0.495347i	−0.968844	−	0.247673i	$-0.920334\pi$
$109$	− 5.17157i	− 0.495347i	0.968844	−	0.247673i	$-0.0796660\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 3.75736i	− 0.353463i	−0.984259	−	0.176731i	$-0.943448\pi$
$113$	− 3.75736i	− 0.353463i	0.984259	−	0.176731i	$-0.0565524\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.14214	−0.196369
$120$	0	0
$121$	3.00000	0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	0.828427i	0.0735110i	0.999324	+	0.0367555i	$0.0117023\pi$
$127$	0.828427i	0.0735110i	−0.999324	+	0.0367555i	$0.988298\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 5.65685i	− 0.494242i	−0.968985	−	0.247121i	$-0.920516\pi$
$131$	− 5.65685i	− 0.494242i	0.968985	−	0.247121i	$-0.0794845\pi$
$132$	0	0
$133$	4.68629i	0.406353i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13.4142i	− 1.14605i	−0.819537	−	0.573027i	$-0.805769\pi$
$137$	− 13.4142i	− 1.14605i	0.819537	−	0.573027i	$-0.194231\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.7279	1.53425	0.767126	−	0.641497i	$-0.221686\pi$
$149$	18.7279	1.53425	0.767126	+	0.641497i	$0.221686\pi$
$150$	0	0
$151$	10.4853i	0.853280i	0.904422	+	0.426640i	$0.140303\pi$
$151$	10.4853i	0.853280i	−0.904422	+	0.426640i	$0.859697\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.17157i	0.415391i
$156$	0	0
$157$	11.6569i	0.930318i	0.885227	+	0.465159i	$0.154003\pi$
$157$	11.6569i	0.930318i	−0.885227	+	0.465159i	$0.845997\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.65685i	0.445823i
$162$	0	0
$163$	9.65685	0.756383	0.378192	−	0.925727i	$-0.376546\pi$
$163$	9.65685	0.756383	0.378192	+	0.925727i	$0.376546\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.4142	1.78015	0.890075	−	0.455814i	$-0.150652\pi$
$173$	23.4142	1.78015	0.890075	+	0.455814i	$0.150652\pi$
$174$	0	0
$175$	3.85786i	0.291627i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3.31371i	− 0.247678i	−0.992302	−	0.123839i	$-0.960479\pi$
$179$	− 3.31371i	− 0.247678i	0.992302	−	0.123839i	$-0.0395207\pi$
$180$	0	0
$181$	− 22.1421i	− 1.64581i	−0.568178	−	0.822906i	$-0.692351\pi$
$181$	− 22.1421i	− 1.64581i	0.568178	−	0.822906i	$-0.307649\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.48528i	0.329764i
$186$	0	0
$187$	−7.31371	−0.534831
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.6569	−1.56703	−0.783517	−	0.621370i	$-0.786577\pi$
$191$	−21.6569	−1.56703	−0.783517	+	0.621370i	$0.786577\pi$
$192$	0	0
$193$	−17.3137	−1.24627	−0.623134	−	0.782115i	$-0.714141\pi$
$193$	−17.3137	−1.24627	−0.623134	+	0.782115i	$0.714141\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.3848	−1.45236	−0.726178	−	0.687507i	$-0.758705\pi$
$197$	−20.3848	−1.45236	−0.726178	+	0.687507i	$0.758705\pi$
$198$	0	0
$199$	0.828427i	0.0587256i	0.999569	+	0.0293628i	$0.00934782\pi$
$199$	0.828427i	0.0587256i	−0.999569	+	0.0293628i	$0.990652\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.82843i	0.198517i
$204$	0	0
$205$	3.17157i	0.221512i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.0000i	1.10674i
$210$	0	0
$211$	−0.686292	−0.0472463	−0.0236231	−	0.999721i	$-0.507520\pi$
$211$	−0.686292	−0.0472463	−0.0236231	+	0.999721i	$0.507520\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.970563	0.0661918
$216$	0	0
$217$	7.31371	0.496487
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.31371	−0.491973
$222$	0	0
$223$	− 21.7990i	− 1.45977i	−0.683571	−	0.729884i	$-0.739574\pi$
$223$	− 21.7990i	− 1.45977i	0.683571	−	0.729884i	$-0.260426\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 21.1716i	− 1.40521i	−0.711582	−	0.702603i	$-0.752021\pi$
$227$	− 21.1716i	− 1.40521i	0.711582	−	0.702603i	$-0.247979\pi$
$228$	0	0
$229$	10.8284i	0.715563i	0.933805	+	0.357781i	$0.116467\pi$
$229$	10.8284i	0.715563i	−0.933805	+	0.357781i	$0.883533\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.89949i	0.124440i	0.998062	+	0.0622200i	$0.0198181\pi$
$233$	1.89949i	0.124440i	−0.998062	+	0.0622200i	$0.980182\pi$
$234$	0	0
$235$	−2.62742	−0.171394
$236$	0	0
$237$	0	0
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−12.9706	−0.835507	−0.417754	−	0.908560i	$-0.637183\pi$
$241$	−12.9706	−0.835507	−0.417754	+	0.908560i	$0.637183\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.69848	0.236288
$246$	0	0
$247$	16.0000i	1.01806i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.51472i	0.474325i	0.971470	+	0.237162i	$0.0762174\pi$
$251$	7.51472i	0.474325i	−0.971470	+	0.237162i	$0.923783\pi$
$252$	0	0
$253$	19.3137i	1.21424i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.89949i	− 0.617514i	−0.951141	−	0.308757i	$-0.900087\pi$
$257$	− 9.89949i	− 0.617514i	0.951141	−	0.308757i	$-0.0999129\pi$
$258$	0	0
$259$	6.34315	0.394144
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.6569	1.82872	0.914360	−	0.404902i	$-0.132694\pi$
$263$	29.6569	1.82872	0.914360	+	0.404902i	$0.132694\pi$
$264$	0	0
$265$	5.31371	0.326419
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.38478	0.267345	0.133672	−	0.991026i	$-0.457323\pi$
$269$	4.38478	0.267345	0.133672	+	0.991026i	$0.457323\pi$
$270$	0	0
$271$	− 18.4853i	− 1.12290i	−0.827510	−	0.561450i	$-0.810244\pi$
$271$	− 18.4853i	− 1.12290i	0.827510	−	0.561450i	$-0.189756\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	13.1716i	0.794276i
$276$	0	0
$277$	− 26.8284i	− 1.61196i	−0.591940	−	0.805982i	$-0.701638\pi$
$277$	− 26.8284i	− 1.61196i	0.591940	−	0.805982i	$-0.298362\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 25.8995i	− 1.54503i	−0.634994	−	0.772517i	$-0.718997\pi$
$281$	− 25.8995i	− 1.54503i	0.634994	−	0.772517i	$-0.281003\pi$
$282$	0	0
$283$	−19.3137	−1.14808	−0.574040	−	0.818827i	$-0.694625\pi$
$283$	−19.3137	−1.14808	−0.574040	+	0.818827i	$0.694625\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.48528	0.264758
$288$	0	0
$289$	10.3137	0.606689
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.8701	−1.45292	−0.726462	−	0.687206i	$-0.758837\pi$
$293$	−24.8701	−1.45292	−0.726462	+	0.687206i	$0.758837\pi$
$294$	0	0
$295$	− 8.00000i	− 0.465778i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	19.3137i	1.11694i
$300$	0	0
$301$	− 1.37258i	− 0.0791144i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.14214i	− 0.122658i
$306$	0	0
$307$	23.3137	1.33058	0.665292	−	0.746583i	$-0.268307\pi$
$307$	23.3137	1.33058	0.665292	+	0.746583i	$0.268307\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.34315	0.132868	0.0664338	−	0.997791i	$-0.478838\pi$
$311$	2.34315	0.132868	0.0664338	+	0.997791i	$0.478838\pi$
$312$	0	0
$313$	13.3137	0.752535	0.376268	−	0.926511i	$-0.377207\pi$
$313$	13.3137	0.752535	0.376268	+	0.926511i	$0.377207\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.75736	0.323366	0.161683	−	0.986843i	$-0.448308\pi$
$317$	5.75736	0.323366	0.161683	+	0.986843i	$0.448308\pi$
$318$	0	0
$319$	9.65685i	0.540680i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	14.6274i	0.813891i
$324$	0	0
$325$	13.1716i	0.730627i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.71573i	0.204855i
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.02944	−0.384059
$336$	0	0
$337$	0.686292	0.0373847	0.0186923	−	0.999825i	$-0.494050\pi$
$337$	0.686292	0.0373847	0.0186923	+	0.999825i	$0.494050\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.9706	1.35223
$342$	0	0
$343$	− 11.0294i	− 0.595534i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.485281i	0.0260513i	0.999915	+	0.0130256i	$0.00414631\pi$
$347$	0.485281i	0.0260513i	−0.999915	+	0.0130256i	$0.995854\pi$
$348$	0	0
$349$	− 0.343146i	− 0.0183682i	−0.999958	−	0.00918409i	$-0.997077\pi$
$349$	− 0.343146i	− 0.0183682i	0.999958	−	0.00918409i	$-0.00292343\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	34.8701i	1.85595i	0.372648	+	0.927973i	$0.378450\pi$
$353$	34.8701i	1.85595i	−0.372648	+	0.927973i	$0.621550\pi$
$354$	0	0
$355$	−7.31371	−0.388171
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.51472	−0.185500	−0.0927499	−	0.995689i	$-0.529566\pi$
$359$	−3.51472	−0.185500	−0.0927499	+	0.995689i	$0.529566\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.34315	−0.122646
$366$	0	0
$367$	− 15.1716i	− 0.791950i	−0.918261	−	0.395975i	$-0.870407\pi$
$367$	− 15.1716i	− 0.791950i	0.918261	−	0.395975i	$-0.129593\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 7.51472i	− 0.390145i
$372$	0	0
$373$	− 18.9706i	− 0.982259i	−0.871087	−	0.491129i	$-0.836584\pi$
$373$	− 18.9706i	− 0.982259i	0.871087	−	0.491129i	$-0.163416\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.65685i	0.497353i
$378$	0	0
$379$	36.2843	1.86380	0.931899	−	0.362718i	$-0.118151\pi$
$379$	36.2843	1.86380	0.931899	+	0.362718i	$0.118151\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.6274	1.56499	0.782494	−	0.622658i	$-0.213947\pi$
$383$	30.6274	1.56499	0.782494	+	0.622658i	$0.213947\pi$
$384$	0	0
$385$	−1.37258	−0.0699533
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−33.5563	−1.70137	−0.850687	−	0.525672i	$-0.823814\pi$
$389$	−33.5563	−1.70137	−0.850687	+	0.525672i	$0.823814\pi$
$390$	0	0
$391$	17.6569i	0.892946i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.14214i	0.309044i
$396$	0	0
$397$	− 14.9706i	− 0.751351i	−0.926751	−	0.375676i	$-0.877411\pi$
$397$	− 14.9706i	− 0.751351i	0.926751	−	0.375676i	$-0.122589\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 10.5858i	− 0.528629i	−0.964437	−	0.264314i	$-0.914854\pi$
$401$	− 10.5858i	− 0.528629i	0.964437	−	0.264314i	$-0.0851457\pi$
$402$	0	0
$403$	24.9706	1.24387
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.6569	1.07349
$408$	0	0
$409$	−21.6569	−1.07086	−0.535431	−	0.844579i	$-0.679851\pi$
$409$	−21.6569	−1.07086	−0.535431	+	0.844579i	$0.679851\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11.3137	−0.556711
$414$	0	0
$415$	− 6.34315i	− 0.311373i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	27.7990i	1.35807i	0.734106	+	0.679035i	$0.237601\pi$
$419$	27.7990i	1.35807i	−0.734106	+	0.679035i	$0.762399\pi$
$420$	0	0
$421$	24.4853i	1.19334i	0.802487	+	0.596670i	$0.203510\pi$
$421$	24.4853i	1.19334i	−0.802487	+	0.596670i	$0.796490\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.0416i	0.584105i
$426$	0	0
$427$	−3.02944	−0.146605
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.5147	0.554644	0.277322	−	0.960777i	$-0.410553\pi$
$431$	11.5147	0.554644	0.277322	+	0.960777i	$0.410553\pi$
$432$	0	0
$433$	−16.6274	−0.799063	−0.399531	−	0.916720i	$-0.630827\pi$
$433$	−16.6274	−0.799063	−0.399531	+	0.916720i	$0.630827\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	38.6274	1.84780
$438$	0	0
$439$	− 36.1421i	− 1.72497i	−0.506083	−	0.862485i	$-0.668907\pi$
$439$	− 36.1421i	− 1.72497i	0.506083	−	0.862485i	$-0.331093\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.4558i	1.58954i	0.606914	+	0.794768i	$0.292407\pi$
$443$	33.4558i	1.58954i	−0.606914	+	0.794768i	$0.707593\pi$
$444$	0	0
$445$	2.20101i	0.104338i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 40.7279i	− 1.92207i	−0.276428	−	0.961035i	$-0.589151\pi$
$449$	− 40.7279i	− 1.92207i	0.276428	−	0.961035i	$-0.410849\pi$
$450$	0	0
$451$	15.3137	0.721094
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.37258	−0.0643477
$456$	0	0
$457$	−24.9706	−1.16807	−0.584037	−	0.811727i	$-0.698528\pi$
$457$	−24.9706	−1.16807	−0.584037	+	0.811727i	$0.698528\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.8995	1.29941	0.649705	−	0.760187i	$-0.274893\pi$
$461$	27.8995	1.29941	0.649705	+	0.760187i	$0.274893\pi$
$462$	0	0
$463$	5.79899i	0.269502i	0.990879	+	0.134751i	$0.0430234\pi$
$463$	5.79899i	0.269502i	−0.990879	+	0.134751i	$0.956977\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.1421i	1.02462i	0.858802	+	0.512308i	$0.171209\pi$
$467$	22.1421i	1.02462i	−0.858802	+	0.512308i	$0.828791\pi$
$468$	0	0
$469$	9.94113i	0.459039i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4.68629i	− 0.215476i
$474$	0	0
$475$	26.3431	1.20871
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.8284	0.677528	0.338764	−	0.940871i	$-0.389991\pi$
$479$	14.8284	0.677528	0.338764	+	0.940871i	$0.389991\pi$
$480$	0	0
$481$	21.6569	0.987468
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.37258	0.0623258
$486$	0	0
$487$	− 23.1716i	− 1.05000i	−0.851101	−	0.525002i	$-0.824065\pi$
$487$	− 23.1716i	− 1.05000i	0.851101	−	0.525002i	$-0.175935\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.6863i	− 0.572524i	−0.958151	−	0.286262i	$-0.907587\pi$
$491$	− 12.6863i	− 0.572524i	0.958151	−	0.286262i	$-0.0924128\pi$
$492$	0	0
$493$	8.82843i	0.397612i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.3431i	0.463953i
$498$	0	0
$499$	−38.6274	−1.72920	−0.864600	−	0.502460i	$-0.832428\pi$
$499$	−38.6274	−1.72920	−0.864600	+	0.502460i	$0.832428\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	23.7990	1.06114	0.530572	−	0.847640i	$-0.321977\pi$
$503$	23.7990	1.06114	0.530572	+	0.847640i	$0.321977\pi$
$504$	0	0
$505$	−8.62742	−0.383915
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−38.7279	−1.71658	−0.858292	−	0.513161i	$-0.828474\pi$
$509$	−38.7279	−1.71658	−0.858292	+	0.513161i	$0.828474\pi$
$510$	0	0
$511$	3.31371i	0.146590i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.17157i	0.227887i
$516$	0	0
$517$	12.6863i	0.557942i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 11.0711i	− 0.485032i	−0.970147	−	0.242516i	$-0.922027\pi$
$521$	− 11.0711i	− 0.485032i	0.970147	−	0.242516i	$-0.0779727\pi$
$522$	0	0
$523$	34.3431	1.50172	0.750860	−	0.660461i	$-0.229639\pi$
$523$	34.3431	1.50172	0.750860	+	0.660461i	$0.229639\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.8284	0.994422
$528$	0	0
$529$	23.6274	1.02728
$530$	0	0
$531$	0	0
$532$	0	0
$533$	15.3137	0.663310
$534$	0	0
$535$	− 6.62742i	− 0.286528i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 17.8579i	− 0.769193i
$540$	0	0
$541$	− 3.79899i	− 0.163331i	−0.996660	−	0.0816657i	$-0.973976\pi$
$541$	− 3.79899i	− 0.163331i	0.996660	−	0.0816657i	$-0.0260240\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.02944i	− 0.129767i
$546$	0	0
$547$	−1.65685	−0.0708420	−0.0354210	−	0.999372i	$-0.511277\pi$
$547$	−1.65685	−0.0708420	−0.0354210	+	0.999372i	$0.511277\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	19.3137	0.822792
$552$	0	0
$553$	8.68629	0.369379
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.07107	−0.214868	−0.107434	−	0.994212i	$-0.534263\pi$
$557$	−5.07107	−0.214868	−0.107434	+	0.994212i	$0.534263\pi$
$558$	0	0
$559$	− 4.68629i	− 0.198209i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.4558i	1.07284i	0.843952	+	0.536418i	$0.180223\pi$
$563$	25.4558i	1.07284i	−0.843952	+	0.536418i	$0.819777\pi$
$564$	0	0
$565$	− 2.20101i	− 0.0925972i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.8995i	0.918075i	0.888417	+	0.459037i	$0.151806\pi$
$569$	21.8995i	0.918075i	−0.888417	+	0.459037i	$0.848194\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	31.7990	1.32611
$576$	0	0
$577$	−6.68629	−0.278354	−0.139177	−	0.990268i	$-0.544446\pi$
$577$	−6.68629	−0.278354	−0.139177	+	0.990268i	$0.544446\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.97056	−0.372162
$582$	0	0
$583$	− 25.6569i	− 1.06260i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.3431i	0.757103i	0.925580	+	0.378551i	$0.123578\pi$
$587$	18.3431i	0.757103i	−0.925580	+	0.378551i	$0.876422\pi$
$588$	0	0
$589$	− 49.9411i	− 2.05779i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 0.928932i	− 0.0381467i	−0.999818	−	0.0190733i	$-0.993928\pi$
$593$	− 0.928932i	− 0.0381467i	0.999818	−	0.0190733i	$-0.00607160\pi$
$594$	0	0
$595$	−1.25483	−0.0514432
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.5147	−1.12422	−0.562110	−	0.827062i	$-0.690010\pi$
$599$	−27.5147	−1.12422	−0.562110	+	0.827062i	$0.690010\pi$
$600$	0	0
$601$	1.31371	0.0535873	0.0267936	−	0.999641i	$-0.491470\pi$
$601$	1.31371	0.0535873	0.0267936	+	0.999641i	$0.491470\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.75736	0.0714468
$606$	0	0
$607$	24.8284i	1.00775i	0.863775	+	0.503877i	$0.168094\pi$
$607$	24.8284i	1.00775i	−0.863775	+	0.503877i	$0.831906\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.6863i	0.513232i
$612$	0	0
$613$	0.343146i	0.0138595i	0.999976	+	0.00692976i	$0.00220583\pi$
$613$	0.343146i	0.0138595i	−0.999976	+	0.00692976i	$0.997794\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	33.4142i	1.34521i	0.740004	+	0.672603i	$0.234824\pi$
$617$	33.4142i	1.34521i	−0.740004	+	0.672603i	$0.765176\pi$
$618$	0	0
$619$	33.9411	1.36421	0.682105	−	0.731255i	$-0.261065\pi$
$619$	33.9411	1.36421	0.682105	+	0.731255i	$0.261065\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.11270	0.124708
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.7990	0.789437
$630$	0	0
$631$	− 18.4853i	− 0.735887i	−0.929848	−	0.367944i	$-0.880062\pi$
$631$	− 18.4853i	− 0.735887i	0.929848	−	0.367944i	$-0.119938\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.485281i	0.0192578i
$636$	0	0
$637$	− 17.8579i	− 0.707554i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 14.8701i	− 0.587332i	−0.955908	−	0.293666i	$-0.905125\pi$
$641$	− 14.8701i	− 0.587332i	0.955908	−	0.293666i	$-0.0948753\pi$
$642$	0	0
$643$	9.65685	0.380829	0.190415	−	0.981704i	$-0.439017\pi$
$643$	9.65685	0.380829	0.190415	+	0.981704i	$0.439017\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.7990	−0.621122	−0.310561	−	0.950553i	$-0.600517\pi$
$647$	−15.7990	−0.621122	−0.310561	+	0.950553i	$0.600517\pi$
$648$	0	0
$649$	−38.6274	−1.51626
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.27208	−0.206312	−0.103156	−	0.994665i	$-0.532894\pi$
$653$	−5.27208	−0.206312	−0.103156	+	0.994665i	$0.532894\pi$
$654$	0	0
$655$	− 3.31371i	− 0.129477i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.97056i	0.349444i	0.984618	+	0.174722i	$0.0559026\pi$
$659$	8.97056i	0.349444i	−0.984618	+	0.174722i	$0.944097\pi$
$660$	0	0
$661$	− 3.65685i	− 0.142235i	−0.997468	−	0.0711176i	$-0.977343\pi$
$661$	− 3.65685i	− 0.142235i	0.997468	−	0.0711176i	$-0.0226566\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.74517i	0.106453i
$666$	0	0
$667$	23.3137	0.902710
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.3431	−0.399293
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.7279	−0.719773	−0.359886	−	0.932996i	$-0.617185\pi$
$677$	−18.7279	−0.719773	−0.359886	+	0.932996i	$0.617185\pi$
$678$	0	0
$679$	− 1.94113i	− 0.0744936i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.1421i	0.847245i	0.905839	+	0.423623i	$0.139242\pi$
$683$	22.1421i	0.847245i	−0.905839	+	0.423623i	$0.860758\pi$
$684$	0	0
$685$	− 7.85786i	− 0.300234i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 25.6569i	− 0.977448i
$690$	0	0
$691$	3.02944	0.115245	0.0576226	−	0.998338i	$-0.481648\pi$
$691$	3.02944	0.115245	0.0576226	+	0.998338i	$0.481648\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.02944	−0.266642
$696$	0	0
$697$	14.0000	0.530288
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.7574	−0.519608	−0.259804	−	0.965661i	$-0.583658\pi$
$701$	−13.7574	−0.519608	−0.259804	+	0.965661i	$0.583658\pi$
$702$	0	0
$703$	− 43.3137i	− 1.63361i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.2010i	0.458866i
$708$	0	0
$709$	48.0833i	1.80580i	0.429846	+	0.902902i	$0.358568\pi$
$709$	48.0833i	1.80580i	−0.429846	+	0.902902i	$0.641432\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 60.2843i	− 2.25766i
$714$	0	0
$715$	−4.68629	−0.175257
$716$	0	0
$717$	0	0
$718$	0	0
$719$	28.4853	1.06232	0.531161	−	0.847271i	$-0.321756\pi$
$719$	28.4853	1.06232	0.531161	+	0.847271i	$0.321756\pi$
$720$	0	0
$721$	7.31371	0.272377
$722$	0	0
$723$	0	0
$724$	0	0
$725$	15.8995	0.590492
$726$	0	0
$727$	8.82843i	0.327428i	0.986508	+	0.163714i	$0.0523475\pi$
$727$	8.82843i	0.327428i	−0.986508	+	0.163714i	$0.947653\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 4.28427i	− 0.158459i
$732$	0	0
$733$	24.4853i	0.904385i	0.891920	+	0.452192i	$0.149358\pi$
$733$	24.4853i	0.904385i	−0.891920	+	0.452192i	$0.850642\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.9411i	1.25024i
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.9706	0.622590	0.311295	−	0.950313i	$-0.399237\pi$
$743$	16.9706	0.622590	0.311295	+	0.950313i	$0.399237\pi$
$744$	0	0
$745$	10.9706	0.401930
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.37258	−0.342467
$750$	0	0
$751$	40.8284i	1.48985i	0.667148	+	0.744925i	$0.267515\pi$
$751$	40.8284i	1.48985i	−0.667148	+	0.744925i	$0.732485\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.14214i	0.223535i
$756$	0	0
$757$	− 0.485281i	− 0.0176379i	−0.999961	−	0.00881893i	$-0.997193\pi$
$757$	− 0.485281i	− 0.0176379i	0.999961	−	0.00881893i	$-0.00280719\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.6985i	0.931569i	0.884898	+	0.465785i	$0.154228\pi$
$761$	25.6985i	0.931569i	−0.884898	+	0.465785i	$0.845772\pi$
$762$	0	0
$763$	−4.28427	−0.155101
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−38.6274	−1.39476
$768$	0	0
$769$	−21.3137	−0.768592	−0.384296	−	0.923210i	$-0.625556\pi$
$769$	−21.3137	−0.768592	−0.384296	+	0.923210i	$0.625556\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.5858	−0.452679	−0.226340	−	0.974048i	$-0.572676\pi$
$773$	−12.5858	−0.452679	−0.226340	+	0.974048i	$0.572676\pi$
$774$	0	0
$775$	− 41.1127i	− 1.47681i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 30.6274i	− 1.09734i
$780$	0	0
$781$	35.3137i	1.26362i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6.82843i	0.243717i
$786$	0	0
$787$	−26.3431	−0.939032	−0.469516	−	0.882924i	$-0.655572\pi$
$787$	−26.3431	−0.939032	−0.469516	+	0.882924i	$0.655572\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.11270	−0.110675
$792$	0	0
$793$	−10.3431	−0.367296
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.0711	0.463001	0.231500	−	0.972835i	$-0.425637\pi$
$797$	13.0711	0.463001	0.231500	+	0.972835i	$0.425637\pi$
$798$	0	0
$799$	11.5980i	0.410307i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.3137i	0.399252i
$804$	0	0
$805$	3.31371i	0.116793i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.6690i	1.50016i	0.661345	+	0.750082i	$0.269986\pi$
$809$	42.6690i	1.50016i	−0.661345	+	0.750082i	$0.730014\pi$
$810$	0	0
$811$	−31.5980	−1.10956	−0.554778	−	0.831999i	$-0.687197\pi$
$811$	−31.5980	−1.10956	−0.554778	+	0.831999i	$0.687197\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.65685	0.198151
$816$	0	0
$817$	−9.37258	−0.327905
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.4142	−0.398359	−0.199179	−	0.979963i	$-0.563828\pi$
$821$	−11.4142	−0.398359	−0.199179	+	0.979963i	$0.563828\pi$
$822$	0	0
$823$	52.4264i	1.82747i	0.406311	+	0.913735i	$0.366815\pi$
$823$	52.4264i	1.82747i	−0.406311	+	0.913735i	$0.633185\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 15.0294i	− 0.522625i	−0.965254	−	0.261312i	$-0.915845\pi$
$827$	− 15.0294i	− 0.522625i	0.965254	−	0.261312i	$-0.0841553\pi$
$828$	0	0
$829$	4.20101i	0.145907i	0.997335	+	0.0729536i	$0.0232425\pi$
$829$	4.20101i	0.145907i	−0.997335	+	0.0729536i	$0.976758\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 16.3259i	− 0.565659i
$834$	0	0
$835$	−3.31371	−0.114676
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.51472	−0.121342	−0.0606708	−	0.998158i	$-0.519324\pi$
$839$	−3.51472	−0.121342	−0.0606708	+	0.998158i	$0.519324\pi$
$840$	0	0
$841$	−17.3431	−0.598040
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.92893	0.100758
$846$	0	0
$847$	− 2.48528i	− 0.0853953i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 52.2843i	− 1.79228i
$852$	0	0
$853$	− 42.2843i	− 1.44779i	−0.689912	−	0.723893i	$-0.742351\pi$
$853$	− 42.2843i	− 1.44779i	0.689912	−	0.723893i	$-0.257649\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 21.8995i	− 0.748072i	−0.927414	−	0.374036i	$-0.877974\pi$
$857$	− 21.8995i	− 0.748072i	0.927414	−	0.374036i	$-0.122026\pi$
$858$	0	0
$859$	1.65685	0.0565311	0.0282656	−	0.999600i	$-0.491002\pi$
$859$	1.65685	0.0565311	0.0282656	+	0.999600i	$0.491002\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.970563	−0.0330383	−0.0165192	−	0.999864i	$-0.505258\pi$
$863$	−0.970563	−0.0330383	−0.0165192	+	0.999864i	$0.505258\pi$
$864$	0	0
$865$	13.7157	0.466349
$866$	0	0
$867$	0	0
$868$	0	0
$869$	29.6569	1.00604
$870$	0	0
$871$	33.9411i	1.15005i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.68629i	0.158426i
$876$	0	0
$877$	− 6.97056i	− 0.235379i	−0.993050	−	0.117690i	$-0.962451\pi$
$877$	− 6.97056i	− 0.235379i	0.993050	−	0.117690i	$-0.0375488\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	12.7279i	0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$	12.7279i	0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$	0	0
$883$	40.2843	1.35567	0.677837	−	0.735212i	$-0.262918\pi$
$883$	40.2843	1.35567	0.677837	+	0.735212i	$0.262918\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.34315	−0.0786751	−0.0393376	−	0.999226i	$-0.512525\pi$
$887$	−2.34315	−0.0786751	−0.0393376	+	0.999226i	$0.512525\pi$
$888$	0	0
$889$	0.686292	0.0230175
$890$	0	0
$891$	0	0
$892$	0	0
$893$	25.3726	0.849061
$894$	0	0
$895$	− 1.94113i	− 0.0648847i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 30.1421i	− 1.00530i
$900$	0	0
$901$	− 23.4558i	− 0.781427i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 12.9706i	− 0.431156i
$906$	0	0
$907$	−14.3431	−0.476256	−0.238128	−	0.971234i	$-0.576534\pi$
$907$	−14.3431	−0.476256	−0.238128	+	0.971234i	$0.576534\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.5980	−0.781836	−0.390918	−	0.920426i	$-0.627842\pi$
$911$	−23.5980	−0.781836	−0.390918	+	0.920426i	$0.627842\pi$
$912$	0	0
$913$	−30.6274	−1.01362
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.68629	−0.154755
$918$	0	0
$919$	− 23.4558i	− 0.773737i	−0.922135	−	0.386868i	$-0.873557\pi$
$919$	− 23.4558i	− 0.773737i	0.922135	−	0.386868i	$-0.126443\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.3137i	1.16236i
$924$	0	0
$925$	− 35.6569i	− 1.17239i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.7279i	0.548825i	0.961612	+	0.274413i	$0.0884834\pi$
$929$	16.7279i	0.548825i	−0.961612	+	0.274413i	$0.911517\pi$
$930$	0	0
$931$	−35.7157	−1.17054
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.28427	−0.140111
$936$	0	0
$937$	56.6274	1.84994	0.924969	−	0.380044i	$-0.124091\pi$
$937$	56.6274	1.84994	0.924969	+	0.380044i	$0.124091\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.2721	0.693450	0.346725	−	0.937967i	$-0.387294\pi$
$941$	21.2721	0.693450	0.346725	+	0.937967i	$0.387294\pi$
$942$	0	0
$943$	− 36.9706i	− 1.20393i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 31.5980i	− 1.02680i	−0.858151	−	0.513398i	$-0.828386\pi$
$947$	− 31.5980i	− 1.02680i	0.858151	−	0.513398i	$-0.171614\pi$
$948$	0	0
$949$	11.3137i	0.367259i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 4.44365i	− 0.143944i	−0.997407	−	0.0719720i	$-0.977071\pi$
$953$	− 4.44365i	− 0.143944i	0.997407	−	0.0719720i	$-0.0229292\pi$
$954$	0	0
$955$	−12.6863	−0.410519
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−11.1127	−0.358848
$960$	0	0
$961$	−46.9411	−1.51423
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.1421	−0.326487
$966$	0	0
$967$	46.0833i	1.48194i	0.671539	+	0.740969i	$0.265633\pi$
$967$	46.0833i	1.48194i	−0.671539	+	0.740969i	$0.734367\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 21.1716i	− 0.679428i	−0.940529	−	0.339714i	$-0.889670\pi$
$971$	− 21.1716i	− 0.679428i	0.940529	−	0.339714i	$-0.110330\pi$
$972$	0	0
$973$	9.94113i	0.318698i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.5269i	0.400771i	0.979717	+	0.200386i	$0.0642195\pi$
$977$	12.5269i	0.400771i	−0.979717	+	0.200386i	$0.935780\pi$
$978$	0	0
$979$	10.6274	0.339654
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.3431	1.09538	0.547688	−	0.836683i	$-0.315508\pi$
$983$	34.3431	1.09538	0.547688	+	0.836683i	$0.315508\pi$
$984$	0	0
$985$	−11.9411	−0.380476
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.3137	−0.359755
$990$	0	0
$991$	− 13.7990i	− 0.438339i	−0.975687	−	0.219170i	$-0.929665\pi$
$991$	− 13.7990i	− 0.438339i	0.975687	−	0.219170i	$-0.0703348\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.485281i	0.0153845i
$996$	0	0
$997$	− 17.5980i	− 0.557334i	−0.960388	−	0.278667i	$-0.910107\pi$
$997$	− 17.5980i	− 0.557334i	0.960388	−	0.278667i	$-0.0898925\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.2.f.g.1151.1		4
3.2	odd	2		2304.2.f.b.1151.3		4
4.3	odd	2		2304.2.f.h.1151.2		4
8.3	odd	2		2304.2.f.b.1151.4		4
8.5	even	2		2304.2.f.a.1151.3		4
12.11	even	2		2304.2.f.a.1151.4		4
16.3	odd	4		1152.2.c.b.1151.2	yes	4
16.5	even	4		1152.2.c.d.1151.3	yes	4
16.11	odd	4		1152.2.c.a.1151.3	yes	4
16.13	even	4		1152.2.c.c.1151.2	yes	4
24.5	odd	2		2304.2.f.h.1151.1		4
24.11	even	2	inner	2304.2.f.g.1151.2		4
48.5	odd	4		1152.2.c.a.1151.2	✓	4
48.11	even	4		1152.2.c.d.1151.2	yes	4
48.29	odd	4		1152.2.c.b.1151.3	yes	4
48.35	even	4		1152.2.c.c.1151.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.c.a.1151.2	✓	4	48.5	odd	4
1152.2.c.a.1151.3	yes	4	16.11	odd	4
1152.2.c.b.1151.2	yes	4	16.3	odd	4
1152.2.c.b.1151.3	yes	4	48.29	odd	4
1152.2.c.c.1151.2	yes	4	16.13	even	4
1152.2.c.c.1151.3	yes	4	48.35	even	4
1152.2.c.d.1151.2	yes	4	48.11	even	4
1152.2.c.d.1151.3	yes	4	16.5	even	4
2304.2.f.a.1151.3		4	8.5	even	2
2304.2.f.a.1151.4		4	12.11	even	2
2304.2.f.b.1151.3		4	3.2	odd	2
2304.2.f.b.1151.4		4	8.3	odd	2
2304.2.f.g.1151.1		4	1.1	even	1	trivial
2304.2.f.g.1151.2		4	24.11	even	2	inner
2304.2.f.h.1151.1		4	24.5	odd	2
2304.2.f.h.1151.2		4	4.3	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:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2304.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.585786 q^{5} -0.828427i q^{7} +O(q^{10})\)	\(q+0.585786 q^{5} -0.828427i q^{7} -2.82843i q^{11} -2.82843i q^{13} -2.58579i q^{17} -5.65685 q^{19} -6.82843 q^{23} -4.65685 q^{25} -3.41421 q^{29} +8.82843i q^{31} -0.485281i q^{35} +7.65685i q^{37} +5.41421i q^{41} +1.65685 q^{43} -4.48528 q^{47} +6.31371 q^{49} +9.07107 q^{53} -1.65685i q^{55} -13.6569i q^{59} -3.65685i q^{61} -1.65685i q^{65} -12.0000 q^{67} -12.4853 q^{71} -4.00000 q^{73} -2.34315 q^{77} +10.4853i q^{79} -10.8284i q^{83} -1.51472i q^{85} +3.75736i q^{89} -2.34315 q^{91} -3.31371 q^{95} +2.34315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5}+O(q^{10}) \) 4 * q + 8 * q^5	\( 4 q + 8 q^{5} - 16 q^{23} + 4 q^{25} - 8 q^{29} - 16 q^{43} + 16 q^{47} - 20 q^{49} + 8 q^{53} - 48 q^{67} - 16 q^{71} - 16 q^{73} - 32 q^{77} - 32 q^{91} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 - 16 * q^23 + 4 * q^25 - 8 * q^29 - 16 * q^43 + 16 * q^47 - 20 * q^49 + 8 * q^53 - 48 * q^67 - 16 * q^71 - 16 * q^73 - 32 * q^77 - 32 * q^91 + 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more