LMFDB - Embedded newform 228.6.p.b.221.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [228,6,Mod(65,228)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(228, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("228.65");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 228 = 2^{2} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	228.p (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:	$36.5675109174$
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			221.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	228.221
Dual form			228.6.p.b.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(13.5000 - 7.79423i) q^{3} +25.0000 q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})$	$q+(13.5000 - 7.79423i) q^{3} +25.0000 q^{7} +(121.500 - 210.444i) q^{9} +(-814.500 - 470.252i) q^{13} +(716.000 - 1401.23i) q^{19} +(337.500 - 194.856i) q^{21} +(-1562.50 + 2706.33i) q^{25} -3788.00i q^{27} -10349.0i q^{31} -13400.9i q^{37} -14661.0 q^{39} +(9561.50 + 16561.0i) q^{43} -16182.0 q^{49} +(-1255.50 - 24497.3i) q^{57} +(-9150.50 + 15849.1i) q^{61} +(3037.50 - 5261.10i) q^{63} +(-54505.5 - 31468.8i) q^{67} +(-39788.5 - 68915.7i) q^{73} +48713.9i q^{75} +(95710.5 - 55258.5i) q^{79} +(-29524.5 - 51137.9i) q^{81} +(-20362.5 - 11756.3i) q^{91} +(-80662.5 - 139712. i) q^{93} +(-110532. + 63815.7i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 27 q^{3} + 50 q^{7} + 243 q^{9}+O(q^{10}) $ 2 * q + 27 * q^3 + 50 * q^7 + 243 * q^9	$ 2 q + 27 q^{3} + 50 q^{7} + 243 q^{9} - 1629 q^{13} + 1432 q^{19} + 675 q^{21} - 3125 q^{25} - 29322 q^{39} + 19123 q^{43} - 32364 q^{49} - 2511 q^{57} - 18301 q^{61} + 6075 q^{63} - 109011 q^{67} - 79577 q^{73} + 191421 q^{79} - 59049 q^{81} - 40725 q^{91} - 161325 q^{93} - 221064 q^{97}+O(q^{100}) $ 2 * q + 27 * q^3 + 50 * q^7 + 243 * q^9 - 1629 * q^13 + 1432 * q^19 + 675 * q^21 - 3125 * q^25 - 29322 * q^39 + 19123 * q^43 - 32364 * q^49 - 2511 * q^57 - 18301 * q^61 + 6075 * q^63 - 109011 * q^67 - 79577 * q^73 + 191421 * q^79 - 59049 * q^81 - 40725 * q^91 - 161325 * q^93 - 221064 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/228\mathbb{Z}\right)^\times$.

$n$	$77$	$97$	$115$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

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
$2$		0			0
$3$	13.5000	−	7.79423i	0.866025	−	0.500000i
$3$	13.5000	−	7.79423i	0.866025	−	0.500000i
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	25.0000			0.192839			0.0964195	−	0.995341i	$-0.469261\pi$
$7$	25.0000			0.192839			0.0964195	+	0.995341i	$0.469261\pi$
$8$		0			0
$9$	121.500	−	210.444i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−814.500	−	470.252i	−1.33670	−	0.771742i	−0.350380	−	0.936608i	$-0.613948\pi$
$13$	−814.500	−	470.252i	−1.33670	−	0.771742i	−0.986316	+	0.164866i	$0.947281\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	716.000	−	1401.23i	0.455018	−	0.890482i
$19$	716.000	−	1401.23i	0.455018	−	0.890482i
$20$		0			0
$21$	337.500	−	194.856i	0.167003	−	0.0964195i
$22$		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
$25$	−1562.50	+	2706.33i	−0.500000	+	0.866025i
$26$		0			0
$27$		−	3788.00i		−	1.00000i
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$		−	10349.0i		−	1.93417i	−0.254456	−	0.967084i	$-0.581897\pi$
$31$		−	10349.0i		−	1.93417i	0.254456	−	0.967084i	$-0.418103\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	13400.9i		−	1.60927i	−0.593770	−	0.804635i	$-0.702361\pi$
$37$		−	13400.9i		−	1.60927i	0.593770	−	0.804635i	$-0.297639\pi$
$38$		0			0
$39$	−14661.0			−1.54348
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	9561.50	+	16561.0i	0.788597	+	1.36589i	0.926827	+	0.375489i	$0.122525\pi$
$43$	9561.50	+	16561.0i	0.788597	+	1.36589i	−0.138230	+	0.990400i	$0.544141\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	−16182.0			−0.962813
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1255.50	−	24497.3i	−0.0511835	−	0.998689i
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−9150.50	+	15849.1i	−0.314862	+	0.545357i	−0.979408	−	0.201890i	$-0.935292\pi$
$61$	−9150.50	+	15849.1i	−0.314862	+	0.545357i	0.664546	+	0.747247i	$0.268625\pi$
$62$		0			0
$63$	3037.50	−	5261.10i	0.0964195	−	0.167003i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−54505.5	−	31468.8i	−1.48338	−	0.856432i	−0.483561	−	0.875310i	$-0.660657\pi$
$67$	−54505.5	−	31468.8i	−1.48338	−	0.856432i	−0.999822	+	0.0188789i	$0.993990\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$		0			0
$73$	−39788.5	−	68915.7i	−0.873877	−	1.51360i	−0.857954	−	0.513727i	$-0.828265\pi$
$73$	−39788.5	−	68915.7i	−0.873877	−	1.51360i	−0.0159232	−	0.999873i	$-0.505069\pi$
$74$		0			0
$75$			48713.9i			1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	95710.5	−	55258.5i	1.72541	−	0.996165i	0.818956	−	0.573856i	$-0.194553\pi$
$79$	95710.5	−	55258.5i	1.72541	−	0.996165i	0.906452	−	0.422309i	$-0.138780\pi$
$80$		0			0
$81$	−29524.5	−	51137.9i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−20362.5	−	11756.3i	−0.257767	−	0.148822i
$92$		0			0
$93$	−80662.5	−	139712.i	−0.967084	−	1.67504i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−110532.	+	63815.7i	−1.19278	+	0.688649i	−0.958935	−	0.283626i	$-0.908463\pi$
$97$	−110532.	+	63815.7i	−1.19278	+	0.688649i	−0.233840	+	0.972275i	$0.575129\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		−	40601.0i		−	0.377089i	−0.982065	−	0.188544i	$-0.939623\pi$
$103$		−	40601.0i		−	0.377089i	0.982065	−	0.188544i	$-0.0603769\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	190602.	−	110044.i	1.53660	−	0.887157i	0.537567	−	0.843221i	$-0.319344\pi$
$109$	190602.	−	110044.i	1.53660	−	0.887157i	0.999034	−	0.0439362i	$-0.0139898\pi$
$110$		0			0
$111$	−104450.	−	180912.i	−0.804635	−	1.39367i
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−197924.	+	114271.i	−1.33670	+	0.771742i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	161051.			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	213561.	+	123300.i	1.17493	+	0.678347i	0.954837	−	0.297131i	$-0.0960299\pi$
$127$	213561.	+	123300.i	1.17493	+	0.678347i	0.220095	+	0.975478i	$0.429363\pi$
$128$		0			0
$129$	258160.	+	149049.i	1.36589	+	0.788597i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	17900.0	−	35030.7i	0.0877453	−	0.171720i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−227328.	+	393745.i	−0.997969	+	1.72853i	−0.443812	+	0.896120i	$0.646374\pi$
$139$	−227328.	+	393745.i	−0.997969	+	1.72853i	−0.554157	+	0.832412i	$0.686959\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−218457.	+	126126.i	−0.833821	+	0.481407i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$			383348.i			1.36820i	0.729387	+	0.684102i	$0.239806\pi$
$151$			383348.i			1.36820i	−0.729387	+	0.684102i	$0.760194\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	235956.	+	408687.i	0.763978	+	1.32325i	0.940785	+	0.339004i	$0.110090\pi$
$157$	235956.	+	408687.i	0.763978	+	1.32325i	−0.176807	+	0.984246i	$0.556577\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	352375.			1.03881			0.519405	−	0.854528i	$-0.326154\pi$
$163$	352375.			1.03881			0.519405	+	0.854528i	$0.326154\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	256627.	+	444491.i	0.691171	+	1.19714i
$170$		0			0
$171$	−207886.	−	320927.i	−0.543671	−	0.839299i
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−39062.5	+	67658.2i	−0.0964195	+	0.167003i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	736014.	+	424938.i	1.66990	+	0.964115i	0.967690	+	0.252142i	$0.0811351\pi$
$181$	736014.	+	424938.i	1.66990	+	0.964115i	0.702207	+	0.711973i	$0.252198\pi$
$182$		0			0
$183$			285284.i			0.629724i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	94699.9i		−	0.192839i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	692990.	−	400098.i	1.33916	−	0.773166i	0.352480	−	0.935820i	$-0.385339\pi$
$193$	692990.	−	400098.i	1.33916	−	0.773166i	0.986683	+	0.162653i	$0.0520053\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	51246.5	−	88761.5i	0.0917343	−	0.158888i	−0.816507	−	0.577336i	$-0.804092\pi$
$199$	51246.5	−	88761.5i	0.0917343	−	0.158888i	0.908241	+	0.418448i	$0.137426\pi$
$200$		0			0
$201$	−981099.			−1.71286
$202$		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
$211$	329174.	−	190048.i	0.509001	−	0.293872i	−0.223422	−	0.974722i	$-0.571723\pi$
$211$	329174.	−	190048.i	0.509001	−	0.293872i	0.732423	+	0.680850i	$0.238389\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	258725.i		−	0.372983i
$218$		0			0
$219$	−1.07429e6	−	620241.i	−1.51360	−	0.873877i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−857540.	+	495101.i	−1.15476	+	0.666701i	−0.950043	−	0.312120i	$-0.898961\pi$
$223$	−857540.	+	495101.i	−1.15476	+	0.666701i	−0.204718	+	0.978821i	$0.565628\pi$
$224$		0			0
$225$	379688.	+	657638.i	0.500000	+	0.866025i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	1.45951e6			1.83915			0.919576	−	0.392913i	$-0.128533\pi$
$229$	1.45951e6			1.83915			0.919576	+	0.392913i	$0.128533\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	861394.	−	1.49198e6i	0.996165	−	1.72541i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	430198.	+	248375.i	0.477119	+	0.275465i	0.719215	−	0.694788i	$-0.244502\pi$
$241$	430198.	+	248375.i	0.477119	+	0.275465i	−0.242096	+	0.970252i	$0.577835\pi$
$242$		0			0
$243$	−797161.	−	460241.i	−0.866025	−	0.500000i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−1.24211e6	+	804601.i	−1.29544	+	0.839147i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		−	335022.i		−	0.310330i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−1.12643e6	−	1.95103e6i	−0.931707	−	1.61376i	−0.780403	−	0.625277i	$-0.784986\pi$
$271$	−1.12643e6	−	1.95103e6i	−0.931707	−	1.61376i	−0.151304	−	0.988487i	$-0.548347\pi$
$272$		0			0
$273$	−366525.			−0.297644
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.48661e6			1.94719			0.973596	−	0.228276i	$-0.0733089\pi$
$277$	2.48661e6			1.94719			0.973596	+	0.228276i	$0.0733089\pi$
$278$		0			0
$279$	−2.17789e6	−	1.25740e6i	−1.67504	−	0.967084i
$280$		0			0
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$	−976.000	−	1690.48i	−0.000724409	−	0.00125471i	0.865663	−	0.500627i	$-0.166897\pi$
$283$	−976.000	−	1690.48i	−0.000724409	−	0.00125471i	−0.866387	+	0.499373i	$0.833564\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−709928.	+	1.22963e6i	−0.500000	+	0.866025i
$290$		0			0
$291$	−994788.	+	1.72302e6i	−0.688649	+	1.19278i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	239037.	+	414025.i	0.152072	+	0.263397i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−2.05176e6	+	1.18458e6i	−1.24245	+	0.717331i	−0.969593	−	0.244723i	$-0.921303\pi$
$307$	−2.05176e6	+	1.18458e6i	−1.24245	+	0.717331i	−0.272860	+	0.962054i	$0.587970\pi$
$308$		0			0
$309$	−316453.	−	548114.i	−0.188544	−	0.326569i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	366949.	−	635574.i	0.211712	−	0.366695i	−0.740539	−	0.672014i	$-0.765430\pi$
$313$	366949.	−	635574.i	0.211712	−	0.366695i	0.952250	+	0.305318i	$0.0987629\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.54531e6	−	1.46954e6i	1.33670	−	0.771742i
$326$		0			0
$327$	1.71542e6	−	2.97119e6i	0.887157	−	1.53660i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	3.94899e6i		−	1.98114i	−0.136998	−	0.990571i	$-0.543745\pi$
$331$		−	3.94899e6i		−	1.98114i	0.136998	−	0.990571i	$-0.456255\pi$
$332$		0			0
$333$	−2.82014e6	−	1.62821e6i	−1.39367	−	0.804635i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	573320.	−	331006.i	0.274993	−	0.158767i	−0.356161	−	0.934424i	$-0.615915\pi$
$337$	573320.	−	331006.i	0.274993	−	0.158767i	0.631155	+	0.775657i	$0.282581\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−824725.			−0.378507
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−3.48766e6			−1.53275			−0.766373	−	0.642396i	$-0.777941\pi$
$349$	−3.48766e6			−1.53275			−0.766373	+	0.642396i	$0.777941\pi$
$350$		0			0
$351$	−1.78131e6	+	3.08532e6i	−0.771742	+	1.33670i
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−1.45079e6	−	2.00656e6i	−0.585916	−	0.810372i
$362$		0			0
$363$	2.17419e6	−	1.25527e6i	0.866025	−	0.500000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.29159e6	−	2.23710e6i	0.500563	−	0.867000i	−0.499437	−	0.866350i	$-0.666460\pi$
$367$	1.29159e6	−	2.23710e6i	0.500563	−	0.867000i	1.00000		0.000650122i	$-0.000206940\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	4.21532e6i		−	1.56877i	−0.620276	−	0.784384i	$-0.712979\pi$
$373$		−	4.21532e6i		−	1.56877i	0.620276	−	0.784384i	$-0.287021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			5.59235e6i			1.99984i	0.0124970	+	0.999922i	$0.496022\pi$
$379$			5.59235e6i			1.99984i	−0.0124970	+	0.999922i	$0.503978\pi$
$380$		0			0
$381$	3.84410e6			1.35669
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.64689e6			1.57719
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.50171e6	+	4.33309e6i	0.796638	+	1.37982i	0.921794	+	0.387681i	$0.126724\pi$
$397$	2.50171e6	+	4.33309e6i	0.796638	+	1.37982i	−0.125156	+	0.992137i	$0.539943\pi$
$398$		0			0
$399$	−31387.5	−	612432.i	−0.00987017	−	0.192586i
$400$		0			0
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−4.86664e6	+	8.42926e6i	−1.49268	+	2.58540i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−5.76755e6	−	3.32990e6i	−1.70484	−	0.984288i	−0.940703	−	0.339231i	$-0.889833\pi$
$409$	−5.76755e6	−	3.32990e6i	−1.70484	−	0.984288i	−0.764134	−	0.645057i	$-0.776834\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			7.08740e6i			1.99594i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	5.99385e6	−	3.46055e6i	1.64816	−	0.951568i	0.670364	−	0.742033i	$-0.266138\pi$
$421$	5.99385e6	−	3.46055e6i	1.64816	−	0.951568i	0.977801	−	0.209536i	$-0.0671953\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−228762.	+	396228.i	−0.0607177	+	0.105166i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$	4.67476e6	+	2.69898e6i	1.19823	+	0.691798i	0.960160	−	0.279450i	$-0.0901522\pi$
$433$	4.67476e6	+	2.69898e6i	1.19823	+	0.691798i	0.238069	+	0.971248i	$0.423486\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−44539.5	+	25714.9i	−0.0110302	+	0.00636830i	−0.505505	−	0.862824i	$-0.668694\pi$
$439$	−44539.5	+	25714.9i	−0.0110302	+	0.00636830i	0.494475	+	0.869192i	$0.335360\pi$
$440$		0			0
$441$	−1.96611e6	+	3.40541e6i	−0.481407	+	0.833821i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.98790e6	+	5.17520e6i	0.684102	+	1.18490i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.18576e6			0.265587			0.132793	−	0.991144i	$-0.457605\pi$
$457$	1.18576e6			0.265587			0.132793	+	0.991144i	$0.457605\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	2.92217e6			0.633510			0.316755	−	0.948507i	$-0.397407\pi$
$463$	2.92217e6			0.633510			0.316755	+	0.948507i	$0.397407\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−1.36264e6	−	786719.i	−0.286054	−	0.165153i
$470$		0			0
$471$	6.37080e6	+	3.67818e6i	1.32325	+	0.763978i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	2.67344e6	+	4.12715e6i	0.543671	+	0.839299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	−6.30179e6	+	1.09150e7i	−1.24194	+	2.15111i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			4.00717e6i			0.765623i	0.923827	+	0.382811i	$0.125044\pi$
$487$			4.00717e6i			0.765623i	−0.923827	+	0.382811i	$0.874956\pi$
$488$		0			0
$489$	4.75706e6	−	2.74649e6i	0.899636	−	0.519405i
$490$		0			0
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.08685e6	−	1.88247e6i	−0.195397	−	0.338437i	0.751634	−	0.659581i	$-0.229266\pi$
$499$	−1.08685e6	−	1.88247e6i	−0.195397	−	0.338437i	−0.947030	+	0.321144i	$0.895933\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	6.92893e6	+	4.00042e6i	1.19714	+	0.691171i
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−994712.	−	1.72289e6i	−0.168518	−	0.291881i
$512$		0			0
$513$	−5.30785e6	−	2.71220e6i	−0.890482	−	0.455018i
$514$		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
$523$	−1.06819e7	−	6.16718e6i	−1.70763	−	0.985898i	−0.937486	−	0.348023i	$-0.886853\pi$
$523$	−1.06819e7	−	6.16718e6i	−1.70763	−	0.985898i	−0.770140	−	0.637875i	$-0.779814\pi$
$524$		0			0
$525$			1.21785e6i			0.192839i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.21817e6	−	5.57404e6i	−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
$541$	1.51118e6	−	2.61743e6i	0.221984	−	0.384488i	−0.733426	−	0.679769i	$-0.762080\pi$
$541$	1.51118e6	−	2.61743e6i	0.221984	−	0.384488i	0.955410	+	0.295281i	$0.0954134\pi$
$542$		0			0
$543$	1.32483e7			1.92823
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.20515e7	−	6.95796e6i	−1.72216	−	0.994292i	−0.914430	−	0.404744i	$-0.867361\pi$
$547$	−1.20515e7	−	6.95796e6i	−1.72216	−	0.994292i	−0.807733	−	0.589548i	$-0.799306\pi$
$548$		0			0
$549$	2.22357e6	+	3.85134e6i	0.314862	+	0.545357i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.39276e6	−	1.38146e6i	0.332726	−	0.192099i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		−	1.79853e7i		−	2.43437i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−738113.	−	1.27845e6i	−0.0964195	−	0.167003i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	5.88780e6			0.755723			0.377862	−	0.925862i	$-0.376660\pi$
$571$	5.88780e6			0.755723			0.377862	+	0.925862i	$0.376660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.72229e6			0.215360			0.107680	−	0.994186i	$-0.465658\pi$
$577$	1.72229e6			0.215360			0.107680	+	0.994186i	$0.465658\pi$
$578$		0			0
$579$	6.23691e6	−	1.08026e7i	0.773166	−	1.33916i
$580$		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
$589$	−1.45013e7	−	7.40989e6i	−1.72234	−	0.880082i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1.59771e6i		−	0.183469i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$			3.11249e6i			0.351498i	0.984435	+	0.175749i	$0.0562346\pi$
$601$			3.11249e6i			0.351498i	−0.984435	+	0.175749i	$0.943765\pi$
$602$		0			0
$603$	−1.32448e7	+	7.64691e6i	−1.48338	+	0.856432i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			1.81550e7i			1.99998i	0.00464665	+	0.999989i	$0.498521\pi$
$607$			1.81550e7i			1.99998i	−0.00464665	+	0.999989i	$0.501479\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−7.91172e6	−	1.37035e7i	−0.850394	−	1.47292i	−0.880853	−	0.473389i	$-0.843031\pi$
$613$	−7.91172e6	−	1.37035e7i	−0.850394	−	1.47292i	0.0304599	−	0.999536i	$-0.490303\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	−1.87828e7			−1.97031			−0.985153	−	0.171676i	$-0.945082\pi$
$619$	−1.87828e7			−1.97031			−0.985153	+	0.171676i	$0.945082\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−4.88281e6	−	8.45728e6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	9.11576e6	−	1.57890e7i	0.911422	−	1.57863i	0.0993658	−	0.995051i	$-0.468319\pi$
$631$	9.11576e6	−	1.57890e7i	0.911422	−	1.57863i	0.812057	−	0.583579i	$-0.198348\pi$
$632$		0			0
$633$	2.96256e6	−	5.13131e6i	0.293872	−	0.509001i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.31802e7	+	7.60961e6i	1.28699	+	0.743043i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	6.69239e6	+	1.15916e7i	0.638342	+	1.10564i	0.985796	+	0.167944i	$0.0537129\pi$
$643$	6.69239e6	+	1.15916e7i	0.638342	+	1.10564i	−0.347454	+	0.937697i	$0.612954\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.01656e6	−	3.49279e6i	−0.186492	−	0.323013i
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.93372e7			−1.74775
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	−1.02418e7	−	5.91313e6i	−0.911748	−	0.526398i	−0.0307548	−	0.999527i	$-0.509791\pi$
$661$	−1.02418e7	−	5.91313e6i	−0.911748	−	0.526398i	−0.880993	+	0.473129i	$0.843124\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−7.71786e6	+	1.33677e7i	−0.666701	+	1.15476i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			7.04614e6i			0.599672i	0.953991	+	0.299836i	$0.0969319\pi$
$673$			7.04614e6i			0.599672i	−0.953991	+	0.299836i	$0.903068\pi$
$674$		0			0
$675$	1.02516e7	+	5.91874e6i	0.866025	+	0.500000i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−2.76330e6	+	1.59539e6i	−0.230014	+	0.132798i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.97033e7	−	1.13757e7i	1.59275	−	0.919576i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.73630e7			1.38335			0.691673	−	0.722211i	$-0.256874\pi$
$691$	1.73630e7			1.38335			0.691673	+	0.722211i	$0.256874\pi$
$692$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$	−1.87777e7	−	9.59503e6i	−1.43303	−	0.732248i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	467806.	−	810263.i	0.0349502	−	0.0605355i	−0.848021	−	0.529962i	$-0.822206\pi$
$709$	467806.	−	810263.i	0.0349502	−	0.0605355i	0.882971	+	0.469427i	$0.155539\pi$
$710$		0			0
$711$		−	2.68556e7i		−	1.99233i
$712$		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
$721$		−	1.01503e6i		−	0.0727174i
$722$		0			0
$723$	7.74357e6			0.550929
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−6.38099e6	−	1.10522e7i	−0.447767	−	0.775555i	0.550474	−	0.834853i	$-0.314447\pi$
$727$	−6.38099e6	−	1.10522e7i	−0.447767	−	0.775555i	−0.998240	+	0.0592978i	$0.981114\pi$
$728$		0			0
$729$	−1.43489e7			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.13028e7			−0.777006			−0.388503	−	0.921448i	$-0.627008\pi$
$733$	−1.13028e7			−0.777006			−0.388503	+	0.921448i	$0.627008\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	8.39325e6	+	1.45375e7i	0.565352	+	0.979218i	0.997017	+	0.0771842i	$0.0245930\pi$
$739$	8.39325e6	+	1.45375e7i	0.565352	+	0.979218i	−0.431665	+	0.902034i	$0.642074\pi$
$740$		0			0
$741$	−1.04973e7	+	2.05434e7i	−0.702314	+	1.37444i
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.25222e7	−	7.22971e6i	−0.810180	−	0.467758i	0.0368381	−	0.999321i	$-0.488271\pi$
$751$	−1.25222e7	−	7.22971e6i	−0.810180	−	0.467758i	−0.847019	+	0.531563i	$0.821605\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	5.64544e6	+	9.77820e6i	0.358062	+	0.620182i	0.987637	−	0.156758i	$-0.0501042\pi$
$757$	5.64544e6	+	9.77820e6i	0.358062	+	0.620182i	−0.629575	+	0.776940i	$0.716771\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	4.76505e6	−	2.75110e6i	0.296317	−	0.171079i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.25009e7	−	2.16521e7i	0.762296	−	1.32034i	−0.179368	−	0.983782i	$-0.557405\pi$
$769$	1.25009e7	−	2.16521e7i	0.762296	−	1.32034i	0.941664	−	0.336554i	$-0.109261\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	2.80078e7	+	1.61703e7i	1.67504	+	0.967084i
$776$		0			0
$777$	−2.61124e6	−	4.52280e6i	−0.155165	−	0.268754i
$778$		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
$787$			3.39531e7i			1.95408i	0.213052	+	0.977041i	$0.431660\pi$
$787$			3.39531e7i			1.95408i	−0.213052	+	0.977041i	$0.568340\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.49062e7	−	8.60608e6i	0.841750	−	0.485984i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		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		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	2.57610e7	+	1.48731e7i	1.37534	+	0.794053i	0.991594	−	0.129386i	$-0.0413007\pi$
$811$	2.57610e7	+	1.48731e7i	1.37534	+	0.794053i	0.383745	+	0.923439i	$0.374634\pi$
$812$		0			0
$813$	−3.04135e7	−	1.75592e7i	−1.61376	−	0.931707i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.00518e7	−	1.54017e6i	1.57513	−	0.0807262i
$818$		0			0
$819$	−4.94809e6	+	2.85678e6i	−0.257767	+	0.148822i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	1.11462e7	−	1.93057e7i	0.573623	−	0.993543i	−0.422567	−	0.906332i	$-0.638871\pi$
$823$	1.11462e7	−	1.93057e7i	0.573623	−	0.993543i	0.996190	−	0.0872118i	$-0.0277957\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$827$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$828$		0			0
$829$			3.58406e7i			1.81130i	0.424030	+	0.905648i	$0.360615\pi$
$829$			3.58406e7i			1.81130i	−0.424030	+	0.905648i	$0.639385\pi$
$830$		0			0
$831$	3.35693e7	−	1.93812e7i	1.68632	−	0.973596i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.92020e7			−1.93417
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	1.02556e7	+	1.77632e7i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	4.02628e6			0.192839
$848$		0			0
$849$	−26352.0	−	15214.3i	−0.00125471	−	0.000724409i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.91096e7	−	3.30987e7i	−0.899245	−	1.55754i	−0.828461	−	0.560047i	$-0.810783\pi$
$853$	−1.91096e7	−	3.30987e7i	−0.899245	−	1.55754i	−0.0707842	−	0.997492i	$-0.522550\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−1.94487e7	+	3.36862e7i	−0.899307	+	1.55765i	−0.0709259	+	0.997482i	$0.522595\pi$
$859$	−1.94487e7	+	3.36862e7i	−0.899307	+	1.55765i	−0.828381	+	0.560164i	$0.810738\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			2.21334e7i			1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	2.95965e7	+	5.12626e7i	1.32189	+	2.28958i
$872$		0			0
$873$			3.10144e7i			1.37730i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.67759e7	−	9.68557e6i	0.736524	−	0.425232i	−0.0842800	−	0.996442i	$-0.526859\pi$
$877$	1.67759e7	−	9.68557e6i	0.736524	−	0.425232i	0.820804	+	0.571210i	$0.193526\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−2.29033e7	+	3.96697e7i	−0.988545	+	1.71221i	−0.363567	+	0.931568i	$0.618441\pi$
$883$	−2.29033e7	+	3.96697e7i	−0.988545	+	1.71221i	−0.624978	+	0.780643i	$0.714892\pi$
$884$		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
$889$	5.33903e6	+	3.08249e6i	0.226573	+	0.130812i
$890$		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
$903$	6.45401e6	+	3.72623e6i	0.263397	+	0.152072i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.78671e7	−	2.18626e7i	1.52842	−	0.882436i	0.528995	−	0.848625i	$-0.322569\pi$
$907$	3.78671e7	−	2.18626e7i	1.52842	−	0.882436i	0.999428	−	0.0338109i	$-0.0107644\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	4.21821e6			0.164755			0.0823777	−	0.996601i	$-0.473749\pi$
$919$	4.21821e6			0.164755			0.0823777	+	0.996601i	$0.473749\pi$
$920$		0			0
$921$	−1.84658e7	+	3.19837e7i	−0.717331	+	1.24245i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.62672e7	+	2.09389e7i	1.39367	+	0.804635i
$926$		0			0
$927$	−8.54424e6	−	4.93302e6i	−0.326569	−	0.188544i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−1.15863e7	+	2.26747e7i	−0.438098	+	0.857368i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.44854e7	−	4.24100e7i	0.911085	−	1.57805i	0.0985501	−	0.995132i	$-0.468580\pi$
$937$	2.44854e7	−	4.24100e7i	0.911085	−	1.57805i	0.812535	−	0.582913i	$-0.198087\pi$
$938$		0			0
$939$		−	1.14403e7i		−	0.423423i
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$			7.48425e7i			2.69763i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−7.84727e7			−2.74101
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.61727e7	−	2.80119e7i	−0.556180	−	0.963332i	−0.997811	−	0.0661347i	$-0.978933\pi$
$967$	−1.61727e7	−	2.80119e7i	−0.556180	−	0.963332i	0.441631	−	0.897197i	$-0.354400\pi$
$968$		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
$973$	−5.68321e6	+	9.84361e6i	−0.192447	+	0.333328i
$974$		0			0
$975$	2.29078e7	−	3.96775e7i	0.771742	−	1.33670i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	5.34814e7i		−	1.77431i
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	5.08328e7	−	2.93483e7i	1.64422	−	0.949290i	0.664908	−	0.746925i	$-0.268471\pi$
$991$	5.08328e7	−	2.93483e7i	1.64422	−	0.949290i	0.979310	−	0.202365i	$-0.0648626\pi$
$992$		0			0
$993$	−3.07793e7	−	5.33113e7i	−0.990571	−	1.71572i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−7.82868e6	+	1.35597e7i	−0.249431	+	0.432027i	−0.963368	−	0.268183i	$-0.913577\pi$
$997$	−7.82868e6	+	1.35597e7i	−0.249431	+	0.432027i	0.713937	+	0.700210i	$0.246910\pi$
$998$		0			0
$999$	−5.07625e7			−1.60927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	228.6.p.b.221.1	yes	2
3.2	odd	2	CM	228.6.p.b.221.1	yes	2
19.8	odd	6	inner	228.6.p.b.65.1	✓	2
57.8	even	6	inner	228.6.p.b.65.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.6.p.b.65.1	✓	2	19.8	odd	6	inner
228.6.p.b.65.1	✓	2	57.8	even	6	inner
228.6.p.b.221.1	yes	2	1.1	even	1	trivial
228.6.p.b.221.1	yes	2	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	228.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(13.5000 - 7.79423i) q^{3} +25.0000 q^{7} +(121.500 - 210.444i) q^{9} +O(q^{10})\)	\(q+(13.5000 - 7.79423i) q^{3} +25.0000 q^{7} +(121.500 - 210.444i) q^{9} +(-814.500 - 470.252i) q^{13} +(716.000 - 1401.23i) q^{19} +(337.500 - 194.856i) q^{21} +(-1562.50 + 2706.33i) q^{25} -3788.00i q^{27} -10349.0i q^{31} -13400.9i q^{37} -14661.0 q^{39} +(9561.50 + 16561.0i) q^{43} -16182.0 q^{49} +(-1255.50 - 24497.3i) q^{57} +(-9150.50 + 15849.1i) q^{61} +(3037.50 - 5261.10i) q^{63} +(-54505.5 - 31468.8i) q^{67} +(-39788.5 - 68915.7i) q^{73} +48713.9i q^{75} +(95710.5 - 55258.5i) q^{79} +(-29524.5 - 51137.9i) q^{81} +(-20362.5 - 11756.3i) q^{91} +(-80662.5 - 139712. i) q^{93} +(-110532. + 63815.7i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 27 q^{3} + 50 q^{7} + 243 q^{9}+O(q^{10}) \) 2 * q + 27 * q^3 + 50 * q^7 + 243 * q^9	\( 2 q + 27 q^{3} + 50 q^{7} + 243 q^{9} - 1629 q^{13} + 1432 q^{19} + 675 q^{21} - 3125 q^{25} - 29322 q^{39} + 19123 q^{43} - 32364 q^{49} - 2511 q^{57} - 18301 q^{61} + 6075 q^{63} - 109011 q^{67} - 79577 q^{73} + 191421 q^{79} - 59049 q^{81} - 40725 q^{91} - 161325 q^{93} - 221064 q^{97}+O(q^{100}) \) 2 * q + 27 * q^3 + 50 * q^7 + 243 * q^9 - 1629 * q^13 + 1432 * q^19 + 675 * q^21 - 3125 * q^25 - 29322 * q^39 + 19123 * q^43 - 32364 * q^49 - 2511 * q^57 - 18301 * q^61 + 6075 * q^63 - 109011 * q^67 - 79577 * q^73 + 191421 * q^79 - 59049 * q^81 - 40725 * q^91 - 161325 * q^93 - 221064 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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