LMFDB - Embedded newform 684.3.y.c.145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("684.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, 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:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			145.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	684.145
Dual form			684.3.y.c.217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.0000 q^{7} +O(q^{10})$	$q-11.0000 q^{7} +(22.5000 + 12.9904i) q^{13} +(-13.0000 - 13.8564i) q^{19} +(12.5000 - 21.6506i) q^{25} -60.6218i q^{31} -57.1577i q^{37} +(-41.5000 - 71.8801i) q^{43} +72.0000 q^{49} +(-60.5000 + 104.789i) q^{61} +(115.500 + 66.6840i) q^{67} +(-71.5000 - 123.842i) q^{73} +(76.5000 - 44.1673i) q^{79} +(-247.500 - 142.894i) q^{91} +(168.000 - 96.9948i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 22 q^{7}+O(q^{10}) $ 2 * q - 22 * q^7	$ 2 q - 22 q^{7} + 45 q^{13} - 26 q^{19} + 25 q^{25} - 83 q^{43} + 144 q^{49} - 121 q^{61} + 231 q^{67} - 143 q^{73} + 153 q^{79} - 495 q^{91} + 336 q^{97}+O(q^{100}) $ 2 * q - 22 * q^7 + 45 * q^13 - 26 * q^19 + 25 * q^25 - 83 * q^43 + 144 * q^49 - 121 * q^61 + 231 * q^67 - 143 * q^73 + 153 * q^79 - 495 * q^91 + 336 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−11.0000			−1.57143			−0.785714	−	0.618590i	$-0.787704\pi$
$7$	−11.0000			−1.57143			−0.785714	+	0.618590i	$0.787704\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	22.5000	+	12.9904i	1.73077	+	0.999260i	0.884615	+	0.466321i	$0.154421\pi$
$13$	22.5000	+	12.9904i	1.73077	+	0.999260i	0.846154	+	0.532939i	$0.178912\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−13.0000	−	13.8564i	−0.684211	−	0.729285i
$19$	−13.0000	−	13.8564i	−0.684211	−	0.729285i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	12.5000	−	21.6506i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		−	60.6218i		−	1.95554i	−0.209677	−	0.977771i	$-0.567241\pi$
$31$		−	60.6218i		−	1.95554i	0.209677	−	0.977771i	$-0.432759\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	57.1577i		−	1.54480i	−0.635135	−	0.772401i	$-0.719056\pi$
$37$		−	57.1577i		−	1.54480i	0.635135	−	0.772401i	$-0.280944\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	−41.5000	−	71.8801i	−0.965116	−	1.67163i	−0.709302	−	0.704904i	$-0.750990\pi$
$43$	−41.5000	−	71.8801i	−0.965116	−	1.67163i	−0.255814	−	0.966726i	$-0.582343\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	72.0000			1.46939
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−60.5000	+	104.789i	−0.991803	+	1.71785i	−0.385246	+	0.922814i	$0.625883\pi$
$61$	−60.5000	+	104.789i	−0.991803	+	1.71785i	−0.606557	+	0.795040i	$0.707450\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	115.500	+	66.6840i	1.72388	+	0.995283i	0.910448	+	0.413624i	$0.135737\pi$
$67$	115.500	+	66.6840i	1.72388	+	0.995283i	0.813433	+	0.581659i	$0.197596\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	−71.5000	−	123.842i	−0.979452	−	1.69646i	−0.664384	−	0.747392i	$-0.731306\pi$
$73$	−71.5000	−	123.842i	−0.979452	−	1.69646i	−0.315068	−	0.949069i	$-0.602027\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	76.5000	−	44.1673i	0.968354	−	0.559080i	0.0696203	−	0.997574i	$-0.477821\pi$
$79$	76.5000	−	44.1673i	0.968354	−	0.559080i	0.898734	+	0.438494i	$0.144488\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−247.500	−	142.894i	−2.71978	−	1.57027i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	168.000	−	96.9948i	1.73196	−	0.999947i	0.860825	−	0.508902i	$-0.169948\pi$
$97$	168.000	−	96.9948i	1.73196	−	0.999947i	0.871134	−	0.491045i	$-0.163385\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$		−	133.368i		−	1.29483i	−0.762136	−	0.647417i	$-0.775849\pi$
$103$		−	133.368i		−	1.29483i	0.762136	−	0.647417i	$-0.224151\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	36.0000	−	20.7846i	0.330275	−	0.190684i	−0.325688	−	0.945477i	$-0.605596\pi$
$109$	36.0000	−	20.7846i	0.330275	−	0.190684i	0.655963	+	0.754793i	$0.272263\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−121.000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	180.000	+	103.923i	1.41732	+	0.818292i	0.996063	−	0.0886483i	$-0.0282547\pi$
$127$	180.000	+	103.923i	1.41732	+	0.818292i	0.421260	+	0.906940i	$0.361588\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	143.000	+	152.420i	1.07519	+	1.14602i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	−114.500	+	198.320i	−0.823741	+	1.42676i	0.0791367	+	0.996864i	$0.474784\pi$
$139$	−114.500	+	198.320i	−0.823741	+	1.42676i	−0.902878	+	0.429898i	$0.858550\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$		−	96.9948i		−	0.642350i	−0.947020	−	0.321175i	$-0.895922\pi$
$151$		−	96.9948i		−	0.642350i	0.947020	−	0.321175i	$-0.104078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−96.5000	−	167.143i	−0.614650	−	1.06460i	−0.990446	−	0.137902i	$-0.955964\pi$
$157$	−96.5000	−	167.143i	−0.614650	−	1.06460i	0.375796	−	0.926702i	$-0.377369\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−299.000			−1.83436			−0.917178	−	0.398478i	$-0.869539\pi$
$163$	−299.000			−1.83436			−0.917178	+	0.398478i	$0.869539\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	253.000	+	438.209i	1.49704	+	2.59295i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−137.500	+	238.157i	−0.785714	+	1.36090i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	156.000	+	90.0666i	0.861878	+	0.497606i	0.864641	−	0.502390i	$-0.167546\pi$
$181$	156.000	+	90.0666i	0.861878	+	0.497606i	−0.00276243	+	0.999996i	$0.500879\pi$
$182$		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			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−262.500	+	151.554i	−1.36010	+	0.785256i	−0.989637	−	0.143590i	$-0.954135\pi$
$193$	−262.500	+	151.554i	−1.36010	+	0.785256i	−0.370466	+	0.928846i	$0.620802\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	54.5000	−	94.3968i	0.273869	−	0.474356i	−0.695980	−	0.718061i	$-0.745030\pi$
$199$	54.5000	−	94.3968i	0.273869	−	0.474356i	0.969849	+	0.243706i	$0.0783631\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	43.5000	−	25.1147i	0.206161	−	0.119027i	−0.393365	−	0.919382i	$-0.628689\pi$
$211$	43.5000	−	25.1147i	0.206161	−	0.119027i	0.599526	+	0.800355i	$0.295356\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			666.840i			3.07299i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−127.500	+	73.6122i	−0.571749	+	0.330099i	−0.757848	−	0.652432i	$-0.773749\pi$
$223$	−127.500	+	73.6122i	−0.571749	+	0.330099i	0.186099	+	0.982531i	$0.440416\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	409.000			1.78603			0.893013	−	0.450031i	$-0.148587\pi$
$229$	409.000			1.78603			0.893013	+	0.450031i	$0.148587\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−46.5000	−	26.8468i	−0.192946	−	0.111397i	0.400415	−	0.916334i	$-0.368866\pi$
$241$	−46.5000	−	26.8468i	−0.192946	−	0.111397i	−0.593361	+	0.804936i	$0.702199\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−112.500	−	480.644i	−0.455466	−	1.94593i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$251$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$			628.734i			2.42755i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	−121.000	−	209.578i	−0.446494	−	0.773351i	0.551661	−	0.834069i	$-0.313994\pi$
$271$	−121.000	−	209.578i	−0.446494	−	0.773351i	−0.998155	+	0.0607176i	$0.980661\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−122.000			−0.440433			−0.220217	−	0.975451i	$-0.570676\pi$
$277$	−122.000			−0.440433			−0.220217	+	0.975451i	$0.570676\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	−229.000	−	396.640i	−0.809187	−	1.40155i	−0.913428	−	0.407001i	$-0.866574\pi$
$283$	−229.000	−	396.640i	−0.809187	−	1.40155i	0.104240	−	0.994552i	$-0.466759\pi$
$284$		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
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	456.500	+	790.681i	1.51661	+	2.62685i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−432.000	+	249.415i	−1.40717	+	0.812428i	−0.995114	−	0.0987325i	$-0.968521\pi$
$307$	−432.000	+	249.415i	−1.40717	+	0.812428i	−0.412052	+	0.911160i	$0.635188\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−71.0000	+	122.976i	−0.226837	+	0.392893i	−0.956869	−	0.290520i	$-0.906172\pi$
$313$	−71.0000	+	122.976i	−0.226837	+	0.392893i	0.730032	+	0.683413i	$0.239505\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	562.500	−	324.760i	1.73077	−	0.999260i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			36.3731i			0.109888i	0.998489	+	0.0549442i	$0.0174981\pi$
$331$			36.3731i			0.109888i	−0.998489	+	0.0549442i	$0.982502\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−565.500	+	326.492i	−1.67804	+	0.968818i	−0.715134	+	0.698988i	$0.753634\pi$
$337$	−565.500	+	326.492i	−1.67804	+	0.968818i	−0.962908	+	0.269830i	$0.913033\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−253.000			−0.737609
$344$		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
$349$	671.000			1.92264			0.961318	−	0.275441i	$-0.0888238\pi$
$349$	671.000			1.92264			0.961318	+	0.275441i	$0.0888238\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		0			0
$361$	−23.0000	+	360.267i	−0.0637119	+	0.997968i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−245.500	+	425.218i	−0.668937	+	1.15863i	0.309264	+	0.950976i	$0.399917\pi$
$367$	−245.500	+	425.218i	−0.668937	+	1.15863i	−0.978202	+	0.207657i	$0.933416\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	263.272i		−	0.705822i	−0.935657	−	0.352911i	$-0.885192\pi$
$373$		−	263.272i		−	0.705822i	0.935657	−	0.352911i	$-0.114808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	448.601i		−	1.18364i	−0.806069	−	0.591822i	$-0.798409\pi$
$379$		−	448.601i		−	1.18364i	0.806069	−	0.591822i	$-0.201591\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−215.500	−	373.257i	−0.542821	−	0.940194i	−0.998741	−	0.0501728i	$-0.984023\pi$
$397$	−215.500	−	373.257i	−0.542821	−	0.940194i	0.455919	−	0.890021i	$-0.349311\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	787.500	−	1363.99i	1.95409	−	3.38459i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−456.000	−	263.272i	−1.11491	−	0.643696i	−0.174817	−	0.984601i	$-0.555933\pi$
$409$	−456.000	−	263.272i	−1.11491	−	0.643696i	−0.940098	+	0.340905i	$0.889267\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	660.000	−	381.051i	1.56770	−	0.905110i	0.571259	−	0.820770i	$-0.306455\pi$
$421$	660.000	−	381.051i	1.56770	−	0.905110i	0.996437	−	0.0843398i	$-0.0268781\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	665.500	−	1152.68i	1.55855	−	2.69948i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	−610.500	−	352.472i	−1.40993	−	0.814024i	−0.414550	−	0.910027i	$-0.636061\pi$
$433$	−610.500	−	352.472i	−1.40993	−	0.814024i	−0.995381	+	0.0960028i	$0.969394\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−448.500	+	258.942i	−1.02164	+	0.589844i	−0.914579	−	0.404408i	$-0.867478\pi$
$439$	−448.500	+	258.942i	−1.02164	+	0.589844i	−0.107062	+	0.994252i	$0.534144\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	767.000			1.67834			0.839168	−	0.543872i	$-0.183042\pi$
$457$	767.000			1.67834			0.839168	+	0.543872i	$0.183042\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$	−397.000			−0.857451			−0.428726	−	0.903435i	$-0.641037\pi$
$463$	−397.000			−0.857451			−0.428726	+	0.903435i	$0.641037\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−1270.50	−	733.524i	−2.70896	−	1.56402i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−462.500	+	108.253i	−0.973684	+	0.227901i
$476$		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
$481$	742.500	−	1286.05i	1.54366	−	2.67370i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			152.420i			0.312978i	0.987680	+	0.156489i	$0.0500176\pi$
$487$			152.420i			0.312978i	−0.987680	+	0.156489i	$0.949982\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	425.500	+	736.988i	0.852705	+	1.47693i	0.878758	+	0.477269i	$0.158373\pi$
$499$	425.500	+	736.988i	0.852705	+	1.47693i	−0.0260521	+	0.999661i	$0.508294\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	786.500	+	1362.26i	1.53914	+	2.66587i
$512$		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		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	580.500	+	335.152i	1.10994	+	0.640826i	0.938815	−	0.344423i	$-0.111925\pi$
$523$	580.500	+	335.152i	1.10994	+	0.640826i	0.171128	+	0.985249i	$0.445259\pi$
$524$		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
$532$		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
$541$	120.500	−	208.712i	0.222736	−	0.385790i	−0.732902	−	0.680334i	$-0.761835\pi$
$541$	120.500	−	208.712i	0.222736	−	0.385790i	0.955638	+	0.294545i	$0.0951680\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	40.5000	+	23.3827i	0.0740402	+	0.0427471i	0.536563	−	0.843860i	$-0.319722\pi$
$547$	40.5000	+	23.3827i	0.0740402	+	0.0427471i	−0.462523	+	0.886607i	$0.653056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−841.500	+	485.840i	−1.52170	+	0.878554i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		−	2156.40i		−	3.85761i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−181.000			−0.316988			−0.158494	−	0.987360i	$-0.550664\pi$
$571$	−181.000			−0.316988			−0.158494	+	0.987360i	$0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−962.000			−1.66724			−0.833622	−	0.552335i	$-0.813737\pi$
$577$	−962.000			−1.66724			−0.833622	+	0.552335i	$0.813737\pi$
$578$		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		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$	−840.000	+	788.083i	−1.42615	+	1.33800i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		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	$0.666667\pi$
$600$		0			0
$601$		−	84.8705i		−	0.141215i	−0.997504	−	0.0706077i	$-0.977506\pi$
$601$		−	84.8705i		−	0.141215i	0.997504	−	0.0706077i	$-0.0224939\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	1155.28i		−	1.90326i	−0.307249	−	0.951629i	$-0.599408\pi$
$607$		−	1155.28i		−	1.90326i	0.307249	−	0.951629i	$-0.400592\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−563.000	−	975.145i	−0.918434	−	1.59077i	−0.801794	−	0.597600i	$-0.796121\pi$
$613$	−563.000	−	975.145i	−0.918434	−	1.59077i	−0.116639	−	0.993174i	$-0.537212\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$	949.000			1.53312			0.766559	−	0.642174i	$-0.221967\pi$
$619$	949.000			1.53312			0.766559	+	0.642174i	$0.221967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	293.500	−	508.357i	0.465135	−	0.805637i	−0.534073	−	0.845438i	$-0.679339\pi$
$631$	293.500	−	508.357i	0.465135	−	0.805637i	0.999208	+	0.0398015i	$0.0126726\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1620.00	+	935.307i	2.54317	+	1.46830i
$638$		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
$643$	618.500	+	1071.27i	0.961897	+	1.66606i	0.717729	+	0.696322i	$0.245181\pi$
$643$	618.500	+	1071.27i	0.961897	+	1.66606i	0.244168	+	0.969733i	$0.421485\pi$
$644$		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			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	1140.00	+	658.179i	1.72466	+	0.995733i	0.908472	+	0.417946i	$0.137250\pi$
$661$	1140.00	+	658.179i	1.72466	+	0.995733i	0.816188	+	0.577787i	$0.196084\pi$
$662$		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
$673$		−	652.983i		−	0.970257i	−0.874443	−	0.485129i	$-0.838773\pi$
$673$		−	652.983i		−	0.970257i	0.874443	−	0.485129i	$-0.161227\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−1848.00	+	1066.94i	−2.72165	+	1.57135i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1318.00			−1.90738			−0.953690	−	0.300790i	$-0.902750\pi$
$691$	−1318.00			−1.90738			−0.953690	+	0.300790i	$0.902750\pi$
$692$		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		0.866025	−	0.500000i	$-0.166667\pi$
$701$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$702$		0			0
$703$	−792.000	+	743.050i	−1.12660	+	1.05697i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	228.500	−	395.774i	0.322285	−	0.558214i	−0.658674	−	0.752428i	$-0.728882\pi$
$709$	228.500	−	395.774i	0.322285	−	0.558214i	0.980959	+	0.194214i	$0.0622158\pi$
$710$		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		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$			1467.05i			2.03474i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	714.500	+	1237.55i	0.982806	+	1.70227i	0.651307	+	0.758815i	$0.274221\pi$
$727$	714.500	+	1237.55i	0.982806	+	1.70227i	0.331499	+	0.943455i	$0.392446\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1034.00			1.41064			0.705321	−	0.708888i	$-0.250803\pi$
$733$	1034.00			1.41064			0.705321	+	0.708888i	$0.250803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−54.5000	−	94.3968i	−0.0737483	−	0.127736i	0.826793	−	0.562506i	$-0.190163\pi$
$739$	−54.5000	−	94.3968i	−0.0737483	−	0.127736i	−0.900541	+	0.434771i	$0.856829\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1291.50	+	745.648i	1.71971	+	0.992873i	0.919441	+	0.393229i	$0.128642\pi$
$751$	1291.50	+	745.648i	1.71971	+	0.992873i	0.800266	+	0.599645i	$0.204691\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−336.500	−	582.835i	−0.444518	−	0.769927i	0.553501	−	0.832849i	$-0.313292\pi$
$757$	−336.500	−	582.835i	−0.444518	−	0.769927i	−0.998018	+	0.0629213i	$0.979958\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−396.000	+	228.631i	−0.519004	+	0.299647i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	431.500	−	747.380i	0.561118	−	0.971885i	−0.436281	−	0.899811i	$-0.643705\pi$
$769$	431.500	−	747.380i	0.561118	−	0.971885i	0.997399	−	0.0720749i	$-0.0229621\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	−1312.50	−	757.772i	−1.69355	−	0.977771i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1449.73i			1.84209i	0.389454	+	0.921046i	$0.372664\pi$
$787$			1449.73i			1.84209i	−0.389454	+	0.921046i	$0.627336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2722.50	+	1571.84i	−3.43317	+	1.98214i
$794$		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			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	504.000	+	290.985i	0.621455	+	0.358797i	0.777435	−	0.628963i	$-0.216520\pi$
$811$	504.000	+	290.985i	0.621455	+	0.358797i	−0.155980	+	0.987760i	$0.549854\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−456.500	+	1509.48i	−0.558752	+	1.84759i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	529.000	−	916.255i	0.642770	−	1.11331i	−0.342041	−	0.939685i	$-0.611118\pi$
$823$	529.000	−	916.255i	0.642770	−	1.11331i	0.984812	−	0.173626i	$-0.0555484\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$827$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$828$		0			0
$829$			1193.38i			1.43955i	0.694210	+	0.719773i	$0.255754\pi$
$829$			1193.38i			1.43955i	−0.694210	+	0.719773i	$0.744246\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1331.00			1.57143
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	588.500	+	1019.31i	0.689918	+	1.19497i	0.971864	+	0.235543i	$0.0756867\pi$
$853$	588.500	+	1019.31i	0.689918	+	1.19497i	−0.281946	+	0.959430i	$0.590980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	−774.500	+	1341.47i	−0.901630	+	1.56167i	−0.0762515	+	0.997089i	$0.524295\pi$
$859$	−774.500	+	1341.47i	−0.901630	+	1.56167i	−0.825378	+	0.564580i	$0.809038\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1732.50	+	3000.78i	1.98909	+	3.44521i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	265.500	−	153.286i	0.302737	−	0.174785i	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	265.500	−	153.286i	0.302737	−	0.174785i	0.643672	+	0.765302i	$0.277410\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	221.500	−	383.649i	0.250849	−	0.434484i	−0.712911	−	0.701255i	$-0.752623\pi$
$883$	221.500	−	383.649i	0.250849	−	0.434484i	0.963760	+	0.266771i	$0.0859568\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−1980.00	−	1143.15i	−2.22722	−	1.28589i
$890$		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
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1560.00	−	900.666i	1.71996	−	0.993017i	0.800992	−	0.598675i	$-0.204306\pi$
$907$	1560.00	−	900.666i	1.71996	−	0.993017i	0.918964	−	0.394342i	$-0.129028\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1837.00			1.99891			0.999456	−	0.0329825i	$-0.0105006\pi$
$919$	1837.00			1.99891			0.999456	+	0.0329825i	$0.0105006\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1237.50	−	714.471i	−1.33784	−	0.772401i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−936.000	−	997.661i	−1.00537	−	1.07160i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−923.500	+	1599.55i	−0.985592	+	1.70710i	−0.346318	+	0.938117i	$0.612568\pi$
$937$	−923.500	+	1599.55i	−0.985592	+	1.70710i	−0.639274	+	0.768979i	$0.720765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$		−	3715.25i		−	3.91491i
$950$		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
$961$	−2714.00			−2.82414
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	893.500	+	1547.59i	0.923992	+	1.60040i	0.793175	+	0.608994i	$0.208427\pi$
$967$	893.500	+	1547.59i	0.923992	+	1.60040i	0.130817	+	0.991407i	$0.458240\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$	1259.50	−	2181.52i	1.29445	−	2.24205i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−823.500	+	475.448i	−0.830979	+	0.479766i	−0.854188	−	0.519965i	$-0.825945\pi$
$991$	−823.500	+	475.448i	−0.830979	+	0.479766i	0.0232089	+	0.999731i	$0.492612\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	743.500	−	1287.78i	0.745737	−	1.29165i	−0.204112	−	0.978947i	$-0.565431\pi$
$997$	743.500	−	1287.78i	0.745737	−	1.29165i	0.949850	−	0.312707i	$-0.101236\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.y.c.145.1	✓	2
3.2	odd	2	CM	684.3.y.c.145.1	✓	2
19.8	odd	6	inner	684.3.y.c.217.1	yes	2
57.8	even	6	inner	684.3.y.c.217.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.y.c.145.1	✓	2	1.1	even	1	trivial
684.3.y.c.145.1	✓	2	3.2	odd	2	CM
684.3.y.c.217.1	yes	2	19.8	odd	6	inner
684.3.y.c.217.1	yes	2	57.8	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-11.0000 q^{7} +O(q^{10})\)	\(q-11.0000 q^{7} +(22.5000 + 12.9904i) q^{13} +(-13.0000 - 13.8564i) q^{19} +(12.5000 - 21.6506i) q^{25} -60.6218i q^{31} -57.1577i q^{37} +(-41.5000 - 71.8801i) q^{43} +72.0000 q^{49} +(-60.5000 + 104.789i) q^{61} +(115.500 + 66.6840i) q^{67} +(-71.5000 - 123.842i) q^{73} +(76.5000 - 44.1673i) q^{79} +(-247.500 - 142.894i) q^{91} +(168.000 - 96.9948i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 22 q^{7}+O(q^{10}) \) 2 * q - 22 * q^7	\( 2 q - 22 q^{7} + 45 q^{13} - 26 q^{19} + 25 q^{25} - 83 q^{43} + 144 q^{49} - 121 q^{61} + 231 q^{67} - 143 q^{73} + 153 q^{79} - 495 q^{91} + 336 q^{97}+O(q^{100}) \) 2 * q - 22 * q^7 + 45 * q^13 - 26 * q^19 + 25 * q^25 - 83 * q^43 + 144 * q^49 - 121 * q^61 + 231 * q^67 - 143 * q^73 + 153 * q^79 - 495 * q^91 + 336 * q^97

Introduction

L-functions

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