LMFDB - Embedded newform 3330.2.e.f.739.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(739,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3330.739");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.e (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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} + 8 x^{13} + 138 x^{12} - 220 x^{11} + 196 x^{10} + 744 x^{9} + 4241 x^{8} + \cdots + 64 $ x^16 - 2x^15 + 2x^14 + 8x^13 + 138x^12 - 220x^11 + 196x^10 + 744x^9 + 4241x^8 - 3018x^7 + 658x^6 - 1584x^5 + 16372x^4 - 18840x^3 + 10952x^2 - 1184*x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			739.6
Root			$-1.28817 - 1.28817i$ of defining polynomial
Character	$\chi$	$=$	3330.739
Dual form			3330.2.e.f.739.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +(-1.28817 + 1.82773i) q^{5} -1.25548i q^{7} +1.00000 q^{8} +(-1.28817 + 1.82773i) q^{10} +5.46829 q^{11} +4.70887 q^{13} -1.25548i q^{14} +1.00000 q^{16} -1.19309 q^{17} -1.22179i q^{19} +(-1.28817 + 1.82773i) q^{20} +5.46829 q^{22} -1.91577 q^{23} +(-1.68122 - 4.70887i) q^{25} +4.70887 q^{26} -1.25548i q^{28} +5.08822i q^{29} -3.43275i q^{31} +1.00000 q^{32} -1.19309 q^{34} +(2.29468 + 1.61728i) q^{35} +(-4.58473 - 3.99753i) q^{37} -1.22179i q^{38} +(-1.28817 + 1.82773i) q^{40} +4.46326 q^{41} +1.34271 q^{43} +5.46829 q^{44} -1.91577 q^{46} -1.90301i q^{47} +5.42377 q^{49} +(-1.68122 - 4.70887i) q^{50} +4.70887 q^{52} +1.11671i q^{53} +(-7.04410 + 9.99458i) q^{55} -1.25548i q^{56} +5.08822i q^{58} +8.99577i q^{59} -6.21190i q^{61} -3.43275i q^{62} +1.00000 q^{64} +(-6.06585 + 8.60657i) q^{65} +4.21117i q^{67} -1.19309 q^{68} +(2.29468 + 1.61728i) q^{70} +12.4170 q^{71} -10.2834i q^{73} +(-4.58473 - 3.99753i) q^{74} -1.22179i q^{76} -6.86533i q^{77} +1.00691i q^{79} +(-1.28817 + 1.82773i) q^{80} +4.46326 q^{82} +0.754493i q^{83} +(1.53691 - 2.18065i) q^{85} +1.34271 q^{86} +5.46829 q^{88} +11.4743i q^{89} -5.91190i q^{91} -1.91577 q^{92} -1.90301i q^{94} +(2.23311 + 1.57388i) q^{95} +16.6814 q^{97} +5.42377 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 2 q^{5} + 16 q^{8} + 2 q^{10} - 2 q^{11} + 16 q^{16} + 38 q^{17} + 2 q^{20} - 2 q^{22} + 20 q^{23} - 4 q^{25} + 16 q^{32} + 38 q^{34} + 10 q^{35} - 4 q^{37} + 2 q^{40} + 6 q^{41}+ \cdots - 18 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 2 * q^5 + 16 * q^8 + 2 * q^10 - 2 * q^11 + 16 * q^16 + 38 * q^17 + 2 * q^20 - 2 * q^22 + 20 * q^23 - 4 * q^25 + 16 * q^32 + 38 * q^34 + 10 * q^35 - 4 * q^37 + 2 * q^40 + 6 * q^41 - 2 * q^43 - 2 * q^44 + 20 * q^46 - 18 * q^49 - 4 * q^50 + 20 * q^55 + 16 * q^64 - 12 * q^65 + 38 * q^68 + 10 * q^70 + 24 * q^71 - 4 * q^74 + 2 * q^80 + 6 * q^82 - 2 * q^86 - 2 * q^88 + 20 * q^92 + 16 * q^95 + 38 * q^97 - 18 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.28817	+	1.82773i	−0.576089	+	0.817387i
$5$	−1.28817	+	1.82773i	−0.576089	+	0.817387i
$6$		0			0
$7$		−	1.25548i		−	0.474527i	−0.971445	−	0.237263i	$-0.923750\pi$
$7$		−	1.25548i		−	0.474527i	0.971445	−	0.237263i	$-0.0762505\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.28817	+	1.82773i	−0.407356	+	0.577980i
$11$	5.46829			1.64875			0.824376	−	0.566043i	$-0.191526\pi$
$11$	5.46829			1.64875			0.824376	+	0.566043i	$0.191526\pi$
$12$		0			0
$13$	4.70887			1.30601			0.653003	−	0.757355i	$-0.273509\pi$
$13$	4.70887			1.30601			0.653003	+	0.757355i	$0.273509\pi$
$14$		−	1.25548i		−	0.335541i
$15$		0			0
$16$	1.00000			0.250000
$17$	−1.19309			−0.289367			−0.144684	−	0.989478i	$-0.546216\pi$
$17$	−1.19309			−0.289367			−0.144684	+	0.989478i	$0.546216\pi$
$18$		0			0
$19$		−	1.22179i		−	0.280298i	−0.990130	−	0.140149i	$-0.955242\pi$
$19$		−	1.22179i		−	0.280298i	0.990130	−	0.140149i	$-0.0447582\pi$
$20$	−1.28817	+	1.82773i	−0.288044	+	0.408694i
$21$		0			0
$22$	5.46829			1.16584
$23$	−1.91577			−0.399466			−0.199733	−	0.979850i	$-0.564007\pi$
$23$	−1.91577			−0.399466			−0.199733	+	0.979850i	$0.564007\pi$
$24$		0			0
$25$	−1.68122	−	4.70887i	−0.336244	−	0.941775i
$26$	4.70887			0.923486
$27$		0			0
$28$		−	1.25548i		−	0.237263i
$29$			5.08822i			0.944858i	0.881369	+	0.472429i	$0.156623\pi$
$29$			5.08822i			0.944858i	−0.881369	+	0.472429i	$0.843377\pi$
$30$		0			0
$31$		−	3.43275i		−	0.616540i	−0.951299	−	0.308270i	$-0.900250\pi$
$31$		−	3.43275i		−	0.616540i	0.951299	−	0.308270i	$-0.0997501\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	−1.19309			−0.204613
$35$	2.29468	+	1.61728i	0.387872	+	0.273370i
$36$		0			0
$37$	−4.58473	−	3.99753i	−0.753725	−	0.657190i
$37$	−4.58473	−	3.99753i	−0.753725	−	0.657190i
$38$		−	1.22179i		−	0.198201i
$39$		0			0
$40$	−1.28817	+	1.82773i	−0.203678	+	0.288990i
$41$	4.46326			0.697043			0.348522	−	0.937301i	$-0.386684\pi$
$41$	4.46326			0.697043			0.348522	+	0.937301i	$0.386684\pi$
$42$		0			0
$43$	1.34271			0.204761			0.102380	−	0.994745i	$-0.467354\pi$
$43$	1.34271			0.204761			0.102380	+	0.994745i	$0.467354\pi$
$44$	5.46829			0.824376
$45$		0			0
$46$	−1.91577			−0.282465
$47$		−	1.90301i		−	0.277583i	−0.990322	−	0.138791i	$-0.955678\pi$
$47$		−	1.90301i		−	0.277583i	0.990322	−	0.138791i	$-0.0443217\pi$
$48$		0			0
$49$	5.42377			0.774824
$50$	−1.68122	−	4.70887i	−0.237760	−	0.665935i
$51$		0			0
$52$	4.70887			0.653003
$53$			1.11671i			0.153392i	0.997055	+	0.0766959i	$0.0244370\pi$
$53$			1.11671i			0.153392i	−0.997055	+	0.0766959i	$0.975563\pi$
$54$		0			0
$55$	−7.04410	+	9.99458i	−0.949827	+	1.34767i
$56$		−	1.25548i		−	0.167771i
$57$		0			0
$58$			5.08822i			0.668116i
$59$			8.99577i			1.17115i	0.810618	+	0.585575i	$0.199131\pi$
$59$			8.99577i			1.17115i	−0.810618	+	0.585575i	$0.800869\pi$
$60$		0			0
$61$		−	6.21190i		−	0.795352i	−0.917526	−	0.397676i	$-0.869817\pi$
$61$		−	6.21190i		−	0.795352i	0.917526	−	0.397676i	$-0.130183\pi$
$62$		−	3.43275i		−	0.435960i
$63$		0			0
$64$	1.00000			0.125000
$65$	−6.06585	+	8.60657i	−0.752376	+	1.06751i
$66$		0			0
$67$			4.21117i			0.514476i	0.966348	+	0.257238i	$0.0828125\pi$
$67$			4.21117i			0.514476i	−0.966348	+	0.257238i	$0.917188\pi$
$68$	−1.19309			−0.144684
$69$		0			0
$70$	2.29468	+	1.61728i	0.274267	+	0.193301i
$71$	12.4170			1.47363			0.736815	−	0.676094i	$-0.236329\pi$
$71$	12.4170			1.47363			0.736815	+	0.676094i	$0.236329\pi$
$72$		0			0
$73$		−	10.2834i		−	1.20359i	−0.798652	−	0.601793i	$-0.794453\pi$
$73$		−	10.2834i		−	1.20359i	0.798652	−	0.601793i	$-0.205547\pi$
$74$	−4.58473	−	3.99753i	−0.532964	−	0.464703i
$75$		0			0
$76$		−	1.22179i		−	0.140149i
$77$		−	6.86533i		−	0.782377i
$78$		0			0
$79$			1.00691i			0.113286i	0.998394	+	0.0566431i	$0.0180397\pi$
$79$			1.00691i			0.113286i	−0.998394	+	0.0566431i	$0.981960\pi$
$80$	−1.28817	+	1.82773i	−0.144022	+	0.204347i
$81$		0			0
$82$	4.46326			0.492884
$83$			0.754493i			0.0828164i	0.999142	+	0.0414082i	$0.0131844\pi$
$83$			0.754493i			0.0828164i	−0.999142	+	0.0414082i	$0.986816\pi$
$84$		0			0
$85$	1.53691	−	2.18065i	0.166701	−	0.236525i
$86$	1.34271			0.144788
$87$		0			0
$88$	5.46829			0.582922
$89$			11.4743i			1.21627i	0.793832	+	0.608137i	$0.208083\pi$
$89$			11.4743i			1.21627i	−0.793832	+	0.608137i	$0.791917\pi$
$90$		0			0
$91$		−	5.91190i		−	0.619735i
$92$	−1.91577			−0.199733
$93$		0			0
$94$		−	1.90301i		−	0.196281i
$95$	2.23311	+	1.57388i	0.229112	+	0.161477i
$96$		0			0
$97$	16.6814			1.69374			0.846868	−	0.531803i	$-0.178485\pi$
$97$	16.6814			1.69374			0.846868	+	0.531803i	$0.178485\pi$
$98$	5.42377			0.547883
$99$		0			0
$100$	−1.68122	−	4.70887i	−0.168122	−	0.470887i
$101$	−3.26214			−0.324595			−0.162298	−	0.986742i	$-0.551890\pi$
$101$	−3.26214			−0.324595			−0.162298	+	0.986742i	$0.551890\pi$
$102$		0			0
$103$	5.19330			0.511711			0.255856	−	0.966715i	$-0.417643\pi$
$103$	5.19330			0.511711			0.255856	+	0.966715i	$0.417643\pi$
$104$	4.70887			0.461743
$105$		0			0
$106$			1.11671i			0.108464i
$107$			8.65950i			0.837146i	0.908183	+	0.418573i	$0.137470\pi$
$107$			8.65950i			0.837146i	−0.908183	+	0.418573i	$0.862530\pi$
$108$		0			0
$109$			2.54677i			0.243936i	0.992534	+	0.121968i	$0.0389205\pi$
$109$			2.54677i			0.243936i	−0.992534	+	0.121968i	$0.961079\pi$
$110$	−7.04410	+	9.99458i	−0.671629	+	0.952945i
$111$		0			0
$112$		−	1.25548i		−	0.118632i
$113$	4.72781			0.444755			0.222378	−	0.974961i	$-0.428618\pi$
$113$	4.72781			0.444755			0.222378	+	0.974961i	$0.428618\pi$
$114$		0			0
$115$	2.46784	−	3.50152i	0.230128	−	0.326518i
$116$			5.08822i			0.472429i
$117$		0			0
$118$			8.99577i			0.828128i
$119$			1.49790i			0.137312i
$120$		0			0
$121$	18.9022			1.71838
$122$		−	6.21190i		−	0.562399i
$123$		0			0
$124$		−	3.43275i		−	0.308270i
$125$	10.7723	+	2.99303i	0.963501	+	0.267704i
$126$		0			0
$127$			8.05067i			0.714381i	0.934031	+	0.357191i	$0.116265\pi$
$127$			8.05067i			0.714381i	−0.934031	+	0.357191i	$0.883735\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	−6.06585	+	8.60657i	−0.532010	+	0.754846i
$131$			12.9999i			1.13581i	0.823094	+	0.567905i	$0.192246\pi$
$131$			12.9999i			1.13581i	−0.823094	+	0.567905i	$0.807754\pi$
$132$		0			0
$133$	−1.53393			−0.133009
$134$			4.21117i			0.363790i
$135$		0			0
$136$	−1.19309			−0.102307
$137$		−	13.5487i		−	1.15754i	−0.815490	−	0.578772i	$-0.803532\pi$
$137$		−	13.5487i		−	1.15754i	0.815490	−	0.578772i	$-0.196468\pi$
$138$		0			0
$139$	15.3257			1.29991			0.649955	−	0.759972i	$-0.274788\pi$
$139$	15.3257			1.29991			0.649955	+	0.759972i	$0.274788\pi$
$140$	2.29468	+	1.61728i	0.193936	+	0.136685i
$141$		0			0
$142$	12.4170			1.04201
$143$	25.7495			2.15328
$144$		0			0
$145$	−9.29991	−	6.55451i	−0.772315	−	0.544322i
$146$		−	10.2834i		−	0.851064i
$147$		0			0
$148$	−4.58473	−	3.99753i	−0.376863	−	0.328595i
$149$	−13.4684			−1.10337			−0.551686	−	0.834052i	$-0.686015\pi$
$149$	−13.4684			−1.10337			−0.551686	+	0.834052i	$0.686015\pi$
$150$		0			0
$151$	−10.3930			−0.845767			−0.422883	−	0.906184i	$-0.638982\pi$
$151$	−10.3930			−0.845767			−0.422883	+	0.906184i	$0.638982\pi$
$152$		−	1.22179i		−	0.0991003i
$153$		0			0
$154$		−	6.86533i		−	0.553224i
$155$	6.27415	+	4.42198i	0.503952	+	0.355182i
$156$		0			0
$157$			14.2297i			1.13565i	0.823148	+	0.567827i	$0.192216\pi$
$157$			14.2297i			1.13565i	−0.823148	+	0.567827i	$0.807784\pi$
$158$			1.00691i			0.0801054i
$159$		0			0
$160$	−1.28817	+	1.82773i	−0.101839	+	0.144495i
$161$			2.40521i			0.189557i
$162$		0			0
$163$	0.455364			0.0356669			0.0178334	−	0.999841i	$-0.494323\pi$
$163$	0.455364			0.0356669			0.0178334	+	0.999841i	$0.494323\pi$
$164$	4.46326			0.348522
$165$		0			0
$166$			0.754493i			0.0585600i
$167$	−0.755752			−0.0584819			−0.0292409	−	0.999572i	$-0.509309\pi$
$167$	−0.755752			−0.0584819			−0.0292409	+	0.999572i	$0.509309\pi$
$168$		0			0
$169$	9.17350			0.705654
$170$	1.53691	−	2.18065i	0.117875	−	0.167248i
$171$		0			0
$172$	1.34271			0.102380
$173$		−	4.41375i		−	0.335571i	−0.985823	−	0.167786i	$-0.946338\pi$
$173$		−	4.41375i		−	0.335571i	0.985823	−	0.167786i	$-0.0536617\pi$
$174$		0			0
$175$	−5.91190	+	2.11074i	−0.446898	+	0.159557i
$176$	5.46829			0.412188
$177$		0			0
$178$			11.4743i			0.860036i
$179$		−	12.6883i		−	0.948368i	−0.880426	−	0.474184i	$-0.842743\pi$
$179$		−	12.6883i		−	0.948368i	0.880426	−	0.474184i	$-0.157257\pi$
$180$		0			0
$181$	13.2444			0.984445			0.492223	−	0.870469i	$-0.336185\pi$
$181$	13.2444			0.984445			0.492223	+	0.870469i	$0.336185\pi$
$182$		−	5.91190i		−	0.438219i
$183$		0			0
$184$	−1.91577			−0.141233
$185$	13.2123	−	3.23016i	0.971391	−	0.237486i
$186$		0			0
$187$	−6.52417			−0.477094
$188$		−	1.90301i		−	0.138791i
$189$		0			0
$190$	2.23311	+	1.57388i	0.162007	+	0.114181i
$191$			7.03990i			0.509389i	0.967022	+	0.254695i	$0.0819749\pi$
$191$			7.03990i			0.509389i	−0.967022	+	0.254695i	$0.918025\pi$
$192$		0			0
$193$	−0.635749			−0.0457622			−0.0228811	−	0.999738i	$-0.507284\pi$
$193$	−0.635749			−0.0457622			−0.0228811	+	0.999738i	$0.507284\pi$
$194$	16.6814			1.19765
$195$		0			0
$196$	5.42377			0.387412
$197$		−	5.18937i		−	0.369727i	−0.982764	−	0.184864i	$-0.940816\pi$
$197$		−	5.18937i		−	0.369727i	0.982764	−	0.184864i	$-0.0591843\pi$
$198$		0			0
$199$			23.8852i			1.69318i	0.532247	+	0.846589i	$0.321348\pi$
$199$			23.8852i			1.69318i	−0.532247	+	0.846589i	$0.678652\pi$
$200$	−1.68122	−	4.70887i	−0.118880	−	0.332968i
$201$		0			0
$202$	−3.26214			−0.229523
$203$	6.38816			0.448361
$204$		0			0
$205$	−5.74945	+	8.15764i	−0.401559	+	0.569754i
$206$	5.19330			0.361834
$207$		0			0
$208$	4.70887			0.326502
$209$		−	6.68111i		−	0.462142i
$210$		0			0
$211$	−26.6260			−1.83301			−0.916505	−	0.400022i	$-0.869002\pi$
$211$	−26.6260			−1.83301			−0.916505	+	0.400022i	$0.869002\pi$
$212$			1.11671i			0.0766959i
$213$		0			0
$214$			8.65950i			0.591952i
$215$	−1.72964	+	2.45411i	−0.117960	+	0.167369i
$216$		0			0
$217$	−4.30975			−0.292565
$218$			2.54677i			0.172489i
$219$		0			0
$220$	−7.04410	+	9.99458i	−0.474913	+	0.673834i
$221$	−5.61812			−0.377915
$222$		0			0
$223$		−	20.0982i		−	1.34588i	−0.739698	−	0.672938i	$-0.765032\pi$
$223$		−	20.0982i		−	1.34588i	0.739698	−	0.672938i	$-0.234968\pi$
$224$		−	1.25548i		−	0.0838853i
$225$		0			0
$226$	4.72781			0.314489
$227$	−15.1621			−1.00635			−0.503173	−	0.864186i	$-0.667834\pi$
$227$	−15.1621			−1.00635			−0.503173	+	0.864186i	$0.667834\pi$
$228$		0			0
$229$	−2.54527			−0.168196			−0.0840980	−	0.996457i	$-0.526801\pi$
$229$	−2.54527			−0.168196			−0.0840980	+	0.996457i	$0.526801\pi$
$230$	2.46784	−	3.50152i	0.162725	−	0.230883i
$231$		0			0
$232$			5.08822i			0.334058i
$233$			13.9212i			0.912008i	0.889978	+	0.456004i	$0.150720\pi$
$233$			13.9212i			0.912008i	−0.889978	+	0.456004i	$0.849280\pi$
$234$		0			0
$235$	3.47820	+	2.45141i	0.226892	+	0.159912i
$236$			8.99577i			0.585575i
$237$		0			0
$238$			1.49790i			0.0970945i
$239$		−	28.8753i		−	1.86779i	−0.357546	−	0.933895i	$-0.616387\pi$
$239$		−	28.8753i		−	1.86779i	0.357546	−	0.933895i	$-0.383613\pi$
$240$		0			0
$241$		−	6.84853i		−	0.441152i	−0.975370	−	0.220576i	$-0.929206\pi$
$241$		−	6.84853i		−	0.441152i	0.975370	−	0.220576i	$-0.0707938\pi$
$242$	18.9022			1.21508
$243$		0			0
$244$		−	6.21190i		−	0.397676i
$245$	−6.98676	+	9.91321i	−0.446367	+	0.633331i
$246$		0			0
$247$		−	5.75326i		−	0.366071i
$248$		−	3.43275i		−	0.217980i
$249$		0			0
$250$	10.7723	+	2.99303i	0.681298	+	0.189296i
$251$			18.7459i			1.18323i	0.806221	+	0.591615i	$0.201509\pi$
$251$			18.7459i			1.18323i	−0.806221	+	0.591615i	$0.798491\pi$
$252$		0			0
$253$	−10.4760			−0.658620
$254$			8.05067i			0.505144i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−17.3509			−1.08232			−0.541158	−	0.840921i	$-0.682014\pi$
$257$	−17.3509			−1.08232			−0.541158	+	0.840921i	$0.682014\pi$
$258$		0			0
$259$	−5.01882	+	5.75604i	−0.311854	+	0.357663i
$260$	−6.06585	+	8.60657i	−0.376188	+	0.533757i
$261$		0			0
$262$			12.9999i			0.803139i
$263$		−	31.3436i		−	1.93273i	−0.257176	−	0.966364i	$-0.582792\pi$
$263$		−	31.3436i		−	1.93273i	0.257176	−	0.966364i	$-0.417208\pi$
$264$		0			0
$265$	−2.04105	−	1.43851i	−0.125380	−	0.0883672i
$266$	−1.53393			−0.0940516
$267$		0			0
$268$			4.21117i			0.257238i
$269$	−20.8503			−1.27126			−0.635631	−	0.771993i	$-0.719260\pi$
$269$	−20.8503			−1.27126			−0.635631	+	0.771993i	$0.719260\pi$
$270$		0			0
$271$	−0.977182			−0.0593596			−0.0296798	−	0.999559i	$-0.509449\pi$
$271$	−0.977182			−0.0593596			−0.0296798	+	0.999559i	$0.509449\pi$
$272$	−1.19309			−0.0723418
$273$		0			0
$274$		−	13.5487i		−	0.818507i
$275$	−9.19339	−	25.7495i	−0.554383	−	1.55275i
$276$		0			0
$277$	−28.0775			−1.68701			−0.843506	−	0.537120i	$-0.819512\pi$
$277$	−28.0775			−1.68701			−0.843506	+	0.537120i	$0.819512\pi$
$278$	15.3257			0.919176
$279$		0			0
$280$	2.29468	+	1.61728i	0.137134	+	0.0966507i
$281$		−	20.9112i		−	1.24746i	−0.781640	−	0.623730i	$-0.785617\pi$
$281$		−	20.9112i		−	1.24746i	0.781640	−	0.623730i	$-0.214383\pi$
$282$		0			0
$283$	−11.7273			−0.697116			−0.348558	−	0.937287i	$-0.613329\pi$
$283$	−11.7273			−0.697116			−0.348558	+	0.937287i	$0.613329\pi$
$284$	12.4170			0.736815
$285$		0			0
$286$	25.7495			1.52260
$287$		−	5.60353i		−	0.330766i
$288$		0			0
$289$	−15.5765			−0.916267
$290$	−9.29991	−	6.55451i	−0.546109	−	0.384894i
$291$		0			0
$292$		−	10.2834i		−	0.601793i
$293$		−	19.0944i		−	1.11550i	−0.830007	−	0.557752i	$-0.811664\pi$
$293$		−	19.0944i		−	1.11550i	0.830007	−	0.557752i	$-0.188336\pi$
$294$		0			0
$295$	−16.4419	−	11.5881i	−0.957283	−	0.674686i
$296$	−4.58473	−	3.99753i	−0.266482	−	0.232352i
$297$		0			0
$298$	−13.4684			−0.780202
$299$	−9.02113			−0.521705
$300$		0			0
$301$		−	1.68574i		−	0.0971645i
$302$	−10.3930			−0.598048
$303$		0			0
$304$		−	1.22179i		−	0.0700745i
$305$	11.3537	+	8.00200i	0.650111	+	0.458193i
$306$		0			0
$307$			31.2854i			1.78555i	0.450500	+	0.892777i	$0.351246\pi$
$307$			31.2854i			1.78555i	−0.450500	+	0.892777i	$0.648754\pi$
$308$		−	6.86533i		−	0.391188i
$309$		0			0
$310$	6.27415	+	4.42198i	0.356348	+	0.251151i
$311$			7.52409i			0.426652i	0.976981	+	0.213326i	$0.0684296\pi$
$311$			7.52409i			0.426652i	−0.976981	+	0.213326i	$0.931570\pi$
$312$		0			0
$313$	−0.0900227			−0.00508838			−0.00254419	−	0.999997i	$-0.500810\pi$
$313$	−0.0900227			−0.00508838			−0.00254419	+	0.999997i	$0.500810\pi$
$314$			14.2297i			0.803029i
$315$		0			0
$316$			1.00691i			0.0566431i
$317$		−	7.86659i		−	0.441832i	−0.975293	−	0.220916i	$-0.929095\pi$
$317$		−	7.86659i		−	0.441832i	0.975293	−	0.220916i	$-0.0709046\pi$
$318$		0			0
$319$			27.8239i			1.55784i
$320$	−1.28817	+	1.82773i	−0.0720111	+	0.102173i
$321$		0			0
$322$			2.40521i			0.134037i
$323$			1.45771i			0.0811090i
$324$		0			0
$325$	−7.91665	−	22.1735i	−0.439137	−	1.22996i
$326$	0.455364			0.0252203
$327$		0			0
$328$	4.46326			0.246442
$329$	−2.38919			−0.131720
$330$		0			0
$331$			21.5849i			1.18641i	0.805051	+	0.593205i	$0.202138\pi$
$331$			21.5849i			1.18641i	−0.805051	+	0.593205i	$0.797862\pi$
$332$			0.754493i			0.0414082i
$333$		0			0
$334$	−0.755752			−0.0413529
$335$	−7.69690	−	5.42472i	−0.420526	−	0.296384i
$336$		0			0
$337$			24.3782i			1.32796i	0.747748	+	0.663982i	$0.231135\pi$
$337$			24.3782i			1.32796i	−0.747748	+	0.663982i	$0.768865\pi$
$338$	9.17350			0.498973
$339$		0			0
$340$	1.53691	−	2.18065i	0.0833505	−	0.118262i
$341$		−	18.7713i		−	1.01652i
$342$		0			0
$343$		−	15.5978i		−	0.842202i
$344$	1.34271			0.0723939
$345$		0			0
$346$		−	4.41375i		−	0.237285i
$347$	−25.4950			−1.36864			−0.684321	−	0.729181i	$-0.739901\pi$
$347$	−25.4950			−1.36864			−0.684321	+	0.729181i	$0.739901\pi$
$348$		0			0
$349$	2.69761			0.144400			0.0721999	−	0.997390i	$-0.476998\pi$
$349$	2.69761			0.144400			0.0721999	+	0.997390i	$0.476998\pi$
$350$	−5.91190	+	2.11074i	−0.316004	+	0.112824i
$351$		0			0
$352$	5.46829			0.291461
$353$	21.7179			1.15592			0.577962	−	0.816063i	$-0.303848\pi$
$353$	21.7179			1.15592			0.577962	+	0.816063i	$0.303848\pi$
$354$		0			0
$355$	−15.9953	+	22.6950i	−0.848942	+	1.20453i
$356$			11.4743i			0.608137i
$357$		0			0
$358$		−	12.6883i		−	0.670597i
$359$	20.3716			1.07517			0.537585	−	0.843210i	$-0.319337\pi$
$359$	20.3716			1.07517			0.537585	+	0.843210i	$0.319337\pi$
$360$		0			0
$361$	17.5072			0.921433
$362$	13.2444			0.696108
$363$		0			0
$364$		−	5.91190i		−	0.309868i
$365$	18.7954	+	13.2469i	0.983796	+	0.693372i
$366$		0			0
$367$		−	31.0972i		−	1.62326i	−0.584172	−	0.811630i	$-0.698581\pi$
$367$		−	31.0972i		−	1.62326i	0.584172	−	0.811630i	$-0.301419\pi$
$368$	−1.91577			−0.0998665
$369$		0			0
$370$	13.2123	−	3.23016i	0.686877	−	0.167928i
$371$	1.40201			0.0727885
$372$		0			0
$373$		−	33.4131i		−	1.73006i	−0.501716	−	0.865032i	$-0.667298\pi$
$373$		−	33.4131i		−	1.73006i	0.501716	−	0.865032i	$-0.332702\pi$
$374$	−6.52417			−0.337357
$375$		0			0
$376$		−	1.90301i		−	0.0981403i
$377$			23.9598i			1.23399i
$378$		0			0
$379$	−24.6191			−1.26460			−0.632298	−	0.774725i	$-0.717888\pi$
$379$	−24.6191			−1.26460			−0.632298	+	0.774725i	$0.717888\pi$
$380$	2.23311	+	1.57388i	0.114556	+	0.0807383i
$381$		0			0
$382$			7.03990i			0.360192i
$383$	−17.5596			−0.897252			−0.448626	−	0.893720i	$-0.648086\pi$
$383$	−17.5596			−0.897252			−0.448626	+	0.893720i	$0.648086\pi$
$384$		0			0
$385$	12.5480	+	8.84373i	0.639505	+	0.450718i
$386$	−0.635749			−0.0323587
$387$		0			0
$388$	16.6814			0.846868
$389$			19.6417i			0.995875i	0.867213	+	0.497937i	$0.165909\pi$
$389$			19.6417i			0.995875i	−0.867213	+	0.497937i	$0.834091\pi$
$390$		0			0
$391$	2.28569			0.115592
$392$	5.42377			0.273942
$393$		0			0
$394$		−	5.18937i		−	0.261437i
$395$	−1.84036	−	1.29707i	−0.0925987	−	0.0652629i
$396$		0			0
$397$		−	4.31480i		−	0.216554i	−0.994121	−	0.108277i	$-0.965467\pi$
$397$		−	4.31480i		−	0.216554i	0.994121	−	0.108277i	$-0.0345333\pi$
$398$			23.8852i			1.19726i
$399$		0			0
$400$	−1.68122	−	4.70887i	−0.0840610	−	0.235444i
$401$		−	9.64980i		−	0.481888i	−0.970539	−	0.240944i	$-0.922543\pi$
$401$		−	9.64980i		−	0.481888i	0.970539	−	0.240944i	$-0.0774570\pi$
$402$		0			0
$403$		−	16.1644i		−	0.805206i
$404$	−3.26214			−0.162298
$405$		0			0
$406$	6.38816			0.317039
$407$	−25.0706	−	21.8596i	−1.24271	−	1.08354i
$408$		0			0
$409$		−	25.5235i		−	1.26206i	−0.775760	−	0.631029i	$-0.782633\pi$
$409$		−	25.5235i		−	1.26206i	0.775760	−	0.631029i	$-0.217367\pi$
$410$	−5.74945	+	8.15764i	−0.283945	+	0.402877i
$411$		0			0
$412$	5.19330			0.255856
$413$	11.2940			0.555742
$414$		0			0
$415$	−1.37901	−	0.971918i	−0.0676930	−	0.0477096i
$416$	4.70887			0.230872
$417$		0			0
$418$		−	6.68111i		−	0.326784i
$419$	22.0969			1.07950			0.539751	−	0.841824i	$-0.318518\pi$
$419$	22.0969			1.07950			0.539751	+	0.841824i	$0.318518\pi$
$420$		0			0
$421$		−	8.20101i		−	0.399693i	−0.979827	−	0.199846i	$-0.935956\pi$
$421$		−	8.20101i		−	0.399693i	0.979827	−	0.199846i	$-0.0640443\pi$
$422$	−26.6260			−1.29613
$423$		0			0
$424$			1.11671i			0.0542322i
$425$	2.00585	+	5.61812i	0.0972979	+	0.272519i
$426$		0			0
$427$	−7.79891			−0.377416
$428$			8.65950i			0.418573i
$429$		0			0
$430$	−1.72964	+	2.45411i	−0.0834106	+	0.118348i
$431$		−	9.83076i		−	0.473531i	−0.971567	−	0.236765i	$-0.923913\pi$
$431$		−	9.83076i		−	0.473531i	0.971567	−	0.236765i	$-0.0760873\pi$
$432$		0			0
$433$			1.61024i			0.0773832i	0.999251	+	0.0386916i	$0.0123190\pi$
$433$			1.61024i			0.0773832i	−0.999251	+	0.0386916i	$0.987681\pi$
$434$	−4.30975			−0.206875
$435$		0			0
$436$			2.54677i			0.121968i
$437$			2.34067i			0.111970i
$438$		0			0
$439$			32.2443i			1.53894i	0.638684	+	0.769469i	$0.279479\pi$
$439$			32.2443i			1.53894i	−0.638684	+	0.769469i	$0.720521\pi$
$440$	−7.04410	+	9.99458i	−0.335815	+	0.476473i
$441$		0			0
$442$	−5.61812			−0.267226
$443$		−	19.0730i		−	0.906186i	−0.891463	−	0.453093i	$-0.850321\pi$
$443$		−	19.0730i		−	0.906186i	0.891463	−	0.453093i	$-0.149679\pi$
$444$		0			0
$445$	−20.9720	−	14.7809i	−0.994168	−	0.700682i
$446$		−	20.0982i		−	0.951679i
$447$		0			0
$448$		−	1.25548i		−	0.0593159i
$449$		−	30.6914i		−	1.44841i	−0.689582	−	0.724207i	$-0.742206\pi$
$449$		−	30.6914i		−	1.44841i	0.689582	−	0.724207i	$-0.257794\pi$
$450$		0			0
$451$	24.4064			1.14925
$452$	4.72781			0.222378
$453$		0			0
$454$	−15.1621			−0.711593
$455$	10.8054	+	7.61555i	0.506564	+	0.357022i
$456$		0			0
$457$	22.4560			1.05045			0.525223	−	0.850965i	$-0.323982\pi$
$457$	22.4560			1.05045			0.525223	+	0.850965i	$0.323982\pi$
$458$	−2.54527			−0.118932
$459$		0			0
$460$	2.46784	−	3.50152i	0.115064	−	0.163259i
$461$			7.37800i			0.343628i	0.985129	+	0.171814i	$0.0549628\pi$
$461$			7.37800i			0.343628i	−0.985129	+	0.171814i	$0.945037\pi$
$462$		0			0
$463$	−6.16145			−0.286347			−0.143173	−	0.989698i	$-0.545731\pi$
$463$	−6.16145			−0.286347			−0.143173	+	0.989698i	$0.545731\pi$
$464$			5.08822i			0.236215i
$465$		0			0
$466$			13.9212i			0.644887i
$467$	−5.37297			−0.248631			−0.124316	−	0.992243i	$-0.539674\pi$
$467$	−5.37297			−0.248631			−0.124316	+	0.992243i	$0.539674\pi$
$468$		0			0
$469$	5.28704			0.244133
$470$	3.47820	+	2.45141i	0.160437	+	0.113075i
$471$		0			0
$472$			8.99577i			0.414064i
$473$	7.34231			0.337600
$474$		0			0
$475$	−5.75326	+	2.05410i	−0.263978	+	0.0942485i
$476$			1.49790i			0.0686562i
$477$		0			0
$478$		−	28.8753i		−	1.32073i
$479$			21.5810i			0.986062i	0.870012	+	0.493031i	$0.164111\pi$
$479$			21.5810i			0.986062i	−0.870012	+	0.493031i	$0.835889\pi$
$480$		0			0
$481$	−21.5889	−	18.8239i	−0.984370	−	0.858294i
$482$		−	6.84853i		−	0.311942i
$483$		0			0
$484$	18.9022			0.859191
$485$	−21.4885	+	30.4891i	−0.975742	+	1.38444i
$486$		0			0
$487$	−0.346315			−0.0156930			−0.00784652	−	0.999969i	$-0.502498\pi$
$487$	−0.346315			−0.0156930			−0.00784652	+	0.999969i	$0.502498\pi$
$488$		−	6.21190i		−	0.281199i
$489$		0			0
$490$	−6.98676	+	9.91321i	−0.315629	+	0.447833i
$491$	−10.6278			−0.479624			−0.239812	−	0.970819i	$-0.577086\pi$
$491$	−10.6278			−0.479624			−0.239812	+	0.970819i	$0.577086\pi$
$492$		0			0
$493$		−	6.07071i		−	0.273411i
$494$		−	5.75326i		−	0.258851i
$495$		0			0
$496$		−	3.43275i		−	0.154135i
$497$		−	15.5893i		−	0.699277i
$498$		0			0
$499$		−	12.0121i		−	0.537736i	−0.963177	−	0.268868i	$-0.913350\pi$
$499$		−	12.0121i		−	0.537736i	0.963177	−	0.268868i	$-0.0866496\pi$
$500$	10.7723	+	2.99303i	0.481751	+	0.133852i
$501$		0			0
$502$			18.7459i			0.836669i
$503$	4.15447			0.185239			0.0926194	−	0.995702i	$-0.470476\pi$
$503$	4.15447			0.185239			0.0926194	+	0.995702i	$0.470476\pi$
$504$		0			0
$505$	4.20220	−	5.96232i	0.186996	−	0.265320i
$506$	−10.4760			−0.465715
$507$		0			0
$508$			8.05067i			0.357191i
$509$	8.30223			0.367990			0.183995	−	0.982927i	$-0.441097\pi$
$509$	8.30223			0.367990			0.183995	+	0.982927i	$0.441097\pi$
$510$		0			0
$511$	−12.9107			−0.571134
$512$	1.00000			0.0441942
$513$		0			0
$514$	−17.3509			−0.765313
$515$	−6.68987	+	9.49197i	−0.294791	+	0.418266i
$516$		0			0
$517$		−	10.4062i		−	0.457665i
$518$	−5.01882	+	5.75604i	−0.220514	+	0.252906i
$519$		0			0
$520$	−6.06585	+	8.60657i	−0.266005	+	0.377423i
$521$	−14.0992			−0.617698			−0.308849	−	0.951111i	$-0.599944\pi$
$521$	−14.0992			−0.617698			−0.308849	+	0.951111i	$0.599944\pi$
$522$		0			0
$523$	29.0802			1.27159			0.635794	−	0.771858i	$-0.280673\pi$
$523$	29.0802			1.27159			0.635794	+	0.771858i	$0.280673\pi$
$524$			12.9999i			0.567905i
$525$		0			0
$526$		−	31.3436i		−	1.36665i
$527$			4.09558i			0.178406i
$528$		0			0
$529$	−19.3298			−0.840427
$530$	−2.04105	−	1.43851i	−0.0886574	−	0.0624851i
$531$		0			0
$532$	−1.53393			−0.0665045
$533$	21.0169			0.910344
$534$		0			0
$535$	−15.8273	−	11.1549i	−0.684272	−	0.482270i
$536$			4.21117i			0.181895i
$537$		0			0
$538$	−20.8503			−0.898918
$539$	29.6587			1.27749
$540$		0			0
$541$			10.2446i			0.440450i	0.975449	+	0.220225i	$0.0706791\pi$
$541$			10.2446i			0.440450i	−0.975449	+	0.220225i	$0.929321\pi$
$542$	−0.977182			−0.0419736
$543$		0			0
$544$	−1.19309			−0.0511533
$545$	−4.65481	−	3.28068i	−0.199390	−	0.140529i
$546$		0			0
$547$	36.6360			1.56644			0.783220	−	0.621744i	$-0.213576\pi$
$547$	36.6360			1.56644			0.783220	+	0.621744i	$0.213576\pi$
$548$		−	13.5487i		−	0.578772i
$549$		0			0
$550$	−9.19339	−	25.7495i	−0.392008	−	1.09796i
$551$	6.21674			0.264842
$552$		0			0
$553$	1.26415			0.0537573
$554$	−28.0775			−1.19290
$555$		0			0
$556$	15.3257			0.649955
$557$	3.19892			0.135543			0.0677714	−	0.997701i	$-0.478411\pi$
$557$	3.19892			0.135543			0.0677714	+	0.997701i	$0.478411\pi$
$558$		0			0
$559$	6.32264			0.267419
$560$	2.29468	+	1.61728i	0.0969680	+	0.0683424i
$561$		0			0
$562$		−	20.9112i		−	0.882087i
$563$	−43.0080			−1.81257			−0.906285	−	0.422668i	$-0.861094\pi$
$563$	−43.0080			−1.81257			−0.906285	+	0.422668i	$0.861094\pi$
$564$		0			0
$565$	−6.09024	+	8.64118i	−0.256218	+	0.363537i
$566$	−11.7273			−0.492936
$567$		0			0
$568$	12.4170			0.521007
$569$		−	19.8692i		−	0.832960i	−0.909145	−	0.416480i	$-0.863264\pi$
$569$		−	19.8692i		−	0.832960i	0.909145	−	0.416480i	$-0.136736\pi$
$570$		0			0
$571$	−16.6100			−0.695105			−0.347553	−	0.937660i	$-0.612987\pi$
$571$	−16.6100			−0.695105			−0.347553	+	0.937660i	$0.612987\pi$
$572$	25.7495			1.07664
$573$		0			0
$574$		−	5.60353i		−	0.233887i
$575$	3.22083	+	9.02113i	0.134318	+	0.376207i
$576$		0			0
$577$	−36.4851			−1.51890			−0.759448	−	0.650568i	$-0.774531\pi$
$577$	−36.4851			−1.51890			−0.759448	+	0.650568i	$0.774531\pi$
$578$	−15.5765			−0.647898
$579$		0			0
$580$	−9.29991	−	6.55451i	−0.386158	−	0.272161i
$581$	0.947251			0.0392986
$582$		0			0
$583$			6.10649i			0.252905i
$584$		−	10.2834i		−	0.425532i
$585$		0			0
$586$		−	19.0944i		−	0.788781i
$587$	−14.3652			−0.592917			−0.296458	−	0.955046i	$-0.595806\pi$
$587$	−14.3652			−0.592917			−0.296458	+	0.955046i	$0.595806\pi$
$588$		0			0
$589$	−4.19411			−0.172815
$590$	−16.4419	−	11.5881i	−0.676901	−	0.477075i
$591$		0			0
$592$	−4.58473	−	3.99753i	−0.188431	−	0.164297i
$593$			19.6724i			0.807850i	0.914792	+	0.403925i	$0.132354\pi$
$593$			19.6724i			0.807850i	−0.914792	+	0.403925i	$0.867646\pi$
$594$		0			0
$595$	−2.73777	−	1.92956i	−0.112237	−	0.0791041i
$596$	−13.4684			−0.551686
$597$		0			0
$598$	−9.02113			−0.368901
$599$	−32.9039			−1.34442			−0.672208	−	0.740362i	$-0.734654\pi$
$599$	−32.9039			−1.34442			−0.672208	+	0.740362i	$0.734654\pi$
$600$		0			0
$601$	43.6508			1.78055			0.890276	−	0.455422i	$-0.150512\pi$
$601$	43.6508			1.78055			0.890276	+	0.455422i	$0.150512\pi$
$602$		−	1.68574i		−	0.0687057i
$603$		0			0
$604$	−10.3930			−0.422883
$605$	−24.3493	+	34.5482i	−0.989940	+	1.40458i
$606$		0			0
$607$	15.4852			0.628523			0.314262	−	0.949336i	$-0.398243\pi$
$607$	15.4852			0.628523			0.314262	+	0.949336i	$0.398243\pi$
$608$		−	1.22179i		−	0.0495502i
$609$		0			0
$610$	11.3537	+	8.00200i	0.459698	+	0.323992i
$611$		−	8.96104i		−	0.362525i
$612$		0			0
$613$			47.5322i			1.91981i	0.280329	+	0.959904i	$0.409556\pi$
$613$			47.5322i			1.91981i	−0.280329	+	0.959904i	$0.590444\pi$
$614$			31.2854i			1.26258i
$615$		0			0
$616$		−	6.86533i		−	0.276612i
$617$			31.3888i			1.26367i	0.775105	+	0.631833i	$0.217697\pi$
$617$			31.3888i			1.26367i	−0.775105	+	0.631833i	$0.782303\pi$
$618$		0			0
$619$	8.12905			0.326734			0.163367	−	0.986565i	$-0.447765\pi$
$619$	8.12905			0.326734			0.163367	+	0.986565i	$0.447765\pi$
$620$	6.27415	+	4.42198i	0.251976	+	0.177591i
$621$		0			0
$622$			7.52409i			0.301688i
$623$	14.4058			0.577155
$624$		0			0
$625$	−19.3470	+	15.8333i	−0.773880	+	0.633332i
$626$	−0.0900227			−0.00359803
$627$		0			0
$628$			14.2297i			0.567827i
$629$	5.47000	+	4.76941i	0.218103	+	0.190169i
$630$		0			0
$631$		−	37.9315i		−	1.51003i	−0.655709	−	0.755014i	$-0.727630\pi$
$631$		−	37.9315i		−	1.51003i	0.655709	−	0.755014i	$-0.272370\pi$
$632$			1.00691i			0.0400527i
$633$		0			0
$634$		−	7.86659i		−	0.312422i
$635$	−14.7145	−	10.3707i	−0.583926	−	0.411547i
$636$		0			0
$637$	25.5399			1.01193
$638$			27.8239i			1.10156i
$639$		0			0
$640$	−1.28817	+	1.82773i	−0.0509195	+	0.0722475i
$641$	36.9793			1.46059			0.730296	−	0.683130i	$-0.239382\pi$
$641$	36.9793			1.46059			0.730296	+	0.683130i	$0.239382\pi$
$642$		0			0
$643$	−8.44944			−0.333213			−0.166607	−	0.986023i	$-0.553281\pi$
$643$	−8.44944			−0.333213			−0.166607	+	0.986023i	$0.553281\pi$
$644$			2.40521i			0.0947786i
$645$		0			0
$646$			1.45771i			0.0573527i
$647$	−4.22924			−0.166269			−0.0831343	−	0.996538i	$-0.526493\pi$
$647$	−4.22924			−0.166269			−0.0831343	+	0.996538i	$0.526493\pi$
$648$		0			0
$649$			49.1915i			1.93093i
$650$	−7.91665	−	22.1735i	−0.310517	−	0.869716i
$651$		0			0
$652$	0.455364			0.0178334
$653$	−40.2219			−1.57400			−0.787002	−	0.616951i	$-0.788368\pi$
$653$	−40.2219			−1.57400			−0.787002	+	0.616951i	$0.788368\pi$
$654$		0			0
$655$	−23.7604	−	16.7462i	−0.928397	−	0.654327i
$656$	4.46326			0.174261
$657$		0			0
$658$	−2.38919			−0.0931404
$659$	−20.4172			−0.795341			−0.397670	−	0.917528i	$-0.630181\pi$
$659$	−20.4172			−0.795341			−0.397670	+	0.917528i	$0.630181\pi$
$660$		0			0
$661$		−	49.8075i		−	1.93729i	−0.248454	−	0.968644i	$-0.579923\pi$
$661$		−	49.8075i		−	1.93729i	0.248454	−	0.968644i	$-0.420077\pi$
$662$			21.5849i			0.838919i
$663$		0			0
$664$			0.754493i			0.0292800i
$665$	1.97597	−	2.80362i	0.0766250	−	0.108720i
$666$		0			0
$667$		−	9.74786i		−	0.377439i
$668$	−0.755752			−0.0292409
$669$		0			0
$670$	−7.69690	−	5.42472i	−0.297357	−	0.209575i
$671$		−	33.9685i		−	1.31134i
$672$		0			0
$673$			30.7799i			1.18648i	0.805026	+	0.593239i	$0.202151\pi$
$673$			30.7799i			1.18648i	−0.805026	+	0.593239i	$0.797849\pi$
$674$			24.3782i			0.939013i
$675$		0			0
$676$	9.17350			0.352827
$677$			36.8784i			1.41735i	0.705533	+	0.708677i	$0.250708\pi$
$677$			36.8784i			1.41735i	−0.705533	+	0.708677i	$0.749292\pi$
$678$		0			0
$679$		−	20.9431i		−	0.803723i
$680$	1.53691	−	2.18065i	0.0589377	−	0.0836242i
$681$		0			0
$682$		−	18.7713i		−	0.718789i
$683$	−22.1763			−0.848552			−0.424276	−	0.905533i	$-0.639471\pi$
$683$	−22.1763			−0.848552			−0.424276	+	0.905533i	$0.639471\pi$
$684$		0			0
$685$	24.7634	+	17.4531i	0.946161	+	0.666848i
$686$		−	15.5978i		−	0.595527i
$687$		0			0
$688$	1.34271			0.0511902
$689$			5.25844i			0.200331i
$690$		0			0
$691$	15.0935			0.574185			0.287093	−	0.957903i	$-0.407311\pi$
$691$	15.0935			0.574185			0.287093	+	0.957903i	$0.407311\pi$
$692$		−	4.41375i		−	0.167786i
$693$		0			0
$694$	−25.4950			−0.967776
$695$	−19.7422	+	28.0113i	−0.748864	+	1.06253i
$696$		0			0
$697$	−5.32507			−0.201701
$698$	2.69761			0.102106
$699$		0			0
$700$	−5.91190	+	2.11074i	−0.223449	+	0.0797784i
$701$		−	31.2390i		−	1.17988i	−0.807447	−	0.589941i	$-0.799151\pi$
$701$		−	31.2390i		−	1.17988i	0.807447	−	0.589941i	$-0.200849\pi$
$702$		0			0
$703$	−4.88414	+	5.60159i	−0.184209	+	0.211268i
$704$	5.46829			0.206094
$705$		0			0
$706$	21.7179			0.817362
$707$			4.09555i			0.154029i
$708$		0			0
$709$		−	23.2007i		−	0.871322i	−0.900111	−	0.435661i	$-0.856515\pi$
$709$		−	23.2007i		−	0.871322i	0.900111	−	0.435661i	$-0.143485\pi$
$710$	−15.9953	+	22.6950i	−0.600293	+	0.851729i
$711$		0			0
$712$			11.4743i			0.430018i
$713$			6.57636i			0.246287i
$714$		0			0
$715$	−33.1698	+	47.0632i	−1.24048	+	1.76006i
$716$		−	12.6883i		−	0.474184i
$717$		0			0
$718$	20.3716			0.760260
$719$	−41.3863			−1.54345			−0.771724	−	0.635958i	$-0.780605\pi$
$719$	−41.3863			−1.54345			−0.771724	+	0.635958i	$0.780605\pi$
$720$		0			0
$721$		−	6.52009i		−	0.242821i
$722$	17.5072			0.651552
$723$		0			0
$724$	13.2444			0.492223
$725$	23.9598	−	8.55441i	0.889844	−	0.317703i
$726$		0			0
$727$	−8.02093			−0.297480			−0.148740	−	0.988876i	$-0.547522\pi$
$727$	−8.02093			−0.297480			−0.148740	+	0.988876i	$0.547522\pi$
$728$		−	5.91190i		−	0.219110i
$729$		0			0
$730$	18.7954	+	13.2469i	0.695649	+	0.490288i
$731$	−1.60197			−0.0592510
$732$		0			0
$733$			43.5044i			1.60687i	0.595391	+	0.803436i	$0.296997\pi$
$733$			43.5044i			1.60687i	−0.595391	+	0.803436i	$0.703003\pi$
$734$		−	31.0972i		−	1.14782i
$735$		0			0
$736$	−1.91577			−0.0706163
$737$			23.0279i			0.848244i
$738$		0			0
$739$	−40.4960			−1.48967			−0.744834	−	0.667249i	$-0.767472\pi$
$739$	−40.4960			−1.48967			−0.744834	+	0.667249i	$0.767472\pi$
$740$	13.2123	−	3.23016i	0.485695	−	0.118743i
$741$		0			0
$742$	1.40201			0.0514692
$743$			33.7617i			1.23860i	0.785156	+	0.619298i	$0.212583\pi$
$743$			33.7617i			1.23860i	−0.785156	+	0.619298i	$0.787417\pi$
$744$		0			0
$745$	17.3496	−	24.6166i	0.635640	−	0.901882i
$746$		−	33.4131i		−	1.22334i
$747$		0			0
$748$	−6.52417			−0.238547
$749$	10.8718			0.397248
$750$		0			0
$751$	−16.9597			−0.618867			−0.309434	−	0.950921i	$-0.600139\pi$
$751$	−16.9597			−0.618867			−0.309434	+	0.950921i	$0.600139\pi$
$752$		−	1.90301i		−	0.0693956i
$753$		0			0
$754$			23.9598i			0.872564i
$755$	13.3879	−	18.9956i	0.487237	−	0.691319i
$756$		0			0
$757$	10.2892			0.373969			0.186984	−	0.982363i	$-0.440129\pi$
$757$	10.2892			0.373969			0.186984	+	0.982363i	$0.440129\pi$
$758$	−24.6191			−0.894204
$759$		0			0
$760$	2.23311	+	1.57388i	0.0810034	+	0.0570906i
$761$	30.3977			1.10192			0.550959	−	0.834533i	$-0.314262\pi$
$761$	30.3977			1.10192			0.550959	+	0.834533i	$0.314262\pi$
$762$		0			0
$763$	3.19741			0.115754
$764$			7.03990i			0.254695i
$765$		0			0
$766$	−17.5596			−0.634453
$767$			42.3600i			1.52953i
$768$		0			0
$769$			46.4643i			1.67554i	0.546019	+	0.837772i	$0.316142\pi$
$769$			46.4643i			1.67554i	−0.546019	+	0.837772i	$0.683858\pi$
$770$	12.5480	+	8.84373i	0.452198	+	0.318706i
$771$		0			0
$772$	−0.635749			−0.0228811
$773$			30.6420i			1.10212i	0.834467	+	0.551058i	$0.185776\pi$
$773$			30.6420i			1.10212i	−0.834467	+	0.551058i	$0.814224\pi$
$774$		0			0
$775$	−16.1644	+	5.77121i	−0.580642	+	0.207308i
$776$	16.6814			0.598826
$777$		0			0
$778$			19.6417i			0.704190i
$779$		−	5.45317i		−	0.195380i
$780$		0			0
$781$	67.8999			2.42965
$782$	2.28569			0.0817361
$783$		0			0
$784$	5.42377			0.193706
$785$	−26.0081	−	18.3303i	−0.928270	−	0.654238i
$786$		0			0
$787$			8.65409i			0.308485i	0.988033	+	0.154243i	$0.0492937\pi$
$787$			8.65409i			0.308485i	−0.988033	+	0.154243i	$0.950706\pi$
$788$		−	5.18937i		−	0.184864i
$789$		0			0
$790$	−1.84036	−	1.29707i	−0.0654772	−	0.0461478i
$791$		−	5.93567i		−	0.211048i
$792$		0			0
$793$		−	29.2511i		−	1.03874i
$794$		−	4.31480i		−	0.153127i
$795$		0			0
$796$			23.8852i			0.846589i
$797$	25.1436			0.890634			0.445317	−	0.895373i	$-0.353091\pi$
$797$	25.1436			0.890634			0.445317	+	0.895373i	$0.353091\pi$
$798$		0			0
$799$			2.27046i			0.0803232i
$800$	−1.68122	−	4.70887i	−0.0594401	−	0.166484i
$801$		0			0
$802$		−	9.64980i		−	0.340746i
$803$		−	56.2328i		−	1.98441i
$804$		0			0
$805$	−4.39609	−	3.09833i	−0.154942	−	0.109202i
$806$		−	16.1644i		−	0.569367i
$807$		0			0
$808$	−3.26214			−0.114762
$809$		−	54.4206i		−	1.91333i	−0.291199	−	0.956663i	$-0.594054\pi$
$809$		−	54.4206i		−	1.91333i	0.291199	−	0.956663i	$-0.405946\pi$
$810$		0			0
$811$	32.8378			1.15309			0.576546	−	0.817065i	$-0.304400\pi$
$811$	32.8378			1.15309			0.576546	+	0.817065i	$0.304400\pi$
$812$	6.38816			0.224180
$813$		0			0
$814$	−25.0706	−	21.8596i	−0.878726	−	0.766180i
$815$	−0.586588	+	0.832284i	−0.0205473	+	0.0291536i
$816$		0			0
$817$		−	1.64051i		−	0.0573941i
$818$		−	25.5235i		−	0.892409i
$819$		0			0
$820$	−5.74945	+	8.15764i	−0.200779	+	0.284877i
$821$	−25.1041			−0.876138			−0.438069	−	0.898941i	$-0.644337\pi$
$821$	−25.1041			−0.876138			−0.438069	+	0.898941i	$0.644337\pi$
$822$		0			0
$823$			11.1202i			0.387627i	0.981038	+	0.193814i	$0.0620857\pi$
$823$			11.1202i			0.387627i	−0.981038	+	0.193814i	$0.937914\pi$
$824$	5.19330			0.180917
$825$		0			0
$826$	11.2940			0.392969
$827$	−7.01483			−0.243930			−0.121965	−	0.992534i	$-0.538920\pi$
$827$	−7.01483			−0.243930			−0.121965	+	0.992534i	$0.538920\pi$
$828$		0			0
$829$		−	12.5028i		−	0.434240i	−0.976145	−	0.217120i	$-0.930334\pi$
$829$		−	12.5028i		−	0.434240i	0.976145	−	0.217120i	$-0.0696662\pi$
$830$	−1.37901	−	0.971918i	−0.0478662	−	0.0337358i
$831$		0			0
$832$	4.70887			0.163251
$833$	−6.47105			−0.224209
$834$		0			0
$835$	0.973540	−	1.38131i	0.0336907	−	0.0478023i
$836$		−	6.68111i		−	0.231071i
$837$		0			0
$838$	22.0969			0.763324
$839$	−44.6267			−1.54069			−0.770343	−	0.637630i	$-0.779915\pi$
$839$	−44.6267			−1.54069			−0.770343	+	0.637630i	$0.779915\pi$
$840$		0			0
$841$	3.11004			0.107243
$842$		−	8.20101i		−	0.282626i
$843$		0			0
$844$	−26.6260			−0.916505
$845$	−11.8171	+	16.7667i	−0.406519	+	0.576793i
$846$		0			0
$847$		−	23.7313i		−	0.815418i
$848$			1.11671i			0.0383479i
$849$		0			0
$850$	2.00585	+	5.61812i	0.0688000	+	0.192700i
$851$	8.78330	+	7.65835i	0.301088	+	0.262525i
$852$		0			0
$853$	3.59757			0.123178			0.0615892	−	0.998102i	$-0.480383\pi$
$853$	3.59757			0.123178			0.0615892	+	0.998102i	$0.480383\pi$
$854$	−7.79891			−0.266873
$855$		0			0
$856$			8.65950i			0.295976i
$857$	−7.66263			−0.261750			−0.130875	−	0.991399i	$-0.541779\pi$
$857$	−7.66263			−0.261750			−0.130875	+	0.991399i	$0.541779\pi$
$858$		0			0
$859$		−	56.3900i		−	1.92400i	−0.273046	−	0.962001i	$-0.588031\pi$
$859$		−	56.3900i		−	1.92400i	0.273046	−	0.962001i	$-0.411969\pi$
$860$	−1.72964	+	2.45411i	−0.0589802	+	0.0836844i
$861$		0			0
$862$		−	9.83076i		−	0.334837i
$863$		−	12.7800i		−	0.435037i	−0.976056	−	0.217518i	$-0.930204\pi$
$863$		−	12.7800i		−	0.435037i	0.976056	−	0.217518i	$-0.0697962\pi$
$864$		0			0
$865$	8.06716	+	5.68568i	0.274292	+	0.193319i
$866$			1.61024i			0.0547182i
$867$		0			0
$868$	−4.30975			−0.146282
$869$			5.50607i			0.186781i
$870$		0			0
$871$			19.8299i			0.671910i
$872$			2.54677i			0.0862444i
$873$		0			0
$874$			2.34067i			0.0791744i
$875$	3.75768	−	13.5244i	0.127033	−	0.457207i
$876$		0			0
$877$		−	40.4607i		−	1.36626i	−0.730297	−	0.683130i	$-0.760618\pi$
$877$		−	40.4607i		−	1.36626i	0.730297	−	0.683130i	$-0.239382\pi$
$878$			32.2443i			1.08819i
$879$		0			0
$880$	−7.04410	+	9.99458i	−0.237457	+	0.336917i
$881$	48.2664			1.62614			0.813069	−	0.582168i	$-0.197795\pi$
$881$	48.2664			1.62614			0.813069	+	0.582168i	$0.197795\pi$
$882$		0			0
$883$	−7.33442			−0.246823			−0.123411	−	0.992356i	$-0.539383\pi$
$883$	−7.33442			−0.246823			−0.123411	+	0.992356i	$0.539383\pi$
$884$	−5.61812			−0.188958
$885$		0			0
$886$		−	19.0730i		−	0.640770i
$887$		−	28.5353i		−	0.958120i	−0.877782	−	0.479060i	$-0.840978\pi$
$887$		−	28.5353i		−	0.958120i	0.877782	−	0.479060i	$-0.159022\pi$
$888$		0			0
$889$	10.1075			0.338993
$890$	−20.9720	−	14.7809i	−0.702983	−	0.495457i
$891$		0			0
$892$		−	20.0982i		−	0.672938i
$893$	−2.32508			−0.0778059
$894$		0			0
$895$	23.1908	+	16.3447i	0.775184	+	0.546344i
$896$		−	1.25548i		−	0.0419426i
$897$		0			0
$898$		−	30.6914i		−	1.02418i
$899$	17.4666			0.582543
$900$		0			0
$901$		−	1.33233i		−	0.0443865i
$902$	24.4064			0.812643
$903$		0			0
$904$	4.72781			0.157245
$905$	−17.0610	+	24.2072i	−0.567128	+	0.804673i
$906$		0			0
$907$	−31.6207			−1.04995			−0.524974	−	0.851118i	$-0.675925\pi$
$907$	−31.6207			−1.04995			−0.524974	+	0.851118i	$0.675925\pi$
$908$	−15.1621			−0.503173
$909$		0			0
$910$	10.8054	+	7.61555i	0.358195	+	0.252453i
$911$			2.67057i			0.0884800i	0.999021	+	0.0442400i	$0.0140866\pi$
$911$			2.67057i			0.0884800i	−0.999021	+	0.0442400i	$0.985913\pi$
$912$		0			0
$913$			4.12579i			0.136544i
$914$	22.4560			0.742777
$915$		0			0
$916$	−2.54527			−0.0840980
$917$	16.3212			0.538972
$918$		0			0
$919$			40.0514i			1.32117i	0.750750	+	0.660587i	$0.229692\pi$
$919$			40.0514i			1.32117i	−0.750750	+	0.660587i	$0.770308\pi$
$920$	2.46784	−	3.50152i	0.0813624	−	0.115442i
$921$		0			0
$922$			7.37800i			0.242981i
$923$	58.4703			1.92457
$924$		0			0
$925$	−11.1159	+	28.3097i	−0.365489	+	0.930816i
$926$	−6.16145			−0.202478
$927$		0			0
$928$			5.08822i			0.167029i
$929$	−47.8279			−1.56918			−0.784592	−	0.620013i	$-0.787127\pi$
$929$	−47.8279			−1.56918			−0.784592	+	0.620013i	$0.787127\pi$
$930$		0			0
$931$		−	6.62671i		−	0.217182i
$932$			13.9212i			0.456004i
$933$		0			0
$934$	−5.37297			−0.175809
$935$	8.40426	−	11.9244i	0.274849	−	0.389971i
$936$		0			0
$937$			44.7707i			1.46260i	0.682058	+	0.731298i	$0.261085\pi$
$937$			44.7707i			1.46260i	−0.682058	+	0.731298i	$0.738915\pi$
$938$	5.28704			0.172628
$939$		0			0
$940$	3.47820	+	2.45141i	0.113446	+	0.0799561i
$941$	−38.1052			−1.24219			−0.621097	−	0.783734i	$-0.713313\pi$
$941$	−38.1052			−1.24219			−0.621097	+	0.783734i	$0.713313\pi$
$942$		0			0
$943$	−8.55058			−0.278445
$944$			8.99577i			0.292787i
$945$		0			0
$946$	7.34231			0.238719
$947$	−53.7661			−1.74716			−0.873582	−	0.486678i	$-0.838209\pi$
$947$	−53.7661			−1.74716			−0.873582	+	0.486678i	$0.838209\pi$
$948$		0			0
$949$		−	48.4234i		−	1.57189i
$950$	−5.75326	+	2.05410i	−0.186660	+	0.0666438i
$951$		0			0
$952$			1.49790i			0.0485473i
$953$		−	12.7882i		−	0.414252i	−0.978314	−	0.207126i	$-0.933589\pi$
$953$		−	12.7882i		−	0.414252i	0.978314	−	0.207126i	$-0.0664109\pi$
$954$		0			0
$955$	−12.8671	−	9.06861i	−0.416368	−	0.293453i
$956$		−	28.8753i		−	0.933895i
$957$		0			0
$958$			21.5810i			0.697251i
$959$	−17.0101			−0.549286
$960$		0			0
$961$	19.2162			0.619878
$962$	−21.5889	−	18.8239i	−0.696055	−	0.606906i
$963$		0			0
$964$		−	6.84853i		−	0.220576i
$965$	0.818954	−	1.16198i	0.0263631	−	0.0374054i
$966$		0			0
$967$	−49.4979			−1.59175			−0.795873	−	0.605463i	$-0.792988\pi$
$967$	−49.4979			−1.59175			−0.795873	+	0.605463i	$0.792988\pi$
$968$	18.9022			0.607540
$969$		0			0
$970$	−21.4885	+	30.4891i	−0.689954	+	0.978946i
$971$	22.7751			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$971$	22.7751			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$972$		0			0
$973$		−	19.2411i		−	0.616843i
$974$	−0.346315			−0.0110967
$975$		0			0
$976$		−	6.21190i		−	0.198838i
$977$	−7.08386			−0.226633			−0.113316	−	0.993559i	$-0.536147\pi$
$977$	−7.08386			−0.226633			−0.113316	+	0.993559i	$0.536147\pi$
$978$		0			0
$979$			62.7449i			2.00534i
$980$	−6.98676	+	9.91321i	−0.223184	+	0.316666i
$981$		0			0
$982$	−10.6278			−0.339146
$983$		−	50.8277i		−	1.62115i	−0.585634	−	0.810576i	$-0.699154\pi$
$983$		−	50.8277i		−	1.62115i	0.585634	−	0.810576i	$-0.300846\pi$
$984$		0			0
$985$	9.48478	+	6.68481i	0.302210	+	0.212996i
$986$		−	6.07071i		−	0.193331i
$987$		0			0
$988$		−	5.75326i		−	0.183036i
$989$	−2.57232			−0.0817949
$990$		0			0
$991$		−	42.8852i		−	1.36229i	−0.732146	−	0.681147i	$-0.761481\pi$
$991$		−	42.8852i		−	1.36229i	0.732146	−	0.681147i	$-0.238519\pi$
$992$		−	3.43275i		−	0.108990i
$993$		0			0
$994$		−	15.5893i		−	0.494464i
$995$	−43.6558	−	30.7683i	−1.38398	−	0.975421i
$996$		0			0
$997$	6.15244			0.194850			0.0974248	−	0.995243i	$-0.468939\pi$
$997$	6.15244			0.194850			0.0974248	+	0.995243i	$0.468939\pi$
$998$		−	12.0121i		−	0.380237i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.e.f.739.6		16
3.2	odd	2		1110.2.e.c.739.14	yes	16
5.4	even	2		3330.2.e.e.739.12		16
15.14	odd	2		1110.2.e.d.739.3	yes	16
37.36	even	2		3330.2.e.e.739.11		16
111.110	odd	2		1110.2.e.d.739.11	yes	16
185.184	even	2	inner	3330.2.e.f.739.5		16
555.554	odd	2		1110.2.e.c.739.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.e.c.739.6	✓	16	555.554	odd	2
1110.2.e.c.739.14	yes	16	3.2	odd	2
1110.2.e.d.739.3	yes	16	15.14	odd	2
1110.2.e.d.739.11	yes	16	111.110	odd	2
3330.2.e.e.739.11		16	37.36	even	2
3330.2.e.e.739.12		16	5.4	even	2
3330.2.e.f.739.5		16	185.184	even	2	inner
3330.2.e.f.739.6		16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(-1.28817 + 1.82773i) q^{5} -1.25548i q^{7} +1.00000 q^{8} +(-1.28817 + 1.82773i) q^{10} +5.46829 q^{11} +4.70887 q^{13} -1.25548i q^{14} +1.00000 q^{16} -1.19309 q^{17} -1.22179i q^{19} +(-1.28817 + 1.82773i) q^{20} +5.46829 q^{22} -1.91577 q^{23} +(-1.68122 - 4.70887i) q^{25} +4.70887 q^{26} -1.25548i q^{28} +5.08822i q^{29} -3.43275i q^{31} +1.00000 q^{32} -1.19309 q^{34} +(2.29468 + 1.61728i) q^{35} +(-4.58473 - 3.99753i) q^{37} -1.22179i q^{38} +(-1.28817 + 1.82773i) q^{40} +4.46326 q^{41} +1.34271 q^{43} +5.46829 q^{44} -1.91577 q^{46} -1.90301i q^{47} +5.42377 q^{49} +(-1.68122 - 4.70887i) q^{50} +4.70887 q^{52} +1.11671i q^{53} +(-7.04410 + 9.99458i) q^{55} -1.25548i q^{56} +5.08822i q^{58} +8.99577i q^{59} -6.21190i q^{61} -3.43275i q^{62} +1.00000 q^{64} +(-6.06585 + 8.60657i) q^{65} +4.21117i q^{67} -1.19309 q^{68} +(2.29468 + 1.61728i) q^{70} +12.4170 q^{71} -10.2834i q^{73} +(-4.58473 - 3.99753i) q^{74} -1.22179i q^{76} -6.86533i q^{77} +1.00691i q^{79} +(-1.28817 + 1.82773i) q^{80} +4.46326 q^{82} +0.754493i q^{83} +(1.53691 - 2.18065i) q^{85} +1.34271 q^{86} +5.46829 q^{88} +11.4743i q^{89} -5.91190i q^{91} -1.91577 q^{92} -1.90301i q^{94} +(2.23311 + 1.57388i) q^{95} +16.6814 q^{97} +5.42377 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 2 q^{5} + 16 q^{8} + 2 q^{10} - 2 q^{11} + 16 q^{16} + 38 q^{17} + 2 q^{20} - 2 q^{22} + 20 q^{23} - 4 q^{25} + 16 q^{32} + 38 q^{34} + 10 q^{35} - 4 q^{37} + 2 q^{40} + 6 q^{41}+ \cdots - 18 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 2 * q^5 + 16 * q^8 + 2 * q^10 - 2 * q^11 + 16 * q^16 + 38 * q^17 + 2 * q^20 - 2 * q^22 + 20 * q^23 - 4 * q^25 + 16 * q^32 + 38 * q^34 + 10 * q^35 - 4 * q^37 + 2 * q^40 + 6 * q^41 - 2 * q^43 - 2 * q^44 + 20 * q^46 - 18 * q^49 - 4 * q^50 + 20 * q^55 + 16 * q^64 - 12 * q^65 + 38 * q^68 + 10 * q^70 + 24 * q^71 - 4 * q^74 + 2 * q^80 + 6 * q^82 - 2 * q^86 - 2 * q^88 + 20 * q^92 + 16 * q^95 + 38 * q^97 - 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more