LMFDB - Embedded newform 912.2.f.a.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(113,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("912.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.f (of order $2$, degree $1$, 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:	$7.28235666434$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	912.113
Dual form			912.2.f.a.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{3} -3.46410i q^{5} -1.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	\(q+(-1.50000 - 0.866025i) q^{3} -3.46410i q^{5} -1.00000 q^{7} +(1.50000 + 2.59808i) q^{9} -3.46410i q^{11} -1.73205i q^{13} +(-3.00000 + 5.19615i) q^{15} -1.73205i q^{17} +(4.00000 - 1.73205i) q^{19} +(1.50000 + 0.866025i) q^{21} -5.19615i q^{23} -7.00000 q^{25} -5.19615i q^{27} -9.00000 q^{29} +10.3923i q^{31} +(-3.00000 + 5.19615i) q^{33} +3.46410i q^{35} +6.92820i q^{37} +(-1.50000 + 2.59808i) q^{39} -2.00000 q^{43} +(9.00000 - 5.19615i) q^{45} +3.46410i q^{47} -6.00000 q^{49} +(-1.50000 + 2.59808i) q^{51} +9.00000 q^{53} -12.0000 q^{55} +(-7.50000 - 0.866025i) q^{57} -3.00000 q^{59} -8.00000 q^{61} +(-1.50000 - 2.59808i) q^{63} -6.00000 q^{65} -8.66025i q^{67} +(-4.50000 + 7.79423i) q^{69} -12.0000 q^{71} +11.0000 q^{73} +(10.5000 + 6.06218i) q^{75} +3.46410i q^{77} -6.92820i q^{79} +(-4.50000 + 7.79423i) q^{81} +10.3923i q^{83} -6.00000 q^{85} +(13.5000 + 7.79423i) q^{87} -6.00000 q^{89} +1.73205i q^{91} +(9.00000 - 15.5885i) q^{93} +(-6.00000 - 13.8564i) q^{95} -13.8564i q^{97} +(9.00000 - 5.19615i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 2 * q^7 + 3 * q^9	$ 2 q - 3 q^{3} - 2 q^{7} + 3 q^{9} - 6 q^{15} + 8 q^{19} + 3 q^{21} - 14 q^{25} - 18 q^{29} - 6 q^{33} - 3 q^{39} - 4 q^{43} + 18 q^{45} - 12 q^{49} - 3 q^{51} + 18 q^{53} - 24 q^{55} - 15 q^{57} - 6 q^{59} - 16 q^{61} - 3 q^{63} - 12 q^{65} - 9 q^{69} - 24 q^{71} + 22 q^{73} + 21 q^{75} - 9 q^{81} - 12 q^{85} + 27 q^{87} - 12 q^{89} + 18 q^{93} - 12 q^{95} + 18 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 2 * q^7 + 3 * q^9 - 6 * q^15 + 8 * q^19 + 3 * q^21 - 14 * q^25 - 18 * q^29 - 6 * q^33 - 3 * q^39 - 4 * q^43 + 18 * q^45 - 12 * q^49 - 3 * q^51 + 18 * q^53 - 24 * q^55 - 15 * q^57 - 6 * q^59 - 16 * q^61 - 3 * q^63 - 12 * q^65 - 9 * q^69 - 24 * q^71 + 22 * q^73 + 21 * q^75 - 9 * q^81 - 12 * q^85 + 27 * q^87 - 12 * q^89 + 18 * q^93 - 12 * q^95 + 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$4$		0			0
$5$		−	3.46410i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	3.46410i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		0			0
$7$	−1.00000			−0.377964			−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000			−0.377964			−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$		−	3.46410i		−	1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$		−	3.46410i		−	1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$		0			0
$13$		−	1.73205i		−	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$		−	1.73205i		−	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$		0			0
$15$	−3.00000	+	5.19615i	−0.774597	+	1.34164i
$16$		0			0
$17$		−	1.73205i		−	0.420084i	−0.977692	−	0.210042i	$-0.932640\pi$
$17$		−	1.73205i		−	0.420084i	0.977692	−	0.210042i	$-0.0673601\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$	1.50000	+	0.866025i	0.327327	+	0.188982i
$22$		0			0
$23$		−	5.19615i		−	1.08347i	−0.840548	−	0.541736i	$-0.817767\pi$
$23$		−	5.19615i		−	1.08347i	0.840548	−	0.541736i	$-0.182233\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$			10.3923i			1.86651i	0.359211	+	0.933257i	$0.383046\pi$
$31$			10.3923i			1.86651i	−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	−3.00000	+	5.19615i	−0.522233	+	0.904534i
$34$		0			0
$35$			3.46410i			0.585540i
$36$		0			0
$37$			6.92820i			1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$			6.92820i			1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	−1.50000	+	2.59808i	−0.240192	+	0.416025i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$	9.00000	−	5.19615i	1.34164	−	0.774597i
$46$		0			0
$47$			3.46410i			0.505291i	0.967559	+	0.252646i	$0.0813007\pi$
$47$			3.46410i			0.505291i	−0.967559	+	0.252646i	$0.918699\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		0			0
$51$	−1.50000	+	2.59808i	−0.210042	+	0.363803i
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$	−12.0000			−1.61808
$56$		0			0
$57$	−7.50000	−	0.866025i	−0.993399	−	0.114708i
$58$		0			0
$59$	−3.00000			−0.390567			−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000			−0.390567			−0.195283	+	0.980747i	$0.562563\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	−1.50000	−	2.59808i	−0.188982	−	0.327327i
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$		−	8.66025i		−	1.05802i	−0.848616	−	0.529009i	$-0.822564\pi$
$67$		−	8.66025i		−	1.05802i	0.848616	−	0.529009i	$-0.177436\pi$
$68$		0			0
$69$	−4.50000	+	7.79423i	−0.541736	+	0.938315i
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	11.0000			1.28745			0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000			1.28745			0.643726	+	0.765256i	$0.277388\pi$
$74$		0			0
$75$	10.5000	+	6.06218i	1.21244	+	0.700000i
$76$		0			0
$77$			3.46410i			0.394771i
$78$		0			0
$79$		−	6.92820i		−	0.779484i	−0.920924	−	0.389742i	$-0.872564\pi$
$79$		−	6.92820i		−	0.779484i	0.920924	−	0.389742i	$-0.127436\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$			10.3923i			1.14070i	0.821401	+	0.570352i	$0.193193\pi$
$83$			10.3923i			1.14070i	−0.821401	+	0.570352i	$0.806807\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$		0			0
$87$	13.5000	+	7.79423i	1.44735	+	0.835629i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$			1.73205i			0.181568i
$92$		0			0
$93$	9.00000	−	15.5885i	0.933257	−	1.61645i
$94$		0			0
$95$	−6.00000	−	13.8564i	−0.615587	−	1.42164i
$96$		0			0
$97$		−	13.8564i		−	1.40690i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$		−	13.8564i		−	1.40690i	0.710742	−	0.703452i	$-0.248359\pi$
$98$		0			0
$99$	9.00000	−	5.19615i	0.904534	−	0.522233i
$100$		0			0
$101$			10.3923i			1.03407i	0.855963	+	0.517036i	$0.172965\pi$
$101$			10.3923i			1.03407i	−0.855963	+	0.517036i	$0.827035\pi$
$102$		0			0
$103$		−	3.46410i		−	0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$		−	3.46410i		−	0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$	3.00000	−	5.19615i	0.292770	−	0.507093i
$106$		0			0
$107$	3.00000			0.290021			0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000			0.290021			0.145010	+	0.989430i	$0.453678\pi$
$108$		0			0
$109$		−	15.5885i		−	1.49310i	−0.665327	−	0.746552i	$-0.731708\pi$
$109$		−	15.5885i		−	1.49310i	0.665327	−	0.746552i	$-0.268292\pi$
$110$		0			0
$111$	6.00000	−	10.3923i	0.569495	−	0.986394i
$112$		0			0
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$	−18.0000			−1.67851
$116$		0			0
$117$	4.50000	−	2.59808i	0.416025	−	0.240192i
$118$		0			0
$119$			1.73205i			0.158777i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$			6.92820i			0.619677i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	3.00000	+	1.73205i	0.264135	+	0.152499i
$130$		0			0
$131$			3.46410i			0.302660i	0.988483	+	0.151330i	$0.0483556\pi$
$131$			3.46410i			0.302660i	−0.988483	+	0.151330i	$0.951644\pi$
$132$		0			0
$133$	−4.00000	+	1.73205i	−0.346844	+	0.150188i
$134$		0			0
$135$	−18.0000			−1.54919
$136$		0			0
$137$		−	8.66025i		−	0.739895i	−0.929053	−	0.369948i	$-0.879376\pi$
$137$		−	8.66025i		−	0.739895i	0.929053	−	0.369948i	$-0.120624\pi$
$138$		0			0
$139$	14.0000			1.18746			0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000			1.18746			0.593732	+	0.804663i	$0.297654\pi$
$140$		0			0
$141$	3.00000	−	5.19615i	0.252646	−	0.437595i
$142$		0			0
$143$	−6.00000			−0.501745
$144$		0			0
$145$			31.1769i			2.58910i
$146$		0			0
$147$	9.00000	+	5.19615i	0.742307	+	0.428571i
$148$		0			0
$149$		−	6.92820i		−	0.567581i	−0.958886	−	0.283790i	$-0.908408\pi$
$149$		−	6.92820i		−	0.567581i	0.958886	−	0.283790i	$-0.0915919\pi$
$150$		0			0
$151$			3.46410i			0.281905i	0.990016	+	0.140952i	$0.0450164\pi$
$151$			3.46410i			0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$		0			0
$153$	4.50000	−	2.59808i	0.363803	−	0.210042i
$154$		0			0
$155$	36.0000			2.89159
$156$		0			0
$157$	4.00000			0.319235			0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000			0.319235			0.159617	+	0.987179i	$0.448974\pi$
$158$		0			0
$159$	−13.5000	−	7.79423i	−1.07062	−	0.618123i
$160$		0			0
$161$			5.19615i			0.409514i
$162$		0			0
$163$	−10.0000			−0.783260			−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000			−0.783260			−0.391630	+	0.920123i	$0.628089\pi$
$164$		0			0
$165$	18.0000	+	10.3923i	1.40130	+	0.809040i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	10.0000			0.769231
$170$		0			0
$171$	10.5000	+	7.79423i	0.802955	+	0.596040i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$	7.00000			0.529150
$176$		0			0
$177$	4.50000	+	2.59808i	0.338241	+	0.195283i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$		−	13.8564i		−	1.02994i	−0.857209	−	0.514969i	$-0.827803\pi$
$181$		−	13.8564i		−	1.02994i	0.857209	−	0.514969i	$-0.172197\pi$
$182$		0			0
$183$	12.0000	+	6.92820i	0.887066	+	0.512148i
$184$		0			0
$185$	24.0000			1.76452
$186$		0			0
$187$	−6.00000			−0.438763
$188$		0			0
$189$			5.19615i			0.377964i
$190$		0			0
$191$			19.0526i			1.37859i	0.724479	+	0.689297i	$0.242081\pi$
$191$			19.0526i			1.37859i	−0.724479	+	0.689297i	$0.757919\pi$
$192$		0			0
$193$		−	3.46410i		−	0.249351i	−0.992198	−	0.124676i	$-0.960211\pi$
$193$		−	3.46410i		−	0.249351i	0.992198	−	0.124676i	$-0.0397891\pi$
$194$		0			0
$195$	9.00000	+	5.19615i	0.644503	+	0.372104i
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−11.0000			−0.779769			−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000			−0.779769			−0.389885	+	0.920864i	$0.627485\pi$
$200$		0			0
$201$	−7.50000	+	12.9904i	−0.529009	+	0.916271i
$202$		0			0
$203$	9.00000			0.631676
$204$		0			0
$205$		0			0
$206$		0			0
$207$	13.5000	−	7.79423i	0.938315	−	0.541736i
$208$		0			0
$209$	−6.00000	−	13.8564i	−0.415029	−	0.958468i
$210$		0			0
$211$		−	12.1244i		−	0.834675i	−0.908752	−	0.417338i	$-0.862963\pi$
$211$		−	12.1244i		−	0.834675i	0.908752	−	0.417338i	$-0.137037\pi$
$212$		0			0
$213$	18.0000	+	10.3923i	1.23334	+	0.712069i
$214$		0			0
$215$			6.92820i			0.472500i
$216$		0			0
$217$		−	10.3923i		−	0.705476i
$218$		0			0
$219$	−16.5000	−	9.52628i	−1.11497	−	0.643726i
$220$		0			0
$221$	−3.00000			−0.201802
$222$		0			0
$223$		−	10.3923i		−	0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$		−	10.3923i		−	0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$		0			0
$225$	−10.5000	−	18.1865i	−0.700000	−	1.21244i
$226$		0			0
$227$	−3.00000			−0.199117			−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000			−0.199117			−0.0995585	+	0.995032i	$0.531743\pi$
$228$		0			0
$229$	−8.00000			−0.528655			−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000			−0.528655			−0.264327	+	0.964433i	$0.585150\pi$
$230$		0			0
$231$	3.00000	−	5.19615i	0.197386	−	0.341882i
$232$		0			0
$233$		−	6.92820i		−	0.453882i	−0.973909	−	0.226941i	$-0.927128\pi$
$233$		−	6.92820i		−	0.453882i	0.973909	−	0.226941i	$-0.0728724\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$	−6.00000	+	10.3923i	−0.389742	+	0.675053i
$238$		0			0
$239$		−	12.1244i		−	0.784259i	−0.919910	−	0.392130i	$-0.871738\pi$
$239$		−	12.1244i		−	0.784259i	0.919910	−	0.392130i	$-0.128262\pi$
$240$		0			0
$241$			13.8564i			0.892570i	0.894891	+	0.446285i	$0.147253\pi$
$241$			13.8564i			0.892570i	−0.894891	+	0.446285i	$0.852747\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$		0			0
$245$			20.7846i			1.32788i
$246$		0			0
$247$	−3.00000	−	6.92820i	−0.190885	−	0.440831i
$248$		0			0
$249$	9.00000	−	15.5885i	0.570352	−	0.987878i
$250$		0			0
$251$		−	13.8564i		−	0.874609i	−0.899314	−	0.437304i	$-0.855933\pi$
$251$		−	13.8564i		−	0.874609i	0.899314	−	0.437304i	$-0.144067\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$		0			0
$255$	9.00000	+	5.19615i	0.563602	+	0.325396i
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$		−	6.92820i		−	0.430498i
$260$		0			0
$261$	−13.5000	−	23.3827i	−0.835629	−	1.44735i
$262$		0			0
$263$		−	3.46410i		−	0.213606i	−0.994280	−	0.106803i	$-0.965939\pi$
$263$		−	3.46410i		−	0.213606i	0.994280	−	0.106803i	$-0.0340614\pi$
$264$		0			0
$265$		−	31.1769i		−	1.91518i
$266$		0			0
$267$	9.00000	+	5.19615i	0.550791	+	0.317999i
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	1.00000			0.0607457			0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000			0.0607457			0.0303728	+	0.999539i	$0.490331\pi$
$272$		0			0
$273$	1.50000	−	2.59808i	0.0907841	−	0.157243i
$274$		0			0
$275$			24.2487i			1.46225i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$	−27.0000	+	15.5885i	−1.61645	+	0.933257i
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$	10.0000			0.594438			0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000			0.594438			0.297219	+	0.954809i	$0.403941\pi$
$284$		0			0
$285$	−3.00000	+	25.9808i	−0.177705	+	1.53897i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	14.0000			0.823529
$290$		0			0
$291$	−12.0000	+	20.7846i	−0.703452	+	1.21842i
$292$		0			0
$293$	21.0000			1.22683			0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000			1.22683			0.613417	+	0.789760i	$0.289795\pi$
$294$		0			0
$295$			10.3923i			0.605063i
$296$		0			0
$297$	−18.0000			−1.04447
$298$		0			0
$299$	−9.00000			−0.520483
$300$		0			0
$301$	2.00000			0.115278
$302$		0			0
$303$	9.00000	−	15.5885i	0.517036	−	0.895533i
$304$		0			0
$305$			27.7128i			1.58683i
$306$		0			0
$307$		−	10.3923i		−	0.593120i	−0.955014	−	0.296560i	$-0.904160\pi$
$307$		−	10.3923i		−	0.593120i	0.955014	−	0.296560i	$-0.0958395\pi$
$308$		0			0
$309$	−3.00000	+	5.19615i	−0.170664	+	0.295599i
$310$		0			0
$311$		−	22.5167i		−	1.27680i	−0.769704	−	0.638401i	$-0.779596\pi$
$311$		−	22.5167i		−	1.27680i	0.769704	−	0.638401i	$-0.220404\pi$
$312$		0			0
$313$	13.0000			0.734803			0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000			0.734803			0.367402	+	0.930062i	$0.380247\pi$
$314$		0			0
$315$	−9.00000	+	5.19615i	−0.507093	+	0.292770i
$316$		0			0
$317$	−3.00000			−0.168497			−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000			−0.168497			−0.0842484	+	0.996445i	$0.526849\pi$
$318$		0			0
$319$			31.1769i			1.74557i
$320$		0			0
$321$	−4.50000	−	2.59808i	−0.251166	−	0.145010i
$322$		0			0
$323$	−3.00000	−	6.92820i	−0.166924	−	0.385496i
$324$		0			0
$325$			12.1244i			0.672538i
$326$		0			0
$327$	−13.5000	+	23.3827i	−0.746552	+	1.29307i
$328$		0			0
$329$		−	3.46410i		−	0.190982i
$330$		0			0
$331$		−	19.0526i		−	1.04722i	−0.851957	−	0.523612i	$-0.824584\pi$
$331$		−	19.0526i		−	1.04722i	0.851957	−	0.523612i	$-0.175416\pi$
$332$		0			0
$333$	−18.0000	+	10.3923i	−0.986394	+	0.569495i
$334$		0			0
$335$	−30.0000			−1.63908
$336$		0			0
$337$			24.2487i			1.32091i	0.750865	+	0.660456i	$0.229637\pi$
$337$			24.2487i			1.32091i	−0.750865	+	0.660456i	$0.770363\pi$
$338$		0			0
$339$	−18.0000	−	10.3923i	−0.977626	−	0.564433i
$340$		0			0
$341$	36.0000			1.94951
$342$		0			0
$343$	13.0000			0.701934
$344$		0			0
$345$	27.0000	+	15.5885i	1.45363	+	0.839254i
$346$		0			0
$347$		−	27.7128i		−	1.48770i	−0.668346	−	0.743851i	$-0.732997\pi$
$347$		−	27.7128i		−	1.48770i	0.668346	−	0.743851i	$-0.267003\pi$
$348$		0			0
$349$	−28.0000			−1.49881			−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000			−1.49881			−0.749403	+	0.662114i	$0.769659\pi$
$350$		0			0
$351$	−9.00000			−0.480384
$352$		0			0
$353$		−	25.9808i		−	1.38282i	−0.722464	−	0.691408i	$-0.756991\pi$
$353$		−	25.9808i		−	1.38282i	0.722464	−	0.691408i	$-0.243009\pi$
$354$		0			0
$355$			41.5692i			2.20627i
$356$		0			0
$357$	1.50000	−	2.59808i	0.0793884	−	0.137505i
$358$		0			0
$359$			19.0526i			1.00556i	0.864416	+	0.502778i	$0.167689\pi$
$359$			19.0526i			1.00556i	−0.864416	+	0.502778i	$0.832311\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$	1.50000	+	0.866025i	0.0787296	+	0.0454545i
$364$		0			0
$365$		−	38.1051i		−	1.99451i
$366$		0			0
$367$	−32.0000			−1.67039			−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000			−1.67039			−0.835193	+	0.549957i	$0.814644\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.00000			−0.467257
$372$		0			0
$373$		−	29.4449i		−	1.52460i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$		−	29.4449i		−	1.52460i	0.647225	−	0.762299i	$-0.275929\pi$
$374$		0			0
$375$	6.00000	−	10.3923i	0.309839	−	0.536656i
$376$		0			0
$377$			15.5885i			0.802846i
$378$		0			0
$379$		−	12.1244i		−	0.622786i	−0.950281	−	0.311393i	$-0.899204\pi$
$379$		−	12.1244i		−	0.622786i	0.950281	−	0.311393i	$-0.100796\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6.00000			0.306586			0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000			0.306586			0.153293	+	0.988181i	$0.451012\pi$
$384$		0			0
$385$	12.0000			0.611577
$386$		0			0
$387$	−3.00000	−	5.19615i	−0.152499	−	0.264135i
$388$		0			0
$389$			3.46410i			0.175637i	0.996136	+	0.0878185i	$0.0279895\pi$
$389$			3.46410i			0.175637i	−0.996136	+	0.0878185i	$0.972010\pi$
$390$		0			0
$391$	−9.00000			−0.455150
$392$		0			0
$393$	3.00000	−	5.19615i	0.151330	−	0.262111i
$394$		0			0
$395$	−24.0000			−1.20757
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$		0			0
$399$	7.50000	+	0.866025i	0.375470	+	0.0433555i
$400$		0			0
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$		0			0
$403$	18.0000			0.896644
$404$		0			0
$405$	27.0000	+	15.5885i	1.34164	+	0.774597i
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$			38.1051i			1.88418i	0.335365	+	0.942088i	$0.391140\pi$
$409$			38.1051i			1.88418i	−0.335365	+	0.942088i	$0.608860\pi$
$410$		0			0
$411$	−7.50000	+	12.9904i	−0.369948	+	0.640768i
$412$		0			0
$413$	3.00000			0.147620
$414$		0			0
$415$	36.0000			1.76717
$416$		0			0
$417$	−21.0000	−	12.1244i	−1.02837	−	0.593732i
$418$		0			0
$419$			10.3923i			0.507697i	0.967244	+	0.253849i	$0.0816965\pi$
$419$			10.3923i			0.507697i	−0.967244	+	0.253849i	$0.918303\pi$
$420$		0			0
$421$		−	12.1244i		−	0.590905i	−0.955357	−	0.295452i	$-0.904530\pi$
$421$		−	12.1244i		−	0.590905i	0.955357	−	0.295452i	$-0.0954704\pi$
$422$		0			0
$423$	−9.00000	+	5.19615i	−0.437595	+	0.252646i
$424$		0			0
$425$			12.1244i			0.588118i
$426$		0			0
$427$	8.00000			0.387147
$428$		0			0
$429$	9.00000	+	5.19615i	0.434524	+	0.250873i
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$		0			0
$435$	27.0000	−	46.7654i	1.29455	−	2.24223i
$436$		0			0
$437$	−9.00000	−	20.7846i	−0.430528	−	0.994263i
$438$		0			0
$439$			20.7846i			0.991995i	0.868324	+	0.495998i	$0.165198\pi$
$439$			20.7846i			0.991995i	−0.868324	+	0.495998i	$0.834802\pi$
$440$		0			0
$441$	−9.00000	−	15.5885i	−0.428571	−	0.742307i
$442$		0			0
$443$		−	17.3205i		−	0.822922i	−0.911427	−	0.411461i	$-0.865019\pi$
$443$		−	17.3205i		−	0.822922i	0.911427	−	0.411461i	$-0.134981\pi$
$444$		0			0
$445$			20.7846i			0.985285i
$446$		0			0
$447$	−6.00000	+	10.3923i	−0.283790	+	0.491539i
$448$		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
$451$		0			0
$452$		0			0
$453$	3.00000	−	5.19615i	0.140952	−	0.244137i
$454$		0			0
$455$	6.00000			0.281284
$456$		0			0
$457$	−25.0000			−1.16945			−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000			−1.16945			−0.584725	+	0.811231i	$0.698798\pi$
$458$		0			0
$459$	−9.00000			−0.420084
$460$		0			0
$461$			13.8564i			0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$			13.8564i			0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$		0			0
$463$	40.0000			1.85896			0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000			1.85896			0.929479	+	0.368875i	$0.120257\pi$
$464$		0			0
$465$	−54.0000	−	31.1769i	−2.50419	−	1.44579i
$466$		0			0
$467$			27.7128i			1.28240i	0.767375	+	0.641198i	$0.221562\pi$
$467$			27.7128i			1.28240i	−0.767375	+	0.641198i	$0.778438\pi$
$468$		0			0
$469$			8.66025i			0.399893i
$470$		0			0
$471$	−6.00000	−	3.46410i	−0.276465	−	0.159617i
$472$		0			0
$473$			6.92820i			0.318559i
$474$		0			0
$475$	−28.0000	+	12.1244i	−1.28473	+	0.556304i
$476$		0			0
$477$	13.5000	+	23.3827i	0.618123	+	1.07062i
$478$		0			0
$479$		−	10.3923i		−	0.474837i	−0.971408	−	0.237418i	$-0.923699\pi$
$479$		−	10.3923i		−	0.474837i	0.971408	−	0.237418i	$-0.0763012\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$		0			0
$483$	4.50000	−	7.79423i	0.204757	−	0.354650i
$484$		0			0
$485$	−48.0000			−2.17957
$486$		0			0
$487$			3.46410i			0.156973i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$			3.46410i			0.156973i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	15.0000	+	8.66025i	0.678323	+	0.391630i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$			15.5885i			0.702069i
$494$		0			0
$495$	−18.0000	−	31.1769i	−0.809040	−	1.40130i
$496$		0			0
$497$	12.0000			0.538274
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	25.9808i		−	1.15842i	−0.815177	−	0.579212i	$-0.803360\pi$
$503$		−	25.9808i		−	1.15842i	0.815177	−	0.579212i	$-0.196640\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$		0			0
$507$	−15.0000	−	8.66025i	−0.666173	−	0.384615i
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$	−11.0000			−0.486611
$512$		0			0
$513$	−9.00000	−	20.7846i	−0.397360	−	0.917663i
$514$		0			0
$515$	−12.0000			−0.528783
$516$		0			0
$517$	12.0000			0.527759
$518$		0			0
$519$	9.00000	+	5.19615i	0.395056	+	0.228086i
$520$		0			0
$521$	24.0000			1.05146			0.525730	−	0.850652i	$-0.323792\pi$
$521$	24.0000			1.05146			0.525730	+	0.850652i	$0.323792\pi$
$522$		0			0
$523$			32.9090i			1.43901i	0.694488	+	0.719504i	$0.255631\pi$
$523$			32.9090i			1.43901i	−0.694488	+	0.719504i	$0.744369\pi$
$524$		0			0
$525$	−10.5000	−	6.06218i	−0.458258	−	0.264575i
$526$		0			0
$527$	18.0000			0.784092
$528$		0			0
$529$	−4.00000			−0.173913
$530$		0			0
$531$	−4.50000	−	7.79423i	−0.195283	−	0.338241i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	10.3923i		−	0.449299i
$536$		0			0
$537$	−18.0000	−	10.3923i	−0.776757	−	0.448461i
$538$		0			0
$539$			20.7846i			0.895257i
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$	−12.0000	+	20.7846i	−0.514969	+	0.891953i
$544$		0			0
$545$	−54.0000			−2.31311
$546$		0			0
$547$		−	24.2487i		−	1.03680i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$		−	24.2487i		−	1.03680i	0.855138	−	0.518400i	$-0.173472\pi$
$548$		0			0
$549$	−12.0000	−	20.7846i	−0.512148	−	0.887066i
$550$		0			0
$551$	−36.0000	+	15.5885i	−1.53365	+	0.664091i
$552$		0			0
$553$			6.92820i			0.294617i
$554$		0			0
$555$	−36.0000	−	20.7846i	−1.52811	−	0.882258i
$556$		0			0
$557$		−	45.0333i		−	1.90812i	−0.299611	−	0.954062i	$-0.596857\pi$
$557$		−	45.0333i		−	1.90812i	0.299611	−	0.954062i	$-0.403143\pi$
$558$		0			0
$559$			3.46410i			0.146516i
$560$		0			0
$561$	9.00000	+	5.19615i	0.379980	+	0.219382i
$562$		0			0
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$		−	41.5692i		−	1.74883i
$566$		0			0
$567$	4.50000	−	7.79423i	0.188982	−	0.327327i
$568$		0			0
$569$	−24.0000			−1.00613			−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000			−1.00613			−0.503066	+	0.864248i	$0.667795\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$	16.5000	−	28.5788i	0.689297	−	1.19390i
$574$		0			0
$575$			36.3731i			1.51686i
$576$		0			0
$577$	25.0000			1.04076			0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000			1.04076			0.520382	+	0.853934i	$0.325790\pi$
$578$		0			0
$579$	−3.00000	+	5.19615i	−0.124676	+	0.215945i
$580$		0			0
$581$		−	10.3923i		−	0.431145i
$582$		0			0
$583$		−	31.1769i		−	1.29122i
$584$		0			0
$585$	−9.00000	−	15.5885i	−0.372104	−	0.644503i
$586$		0			0
$587$		−	13.8564i		−	0.571915i	−0.958242	−	0.285958i	$-0.907688\pi$
$587$		−	13.8564i		−	0.571915i	0.958242	−	0.285958i	$-0.0923116\pi$
$588$		0			0
$589$	18.0000	+	41.5692i	0.741677	+	1.71283i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			20.7846i			0.853522i	0.904365	+	0.426761i	$0.140345\pi$
$593$			20.7846i			0.853522i	−0.904365	+	0.426761i	$0.859655\pi$
$594$		0			0
$595$	6.00000			0.245976
$596$		0			0
$597$	16.5000	+	9.52628i	0.675300	+	0.389885i
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$			10.3923i			0.423911i	0.977279	+	0.211955i	$0.0679832\pi$
$601$			10.3923i			0.423911i	−0.977279	+	0.211955i	$0.932017\pi$
$602$		0			0
$603$	22.5000	−	12.9904i	0.916271	−	0.529009i
$604$		0			0
$605$			3.46410i			0.140836i
$606$		0			0
$607$		−	27.7128i		−	1.12483i	−0.826856	−	0.562414i	$-0.809873\pi$
$607$		−	27.7128i		−	1.12483i	0.826856	−	0.562414i	$-0.190127\pi$
$608$		0			0
$609$	−13.5000	−	7.79423i	−0.547048	−	0.315838i
$610$		0			0
$611$	6.00000			0.242734
$612$		0			0
$613$	−2.00000			−0.0807792			−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000			−0.0807792			−0.0403896	+	0.999184i	$0.512860\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			48.4974i			1.95243i	0.216799	+	0.976216i	$0.430439\pi$
$617$			48.4974i			1.95243i	−0.216799	+	0.976216i	$0.569561\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		0			0
$621$	−27.0000			−1.08347
$622$		0			0
$623$	6.00000			0.240385
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$	−3.00000	+	25.9808i	−0.119808	+	1.03757i
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$	−10.5000	+	18.1865i	−0.417338	+	0.722850i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			10.3923i			0.411758i
$638$		0			0
$639$	−18.0000	−	31.1769i	−0.712069	−	1.23334i
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$	28.0000			1.10421			0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000			1.10421			0.552106	+	0.833774i	$0.313824\pi$
$644$		0			0
$645$	6.00000	−	10.3923i	0.236250	−	0.409197i
$646$		0			0
$647$			1.73205i			0.0680939i	0.999420	+	0.0340470i	$0.0108396\pi$
$647$			1.73205i			0.0680939i	−0.999420	+	0.0340470i	$0.989160\pi$
$648$		0			0
$649$			10.3923i			0.407934i
$650$		0			0
$651$	−9.00000	+	15.5885i	−0.352738	+	0.610960i
$652$		0			0
$653$			17.3205i			0.677804i	0.940822	+	0.338902i	$0.110055\pi$
$653$			17.3205i			0.677804i	−0.940822	+	0.338902i	$0.889945\pi$
$654$		0			0
$655$	12.0000			0.468879
$656$		0			0
$657$	16.5000	+	28.5788i	0.643726	+	1.11497i
$658$		0			0
$659$	−33.0000			−1.28550			−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000			−1.28550			−0.642749	+	0.766077i	$0.722206\pi$
$660$		0			0
$661$		−	22.5167i		−	0.875797i	−0.899025	−	0.437898i	$-0.855723\pi$
$661$		−	22.5167i		−	0.875797i	0.899025	−	0.437898i	$-0.144277\pi$
$662$		0			0
$663$	4.50000	+	2.59808i	0.174766	+	0.100901i
$664$		0			0
$665$	6.00000	+	13.8564i	0.232670	+	0.537328i
$666$		0			0
$667$			46.7654i			1.81076i
$668$		0			0
$669$	−9.00000	+	15.5885i	−0.347960	+	0.602685i
$670$		0			0
$671$			27.7128i			1.06984i
$672$		0			0
$673$		−	48.4974i		−	1.86944i	−0.355387	−	0.934719i	$-0.615651\pi$
$673$		−	48.4974i		−	1.86944i	0.355387	−	0.934719i	$-0.384349\pi$
$674$		0			0
$675$			36.3731i			1.40000i
$676$		0			0
$677$	15.0000			0.576497			0.288248	−	0.957556i	$-0.406927\pi$
$677$	15.0000			0.576497			0.288248	+	0.957556i	$0.406927\pi$
$678$		0			0
$679$			13.8564i			0.531760i
$680$		0			0
$681$	4.50000	+	2.59808i	0.172440	+	0.0995585i
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$	−30.0000			−1.14624
$686$		0			0
$687$	12.0000	+	6.92820i	0.457829	+	0.264327i
$688$		0			0
$689$		−	15.5885i		−	0.593873i
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		0			0
$693$	−9.00000	+	5.19615i	−0.341882	+	0.197386i
$694$		0			0
$695$		−	48.4974i		−	1.83961i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−6.00000	+	10.3923i	−0.226941	+	0.393073i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	12.0000	+	27.7128i	0.452589	+	1.04521i
$704$		0			0
$705$	−18.0000	−	10.3923i	−0.677919	−	0.391397i
$706$		0			0
$707$		−	10.3923i		−	0.390843i
$708$		0			0
$709$	−4.00000			−0.150223			−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000			−0.150223			−0.0751116	+	0.997175i	$0.523931\pi$
$710$		0			0
$711$	18.0000	−	10.3923i	0.675053	−	0.389742i
$712$		0			0
$713$	54.0000			2.02232
$714$		0			0
$715$			20.7846i			0.777300i
$716$		0			0
$717$	−10.5000	+	18.1865i	−0.392130	+	0.679189i
$718$		0			0
$719$		−	8.66025i		−	0.322973i	−0.986875	−	0.161486i	$-0.948371\pi$
$719$		−	8.66025i		−	0.322973i	0.986875	−	0.161486i	$-0.0516288\pi$
$720$		0			0
$721$			3.46410i			0.129010i
$722$		0			0
$723$	12.0000	−	20.7846i	0.446285	−	0.772988i
$724$		0			0
$725$	63.0000			2.33976
$726$		0			0
$727$	7.00000			0.259616			0.129808	−	0.991539i	$-0.458564\pi$
$727$	7.00000			0.259616			0.129808	+	0.991539i	$0.458564\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$			3.46410i			0.128124i
$732$		0			0
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$		0			0
$735$	18.0000	−	31.1769i	0.663940	−	1.14998i
$736$		0			0
$737$	−30.0000			−1.10506
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$		0			0
$741$	−1.50000	+	12.9904i	−0.0551039	+	0.477214i
$742$		0			0
$743$	30.0000			1.10059			0.550297	−	0.834969i	$-0.314515\pi$
$743$	30.0000			1.10059			0.550297	+	0.834969i	$0.314515\pi$
$744$		0			0
$745$	−24.0000			−0.879292
$746$		0			0
$747$	−27.0000	+	15.5885i	−0.987878	+	0.570352i
$748$		0			0
$749$	−3.00000			−0.109618
$750$		0			0
$751$		−	6.92820i		−	0.252814i	−0.991978	−	0.126407i	$-0.959656\pi$
$751$		−	6.92820i		−	0.252814i	0.991978	−	0.126407i	$-0.0403445\pi$
$752$		0			0
$753$	−12.0000	+	20.7846i	−0.437304	+	0.757433i
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$		0			0
$759$	27.0000	+	15.5885i	0.980038	+	0.565825i
$760$		0			0
$761$			8.66025i			0.313934i	0.987604	+	0.156967i	$0.0501716\pi$
$761$			8.66025i			0.313934i	−0.987604	+	0.156967i	$0.949828\pi$
$762$		0			0
$763$			15.5885i			0.564340i
$764$		0			0
$765$	−9.00000	−	15.5885i	−0.325396	−	0.563602i
$766$		0			0
$767$			5.19615i			0.187622i
$768$		0			0
$769$	49.0000			1.76699			0.883493	−	0.468445i	$-0.155186\pi$
$769$	49.0000			1.76699			0.883493	+	0.468445i	$0.155186\pi$
$770$		0			0
$771$	9.00000	+	5.19615i	0.324127	+	0.187135i
$772$		0			0
$773$	3.00000			0.107903			0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000			0.107903			0.0539513	+	0.998544i	$0.482818\pi$
$774$		0			0
$775$		−	72.7461i		−	2.61312i
$776$		0			0
$777$	−6.00000	+	10.3923i	−0.215249	+	0.372822i
$778$		0			0
$779$		0			0
$780$		0			0
$781$			41.5692i			1.48746i
$782$		0			0
$783$			46.7654i			1.67126i
$784$		0			0
$785$		−	13.8564i		−	0.494556i
$786$		0			0
$787$			22.5167i			0.802632i	0.915940	+	0.401316i	$0.131447\pi$
$787$			22.5167i			0.802632i	−0.915940	+	0.401316i	$0.868553\pi$
$788$		0			0
$789$	−3.00000	+	5.19615i	−0.106803	+	0.184988i
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$			13.8564i			0.492055i
$794$		0			0
$795$	−27.0000	+	46.7654i	−0.957591	+	1.65860i
$796$		0			0
$797$	21.0000			0.743858			0.371929	−	0.928261i	$-0.378696\pi$
$797$	21.0000			0.743858			0.371929	+	0.928261i	$0.378696\pi$
$798$		0			0
$799$	6.00000			0.212265
$800$		0			0
$801$	−9.00000	−	15.5885i	−0.317999	−	0.550791i
$802$		0			0
$803$		−	38.1051i		−	1.34470i
$804$		0			0
$805$	18.0000			0.634417
$806$		0			0
$807$	27.0000	+	15.5885i	0.950445	+	0.548740i
$808$		0			0
$809$		−	32.9090i		−	1.15702i	−0.815676	−	0.578509i	$-0.803635\pi$
$809$		−	32.9090i		−	1.15702i	0.815676	−	0.578509i	$-0.196365\pi$
$810$		0			0
$811$		−	46.7654i		−	1.64215i	−0.570817	−	0.821077i	$-0.693374\pi$
$811$		−	46.7654i		−	1.64215i	0.570817	−	0.821077i	$-0.306626\pi$
$812$		0			0
$813$	−1.50000	−	0.866025i	−0.0526073	−	0.0303728i
$814$		0			0
$815$			34.6410i			1.21342i
$816$		0			0
$817$	−8.00000	+	3.46410i	−0.279885	+	0.121194i
$818$		0			0
$819$	−4.50000	+	2.59808i	−0.157243	+	0.0907841i
$820$		0			0
$821$			17.3205i			0.604490i	0.953230	+	0.302245i	$0.0977361\pi$
$821$			17.3205i			0.604490i	−0.953230	+	0.302245i	$0.902264\pi$
$822$		0			0
$823$	23.0000			0.801730			0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000			0.801730			0.400865	+	0.916137i	$0.368710\pi$
$824$		0			0
$825$	21.0000	−	36.3731i	0.731126	−	1.26635i
$826$		0			0
$827$	−33.0000			−1.14752			−0.573761	−	0.819023i	$-0.694516\pi$
$827$	−33.0000			−1.14752			−0.573761	+	0.819023i	$0.694516\pi$
$828$		0			0
$829$			1.73205i			0.0601566i	0.999548	+	0.0300783i	$0.00957567\pi$
$829$			1.73205i			0.0601566i	−0.999548	+	0.0300783i	$0.990424\pi$
$830$		0			0
$831$	3.00000	+	1.73205i	0.104069	+	0.0600842i
$832$		0			0
$833$			10.3923i			0.360072i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	54.0000			1.86651
$838$		0			0
$839$	42.0000			1.45000			0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000			1.45000			0.725001	+	0.688748i	$0.241839\pi$
$840$		0			0
$841$	52.0000			1.79310
$842$		0			0
$843$	−18.0000	−	10.3923i	−0.619953	−	0.357930i
$844$		0			0
$845$		−	34.6410i		−	1.19169i
$846$		0			0
$847$	1.00000			0.0343604
$848$		0			0
$849$	−15.0000	−	8.66025i	−0.514799	−	0.297219i
$850$		0			0
$851$	36.0000			1.23406
$852$		0			0
$853$	14.0000			0.479351			0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000			0.479351			0.239675	+	0.970853i	$0.422959\pi$
$854$		0			0
$855$	27.0000	−	36.3731i	0.923381	−	1.24393i
$856$		0			0
$857$	24.0000			0.819824			0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000			0.819824			0.409912	+	0.912125i	$0.365559\pi$
$858$		0			0
$859$	50.0000			1.70598			0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000			1.70598			0.852989	+	0.521929i	$0.174787\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$			20.7846i			0.706698i
$866$		0			0
$867$	−21.0000	−	12.1244i	−0.713197	−	0.411765i
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	−15.0000			−0.508256
$872$		0			0
$873$	36.0000	−	20.7846i	1.21842	−	0.703452i
$874$		0			0
$875$		−	6.92820i		−	0.234216i
$876$		0			0
$877$		−	53.6936i		−	1.81310i	−0.422095	−	0.906552i	$-0.638705\pi$
$877$		−	53.6936i		−	1.81310i	0.422095	−	0.906552i	$-0.361295\pi$
$878$		0			0
$879$	−31.5000	−	18.1865i	−1.06247	−	0.613417i
$880$		0			0
$881$		−	34.6410i		−	1.16709i	−0.812082	−	0.583543i	$-0.801666\pi$
$881$		−	34.6410i		−	1.16709i	0.812082	−	0.583543i	$-0.198334\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$	9.00000	−	15.5885i	0.302532	−	0.524000i
$886$		0			0
$887$	−36.0000			−1.20876			−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000			−1.20876			−0.604381	+	0.796696i	$0.706579\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	27.0000	+	15.5885i	0.904534	+	0.522233i
$892$		0			0
$893$	6.00000	+	13.8564i	0.200782	+	0.463687i
$894$		0			0
$895$		−	41.5692i		−	1.38951i
$896$		0			0
$897$	13.5000	+	7.79423i	0.450752	+	0.260242i
$898$		0			0
$899$		−	93.5307i		−	3.11942i
$900$		0			0
$901$		−	15.5885i		−	0.519327i
$902$		0			0
$903$	−3.00000	−	1.73205i	−0.0998337	−	0.0576390i
$904$		0			0
$905$	−48.0000			−1.59557
$906$		0			0
$907$		−	8.66025i		−	0.287559i	−0.989610	−	0.143780i	$-0.954074\pi$
$907$		−	8.66025i		−	0.287559i	0.989610	−	0.143780i	$-0.0459256\pi$
$908$		0			0
$909$	−27.0000	+	15.5885i	−0.895533	+	0.517036i
$910$		0			0
$911$	30.0000			0.993944			0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000			0.993944			0.496972	+	0.867766i	$0.334445\pi$
$912$		0			0
$913$	36.0000			1.19143
$914$		0			0
$915$	24.0000	−	41.5692i	0.793416	−	1.37424i
$916$		0			0
$917$		−	3.46410i		−	0.114395i
$918$		0			0
$919$	11.0000			0.362857			0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000			0.362857			0.181428	+	0.983404i	$0.441928\pi$
$920$		0			0
$921$	−9.00000	+	15.5885i	−0.296560	+	0.513657i
$922$		0			0
$923$			20.7846i			0.684134i
$924$		0			0
$925$		−	48.4974i		−	1.59459i
$926$		0			0
$927$	9.00000	−	5.19615i	0.295599	−	0.170664i
$928$		0			0
$929$		−	15.5885i		−	0.511441i	−0.966751	−	0.255720i	$-0.917687\pi$
$929$		−	15.5885i		−	0.511441i	0.966751	−	0.255720i	$-0.0823126\pi$
$930$		0			0
$931$	−24.0000	+	10.3923i	−0.786568	+	0.340594i
$932$		0			0
$933$	−19.5000	+	33.7750i	−0.638401	+	1.10574i
$934$		0			0
$935$			20.7846i			0.679729i
$936$		0			0
$937$	−7.00000			−0.228680			−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000			−0.228680			−0.114340	+	0.993442i	$0.536475\pi$
$938$		0			0
$939$	−19.5000	−	11.2583i	−0.636358	−	0.367402i
$940$		0			0
$941$	−39.0000			−1.27136			−0.635682	−	0.771951i	$-0.719281\pi$
$941$	−39.0000			−1.27136			−0.635682	+	0.771951i	$0.719281\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	18.0000			0.585540
$946$		0			0
$947$			31.1769i			1.01311i	0.862207	+	0.506557i	$0.169082\pi$
$947$			31.1769i			1.01311i	−0.862207	+	0.506557i	$0.830918\pi$
$948$		0			0
$949$		−	19.0526i		−	0.618472i
$950$		0			0
$951$	4.50000	+	2.59808i	0.145922	+	0.0842484i
$952$		0			0
$953$	54.0000			1.74923			0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000			1.74923			0.874616	+	0.484817i	$0.161114\pi$
$954$		0			0
$955$	66.0000			2.13571
$956$		0			0
$957$	27.0000	−	46.7654i	0.872786	−	1.51171i
$958$		0			0
$959$			8.66025i			0.279654i
$960$		0			0
$961$	−77.0000			−2.48387
$962$		0			0
$963$	4.50000	+	7.79423i	0.145010	+	0.251166i
$964$		0			0
$965$	−12.0000			−0.386294
$966$		0			0
$967$	−16.0000			−0.514525			−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000			−0.514525			−0.257263	+	0.966342i	$0.582821\pi$
$968$		0			0
$969$	−1.50000	+	12.9904i	−0.0481869	+	0.417311i
$970$		0			0
$971$	48.0000			1.54039			0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000			1.54039			0.770197	+	0.637806i	$0.220158\pi$
$972$		0			0
$973$	−14.0000			−0.448819
$974$		0			0
$975$	10.5000	−	18.1865i	0.336269	−	0.582435i
$976$		0			0
$977$	30.0000			0.959785			0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000			0.959785			0.479893	+	0.877327i	$0.340676\pi$
$978$		0			0
$979$			20.7846i			0.664279i
$980$		0			0
$981$	40.5000	−	23.3827i	1.29307	−	0.746552i
$982$		0			0
$983$	60.0000			1.91370			0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000			1.91370			0.956851	+	0.290578i	$0.0938475\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−3.00000	+	5.19615i	−0.0954911	+	0.165395i
$988$		0			0
$989$			10.3923i			0.330456i
$990$		0			0
$991$			3.46410i			0.110041i	0.998485	+	0.0550204i	$0.0175224\pi$
$991$			3.46410i			0.110041i	−0.998485	+	0.0550204i	$0.982478\pi$
$992$		0			0
$993$	−16.5000	+	28.5788i	−0.523612	+	0.906922i
$994$		0			0
$995$			38.1051i			1.20801i
$996$		0			0
$997$	−38.0000			−1.20347			−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000			−1.20347			−0.601736	+	0.798695i	$0.705524\pi$
$998$		0			0
$999$	36.0000			1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.f.a.113.1		2
3.2	odd	2		912.2.f.e.113.1		2
4.3	odd	2		114.2.b.b.113.2	yes	2
12.11	even	2		114.2.b.c.113.2	yes	2
19.18	odd	2		912.2.f.e.113.2		2
57.56	even	2	inner	912.2.f.a.113.2		2
76.75	even	2		114.2.b.c.113.1	yes	2
228.227	odd	2		114.2.b.b.113.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.b.b.113.1	✓	2	228.227	odd	2
114.2.b.b.113.2	yes	2	4.3	odd	2
114.2.b.c.113.1	yes	2	76.75	even	2
114.2.b.c.113.2	yes	2	12.11	even	2
912.2.f.a.113.1		2	1.1	even	1	trivial
912.2.f.a.113.2		2	57.56	even	2	inner
912.2.f.e.113.1		2	3.2	odd	2
912.2.f.e.113.2		2	19.18	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{3} -3.46410i q^{5} -1.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 0.866025i) q^{3} -3.46410i q^{5} -1.00000 q^{7} +(1.50000 + 2.59808i) q^{9} -3.46410i q^{11} -1.73205i q^{13} +(-3.00000 + 5.19615i) q^{15} -1.73205i q^{17} +(4.00000 - 1.73205i) q^{19} +(1.50000 + 0.866025i) q^{21} -5.19615i q^{23} -7.00000 q^{25} -5.19615i q^{27} -9.00000 q^{29} +10.3923i q^{31} +(-3.00000 + 5.19615i) q^{33} +3.46410i q^{35} +6.92820i q^{37} +(-1.50000 + 2.59808i) q^{39} -2.00000 q^{43} +(9.00000 - 5.19615i) q^{45} +3.46410i q^{47} -6.00000 q^{49} +(-1.50000 + 2.59808i) q^{51} +9.00000 q^{53} -12.0000 q^{55} +(-7.50000 - 0.866025i) q^{57} -3.00000 q^{59} -8.00000 q^{61} +(-1.50000 - 2.59808i) q^{63} -6.00000 q^{65} -8.66025i q^{67} +(-4.50000 + 7.79423i) q^{69} -12.0000 q^{71} +11.0000 q^{73} +(10.5000 + 6.06218i) q^{75} +3.46410i q^{77} -6.92820i q^{79} +(-4.50000 + 7.79423i) q^{81} +10.3923i q^{83} -6.00000 q^{85} +(13.5000 + 7.79423i) q^{87} -6.00000 q^{89} +1.73205i q^{91} +(9.00000 - 15.5885i) q^{93} +(-6.00000 - 13.8564i) q^{95} -13.8564i q^{97} +(9.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 2 * q^7 + 3 * q^9	\( 2 q - 3 q^{3} - 2 q^{7} + 3 q^{9} - 6 q^{15} + 8 q^{19} + 3 q^{21} - 14 q^{25} - 18 q^{29} - 6 q^{33} - 3 q^{39} - 4 q^{43} + 18 q^{45} - 12 q^{49} - 3 q^{51} + 18 q^{53} - 24 q^{55} - 15 q^{57} - 6 q^{59} - 16 q^{61} - 3 q^{63} - 12 q^{65} - 9 q^{69} - 24 q^{71} + 22 q^{73} + 21 q^{75} - 9 q^{81} - 12 q^{85} + 27 q^{87} - 12 q^{89} + 18 q^{93} - 12 q^{95} + 18 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 2 * q^7 + 3 * q^9 - 6 * q^15 + 8 * q^19 + 3 * q^21 - 14 * q^25 - 18 * q^29 - 6 * q^33 - 3 * q^39 - 4 * q^43 + 18 * q^45 - 12 * q^49 - 3 * q^51 + 18 * q^53 - 24 * q^55 - 15 * q^57 - 6 * q^59 - 16 * q^61 - 3 * q^63 - 12 * q^65 - 9 * q^69 - 24 * q^71 + 22 * q^73 + 21 * q^75 - 9 * q^81 - 12 * q^85 + 27 * q^87 - 12 * q^89 + 18 * q^93 - 12 * q^95 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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