LMFDB - Embedded newform 912.2.ch.c.815.1

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			815.1
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	912.815
Dual form			912.2.ch.c.47.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.11334 - 1.32683i) q^{3} +(0.0714517 - 0.0412527i) q^{7} +(-0.520945 - 2.95442i) q^{9} +O(q^{10})$	$q+(1.11334 - 1.32683i) q^{3} +(0.0714517 - 0.0412527i) q^{7} +(-0.520945 - 2.95442i) q^{9} +(3.25490 - 2.73119i) q^{13} +(-3.50000 - 2.59808i) q^{19} +(0.0248149 - 0.140732i) q^{21} +(3.83022 - 3.21394i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(1.84864 - 1.06731i) q^{31} +5.08647 q^{37} -7.35943i q^{39} +(-3.31180 - 9.09911i) q^{43} +(-3.49660 + 6.05628i) q^{49} +(-7.34389 + 1.75135i) q^{57} +(14.5890 + 5.30996i) q^{61} +(-0.159100 - 0.189608i) q^{63} +(-7.51501 + 1.32510i) q^{67} +(-3.56805 - 2.99395i) q^{73} -8.66025i q^{75} +(-6.87598 + 8.19448i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(0.119900 - 0.329421i) q^{91} +(0.642026 - 3.64111i) q^{93} +(-0.868241 + 4.92404i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 21 q^{13} - 21 q^{19} - 27 q^{21} - 27 q^{27} + 15 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 15 q^{67} + 21 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93}+O(q^{100}) $ 6 * q + 21 * q^13 - 21 * q^19 - 27 * q^21 - 27 * q^27 + 15 * q^43 + 21 * q^49 + 42 * q^61 + 36 * q^63 - 15 * q^67 + 21 * 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 $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$e\left(\frac{5}{9}\right)$	$1$	$-1$	$-1$

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
$3$	1.11334	−	1.32683i	0.642788	−	0.766044i
$3$	1.11334	−	1.32683i	0.642788	−	0.766044i
$4$		0			0
$5$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$5$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$6$		0			0
$7$	0.0714517	−	0.0412527i	0.0270062	−	0.0155920i	−0.486436	−	0.873716i	$-0.661703\pi$
$7$	0.0714517	−	0.0412527i	0.0270062	−	0.0155920i	0.513442	+	0.858124i	$0.328370\pi$
$8$		0			0
$9$	−0.520945	−	2.95442i	−0.173648	−	0.984808i
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	3.25490	−	2.73119i	0.902747	−	0.757495i	−0.0679785	−	0.997687i	$-0.521655\pi$
$13$	3.25490	−	2.73119i	0.902747	−	0.757495i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$17$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$18$		0			0
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$19$	−3.50000	−	2.59808i	−0.802955	−	0.596040i
$20$		0			0
$21$	0.0248149	−	0.140732i	0.00541506	−	0.0307103i
$22$		0			0
$23$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$23$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$24$		0			0
$25$	3.83022	−	3.21394i	0.766044	−	0.642788i
$26$		0			0
$27$	−4.50000	−	2.59808i	−0.866025	−	0.500000i
$28$		0			0
$29$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$29$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$30$		0			0
$31$	1.84864	−	1.06731i	0.332026	−	0.191695i	−0.324714	−	0.945812i	$-0.605268\pi$
$31$	1.84864	−	1.06731i	0.332026	−	0.191695i	0.656740	+	0.754117i	$0.271935\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.08647			0.836210			0.418105	−	0.908399i	$-0.362694\pi$
$37$	5.08647			0.836210			0.418105	+	0.908399i	$0.362694\pi$
$38$		0			0
$39$		−	7.35943i		−	1.17845i
$40$		0			0
$41$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$41$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$42$		0			0
$43$	−3.31180	−	9.09911i	−0.505045	−	1.38760i	−0.886292	−	0.463127i	$-0.846727\pi$
$43$	−3.31180	−	9.09911i	−0.505045	−	1.38760i	0.381246	−	0.924473i	$-0.375495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$47$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$48$		0			0
$49$	−3.49660	+	6.05628i	−0.499514	+	0.865183i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$53$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−7.34389	+	1.75135i	−0.972722	+	0.231972i
$58$		0			0
$59$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$59$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$60$		0			0
$61$	14.5890	+	5.30996i	1.86793	+	0.679871i	0.971671	+	0.236338i	$0.0759472\pi$
$61$	14.5890	+	5.30996i	1.86793	+	0.679871i	0.896258	+	0.443533i	$0.146275\pi$
$62$		0			0
$63$	−0.159100	−	0.189608i	−0.0200447	−	0.0238884i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−7.51501	+	1.32510i	−0.918105	+	0.161887i	−0.612682	−	0.790330i	$-0.709909\pi$
$67$	−7.51501	+	1.32510i	−0.918105	+	0.161887i	−0.305424	+	0.952217i	$0.598798\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$71$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$72$		0			0
$73$	−3.56805	−	2.99395i	−0.417608	−	0.350415i	0.409644	−	0.912245i	$-0.365653\pi$
$73$	−3.56805	−	2.99395i	−0.417608	−	0.350415i	−0.827252	+	0.561830i	$0.810097\pi$
$74$		0			0
$75$		−	8.66025i		−	1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.87598	+	8.19448i	−0.773608	+	0.921951i	−0.998626	−	0.0524041i	$-0.983312\pi$
$79$	−6.87598	+	8.19448i	−0.773608	+	0.921951i	0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	−8.45723	+	3.07818i	−0.939693	+	0.342020i
$82$		0			0
$83$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$83$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$89$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$90$		0			0
$91$	0.119900	−	0.329421i	0.0125689	−	0.0345327i
$92$		0			0
$93$	0.642026	−	3.64111i	0.0665750	−	0.377566i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.868241	+	4.92404i	−0.0881565	+	0.499960i	0.908474	+	0.417941i	$0.137248\pi$
$97$	−0.868241	+	4.92404i	−0.0881565	+	0.499960i	−0.996631	+	0.0820195i	$0.973863\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$101$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$102$		0			0
$103$	−3.10488	−	1.79261i	−0.305933	−	0.176631i	0.339172	−	0.940724i	$-0.389853\pi$
$103$	−3.10488	−	1.79261i	−0.305933	−	0.176631i	−0.645105	+	0.764094i	$0.723187\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$	15.9748	−	5.81434i	1.53011	−	0.556913i	0.566458	−	0.824090i	$-0.308313\pi$
$109$	15.9748	−	5.81434i	1.53011	−	0.556913i	0.963647	+	0.267177i	$0.0860909\pi$
$110$		0			0
$111$	5.66297	−	6.74887i	0.537505	−	0.640574i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−9.76470	−	8.19356i	−0.902747	−	0.757495i
$118$		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
$127$	14.4734	+	17.2488i	1.28431	+	1.53058i	0.676142	+	0.736771i	$0.263650\pi$
$127$	14.4734	+	17.2488i	1.28431	+	1.53058i	0.608167	+	0.793809i	$0.291905\pi$
$128$		0			0
$129$	−15.7601	−	5.73621i	−1.38760	−	0.505045i
$130$		0			0
$131$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$131$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$132$		0			0
$133$	−0.357259	−	0.0412527i	−0.0309783	−	0.00357706i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$137$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$138$		0			0
$139$	6.94444	+	8.27606i	0.589020	+	0.701966i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	6.94444	+	8.27606i	0.589020	+	0.701966i	−0.386398	+	0.922332i	$0.626281\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.14274	+	11.3821i	0.341688	+	0.938779i
$148$		0			0
$149$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$149$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$150$		0			0
$151$			8.66025i			0.704761i	0.935857	+	0.352381i	$0.114628\pi$
$151$			8.66025i			0.704761i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.47653	−	0.901383i	0.197649	−	0.0719382i	−0.241299	−	0.970451i	$-0.577574\pi$
$157$	2.47653	−	0.901383i	0.197649	−	0.0719382i	0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	22.1031	+	12.7612i	1.73125	+	0.999538i	0.880821	+	0.473450i	$0.156991\pi$
$163$	22.1031	+	12.7612i	1.73125	+	0.999538i	0.850430	+	0.526088i	$0.176342\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$167$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$168$		0			0
$169$	0.877574	−	4.97697i	0.0675057	−	0.382844i
$170$		0			0
$171$	−5.85251	+	11.6939i	−0.447553	+	0.894258i
$172$		0			0
$173$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$173$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$174$		0			0
$175$	0.141092	−	0.387648i	0.0106656	−	0.0293035i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	1.21554	+	6.89365i	0.0903502	+	0.512401i	0.996073	+	0.0885316i	$0.0282174\pi$
$181$	1.21554	+	6.89365i	0.0903502	+	0.512401i	−0.905723	+	0.423870i	$0.860671\pi$
$182$		0			0
$183$	23.2879	−	13.4453i	1.72149	−	0.993904i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.428710			−0.0311841
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−20.6407	−	17.3196i	−1.48575	−	1.24669i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−20.6407	−	17.3196i	−1.48575	−	1.24669i	−0.585979	−	0.810326i	$-0.699290\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−27.7481	+	4.89274i	−1.96701	+	0.346837i	−0.974576	+	0.224055i	$0.928070\pi$
$199$	−27.7481	+	4.89274i	−1.96701	+	0.346837i	−0.992434	+	0.122782i	$0.960818\pi$
$200$		0			0
$201$	−6.60859	+	11.4464i	−0.466134	+	0.807368i
$202$		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
$211$	19.6643	+	3.46735i	1.35375	+	0.238702i	0.803005	−	0.595973i	$-0.203233\pi$
$211$	19.6643	+	3.46735i	1.35375	+	0.238702i	0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.0880590	−	0.152523i	0.00597784	−	0.0103539i
$218$		0			0
$219$	−7.94491	+	1.40090i	−0.536867	+	0.0946642i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	8.87851	+	24.3935i	0.594549	+	1.63351i	0.761961	+	0.647623i	$0.224237\pi$
$223$	8.87851	+	24.3935i	0.594549	+	1.63351i	−0.167412	+	0.985887i	$0.553541\pi$
$224$		0			0
$225$	−11.4907	−	9.64181i	−0.766044	−	0.642788i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	24.2891			1.60507			0.802535	−	0.596606i	$-0.203484\pi$
$229$	24.2891			1.60507			0.802535	+	0.596606i	$0.203484\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$233$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.21735	+	18.2465i	0.208989	+	1.18524i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−21.8614	+	18.3439i	−1.40822	+	1.18164i	−0.450910	+	0.892570i	$0.648900\pi$
$241$	−21.8614	+	18.3439i	−1.40822	+	1.18164i	−0.957309	+	0.289066i	$0.906655\pi$
$242$		0			0
$243$	−5.33157	+	14.6484i	−0.342020	+	0.939693i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−18.4880	+	1.10267i	−1.17636	+	0.0701613i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$251$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$257$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$258$		0			0
$259$	0.363437	−	0.209830i	0.0225829	−	0.0130382i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$263$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$269$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$270$		0			0
$271$	−5.33157	−	14.6484i	−0.323870	−	0.889824i	−0.989628	−	0.143657i	$-0.954114\pi$
$271$	−5.33157	−	14.6484i	−0.323870	−	0.889824i	0.665758	−	0.746168i	$-0.268108\pi$
$272$		0			0
$273$	−0.303596	−	0.525844i	−0.0183745	−	0.0318255i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.5000	+	26.8468i	−0.931305	+	1.61307i	−0.150210	+	0.988654i	$0.547995\pi$
$277$	−15.5000	+	26.8468i	−0.931305	+	1.61307i	−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$	−4.11633	−	4.90566i	−0.246438	−	0.293694i
$280$		0			0
$281$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$281$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$282$		0			0
$283$	22.1746	+	3.90998i	1.31814	+	0.232424i	0.788100	−	0.615547i	$-0.211065\pi$
$283$	22.1746	+	3.90998i	1.31814	+	0.232424i	0.530042	+	0.847971i	$0.322176\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.9748	−	5.81434i	−0.939693	−	0.342020i
$290$		0			0
$291$	5.56670	+	6.63414i	0.326326	+	0.388900i
$292$		0			0
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.611997	−	0.513526i	−0.0352749	−	0.0295992i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.11334	+	1.32683i	−0.0635417	+	0.0757261i	−0.796879	−	0.604139i	$-0.793517\pi$
$307$	−1.11334	+	1.32683i	−0.0635417	+	0.0757261i	0.733337	+	0.679865i	$0.237962\pi$
$308$		0			0
$309$	−5.83527	+	2.12387i	−0.331957	+	0.120823i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	6.07769	+	34.4683i	0.343531	+	1.94826i	0.316387	+	0.948630i	$0.397530\pi$
$313$	6.07769	+	34.4683i	0.343531	+	1.94826i	0.0271446	+	0.999632i	$0.491359\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$317$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	3.68913	−	20.9221i	0.204636	−	1.16055i
$326$		0			0
$327$	10.0707	−	27.6691i	0.556913	−	1.53011i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−24.6871	−	14.2531i	−1.35692	−	0.783420i	−0.367716	−	0.929938i	$-0.619860\pi$
$331$	−24.6871	−	14.2531i	−1.35692	−	0.783420i	−0.989208	+	0.146518i	$0.953193\pi$
$332$		0			0
$333$	−2.64977	−	15.0276i	−0.145206	−	0.823506i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	25.5694	−	9.30650i	1.39285	−	0.506957i	0.466805	−	0.884361i	$-0.345405\pi$
$337$	25.5694	−	9.30650i	1.39285	−	0.506957i	0.926049	+	0.377403i	$0.123183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.15451i			0.0623379i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$347$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$348$		0			0
$349$	15.8418	+	27.4389i	0.847994	+	1.46877i	0.882996	+	0.469381i	$0.155523\pi$
$349$	15.8418	+	27.4389i	0.847994	+	1.46877i	−0.0350017	+	0.999387i	$0.511144\pi$
$350$		0			0
$351$	−21.7429	+	3.83386i	−1.16055	+	0.204636i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$359$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$360$		0			0
$361$	5.50000	+	18.1865i	0.289474	+	0.957186i
$362$		0			0
$363$	18.7631	+	3.30844i	0.984808	+	0.173648i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−6.78534	−	8.08645i	−0.354192	−	0.422109i	0.559301	−	0.828965i	$-0.311070\pi$
$367$	−6.78534	−	8.08645i	−0.354192	−	0.422109i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−19.0000	−	32.9090i	−0.983783	−	1.70396i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−19.0000	−	32.9090i	−0.983783	−	1.70396i	−0.336557	−	0.941663i	$-0.609263\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	38.5433i		−	1.97983i	−0.141648	−	0.989917i	$-0.545240\pi$
$379$		−	38.5433i		−	1.97983i	0.141648	−	0.989917i	$-0.454760\pi$
$380$		0			0
$381$	39.0000			1.99803
$382$		0			0
$383$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$383$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−25.1573	+	14.5246i	−1.27882	+	0.738327i
$388$		0			0
$389$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$389$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	4.31356	−	24.4634i	0.216491	−	1.22778i	−0.661809	−	0.749673i	$-0.730211\pi$
$397$	4.31356	−	24.4634i	0.216491	−	1.22778i	0.878300	−	0.478110i	$-0.158678\pi$
$398$		0			0
$399$	−0.452486	+	0.428092i	−0.0226526	+	0.0214314i
$400$		0			0
$401$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$401$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$402$		0			0
$403$	3.10211	−	8.52298i	0.154527	−	0.424560i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−6.59863	−	37.4227i	−0.326281	−	1.85043i	−0.500514	−	0.865729i	$-0.666856\pi$
$409$	−6.59863	−	37.4227i	−0.326281	−	1.85043i	0.174232	−	0.984705i	$-0.444256\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	18.7124			0.916352
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−31.4078	−	26.3543i	−1.53072	−	1.28443i	−0.792492	−	0.609882i	$-0.791217\pi$
$421$	−31.4078	−	26.3543i	−1.53072	−	1.28443i	−0.738231	−	0.674548i	$-0.764339\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.26146	−	0.222429i	0.0610463	−	0.0107641i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$431$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$432$		0			0
$433$	−26.5484	−	9.66284i	−1.27584	−	0.464367i	−0.386784	−	0.922170i	$-0.626414\pi$
$433$	−26.5484	−	9.66284i	−1.27584	−	0.464367i	−0.889053	+	0.457804i	$0.848636\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−19.4206	−	3.42437i	−0.926893	−	0.163436i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−19.4206	−	3.42437i	−0.926893	−	0.163436i	−0.616665	+	0.787226i	$0.711517\pi$
$440$		0			0
$441$	19.7144	+	7.17544i	0.938779	+	0.341688i
$442$		0			0
$443$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$443$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	11.4907	+	9.64181i	0.539879	+	0.453012i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	34.3806			1.60826			0.804129	−	0.594455i	$-0.202632\pi$
$457$	34.3806			1.60826			0.804129	+	0.594455i	$0.202632\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$461$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$462$		0			0
$463$	25.0859	−	14.4833i	1.16584	−	0.673098i	0.213144	−	0.977021i	$-0.431630\pi$
$463$	25.0859	−	14.4833i	1.16584	−	0.673098i	0.952697	+	0.303923i	$0.0982964\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	−0.482297	+	0.404695i	−0.0222704	+	0.0186871i
$470$		0			0
$471$	1.56124	−	4.28947i	0.0719382	−	0.197649i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−21.7558	+	1.29757i	−0.998226	+	0.0595368i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$479$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$480$		0			0
$481$	16.5559	−	13.8921i	0.754886	−	0.633424i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	31.5000	−	18.1865i	1.42740	−	0.824110i	0.430486	−	0.902597i	$-0.358342\pi$
$487$	31.5000	−	18.1865i	1.42740	−	0.824110i	0.996915	+	0.0784867i	$0.0250088\pi$
$488$		0			0
$489$	41.5403	−	15.1194i	1.87852	−	0.683724i
$490$		0			0
$491$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$491$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−13.7634	−	37.8145i	−0.616132	−	1.69281i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−13.7634	−	37.8145i	−0.616132	−	1.69281i	0.100126	−	0.994975i	$-0.468075\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$503$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.62654	−	6.70545i	−0.249884	−	0.297800i
$508$		0			0
$509$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$509$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$510$		0			0
$511$	−0.378452	−	0.0667312i	−0.0167417	−	0.00295202i
$512$		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	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	15.4599	−	2.72600i	0.676015	−	0.119200i	0.174908	−	0.984585i	$-0.444037\pi$
$523$	15.4599	−	2.72600i	0.676015	−	0.119200i	0.501107	+	0.865385i	$0.332926\pi$
$524$		0			0
$525$	−0.357259	−	0.618790i	−0.0155920	−	0.0270062i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−17.6190	−	14.7841i	−0.766044	−	0.642788i
$530$		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
$541$	6.89234	+	39.0884i	0.296325	+	1.68054i	0.661768	+	0.749708i	$0.269806\pi$
$541$	6.89234	+	39.0884i	0.296325	+	1.68054i	−0.365444	+	0.930834i	$0.619083\pi$
$542$		0			0
$543$	10.5000	+	6.06218i	0.450598	+	0.260153i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	15.1471	−	41.6163i	0.647642	−	1.77938i	0.0213785	−	0.999771i	$-0.493195\pi$
$547$	15.1471	−	41.6163i	0.647642	−	1.77938i	0.626264	−	0.779611i	$-0.284583\pi$
$548$		0			0
$549$	8.08781	−	45.8683i	0.345179	−	1.95761i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.153257	+	0.869162i	−0.00651714	+	0.0369605i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$557$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$558$		0			0
$559$	−35.6309	−	20.5715i	−1.50703	−	0.870083i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.477301	+	0.568825i	−0.0200447	+	0.0238884i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$			7.93697i			0.332152i	0.986113	+	0.166076i	$0.0531097\pi$
$571$			7.93697i			0.332152i	−0.986113	+	0.166076i	$0.946890\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−5.50000	−	9.52628i	−0.228968	−	0.396584i	0.728535	−	0.685009i	$-0.240202\pi$
$577$	−5.50000	−	9.52628i	−0.228968	−	0.396584i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	−45.9602	+	8.10403i	−1.91004	+	0.336792i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$587$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$588$		0			0
$589$	−9.24320	−	1.06731i	−0.380860	−	0.0439779i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$593$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−24.4013	+	42.2642i	−0.998677	+	1.72976i
$598$		0			0
$599$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$599$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$600$		0			0
$601$	5.10725	+	8.84601i	0.208329	+	0.360836i	0.951188	−	0.308611i	$-0.0998642\pi$
$601$	5.10725	+	8.84601i	0.208329	+	0.360836i	−0.742859	+	0.669448i	$0.766531\pi$
$602$		0			0
$603$	7.82981	+	21.5122i	0.318855	+	0.876046i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			47.3533i			1.92201i	0.276531	+	0.961005i	$0.410815\pi$
$607$			47.3533i			1.92201i	−0.276531	+	0.961005i	$0.589185\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−44.1656	+	16.0749i	−1.78383	+	0.649261i	−0.784245	+	0.620451i	$0.786949\pi$
$613$	−44.1656	+	16.0749i	−1.78383	+	0.649261i	−0.999585	+	0.0288097i	$0.990828\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$617$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$618$		0			0
$619$	−43.0929	−	24.8797i	−1.73205	−	1.00000i	−0.866226	−	0.499653i	$-0.833461\pi$
$619$	−43.0929	−	24.8797i	−1.73205	−	1.00000i	−0.865825	−	0.500347i	$-0.833206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.34120	−	24.6202i	0.173648	−	0.984808i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−16.2604	+	44.6751i	−0.647317	+	1.77849i	−0.0199047	+	0.999802i	$0.506336\pi$
$631$	−16.2604	+	44.6751i	−0.647317	+	1.77849i	−0.627412	+	0.778687i	$0.715886\pi$
$632$		0			0
$633$	26.4937	−	22.2308i	1.05303	−	0.883596i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	5.15976	+	29.2624i	0.204437	+	1.15942i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$641$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$642$		0			0
$643$	31.8787	−	37.9915i	1.25717	−	1.49824i	0.468449	−	0.883491i	$-0.344813\pi$
$643$	31.8787	−	37.9915i	1.25717	−	1.49824i	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
$651$	−0.104332	−	0.286649i	−0.00408908	−	0.0112347i
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6.98663	+	12.1012i	−0.272575	+	0.472113i
$658$		0			0
$659$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$659$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$660$		0			0
$661$	−35.7083	−	12.9968i	−1.38889	−	0.505516i	−0.464031	−	0.885819i	$-0.653597\pi$
$661$	−35.7083	−	12.9968i	−1.38889	−	0.505516i	−0.924862	+	0.380303i	$0.875820\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	42.2508	+	15.3780i	1.63351	+	0.594549i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.6977	+	25.4572i	−0.566557	+	0.981305i	0.430346	+	0.902664i	$0.358391\pi$
$673$	−14.6977	+	25.4572i	−0.566557	+	0.981305i	−0.996903	+	0.0786409i	$0.974942\pi$
$674$		0			0
$675$	−25.5861	+	4.51151i	−0.984808	+	0.173648i
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	0.141092	+	0.387648i	0.00541463	+	0.0148766i
$680$		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
$687$	27.0421	−	32.2275i	1.03172	−	1.22955i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−16.5000	+	9.52628i	−0.627690	+	0.362397i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−16.5000	+	9.52628i	−0.627690	+	0.362397i	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		0.173648	−	0.984808i	$-0.444444\pi$
$701$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$702$		0			0
$703$	−17.8026	−	13.2150i	−0.671439	−	0.498414i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−28.5431	+	23.9505i	−1.07196	+	0.899479i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−28.5431	+	23.9505i	−1.07196	+	0.899479i	−0.0767291	+	0.997052i	$0.524448\pi$
$710$		0			0
$711$	27.7920	+	16.0457i	1.04228	+	0.601760i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$719$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$720$		0			0
$721$	−0.295799			−0.0110161
$722$		0			0
$723$			49.4294i			1.83830i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	4.87304	+	13.3886i	0.180731	+	0.496555i	0.996666	−	0.0815889i	$-0.0259995\pi$
$727$	4.87304	+	13.3886i	0.180731	+	0.496555i	−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
$733$	−3.50000	+	6.06218i	−0.129275	+	0.223912i	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−3.50000	+	6.06218i	−0.129275	+	0.223912i	0.794121	+	0.607760i	$0.207932\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	49.3285	+	8.69794i	1.81458	+	0.319959i	0.974818	−	0.223001i	$-0.0715853\pi$
$739$	49.3285	+	8.69794i	1.81458	+	0.319959i	0.839759	+	0.542960i	$0.182696\pi$
$740$		0			0
$741$	−19.1204	+	25.7580i	−0.702404	+	0.946244i
$742$		0			0
$743$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$743$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	33.5435	−	5.91463i	1.22402	−	0.215828i	0.475965	−	0.879464i	$-0.342099\pi$
$751$	33.5435	−	5.91463i	1.22402	−	0.215828i	0.748056	+	0.663636i	$0.230988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−40.0453	−	33.6020i	−1.45547	−	1.22129i	−0.928461	−	0.371429i	$-0.878868\pi$
$757$	−40.0453	−	33.6020i	−1.45547	−	1.22129i	−0.527011	−	0.849858i	$-0.676688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	0.901568	−	1.07445i	0.0326390	−	0.0388976i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.96544	+	16.8179i	0.106937	+	0.606467i	0.990429	+	0.138022i	$0.0440745\pi$
$769$	2.96544	+	16.8179i	0.106937	+	0.606467i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$773$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$774$		0			0
$775$	3.65043	−	10.0295i	0.131127	−	0.360269i
$776$		0			0
$777$	0.126220	−	0.715831i	0.00452813	−	0.0256803i
$778$		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
$787$	−19.4062	−	11.2042i	−0.691755	−	0.399385i	0.112514	−	0.993650i	$-0.464110\pi$
$787$	−19.4062	−	11.2042i	−0.691755	−	0.399385i	−0.804269	+	0.594265i	$0.797443\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	61.9882	−	22.5619i	2.20127	−	0.801195i
$794$		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	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	27.8335	+	33.1707i	0.977367	+	1.16478i	0.986324	+	0.164821i	$0.0527046\pi$
$811$	27.8335	+	33.1707i	0.977367	+	1.16478i	−0.00895645	+	0.999960i	$0.502851\pi$
$812$		0			0
$813$	−25.3717	−	9.23454i	−0.889824	−	0.323870i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−12.0489	+	40.4512i	−0.421536	+	1.41521i
$818$		0			0
$819$	−1.03571	−	0.182624i	−0.0361907	−	0.00638139i
$820$		0			0
$821$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$821$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$822$		0			0
$823$	36.7402	+	43.7853i	1.28068	+	1.52626i	0.707099	+	0.707115i	$0.250004\pi$
$823$	36.7402	+	43.7853i	1.28068	+	1.52626i	0.573586	+	0.819146i	$0.305552\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$827$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$828$		0			0
$829$	−28.7524	−	49.8007i	−0.998613	−	1.72965i	−0.544900	−	0.838501i	$-0.683433\pi$
$829$	−28.7524	−	49.8007i	−0.998613	−	1.72965i	−0.453713	−	0.891148i	$-0.649901\pi$
$830$		0			0
$831$	18.3643	+	50.4555i	0.637050	+	1.75028i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−11.0918			−0.383390
$838$		0			0
$839$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$839$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$840$		0			0
$841$	−27.2511	+	9.91858i	−0.939693	+	0.342020i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	0.785969	+	0.453779i	0.0270062	+	0.0155920i
$848$		0			0
$849$	29.8757	−	25.0687i	1.02533	−	0.860356i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−2.93371	+	16.6379i	−0.100448	+	0.569670i	0.892493	+	0.451061i	$0.148954\pi$
$853$	−2.93371	+	16.6379i	−0.100448	+	0.569670i	−0.992941	+	0.118609i	$0.962157\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$857$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$858$		0			0
$859$	−16.8494	+	46.2934i	−0.574895	+	1.57951i	0.221777	+	0.975097i	$0.428814\pi$
$859$	−16.8494	+	46.2934i	−0.574895	+	1.57951i	−0.796672	+	0.604412i	$0.793408\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		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
$871$	−20.8415	+	24.8380i	−0.706188	+	0.841602i
$872$		0			0
$873$	15.0000			0.507673
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−43.8323	−	36.7797i	−1.48011	−	1.24196i	−0.906064	−	0.423141i	$-0.860927\pi$
$877$	−43.8323	−	36.7797i	−1.48011	−	1.24196i	−0.574049	−	0.818821i	$-0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	49.4910	−	8.72659i	1.66550	−	0.293673i	0.740055	−	0.672546i	$-0.234799\pi$
$883$	49.4910	−	8.72659i	1.66550	−	0.293673i	0.925449	+	0.378873i	$0.123688\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$887$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$888$		0			0
$889$	1.74571	+	0.635386i	0.0585492	+	0.0213102i
$890$		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
$903$	−1.36272	+	0.240285i	−0.0453485	+	0.00799617i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	4.14677	+	11.3932i	0.137691	+	0.378304i	0.989304	−	0.145868i	$-0.0465973\pi$
$907$	4.14677	+	11.3932i	0.137691	+	0.378304i	−0.851613	+	0.524171i	$0.824375\pi$
$908$		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
$919$	−39.6607	+	22.8981i	−1.30828	+	0.755338i	−0.981810	−	0.189867i	$-0.939194\pi$
$919$	−39.6607	+	22.8981i	−1.30828	+	0.755338i	−0.326475	+	0.945206i	$0.605861\pi$
$920$		0			0
$921$	0.520945	+	2.95442i	0.0171657	+	0.0973516i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	19.4823	−	16.3476i	0.640574	−	0.537505i
$926$		0			0
$927$	−3.67864	+	10.1070i	−0.120823	+	0.331957i
$928$		0			0
$929$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$929$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$930$		0			0
$931$	27.9728	−	12.1126i	0.916771	−	0.396973i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	42.5492	−	35.7030i	1.39002	−	1.16637i	0.424691	−	0.905338i	$-0.360383\pi$
$937$	42.5492	−	35.7030i	1.39002	−	1.16637i	0.965331	−	0.261029i	$-0.0840619\pi$
$938$		0			0
$939$	52.5000	+	30.3109i	1.71327	+	0.989158i
$940$		0			0
$941$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$941$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$947$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$948$		0			0
$949$	−19.7907			−0.642432
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$953$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−13.2217	+	22.9006i	−0.426506	+	0.738730i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	29.4677	+	5.19594i	0.947616	+	0.167090i	0.626038	−	0.779793i	$-0.284676\pi$
$967$	29.4677	+	5.19594i	0.947616	+	0.167090i	0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$971$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$972$		0			0
$973$	0.837602	+	0.304862i	0.0268523	+	0.00977343i
$974$		0			0
$975$	−23.6528	−	28.1883i	−0.757495	−	0.902747i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−25.5000	−	44.1673i	−0.814152	−	1.41015i
$982$		0			0
$983$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$983$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	36.8744	−	43.9452i	1.17135	−	1.39596i	0.270011	−	0.962857i	$-0.412973\pi$
$991$	36.8744	−	43.9452i	1.17135	−	1.39596i	0.901342	−	0.433108i	$-0.142583\pi$
$992$		0			0
$993$	−46.3965	+	16.8870i	−1.47235	+	0.535891i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	10.3616	+	58.7635i	0.328155	+	1.86106i	0.486507	+	0.873677i	$0.338271\pi$
$997$	10.3616	+	58.7635i	0.328155	+	1.86106i	−0.158352	+	0.987383i	$0.550618\pi$
$998$		0			0
$999$	−22.8891	−	13.2150i	−0.724179	−	0.418105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.ch.c.815.1	yes	6
3.2	odd	2	CM	912.2.ch.c.815.1	yes	6
4.3	odd	2		912.2.ch.d.815.1	yes	6
12.11	even	2		912.2.ch.d.815.1	yes	6
19.9	even	9		912.2.ch.d.47.1	yes	6
57.47	odd	18		912.2.ch.d.47.1	yes	6
76.47	odd	18	inner	912.2.ch.c.47.1	✓	6
228.47	even	18	inner	912.2.ch.c.47.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.2.ch.c.47.1	✓	6	76.47	odd	18	inner
912.2.ch.c.47.1	✓	6	228.47	even	18	inner
912.2.ch.c.815.1	yes	6	1.1	even	1	trivial
912.2.ch.c.815.1	yes	6	3.2	odd	2	CM
912.2.ch.d.47.1	yes	6	19.9	even	9
912.2.ch.d.47.1	yes	6	57.47	odd	18
912.2.ch.d.815.1	yes	6	4.3	odd	2
912.2.ch.d.815.1	yes	6	12.11	even	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.ch (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(1.11334 - 1.32683i) q^{3} +(0.0714517 - 0.0412527i) q^{7} +(-0.520945 - 2.95442i) q^{9} +O(q^{10})\)	\(q+(1.11334 - 1.32683i) q^{3} +(0.0714517 - 0.0412527i) q^{7} +(-0.520945 - 2.95442i) q^{9} +(3.25490 - 2.73119i) q^{13} +(-3.50000 - 2.59808i) q^{19} +(0.0248149 - 0.140732i) q^{21} +(3.83022 - 3.21394i) q^{25} +(-4.50000 - 2.59808i) q^{27} +(1.84864 - 1.06731i) q^{31} +5.08647 q^{37} -7.35943i q^{39} +(-3.31180 - 9.09911i) q^{43} +(-3.49660 + 6.05628i) q^{49} +(-7.34389 + 1.75135i) q^{57} +(14.5890 + 5.30996i) q^{61} +(-0.159100 - 0.189608i) q^{63} +(-7.51501 + 1.32510i) q^{67} +(-3.56805 - 2.99395i) q^{73} -8.66025i q^{75} +(-6.87598 + 8.19448i) q^{79} +(-8.45723 + 3.07818i) q^{81} +(0.119900 - 0.329421i) q^{91} +(0.642026 - 3.64111i) q^{93} +(-0.868241 + 4.92404i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 21 q^{13} - 21 q^{19} - 27 q^{21} - 27 q^{27} + 15 q^{43} + 21 q^{49} + 42 q^{61} + 36 q^{63} - 15 q^{67} + 21 q^{73} + 12 q^{79} - 48 q^{91} - 54 q^{93}+O(q^{100}) \) 6 * q + 21 * q^13 - 21 * q^19 - 27 * q^21 - 27 * q^27 + 15 * q^43 + 21 * q^49 + 42 * q^61 + 36 * q^63 - 15 * q^67 + 21 * 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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