LMFDB - Embedded newform 3672.1.dv.a.2339.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3672,1,Mod(683,3672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3672, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 24, 8, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3672.683");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3672 = 2^{3} \cdot 3^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3672.dv (of order $48$, degree $16$, 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:	$1.83256672639$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1224)
Projective image:	$D_{48}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{48} - \cdots)$

Embedding invariants

Embedding label			2339.1
Root			$0.608761 + 0.793353i$ of defining polynomial
Character	$\chi$	$=$	3672.2339
Dual form			3672.1.dv.a.1763.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.130526 + 0.991445i) q^{2} +(-0.965926 + 0.258819i) q^{4} +(-0.382683 - 0.923880i) q^{8} +(-0.793353 - 0.391239i) q^{11} +(0.866025 - 0.500000i) q^{16} +(0.991445 - 0.130526i) q^{17} +(0.198092 - 0.478235i) q^{19} +(0.284338 - 0.837633i) q^{22} +(0.793353 + 0.608761i) q^{25} +(0.608761 + 0.793353i) q^{32} +(0.258819 + 0.965926i) q^{34} +(0.500000 + 0.133975i) q^{38} +(1.34861 + 1.18270i) q^{41} +(1.20711 - 1.57313i) q^{43} +(0.867580 + 0.172572i) q^{44} +(-0.793353 + 0.608761i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(1.57313 + 0.207107i) q^{59} +(-0.707107 + 0.707107i) q^{64} +(0.226078 + 0.130526i) q^{67} +(-0.923880 + 0.382683i) q^{68} +(-0.128293 + 0.0255190i) q^{73} +(-0.0675653 + 0.513210i) q^{76} +(-0.996552 + 1.49144i) q^{82} +(-0.758819 + 0.0999004i) q^{83} +(1.71723 + 0.991445i) q^{86} +(-0.0578541 + 0.882683i) q^{88} +(1.30656 - 1.30656i) q^{89} +(-0.483342 + 0.423880i) q^{97} +(-0.707107 - 0.707107i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{38} + 8 q^{43} - 8 q^{50} - 8 q^{83}+O(q^{100}) $ 16 * q + 8 * q^38 + 8 * q^43 - 8 * q^50 - 8 * q^83

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3672\mathbb{Z}\right)^\times$.

$n$	$137$	$649$	$919$	$1837$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{3}{16}\right)$	$-1$	$-1$

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 $

$2$	0.130526	+	0.991445i	0.130526	+	0.991445i
$2$	0.130526	+	0.991445i	0.130526	+	0.991445i
$3$		0			0
$3$		0			0
$4$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$5$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$5$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$6$		0			0
$7$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$7$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$8$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$9$		0			0
$10$		0			0
$11$	−0.793353	−	0.391239i	−0.793353	−	0.391239i		−	1.00000i	$-0.5\pi$
$11$	−0.793353	−	0.391239i	−0.793353	−	0.391239i	−0.793353	+	0.608761i	$0.791667\pi$
$12$		0			0
$13$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$13$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$14$		0			0
$15$		0			0
$16$	0.866025	−	0.500000i	0.866025	−	0.500000i
$17$	0.991445	−	0.130526i	0.991445	−	0.130526i
$17$	0.991445	−	0.130526i	0.991445	−	0.130526i
$18$		0			0
$19$	0.198092	−	0.478235i	0.198092	−	0.478235i	−0.793353	−	0.608761i	$-0.791667\pi$
$19$	0.198092	−	0.478235i	0.198092	−	0.478235i	0.991445	+	0.130526i	$0.0416667\pi$
$20$		0			0
$21$		0			0
$22$	0.284338	−	0.837633i	0.284338	−	0.837633i
$23$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$23$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$24$		0			0
$25$	0.793353	+	0.608761i	0.793353	+	0.608761i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$29$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$30$		0			0
$31$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$31$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$32$	0.608761	+	0.793353i	0.608761	+	0.793353i
$33$		0			0
$34$	0.258819	+	0.965926i	0.258819	+	0.965926i
$35$		0			0
$36$		0			0
$37$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$37$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$38$	0.500000	+	0.133975i	0.500000	+	0.133975i
$39$		0			0
$40$		0			0
$41$	1.34861	+	1.18270i	1.34861	+	1.18270i	0.965926	+	0.258819i	$0.0833333\pi$
$41$	1.34861	+	1.18270i	1.34861	+	1.18270i	0.382683	+	0.923880i	$0.375000\pi$
$42$		0			0
$43$	1.20711	−	1.57313i	1.20711	−	1.57313i	0.500000	−	0.866025i	$-0.333333\pi$
$43$	1.20711	−	1.57313i	1.20711	−	1.57313i	0.707107	−	0.707107i	$-0.250000\pi$
$44$	0.867580	+	0.172572i	0.867580	+	0.172572i
$45$		0			0
$46$		0			0
$47$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$47$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$48$		0			0
$49$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$50$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$51$		0			0
$52$		0			0
$53$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$53$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.57313	+	0.207107i	1.57313	+	0.207107i	0.866025	−	0.500000i	$-0.166667\pi$
$59$	1.57313	+	0.207107i	1.57313	+	0.207107i	0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$61$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$65$		0			0
$66$		0			0
$67$	0.226078	+	0.130526i	0.226078	+	0.130526i	0.608761	−	0.793353i	$-0.291667\pi$
$67$	0.226078	+	0.130526i	0.226078	+	0.130526i	−0.382683	+	0.923880i	$0.625000\pi$
$68$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$69$		0			0
$70$		0			0
$71$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$71$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$72$		0			0
$73$	−0.128293	+	0.0255190i	−0.128293	+	0.0255190i	−0.258819	−	0.965926i	$-0.583333\pi$
$73$	−0.128293	+	0.0255190i	−0.128293	+	0.0255190i	0.130526	+	0.991445i	$0.458333\pi$
$74$		0			0
$75$		0			0
$76$	−0.0675653	+	0.513210i	−0.0675653	+	0.513210i
$77$		0			0
$78$		0			0
$79$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$79$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$80$		0			0
$81$		0			0
$82$	−0.996552	+	1.49144i	−0.996552	+	1.49144i
$83$	−0.758819	+	0.0999004i	−0.758819	+	0.0999004i	−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−0.758819	+	0.0999004i	−0.758819	+	0.0999004i	−0.258819	+	0.965926i	$0.583333\pi$
$84$		0			0
$85$		0			0
$86$	1.71723	+	0.991445i	1.71723	+	0.991445i
$87$		0			0
$88$	−0.0578541	+	0.882683i	−0.0578541	+	0.882683i
$89$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$89$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.483342	+	0.423880i	−0.483342	+	0.423880i	−0.866025	−	0.500000i	$-0.833333\pi$
$97$	−0.483342	+	0.423880i	−0.483342	+	0.423880i	0.382683	+	0.923880i	$0.375000\pi$
$98$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$99$		0			0
$100$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.369474	+	1.85747i	0.369474	+	1.85747i	0.500000	+	0.866025i	$0.333333\pi$
$107$	0.369474	+	1.85747i	0.369474	+	1.85747i	−0.130526	+	0.991445i	$0.541667\pi$
$108$		0			0
$109$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$109$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.0255190	+	0.389345i	0.0255190	+	0.389345i	0.991445	+	0.130526i	$0.0416667\pi$
$113$	0.0255190	+	0.389345i	0.0255190	+	0.389345i	−0.965926	+	0.258819i	$0.916667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$			1.58671i			1.58671i
$119$		0			0
$120$		0			0
$121$	−0.132420	−	0.172572i	−0.132420	−	0.172572i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$127$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$128$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$129$		0			0
$130$		0			0
$131$	0.534534	−	1.57469i	0.534534	−	1.57469i	−0.258819	−	0.965926i	$-0.583333\pi$
$131$	0.534534	−	1.57469i	0.534534	−	1.57469i	0.793353	−	0.608761i	$-0.208333\pi$
$132$		0			0
$133$		0			0
$134$	−0.0999004	+	0.241181i	−0.0999004	+	0.241181i
$135$		0			0
$136$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$137$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$137$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$138$		0			0
$139$	−1.18270	+	0.583242i	−1.18270	+	0.583242i	−0.923880	−	0.382683i	$-0.875000\pi$
$139$	−1.18270	+	0.583242i	−1.18270	+	0.583242i	−0.258819	+	0.965926i	$0.583333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−0.0420463	−	0.123864i	−0.0420463	−	0.123864i
$147$		0			0
$148$		0			0
$149$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$149$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$150$		0			0
$151$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$151$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$152$	−0.517638			−0.517638
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$157$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.382683	−	1.92388i	0.382683	−	1.92388i		−	1.00000i	$-0.5\pi$
$163$	0.382683	−	1.92388i	0.382683	−	1.92388i	0.382683	−	0.923880i	$-0.375000\pi$
$164$	−1.60876	−	0.793353i	−1.60876	−	0.793353i
$165$		0			0
$166$	−0.198092	−	0.739288i	−0.198092	−	0.739288i
$167$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$167$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$168$		0			0
$169$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$170$		0			0
$171$		0			0
$172$	−0.758819	+	1.83195i	−0.758819	+	1.83195i
$173$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$173$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$174$		0			0
$175$		0			0
$176$	−0.882683	+	0.0578541i	−0.882683	+	0.0578541i
$177$		0			0
$178$	1.46593	+	1.12484i	1.46593	+	1.12484i
$179$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$179$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$180$		0			0
$181$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$181$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.837633	−	0.284338i	−0.837633	−	0.284338i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$191$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$192$		0			0
$193$	0.0983454	+	1.50046i	0.0983454	+	1.50046i	0.707107	+	0.707107i	$0.250000\pi$
$193$	0.0983454	+	1.50046i	0.0983454	+	1.50046i	−0.608761	+	0.793353i	$0.708333\pi$
$194$	−0.483342	−	0.423880i	−0.483342	−	0.423880i
$195$		0			0
$196$	0.608761	−	0.793353i	0.608761	−	0.793353i
$197$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$197$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$198$		0			0
$199$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$199$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$200$	0.258819	−	0.965926i	0.258819	−	0.965926i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.344261	+	0.301908i	−0.344261	+	0.301908i
$210$		0			0
$211$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.130526	−	0.991445i	$-0.458333\pi$
$211$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.965926	−	0.258819i	$-0.0833333\pi$
$212$		0			0
$213$		0			0
$214$	−1.79335	+	0.608761i	−1.79335	+	0.608761i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$223$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$224$		0			0
$225$		0			0
$226$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$227$	−0.665060	+	1.34861i	−0.665060	+	1.34861i	0.258819	+	0.965926i	$0.416667\pi$
$227$	−0.665060	+	1.34861i	−0.665060	+	1.34861i	−0.923880	+	0.382683i	$0.875000\pi$
$228$		0			0
$229$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$229$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.29335	−	0.257264i	1.29335	−	0.257264i	0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.29335	−	0.257264i	1.29335	−	0.257264i	0.793353	+	0.608761i	$0.208333\pi$
$234$		0			0
$235$		0			0
$236$	−1.57313	+	0.207107i	−1.57313	+	0.207107i
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	−0.130526	+	1.99144i	−0.130526	+	1.99144i			1.00000i	$0.5\pi$
$241$	−0.130526	+	1.99144i	−0.130526	+	1.99144i	−0.130526	+	0.991445i	$0.541667\pi$
$242$	0.153812	−	0.153812i	0.153812	−	0.153812i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.860919	−	0.860919i	−0.860919	−	0.860919i	0.130526	−	0.991445i	$-0.458333\pi$
$251$	−0.860919	−	0.860919i	−0.860919	−	0.860919i	−0.991445	+	0.130526i	$0.958333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.500000	−	0.866025i	0.500000	−	0.866025i
$257$	−0.965926	+	0.741181i	−0.965926	+	0.741181i	−0.965926	−	0.258819i	$-0.916667\pi$
$257$	−0.965926	+	0.741181i	−0.965926	+	0.741181i			1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.63099	+	0.324423i	1.63099	+	0.324423i
$263$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$263$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.252157	−	0.0675653i	−0.252157	−	0.0675653i
$269$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$269$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	0.793353	−	0.608761i	0.793353	−	0.608761i
$273$		0			0
$274$	1.17604	+	1.53264i	1.17604	+	1.53264i
$275$	−0.391239	−	0.793353i	−0.391239	−	0.793353i
$276$		0			0
$277$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$277$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$278$	−0.732626	−	1.09645i	−0.732626	−	1.09645i
$279$		0			0
$280$		0			0
$281$	−1.12197	−	0.860919i	−1.12197	−	0.860919i	−0.130526	−	0.991445i	$-0.541667\pi$
$281$	−1.12197	−	0.860919i	−1.12197	−	0.860919i	−0.991445	+	0.130526i	$0.958333\pi$
$282$		0			0
$283$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i	−0.608761	−	0.793353i	$-0.708333\pi$
$283$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i	−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.965926	−	0.258819i	0.965926	−	0.258819i
$290$		0			0
$291$		0			0
$292$	0.117317	−	0.0578541i	0.117317	−	0.0578541i
$293$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$293$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.0675653	−	0.513210i	−0.0675653	−	0.513210i
$305$		0			0
$306$		0			0
$307$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$307$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$311$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$312$		0			0
$313$	−0.608761	−	1.79335i	−0.608761	−	1.79335i	−0.608761	−	0.793353i	$-0.708333\pi$
$313$	−0.608761	−	1.79335i	−0.608761	−	1.79335i		−	1.00000i	$-0.5\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$317$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.133975	−	0.500000i	0.133975	−	0.500000i
$324$		0			0
$325$		0			0
$326$	1.95737	+	0.128293i	1.95737	+	0.128293i
$327$		0			0
$328$	0.576581	−	1.69855i	0.576581	−	1.69855i
$329$		0			0
$330$		0			0
$331$	−1.46593	−	1.12484i	−1.46593	−	1.12484i	−0.965926	−	0.258819i	$-0.916667\pi$
$331$	−1.46593	−	1.12484i	−1.46593	−	1.12484i	−0.500000	−	0.866025i	$-0.666667\pi$
$332$	0.707107	−	0.292893i	0.707107	−	0.292893i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.583242	−	1.18270i	−0.583242	−	1.18270i	−0.965926	−	0.258819i	$-0.916667\pi$
$337$	−0.583242	−	1.18270i	−0.583242	−	1.18270i	0.382683	−	0.923880i	$-0.375000\pi$
$338$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−1.91532	−	0.513210i	−1.91532	−	0.513210i
$345$		0			0
$346$		0			0
$347$	0.0983454	+	0.0862466i	0.0983454	+	0.0862466i	0.707107	−	0.707107i	$-0.250000\pi$
$347$	0.0983454	+	0.0862466i	0.0983454	+	0.0862466i	−0.608761	+	0.793353i	$0.708333\pi$
$348$		0			0
$349$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$349$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$350$		0			0
$351$		0			0
$352$	−0.172572	−	0.867580i	−0.172572	−	0.867580i
$353$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i	−0.991445	−	0.130526i	$-0.958333\pi$
$353$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i	0.923880	+	0.382683i	$0.125000\pi$
$354$		0			0
$355$		0			0
$356$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$357$		0			0
$358$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$359$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$359$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$360$		0			0
$361$	0.517638	+	0.517638i	0.517638	+	0.517638i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$367$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$373$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$374$	0.172572	−	0.867580i	0.172572	−	0.867580i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.95737	+	0.389345i	−1.95737	+	0.389345i	−0.965926	+	0.258819i	$0.916667\pi$
$379$	−1.95737	+	0.389345i	−1.95737	+	0.389345i	−0.991445	+	0.130526i	$0.958333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$383$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$384$		0			0
$385$		0			0
$386$	−1.47479	+	0.293353i	−1.47479	+	0.293353i
$387$		0			0
$388$	0.357164	−	0.534534i	0.357164	−	0.534534i
$389$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$389$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$390$		0			0
$391$		0			0
$392$	0.866025	+	0.500000i	0.866025	+	0.500000i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$397$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$398$		0			0
$399$		0			0
$400$	0.991445	+	0.130526i	0.991445	+	0.130526i
$401$	−0.423880	+	0.483342i	−0.423880	+	0.483342i	−0.923880	−	0.382683i	$-0.875000\pi$
$401$	−0.423880	+	0.483342i	−0.423880	+	0.483342i	0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.793353	−	1.37413i	0.793353	−	1.37413i	−0.130526	−	0.991445i	$-0.541667\pi$
$409$	0.793353	−	1.37413i	0.793353	−	1.37413i	0.923880	−	0.382683i	$-0.125000\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−0.344261	−	0.301908i	−0.344261	−	0.301908i
$419$	−0.0255190	−	0.389345i	−0.0255190	−	0.389345i	−0.991445	−	0.130526i	$-0.958333\pi$
$419$	−0.0255190	−	0.389345i	−0.0255190	−	0.389345i	0.965926	−	0.258819i	$-0.0833333\pi$
$420$		0			0
$421$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$421$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$422$	1.38268	+	0.923880i	1.38268	+	0.923880i
$423$		0			0
$424$		0			0
$425$	0.866025	+	0.500000i	0.866025	+	0.500000i
$426$		0			0
$427$		0			0
$428$	−0.837633	−	1.69855i	−0.837633	−	1.69855i
$429$		0			0
$430$		0			0
$431$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$431$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$432$		0			0
$433$	1.78480	−	0.739288i	1.78480	−	0.739288i	0.793353	−	0.608761i	$-0.208333\pi$
$433$	1.78480	−	0.739288i	1.78480	−	0.739288i	0.991445	−	0.130526i	$-0.0416667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$439$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.226078	−	0.130526i	0.226078	−	0.130526i	−0.382683	−	0.923880i	$-0.625000\pi$
$443$	0.226078	−	0.130526i	0.226078	−	0.130526i	0.608761	+	0.793353i	$0.291667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	0.130526	+	0.991445i	$0.458333\pi$
$449$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$	−0.607206	−	1.46593i	−0.607206	−	1.46593i
$452$	−0.125419	−	0.369474i	−0.125419	−	0.369474i
$453$		0			0
$454$	−1.42388	−	0.483342i	−1.42388	−	0.483342i
$455$		0			0
$456$		0			0
$457$	−0.252157	−	1.91532i	−0.252157	−	1.91532i	−0.382683	−	0.923880i	$-0.625000\pi$
$457$	−0.252157	−	1.91532i	−0.252157	−	1.91532i	0.130526	−	0.991445i	$-0.458333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$461$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$462$		0			0
$463$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$463$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$464$		0			0
$465$		0			0
$466$	0.423880	+	1.24871i	0.423880	+	1.24871i
$467$	−0.739288	−	1.78480i	−0.739288	−	1.78480i	−0.608761	−	0.793353i	$-0.708333\pi$
$467$	−0.739288	−	1.78480i	−0.739288	−	1.78480i	−0.130526	−	0.991445i	$-0.541667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.410670	−	1.53264i	−0.410670	−	1.53264i
$473$	−1.57313	+	0.775783i	−1.57313	+	0.775783i
$474$		0			0
$475$	0.448288	−	0.258819i	0.448288	−	0.258819i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$479$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$480$		0			0
$481$		0			0
$482$	−1.99144	+	0.130526i	−1.99144	+	0.130526i
$483$		0			0
$484$	0.172572	+	0.132420i	0.172572	+	0.132420i
$485$		0			0
$486$		0			0
$487$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$487$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.17604	−	1.53264i	−1.17604	−	1.53264i	−0.793353	−	0.608761i	$-0.791667\pi$
$491$	−1.17604	−	1.53264i	−1.17604	−	1.53264i	−0.382683	−	0.923880i	$-0.625000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.130526	+	1.99144i	0.130526	+	1.99144i	0.130526	+	0.991445i	$0.458333\pi$
$499$	0.130526	+	1.99144i	0.130526	+	1.99144i			1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$	0.741181	−	0.965926i	0.741181	−	0.965926i
$503$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$503$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$504$		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
$511$		0			0
$512$	0.923880	+	0.382683i	0.923880	+	0.382683i
$513$		0			0
$514$	−0.860919	−	0.860919i	−0.860919	−	0.860919i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.65938	−	1.10876i	1.65938	−	1.10876i	0.793353	−	0.608761i	$-0.208333\pi$
$521$	1.65938	−	1.10876i	1.65938	−	1.10876i	0.866025	−	0.500000i	$-0.166667\pi$
$522$		0			0
$523$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$523$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$524$	−0.108761	+	1.65938i	−0.108761	+	1.65938i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.991445	−	0.130526i	0.991445	−	0.130526i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.0340742	−	0.258819i	0.0340742	−	0.258819i
$537$		0			0
$538$		0			0
$539$	0.867580	−	0.172572i	0.867580	−	0.172572i
$540$		0			0
$541$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$541$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$542$		0			0
$543$		0			0
$544$	0.707107	+	0.707107i	0.707107	+	0.707107i
$545$		0			0
$546$		0			0
$547$	−0.0420463	+	0.641502i	−0.0420463	+	0.641502i	0.923880	+	0.382683i	$0.125000\pi$
$547$	−0.0420463	+	0.641502i	−0.0420463	+	0.641502i	−0.965926	+	0.258819i	$0.916667\pi$
$548$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$549$		0			0
$550$	0.735499	−	0.491445i	0.735499	−	0.491445i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.991445	−	0.869474i	0.991445	−	0.869474i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.707107	−	1.22474i	0.707107	−	1.22474i
$563$	−1.57313	+	1.20711i	−1.57313	+	1.20711i	−0.707107	+	0.707107i	$0.750000\pi$
$563$	−1.57313	+	1.20711i	−1.57313	+	1.20711i	−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$567$		0			0
$568$		0			0
$569$	−1.20711	+	1.57313i	−1.20711	+	1.57313i	−0.500000	+	0.866025i	$0.666667\pi$
$569$	−1.20711	+	1.57313i	−1.20711	+	1.57313i	−0.707107	+	0.707107i	$0.750000\pi$
$570$		0			0
$571$	1.34861	+	1.18270i	1.34861	+	1.18270i	0.965926	+	0.258819i	$0.0833333\pi$
$571$	1.34861	+	1.18270i	1.34861	+	1.18270i	0.382683	+	0.923880i	$0.375000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$577$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$578$	0.382683	+	0.923880i	0.382683	+	0.923880i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.0726721	+	0.108761i	0.0726721	+	0.108761i
$585$		0			0
$586$		0			0
$587$	1.37413	+	1.05441i	1.37413	+	1.05441i	0.991445	+	0.130526i	$0.0416667\pi$
$587$	1.37413	+	1.05441i	1.37413	+	1.05441i	0.382683	+	0.923880i	$0.375000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$593$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$599$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$600$		0			0
$601$	−1.69855	−	0.837633i	−1.69855	−	0.837633i	−0.991445	−	0.130526i	$-0.958333\pi$
$601$	−1.69855	−	0.837633i	−1.69855	−	0.837633i	−0.707107	−	0.707107i	$-0.750000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$607$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$608$	0.500000	−	0.133975i	0.500000	−	0.133975i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	0.226078	+	1.71723i	0.226078	+	1.71723i
$615$		0			0
$616$		0			0
$617$	−0.123864	−	0.0420463i	−0.123864	−	0.0420463i	0.258819	−	0.965926i	$-0.416667\pi$
$617$	−0.123864	−	0.0420463i	−0.123864	−	0.0420463i	−0.382683	+	0.923880i	$0.625000\pi$
$618$		0			0
$619$	−0.206647	−	0.608761i	−0.206647	−	0.608761i	0.793353	−	0.608761i	$-0.208333\pi$
$619$	−0.206647	−	0.608761i	−0.206647	−	0.608761i	−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.258819	+	0.965926i	0.258819	+	0.965926i
$626$	1.69855	−	0.837633i	1.69855	−	0.837633i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.31587	−	1.50046i	−1.31587	−	1.50046i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$	−1.31587	−	1.50046i	−1.31587	−	1.50046i	−0.608761	−	0.793353i	$-0.708333\pi$
$642$		0			0
$643$	−0.0578541	−	0.117317i	−0.0578541	−	0.117317i	0.866025	−	0.500000i	$-0.166667\pi$
$643$	−0.0578541	−	0.117317i	−0.0578541	−	0.117317i	−0.923880	+	0.382683i	$0.875000\pi$
$644$		0			0
$645$		0			0
$646$	0.513210	+	0.0675653i	0.513210	+	0.0675653i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−1.16702	−	0.779779i	−1.16702	−	0.779779i
$650$		0			0
$651$		0			0
$652$	0.128293	+	1.95737i	0.128293	+	1.95737i
$653$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$653$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$654$		0			0
$655$		0			0
$656$	1.75928	+	0.349942i	1.75928	+	0.349942i
$657$		0			0
$658$		0			0
$659$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$659$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$660$		0			0
$661$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$661$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$662$	0.923880	−	1.60021i	0.923880	−	1.60021i
$663$		0			0
$664$	0.382683	+	0.662827i	0.382683	+	0.662827i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.05217	+	0.357164i	−1.05217	+	0.357164i	−0.793353	−	0.608761i	$-0.791667\pi$
$673$	−1.05217	+	0.357164i	−1.05217	+	0.357164i	−0.258819	+	0.965926i	$0.583333\pi$
$674$	1.09645	−	0.732626i	1.09645	−	0.732626i
$675$		0			0
$676$	0.707107	−	0.707107i	0.707107	−	0.707107i
$677$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$677$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.0726721	−	0.108761i	0.0726721	−	0.108761i	−0.793353	−	0.608761i	$-0.791667\pi$
$683$	0.0726721	−	0.108761i	0.0726721	−	0.108761i	0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.258819	−	1.96593i	0.258819	−	1.96593i
$689$		0			0
$690$		0			0
$691$	−0.491445	+	0.996552i	−0.491445	+	0.996552i	0.500000	+	0.866025i	$0.333333\pi$
$691$	−0.491445	+	0.996552i	−0.491445	+	0.996552i	−0.991445	+	0.130526i	$0.958333\pi$
$692$		0			0
$693$		0			0
$694$	−0.0726721	+	0.108761i	−0.0726721	+	0.108761i
$695$		0			0
$696$		0			0
$697$	1.49144	+	0.996552i	1.49144	+	0.996552i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$	0.837633	−	0.284338i	0.837633	−	0.284338i
$705$		0			0
$706$	−0.258819	−	0.0340742i	−0.258819	−	0.0340742i
$707$		0			0
$708$		0			0
$709$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$709$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$710$		0			0
$711$		0			0
$712$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.12197	−	0.860919i	1.12197	−	0.860919i
$717$		0			0
$718$		0			0
$719$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$719$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$720$		0			0
$721$		0			0
$722$	−0.445644	+	0.580775i	−0.445644	+	0.580775i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$727$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.991445	−	1.71723i	0.991445	−	1.71723i
$732$		0			0
$733$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$733$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.128293	−	0.192004i	−0.128293	−	0.192004i
$738$		0			0
$739$	−1.60021	+	0.662827i	−1.60021	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$739$	−1.60021	+	0.662827i	−1.60021	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$743$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	0.882683	+	0.0578541i	0.882683	+	0.0578541i
$749$		0			0
$750$		0			0
$751$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$751$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$757$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$758$	−0.641502	−	1.88981i	−0.641502	−	1.88981i
$759$		0			0
$760$		0			0
$761$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$761$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$769$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$		0			0
$772$	−0.483342	−	1.42388i	−0.483342	−	1.42388i
$773$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$773$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$774$		0			0
$775$		0			0
$776$	0.576581	+	0.284338i	0.576581	+	0.284338i
$777$		0			0
$778$		0			0
$779$	0.832756	−	0.410670i	0.832756	−	0.410670i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$785$		0			0
$786$		0			0
$787$	−0.125419	+	0.369474i	−0.125419	+	0.369474i	−0.991445	−	0.130526i	$-0.958333\pi$
$787$	−0.125419	+	0.369474i	−0.125419	+	0.369474i	0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$797$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$798$		0			0
$799$		0			0
$800$			1.00000i			1.00000i
$801$		0			0
$802$	−0.534534	−	0.357164i	−0.534534	−	0.357164i
$803$	0.111766	+	0.0299475i	0.111766	+	0.0299475i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.29335	−	0.257264i	−1.29335	−	0.257264i	−0.500000	−	0.866025i	$-0.666667\pi$
$809$	−1.29335	−	0.257264i	−1.29335	−	0.257264i	−0.793353	+	0.608761i	$0.791667\pi$
$810$		0			0
$811$	0.369474	+	1.85747i	0.369474	+	1.85747i	0.500000	+	0.866025i	$0.333333\pi$
$811$	0.369474	+	1.85747i	0.369474	+	1.85747i	−0.130526	+	0.991445i	$0.541667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.513210	−	0.888905i	−0.513210	−	0.888905i
$818$	1.46593	+	0.607206i	1.46593	+	0.607206i
$819$		0			0
$820$		0			0
$821$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$821$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$822$		0			0
$823$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$823$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.63099	+	1.08979i	−1.63099	+	1.08979i	−0.707107	+	0.707107i	$0.750000\pi$
$827$	−1.63099	+	1.08979i	−1.63099	+	1.08979i	−0.923880	+	0.382683i	$0.875000\pi$
$828$		0			0
$829$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$829$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$834$		0			0
$835$		0			0
$836$	0.254391	−	0.380722i	0.254391	−	0.380722i
$837$		0			0
$838$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$839$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$839$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$840$		0			0
$841$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$842$		0			0
$843$		0			0
$844$	−0.735499	+	1.49144i	−0.735499	+	1.49144i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$851$		0			0
$852$		0			0
$853$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$853$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$854$		0			0
$855$		0			0
$856$	1.57469	−	1.05217i	1.57469	−	1.05217i
$857$	1.05217	−	0.357164i	1.05217	−	0.357164i	0.258819	−	0.965926i	$-0.416667\pi$
$857$	1.05217	−	0.357164i	1.05217	−	0.357164i	0.793353	+	0.608761i	$0.208333\pi$
$858$		0			0
$859$	−1.20711	−	0.158919i	−1.20711	−	0.158919i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.20711	−	0.158919i	−1.20711	−	0.158919i	−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$	0.965926	+	1.67303i	0.965926	+	1.67303i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$877$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.923880	−	0.617317i	−0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$881$	−0.923880	−	0.617317i	−0.923880	−	0.617317i	−0.923880	+	0.382683i	$0.875000\pi$
$882$		0			0
$883$			1.21752i			1.21752i	0.793353	+	0.608761i	$0.208333\pi$
$883$			1.21752i			1.21752i	−0.793353	+	0.608761i	$0.791667\pi$
$884$		0			0
$885$		0			0
$886$	0.158919	+	0.207107i	0.158919	+	0.207107i
$887$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$887$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.88981	−	0.123864i	−1.88981	−	0.123864i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	1.37413	−	0.793353i	1.37413	−	0.793353i
$903$		0			0
$904$	0.349942	−	0.172572i	0.349942	−	0.172572i
$905$		0			0
$906$		0			0
$907$	1.60876	+	0.793353i	1.60876	+	0.793353i	1.00000			$0$
$907$	1.60876	+	0.793353i	1.60876	+	0.793353i	0.608761	+	0.793353i	$0.291667\pi$
$908$	0.293353	−	1.47479i	0.293353	−	1.47479i
$909$		0			0
$910$		0			0
$911$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$911$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$912$		0			0
$913$	0.641097	+	0.217623i	0.641097	+	0.217623i
$914$	1.86603	−	0.500000i	1.86603	−	0.500000i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.75928	−	0.867580i	−1.75928	−	0.867580i	−0.965926	−	0.258819i	$-0.916667\pi$
$929$	−1.75928	−	0.867580i	−1.75928	−	0.867580i	−0.793353	−	0.608761i	$-0.791667\pi$
$930$		0			0
$931$	0.133975	+	0.500000i	0.133975	+	0.500000i
$932$	−1.18270	+	0.583242i	−1.18270	+	0.583242i
$933$		0			0
$934$	1.67303	−	0.965926i	1.67303	−	0.965926i
$935$		0			0
$936$		0			0
$937$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$937$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$941$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$942$		0			0
$943$		0			0
$944$	1.46593	−	0.607206i	1.46593	−	0.607206i
$945$		0			0
$946$	−0.974481	−	1.45841i	−0.974481	−	1.45841i
$947$	0.423880	+	0.483342i	0.423880	+	0.483342i	0.923880	−	0.382683i	$-0.125000\pi$
$947$	0.423880	+	0.483342i	0.423880	+	0.483342i	−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$	0.315118	+	0.410670i	0.315118	+	0.410670i
$951$		0			0
$952$		0			0
$953$		−	1.58671i		−	1.58671i	−0.608761	−	0.793353i	$-0.708333\pi$
$953$		−	1.58671i		−	1.58671i	0.608761	−	0.793353i	$-0.291667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$962$		0			0
$963$		0			0
$964$	−0.389345	−	1.95737i	−0.389345	−	1.95737i
$965$		0			0
$966$		0			0
$967$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$967$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$968$	−0.108761	+	0.188380i	−0.108761	+	0.188380i
$969$		0			0
$970$		0			0
$971$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$971$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.258819	−	0.0340742i	−0.258819	−	0.0340742i		−	1.00000i	$-0.5\pi$
$977$	−0.258819	−	0.0340742i	−0.258819	−	0.0340742i	−0.258819	+	0.965926i	$0.583333\pi$
$978$		0			0
$979$	−1.54774	+	0.525388i	−1.54774	+	0.525388i
$980$		0			0
$981$		0			0
$982$	1.36603	−	1.36603i	1.36603	−	1.36603i
$983$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$983$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$991$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$997$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$998$	−1.95737	+	0.389345i	−1.95737	+	0.389345i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3672.1.dv.a.2339.1		16
3.2	odd	2		1224.1.cx.a.707.1	✓	16
8.3	odd	2	CM	3672.1.dv.a.2339.1		16
9.2	odd	6		3672.1.dv.b.1115.1		16
9.7	even	3		1224.1.cx.b.299.1	yes	16
17.12	odd	16		3672.1.dv.b.2987.1		16
24.11	even	2		1224.1.cx.a.707.1	✓	16
51.29	even	16		1224.1.cx.b.131.1	yes	16
72.11	even	6		3672.1.dv.b.1115.1		16
72.43	odd	6		1224.1.cx.b.299.1	yes	16
136.131	even	16		3672.1.dv.b.2987.1		16
153.29	even	48	inner	3672.1.dv.a.1763.1		16
153.97	odd	48		1224.1.cx.a.947.1	yes	16
408.131	odd	16		1224.1.cx.b.131.1	yes	16
1224.403	even	48		1224.1.cx.a.947.1	yes	16
1224.947	odd	48	inner	3672.1.dv.a.1763.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1224.1.cx.a.707.1	✓	16	3.2	odd	2
1224.1.cx.a.707.1	✓	16	24.11	even	2
1224.1.cx.a.947.1	yes	16	153.97	odd	48
1224.1.cx.a.947.1	yes	16	1224.403	even	48
1224.1.cx.b.131.1	yes	16	51.29	even	16
1224.1.cx.b.131.1	yes	16	408.131	odd	16
1224.1.cx.b.299.1	yes	16	9.7	even	3
1224.1.cx.b.299.1	yes	16	72.43	odd	6
3672.1.dv.a.1763.1		16	153.29	even	48	inner
3672.1.dv.a.1763.1		16	1224.947	odd	48	inner
3672.1.dv.a.2339.1		16	1.1	even	1	trivial
3672.1.dv.a.2339.1		16	8.3	odd	2	CM
3672.1.dv.b.1115.1		16	9.2	odd	6
3672.1.dv.b.1115.1		16	72.11	even	6
3672.1.dv.b.2987.1		16	17.12	odd	16
3672.1.dv.b.2987.1		16	136.131	even	16

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:	\( N \)	\(=\)	\( 3672 = 2^{3} \cdot 3^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3672.dv (of order \(48\), degree \(16\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.130526 + 0.991445i) q^{2} +(-0.965926 + 0.258819i) q^{4} +(-0.382683 - 0.923880i) q^{8} +(-0.793353 - 0.391239i) q^{11} +(0.866025 - 0.500000i) q^{16} +(0.991445 - 0.130526i) q^{17} +(0.198092 - 0.478235i) q^{19} +(0.284338 - 0.837633i) q^{22} +(0.793353 + 0.608761i) q^{25} +(0.608761 + 0.793353i) q^{32} +(0.258819 + 0.965926i) q^{34} +(0.500000 + 0.133975i) q^{38} +(1.34861 + 1.18270i) q^{41} +(1.20711 - 1.57313i) q^{43} +(0.867580 + 0.172572i) q^{44} +(-0.793353 + 0.608761i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(1.57313 + 0.207107i) q^{59} +(-0.707107 + 0.707107i) q^{64} +(0.226078 + 0.130526i) q^{67} +(-0.923880 + 0.382683i) q^{68} +(-0.128293 + 0.0255190i) q^{73} +(-0.0675653 + 0.513210i) q^{76} +(-0.996552 + 1.49144i) q^{82} +(-0.758819 + 0.0999004i) q^{83} +(1.71723 + 0.991445i) q^{86} +(-0.0578541 + 0.882683i) q^{88} +(1.30656 - 1.30656i) q^{89} +(-0.483342 + 0.423880i) q^{97} +(-0.707107 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{38} + 8 q^{43} - 8 q^{50} - 8 q^{83}+O(q^{100}) \) 16 * q + 8 * q^38 + 8 * q^43 - 8 * q^50 - 8 * q^83

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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