LMFDB - Embedded newform 117.3.bd.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,3,Mod(19,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("117.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	117.bd (of order $12$, degree $4$, 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:	$3.18801909302$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			19.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	117.19
Dual form			117.3.bd.a.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.46410 - 2.00000i) q^{4} +(1.42820 - 5.33013i) q^{7} +(12.9904 + 0.500000i) q^{13} +(8.00000 - 13.8564i) q^{16} +(-1.16987 - 0.313467i) q^{19} +25.0000i q^{25} +(-5.71281 - 21.3205i) q^{28} +(-36.8109 + 36.8109i) q^{31} +(-65.0788 + 17.4378i) q^{37} +(-19.5000 + 11.2583i) q^{43} +(16.0648 + 9.27499i) q^{49} +(46.0000 - 24.2487i) q^{52} +(56.2917 + 97.5000i) q^{61} -64.0000i q^{64} +(-22.0289 - 82.2128i) q^{67} +(-56.7776 - 56.7776i) q^{73} +(-4.67949 + 1.25387i) q^{76} +157.617 q^{79} +(21.2180 - 68.5263i) q^{91} +(180.495 + 48.3634i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{7} + 32 q^{16} - 22 q^{19} + 88 q^{28} - 26 q^{31} - 146 q^{37} - 78 q^{43} + 234 q^{49} + 184 q^{52} - 244 q^{67} - 286 q^{73} - 88 q^{76} + 362 q^{91} + 334 q^{97}+O(q^{100}) $ 4 * q - 22 * q^7 + 32 * q^16 - 22 * q^19 + 88 * q^28 - 26 * q^31 - 146 * q^37 - 78 * q^43 + 234 * q^49 + 184 * q^52 - 244 * q^67 - 286 * q^73 - 88 * q^76 + 362 * q^91 + 334 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{5}{12}\right)$	$1$

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		0.965926	−	0.258819i	$-0.0833333\pi$
$2$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$3$		0			0
$3$		0			0
$4$	3.46410	−	2.00000i	0.866025	−	0.500000i
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$	1.42820	−	5.33013i	0.204029	−	0.761447i	−0.785714	−	0.618590i	$-0.787704\pi$
$7$	1.42820	−	5.33013i	0.204029	−	0.761447i	0.989743	−	0.142857i	$-0.0456289\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$11$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$12$		0			0
$13$	12.9904	+	0.500000i	0.999260	+	0.0384615i
$13$	12.9904	+	0.500000i	0.999260	+	0.0384615i
$14$		0			0
$15$		0			0
$16$	8.00000	−	13.8564i	0.500000	−	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−1.16987	−	0.313467i	−0.0615723	−	0.0164982i	0.227901	−	0.973684i	$-0.426814\pi$
$19$	−1.16987	−	0.313467i	−0.0615723	−	0.0164982i	−0.289474	+	0.957186i	$0.593480\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$			25.0000i			1.00000i
$26$		0			0
$27$		0			0
$28$	−5.71281	−	21.3205i	−0.204029	−	0.761447i
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	−36.8109	+	36.8109i	−1.18745	+	1.18745i	−0.209677	+	0.977771i	$0.567241\pi$
$31$	−36.8109	+	36.8109i	−1.18745	+	1.18745i	−0.977771	+	0.209677i	$0.932759\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−65.0788	+	17.4378i	−1.75889	+	0.471292i	−0.986486	−	0.163843i	$-0.947611\pi$
$37$	−65.0788	+	17.4378i	−1.75889	+	0.471292i	−0.772401	+	0.635135i	$0.780944\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$41$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$42$		0			0
$43$	−19.5000	+	11.2583i	−0.453488	+	0.261822i	−0.709302	−	0.704904i	$-0.750990\pi$
$43$	−19.5000	+	11.2583i	−0.453488	+	0.261822i	0.255814	+	0.966726i	$0.417657\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$		0			0
$49$	16.0648	+	9.27499i	0.327852	+	0.189286i
$50$		0			0
$51$		0			0
$52$	46.0000	−	24.2487i	0.884615	−	0.466321i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$59$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$60$		0			0
$61$	56.2917	+	97.5000i	0.922814	+	1.59836i	0.795040	+	0.606557i	$0.207450\pi$
$61$	56.2917	+	97.5000i	0.922814	+	1.59836i	0.127774	+	0.991803i	$0.459217\pi$
$62$		0			0
$63$		0			0
$64$		−	64.0000i		−	1.00000i
$65$		0			0
$66$		0			0
$67$	−22.0289	−	82.2128i	−0.328789	−	1.22706i	−0.910448	−	0.413624i	$-0.864263\pi$
$67$	−22.0289	−	82.2128i	−0.328789	−	1.22706i	0.581659	−	0.813433i	$-0.302404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$71$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$72$		0			0
$73$	−56.7776	−	56.7776i	−0.777775	−	0.777775i	0.201677	−	0.979452i	$-0.435361\pi$
$73$	−56.7776	−	56.7776i	−0.777775	−	0.777775i	−0.979452	+	0.201677i	$0.935361\pi$
$74$		0			0
$75$		0			0
$76$	−4.67949	+	1.25387i	−0.0615723	+	0.0164982i
$77$		0			0
$78$		0			0
$79$	157.617			1.99515			0.997574	−	0.0696203i	$-0.0221788\pi$
$79$	157.617			1.99515			0.997574	+	0.0696203i	$0.0221788\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$89$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$90$		0			0
$91$	21.2180	−	68.5263i	0.233164	−	0.753036i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	180.495	+	48.3634i	1.86077	+	0.498592i	0.999947	−	0.0103093i	$-0.00328160\pi$
$97$	180.495	+	48.3634i	1.86077	+	0.498592i	0.860825	+	0.508902i	$0.169948\pi$
$98$		0			0
$99$		0			0
$100$	50.0000	+	86.6025i	0.500000	+	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		−	202.650i		−	1.96748i	−0.179612	−	0.983738i	$-0.557484\pi$
$103$		−	202.650i		−	1.96748i	0.179612	−	0.983738i	$-0.442516\pi$
$104$		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
$109$	−153.772	+	153.772i	−1.41076	+	1.41076i	−0.655963	+	0.754793i	$0.727737\pi$
$109$	−153.772	+	153.772i	−1.41076	+	1.41076i	−0.754793	+	0.655963i	$0.772263\pi$
$110$		0			0
$111$		0			0
$112$	−62.4308	−	62.4308i	−0.557418	−	0.557418i
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−104.789	+	60.5000i	−0.866025	+	0.500000i
$122$		0			0
$123$		0			0
$124$	−53.8949	+	201.138i	−0.434636	+	1.62208i
$125$		0			0
$126$		0			0
$127$	−219.104	−	126.500i	−1.72523	−	0.996063i	−0.906940	−	0.421260i	$-0.861588\pi$
$127$	−219.104	−	126.500i	−1.72523	−	0.996063i	−0.818292	−	0.574803i	$-0.805079\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−3.34163	+	5.78788i	−0.0251251	+	0.0435179i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$137$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$138$		0			0
$139$	114.500	+	198.320i	0.823741	+	1.42676i	0.902878	+	0.429898i	$0.141450\pi$
$139$	114.500	+	198.320i	0.823741	+	1.42676i	−0.0791367	+	0.996864i	$0.525216\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−190.564	+	190.564i	−1.28760	+	1.28760i
$149$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$149$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$150$		0			0
$151$	−191.497	−	191.497i	−1.26819	−	1.26819i	−0.947020	−	0.321175i	$-0.895922\pi$
$151$	−191.497	−	191.497i	−1.26819	−	1.26819i	−0.321175	−	0.947020i	$-0.604078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	193.000			1.22930			0.614650	−	0.788800i	$-0.289297\pi$
$157$	193.000			1.22930			0.614650	+	0.788800i	$0.289297\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	30.9468	−	115.495i	0.189857	−	0.708557i	−0.803681	−	0.595060i	$-0.797128\pi$
$163$	30.9468	−	115.495i	0.189857	−	0.708557i	0.993538	−	0.113497i	$-0.0362052\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$167$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$168$		0			0
$169$	168.500	+	12.9904i	0.997041	+	0.0768662i
$170$		0			0
$171$		0			0
$172$	−45.0333	+	78.0000i	−0.261822	+	0.453488i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	133.253	+	35.7051i	0.761447	+	0.204029i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		−	180.133i		−	0.995212i	−0.867403	−	0.497606i	$-0.834213\pi$
$181$		−	180.133i		−	0.995212i	0.867403	−	0.497606i	$-0.165787\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−147.213	+	39.4456i	−0.762761	+	0.204381i	−0.619171	−	0.785256i	$-0.712531\pi$
$193$	−147.213	+	39.4456i	−0.762761	+	0.204381i	−0.143590	+	0.989637i	$0.545865\pi$
$194$		0			0
$195$		0			0
$196$	74.1999			0.378571
$197$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$197$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$	94.3968	−	54.5000i	0.474356	−	0.273869i	−0.243706	−	0.969849i	$-0.578363\pi$
$199$	94.3968	−	54.5000i	0.474356	−	0.273869i	0.718061	+	0.695980i	$0.245030\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	110.851	−	176.000i	0.532939	−	0.846154i
$209$		0			0
$210$		0			0
$211$	168.875	−	292.500i	0.800355	−	1.38626i	−0.119027	−	0.992891i	$-0.537978\pi$
$211$	168.875	−	292.500i	0.800355	−	1.38626i	0.919382	−	0.393365i	$-0.128689\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	143.633	+	248.780i	0.661905	+	1.14645i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−115.112	−	429.604i	−0.516198	−	1.92648i	−0.330099	−	0.943946i	$-0.607082\pi$
$223$	−115.112	−	429.604i	−0.516198	−	1.92648i	−0.186099	−	0.982531i	$-0.559584\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$227$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$	215.631	+	215.631i	0.941619	+	0.941619i	0.998387	−	0.0567686i	$-0.0180797\pi$
$229$	215.631	+	215.631i	0.941619	+	0.941619i	−0.0567686	+	0.998387i	$0.518080\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	18.6635	−	69.6532i	0.0774420	−	0.289017i	−0.916334	−	0.400415i	$-0.868866\pi$
$241$	18.6635	−	69.6532i	0.0774420	−	0.289017i	0.993776	+	0.111397i	$0.0355327\pi$
$242$		0			0
$243$		0			0
$244$	390.000	+	225.167i	1.59836	+	0.922814i
$245$		0			0
$246$		0			0
$247$	−15.0404	−	4.65699i	−0.0608922	−	0.0188542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	−	221.703i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$			371.783i			1.43546i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−240.736	−	240.736i	−0.898268	−	0.898268i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−512.987	+	137.454i	−1.89294	+	0.507212i	−0.894786	+	0.446494i	$0.852672\pi$
$271$	−512.987	+	137.454i	−1.89294	+	0.507212i	−0.998155	+	0.0607176i	$0.980661\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−468.000	+	270.200i	−1.68953	+	0.975451i	−0.734657	+	0.678439i	$0.762657\pi$
$277$	−468.000	+	270.200i	−1.68953	+	0.975451i	−0.954874	+	0.297012i	$0.904010\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$	−51.0955	−	29.5000i	−0.180549	−	0.104240i	0.407001	−	0.913428i	$-0.366574\pi$
$283$	−51.0955	−	29.5000i	−0.180549	−	0.104240i	−0.587551	+	0.809187i	$0.699908\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−144.500	+	250.281i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$	−310.238	−	83.1281i	−1.06246	−	0.284685i
$293$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$293$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	32.1584	+	120.017i	0.106838	+	0.398726i
$302$		0			0
$303$		0			0
$304$	−13.7025	+	13.7025i	−0.0450740	+	0.0450740i
$305$		0			0
$306$		0			0
$307$	335.811	+	335.811i	1.09385	+	1.09385i	0.995114	+	0.0987325i	$0.0314788\pi$
$307$	335.811	+	335.811i	1.09385	+	1.09385i	0.0987325	+	0.995114i	$0.468521\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−427.817			−1.36683			−0.683413	−	0.730032i	$-0.739505\pi$
$313$	−427.817			−1.36683			−0.683413	+	0.730032i	$0.739505\pi$
$314$		0			0
$315$		0			0
$316$	546.000	−	315.233i	1.72785	−	0.997574i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−12.5000	+	324.760i	−0.0384615	+	0.999260i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−199.187	−	53.3719i	−0.601772	−	0.161244i	−0.0549442	−	0.998489i	$-0.517498\pi$
$331$	−199.187	−	53.3719i	−0.601772	−	0.161244i	−0.546828	+	0.837245i	$0.684165\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			167.000i			0.495549i	0.968818	+	0.247774i	$0.0796992\pi$
$337$			167.000i			0.495549i	−0.968818	+	0.247774i	$0.920301\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	263.575	−	263.575i	0.768440	−	0.768440i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	347.129	−	93.0129i	0.994638	−	0.266513i	0.275441	−	0.961318i	$-0.411176\pi$
$349$	347.129	−	93.0129i	0.994638	−	0.266513i	0.719198	+	0.694805i	$0.244510\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$353$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$	−311.365	−	179.767i	−0.862506	−	0.497968i
$362$		0			0
$363$		0			0
$364$	−63.5514	−	279.818i	−0.174592	−	0.768731i
$365$		0			0
$366$		0			0
$367$	113.500	−	196.588i	0.309264	−	0.535661i	−0.668937	−	0.743319i	$-0.733251\pi$
$367$	113.500	−	196.588i	0.309264	−	0.535661i	0.978202	+	0.207657i	$0.0665839\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−236.425	−	409.500i	−0.633847	−	1.09786i	−0.986758	−	0.162198i	$-0.948142\pi$
$373$	−236.425	−	409.500i	−0.633847	−	1.09786i	0.352911	−	0.935657i	$-0.385192\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	193.920	+	723.721i	0.511664	+	1.90955i	0.402165	+	0.915567i	$0.368258\pi$
$379$	193.920	+	723.721i	0.511664	+	1.90955i	0.109499	+	0.993987i	$0.465075\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$383$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	721.979	−	193.454i	1.86077	−	0.498592i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	152.420	−	568.838i	0.383929	−	1.43284i	−0.455919	−	0.890021i	$-0.650689\pi$
$397$	152.420	−	568.838i	0.383929	−	1.43284i	0.839848	−	0.542821i	$-0.182644\pi$
$398$		0			0
$399$		0			0
$400$	346.410	+	200.000i	0.866025	+	0.500000i
$401$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$401$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$402$		0			0
$403$	−496.593	+	459.782i	−1.23224	+	1.14090i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	647.772	+	173.570i	1.58379	+	0.424376i	0.940098	−	0.340905i	$-0.110733\pi$
$409$	647.772	+	173.570i	1.58379	+	0.424376i	0.643696	+	0.765281i	$0.277400\pi$
$410$		0			0
$411$		0			0
$412$	−405.300	−	702.000i	−0.983738	−	1.70388i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	586.044	−	586.044i	1.39203	−	1.39203i	0.571259	−	0.820770i	$-0.306455\pi$
$421$	586.044	−	586.044i	1.39203	−	1.39203i	0.820770	−	0.571259i	$-0.193545\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	600.083	−	160.792i	1.40535	−	0.376562i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$431$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$432$		0			0
$433$	310.903	−	179.500i	0.718021	−	0.414550i	−0.0960028	−	0.995381i	$-0.530606\pi$
$433$	310.903	−	179.500i	0.718021	−	0.414550i	0.814024	+	0.580831i	$0.197272\pi$
$434$		0			0
$435$		0			0
$436$	−225.138	+	840.228i	−0.516373	+	1.92713i
$437$		0			0
$438$		0			0
$439$	−448.500	−	258.942i	−1.02164	−	0.589844i	−0.107062	−	0.994252i	$-0.534144\pi$
$439$	−448.500	−	258.942i	−1.02164	−	0.589844i	−0.914579	+	0.404408i	$0.867478\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−341.128	−	91.4050i	−0.761447	−	0.204029i
$449$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$449$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	49.3954	+	184.346i	0.108086	+	0.403383i	0.998677	−	0.0514223i	$-0.0163755\pi$
$457$	49.3954	+	184.346i	0.108086	+	0.403383i	−0.890591	+	0.454805i	$0.849709\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$461$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$462$		0			0
$463$	424.261	+	424.261i	0.916330	+	0.916330i	0.996760	−	0.0804300i	$-0.0256293\pi$
$463$	424.261	+	424.261i	0.916330	+	0.916330i	−0.0804300	+	0.996760i	$0.525629\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−469.666			−1.00142
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	7.83666	−	29.2468i	0.0164982	−	0.0615723i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$479$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$480$		0			0
$481$	−854.118	+	193.985i	−1.77571	+	0.403294i
$482$		0			0
$483$		0			0
$484$	−242.000	+	419.156i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−761.163	−	203.953i	−1.56296	−	0.418795i	−0.629363	−	0.777111i	$-0.716684\pi$
$487$	−761.163	−	203.953i	−1.56296	−	0.418795i	−0.933600	+	0.358316i	$0.883351\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	215.580	+	804.554i	0.434636	+	1.62208i
$497$		0			0
$498$		0			0
$499$	511.831	−	511.831i	1.02571	−	1.02571i	0.0260521	−	0.999661i	$-0.491706\pi$
$499$	511.831	−	511.831i	1.02571	−	1.02571i	0.999661	−	0.0260521i	$-0.00829358\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1012.00			−1.99213
$509$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$509$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$510$		0			0
$511$	−383.722	+	221.542i	−0.750923	+	0.433545i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−491.000	+	850.437i	−0.938815	+	1.62607i	−0.171128	+	0.985249i	$0.554741\pi$
$523$	−491.000	+	850.437i	−0.938815	+	1.62607i	−0.767686	+	0.640826i	$0.778592\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	−	458.127i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$			26.7331i			0.0502501i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	28.4392	+	28.4392i	0.0525678	+	0.0525678i	0.732902	−	0.680334i	$-0.238165\pi$
$541$	28.4392	+	28.4392i	0.0525678	+	0.0525678i	−0.680334	+	0.732902i	$0.738165\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	587.000			1.07313			0.536563	−	0.843860i	$-0.319722\pi$
$547$	587.000			1.07313			0.536563	+	0.843860i	$0.319722\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	225.109	−	840.117i	0.407068	−	1.51920i
$554$		0			0
$555$		0			0
$556$	793.279	+	458.000i	1.42676	+	0.823741i
$557$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$557$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$558$		0			0
$559$	−258.942	+	136.500i	−0.463223	+	0.244186i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$			886.000i			1.55166i	0.630940	+	0.775832i	$0.282670\pi$
$571$			886.000i			1.55166i	−0.630940	+	0.775832i	$0.717330\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	799.697	−	799.697i	1.38596	−	1.38596i	0.552335	−	0.833622i	$-0.313737\pi$
$577$	799.697	−	799.697i	1.38596	−	1.38596i	0.833622	−	0.552335i	$-0.186263\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$587$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$588$		0			0
$589$	54.6030	−	31.5251i	0.0927047	−	0.0535231i
$590$		0			0
$591$		0			0
$592$	−279.005	+	1041.26i	−0.471292	+	1.75889i
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	540.400	−	936.000i	0.899168	−	1.55740i	0.0706077	−	0.997504i	$-0.477506\pi$
$601$	540.400	−	936.000i	0.899168	−	1.55740i	0.828560	−	0.559900i	$-0.189161\pi$
$602$		0			0
$603$		0			0
$604$	−1046.36	−	280.372i	−1.73239	−	0.464192i
$605$		0			0
$606$		0			0
$607$	−407.000	−	704.945i	−0.670511	−	1.16136i	−0.977759	−	0.209729i	$-0.932742\pi$
$607$	−407.000	−	704.945i	−0.670511	−	1.16136i	0.307249	−	0.951629i	$-0.400592\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−196.671	−	733.987i	−0.320834	−	1.19737i	−0.918434	−	0.395574i	$-0.870546\pi$
$613$	−196.671	−	733.987i	−0.320834	−	1.19737i	0.597600	−	0.801794i	$-0.296121\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$617$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$618$		0			0
$619$	872.006	+	872.006i	1.40873	+	1.40873i	0.766559	+	0.642174i	$0.221967\pi$
$619$	872.006	+	872.006i	1.40873	+	1.40873i	0.642174	+	0.766559i	$0.278033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−625.000			−1.00000
$626$		0			0
$627$		0			0
$628$	668.572	−	386.000i	1.06460	−	0.614650i
$629$		0			0
$630$		0			0
$631$	221.586	−	826.972i	0.351167	−	1.31057i	−0.534073	−	0.845438i	$-0.679339\pi$
$631$	221.586	−	826.972i	0.351167	−	1.31057i	0.885240	−	0.465135i	$-0.153994\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	204.050	+	128.518i	0.320329	+	0.201755i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	1242.04	+	332.803i	1.93163	+	0.517579i	0.969733	+	0.244168i	$0.0785148\pi$
$643$	1242.04	+	332.803i	1.93163	+	0.517579i	0.961897	+	0.273411i	$0.0881518\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−123.787	−	461.979i	−0.189857	−	0.708557i
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−215.262	+	57.6793i	−0.325661	+	0.0872607i	−0.417946	−	0.908472i	$-0.637250\pi$
$661$	−215.262	+	57.6793i	−0.325661	+	0.0872607i	0.0922844	+	0.995733i	$0.470583\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	565.500	+	326.492i	0.840267	+	0.485129i	0.857355	−	0.514725i	$-0.172106\pi$
$673$	565.500	+	326.492i	0.840267	+	0.485129i	−0.0170877	+	0.999854i	$0.505439\pi$
$674$		0			0
$675$		0			0
$676$	609.682	−	292.000i	0.901896	−	0.431953i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	515.567	−	892.988i	0.759303	−	1.31515i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$683$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$			360.267i			0.523643i
$689$		0			0
$690$		0			0
$691$	301.654	+	1125.79i	0.436547	+	1.62922i	0.737337	+	0.675525i	$0.236083\pi$
$691$	301.654	+	1125.79i	0.436547	+	1.62922i	−0.300790	+	0.953690i	$0.597250\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	533.013	−	142.820i	0.761447	−	0.204029i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	81.6001			0.116074
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	304.972	−	1138.17i	0.430143	−	1.60532i	−0.322285	−	0.946643i	$-0.604451\pi$
$709$	304.972	−	1138.17i	0.430143	−	1.60532i	0.752428	−	0.658674i	$-0.228882\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−1080.15	−	289.425i	−1.49813	−	0.401422i
$722$		0			0
$723$		0			0
$724$	−360.267	−	624.000i	−0.497606	−	0.861878i
$725$		0			0
$726$		0			0
$727$			947.000i			1.30261i	0.758815	+	0.651307i	$0.225779\pi$
$727$			947.000i			1.30261i	−0.758815	+	0.651307i	$0.774221\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−896.428	+	896.428i	−1.22296	+	1.22296i	−0.256381	+	0.966576i	$0.582530\pi$
$733$	−896.428	+	896.428i	−1.22296	+	1.22296i	−0.966576	+	0.256381i	$0.917470\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1402.49	+	375.795i	−1.89782	+	0.508519i	−0.900541	+	0.434771i	$0.856829\pi$
$739$	−1402.49	+	375.795i	−1.89782	+	0.508519i	−0.997277	+	0.0737483i	$0.976504\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$743$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−780.000	−	450.333i	−1.03862	−	0.599645i	−0.119174	−	0.992873i	$-0.538025\pi$
$751$	−780.000	−	450.333i	−1.03862	−	0.599645i	−0.919441	+	0.393229i	$0.871358\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−630.466	+	1092.00i	−0.832849	+	1.44254i	0.0629213	+	0.998018i	$0.479958\pi$
$757$	−630.466	+	1092.00i	−0.832849	+	1.44254i	−0.895770	+	0.444518i	$0.853375\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$761$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$762$		0			0
$763$	600.008	+	1039.24i	0.786380	+	1.36205i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−260.454	−	972.029i	−0.338692	−	1.26402i	−0.899811	−	0.436281i	$-0.856295\pi$
$769$	−260.454	−	972.029i	−0.338692	−	1.26402i	0.561118	−	0.827736i	$-0.310371\pi$
$770$		0			0
$771$		0			0
$772$	−431.069	+	431.069i	−0.558380	+	0.558380i
$773$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$773$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$774$		0			0
$775$	−920.272	−	920.272i	−1.18745	−	1.18745i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	257.036	−	148.400i	0.327852	−	0.189286i
$785$		0			0
$786$		0			0
$787$	−403.495	+	1505.86i	−0.512700	+	1.91342i	−0.123246	+	0.992376i	$0.539331\pi$
$787$	−403.495	+	1505.86i	−0.512700	+	1.91342i	−0.389454	+	0.921046i	$0.627336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	682.500	+	1294.71i	0.860656	+	1.63267i
$794$		0			0
$795$		0			0
$796$	218.000	−	377.587i	0.273869	−	0.474356i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$	120.411	−	120.411i	0.148472	−	0.148472i	−0.628963	−	0.777435i	$-0.716520\pi$
$811$	120.411	−	120.411i	0.148472	−	0.148472i	0.777435	+	0.628963i	$0.216520\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	26.3416	−	7.05822i	0.0322419	−	0.00863919i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$821$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$822$		0			0
$823$	−1092.00	+	630.466i	−1.32685	+	0.766059i	−0.984812	−	0.173626i	$-0.944452\pi$
$823$	−1092.00	+	630.466i	−1.32685	+	0.766059i	−0.342041	+	0.939685i	$0.611118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	−996.795	−	575.500i	−1.20241	−	0.694210i	−0.241317	−	0.970446i	$-0.577579\pi$
$829$	−996.795	−	575.500i	−1.20241	−	0.694210i	−0.961090	+	0.276236i	$0.910913\pi$
$830$		0			0
$831$		0			0
$832$	32.0000	−	831.384i	0.0384615	−	0.999260i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$839$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$840$		0			0
$841$	420.500	+	728.327i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		−	1351.00i		−	1.60071i
$845$		0			0
$846$		0			0
$847$	172.813	+	644.945i	0.204029	+	0.761447i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	577.894	+	577.894i	0.677484	+	0.677484i	0.959430	−	0.281946i	$-0.0909799\pi$
$853$	577.894	+	577.894i	0.677484	+	0.677484i	−0.281946	+	0.959430i	$0.590980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	743.050			0.865017			0.432509	−	0.901630i	$-0.357629\pi$
$859$	743.050			0.865017			0.432509	+	0.901630i	$0.357629\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	995.121	+	574.533i	1.14645	+	0.661905i
$869$		0			0
$870$		0			0
$871$	−245.057	−	1078.99i	−0.281351	−	1.23879i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1534.67	+	411.214i	1.74991	+	0.468887i	0.984607	−	0.174785i	$-0.0559231\pi$
$877$	1534.67	+	411.214i	1.74991	+	0.468887i	0.765302	+	0.643672i	$0.222590\pi$
$878$		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
$883$		−	1259.00i		−	1.42582i	−0.701255	−	0.712911i	$-0.747377\pi$
$883$		−	1259.00i		−	1.42582i	0.701255	−	0.712911i	$-0.252623\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−987.187	+	987.187i	−1.11045	+	1.11045i
$890$		0			0
$891$		0			0
$892$	−1257.97	−	1257.97i	−1.41028	−	1.41028i
$893$		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
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−185.329	−	107.000i	−0.204332	−	0.117971i	0.394342	−	0.918964i	$-0.370972\pi$
$907$	−185.329	−	107.000i	−0.204332	−	0.117971i	−0.598675	+	0.800992i	$0.704306\pi$
$908$		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
$916$	1178.23	+	315.705i	1.28628	+	0.344656i
$917$		0			0
$918$		0			0
$919$	810.600	+	1404.00i	0.882045	+	1.52775i	0.849063	+	0.528292i	$0.177167\pi$
$919$	810.600	+	1404.00i	0.882045	+	1.52775i	0.0329825	+	0.999456i	$0.489499\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−435.946	−	1626.97i	−0.471292	−	1.75889i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$929$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$930$		0			0
$931$	−15.8863	−	15.8863i	−0.0170637	−	0.0170637i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1441.07			1.53796			0.768979	−	0.639274i	$-0.220765\pi$
$937$	1441.07			1.53796			0.768979	+	0.639274i	$0.220765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$947$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$948$		0			0
$949$	−709.173	−	765.951i	−0.747285	−	0.807114i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	1749.08i		−	1.82007i
$962$		0			0
$963$		0			0
$964$	−74.6541	−	278.613i	−0.0774420	−	0.289017i
$965$		0			0
$966$		0			0
$967$	178.103	−	178.103i	0.184181	−	0.184181i	−0.608994	−	0.793175i	$-0.708427\pi$
$967$	178.103	−	178.103i	0.184181	−	0.184181i	0.793175	+	0.608994i	$0.208427\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$	1220.60	−	327.059i	1.25447	−	0.336134i
$974$		0			0
$975$		0			0
$976$	1801.33			1.84563
$977$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$977$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−61.4153	+	13.9485i	−0.0621613	+	0.0141179i
$989$		0			0
$990$		0			0
$991$	23.0000	−	39.8372i	0.0232089	−	0.0401990i	−0.854188	−	0.519965i	$-0.825945\pi$
$991$	23.0000	−	39.8372i	0.0232089	−	0.0401990i	0.877397	+	0.479766i	$0.159278\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	743.500	+	1287.78i	0.745737	+	1.29165i	0.949850	+	0.312707i	$0.101236\pi$
$997$	743.500	+	1287.78i	0.745737	+	1.29165i	−0.204112	+	0.978947i	$0.565431\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.3.bd.a.19.1	✓	4
3.2	odd	2	CM	117.3.bd.a.19.1	✓	4
13.11	odd	12	inner	117.3.bd.a.37.1	yes	4
39.11	even	12	inner	117.3.bd.a.37.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.3.bd.a.19.1	✓	4	1.1	even	1	trivial
117.3.bd.a.19.1	✓	4	3.2	odd	2	CM
117.3.bd.a.37.1	yes	4	13.11	odd	12	inner
117.3.bd.a.37.1	yes	4	39.11	even	12	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	117.bd (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(3.46410 - 2.00000i) q^{4} +(1.42820 - 5.33013i) q^{7} +(12.9904 + 0.500000i) q^{13} +(8.00000 - 13.8564i) q^{16} +(-1.16987 - 0.313467i) q^{19} +25.0000i q^{25} +(-5.71281 - 21.3205i) q^{28} +(-36.8109 + 36.8109i) q^{31} +(-65.0788 + 17.4378i) q^{37} +(-19.5000 + 11.2583i) q^{43} +(16.0648 + 9.27499i) q^{49} +(46.0000 - 24.2487i) q^{52} +(56.2917 + 97.5000i) q^{61} -64.0000i q^{64} +(-22.0289 - 82.2128i) q^{67} +(-56.7776 - 56.7776i) q^{73} +(-4.67949 + 1.25387i) q^{76} +157.617 q^{79} +(21.2180 - 68.5263i) q^{91} +(180.495 + 48.3634i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{7} + 32 q^{16} - 22 q^{19} + 88 q^{28} - 26 q^{31} - 146 q^{37} - 78 q^{43} + 234 q^{49} + 184 q^{52} - 244 q^{67} - 286 q^{73} - 88 q^{76} + 362 q^{91} + 334 q^{97}+O(q^{100}) \) 4 * q - 22 * q^7 + 32 * q^16 - 22 * q^19 + 88 * q^28 - 26 * q^31 - 146 * q^37 - 78 * q^43 + 234 * q^49 + 184 * q^52 - 244 * q^67 - 286 * q^73 - 88 * q^76 + 362 * q^91 + 334 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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