LMFDB - Embedded newform 1449.1.bq.b.307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,1,Mod(55,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 10]))

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

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

chi := DirichletCharacter("1449.55");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1449.bq (of order $22$, degree $10$, 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:	$0.723145203305$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
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:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} - \cdots)$

Embedding invariants

Embedding label			307.1
Root			$0.281733 - 0.959493i$ of defining polynomial
Character	$\chi$	$=$	1449.307
Dual form			1449.1.bq.b.118.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0801894 + 0.557730i) q^{2} +(0.654861 + 0.192284i) q^{4} +(-0.841254 - 0.540641i) q^{7} +(-0.393828 + 0.862362i) q^{8} +O(q^{10})$	$q+(-0.0801894 + 0.557730i) q^{2} +(0.654861 + 0.192284i) q^{4} +(-0.841254 - 0.540641i) q^{7} +(-0.393828 + 0.862362i) q^{8} +(0.258908 + 1.80075i) q^{11} +(0.368991 - 0.425839i) q^{14} +(0.124777 + 0.0801894i) q^{16} -1.02509 q^{22} +(-0.755750 + 0.654861i) q^{23} +(-0.142315 + 0.989821i) q^{25} +(-0.446947 - 0.515804i) q^{28} +(1.89945 - 0.557730i) q^{29} +(-0.675560 + 0.779638i) q^{32} +(1.10181 - 1.27155i) q^{37} +(0.544078 + 1.19136i) q^{43} +(-0.176707 + 1.22902i) q^{44} +(-0.304632 - 0.474017i) q^{46} +(0.415415 + 0.909632i) q^{49} +(-0.540641 - 0.158746i) q^{50} +(-1.66538 - 1.07028i) q^{53} +(0.797537 - 0.512546i) q^{56} +(0.158746 + 1.10411i) q^{58} +(-0.283524 - 0.327204i) q^{64} +(0.273100 - 1.89945i) q^{67} +(0.258908 - 1.80075i) q^{71} +(0.620830 + 0.716476i) q^{74} +(0.755750 - 1.65486i) q^{77} +(-0.239446 + 0.153882i) q^{79} +(-0.708089 + 0.207914i) q^{86} +(-1.65486 - 0.485911i) q^{88} +(-0.620830 + 0.283524i) q^{92} +(-0.540641 + 0.158746i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{4} + 2 q^{7}+O(q^{10}) $ 20 * q + 2 * q^4 + 2 * q^7	$ 20 q + 2 q^{4} + 2 q^{7} - 24 q^{16} - 2 q^{25} - 2 q^{28} + 4 q^{37} + 4 q^{43} - 2 q^{49} + 22 q^{58} + 2 q^{64} - 4 q^{67} - 4 q^{79} - 22 q^{88}+O(q^{100}) $ 20 * q + 2 * q^4 + 2 * q^7 - 24 * q^16 - 2 * q^25 - 2 * q^28 + 4 * q^37 + 4 * q^43 - 2 * q^49 + 22 * q^58 + 2 * q^64 - 4 * q^67 - 4 * q^79 - 22 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times$.

$n$	$442$	$829$	$1289$
$\chi(n)$	$e\left(\frac{3}{11}\right)$	$-1$	$1$

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 $

$2$	−0.0801894	+	0.557730i	−0.0801894	+	0.557730i	0.909632	+	0.415415i	$0.136364\pi$
$2$	−0.0801894	+	0.557730i	−0.0801894	+	0.557730i	−0.989821	+	0.142315i	$0.954545\pi$
$3$		0			0
$3$		0			0
$4$	0.654861	+	0.192284i	0.654861	+	0.192284i
$5$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$5$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$6$		0			0
$7$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$7$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$8$	−0.393828	+	0.862362i	−0.393828	+	0.862362i
$9$		0			0
$10$		0			0
$11$	0.258908	+	1.80075i	0.258908	+	1.80075i	0.540641	+	0.841254i	$0.318182\pi$
$11$	0.258908	+	1.80075i	0.258908	+	1.80075i	−0.281733	+	0.959493i	$0.590909\pi$
$12$		0			0
$13$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$13$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$14$	0.368991	−	0.425839i	0.368991	−	0.425839i
$15$		0			0
$16$	0.124777	+	0.0801894i	0.124777	+	0.0801894i
$17$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$17$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$18$		0			0
$19$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$19$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$20$		0			0
$21$		0			0
$22$	−1.02509			−1.02509
$23$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$23$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$24$		0			0
$25$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$26$		0			0
$27$		0			0
$28$	−0.446947	−	0.515804i	−0.446947	−	0.515804i
$29$	1.89945	−	0.557730i	1.89945	−	0.557730i	0.909632	−	0.415415i	$-0.136364\pi$
$29$	1.89945	−	0.557730i	1.89945	−	0.557730i	0.989821	−	0.142315i	$-0.0454545\pi$
$30$		0			0
$31$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$31$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$32$	−0.675560	+	0.779638i	−0.675560	+	0.779638i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$37$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$41$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$42$		0			0
$43$	0.544078	+	1.19136i	0.544078	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$43$	0.544078	+	1.19136i	0.544078	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$44$	−0.176707	+	1.22902i	−0.176707	+	1.22902i
$45$		0			0
$46$	−0.304632	−	0.474017i	−0.304632	−	0.474017i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	0.415415	+	0.909632i	0.415415	+	0.909632i
$50$	−0.540641	−	0.158746i	−0.540641	−	0.158746i
$51$		0			0
$52$		0			0
$53$	−1.66538	−	1.07028i	−1.66538	−	1.07028i	−0.909632	−	0.415415i	$-0.863636\pi$
$53$	−1.66538	−	1.07028i	−1.66538	−	1.07028i	−0.755750	−	0.654861i	$-0.772727\pi$
$54$		0			0
$55$		0			0
$56$	0.797537	−	0.512546i	0.797537	−	0.512546i
$57$		0			0
$58$	0.158746	+	1.10411i	0.158746	+	1.10411i
$59$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$59$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$60$		0			0
$61$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$61$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$62$		0			0
$63$		0			0
$64$	−0.283524	−	0.327204i	−0.283524	−	0.327204i
$65$		0			0
$66$		0			0
$67$	0.273100	−	1.89945i	0.273100	−	1.89945i	−0.142315	−	0.989821i	$-0.545455\pi$
$67$	0.273100	−	1.89945i	0.273100	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.258908	−	1.80075i	0.258908	−	1.80075i	−0.281733	−	0.959493i	$-0.590909\pi$
$71$	0.258908	−	1.80075i	0.258908	−	1.80075i	0.540641	−	0.841254i	$-0.318182\pi$
$72$		0			0
$73$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$73$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$74$	0.620830	+	0.716476i	0.620830	+	0.716476i
$75$		0			0
$76$		0			0
$77$	0.755750	−	1.65486i	0.755750	−	1.65486i
$78$		0			0
$79$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$79$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$83$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$84$		0			0
$85$		0			0
$86$	−0.708089	+	0.207914i	−0.708089	+	0.207914i
$87$		0			0
$88$	−1.65486	−	0.485911i	−1.65486	−	0.485911i
$89$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$89$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$90$		0			0
$91$		0			0
$92$	−0.620830	+	0.283524i	−0.620830	+	0.283524i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$97$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$98$	−0.540641	+	0.158746i	−0.540641	+	0.158746i
$99$		0			0
$100$	−0.283524	+	0.620830i	−0.283524	+	0.620830i
$101$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$101$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$102$		0			0
$103$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$103$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$104$		0			0
$105$		0			0
$106$	0.730471	−	0.843008i	0.730471	−	0.843008i
$107$	−0.449181	+	0.983568i	−0.449181	+	0.983568i	0.540641	+	0.841254i	$0.318182\pi$
$107$	−0.449181	+	0.983568i	−0.449181	+	0.983568i	−0.989821	+	0.142315i	$0.954545\pi$
$108$		0			0
$109$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−0.841254	+	0.540641i	$0.818182\pi$
$109$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−1.00000			$1.00000\pi$
$110$		0			0
$111$		0			0
$112$	−0.0616156	−	0.134919i	−0.0616156	−	0.134919i
$113$	0.215109	−	1.49611i	0.215109	−	1.49611i	−0.540641	−	0.841254i	$-0.681818\pi$
$113$	0.215109	−	1.49611i	0.215109	−	1.49611i	0.755750	−	0.654861i	$-0.227273\pi$
$114$		0			0
$115$		0			0
$116$	1.35112			1.35112
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.21616	+	0.650724i	−2.21616	+	0.650724i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$127$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$128$	−0.662618	+	0.425839i	−0.662618	+	0.425839i
$129$		0			0
$130$		0			0
$131$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$131$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$132$		0			0
$133$		0			0
$134$	1.03748	+	0.304632i	1.03748	+	0.304632i
$135$		0			0
$136$		0			0
$137$	0.563465			0.563465			0.281733	−	0.959493i	$-0.409091\pi$
$137$	0.563465			0.563465			0.281733	+	0.959493i	$0.409091\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	0.983568	+	0.288802i	0.983568	+	0.288802i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.966031	−	0.620830i	0.966031	−	0.620830i
$149$	−0.215109	−	1.49611i	−0.215109	−	1.49611i	−0.755750	−	0.654861i	$-0.772727\pi$
$149$	−0.215109	−	1.49611i	−0.215109	−	1.49611i	0.540641	−	0.841254i	$-0.318182\pi$
$150$		0			0
$151$	0.239446	−	0.153882i	0.239446	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$151$	0.239446	−	0.153882i	0.239446	−	0.153882i	0.654861	+	0.755750i	$0.272727\pi$
$152$		0			0
$153$		0			0
$154$	0.862362	+	0.554206i	0.862362	+	0.554206i
$155$		0			0
$156$		0			0
$157$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$157$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$158$	−0.0666238	−	0.145886i	−0.0666238	−	0.145886i
$159$		0			0
$160$		0			0
$161$	0.989821	−	0.142315i	0.989821	−	0.142315i
$162$		0			0
$163$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$163$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$167$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$168$		0			0
$169$	0.415415	−	0.909632i	0.415415	−	0.909632i
$170$		0			0
$171$		0			0
$172$	0.127214	+	0.884796i	0.127214	+	0.884796i
$173$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$173$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$174$		0			0
$175$	0.654861	−	0.755750i	0.654861	−	0.755750i
$176$	−0.112095	+	0.245454i	−0.112095	+	0.245454i
$177$		0			0
$178$		0			0
$179$	−0.989821	−	1.14231i	−0.989821	−	1.14231i	−0.989821	−	0.142315i	$-0.954545\pi$
$179$	−0.989821	−	1.14231i	−0.989821	−	1.14231i		−	1.00000i	$-0.5\pi$
$180$		0			0
$181$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$181$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$182$		0			0
$183$		0			0
$184$	−0.267092	−	0.909632i	−0.267092	−	0.909632i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.909632	+	0.584585i	0.909632	+	0.584585i	0.909632	−	0.415415i	$-0.136364\pi$
$191$	0.909632	+	0.584585i	0.909632	+	0.584585i			1.00000i	$0.5\pi$
$192$		0			0
$193$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$193$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$194$		0			0
$195$		0			0
$196$	0.0971309	+	0.675560i	0.0971309	+	0.675560i
$197$	1.27155	−	0.817178i	1.27155	−	0.817178i	0.281733	−	0.959493i	$-0.409091\pi$
$197$	1.27155	−	0.817178i	1.27155	−	0.817178i	0.989821	+	0.142315i	$0.0454545\pi$
$198$		0			0
$199$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$199$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$200$	−0.797537	−	0.512546i	−0.797537	−	0.512546i
$201$		0			0
$202$		0			0
$203$	−1.89945	−	0.557730i	−1.89945	−	0.557730i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.654861	−	0.755750i	$-0.272727\pi$
$211$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.142315	+	0.989821i	$0.454545\pi$
$212$	−0.884796	−	1.02111i	−0.884796	−	1.02111i
$213$		0			0
$214$	−0.512546	−	0.329393i	−0.512546	−	0.329393i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.153882	−	1.07028i	−0.153882	−	1.07028i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$223$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$224$	0.989821	−	0.290638i	0.989821	−	0.290638i
$225$		0			0
$226$	0.817178	+	0.239945i	0.817178	+	0.239945i
$227$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$227$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−0.267092	+	1.85767i	−0.267092	+	1.85767i
$233$	0.449181	+	0.983568i	0.449181	+	0.983568i	0.989821	+	0.142315i	$0.0454545\pi$
$233$	0.449181	+	0.983568i	0.449181	+	0.983568i	−0.540641	+	0.841254i	$0.681818\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.708089	−	0.817178i	0.708089	−	0.817178i	−0.281733	−	0.959493i	$-0.590909\pi$
$239$	0.708089	−	0.817178i	0.708089	−	0.817178i	0.989821	+	0.142315i	$0.0454545\pi$
$240$		0			0
$241$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$241$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$242$	−0.185215	−	1.28820i	−0.185215	−	1.28820i
$243$		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			0		0.142315	−	0.989821i	$-0.454545\pi$
$251$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$252$		0			0
$253$	−1.37491	−	1.19136i	−1.37491	−	1.19136i
$254$	−0.737982			−0.737982
$255$		0			0
$256$	−0.364223	−	0.797537i	−0.364223	−	0.797537i
$257$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$257$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$258$		0			0
$259$	−1.61435	+	0.474017i	−1.61435	+	0.474017i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.474017	−	0.304632i	0.474017	−	0.304632i	−0.281733	−	0.959493i	$-0.590909\pi$
$263$	0.474017	−	0.304632i	0.474017	−	0.304632i	0.755750	+	0.654861i	$0.227273\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.544078	−	1.19136i	0.544078	−	1.19136i
$269$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$269$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$270$		0			0
$271$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$271$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$272$		0			0
$273$		0			0
$274$	−0.0451840	+	0.314261i	−0.0451840	+	0.314261i
$275$	−1.81926			−1.81926
$276$		0			0
$277$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$277$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.755750	−	0.654861i	$-0.772727\pi$
$281$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.540641	−	0.841254i	$-0.681818\pi$
$282$		0			0
$283$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$283$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$284$	0.515804	−	1.12945i	0.515804	−	1.12945i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.841254	−	0.540641i	0.841254	−	0.540641i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$293$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$294$		0			0
$295$		0			0
$296$	0.662618	+	1.45093i	0.662618	+	1.45093i
$297$		0			0
$298$	0.851677			0.851677
$299$		0			0
$300$		0			0
$301$	0.186393	−	1.29639i	0.186393	−	1.29639i
$302$	0.0666238	+	0.145886i	0.0666238	+	0.145886i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$307$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$308$	0.813115	−	0.938384i	0.813115	−	0.938384i
$309$		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			0		0.654861	−	0.755750i	$-0.272727\pi$
$313$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$314$		0			0
$315$		0			0
$316$	−0.186393	+	0.0547299i	−0.186393	+	0.0547299i
$317$	0.708089	+	0.817178i	0.708089	+	0.817178i	0.989821	−	0.142315i	$-0.0454545\pi$
$317$	0.708089	+	0.817178i	0.708089	+	0.817178i	−0.281733	+	0.959493i	$0.590909\pi$
$318$		0			0
$319$	1.49611	+	3.27603i	1.49611	+	3.27603i
$320$		0			0
$321$		0			0
$322$			0.563465i			0.563465i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	0.153882	+	0.0451840i	0.153882	+	0.0451840i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.857685	+	0.989821i	−0.857685	+	0.989821i	0.142315	+	0.989821i	$0.454545\pi$
$331$	−0.857685	+	0.989821i	−0.857685	+	0.989821i	−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	0.654861	+	0.755750i	$0.272727\pi$
$337$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	−1.00000			$\pi$
$338$	0.474017	+	0.304632i	0.474017	+	0.304632i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.142315	−	0.989821i	0.142315	−	0.989821i
$344$	−1.24166			−1.24166
$345$		0			0
$346$		0			0
$347$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	0.755750	+	0.654861i	$0.227273\pi$
$347$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	−0.909632	+	0.415415i	$0.863636\pi$
$348$		0			0
$349$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$349$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$350$	0.368991	+	0.425839i	0.368991	+	0.425839i
$351$		0			0
$352$	−1.57884	−	1.01466i	−1.57884	−	1.01466i
$353$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$353$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.716476	−	0.460451i	0.716476	−	0.460451i
$359$	−0.989821	+	1.14231i	−0.989821	+	1.14231i			1.00000i	$0.5\pi$
$359$	−0.989821	+	1.14231i	−0.989821	+	1.14231i	−0.989821	+	0.142315i	$0.954545\pi$
$360$		0			0
$361$	0.841254	+	0.540641i	0.841254	+	0.540641i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	−0.146813	+	0.0211086i	−0.146813	+	0.0211086i
$369$		0			0
$370$		0			0
$371$	0.822373	+	1.80075i	0.822373	+	1.80075i
$372$		0			0
$373$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$373$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i	0.959493	−	0.281733i	$-0.0909091\pi$
$379$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	−0.398983	+	0.460451i	−0.398983	+	0.460451i
$383$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$383$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$384$		0			0
$385$		0			0
$386$	0.306569	+	0.353799i	0.306569	+	0.353799i
$387$		0			0
$388$		0			0
$389$	−0.258908	+	1.80075i	−0.258908	+	1.80075i	0.281733	+	0.959493i	$0.409091\pi$
$389$	−0.258908	+	1.80075i	−0.258908	+	1.80075i	−0.540641	+	0.841254i	$0.681818\pi$
$390$		0			0
$391$		0			0
$392$	−0.948034			−0.948034
$393$		0			0
$394$	0.353799	+	0.774713i	0.353799	+	0.774713i
$395$		0			0
$396$		0			0
$397$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$397$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$398$		0			0
$399$		0			0
$400$	−0.0971309	+	0.112095i	−0.0971309	+	0.112095i
$401$	0.909632	−	0.584585i	0.909632	−	0.584585i		−	1.00000i	$-0.5\pi$
$401$	0.909632	−	0.584585i	0.909632	−	0.584585i	0.909632	+	0.415415i	$0.136364\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.463379	−	1.01466i	0.463379	−	1.01466i
$407$	2.57501	+	1.65486i	2.57501	+	1.65486i
$408$		0			0
$409$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$409$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$410$		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		−0.654861	−	0.755750i	$-0.727273\pi$
$419$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$420$		0			0
$421$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$421$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$422$	−0.194474	+	0.425839i	−0.194474	+	0.425839i
$423$		0			0
$424$	1.57884	−	1.01466i	1.57884	−	1.01466i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.483276	+	0.557730i	−0.483276	+	0.557730i
$429$		0			0
$430$		0			0
$431$	1.45027	−	0.425839i	1.45027	−	0.425839i	0.540641	−	0.841254i	$-0.318182\pi$
$431$	1.45027	−	0.425839i	1.45027	−	0.425839i	0.909632	+	0.415415i	$0.136364\pi$
$432$		0			0
$433$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$433$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$434$		0			0
$435$		0			0
$436$	−1.30972			−1.30972
$437$		0			0
$438$		0			0
$439$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$439$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.74557	+	0.512546i	−1.74557	+	0.512546i	−0.989821	−	0.142315i	$-0.954545\pi$
$443$	−1.74557	+	0.512546i	−1.74557	+	0.512546i	−0.755750	+	0.654861i	$0.772727\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.0616156	+	0.428546i	0.0616156	+	0.428546i
$449$	0.153882	+	1.07028i	0.153882	+	1.07028i	0.909632	+	0.415415i	$0.136364\pi$
$449$	0.153882	+	1.07028i	0.153882	+	1.07028i	−0.755750	+	0.654861i	$0.772727\pi$
$450$		0			0
$451$		0			0
$452$	0.428546	−	0.938384i	0.428546	−	0.938384i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.118239	+	0.258908i	0.118239	+	0.258908i	0.959493	−	0.281733i	$-0.0909091\pi$
$457$	0.118239	+	0.258908i	0.118239	+	0.258908i	−0.841254	+	0.540641i	$0.818182\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$463$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$464$	0.281733	+	0.0827241i	0.281733	+	0.0827241i
$465$		0			0
$466$	−0.584585	+	0.171650i	−0.584585	+	0.171650i
$467$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$467$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$468$		0			0
$469$	−1.25667	+	1.45027i	−1.25667	+	1.45027i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−2.00448	+	1.28820i	−2.00448	+	1.28820i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.398983	+	0.460451i	0.398983	+	0.460451i
$479$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$479$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.57640			−1.57640
$485$		0			0
$486$		0			0
$487$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.415415	−	0.909632i	$-0.636364\pi$
$487$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.841254	+	0.540641i	$0.818182\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.234072	+	0.512546i	−0.234072	+	0.512546i	−0.989821	−	0.142315i	$-0.954545\pi$
$491$	−0.234072	+	0.512546i	−0.234072	+	0.512546i	0.755750	+	0.654861i	$0.227273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.19136	+	1.37491i	−1.19136	+	1.37491i
$498$		0			0
$499$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.959493	+	0.281733i	$0.0909091\pi$
$499$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$503$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$504$		0			0
$505$		0			0
$506$	0.774713	−	0.671292i	0.774713	−	0.671292i
$507$		0			0
$508$	−0.127214	+	0.884796i	−0.127214	+	0.884796i
$509$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$509$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$		0			0
$512$	−0.281733	+	0.0827241i	−0.281733	+	0.0827241i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.134919	−	0.938384i	−0.134919	−	0.938384i
$519$		0			0
$520$		0			0
$521$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$521$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$522$		0			0
$523$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$523$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$524$		0			0
$525$		0			0
$526$	0.131891	+	0.288802i	0.131891	+	0.288802i
$527$		0			0
$528$		0			0
$529$	0.142315	−	0.989821i	0.142315	−	0.989821i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	1.53046	+	0.983568i	1.53046	+	0.983568i
$537$		0			0
$538$		0			0
$539$	−1.53046	+	0.983568i	−1.53046	+	0.983568i
$540$		0			0
$541$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$541$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$547$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$548$	0.368991	+	0.108346i	0.368991	+	0.108346i
$549$		0			0
$550$	0.145886	−	1.01466i	0.145886	−	1.01466i
$551$		0			0
$552$		0			0
$553$	0.284630			0.284630
$554$	−0.134919	+	0.938384i	−0.134919	+	0.938384i
$555$		0			0
$556$		0			0
$557$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$557$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.938384	−	0.603063i	0.938384	−	0.603063i
$563$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$563$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.45093	+	0.932456i	1.45093	+	0.932456i
$569$	1.03748	−	0.304632i	1.03748	−	0.304632i	0.281733	−	0.959493i	$-0.409091\pi$
$569$	1.03748	−	0.304632i	1.03748	−	0.304632i	0.755750	+	0.654861i	$0.227273\pi$
$570$		0			0
$571$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$571$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$576$		0			0
$577$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$577$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$578$	0.234072	+	0.512546i	0.234072	+	0.512546i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.49611	−	3.27603i	1.49611	−	3.27603i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$587$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.239446	−	0.0703076i	0.239446	−	0.0703076i
$593$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$593$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$594$		0			0
$595$		0			0
$596$	0.146813	−	1.02111i	0.146813	−	1.02111i
$597$		0			0
$598$		0			0
$599$	−0.563465			−0.563465			−0.281733	−	0.959493i	$-0.590909\pi$
$599$	−0.563465			−0.563465			−0.281733	+	0.959493i	$0.590909\pi$
$600$		0			0
$601$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$601$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$602$	0.708089	+	0.207914i	0.708089	+	0.207914i
$603$		0			0
$604$	0.186393	−	0.0547299i	0.186393	−	0.0547299i
$605$		0			0
$606$		0			0
$607$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$607$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$613$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$614$		0			0
$615$		0			0
$616$	1.12945	+	1.30346i	1.12945	+	1.30346i
$617$	0.540641	+	0.158746i	0.540641	+	0.158746i	0.540641	−	0.841254i	$-0.318182\pi$
$617$	0.540641	+	0.158746i	0.540641	+	0.158746i			1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$619$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$631$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$632$	−0.0384020	−	0.267092i	−0.0384020	−	0.267092i
$633$		0			0
$634$	−0.512546	+	0.329393i	−0.512546	+	0.329393i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−1.94711	+	0.571724i	−1.94711	+	0.571724i
$639$		0			0
$640$		0			0
$641$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$641$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	0.675560	+	0.0971309i	0.675560	+	0.0971309i
$645$		0			0
$646$		0			0
$647$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$647$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.0806993	−	0.176707i	0.0806993	−	0.176707i
$653$	0.368991	−	0.425839i	0.368991	−	0.425839i	−0.540641	−	0.841254i	$-0.681818\pi$
$653$	0.368991	−	0.425839i	0.368991	−	0.425839i	0.909632	+	0.415415i	$0.136364\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.822373	+	1.80075i	−0.822373	+	1.80075i	−0.281733	+	0.959493i	$0.590909\pi$
$659$	−0.822373	+	1.80075i	−0.822373	+	1.80075i	−0.540641	+	0.841254i	$0.681818\pi$
$660$		0			0
$661$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$661$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$662$	−0.483276	−	0.557730i	−0.483276	−	0.557730i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.07028	+	1.66538i	−1.07028	+	1.66538i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−0.841254	+	0.540641i	$0.818182\pi$
$673$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−1.00000			$1.00000\pi$
$674$	−0.393828	−	0.253098i	−0.393828	−	0.253098i
$675$		0			0
$676$	0.446947	−	0.515804i	0.446947	−	0.515804i
$677$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$677$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.909632	−	0.584585i	−0.909632	−	0.584585i		−	1.00000i	$-0.5\pi$
$683$	−0.909632	−	0.584585i	−0.909632	−	0.584585i	−0.909632	+	0.415415i	$0.863636\pi$
$684$		0			0
$685$		0			0
$686$	0.540641	+	0.158746i	0.540641	+	0.158746i
$687$		0			0
$688$	−0.0276463	+	0.192284i	−0.0276463	+	0.192284i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	−0.584585	−	0.171650i	−0.584585	−	0.171650i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.574161	−	0.368991i	0.574161	−	0.368991i
$701$	0.0801894	+	0.557730i	0.0801894	+	0.557730i	0.989821	+	0.142315i	$0.0454545\pi$
$701$	0.0801894	+	0.557730i	0.0801894	+	0.557730i	−0.909632	+	0.415415i	$0.863636\pi$
$702$		0			0
$703$		0			0
$704$	0.515804	−	0.595270i	0.515804	−	0.595270i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.654861	−	0.755750i	$-0.272727\pi$
$709$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.142315	+	0.989821i	$0.454545\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.428546	−	0.938384i	−0.428546	−	0.938384i
$717$		0			0
$718$	−0.557730	−	0.643655i	−0.557730	−	0.643655i
$719$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$719$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$720$		0			0
$721$		0			0
$722$	−0.368991	+	0.425839i	−0.368991	+	0.425839i
$723$		0			0
$724$		0			0
$725$	0.281733	+	1.95949i	0.281733	+	1.95949i
$726$		0			0
$727$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$727$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$733$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$734$		0			0
$735$		0			0
$736$		−	1.03161i		−	1.03161i
$737$	3.49114			3.49114
$738$		0			0
$739$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$739$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$740$		0			0
$741$		0			0
$742$	−1.07028	+	0.314261i	−1.07028	+	0.314261i
$743$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	0.281733	−	0.959493i	$-0.409091\pi$
$743$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	−0.755750	+	0.654861i	$0.772727\pi$
$744$		0			0
$745$		0			0
$746$	0.393828	−	0.253098i	0.393828	−	0.253098i
$747$		0			0
$748$		0			0
$749$	0.909632	−	0.584585i	0.909632	−	0.584585i
$750$		0			0
$751$	0.118239	−	0.258908i	0.118239	−	0.258908i	−0.841254	−	0.540641i	$-0.818182\pi$
$751$	0.118239	−	0.258908i	0.118239	−	0.258908i	0.959493	+	0.281733i	$0.0909091\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.118239	−	0.822373i	0.118239	−	0.822373i	−0.841254	−	0.540641i	$-0.818182\pi$
$757$	0.118239	−	0.822373i	0.118239	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$758$	0.160379			0.160379
$759$		0			0
$760$		0			0
$761$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$761$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$762$		0			0
$763$	1.84125	+	0.540641i	1.84125	+	0.540641i
$764$	0.483276	+	0.557730i	0.483276	+	0.557730i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$769$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$770$		0			0
$771$		0			0
$772$	0.477031	−	0.306569i	0.477031	−	0.306569i
$773$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$773$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.983568	−	0.288802i	−0.983568	−	0.288802i
$779$		0			0
$780$		0			0
$781$	3.30972			3.30972
$782$		0			0
$783$		0			0
$784$	−0.0211086	+	0.146813i	−0.0211086	+	0.146813i
$785$		0			0
$786$		0			0
$787$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$787$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$788$	0.989821	−	0.290638i	0.989821	−	0.290638i
$789$		0			0
$790$		0			0
$791$	−0.989821	+	1.14231i	−0.989821	+	1.14231i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$797$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$798$		0			0
$799$		0			0
$800$	−0.675560	−	0.779638i	−0.675560	−	0.779638i
$801$		0			0
$802$	0.253098	+	0.554206i	0.253098	+	0.554206i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.89945	−	0.557730i	−1.89945	−	0.557730i	−0.989821	−	0.142315i	$-0.954545\pi$
$809$	−1.89945	−	0.557730i	−1.89945	−	0.557730i	−0.909632	−	0.415415i	$-0.863636\pi$
$810$		0			0
$811$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$811$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$812$	−1.13663	−	0.730471i	−1.13663	−	0.730471i
$813$		0			0
$814$	−1.12945	+	1.30346i	−1.12945	+	1.30346i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.53046	−	0.983568i	−1.53046	−	0.983568i	−0.989821	−	0.142315i	$-0.954545\pi$
$821$	−1.53046	−	0.983568i	−1.53046	−	0.983568i	−0.540641	−	0.841254i	$-0.681818\pi$
$822$		0			0
$823$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$823$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.08128			1.08128			0.540641	−	0.841254i	$-0.318182\pi$
$827$	1.08128			1.08128			0.540641	+	0.841254i	$0.318182\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$839$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$840$		0			0
$841$	2.45561	−	1.57812i	2.45561	−	1.57812i
$842$	0.483276	−	0.557730i	0.483276	−	0.557730i
$843$		0			0
$844$	0.477031	+	0.306569i	0.477031	+	0.306569i
$845$		0			0
$846$		0			0
$847$	2.21616	+	0.650724i	2.21616	+	0.650724i
$848$	−0.121977	−	0.267092i	−0.121977	−	0.267092i
$849$		0			0
$850$		0			0
$851$			1.68251i			1.68251i
$852$		0			0
$853$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$853$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$854$		0			0
$855$		0			0
$856$	−0.671292	−	0.774713i	−0.671292	−	0.774713i
$857$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$857$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$858$		0			0
$859$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$859$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$860$		0			0
$861$		0			0
$862$	0.121206	+	0.843008i	0.121206	+	0.843008i
$863$	−0.258908	−	1.80075i	−0.258908	−	1.80075i	−0.540641	−	0.841254i	$-0.681818\pi$
$863$	−0.258908	−	1.80075i	−0.258908	−	1.80075i	0.281733	−	0.959493i	$-0.409091\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.339098	−	0.391340i	−0.339098	−	0.391340i
$870$		0			0
$871$		0			0
$872$	0.258908	−	1.80075i	0.258908	−	1.80075i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$877$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$881$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$882$		0			0
$883$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$883$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$884$		0			0
$885$		0			0
$886$	−0.145886	−	1.01466i	−0.145886	−	1.01466i
$887$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$887$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$888$		0			0
$889$	0.544078	−	1.19136i	0.544078	−	1.19136i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.787655			0.787655
$897$		0			0
$898$	−0.609264			−0.609264
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.20548	+	0.774713i	1.20548	+	0.774713i
$905$		0			0
$906$		0			0
$907$	1.61435	−	1.03748i	1.61435	−	1.03748i	0.654861	−	0.755750i	$-0.272727\pi$
$907$	1.61435	−	1.03748i	1.61435	−	1.03748i	0.959493	−	0.281733i	$-0.0909091\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.368991	−	0.425839i	0.368991	−	0.425839i	−0.540641	−	0.841254i	$-0.681818\pi$
$911$	0.368991	−	0.425839i	0.368991	−	0.425839i	0.909632	+	0.415415i	$0.136364\pi$
$912$		0			0
$913$		0			0
$914$	−0.153882	+	0.0451840i	−0.153882	+	0.0451840i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$919$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.10181	+	1.27155i	1.10181	+	1.27155i
$926$	1.03748	−	0.304632i	1.03748	−	0.304632i
$927$		0			0
$928$	−0.848368	+	1.85767i	−0.848368	+	1.85767i
$929$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$929$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$930$		0			0
$931$		0			0
$932$	0.105026	+	0.730471i	0.105026	+	0.730471i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$937$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$938$	−0.708089	−	0.817178i	−0.708089	−	0.817178i
$939$		0			0
$940$		0			0
$941$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$941$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.557730	−	1.22126i	−0.557730	−	1.22126i
$947$	1.03748	+	0.304632i	1.03748	+	0.304632i	0.755750	−	0.654861i	$-0.227273\pi$
$947$	1.03748	+	0.304632i	1.03748	+	0.304632i	0.281733	+	0.959493i	$0.409091\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.909632	+	0.584585i	−0.909632	+	0.584585i	−0.909632	−	0.415415i	$-0.863636\pi$
$953$	−0.909632	+	0.584585i	−0.909632	+	0.584585i			1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	0.620830	−	0.398983i	0.620830	−	0.398983i
$957$		0			0
$958$		0			0
$959$	−0.474017	−	0.304632i	−0.474017	−	0.304632i
$960$		0			0
$961$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$967$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$968$	0.311626	−	2.16741i	0.311626	−	2.16741i
$969$		0			0
$970$		0			0
$971$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$971$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$972$		0			0
$973$		0			0
$974$	0.306569	−	0.671292i	0.306569	−	0.671292i
$975$		0			0
$976$		0			0
$977$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$977$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−0.267092	−	0.171650i	−0.267092	−	0.171650i
$983$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$983$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.19136	−	0.544078i	−1.19136	−	0.544078i
$990$		0			0
$991$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$991$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$992$		0			0
$993$		0			0
$994$	−0.671292	−	0.774713i	−0.671292	−	0.774713i
$995$		0			0
$996$		0			0
$997$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$997$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$998$	−0.708089	+	0.817178i	−0.708089	+	0.817178i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.1.bq.b.307.1	yes	20
3.2	odd	2	inner	1449.1.bq.b.307.2	yes	20
7.6	odd	2	CM	1449.1.bq.b.307.1	yes	20
21.20	even	2	inner	1449.1.bq.b.307.2	yes	20
23.3	even	11	inner	1449.1.bq.b.118.1	✓	20
69.26	odd	22	inner	1449.1.bq.b.118.2	yes	20
161.118	odd	22	inner	1449.1.bq.b.118.1	✓	20
483.440	even	22	inner	1449.1.bq.b.118.2	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1449.1.bq.b.118.1	✓	20	23.3	even	11	inner
1449.1.bq.b.118.1	✓	20	161.118	odd	22	inner
1449.1.bq.b.118.2	yes	20	69.26	odd	22	inner
1449.1.bq.b.118.2	yes	20	483.440	even	22	inner
1449.1.bq.b.307.1	yes	20	1.1	even	1	trivial
1449.1.bq.b.307.1	yes	20	7.6	odd	2	CM
1449.1.bq.b.307.2	yes	20	3.2	odd	2	inner
1449.1.bq.b.307.2	yes	20	21.20	even	2	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1449.bq (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.0801894 + 0.557730i) q^{2} +(0.654861 + 0.192284i) q^{4} +(-0.841254 - 0.540641i) q^{7} +(-0.393828 + 0.862362i) q^{8} +O(q^{10})\)	\(q+(-0.0801894 + 0.557730i) q^{2} +(0.654861 + 0.192284i) q^{4} +(-0.841254 - 0.540641i) q^{7} +(-0.393828 + 0.862362i) q^{8} +(0.258908 + 1.80075i) q^{11} +(0.368991 - 0.425839i) q^{14} +(0.124777 + 0.0801894i) q^{16} -1.02509 q^{22} +(-0.755750 + 0.654861i) q^{23} +(-0.142315 + 0.989821i) q^{25} +(-0.446947 - 0.515804i) q^{28} +(1.89945 - 0.557730i) q^{29} +(-0.675560 + 0.779638i) q^{32} +(1.10181 - 1.27155i) q^{37} +(0.544078 + 1.19136i) q^{43} +(-0.176707 + 1.22902i) q^{44} +(-0.304632 - 0.474017i) q^{46} +(0.415415 + 0.909632i) q^{49} +(-0.540641 - 0.158746i) q^{50} +(-1.66538 - 1.07028i) q^{53} +(0.797537 - 0.512546i) q^{56} +(0.158746 + 1.10411i) q^{58} +(-0.283524 - 0.327204i) q^{64} +(0.273100 - 1.89945i) q^{67} +(0.258908 - 1.80075i) q^{71} +(0.620830 + 0.716476i) q^{74} +(0.755750 - 1.65486i) q^{77} +(-0.239446 + 0.153882i) q^{79} +(-0.708089 + 0.207914i) q^{86} +(-1.65486 - 0.485911i) q^{88} +(-0.620830 + 0.283524i) q^{92} +(-0.540641 + 0.158746i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{4} + 2 q^{7}+O(q^{10}) \) 20 * q + 2 * q^4 + 2 * q^7	\( 20 q + 2 q^{4} + 2 q^{7} - 24 q^{16} - 2 q^{25} - 2 q^{28} + 4 q^{37} + 4 q^{43} - 2 q^{49} + 22 q^{58} + 2 q^{64} - 4 q^{67} - 4 q^{79} - 22 q^{88}+O(q^{100}) \) 20 * q + 2 * q^4 + 2 * q^7 - 24 * q^16 - 2 * q^25 - 2 * q^28 + 4 * q^37 + 4 * q^43 - 2 * q^49 + 22 * q^58 + 2 * q^64 - 4 * q^67 - 4 * q^79 - 22 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more