LMFDB - Embedded newform 1872.2.by.i.1297.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(433,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("1872.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.by (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:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1297.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1872.1297
Dual form			1872.2.by.i.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{5} +O(q^{10})$	$q+3.00000i q^{5} +(-2.50000 + 2.59808i) q^{13} +(-2.59808 + 4.50000i) q^{17} +(-6.00000 - 3.46410i) q^{19} -4.00000 q^{25} +(-2.59808 - 4.50000i) q^{29} -6.92820i q^{31} +(-1.50000 + 0.866025i) q^{37} +(7.79423 - 4.50000i) q^{41} +(-2.00000 + 3.46410i) q^{43} +12.0000i q^{47} +(-3.50000 - 6.06218i) q^{49} +5.19615 q^{53} +(-10.3923 - 6.00000i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-7.79423 - 7.50000i) q^{65} +(-12.0000 + 6.92820i) q^{67} +(10.3923 + 6.00000i) q^{71} +8.66025i q^{73} -4.00000 q^{79} -12.0000i q^{83} +(-13.5000 - 7.79423i) q^{85} +(5.19615 - 3.00000i) q^{89} +(10.3923 - 18.0000i) q^{95} +(-12.0000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 10 q^{13} - 24 q^{19} - 16 q^{25} - 6 q^{37} - 8 q^{43} - 14 q^{49} + 10 q^{61} - 48 q^{67} - 16 q^{79} - 54 q^{85} - 48 q^{97}+O(q^{100}) $ 4 * q - 10 * q^13 - 24 * q^19 - 16 * q^25 - 6 * q^37 - 8 * q^43 - 14 * q^49 + 10 * q^61 - 48 * q^67 - 16 * q^79 - 54 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			3.00000i			1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$			3.00000i			1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−2.50000	+	2.59808i	−0.693375	+	0.720577i
$13$	−2.50000	+	2.59808i	−0.693375	+	0.720577i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.59808	+	4.50000i	−0.630126	+	1.09141i	0.357400	+	0.933952i	$0.383663\pi$
$17$	−2.59808	+	4.50000i	−0.630126	+	1.09141i	−0.987526	+	0.157459i	$0.949670\pi$
$18$		0			0
$19$	−6.00000	−	3.46410i	−1.37649	−	0.794719i	−0.384759	−	0.923017i	$-0.625715\pi$
$19$	−6.00000	−	3.46410i	−1.37649	−	0.794719i	−0.991736	+	0.128298i	$0.959049\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.59808	−	4.50000i	−0.482451	−	0.835629i	0.517346	−	0.855776i	$-0.326920\pi$
$29$	−2.59808	−	4.50000i	−0.482451	−	0.835629i	−0.999797	+	0.0201471i	$0.993587\pi$
$30$		0			0
$31$		−	6.92820i		−	1.24434i	−0.782881	−	0.622171i	$-0.786251\pi$
$31$		−	6.92820i		−	1.24434i	0.782881	−	0.622171i	$-0.213749\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.50000	+	0.866025i	−0.246598	+	0.142374i	−0.618206	−	0.786016i	$-0.712140\pi$
$37$	−1.50000	+	0.866025i	−0.246598	+	0.142374i	0.371607	+	0.928390i	$0.378807\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.79423	−	4.50000i	1.21725	−	0.702782i	0.252924	−	0.967486i	$-0.418608\pi$
$41$	7.79423	−	4.50000i	1.21725	−	0.702782i	0.964330	+	0.264704i	$0.0852743\pi$
$42$		0			0
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	−0.977261	−	0.212041i	$-0.931989\pi$
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	0.672264	+	0.740312i	$0.265322\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.0000i			1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$			12.0000i			1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$		0			0
$49$	−3.50000	−	6.06218i	−0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.19615			0.713746			0.356873	−	0.934153i	$-0.383843\pi$
$53$	5.19615			0.713746			0.356873	+	0.934153i	$0.383843\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.3923	−	6.00000i	−1.35296	−	0.781133i	−0.364299	−	0.931282i	$-0.618692\pi$
$59$	−10.3923	−	6.00000i	−1.35296	−	0.781133i	−0.988663	+	0.150148i	$0.952025\pi$
$60$		0			0
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	$-0.729619\pi$
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	0.980507	+	0.196485i	$0.0629528\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.79423	−	7.50000i	−0.966755	−	0.930261i
$66$		0			0
$67$	−12.0000	+	6.92820i	−1.46603	+	0.846415i	−0.999279	−	0.0379722i	$-0.987910\pi$
$67$	−12.0000	+	6.92820i	−1.46603	+	0.846415i	−0.466755	+	0.884387i	$0.654577\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.3923	+	6.00000i	1.23334	+	0.712069i	0.967725	−	0.252010i	$-0.0810916\pi$
$71$	10.3923	+	6.00000i	1.23334	+	0.712069i	0.265615	+	0.964079i	$0.414425\pi$
$72$		0			0
$73$			8.66025i			1.01361i	0.862062	+	0.506803i	$0.169173\pi$
$73$			8.66025i			1.01361i	−0.862062	+	0.506803i	$0.830827\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	−13.5000	−	7.79423i	−1.46428	−	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.19615	−	3.00000i	0.550791	−	0.317999i	−0.198650	−	0.980071i	$-0.563656\pi$
$89$	5.19615	−	3.00000i	0.550791	−	0.317999i	0.749441	+	0.662071i	$0.230322\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.3923	−	18.0000i	1.06623	−	1.84676i
$96$		0			0
$97$	−12.0000	−	6.92820i	−1.21842	−	0.703452i	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−12.0000	−	6.92820i	−1.21842	−	0.703452i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.59808	−	4.50000i	−0.258518	−	0.447767i	0.707327	−	0.706887i	$-0.249901\pi$
$101$	−2.59808	−	4.50000i	−0.258518	−	0.447767i	−0.965845	+	0.259120i	$0.916568\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$		−	6.92820i		−	0.663602i	−0.943349	−	0.331801i	$-0.892344\pi$
$109$		−	6.92820i		−	0.663602i	0.943349	−	0.331801i	$-0.107656\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.79423	+	13.5000i	−0.733219	+	1.26997i	0.222281	+	0.974983i	$0.428650\pi$
$113$	−7.79423	+	13.5000i	−0.733219	+	1.26997i	−0.955500	+	0.294990i	$0.904684\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.00000i			0.268328i
$126$		0			0
$127$	−4.00000	−	6.92820i	−0.354943	−	0.614779i	0.632166	−	0.774833i	$-0.282166\pi$
$127$	−4.00000	−	6.92820i	−0.354943	−	0.614779i	−0.987108	+	0.160055i	$0.948833\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−20.7846			−1.81596			−0.907980	−	0.419014i	$-0.862376\pi$
$131$	−20.7846			−1.81596			−0.907980	+	0.419014i	$0.862376\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.79423	+	4.50000i	0.665906	+	0.384461i	0.794524	−	0.607233i	$-0.207721\pi$
$137$	7.79423	+	4.50000i	0.665906	+	0.384461i	−0.128618	+	0.991694i	$0.541054\pi$
$138$		0			0
$139$	−8.00000	+	13.8564i	−0.678551	+	1.17529i	0.296866	+	0.954919i	$0.404058\pi$
$139$	−8.00000	+	13.8564i	−0.678551	+	1.17529i	−0.975417	+	0.220366i	$0.929275\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	13.5000	−	7.79423i	1.12111	−	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.59808	+	1.50000i	0.212843	+	0.122885i	0.602632	−	0.798019i	$-0.294119\pi$
$149$	2.59808	+	1.50000i	0.212843	+	0.122885i	−0.389789	+	0.920904i	$0.627452\pi$
$150$		0			0
$151$		−	13.8564i		−	1.12762i	−0.825905	−	0.563809i	$-0.809335\pi$
$151$		−	13.8564i		−	1.12762i	0.825905	−	0.563809i	$-0.190665\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	20.7846			1.66946
$156$		0			0
$157$	−7.00000			−0.558661			−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000			−0.558661			−0.279330	+	0.960195i	$0.590112\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.3923	+	6.00000i	−0.804181	+	0.464294i	−0.844931	−	0.534875i	$-0.820359\pi$
$167$	−10.3923	+	6.00000i	−0.804181	+	0.464294i	0.0407502	+	0.999169i	$0.487025\pi$
$168$		0			0
$169$	−0.500000	−	12.9904i	−0.0384615	−	0.999260i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.3923	+	18.0000i	−0.790112	+	1.36851i	0.135785	+	0.990738i	$0.456644\pi$
$173$	−10.3923	+	18.0000i	−0.790112	+	1.36851i	−0.925897	+	0.377776i	$0.876689\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.3923	+	18.0000i	0.776757	+	1.34538i	0.933801	+	0.357792i	$0.116470\pi$
$179$	10.3923	+	18.0000i	0.776757	+	1.34538i	−0.157044	+	0.987592i	$0.550196\pi$
$180$		0			0
$181$	−1.00000			−0.0743294			−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000			−0.0743294			−0.0371647	+	0.999309i	$0.511833\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.59808	−	4.50000i	−0.191014	−	0.330847i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.3923	+	18.0000i	−0.751961	+	1.30243i	0.194910	+	0.980821i	$0.437558\pi$
$191$	−10.3923	+	18.0000i	−0.751961	+	1.30243i	−0.946871	+	0.321613i	$0.895775\pi$
$192$		0			0
$193$	−4.50000	+	2.59808i	−0.323917	+	0.187014i	−0.653137	−	0.757240i	$-0.726548\pi$
$193$	−4.50000	+	2.59808i	−0.323917	+	0.187014i	0.329220	+	0.944253i	$0.393214\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.19615	−	3.00000i	0.370211	−	0.213741i	−0.303340	−	0.952882i	$-0.598102\pi$
$197$	5.19615	−	3.00000i	0.370211	−	0.213741i	0.673550	+	0.739141i	$0.264768\pi$
$198$		0			0
$199$	−4.00000	+	6.92820i	−0.283552	+	0.491127i	−0.972257	−	0.233915i	$-0.924846\pi$
$199$	−4.00000	+	6.92820i	−0.283552	+	0.491127i	0.688705	+	0.725042i	$0.258180\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	13.5000	+	23.3827i	0.942881	+	1.63312i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	0.926620	−	0.375999i	$-0.122700\pi$
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	−0.788935	+	0.614477i	$0.789367\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−10.3923	−	6.00000i	−0.708749	−	0.409197i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.19615	−	18.0000i	−0.349531	−	1.21081i
$222$		0			0
$223$	18.0000	−	10.3923i	1.20537	−	0.695920i	0.243625	−	0.969870i	$-0.421663\pi$
$223$	18.0000	−	10.3923i	1.20537	−	0.695920i	0.961744	+	0.273949i	$0.0883300\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.3923	+	6.00000i	0.689761	+	0.398234i	0.803523	−	0.595274i	$-0.202957\pi$
$227$	10.3923	+	6.00000i	0.689761	+	0.398234i	−0.113761	+	0.993508i	$0.536290\pi$
$228$		0			0
$229$			20.7846i			1.37349i	0.726900	+	0.686743i	$0.240960\pi$
$229$			20.7846i			1.37349i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−36.0000			−2.34838
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	7.50000	+	4.33013i	0.483117	+	0.278928i	0.721715	−	0.692191i	$-0.243354\pi$
$241$	7.50000	+	4.33013i	0.483117	+	0.278928i	−0.238597	+	0.971119i	$0.576688\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18.1865	−	10.5000i	1.16190	−	0.670820i
$246$		0			0
$247$	24.0000	−	6.92820i	1.52708	−	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.3923	+	18.0000i	−0.655956	+	1.13615i	0.325697	+	0.945474i	$0.394401\pi$
$251$	−10.3923	+	18.0000i	−0.655956	+	1.13615i	−0.981653	+	0.190676i	$0.938932\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.79423	−	13.5000i	−0.486191	−	0.842107i	0.513683	−	0.857980i	$-0.328281\pi$
$257$	−7.79423	−	13.5000i	−0.486191	−	0.842107i	−0.999874	+	0.0158730i	$0.994947\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−10.3923	−	18.0000i	−0.640817	−	1.10993i	−0.985251	−	0.171117i	$-0.945262\pi$
$263$	−10.3923	−	18.0000i	−0.640817	−	1.10993i	0.344434	−	0.938811i	$-0.388071\pi$
$264$		0			0
$265$			15.5885i			0.957591i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−10.3923	+	18.0000i	−0.633630	+	1.09748i	0.353174	+	0.935558i	$0.385102\pi$
$269$	−10.3923	+	18.0000i	−0.633630	+	1.09748i	−0.986804	+	0.161922i	$0.948231\pi$
$270$		0			0
$271$	18.0000	−	10.3923i	1.09342	−	0.631288i	0.158937	−	0.987289i	$-0.449193\pi$
$271$	18.0000	−	10.3923i	1.09342	−	0.631288i	0.934485	+	0.356001i	$0.115860\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	9.50000	−	16.4545i	0.570800	−	0.988654i	−0.425684	−	0.904872i	$-0.639967\pi$
$277$	9.50000	−	16.4545i	0.570800	−	0.988654i	0.996484	−	0.0837823i	$-0.0267000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.00000i			0.178965i	0.995988	+	0.0894825i	$0.0285213\pi$
$281$			3.00000i			0.178965i	−0.995988	+	0.0894825i	$0.971479\pi$
$282$		0			0
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	0.999608	−	0.0280052i	$-0.00891551\pi$
$283$	8.00000	+	13.8564i	0.475551	+	0.823678i	−0.524057	+	0.851683i	$0.675582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.00000	−	8.66025i	−0.294118	−	0.509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.79423	+	4.50000i	0.455344	+	0.262893i	0.710084	−	0.704117i	$-0.248657\pi$
$293$	7.79423	+	4.50000i	0.455344	+	0.262893i	−0.254741	+	0.967009i	$0.581990\pi$
$294$		0			0
$295$	18.0000	−	31.1769i	1.04800	−	1.81519i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.9904	+	7.50000i	0.743827	+	0.429449i
$306$		0			0
$307$			13.8564i			0.790827i	0.918503	+	0.395413i	$0.129399\pi$
$307$			13.8564i			0.790827i	−0.918503	+	0.395413i	$0.870601\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	20.7846			1.17859			0.589294	−	0.807919i	$-0.299406\pi$
$311$	20.7846			1.17859			0.589294	+	0.807919i	$0.299406\pi$
$312$		0			0
$313$	22.0000			1.24351			0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000			1.24351			0.621757	+	0.783210i	$0.286419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9.00000i		−	0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$		−	9.00000i		−	0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	31.1769	−	18.0000i	1.73473	−	1.00155i
$324$		0			0
$325$	10.0000	−	10.3923i	0.554700	−	0.576461i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.0000	−	6.92820i	−0.659580	−	0.380808i	0.132537	−	0.991178i	$-0.457688\pi$
$331$	−12.0000	−	6.92820i	−0.659580	−	0.380808i	−0.792117	+	0.610370i	$0.791021\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−20.7846	−	36.0000i	−1.13558	−	1.96689i
$336$		0			0
$337$	7.00000			0.381314			0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000			0.381314			0.190657	+	0.981657i	$0.438938\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	−6.00000	+	3.46410i	−0.321173	+	0.185429i	−0.651915	−	0.758292i	$-0.726034\pi$
$349$	−6.00000	+	3.46410i	−0.321173	+	0.185429i	0.330743	+	0.943721i	$0.392701\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12.9904	+	7.50000i	−0.691408	+	0.399185i	−0.804139	−	0.594441i	$-0.797373\pi$
$353$	−12.9904	+	7.50000i	−0.691408	+	0.399185i	0.112731	+	0.993626i	$0.464040\pi$
$354$		0			0
$355$	−18.0000	+	31.1769i	−0.955341	+	1.65470i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	12.0000i		−	0.633336i	−0.948536	−	0.316668i	$-0.897436\pi$
$359$		−	12.0000i		−	0.633336i	0.948536	−	0.316668i	$-0.102564\pi$
$360$		0			0
$361$	14.5000	+	25.1147i	0.763158	+	1.32183i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−25.9808			−1.35990
$366$		0			0
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	0.893873	+	0.448320i	$0.147978\pi$
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	−0.0586798	+	0.998277i	$0.518689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	11.5000	−	19.9186i	0.595447	−	1.03135i	−0.398036	−	0.917370i	$-0.630308\pi$
$373$	11.5000	−	19.9186i	0.595447	−	1.03135i	0.993484	−	0.113975i	$-0.0363585\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	18.1865	+	4.50000i	0.936654	+	0.231762i
$378$		0			0
$379$	12.0000	−	6.92820i	0.616399	−	0.355878i	−0.159067	−	0.987268i	$-0.550849\pi$
$379$	12.0000	−	6.92820i	0.616399	−	0.355878i	0.775466	+	0.631390i	$0.217515\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.19615			0.263455			0.131728	−	0.991286i	$-0.457948\pi$
$389$	5.19615			0.263455			0.131728	+	0.991286i	$0.457948\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	12.0000i		−	0.603786i
$396$		0			0
$397$	6.00000	+	3.46410i	0.301131	+	0.173858i	0.642951	−	0.765907i	$-0.277710\pi$
$397$	6.00000	+	3.46410i	0.301131	+	0.173858i	−0.341820	+	0.939766i	$0.611043\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.59808	+	1.50000i	−0.129742	+	0.0749064i	−0.563466	−	0.826139i	$-0.690532\pi$
$401$	−2.59808	+	1.50000i	−0.129742	+	0.0749064i	0.433724	+	0.901046i	$0.357199\pi$
$402$		0			0
$403$	18.0000	+	17.3205i	0.896644	+	0.862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	0.127609	−	0.991825i	$-0.459270\pi$
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	−0.795141	+	0.606425i	$0.792603\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	36.0000			1.76717
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.3923	+	18.0000i	0.507697	+	0.879358i	0.999960	+	0.00891102i	$0.00283650\pi$
$419$	10.3923	+	18.0000i	0.507697	+	0.879358i	−0.492263	+	0.870447i	$0.663830\pi$
$420$		0			0
$421$		−	15.5885i		−	0.759735i	−0.925041	−	0.379867i	$-0.875970\pi$
$421$		−	15.5885i		−	0.759735i	0.925041	−	0.379867i	$-0.124030\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.3923	−	18.0000i	0.504101	−	0.873128i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	10.3923	−	6.00000i	0.500580	−	0.289010i	−0.228373	−	0.973574i	$-0.573341\pi$
$431$	10.3923	−	6.00000i	0.500580	−	0.289010i	0.728953	+	0.684564i	$0.240007\pi$
$432$		0			0
$433$	−9.50000	+	16.4545i	−0.456541	+	0.790752i	−0.998775	−	0.0494752i	$-0.984245\pi$
$433$	−9.50000	+	16.4545i	−0.456541	+	0.790752i	0.542234	+	0.840227i	$0.317578\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−2.00000	−	3.46410i	−0.0954548	−	0.165333i	0.814344	−	0.580383i	$-0.197097\pi$
$439$	−2.00000	−	3.46410i	−0.0954548	−	0.165333i	−0.909798	+	0.415051i	$0.863764\pi$
$440$		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
$445$	9.00000	+	15.5885i	0.426641	+	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−25.9808	−	15.0000i	−1.22611	−	0.707894i	−0.259895	−	0.965637i	$-0.583688\pi$
$449$	−25.9808	−	15.0000i	−1.22611	−	0.707894i	−0.966213	+	0.257743i	$0.917021\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	10.5000	−	6.06218i	0.491169	−	0.283577i	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	10.5000	−	6.06218i	0.491169	−	0.283577i	0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.1865	+	10.5000i	0.847031	+	0.489034i	0.859648	−	0.510887i	$-0.170683\pi$
$461$	18.1865	+	10.5000i	0.847031	+	0.489034i	−0.0126168	+	0.999920i	$0.504016\pi$
$462$		0			0
$463$		−	6.92820i		−	0.321981i	−0.986956	−	0.160990i	$-0.948531\pi$
$463$		−	6.92820i		−	0.321981i	0.986956	−	0.160990i	$-0.0514688\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−41.5692			−1.92359			−0.961797	−	0.273764i	$-0.911731\pi$
$467$	−41.5692			−1.92359			−0.961797	+	0.273764i	$0.911731\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	24.0000	+	13.8564i	1.10120	+	0.635776i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−20.7846	+	12.0000i	−0.949673	+	0.548294i	−0.892979	−	0.450098i	$-0.851389\pi$
$479$	−20.7846	+	12.0000i	−0.949673	+	0.548294i	−0.0566937	+	0.998392i	$0.518056\pi$
$480$		0			0
$481$	1.50000	−	6.06218i	0.0683941	−	0.276412i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.7846	−	36.0000i	0.943781	−	1.63468i
$486$		0			0
$487$	6.00000	+	3.46410i	0.271886	+	0.156973i	0.629744	−	0.776802i	$-0.283160\pi$
$487$	6.00000	+	3.46410i	0.271886	+	0.156973i	−0.357858	+	0.933776i	$0.616493\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.3923	+	18.0000i	0.468998	+	0.812329i	0.999372	−	0.0354353i	$-0.0112818\pi$
$491$	10.3923	+	18.0000i	0.468998	+	0.812329i	−0.530374	+	0.847764i	$0.677948\pi$
$492$		0			0
$493$	27.0000			1.21602
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−10.3923	+	18.0000i	−0.463370	+	0.802580i	−0.999126	−	0.0417923i	$-0.986693\pi$
$503$	−10.3923	+	18.0000i	−0.463370	+	0.802580i	0.535756	+	0.844373i	$0.320027\pi$
$504$		0			0
$505$	13.5000	−	7.79423i	0.600742	−	0.346839i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12.9904	−	7.50000i	0.575789	−	0.332432i	−0.183669	−	0.982988i	$-0.558798\pi$
$509$	12.9904	−	7.50000i	0.575789	−	0.332432i	0.759458	+	0.650556i	$0.225464\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			12.0000i			0.528783i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.5885			−0.682943			−0.341471	−	0.939892i	$-0.610925\pi$
$521$	−15.5885			−0.682943			−0.341471	+	0.939892i	$0.610925\pi$
$522$		0			0
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	0.940129	−	0.340818i	$-0.110704\pi$
$523$	4.00000	+	6.92820i	0.174908	+	0.302949i	−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	31.1769	+	18.0000i	1.35809	+	0.784092i
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.79423	+	31.5000i	−0.337606	+	1.36442i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$			32.9090i			1.41487i	0.706780	+	0.707433i	$0.250147\pi$
$541$			32.9090i			1.41487i	−0.706780	+	0.707433i	$0.749853\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	20.7846			0.890315
$546$		0			0
$547$	4.00000			0.171028			0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000			0.171028			0.0855138	+	0.996337i	$0.472747\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			36.0000i			1.53365i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−38.9711	+	22.5000i	−1.65126	+	0.953356i	−0.674705	+	0.738087i	$0.735729\pi$
$557$	−38.9711	+	22.5000i	−1.65126	+	0.953356i	−0.976555	+	0.215268i	$0.930937\pi$
$558$		0			0
$559$	−4.00000	−	13.8564i	−0.169182	−	0.586064i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.3923	+	18.0000i	−0.437983	+	0.758610i	−0.997534	−	0.0701867i	$-0.977640\pi$
$563$	−10.3923	+	18.0000i	−0.437983	+	0.758610i	0.559550	+	0.828796i	$0.310974\pi$
$564$		0			0
$565$	−40.5000	−	23.3827i	−1.70385	−	0.983717i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	20.7846	+	36.0000i	0.871336	+	1.50920i	0.860615	+	0.509256i	$0.170079\pi$
$569$	20.7846	+	36.0000i	0.871336	+	1.50920i	0.0107211	+	0.999943i	$0.496587\pi$
$570$		0			0
$571$	−16.0000			−0.669579			−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000			−0.669579			−0.334790	+	0.942293i	$0.608665\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	43.3013i		−	1.80266i	−0.433138	−	0.901328i	$-0.642594\pi$
$577$		−	43.3013i		−	1.80266i	0.433138	−	0.901328i	$-0.357406\pi$
$578$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	−24.0000	+	41.5692i	−0.988903	+	1.71283i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.00000i			0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$			9.00000i			0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.7846			0.849236			0.424618	−	0.905373i	$-0.360408\pi$
$599$	20.7846			0.849236			0.424618	+	0.905373i	$0.360408\pi$
$600$		0			0
$601$	17.5000	+	30.3109i	0.713840	+	1.23641i	0.963405	+	0.268049i	$0.0863789\pi$
$601$	17.5000	+	30.3109i	0.713840	+	1.23641i	−0.249565	+	0.968358i	$0.580288\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−28.5788	−	16.5000i	−1.16190	−	0.670820i
$606$		0			0
$607$	−8.00000	+	13.8564i	−0.324710	+	0.562414i	−0.981454	−	0.191700i	$-0.938600\pi$
$607$	−8.00000	+	13.8564i	−0.324710	+	0.562414i	0.656744	+	0.754114i	$0.271933\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−31.1769	−	30.0000i	−1.26128	−	1.21367i
$612$		0			0
$613$	34.5000	−	19.9186i	1.39344	−	0.804504i	0.399747	−	0.916625i	$-0.369098\pi$
$613$	34.5000	−	19.9186i	1.39344	−	0.804504i	0.993695	+	0.112121i	$0.0357645\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.1865	−	10.5000i	−0.732162	−	0.422714i	0.0870504	−	0.996204i	$-0.472256\pi$
$617$	−18.1865	−	10.5000i	−0.732162	−	0.422714i	−0.819213	+	0.573490i	$0.805589\pi$
$618$		0			0
$619$			20.7846i			0.835404i	0.908584	+	0.417702i	$0.137164\pi$
$619$			20.7846i			0.835404i	−0.908584	+	0.417702i	$0.862836\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	9.00000i		−	0.358854i
$630$		0			0
$631$	−12.0000	−	6.92820i	−0.477712	−	0.275807i	0.241750	−	0.970339i	$-0.422279\pi$
$631$	−12.0000	−	6.92820i	−0.477712	−	0.275807i	−0.719463	+	0.694531i	$0.755612\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	20.7846	−	12.0000i	0.824812	−	0.476205i
$636$		0			0
$637$	24.5000	+	6.06218i	0.970725	+	0.240192i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.59808	−	4.50000i	0.102618	−	0.177739i	−0.810145	−	0.586230i	$-0.800611\pi$
$641$	2.59808	−	4.50000i	0.102618	−	0.177739i	0.912762	+	0.408491i	$0.133945\pi$
$642$		0			0
$643$	−30.0000	−	17.3205i	−1.18308	−	0.683054i	−0.226358	−	0.974044i	$-0.572682\pi$
$643$	−30.0000	−	17.3205i	−1.18308	−	0.683054i	−0.956726	+	0.290990i	$0.906015\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10.3923	−	18.0000i	−0.408564	−	0.707653i	0.586165	−	0.810191i	$-0.300637\pi$
$647$	−10.3923	−	18.0000i	−0.408564	−	0.707653i	−0.994729	+	0.102538i	$0.967304\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.3923	+	18.0000i	0.406682	+	0.704394i	0.994516	−	0.104588i	$-0.0333523\pi$
$653$	10.3923	+	18.0000i	0.406682	+	0.704394i	−0.587833	+	0.808982i	$0.700019\pi$
$654$		0			0
$655$		−	62.3538i		−	2.43637i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.7846	−	36.0000i	0.809653	−	1.40236i	−0.103451	−	0.994635i	$-0.532988\pi$
$659$	20.7846	−	36.0000i	0.809653	−	1.40236i	0.913104	−	0.407726i	$-0.133678\pi$
$660$		0			0
$661$	4.50000	−	2.59808i	0.175030	−	0.101053i	−0.409926	−	0.912119i	$-0.634445\pi$
$661$	4.50000	−	2.59808i	0.175030	−	0.101053i	0.584955	+	0.811065i	$0.301112\pi$
$662$		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
$673$	−2.50000	−	4.33013i	−0.0963679	−	0.166914i	0.813811	−	0.581130i	$-0.197389\pi$
$673$	−2.50000	−	4.33013i	−0.0963679	−	0.166914i	−0.910179	+	0.414216i	$0.864056\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.7846			0.798817			0.399409	−	0.916773i	$-0.369215\pi$
$677$	20.7846			0.798817			0.399409	+	0.916773i	$0.369215\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.3923	+	6.00000i	0.397650	+	0.229584i	0.685470	−	0.728101i	$-0.259597\pi$
$683$	10.3923	+	6.00000i	0.397650	+	0.229584i	−0.287819	+	0.957685i	$0.592930\pi$
$684$		0			0
$685$	−13.5000	+	23.3827i	−0.515808	+	0.893407i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.9904	+	13.5000i	−0.494894	+	0.514309i
$690$		0			0
$691$	−12.0000	+	6.92820i	−0.456502	+	0.263561i	−0.710572	−	0.703624i	$-0.751564\pi$
$691$	−12.0000	+	6.92820i	−0.456502	+	0.263561i	0.254071	+	0.967186i	$0.418230\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−41.5692	−	24.0000i	−1.57681	−	0.910372i
$696$		0			0
$697$			46.7654i			1.77136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.7846			0.785024			0.392512	−	0.919747i	$-0.371606\pi$
$701$	20.7846			0.785024			0.392512	+	0.919747i	$0.371606\pi$
$702$		0			0
$703$	12.0000			0.452589
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−22.5000	−	12.9904i	−0.845005	−	0.487864i	0.0139572	−	0.999903i	$-0.495557\pi$
$709$	−22.5000	−	12.9904i	−0.845005	−	0.487864i	−0.858962	+	0.512039i	$0.828890\pi$
$710$		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
$719$	−10.3923	+	18.0000i	−0.387568	+	0.671287i	−0.992122	−	0.125277i	$-0.960018\pi$
$719$	−10.3923	+	18.0000i	−0.387568	+	0.671287i	0.604554	+	0.796564i	$0.293351\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	10.3923	+	18.0000i	0.385961	+	0.668503i
$726$		0			0
$727$	32.0000			1.18681			0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000			1.18681			0.593407	+	0.804902i	$0.297782\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10.3923	−	18.0000i	−0.384373	−	0.665754i
$732$		0			0
$733$			8.66025i			0.319874i	0.987127	+	0.159937i	$0.0511291\pi$
$733$			8.66025i			0.319874i	−0.987127	+	0.159937i	$0.948871\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	36.0000	−	20.7846i	1.32428	−	0.764574i	0.339873	−	0.940471i	$-0.389616\pi$
$739$	36.0000	−	20.7846i	1.32428	−	0.764574i	0.984409	+	0.175897i	$0.0562826\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	20.7846	−	12.0000i	0.762513	−	0.440237i	−0.0676840	−	0.997707i	$-0.521561\pi$
$743$	20.7846	−	12.0000i	0.762513	−	0.440237i	0.830197	+	0.557470i	$0.188228\pi$
$744$		0			0
$745$	−4.50000	+	7.79423i	−0.164867	+	0.285558i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	0.988761	−	0.149505i	$-0.0477681\pi$
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	−0.623856	+	0.781540i	$0.714435\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	41.5692			1.51286
$756$		0			0
$757$	−5.00000	−	8.66025i	−0.181728	−	0.314762i	0.760741	−	0.649056i	$-0.224836\pi$
$757$	−5.00000	−	8.66025i	−0.181728	−	0.314762i	−0.942469	+	0.334293i	$0.891502\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.19615	+	3.00000i	0.188360	+	0.108750i	0.591215	−	0.806514i	$-0.298649\pi$
$761$	5.19615	+	3.00000i	0.188360	+	0.108750i	−0.402854	+	0.915264i	$0.631982\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	41.5692	−	12.0000i	1.50098	−	0.433295i
$768$		0			0
$769$	−12.0000	+	6.92820i	−0.432731	+	0.249837i	−0.700509	−	0.713643i	$-0.747044\pi$
$769$	−12.0000	+	6.92820i	−0.432731	+	0.249837i	0.267778	+	0.963481i	$0.413711\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−36.3731	−	21.0000i	−1.30825	−	0.755318i	−0.326445	−	0.945216i	$-0.605851\pi$
$773$	−36.3731	−	21.0000i	−1.30825	−	0.755318i	−0.981804	+	0.189899i	$0.939184\pi$
$774$		0			0
$775$			27.7128i			0.995474i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−62.3538			−2.23406
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	21.0000i		−	0.749522i
$786$		0			0
$787$	−42.0000	−	24.2487i	−1.49714	−	0.864373i	−0.497144	−	0.867668i	$-0.665618\pi$
$787$	−42.0000	−	24.2487i	−1.49714	−	0.864373i	−0.999995	+	0.00329499i	$0.998951\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5.00000	+	17.3205i	0.177555	+	0.615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.3923	−	18.0000i	0.368114	−	0.637593i	−0.621156	−	0.783687i	$-0.713337\pi$
$797$	10.3923	−	18.0000i	0.368114	−	0.637593i	0.989271	+	0.146094i	$0.0466702\pi$
$798$		0			0
$799$	−54.0000	−	31.1769i	−1.91038	−	1.10296i
$800$		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
$809$	−7.79423	−	13.5000i	−0.274030	−	0.474635i	0.695860	−	0.718178i	$-0.255024\pi$
$809$	−7.79423	−	13.5000i	−0.274030	−	0.474635i	−0.969890	+	0.243543i	$0.921690\pi$
$810$		0			0
$811$		−	6.92820i		−	0.243282i	−0.992574	−	0.121641i	$-0.961184\pi$
$811$		−	6.92820i		−	0.243282i	0.992574	−	0.121641i	$-0.0388157\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	24.0000	−	13.8564i	0.839654	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−46.7654	+	27.0000i	−1.63212	+	0.942306i	−0.648686	+	0.761056i	$0.724681\pi$
$821$	−46.7654	+	27.0000i	−1.63212	+	0.942306i	−0.983437	+	0.181250i	$0.941986\pi$
$822$		0			0
$823$	−26.0000	+	45.0333i	−0.906303	+	1.56976i	−0.0871445	+	0.996196i	$0.527774\pi$
$823$	−26.0000	+	45.0333i	−0.906303	+	1.56976i	−0.819159	+	0.573567i	$0.805559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		0			0
$829$	−18.5000	−	32.0429i	−0.642532	−	1.11290i	−0.984866	−	0.173319i	$-0.944551\pi$
$829$	−18.5000	−	32.0429i	−0.642532	−	1.11290i	0.342334	−	0.939578i	$-0.388783\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	36.3731			1.26025
$834$		0			0
$835$	−18.0000	−	31.1769i	−0.622916	−	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	41.5692	+	24.0000i	1.43513	+	0.828572i	0.997506	−	0.0705865i	$-0.0224871\pi$
$839$	41.5692	+	24.0000i	1.43513	+	0.828572i	0.437623	+	0.899158i	$0.355820\pi$
$840$		0			0
$841$	1.00000	−	1.73205i	0.0344828	−	0.0597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	38.9711	−	1.50000i	1.34065	−	0.0516016i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	36.3731i		−	1.24539i	−0.782465	−	0.622695i	$-0.786038\pi$
$853$		−	36.3731i		−	1.24539i	0.782465	−	0.622695i	$-0.213962\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−25.9808			−0.887486			−0.443743	−	0.896154i	$-0.646350\pi$
$857$	−25.9808			−0.887486			−0.443743	+	0.896154i	$0.646350\pi$
$858$		0			0
$859$	−40.0000			−1.36478			−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000			−1.36478			−0.682391	+	0.730987i	$0.739060\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$	−54.0000	−	31.1769i	−1.83606	−	1.06005i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	12.0000	−	48.4974i	0.406604	−	1.64327i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−16.5000	−	9.52628i	−0.557165	−	0.321680i	0.194842	−	0.980835i	$-0.437581\pi$
$877$	−16.5000	−	9.52628i	−0.557165	−	0.321680i	−0.752007	+	0.659155i	$0.770914\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.9904	−	22.5000i	−0.437657	−	0.758044i	0.559851	−	0.828593i	$-0.310858\pi$
$881$	−12.9904	−	22.5000i	−0.437657	−	0.758044i	−0.997508	+	0.0705489i	$0.977525\pi$
$882$		0			0
$883$	8.00000			0.269221			0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000			0.269221			0.134611	+	0.990899i	$0.457022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.3923	+	18.0000i	0.348939	+	0.604381i	0.986061	−	0.166382i	$-0.0532086\pi$
$887$	10.3923	+	18.0000i	0.348939	+	0.604381i	−0.637122	+	0.770763i	$0.719875\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	41.5692	−	72.0000i	1.39106	−	2.40939i
$894$		0			0
$895$	−54.0000	+	31.1769i	−1.80502	+	1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−31.1769	+	18.0000i	−1.03981	+	0.600334i
$900$		0			0
$901$	−13.5000	+	23.3827i	−0.449750	+	0.778990i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	3.00000i		−	0.0997234i
$906$		0			0
$907$	20.0000	+	34.6410i	0.664089	+	1.15024i	0.979531	+	0.201291i	$0.0645138\pi$
$907$	20.0000	+	34.6410i	0.664089	+	1.15024i	−0.315442	+	0.948945i	$0.602153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	20.7846			0.688625			0.344312	−	0.938855i	$-0.388112\pi$
$911$	20.7846			0.688625			0.344312	+	0.938855i	$0.388112\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	−0.703378	−	0.710816i	$-0.748326\pi$
$919$	8.00000	−	13.8564i	0.263896	−	0.457081i	0.967274	+	0.253735i	$0.0816592\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−41.5692	+	12.0000i	−1.36827	+	0.394985i
$924$		0			0
$925$	6.00000	−	3.46410i	0.197279	−	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	23.3827	+	13.5000i	0.767161	+	0.442921i	0.831861	−	0.554984i	$-0.187276\pi$
$929$	23.3827	+	13.5000i	0.767161	+	0.442921i	−0.0646999	+	0.997905i	$0.520609\pi$
$930$		0			0
$931$			48.4974i			1.58944i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−53.0000			−1.73143			−0.865717	−	0.500533i	$-0.833137\pi$
$937$	−53.0000			−1.73143			−0.865717	+	0.500533i	$0.833137\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			18.0000i			0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$			18.0000i			0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−20.7846	+	12.0000i	−0.675409	+	0.389948i	−0.798123	−	0.602494i	$-0.794174\pi$
$947$	−20.7846	+	12.0000i	−0.675409	+	0.389948i	0.122714	+	0.992442i	$0.460840\pi$
$948$		0			0
$949$	−22.5000	−	21.6506i	−0.730381	−	0.702809i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.7846	−	36.0000i	0.673280	−	1.16615i	−0.303689	−	0.952771i	$-0.598218\pi$
$953$	20.7846	−	36.0000i	0.673280	−	1.16615i	0.976969	−	0.213383i	$-0.0684483\pi$
$954$		0			0
$955$	−54.0000	−	31.1769i	−1.74740	−	1.00886i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−17.0000			−0.548387
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−7.79423	−	13.5000i	−0.250905	−	0.434580i
$966$		0			0
$967$			6.92820i			0.222796i	0.993776	+	0.111398i	$0.0355328\pi$
$967$			6.92820i			0.222796i	−0.993776	+	0.111398i	$0.964467\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.7846	−	36.0000i	0.667010	−	1.15529i	−0.311726	−	0.950172i	$-0.600907\pi$
$971$	20.7846	−	36.0000i	0.667010	−	1.15529i	0.978736	−	0.205123i	$-0.0657595\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.59808	−	1.50000i	0.0831198	−	0.0479893i	−0.457864	−	0.889022i	$-0.651385\pi$
$977$	2.59808	−	1.50000i	0.0831198	−	0.0479893i	0.540984	+	0.841033i	$0.318052\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$			36.0000i			1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$			36.0000i			1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$		0			0
$985$	9.00000	+	15.5885i	0.286764	+	0.496690i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	0.968864	+	0.247592i	$0.0796392\pi$
$991$	22.0000	+	38.1051i	0.698853	+	1.21045i	−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−20.7846	−	12.0000i	−0.658916	−	0.380426i
$996$		0			0
$997$	6.50000	−	11.2583i	0.205857	−	0.356555i	−0.744548	−	0.667568i	$-0.767335\pi$
$997$	6.50000	−	11.2583i	0.205857	−	0.356555i	0.950405	+	0.311014i	$0.100668\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.by.i.1297.2		4
3.2	odd	2	inner	1872.2.by.i.1297.1		4
4.3	odd	2		234.2.l.b.127.1	✓	4
12.11	even	2		234.2.l.b.127.2	yes	4
13.4	even	6	inner	1872.2.by.i.433.1		4
39.17	odd	6	inner	1872.2.by.i.433.2		4
52.3	odd	6		3042.2.b.m.1351.2		4
52.11	even	12		3042.2.a.u.1.1		2
52.15	even	12		3042.2.a.t.1.1		2
52.23	odd	6		3042.2.b.m.1351.4		4
52.43	odd	6		234.2.l.b.199.1	yes	4
156.11	odd	12		3042.2.a.t.1.2		2
156.23	even	6		3042.2.b.m.1351.1		4
156.95	even	6		234.2.l.b.199.2	yes	4
156.107	even	6		3042.2.b.m.1351.3		4
156.119	odd	12		3042.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.l.b.127.1	✓	4	4.3	odd	2
234.2.l.b.127.2	yes	4	12.11	even	2
234.2.l.b.199.1	yes	4	52.43	odd	6
234.2.l.b.199.2	yes	4	156.95	even	6
1872.2.by.i.433.1		4	13.4	even	6	inner
1872.2.by.i.433.2		4	39.17	odd	6	inner
1872.2.by.i.1297.1		4	3.2	odd	2	inner
1872.2.by.i.1297.2		4	1.1	even	1	trivial
3042.2.a.t.1.1		2	52.15	even	12
3042.2.a.t.1.2		2	156.11	odd	12
3042.2.a.u.1.1		2	52.11	even	12
3042.2.a.u.1.2		2	156.119	odd	12
3042.2.b.m.1351.1		4	156.23	even	6
3042.2.b.m.1351.2		4	52.3	odd	6
3042.2.b.m.1351.3		4	156.107	even	6
3042.2.b.m.1351.4		4	52.23	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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.by (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{5} +O(q^{10})\)	\(q+3.00000i q^{5} +(-2.50000 + 2.59808i) q^{13} +(-2.59808 + 4.50000i) q^{17} +(-6.00000 - 3.46410i) q^{19} -4.00000 q^{25} +(-2.59808 - 4.50000i) q^{29} -6.92820i q^{31} +(-1.50000 + 0.866025i) q^{37} +(7.79423 - 4.50000i) q^{41} +(-2.00000 + 3.46410i) q^{43} +12.0000i q^{47} +(-3.50000 - 6.06218i) q^{49} +5.19615 q^{53} +(-10.3923 - 6.00000i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-7.79423 - 7.50000i) q^{65} +(-12.0000 + 6.92820i) q^{67} +(10.3923 + 6.00000i) q^{71} +8.66025i q^{73} -4.00000 q^{79} -12.0000i q^{83} +(-13.5000 - 7.79423i) q^{85} +(5.19615 - 3.00000i) q^{89} +(10.3923 - 18.0000i) q^{95} +(-12.0000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 10 q^{13} - 24 q^{19} - 16 q^{25} - 6 q^{37} - 8 q^{43} - 14 q^{49} + 10 q^{61} - 48 q^{67} - 16 q^{79} - 54 q^{85} - 48 q^{97}+O(q^{100}) \) 4 * q - 10 * q^13 - 24 * q^19 - 16 * q^25 - 6 * q^37 - 8 * q^43 - 14 * q^49 + 10 * q^61 - 48 * q^67 - 16 * q^79 - 54 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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