LMFDB - Embedded newform 1035.1.bd.a.244.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,1,Mod(19,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 15]))

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

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

chi := DirichletCharacter("1035.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1035 = 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1035.bd (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:	$0.516532288075$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
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			244.1
Root			$-0.841254 + 0.540641i$ of defining polynomial
Character	$\chi$	$=$	1035.244
Dual form			1035.1.bd.a.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.425839 - 1.45027i) q^{2} +(-1.08070 + 0.694523i) q^{4} +(-0.142315 - 0.989821i) q^{5} +(0.325137 + 0.281733i) q^{8} +(-1.37491 + 0.627899i) q^{10} +(-0.263521 + 0.577031i) q^{16} +(-1.41542 - 0.909632i) q^{17} +(-0.983568 - 1.53046i) q^{19} +(0.841254 + 0.970858i) q^{20} +(0.959493 + 0.281733i) q^{23} +(-0.959493 + 0.281733i) q^{25} +(0.544078 - 0.627899i) q^{31} +(1.37491 + 0.197682i) q^{32} +(-0.716476 + 2.44009i) q^{34} +(-1.80075 + 2.07817i) q^{38} +(0.232593 - 0.361922i) q^{40} -1.51150i q^{46} -0.563465i q^{47} +(0.654861 + 0.755750i) q^{49} +(0.817178 + 1.27155i) q^{50} +(-0.797176 + 1.74557i) q^{53} +(-0.425839 - 0.368991i) q^{61} +(-1.14231 - 0.521678i) q^{62} +(-0.208518 - 1.45027i) q^{64} +2.16140 q^{68} +(2.12588 + 0.970858i) q^{76} +(1.80075 - 0.822373i) q^{79} +(0.608660 + 0.178719i) q^{80} +(0.186393 - 1.29639i) q^{83} +(-0.698939 + 1.53046i) q^{85} +(-1.23259 + 0.361922i) q^{92} +(-0.817178 + 0.239945i) q^{94} +(-1.37491 + 1.19136i) q^{95} +(0.817178 - 1.27155i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{4} - q^{5} + 11 q^{8} - q^{16} - 9 q^{17} - q^{20} + q^{23} - q^{25} + 2 q^{31} - 11 q^{34} - 11 q^{40} + q^{49} - 2 q^{53} - 11 q^{62} + 10 q^{64} + 2 q^{68} + 11 q^{76} + 10 q^{80} - 2 q^{83}+ \cdots + q^{92}+O(q^{100}) $ 10 * q - q^4 - q^5 + 11 * q^8 - q^16 - 9 * q^17 - q^20 + q^23 - q^25 + 2 * q^31 - 11 * q^34 - 11 * q^40 + q^49 - 2 * q^53 - 11 * q^62 + 10 * q^64 + 2 * q^68 + 11 * q^76 + 10 * q^80 - 2 * q^83 + 2 * q^85 + q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1035\mathbb{Z}\right)^\times$.

$n$	$461$	$622$	$856$
$\chi(n)$	$1$	$-1$	$e\left(\frac{21}{22}\right)$

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 $

$2$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$2$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$3$		0			0
$3$		0			0
$4$	−1.08070	+	0.694523i	−1.08070	+	0.694523i
$5$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$5$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$6$		0			0
$7$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$7$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$8$	0.325137	+	0.281733i	0.325137	+	0.281733i
$9$		0			0
$10$	−1.37491	+	0.627899i	−1.37491	+	0.627899i
$11$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$11$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$12$		0			0
$13$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$13$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$14$		0			0
$15$		0			0
$16$	−0.263521	+	0.577031i	−0.263521	+	0.577031i
$17$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−0.415415	−	0.909632i	$-0.636364\pi$
$17$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−1.00000			$\pi$
$18$		0			0
$19$	−0.983568	−	1.53046i	−0.983568	−	1.53046i	−0.841254	−	0.540641i	$-0.818182\pi$
$19$	−0.983568	−	1.53046i	−0.983568	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$20$	0.841254	+	0.970858i	0.841254	+	0.970858i
$21$		0			0
$22$		0			0
$23$	0.959493	+	0.281733i	0.959493	+	0.281733i
$23$	0.959493	+	0.281733i	0.959493	+	0.281733i
$24$		0			0
$25$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$29$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$30$		0			0
$31$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$31$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$32$	1.37491	+	0.197682i	1.37491	+	0.197682i
$33$		0			0
$34$	−0.716476	+	2.44009i	−0.716476	+	2.44009i
$35$		0			0
$36$		0			0
$37$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$37$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$38$	−1.80075	+	2.07817i	−1.80075	+	2.07817i
$39$		0			0
$40$	0.232593	−	0.361922i	0.232593	−	0.361922i
$41$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$41$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$42$		0			0
$43$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$43$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$44$		0			0
$45$		0			0
$46$		−	1.51150i		−	1.51150i
$47$		−	0.563465i		−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$47$		−	0.563465i		−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$48$		0			0
$49$	0.654861	+	0.755750i	0.654861	+	0.755750i
$50$	0.817178	+	1.27155i	0.817178	+	1.27155i
$51$		0			0
$52$		0			0
$53$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$53$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$59$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$60$		0			0
$61$	−0.425839	−	0.368991i	−0.425839	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$61$	−0.425839	−	0.368991i	−0.425839	−	0.368991i	−0.841254	+	0.540641i	$0.818182\pi$
$62$	−1.14231	−	0.521678i	−1.14231	−	0.521678i
$63$		0			0
$64$	−0.208518	−	1.45027i	−0.208518	−	1.45027i
$65$		0			0
$66$		0			0
$67$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$67$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$68$	2.16140			2.16140
$69$		0			0
$70$		0			0
$71$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$71$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$72$		0			0
$73$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$73$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$74$		0			0
$75$		0			0
$76$	2.12588	+	0.970858i	2.12588	+	0.970858i
$77$		0			0
$78$		0			0
$79$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$79$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$80$	0.608660	+	0.178719i	0.608660	+	0.178719i
$81$		0			0
$82$		0			0
$83$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$83$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$84$		0			0
$85$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$89$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$90$		0			0
$91$		0			0
$92$	−1.23259	+	0.361922i	−1.23259	+	0.361922i
$93$		0			0
$94$	−0.817178	+	0.239945i	−0.817178	+	0.239945i
$95$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$96$		0			0
$97$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$97$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$98$	0.817178	−	1.27155i	0.817178	−	1.27155i
$99$		0			0
$100$	0.841254	−	0.970858i	0.841254	−	0.970858i
$101$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$101$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$102$		0			0
$103$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$103$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$104$		0			0
$105$		0			0
$106$	2.87102	+	0.412791i	2.87102	+	0.412791i
$107$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$107$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$108$		0			0
$109$	1.07028	−	1.66538i	1.07028	−	1.66538i	0.415415	−	0.909632i	$-0.363636\pi$
$109$	1.07028	−	1.66538i	1.07028	−	1.66538i	0.654861	−	0.755750i	$-0.272727\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$113$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$114$		0			0
$115$	0.142315	−	0.989821i	0.142315	−	0.989821i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.841254	+	0.540641i	0.841254	+	0.540641i
$122$	−0.353799	+	0.774713i	−0.353799	+	0.774713i
$123$		0			0
$124$	−0.151894	+	1.05645i	−0.151894	+	1.05645i
$125$	0.415415	+	0.909632i	0.415415	+	0.909632i
$126$		0			0
$127$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$127$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$128$	−0.750975	+	0.342959i	−0.750975	+	0.342959i
$129$		0			0
$130$		0			0
$131$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$131$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.203930	−	0.694523i	−0.203930	−	0.694523i
$137$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$137$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$138$		0			0
$139$	1.91899			1.91899			0.959493	−	0.281733i	$-0.0909091\pi$
$139$	1.91899			1.91899			0.959493	+	0.281733i	$0.0909091\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$149$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$150$		0			0
$151$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	−0.841254	−	0.540641i	$-0.818182\pi$
$151$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$152$	0.111387	−	0.774713i	0.111387	−	0.774713i
$153$		0			0
$154$		0			0
$155$	−0.698939	−	0.449181i	−0.698939	−	0.449181i
$156$		0			0
$157$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$157$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$158$	−1.95949	−	2.26138i	−1.95949	−	2.26138i
$159$		0			0
$160$		−	1.38905i		−	1.38905i
$161$		0			0
$162$		0			0
$163$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$163$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$164$		0			0
$165$		0			0
$166$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$167$	1.07028	−	1.66538i	1.07028	−	1.66538i	0.415415	−	0.909632i	$-0.363636\pi$
$167$	1.07028	−	1.66538i	1.07028	−	1.66538i	0.654861	−	0.755750i	$-0.272727\pi$
$168$		0			0
$169$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$170$	2.51722	+	0.361922i	2.51722	+	0.361922i
$171$		0			0
$172$		0			0
$173$	0.158746	−	0.540641i	0.158746	−	0.540641i	−0.841254	−	0.540641i	$-0.818182\pi$
$173$	0.158746	−	0.540641i	0.158746	−	0.540641i	1.00000			$0$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$179$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$180$		0			0
$181$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$181$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	0.142315	+	0.989821i	$0.454545\pi$
$182$		0			0
$183$		0			0
$184$	0.232593	+	0.361922i	0.232593	+	0.361922i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.391340	+	0.608936i	0.391340	+	0.608936i
$189$		0			0
$190$	2.31329	+	1.48666i	2.31329	+	1.48666i
$191$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$191$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$192$		0			0
$193$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$193$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$194$		0			0
$195$		0			0
$196$	−1.23259	−	0.361922i	−1.23259	−	0.361922i
$197$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$197$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$198$		0			0
$199$	−1.14231	−	0.989821i	−1.14231	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$199$	−1.14231	−	0.989821i	−1.14231	−	0.989821i	−1.00000			$\pi$
$200$	−0.391340	−	0.178719i	−0.391340	−	0.178719i
$201$		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			0
$210$		0			0
$211$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$211$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−1.00000			$\pi$
$212$	−0.350833	−	2.44009i	−0.350833	−	2.44009i
$213$		0			0
$214$	0.391340	+	0.178719i	0.391340	+	0.178719i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.87102	−	0.843008i	−2.87102	−	0.843008i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$223$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$224$		0			0
$225$		0			0
$226$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$227$	1.25667	+	1.45027i	1.25667	+	1.45027i	0.841254	+	0.540641i	$0.181818\pi$
$227$	1.25667	+	1.45027i	1.25667	+	1.45027i	0.415415	+	0.909632i	$0.363636\pi$
$228$		0			0
$229$		−	1.51150i		−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$229$		−	1.51150i		−	1.51150i	0.654861	−	0.755750i	$-0.272727\pi$
$230$	−1.49611	+	0.215109i	−1.49611	+	0.215109i
$231$		0			0
$232$		0			0
$233$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.654861	−	0.755750i	$-0.272727\pi$
$233$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$234$		0			0
$235$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$239$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$240$		0			0
$241$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$241$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$242$	0.425839	−	1.45027i	0.425839	−	1.45027i
$243$		0			0
$244$	0.716476	+	0.103014i	0.716476	+	0.103014i
$245$	0.654861	−	0.755750i	0.654861	−	0.755750i
$246$		0			0
$247$		0			0
$248$	0.353799	−	0.0508687i	0.353799	−	0.0508687i
$249$		0			0
$250$	1.14231	−	0.989821i	1.14231	−	0.989821i
$251$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$251$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.142315	−	0.164240i	−0.142315	−	0.164240i
$257$	0.584585	+	0.909632i	0.584585	+	0.909632i	1.00000			$0$
$257$	0.584585	+	0.909632i	0.584585	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$263$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$264$		0			0
$265$	1.84125	+	0.540641i	1.84125	+	0.540641i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$269$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$270$		0			0
$271$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$271$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$272$	0.897877	−	0.577031i	0.897877	−	0.577031i
$273$		0			0
$274$	−0.121206	−	0.412791i	−0.121206	−	0.412791i
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	−0.817178	−	2.78305i	−0.817178	−	2.78305i
$279$		0			0
$280$		0			0
$281$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$281$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$282$		0			0
$283$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$283$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.760554	+	1.66538i	0.760554	+	1.66538i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.10181	+	0.708089i	1.10181	+	0.708089i	0.959493	−	0.281733i	$-0.0909091\pi$
$293$	1.10181	+	0.708089i	1.10181	+	0.708089i	0.142315	+	0.989821i	$0.454545\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−1.92195	+	1.66538i	−1.92195	+	1.66538i
$303$		0			0
$304$	1.14231	−	0.164240i	1.14231	−	0.164240i
$305$	−0.304632	+	0.474017i	−0.304632	+	0.474017i
$306$		0			0
$307$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$307$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$308$		0			0
$309$		0			0
$310$	−0.353799	+	1.20493i	−0.353799	+	1.20493i
$311$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$311$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$312$		0			0
$313$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$313$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$314$		0			0
$315$		0			0
$316$	−1.37491	+	2.13940i	−1.37491	+	2.13940i
$317$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$317$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$318$		0			0
$319$		0			0
$320$	−1.40584	+	0.412791i	−1.40584	+	0.412791i
$321$		0			0
$322$		0			0
$323$			3.06092i			3.06092i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$331$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$332$	0.698939	+	1.53046i	0.698939	+	1.53046i
$333$		0			0
$334$	−2.87102	−	0.843008i	−2.87102	−	0.843008i
$335$		0			0
$336$		0			0
$337$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$337$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$338$	1.37491	+	0.627899i	1.37491	+	0.627899i
$339$		0			0
$340$	−0.307599	−	2.13940i	−0.307599	−	2.13940i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.851677			−0.851677
$347$	0.512546	+	1.74557i	0.512546	+	1.74557i	0.654861	+	0.755750i	$0.272727\pi$
$347$	0.512546	+	1.74557i	0.512546	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$348$		0			0
$349$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$349$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.37491	+	1.19136i	1.37491	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$353$	1.37491	+	1.19136i	1.37491	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$359$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$360$		0			0
$361$	−0.959493	+	2.10100i	−0.959493	+	2.10100i
$362$	1.37491	+	0.883600i	1.37491	+	0.883600i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−0.415415	+	0.479414i	−0.415415	+	0.479414i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$373$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$374$		0			0
$375$		0			0
$376$	0.158746	−	0.183203i	0.158746	−	0.183203i
$377$		0			0
$378$		0			0
$379$	0.425839	−	1.45027i	0.425839	−	1.45027i	−0.415415	−	0.909632i	$-0.636364\pi$
$379$	0.425839	−	1.45027i	0.425839	−	1.45027i	0.841254	−	0.540641i	$-0.181818\pi$
$380$	0.658432	−	2.24241i	0.658432	−	2.24241i
$381$		0			0
$382$		0			0
$383$	−0.857685	+	0.989821i	−0.857685	+	0.989821i	0.142315	+	0.989821i	$0.454545\pi$
$383$	−0.857685	+	0.989821i	−0.857685	+	0.989821i	−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$389$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$390$		0			0
$391$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$392$			0.430218i			0.430218i
$393$		0			0
$394$	1.95949	+	2.26138i	1.95949	+	2.26138i
$395$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$396$		0			0
$397$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$397$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$398$	−0.949069	+	2.07817i	−0.949069	+	2.07817i
$399$		0			0
$400$	0.0902783	−	0.627899i	0.0902783	−	0.627899i
$401$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$401$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.239446	+	1.66538i	0.239446	+	1.66538i	0.654861	+	0.755750i	$0.272727\pi$
$409$	0.239446	+	1.66538i	0.239446	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.30972			−1.30972
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$419$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$420$		0			0
$421$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$421$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−1.00000			$\pi$
$422$	1.92195	+	1.66538i	1.92195	+	1.66538i
$423$		0			0
$424$	−0.750975	+	0.342959i	−0.750975	+	0.342959i
$425$	1.61435	+	0.474017i	1.61435	+	0.474017i
$426$		0			0
$427$		0			0
$428$	0.0520365	−	0.361922i	0.0520365	−	0.361922i
$429$		0			0
$430$		0			0
$431$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$431$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$432$		0			0
$433$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$433$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$434$		0			0
$435$		0			0
$436$			2.54311i			2.54311i
$437$	−0.512546	−	1.74557i	−0.512546	−	1.74557i
$438$		0			0
$439$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$439$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.415415	+	0.909632i	$0.636364\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.584585	−	0.909632i	0.584585	−	0.909632i	−0.415415	−	0.909632i	$-0.636364\pi$
$443$	0.584585	−	0.909632i	0.584585	−	0.909632i	1.00000			$0$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$449$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$450$		0			0
$451$		0			0
$452$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$453$		0			0
$454$	1.56815	−	2.44009i	1.56815	−	2.44009i
$455$		0			0
$456$		0			0
$457$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$457$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$458$	−2.19209	+	0.643655i	−2.19209	+	0.643655i
$459$		0			0
$460$	0.533654	+	1.16854i	0.533654	+	1.16854i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$463$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$464$		0			0
$465$		0			0
$466$	−2.51722	−	1.61772i	−2.51722	−	1.61772i
$467$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$467$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$468$		0			0
$469$		0			0
$470$	0.353799	+	0.774713i	0.353799	+	0.774713i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.37491	+	1.19136i	1.37491	+	1.19136i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$479$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$480$		0			0
$481$		0			0
$482$	1.63436			1.63436
$483$		0			0
$484$	−1.28463			−1.28463
$485$		0			0
$486$		0			0
$487$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$487$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$488$	−0.0344989	−	0.239945i	−0.0344989	−	0.239945i
$489$		0			0
$490$	−1.37491	−	0.627899i	−1.37491	−	0.627899i
$491$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$491$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.218941	+	0.479414i	0.218941	+	0.479414i
$497$		0			0
$498$		0			0
$499$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$499$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$500$	−1.08070	−	0.694523i	−1.08070	−	0.694523i
$501$		0			0
$502$		0			0
$503$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$503$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$509$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$510$		0			0
$511$		0			0
$512$	−0.623933	+	0.970858i	−0.623933	+	0.970858i
$513$		0			0
$514$	1.07028	−	1.23516i	1.07028	−	1.23516i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$521$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$522$		0			0
$523$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$523$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$524$		0			0
$525$		0			0
$526$	0.949069	−	0.822373i	0.949069	−	0.822373i
$527$	−1.34125	+	0.393828i	−1.34125	+	0.393828i
$528$		0			0
$529$	0.841254	+	0.540641i	0.841254	+	0.540641i
$530$		−	2.90055i		−	2.90055i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.239446	+	0.153882i	0.239446	+	0.153882i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.654861	−	0.755750i	$-0.272727\pi$
$541$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.142315	+	0.989821i	$0.454545\pi$
$542$	2.63843	−	1.20493i	2.63843	−	1.20493i
$543$		0			0
$544$	−1.76625	−	1.53046i	−1.76625	−	1.53046i
$545$	−1.80075	−	0.822373i	−1.80075	−	0.822373i
$546$		0			0
$547$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$547$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$548$	−0.307599	+	0.197682i	−0.307599	+	0.197682i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−2.07385	+	1.33278i	−2.07385	+	1.33278i
$557$	0.118239	+	0.822373i	0.118239	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$557$	0.118239	+	0.822373i	0.118239	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.84125	−	0.540641i	−1.84125	−	0.540641i	−0.841254	−	0.540641i	$-0.818182\pi$
$563$	−1.84125	−	0.540641i	−1.84125	−	0.540641i	−1.00000			$\pi$
$564$		0			0
$565$	−0.544078	−	1.19136i	−0.544078	−	1.19136i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$569$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$570$		0			0
$571$	0.304632	+	0.474017i	0.304632	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$571$	0.304632	+	0.474017i	0.304632	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−1.00000
$576$		0			0
$577$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$577$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$578$	2.09138	−	1.81219i	2.09138	−	1.81219i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.557730	−	1.89945i	0.557730	−	1.89945i
$587$	0.512546	−	1.74557i	0.512546	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$587$	0.512546	−	1.74557i	0.512546	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$588$		0			0
$589$	−1.49611	−	0.215109i	−1.49611	−	0.215109i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.80075	−	0.258908i	1.80075	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$593$	1.80075	−	0.258908i	1.80075	−	0.258908i	0.959493	+	0.281733i	$0.0909091\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$601$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$602$		0			0
$603$		0			0
$604$	1.81828	+	1.16854i	1.81828	+	1.16854i
$605$	0.415415	−	0.909632i	0.415415	−	0.909632i
$606$		0			0
$607$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$607$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$608$	−1.04977	−	2.29868i	−1.04977	−	2.29868i
$609$		0			0
$610$	0.817178	+	0.239945i	0.817178	+	0.239945i
$611$		0			0
$612$		0			0
$613$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$613$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$617$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$618$		0			0
$619$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$619$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$620$	1.06731			1.06731
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.841254	−	0.540641i	0.841254	−	0.540641i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.142315	−	0.989821i	$-0.454545\pi$
$631$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$632$	0.817178	+	0.239945i	0.817178	+	0.239945i
$633$		0			0
$634$	0.678936	+	1.48666i	0.678936	+	1.48666i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.446343	+	0.694523i	0.446343	+	0.694523i
$641$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$641$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	4.43918	−	1.30346i	4.43918	−	1.30346i
$647$	0.817178	−	0.708089i	0.817178	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$647$	0.817178	−	0.708089i	0.817178	−	0.708089i	0.959493	+	0.281733i	$0.0909091\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.95949	−	0.281733i	−1.95949	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$653$	−1.95949	−	0.281733i	−1.95949	−	0.281733i	−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$659$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$660$		0			0
$661$	−0.983568	+	1.53046i	−0.983568	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$661$	−0.983568	+	1.53046i	−0.983568	+	1.53046i	−0.841254	+	0.540641i	$0.818182\pi$
$662$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$663$		0			0
$664$	0.425839	−	0.368991i	0.425839	−	0.368991i
$665$		0			0
$666$		0			0
$667$		0			0
$668$			2.54311i			2.54311i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$673$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$674$		0			0
$675$		0			0
$676$	0.182822	−	1.27155i	0.182822	−	1.27155i
$677$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$677$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$678$		0			0
$679$		0			0
$680$	−0.658432	+	0.300696i	−0.658432	+	0.300696i
$681$		0			0
$682$		0			0
$683$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$683$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$684$		0			0
$685$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.30972			1.30972			0.654861	−	0.755750i	$-0.272727\pi$
$691$	1.30972			1.30972			0.654861	+	0.755750i	$0.272727\pi$
$692$	0.203930	+	0.694523i	0.203930	+	0.694523i
$693$		0			0
$694$	2.31329	−	1.48666i	2.31329	−	1.48666i
$695$	−0.273100	−	1.89945i	−0.273100	−	1.89945i
$696$		0			0
$697$		0			0
$698$	1.49611	+	1.29639i	1.49611	+	1.29639i
$699$		0			0
$700$		0			0
$701$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$701$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.14231	−	2.50132i	1.14231	−	2.50132i
$707$		0			0
$708$		0			0
$709$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$709$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.698939	−	0.449181i	0.698939	−	0.449181i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$719$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$720$		0			0
$721$		0			0
$722$	3.45561	+	0.496841i	3.45561	+	0.496841i
$723$		0			0
$724$	0.391340	−	1.33278i	0.391340	−	1.33278i
$725$		0			0
$726$		0			0
$727$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$727$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$733$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$734$		0			0
$735$		0			0
$736$	1.26352	+	0.577031i	1.26352	+	0.577031i
$737$		0			0
$738$		0			0
$739$	−0.857685	−	0.989821i	−0.857685	−	0.989821i	0.142315	−	0.989821i	$-0.454545\pi$
$739$	−0.857685	−	0.989821i	−0.857685	−	0.989821i	−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.544078	−	1.19136i	0.544078	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$743$	0.544078	−	1.19136i	0.544078	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.817178	+	0.708089i	0.817178	+	0.708089i	0.959493	−	0.281733i	$-0.0909091\pi$
$751$	0.817178	+	0.708089i	0.817178	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$752$	0.325137	+	0.148485i	0.325137	+	0.148485i
$753$		0			0
$754$		0			0
$755$	−1.41542	+	0.909632i	−1.41542	+	0.909632i
$756$		0			0
$757$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$757$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$758$	−2.28463			−2.28463
$759$		0			0
$760$	−0.782679			−0.782679
$761$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$761$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1.80075	+	0.822373i	1.80075	+	0.822373i
$767$		0			0
$768$		0			0
$769$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$769$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.118239	−	0.822373i	0.118239	−	0.822373i	−0.841254	−	0.540641i	$-0.818182\pi$
$773$	0.118239	−	0.822373i	0.118239	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$774$		0			0
$775$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−1.37491	+	2.13940i	−1.37491	+	2.13940i
$783$		0			0
$784$	−0.608660	+	0.178719i	−0.608660	+	0.178719i
$785$		0			0
$786$		0			0
$787$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$787$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$788$	1.37491	−	2.13940i	1.37491	−	2.13940i
$789$		0			0
$790$	−1.95949	+	2.26138i	−1.95949	+	2.26138i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.92195	+	0.276335i	1.92195	+	0.276335i
$797$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$797$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$798$		0			0
$799$	−0.512546	+	0.797537i	−0.512546	+	0.797537i
$800$	−1.37491	+	0.197682i	−1.37491	+	0.197682i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$809$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$810$		0			0
$811$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	0.415415	−	0.909632i	$-0.363636\pi$
$811$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	2.31329	−	1.05645i	2.31329	−	1.05645i
$819$		0			0
$820$		0			0
$821$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$821$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$822$		0			0
$823$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$823$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$827$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$828$		0			0
$829$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$829$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$830$	0.557730	+	1.89945i	0.557730	+	1.89945i
$831$		0			0
$832$		0			0
$833$	−0.239446	−	1.66538i	−0.239446	−	1.66538i
$834$		0			0
$835$	−1.80075	−	0.822373i	−1.80075	−	0.822373i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$839$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$840$		0			0
$841$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$842$	−0.391340	+	2.72183i	−0.391340	+	2.72183i
$843$		0			0
$844$	0.897877	−	1.96608i	0.897877	−	1.96608i
$845$	0.841254	+	0.540641i	0.841254	+	0.540641i
$846$		0			0
$847$		0			0
$848$	−0.797176	−	0.919990i	−0.797176	−	0.919990i
$849$		0			0
$850$		−	2.54311i		−	2.54311i
$851$		0			0
$852$		0			0
$853$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$853$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$854$		0			0
$855$		0			0
$856$	−0.121206	+	0.0174268i	−0.121206	+	0.0174268i
$857$	−0.983568	+	1.53046i	−0.983568	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$857$	−0.983568	+	1.53046i	−0.983568	+	1.53046i	−0.841254	+	0.540641i	$0.818182\pi$
$858$		0			0
$859$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$859$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.959493	+	0.281733i	$0.909091\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$863$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$864$		0			0
$865$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.817178	−	0.239945i	0.817178	−	0.239945i
$873$		0			0
$874$	−2.31329	+	1.48666i	−2.31329	+	1.48666i
$875$		0			0
$876$		0			0
$877$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$877$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$878$	1.07028	+	1.66538i	1.07028	+	1.66538i
$879$		0			0
$880$		0			0
$881$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$881$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$882$		0			0
$883$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$883$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$884$		0			0
$885$		0			0
$886$	−1.56815	−	0.460451i	−1.56815	−	0.460451i
$887$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$887$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.862362	+	0.554206i	−0.862362	+	0.554206i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	2.71616	−	1.74557i	2.71616	−	1.74557i
$902$		0			0
$903$		0			0
$904$	0.512546	+	0.234072i	0.512546	+	0.234072i
$905$	0.817178	+	0.708089i	0.817178	+	0.708089i
$906$		0			0
$907$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$907$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$908$	−2.36533	−	0.694523i	−2.36533	−	0.694523i
$909$		0			0
$910$		0			0
$911$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$911$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.04977	+	1.63348i	1.04977	+	1.63348i
$917$		0			0
$918$		0			0
$919$			1.08128i			1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$919$			1.08128i			1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$920$	0.325137	−	0.281733i	0.325137	−	0.281733i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$929$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$930$		0			0
$931$	0.512546	−	1.74557i	0.512546	−	1.74557i
$932$	−0.716476	+	2.44009i	−0.716476	+	2.44009i
$933$		0			0
$934$	−2.51722	−	0.361922i	−2.51722	−	0.361922i
$935$		0			0
$936$		0			0
$937$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$937$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$938$		0			0
$939$		0			0
$940$	0.547045	−	0.474017i	0.547045	−	0.474017i
$941$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$941$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.304632	+	0.474017i	0.304632	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$947$	0.304632	+	0.474017i	0.304632	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$948$		0			0
$949$		0			0
$950$	1.14231	−	2.50132i	1.14231	−	2.50132i
$951$		0			0
$952$		0			0
$953$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$953$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.0440780	+	0.306569i	0.0440780	+	0.306569i
$962$		0			0
$963$		0			0
$964$	−0.391340	−	1.33278i	−0.391340	−	1.33278i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.121206	+	0.412791i	0.121206	+	0.412791i
$969$		0			0
$970$		0			0
$971$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$971$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.325137	−	0.148485i	0.325137	−	0.148485i
$977$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	0.142315	−	0.989821i	$-0.454545\pi$
$977$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	−0.415415	+	0.909632i	$0.636364\pi$
$978$		0			0
$979$		0			0
$980$	−0.182822	+	1.27155i	−0.182822	+	1.27155i
$981$		0			0
$982$		0			0
$983$	0.239446	+	0.153882i	0.239446	+	0.153882i	0.654861	−	0.755750i	$-0.272727\pi$
$983$	0.239446	+	0.153882i	0.239446	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$984$		0			0
$985$	1.07028	+	1.66538i	1.07028	+	1.66538i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$991$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$992$	0.872182	−	0.755750i	0.872182	−	0.755750i
$993$		0			0
$994$		0			0
$995$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$996$		0			0
$997$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$997$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$998$	0.425839	+	0.0612263i	0.425839	+	0.0612263i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1035.1.bd.a.244.1	✓	10
3.2	odd	2		1035.1.bd.b.244.1	yes	10
5.4	even	2		1035.1.bd.b.244.1	yes	10
15.14	odd	2	CM	1035.1.bd.a.244.1	✓	10
23.5	odd	22		1035.1.bd.b.649.1	yes	10
69.5	even	22	inner	1035.1.bd.a.649.1	yes	10
115.74	odd	22	inner	1035.1.bd.a.649.1	yes	10
345.74	even	22		1035.1.bd.b.649.1	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1035.1.bd.a.244.1	✓	10	1.1	even	1	trivial
1035.1.bd.a.244.1	✓	10	15.14	odd	2	CM
1035.1.bd.a.649.1	yes	10	69.5	even	22	inner
1035.1.bd.a.649.1	yes	10	115.74	odd	22	inner
1035.1.bd.b.244.1	yes	10	3.2	odd	2
1035.1.bd.b.244.1	yes	10	5.4	even	2
1035.1.bd.b.649.1	yes	10	23.5	odd	22
1035.1.bd.b.649.1	yes	10	345.74	even	22

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:	\( N \)	\(=\)	\( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1035.bd (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.425839 - 1.45027i) q^{2} +(-1.08070 + 0.694523i) q^{4} +(-0.142315 - 0.989821i) q^{5} +(0.325137 + 0.281733i) q^{8} +(-1.37491 + 0.627899i) q^{10} +(-0.263521 + 0.577031i) q^{16} +(-1.41542 - 0.909632i) q^{17} +(-0.983568 - 1.53046i) q^{19} +(0.841254 + 0.970858i) q^{20} +(0.959493 + 0.281733i) q^{23} +(-0.959493 + 0.281733i) q^{25} +(0.544078 - 0.627899i) q^{31} +(1.37491 + 0.197682i) q^{32} +(-0.716476 + 2.44009i) q^{34} +(-1.80075 + 2.07817i) q^{38} +(0.232593 - 0.361922i) q^{40} -1.51150i q^{46} -0.563465i q^{47} +(0.654861 + 0.755750i) q^{49} +(0.817178 + 1.27155i) q^{50} +(-0.797176 + 1.74557i) q^{53} +(-0.425839 - 0.368991i) q^{61} +(-1.14231 - 0.521678i) q^{62} +(-0.208518 - 1.45027i) q^{64} +2.16140 q^{68} +(2.12588 + 0.970858i) q^{76} +(1.80075 - 0.822373i) q^{79} +(0.608660 + 0.178719i) q^{80} +(0.186393 - 1.29639i) q^{83} +(-0.698939 + 1.53046i) q^{85} +(-1.23259 + 0.361922i) q^{92} +(-0.817178 + 0.239945i) q^{94} +(-1.37491 + 1.19136i) q^{95} +(0.817178 - 1.27155i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{4} - q^{5} + 11 q^{8} - q^{16} - 9 q^{17} - q^{20} + q^{23} - q^{25} + 2 q^{31} - 11 q^{34} - 11 q^{40} + q^{49} - 2 q^{53} - 11 q^{62} + 10 q^{64} + 2 q^{68} + 11 q^{76} + 10 q^{80} - 2 q^{83}+ \cdots + q^{92}+O(q^{100}) \) 10 * q - q^4 - q^5 + 11 * q^8 - q^16 - 9 * q^17 - q^20 + q^23 - q^25 + 2 * q^31 - 11 * q^34 - 11 * q^40 + q^49 - 2 * q^53 - 11 * q^62 + 10 * q^64 + 2 * q^68 + 11 * q^76 + 10 * q^80 - 2 * q^83 + 2 * q^85 + q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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