LMFDB - Embedded newform 392.2.m.d.227.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,2,Mod(19,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

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

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

chi := DirichletCharacter("392.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	392.m (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:	$3.13013575923$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			227.1
Root			$0.662827 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	392.227
Dual form			392.2.m.d.19.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.22474i) q^{2} +(-2.92586 - 1.68925i) q^{3} +(-1.00000 + 1.73205i) q^{4} +4.77791i q^{6} +2.82843 q^{8} +(4.20711 + 7.28692i) q^{9} +O(q^{10})$	\(q+(-0.707107 - 1.22474i) q^{2} +(-2.92586 - 1.68925i) q^{3} +(-1.00000 + 1.73205i) q^{4} +4.77791i q^{6} +2.82843 q^{8} +(4.20711 + 7.28692i) q^{9} +(-1.12132 + 1.94218i) q^{11} +(5.85172 - 3.37849i) q^{12} +(-2.00000 - 3.46410i) q^{16} +(2.92586 + 1.68925i) q^{17} +(5.94975 - 10.3053i) q^{18} +(-1.21193 + 0.699709i) q^{19} +3.17157 q^{22} +(-8.27558 - 4.77791i) q^{24} +(2.50000 - 4.33013i) q^{25} -18.2919i q^{27} +(-2.82843 + 4.89898i) q^{32} +(6.56165 - 3.78837i) q^{33} -4.77791i q^{34} -16.8284 q^{36} +(1.71393 + 0.989538i) q^{38} +8.15640i q^{41} +13.0711 q^{43} +(-2.24264 - 3.88437i) q^{44} +13.5140i q^{48} -7.07107 q^{50} +(-5.70711 - 9.88500i) q^{51} +(-22.4029 + 12.9343i) q^{54} +4.72792 q^{57} +(9.48751 + 5.47762i) q^{59} +8.00000 q^{64} +(-9.27958 - 5.35757i) q^{66} +(4.24264 - 7.34847i) q^{67} +(-5.85172 + 3.37849i) q^{68} +(11.8995 + 20.6105i) q^{72} +(12.9154 + 7.45669i) q^{73} +(-14.6293 + 8.44623i) q^{75} -2.79884i q^{76} +(-18.2782 + 31.6587i) q^{81} +(9.98951 - 5.76745i) q^{82} +17.7122i q^{83} +(-9.24264 - 16.0087i) q^{86} +(-3.17157 + 5.49333i) q^{88} +(10.4915 - 6.05728i) q^{89} +(16.5512 - 9.55582i) q^{96} -15.7331i q^{97} -18.8701 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} + 28 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 + 28 * q^9	$ 8 q - 8 q^{4} + 28 q^{9} + 8 q^{11} - 16 q^{16} + 8 q^{18} + 48 q^{22} + 20 q^{25} - 112 q^{36} + 48 q^{43} + 16 q^{44} - 40 q^{51} - 64 q^{57} + 64 q^{64} + 16 q^{72} - 84 q^{81} - 40 q^{86} - 48 q^{88} + 64 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 + 28 * q^9 + 8 * q^11 - 16 * q^16 + 8 * q^18 + 48 * q^22 + 20 * q^25 - 112 * q^36 + 48 * q^43 + 16 * q^44 - 40 * q^51 - 64 * q^57 + 64 * q^64 + 16 * q^72 - 84 * q^81 - 40 * q^86 - 48 * q^88 + 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$	−0.707107	−	1.22474i	−0.500000	−	0.866025i
$2$	−0.707107	−	1.22474i	−0.500000	−	0.866025i
$3$	−2.92586	−	1.68925i	−1.68925	−	0.975287i	−0.955094	−	0.296302i	$-0.904247\pi$
$3$	−2.92586	−	1.68925i	−1.68925	−	0.975287i	−0.734152	−	0.678985i	$-0.762420\pi$
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$			4.77791i			1.95057i
$7$		0			0
$7$		0			0
$8$	2.82843			1.00000
$9$	4.20711	+	7.28692i	1.40237	+	2.42897i
$10$		0			0
$11$	−1.12132	+	1.94218i	−0.338091	+	0.585590i	−0.984074	−	0.177762i	$-0.943114\pi$
$11$	−1.12132	+	1.94218i	−0.338091	+	0.585590i	0.645983	+	0.763352i	$0.276448\pi$
$12$	5.85172	−	3.37849i	1.68925	−	0.975287i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	2.92586	+	1.68925i	0.709625	+	0.409702i	0.810922	−	0.585154i	$-0.198966\pi$
$17$	2.92586	+	1.68925i	0.709625	+	0.409702i	−0.101297	+	0.994856i	$0.532299\pi$
$18$	5.94975	−	10.3053i	1.40237	−	2.42897i
$19$	−1.21193	+	0.699709i	−0.278036	+	0.160524i	−0.632534	−	0.774533i	$-0.717985\pi$
$19$	−1.21193	+	0.699709i	−0.278036	+	0.160524i	0.354498	+	0.935057i	$0.384652\pi$
$20$		0			0
$21$		0			0
$22$	3.17157			0.676182
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−8.27558	−	4.77791i	−1.68925	−	0.975287i
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		−	18.2919i		−	3.52027i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	−2.82843	+	4.89898i	−0.500000	+	0.866025i
$33$	6.56165	−	3.78837i	1.14224	−	0.659471i
$34$		−	4.77791i		−	0.819405i
$35$		0			0
$36$	−16.8284			−2.80474
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	1.71393	+	0.989538i	0.278036	+	0.160524i
$39$		0			0
$40$		0			0
$41$			8.15640i			1.27382i	0.770940	+	0.636908i	$0.219787\pi$
$41$			8.15640i			1.27382i	−0.770940	+	0.636908i	$0.780213\pi$
$42$		0			0
$43$	13.0711			1.99332			0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	13.0711			1.99332			0.996660	+	0.0816682i	$0.0260248\pi$
$44$	−2.24264	−	3.88437i	−0.338091	−	0.585590i
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$			13.5140i			1.95057i
$49$		0			0
$50$	−7.07107			−1.00000
$51$	−5.70711	−	9.88500i	−0.799155	−	1.38418i
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	−22.4029	+	12.9343i	−3.04865	+	1.76014i
$55$		0			0
$56$		0			0
$57$	4.72792			0.626229
$58$		0			0
$59$	9.48751	+	5.47762i	1.23517	+	0.713125i	0.968103	−	0.250553i	$-0.0806124\pi$
$59$	9.48751	+	5.47762i	1.23517	+	0.713125i	0.267066	+	0.963678i	$0.413946\pi$
$60$		0			0
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$		0			0
$66$	−9.27958	−	5.35757i	−1.14224	−	0.659471i
$67$	4.24264	−	7.34847i	0.518321	−	0.897758i	−0.481452	−	0.876472i	$-0.659891\pi$
$67$	4.24264	−	7.34847i	0.518321	−	0.897758i	0.999773	−	0.0212861i	$-0.00677610\pi$
$68$	−5.85172	+	3.37849i	−0.709625	+	0.409702i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	11.8995	+	20.6105i	1.40237	+	2.42897i
$73$	12.9154	+	7.45669i	1.51163	+	0.872740i	0.999908	+	0.0135893i	$0.00432573\pi$
$73$	12.9154	+	7.45669i	1.51163	+	0.872740i	0.511722	+	0.859151i	$0.329008\pi$
$74$		0			0
$75$	−14.6293	+	8.44623i	−1.68925	+	0.975287i
$76$		−	2.79884i		−	0.321048i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−18.2782	+	31.6587i	−2.03091	+	3.51764i
$82$	9.98951	−	5.76745i	1.10316	−	0.636908i
$83$			17.7122i			1.94417i	0.234631	+	0.972085i	$0.424612\pi$
$83$			17.7122i			1.94417i	−0.234631	+	0.972085i	$0.575388\pi$
$84$		0			0
$85$		0			0
$86$	−9.24264	−	16.0087i	−0.996660	−	1.72627i
$87$		0			0
$88$	−3.17157	+	5.49333i	−0.338091	+	0.585590i
$89$	10.4915	−	6.05728i	1.11210	−	0.642070i	0.172726	−	0.984970i	$-0.444742\pi$
$89$	10.4915	−	6.05728i	1.11210	−	0.642070i	0.939372	+	0.342900i	$0.111409\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	16.5512	−	9.55582i	1.68925	−	0.975287i
$97$		−	15.7331i		−	1.59746i	−0.601690	−	0.798730i	$-0.705506\pi$
$97$		−	15.7331i		−	1.59746i	0.601690	−	0.798730i	$-0.294494\pi$
$98$		0			0
$99$	−18.8701			−1.89651
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$	−8.07107	+	13.9795i	−0.799155	+	1.38418i
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	0.973814	−	0.227345i	$-0.0730044\pi$
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	−0.683793	+	0.729676i	$0.739671\pi$
$108$	31.6825	+	18.2919i	3.04865	+	1.76014i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−20.7279			−1.94992			−0.974959	−	0.222383i	$-0.928617\pi$
$113$	−20.7279			−1.94992			−0.974959	+	0.222383i	$0.928617\pi$
$114$	−3.34315	−	5.79050i	−0.313114	−	0.542330i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	15.4930i		−	1.42625i
$119$		0			0
$120$		0			0
$121$	2.98528	+	5.17066i	0.271389	+	0.470060i
$122$		0			0
$123$	13.7782	−	23.8645i	1.24234	−	2.15179i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−5.65685	−	9.79796i	−0.500000	−	0.866025i
$129$	−38.2441	−	22.0803i	−3.36721	−	1.94406i
$130$		0			0
$131$	−5.34972	+	3.08866i	−0.467407	+	0.269858i	−0.715154	−	0.698967i	$-0.753643\pi$
$131$	−5.34972	+	3.08866i	−0.467407	+	0.269858i	0.247746	+	0.968825i	$0.420310\pi$
$132$			15.1535i			1.31894i
$133$		0			0
$134$	−12.0000			−1.03664
$135$		0			0
$136$	8.27558	+	4.77791i	0.709625	+	0.409702i
$137$	5.87868	−	10.1822i	0.502249	−	0.869922i	−0.497747	−	0.867322i	$-0.665839\pi$
$137$	5.87868	−	10.1822i	0.502249	−	0.869922i	0.999997	−	0.00259945i	$-0.000827431\pi$
$138$		0			0
$139$			0.579658i			0.0491659i	0.999698	+	0.0245830i	$0.00782579\pi$
$139$			0.579658i			0.0491659i	−0.999698	+	0.0245830i	$0.992174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	16.8284	−	29.1477i	1.40237	−	2.42897i
$145$		0			0
$146$		−	21.0907i		−	1.74548i
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	20.6890	+	11.9448i	1.68925	+	0.975287i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−3.42786	+	1.97908i	−0.278036	+	0.160524i
$153$			28.4274i			2.29822i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	51.6985			4.06182
$163$	9.70711	+	16.8132i	0.760319	+	1.31691i	0.942686	+	0.333681i	$0.108291\pi$
$163$	9.70711	+	16.8132i	0.760319	+	1.31691i	−0.182367	+	0.983231i	$0.558376\pi$
$164$	−14.1273	−	8.15640i	−1.10316	−	0.636908i
$165$		0			0
$166$	21.6930	−	12.5244i	1.68370	−	0.972085i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	−10.1974	−	5.88750i	−0.779818	−	0.450228i
$172$	−13.0711	+	22.6398i	−0.996660	+	1.72627i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$	8.97056			0.676182
$177$	−18.5061	−	32.0535i	−1.39100	−	2.40929i
$178$	−14.8372	−	8.56628i	−1.11210	−	0.642070i
$179$	−9.00000	+	15.5885i	−0.672692	+	1.16514i	0.304446	+	0.952529i	$0.401529\pi$
$179$	−9.00000	+	15.5885i	−0.672692	+	1.16514i	−0.977138	+	0.212607i	$0.931805\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.56165	+	3.78837i	−0.479836	+	0.277033i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−23.4069	−	13.5140i	−1.68925	−	0.975287i
$193$	1.77817	−	3.07989i	0.127996	−	0.221695i	−0.794904	−	0.606735i	$-0.792479\pi$
$193$	1.77817	−	3.07989i	0.127996	−	0.221695i	0.922900	+	0.385040i	$0.125812\pi$
$194$	−19.2691	+	11.1250i	−1.38344	+	0.798730i
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	13.3431	+	23.1110i	0.948256	+	1.64243i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	7.07107	−	12.2474i	0.500000	−	0.866025i
$201$	−24.8268	+	14.3337i	−1.75114	+	1.01102i
$202$		0			0
$203$		0			0
$204$	22.8284			1.59831
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.13839i		−	0.217087i
$210$		0			0
$211$	25.4558			1.75245			0.876226	−	0.481900i	$-0.160053\pi$
$211$	25.4558			1.75245			0.876226	+	0.481900i	$0.160053\pi$
$212$		0			0
$213$		0			0
$214$	4.24264	−	7.34847i	0.290021	−	0.502331i
$215$		0			0
$216$		−	51.7373i		−	3.52027i
$217$		0			0
$218$		0			0
$219$	−25.1924	−	43.6345i	−1.70234	−	2.94855i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	42.0711			2.80474
$226$	14.6569	+	25.3864i	0.974959	+	1.68868i
$227$	−12.2054	−	7.04681i	−0.810104	−	0.467714i	0.0368883	−	0.999319i	$-0.488255\pi$
$227$	−12.2054	−	7.04681i	−0.810104	−	0.467714i	−0.846992	+	0.531606i	$0.821589\pi$
$228$	−4.72792	+	8.18900i	−0.313114	+	0.542330i
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.6066	+	21.8353i	0.825886	+	1.43048i	0.901240	+	0.433320i	$0.142658\pi$
$233$	12.6066	+	21.8353i	0.825886	+	1.43048i	−0.0753544	+	0.997157i	$0.524009\pi$
$234$		0			0
$235$		0			0
$236$	−18.9750	+	10.9552i	−1.23517	+	0.713125i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	22.1950	+	12.8143i	1.42970	+	0.825439i	0.997097	−	0.0761440i	$-0.0242609\pi$
$241$	22.1950	+	12.8143i	1.42970	+	0.825439i	0.432606	+	0.901583i	$0.357594\pi$
$242$	4.22183	−	7.31242i	0.271389	−	0.470060i
$243$	59.4351	−	34.3149i	3.81276	−	2.20130i
$244$		0			0
$245$		0			0
$246$	−38.9706			−2.48467
$247$		0			0
$248$		0			0
$249$	29.9203	−	51.8235i	1.89612	−	3.28418i
$250$		0			0
$251$			26.4483i			1.66940i	0.550704	+	0.834700i	$0.314359\pi$
$251$			26.4483i			1.66940i	−0.550704	+	0.834700i	$0.685641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	27.7526	−	16.0230i	1.73116	−	0.999486i	0.849549	−	0.527510i	$-0.176874\pi$
$257$	27.7526	−	16.0230i	1.73116	−	0.999486i	0.881611	−	0.471976i	$-0.156459\pi$
$258$			62.4524i			3.88812i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	7.56565	+	4.36803i	0.467407	+	0.269858i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	18.5592	−	10.7151i	1.14224	−	0.659471i
$265$		0			0
$266$		0			0
$267$	−40.9289			−2.50481
$268$	8.48528	+	14.6969i	0.518321	+	0.897758i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		−	13.5140i		−	0.819405i
$273$		0			0
$274$	−16.6274			−1.00450
$275$	5.60660	+	9.71092i	0.338091	+	0.585590i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	0.709933	−	0.409880i	0.0425789	−	0.0245830i
$279$		0			0
$280$		0			0
$281$	7.27208			0.433816			0.216908	−	0.976192i	$-0.430403\pi$
$281$	7.27208			0.433816			0.216908	+	0.976192i	$0.430403\pi$
$282$		0			0
$283$	−26.0387	−	15.0334i	−1.54784	−	0.893645i	−0.998307	−	0.0581728i	$-0.981473\pi$
$283$	−26.0387	−	15.0334i	−1.54784	−	0.893645i	−0.549532	−	0.835472i	$-0.685194\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−47.5980			−2.80474
$289$	−2.79289	−	4.83743i	−0.164288	−	0.284555i
$290$		0			0
$291$	−26.5772	+	46.0330i	−1.55798	+	2.69850i
$292$	−25.8307	+	14.9134i	−1.51163	+	0.872740i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	35.5262	+	20.5111i	2.06144	+	1.19017i
$298$		0			0
$299$		0			0
$300$		−	33.7849i		−	1.95057i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	4.84772	+	2.79884i	0.278036	+	0.160524i
$305$		0			0
$306$	34.8163	−	20.1012i	1.99031	−	1.14911i
$307$		−	20.8506i		−	1.19001i	−0.803723	−	0.595004i	$-0.797151\pi$
$307$		−	20.8506i		−	1.19001i	0.803723	−	0.595004i	$-0.202849\pi$
$308$		0			0
$309$		0			0
$310$		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
$313$	−30.4705	+	17.5922i	−1.72230	+	0.994368i	−0.808159	+	0.588964i	$0.799536\pi$
$313$	−30.4705	+	17.5922i	−1.72230	+	0.994368i	−0.914138	+	0.405404i	$0.867131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		−	20.2710i		−	1.13141i
$322$		0			0
$323$	−4.72792			−0.263069
$324$	−36.5563	−	63.3175i	−2.03091	−	3.51764i
$325$		0			0
$326$	13.7279	−	23.7775i	0.760319	−	1.31691i
$327$		0			0
$328$			23.0698i			1.27382i
$329$		0			0
$330$		0			0
$331$	−0.192388	−	0.333226i	−0.0105746	−	0.0183158i	0.860690	−	0.509130i	$-0.170033\pi$
$331$	−0.192388	−	0.333226i	−0.0105746	−	0.0183158i	−0.871264	+	0.490814i	$0.836699\pi$
$332$	−30.6785	−	17.7122i	−1.68370	−	0.972085i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−33.9411			−1.84889			−0.924445	−	0.381314i	$-0.875472\pi$
$337$	−33.9411			−1.84889			−0.924445	+	0.381314i	$0.875472\pi$
$338$	9.19239	+	15.9217i	0.500000	+	0.866025i
$339$	60.6470	+	35.0146i	3.29389	+	1.90173i
$340$		0			0
$341$		0			0
$342$			16.6524i			0.900457i
$343$		0			0
$344$	36.9706			1.99332
$345$		0			0
$346$		0			0
$347$	−15.1213	+	26.1909i	−0.811755	+	1.40600i	0.0998797	+	0.995000i	$0.468154\pi$
$347$	−15.1213	+	26.1909i	−0.811755	+	1.40600i	−0.911635	+	0.411001i	$0.865179\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	−6.34315	−	10.9867i	−0.338091	−	0.585590i
$353$	−28.0467	−	16.1928i	−1.49277	−	0.861853i	−0.492808	−	0.870138i	$-0.664030\pi$
$353$	−28.0467	−	16.1928i	−1.49277	−	0.861853i	−0.999966	+	0.00828457i	$0.997363\pi$
$354$	−26.1716	+	45.3305i	−1.39100	+	2.40929i
$355$		0			0
$356$			24.2291i			1.28414i
$357$		0			0
$358$	25.4558			1.34538
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−8.52082	+	14.7585i	−0.448464	+	0.776762i
$362$		0			0
$363$		−	20.1715i		−	1.05873i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$	−59.4351	+	34.3149i	−3.09407	+	1.78636i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$	9.27958	+	5.35757i	0.479836	+	0.277033i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	32.8701			1.68842			0.844211	−	0.536011i	$-0.180070\pi$
$379$	32.8701			1.68842			0.844211	+	0.536011i	$0.180070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$			38.2233i			1.95057i
$385$		0			0
$386$	−5.02944			−0.255992
$387$	54.9914	+	95.2479i	2.79537	+	4.84172i
$388$	27.2506	+	15.7331i	1.38344	+	0.798730i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	20.8701			1.05276
$394$		0			0
$395$		0			0
$396$	18.8701	−	32.6839i	0.948256	−	1.64243i
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$	35.1103	+	20.2710i	1.75114	+	1.01102i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−16.1421	−	27.9590i	−0.799155	−	1.38418i
$409$	−6.35372	−	3.66832i	−0.314171	−	0.181387i	0.334620	−	0.942353i	$-0.391392\pi$
$409$	−6.35372	−	3.66832i	−0.314171	−	0.181387i	−0.648792	+	0.760966i	$0.724725\pi$
$410$		0			0
$411$	−34.4004	+	19.8611i	−1.69685	+	0.979675i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.979185	−	1.69600i	0.0479509	−	0.0830534i
$418$	−3.84373	+	2.21918i	−0.188003	+	0.108544i
$419$			2.55873i			0.125002i	0.998045	+	0.0625011i	$0.0199077\pi$
$419$			2.55873i			0.125002i	−0.998045	+	0.0625011i	$0.980092\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−18.0000	−	31.1769i	−0.876226	−	1.51767i
$423$		0			0
$424$		0			0
$425$	14.6293	−	8.44623i	0.709625	−	0.409702i
$426$		0			0
$427$		0			0
$428$	−12.0000			−0.580042
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	−63.3649	+	36.5838i	−3.04865	+	1.76014i
$433$		−	41.6018i		−	1.99925i	−0.0273152	−	0.999627i	$-0.508696\pi$
$433$		−	41.6018i		−	1.99925i	0.0273152	−	0.999627i	$-0.491304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−35.6274	+	61.7085i	−1.70234	+	2.94855i
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.41421	−	2.44949i	−0.0671913	−	0.116379i	0.830473	−	0.557059i	$-0.188070\pi$
$443$	−1.41421	−	2.44949i	−0.0671913	−	0.116379i	−0.897664	+	0.440681i	$0.854737\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.65685			0.266963			0.133482	−	0.991051i	$-0.457384\pi$
$449$	5.65685			0.266963			0.133482	+	0.991051i	$0.457384\pi$
$450$	−29.7487	−	51.5263i	−1.40237	−	2.42897i
$451$	−15.8412	−	9.14594i	−0.745935	−	0.430666i
$452$	20.7279	−	35.9018i	0.974959	−	1.68868i
$453$		0			0
$454$			19.9314i			0.935427i
$455$		0			0
$456$	13.3726			0.626229
$457$	−21.1924	−	36.7063i	−0.991338	−	1.71705i	−0.609408	−	0.792857i	$-0.708593\pi$
$457$	−21.1924	−	36.7063i	−0.991338	−	1.71705i	−0.381930	−	0.924191i	$-0.624740\pi$
$458$		0			0
$459$	30.8995	−	53.5195i	1.44226	−	2.49808i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	17.8284	−	30.8797i	0.825886	−	1.43048i
$467$	34.3143	−	19.8114i	1.58787	−	0.916760i	0.594218	−	0.804304i	$-0.297462\pi$
$467$	34.3143	−	19.8114i	1.58787	−	0.916760i	0.993657	−	0.112456i	$-0.0358717\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	26.8347	+	15.4930i	1.23517	+	0.713125i
$473$	−14.6569	+	25.3864i	−0.673923	+	1.16727i
$474$		0			0
$475$			6.99709i			0.321048i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		−	36.2442i		−	1.65088i
$483$		0			0
$484$	−11.9411			−0.542778
$485$		0			0
$486$	−84.0539	−	48.5285i	−3.81276	−	2.20130i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		−	65.5908i		−	2.96612i
$490$		0			0
$491$	−14.1421			−0.638226			−0.319113	−	0.947717i	$-0.603385\pi$
$491$	−14.1421			−0.638226			−0.319113	+	0.947717i	$0.603385\pi$
$492$	27.5563	+	47.7290i	1.24234	+	2.15179i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−84.6274			−3.79225
$499$	−21.2132	−	36.7423i	−0.949633	−	1.64481i	−0.746197	−	0.665725i	$-0.768122\pi$
$499$	−21.2132	−	36.7423i	−0.949633	−	1.64481i	−0.203436	−	0.979088i	$-0.565211\pi$
$500$		0			0
$501$		0			0
$502$	32.3924	−	18.7018i	1.44574	−	0.834700i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	38.0362	+	21.9602i	1.68925	+	0.975287i
$508$		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
$511$		0			0
$512$	22.6274			1.00000
$513$	12.7990	+	22.1685i	0.565089	+	0.978763i
$514$	−39.2481	−	22.6599i	−1.73116	−	0.999486i
$515$		0			0
$516$	76.4882	−	44.1605i	3.36721	−	1.94406i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.1974	−	5.88750i	−0.446758	−	0.257936i	0.259702	−	0.965689i	$-0.416376\pi$
$521$	−10.1974	−	5.88750i	−0.446758	−	0.257936i	−0.706460	+	0.707753i	$0.749709\pi$
$522$		0			0
$523$	7.77359	−	4.48808i	0.339915	−	0.196250i	−0.320319	−	0.947310i	$-0.603790\pi$
$523$	7.77359	−	4.48808i	0.339915	−	0.196250i	0.660235	+	0.751060i	$0.270457\pi$
$524$		−	12.3547i		−	0.539716i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−26.2466	−	15.1535i	−1.14224	−	0.659471i
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$			92.1797i			4.00026i
$532$		0			0
$533$		0			0
$534$	28.9411	+	50.1275i	1.25240	+	2.16923i
$535$		0			0
$536$	12.0000	−	20.7846i	0.518321	−	0.897758i
$537$	52.6655	−	30.4064i	2.27268	−	1.31213i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$	−16.5512	+	9.55582i	−0.709625	+	0.409702i
$545$		0			0
$546$		0			0
$547$	−26.5269			−1.13421			−0.567104	−	0.823646i	$-0.691936\pi$
$547$	−26.5269			−1.13421			−0.567104	+	0.823646i	$0.691936\pi$
$548$	11.7574	+	20.3643i	0.502249	+	0.869922i
$549$		0			0
$550$	7.92893	−	13.7333i	0.338091	−	0.585590i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.00400	−	0.579658i	−0.0425789	−	0.0245830i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	25.5980			1.08075
$562$	−5.14214	−	8.90644i	−0.216908	−	0.375696i
$563$	19.4770	+	11.2451i	0.820859	+	0.473923i	0.850713	−	0.525631i	$-0.176171\pi$
$563$	19.4770	+	11.2451i	0.820859	+	0.473923i	−0.0298537	+	0.999554i	$0.509504\pi$
$564$		0			0
$565$		0			0
$566$			42.5210i			1.78729i
$567$		0			0
$568$		0			0
$569$	−11.3137	−	19.5959i	−0.474295	−	0.821504i	0.525271	−	0.850935i	$-0.323964\pi$
$569$	−11.3137	−	19.5959i	−0.474295	−	0.821504i	−0.999567	+	0.0294311i	$0.990630\pi$
$570$		0			0
$571$	22.7782	−	39.4530i	0.953237	−	1.65105i	0.214885	−	0.976639i	$-0.431062\pi$
$571$	22.7782	−	39.4530i	0.953237	−	1.65105i	0.738352	−	0.674415i	$-0.235604\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	33.6569	+	58.2954i	1.40237	+	2.42897i
$577$	38.4521	+	22.2003i	1.60078	+	0.924211i	0.991331	+	0.131385i	$0.0419425\pi$
$577$	38.4521	+	22.2003i	1.60078	+	0.924211i	0.609449	+	0.792826i	$0.291391\pi$
$578$	−3.94975	+	6.84116i	−0.164288	+	0.284555i
$579$	−10.4054	+	6.00755i	−0.432433	+	0.249665i
$580$		0			0
$581$		0			0
$582$	75.1716			3.11596
$583$		0			0
$584$	36.5302	+	21.0907i	1.51163	+	0.872740i
$585$		0			0
$586$		0			0
$587$		−	46.7192i		−	1.92831i	−0.265341	−	0.964155i	$-0.585484\pi$
$587$		−	46.7192i		−	1.92831i	0.265341	−	0.964155i	$-0.414516\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	42.1740	−	24.3492i	1.73188	−	0.999900i	0.858871	−	0.512192i	$-0.171166\pi$
$593$	42.1740	−	24.3492i	1.73188	−	0.999900i	0.873007	−	0.487708i	$-0.162167\pi$
$594$		−	58.0140i		−	2.38034i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	−41.3779	+	23.8896i	−1.68925	+	0.975287i
$601$			36.0041i			1.46864i	0.678804	+	0.734319i	$0.262498\pi$
$601$			36.0041i			1.46864i	−0.678804	+	0.734319i	$0.737502\pi$
$602$		0			0
$603$	71.3970			2.90751
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		−	7.91630i		−	0.321048i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−49.2376	−	28.4274i	−1.99031	−	1.14911i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−25.5367	+	14.7436i	−1.03058	+	0.595004i
$615$		0			0
$616$		0			0
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	−25.3287	−	14.6236i	−1.01805	−	0.587770i	−0.104510	−	0.994524i	$-0.533328\pi$
$619$	−25.3287	−	14.6236i	−1.01805	−	0.587770i	−0.913538	+	0.406753i	$0.866661\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$	43.0918	+	24.8791i	1.72230	+	0.994368i
$627$	−5.30152	+	9.18249i	−0.211722	+	0.366713i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−74.4803	−	43.0012i	−2.96032	−	1.70914i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	14.1421	−	24.4949i	0.558581	−	0.967490i	−0.439034	−	0.898470i	$-0.644679\pi$
$641$	14.1421	−	24.4949i	0.558581	−	0.967490i	0.997615	−	0.0690201i	$-0.0219873\pi$
$642$	−24.8268	+	14.3337i	−0.979834	+	0.565707i
$643$			11.2948i			0.445423i	0.974884	+	0.222712i	$0.0714908\pi$
$643$			11.2948i			0.445423i	−0.974884	+	0.222712i	$0.928509\pi$
$644$		0			0
$645$		0			0
$646$	3.34315	+	5.79050i	0.131534	+	0.227824i
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$	−51.6985	+	89.5444i	−2.03091	+	3.51764i
$649$	−21.2771	+	12.2843i	−0.835199	+	0.482202i
$650$		0			0
$651$		0			0
$652$	−38.8284			−1.52064
$653$		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
$656$	28.2546	−	16.3128i	1.10316	−	0.636908i
$657$			125.484i			4.89561i
$658$		0			0
$659$	21.2721			0.828643			0.414321	−	0.910131i	$-0.364019\pi$
$659$	21.2721			0.828643			0.414321	+	0.910131i	$0.364019\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	−0.272078	+	0.471253i	−0.0105746	+	0.0183158i
$663$		0			0
$664$			50.0977i			1.94417i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−28.9289			−1.11513			−0.557564	−	0.830134i	$-0.688264\pi$
$673$	−28.9289			−1.11513			−0.557564	+	0.830134i	$0.688264\pi$
$674$	24.0000	+	41.5692i	0.924445	+	1.60119i
$675$	−79.2062	−	45.7297i	−3.04865	−	1.76014i
$676$	13.0000	−	22.5167i	0.500000	−	0.866025i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		−	99.0362i		−	3.80346i
$679$		0			0
$680$		0			0
$681$	23.8076	+	41.2360i	0.912310	+	1.58017i
$682$		0			0
$683$	−15.5563	+	26.9444i	−0.595247	+	1.03100i	0.398265	+	0.917270i	$0.369613\pi$
$683$	−15.5563	+	26.9444i	−0.595247	+	1.03100i	−0.993512	+	0.113728i	$0.963721\pi$
$684$	20.3949	−	11.7750i	0.779818	−	0.450228i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−26.1421	−	45.2795i	−0.996660	−	1.72627i
$689$		0			0
$690$		0			0
$691$	35.6123	−	20.5608i	1.35476	−	0.782169i	0.365845	−	0.930676i	$-0.380780\pi$
$691$	35.6123	−	20.5608i	1.35476	−	0.782169i	0.988911	+	0.148507i	$0.0474466\pi$
$692$		0			0
$693$		0			0
$694$	42.7696			1.62351
$695$		0			0
$696$		0			0
$697$	−13.7782	+	23.8645i	−0.521886	+	0.903932i
$698$		0			0
$699$		−	85.1826i		−	3.22190i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−8.97056	+	15.5375i	−0.338091	+	0.585590i
$705$		0			0
$706$			45.8000i			1.72371i
$707$		0			0
$708$	74.0244			2.78201
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	29.6745	−	17.1326i	1.11210	−	0.642070i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−18.0000	−	31.1769i	−0.672692	−	1.16514i
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$	24.1005			0.896928
$723$	−43.2929	−	74.9855i	−1.61008	−	2.78874i
$724$		0			0
$725$		0			0
$726$	−24.7049	+	14.2634i	−0.916887	+	0.529365i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−122.196			−4.52578
$730$		0			0
$731$	38.2441	+	22.0803i	1.41451	+	0.816668i
$732$		0			0
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.51472	+	16.4800i	0.350479	+	0.607048i
$738$	84.0539	+	48.5285i	3.09407	+	1.78636i
$739$	2.97918	−	5.16010i	0.109591	−	0.189817i	−0.806014	−	0.591897i	$-0.798379\pi$
$739$	2.97918	−	5.16010i	0.109591	−	0.189817i	0.915605	+	0.402080i	$0.131713\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−129.068	+	74.5172i	−4.72234	+	2.72644i
$748$		−	15.1535i		−	0.554066i
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	44.6777	−	77.3840i	1.62814	−	2.82003i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−23.2426	−	40.2574i	−0.844211	−	1.46222i
$759$		0			0
$760$		0			0
$761$	−39.4561	+	22.7800i	−1.43028	+	0.825773i	−0.997142	−	0.0755541i	$-0.975927\pi$
$761$	−39.4561	+	22.7800i	−1.43028	+	0.825773i	−0.433139	+	0.901327i	$0.642594\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	46.8138	−	27.0279i	1.68925	−	0.975287i
$769$			55.4553i			1.99977i	0.0151869	+	0.999885i	$0.495166\pi$
$769$			55.4553i			1.99977i	−0.0151869	+	0.999885i	$0.504834\pi$
$770$		0			0
$771$	−108.267			−3.89914
$772$	3.55635	+	6.15978i	0.127996	+	0.221695i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$	77.7696	−	134.701i	2.79537	−	4.84172i
$775$		0			0
$776$		−	44.5001i		−	1.59746i
$777$		0			0
$778$		0			0
$779$	−5.70711	−	9.88500i	−0.204478	−	0.354167i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	−14.7574	−	25.5605i	−0.526378	−	0.911713i
$787$	−48.4416	−	27.9678i	−1.72676	−	0.996943i	−0.902446	−	0.430804i	$-0.858230\pi$
$787$	−48.4416	−	27.9678i	−1.72676	−	0.996943i	−0.824310	−	0.566139i	$-0.808437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−53.3726			−1.89651
$793$		0			0
$794$		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
$800$	14.1421	+	24.4949i	0.500000	+	0.866025i
$801$	88.2778	+	50.9672i	3.11914	+	1.80084i
$802$	4.24264	−	7.34847i	0.149813	−	0.259483i
$803$	−28.9645	+	16.7227i	−1.02214	+	0.590131i
$804$		−	57.3349i		−	2.02205i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.1213	+	38.3153i	−0.777744	+	1.34709i	0.155495	+	0.987837i	$0.450303\pi$
$809$	−22.1213	+	38.3153i	−0.777744	+	1.34709i	−0.933239	+	0.359256i	$0.883031\pi$
$810$		0			0
$811$		−	18.8715i		−	0.662669i	−0.943513	−	0.331335i	$-0.892501\pi$
$811$		−	18.8715i		−	0.662669i	0.943513	−	0.331335i	$-0.107499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−22.8284	+	39.5400i	−0.799155	+	1.38418i
$817$	−15.8412	+	9.14594i	−0.554215	+	0.319976i
$818$			10.3756i			0.362774i
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	48.6495	+	28.0878i	1.69685	+	0.979675i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		−	37.8837i		−	1.31894i
$826$		0			0
$827$	−54.0000			−1.87776			−0.938882	−	0.344239i	$-0.888137\pi$
$827$	−54.0000			−1.87776			−0.938882	+	0.344239i	$0.888137\pi$
$828$		0			0
$829$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$829$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$	−2.76955			−0.0959018
$835$		0			0
$836$	5.43585	+	3.13839i	0.188003	+	0.108544i
$837$		0			0
$838$	3.13380	−	1.80930i	0.108255	−	0.0625011i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−21.2771	−	12.2843i	−0.732822	−	0.423095i
$844$	−25.4558	+	44.0908i	−0.876226	+	1.51767i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	50.7904	+	87.9715i	1.74312	+	3.01917i
$850$	−20.6890	−	11.9448i	−0.709625	−	0.409702i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	8.48528	+	14.6969i	0.290021	+	0.502331i
$857$	−0.207935	−	0.120051i	−0.00710291	−	0.00410087i	0.496444	−	0.868069i	$-0.334639\pi$
$857$	−0.207935	−	0.120051i	−0.00710291	−	0.00410087i	−0.503547	+	0.863968i	$0.667972\pi$
$858$		0			0
$859$	43.5938	−	25.1689i	1.48740	−	0.858752i	0.487506	−	0.873120i	$-0.337907\pi$
$859$	43.5938	−	25.1689i	1.48740	−	0.858752i	0.999897	+	0.0143672i	$0.00457339\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$	89.6116	+	51.7373i	3.04865	+	1.76014i
$865$		0			0
$866$	−50.9516	+	29.4169i	−1.73140	+	0.999627i
$867$			18.8715i			0.640911i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	114.646	−	66.1910i	3.88019	−	2.24023i
$874$		0			0
$875$		0			0
$876$	100.770			3.40469
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	5.01801i		−	0.169061i	−0.996421	−	0.0845306i	$-0.973061\pi$
$881$		−	5.01801i		−	0.169061i	0.996421	−	0.0845306i	$-0.0269391\pi$
$882$		0			0
$883$	2.00000			0.0673054			0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000			0.0673054			0.0336527	+	0.999434i	$0.489286\pi$
$884$		0			0
$885$		0			0
$886$	−2.00000	+	3.46410i	−0.0671913	+	0.116379i
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−40.9914	−	70.9991i	−1.37326	−	2.37856i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−4.00000	−	6.92820i	−0.133482	−	0.231197i
$899$		0			0
$900$	−42.0711	+	72.8692i	−1.40237	+	2.42897i
$901$		0			0
$902$			25.8686i			0.861331i
$903$		0			0
$904$	−58.6274			−1.94992
$905$		0			0
$906$		0			0
$907$	5.00000	−	8.66025i	0.166022	−	0.287559i	−0.770996	−	0.636841i	$-0.780241\pi$
$907$	5.00000	−	8.66025i	0.166022	−	0.287559i	0.937018	+	0.349281i	$0.113574\pi$
$908$	24.4109	−	14.0936i	0.810104	−	0.467714i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−9.45584	−	16.3780i	−0.313114	−	0.542330i
$913$	−34.4004	−	19.8611i	−1.13849	−	0.657306i
$914$	−29.9706	+	51.9105i	−0.991338	+	1.71705i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−87.3970			−2.88453
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$	−35.2218	+	61.0060i	−1.16060	+	2.01022i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−52.5794	+	30.3567i	−1.72507	+	0.995971i	−0.817700	+	0.575644i	$0.804751\pi$
$929$	−52.5794	+	30.3567i	−1.72507	+	0.995971i	−0.907373	+	0.420327i	$0.861915\pi$
$930$		0			0
$931$		0			0
$932$	−50.4264			−1.65177
$933$		0			0
$934$	−48.5277	−	28.0175i	−1.58787	−	0.916760i
$935$		0			0
$936$		0			0
$937$			34.0250i			1.11155i	0.831333	+	0.555775i	$0.187578\pi$
$937$			34.0250i			1.11155i	−0.831333	+	0.555775i	$0.812422\pi$
$938$		0			0
$939$	118.870			3.87918
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		−	43.8210i		−	1.42625i
$945$		0			0
$946$	41.4558			1.34785
$947$	−8.39340	−	14.5378i	−0.272749	−	0.472415i	0.696816	−	0.717250i	$-0.254599\pi$
$947$	−8.39340	−	14.5378i	−0.272749	−	0.472415i	−0.969565	+	0.244835i	$0.921266\pi$
$948$		0			0
$949$		0			0
$950$	8.56965	−	4.94769i	0.278036	−	0.160524i
$951$		0			0
$952$		0			0
$953$	45.2548			1.46595			0.732974	−	0.680257i	$-0.238132\pi$
$953$	45.2548			1.46595			0.732974	+	0.680257i	$0.238132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$	−25.2426	+	43.7215i	−0.813433	+	1.40891i
$964$	−44.3899	+	25.6285i	−1.42970	+	0.825439i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	8.44365	+	14.6248i	0.271389	+	0.470060i
$969$	13.8332	+	7.98663i	0.444388	+	0.256567i
$970$		0			0
$971$	−32.8944	+	18.9916i	−1.05563	+	0.609469i	−0.924221	−	0.381859i	$-0.875284\pi$
$971$	−32.8944	+	18.9916i	−1.05563	+	0.609469i	−0.131411	+	0.991328i	$0.541951\pi$
$972$			137.259i			4.40260i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.8787	−	34.4309i	0.635975	−	1.10154i	−0.350332	−	0.936625i	$-0.613931\pi$
$977$	19.8787	−	34.4309i	0.635975	−	1.10154i	0.986308	−	0.164916i	$-0.0527353\pi$
$978$	−80.3320	+	46.3797i	−2.56873	+	1.48306i
$979$			27.1686i			0.868312i
$980$		0			0
$981$		0			0
$982$	10.0000	+	17.3205i	0.319113	+	0.552720i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$	38.9706	−	67.4990i	1.24234	−	2.15179i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$			1.29996i			0.0412531i
$994$		0			0
$995$		0			0
$996$	59.8406	+	103.647i	1.89612	+	3.28418i
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$	−30.0000	+	51.9615i	−0.949633	+	1.64481i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.m.d.227.1		8
4.3	odd	2		1568.2.q.f.815.4		8
7.2	even	3	inner	392.2.m.d.19.2		8
7.3	odd	6		392.2.e.a.195.3	✓	4
7.4	even	3		392.2.e.a.195.4	yes	4
7.5	odd	6	inner	392.2.m.d.19.1		8
7.6	odd	2	inner	392.2.m.d.227.2		8
8.3	odd	2	CM	392.2.m.d.227.1		8
8.5	even	2		1568.2.q.f.815.4		8
28.3	even	6		1568.2.e.a.783.4		4
28.11	odd	6		1568.2.e.a.783.1		4
28.19	even	6		1568.2.q.f.1391.4		8
28.23	odd	6		1568.2.q.f.1391.1		8
28.27	even	2		1568.2.q.f.815.1		8
56.3	even	6		392.2.e.a.195.3	✓	4
56.5	odd	6		1568.2.q.f.1391.4		8
56.11	odd	6		392.2.e.a.195.4	yes	4
56.13	odd	2		1568.2.q.f.815.1		8
56.19	even	6	inner	392.2.m.d.19.1		8
56.27	even	2	inner	392.2.m.d.227.2		8
56.37	even	6		1568.2.q.f.1391.1		8
56.45	odd	6		1568.2.e.a.783.4		4
56.51	odd	6	inner	392.2.m.d.19.2		8
56.53	even	6		1568.2.e.a.783.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.e.a.195.3	✓	4	7.3	odd	6
392.2.e.a.195.3	✓	4	56.3	even	6
392.2.e.a.195.4	yes	4	7.4	even	3
392.2.e.a.195.4	yes	4	56.11	odd	6
392.2.m.d.19.1		8	7.5	odd	6	inner
392.2.m.d.19.1		8	56.19	even	6	inner
392.2.m.d.19.2		8	7.2	even	3	inner
392.2.m.d.19.2		8	56.51	odd	6	inner
392.2.m.d.227.1		8	1.1	even	1	trivial
392.2.m.d.227.1		8	8.3	odd	2	CM
392.2.m.d.227.2		8	7.6	odd	2	inner
392.2.m.d.227.2		8	56.27	even	2	inner
1568.2.e.a.783.1		4	28.11	odd	6
1568.2.e.a.783.1		4	56.53	even	6
1568.2.e.a.783.4		4	28.3	even	6
1568.2.e.a.783.4		4	56.45	odd	6
1568.2.q.f.815.1		8	28.27	even	2
1568.2.q.f.815.1		8	56.13	odd	2
1568.2.q.f.815.4		8	4.3	odd	2
1568.2.q.f.815.4		8	8.5	even	2
1568.2.q.f.1391.1		8	28.23	odd	6
1568.2.q.f.1391.1		8	56.37	even	6
1568.2.q.f.1391.4		8	28.19	even	6
1568.2.q.f.1391.4		8	56.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-2.92586 - 1.68925i) q^{3} +(-1.00000 + 1.73205i) q^{4} +4.77791i q^{6} +2.82843 q^{8} +(4.20711 + 7.28692i) q^{9} +O(q^{10})\)	\(q+(-0.707107 - 1.22474i) q^{2} +(-2.92586 - 1.68925i) q^{3} +(-1.00000 + 1.73205i) q^{4} +4.77791i q^{6} +2.82843 q^{8} +(4.20711 + 7.28692i) q^{9} +(-1.12132 + 1.94218i) q^{11} +(5.85172 - 3.37849i) q^{12} +(-2.00000 - 3.46410i) q^{16} +(2.92586 + 1.68925i) q^{17} +(5.94975 - 10.3053i) q^{18} +(-1.21193 + 0.699709i) q^{19} +3.17157 q^{22} +(-8.27558 - 4.77791i) q^{24} +(2.50000 - 4.33013i) q^{25} -18.2919i q^{27} +(-2.82843 + 4.89898i) q^{32} +(6.56165 - 3.78837i) q^{33} -4.77791i q^{34} -16.8284 q^{36} +(1.71393 + 0.989538i) q^{38} +8.15640i q^{41} +13.0711 q^{43} +(-2.24264 - 3.88437i) q^{44} +13.5140i q^{48} -7.07107 q^{50} +(-5.70711 - 9.88500i) q^{51} +(-22.4029 + 12.9343i) q^{54} +4.72792 q^{57} +(9.48751 + 5.47762i) q^{59} +8.00000 q^{64} +(-9.27958 - 5.35757i) q^{66} +(4.24264 - 7.34847i) q^{67} +(-5.85172 + 3.37849i) q^{68} +(11.8995 + 20.6105i) q^{72} +(12.9154 + 7.45669i) q^{73} +(-14.6293 + 8.44623i) q^{75} -2.79884i q^{76} +(-18.2782 + 31.6587i) q^{81} +(9.98951 - 5.76745i) q^{82} +17.7122i q^{83} +(-9.24264 - 16.0087i) q^{86} +(-3.17157 + 5.49333i) q^{88} +(10.4915 - 6.05728i) q^{89} +(16.5512 - 9.55582i) q^{96} -15.7331i q^{97} -18.8701 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} + 28 q^{9}+O(q^{10}) \) 8 * q - 8 * q^4 + 28 * q^9	\( 8 q - 8 q^{4} + 28 q^{9} + 8 q^{11} - 16 q^{16} + 8 q^{18} + 48 q^{22} + 20 q^{25} - 112 q^{36} + 48 q^{43} + 16 q^{44} - 40 q^{51} - 64 q^{57} + 64 q^{64} + 16 q^{72} - 84 q^{81} - 40 q^{86} - 48 q^{88} + 64 q^{99}+O(q^{100}) \) 8 * q - 8 * q^4 + 28 * q^9 + 8 * q^11 - 16 * q^16 + 8 * q^18 + 48 * q^22 + 20 * q^25 - 112 * q^36 + 48 * q^43 + 16 * q^44 - 40 * q^51 - 64 * q^57 + 64 * q^64 + 16 * q^72 - 84 * q^81 - 40 * q^86 - 48 * q^88 + 64 * q^99

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