LMFDB - Embedded newform 912.2.ch.a.671.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(47,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 0, 9, 8]))

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

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

chi := DirichletCharacter("912.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.ch (of order $18$, degree $6$, 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:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			671.1
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	912.671
Dual form			912.2.ch.a.719.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.592396 + 1.62760i) q^{3} +(1.63429 + 0.943555i) q^{7} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})$	$q+(-0.592396 + 1.62760i) q^{3} +(1.63429 + 0.943555i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(-6.08512 + 2.21480i) q^{13} +(-0.500000 + 4.33013i) q^{19} +(-2.50387 + 2.10100i) q^{21} +(-4.69846 + 1.71010i) q^{25} +(4.50000 - 2.59808i) q^{27} +(4.66772 + 2.69491i) q^{31} -3.20708 q^{37} -11.2162i q^{39} +(-12.9042 + 2.27536i) q^{43} +(-1.71941 - 2.97810i) q^{49} +(-6.75150 - 3.37895i) q^{57} +(-0.762641 + 4.32515i) q^{61} +(-1.93629 - 5.31991i) q^{63} +(1.42514 - 1.69842i) q^{67} +(-14.4547 - 5.26108i) q^{73} -8.66025i q^{75} +(-5.09879 + 14.0088i) q^{79} +(1.56283 + 8.86327i) q^{81} +(-12.0346 - 2.12203i) q^{91} +(-7.15136 + 6.00070i) q^{93} +(14.5548 - 12.2130i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 15 q^{13} - 3 q^{19} + 9 q^{21} + 27 q^{27} - 39 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 33 q^{67} - 51 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93}+O(q^{100}) $ 6 * q - 15 * q^13 - 3 * q^19 + 9 * q^21 + 27 * q^27 - 39 * q^43 + 21 * q^49 + 42 * q^61 + 36 * q^63 - 33 * q^67 - 51 * q^73 + 12 * q^79 - 48 * q^91 - 54 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{7}{9}\right)$	$1$	$-1$	$-1$

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
$3$	−0.592396	+	1.62760i	−0.342020	+	0.939693i
$3$	−0.592396	+	1.62760i	−0.342020	+	0.939693i
$4$		0			0
$5$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$5$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$6$		0			0
$7$	1.63429	+	0.943555i	0.617702	+	0.356630i	0.775974	−	0.630765i	$-0.217259\pi$
$7$	1.63429	+	0.943555i	0.617702	+	0.356630i	−0.158272	+	0.987396i	$0.550592\pi$
$8$		0			0
$9$	−2.29813	−	1.92836i	−0.766044	−	0.642788i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	−6.08512	+	2.21480i	−1.68771	+	0.614276i	−0.994334	−	0.106301i	$-0.966099\pi$
$13$	−6.08512	+	2.21480i	−1.68771	+	0.614276i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$17$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$18$		0			0
$19$	−0.500000	+	4.33013i	−0.114708	+	0.993399i
$19$	−0.500000	+	4.33013i	−0.114708	+	0.993399i
$20$		0			0
$21$	−2.50387	+	2.10100i	−0.546389	+	0.458475i
$22$		0			0
$23$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$23$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$24$		0			0
$25$	−4.69846	+	1.71010i	−0.939693	+	0.342020i
$26$		0			0
$27$	4.50000	−	2.59808i	0.866025	−	0.500000i
$28$		0			0
$29$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$29$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$30$		0			0
$31$	4.66772	+	2.69491i	0.838347	+	0.484020i	0.856702	−	0.515812i	$-0.172510\pi$
$31$	4.66772	+	2.69491i	0.838347	+	0.484020i	−0.0183550	+	0.999832i	$0.505843\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.20708			−0.527241			−0.263620	−	0.964626i	$-0.584917\pi$
$37$	−3.20708			−0.527241			−0.263620	+	0.964626i	$0.584917\pi$
$38$		0			0
$39$		−	11.2162i		−	1.79602i
$40$		0			0
$41$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$41$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$42$		0			0
$43$	−12.9042	+	2.27536i	−1.96787	+	0.346989i	−0.976631	+	0.214921i	$0.931050\pi$
$43$	−12.9042	+	2.27536i	−1.96787	+	0.346989i	−0.991241	+	0.132068i	$0.957838\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$47$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$48$		0			0
$49$	−1.71941	−	2.97810i	−0.245630	−	0.425443i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$53$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.75150	−	3.37895i	−0.894258	−	0.447553i
$58$		0			0
$59$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$59$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$60$		0			0
$61$	−0.762641	+	4.32515i	−0.0976462	+	0.553779i	0.896258	+	0.443533i	$0.146275\pi$
$61$	−0.762641	+	4.32515i	−0.0976462	+	0.553779i	−0.993904	+	0.110246i	$0.964836\pi$
$62$		0			0
$63$	−1.93629	−	5.31991i	−0.243949	−	0.670246i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.42514	−	1.69842i	0.174109	−	0.207495i	−0.671932	−	0.740613i	$-0.734535\pi$
$67$	1.42514	−	1.69842i	0.174109	−	0.207495i	0.846041	+	0.533118i	$0.178980\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$71$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$72$		0			0
$73$	−14.4547	−	5.26108i	−1.69180	−	0.615763i	−0.696946	−	0.717124i	$-0.745458\pi$
$73$	−14.4547	−	5.26108i	−1.69180	−	0.615763i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$		−	8.66025i		−	1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−5.09879	+	14.0088i	−0.573659	+	1.57612i	0.225018	+	0.974355i	$0.427756\pi$
$79$	−5.09879	+	14.0088i	−0.573659	+	1.57612i	−0.798677	+	0.601760i	$0.794466\pi$
$80$		0			0
$81$	1.56283	+	8.86327i	0.173648	+	0.984808i
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$89$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$90$		0			0
$91$	−12.0346	−	2.12203i	−1.26157	−	0.222449i
$92$		0			0
$93$	−7.15136	+	6.00070i	−0.741561	+	0.622244i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	14.5548	−	12.2130i	1.47782	−	1.24004i	0.569346	−	0.822098i	$-0.307196\pi$
$97$	14.5548	−	12.2130i	1.47782	−	1.24004i	0.908474	−	0.417941i	$-0.137248\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$101$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$102$		0			0
$103$	17.5783	−	10.1488i	1.73204	−	0.999995i	0.864507	−	0.502621i	$-0.167631\pi$
$103$	17.5783	−	10.1488i	1.73204	−	0.999995i	0.867536	−	0.497374i	$-0.165702\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	3.29932	+	18.7113i	0.316017	+	1.79222i	0.566458	+	0.824090i	$0.308313\pi$
$109$	3.29932	+	18.7113i	0.316017	+	1.79222i	−0.250441	+	0.968132i	$0.580576\pi$
$110$		0			0
$111$	1.89986	−	5.21983i	0.180327	−	0.495444i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	18.2554	+	6.64441i	1.68771	+	0.614276i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	4.14677	+	11.3932i	0.367967	+	1.01098i	0.976134	+	0.217171i	$0.0696829\pi$
$127$	4.14677	+	11.3932i	0.367967	+	1.01098i	−0.608167	+	0.793809i	$0.708095\pi$
$128$		0			0
$129$	3.94104	−	22.3507i	0.346989	−	1.96787i
$130$		0			0
$131$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$131$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$132$		0			0
$133$	−4.90286	+	6.60489i	−0.425132	+	0.572716i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$137$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$138$		0			0
$139$	6.41787	+	17.6330i	0.544357	+	1.49561i	0.841223	+	0.540689i	$0.181836\pi$
$139$	6.41787	+	17.6330i	0.544357	+	1.49561i	−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.86571	−	1.03428i	0.483796	−	0.0853063i
$148$		0			0
$149$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$149$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$150$		0			0
$151$			15.5885i			1.26857i	0.773099	+	0.634285i	$0.218706\pi$
$151$			15.5885i			1.26857i	−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.05004	−	5.95507i	−0.0838023	−	0.475267i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−1.05004	−	5.95507i	−0.0838023	−	0.475267i	0.913806	−	0.406150i	$-0.133129\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3.17634	+	1.83386i	−0.248790	+	0.143639i	−0.619210	−	0.785225i	$-0.712547\pi$
$163$	−3.17634	+	1.83386i	−0.248790	+	0.143639i	0.370420	+	0.928864i	$0.379214\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$167$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$168$		0			0
$169$	22.1648	−	18.5985i	1.70498	−	1.43065i
$170$		0			0
$171$	9.49912	−	8.98703i	0.726416	−	0.687255i
$172$		0			0
$173$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$173$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$174$		0			0
$175$	−9.29220	−	1.63847i	−0.702425	−	0.123856i
$176$		0			0
$177$		0			0
$178$		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
$181$	14.5548	+	12.2130i	1.08185	+	0.907783i	0.996073	−	0.0885316i	$-0.0282174\pi$
$181$	14.5548	+	12.2130i	1.08185	+	0.907783i	0.0857797	+	0.996314i	$0.472662\pi$
$182$		0			0
$183$	−6.58781	−	3.80347i	−0.486985	−	0.281161i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	9.80571			0.713261
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	15.2995	+	5.56855i	1.10128	+	0.400833i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	15.2995	+	5.56855i	1.10128	+	0.400833i	0.273492	+	0.961874i	$0.411821\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	−13.2747	+	15.8201i	−0.941016	+	1.12146i	0.0514178	+	0.998677i	$0.483626\pi$
$199$	−13.2747	+	15.8201i	−0.941016	+	1.12146i	−0.992434	+	0.122782i	$0.960818\pi$
$200$		0			0
$201$	1.92009	+	3.32570i	0.135433	+	0.234577i
$202$		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
$211$	18.5510	+	22.1082i	1.27710	+	1.52199i	0.726359	+	0.687315i	$0.241211\pi$
$211$	18.5510	+	22.1082i	1.27710	+	1.52199i	0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	5.08559	+	8.80850i	0.345232	+	0.597960i
$218$		0			0
$219$	17.1258	−	20.4098i	1.15726	−	1.37916i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−19.0483	+	3.35873i	−1.27557	+	0.224917i	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	−19.0483	+	3.35873i	−1.27557	+	0.224917i	−0.505471	+	0.862844i	$0.668681\pi$
$224$		0			0
$225$	14.0954	+	5.13030i	0.939693	+	0.342020i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	30.2131			1.99654			0.998268	−	0.0588329i	$-0.0187379\pi$
$229$	30.2131			1.99654			0.998268	+	0.0588329i	$0.0187379\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$233$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.7802	−	16.5975i	−1.28486	−	1.07813i
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−21.2690	+	7.74130i	−1.37006	+	0.498661i	−0.919150	−	0.393909i	$-0.871123\pi$
$241$	−21.2690	+	7.74130i	−1.37006	+	0.498661i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	−15.3516	−	2.70691i	−0.984808	−	0.173648i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.54782	−	27.4568i	−0.416628	−	1.74703i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$251$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$257$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$258$		0			0
$259$	−5.24129	−	3.02606i	−0.325678	−	0.188030i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$263$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$269$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$270$		0			0
$271$	32.4090	−	5.71458i	1.96871	−	0.347136i	0.979079	−	0.203479i	$-0.0652250\pi$
$271$	32.4090	−	5.71458i	1.96871	−	0.347136i	0.989628	−	0.143657i	$-0.0458861\pi$
$272$		0			0
$273$	10.5831	−	18.3304i	0.640516	−	1.10941i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.50000	+	4.33013i	0.150210	+	0.260172i	0.931305	−	0.364241i	$-0.118672\pi$
$277$	2.50000	+	4.33013i	0.150210	+	0.260172i	−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	−5.53028	−	15.1943i	−0.331089	−	0.909660i
$280$		0			0
$281$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$281$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$282$		0			0
$283$	−21.1535	−	25.2097i	−1.25744	−	1.49856i	−0.788100	−	0.615547i	$-0.788935\pi$
$283$	−21.1535	−	25.2097i	−1.25744	−	1.49856i	−0.469344	−	0.883016i	$-0.655509\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.95202	−	16.7417i	0.173648	−	0.984808i
$290$		0			0
$291$	11.2555	+	30.9243i	0.659811	+	1.81282i
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−23.2361	−	8.45724i	−1.33930	−	0.487467i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.0707	−	27.6691i	0.574767	−	1.57916i	−0.222112	−	0.975021i	$-0.571295\pi$
$307$	10.0707	−	27.6691i	0.574767	−	1.57916i	0.796879	−	0.604139i	$-0.206483\pi$
$308$		0			0
$309$	6.10488	+	34.6225i	0.347295	+	1.96961i
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−9.95858	−	8.35624i	−0.562892	−	0.472323i	0.316387	−	0.948630i	$-0.397530\pi$
$313$	−9.95858	−	8.35624i	−0.562892	−	0.472323i	−0.879279	+	0.476308i	$0.841975\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$317$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	24.8032	−	20.8123i	1.37583	−	1.15446i
$326$		0			0
$327$	−32.4090	−	5.71458i	−1.79222	−	0.316017i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.32279	−	3.65046i	0.347532	−	0.200648i	−0.316066	−	0.948737i	$-0.602362\pi$
$331$	6.32279	−	3.65046i	0.347532	−	0.200648i	0.663598	+	0.748090i	$0.269029\pi$
$332$		0			0
$333$	7.37030	+	6.18442i	0.403890	+	0.338904i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.39481	+	19.2529i	0.184927	+	1.04877i	0.926049	+	0.377403i	$0.123183\pi$
$337$	3.39481	+	19.2529i	0.184927	+	1.04877i	−0.741122	+	0.671370i	$0.765706\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	19.6992i		−	1.06366i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$347$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$348$		0			0
$349$	5.77110	−	9.99583i	0.308920	−	0.535065i	−0.669207	−	0.743076i	$-0.733366\pi$
$349$	5.77110	−	9.99583i	0.308920	−	0.535065i	0.978126	+	0.208012i	$0.0666992\pi$
$350$		0			0
$351$	−21.6288	+	25.7762i	−1.15446	+	1.37583i
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$359$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$360$		0			0
$361$	−18.5000	−	4.33013i	−0.973684	−	0.227901i
$362$		0			0
$363$	12.2467	+	14.5951i	0.642788	+	0.766044i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	10.8630	+	29.8459i	0.567045	+	1.55794i	0.809093	+	0.587680i	$0.199959\pi$
$367$	10.8630	+	29.8459i	0.567045	+	1.55794i	−0.242048	+	0.970264i	$0.577819\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.336557	+	0.941663i	$0.609263\pi$
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	1.26158i		−	0.0648029i	−0.999475	−	0.0324014i	$-0.989684\pi$
$379$		−	1.26158i		−	0.0648029i	0.999475	−	0.0324014i	$-0.0103155\pi$
$380$		0			0
$381$	−21.0000			−1.07586
$382$		0			0
$383$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$383$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	34.0433	+	19.6549i	1.73052	+	0.999115i
$388$		0			0
$389$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$389$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−20.2028	+	16.9522i	−1.01395	+	0.850805i	−0.988855	−	0.148880i	$-0.952433\pi$
$397$	−20.2028	+	16.9522i	−1.01395	+	0.850805i	−0.0250943	+	0.999685i	$0.507989\pi$
$398$		0			0
$399$	−7.84565	−	11.8926i	−0.392774	−	0.595373i
$400$		0			0
$401$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$401$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$402$		0			0
$403$	−34.3723	−	6.06077i	−1.71221	−	0.301908i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−29.1097	−	24.4259i	−1.43938	−	1.20778i	−0.939895	−	0.341463i	$-0.889078\pi$
$409$	−29.1097	−	24.4259i	−1.43938	−	1.20778i	−0.499486	−	0.866322i	$-0.666478\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−32.5012			−1.59159
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−17.8542	−	6.49838i	−0.870159	−	0.316712i	−0.131927	−	0.991259i	$-0.542117\pi$
$421$	−17.8542	−	6.49838i	−0.870159	−	0.316712i	−0.738231	+	0.674548i	$0.764339\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−5.32739	+	6.34894i	−0.257811	+	0.307247i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$431$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$432$		0			0
$433$	−2.79520	+	15.8523i	−0.134329	+	0.761815i	0.840996	+	0.541041i	$0.181970\pi$
$433$	−2.79520	+	15.8523i	−0.134329	+	0.761815i	−0.975325	+	0.220774i	$0.929142\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	21.2046	+	25.2706i	1.01204	+	1.20610i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	21.2046	+	25.2706i	1.01204	+	1.20610i	0.0336266	+	0.999434i	$0.489294\pi$
$440$		0			0
$441$	−1.79143	+	10.1597i	−0.0853063	+	0.483796i
$442$		0			0
$443$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$443$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−25.3717	−	9.23454i	−1.19207	−	0.433877i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−23.6144			−1.10464			−0.552318	−	0.833633i	$-0.686257\pi$
$457$	−23.6144			−1.10464			−0.552318	+	0.833633i	$0.686257\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$461$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$462$		0			0
$463$	22.7878	+	13.1565i	1.05904	+	0.611435i	0.925166	−	0.379563i	$-0.123926\pi$
$463$	22.7878	+	13.1565i	1.05904	+	0.611435i	0.133871	+	0.990999i	$0.457259\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		0			0
$469$	3.93165	−	1.43100i	0.181547	−	0.0660776i
$470$		0			0
$471$	10.3145	+	1.81872i	0.475267	+	0.0838023i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−5.05572	−	21.2000i	−0.231972	−	0.972722i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$479$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$480$		0			0
$481$	19.5155	−	7.10305i	0.889829	−	0.323871i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	34.5000	+	19.9186i	1.56334	+	0.902597i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$	34.5000	+	19.9186i	1.56334	+	0.902597i	0.566429	+	0.824110i	$0.308325\pi$
$488$		0			0
$489$	−1.10313	−	6.25616i	−0.0498853	−	0.282913i
$490$		0			0
$491$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$491$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−25.7035	+	4.53222i	−1.15065	+	0.202890i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−25.7035	+	4.53222i	−1.15065	+	0.202890i	−0.434389	+	0.900725i	$0.643036\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$503$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	17.1404	+	47.0930i	0.761233	+	2.09147i
$508$		0			0
$509$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$509$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$510$		0			0
$511$	−18.6590	−	22.2369i	−0.825425	−	0.983704i
$512$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	25.4800	−	30.3659i	1.11416	−	1.32781i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	25.4800	−	30.3659i	1.11416	−	1.32781i	0.939254	−	0.343222i	$-0.111518\pi$
$524$		0			0
$525$	8.17143	−	14.1533i	0.356630	−	0.617702i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	21.6129	+	7.86646i	0.939693	+	0.342020i
$530$		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
$541$	−34.9281	−	29.3082i	−1.50168	−	1.26006i	−0.878274	−	0.478157i	$-0.841305\pi$
$541$	−34.9281	−	29.3082i	−1.50168	−	1.26006i	−0.623404	−	0.781900i	$-0.714251\pi$
$542$		0			0
$543$	−28.5000	+	16.4545i	−1.22305	+	0.706129i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−35.9131	−	6.33245i	−1.53553	−	0.270756i	−0.659018	−	0.752128i	$-0.729028\pi$
$547$	−35.9131	−	6.33245i	−1.53553	−	0.270756i	−0.876517	+	0.481371i	$0.840139\pi$
$548$		0			0
$549$	10.0931	−	8.46913i	0.430764	−	0.361453i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−21.5510	+	18.0834i	−0.916441	+	0.768985i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$557$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$558$		0			0
$559$	73.4842	−	42.4261i	3.10805	−	1.79443i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−5.80887	+	15.9597i	−0.243949	+	0.670246i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		−	24.2129i		−	1.01328i	−0.862158	−	0.506640i	$-0.830887\pi$
$571$		−	24.2129i		−	1.01328i	0.862158	−	0.506640i	$-0.169113\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−17.5000	+	30.3109i	−0.728535	+	1.26186i	0.228968	+	0.973434i	$0.426465\pi$
$577$	−17.5000	+	30.3109i	−0.728535	+	1.26186i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	−18.1267	+	21.6026i	−0.753320	+	0.897772i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$587$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$588$		0			0
$589$	−14.0032	+	18.8644i	−0.576990	+	0.777292i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$593$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−17.8849	−	30.9776i	−0.731980	−	1.26783i
$598$		0			0
$599$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$599$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$600$		0			0
$601$	−22.7263	+	39.3631i	−0.927024	+	1.60565i	−0.138751	+	0.990327i	$0.544309\pi$
$601$	−22.7263	+	39.3631i	−0.927024	+	1.60565i	−0.788273	+	0.615325i	$0.789025\pi$
$602$		0			0
$603$	−6.55035	+	1.15500i	−0.266751	+	0.0470353i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	49.1579i		−	1.99526i	−0.0688294	−	0.997628i	$-0.521926\pi$
$607$		−	49.1579i		−	1.99526i	0.0688294	−	0.997628i	$-0.478074\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.42498	−	36.4379i	−0.259503	−	1.47171i	−0.784245	−	0.620451i	$-0.786949\pi$
$613$	−6.42498	−	36.4379i	−0.259503	−	1.47171i	0.524742	−	0.851261i	$-0.324162\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$617$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$618$		0			0
$619$	−40.4882	+	23.3759i	−1.62736	+	0.939556i	−0.642481	+	0.766302i	$0.722095\pi$
$619$	−40.4882	+	23.3759i	−1.62736	+	0.939556i	−0.984877	+	0.173254i	$0.944572\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	19.1511	−	16.0697i	0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−46.1464	−	8.13685i	−1.83706	−	0.323923i	−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−46.1464	−	8.13685i	−1.83706	−	0.323923i	−0.981156	+	0.193216i	$0.938108\pi$
$632$		0			0
$633$	−46.9727	+	17.0967i	−1.86700	+	0.679532i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	17.0587	+	14.3140i	0.675891	+	0.567140i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$641$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$642$		0			0
$643$	14.6992	−	40.3857i	0.579679	−	1.59265i	−0.209044	−	0.977906i	$-0.567035\pi$
$643$	14.6992	−	40.3857i	0.579679	−	1.59265i	0.788723	−	0.614749i	$-0.210743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−17.3494	+	3.05916i	−0.679975	+	0.119898i
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	23.0736	+	39.9646i	0.900186	+	1.55917i
$658$		0			0
$659$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$659$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$660$		0			0
$661$	6.59863	−	37.4227i	0.256657	−	1.45557i	−0.535126	−	0.844772i	$-0.679736\pi$
$661$	6.59863	−	37.4227i	0.256657	−	1.45557i	0.791783	−	0.610802i	$-0.209153\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	5.81749	−	32.9926i	0.224917	−	1.27557i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	21.1227	+	36.5856i	0.814221	+	1.41027i	0.909886	+	0.414859i	$0.136169\pi$
$673$	21.1227	+	36.5856i	0.814221	+	1.41027i	−0.0956642	+	0.995414i	$0.530497\pi$
$674$		0			0
$675$	−16.7001	+	19.9024i	−0.642788	+	0.766044i
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	35.3104	−	6.22617i	1.35509	−	0.238939i
$680$		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
$687$	−17.8981	+	49.1746i	−0.682855	+	1.87613i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	28.5000	+	16.4545i	1.08419	+	0.625958i	0.932024	−	0.362397i	$-0.118041\pi$
$691$	28.5000	+	16.4545i	1.08419	+	0.625958i	0.152167	+	0.988355i	$0.451375\pi$
$692$		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		0.766044	−	0.642788i	$-0.222222\pi$
$701$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$702$		0			0
$703$	1.60354	−	13.8871i	0.0604787	−	0.523761i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	41.2904	−	15.0285i	1.55070	−	0.564407i	0.582115	−	0.813107i	$-0.302225\pi$
$709$	41.2904	−	15.0285i	1.55070	−	0.564407i	0.968581	+	0.248700i	$0.0800032\pi$
$710$		0			0
$711$	38.7318	−	22.3618i	1.45256	−	0.838633i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$719$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$720$		0			0
$721$	38.3040			1.42651
$722$		0			0
$723$		−	39.2033i		−	1.45799i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−48.0048	+	8.46454i	−1.78040	+	0.313932i	−0.964465	−	0.264211i	$-0.914888\pi$
$727$	−48.0048	+	8.46454i	−1.78040	+	0.313932i	−0.815935	+	0.578144i	$0.803777\pi$
$728$		0			0
$729$	13.5000	−	23.3827i	0.500000	−	0.866025i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−21.5000	−	37.2391i	−0.794121	−	1.37546i	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−21.5000	−	37.2391i	−0.794121	−	1.37546i	0.129275	−	0.991609i	$-0.458735\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−34.9035	−	41.5964i	−1.28395	−	1.53015i	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−34.9035	−	41.5964i	−1.28395	−	1.53015i	−0.603412	−	0.797430i	$-0.706193\pi$
$740$		0			0
$741$	48.5674	+	5.60808i	1.78417	+	0.206018i
$742$		0			0
$743$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$743$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	30.9839	−	36.9251i	1.13062	−	1.34742i	0.200698	−	0.979653i	$-0.435679\pi$
$751$	30.9839	−	36.9251i	1.13062	−	1.34742i	0.929919	−	0.367764i	$-0.119877\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	48.0096	+	17.4740i	1.74494	+	0.635105i	0.999505	−	0.0314762i	$-0.0100208\pi$
$757$	48.0096	+	17.4740i	1.74494	+	0.635105i	0.745432	+	0.666581i	$0.232243\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−12.2632	+	33.6928i	−0.443956	+	1.21976i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	42.0795	+	35.3089i	1.51743	+	1.27327i	0.847432	+	0.530904i	$0.178148\pi$
$769$	42.0795	+	35.3089i	1.51743	+	1.27327i	0.669994	+	0.742367i	$0.266297\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$773$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$774$		0			0
$775$	−26.5397	−	4.67966i	−0.953333	−	0.168098i
$776$		0			0
$777$	8.03012	−	6.73807i	0.288079	−	0.241727i
$778$		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
$787$	47.2397	−	27.2738i	1.68391	−	0.972208i	0.724897	−	0.688858i	$-0.241887\pi$
$787$	47.2397	−	27.2738i	1.68391	−	0.972208i	0.959017	−	0.283350i	$-0.0914459\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−4.93860	−	28.0082i	−0.175375	−	0.994599i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	18.3643	+	50.4555i	0.644857	+	1.77173i	0.635901	+	0.771771i	$0.280629\pi$
$811$	18.3643	+	50.4555i	0.644857	+	1.77173i	0.00895645	+	0.999960i	$0.497149\pi$
$812$		0			0
$813$	−9.89795	+	56.1340i	−0.347136	+	1.96871i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3.40049	−	57.0145i	−0.118968	−	1.99469i
$818$		0			0
$819$	23.5651	+	28.0838i	0.823432	+	0.981328i
$820$		0			0
$821$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$821$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$822$		0			0
$823$	11.2555	+	30.9243i	0.392343	+	1.07795i	0.965929	+	0.258808i	$0.0833297\pi$
$823$	11.2555	+	30.9243i	0.392343	+	1.07795i	−0.573586	+	0.819146i	$0.694448\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$827$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$828$		0			0
$829$	27.5369	−	47.6953i	0.956396	−	1.65653i	0.225255	−	0.974300i	$-0.427679\pi$
$829$	27.5369	−	47.6953i	0.956396	−	1.65653i	0.731141	−	0.682226i	$-0.238988\pi$
$830$		0			0
$831$	−8.52869	+	1.50384i	−0.295857	+	0.0521675i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	28.0063			0.968040
$838$		0			0
$839$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$839$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$840$		0			0
$841$	5.03580	+	28.5594i	0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	17.9771	−	10.3791i	0.617702	−	0.356630i
$848$		0			0
$849$	53.5625	−	19.4951i	1.83826	−	0.669072i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−37.4473	+	31.4220i	−1.28217	+	1.07587i	−0.289229	+	0.957260i	$0.593399\pi$
$853$	−37.4473	+	31.4220i	−1.28217	+	1.07587i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$857$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$858$		0			0
$859$	34.8908	+	6.15220i	1.19046	+	0.209910i	0.733571	−	0.679613i	$-0.237852\pi$
$859$	34.8908	+	6.15220i	1.19046	+	0.209910i	0.456889	+	0.889524i	$0.348964\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$863$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	25.5000	+	14.7224i	0.866025	+	0.500000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−4.91051	+	13.4915i	−0.166386	+	0.457143i
$872$		0			0
$873$	−57.0000			−1.92916
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−45.6095	−	16.6005i	−1.54012	−	0.560559i	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−45.6095	−	16.6005i	−1.54012	−	0.560559i	−0.966075	+	0.258261i	$0.916850\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	−37.3295	+	44.4876i	−1.25624	+	1.49713i	−0.465400	+	0.885100i	$0.654090\pi$
$883$	−37.3295	+	44.4876i	−1.25624	+	1.49713i	−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$887$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$888$		0			0
$889$	−3.97307	+	22.5324i	−0.133252	+	0.755712i
$890$		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
$903$	27.5299	−	32.8089i	0.916138	−	1.09181i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−56.2893	+	9.92533i	−1.86906	+	0.329565i	−0.989304	−	0.145868i	$-0.953403\pi$
$907$	−56.2893	+	9.92533i	−1.86906	+	0.329565i	−0.879752	+	0.475433i	$0.842292\pi$
$908$		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
$919$	8.26366	+	4.77103i	0.272593	+	0.157382i	0.630065	−	0.776542i	$-0.283028\pi$
$919$	8.26366	+	4.77103i	0.272593	+	0.157382i	−0.357472	+	0.933924i	$0.616361\pi$
$920$		0			0
$921$	39.0683	+	32.7822i	1.28734	+	1.08021i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	15.0684	−	5.48443i	0.495444	−	0.180327i
$926$		0			0
$927$	−59.9680	−	10.5740i	−1.96961	−	0.347295i
$928$		0			0
$929$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$929$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$930$		0			0
$931$	13.7553	−	5.95620i	0.450810	−	0.195207i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	40.7720	−	14.8398i	1.33196	−	0.484795i	0.424691	−	0.905338i	$-0.360383\pi$
$937$	40.7720	−	14.8398i	1.33196	−	0.484795i	0.907273	+	0.420543i	$0.138160\pi$
$938$		0			0
$939$	19.5000	−	11.2583i	0.636358	−	0.367402i
$940$		0			0
$941$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$941$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$947$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$948$		0			0
$949$	99.6109			3.23351
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$953$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.974937	−	1.68864i	−0.0314496	−	0.0544723i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	37.2610	+	44.4060i	1.19823	+	1.42800i	0.876656	+	0.481117i	$0.159769\pi$
$967$	37.2610	+	44.4060i	1.19823	+	1.42800i	0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$971$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$972$		0			0
$973$	−6.14903	+	34.8729i	−0.197129	+	1.11797i
$974$		0			0
$975$	19.1808	+	52.6987i	0.614276	+	1.68771i
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	28.5000	−	49.3634i	0.909935	−	1.57605i
$982$		0			0
$983$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$983$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−12.1457	+	33.3700i	−0.385820	+	1.06003i	0.583044	+	0.812440i	$0.301861\pi$
$991$	−12.1457	+	33.3700i	−0.385820	+	1.06003i	−0.968864	+	0.247592i	$0.920361\pi$
$992$		0			0
$993$	2.19588	+	12.4535i	0.0696842	+	0.395199i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	24.8350	+	20.8391i	0.786533	+	0.659980i	0.944885	−	0.327403i	$-0.106173\pi$
$997$	24.8350	+	20.8391i	0.786533	+	0.659980i	−0.158352	+	0.987383i	$0.550618\pi$
$998$		0			0
$999$	−14.4319	+	8.33224i	−0.456604	+	0.263620i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.ch.a.671.1	✓	6
3.2	odd	2	CM	912.2.ch.a.671.1	✓	6
4.3	odd	2		912.2.ch.b.671.1	yes	6
12.11	even	2		912.2.ch.b.671.1	yes	6
19.16	even	9		912.2.ch.b.719.1	yes	6
57.35	odd	18		912.2.ch.b.719.1	yes	6
76.35	odd	18	inner	912.2.ch.a.719.1	yes	6
228.35	even	18	inner	912.2.ch.a.719.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.ch.a.671.1	✓	6	1.1	even	1	trivial
912.2.ch.a.671.1	✓	6	3.2	odd	2	CM
912.2.ch.a.719.1	yes	6	76.35	odd	18	inner
912.2.ch.a.719.1	yes	6	228.35	even	18	inner
912.2.ch.b.671.1	yes	6	4.3	odd	2
912.2.ch.b.671.1	yes	6	12.11	even	2
912.2.ch.b.719.1	yes	6	19.16	even	9
912.2.ch.b.719.1	yes	6	57.35	odd	18

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.ch (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.592396 + 1.62760i) q^{3} +(1.63429 + 0.943555i) q^{7} +(-2.29813 - 1.92836i) q^{9} +O(q^{10})\)	\(q+(-0.592396 + 1.62760i) q^{3} +(1.63429 + 0.943555i) q^{7} +(-2.29813 - 1.92836i) q^{9} +(-6.08512 + 2.21480i) q^{13} +(-0.500000 + 4.33013i) q^{19} +(-2.50387 + 2.10100i) q^{21} +(-4.69846 + 1.71010i) q^{25} +(4.50000 - 2.59808i) q^{27} +(4.66772 + 2.69491i) q^{31} -3.20708 q^{37} -11.2162i q^{39} +(-12.9042 + 2.27536i) q^{43} +(-1.71941 - 2.97810i) q^{49} +(-6.75150 - 3.37895i) q^{57} +(-0.762641 + 4.32515i) q^{61} +(-1.93629 - 5.31991i) q^{63} +(1.42514 - 1.69842i) q^{67} +(-14.4547 - 5.26108i) q^{73} -8.66025i q^{75} +(-5.09879 + 14.0088i) q^{79} +(1.56283 + 8.86327i) q^{81} +(-12.0346 - 2.12203i) q^{91} +(-7.15136 + 6.00070i) q^{93} +(14.5548 - 12.2130i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 15 q^{13} - 3 q^{19} + 9 q^{21} + 27 q^{27} - 39 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 33 q^{67} - 51 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93}+O(q^{100}) \) 6 * q - 15 * q^13 - 3 * q^19 + 9 * q^21 + 27 * q^27 - 39 * q^43 + 21 * q^49 + 42 * q^61 + 36 * q^63 - 33 * q^67 - 51 * q^73 + 12 * q^79 - 48 * q^91 - 54 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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