LMFDB - Embedded newform 3456.2.i.e.2305.5

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:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	10.0.8528759163648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 $ x^10 - 2x^9 + x^8 + 9x^6 - 36x^5 + 27x^4 + 27x^2 - 162x + 243
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2305.5
Root			$1.06839 + 1.36328i$ of defining polynomial
Character	$\chi$	$=$	3456.2305
Dual form			3456.2.i.e.1153.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.34011 - 2.32114i) q^{5} +(-2.48656 - 4.30684i) q^{7} +O(q^{10})$	$q+(1.34011 - 2.32114i) q^{5} +(-2.48656 - 4.30684i) q^{7} +(-1.26947 - 2.19879i) q^{11} +(2.21483 - 3.83620i) q^{13} +2.43417 q^{17} -4.18361 q^{19} +(-0.570641 + 0.988379i) q^{23} +(-1.09180 - 1.89106i) q^{25} +(-3.00434 - 5.20366i) q^{29} +(2.65303 - 4.59518i) q^{31} -13.3291 q^{35} +0.241556 q^{37} +(3.21105 - 5.56170i) q^{41} +(5.57273 + 9.65224i) q^{43} +(-2.37728 - 4.11757i) q^{47} +(-8.86592 + 15.3562i) q^{49} +9.38345 q^{53} -6.80494 q^{55} +(5.40906 - 9.36876i) q^{59} +(4.16044 + 7.20610i) q^{61} +(-5.93625 - 10.2819i) q^{65} +(1.13269 - 1.96188i) q^{67} -4.52639 q^{71} -3.34728 q^{73} +(-6.31322 + 10.9348i) q^{77} +(3.14159 + 5.44139i) q^{79} +(-0.738248 - 1.27868i) q^{83} +(3.26206 - 5.65005i) q^{85} -14.8823 q^{89} -22.0292 q^{91} +(-5.60651 + 9.71076i) q^{95} +(5.89884 + 10.2171i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{7}+O(q^{10}) $ 10 * q - 4 * q^7	$ 10 q - 4 q^{7} - q^{11} + 6 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{23} + q^{25} + 4 q^{29} - 8 q^{31} - 24 q^{35} - 20 q^{37} + 5 q^{41} + 13 q^{43} + 6 q^{47} + 3 q^{49} + 12 q^{55} - 13 q^{59} + 10 q^{61} + 17 q^{67} - 8 q^{71} - 34 q^{73} - 8 q^{77} - 6 q^{79} + 12 q^{83} + 18 q^{85} - 44 q^{89} - 36 q^{91} + 6 q^{95} + 27 q^{97}+O(q^{100}) $ 10 * q - 4 * q^7 - q^11 + 6 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^23 + q^25 + 4 * q^29 - 8 * q^31 - 24 * q^35 - 20 * q^37 + 5 * q^41 + 13 * q^43 + 6 * q^47 + 3 * q^49 + 12 * q^55 - 13 * q^59 + 10 * q^61 + 17 * q^67 - 8 * q^71 - 34 * q^73 - 8 * q^77 - 6 * q^79 + 12 * q^83 + 18 * q^85 - 44 * q^89 - 36 * q^91 + 6 * q^95 + 27 * 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
$5$	1.34011	−	2.32114i	0.599317	−	1.03805i	−0.393605	−	0.919279i	$-0.628772\pi$
$5$	1.34011	−	2.32114i	0.599317	−	1.03805i	0.992922	−	0.118767i	$-0.0378943\pi$
$6$		0			0
$7$	−2.48656	−	4.30684i	−0.939830	−	1.62783i	−0.765786	−	0.643095i	$-0.777650\pi$
$7$	−2.48656	−	4.30684i	−0.939830	−	1.62783i	−0.174043	−	0.984738i	$-0.555683\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.26947	−	2.19879i	−0.382760	−	0.662960i	0.608696	−	0.793404i	$-0.291693\pi$
$11$	−1.26947	−	2.19879i	−0.382760	−	0.662960i	−0.991456	+	0.130444i	$0.958360\pi$
$12$		0			0
$13$	2.21483	−	3.83620i	0.614284	−	1.06397i	−0.376225	−	0.926528i	$-0.622778\pi$
$13$	2.21483	−	3.83620i	0.614284	−	1.06397i	0.990510	−	0.137443i	$-0.0438885\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.43417			0.590373			0.295186	−	0.955440i	$-0.404618\pi$
$17$	2.43417			0.590373			0.295186	+	0.955440i	$0.404618\pi$
$18$		0			0
$19$	−4.18361			−0.959786			−0.479893	−	0.877327i	$-0.659324\pi$
$19$	−4.18361			−0.959786			−0.479893	+	0.877327i	$0.659324\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.570641	+	0.988379i	−0.118987	+	0.206091i	−0.919366	−	0.393402i	$-0.871298\pi$
$23$	−0.570641	+	0.988379i	−0.118987	+	0.206091i	0.800380	+	0.599494i	$0.204631\pi$
$24$		0			0
$25$	−1.09180	−	1.89106i	−0.218361	−	0.378212i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.00434	−	5.20366i	−0.557891	−	0.966296i	−0.997672	−	0.0681908i	$-0.978277\pi$
$29$	−3.00434	−	5.20366i	−0.557891	−	0.966296i	0.439781	−	0.898105i	$-0.355056\pi$
$30$		0			0
$31$	2.65303	−	4.59518i	0.476498	−	0.825319i	−0.523139	−	0.852247i	$-0.675239\pi$
$31$	2.65303	−	4.59518i	0.476498	−	0.825319i	0.999637	+	0.0269282i	$0.00857256\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−13.3291			−2.25302
$36$		0			0
$37$	0.241556			0.0397115			0.0198558	−	0.999803i	$-0.493679\pi$
$37$	0.241556			0.0397115			0.0198558	+	0.999803i	$0.493679\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.21105	−	5.56170i	0.501482	−	0.868592i	−0.498517	−	0.866880i	$-0.666122\pi$
$41$	3.21105	−	5.56170i	0.501482	−	0.868592i	0.999999	−	0.00171211i	$-0.000544983\pi$
$42$		0			0
$43$	5.57273	+	9.65224i	0.849833	+	1.47195i	0.881357	+	0.472451i	$0.156630\pi$
$43$	5.57273	+	9.65224i	0.849833	+	1.47195i	−0.0315245	+	0.999503i	$0.510036\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.37728	−	4.11757i	−0.346762	−	0.600609i	0.638910	−	0.769281i	$-0.279385\pi$
$47$	−2.37728	−	4.11757i	−0.346762	−	0.600609i	−0.985672	+	0.168672i	$0.946052\pi$
$48$		0			0
$49$	−8.86592	+	15.3562i	−1.26656	+	2.19375i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.38345			1.28892			0.644458	−	0.764640i	$-0.277083\pi$
$53$	9.38345			1.28892			0.644458	+	0.764640i	$0.277083\pi$
$54$		0			0
$55$	−6.80494			−0.917578
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.40906	−	9.36876i	0.704199	−	1.21971i	−0.262781	−	0.964856i	$-0.584640\pi$
$59$	5.40906	−	9.36876i	0.704199	−	1.21971i	0.966980	−	0.254853i	$-0.0820272\pi$
$60$		0			0
$61$	4.16044	+	7.20610i	0.532690	+	0.922646i	0.999271	+	0.0381677i	$0.0121521\pi$
$61$	4.16044	+	7.20610i	0.532690	+	0.922646i	−0.466581	+	0.884478i	$0.654515\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.93625	−	10.2819i	−0.736302	−	1.27531i
$66$		0			0
$67$	1.13269	−	1.96188i	0.138380	−	0.239682i	−0.788503	−	0.615030i	$-0.789144\pi$
$67$	1.13269	−	1.96188i	0.138380	−	0.239682i	0.926884	+	0.375349i	$0.122477\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.52639			−0.537183			−0.268592	−	0.963254i	$-0.586558\pi$
$71$	−4.52639			−0.537183			−0.268592	+	0.963254i	$0.586558\pi$
$72$		0			0
$73$	−3.34728			−0.391769			−0.195885	−	0.980627i	$-0.562758\pi$
$73$	−3.34728			−0.391769			−0.195885	+	0.980627i	$0.562758\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.31322	+	10.9348i	−0.719459	+	1.24614i
$78$		0			0
$79$	3.14159	+	5.44139i	0.353457	+	0.612205i	0.986853	−	0.161623i	$-0.0516729\pi$
$79$	3.14159	+	5.44139i	0.353457	+	0.612205i	−0.633396	+	0.773828i	$0.718340\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.738248	−	1.27868i	−0.0810332	−	0.140354i	0.822661	−	0.568533i	$-0.192489\pi$
$83$	−0.738248	−	1.27868i	−0.0810332	−	0.140354i	−0.903694	+	0.428179i	$0.859155\pi$
$84$		0			0
$85$	3.26206	−	5.65005i	0.353820	−	0.612834i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.8823			−1.57752			−0.788762	−	0.614698i	$-0.789278\pi$
$89$	−14.8823			−1.57752			−0.788762	+	0.614698i	$0.789278\pi$
$90$		0			0
$91$	−22.0292			−2.30929
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.60651	+	9.71076i	−0.575216	+	0.996303i
$96$		0			0
$97$	5.89884	+	10.2171i	0.598937	+	1.03739i	0.992978	+	0.118296i	$0.0377432\pi$
$97$	5.89884	+	10.2171i	0.598937	+	1.03739i	−0.394042	+	0.919093i	$0.628923\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.81211	+	10.0669i	0.578326	+	1.00169i	0.995671	+	0.0929424i	$0.0296272\pi$
$101$	5.81211	+	10.0669i	0.578326	+	1.00169i	−0.417345	+	0.908748i	$0.637039\pi$
$102$		0			0
$103$	−9.54962	+	16.5404i	−0.940952	+	1.62978i	−0.177292	+	0.984158i	$0.556734\pi$
$103$	−9.54962	+	16.5404i	−0.940952	+	1.62978i	−0.763660	+	0.645618i	$0.776600\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.99694			0.483072			0.241536	−	0.970392i	$-0.422349\pi$
$107$	4.99694			0.483072			0.241536	+	0.970392i	$0.422349\pi$
$108$		0			0
$109$	−2.15690			−0.206594			−0.103297	−	0.994651i	$-0.532939\pi$
$109$	−2.15690			−0.206594			−0.103297	+	0.994651i	$0.532939\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.2451	+	17.7451i	−0.963782	+	1.66932i	−0.250929	+	0.968005i	$0.580736\pi$
$113$	−10.2451	+	17.7451i	−0.963782	+	1.66932i	−0.712853	+	0.701314i	$0.752597\pi$
$114$		0			0
$115$	1.52945	+	2.64908i	0.142622	+	0.247028i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.05270	−	10.4836i	−0.554850	−	0.961028i
$120$		0			0
$121$	2.27688	−	3.94368i	0.206989	−	0.358516i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.54856			0.675164
$126$		0			0
$127$	−9.71867			−0.862392			−0.431196	−	0.902258i	$-0.641908\pi$
$127$	−9.71867			−0.862392			−0.431196	+	0.902258i	$0.641908\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.49219	−	6.04865i	0.305114	−	0.528473i	−0.672173	−	0.740394i	$-0.734639\pi$
$131$	3.49219	−	6.04865i	0.305114	−	0.528473i	0.977287	+	0.211921i	$0.0679721\pi$
$132$		0			0
$133$	10.4028	+	18.0181i	0.902035	+	1.56237i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.2072	−	19.4114i	−0.957493	−	1.65843i	−0.728558	−	0.684984i	$-0.759809\pi$
$137$	−11.2072	−	19.4114i	−0.957493	−	1.65843i	−0.228935	−	0.973442i	$-0.573524\pi$
$138$		0			0
$139$	−2.56822	+	4.44830i	−0.217834	+	0.377299i	−0.954146	−	0.299343i	$-0.903232\pi$
$139$	−2.56822	+	4.44830i	−0.217834	+	0.377299i	0.736312	+	0.676643i	$0.236566\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.2467			−0.940494
$144$		0			0
$145$	−16.1046			−1.33741
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.08562	−	1.88034i	0.0889372	−	0.154044i	−0.818125	−	0.575040i	$-0.804986\pi$
$149$	1.08562	−	1.88034i	0.0889372	−	0.154044i	0.907062	+	0.420997i	$0.138320\pi$
$150$		0			0
$151$	3.10508	+	5.37816i	0.252688	+	0.437669i	0.964265	−	0.264940i	$-0.0853520\pi$
$151$	3.10508	+	5.37816i	0.252688	+	0.437669i	−0.711577	+	0.702608i	$0.752019\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.11072	−	12.3161i	−0.571147	−	0.989255i
$156$		0			0
$157$	7.97578	−	13.8144i	0.636536	−	1.10251i	−0.349651	−	0.936880i	$-0.613700\pi$
$157$	7.97578	−	13.8144i	0.636536	−	1.10251i	0.986187	−	0.165633i	$-0.0529667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.67572			0.447310
$162$		0			0
$163$	4.72991			0.370475			0.185237	−	0.982694i	$-0.440695\pi$
$163$	4.72991			0.370475			0.185237	+	0.982694i	$0.440695\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.18611	−	2.05440i	0.0917836	−	0.158974i	−0.816478	−	0.577376i	$-0.804077\pi$
$167$	1.18611	−	2.05440i	0.0917836	−	0.158974i	0.908262	+	0.418403i	$0.137410\pi$
$168$		0			0
$169$	−3.31097	−	5.73477i	−0.254690	−	0.441137i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.19528	−	2.07028i	−0.0908752	−	0.157400i	0.817004	−	0.576631i	$-0.195633\pi$
$173$	−1.19528	−	2.07028i	−0.0908752	−	0.157400i	−0.907880	+	0.419231i	$0.862300\pi$
$174$		0			0
$175$	−5.42967	+	9.40446i	−0.410444	+	0.710910i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.19567			−0.0893689			−0.0446845	−	0.999001i	$-0.514228\pi$
$179$	−1.19567			−0.0893689			−0.0446845	+	0.999001i	$0.514228\pi$
$180$		0			0
$181$	5.17390			0.384573			0.192287	−	0.981339i	$-0.438410\pi$
$181$	5.17390			0.384573			0.192287	+	0.981339i	$0.438410\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.323712	−	0.560686i	0.0237998	−	0.0412224i
$186$		0			0
$187$	−3.09011	−	5.35222i	−0.225971	−	0.391393i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.654726	+	1.13402i	0.0473743	+	0.0820548i	0.888740	−	0.458411i	$-0.151581\pi$
$191$	0.654726	+	1.13402i	0.0473743	+	0.0820548i	−0.841366	+	0.540466i	$0.818248\pi$
$192$		0			0
$193$	10.0726	−	17.4463i	0.725043	−	1.25581i	−0.233913	−	0.972257i	$-0.575153\pi$
$193$	10.0726	−	17.4463i	0.725043	−	1.25581i	0.958956	−	0.283554i	$-0.0915135\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.46023			0.531520			0.265760	−	0.964039i	$-0.414377\pi$
$197$	7.46023			0.531520			0.265760	+	0.964039i	$0.414377\pi$
$198$		0			0
$199$	3.47361			0.246238			0.123119	−	0.992392i	$-0.460710\pi$
$199$	3.47361			0.246238			0.123119	+	0.992392i	$0.460710\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.9409	+	25.8784i	−1.04865	+	1.81631i
$204$		0			0
$205$	−8.60634	−	14.9066i	−0.601093	−	1.04112i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.31097	+	9.19888i	0.367368	+	0.636300i
$210$		0			0
$211$	−4.38878	+	7.60159i	−0.302136	+	0.523315i	−0.976620	−	0.214975i	$-0.931033\pi$
$211$	−4.38878	+	7.60159i	−0.302136	+	0.523315i	0.674483	+	0.738290i	$0.264366\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	29.8723			2.03728
$216$		0			0
$217$	−26.3876			−1.79131
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.39128	−	9.33797i	0.362657	−	0.628140i
$222$		0			0
$223$	−7.42411	−	12.8589i	−0.497155	−	0.861098i	0.502840	−	0.864380i	$-0.332289\pi$
$223$	−7.42411	−	12.8589i	−0.497155	−	0.861098i	−0.999995	+	0.00328197i	$0.998955\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.49314	+	16.4426i	0.630082	+	1.09133i	0.987535	+	0.157403i	$0.0503121\pi$
$227$	9.49314	+	16.4426i	0.630082	+	1.09133i	−0.357453	+	0.933931i	$0.616355\pi$
$228$		0			0
$229$	−1.17194	+	2.02986i	−0.0774441	+	0.134137i	−0.902147	−	0.431430i	$-0.858009\pi$
$229$	−1.17194	+	2.02986i	−0.0774441	+	0.134137i	0.824702	+	0.565567i	$0.191343\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.0912			−0.792120			−0.396060	−	0.918225i	$-0.629623\pi$
$233$	−12.0912			−0.792120			−0.396060	+	0.918225i	$0.629623\pi$
$234$		0			0
$235$	−12.7433			−0.831281
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.54375	−	13.0662i	0.487965	−	0.845180i	−0.511939	−	0.859022i	$-0.671073\pi$
$239$	7.54375	−	13.0662i	0.487965	−	0.845180i	0.999904	+	0.0138417i	$0.00440610\pi$
$240$		0			0
$241$	1.01095	+	1.75101i	0.0651208	+	0.112793i	0.896748	−	0.442542i	$-0.145923\pi$
$241$	1.01095	+	1.75101i	0.0651208	+	0.112793i	−0.831627	+	0.555335i	$0.812590\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	23.7627	+	41.1581i	1.51814	+	2.62950i
$246$		0			0
$247$	−9.26600	+	16.0492i	−0.589581	+	1.02118i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.7796			−1.18536			−0.592679	−	0.805438i	$-0.701930\pi$
$251$	−18.7796			−1.18536			−0.592679	+	0.805438i	$0.701930\pi$
$252$		0			0
$253$	2.89765			0.182174
$254$		0			0
$255$		0			0
$256$		0			0
$257$	11.5006	−	19.9197i	0.717389	−	1.24255i	−0.244642	−	0.969613i	$-0.578671\pi$
$257$	11.5006	−	19.9197i	0.717389	−	1.24255i	0.962031	−	0.272940i	$-0.0879961\pi$
$258$		0			0
$259$	−0.600642	−	1.04034i	−0.0373221	−	0.0646437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−9.77774	−	16.9356i	−0.602922	−	1.04429i	−0.992376	−	0.123245i	$-0.960670\pi$
$263$	−9.77774	−	16.9356i	−0.602922	−	1.04429i	0.389455	−	0.921046i	$-0.372663\pi$
$264$		0			0
$265$	12.5749	−	21.7803i	0.772469	−	1.33796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	13.2107			0.805468			0.402734	−	0.915317i	$-0.368060\pi$
$269$	13.2107			0.805468			0.402734	+	0.915317i	$0.368060\pi$
$270$		0			0
$271$	8.63055			0.524268			0.262134	−	0.965031i	$-0.415574\pi$
$271$	8.63055			0.524268			0.262134	+	0.965031i	$0.415574\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.77203	+	4.80130i	−0.167160	+	0.289529i
$276$		0			0
$277$	−1.21650	−	2.10705i	−0.0730927	−	0.126600i	0.827163	−	0.561963i	$-0.189954\pi$
$277$	−1.21650	−	2.10705i	−0.0730927	−	0.126600i	−0.900255	+	0.435362i	$0.856620\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.75715	+	8.23962i	0.283787	+	0.491534i	0.972314	−	0.233676i	$-0.0750756\pi$
$281$	4.75715	+	8.23962i	0.283787	+	0.491534i	−0.688527	+	0.725211i	$0.741742\pi$
$282$		0			0
$283$	5.27042	−	9.12864i	0.313294	−	0.542641i	−0.665779	−	0.746149i	$-0.731901\pi$
$283$	5.27042	−	9.12864i	0.313294	−	0.542641i	0.979073	+	0.203507i	$0.0652341\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−31.9378			−1.88523
$288$		0			0
$289$	−11.0748			−0.651460
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.10888	+	5.38474i	−0.181623	+	0.314580i	−0.942433	−	0.334394i	$-0.891468\pi$
$293$	−3.10888	+	5.38474i	−0.181623	+	0.314580i	0.760811	+	0.648974i	$0.224802\pi$
$294$		0			0
$295$	−14.4975	−	25.1104i	−0.844077	−	1.46198i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.52775	+	4.37819i	0.146184	+	0.253197i
$300$		0			0
$301$	27.7138	−	48.0017i	1.59740	−	2.76677i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	22.3018			1.27700
$306$		0			0
$307$	−2.45594			−0.140168			−0.0700839	−	0.997541i	$-0.522327\pi$
$307$	−2.45594			−0.140168			−0.0700839	+	0.997541i	$0.522327\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.06337	−	3.57386i	0.117003	−	0.202655i	−0.801576	−	0.597893i	$-0.796005\pi$
$311$	2.06337	−	3.57386i	0.117003	−	0.202655i	0.918579	+	0.395238i	$0.129338\pi$
$312$		0			0
$313$	2.66061	+	4.60831i	0.150386	+	0.260477i	0.931370	−	0.364075i	$-0.118615\pi$
$313$	2.66061	+	4.60831i	0.150386	+	0.260477i	−0.780983	+	0.624552i	$0.785282\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.1291	−	17.5440i	−0.568904	−	0.985371i	−0.996675	−	0.0814836i	$-0.974034\pi$
$317$	−10.1291	−	17.5440i	−0.568904	−	0.985371i	0.427770	−	0.903887i	$-0.359299\pi$
$318$		0			0
$319$	−7.62784	+	13.2118i	−0.427077	+	0.739719i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−10.1836			−0.566631
$324$		0			0
$325$	−9.67266			−0.536543
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.8225	+	20.4771i	−0.651794	+	1.12894i
$330$		0			0
$331$	4.88990	+	8.46955i	0.268773	+	0.465529i	0.968545	−	0.248837i	$-0.0800485\pi$
$331$	4.88990	+	8.46955i	0.268773	+	0.465529i	−0.699772	+	0.714366i	$0.746715\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.03587	−	5.25828i	−0.165867	−	0.287290i
$336$		0			0
$337$	−7.61501	+	13.1896i	−0.414816	+	0.718482i	−0.995409	−	0.0957115i	$-0.969487\pi$
$337$	−7.61501	+	13.1896i	−0.414816	+	0.718482i	0.580593	+	0.814194i	$0.302821\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−13.4718			−0.729538
$342$		0			0
$343$	53.3706			2.88174
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.12881	−	5.41925i	0.167963	−	0.290921i	−0.769741	−	0.638357i	$-0.779614\pi$
$347$	3.12881	−	5.41925i	0.167963	−	0.290921i	0.937704	+	0.347436i	$0.112948\pi$
$348$		0			0
$349$	−10.0277	−	17.3685i	−0.536772	−	0.929716i	−0.999075	−	0.0429941i	$-0.986310\pi$
$349$	−10.0277	−	17.3685i	−0.536772	−	0.929716i	0.462304	−	0.886722i	$-0.347023\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.26961	+	16.0554i	0.493372	+	0.854545i	0.999971	−	0.00763681i	$-0.00243089\pi$
$353$	9.26961	+	16.0554i	0.493372	+	0.854545i	−0.506599	+	0.862182i	$0.669098\pi$
$354$		0			0
$355$	−6.06587	+	10.5064i	−0.321943	+	0.557621i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.2838			−1.33443			−0.667213	−	0.744867i	$-0.732513\pi$
$359$	−25.2838			−1.33443			−0.667213	+	0.744867i	$0.732513\pi$
$360$		0			0
$361$	−1.49741			−0.0788112
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.48573	+	7.76951i	−0.234794	+	0.406675i
$366$		0			0
$367$	−6.32458	−	10.9545i	−0.330141	−	0.571820i	0.652399	−	0.757876i	$-0.273763\pi$
$367$	−6.32458	−	10.9545i	−0.330141	−	0.571820i	−0.982539	+	0.186056i	$0.940429\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−23.3325	−	40.4130i	−1.21136	−	2.09814i
$372$		0			0
$373$	−9.39906	+	16.2796i	−0.486665	+	0.842928i	−0.999882	−	0.0153303i	$-0.995120\pi$
$373$	−9.39906	+	16.2796i	−0.486665	+	0.842928i	0.513218	+	0.858258i	$0.328453\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−26.6164			−1.37081
$378$		0			0
$379$	2.15057			0.110467			0.0552337	−	0.998473i	$-0.482410\pi$
$379$	2.15057			0.110467			0.0552337	+	0.998473i	$0.482410\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.2847	−	28.2060i	0.832111	−	1.44126i	−0.0642505	−	0.997934i	$-0.520466\pi$
$383$	16.2847	−	28.2060i	0.832111	−	1.44126i	0.896361	−	0.443324i	$-0.146201\pi$
$384$		0			0
$385$	16.9209	+	29.3078i	0.862367	+	1.49366i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.9569	−	20.7100i	−0.606241	−	1.05004i	−0.991854	−	0.127380i	$-0.959343\pi$
$389$	−11.9569	−	20.7100i	−0.606241	−	1.05004i	0.385613	−	0.922661i	$-0.373990\pi$
$390$		0			0
$391$	−1.38904	+	2.40588i	−0.0702466	+	0.121671i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.8403			0.847329
$396$		0			0
$397$	4.42557			0.222113			0.111056	−	0.993814i	$-0.464577\pi$
$397$	4.42557			0.222113			0.111056	+	0.993814i	$0.464577\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.04462	−	5.27344i	0.152041	−	0.263343i	−0.779936	−	0.625859i	$-0.784749\pi$
$401$	3.04462	−	5.27344i	0.152041	−	0.263343i	0.931978	+	0.362515i	$0.118082\pi$
$402$		0			0
$403$	−11.7520	−	20.3551i	−0.585411	−	1.01396i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.306648	−	0.531130i	−0.0152000	−	0.0263272i
$408$		0			0
$409$	−1.23061	+	2.13148i	−0.0608497	+	0.105395i	−0.894845	−	0.446376i	$-0.852714\pi$
$409$	−1.23061	+	2.13148i	−0.0608497	+	0.105395i	0.833996	+	0.551771i	$0.186048\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−53.7997			−2.64731
$414$		0			0
$415$	−3.95734			−0.194258
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.75667	−	16.8991i	0.476645	−	0.825573i	−0.522997	−	0.852334i	$-0.675186\pi$
$419$	9.75667	−	16.8991i	0.476645	−	0.825573i	0.999642	+	0.0267617i	$0.00851954\pi$
$420$		0			0
$421$	9.52846	+	16.5038i	0.464389	+	0.804345i	0.999174	−	0.0406435i	$-0.0129408\pi$
$421$	9.52846	+	16.5038i	0.464389	+	0.804345i	−0.534785	+	0.844988i	$0.679607\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.65764	−	4.60316i	−0.128914	−	0.223286i
$426$		0			0
$427$	20.6903	−	35.8367i	1.00128	−	1.73426i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	23.6111			1.13731			0.568654	−	0.822577i	$-0.307464\pi$
$431$	23.6111			1.13731			0.568654	+	0.822577i	$0.307464\pi$
$432$		0			0
$433$	−2.09660			−0.100756			−0.0503780	−	0.998730i	$-0.516043\pi$
$433$	−2.09660			−0.100756			−0.0503780	+	0.998730i	$0.516043\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.38734	−	4.13499i	0.114202	−	0.197804i
$438$		0			0
$439$	−4.38026	−	7.58684i	−0.209059	−	0.362100i	0.742360	−	0.670002i	$-0.233707\pi$
$439$	−4.38026	−	7.58684i	−0.209059	−	0.362100i	−0.951418	+	0.307901i	$0.900373\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.51251	−	7.81590i	−0.214396	−	0.371345i	0.738690	−	0.674046i	$-0.235445\pi$
$443$	−4.51251	−	7.81590i	−0.214396	−	0.371345i	−0.953086	+	0.302701i	$0.902112\pi$
$444$		0			0
$445$	−19.9440	+	34.5440i	−0.945437	+	1.63754i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.54295			0.120009			0.0600047	−	0.998198i	$-0.480888\pi$
$449$	2.54295			0.120009			0.0600047	+	0.998198i	$0.480888\pi$
$450$		0			0
$451$	−16.3054			−0.767789
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−29.5216	+	51.1330i	−1.38400	+	2.39715i
$456$		0			0
$457$	12.9090	+	22.3590i	0.603855	+	1.04591i	0.992231	+	0.124407i	$0.0397029\pi$
$457$	12.9090	+	22.3590i	0.603855	+	1.04591i	−0.388376	+	0.921501i	$0.626964\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.4517	+	21.5669i	0.579932	+	1.00447i	0.995486	+	0.0949039i	$0.0302544\pi$
$461$	12.4517	+	21.5669i	0.579932	+	1.00447i	−0.415554	+	0.909568i	$0.636412\pi$
$462$		0			0
$463$	−11.0655	+	19.1660i	−0.514258	+	0.890721i	0.485605	+	0.874178i	$0.338599\pi$
$463$	−11.0655	+	19.1660i	−0.514258	+	0.890721i	−0.999863	+	0.0165428i	$0.994734\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.7135			1.14361			0.571803	−	0.820391i	$-0.306244\pi$
$467$	24.7135			1.14361			0.571803	+	0.820391i	$0.306244\pi$
$468$		0			0
$469$	−11.2660			−0.520215
$470$		0			0
$471$		0			0
$472$		0			0
$473$	14.1488	−	24.5065i	0.650564	−	1.12681i
$474$		0			0
$475$	4.56768	+	7.91146i	0.209580	+	0.363003i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.6023	+	33.9522i	0.895653	+	1.55132i	0.832995	+	0.553281i	$0.186624\pi$
$479$	19.6023	+	33.9522i	0.895653	+	1.55132i	0.0626578	+	0.998035i	$0.480042\pi$
$480$		0			0
$481$	0.535006	−	0.926657i	0.0243942	−	0.0422519i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	31.6204			1.43581
$486$		0			0
$487$	−29.3140			−1.32835			−0.664173	−	0.747579i	$-0.731216\pi$
$487$	−29.3140			−1.32835			−0.664173	+	0.747579i	$0.731216\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−2.57211	+	4.45503i	−0.116078	+	0.201052i	−0.918210	−	0.396094i	$-0.870365\pi$
$491$	−2.57211	+	4.45503i	−0.116078	+	0.201052i	0.802132	+	0.597146i	$0.203699\pi$
$492$		0			0
$493$	−7.31306	−	12.6666i	−0.329364	−	0.570474i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	11.2551	+	19.4944i	0.504861	+	0.874445i
$498$		0			0
$499$	17.5877	−	30.4627i	0.787332	−	1.36370i	−0.140263	−	0.990114i	$-0.544795\pi$
$499$	17.5877	−	30.4627i	0.787332	−	1.36370i	0.927596	−	0.373585i	$-0.121872\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−13.2366			−0.590192			−0.295096	−	0.955468i	$-0.595352\pi$
$503$	−13.2366			−0.590192			−0.295096	+	0.955468i	$0.595352\pi$
$504$		0			0
$505$	31.1555			1.38640
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.8738	−	27.4942i	0.703593	−	1.21866i	−0.263604	−	0.964631i	$-0.584911\pi$
$509$	15.8738	−	27.4942i	0.703593	−	1.21866i	0.967197	−	0.254028i	$-0.0817554\pi$
$510$		0			0
$511$	8.32319	+	14.4162i	0.368196	+	0.637735i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	25.5951	+	44.3321i	1.12786	+	1.95350i
$516$		0			0
$517$	−6.03578	+	10.4543i	−0.265453	+	0.459778i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.83339			0.124133			0.0620666	−	0.998072i	$-0.480231\pi$
$521$	2.83339			0.124133			0.0620666	+	0.998072i	$0.480231\pi$
$522$		0			0
$523$	32.0242			1.40032			0.700160	−	0.713986i	$-0.253112\pi$
$523$	32.0242			1.40032			0.700160	+	0.713986i	$0.253112\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.45792	−	11.1854i	0.281311	−	0.487246i
$528$		0			0
$529$	10.8487	+	18.7906i	0.471684	+	0.816981i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−14.2239	−	24.6365i	−0.616105	−	1.06713i
$534$		0			0
$535$	6.69646	−	11.5986i	0.289513	−	0.501452i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	45.0201			1.93915
$540$		0			0
$541$	−3.84546			−0.165329			−0.0826646	−	0.996577i	$-0.526343\pi$
$541$	−3.84546			−0.165329			−0.0826646	+	0.996577i	$0.526343\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.89049	+	5.00648i	−0.123815	+	0.214454i
$546$		0			0
$547$	22.4749	+	38.9277i	0.960958	+	1.66443i	0.720102	+	0.693868i	$0.244095\pi$
$547$	22.4749	+	38.9277i	0.960958	+	1.66443i	0.240856	+	0.970561i	$0.422572\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	12.5690	+	21.7701i	0.535456	+	0.927437i
$552$		0			0
$553$	15.6235	−	27.0607i	0.664378	−	1.15074i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−35.8629			−1.51956			−0.759780	−	0.650180i	$-0.774693\pi$
$557$	−35.8629			−1.51956			−0.759780	+	0.650180i	$0.774693\pi$
$558$		0			0
$559$	49.3706			2.08816
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.11092	+	8.85237i	−0.215400	+	0.373083i	−0.953396	−	0.301721i	$-0.902439\pi$
$563$	−5.11092	+	8.85237i	−0.215400	+	0.373083i	0.737997	+	0.674805i	$0.235772\pi$
$564$		0			0
$565$	27.4593	+	47.5609i	1.15522	+	2.00090i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.4319	−	19.8006i	−0.479250	−	0.830085i	0.520467	−	0.853882i	$-0.325758\pi$
$569$	−11.4319	−	19.8006i	−0.479250	−	0.830085i	−0.999717	+	0.0237967i	$0.992425\pi$
$570$		0			0
$571$	−1.63442	+	2.83091i	−0.0683985	+	0.118470i	−0.898197	−	0.439594i	$-0.855122\pi$
$571$	−1.63442	+	2.83091i	−0.0683985	+	0.118470i	0.829798	+	0.558064i	$0.188456\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.49211			0.103928
$576$		0			0
$577$	−40.2432			−1.67535			−0.837673	−	0.546172i	$-0.816084\pi$
$577$	−40.2432			−1.67535			−0.837673	+	0.546172i	$0.816084\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.67139	+	6.35903i	−0.152315	+	0.263817i
$582$		0			0
$583$	−11.9120	−	20.6322i	−0.493346	−	0.854500i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.3312	−	23.0903i	−0.550236	−	0.953037i	−0.998257	−	0.0590141i	$-0.981204\pi$
$587$	−13.3312	−	23.0903i	−0.550236	−	0.953037i	0.448021	−	0.894023i	$-0.352129\pi$
$588$		0			0
$589$	−11.0992	+	19.2244i	−0.457336	+	0.792129i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−36.4349			−1.49620			−0.748101	−	0.663585i	$-0.769034\pi$
$593$	−36.4349			−1.49620			−0.748101	+	0.663585i	$0.769034\pi$
$594$		0			0
$595$	−32.4452			−1.33012
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.51357	−	11.2818i	0.266137	−	0.460963i	−0.701724	−	0.712449i	$-0.747586\pi$
$599$	6.51357	−	11.2818i	0.266137	−	0.460963i	0.967861	+	0.251486i	$0.0809193\pi$
$600$		0			0
$601$	−23.1094	−	40.0267i	−0.942653	−	1.63272i	−0.760384	−	0.649473i	$-0.774989\pi$
$601$	−23.1094	−	40.0267i	−0.942653	−	1.63272i	−0.182268	−	0.983249i	$-0.558344\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.10256	−	10.5699i	−0.248104	−	0.429729i
$606$		0			0
$607$	6.32515	−	10.9555i	0.256730	−	0.444669i	−0.708634	−	0.705576i	$-0.750688\pi$
$607$	6.32515	−	10.9555i	0.256730	−	0.444669i	0.965364	+	0.260907i	$0.0840215\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.0611			−0.852041
$612$		0			0
$613$	−16.2013			−0.654365			−0.327182	−	0.944961i	$-0.606099\pi$
$613$	−16.2013			−0.654365			−0.327182	+	0.944961i	$0.606099\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.54706	−	4.41164i	0.102541	−	0.177606i	−0.810190	−	0.586167i	$-0.800636\pi$
$617$	2.54706	−	4.41164i	0.102541	−	0.177606i	0.912731	+	0.408561i	$0.133969\pi$
$618$		0			0
$619$	12.5926	+	21.8111i	0.506140	+	0.876660i	0.999975	+	0.00710457i	$0.00226147\pi$
$619$	12.5926	+	21.8111i	0.506140	+	0.876660i	−0.493835	+	0.869556i	$0.664405\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	37.0058	+	64.0959i	1.48260	+	2.56795i
$624$		0			0
$625$	15.5749	−	26.9766i	0.622998	−	1.07906i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.587987			0.0234446
$630$		0			0
$631$	18.4509			0.734518			0.367259	−	0.930119i	$-0.380296\pi$
$631$	18.4509			0.734518			0.367259	+	0.930119i	$0.380296\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.0241	+	22.5584i	−0.516846	+	0.895204i
$636$		0			0
$637$	39.2731	+	68.0230i	1.55606	+	2.69517i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.2431	+	29.8659i	0.681062	+	1.17963i	0.974657	+	0.223704i	$0.0718149\pi$
$641$	17.2431	+	29.8659i	0.681062	+	1.17963i	−0.293595	+	0.955930i	$0.594852\pi$
$642$		0			0
$643$	5.28665	−	9.15675i	0.208485	−	0.361107i	−0.742752	−	0.669566i	$-0.766480\pi$
$643$	5.28665	−	9.15675i	0.208485	−	0.361107i	0.951238	+	0.308459i	$0.0998133\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.5011			0.570098			0.285049	−	0.958513i	$-0.407990\pi$
$647$	14.5011			0.570098			0.285049	+	0.958513i	$0.407990\pi$
$648$		0			0
$649$	−27.4666			−1.07816
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.29761	−	5.71162i	0.129045	−	0.223513i	−0.794262	−	0.607576i	$-0.792142\pi$
$653$	3.29761	−	5.71162i	0.129045	−	0.223513i	0.923307	+	0.384063i	$0.125475\pi$
$654$		0			0
$655$	−9.35986	−	16.2117i	−0.365720	−	0.633445i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.5531	+	26.9387i	0.605861	+	1.04938i	0.991915	+	0.126906i	$0.0405047\pi$
$659$	15.5531	+	26.9387i	0.605861	+	1.04938i	−0.386053	+	0.922476i	$0.626162\pi$
$660$		0			0
$661$	17.1695	−	29.7384i	0.667816	−	1.15669i	−0.310698	−	0.950509i	$-0.600563\pi$
$661$	17.1695	−	29.7384i	0.667816	−	1.15669i	0.978514	−	0.206182i	$-0.0661040\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	55.7636			2.16242
$666$		0			0
$667$	6.85759			0.265527
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.5631	−	18.2959i	0.407785	−	0.706304i
$672$		0			0
$673$	6.32809	+	10.9606i	0.243930	+	0.422499i	0.961830	−	0.273647i	$-0.0882299\pi$
$673$	6.32809	+	10.9606i	0.243930	+	0.422499i	−0.717900	+	0.696146i	$0.754897\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.78910	−	11.7591i	−0.260926	−	0.451938i	0.705562	−	0.708648i	$-0.250695\pi$
$677$	−6.78910	−	11.7591i	−0.260926	−	0.451938i	−0.966488	+	0.256710i	$0.917361\pi$
$678$		0			0
$679$	29.3356	−	50.8107i	1.12580	−	1.94994i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3.85174			−0.147383			−0.0736914	−	0.997281i	$-0.523478\pi$
$683$	−3.85174			−0.147383			−0.0736914	+	0.997281i	$0.523478\pi$
$684$		0			0
$685$	−60.0755			−2.29537
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20.7828	−	35.9968i	0.791761	−	1.37137i
$690$		0			0
$691$	20.3658	+	35.2746i	0.774752	+	1.34191i	0.934934	+	0.354822i	$0.115458\pi$
$691$	20.3658	+	35.2746i	0.774752	+	1.34191i	−0.160182	+	0.987087i	$0.551208\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.88342	+	11.9224i	0.261103	+	0.452244i
$696$		0			0
$697$	7.81624	−	13.5381i	0.296061	−	0.512793i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	17.5260			0.661948			0.330974	−	0.943640i	$-0.392623\pi$
$701$	17.5260			0.661948			0.330974	+	0.943640i	$0.392623\pi$
$702$		0			0
$703$	−1.01057			−0.0381146
$704$		0			0
$705$		0			0
$706$		0			0
$707$	28.9043	−	50.0636i	1.08706	−	1.88284i
$708$		0			0
$709$	−11.5020	−	19.9220i	−0.431965	−	0.748185i	0.565078	−	0.825038i	$-0.308846\pi$
$709$	−11.5020	−	19.9220i	−0.431965	−	0.748185i	−0.997042	+	0.0768527i	$0.975513\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.02786	+	5.24440i	0.113394	+	0.196404i
$714$		0			0
$715$	−15.0718	+	26.1051i	−0.563654	+	0.976277i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.6172			0.433250			0.216625	−	0.976255i	$-0.430495\pi$
$719$	11.6172			0.433250			0.216625	+	0.976255i	$0.430495\pi$
$720$		0			0
$721$	94.9826			3.53734
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.56030	+	11.3628i	−0.243643	+	0.422003i
$726$		0			0
$727$	23.5416	+	40.7752i	0.873109	+	1.51227i	0.858763	+	0.512373i	$0.171233\pi$
$727$	23.5416	+	40.7752i	0.873109	+	1.51227i	0.0143459	+	0.999897i	$0.495433\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	13.5650	+	23.4952i	0.501718	+	0.869001i
$732$		0			0
$733$	12.5946	−	21.8145i	0.465193	−	0.805737i	−0.534018	−	0.845473i	$-0.679318\pi$
$733$	12.5946	−	21.8145i	0.465193	−	0.805737i	0.999210	+	0.0397362i	$0.0126518\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.75168			−0.211866
$738$		0			0
$739$	−9.73735			−0.358194			−0.179097	−	0.983831i	$-0.557318\pi$
$739$	−9.73735			−0.358194			−0.179097	+	0.983831i	$0.557318\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.0084	+	31.1915i	−0.660666	+	1.14431i	0.319775	+	0.947493i	$0.396393\pi$
$743$	−18.0084	+	31.1915i	−0.660666	+	1.14431i	−0.980441	+	0.196813i	$0.936941\pi$
$744$		0			0
$745$	−2.90970	−	5.03974i	−0.106603	−	0.184642i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12.4252	−	21.5210i	−0.454006	−	0.786361i
$750$		0			0
$751$	14.3734	−	24.8954i	0.524493	−	0.908448i	−0.475101	−	0.879931i	$-0.657588\pi$
$751$	14.3734	−	24.8954i	0.524493	−	0.908448i	0.999593	−	0.0285164i	$-0.00907828\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.6446			0.605761
$756$		0			0
$757$	−53.3478			−1.93896			−0.969479	−	0.245174i	$-0.921155\pi$
$757$	−53.3478			−1.93896			−0.969479	+	0.245174i	$0.921155\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.4580	+	23.3100i	−0.487853	+	0.844986i	−0.999902	−	0.0139702i	$-0.995553\pi$
$761$	−13.4580	+	23.3100i	−0.487853	+	0.844986i	0.512050	+	0.858956i	$0.328886\pi$
$762$		0			0
$763$	5.36326	+	9.28943i	0.194163	+	0.336300i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−23.9603	−	41.5005i	−0.865157	−	1.49850i
$768$		0			0
$769$	−4.56541	+	7.90752i	−0.164633	+	0.285152i	−0.936525	−	0.350601i	$-0.885977\pi$
$769$	−4.56541	+	7.90752i	−0.164633	+	0.285152i	0.771892	+	0.635754i	$0.219311\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	20.5628			0.739591			0.369796	−	0.929113i	$-0.379428\pi$
$773$	20.5628			0.739591			0.369796	+	0.929113i	$0.379428\pi$
$774$		0			0
$775$	−11.5864			−0.416194
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−13.4338	+	23.2680i	−0.481315	+	0.833663i
$780$		0			0
$781$	5.74612	+	9.95257i	0.205612	+	0.356131i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−21.3769	−	37.0258i	−0.762974	−	1.32151i
$786$		0			0
$787$	−3.17998	+	5.50789i	−0.113354	+	0.196335i	−0.917121	−	0.398610i	$-0.869493\pi$
$787$	−3.17998	+	5.50789i	−0.113354	+	0.196335i	0.803767	+	0.594945i	$0.202826\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	101.900			3.62316
$792$		0			0
$793$	36.8587			1.30889
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.3241	−	17.8819i	0.365699	−	0.633409i	−0.623189	−	0.782071i	$-0.714163\pi$
$797$	10.3241	−	17.8819i	0.365699	−	0.633409i	0.988888	+	0.148662i	$0.0474967\pi$
$798$		0			0
$799$	−5.78670	−	10.0229i	−0.204719	−	0.354583i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.24928	+	7.35996i	0.149954	+	0.259727i
$804$		0			0
$805$	7.60611	−	13.1742i	0.268080	−	0.464328i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.2805			1.59198			0.795989	−	0.605311i	$-0.206951\pi$
$809$	45.2805			1.59198			0.795989	+	0.605311i	$0.206951\pi$
$810$		0			0
$811$	4.29363			0.150770			0.0753848	−	0.997155i	$-0.475981\pi$
$811$	4.29363			0.150770			0.0753848	+	0.997155i	$0.475981\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	6.33861	−	10.9788i	0.222032	−	0.384570i
$816$		0			0
$817$	−23.3141	−	40.3812i	−0.815657	−	1.41276i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.49228	+	14.7091i	0.296383	+	0.513350i	0.975306	−	0.220860i	$-0.0708863\pi$
$821$	8.49228	+	14.7091i	0.296383	+	0.513350i	−0.678923	+	0.734210i	$0.737553\pi$
$822$		0			0
$823$	−2.98907	+	5.17721i	−0.104192	+	0.180466i	−0.913408	−	0.407045i	$-0.866559\pi$
$823$	−2.98907	+	5.17721i	−0.104192	+	0.180466i	0.809216	+	0.587512i	$0.199892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	11.4590			0.398469			0.199234	−	0.979952i	$-0.436155\pi$
$827$	11.4590			0.398469			0.199234	+	0.979952i	$0.436155\pi$
$828$		0			0
$829$	−34.5859			−1.20122			−0.600609	−	0.799543i	$-0.705075\pi$
$829$	−34.5859			−1.20122			−0.600609	+	0.799543i	$0.705075\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.5811	+	37.3796i	−0.747742	+	1.29513i
$834$		0			0
$835$	−3.17903	−	5.50625i	−0.110015	−	0.190551i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.1093	−	20.9740i	−0.418060	−	0.724102i	0.577684	−	0.816261i	$-0.303957\pi$
$839$	−12.1093	−	20.9740i	−0.418060	−	0.724102i	−0.995744	+	0.0921587i	$0.970623\pi$
$840$		0			0
$841$	−3.55207	+	6.15236i	−0.122485	+	0.212150i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−17.7483			−0.610561
$846$		0			0
$847$	−22.6464			−0.778139
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.137842	+	0.238749i	−0.00472515	+	0.00818420i
$852$		0			0
$853$	−26.7241	−	46.2875i	−0.915016	−	1.58485i	−0.806878	−	0.590719i	$-0.798844\pi$
$853$	−26.7241	−	46.2875i	−0.915016	−	1.58485i	−0.108138	−	0.994136i	$-0.534489\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.79836	−	3.11486i	−0.0614310	−	0.106402i	0.833674	−	0.552256i	$-0.186233\pi$
$857$	−1.79836	−	3.11486i	−0.0614310	−	0.106402i	−0.895105	+	0.445855i	$0.852900\pi$
$858$		0			0
$859$	15.9691	−	27.6594i	0.544860	−	0.943726i	−0.453755	−	0.891126i	$-0.649916\pi$
$859$	15.9691	−	27.6594i	0.544860	−	0.943726i	0.998616	−	0.0525994i	$-0.0167506\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.20625			0.0751016			0.0375508	−	0.999295i	$-0.488044\pi$
$863$	2.20625			0.0751016			0.0375508	+	0.999295i	$0.488044\pi$
$864$		0			0
$865$	−6.40722			−0.217852
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.97632	−	13.8154i	0.270578	−	0.468655i
$870$		0			0
$871$	−5.01744	−	8.69047i	−0.170010	−	0.294465i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.7699	−	32.5105i	−0.634539	−	1.09905i
$876$		0			0
$877$	3.07129	−	5.31962i	0.103710	−	0.179631i	−0.809500	−	0.587119i	$-0.800262\pi$
$877$	3.07129	−	5.31962i	0.103710	−	0.179631i	0.913210	+	0.407488i	$0.133595\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	38.6422			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$881$	38.6422			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$882$		0			0
$883$	38.4651			1.29446			0.647228	−	0.762297i	$-0.275928\pi$
$883$	38.4651			1.29446			0.647228	+	0.762297i	$0.275928\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.1015	−	43.4770i	0.842825	−	1.45982i	−0.0446713	−	0.999002i	$-0.514224\pi$
$887$	25.1015	−	43.4770i	0.842825	−	1.45982i	0.887497	−	0.460814i	$-0.152443\pi$
$888$		0			0
$889$	24.1660	+	41.8568i	0.810502	+	1.40383i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9.94561	+	17.2263i	0.332817	+	0.576456i
$894$		0			0
$895$	−1.60234	+	2.77533i	−0.0535603	+	0.0927691i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−31.8824			−1.06334
$900$		0			0
$901$	22.8409			0.760941
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.93361	−	12.0094i	0.230481	−	0.399205i
$906$		0			0
$907$	−28.5437	−	49.4391i	−0.947777	−	1.64160i	−0.750093	−	0.661332i	$-0.769991\pi$
$907$	−28.5437	−	49.4391i	−0.947777	−	1.64160i	−0.197684	−	0.980266i	$-0.563342\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.15893	−	14.1317i	−0.270317	−	0.468204i	0.698626	−	0.715487i	$-0.253795\pi$
$911$	−8.15893	−	14.1317i	−0.270317	−	0.468204i	−0.968943	+	0.247284i	$0.920462\pi$
$912$		0			0
$913$	−1.87437	+	3.24650i	−0.0620326	+	0.107444i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−34.7341			−1.14702
$918$		0			0
$919$	25.5992			0.844441			0.422220	−	0.906493i	$-0.361251\pi$
$919$	25.5992			0.844441			0.422220	+	0.906493i	$0.361251\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.0252	+	17.3641i	−0.329983	+	0.571548i
$924$		0			0
$925$	−0.263732	−	0.456797i	−0.00867145	−	0.0150194i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.3480	+	21.3874i	0.405126	+	0.701699i	0.994336	−	0.106281i	$-0.0338944\pi$
$929$	12.3480	+	21.3874i	0.405126	+	0.701699i	−0.589210	+	0.807980i	$0.700561\pi$
$930$		0			0
$931$	37.0915	−	64.2444i	1.21563	−	2.10553i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16.5644			−0.541713
$936$		0			0
$937$	−27.3574			−0.893728			−0.446864	−	0.894602i	$-0.647459\pi$
$937$	−27.3574			−0.893728			−0.446864	+	0.894602i	$0.647459\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.2953	+	21.2962i	−0.400817	+	0.694235i	−0.993825	−	0.110962i	$-0.964607\pi$
$941$	−12.2953	+	21.2962i	−0.400817	+	0.694235i	0.593008	+	0.805197i	$0.297940\pi$
$942$		0			0
$943$	3.66472	+	6.34747i	0.119340	+	0.206702i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.0107	+	17.3391i	0.325305	+	0.563445i	0.981574	−	0.191082i	$-0.0611997\pi$
$947$	10.0107	+	17.3391i	0.325305	+	0.563445i	−0.656269	+	0.754527i	$0.727866\pi$
$948$		0			0
$949$	−7.41366	+	12.8408i	−0.240658	+	0.416831i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.59573			0.246050			0.123025	−	0.992404i	$-0.460741\pi$
$953$	7.59573			0.246050			0.123025	+	0.992404i	$0.460741\pi$
$954$		0			0
$955$	3.50963			0.113569
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−55.7345	+	96.5350i	−1.79976	+	3.11728i
$960$		0			0
$961$	1.42287	+	2.46448i	0.0458990	+	0.0794994i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−26.9969	−	46.7600i	−0.869061	−	1.50526i
$966$		0			0
$967$	−10.2203	+	17.7020i	−0.328662	+	0.569259i	−0.982247	−	0.187594i	$-0.939931\pi$
$967$	−10.2203	+	17.7020i	−0.328662	+	0.569259i	0.653585	+	0.756853i	$0.273264\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	47.8189			1.53458			0.767290	−	0.641300i	$-0.221605\pi$
$971$	47.8189			1.53458			0.767290	+	0.641300i	$0.221605\pi$
$972$		0			0
$973$	25.5441			0.818907
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5.33054	−	9.23277i	0.170539	−	0.295382i	−0.768069	−	0.640367i	$-0.778782\pi$
$977$	5.33054	−	9.23277i	0.170539	−	0.295382i	0.938608	+	0.344984i	$0.112116\pi$
$978$		0			0
$979$	18.8927	+	32.7231i	0.603813	+	1.04584i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11.9147	−	20.6369i	−0.380020	−	0.658214i	0.611045	−	0.791596i	$-0.290750\pi$
$983$	−11.9147	−	20.6369i	−0.380020	−	0.658214i	−0.991065	+	0.133382i	$0.957416\pi$
$984$		0			0
$985$	9.99755	−	17.3163i	0.318549	−	0.551742i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.7201			−0.404476
$990$		0			0
$991$	11.3401			0.360231			0.180116	−	0.983645i	$-0.442353\pi$
$991$	11.3401			0.360231			0.180116	+	0.983645i	$0.442353\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.65503	−	8.06276i	0.147575	−	0.255607i
$996$		0			0
$997$	16.0454	+	27.7914i	0.508163	+	0.880164i	0.999955	+	0.00945137i	$0.00300851\pi$
$997$	16.0454	+	27.7914i	0.508163	+	0.880164i	−0.491793	+	0.870712i	$0.663658\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3456.2.i.e.2305.5		10
3.2	odd	2		1152.2.i.e.769.1	yes	10
4.3	odd	2		3456.2.i.h.2305.5		10
8.3	odd	2		3456.2.i.g.2305.1		10
8.5	even	2		3456.2.i.f.2305.1		10
9.2	odd	6		1152.2.i.e.385.1	✓	10
9.7	even	3	inner	3456.2.i.e.1153.5		10
12.11	even	2		1152.2.i.h.769.5	yes	10
24.5	odd	2		1152.2.i.g.769.5	yes	10
24.11	even	2		1152.2.i.f.769.1	yes	10
36.7	odd	6		3456.2.i.h.1153.5		10
36.11	even	6		1152.2.i.h.385.5	yes	10
72.11	even	6		1152.2.i.f.385.1	yes	10
72.29	odd	6		1152.2.i.g.385.5	yes	10
72.43	odd	6		3456.2.i.g.1153.1		10
72.61	even	6		3456.2.i.f.1153.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.2.i.e.385.1	✓	10	9.2	odd	6
1152.2.i.e.769.1	yes	10	3.2	odd	2
1152.2.i.f.385.1	yes	10	72.11	even	6
1152.2.i.f.769.1	yes	10	24.11	even	2
1152.2.i.g.385.5	yes	10	72.29	odd	6
1152.2.i.g.769.5	yes	10	24.5	odd	2
1152.2.i.h.385.5	yes	10	36.11	even	6
1152.2.i.h.769.5	yes	10	12.11	even	2
3456.2.i.e.1153.5		10	9.7	even	3	inner
3456.2.i.e.2305.5		10	1.1	even	1	trivial
3456.2.i.f.1153.1		10	72.61	even	6
3456.2.i.f.2305.1		10	8.5	even	2
3456.2.i.g.1153.1		10	72.43	odd	6
3456.2.i.g.2305.1		10	8.3	odd	2
3456.2.i.h.1153.5		10	36.7	odd	6
3456.2.i.h.2305.5		10	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.34011 - 2.32114i) q^{5} +(-2.48656 - 4.30684i) q^{7} +O(q^{10})\)	\(q+(1.34011 - 2.32114i) q^{5} +(-2.48656 - 4.30684i) q^{7} +(-1.26947 - 2.19879i) q^{11} +(2.21483 - 3.83620i) q^{13} +2.43417 q^{17} -4.18361 q^{19} +(-0.570641 + 0.988379i) q^{23} +(-1.09180 - 1.89106i) q^{25} +(-3.00434 - 5.20366i) q^{29} +(2.65303 - 4.59518i) q^{31} -13.3291 q^{35} +0.241556 q^{37} +(3.21105 - 5.56170i) q^{41} +(5.57273 + 9.65224i) q^{43} +(-2.37728 - 4.11757i) q^{47} +(-8.86592 + 15.3562i) q^{49} +9.38345 q^{53} -6.80494 q^{55} +(5.40906 - 9.36876i) q^{59} +(4.16044 + 7.20610i) q^{61} +(-5.93625 - 10.2819i) q^{65} +(1.13269 - 1.96188i) q^{67} -4.52639 q^{71} -3.34728 q^{73} +(-6.31322 + 10.9348i) q^{77} +(3.14159 + 5.44139i) q^{79} +(-0.738248 - 1.27868i) q^{83} +(3.26206 - 5.65005i) q^{85} -14.8823 q^{89} -22.0292 q^{91} +(-5.60651 + 9.71076i) q^{95} +(5.89884 + 10.2171i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{7}+O(q^{10}) \) 10 * q - 4 * q^7	\( 10 q - 4 q^{7} - q^{11} + 6 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{23} + q^{25} + 4 q^{29} - 8 q^{31} - 24 q^{35} - 20 q^{37} + 5 q^{41} + 13 q^{43} + 6 q^{47} + 3 q^{49} + 12 q^{55} - 13 q^{59} + 10 q^{61} + 17 q^{67} - 8 q^{71} - 34 q^{73} - 8 q^{77} - 6 q^{79} + 12 q^{83} + 18 q^{85} - 44 q^{89} - 36 q^{91} + 6 q^{95} + 27 q^{97}+O(q^{100}) \) 10 * q - 4 * q^7 - q^11 + 6 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^23 + q^25 + 4 * q^29 - 8 * q^31 - 24 * q^35 - 20 * q^37 + 5 * q^41 + 13 * q^43 + 6 * q^47 + 3 * q^49 + 12 * q^55 - 13 * q^59 + 10 * q^61 + 17 * q^67 - 8 * q^71 - 34 * q^73 - 8 * q^77 - 6 * q^79 + 12 * q^83 + 18 * q^85 - 44 * q^89 - 36 * q^91 + 6 * q^95 + 27 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more