LMFDB - Embedded newform 2664.2.r.n.1009.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2664,2,Mod(433,2664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2664.433"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2664 = 2^{3} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2664.r (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,-3,0,3,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$21.2721470985$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4 $ x^10 - x^9 + 9x^8 + 4x^7 + 54x^6 + 6x^5 + 98x^4 - 8x^3 + 148x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 296)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1009.4
Root			$1.50928 + 2.61415i$ of defining polynomial
Character	$\chi$	$=$	2664.1009
Dual form			2664.2.r.n.433.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.717824 + 1.24331i) q^{5} +(-0.0542045 - 0.0938850i) q^{7} +4.03713 q^{11} +(-1.19743 - 2.07401i) q^{13} +(-0.500000 + 0.866025i) q^{17} +(-1.82114 - 3.15430i) q^{19} +3.53386 q^{23} +(1.46946 - 2.54517i) q^{25} -1.30484 q^{29} +4.03713 q^{31} +(0.0778186 - 0.134786i) q^{35} +(5.87256 + 1.58526i) q^{37} +(-1.53381 - 2.65663i) q^{41} +2.43565 q^{43} -5.40706 q^{47} +(3.49412 - 6.05200i) q^{49} +(3.91430 - 6.77977i) q^{53} +(2.89795 + 5.01939i) q^{55} +(-3.83970 + 6.65056i) q^{59} +(1.56340 + 2.70789i) q^{61} +(1.71909 - 2.97754i) q^{65} +(0.614512 + 1.06437i) q^{67} +(-1.69648 - 2.93839i) q^{71} +5.76541 q^{73} +(-0.218831 - 0.379026i) q^{77} +(6.34279 + 10.9860i) q^{79} +(-1.01656 + 1.76074i) q^{83} -1.43565 q^{85} +(-3.53381 + 6.12073i) q^{89} +(-0.129812 + 0.224841i) q^{91} +(2.61451 - 4.52847i) q^{95} -13.0250 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{5} + 3 q^{7} - 16 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} + 12 q^{23} - 12 q^{25} - 16 q^{31} - 5 q^{35} - 12 q^{37} - 9 q^{41} + 4 q^{43} - 32 q^{47} - 18 q^{49} - 5 q^{53} + 4 q^{55} + 5 q^{59}+ \cdots - 36 q^{97}+O(q^{100}) $ 10 * q - 3 * q^5 + 3 * q^7 - 16 * q^11 + q^13 - 5 * q^17 - 3 * q^19 + 12 * q^23 - 12 * q^25 - 16 * q^31 - 5 * q^35 - 12 * q^37 - 9 * q^41 + 4 * q^43 - 32 * q^47 - 18 * q^49 - 5 * q^53 + 4 * q^55 + 5 * q^59 + 15 * q^61 - 23 * q^65 + q^67 + 17 * q^71 - 36 * q^73 + 4 * q^77 + 21 * q^79 + 15 * q^83 + 6 * q^85 - 29 * q^89 - q^91 + 21 * q^95 - 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

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
$5$	0.717824	+	1.24331i	0.321021	+	0.556024i	0.980699	−	0.195524i	$-0.0626407\pi$
$5$	0.717824	+	1.24331i	0.321021	+	0.556024i	−0.659678	+	0.751548i	$0.729307\pi$
$6$		0			0
$7$	−0.0542045	−	0.0938850i	−0.0204874	−	0.0354852i	0.855600	−	0.517638i	$-0.173188\pi$
$7$	−0.0542045	−	0.0938850i	−0.0204874	−	0.0354852i	−0.876087	+	0.482152i	$0.839855\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.03713			1.21724			0.608620	−	0.793462i	$-0.291723\pi$
$11$	4.03713			1.21724			0.608620	+	0.793462i	$0.291723\pi$
$12$		0			0
$13$	−1.19743	−	2.07401i	−0.332107	−	0.575226i	0.650818	−	0.759234i	$-0.274426\pi$
$13$	−1.19743	−	2.07401i	−0.332107	−	0.575226i	−0.982925	+	0.184008i	$0.941093\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.500000	+	0.866025i	−0.121268	+	0.210042i	−0.920268	−	0.391289i	$-0.872029\pi$
$17$	−0.500000	+	0.866025i	−0.121268	+	0.210042i	0.799000	+	0.601331i	$0.205363\pi$
$18$		0			0
$19$	−1.82114	−	3.15430i	−0.417797	−	0.723646i	0.577920	−	0.816093i	$-0.303864\pi$
$19$	−1.82114	−	3.15430i	−0.417797	−	0.723646i	−0.995718	+	0.0924470i	$0.970531\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.53386			0.736862			0.368431	−	0.929655i	$-0.379895\pi$
$23$	3.53386			0.736862			0.368431	+	0.929655i	$0.379895\pi$
$24$		0			0
$25$	1.46946	−	2.54517i	0.293891	−	0.509035i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.30484			−0.242303			−0.121151	−	0.992634i	$-0.538659\pi$
$29$	−1.30484			−0.242303			−0.121151	+	0.992634i	$0.538659\pi$
$30$		0			0
$31$	4.03713			0.725090			0.362545	−	0.931966i	$-0.381908\pi$
$31$	4.03713			0.725090			0.362545	+	0.931966i	$0.381908\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.0778186	−	0.134786i	0.0131538	−	0.0227830i
$36$		0			0
$37$	5.87256	+	1.58526i	0.965443	+	0.260616i
$37$	5.87256	+	1.58526i	0.965443	+	0.260616i
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.53381	−	2.65663i	−0.239541	−	0.414896i	0.721042	−	0.692891i	$-0.243663\pi$
$41$	−1.53381	−	2.65663i	−0.239541	−	0.414896i	−0.960583	+	0.277995i	$0.910330\pi$
$42$		0			0
$43$	2.43565			0.371433			0.185716	−	0.982603i	$-0.440539\pi$
$43$	2.43565			0.371433			0.185716	+	0.982603i	$0.440539\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.40706			−0.788701			−0.394350	−	0.918960i	$-0.629030\pi$
$47$	−5.40706			−0.788701			−0.394350	+	0.918960i	$0.629030\pi$
$48$		0			0
$49$	3.49412	−	6.05200i	0.499161	−	0.864571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.91430	−	6.77977i	0.537671	−	0.931273i	−0.461358	−	0.887214i	$-0.652638\pi$
$53$	3.91430	−	6.77977i	0.537671	−	0.931273i	0.999029	−	0.0440590i	$-0.0140290\pi$
$54$		0			0
$55$	2.89795	+	5.01939i	0.390759	+	0.676815i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.83970	+	6.65056i	−0.499886	+	0.865829i	−1.00000		0.000131117i	$-0.999958\pi$
$59$	−3.83970	+	6.65056i	−0.499886	+	0.865829i	0.500114	+	0.865960i	$0.333292\pi$
$60$		0			0
$61$	1.56340	+	2.70789i	0.200173	+	0.346710i	0.948584	−	0.316525i	$-0.102516\pi$
$61$	1.56340	+	2.70789i	0.200173	+	0.346710i	−0.748411	+	0.663235i	$0.769183\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.71909	−	2.97754i	0.213226	−	0.369319i
$66$		0			0
$67$	0.614512	+	1.06437i	0.0750746	+	0.130033i	0.901119	−	0.433572i	$-0.142747\pi$
$67$	0.614512	+	1.06437i	0.0750746	+	0.130033i	−0.826044	+	0.563605i	$0.809414\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.69648	−	2.93839i	−0.201335	−	0.348722i	0.747624	−	0.664122i	$-0.231195\pi$
$71$	−1.69648	−	2.93839i	−0.201335	−	0.348722i	−0.948959	+	0.315400i	$0.897861\pi$
$72$		0			0
$73$	5.76541			0.674790			0.337395	−	0.941363i	$-0.390454\pi$
$73$	5.76541			0.674790			0.337395	+	0.941363i	$0.390454\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.218831	−	0.379026i	−0.0249381	−	0.0431940i
$78$		0			0
$79$	6.34279	+	10.9860i	0.713620	+	1.23603i	0.963489	+	0.267747i	$0.0862790\pi$
$79$	6.34279	+	10.9860i	0.713620	+	1.23603i	−0.249869	+	0.968280i	$0.580388\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.01656	+	1.76074i	−0.111582	+	0.193266i	−0.916408	−	0.400245i	$-0.868925\pi$
$83$	−1.01656	+	1.76074i	−0.111582	+	0.193266i	0.804826	+	0.593511i	$0.202258\pi$
$84$		0			0
$85$	−1.43565			−0.155718
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.53381	+	6.12073i	−0.374583	+	0.648797i	−0.990265	−	0.139198i	$-0.955547\pi$
$89$	−3.53381	+	6.12073i	−0.374583	+	0.648797i	0.615682	+	0.787995i	$0.288881\pi$
$90$		0			0
$91$	−0.129812	+	0.224841i	−0.0136080	+	0.0235697i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.61451	−	4.52847i	0.268243	−	0.464611i
$96$		0			0
$97$	−13.0250			−1.32249			−0.661246	−	0.750169i	$-0.729972\pi$
$97$	−13.0250			−1.32249			−0.661246	+	0.750169i	$0.729972\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.25761			0.722159			0.361080	−	0.932535i	$-0.382408\pi$
$101$	7.25761			0.722159			0.361080	+	0.932535i	$0.382408\pi$
$102$		0			0
$103$	11.0724			1.09099			0.545496	−	0.838114i	$-0.316341\pi$
$103$	11.0724			1.09099			0.545496	+	0.838114i	$0.316341\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.05752	+	8.75989i	0.488929	+	0.846851i	0.999919	−	0.0127364i	$-0.00405423\pi$
$107$	5.05752	+	8.75989i	0.488929	+	0.846851i	−0.510989	+	0.859587i	$0.670721\pi$
$108$		0			0
$109$	8.72114	−	15.1055i	0.835334	−	1.44684i	−0.0584239	−	0.998292i	$-0.518607\pi$
$109$	8.72114	−	15.1055i	0.835334	−	1.44684i	0.893758	−	0.448549i	$-0.148059\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.10253	−	8.83785i	0.480006	−	0.831395i	−0.519731	−	0.854330i	$-0.673968\pi$
$113$	5.10253	−	8.83785i	0.480006	−	0.831395i	0.999737	+	0.0229353i	$0.00730118\pi$
$114$		0			0
$115$	2.53669	+	4.39368i	0.236548	+	0.409713i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.108409			0.00993784
$120$		0			0
$121$	5.29840			0.481673
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.3975			1.01942
$126$		0			0
$127$	−4.43050	+	7.67384i	−0.393143	+	0.680944i	−0.992862	−	0.119267i	$-0.961946\pi$
$127$	−4.43050	+	7.67384i	−0.393143	+	0.680944i	0.599719	+	0.800210i	$0.295279\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.88786	+	13.6622i	−0.689165	+	1.19367i	0.282943	+	0.959137i	$0.408689\pi$
$131$	−7.88786	+	13.6622i	−0.689165	+	1.19367i	−0.972108	+	0.234533i	$0.924644\pi$
$132$		0			0
$133$	−0.197428	+	0.341955i	−0.0171191	+	0.0296512i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	20.7904			1.77625			0.888124	−	0.459604i	$-0.152009\pi$
$137$	20.7904			1.77625			0.888124	+	0.459604i	$0.152009\pi$
$138$		0			0
$139$	5.12898	−	8.88365i	0.435034	−	0.753501i	−0.562264	−	0.826958i	$-0.690070\pi$
$139$	5.12898	−	8.88365i	0.435034	−	0.753501i	0.997298	+	0.0734564i	$0.0234030\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.83417	−	8.37303i	−0.404254	−	0.700188i
$144$		0			0
$145$	−0.936646	−	1.62232i	−0.0777842	−	0.134726i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.78328			−0.473784			−0.236892	−	0.971536i	$-0.576129\pi$
$149$	−5.78328			−0.473784			−0.236892	+	0.971536i	$0.576129\pi$
$150$		0			0
$151$	−0.757735	−	1.31244i	−0.0616636	−	0.106805i	0.833546	−	0.552451i	$-0.186307\pi$
$151$	−0.757735	−	1.31244i	−0.0616636	−	0.106805i	−0.895209	+	0.445646i	$0.852974\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.89795	+	5.01939i	0.232769	+	0.403167i
$156$		0			0
$157$	1.99905	−	3.46246i	0.159542	−	0.276334i	−0.775162	−	0.631763i	$-0.782332\pi$
$157$	1.99905	−	3.46246i	0.159542	−	0.276334i	0.934703	+	0.355429i	$0.115665\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.191551	−	0.331777i	−0.0150964	−	0.0261477i
$162$		0			0
$163$	7.03513	−	12.1852i	0.551034	−	0.954418i	−0.447167	−	0.894451i	$-0.647567\pi$
$163$	7.03513	−	12.1852i	0.551034	−	0.954418i	0.998200	−	0.0599675i	$-0.0190997\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	11.5904	+	20.0751i	0.896891	+	1.55346i	0.831447	+	0.555604i	$0.187513\pi$
$167$	11.5904	+	20.0751i	0.896891	+	1.55346i	0.0654437	+	0.997856i	$0.479154\pi$
$168$		0			0
$169$	3.63233	−	6.29139i	0.279410	−	0.483953i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.653417	−	1.13175i	0.0496784	−	0.0860454i	−0.840117	−	0.542405i	$-0.817514\pi$
$173$	0.653417	−	1.13175i	0.0496784	−	0.0860454i	0.889795	+	0.456360i	$0.150847\pi$
$174$		0			0
$175$	−0.318605			−0.0240843
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.30493			−0.396509			−0.198255	−	0.980151i	$-0.563527\pi$
$179$	−5.30493			−0.396509			−0.198255	+	0.980151i	$0.563527\pi$
$180$		0			0
$181$	11.2888	+	19.5528i	0.839091	+	1.45335i	0.890655	+	0.454679i	$0.150246\pi$
$181$	11.2888	+	19.5528i	0.839091	+	1.45335i	−0.0515644	+	0.998670i	$0.516421\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.24449	+	8.43934i	0.165019	+	0.620473i
$186$		0			0
$187$	−2.01856	+	3.49626i	−0.147612	+	0.255672i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.79127			−0.129612			−0.0648059	−	0.997898i	$-0.520643\pi$
$191$	−1.79127			−0.129612			−0.0648059	+	0.997898i	$0.520643\pi$
$192$		0			0
$193$	−18.6998			−1.34604			−0.673020	−	0.739624i	$-0.735003\pi$
$193$	−18.6998			−1.34604			−0.673020	+	0.739624i	$0.735003\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	13.1633	−	22.7995i	0.937848	−	1.62440i	0.168373	−	0.985723i	$-0.446149\pi$
$197$	13.1633	−	22.7995i	0.937848	−	1.62440i	0.769475	−	0.638677i	$-0.220518\pi$
$198$		0			0
$199$	9.76742			0.692394			0.346197	−	0.938162i	$-0.387473\pi$
$199$	9.76742			0.692394			0.346197	+	0.938162i	$0.387473\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.0707282	+	0.122505i	0.00496415	+	0.00859815i
$204$		0			0
$205$	2.20201	−	3.81399i	0.153795	−	0.266381i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.35216	−	12.7343i	−0.508560	−	0.880851i
$210$		0			0
$211$	−23.3974			−1.61074			−0.805371	−	0.592771i	$-0.798034\pi$
$211$	−23.3974			−1.61074			−0.805371	+	0.592771i	$0.798034\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.74837	+	3.02826i	0.119238	+	0.206526i
$216$		0			0
$217$	−0.218831	−	0.379026i	−0.0148552	−	0.0257299i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.39486			0.161095
$222$		0			0
$223$	4.83880			0.324030			0.162015	−	0.986788i	$-0.448201\pi$
$223$	4.83880			0.324030			0.162015	+	0.986788i	$0.448201\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.87666	−	17.1069i	−0.655537	−	1.13542i	−0.981759	−	0.190130i	$-0.939109\pi$
$227$	−9.87666	−	17.1069i	−0.655537	−	1.13542i	0.326222	−	0.945293i	$-0.394224\pi$
$228$		0			0
$229$	−3.13629	−	5.43222i	−0.207252	−	0.358971i	0.743596	−	0.668629i	$-0.233119\pi$
$229$	−3.13629	−	5.43222i	−0.207252	−	0.358971i	−0.950848	+	0.309658i	$0.899785\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−8.87583			−0.581475			−0.290738	−	0.956803i	$-0.593901\pi$
$233$	−8.87583			−0.581475			−0.290738	+	0.956803i	$0.593901\pi$
$234$		0			0
$235$	−3.88132	−	6.72264i	−0.253189	−	0.438537i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.44906	−	4.24190i	0.158417	−	0.274385i	−0.775881	−	0.630879i	$-0.782694\pi$
$239$	2.44906	−	4.24190i	0.158417	−	0.274385i	0.934298	+	0.356493i	$0.116028\pi$
$240$		0			0
$241$	7.53417	+	13.0496i	0.485319	+	0.840597i	0.999858	−	0.0168701i	$-0.00537019\pi$
$241$	7.53417	+	13.0496i	0.485319	+	0.840597i	−0.514539	+	0.857467i	$0.672037\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.0327			0.640964
$246$		0			0
$247$	−4.36136	+	7.55409i	−0.277507	+	0.480656i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.68341			0.358734			0.179367	−	0.983782i	$-0.442595\pi$
$251$	5.68341			0.358734			0.179367	+	0.983782i	$0.442595\pi$
$252$		0			0
$253$	14.2667			0.896937
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.55621	+	9.62363i	−0.346587	+	0.600306i	−0.985641	−	0.168856i	$-0.945993\pi$
$257$	−5.55621	+	9.62363i	−0.346587	+	0.600306i	0.639054	+	0.769162i	$0.279326\pi$
$258$		0			0
$259$	−0.169487	−	0.637273i	−0.0105314	−	0.0395982i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.75369	−	13.4298i	−0.478113	−	0.828116i	0.521572	−	0.853207i	$-0.325346\pi$
$263$	−7.75369	−	13.4298i	−0.478113	−	0.828116i	−0.999685	+	0.0250910i	$0.992012\pi$
$264$		0			0
$265$	11.2391			0.690414
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−31.4086			−1.91502			−0.957509	−	0.288402i	$-0.906876\pi$
$269$	−31.4086			−1.91502			−0.957509	+	0.288402i	$0.906876\pi$
$270$		0			0
$271$	−1.39579	+	2.41759i	−0.0847884	+	0.146858i	−0.905301	−	0.424771i	$-0.860355\pi$
$271$	−1.39579	+	2.41759i	−0.0847884	+	0.146858i	0.820513	+	0.571628i	$0.193688\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.93238	−	10.2752i	0.357736	−	0.619617i
$276$		0			0
$277$	4.46232	+	7.72896i	0.268115	+	0.464388i	0.968375	−	0.249499i	$-0.0802660\pi$
$277$	4.46232	+	7.72896i	0.268115	+	0.464388i	−0.700260	+	0.713888i	$0.746933\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.47760	−	2.55928i	0.0881462	−	0.152674i	−0.818581	−	0.574391i	$-0.805239\pi$
$281$	1.47760	−	2.55928i	0.0881462	−	0.152674i	0.906728	+	0.421717i	$0.138572\pi$
$282$		0			0
$283$	10.0001	+	17.3207i	0.594444	+	1.02961i	0.993625	+	0.112736i	$0.0359613\pi$
$283$	10.0001	+	17.3207i	0.594444	+	1.02961i	−0.399181	+	0.916872i	$0.630705\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.166279	+	0.288003i	−0.00981512	+	0.0170003i
$288$		0			0
$289$	8.00000	+	13.8564i	0.470588	+	0.815083i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7.20201	−	12.4742i	−0.420746	−	0.728753i	0.575267	−	0.817966i	$-0.304898\pi$
$293$	−7.20201	−	12.4742i	−0.420746	−	0.728753i	−0.996013	+	0.0892127i	$0.971565\pi$
$294$		0			0
$295$	−11.0249			−0.641896
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.23155	−	7.32925i	−0.244717	−	0.423862i
$300$		0			0
$301$	−0.132023	−	0.228671i	−0.00760969	−	0.0131804i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.24449	+	3.88758i	−0.128519	+	0.222602i
$306$		0			0
$307$	−10.7635			−0.614306			−0.307153	−	0.951660i	$-0.599376\pi$
$307$	−10.7635			−0.614306			−0.307153	+	0.951660i	$0.599376\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.1266	+	22.7359i	−0.744339	+	1.28923i	0.206164	+	0.978517i	$0.433902\pi$
$311$	−13.1266	+	22.7359i	−0.744339	+	1.28923i	−0.950503	+	0.310715i	$0.899431\pi$
$312$		0			0
$313$	16.0661	−	27.8273i	0.908110	−	1.57289i	0.0914237	−	0.995812i	$-0.470858\pi$
$313$	16.0661	−	27.8273i	0.908110	−	1.57289i	0.816687	−	0.577081i	$-0.195808\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.37219	−	4.10876i	0.133236	−	0.230771i	−0.791687	−	0.610928i	$-0.790797\pi$
$317$	2.37219	−	4.10876i	0.133236	−	0.230771i	0.924922	+	0.380157i	$0.124130\pi$
$318$		0			0
$319$	−5.26781			−0.294940
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.64227			0.202661
$324$		0			0
$325$	−7.03827			−0.390413
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.293087	+	0.507642i	0.0161584	+	0.0279872i
$330$		0			0
$331$	8.77399	−	15.1970i	0.482262	−	0.835302i	−0.517531	−	0.855665i	$-0.673149\pi$
$331$	8.77399	−	15.1970i	0.482262	−	0.835302i	0.999793	+	0.0203626i	$0.00648205\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.882224	+	1.52806i	−0.0482010	+	0.0834866i
$336$		0			0
$337$	8.90837	+	15.4297i	0.485270	+	0.840512i	0.999857	−	0.0169263i	$-0.00538805\pi$
$337$	8.90837	+	15.4297i	0.485270	+	0.840512i	−0.514587	+	0.857438i	$0.672055\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.2984			0.882608
$342$		0			0
$343$	−1.51645			−0.0818807
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.476577			0.0255840			0.0127920	−	0.999918i	$-0.495928\pi$
$347$	0.476577			0.0255840			0.0127920	+	0.999918i	$0.495928\pi$
$348$		0			0
$349$	−4.54622	+	7.87429i	−0.243354	+	0.421501i	−0.961667	−	0.274218i	$-0.911581\pi$
$349$	−4.54622	+	7.87429i	−0.243354	+	0.421501i	0.718314	+	0.695719i	$0.244914\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.1029	−	29.6231i	0.910296	−	1.57668i	0.0966487	−	0.995319i	$-0.469188\pi$
$353$	17.1029	−	29.6231i	0.910296	−	1.57668i	0.813647	−	0.581360i	$-0.197479\pi$
$354$		0			0
$355$	2.43555	−	4.21849i	0.129265	−	0.223894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	27.9188			1.47350			0.736750	−	0.676166i	$-0.236360\pi$
$359$	27.9188			1.47350			0.736750	+	0.676166i	$0.236360\pi$
$360$		0			0
$361$	2.86692	−	4.96566i	0.150891	−	0.261350i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.13855	+	7.16818i	0.216622	+	0.375200i
$366$		0			0
$367$	−9.73959	−	16.8695i	−0.508402	−	0.880578i	−0.999953	−	0.00972935i	$-0.996903\pi$
$367$	−9.73959	−	16.8695i	−0.508402	−	0.880578i	0.491550	−	0.870849i	$-0.336430\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.848691			−0.0440619
$372$		0			0
$373$	−7.34990	−	12.7304i	−0.380564	−	0.659155i	0.610579	−	0.791955i	$-0.290937\pi$
$373$	−7.34990	−	12.7304i	−0.380564	−	0.659155i	−0.991143	+	0.132800i	$0.957603\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.56245	+	2.70624i	0.0804703	+	0.139379i
$378$		0			0
$379$	−18.4137	+	31.8935i	−0.945848	+	1.63826i	−0.191803	+	0.981433i	$0.561434\pi$
$379$	−18.4137	+	31.8935i	−0.945848	+	1.63826i	−0.754045	+	0.656823i	$0.771900\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.3167	+	29.9934i	0.884842	+	1.53259i	0.845895	+	0.533349i	$0.179067\pi$
$383$	17.3167	+	29.9934i	0.884842	+	1.53259i	0.0389466	+	0.999241i	$0.487600\pi$
$384$		0			0
$385$	0.314164	−	0.544148i	0.0160113	−	0.0277323i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.58912	−	4.48449i	−0.131274	−	0.227373i	0.792894	−	0.609359i	$-0.208573\pi$
$389$	−2.58912	−	4.48449i	−0.131274	−	0.227373i	−0.924168	+	0.381987i	$0.875240\pi$
$390$		0			0
$391$	−1.76693	+	3.06042i	−0.0893576	+	0.154772i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−9.10602	+	15.7721i	−0.458174	+	0.793580i
$396$		0			0
$397$	−35.1077			−1.76201			−0.881003	−	0.473111i	$-0.843131\pi$
$397$	−35.1077			−1.76201			−0.881003	+	0.473111i	$0.843131\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−33.7787			−1.68683			−0.843414	−	0.537265i	$-0.819458\pi$
$401$	−33.7787			−1.68683			−0.843414	+	0.537265i	$0.819458\pi$
$402$		0			0
$403$	−4.83417	−	8.37303i	−0.240807	−	0.417090i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.7083	+	6.39991i	1.17518	+	0.317232i
$408$		0			0
$409$	12.3669	−	21.4202i	0.611506	−	1.05916i	−0.379481	−	0.925200i	$-0.623897\pi$
$409$	12.3669	−	21.4202i	0.611506	−	1.05916i	0.990987	−	0.133960i	$-0.0427694\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.832516			0.0409655
$414$		0			0
$415$	−2.91885			−0.143281
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.96584	+	15.5293i	−0.438010	+	0.758655i	−0.997536	−	0.0701574i	$-0.977650\pi$
$419$	−8.96584	+	15.5293i	−0.438010	+	0.758655i	0.559526	+	0.828813i	$0.310983\pi$
$420$		0			0
$421$	−19.6597			−0.958157			−0.479078	−	0.877772i	$-0.659029\pi$
$421$	−19.6597			−0.958157			−0.479078	+	0.877772i	$0.659029\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.46946	+	2.54517i	0.0712791	+	0.123459i
$426$		0			0
$427$	0.169487	−	0.293560i	0.00820204	−	0.0142064i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.6767	+	21.9566i	0.610614	+	1.05761i	0.991137	+	0.132843i	$0.0424106\pi$
$431$	12.6767	+	21.9566i	0.610614	+	1.05761i	−0.380523	+	0.924771i	$0.624256\pi$
$432$		0			0
$433$	−1.83825			−0.0883407			−0.0441703	−	0.999024i	$-0.514064\pi$
$433$	−1.83825			−0.0883407			−0.0441703	+	0.999024i	$0.514064\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.43565	−	11.1469i	−0.307859	−	0.533227i
$438$		0			0
$439$	11.9705	+	20.7335i	0.571321	+	0.989557i	0.996431	+	0.0844155i	$0.0269023\pi$
$439$	11.9705	+	20.7335i	0.571321	+	0.989557i	−0.425109	+	0.905142i	$0.639764\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−31.8299			−1.51228			−0.756142	−	0.654407i	$-0.772918\pi$
$443$	−31.8299			−1.51228			−0.756142	+	0.654407i	$0.772918\pi$
$444$		0			0
$445$	−10.1466			−0.480996
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.07159	+	3.58810i	0.0977645	+	0.169333i	0.910759	−	0.412938i	$-0.135497\pi$
$449$	2.07159	+	3.58810i	0.0977645	+	0.169333i	−0.812995	+	0.582271i	$0.802164\pi$
$450$		0			0
$451$	−6.19218	−	10.7252i	−0.291578	−	0.505028i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.372729			−0.0174738
$456$		0			0
$457$	−11.0014	−	19.0549i	−0.514622	−	0.891352i	−0.999856	−	0.0169673i	$-0.994599\pi$
$457$	−11.0014	−	19.0549i	−0.514622	−	0.891352i	0.485234	−	0.874384i	$-0.338734\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.60614	+	7.97807i	−0.214529	+	0.371576i	−0.953127	−	0.302571i	$-0.902155\pi$
$461$	−4.60614	+	7.97807i	−0.214529	+	0.371576i	0.738597	+	0.674147i	$0.235488\pi$
$462$		0			0
$463$	−19.5846	−	33.9215i	−0.910174	−	1.57647i	−0.813817	−	0.581121i	$-0.802614\pi$
$463$	−19.5846	−	33.9215i	−0.910174	−	1.57647i	−0.0963566	−	0.995347i	$-0.530719\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−29.2572			−1.35386			−0.676932	−	0.736046i	$-0.736691\pi$
$467$	−29.2572			−1.35386			−0.676932	+	0.736046i	$0.736691\pi$
$468$		0			0
$469$	0.0666187	−	0.115387i	0.00307616	−	0.00532807i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	9.83303			0.452123
$474$		0			0
$475$	−10.7043			−0.491148
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.00111	−	1.73397i	0.0457418	−	0.0792272i	−0.842248	−	0.539090i	$-0.818768\pi$
$479$	1.00111	−	1.73397i	0.0457418	−	0.0792272i	0.887990	+	0.459863i	$0.152102\pi$
$480$		0			0
$481$	−3.74412	−	14.0780i	−0.170717	−	0.641900i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.34969	−	16.1941i	−0.424547	−	0.735338i
$486$		0			0
$487$	9.26318			0.419755			0.209877	−	0.977728i	$-0.432694\pi$
$487$	9.26318			0.419755			0.209877	+	0.977728i	$0.432694\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−38.7415			−1.74838			−0.874189	−	0.485587i	$-0.838606\pi$
$491$	−38.7415			−1.74838			−0.874189	+	0.485587i	$0.838606\pi$
$492$		0			0
$493$	0.652420	−	1.13002i	0.0293835	−	0.0508937i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.183913	+	0.318547i	−0.00824965	+	0.0142888i
$498$		0			0
$499$	11.6970	+	20.2597i	0.523628	+	0.906951i	0.999622	+	0.0275019i	$0.00875523\pi$
$499$	11.6970	+	20.2597i	0.523628	+	0.906951i	−0.475994	+	0.879449i	$0.657911\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.81104	+	10.0650i	−0.259101	+	0.448777i	−0.966001	−	0.258537i	$-0.916760\pi$
$503$	−5.81104	+	10.0650i	−0.259101	+	0.448777i	0.706900	+	0.707313i	$0.250093\pi$
$504$		0			0
$505$	5.20969	+	9.02345i	0.231828	+	0.401538i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.0681	+	27.8307i	−0.712205	+	1.23358i	0.251822	+	0.967774i	$0.418970\pi$
$509$	−16.0681	+	27.8307i	−0.712205	+	1.23358i	−0.964028	+	0.265802i	$0.914363\pi$
$510$		0			0
$511$	−0.312511	−	0.541285i	−0.0138247	−	0.0239451i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.94801	+	13.7664i	0.350231	+	0.606618i
$516$		0			0
$517$	−21.8290			−0.960038
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.97383	+	17.2752i	0.436961	+	0.756839i	0.997453	−	0.0713204i	$-0.0227213\pi$
$521$	9.97383	+	17.2752i	0.436961	+	0.756839i	−0.560492	+	0.828160i	$0.689388\pi$
$522$		0			0
$523$	7.64601	+	13.2433i	0.334336	+	0.579088i	0.983357	−	0.181683i	$-0.0581545\pi$
$523$	7.64601	+	13.2433i	0.334336	+	0.579088i	−0.649021	+	0.760771i	$0.724821\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.01856	+	3.49626i	−0.0879300	+	0.152299i
$528$		0			0
$529$	−10.5118			−0.457035
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.67325	+	6.36225i	−0.159106	+	0.275580i
$534$		0			0
$535$	−7.26083	+	12.5761i	−0.313913	+	0.543713i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	14.1062	−	24.4327i	0.607598	−	1.05239i
$540$		0			0
$541$	−12.3482			−0.530889			−0.265444	−	0.964126i	$-0.585519\pi$
$541$	−12.3482			−0.530889			−0.265444	+	0.964126i	$0.585519\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	25.0410			1.07264
$546$		0			0
$547$	−14.3317			−0.612781			−0.306390	−	0.951906i	$-0.599121\pi$
$547$	−14.3317			−0.612781			−0.306390	+	0.951906i	$0.599121\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.37629	+	4.11586i	0.101233	+	0.175341i
$552$		0			0
$553$	0.687616	−	1.19099i	0.0292404	−	0.0506459i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.24159	−	14.2748i	0.349207	−	0.604845i	−0.636902	−	0.770945i	$-0.719784\pi$
$557$	8.24159	−	14.2748i	0.349207	−	0.604845i	0.986109	+	0.166100i	$0.0531176\pi$
$558$		0			0
$559$	−2.91651	−	5.05155i	−0.123355	−	0.213658i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.0160			−1.01215			−0.506076	−	0.862489i	$-0.668905\pi$
$563$	−24.0160			−1.01215			−0.506076	+	0.862489i	$0.668905\pi$
$564$		0			0
$565$	14.6509			0.616368
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−21.2829			−0.892224			−0.446112	−	0.894977i	$-0.647192\pi$
$569$	−21.2829			−0.892224			−0.446112	+	0.894977i	$0.647192\pi$
$570$		0			0
$571$	−18.4543	+	31.9638i	−0.772289	+	1.33764i	0.164017	+	0.986458i	$0.447555\pi$
$571$	−18.4543	+	31.9638i	−0.772289	+	1.33764i	−0.936306	+	0.351186i	$0.885778\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.19286	−	8.99430i	0.216557	−	0.375088i
$576$		0			0
$577$	−4.89321	+	8.47529i	−0.203707	+	0.352831i	−0.949720	−	0.313101i	$-0.898632\pi$
$577$	−4.89321	+	8.47529i	−0.203707	+	0.352831i	0.746013	+	0.665931i	$0.231966\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.220409			0.00914410
$582$		0			0
$583$	15.8025	−	27.3708i	0.654474	−	1.13358i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11.6730	−	20.2183i	−0.481798	−	0.834498i	0.517984	−	0.855390i	$-0.326683\pi$
$587$	−11.6730	−	20.2183i	−0.481798	−	0.834498i	−0.999782	+	0.0208922i	$0.993349\pi$
$588$		0			0
$589$	−7.35216	−	12.7343i	−0.302941	−	0.524708i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−24.5229			−1.00703			−0.503517	−	0.863985i	$-0.667961\pi$
$593$	−24.5229			−1.00703			−0.503517	+	0.863985i	$0.667961\pi$
$594$		0			0
$595$	0.0778186	+	0.134786i	0.00319025	+	0.00552568i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−7.24659	−	12.5515i	−0.296088	−	0.512839i	0.679150	−	0.734000i	$-0.262349\pi$
$599$	−7.24659	−	12.5515i	−0.296088	−	0.512839i	−0.975237	+	0.221161i	$0.929015\pi$
$600$		0			0
$601$	−12.7216	+	22.0344i	−0.518923	+	0.898801i	0.480835	+	0.876811i	$0.340334\pi$
$601$	−12.7216	+	22.0344i	−0.518923	+	0.898801i	−0.999758	+	0.0219902i	$0.993000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.80332	+	6.58755i	0.154627	+	0.267822i
$606$		0			0
$607$	−12.3718	+	21.4286i	−0.502157	+	0.869762i	0.497840	+	0.867269i	$0.334127\pi$
$607$	−12.3718	+	21.4286i	−0.502157	+	0.869762i	−0.999997	+	0.00249282i	$0.999207\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	6.47456	+	11.2143i	0.261933	+	0.453681i
$612$		0			0
$613$	−14.4675	+	25.0584i	−0.584336	+	1.01210i	0.410622	+	0.911806i	$0.365312\pi$
$613$	−14.4675	+	25.0584i	−0.584336	+	1.01210i	−0.994958	+	0.100294i	$0.968022\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.99021	+	10.3753i	−0.241157	+	0.417696i	−0.961044	−	0.276395i	$-0.910860\pi$
$617$	−5.99021	+	10.3753i	−0.241157	+	0.417696i	0.719887	+	0.694091i	$0.244193\pi$
$618$		0			0
$619$	27.5003			1.10533			0.552664	−	0.833404i	$-0.313611\pi$
$619$	27.5003			1.10533			0.552664	+	0.833404i	$0.313611\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.766193			0.0306969
$624$		0			0
$625$	0.834113	+	1.44473i	0.0333645	+	0.0577890i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.30916	+	4.29315i	−0.171817	+	0.171179i
$630$		0			0
$631$	11.3878	−	19.7243i	0.453341	−	0.785210i	−0.545250	−	0.838274i	$-0.683565\pi$
$631$	11.3878	−	19.7243i	0.453341	−	0.785210i	0.998591	+	0.0530635i	$0.0168986\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.7213			−0.504828
$636$		0			0
$637$	−16.7358			−0.663098
$638$		0			0
$639$		0			0
$640$		0			0
$641$	23.8026	−	41.2273i	0.940146	−	1.62838i	0.174956	−	0.984576i	$-0.444022\pi$
$641$	23.8026	−	41.2273i	0.940146	−	1.62838i	0.765190	−	0.643804i	$-0.222645\pi$
$642$		0			0
$643$	12.8931			0.508454			0.254227	−	0.967145i	$-0.418179\pi$
$643$	12.8931			0.508454			0.254227	+	0.967145i	$0.418179\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9.54728	−	16.5364i	−0.375342	−	0.650112i	0.615036	−	0.788499i	$-0.289141\pi$
$647$	−9.54728	−	16.5364i	−0.375342	−	0.650112i	−0.990378	+	0.138387i	$0.955808\pi$
$648$		0			0
$649$	−15.5014	+	26.8492i	−0.608482	+	1.05392i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.61858	+	13.1958i	0.298138	+	0.516390i	0.975710	−	0.219066i	$-0.0703011\pi$
$653$	7.61858	+	13.1958i	0.298138	+	0.516390i	−0.677572	+	0.735456i	$0.736968\pi$
$654$		0			0
$655$	−22.6484			−0.884946
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.2087	−	31.5384i	−0.709310	−	1.22856i	−0.965113	−	0.261832i	$-0.915673\pi$
$659$	−18.2087	−	31.5384i	−0.709310	−	1.22856i	0.255803	−	0.966729i	$-0.417660\pi$
$660$		0			0
$661$	−3.51949	−	6.09594i	−0.136892	−	0.237105i	0.789426	−	0.613845i	$-0.210378\pi$
$661$	−3.51949	−	6.09594i	−0.136892	−	0.237105i	−0.926319	+	0.376741i	$0.877045\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.566873			−0.0219824
$666$		0			0
$667$	−4.61113			−0.178544
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.31165	+	10.9321i	0.243659	+	0.422029i
$672$		0			0
$673$	−7.20694	−	12.4828i	−0.277807	−	0.481176i	0.693033	−	0.720906i	$-0.256274\pi$
$673$	−7.20694	−	12.4828i	−0.277807	−	0.481176i	−0.970839	+	0.239731i	$0.922941\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.20640			0.276964			0.138482	−	0.990365i	$-0.455778\pi$
$677$	7.20640			0.276964			0.138482	+	0.990365i	$0.455778\pi$
$678$		0			0
$679$	0.706016	+	1.22286i	0.0270944	+	0.0469289i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.20054	+	14.2038i	−0.313785	+	0.543491i	−0.979178	−	0.203001i	$-0.934930\pi$
$683$	−8.20054	+	14.2038i	−0.313785	+	0.543491i	0.665394	+	0.746493i	$0.268264\pi$
$684$		0			0
$685$	14.9239	+	25.8489i	0.570212	+	0.987637i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.7484			−0.714256
$690$		0			0
$691$	−14.8159	+	25.6619i	−0.563624	+	0.976225i	0.433552	+	0.901128i	$0.357260\pi$
$691$	−14.8159	+	25.6619i	−0.563624	+	0.976225i	−0.997176	+	0.0750968i	$0.976073\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14.7268			0.558620
$696$		0			0
$697$	3.06762			0.116194
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12.6580	+	21.9243i	−0.478086	+	0.828069i	−0.999684	−	0.0251219i	$-0.992003\pi$
$701$	−12.6580	+	21.9243i	−0.478086	+	0.828069i	0.521598	+	0.853191i	$0.325336\pi$
$702$		0			0
$703$	−5.69433	−	21.4108i	−0.214766	−	0.807523i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.393395	−	0.681381i	−0.0147952	−	0.0256260i
$708$		0			0
$709$	43.4575			1.63208			0.816040	−	0.577996i	$-0.196165\pi$
$709$	43.4575			1.63208			0.816040	+	0.577996i	$0.196165\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	14.2667			0.534291
$714$		0			0
$715$	6.94017	−	12.0207i	0.259548	−	0.449550i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	20.3150	−	35.1865i	0.757620	−	1.31224i	−0.186441	−	0.982466i	$-0.559695\pi$
$719$	20.3150	−	35.1865i	0.757620	−	1.31224i	0.944061	−	0.329771i	$-0.106971\pi$
$720$		0			0
$721$	−0.600172	−	1.03953i	−0.0223516	−	0.0387140i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.91741	+	3.32104i	−0.0712106	+	0.123340i
$726$		0			0
$727$	−0.208149	−	0.360525i	−0.00771984	−	0.0133711i	0.862140	−	0.506671i	$-0.169124\pi$
$727$	−0.208149	−	0.360525i	−0.00771984	−	0.0133711i	−0.869860	+	0.493300i	$0.835791\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.21782	+	2.10933i	−0.0450429	+	0.0780165i
$732$		0			0
$733$	13.5615	+	23.4891i	0.500904	+	0.867591i	0.999999	+	0.00104386i	$0.000332271\pi$
$733$	13.5615	+	23.4891i	0.500904	+	0.867591i	−0.499096	+	0.866547i	$0.666334\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.48086	+	4.29698i	0.0913838	+	0.158281i
$738$		0			0
$739$	−43.7595			−1.60972			−0.804859	−	0.593466i	$-0.797759\pi$
$739$	−43.7595			−1.60972			−0.804859	+	0.593466i	$0.797759\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.96049	−	15.5200i	−0.328728	−	0.569374i	0.653531	−	0.756899i	$-0.273287\pi$
$743$	−8.96049	−	15.5200i	−0.328728	−	0.569374i	−0.982260	+	0.187525i	$0.939953\pi$
$744$		0			0
$745$	−4.15138	−	7.19039i	−0.152095	−	0.263436i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.548281	−	0.949651i	0.0200338	−	0.0346995i
$750$		0			0
$751$	45.0987			1.64567			0.822837	−	0.568277i	$-0.192390\pi$
$751$	45.0987			1.64567			0.822837	+	0.568277i	$0.192390\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.08784	−	1.88420i	0.0395906	−	0.0685729i
$756$		0			0
$757$	0.495673	−	0.858531i	0.0180156	−	0.0312039i	−0.856877	−	0.515521i	$-0.827598\pi$
$757$	0.495673	−	0.858531i	0.0180156	−	0.0312039i	0.874893	+	0.484317i	$0.160932\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.9423	+	18.9526i	−0.396658	+	0.687031i	−0.993311	−	0.115468i	$-0.963163\pi$
$761$	−10.9423	+	18.9526i	−0.396658	+	0.687031i	0.596654	+	0.802499i	$0.296497\pi$
$762$		0			0
$763$	−1.89090			−0.0684552
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.3911			0.664062
$768$		0			0
$769$	−55.1741			−1.98963			−0.994814	−	0.101714i	$-0.967567\pi$
$769$	−55.1741			−1.98963			−0.994814	+	0.101714i	$0.967567\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−21.3288	−	36.9426i	−0.767144	−	1.32873i	−0.939106	−	0.343629i	$-0.888344\pi$
$773$	−21.3288	−	36.9426i	−0.767144	−	1.32873i	0.171962	−	0.985104i	$-0.444990\pi$
$774$		0			0
$775$	5.93238	−	10.2752i	0.213098	−	0.369096i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.58655	+	9.67618i	−0.200159	+	0.346685i
$780$		0			0
$781$	−6.84890	−	11.8626i	−0.245073	−	0.424479i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.73987			0.204865
$786$		0			0
$787$	31.3011			1.11576			0.557882	−	0.829920i	$-0.311614\pi$
$787$	31.3011			1.11576			0.557882	+	0.829920i	$0.311614\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.10632			−0.0393363
$792$		0			0
$793$	3.74412	−	6.48501i	0.132958	−	0.230289i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.45442	−	2.51912i	0.0515181	−	0.0892320i	−0.839116	−	0.543952i	$-0.816927\pi$
$797$	1.45442	−	2.51912i	0.0515181	−	0.0892320i	0.890634	+	0.454720i	$0.150261\pi$
$798$		0			0
$799$	2.70353	−	4.68265i	0.0956440	−	0.165660i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	23.2757			0.821382
$804$		0			0
$805$	0.275000	−	0.476315i	0.00969249	−	0.0167879i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.20884	+	14.2181i	0.288607	+	0.499882i	0.973478	−	0.228783i	$-0.0734745\pi$
$809$	8.20884	+	14.2181i	0.288607	+	0.499882i	−0.684870	+	0.728665i	$0.740141\pi$
$810$		0			0
$811$	−19.6948	−	34.1124i	−0.691577	−	1.19785i	−0.971321	−	0.237772i	$-0.923583\pi$
$811$	−19.6948	−	34.1124i	−0.691577	−	1.19785i	0.279743	−	0.960075i	$-0.409751\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	20.1999			0.707573
$816$		0			0
$817$	−4.43565	−	7.68277i	−0.155184	−	0.268786i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.3763	−	33.5607i	−0.676237	−	1.17128i	−0.976106	−	0.217297i	$-0.930276\pi$
$821$	−19.3763	−	33.5607i	−0.676237	−	1.17128i	0.299868	−	0.953981i	$-0.403057\pi$
$822$		0			0
$823$	1.34259	−	2.32543i	0.0467996	−	0.0810593i	−0.841677	−	0.539982i	$-0.818431\pi$
$823$	1.34259	−	2.32543i	0.0467996	−	0.0810593i	0.888476	+	0.458922i	$0.151764\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.04014	−	6.99772i	−0.140489	−	0.243335i	0.787192	−	0.616708i	$-0.211534\pi$
$827$	−4.04014	−	6.99772i	−0.140489	−	0.243335i	−0.927681	+	0.373374i	$0.878201\pi$
$828$		0			0
$829$	20.5826	−	35.6502i	0.714865	−	1.23818i	−0.248147	−	0.968722i	$-0.579822\pi$
$829$	20.5826	−	35.6502i	0.714865	−	1.23818i	0.963012	−	0.269459i	$-0.0868450\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.49412	+	6.05200i	0.121064	+	0.209689i
$834$		0			0
$835$	−16.6397	+	28.8208i	−0.575841	+	0.997386i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	3.60297	−	6.24052i	0.124388	−	0.215447i	−0.797105	−	0.603840i	$-0.793637\pi$
$839$	3.60297	−	6.24052i	0.124388	−	0.215447i	0.921494	+	0.388393i	$0.126970\pi$
$840$		0			0
$841$	−27.2974			−0.941289
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.4295			0.358786
$846$		0			0
$847$	−0.287197	−	0.497441i	−0.00986822	−	0.0170923i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20.7528	+	5.60210i	0.711398	+	0.192038i
$852$		0			0
$853$	−14.7749	+	25.5909i	−0.505883	+	0.876215i	0.494094	+	0.869409i	$0.335500\pi$
$853$	−14.7749	+	25.5909i	−0.505883	+	0.876215i	−0.999977	+	0.00680647i	$0.997833\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.19030			−0.143138			−0.0715689	−	0.997436i	$-0.522801\pi$
$857$	−4.19030			−0.143138			−0.0715689	+	0.997436i	$0.522801\pi$
$858$		0			0
$859$	−4.17824			−0.142560			−0.0712799	−	0.997456i	$-0.522708\pi$
$859$	−4.17824			−0.142560			−0.0712799	+	0.997456i	$0.522708\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.6916	−	32.3748i	0.636269	−	1.10205i	−0.349976	−	0.936759i	$-0.613810\pi$
$863$	18.6916	−	32.3748i	0.636269	−	1.10205i	0.986245	−	0.165291i	$-0.0528563\pi$
$864$		0			0
$865$	1.87615			0.0637911
$866$		0			0
$867$		0			0
$868$		0			0
$869$	25.6067	+	44.3521i	0.868647	+	1.50454i
$870$		0			0
$871$	1.47167	−	2.54900i	0.0498656	−	0.0863697i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.617795	−	1.07005i	−0.0208853	−	0.0361744i
$876$		0			0
$877$	24.4772			0.826535			0.413267	−	0.910610i	$-0.364387\pi$
$877$	24.4772			0.826535			0.413267	+	0.910610i	$0.364387\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.7602	−	22.1014i	−0.429904	−	0.744615i	0.566961	−	0.823745i	$-0.308119\pi$
$881$	−12.7602	−	22.1014i	−0.429904	−	0.744615i	−0.996864	+	0.0791300i	$0.974786\pi$
$882$		0			0
$883$	−23.4144	−	40.5549i	−0.787957	−	1.36478i	−0.927217	−	0.374526i	$-0.877806\pi$
$883$	−23.4144	−	40.5549i	−0.787957	−	1.36478i	0.139260	−	0.990256i	$-0.455528\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	11.0809			0.372059			0.186029	−	0.982544i	$-0.440438\pi$
$887$	11.0809			0.372059			0.186029	+	0.982544i	$0.440438\pi$
$888$		0			0
$889$	0.960611			0.0322179
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9.84700	+	17.0555i	0.329517	+	0.570741i
$894$		0			0
$895$	−3.80801	−	6.59567i	−0.127288	−	0.220469i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.26781			−0.175691
$900$		0			0
$901$	3.91430	+	6.77977i	0.130404	+	0.225867i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−16.2068	+	28.0710i	−0.538731	+	0.933110i
$906$		0			0
$907$	5.50863	+	9.54122i	0.182911	+	0.316811i	0.942871	−	0.333159i	$-0.108115\pi$
$907$	5.50863	+	9.54122i	0.182911	+	0.316811i	−0.759960	+	0.649970i	$0.774781\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	43.0485			1.42626			0.713130	−	0.701031i	$-0.247277\pi$
$911$	43.0485			1.42626			0.713130	+	0.701031i	$0.247277\pi$
$912$		0			0
$913$	−4.10399	+	7.10831i	−0.135822	+	0.235251i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.71023			0.0564768
$918$		0			0
$919$	−25.4727			−0.840266			−0.420133	−	0.907463i	$-0.638017\pi$
$919$	−25.4727			−0.840266			−0.420133	+	0.907463i	$0.638017\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.06282	+	7.03701i	−0.133729	+	0.231626i
$924$		0			0
$925$	12.6642	−	12.6172i	0.416398	−	0.414851i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.40048	−	4.15775i	−0.0787571	−	0.136411i	0.823957	−	0.566653i	$-0.191762\pi$
$929$	−2.40048	−	4.15775i	−0.0787571	−	0.136411i	−0.902714	+	0.430241i	$0.858428\pi$
$930$		0			0
$931$	−25.4531			−0.834192
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.79590			−0.189546
$936$		0			0
$937$	0.799339	−	1.38450i	0.0261133	−	0.0452295i	−0.852673	−	0.522444i	$-0.825020\pi$
$937$	0.799339	−	1.38450i	0.0261133	−	0.0452295i	0.878787	+	0.477215i	$0.158354\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23.5581	+	40.8038i	−0.767972	+	1.33017i	0.170690	+	0.985325i	$0.445400\pi$
$941$	−23.5581	+	40.8038i	−0.767972	+	1.33017i	−0.938661	+	0.344841i	$0.887933\pi$
$942$		0			0
$943$	−5.42027	−	9.38818i	−0.176508	−	0.305721i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.0133155	+	0.0230631i	−0.000432695	+	0.000749451i	−0.866242	−	0.499625i	$-0.833471\pi$
$947$	−0.0133155	+	0.0230631i	−0.000432695	+	0.000749451i	0.865809	+	0.500375i	$0.166804\pi$
$948$		0			0
$949$	−6.90366	−	11.9575i	−0.224102	−	0.388157i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.59727	+	4.49861i	−0.0841340	+	0.145724i	−0.905022	−	0.425365i	$-0.860146\pi$
$953$	−2.59727	+	4.49861i	−0.0841340	+	0.145724i	0.820888	+	0.571089i	$0.193479\pi$
$954$		0			0
$955$	−1.28582	−	2.22710i	−0.0416081	−	0.0720673i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.12694	−	1.95191i	−0.0363907	−	0.0630305i
$960$		0			0
$961$	−14.7016			−0.474245
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−13.4232	−	23.2496i	−0.432107	−	0.748431i
$966$		0			0
$967$	16.0381	+	27.7788i	0.515751	+	0.893308i	0.999833	+	0.0182846i	$0.00582049\pi$
$967$	16.0381	+	27.7788i	0.515751	+	0.893308i	−0.484081	+	0.875023i	$0.660846\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.02675	−	13.9027i	0.257591	−	0.446160i	−0.708005	−	0.706207i	$-0.750405\pi$
$971$	8.02675	−	13.9027i	0.257591	−	0.446160i	0.965596	+	0.260047i	$0.0837380\pi$
$972$		0			0
$973$	−1.11205			−0.0356508
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.4120	+	37.0867i	−0.685031	+	1.18651i	0.288397	+	0.957511i	$0.406878\pi$
$977$	−21.4120	+	37.0867i	−0.685031	+	1.18651i	−0.973427	+	0.228997i	$0.926455\pi$
$978$		0			0
$979$	−14.2664	+	24.7102i	−0.455957	+	0.789741i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.5524	−	26.9375i	0.496044	−	0.859174i	−0.503945	−	0.863736i	$-0.668119\pi$
$983$	15.5524	−	26.9375i	0.496044	−	0.859174i	0.999990	+	0.00456159i	$0.00145200\pi$
$984$		0			0
$985$	37.7958			1.20427
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.60725			0.273695
$990$		0			0
$991$	13.1784			0.418624			0.209312	−	0.977849i	$-0.432878\pi$
$991$	13.1784			0.418624			0.209312	+	0.977849i	$0.432878\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	7.01129	+	12.1439i	0.222273	+	0.384988i
$996$		0			0
$997$	10.3616	−	17.9468i	0.328155	−	0.568382i	−0.653991	−	0.756503i	$-0.726906\pi$
$997$	10.3616	−	17.9468i	0.328155	−	0.568382i	0.982146	+	0.188121i	$0.0602397\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2664.2.r.n.1009.4		10
3.2	odd	2		296.2.i.c.121.2	✓	10
12.11	even	2		592.2.i.h.417.4		10
37.26	even	3	inner	2664.2.r.n.433.4		10
111.26	odd	6		296.2.i.c.137.2	yes	10
444.359	even	6		592.2.i.h.433.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.i.c.121.2	✓	10	3.2	odd	2
296.2.i.c.137.2	yes	10	111.26	odd	6
592.2.i.h.417.4		10	12.11	even	2
592.2.i.h.433.4		10	444.359	even	6
2664.2.r.n.433.4		10	37.26	even	3	inner
2664.2.r.n.1009.4		10	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2664.r (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.717824 + 1.24331i) q^{5} +(-0.0542045 - 0.0938850i) q^{7} +4.03713 q^{11} +(-1.19743 - 2.07401i) q^{13} +(-0.500000 + 0.866025i) q^{17} +(-1.82114 - 3.15430i) q^{19} +3.53386 q^{23} +(1.46946 - 2.54517i) q^{25} -1.30484 q^{29} +4.03713 q^{31} +(0.0778186 - 0.134786i) q^{35} +(5.87256 + 1.58526i) q^{37} +(-1.53381 - 2.65663i) q^{41} +2.43565 q^{43} -5.40706 q^{47} +(3.49412 - 6.05200i) q^{49} +(3.91430 - 6.77977i) q^{53} +(2.89795 + 5.01939i) q^{55} +(-3.83970 + 6.65056i) q^{59} +(1.56340 + 2.70789i) q^{61} +(1.71909 - 2.97754i) q^{65} +(0.614512 + 1.06437i) q^{67} +(-1.69648 - 2.93839i) q^{71} +5.76541 q^{73} +(-0.218831 - 0.379026i) q^{77} +(6.34279 + 10.9860i) q^{79} +(-1.01656 + 1.76074i) q^{83} -1.43565 q^{85} +(-3.53381 + 6.12073i) q^{89} +(-0.129812 + 0.224841i) q^{91} +(2.61451 - 4.52847i) q^{95} -13.0250 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{5} + 3 q^{7} - 16 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} + 12 q^{23} - 12 q^{25} - 16 q^{31} - 5 q^{35} - 12 q^{37} - 9 q^{41} + 4 q^{43} - 32 q^{47} - 18 q^{49} - 5 q^{53} + 4 q^{55} + 5 q^{59}+ \cdots - 36 q^{97}+O(q^{100}) \) 10 * q - 3 * q^5 + 3 * q^7 - 16 * q^11 + q^13 - 5 * q^17 - 3 * q^19 + 12 * q^23 - 12 * q^25 - 16 * q^31 - 5 * q^35 - 12 * q^37 - 9 * q^41 + 4 * q^43 - 32 * q^47 - 18 * q^49 - 5 * q^53 + 4 * q^55 + 5 * q^59 + 15 * q^61 - 23 * q^65 + q^67 + 17 * q^71 - 36 * q^73 + 4 * q^77 + 21 * q^79 + 15 * q^83 + 6 * q^85 - 29 * q^89 - q^91 + 21 * q^95 - 36 * q^97

Introduction

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