LMFDB - Embedded newform 3456.2.i.l.2305.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1153,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("3456.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3456 = 2^{7} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3456.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.5962989386$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} + \cdots + 729 $ x^12 - 2x^11 + 3x^10 - 8x^9 + 22x^8 - 42x^7 + 51x^6 - 126x^5 + 198x^4 - 216x^3 + 243x^2 - 486*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2305.4
Root			$-1.28252 - 1.16410i$ of defining polynomial
Character	$\chi$	$=$	3456.2305
Dual form			3456.2.i.l.1153.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.05471 - 1.82681i) q^{5} +(-1.43914 - 2.49267i) q^{7} +O(q^{10})$	$q+(1.05471 - 1.82681i) q^{5} +(-1.43914 - 2.49267i) q^{7} +(1.21325 + 2.10141i) q^{11} +(3.30008 - 5.71590i) q^{13} +7.56848 q^{17} +6.25779 q^{19} +(-2.63611 + 4.56587i) q^{23} +(0.275172 + 0.476612i) q^{25} +(1.57821 + 2.73353i) q^{29} +(1.79039 - 3.10104i) q^{31} -6.07151 q^{35} -6.70957 q^{37} +(1.74537 - 3.02306i) q^{41} +(3.12570 + 5.41388i) q^{43} +(1.32972 + 2.30314i) q^{47} +(-0.642255 + 1.11242i) q^{49} -0.953009 q^{53} +5.11850 q^{55} +(-4.84757 + 8.39624i) q^{59} +(2.57821 + 4.46558i) q^{61} +(-6.96125 - 12.0572i) q^{65} +(0.949546 - 1.64466i) q^{67} +5.82491 q^{71} -5.01222 q^{73} +(3.49207 - 6.04844i) q^{77} +(-6.49996 - 11.2583i) q^{79} +(-1.54502 - 2.67606i) q^{83} +(7.98256 - 13.8262i) q^{85} +2.95301 q^{89} -18.9971 q^{91} +(6.60015 - 11.4318i) q^{95} +(-5.51242 - 9.54779i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{5} + 6 q^{7}+O(q^{10}) $ 12 * q + 2 * q^5 + 6 * q^7	$ 12 q + 2 q^{5} + 6 q^{7} - 4 q^{11} + 10 q^{13} - 4 q^{17} + 4 q^{19} - 8 q^{23} - 14 q^{25} + 2 q^{29} + 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} + 14 q^{47} - 18 q^{49} - 24 q^{53} - 16 q^{55} - 6 q^{59} + 14 q^{61} + 8 q^{65} + 4 q^{67} + 28 q^{71} + 60 q^{73} - 2 q^{77} + 16 q^{79} - 24 q^{83} + 16 q^{85} + 48 q^{89} - 52 q^{91} + 20 q^{95} - 14 q^{97}+O(q^{100}) $ 12 * q + 2 * q^5 + 6 * q^7 - 4 * q^11 + 10 * q^13 - 4 * q^17 + 4 * q^19 - 8 * q^23 - 14 * q^25 + 2 * q^29 + 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 + 14 * q^47 - 18 * q^49 - 24 * q^53 - 16 * q^55 - 6 * q^59 + 14 * q^61 + 8 * q^65 + 4 * q^67 + 28 * q^71 + 60 * q^73 - 2 * q^77 + 16 * q^79 - 24 * q^83 + 16 * q^85 + 48 * q^89 - 52 * q^91 + 20 * q^95 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$.

$n$	$2053$	$2431$	$2945$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.05471	−	1.82681i	0.471681	−	0.816975i	−0.527794	−	0.849372i	$-0.676981\pi$
$5$	1.05471	−	1.82681i	0.471681	−	0.816975i	0.999475	+	0.0323971i	$0.0103141\pi$
$6$		0			0
$7$	−1.43914	−	2.49267i	−0.543944	−	0.942139i	−0.998673	−	0.0515089i	$-0.983597\pi$
$7$	−1.43914	−	2.49267i	−0.543944	−	0.942139i	0.454728	−	0.890630i	$-0.349736\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.21325	+	2.10141i	0.365808	+	0.633598i	0.988905	−	0.148546i	$-0.0474594\pi$
$11$	1.21325	+	2.10141i	0.365808	+	0.633598i	−0.623097	+	0.782144i	$0.714126\pi$
$12$		0			0
$13$	3.30008	−	5.71590i	0.915277	−	1.58531i	0.108781	−	0.994066i	$-0.465305\pi$
$13$	3.30008	−	5.71590i	0.915277	−	1.58531i	0.806496	−	0.591240i	$-0.201361\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.56848			1.83563			0.917813	−	0.397013i	$-0.129953\pi$
$17$	7.56848			1.83563			0.917813	+	0.397013i	$0.129953\pi$
$18$		0			0
$19$	6.25779			1.43563			0.717817	−	0.696231i	$-0.245141\pi$
$19$	6.25779			1.43563			0.717817	+	0.696231i	$0.245141\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.63611	+	4.56587i	−0.549666	+	0.952050i	0.448631	+	0.893717i	$0.351912\pi$
$23$	−2.63611	+	4.56587i	−0.549666	+	0.952050i	−0.998297	+	0.0583329i	$0.981422\pi$
$24$		0			0
$25$	0.275172	+	0.476612i	0.0550344	+	0.0953223i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.57821	+	2.73353i	0.293065	+	0.507604i	0.974533	−	0.224244i	$-0.0719913\pi$
$29$	1.57821	+	2.73353i	0.293065	+	0.507604i	−0.681468	+	0.731848i	$0.738658\pi$
$30$		0			0
$31$	1.79039	−	3.10104i	0.321563	−	0.556964i	−0.659248	−	0.751926i	$-0.729125\pi$
$31$	1.79039	−	3.10104i	0.321563	−	0.556964i	0.980811	+	0.194962i	$0.0624584\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.07151			−1.02627
$36$		0			0
$37$	−6.70957			−1.10305			−0.551524	−	0.834159i	$-0.685953\pi$
$37$	−6.70957			−1.10305			−0.551524	+	0.834159i	$0.685953\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.74537	−	3.02306i	0.272580	−	0.472123i	−0.696941	−	0.717128i	$-0.745456\pi$
$41$	1.74537	−	3.02306i	0.272580	−	0.472123i	0.969522	+	0.245005i	$0.0787896\pi$
$42$		0			0
$43$	3.12570	+	5.41388i	0.476665	+	0.825608i	0.999642	−	0.0267383i	$-0.00851207\pi$
$43$	3.12570	+	5.41388i	0.476665	+	0.825608i	−0.522977	+	0.852347i	$0.675179\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.32972	+	2.30314i	0.193960	+	0.335948i	0.946559	−	0.322531i	$-0.104534\pi$
$47$	1.32972	+	2.30314i	0.193960	+	0.335948i	−0.752599	+	0.658479i	$0.771200\pi$
$48$		0			0
$49$	−0.642255	+	1.11242i	−0.0917508	+	0.158917i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.953009			−0.130906			−0.0654529	−	0.997856i	$-0.520849\pi$
$53$	−0.953009			−0.130906			−0.0654529	+	0.997856i	$0.520849\pi$
$54$		0			0
$55$	5.11850			0.690178
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.84757	+	8.39624i	−0.631100	+	1.09310i	0.356227	+	0.934400i	$0.384063\pi$
$59$	−4.84757	+	8.39624i	−0.631100	+	1.09310i	−0.987327	+	0.158698i	$0.949270\pi$
$60$		0			0
$61$	2.57821	+	4.46558i	0.330105	+	0.571759i	0.982532	−	0.186092i	$-0.0595824\pi$
$61$	2.57821	+	4.46558i	0.330105	+	0.571759i	−0.652427	+	0.757852i	$0.726249\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.96125	−	12.0572i	−0.863437	−	1.49552i
$66$		0			0
$67$	0.949546	−	1.64466i	0.116005	−	0.200927i	−0.802176	−	0.597088i	$-0.796324\pi$
$67$	0.949546	−	1.64466i	0.116005	−	0.200927i	0.918181	+	0.396161i	$0.129658\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.82491			0.691290			0.345645	−	0.938365i	$-0.387660\pi$
$71$	5.82491			0.691290			0.345645	+	0.938365i	$0.387660\pi$
$72$		0			0
$73$	−5.01222			−0.586636			−0.293318	−	0.956015i	$-0.594760\pi$
$73$	−5.01222			−0.586636			−0.293318	+	0.956015i	$0.594760\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.49207	−	6.04844i	0.397958	−	0.689284i
$78$		0			0
$79$	−6.49996	−	11.2583i	−0.731303	−	1.26665i	−0.956327	−	0.292301i	$-0.905579\pi$
$79$	−6.49996	−	11.2583i	−0.731303	−	1.26665i	0.225024	−	0.974353i	$-0.427754\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.54502	−	2.67606i	−0.169588	−	0.293735i	0.768687	−	0.639625i	$-0.220910\pi$
$83$	−1.54502	−	2.67606i	−0.169588	−	0.293735i	−0.938275	+	0.345890i	$0.887577\pi$
$84$		0			0
$85$	7.98256	−	13.8262i	0.865830	−	1.49966i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.95301			0.313018			0.156509	−	0.987677i	$-0.449976\pi$
$89$	2.95301			0.313018			0.156509	+	0.987677i	$0.449976\pi$
$90$		0			0
$91$	−18.9971			−1.99144
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.60015	−	11.4318i	0.677161	−	1.17288i
$96$		0			0
$97$	−5.51242	−	9.54779i	−0.559702	−	0.969432i	−0.997521	−	0.0703686i	$-0.977582\pi$
$97$	−5.51242	−	9.54779i	−0.559702	−	0.969432i	0.437820	−	0.899063i	$-0.355751\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.82357	−	3.15852i	−0.181452	−	0.314284i	0.760923	−	0.648842i	$-0.224746\pi$
$101$	−1.82357	−	3.15852i	−0.181452	−	0.314284i	−0.942375	+	0.334558i	$0.891413\pi$
$102$		0			0
$103$	5.43914	−	9.42087i	0.535935	−	0.928266i	−0.463183	−	0.886263i	$-0.653293\pi$
$103$	5.43914	−	9.42087i	0.535935	−	0.928266i	0.999117	−	0.0420031i	$-0.0133739\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.94922			0.575133			0.287567	−	0.957761i	$-0.407154\pi$
$107$	5.94922			0.575133			0.287567	+	0.957761i	$0.407154\pi$
$108$		0			0
$109$	9.87113			0.945483			0.472741	−	0.881201i	$-0.343265\pi$
$109$	9.87113			0.945483			0.472741	+	0.881201i	$0.343265\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.51518	−	13.0167i	0.706969	−	1.22451i	−0.259008	−	0.965875i	$-0.583396\pi$
$113$	7.51518	−	13.0167i	0.706969	−	1.22451i	0.965976	−	0.258630i	$-0.0832711\pi$
$114$		0			0
$115$	5.56066	+	9.63134i	0.518534	+	0.898128i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−10.8921	−	18.8657i	−0.998478	−	1.72942i
$120$		0			0
$121$	2.55606	−	4.42722i	0.232369	−	0.402475i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.7080			1.04720
$126$		0			0
$127$	−8.07789			−0.716797			−0.358398	−	0.933569i	$-0.616677\pi$
$127$	−8.07789			−0.716797			−0.358398	+	0.933569i	$0.616677\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.1329	+	19.2827i	−0.972684	+	1.68474i	−0.285309	+	0.958435i	$0.592096\pi$
$131$	−11.1329	+	19.2827i	−0.972684	+	1.68474i	−0.687375	+	0.726303i	$0.741237\pi$
$132$		0			0
$133$	−9.00584	−	15.5986i	−0.780905	−	1.35257i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.62365	+	8.00839i	0.395025	+	0.684203i	0.993104	−	0.117234i	$-0.0374026\pi$
$137$	4.62365	+	8.00839i	0.395025	+	0.684203i	−0.598079	+	0.801437i	$0.704069\pi$
$138$		0			0
$139$	4.20256	−	7.27905i	0.356456	−	0.617401i	−0.630910	−	0.775856i	$-0.717318\pi$
$139$	4.20256	−	7.27905i	0.356456	−	0.617401i	0.987366	+	0.158456i	$0.0506515\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.0152			1.33926
$144$		0			0
$145$	6.65820			0.552934
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.17836	+	10.7012i	−0.506151	+	0.876679i	0.493824	+	0.869562i	$0.335599\pi$
$149$	−6.17836	+	10.7012i	−0.506151	+	0.876679i	−0.999975	+	0.00711709i	$0.997735\pi$
$150$		0			0
$151$	4.91424	+	8.51171i	0.399915	+	0.692673i	0.993715	−	0.111940i	$-0.0357063\pi$
$151$	4.91424	+	8.51171i	0.399915	+	0.692673i	−0.593800	+	0.804613i	$0.702373\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.77668	−	6.54141i	−0.303350	−	0.525418i
$156$		0			0
$157$	−0.108083	+	0.187206i	−0.00862598	+	0.0149406i	−0.870306	−	0.492511i	$-0.836079\pi$
$157$	−0.108083	+	0.187206i	−0.00862598	+	0.0149406i	0.861680	+	0.507452i	$0.169412\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.1749			1.19595
$162$		0			0
$163$	−5.78116			−0.452815			−0.226408	−	0.974033i	$-0.572698\pi$
$163$	−5.78116			−0.452815			−0.226408	+	0.974033i	$0.572698\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.818646	−	1.41794i	0.0633487	−	0.109723i	−0.832612	−	0.553857i	$-0.813155\pi$
$167$	0.818646	−	1.41794i	0.0633487	−	0.109723i	0.895960	+	0.444134i	$0.146489\pi$
$168$		0			0
$169$	−15.2810	−	26.4675i	−1.17546	−	2.03596i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.99228	+	5.18278i	0.227499	+	0.394040i	0.957066	−	0.289869i	$-0.0936119\pi$
$173$	2.99228	+	5.18278i	0.227499	+	0.394040i	−0.729567	+	0.683909i	$0.760279\pi$
$174$		0			0
$175$	0.792022	−	1.37182i	0.0598713	−	0.103700i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.51313			−0.711045			−0.355522	−	0.934668i	$-0.615697\pi$
$179$	−9.51313			−0.711045			−0.355522	+	0.934668i	$0.615697\pi$
$180$		0			0
$181$	−23.7526			−1.76551			−0.882757	−	0.469830i	$-0.844315\pi$
$181$	−23.7526			−1.76551			−0.882757	+	0.469830i	$0.844315\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.07666	+	12.2571i	−0.520286	+	0.901162i
$186$		0			0
$187$	9.18244	+	15.9045i	0.671487	+	1.16305i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.14164	+	7.17352i	0.299678	+	0.519058i	0.976062	−	0.217491i	$-0.0697873\pi$
$191$	4.14164	+	7.17352i	0.299678	+	0.519058i	−0.676384	+	0.736549i	$0.736454\pi$
$192$		0			0
$193$	5.00315	−	8.66572i	0.360135	−	0.623772i	−0.627848	−	0.778336i	$-0.716064\pi$
$193$	5.00315	−	8.66572i	0.360135	−	0.623772i	0.987983	+	0.154564i	$0.0493973\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.556259			−0.0396318			−0.0198159	−	0.999804i	$-0.506308\pi$
$197$	−0.556259			−0.0396318			−0.0198159	+	0.999804i	$0.506308\pi$
$198$		0			0
$199$	−21.5526			−1.52782			−0.763912	−	0.645320i	$-0.776724\pi$
$199$	−21.5526			−1.52782			−0.763912	+	0.645320i	$0.776724\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.54252	−	7.86788i	0.318823	−	0.552217i
$204$		0			0
$205$	−3.68171	−	6.37691i	−0.257142	−	0.445383i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.59225	+	13.1502i	0.525167	+	0.909615i
$210$		0			0
$211$	−8.32984	+	14.4277i	−0.573450	+	0.993245i	0.422758	+	0.906243i	$0.361062\pi$
$211$	−8.32984	+	14.4277i	−0.573450	+	0.993245i	−0.996208	+	0.0870022i	$0.972271\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13.1868			0.899335
$216$		0			0
$217$	−10.3065			−0.699650
$218$		0			0
$219$		0			0
$220$		0			0
$221$	24.9766	−	43.2607i	1.68011	−	2.91003i
$222$		0			0
$223$	4.49251	+	7.78126i	0.300841	+	0.521072i	0.976327	−	0.216301i	$-0.0693993\pi$
$223$	4.49251	+	7.78126i	0.300841	+	0.521072i	−0.675486	+	0.737373i	$0.736066\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.32448	−	9.22226i	−0.353398	−	0.612103i	0.633445	−	0.773788i	$-0.281640\pi$
$227$	−5.32448	−	9.22226i	−0.353398	−	0.612103i	−0.986842	+	0.161685i	$0.948307\pi$
$228$		0			0
$229$	−5.50786	+	9.53988i	−0.363969	+	0.630413i	−0.988610	−	0.150499i	$-0.951912\pi$
$229$	−5.50786	+	9.53988i	−0.363969	+	0.630413i	0.624641	+	0.780912i	$0.285245\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−27.2595			−1.78583			−0.892915	−	0.450225i	$-0.851344\pi$
$233$	−27.2595			−1.78583			−0.892915	+	0.450225i	$0.851344\pi$
$234$		0			0
$235$	5.60988			0.365948
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.82295	−	10.0856i	0.376655	−	0.652386i	−0.613918	−	0.789370i	$-0.710407\pi$
$239$	5.82295	−	10.0856i	0.376655	−	0.652386i	0.990573	+	0.136984i	$0.0437407\pi$
$240$		0			0
$241$	9.83554	+	17.0356i	0.633563	+	1.09736i	0.986818	+	0.161835i	$0.0517414\pi$
$241$	9.83554	+	17.0356i	0.633563	+	1.09736i	−0.353255	+	0.935527i	$0.614925\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.35479	+	2.34656i	0.0865542	+	0.149916i
$246$		0			0
$247$	20.6512	−	35.7689i	1.31400	−	2.27592i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−23.3802			−1.47575			−0.737873	−	0.674940i	$-0.764170\pi$
$251$	−23.3802			−1.47575			−0.737873	+	0.674940i	$0.764170\pi$
$252$		0			0
$253$	−12.7930			−0.804289
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.89037	−	3.27421i	0.117918	−	0.204240i	−0.801024	−	0.598632i	$-0.795711\pi$
$257$	1.89037	−	3.27421i	0.117918	−	0.204240i	0.918942	+	0.394392i	$0.129045\pi$
$258$		0			0
$259$	9.65603	+	16.7247i	0.599996	+	1.03922i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.3638	+	17.9507i	0.639060	+	1.10689i	0.985639	+	0.168864i	$0.0540100\pi$
$263$	10.3638	+	17.9507i	0.639060	+	1.10689i	−0.346579	+	0.938021i	$0.612657\pi$
$264$		0			0
$265$	−1.00515	+	1.74097i	−0.0617458	+	0.106947i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.965772			−0.0588842			−0.0294421	−	0.999566i	$-0.509373\pi$
$269$	−0.965772			−0.0588842			−0.0294421	+	0.999566i	$0.509373\pi$
$270$		0			0
$271$	−4.28219			−0.260124			−0.130062	−	0.991506i	$-0.541518\pi$
$271$	−4.28219			−0.260124			−0.130062	+	0.991506i	$0.541518\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.667703	+	1.15650i	−0.0402640	+	0.0697393i
$276$		0			0
$277$	4.83619	+	8.37653i	0.290579	+	0.503297i	0.973947	−	0.226777i	$-0.0728189\pi$
$277$	4.83619	+	8.37653i	0.290579	+	0.503297i	−0.683368	+	0.730074i	$0.739486\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.34848	−	12.7279i	−0.438373	−	0.759285i	0.559191	−	0.829039i	$-0.311112\pi$
$281$	−7.34848	−	12.7279i	−0.438373	−	0.759285i	−0.997564	+	0.0697540i	$0.977779\pi$
$282$		0			0
$283$	−7.90174	+	13.6862i	−0.469710	+	0.813561i	−0.999400	−	0.0346299i	$-0.988975\pi$
$283$	−7.90174	+	13.6862i	−0.469710	+	0.813561i	0.529690	+	0.848191i	$0.322308\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−10.0473			−0.593074
$288$		0			0
$289$	40.2819			2.36952
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.99866	−	5.19384i	0.175184	−	0.303427i	−0.765041	−	0.643981i	$-0.777281\pi$
$293$	2.99866	−	5.19384i	0.175184	−	0.303427i	0.940225	+	0.340554i	$0.110615\pi$
$294$		0			0
$295$	10.2256	+	17.7112i	0.595356	+	1.03119i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	17.3987	+	30.1355i	1.00619	+	1.74278i
$300$		0			0
$301$	8.99666	−	15.5827i	0.518559	−	0.898170i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.8770			0.622818
$306$		0			0
$307$	−11.7568			−0.670994			−0.335497	−	0.942041i	$-0.608904\pi$
$307$	−11.7568			−0.670994			−0.335497	+	0.942041i	$0.608904\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	16.1797	−	28.0240i	0.917465	−	1.58910i	0.114213	−	0.993456i	$-0.463565\pi$
$311$	16.1797	−	28.0240i	0.917465	−	1.58910i	0.803252	−	0.595640i	$-0.203101\pi$
$312$		0			0
$313$	−14.6062	−	25.2987i	−0.825591	−	1.42997i	−0.901466	−	0.432849i	$-0.857508\pi$
$313$	−14.6062	−	25.2987i	−0.825591	−	1.42997i	0.0758750	−	0.997117i	$-0.475825\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.7011	−	23.7310i	−0.769530	−	1.33286i	−0.937818	−	0.347127i	$-0.887157\pi$
$317$	−13.7011	−	23.7310i	−0.769530	−	1.33286i	0.168288	−	0.985738i	$-0.446176\pi$
$318$		0			0
$319$	−3.82951	+	6.63291i	−0.214411	+	0.371371i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	47.3619			2.63529
$324$		0			0
$325$	3.63235			0.201487
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.82731	−	6.62910i	0.211007	−	0.365474i
$330$		0			0
$331$	−1.08930	−	1.88673i	−0.0598735	−	0.103704i	0.834535	−	0.550955i	$-0.185736\pi$
$331$	−1.08930	−	1.88673i	−0.0598735	−	0.103704i	−0.894408	+	0.447251i	$0.852403\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.00299	−	3.46928i	−0.109435	−	0.189547i
$336$		0			0
$337$	−0.715700	+	1.23963i	−0.0389866	+	0.0675268i	−0.884860	−	0.465856i	$-0.845746\pi$
$337$	−0.715700	+	1.23963i	−0.0389866	+	0.0675268i	0.845874	+	0.533383i	$0.179080\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.68874			0.470522
$342$		0			0
$343$	−16.4508			−0.888259
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.1223	−	22.7285i	0.704441	−	1.22013i	−0.262452	−	0.964945i	$-0.584531\pi$
$347$	13.1223	−	22.7285i	0.704441	−	1.22013i	0.966893	−	0.255183i	$-0.0821357\pi$
$348$		0			0
$349$	17.1673	+	29.7346i	0.918944	+	1.59166i	0.801022	+	0.598635i	$0.204290\pi$
$349$	17.1673	+	29.7346i	0.918944	+	1.59166i	0.117923	+	0.993023i	$0.462377\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.28933	+	3.96523i	0.121849	+	0.211048i	0.920497	−	0.390750i	$-0.127784\pi$
$353$	2.28933	+	3.96523i	0.121849	+	0.211048i	−0.798648	+	0.601798i	$0.794451\pi$
$354$		0			0
$355$	6.14359	−	10.6410i	0.326068	−	0.564766i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.0591			−0.689231			−0.344616	−	0.938744i	$-0.611991\pi$
$359$	−13.0591			−0.689231			−0.344616	+	0.938744i	$0.611991\pi$
$360$		0			0
$361$	20.1599			1.06105
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−5.28644	+	9.15639i	−0.276705	+	0.479267i
$366$		0			0
$367$	1.70626	+	2.95533i	0.0890660	+	0.154267i	0.907117	−	0.420879i	$-0.138278\pi$
$367$	1.70626	+	2.95533i	0.0890660	+	0.154267i	−0.818051	+	0.575146i	$0.804945\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.37151	+	2.37553i	0.0712055	+	0.123332i
$372$		0			0
$373$	6.42179	−	11.1229i	0.332508	−	0.575921i	−0.650495	−	0.759511i	$-0.725439\pi$
$373$	6.42179	−	11.1229i	0.332508	−	0.575921i	0.983003	+	0.183590i	$0.0587719\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.8328			1.07294
$378$		0			0
$379$	12.1642			0.624833			0.312417	−	0.949945i	$-0.398862\pi$
$379$	12.1642			0.624833			0.312417	+	0.949945i	$0.398862\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−8.84072	+	15.3126i	−0.451740	+	0.782436i	−0.998494	−	0.0548568i	$-0.982530\pi$
$383$	−8.84072	+	15.3126i	−0.451740	+	0.782436i	0.546755	+	0.837293i	$0.315863\pi$
$384$		0			0
$385$	−7.36625	−	12.7587i	−0.375419	−	0.650244i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.77658	+	6.54123i	0.191480	+	0.331654i	0.945741	−	0.324921i	$-0.105338\pi$
$389$	3.77658	+	6.54123i	0.191480	+	0.331654i	−0.754261	+	0.656575i	$0.772005\pi$
$390$		0			0
$391$	−19.9513	+	34.5567i	−1.00898	+	1.74761i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−27.4223			−1.37977
$396$		0			0
$397$	7.97075			0.400040			0.200020	−	0.979792i	$-0.435899\pi$
$397$	7.97075			0.400040			0.200020	+	0.979792i	$0.435899\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.53771	−	6.12750i	0.176665	−	0.305993i	−0.764071	−	0.645132i	$-0.776802\pi$
$401$	3.53771	−	6.12750i	0.176665	−	0.305993i	0.940736	+	0.339139i	$0.110136\pi$
$402$		0			0
$403$	−11.8168	−	20.4674i	−0.588639	−	1.01955i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−8.14038	−	14.0995i	−0.403503	−	0.698889i
$408$		0			0
$409$	−16.0499	+	27.7993i	−0.793619	+	1.37459i	0.130094	+	0.991502i	$0.458472\pi$
$409$	−16.0499	+	27.7993i	−0.793619	+	1.37459i	−0.923713	+	0.383086i	$0.874861\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	27.9054			1.37313
$414$		0			0
$415$	−6.51820			−0.319966
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.63571	−	9.76133i	0.275322	−	0.476872i	−0.694894	−	0.719112i	$-0.744549\pi$
$419$	5.63571	−	9.76133i	0.275322	−	0.476872i	0.970216	+	0.242240i	$0.0778821\pi$
$420$		0			0
$421$	4.82872	+	8.36359i	0.235337	+	0.407616i	0.959371	−	0.282149i	$-0.0910471\pi$
$421$	4.82872	+	8.36359i	0.235337	+	0.407616i	−0.724033	+	0.689765i	$0.757714\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.08263	+	3.60723i	0.101023	+	0.174976i
$426$		0			0
$427$	7.42081	−	12.8532i	0.359118	−	0.622011i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−40.1842			−1.93560			−0.967802	−	0.251711i	$-0.919007\pi$
$431$	−40.1842			−1.93560			−0.967802	+	0.251711i	$0.919007\pi$
$432$		0			0
$433$	16.1510			0.776168			0.388084	−	0.921624i	$-0.373137\pi$
$433$	16.1510			0.776168			0.388084	+	0.921624i	$0.373137\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−16.4962	+	28.5723i	−0.789120	+	1.36680i
$438$		0			0
$439$	−9.25383	−	16.0281i	−0.441661	−	0.764980i	0.556152	−	0.831081i	$-0.312277\pi$
$439$	−9.25383	−	16.0281i	−0.441661	−	0.764980i	−0.997813	+	0.0661011i	$0.978944\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.72586	−	2.98927i	−0.0819979	−	0.142025i	0.822110	−	0.569329i	$-0.192797\pi$
$443$	−1.72586	−	2.98927i	−0.0819979	−	0.142025i	−0.904108	+	0.427304i	$0.859463\pi$
$444$		0			0
$445$	3.11457	−	5.39459i	0.147645	−	0.255728i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.83790			−0.464279			−0.232140	−	0.972682i	$-0.574573\pi$
$449$	−9.83790			−0.464279			−0.232140	+	0.972682i	$0.574573\pi$
$450$		0			0
$451$	8.47025			0.398848
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−20.0364	+	34.7041i	−0.939323	+	1.62696i
$456$		0			0
$457$	0.0111990	+	0.0193973i	0.000523869	+	0.000907367i	0.866287	−	0.499546i	$-0.166500\pi$
$457$	0.0111990	+	0.0193973i	0.000523869	+	0.000907367i	−0.865763	+	0.500454i	$0.833167\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	7.50986	+	13.0075i	0.349769	+	0.605818i	0.986208	−	0.165509i	$-0.0529267\pi$
$461$	7.50986	+	13.0075i	0.349769	+	0.605818i	−0.636439	+	0.771327i	$0.719593\pi$
$462$		0			0
$463$	16.2691	−	28.1790i	0.756091	−	1.30959i	−0.188738	−	0.982027i	$-0.560440\pi$
$463$	16.2691	−	28.1790i	0.756091	−	1.30959i	0.944830	−	0.327561i	$-0.106227\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.2983			−0.476550			−0.238275	−	0.971198i	$-0.576582\pi$
$467$	−10.2983			−0.476550			−0.238275	+	0.971198i	$0.576582\pi$
$468$		0			0
$469$	−5.46612			−0.252402
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.58450	+	13.1367i	−0.348736	+	0.604028i
$474$		0			0
$475$	1.72197	+	2.98253i	0.0790093	+	0.136848i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.79020	+	6.56481i	0.173178	+	0.299954i	0.939529	−	0.342468i	$-0.111263\pi$
$479$	3.79020	+	6.56481i	0.173178	+	0.299954i	−0.766351	+	0.642422i	$0.777930\pi$
$480$		0			0
$481$	−22.1421	+	38.3513i	−1.00959	+	1.74867i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−23.2560			−1.05600
$486$		0			0
$487$	−20.6214			−0.934444			−0.467222	−	0.884140i	$-0.654745\pi$
$487$	−20.6214			−0.934444			−0.467222	+	0.884140i	$0.654745\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.98229	−	6.89752i	0.179718	−	0.311281i	−0.762066	−	0.647499i	$-0.775815\pi$
$491$	3.98229	−	6.89752i	0.179718	−	0.311281i	0.941784	+	0.336219i	$0.109148\pi$
$492$		0			0
$493$	11.9446	+	20.6887i	0.537959	+	0.931772i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.38287	−	14.5196i	−0.376023	−	0.651291i
$498$		0			0
$499$	10.5911	−	18.3444i	0.474125	−	0.821208i	−0.525436	−	0.850833i	$-0.676098\pi$
$499$	10.5911	−	18.3444i	0.474125	−	0.821208i	0.999561	+	0.0296248i	$0.00943125\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.0876			0.851073			0.425536	−	0.904941i	$-0.360085\pi$
$503$	19.0876			0.851073			0.425536	+	0.904941i	$0.360085\pi$
$504$		0			0
$505$	−7.69336			−0.342350
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−18.3596	+	31.7998i	−0.813776	+	1.40950i	0.0964283	+	0.995340i	$0.469258\pi$
$509$	−18.3596	+	31.7998i	−0.813776	+	1.40950i	−0.910204	+	0.414161i	$0.864075\pi$
$510$		0			0
$511$	7.21330	+	12.4938i	0.319097	+	0.552693i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11.4734	−	19.8726i	−0.505580	−	0.875690i
$516$		0			0
$517$	−3.22656	+	5.58857i	−0.141904	+	0.245785i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20.5620			−0.900839			−0.450419	−	0.892817i	$-0.648725\pi$
$521$	−20.5620			−0.900839			−0.450419	+	0.892817i	$0.648725\pi$
$522$		0			0
$523$	−16.2754			−0.711672			−0.355836	−	0.934548i	$-0.615804\pi$
$523$	−16.2754			−0.711672			−0.355836	+	0.934548i	$0.615804\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	13.5505	−	23.4702i	0.590270	−	1.02238i
$528$		0			0
$529$	−2.39812	−	4.15367i	−0.104266	−	0.180594i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11.5197	−	19.9527i	−0.498973	−	0.864246i
$534$		0			0
$535$	6.27471	−	10.8681i	0.271279	−	0.469870i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3.11686			−0.134253
$540$		0			0
$541$	−3.19402			−0.137322			−0.0686609	−	0.997640i	$-0.521873\pi$
$541$	−3.19402			−0.137322			−0.0686609	+	0.997640i	$0.521873\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.4112	−	18.0327i	0.445966	−	0.772436i
$546$		0			0
$547$	−7.43936	−	12.8854i	−0.318084	−	0.550938i	0.662004	−	0.749500i	$-0.269706\pi$
$547$	−7.43936	−	12.8854i	−0.318084	−	0.550938i	−0.980088	+	0.198562i	$0.936373\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.87608	+	17.1059i	0.420735	+	0.728734i
$552$		0			0
$553$	−18.7087	+	32.4045i	−0.795576	+	1.37798i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.76686			0.244350			0.122175	−	0.992509i	$-0.461013\pi$
$557$	5.76686			0.244350			0.122175	+	0.992509i	$0.461013\pi$
$558$		0			0
$559$	41.2602			1.74512
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.76738	+	8.25734i	−0.200921	+	0.348005i	−0.948825	−	0.315801i	$-0.897727\pi$
$563$	−4.76738	+	8.25734i	−0.200921	+	0.348005i	0.747904	+	0.663806i	$0.231060\pi$
$564$		0			0
$565$	−15.8527	−	27.4576i	−0.666927	−	1.15515i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.27200	+	7.39933i	0.179092	+	0.310196i	0.941570	−	0.336818i	$-0.109351\pi$
$569$	4.27200	+	7.39933i	0.179092	+	0.310196i	−0.762478	+	0.647014i	$0.776017\pi$
$570$		0			0
$571$	18.1444	−	31.4270i	0.759318	−	1.31518i	−0.183881	−	0.982949i	$-0.558866\pi$
$571$	18.1444	−	31.4270i	0.759318	−	1.31518i	0.943199	−	0.332229i	$-0.107801\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.90153			−0.121002
$576$		0			0
$577$	−6.11652			−0.254634			−0.127317	−	0.991862i	$-0.540637\pi$
$577$	−6.11652			−0.254634			−0.127317	+	0.991862i	$0.540637\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.44701	+	7.70245i	−0.184493	+	0.319551i
$582$		0			0
$583$	−1.15624	−	2.00266i	−0.0478864	−	0.0829417i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.92500	−	3.33419i	−0.0794532	−	0.137617i	0.823561	−	0.567228i	$-0.191984\pi$
$587$	−1.92500	−	3.33419i	−0.0794532	−	0.137617i	−0.903014	+	0.429611i	$0.858651\pi$
$588$		0			0
$589$	11.2039	−	19.4057i	0.461647	−	0.799597i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16.6462			−0.683579			−0.341789	−	0.939777i	$-0.611033\pi$
$593$	−16.6462			−0.683579			−0.341789	+	0.939777i	$0.611033\pi$
$594$		0			0
$595$	−45.9521			−1.88385
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−17.8791	+	30.9675i	−0.730520	+	1.26530i	0.226141	+	0.974095i	$0.427389\pi$
$599$	−17.8791	+	30.9675i	−0.730520	+	1.26530i	−0.956661	+	0.291203i	$0.905944\pi$
$600$		0			0
$601$	6.77202	+	11.7295i	0.276236	+	0.478455i	0.970446	−	0.241317i	$-0.0775794\pi$
$601$	6.77202	+	11.7295i	0.276236	+	0.478455i	−0.694210	+	0.719773i	$0.744246\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−5.39181	−	9.33888i	−0.219208	−	0.379679i
$606$		0			0
$607$	14.0131	−	24.2714i	0.568775	−	0.985147i	−0.427913	−	0.903820i	$-0.640751\pi$
$607$	14.0131	−	24.2714i	0.568775	−	0.985147i	0.996688	−	0.0813266i	$-0.0259157\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17.5527			0.710107
$612$		0			0
$613$	−37.6283			−1.51979			−0.759897	−	0.650043i	$-0.774751\pi$
$613$	−37.6283			−1.51979			−0.759897	+	0.650043i	$0.774751\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−5.57241	+	9.65169i	−0.224337	+	0.388562i	−0.956120	−	0.292975i	$-0.905355\pi$
$617$	−5.57241	+	9.65169i	−0.224337	+	0.388562i	0.731784	+	0.681537i	$0.238688\pi$
$618$		0			0
$619$	9.54119	+	16.5258i	0.383493	+	0.664229i	0.991559	−	0.129658i	$-0.0413878\pi$
$619$	9.54119	+	16.5258i	0.383493	+	0.664229i	−0.608066	+	0.793886i	$0.708054\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.24980	−	7.36086i	−0.170265	−	0.294907i
$624$		0			0
$625$	10.9727	−	19.0053i	0.438908	−	0.760211i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−50.7813			−2.02478
$630$		0			0
$631$	29.7049			1.18253			0.591267	−	0.806476i	$-0.298628\pi$
$631$	29.7049			1.18253			0.591267	+	0.806476i	$0.298628\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8.51984	+	14.7568i	−0.338099	+	0.585605i
$636$		0			0
$637$	4.23898	+	7.34214i	0.167955	+	0.290906i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.60138	+	16.6301i	0.379232	+	0.656848i	0.990951	−	0.134227i	$-0.0428551\pi$
$641$	9.60138	+	16.6301i	0.379232	+	0.656848i	−0.611719	+	0.791075i	$0.709522\pi$
$642$		0			0
$643$	3.52181	−	6.09995i	0.138887	−	0.240559i	−0.788189	−	0.615434i	$-0.788981\pi$
$643$	3.52181	−	6.09995i	0.138887	−	0.240559i	0.927075	+	0.374875i	$0.122314\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	36.8474			1.44862			0.724311	−	0.689473i	$-0.242158\pi$
$647$	36.8474			1.44862			0.724311	+	0.689473i	$0.242158\pi$
$648$		0			0
$649$	−23.5252			−0.923446
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−20.6354	+	35.7416i	−0.807526	+	1.39868i	0.107046	+	0.994254i	$0.465861\pi$
$653$	−20.6354	+	35.7416i	−0.807526	+	1.39868i	−0.914572	+	0.404422i	$0.867473\pi$
$654$		0			0
$655$	23.4839	+	40.6754i	0.917593	+	1.58932i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.79606	+	13.5032i	0.303691	+	0.526009i	0.976969	−	0.213381i	$-0.0684475\pi$
$659$	7.79606	+	13.5032i	0.303691	+	0.526009i	−0.673278	+	0.739390i	$0.735114\pi$
$660$		0			0
$661$	−0.273228	+	0.473246i	−0.0106274	+	0.0184071i	−0.871290	−	0.490768i	$-0.836716\pi$
$661$	−0.273228	+	0.473246i	−0.0106274	+	0.0184071i	0.860663	+	0.509175i	$0.170050\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−37.9942			−1.47335
$666$		0			0
$667$	−16.6413			−0.644353
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.25601	+	10.8357i	−0.241510	+	0.418308i
$672$		0			0
$673$	7.71994	+	13.3713i	0.297582	+	0.515427i	0.975582	−	0.219635i	$-0.0704866\pi$
$673$	7.71994	+	13.3713i	0.297582	+	0.515427i	−0.678000	+	0.735061i	$0.737153\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.7394	+	35.9216i	0.797079	+	1.38058i	0.921511	+	0.388353i	$0.126956\pi$
$677$	20.7394	+	35.9216i	0.797079	+	1.38058i	−0.124432	+	0.992228i	$0.539711\pi$
$678$		0			0
$679$	−15.8663	+	27.4812i	−0.608893	+	1.05463i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−7.83506			−0.299800			−0.149900	−	0.988701i	$-0.547895\pi$
$683$	−7.83506			−0.299800			−0.149900	+	0.988701i	$0.547895\pi$
$684$		0			0
$685$	19.5064			0.745303
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.14500	+	5.44730i	−0.119815	+	0.207526i
$690$		0			0
$691$	11.8366	+	20.5016i	0.450286	+	0.779918i	0.998404	−	0.0564831i	$-0.0179887\pi$
$691$	11.8366	+	20.5016i	0.450286	+	0.779918i	−0.548118	+	0.836401i	$0.684655\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.86497	−	15.3546i	−0.336267	−	0.582432i
$696$		0			0
$697$	13.2098	−	22.8800i	0.500356	−	0.866642i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7.14909			0.270017			0.135009	−	0.990844i	$-0.456894\pi$
$701$	7.14909			0.270017			0.135009	+	0.990844i	$0.456894\pi$
$702$		0			0
$703$	−41.9871			−1.58357
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5.24876	+	9.09111i	−0.197400	+	0.341906i
$708$		0			0
$709$	19.3826	+	33.5716i	0.727928	+	1.26081i	0.957757	+	0.287577i	$0.0928498\pi$
$709$	19.3826	+	33.5716i	0.727928	+	1.26081i	−0.229830	+	0.973231i	$0.573817\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	9.43931	+	16.3494i	0.353505	+	0.612289i
$714$		0			0
$715$	16.8914	−	29.2568i	0.631704	−	1.09414i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−38.3429			−1.42995			−0.714975	−	0.699151i	$-0.753562\pi$
$719$	−38.3429			−1.42995			−0.714975	+	0.699151i	$0.753562\pi$
$720$		0			0
$721$	−31.3108			−1.16607
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.868556	+	1.50438i	−0.0322574	+	0.0558714i
$726$		0			0
$727$	21.9734	+	38.0591i	0.814950	+	1.41154i	0.909364	+	0.416002i	$0.136569\pi$
$727$	21.9734	+	38.0591i	0.814950	+	1.41154i	−0.0944136	+	0.995533i	$0.530098\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	23.6568	+	40.9748i	0.874979	+	1.51551i
$732$		0			0
$733$	−24.7222	+	42.8202i	−0.913137	+	1.58160i	−0.103531	+	0.994626i	$0.533014\pi$
$733$	−24.7222	+	42.8202i	−0.913137	+	1.58160i	−0.809606	+	0.586973i	$0.800319\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.60814			0.169743
$738$		0			0
$739$	9.25073			0.340294			0.170147	−	0.985419i	$-0.445576\pi$
$739$	9.25073			0.340294			0.170147	+	0.985419i	$0.445576\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.6473	+	35.7621i	−0.757475	+	1.31198i	0.186660	+	0.982425i	$0.440234\pi$
$743$	−20.6473	+	35.7621i	−0.757475	+	1.31198i	−0.944135	+	0.329560i	$0.893100\pi$
$744$		0			0
$745$	13.0328	+	22.5734i	0.477483	+	0.827025i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8.56178	−	14.8294i	−0.312840	−	0.541856i
$750$		0			0
$751$	−10.2101	+	17.6845i	−0.372573	+	0.645315i	−0.989961	−	0.141344i	$-0.954858\pi$
$751$	−10.2101	+	17.6845i	−0.372573	+	0.645315i	0.617388	+	0.786659i	$0.288191\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20.7324			0.754529
$756$		0			0
$757$	7.74944			0.281658			0.140829	−	0.990034i	$-0.455023\pi$
$757$	7.74944			0.281658			0.140829	+	0.990034i	$0.455023\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.3416	+	23.1084i	−0.483634	+	0.837679i	−0.999823	−	0.0187955i	$-0.994017\pi$
$761$	−13.3416	+	23.1084i	−0.483634	+	0.837679i	0.516189	+	0.856475i	$0.327350\pi$
$762$		0			0
$763$	−14.2059	−	24.6054i	−0.514290	−	0.890776i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	31.9947	+	55.4165i	1.15526	+	2.00097i
$768$		0			0
$769$	−24.5226	+	42.4744i	−0.884307	+	1.53166i	−0.0378010	+	0.999285i	$0.512035\pi$
$769$	−24.5226	+	42.4744i	−0.884307	+	1.53166i	−0.846506	+	0.532379i	$0.821298\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.1406			−0.436665			−0.218333	−	0.975874i	$-0.570062\pi$
$773$	−12.1406			−0.436665			−0.218333	+	0.975874i	$0.570062\pi$
$774$		0			0
$775$	1.97066			0.0707881
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.9221	−	18.9177i	0.391326	−	0.677796i
$780$		0			0
$781$	7.06706	+	12.2405i	0.252879	+	0.438000i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.227993	+	0.394895i	0.00813742	+	0.0140944i
$786$		0			0
$787$	24.2553	−	42.0114i	0.864608	−	1.49754i	−0.00282812	−	0.999996i	$-0.500900\pi$
$787$	24.2553	−	42.0114i	0.864608	−	1.49754i	0.867436	−	0.497549i	$-0.165766\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−43.2616			−1.53821
$792$		0			0
$793$	34.0331			1.20855
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2.54989	−	4.41654i	0.0903218	−	0.156442i	−0.817325	−	0.576177i	$-0.804544\pi$
$797$	2.54989	−	4.41654i	0.0903218	−	0.156442i	0.907647	+	0.419735i	$0.137877\pi$
$798$		0			0
$799$	10.0640	+	17.4313i	0.356037	+	0.616675i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.08107	−	10.5327i	−0.214596	−	0.371692i
$804$		0			0
$805$	16.0051	−	27.7217i	0.564107	−	0.977063i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	38.1416			1.34099			0.670494	−	0.741915i	$-0.266082\pi$
$809$	38.1416			1.34099			0.670494	+	0.741915i	$0.266082\pi$
$810$		0			0
$811$	−2.88343			−0.101251			−0.0506255	−	0.998718i	$-0.516122\pi$
$811$	−2.88343			−0.101251			−0.0506255	+	0.998718i	$0.516122\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.09745	+	10.5611i	−0.213584	+	0.369939i
$816$		0			0
$817$	19.5600	+	33.8789i	0.684317	+	1.18527i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.84327	+	10.1208i	0.203931	+	0.353220i	0.949792	−	0.312883i	$-0.101295\pi$
$821$	5.84327	+	10.1208i	0.203931	+	0.353220i	−0.745860	+	0.666102i	$0.767961\pi$
$822$		0			0
$823$	5.91203	−	10.2399i	0.206080	−	0.356942i	−0.744396	−	0.667738i	$-0.767262\pi$
$823$	5.91203	−	10.2399i	0.206080	−	0.356942i	0.950476	+	0.310797i	$0.100596\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	27.5265			0.957192			0.478596	−	0.878035i	$-0.341146\pi$
$827$	27.5265			0.957192			0.478596	+	0.878035i	$0.341146\pi$
$828$		0			0
$829$	−27.7398			−0.963443			−0.481721	−	0.876324i	$-0.659988\pi$
$829$	−27.7398			−0.963443			−0.481721	+	0.876324i	$0.659988\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−4.86090	+	8.41932i	−0.168420	+	0.291712i
$834$		0			0
$835$	−1.72687	−	2.99102i	−0.0597608	−	0.103509i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.7349	−	22.0574i	−0.439656	−	0.761507i	0.558007	−	0.829837i	$-0.311566\pi$
$839$	−12.7349	−	22.0574i	−0.439656	−	0.761507i	−0.997663	+	0.0683295i	$0.978233\pi$
$840$		0			0
$841$	9.51853	−	16.4866i	0.328225	−	0.568503i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−64.4682			−2.21777
$846$		0			0
$847$	−14.7141			−0.505583
$848$		0			0
$849$		0			0
$850$		0			0
$851$	17.6872	−	30.6351i	0.606308	−	1.05016i
$852$		0			0
$853$	15.2819	+	26.4690i	0.523241	+	0.906280i	0.999634	+	0.0270477i	$0.00861061\pi$
$853$	15.2819	+	26.4690i	0.523241	+	0.906280i	−0.476393	+	0.879232i	$0.658056\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.69252	−	13.3238i	−0.262772	−	0.455134i	0.704206	−	0.709996i	$-0.251303\pi$
$857$	−7.69252	−	13.3238i	−0.262772	−	0.455134i	−0.966977	+	0.254862i	$0.917970\pi$
$858$		0			0
$859$	9.19471	−	15.9257i	0.313720	−	0.543378i	−0.665445	−	0.746447i	$-0.731758\pi$
$859$	9.19471	−	15.9257i	0.313720	−	0.543378i	0.979164	+	0.203069i	$0.0650915\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−35.5698			−1.21081			−0.605404	−	0.795918i	$-0.706989\pi$
$863$	−35.5698			−1.21081			−0.605404	+	0.795918i	$0.706989\pi$
$864$		0			0
$865$	12.6240			0.429227
$866$		0			0
$867$		0			0
$868$		0			0
$869$	15.7721	−	27.3181i	0.535033	−	0.926704i
$870$		0			0
$871$	−6.26715	−	10.8550i	−0.212354	−	0.367808i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16.8495	−	29.1842i	−0.569616	−	0.986605i
$876$		0			0
$877$	11.1127	−	19.2477i	0.375249	−	0.649950i	−0.615115	−	0.788437i	$-0.710891\pi$
$877$	11.1127	−	19.2477i	0.375249	−	0.649950i	0.990364	+	0.138487i	$0.0442239\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−38.5586			−1.29907			−0.649536	−	0.760331i	$-0.725037\pi$
$881$	−38.5586			−1.29907			−0.649536	+	0.760331i	$0.725037\pi$
$882$		0			0
$883$	10.4984			0.353298			0.176649	−	0.984274i	$-0.443474\pi$
$883$	10.4984			0.353298			0.176649	+	0.984274i	$0.443474\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	23.0186	−	39.8693i	0.772888	−	1.33868i	−0.163087	−	0.986612i	$-0.552145\pi$
$887$	23.0186	−	39.8693i	0.772888	−	1.33868i	0.935974	−	0.352069i	$-0.114522\pi$
$888$		0			0
$889$	11.6252	+	20.1355i	0.389898	+	0.675322i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.32111	+	14.4126i	0.278455	+	0.482299i
$894$		0			0
$895$	−10.0336	+	17.3787i	−0.335386	+	0.580906i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.3024			0.376956
$900$		0			0
$901$	−7.21283			−0.240294
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−25.0521	+	43.3914i	−0.832759	+	1.44238i
$906$		0			0
$907$	−8.68354	−	15.0403i	−0.288332	−	0.499406i	0.685080	−	0.728468i	$-0.259767\pi$
$907$	−8.68354	−	15.0403i	−0.288332	−	0.499406i	−0.973412	+	0.229062i	$0.926434\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.29095	−	2.23599i	−0.0427711	−	0.0740817i	0.843847	−	0.536583i	$-0.180285\pi$
$911$	−1.29095	−	2.23599i	−0.0427711	−	0.0740817i	−0.886618	+	0.462502i	$0.846952\pi$
$912$		0			0
$913$	3.74899	−	6.49344i	0.124073	−	0.214902i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	64.0871			2.11634
$918$		0			0
$919$	34.8909			1.15095			0.575473	−	0.817821i	$-0.304818\pi$
$919$	34.8909			1.15095			0.575473	+	0.817821i	$0.304818\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	19.2226	−	33.2946i	0.632721	−	1.09591i
$924$		0			0
$925$	−1.84629	−	3.19786i	−0.0607055	−	0.105145i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.38201	−	7.58986i	−0.143769	−	0.249015i	0.785144	−	0.619313i	$-0.212589\pi$
$929$	−4.38201	−	7.58986i	−0.143769	−	0.249015i	−0.928913	+	0.370298i	$0.879256\pi$
$930$		0			0
$931$	−4.01910	+	6.96128i	−0.131721	+	0.228147i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	38.7393			1.26691
$936$		0			0
$937$	39.6212			1.29437			0.647184	−	0.762334i	$-0.275946\pi$
$937$	39.6212			1.29437			0.647184	+	0.762334i	$0.275946\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.5675	−	23.4995i	0.442287	−	0.766063i	−0.555572	−	0.831468i	$-0.687501\pi$
$941$	13.5675	−	23.4995i	0.442287	−	0.766063i	0.997859	+	0.0654053i	$0.0208340\pi$
$942$		0			0
$943$	9.20195	+	15.9382i	0.299657	+	0.519020i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.06166	+	1.83886i	0.0344995	+	0.0597548i	0.882760	−	0.469825i	$-0.155683\pi$
$947$	1.06166	+	1.83886i	0.0344995	+	0.0597548i	−0.848260	+	0.529580i	$0.822350\pi$
$948$		0			0
$949$	−16.5407	+	28.6494i	−0.536934	+	0.929998i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.523434			0.0169557			0.00847784	−	0.999964i	$-0.497301\pi$
$953$	0.523434			0.0169557			0.00847784	+	0.999964i	$0.497301\pi$
$954$		0			0
$955$	17.4729			0.565410
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13.3082	−	23.0504i	0.429743	−	0.744337i
$960$		0			0
$961$	9.08902	+	15.7426i	0.293194	+	0.507827i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10.5538	−	18.2796i	−0.339737	−	0.588442i
$966$		0			0
$967$	−2.74196	+	4.74922i	−0.0881756	+	0.152725i	−0.906740	−	0.421690i	$-0.861437\pi$
$967$	−2.74196	+	4.74922i	−0.0881756	+	0.152725i	0.818564	+	0.574415i	$0.194770\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.813501			0.0261065			0.0130532	−	0.999915i	$-0.495845\pi$
$971$	0.813501			0.0261065			0.0130532	+	0.999915i	$0.495845\pi$
$972$		0			0
$973$	−24.1923			−0.775570
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.85335	+	17.0665i	−0.315237	+	0.546006i	−0.979488	−	0.201504i	$-0.935417\pi$
$977$	−9.85335	+	17.0665i	−0.315237	+	0.546006i	0.664251	+	0.747509i	$0.268751\pi$
$978$		0			0
$979$	3.58273	+	6.20547i	0.114505	+	0.198328i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−27.9588	−	48.4261i	−0.891747	−	1.54455i	−0.837779	−	0.546009i	$-0.816146\pi$
$983$	−27.9588	−	48.4261i	−0.891747	−	1.54455i	−0.0539681	−	0.998543i	$-0.517187\pi$
$984$		0			0
$985$	−0.586692	+	1.01618i	−0.0186936	+	0.0323782i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−32.9588			−1.04803
$990$		0			0
$991$	−22.4053			−0.711727			−0.355863	−	0.934538i	$-0.615813\pi$
$991$	−22.4053			−0.711727			−0.355863	+	0.934538i	$0.615813\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−22.7318	+	39.3726i	−0.720646	+	1.24819i
$996$		0			0
$997$	16.6132	+	28.7749i	0.526146	+	0.911311i	0.999536	+	0.0304585i	$0.00969675\pi$
$997$	16.6132	+	28.7749i	0.526146	+	0.911311i	−0.473390	+	0.880853i	$0.656970\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.i.l.2305.4		12
3.2	odd	2		1152.2.i.k.769.3	yes	12
4.3	odd	2		3456.2.i.k.2305.4		12
8.3	odd	2		3456.2.i.i.2305.3		12
8.5	even	2		3456.2.i.j.2305.3		12
9.2	odd	6		1152.2.i.k.385.3	yes	12
9.7	even	3	inner	3456.2.i.l.1153.4		12
12.11	even	2		1152.2.i.i.769.4	yes	12
24.5	odd	2		1152.2.i.j.769.4	yes	12
24.11	even	2		1152.2.i.l.769.3	yes	12
36.7	odd	6		3456.2.i.k.1153.4		12
36.11	even	6		1152.2.i.i.385.4	✓	12
72.11	even	6		1152.2.i.l.385.3	yes	12
72.29	odd	6		1152.2.i.j.385.4	yes	12
72.43	odd	6		3456.2.i.i.1153.3		12
72.61	even	6		3456.2.i.j.1153.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.i.i.385.4	✓	12	36.11	even	6
1152.2.i.i.769.4	yes	12	12.11	even	2
1152.2.i.j.385.4	yes	12	72.29	odd	6
1152.2.i.j.769.4	yes	12	24.5	odd	2
1152.2.i.k.385.3	yes	12	9.2	odd	6
1152.2.i.k.769.3	yes	12	3.2	odd	2
1152.2.i.l.385.3	yes	12	72.11	even	6
1152.2.i.l.769.3	yes	12	24.11	even	2
3456.2.i.i.1153.3		12	72.43	odd	6
3456.2.i.i.2305.3		12	8.3	odd	2
3456.2.i.j.1153.3		12	72.61	even	6
3456.2.i.j.2305.3		12	8.5	even	2
3456.2.i.k.1153.4		12	36.7	odd	6
3456.2.i.k.2305.4		12	4.3	odd	2
3456.2.i.l.1153.4		12	9.7	even	3	inner
3456.2.i.l.2305.4		12	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:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3456.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.05471 - 1.82681i) q^{5} +(-1.43914 - 2.49267i) q^{7} +O(q^{10})\)	\(q+(1.05471 - 1.82681i) q^{5} +(-1.43914 - 2.49267i) q^{7} +(1.21325 + 2.10141i) q^{11} +(3.30008 - 5.71590i) q^{13} +7.56848 q^{17} +6.25779 q^{19} +(-2.63611 + 4.56587i) q^{23} +(0.275172 + 0.476612i) q^{25} +(1.57821 + 2.73353i) q^{29} +(1.79039 - 3.10104i) q^{31} -6.07151 q^{35} -6.70957 q^{37} +(1.74537 - 3.02306i) q^{41} +(3.12570 + 5.41388i) q^{43} +(1.32972 + 2.30314i) q^{47} +(-0.642255 + 1.11242i) q^{49} -0.953009 q^{53} +5.11850 q^{55} +(-4.84757 + 8.39624i) q^{59} +(2.57821 + 4.46558i) q^{61} +(-6.96125 - 12.0572i) q^{65} +(0.949546 - 1.64466i) q^{67} +5.82491 q^{71} -5.01222 q^{73} +(3.49207 - 6.04844i) q^{77} +(-6.49996 - 11.2583i) q^{79} +(-1.54502 - 2.67606i) q^{83} +(7.98256 - 13.8262i) q^{85} +2.95301 q^{89} -18.9971 q^{91} +(6.60015 - 11.4318i) q^{95} +(-5.51242 - 9.54779i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{5} + 6 q^{7}+O(q^{10}) \) 12 * q + 2 * q^5 + 6 * q^7	\( 12 q + 2 q^{5} + 6 q^{7} - 4 q^{11} + 10 q^{13} - 4 q^{17} + 4 q^{19} - 8 q^{23} - 14 q^{25} + 2 q^{29} + 8 q^{31} - 8 q^{35} + 2 q^{41} - 2 q^{43} + 14 q^{47} - 18 q^{49} - 24 q^{53} - 16 q^{55} - 6 q^{59} + 14 q^{61} + 8 q^{65} + 4 q^{67} + 28 q^{71} + 60 q^{73} - 2 q^{77} + 16 q^{79} - 24 q^{83} + 16 q^{85} + 48 q^{89} - 52 q^{91} + 20 q^{95} - 14 q^{97}+O(q^{100}) \) 12 * q + 2 * q^5 + 6 * q^7 - 4 * q^11 + 10 * q^13 - 4 * q^17 + 4 * q^19 - 8 * q^23 - 14 * q^25 + 2 * q^29 + 8 * q^31 - 8 * q^35 + 2 * q^41 - 2 * q^43 + 14 * q^47 - 18 * q^49 - 24 * q^53 - 16 * q^55 - 6 * q^59 + 14 * q^61 + 8 * q^65 + 4 * q^67 + 28 * q^71 + 60 * q^73 - 2 * q^77 + 16 * q^79 - 24 * q^83 + 16 * q^85 + 48 * q^89 - 52 * q^91 + 20 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more