LMFDB - Embedded newform 1424.1.bt.a.959.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1424,1,Mod(47,1424)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1424, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 0, 27]))

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

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

chi := DirichletCharacter("1424.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1424 = 2^{4} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1424.bt (of order $44$, degree $20$, 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:	$0.710668577989$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} - \cdots)$

Embedding invariants

Embedding label			959.1
Root			$-0.755750 - 0.654861i$ of defining polynomial
Character	$\chi$	$=$	1424.959
Dual form			1424.1.bt.a.1375.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.983568 + 0.449181i) q^{5} +(-0.540641 - 0.841254i) q^{9} +O(q^{10})$	$q+(0.983568 + 0.449181i) q^{5} +(-0.540641 - 0.841254i) q^{9} +(0.373128 + 0.203743i) q^{13} +(1.29639 - 0.186393i) q^{17} +(0.110783 + 0.127850i) q^{25} +(0.133682 - 0.0498610i) q^{29} +(0.847507 + 0.847507i) q^{37} +(-0.841254 + 0.459359i) q^{41} +(-0.153882 - 1.07028i) q^{45} +(-0.755750 + 0.654861i) q^{49} +(-0.0801894 - 0.273100i) q^{53} +(0.0498610 + 0.697148i) q^{61} +(0.275479 + 0.367998i) q^{65} +(-1.61435 - 1.03748i) q^{73} +(-0.415415 + 0.909632i) q^{81} +(1.35881 + 0.398983i) q^{85} +(-0.415415 - 0.909632i) q^{89} +(0.234072 - 0.512546i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 2 q^{13} - 2 q^{25} - 2 q^{29} - 2 q^{37} + 2 q^{41} + 2 q^{61} - 4 q^{73} + 2 q^{81} + 2 q^{89}+O(q^{100}) $ 20 * q + 2 * q^13 - 2 * q^25 - 2 * q^29 - 2 * q^37 + 2 * q^41 + 2 * q^61 - 4 * q^73 + 2 * q^81 + 2 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1424\mathbb{Z}\right)^\times$.

$n$	$357$	$1247$	$1249$
$\chi(n)$	$1$	$-1$	$e\left(\frac{29}{44}\right)$

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		0.479249	−	0.877679i	$-0.340909\pi$
$3$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$4$		0			0
$5$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$5$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$6$		0			0
$7$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$7$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$8$		0			0
$9$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$10$		0			0
$11$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$11$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$12$		0			0
$13$	0.373128	+	0.203743i	0.373128	+	0.203743i	0.654861	−	0.755750i	$-0.272727\pi$
$13$	0.373128	+	0.203743i	0.373128	+	0.203743i	−0.281733	+	0.959493i	$0.590909\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.540641	−	0.841254i	$-0.318182\pi$
$17$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.755750	+	0.654861i	$0.227273\pi$
$18$		0			0
$19$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$19$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$23$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$24$		0			0
$25$	0.110783	+	0.127850i	0.110783	+	0.127850i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.133682	−	0.0498610i	0.133682	−	0.0498610i	−0.281733	−	0.959493i	$-0.590909\pi$
$29$	0.133682	−	0.0498610i	0.133682	−	0.0498610i	0.415415	+	0.909632i	$0.363636\pi$
$30$		0			0
$31$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$31$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.847507	+	0.847507i	0.847507	+	0.847507i	0.989821	−	0.142315i	$-0.0454545\pi$
$37$	0.847507	+	0.847507i	0.847507	+	0.847507i	−0.142315	+	0.989821i	$0.545455\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.841254	+	0.459359i	−0.841254	+	0.459359i	−0.841254	−	0.540641i	$-0.818182\pi$
$41$	−0.841254	+	0.459359i	−0.841254	+	0.459359i			1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$43$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$44$		0			0
$45$	−0.153882	−	1.07028i	−0.153882	−	1.07028i
$46$		0			0
$47$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$47$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$48$		0			0
$49$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	0.909632	−	0.415415i	$-0.136364\pi$
$53$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	−0.989821	+	0.142315i	$0.954545\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$59$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$60$		0			0
$61$	0.0498610	+	0.697148i	0.0498610	+	0.697148i	0.959493	+	0.281733i	$0.0909091\pi$
$61$	0.0498610	+	0.697148i	0.0498610	+	0.697148i	−0.909632	+	0.415415i	$0.863636\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.275479	+	0.367998i	0.275479	+	0.367998i
$66$		0			0
$67$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$67$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$71$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$72$		0			0
$73$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$73$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$79$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$80$		0			0
$81$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$82$		0			0
$83$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$83$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$84$		0			0
$85$	1.35881	+	0.398983i	1.35881	+	0.398983i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$89$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.234072	−	0.512546i	0.234072	−	0.512546i	−0.755750	−	0.654861i	$-0.772727\pi$
$97$	0.234072	−	0.512546i	0.234072	−	0.512546i	0.989821	+	0.142315i	$0.0454545\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.24123	−	1.24123i	1.24123	−	1.24123i	0.281733	−	0.959493i	$-0.409091\pi$
$101$	1.24123	−	1.24123i	1.24123	−	1.24123i	0.959493	−	0.281733i	$-0.0909091\pi$
$102$		0			0
$103$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$103$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$107$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$108$		0			0
$109$	1.45027	+	1.25667i	1.45027	+	1.25667i	0.909632	+	0.415415i	$0.136364\pi$
$109$	1.45027	+	1.25667i	1.45027	+	1.25667i	0.540641	+	0.841254i	$0.318182\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.959493	−	1.28173i	−0.959493	−	1.28173i	−0.959493	−	0.281733i	$-0.909091\pi$
$113$	−0.959493	−	1.28173i	−0.959493	−	1.28173i		−	1.00000i	$-0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.0303285	−	0.424047i	−0.0303285	−	0.424047i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.253098	−	0.861971i	−0.253098	−	0.861971i
$126$		0			0
$127$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$127$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$131$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.841254	+	0.540641i	$0.818182\pi$
$137$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.909632	+	0.415415i	$0.863636\pi$
$138$		0			0
$139$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$139$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.153882	+	0.0110059i	0.153882	+	0.0110059i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.32505	+	0.494217i	−1.32505	+	0.494217i	−0.909632	−	0.415415i	$-0.863636\pi$
$149$	−1.32505	+	0.494217i	−1.32505	+	0.494217i	−0.415415	+	0.909632i	$0.636364\pi$
$150$		0			0
$151$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$151$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$152$		0			0
$153$	−0.857685	−	0.989821i	−0.857685	−	0.989821i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.281733	+	0.0405070i	0.281733	+	0.0405070i	0.281733	−	0.959493i	$-0.409091\pi$
$157$	0.281733	+	0.0405070i	0.281733	+	0.0405070i			1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$163$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$167$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$168$		0			0
$169$	−0.442928	−	0.689209i	−0.442928	−	0.689209i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$173$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−0.898064	+	1.64468i	−0.898064	+	1.64468i	−0.142315	+	0.989821i	$0.545455\pi$
$181$	−0.898064	+	1.64468i	−0.898064	+	1.64468i	−0.755750	+	0.654861i	$0.772727\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.452897	+	1.21426i	0.452897	+	1.21426i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$191$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$192$		0			0
$193$	−1.50013	+	1.12299i	−1.50013	+	1.12299i	−0.540641	+	0.841254i	$0.681818\pi$
$193$	−1.50013	+	1.12299i	−1.50013	+	1.12299i	−0.959493	+	0.281733i	$0.909091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.682956	−	0.148568i	−0.682956	−	0.148568i	−0.142315	−	0.989821i	$-0.545455\pi$
$197$	−0.682956	−	0.148568i	−0.682956	−	0.148568i	−0.540641	+	0.841254i	$0.681818\pi$
$198$		0			0
$199$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$199$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.03377	+	0.0739364i	−1.03377	+	0.0739364i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$211$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.521696	+	0.194583i	0.521696	+	0.194583i
$222$		0			0
$223$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$223$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$224$		0			0
$225$	0.0476607	−	0.162317i	0.0476607	−	0.162317i
$226$		0			0
$227$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$227$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$228$		0			0
$229$	−0.559521	+	1.50013i	−0.559521	+	1.50013i	0.281733	+	0.959493i	$0.409091\pi$
$229$	−0.559521	+	1.50013i	−0.559521	+	1.50013i	−0.841254	+	0.540641i	$0.818182\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.51150i		−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$233$		−	1.51150i		−	1.51150i	0.654861	−	0.755750i	$-0.272727\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$239$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$240$		0			0
$241$	0.847507	−	1.13214i	0.847507	−	1.13214i	−0.142315	−	0.989821i	$-0.545455\pi$
$241$	0.847507	−	1.13214i	0.847507	−	1.13214i	0.989821	−	0.142315i	$-0.0454545\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.03748	+	0.304632i	−1.03748	+	0.304632i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$251$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.07028	+	1.66538i	−1.07028	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$257$	−1.07028	+	1.66538i	−1.07028	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.114220	−	0.0855040i	−0.114220	−	0.0855040i
$262$		0			0
$263$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$263$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$264$		0			0
$265$	0.0437995	−	0.304632i	0.0437995	−	0.304632i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.215109	−	1.49611i	0.215109	−	1.49611i	−0.540641	−	0.841254i	$-0.681818\pi$
$269$	0.215109	−	1.49611i	0.215109	−	1.49611i	0.755750	−	0.654861i	$-0.227273\pi$
$270$		0			0
$271$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$271$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.983568	−	1.53046i	0.983568	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$277$	0.983568	−	1.53046i	0.983568	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.936593	−	0.203743i	0.936593	−	0.203743i	0.281733	−	0.959493i	$-0.409091\pi$
$281$	0.936593	−	0.203743i	0.936593	−	0.203743i	0.654861	+	0.755750i	$0.272727\pi$
$282$		0			0
$283$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$283$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.686393	−	0.201543i	0.686393	−	0.201543i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.0855040	−	0.114220i	0.0855040	−	0.114220i	−0.755750	−	0.654861i	$-0.772727\pi$
$293$	0.0855040	−	0.114220i	0.0855040	−	0.114220i	0.841254	+	0.540641i	$0.181818\pi$
$294$		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			0
$305$	−0.264103	+	0.708089i	−0.264103	+	0.708089i
$306$		0			0
$307$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$307$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$311$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$312$		0			0
$313$	0.133682	+	0.0498610i	0.133682	+	0.0498610i	0.415415	−	0.909632i	$-0.363636\pi$
$313$	0.133682	+	0.0498610i	0.133682	+	0.0498610i	−0.281733	+	0.959493i	$0.590909\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$317$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.0152876	+	0.0702757i	0.0152876	+	0.0702757i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$331$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$332$		0			0
$333$	0.254771	−	1.17116i	0.254771	−	1.17116i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.90963	+	0.415415i	1.90963	+	0.415415i	1.00000			$0$
$337$	1.90963	+	0.415415i	1.90963	+	0.415415i	0.909632	+	0.415415i	$0.136364\pi$
$338$		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		−0.540641	−	0.841254i	$-0.681818\pi$
$347$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$348$		0			0
$349$	0.697148	+	1.86912i	0.697148	+	1.86912i	0.415415	+	0.909632i	$0.363636\pi$
$349$	0.697148	+	1.86912i	0.697148	+	1.86912i	0.281733	+	0.959493i	$0.409091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.841254	−	1.54064i	0.841254	−	1.54064i		−	1.00000i	$-0.5\pi$
$353$	0.841254	−	1.54064i	0.841254	−	1.54064i	0.841254	−	0.540641i	$-0.181818\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$359$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$360$		0			0
$361$	0.909632	+	0.415415i	0.909632	+	0.415415i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.12181	−	1.74557i	−1.12181	−	1.74557i
$366$		0			0
$367$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$367$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$368$		0			0
$369$	0.841254	+	0.459359i	0.841254	+	0.459359i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$373$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.0600395	+	0.00863238i	0.0600395	+	0.00863238i
$378$		0			0
$379$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$379$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$383$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.98982	+	0.142315i	1.98982	+	0.142315i	1.00000			$0$
$389$	1.98982	+	0.142315i	1.98982	+	0.142315i	0.989821	+	0.142315i	$0.0454545\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.71524	+	0.936593i	−1.71524	+	0.936593i	−0.755750	+	0.654861i	$0.772727\pi$
$397$	−1.71524	+	0.936593i	−1.71524	+	0.936593i	−0.959493	+	0.281733i	$0.909091\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.0801894	+	0.557730i	0.0801894	+	0.557730i	0.989821	+	0.142315i	$0.0454545\pi$
$401$	0.0801894	+	0.557730i	0.0801894	+	0.557730i	−0.909632	+	0.415415i	$0.863636\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.817178	+	0.708089i	−0.817178	+	0.708089i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	−0.989821	−	0.142315i	$-0.954545\pi$
$409$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	0.755750	−	0.654861i	$-0.227273\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$419$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$420$		0			0
$421$	1.17116	+	1.56449i	1.17116	+	1.56449i	0.755750	+	0.654861i	$0.227273\pi$
$421$	1.17116	+	1.56449i	1.17116	+	1.56449i	0.415415	+	0.909632i	$0.363636\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.167448	+	0.145095i	0.167448	+	0.145095i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$431$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$432$		0			0
$433$	−0.847507	+	0.847507i	−0.847507	+	0.847507i	−0.989821	−	0.142315i	$-0.954545\pi$
$433$	−0.847507	+	0.847507i	−0.847507	+	0.847507i	0.142315	+	0.989821i	$0.454545\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$439$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$440$		0			0
$441$	0.959493	+	0.281733i	0.959493	+	0.281733i
$442$		0			0
$443$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$443$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$444$		0			0
$445$		−	1.08128i		−	1.08128i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$449$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.677760	−	0.677760i	0.677760	−	0.677760i	−0.281733	−	0.959493i	$-0.590909\pi$
$457$	0.677760	−	0.677760i	0.677760	−	0.677760i	0.959493	+	0.281733i	$0.0909091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.53046	−	0.983568i	−1.53046	−	0.983568i	−0.989821	−	0.142315i	$-0.954545\pi$
$461$	−1.53046	−	0.983568i	−1.53046	−	0.983568i	−0.540641	−	0.841254i	$-0.681818\pi$
$462$		0			0
$463$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$463$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$467$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.186393	+	0.215109i	−0.186393	+	0.215109i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	0.143555	+	0.488902i	0.143555	+	0.488902i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.460451	−	0.398983i	0.460451	−	0.398983i
$486$		0			0
$487$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$487$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$491$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$492$		0			0
$493$	0.164011	−	0.0895567i	0.164011	−	0.0895567i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$499$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$503$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$504$		0			0
$505$	1.77836	−	0.663296i	1.77836	−	0.663296i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.368991	−	0.425839i	−0.368991	−	0.425839i	0.540641	−	0.841254i	$-0.318182\pi$
$509$	−0.368991	−	0.425839i	−0.368991	−	0.425839i	−0.909632	+	0.415415i	$0.863636\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.71524	+	0.936593i	1.71524	+	0.936593i	0.959493	+	0.281733i	$0.0909091\pi$
$521$	1.71524	+	0.936593i	1.71524	+	0.936593i	0.755750	+	0.654861i	$0.227273\pi$
$522$		0			0
$523$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$523$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.909632	−	0.415415i	−0.909632	−	0.415415i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.407487			−0.407487
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.654861	+	1.75575i	0.654861	+	1.75575i	0.654861	+	0.755750i	$0.272727\pi$
$541$	0.654861	+	1.75575i	0.654861	+	1.75575i			1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.861971	+	1.88745i	0.861971	+	1.88745i
$546$		0			0
$547$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$547$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$548$		0			0
$549$	0.559521	−	0.418852i	0.559521	−	0.418852i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.415415	−	1.90963i	0.415415	−	1.90963i		−	1.00000i	$-0.5\pi$
$557$	0.415415	−	1.90963i	0.415415	−	1.90963i	0.415415	−	0.909632i	$-0.363636\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$563$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$564$		0			0
$565$	−0.367998	−	1.69166i	−0.367998	−	1.69166i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.0498610	+	0.697148i	−0.0498610	+	0.697148i	0.909632	+	0.415415i	$0.136364\pi$
$569$	−0.0498610	+	0.697148i	−0.0498610	+	0.697148i	−0.959493	+	0.281733i	$0.909091\pi$
$570$		0			0
$571$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.32505	+	0.494217i	1.32505	+	0.494217i	0.909632	−	0.415415i	$-0.136364\pi$
$577$	1.32505	+	0.494217i	1.32505	+	0.494217i	0.415415	+	0.909632i	$0.363636\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	0.160644	−	0.430703i	0.160644	−	0.430703i
$586$		0			0
$587$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$587$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.334961	−	0.613435i	−0.334961	−	0.613435i	0.654861	−	0.755750i	$-0.272727\pi$
$593$	−0.334961	−	0.613435i	−0.334961	−	0.613435i	−0.989821	+	0.142315i	$0.954545\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$599$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$600$		0			0
$601$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$601$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.983568	+	0.449181i	−0.983568	+	0.449181i
$606$		0			0
$607$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$607$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$613$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	−0.989821	+	0.142315i	$0.954545\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.340335	+	0.254771i	0.340335	+	0.254771i	0.755750	−	0.654861i	$-0.227273\pi$
$617$	0.340335	+	0.254771i	0.340335	+	0.254771i	−0.415415	+	0.909632i	$0.636364\pi$
$618$		0			0
$619$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$619$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.162317	−	1.12894i	0.162317	−	1.12894i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.25667	+	0.940730i	1.25667	+	0.940730i
$630$		0			0
$631$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$631$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.415415	+	0.0903680i	−0.415415	+	0.0903680i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$641$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$642$		0			0
$643$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$643$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$647$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.767317	−	1.40524i	−0.767317	−	1.40524i	−0.909632	−	0.415415i	$-0.863636\pi$
$653$	−0.767317	−	1.40524i	−0.767317	−	1.40524i	0.142315	−	0.989821i	$-0.454545\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			1.91899i			1.91899i
$658$		0			0
$659$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$659$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$660$		0			0
$661$	0.244250	−	0.654861i	0.244250	−	0.654861i	−0.755750	−	0.654861i	$-0.772727\pi$
$661$	0.244250	−	0.654861i	0.244250	−	0.654861i	1.00000			$0$
$662$		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
$673$	1.66538	−	1.07028i	1.66538	−	1.07028i	0.755750	−	0.654861i	$-0.227273\pi$
$673$	1.66538	−	1.07028i	1.66538	−	1.07028i	0.909632	−	0.415415i	$-0.136364\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.100889	+	1.41061i	−0.100889	+	1.41061i	0.654861	+	0.755750i	$0.272727\pi$
$677$	−0.100889	+	1.41061i	−0.100889	+	1.41061i	−0.755750	+	0.654861i	$0.772727\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$683$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$684$		0			0
$685$	−2.15156	+	0.153882i	−2.15156	+	0.153882i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.0257214	−	0.118239i	0.0257214	−	0.118239i
$690$		0			0
$691$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$691$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.00497	+	0.752312i	−1.00497	+	0.752312i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$701$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.203743	−	0.373128i	0.203743	−	0.373128i	−0.755750	−	0.654861i	$-0.772727\pi$
$709$	0.203743	−	0.373128i	0.203743	−	0.373128i	0.959493	+	0.281733i	$0.0909091\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$719$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.0211844	+	0.0115676i	0.0211844	+	0.0115676i
$726$		0			0
$727$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$727$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$728$		0			0
$729$	0.989821	−	0.142315i	0.989821	−	0.142315i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.822373	+	0.118239i	0.822373	+	0.118239i	0.540641	−	0.841254i	$-0.318182\pi$
$733$	0.822373	+	0.118239i	0.822373	+	0.118239i	0.281733	+	0.959493i	$0.409091\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$739$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$743$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$744$		0			0
$745$	−1.52527	−	0.109089i	−1.52527	−	0.109089i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$751$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.258908	−	1.80075i	−0.258908	−	1.80075i	−0.540641	−	0.841254i	$-0.681818\pi$
$757$	−0.258908	−	1.80075i	−0.258908	−	1.80075i	0.281733	−	0.959493i	$-0.409091\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.215109	+	0.186393i	−0.215109	+	0.186393i	−0.755750	−	0.654861i	$-0.772727\pi$
$761$	−0.215109	+	0.186393i	−0.215109	+	0.186393i	0.540641	+	0.841254i	$0.318182\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.398983	−	1.35881i	−0.398983	−	1.35881i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.989821	−	1.14231i	0.989821	−	1.14231i		−	1.00000i	$-0.5\pi$
$769$	0.989821	−	1.14231i	0.989821	−	1.14231i	0.989821	−	0.142315i	$-0.0454545\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.0683785	+	0.956056i	0.0683785	+	0.956056i	0.909632	+	0.415415i	$0.136364\pi$
$773$	0.0683785	+	0.956056i	0.0683785	+	0.956056i	−0.841254	+	0.540641i	$0.818182\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.258908	+	0.166390i	0.258908	+	0.166390i
$786$		0			0
$787$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$787$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.123435	+	0.270284i	−0.123435	+	0.270284i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$797$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.540641	+	0.841254i	−0.540641	+	0.841254i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$809$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$810$		0			0
$811$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$811$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.989821	+	0.142315i	$0.0454545\pi$
$821$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$822$		0			0
$823$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$823$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$827$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$828$		0			0
$829$	0.100889	+	1.41061i	0.100889	+	1.41061i	0.755750	+	0.654861i	$0.227273\pi$
$829$	0.100889	+	1.41061i	0.100889	+	1.41061i	−0.654861	+	0.755750i	$0.727273\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.857685	+	0.989821i	−0.857685	+	0.989821i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$839$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$840$		0			0
$841$	−0.740365	+	0.641530i	−0.740365	+	0.641530i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.126070	−	0.876838i	−0.126070	−	0.876838i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$853$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.19550	+	0.0855040i	1.19550	+	0.0855040i	0.654861	−	0.755750i	$-0.272727\pi$
$857$	1.19550	+	0.0855040i	1.19550	+	0.0855040i	0.540641	+	0.841254i	$0.318182\pi$
$858$		0			0
$859$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$859$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$863$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$864$		0			0
$865$	−1.28820	−	1.48666i	−1.28820	−	1.48666i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.54064	−	0.841254i	−1.54064	−	0.841254i	−0.540641	−	0.841254i	$-0.681818\pi$
$877$	−1.54064	−	0.841254i	−1.54064	−	0.841254i	−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.708089	−	1.10181i	−0.708089	−	1.10181i	−0.989821	−	0.142315i	$-0.954545\pi$
$881$	−0.708089	−	1.10181i	−0.708089	−	1.10181i	0.281733	−	0.959493i	$-0.409091\pi$
$882$		0			0
$883$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$883$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$887$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$888$		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
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.154861	−	0.339098i	−0.154861	−	0.339098i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.62207	+	1.21426i	−1.62207	+	1.21426i
$906$		0			0
$907$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$907$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$908$		0			0
$909$	−1.71524	−	0.373128i	−1.71524	−	0.373128i
$910$		0			0
$911$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$911$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$919$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.0144647	+	0.202243i	−0.0144647	+	0.202243i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$929$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.158746	+	0.540641i	−0.158746	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$937$	−0.158746	+	0.540641i	−0.158746	+	0.540641i	−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.418852	+	1.12299i	−0.418852	+	1.12299i	0.540641	+	0.841254i	$0.318182\pi$
$941$	−0.418852	+	1.12299i	−0.418852	+	1.12299i	−0.959493	+	0.281733i	$0.909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$947$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$948$		0			0
$949$	−0.390981	−	0.716028i	−0.390981	−	0.716028i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.05195	−	1.40524i	1.05195	−	1.40524i	0.142315	−	0.989821i	$-0.454545\pi$
$953$	1.05195	−	1.40524i	1.05195	−	1.40524i	0.909632	−	0.415415i	$-0.136364\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.97991	+	0.430703i	−1.97991	+	0.430703i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$971$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$977$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.273100	−	1.89945i	0.273100	−	1.89945i
$982$		0			0
$983$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$983$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$984$		0			0
$985$	−0.605000	−	0.452897i	−0.605000	−	0.452897i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$991$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.512546	−	0.234072i	0.512546	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$997$	0.512546	−	0.234072i	0.512546	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1424.1.bt.a.959.1	✓	20
4.3	odd	2	CM	1424.1.bt.a.959.1	✓	20
89.40	even	44	inner	1424.1.bt.a.1375.1	yes	20
356.307	odd	44	inner	1424.1.bt.a.1375.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1424.1.bt.a.959.1	✓	20	1.1	even	1	trivial
1424.1.bt.a.959.1	✓	20	4.3	odd	2	CM
1424.1.bt.a.1375.1	yes	20	89.40	even	44	inner
1424.1.bt.a.1375.1	yes	20	356.307	odd	44	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1424 = 2^{4} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1424.bt (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.983568 + 0.449181i) q^{5} +(-0.540641 - 0.841254i) q^{9} +O(q^{10})\)	\(q+(0.983568 + 0.449181i) q^{5} +(-0.540641 - 0.841254i) q^{9} +(0.373128 + 0.203743i) q^{13} +(1.29639 - 0.186393i) q^{17} +(0.110783 + 0.127850i) q^{25} +(0.133682 - 0.0498610i) q^{29} +(0.847507 + 0.847507i) q^{37} +(-0.841254 + 0.459359i) q^{41} +(-0.153882 - 1.07028i) q^{45} +(-0.755750 + 0.654861i) q^{49} +(-0.0801894 - 0.273100i) q^{53} +(0.0498610 + 0.697148i) q^{61} +(0.275479 + 0.367998i) q^{65} +(-1.61435 - 1.03748i) q^{73} +(-0.415415 + 0.909632i) q^{81} +(1.35881 + 0.398983i) q^{85} +(-0.415415 - 0.909632i) q^{89} +(0.234072 - 0.512546i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 2 q^{13} - 2 q^{25} - 2 q^{29} - 2 q^{37} + 2 q^{41} + 2 q^{61} - 4 q^{73} + 2 q^{81} + 2 q^{89}+O(q^{100}) \) 20 * q + 2 * q^13 - 2 * q^25 - 2 * q^29 - 2 * q^37 + 2 * q^41 + 2 * q^61 - 4 * q^73 + 2 * q^81 + 2 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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