LMFDB - Embedded newform 1568.2.i.b.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(961,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1568.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.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:	$12.5205430369$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1568.961
Dual form			1568.2.i.b.1537.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} -4.00000 q^{13} +(1.00000 + 1.73205i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(4.00000 - 6.92820i) q^{23} +(2.50000 + 4.33013i) q^{25} -4.00000 q^{27} +2.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(-4.00000 + 6.92820i) q^{33} +(-5.00000 + 8.66025i) q^{37} +(4.00000 + 6.92820i) q^{39} -10.0000 q^{41} -4.00000 q^{43} +(2.00000 - 3.46410i) q^{47} +(2.00000 - 3.46410i) q^{51} +(1.00000 + 1.73205i) q^{53} +12.0000 q^{57} +(5.00000 + 8.66025i) q^{59} +(4.00000 - 6.92820i) q^{61} +(-4.00000 - 6.92820i) q^{67} -16.0000 q^{69} +(3.00000 + 5.19615i) q^{73} +(5.00000 - 8.66025i) q^{75} +(-8.00000 + 13.8564i) q^{79} +(5.50000 + 9.52628i) q^{81} -2.00000 q^{83} +(-2.00000 - 3.46410i) q^{87} +(-9.00000 + 15.5885i) q^{89} +(-4.00000 + 6.92820i) q^{93} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^9	$ 2 q - 2 q^{3} - q^{9} - 4 q^{11} - 8 q^{13} + 2 q^{17} - 6 q^{19} + 8 q^{23} + 5 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} - 8 q^{33} - 10 q^{37} + 8 q^{39} - 20 q^{41} - 8 q^{43} + 4 q^{47} + 4 q^{51} + 2 q^{53} + 24 q^{57} + 10 q^{59} + 8 q^{61} - 8 q^{67} - 32 q^{69} + 6 q^{73} + 10 q^{75} - 16 q^{79} + 11 q^{81} - 4 q^{83} - 4 q^{87} - 18 q^{89} - 8 q^{93} - 4 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^9 - 4 * q^11 - 8 * q^13 + 2 * q^17 - 6 * q^19 + 8 * q^23 + 5 * q^25 - 8 * q^27 + 4 * q^29 - 4 * q^31 - 8 * q^33 - 10 * q^37 + 8 * q^39 - 20 * q^41 - 8 * q^43 + 4 * q^47 + 4 * q^51 + 2 * q^53 + 24 * q^57 + 10 * q^59 + 8 * q^61 - 8 * q^67 - 32 * q^69 + 6 * q^73 + 10 * q^75 - 16 * q^79 + 11 * q^81 - 4 * q^83 - 4 * q^87 - 18 * q^89 - 8 * q^93 - 4 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	−0.995782	−	0.0917517i	$-0.970753\pi$
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	0.418432	−	0.908248i	$-0.362580\pi$
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	−0.992361	−	0.123371i	$-0.960630\pi$
$11$	−2.00000	−	3.46410i	−0.603023	−	1.04447i	0.389338	−	0.921095i	$-0.372704\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	0.961436	−	0.275029i	$-0.0886875\pi$
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	−0.718900	+	0.695113i	$0.755354\pi$
$18$		0			0
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	0.284157	+	0.958778i	$0.408286\pi$
$19$	−3.00000	+	5.19615i	−0.688247	+	1.19208i	−0.972404	+	0.233301i	$0.925047\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.00000	−	6.92820i	0.834058	−	1.44463i	−0.0607377	−	0.998154i	$-0.519345\pi$
$23$	4.00000	−	6.92820i	0.834058	−	1.44463i	0.894795	−	0.446476i	$-0.147321\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	−4.00000	+	6.92820i	−0.696311	+	1.20605i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	0.0821995	+	0.996616i	$0.473806\pi$
$37$	−5.00000	+	8.66025i	−0.821995	+	1.42374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$	4.00000	+	6.92820i	0.640513	+	1.10940i
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	−0.682489	−	0.730896i	$-0.739102\pi$
$47$	2.00000	−	3.46410i	0.291730	−	0.505291i	0.974219	+	0.225605i	$0.0724358\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	2.00000	−	3.46410i	0.280056	−	0.485071i
$52$		0			0
$53$	1.00000	+	1.73205i	0.137361	+	0.237915i	0.926497	−	0.376303i	$-0.122805\pi$
$53$	1.00000	+	1.73205i	0.137361	+	0.237915i	−0.789136	+	0.614218i	$0.789471\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	12.0000			1.58944
$58$		0			0
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	0.982894	+	0.184172i	$0.0589603\pi$
$59$	5.00000	+	8.66025i	0.650945	+	1.12747i	−0.331949	+	0.943297i	$0.607706\pi$
$60$		0			0
$61$	4.00000	−	6.92820i	0.512148	−	0.887066i	−0.487753	−	0.872982i	$-0.662183\pi$
$61$	4.00000	−	6.92820i	0.512148	−	0.887066i	0.999901	−	0.0140840i	$-0.00448323\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	0.511237	−	0.859440i	$-0.329187\pi$
$67$	−4.00000	−	6.92820i	−0.488678	−	0.846415i	−0.999915	+	0.0130248i	$0.995854\pi$
$68$		0			0
$69$	−16.0000			−1.92617
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	3.00000	+	5.19615i	0.351123	+	0.608164i	0.986447	−	0.164083i	$-0.0524664\pi$
$73$	3.00000	+	5.19615i	0.351123	+	0.608164i	−0.635323	+	0.772246i	$0.719133\pi$
$74$		0			0
$75$	5.00000	−	8.66025i	0.577350	−	1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000	+	13.8564i	−0.900070	+	1.55897i	−0.0726692	+	0.997356i	$0.523152\pi$
$79$	−8.00000	+	13.8564i	−0.900070	+	1.55897i	−0.827401	+	0.561611i	$0.810182\pi$
$80$		0			0
$81$	5.50000	+	9.52628i	0.611111	+	1.05848i
$82$		0			0
$83$	−2.00000			−0.219529			−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000			−0.219529			−0.109764	+	0.993958i	$0.535010\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.00000	−	3.46410i	−0.214423	−	0.371391i
$88$		0			0
$89$	−9.00000	+	15.5885i	−0.953998	+	1.65237i	−0.217354	+	0.976093i	$0.569742\pi$
$89$	−9.00000	+	15.5885i	−0.953998	+	1.65237i	−0.736644	+	0.676280i	$0.763591\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−4.00000	+	6.92820i	−0.414781	+	0.718421i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	−0.947576	−	0.319531i	$-0.896475\pi$
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	0.750510	+	0.660859i	$0.229808\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000	+	13.8564i	−0.773389	+	1.33955i	0.162306	+	0.986740i	$0.448107\pi$
$107$	−8.00000	+	13.8564i	−0.773389	+	1.33955i	−0.935695	+	0.352809i	$0.885227\pi$
$108$		0			0
$109$	−5.00000	−	8.66025i	−0.478913	−	0.829502i	0.520794	−	0.853682i	$-0.325636\pi$
$109$	−5.00000	−	8.66025i	−0.478913	−	0.829502i	−0.999708	+	0.0241802i	$0.992302\pi$
$110$		0			0
$111$	20.0000			1.89832
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.00000	−	3.46410i	0.184900	−	0.320256i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$	10.0000	+	17.3205i	0.901670	+	1.56174i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$	4.00000	+	6.92820i	0.352180	+	0.609994i
$130$		0			0
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	0.993109	+	0.117198i	$0.0373911\pi$
$137$	7.00000	+	12.1244i	0.598050	+	1.03585i	−0.395058	+	0.918656i	$0.629276\pi$
$138$		0			0
$139$	10.0000			0.848189			0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000			0.848189			0.424094	+	0.905618i	$0.360592\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$	8.00000	+	13.8564i	0.668994	+	1.15873i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	−8.00000	−	13.8564i	−0.651031	−	1.12762i	−0.982873	−	0.184284i	$-0.941004\pi$
$151$	−8.00000	−	13.8564i	−0.651031	−	1.12762i	0.331842	−	0.943335i	$-0.392330\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−6.00000	−	10.3923i	−0.478852	−	0.829396i	0.520854	−	0.853646i	$-0.325614\pi$
$157$	−6.00000	−	10.3923i	−0.478852	−	0.829396i	−0.999706	+	0.0242497i	$0.992280\pi$
$158$		0			0
$159$	2.00000	−	3.46410i	0.158610	−	0.274721i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	−0.777007	−	0.629492i	$-0.783263\pi$
$163$	2.00000	−	3.46410i	0.156652	−	0.271329i	0.933659	+	0.358162i	$0.116597\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−20.0000			−1.54765			−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000			−1.54765			−0.773823	+	0.633402i	$0.781658\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	−3.00000	−	5.19615i	−0.229416	−	0.397360i
$172$		0			0
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	−0.542583	−	0.840002i	$-0.682554\pi$
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	0.998755	+	0.0498898i	$0.0158870\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	10.0000	−	17.3205i	0.751646	−	1.30189i
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	−12.0000			−0.891953			−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000			−0.891953			−0.445976	+	0.895045i	$0.647144\pi$
$182$		0			0
$183$	−16.0000			−1.18275
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.00000	−	6.92820i	0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−7.00000	−	12.1244i	−0.503871	−	0.872730i	−0.999990	−	0.00447566i	$-0.998575\pi$
$193$	−7.00000	−	12.1244i	−0.503871	−	0.872730i	0.496119	−	0.868255i	$-0.334758\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	2.00000	+	3.46410i	0.141776	+	0.245564i	0.928166	−	0.372168i	$-0.121385\pi$
$199$	2.00000	+	3.46410i	0.141776	+	0.245564i	−0.786389	+	0.617731i	$0.788052\pi$
$200$		0			0
$201$	−8.00000	+	13.8564i	−0.564276	+	0.977356i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	4.00000	+	6.92820i	0.278019	+	0.481543i
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	6.00000	−	10.3923i	0.405442	−	0.702247i
$220$		0			0
$221$	−4.00000	−	6.92820i	−0.269069	−	0.466041i
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		0			0
$225$	−5.00000			−0.333333
$226$		0			0
$227$	−11.0000	−	19.0526i	−0.730096	−	1.26456i	−0.956842	−	0.290609i	$-0.906142\pi$
$227$	−11.0000	−	19.0526i	−0.730096	−	1.26456i	0.226746	−	0.973954i	$-0.427191\pi$
$228$		0			0
$229$	10.0000	−	17.3205i	0.660819	−	1.14457i	−0.319582	−	0.947559i	$-0.603543\pi$
$229$	10.0000	−	17.3205i	0.660819	−	1.14457i	0.980401	−	0.197013i	$-0.0631241\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	−0.750867	−	0.660454i	$-0.770364\pi$
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	0.947403	+	0.320043i	$0.103697\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	32.0000			2.07862
$238$		0			0
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$		0			0
$241$	−7.00000	−	12.1244i	−0.450910	−	0.780998i	0.547533	−	0.836784i	$-0.315567\pi$
$241$	−7.00000	−	12.1244i	−0.450910	−	0.780998i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$	5.00000	−	8.66025i	0.320750	−	0.555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	12.0000	−	20.7846i	0.763542	−	1.32249i
$248$		0			0
$249$	2.00000	+	3.46410i	0.126745	+	0.219529i
$250$		0			0
$251$	10.0000			0.631194			0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000			0.631194			0.315597	+	0.948893i	$0.397795\pi$
$252$		0			0
$253$	−32.0000			−2.01182
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.00000	+	1.73205i	−0.0623783	+	0.108042i	−0.895528	−	0.445005i	$-0.853202\pi$
$257$	−1.00000	+	1.73205i	−0.0623783	+	0.108042i	0.833150	+	0.553047i	$0.186535\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.00000	+	1.73205i	−0.0618984	+	0.107211i
$262$		0			0
$263$	−8.00000	−	13.8564i	−0.493301	−	0.854423i	0.506669	−	0.862141i	$-0.330877\pi$
$263$	−8.00000	−	13.8564i	−0.493301	−	0.854423i	−0.999970	+	0.00771799i	$0.997543\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	36.0000			2.20316
$268$		0			0
$269$	6.00000	+	10.3923i	0.365826	+	0.633630i	0.988908	−	0.148527i	$-0.0474530\pi$
$269$	6.00000	+	10.3923i	0.365826	+	0.633630i	−0.623082	+	0.782157i	$0.714120\pi$
$270$		0			0
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	−0.999870	−	0.0161307i	$-0.994865\pi$
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	0.513905	+	0.857847i	$0.328199\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.0000	−	17.3205i	0.603023	−	1.04447i
$276$		0			0
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	0.575408	−	0.817867i	$-0.304843\pi$
$277$	−7.00000	−	12.1244i	−0.420589	−	0.728482i	−0.995997	+	0.0893846i	$0.971510\pi$
$278$		0			0
$279$	4.00000			0.239474
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$	5.00000	+	8.66025i	0.297219	+	0.514799i	0.975499	−	0.220005i	$-0.0706075\pi$
$283$	5.00000	+	8.66025i	0.297219	+	0.514799i	−0.678280	+	0.734804i	$0.737274\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$	2.00000	+	3.46410i	0.117242	+	0.203069i
$292$		0			0
$293$	8.00000			0.467365			0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000			0.467365			0.233682	+	0.972313i	$0.424922\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	8.00000	+	13.8564i	0.464207	+	0.804030i
$298$		0			0
$299$	−16.0000	+	27.7128i	−0.925304	+	1.60267i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10.0000			−0.570730			−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000			−0.570730			−0.285365	+	0.958419i	$0.592115\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	−12.0000	−	20.7846i	−0.680458	−	1.17859i	−0.974841	−	0.222900i	$-0.928448\pi$
$311$	−12.0000	−	20.7846i	−0.680458	−	1.17859i	0.294384	−	0.955687i	$-0.404886\pi$
$312$		0			0
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	−0.220006	−	0.975499i	$-0.570608\pi$
$313$	13.0000	−	22.5167i	0.734803	−	1.27272i	0.954810	−	0.297218i	$-0.0960589\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	17.0000	−	29.4449i	0.954815	−	1.65379i	0.220024	−	0.975494i	$-0.429386\pi$
$317$	17.0000	−	29.4449i	0.954815	−	1.65379i	0.734791	−	0.678294i	$-0.237280\pi$
$318$		0			0
$319$	−4.00000	−	6.92820i	−0.223957	−	0.387905i
$320$		0			0
$321$	32.0000			1.78607
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$	−10.0000	−	17.3205i	−0.554700	−	0.960769i
$326$		0			0
$327$	−10.0000	+	17.3205i	−0.553001	+	0.957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	−0.982470	−	0.186421i	$-0.940311\pi$
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	0.652680	+	0.757634i	$0.273645\pi$
$332$		0			0
$333$	−5.00000	−	8.66025i	−0.273998	−	0.474579i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.0000			−0.980522			−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000			−0.980522			−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$	−6.00000	−	10.3923i	−0.325875	−	0.564433i
$340$		0			0
$341$	−8.00000	+	13.8564i	−0.433224	+	0.750366i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	0.658824	−	0.752297i	$-0.271054\pi$
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	−0.980921	+	0.194409i	$0.937721\pi$
$348$		0			0
$349$	20.0000			1.07058			0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000			1.07058			0.535288	+	0.844670i	$0.320203\pi$
$350$		0			0
$351$	16.0000			0.854017
$352$		0			0
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	0.520689	−	0.853746i	$-0.325675\pi$
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	−0.999711	+	0.0240566i	$0.992342\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.00000	+	13.8564i	−0.422224	+	0.731313i	−0.996157	−	0.0875892i	$-0.972084\pi$
$359$	−8.00000	+	13.8564i	−0.422224	+	0.731313i	0.573933	+	0.818902i	$0.305417\pi$
$360$		0			0
$361$	−8.50000	−	14.7224i	−0.447368	−	0.774865i
$362$		0			0
$363$	10.0000			0.524864
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	0.951336	−	0.308155i	$-0.0997115\pi$
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	−0.742538	+	0.669804i	$0.766378\pi$
$368$		0			0
$369$	5.00000	−	8.66025i	0.260290	−	0.450835i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−3.00000	+	5.19615i	−0.155334	+	0.269047i	−0.933181	−	0.359408i	$-0.882979\pi$
$373$	−3.00000	+	5.19615i	−0.155334	+	0.269047i	0.777847	+	0.628454i	$0.216312\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.00000			−0.412021
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$	−16.0000	−	27.7128i	−0.819705	−	1.41977i
$382$		0			0
$383$	10.0000	−	17.3205i	0.510976	−	0.885037i	−0.488943	−	0.872316i	$-0.662617\pi$
$383$	10.0000	−	17.3205i	0.510976	−	0.885037i	0.999919	−	0.0127209i	$-0.00404928\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.00000	−	3.46410i	0.101666	−	0.176090i
$388$		0			0
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	0.932002	−	0.362454i	$-0.118061\pi$
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	−0.779895	+	0.625910i	$0.784728\pi$
$390$		0			0
$391$	16.0000			0.809155
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$		0			0
$396$		0			0
$397$	14.0000	−	24.2487i	0.702640	−	1.21701i	−0.264897	−	0.964277i	$-0.585338\pi$
$397$	14.0000	−	24.2487i	0.702640	−	1.21701i	0.967537	−	0.252731i	$-0.0813288\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.00000	−	1.73205i	0.0499376	−	0.0864945i	−0.839976	−	0.542623i	$-0.817431\pi$
$401$	1.00000	−	1.73205i	0.0499376	−	0.0864945i	0.889914	+	0.456129i	$0.150764\pi$
$402$		0			0
$403$	8.00000	+	13.8564i	0.398508	+	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	40.0000			1.98273
$408$		0			0
$409$	1.00000	+	1.73205i	0.0494468	+	0.0856444i	0.889689	−	0.456566i	$-0.150921\pi$
$409$	1.00000	+	1.73205i	0.0494468	+	0.0856444i	−0.840243	+	0.542211i	$0.817588\pi$
$410$		0			0
$411$	14.0000	−	24.2487i	0.690569	−	1.19610i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−10.0000	−	17.3205i	−0.489702	−	0.848189i
$418$		0			0
$419$	−6.00000			−0.293119			−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000			−0.293119			−0.146560	+	0.989202i	$0.546820\pi$
$420$		0			0
$421$	−18.0000			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$422$		0			0
$423$	2.00000	+	3.46410i	0.0972433	+	0.168430i
$424$		0			0
$425$	−5.00000	+	8.66025i	−0.242536	+	0.420084i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	16.0000	−	27.7128i	0.772487	−	1.33799i
$430$		0			0
$431$	4.00000	+	6.92820i	0.192673	+	0.333720i	0.946135	−	0.323772i	$-0.104951\pi$
$431$	4.00000	+	6.92820i	0.192673	+	0.333720i	−0.753462	+	0.657491i	$0.771618\pi$
$432$		0			0
$433$	22.0000			1.05725			0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000			1.05725			0.528626	+	0.848855i	$0.322707\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	24.0000	+	41.5692i	1.14808	+	1.98853i
$438$		0			0
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	−0.423556	−	0.905870i	$-0.639218\pi$
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	0.996284	−	0.0861252i	$-0.0274485\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	0.426414	+	0.904528i	$0.359777\pi$
$443$	−12.0000	+	20.7846i	−0.570137	+	0.987507i	−0.996551	+	0.0829786i	$0.973557\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	12.0000			0.567581
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	20.0000	+	34.6410i	0.941763	+	1.63118i
$452$		0			0
$453$	−16.0000	+	27.7128i	−0.751746	+	1.30206i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	0.485299	+	0.874348i	$0.338711\pi$
$457$	−11.0000	+	19.0526i	−0.514558	+	0.891241i	−0.999857	+	0.0168929i	$0.994623\pi$
$458$		0			0
$459$	−4.00000	−	6.92820i	−0.186704	−	0.323381i
$460$		0			0
$461$	28.0000			1.30409			0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000			1.30409			0.652045	+	0.758180i	$0.273911\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.00000	−	15.5885i	0.416470	−	0.721348i	−0.579111	−	0.815249i	$-0.696600\pi$
$467$	9.00000	−	15.5885i	0.416470	−	0.721348i	0.995582	+	0.0939008i	$0.0299336\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−12.0000	+	20.7846i	−0.552931	+	0.957704i
$472$		0			0
$473$	8.00000	+	13.8564i	0.367840	+	0.637118i
$474$		0			0
$475$	−30.0000			−1.37649
$476$		0			0
$477$	−2.00000			−0.0915737
$478$		0			0
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	0.969920	−	0.243426i	$-0.0782712\pi$
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	−0.695773	+	0.718262i	$0.744938\pi$
$480$		0			0
$481$	20.0000	−	34.6410i	0.911922	−	1.57949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	12.0000	+	20.7846i	0.543772	+	0.941841i	0.998683	+	0.0513038i	$0.0163377\pi$
$487$	12.0000	+	20.7846i	0.543772	+	0.941841i	−0.454911	+	0.890537i	$0.650329\pi$
$488$		0			0
$489$	−8.00000			−0.361773
$490$		0			0
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$	2.00000	+	3.46410i	0.0900755	+	0.156015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4.00000	+	6.92820i	−0.179065	+	0.310149i	−0.941560	−	0.336844i	$-0.890640\pi$
$499$	−4.00000	+	6.92820i	−0.179065	+	0.310149i	0.762496	+	0.646993i	$0.223974\pi$
$500$		0			0
$501$	20.0000	+	34.6410i	0.893534	+	1.54765i
$502$		0			0
$503$	16.0000			0.713405			0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000			0.713405			0.356702	+	0.934218i	$0.383901\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.00000	−	5.19615i	−0.133235	−	0.230769i
$508$		0			0
$509$	−10.0000	+	17.3205i	−0.443242	+	0.767718i	−0.997928	−	0.0643419i	$-0.979505\pi$
$509$	−10.0000	+	17.3205i	−0.443242	+	0.767718i	0.554686	+	0.832060i	$0.312839\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	12.0000	−	20.7846i	0.529813	−	0.917663i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−16.0000			−0.703679
$518$		0			0
$519$	−24.0000			−1.05348
$520$		0			0
$521$	−19.0000	−	32.9090i	−0.832405	−	1.44177i	−0.896126	−	0.443800i	$-0.853630\pi$
$521$	−19.0000	−	32.9090i	−0.832405	−	1.44177i	0.0637207	−	0.997968i	$-0.479703\pi$
$522$		0			0
$523$	−7.00000	+	12.1244i	−0.306089	+	0.530161i	−0.977503	−	0.210921i	$-0.932354\pi$
$523$	−7.00000	+	12.1244i	−0.306089	+	0.530161i	0.671414	+	0.741082i	$0.265687\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.00000	−	6.92820i	0.174243	−	0.301797i
$528$		0			0
$529$	−20.5000	−	35.5070i	−0.891304	−	1.54378i
$530$		0			0
$531$	−10.0000			−0.433963
$532$		0			0
$533$	40.0000			1.73259
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−15.0000	+	25.9808i	−0.644900	+	1.11700i	0.339424	+	0.940633i	$0.389768\pi$
$541$	−15.0000	+	25.9808i	−0.644900	+	1.11700i	−0.984325	+	0.176367i	$0.943566\pi$
$542$		0			0
$543$	12.0000	+	20.7846i	0.514969	+	0.891953i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		0			0
$549$	4.00000	+	6.92820i	0.170716	+	0.295689i
$550$		0			0
$551$	−6.00000	+	10.3923i	−0.255609	+	0.442727i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.0000	−	25.9808i	−0.635570	−	1.10084i	−0.986394	−	0.164399i	$-0.947432\pi$
$557$	−15.0000	−	25.9808i	−0.635570	−	1.10084i	0.350824	−	0.936442i	$-0.385902\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$	−16.0000			−0.675521
$562$		0			0
$563$	23.0000	+	39.8372i	0.969334	+	1.67894i	0.697489	+	0.716596i	$0.254301\pi$
$563$	23.0000	+	39.8372i	0.969334	+	1.67894i	0.271846	+	0.962341i	$0.412366\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	0.821138	−	0.570730i	$-0.193340\pi$
$571$	−2.00000	−	3.46410i	−0.0836974	−	0.144968i	−0.904835	+	0.425762i	$0.860006\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	40.0000			1.66812
$576$		0			0
$577$	7.00000	+	12.1244i	0.291414	+	0.504744i	0.974144	−	0.225927i	$-0.0725410\pi$
$577$	7.00000	+	12.1244i	0.291414	+	0.504744i	−0.682730	+	0.730670i	$0.739208\pi$
$578$		0			0
$579$	−14.0000	+	24.2487i	−0.581820	+	1.00774i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.00000	−	6.92820i	0.165663	−	0.286937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$		0			0
$589$	24.0000			0.988903
$590$		0			0
$591$	−6.00000	−	10.3923i	−0.246807	−	0.427482i
$592$		0			0
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	−0.797831	−	0.602881i	$-0.794019\pi$
$593$	3.00000	−	5.19615i	0.123195	−	0.213380i	0.921026	+	0.389501i	$0.127353\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.00000	−	6.92820i	0.163709	−	0.283552i
$598$		0			0
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	0.999938	−	0.0111569i	$-0.00355143\pi$
$599$	12.0000	+	20.7846i	0.490307	+	0.849236i	−0.509631	+	0.860393i	$0.670218\pi$
$600$		0			0
$601$	−46.0000			−1.87638			−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000			−1.87638			−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	8.00000			0.325785
$604$		0			0
$605$		0			0
$606$		0			0
$607$	24.0000	−	41.5692i	0.974130	−	1.68724i	0.291353	−	0.956616i	$-0.405895\pi$
$607$	24.0000	−	41.5692i	0.974130	−	1.68724i	0.682777	−	0.730627i	$-0.260772\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.00000	+	13.8564i	−0.323645	+	0.560570i
$612$		0			0
$613$	11.0000	+	19.0526i	0.444286	+	0.769526i	0.998002	−	0.0631797i	$-0.0201241\pi$
$613$	11.0000	+	19.0526i	0.444286	+	0.769526i	−0.553716	+	0.832705i	$0.686791\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$		0			0
$619$	−11.0000	−	19.0526i	−0.442127	−	0.765787i	0.555720	−	0.831370i	$-0.312443\pi$
$619$	−11.0000	−	19.0526i	−0.442127	−	0.765787i	−0.997847	+	0.0655827i	$0.979109\pi$
$620$		0			0
$621$	−16.0000	+	27.7128i	−0.642058	+	1.11208i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$	−24.0000	−	41.5692i	−0.958468	−	1.66011i
$628$		0			0
$629$	−20.0000			−0.797452
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$		0			0
$633$	8.00000	+	13.8564i	0.317971	+	0.550743i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.0000	+	29.4449i	0.671460	+	1.16300i	0.977490	+	0.210981i	$0.0676657\pi$
$641$	17.0000	+	29.4449i	0.671460	+	1.16300i	−0.306031	+	0.952022i	$0.599001\pi$
$642$		0			0
$643$	34.0000			1.34083			0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000			1.34083			0.670415	+	0.741987i	$0.266116\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.00000	−	10.3923i	−0.235884	−	0.408564i	0.723645	−	0.690172i	$-0.242465\pi$
$647$	−6.00000	−	10.3923i	−0.235884	−	0.408564i	−0.959529	+	0.281609i	$0.909132\pi$
$648$		0			0
$649$	20.0000	−	34.6410i	0.785069	−	1.35978i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.0000	−	32.9090i	0.743527	−	1.28783i	−0.207352	−	0.978266i	$-0.566485\pi$
$653$	19.0000	−	32.9090i	0.743527	−	1.28783i	0.950880	−	0.309561i	$-0.100182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$	−12.0000	−	20.7846i	−0.466746	−	0.808428i	0.532533	−	0.846410i	$-0.321240\pi$
$661$	−12.0000	−	20.7846i	−0.466746	−	0.808428i	−0.999278	+	0.0379819i	$0.987907\pi$
$662$		0			0
$663$	−8.00000	+	13.8564i	−0.310694	+	0.538138i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8.00000	−	13.8564i	0.309761	−	0.536522i
$668$		0			0
$669$	24.0000	+	41.5692i	0.927894	+	1.60716i
$670$		0			0
$671$	−32.0000			−1.23535
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		0			0
$675$	−10.0000	−	17.3205i	−0.384900	−	0.666667i
$676$		0			0
$677$	−18.0000	+	31.1769i	−0.691796	+	1.19823i	0.279453	+	0.960159i	$0.409847\pi$
$677$	−18.0000	+	31.1769i	−0.691796	+	1.19823i	−0.971249	+	0.238067i	$0.923486\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−22.0000	+	38.1051i	−0.843042	+	1.46019i
$682$		0			0
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	0.539750	−	0.841825i	$-0.318519\pi$
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	−0.998917	+	0.0465244i	$0.985185\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−40.0000			−1.52610
$688$		0			0
$689$	−4.00000	−	6.92820i	−0.152388	−	0.263944i
$690$		0			0
$691$	−11.0000	+	19.0526i	−0.418460	+	0.724793i	−0.995785	−	0.0917209i	$-0.970763\pi$
$691$	−11.0000	+	19.0526i	−0.418460	+	0.724793i	0.577325	+	0.816514i	$0.304097\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−10.0000	−	17.3205i	−0.378777	−	0.656061i
$698$		0			0
$699$	−12.0000			−0.453882
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$	−30.0000	−	51.9615i	−1.13147	−	1.95977i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	−0.999908	−	0.0135434i	$-0.995689\pi$
$709$	−13.0000	+	22.5167i	−0.488225	+	0.845631i	0.511683	+	0.859174i	$0.329022\pi$
$710$		0			0
$711$	−8.00000	−	13.8564i	−0.300023	−	0.519656i
$712$		0			0
$713$	−32.0000			−1.19841
$714$		0			0
$715$		0			0
$716$		0			0
$717$	24.0000	+	41.5692i	0.896296	+	1.55243i
$718$		0			0
$719$	−18.0000	+	31.1769i	−0.671287	+	1.16270i	0.306253	+	0.951950i	$0.400925\pi$
$719$	−18.0000	+	31.1769i	−0.671287	+	1.16270i	−0.977539	+	0.210752i	$0.932409\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−14.0000	+	24.2487i	−0.520666	+	0.901819i
$724$		0			0
$725$	5.00000	+	8.66025i	0.185695	+	0.321634i
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−4.00000	−	6.92820i	−0.147945	−	0.256249i
$732$		0			0
$733$	−4.00000	+	6.92820i	−0.147743	+	0.255899i	−0.930393	−	0.366563i	$-0.880534\pi$
$733$	−4.00000	+	6.92820i	−0.147743	+	0.255899i	0.782650	+	0.622462i	$0.213868\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.0000	+	27.7128i	−0.589368	+	1.02081i
$738$		0			0
$739$	2.00000	+	3.46410i	0.0735712	+	0.127429i	0.900464	−	0.434930i	$-0.143227\pi$
$739$	2.00000	+	3.46410i	0.0735712	+	0.127429i	−0.826893	+	0.562360i	$0.809894\pi$
$740$		0			0
$741$	−48.0000			−1.76332
$742$		0			0
$743$	−16.0000			−0.586983			−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000			−0.586983			−0.293492	+	0.955962i	$0.594817\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.00000	−	1.73205i	0.0365881	−	0.0633724i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	12.0000	−	20.7846i	0.437886	−	0.758441i	−0.559640	−	0.828736i	$-0.689061\pi$
$751$	12.0000	−	20.7846i	0.437886	−	0.758441i	0.997526	+	0.0702946i	$0.0223939\pi$
$752$		0			0
$753$	−10.0000	−	17.3205i	−0.364420	−	0.631194i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$	32.0000	+	55.4256i	1.16153	+	2.01182i
$760$		0			0
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	0.454935	+	0.890525i	$0.349663\pi$
$761$	−15.0000	+	25.9808i	−0.543750	+	0.941802i	−0.998684	+	0.0512772i	$0.983671\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−20.0000	−	34.6410i	−0.722158	−	1.25081i
$768$		0			0
$769$	6.00000			0.216366			0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000			0.216366			0.108183	+	0.994131i	$0.465497\pi$
$770$		0			0
$771$	4.00000			0.144056
$772$		0			0
$773$	12.0000	+	20.7846i	0.431610	+	0.747570i	0.997012	−	0.0772449i	$-0.0246123\pi$
$773$	12.0000	+	20.7846i	0.431610	+	0.747570i	−0.565402	+	0.824815i	$0.691279\pi$
$774$		0			0
$775$	10.0000	−	17.3205i	0.359211	−	0.622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	30.0000	−	51.9615i	1.07486	−	1.86171i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−8.00000			−0.285897
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−23.0000	−	39.8372i	−0.819861	−	1.42004i	−0.905784	−	0.423740i	$-0.860717\pi$
$787$	−23.0000	−	39.8372i	−0.819861	−	1.42004i	0.0859225	−	0.996302i	$-0.472616\pi$
$788$		0			0
$789$	−16.0000	+	27.7128i	−0.569615	+	0.986602i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−16.0000	+	27.7128i	−0.568177	+	0.984111i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20.0000			0.708436			0.354218	−	0.935163i	$-0.384747\pi$
$797$	20.0000			0.708436			0.354218	+	0.935163i	$0.384747\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$	−9.00000	−	15.5885i	−0.317999	−	0.550791i
$802$		0			0
$803$	12.0000	−	20.7846i	0.423471	−	0.733473i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	12.0000	−	20.7846i	0.422420	−	0.731653i
$808$		0			0
$809$	13.0000	+	22.5167i	0.457056	+	0.791644i	0.998804	−	0.0488972i	$-0.0155707\pi$
$809$	13.0000	+	22.5167i	0.457056	+	0.791644i	−0.541748	+	0.840541i	$0.682237\pi$
$810$		0			0
$811$	−34.0000			−1.19390			−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000			−1.19390			−0.596951	+	0.802278i	$0.703621\pi$
$812$		0			0
$813$	32.0000			1.12229
$814$		0			0
$815$		0			0
$816$		0			0
$817$	12.0000	−	20.7846i	0.419827	−	0.727161i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000	−	1.73205i	0.0349002	−	0.0604490i	−0.848048	−	0.529920i	$-0.822222\pi$
$821$	1.00000	−	1.73205i	0.0349002	−	0.0604490i	0.882948	+	0.469471i	$0.155555\pi$
$822$		0			0
$823$	4.00000	+	6.92820i	0.139431	+	0.241502i	0.927281	−	0.374365i	$-0.122139\pi$
$823$	4.00000	+	6.92820i	0.139431	+	0.241502i	−0.787850	+	0.615867i	$0.788806\pi$
$824$		0			0
$825$	−40.0000			−1.39262
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−12.0000	−	20.7846i	−0.416777	−	0.721879i	0.578836	−	0.815444i	$-0.303507\pi$
$829$	−12.0000	−	20.7846i	−0.416777	−	0.721879i	−0.995613	+	0.0935647i	$0.970174\pi$
$830$		0			0
$831$	−14.0000	+	24.2487i	−0.485655	+	0.841178i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	8.00000	+	13.8564i	0.276520	+	0.478947i
$838$		0			0
$839$	52.0000			1.79524			0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000			1.79524			0.897620	+	0.440771i	$0.145295\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	6.00000	+	10.3923i	0.206651	+	0.357930i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	10.0000	−	17.3205i	0.343199	−	0.594438i
$850$		0			0
$851$	40.0000	+	69.2820i	1.37118	+	2.37496i
$852$		0			0
$853$	−36.0000			−1.23262			−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000			−1.23262			−0.616308	+	0.787505i	$0.711372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.00000	+	15.5885i	0.307434	+	0.532492i	0.977800	−	0.209539i	$-0.0671963\pi$
$857$	9.00000	+	15.5885i	0.307434	+	0.532492i	−0.670366	+	0.742030i	$0.733863\pi$
$858$		0			0
$859$	−13.0000	+	22.5167i	−0.443554	+	0.768259i	−0.997950	−	0.0639945i	$-0.979616\pi$
$859$	−13.0000	+	22.5167i	−0.443554	+	0.768259i	0.554396	+	0.832253i	$0.312949\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0000	−	41.5692i	0.816970	−	1.41503i	−0.0909355	−	0.995857i	$-0.528986\pi$
$863$	24.0000	−	41.5692i	0.816970	−	1.41503i	0.907905	−	0.419176i	$-0.137681\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−26.0000			−0.883006
$868$		0			0
$869$	64.0000			2.17105
$870$		0			0
$871$	16.0000	+	27.7128i	0.542139	+	0.939013i
$872$		0			0
$873$	1.00000	−	1.73205i	0.0338449	−	0.0586210i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	−0.723298	−	0.690536i	$-0.757375\pi$
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	0.959671	+	0.281126i	$0.0907079\pi$
$878$		0			0
$879$	−8.00000	−	13.8564i	−0.269833	−	0.467365i
$880$		0			0
$881$	−38.0000			−1.28025			−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000			−1.28025			−0.640126	+	0.768270i	$0.721118\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.0000	−	24.2487i	0.470074	−	0.814192i	−0.529340	−	0.848410i	$-0.677561\pi$
$887$	14.0000	−	24.2487i	0.470074	−	0.814192i	0.999414	+	0.0342175i	$0.0108939\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	22.0000	−	38.1051i	0.737028	−	1.27657i
$892$		0			0
$893$	12.0000	+	20.7846i	0.401565	+	0.695530i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	64.0000			2.13690
$898$		0			0
$899$	−4.00000	−	6.92820i	−0.133407	−	0.231069i
$900$		0			0
$901$	−2.00000	+	3.46410i	−0.0666297	+	0.115406i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	0.999334	+	0.0364935i	$0.0116188\pi$
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	−0.468063	+	0.883695i	$0.655048\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$	4.00000	+	6.92820i	0.132381	+	0.229290i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−20.0000	+	34.6410i	−0.659739	+	1.14270i	0.320944	+	0.947098i	$0.396000\pi$
$919$	−20.0000	+	34.6410i	−0.659739	+	1.14270i	−0.980683	+	0.195603i	$0.937333\pi$
$920$		0			0
$921$	10.0000	+	17.3205i	0.329511	+	0.570730i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−50.0000			−1.64399
$926$		0			0
$927$	−2.00000	−	3.46410i	−0.0656886	−	0.113776i
$928$		0			0
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	−0.679770	−	0.733426i	$-0.737920\pi$
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	0.975050	+	0.221985i	$0.0712536\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−24.0000	+	41.5692i	−0.785725	+	1.36092i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$	−52.0000			−1.69696
$940$		0			0
$941$	20.0000	+	34.6410i	0.651981	+	1.12926i	0.982641	+	0.185515i	$0.0593953\pi$
$941$	20.0000	+	34.6410i	0.651981	+	1.12926i	−0.330660	+	0.943750i	$0.607271\pi$
$942$		0			0
$943$	−40.0000	+	69.2820i	−1.30258	+	2.25613i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.00000	+	3.46410i	−0.0649913	+	0.112568i	−0.896690	−	0.442659i	$-0.854035\pi$
$947$	−2.00000	+	3.46410i	−0.0649913	+	0.112568i	0.831699	+	0.555227i	$0.187369\pi$
$948$		0			0
$949$	−12.0000	−	20.7846i	−0.389536	−	0.674697i
$950$		0			0
$951$	−68.0000			−2.20505
$952$		0			0
$953$	−22.0000			−0.712650			−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000			−0.712650			−0.356325	+	0.934362i	$0.615970\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−8.00000	+	13.8564i	−0.258603	+	0.447914i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$		0			0
$963$	−8.00000	−	13.8564i	−0.257796	−	0.446516i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$		0			0
$969$	12.0000	+	20.7846i	0.385496	+	0.667698i
$970$		0			0
$971$	21.0000	−	36.3731i	0.673922	−	1.16727i	−0.302861	−	0.953035i	$-0.597942\pi$
$971$	21.0000	−	36.3731i	0.673922	−	1.16727i	0.976783	−	0.214232i	$-0.0687250\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−20.0000	+	34.6410i	−0.640513	+	1.10940i
$976$		0			0
$977$	−13.0000	−	22.5167i	−0.415907	−	0.720372i	0.579616	−	0.814890i	$-0.303202\pi$
$977$	−13.0000	−	22.5167i	−0.415907	−	0.720372i	−0.995523	+	0.0945177i	$0.969869\pi$
$978$		0			0
$979$	72.0000			2.30113
$980$		0			0
$981$	10.0000			0.319275
$982$		0			0
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	−0.996138	−	0.0878058i	$-0.972015\pi$
$983$	−18.0000	−	31.1769i	−0.574111	−	0.994389i	0.422027	−	0.906583i	$-0.361319\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000	+	27.7128i	−0.508770	+	0.881216i
$990$		0			0
$991$	−20.0000	−	34.6410i	−0.635321	−	1.10041i	−0.986447	−	0.164080i	$-0.947534\pi$
$991$	−20.0000	−	34.6410i	−0.635321	−	1.10041i	0.351126	−	0.936328i	$-0.385799\pi$
$992$		0			0
$993$	24.0000			0.761617
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−4.00000	−	6.92820i	−0.126681	−	0.219418i	0.795708	−	0.605681i	$-0.207099\pi$
$997$	−4.00000	−	6.92820i	−0.126681	−	0.219418i	−0.922389	+	0.386263i	$0.873766\pi$
$998$		0			0
$999$	20.0000	−	34.6410i	0.632772	−	1.09599i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.2.i.b.961.1		2
4.3	odd	2		1568.2.i.k.961.1		2
7.2	even	3		224.2.a.b.1.1	yes	1
7.3	odd	6		1568.2.i.j.1537.1		2
7.4	even	3	inner	1568.2.i.b.1537.1		2
7.5	odd	6		1568.2.a.b.1.1		1
7.6	odd	2		1568.2.i.j.961.1		2
21.2	odd	6		2016.2.a.g.1.1		1
28.3	even	6		1568.2.i.c.1537.1		2
28.11	odd	6		1568.2.i.k.1537.1		2
28.19	even	6		1568.2.a.h.1.1		1
28.23	odd	6		224.2.a.a.1.1	✓	1
28.27	even	2		1568.2.i.c.961.1		2
35.9	even	6		5600.2.a.c.1.1		1
56.5	odd	6		3136.2.a.y.1.1		1
56.19	even	6		3136.2.a.f.1.1		1
56.37	even	6		448.2.a.b.1.1		1
56.51	odd	6		448.2.a.f.1.1		1
84.23	even	6		2016.2.a.e.1.1		1
112.37	even	12		1792.2.b.b.897.1		2
112.51	odd	12		1792.2.b.f.897.1		2
112.93	even	12		1792.2.b.b.897.2		2
112.107	odd	12		1792.2.b.f.897.2		2
140.79	odd	6		5600.2.a.t.1.1		1
168.107	even	6		4032.2.a.p.1.1		1
168.149	odd	6		4032.2.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.a.a.1.1	✓	1	28.23	odd	6
224.2.a.b.1.1	yes	1	7.2	even	3
448.2.a.b.1.1		1	56.37	even	6
448.2.a.f.1.1		1	56.51	odd	6
1568.2.a.b.1.1		1	7.5	odd	6
1568.2.a.h.1.1		1	28.19	even	6
1568.2.i.b.961.1		2	1.1	even	1	trivial
1568.2.i.b.1537.1		2	7.4	even	3	inner
1568.2.i.c.961.1		2	28.27	even	2
1568.2.i.c.1537.1		2	28.3	even	6
1568.2.i.j.961.1		2	7.6	odd	2
1568.2.i.j.1537.1		2	7.3	odd	6
1568.2.i.k.961.1		2	4.3	odd	2
1568.2.i.k.1537.1		2	28.11	odd	6
1792.2.b.b.897.1		2	112.37	even	12
1792.2.b.b.897.2		2	112.93	even	12
1792.2.b.f.897.1		2	112.51	odd	12
1792.2.b.f.897.2		2	112.107	odd	12
2016.2.a.e.1.1		1	84.23	even	6
2016.2.a.g.1.1		1	21.2	odd	6
3136.2.a.f.1.1		1	56.19	even	6
3136.2.a.y.1.1		1	56.5	odd	6
4032.2.a.p.1.1		1	168.107	even	6
4032.2.a.z.1.1		1	168.149	odd	6
5600.2.a.c.1.1		1	35.9	even	6
5600.2.a.t.1.1		1	140.79	odd	6

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} +(-2.00000 - 3.46410i) q^{11} -4.00000 q^{13} +(1.00000 + 1.73205i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(4.00000 - 6.92820i) q^{23} +(2.50000 + 4.33013i) q^{25} -4.00000 q^{27} +2.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(-4.00000 + 6.92820i) q^{33} +(-5.00000 + 8.66025i) q^{37} +(4.00000 + 6.92820i) q^{39} -10.0000 q^{41} -4.00000 q^{43} +(2.00000 - 3.46410i) q^{47} +(2.00000 - 3.46410i) q^{51} +(1.00000 + 1.73205i) q^{53} +12.0000 q^{57} +(5.00000 + 8.66025i) q^{59} +(4.00000 - 6.92820i) q^{61} +(-4.00000 - 6.92820i) q^{67} -16.0000 q^{69} +(3.00000 + 5.19615i) q^{73} +(5.00000 - 8.66025i) q^{75} +(-8.00000 + 13.8564i) q^{79} +(5.50000 + 9.52628i) q^{81} -2.00000 q^{83} +(-2.00000 - 3.46410i) q^{87} +(-9.00000 + 15.5885i) q^{89} +(-4.00000 + 6.92820i) q^{93} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^9	\( 2 q - 2 q^{3} - q^{9} - 4 q^{11} - 8 q^{13} + 2 q^{17} - 6 q^{19} + 8 q^{23} + 5 q^{25} - 8 q^{27} + 4 q^{29} - 4 q^{31} - 8 q^{33} - 10 q^{37} + 8 q^{39} - 20 q^{41} - 8 q^{43} + 4 q^{47} + 4 q^{51} + 2 q^{53} + 24 q^{57} + 10 q^{59} + 8 q^{61} - 8 q^{67} - 32 q^{69} + 6 q^{73} + 10 q^{75} - 16 q^{79} + 11 q^{81} - 4 q^{83} - 4 q^{87} - 18 q^{89} - 8 q^{93} - 4 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^9 - 4 * q^11 - 8 * q^13 + 2 * q^17 - 6 * q^19 + 8 * q^23 + 5 * q^25 - 8 * q^27 + 4 * q^29 - 4 * q^31 - 8 * q^33 - 10 * q^37 + 8 * q^39 - 20 * q^41 - 8 * q^43 + 4 * q^47 + 4 * q^51 + 2 * q^53 + 24 * q^57 + 10 * q^59 + 8 * q^61 - 8 * q^67 - 32 * q^69 + 6 * q^73 + 10 * q^75 - 16 * q^79 + 11 * q^81 - 4 * q^83 - 4 * q^87 - 18 * q^89 - 8 * q^93 - 4 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more