LMFDB - Embedded newform 912.2.bn.e.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(65,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

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

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

chi := DirichletCharacter("912.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.bn (of order $6$, 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:	$7.28235666434$
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 228)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			65.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	912.65
Dual form			912.2.bn.e.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 0.866025i) q^{3} +5.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})$	$q+(1.50000 + 0.866025i) q^{3} +5.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +(-4.50000 + 2.59808i) q^{13} +(4.00000 + 1.73205i) q^{19} +(7.50000 + 4.33013i) q^{21} +(-2.50000 - 4.33013i) q^{25} +5.19615i q^{27} -8.66025i q^{31} -5.19615i q^{37} -9.00000 q^{39} +(-6.50000 + 11.2583i) q^{43} +18.0000 q^{49} +(4.50000 + 6.06218i) q^{57} +(-0.500000 - 0.866025i) q^{61} +(7.50000 + 12.9904i) q^{63} +(10.5000 - 6.06218i) q^{67} +(-8.50000 + 14.7224i) q^{73} -8.66025i q^{75} +(4.50000 + 2.59808i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-22.5000 + 12.9904i) q^{91} +(7.50000 - 12.9904i) q^{93} +(-12.0000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 10 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 10 * q^7 + 3 * q^9	$ 2 q + 3 q^{3} + 10 q^{7} + 3 q^{9} - 9 q^{13} + 8 q^{19} + 15 q^{21} - 5 q^{25} - 18 q^{39} - 13 q^{43} + 36 q^{49} + 9 q^{57} - q^{61} + 15 q^{63} + 21 q^{67} - 17 q^{73} + 9 q^{79} - 9 q^{81} - 45 q^{91} + 15 q^{93} - 24 q^{97}+O(q^{100}) $ 2 * q + 3 * q^3 + 10 * q^7 + 3 * q^9 - 9 * q^13 + 8 * q^19 + 15 * q^21 - 5 * q^25 - 18 * q^39 - 13 * q^43 + 36 * q^49 + 9 * q^57 - q^61 + 15 * q^63 + 21 * q^67 - 17 * q^73 + 9 * q^79 - 9 * q^81 - 45 * q^91 + 15 * q^93 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$-1$	$1$

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
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	5.00000			1.88982			0.944911	−	0.327327i	$-0.106148\pi$
$7$	5.00000			1.88982			0.944911	+	0.327327i	$0.106148\pi$
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−4.50000	+	2.59808i	−1.24808	+	0.720577i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−4.50000	+	2.59808i	−1.24808	+	0.720577i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$19$	4.00000	+	1.73205i	0.917663	+	0.397360i
$20$		0			0
$21$	7.50000	+	4.33013i	1.63663	+	0.944911i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$26$		0			0
$27$			5.19615i			1.00000i
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$		−	8.66025i		−	1.55543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$		−	8.66025i		−	1.55543i	0.628619	−	0.777714i	$-0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	5.19615i		−	0.854242i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$		−	5.19615i		−	0.854242i	0.904194	−	0.427121i	$-0.140472\pi$
$38$		0			0
$39$	−9.00000			−1.44115
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−6.50000	+	11.2583i	−0.991241	+	1.71688i	−0.381246	+	0.924473i	$0.624505\pi$
$43$	−6.50000	+	11.2583i	−0.991241	+	1.71688i	−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	18.0000			2.57143
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.50000	+	6.06218i	0.596040	+	0.802955i
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	−0.500000	−	0.866025i	−0.0640184	−	0.110883i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	7.50000	+	12.9904i	0.944911	+	1.63663i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	10.5000	−	6.06218i	1.28278	−	0.740613i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	10.5000	−	6.06218i	1.28278	−	0.740613i	0.977356	+	0.211604i	$0.0678686\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	−8.50000	+	14.7224i	−0.994850	+	1.72313i	−0.409644	+	0.912245i	$0.634347\pi$
$73$	−8.50000	+	14.7224i	−0.994850	+	1.72313i	−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$		−	8.66025i		−	1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.50000	+	2.59808i	0.506290	+	0.292306i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	4.50000	+	2.59808i	0.506290	+	0.292306i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−22.5000	+	12.9904i	−2.35864	+	1.36176i
$92$		0			0
$93$	7.50000	−	12.9904i	0.777714	−	1.34704i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.0000	−	6.92820i	−1.21842	−	0.703452i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−12.0000	−	6.92820i	−1.21842	−	0.703452i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		−	19.0526i		−	1.87730i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$		−	19.0526i		−	1.87730i	0.344865	−	0.938652i	$-0.387925\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−18.0000	−	10.3923i	−1.72409	−	0.995402i	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−18.0000	−	10.3923i	−1.72409	−	0.995402i	−0.814152	−	0.580651i	$-0.802798\pi$
$110$		0			0
$111$	4.50000	−	7.79423i	0.427121	−	0.739795i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−13.5000	−	7.79423i	−1.24808	−	0.720577i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	9.00000	−	5.19615i	0.798621	−	0.461084i	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	9.00000	−	5.19615i	0.798621	−	0.461084i	0.842989	+	0.537931i	$0.180794\pi$
$128$		0			0
$129$	−19.5000	+	11.2583i	−1.71688	+	0.991241i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	20.0000	+	8.66025i	1.73422	+	0.750939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	27.0000	+	15.5885i	2.22692	+	1.28571i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		−	24.2487i		−	1.97333i	−0.162758	−	0.986666i	$-0.552039\pi$
$151$		−	24.2487i		−	1.97333i	0.162758	−	0.986666i	$-0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	5.50000	−	9.52628i	0.438948	−	0.760280i	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	5.50000	−	9.52628i	0.438948	−	0.760280i	0.997609	+	0.0691164i	$0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−25.0000			−1.95815			−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−25.0000			−1.95815			−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	7.00000	−	12.1244i	0.538462	−	0.932643i
$170$		0			0
$171$	1.50000	+	12.9904i	0.114708	+	0.993399i
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−12.5000	−	21.6506i	−0.944911	−	1.63663i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−6.00000	+	3.46410i	−0.445976	+	0.257485i	−0.706129	−	0.708083i	$-0.749560\pi$
$181$	−6.00000	+	3.46410i	−0.445976	+	0.257485i	0.260153	+	0.965567i	$0.416227\pi$
$182$		0			0
$183$		−	1.73205i		−	0.128037i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			25.9808i			1.88982i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	−10.5000	−	6.06218i	−0.755807	−	0.436365i	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	−10.5000	−	6.06218i	−0.755807	−	0.436365i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	8.50000	+	14.7224i	0.602549	+	1.04365i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	8.50000	+	14.7224i	0.602549	+	1.04365i	−0.389885	+	0.920864i	$0.627485\pi$
$200$		0			0
$201$	21.0000			1.48123
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.50000	+	0.866025i	0.103264	+	0.0596196i	0.550743	−	0.834675i	$-0.314345\pi$
$211$	1.50000	+	0.866025i	0.103264	+	0.0596196i	−0.447478	+	0.894295i	$0.647678\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	43.3013i		−	2.93948i
$218$		0			0
$219$	−25.5000	+	14.7224i	−1.72313	+	0.994850i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−25.5000	−	14.7224i	−1.70761	−	0.985887i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−25.5000	−	14.7224i	−1.70761	−	0.985887i	−0.770097	−	0.637927i	$-0.779792\pi$
$224$		0			0
$225$	7.50000	−	12.9904i	0.500000	−	0.866025i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	7.00000			0.462573			0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000			0.462573			0.231287	+	0.972886i	$0.425707\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.50000	+	7.79423i	0.292306	+	0.506290i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1.50000	+	0.866025i	−0.0966235	+	0.0557856i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−1.50000	+	0.866025i	−0.0966235	+	0.0557856i	0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	−13.5000	+	7.79423i	−0.866025	+	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−22.5000	+	2.59808i	−1.43164	+	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		−	25.9808i		−	1.61437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−14.0000	+	24.2487i	−0.850439	+	1.47300i	0.0303728	+	0.999539i	$0.490331\pi$
$271$	−14.0000	+	24.2487i	−0.850439	+	1.47300i	−0.880812	+	0.473466i	$0.843003\pi$
$272$		0			0
$273$	−45.0000			−2.72352
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−26.0000			−1.56219			−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000			−1.56219			−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	22.5000	−	12.9904i	1.34704	−	0.777714i
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$	16.0000	−	27.7128i	0.951101	−	1.64736i	0.208053	−	0.978117i	$-0.433287\pi$
$283$	16.0000	−	27.7128i	0.951101	−	1.64736i	0.743048	−	0.669238i	$-0.233379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.50000	−	14.7224i	−0.500000	−	0.866025i
$290$		0			0
$291$	−12.0000	−	20.7846i	−0.703452	−	1.21842i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−32.5000	+	56.2917i	−1.87327	+	3.24460i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	27.0000	+	15.5885i	1.54097	+	0.889680i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$	27.0000	+	15.5885i	1.54097	+	0.889680i	0.542194	+	0.840254i	$0.317594\pi$
$308$		0			0
$309$	16.5000	−	28.5788i	0.938652	−	1.62579i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−11.0000	−	19.0526i	−0.621757	−	1.07691i	−0.989158	−	0.146852i	$-0.953086\pi$
$313$	−11.0000	−	19.0526i	−0.621757	−	1.07691i	0.367402	−	0.930062i	$-0.380247\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	22.5000	+	12.9904i	1.24808	+	0.720577i
$326$		0			0
$327$	−18.0000	−	31.1769i	−0.995402	−	1.72409i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			36.3731i			1.99924i	0.0274825	+	0.999622i	$0.491251\pi$
$331$			36.3731i			1.99924i	−0.0274825	+	0.999622i	$0.508749\pi$
$332$		0			0
$333$	13.5000	−	7.79423i	0.739795	−	0.427121i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	19.5000	+	11.2583i	1.06223	+	0.613280i	0.926049	−	0.377403i	$-0.123183\pi$
$337$	19.5000	+	11.2583i	1.06223	+	0.613280i	0.136184	+	0.990684i	$0.456516\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	55.0000			2.96972
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−37.0000			−1.98056			−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−37.0000			−1.98056			−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$	−13.5000	−	23.3827i	−0.720577	−	1.24808i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$	16.5000	+	9.52628i	0.866025	+	0.500000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.809093	+	0.587680i	$0.199959\pi$
$367$	17.5000	+	30.3109i	0.913493	+	1.58222i	0.104399	+	0.994535i	$0.466708\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	6.92820i		−	0.358729i	−0.983783	−	0.179364i	$-0.942596\pi$
$373$		−	6.92820i		−	0.358729i	0.983783	−	0.179364i	$-0.0574041\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		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
$381$	18.0000			0.922168
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−39.0000			−1.98248
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−17.5000	+	30.3109i	−0.878300	+	1.52126i	−0.0250943	+	0.999685i	$0.507989\pi$
$397$	−17.5000	+	30.3109i	−0.878300	+	1.52126i	−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$	22.5000	+	30.3109i	1.12641	+	1.51744i
$400$		0			0
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	22.5000	+	38.9711i	1.12080	+	1.94129i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	12.0000	−	6.92820i	0.593362	−	0.342578i	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	12.0000	−	6.92820i	0.593362	−	0.342578i	0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			12.1244i			0.593732i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	30.0000	+	17.3205i	1.46211	+	0.844150i	0.999109	−	0.0422075i	$-0.0134391\pi$
$421$	30.0000	+	17.3205i	1.46211	+	0.844150i	0.463002	+	0.886357i	$0.346772\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.50000	−	4.33013i	−0.120983	−	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$		0			0
$433$	−16.5000	+	9.52628i	−0.792939	+	0.457804i	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−16.5000	+	9.52628i	−0.792939	+	0.457804i	0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	34.5000	+	19.9186i	1.64660	+	0.950662i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	27.0000	+	46.7654i	1.28571	+	2.22692i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	21.0000	−	36.3731i	0.986666	−	1.70896i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	41.0000			1.91790			0.958950	−	0.283577i	$-0.0915211\pi$
$457$	41.0000			1.91790			0.958950	+	0.283577i	$0.0915211\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	−23.0000			−1.06890			−0.534450	−	0.845200i	$-0.679481\pi$
$463$	−23.0000			−1.06890			−0.534450	+	0.845200i	$0.679481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	52.5000	−	30.3109i	2.42422	−	1.39963i
$470$		0			0
$471$	16.5000	−	9.52628i	0.760280	−	0.438948i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−2.50000	−	21.6506i	−0.114708	−	0.993399i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	13.5000	+	23.3827i	0.615547	+	1.06616i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	3.46410i		−	0.156973i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$		−	3.46410i		−	0.156973i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$		0			0
$489$	−37.5000	−	21.6506i	−1.69581	−	0.979076i
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	21.5000	−	37.2391i	0.962472	−	1.66705i	0.246214	−	0.969216i	$-0.420813\pi$
$499$	21.5000	−	37.2391i	0.962472	−	1.66705i	0.716258	−	0.697835i	$-0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	21.0000	−	12.1244i	0.932643	−	0.538462i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−42.5000	+	73.6122i	−1.88009	+	3.25641i
$512$		0			0
$513$	−9.00000	+	20.7846i	−0.397360	+	0.917663i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−13.5000	+	7.79423i	−0.590314	+	0.340818i	−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−13.5000	+	7.79423i	−0.590314	+	0.340818i	0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$		−	43.3013i		−	1.88982i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	0.365444	−	0.930834i	$-0.380917\pi$
$542$		0			0
$543$	−12.0000			−0.514969
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.5000	+	23.3827i	−1.73165	+	0.999771i	−0.855138	+	0.518400i	$0.826528\pi$
$547$	−40.5000	+	23.3827i	−1.73165	+	0.999771i	−0.876517	+	0.481371i	$0.840139\pi$
$548$		0			0
$549$	1.50000	−	2.59808i	0.0640184	−	0.110883i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	22.5000	+	12.9904i	0.956797	+	0.552407i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		−	67.5500i		−	2.85706i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.5000	+	38.9711i	−0.944911	+	1.63663i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	31.0000			1.29731			0.648655	−	0.761083i	$-0.275332\pi$
$571$	31.0000			1.29731			0.648655	+	0.761083i	$0.275332\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	46.0000			1.91501			0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000			1.91501			0.957503	+	0.288425i	$0.0931316\pi$
$578$		0			0
$579$	−10.5000	−	18.1865i	−0.436365	−	0.755807i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$	15.0000	−	34.6410i	0.618064	−	1.42736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			29.4449i			1.20510i
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$			1.73205i			0.0706518i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$			1.73205i			0.0706518i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$	31.5000	+	18.1865i	1.28278	+	0.740613i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			39.8372i			1.61694i	0.588537	+	0.808470i	$0.299704\pi$
$607$			39.8372i			1.61694i	−0.588537	+	0.808470i	$0.700296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−5.00000	+	8.66025i	−0.201948	+	0.349784i	−0.949156	−	0.314806i	$-0.898061\pi$
$613$	−5.00000	+	8.66025i	−0.201948	+	0.349784i	0.747208	+	0.664590i	$0.231394\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	17.0000			0.683288			0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000			0.683288			0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	21.5000	+	37.2391i	0.855901	+	1.48246i	0.875806	+	0.482663i	$0.160330\pi$
$631$	21.5000	+	37.2391i	0.855901	+	1.48246i	−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$	1.50000	+	2.59808i	0.0596196	+	0.103264i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−81.0000	+	46.7654i	−3.20934	+	1.85291i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−3.50000	+	6.06218i	−0.138027	+	0.239069i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$	−3.50000	+	6.06218i	−0.138027	+	0.239069i	0.788723	+	0.614749i	$0.210743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	37.5000	−	64.9519i	1.46974	−	2.54567i
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−51.0000			−1.98970
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−30.0000	+	17.3205i	−1.16686	+	0.673690i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−30.0000	+	17.3205i	−1.16686	+	0.673690i	−0.213925	+	0.976850i	$0.568625\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−25.5000	−	44.1673i	−0.985887	−	1.70761i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			50.2295i			1.93620i	0.250557	+	0.968102i	$0.419386\pi$
$673$			50.2295i			1.93620i	−0.250557	+	0.968102i	$0.580614\pi$
$674$		0			0
$675$	22.5000	−	12.9904i	0.866025	−	0.500000i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−60.0000	−	34.6410i	−2.30259	−	1.32940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	10.5000	+	6.06218i	0.400600	+	0.231287i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$701$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$	9.00000	−	20.7846i	0.339441	−	0.783906i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	0.995228	+	0.0975728i	$0.0311079\pi$
$709$	15.5000	+	26.8468i	0.582115	+	1.00825i	−0.413114	+	0.910679i	$0.635559\pi$
$710$		0			0
$711$			15.5885i			0.584613i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		−	95.2628i		−	3.54777i
$722$		0			0
$723$	−3.00000			−0.111571
$724$		0			0
$725$		0			0
$726$		0			0
$727$	2.50000	−	4.33013i	0.0927199	−	0.160596i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	2.50000	−	4.33013i	0.0927199	−	0.160596i	0.908655	+	0.417548i	$0.137111\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	50.0000			1.84679			0.923396	−	0.383849i	$-0.125402\pi$
$733$	50.0000			1.84679			0.923396	+	0.383849i	$0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	18.5000	−	32.0429i	0.680534	−	1.17872i	0.974818	−	0.223001i	$-0.0715853\pi$
$740$		0			0
$741$	−36.0000	−	15.5885i	−1.32249	−	0.572656i
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	31.5000	−	18.1865i	1.14945	−	0.663636i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	31.5000	−	18.1865i	1.14945	−	0.663636i	0.948753	+	0.316017i	$0.102346\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	14.5000	−	25.1147i	0.527011	−	0.912811i	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	14.5000	−	25.1147i	0.527011	−	0.912811i	0.999505	−	0.0314762i	$-0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−90.0000	−	51.9615i	−3.25822	−	1.88113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−24.5000	−	42.4352i	−0.883493	−	1.53025i	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−24.5000	−	42.4352i	−0.883493	−	1.53025i	−0.0360609	−	0.999350i	$-0.511481\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−37.5000	+	21.6506i	−1.34704	+	0.777714i
$776$		0			0
$777$	22.5000	−	38.9711i	0.807183	−	1.39808i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			46.7654i			1.66701i	0.552515	+	0.833503i	$0.313668\pi$
$787$			46.7654i			1.66701i	−0.552515	+	0.833503i	$0.686332\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4.50000	+	2.59808i	0.159800	+	0.0922604i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	9.00000	−	5.19615i	0.316033	−	0.182462i	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	9.00000	−	5.19615i	0.316033	−	0.182462i	0.649623	+	0.760257i	$0.274927\pi$
$812$		0			0
$813$	−42.0000	+	24.2487i	−1.47300	+	0.850439i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−45.5000	+	33.7750i	−1.59184	+	1.18164i
$818$		0			0
$819$	−67.5000	−	38.9711i	−2.35864	−	1.36176i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	26.0000	+	45.0333i	0.906303	+	1.56976i	0.819159	+	0.573567i	$0.194441\pi$
$823$	26.0000	+	45.0333i	0.906303	+	1.56976i	0.0871445	+	0.996196i	$0.472226\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$827$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$828$		0			0
$829$			22.5167i			0.782036i	0.920383	+	0.391018i	$0.127877\pi$
$829$			22.5167i			0.782036i	−0.920383	+	0.391018i	$0.872123\pi$
$830$		0			0
$831$	−39.0000	−	22.5167i	−1.35290	−	0.781094i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	45.0000			1.55543
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	55.0000			1.88982
$848$		0			0
$849$	48.0000	−	27.7128i	1.64736	−	0.951101i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−11.5000	+	19.9186i	−0.393753	+	0.681999i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−11.5000	+	19.9186i	−0.393753	+	0.681999i	0.599189	+	0.800608i	$0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	6.50000	+	11.2583i	0.221777	+	0.384129i	0.955348	−	0.295484i	$-0.0954809\pi$
$859$	6.50000	+	11.2583i	0.221777	+	0.384129i	−0.733571	+	0.679613i	$0.762148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−31.5000	+	54.5596i	−1.06734	+	1.84868i
$872$		0			0
$873$		−	41.5692i		−	1.40690i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.50000	−	2.59808i	−0.151954	−	0.0877308i	0.422095	−	0.906552i	$-0.361295\pi$
$877$	−4.50000	−	2.59808i	−0.151954	−	0.0877308i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−23.5000	−	40.7032i	−0.790838	−	1.36977i	−0.925449	−	0.378873i	$-0.876312\pi$
$883$	−23.5000	−	40.7032i	−0.790838	−	1.36977i	0.134611	−	0.990899i	$-0.457022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	45.0000	−	25.9808i	1.50925	−	0.871367i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−97.5000	+	56.2917i	−3.24460	+	1.87327i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−39.0000	−	22.5167i	−1.29497	−	0.747653i	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	−39.0000	−	22.5167i	−1.29497	−	0.747653i	−0.979531	+	0.201291i	$0.935486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.00000			−0.0329870			−0.0164935	−	0.999864i	$-0.505250\pi$
$919$	−1.00000			−0.0329870			−0.0164935	+	0.999864i	$0.505250\pi$
$920$		0			0
$921$	27.0000	+	46.7654i	0.889680	+	1.54097i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−22.5000	+	12.9904i	−0.739795	+	0.427121i
$926$		0			0
$927$	49.5000	−	28.5788i	1.62579	−	0.938652i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	72.0000	+	31.1769i	2.35970	+	1.02178i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	30.5000	+	52.8275i	0.996392	+	1.72580i	0.571700	+	0.820463i	$0.306284\pi$
$937$	30.5000	+	52.8275i	0.996392	+	1.72580i	0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$		−	38.1051i		−	1.24351i
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		−	88.3346i		−	2.86746i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−44.0000			−1.41935
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	30.5000	−	52.8275i	0.980814	−	1.69882i	0.321578	−	0.946883i	$-0.395787\pi$
$967$	30.5000	−	52.8275i	0.980814	−	1.69882i	0.659236	−	0.751936i	$-0.270880\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$	17.5000	+	30.3109i	0.561024	+	0.971722i
$974$		0			0
$975$	22.5000	+	38.9711i	0.720577	+	1.24808i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	62.3538i		−	1.99080i
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	13.5000	+	7.79423i	0.428842	+	0.247592i	0.698853	−	0.715265i	$-0.253694\pi$
$991$	13.5000	+	7.79423i	0.428842	+	0.247592i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$	−31.5000	+	54.5596i	−0.999622	+	1.73140i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	29.5000	+	51.0955i	0.934274	+	1.61821i	0.775923	+	0.630828i	$0.217285\pi$
$997$	29.5000	+	51.0955i	0.934274	+	1.61821i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	27.0000			0.854242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.bn.e.65.1		2
3.2	odd	2	CM	912.2.bn.e.65.1		2
4.3	odd	2		228.2.p.a.65.1	✓	2
12.11	even	2		228.2.p.a.65.1	✓	2
19.12	odd	6	inner	912.2.bn.e.449.1		2
57.50	even	6	inner	912.2.bn.e.449.1		2
76.31	even	6		228.2.p.a.221.1	yes	2
228.107	odd	6		228.2.p.a.221.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.p.a.65.1	✓	2	4.3	odd	2
228.2.p.a.65.1	✓	2	12.11	even	2
228.2.p.a.221.1	yes	2	76.31	even	6
228.2.p.a.221.1	yes	2	228.107	odd	6
912.2.bn.e.65.1		2	1.1	even	1	trivial
912.2.bn.e.65.1		2	3.2	odd	2	CM
912.2.bn.e.449.1		2	19.12	odd	6	inner
912.2.bn.e.449.1		2	57.50	even	6	inner

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.bn (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 0.866025i) q^{3} +5.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\)	\(q+(1.50000 + 0.866025i) q^{3} +5.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +(-4.50000 + 2.59808i) q^{13} +(4.00000 + 1.73205i) q^{19} +(7.50000 + 4.33013i) q^{21} +(-2.50000 - 4.33013i) q^{25} +5.19615i q^{27} -8.66025i q^{31} -5.19615i q^{37} -9.00000 q^{39} +(-6.50000 + 11.2583i) q^{43} +18.0000 q^{49} +(4.50000 + 6.06218i) q^{57} +(-0.500000 - 0.866025i) q^{61} +(7.50000 + 12.9904i) q^{63} +(10.5000 - 6.06218i) q^{67} +(-8.50000 + 14.7224i) q^{73} -8.66025i q^{75} +(4.50000 + 2.59808i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-22.5000 + 12.9904i) q^{91} +(7.50000 - 12.9904i) q^{93} +(-12.0000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 10 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 10 * q^7 + 3 * q^9	\( 2 q + 3 q^{3} + 10 q^{7} + 3 q^{9} - 9 q^{13} + 8 q^{19} + 15 q^{21} - 5 q^{25} - 18 q^{39} - 13 q^{43} + 36 q^{49} + 9 q^{57} - q^{61} + 15 q^{63} + 21 q^{67} - 17 q^{73} + 9 q^{79} - 9 q^{81} - 45 q^{91} + 15 q^{93} - 24 q^{97}+O(q^{100}) \) 2 * q + 3 * q^3 + 10 * q^7 + 3 * q^9 - 9 * q^13 + 8 * q^19 + 15 * q^21 - 5 * q^25 - 18 * q^39 - 13 * q^43 + 36 * q^49 + 9 * q^57 - q^61 + 15 * q^63 + 21 * q^67 - 17 * q^73 + 9 * q^79 - 9 * q^81 - 45 * q^91 + 15 * q^93 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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