LMFDB - Embedded newform 712.1.w.a.235.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [712,1,Mod(11,712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(712, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 21]))

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

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

chi := DirichletCharacter("712.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 712 = 2^{3} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	712.w (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:	$0.355334288995$
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:	$\Z[a_1, a_2]$
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			235.1
Root			$0.654861 - 0.755750i$ of defining polynomial
Character	$\chi$	$=$	712.235
Dual form			712.1.w.a.203.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.142315 + 0.989821i) q^{2} +(-0.425839 - 1.45027i) q^{3} +(-0.959493 - 0.281733i) q^{4} +(1.49611 - 0.215109i) q^{6} +(0.415415 - 0.909632i) q^{8} +(-1.08070 + 0.694523i) q^{9} +O(q^{10})$	\(q+(-0.142315 + 0.989821i) q^{2} +(-0.425839 - 1.45027i) q^{3} +(-0.959493 - 0.281733i) q^{4} +(1.49611 - 0.215109i) q^{6} +(0.415415 - 0.909632i) q^{8} +(-1.08070 + 0.694523i) q^{9} +(-0.345139 - 0.755750i) q^{11} +1.51150i q^{12} +(0.841254 + 0.540641i) q^{16} +(-0.0405070 - 0.281733i) q^{17} +(-0.533654 - 1.16854i) q^{18} +(-0.983568 - 1.53046i) q^{19} +(0.797176 - 0.234072i) q^{22} +(-1.49611 - 0.215109i) q^{24} +(-0.654861 - 0.755750i) q^{25} +(0.325137 + 0.281733i) q^{27} +(-0.654861 + 0.755750i) q^{32} +(-0.949069 + 0.822373i) q^{33} +0.284630 q^{34} +(1.23259 - 0.361922i) q^{36} +(1.65486 - 0.755750i) q^{38} +(1.80075 - 0.822373i) q^{43} +(0.118239 + 0.822373i) q^{44} +(0.425839 - 1.45027i) q^{48} +(0.654861 + 0.755750i) q^{49} +(0.841254 - 0.540641i) q^{50} +(-0.391340 + 0.178719i) q^{51} +(-0.325137 + 0.281733i) q^{54} +(-1.80075 + 2.07817i) q^{57} +(-0.557730 + 1.89945i) q^{59} +(-0.654861 - 0.755750i) q^{64} +(-0.678936 - 1.05645i) q^{66} +(1.61435 - 0.474017i) q^{67} +(-0.0405070 + 0.281733i) q^{68} +(0.182822 + 1.27155i) q^{72} +(1.10181 + 0.708089i) q^{73} +(-0.817178 + 1.27155i) q^{75} +(0.512546 + 1.74557i) q^{76} +(-0.263521 + 0.577031i) q^{81} +(-0.557730 + 0.0801894i) q^{83} +(0.557730 + 1.89945i) q^{86} -0.830830 q^{88} +(-0.654861 + 0.755750i) q^{89} +(1.37491 + 0.627899i) q^{96} +(0.698939 - 1.53046i) q^{97} +(-0.841254 + 0.540641i) q^{98} +(0.897877 + 0.577031i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} - q^{4} - q^{8} - q^{9}+O(q^{10}) $ 10 * q - q^2 - q^4 - q^8 - q^9	$ 10 q - q^{2} - q^{4} - q^{8} - q^{9} - 9 q^{11} - q^{16} - 9 q^{17} - q^{18} + 2 q^{22} - q^{25} + 11 q^{27} - q^{32} + 2 q^{34} - q^{36} + 11 q^{38} + 2 q^{44} + q^{49} - q^{50} - 11 q^{54} - q^{64} + 2 q^{67} - 9 q^{68} + 10 q^{72} + 2 q^{73} - q^{81} + 2 q^{88} - q^{89} - 2 q^{97} + q^{98} - 9 q^{99}+O(q^{100}) $ 10 * q - q^2 - q^4 - q^8 - q^9 - 9 * q^11 - q^16 - 9 * q^17 - q^18 + 2 * q^22 - q^25 + 11 * q^27 - q^32 + 2 * q^34 - q^36 + 11 * q^38 + 2 * q^44 + q^49 - q^50 - 11 * q^54 - q^64 + 2 * q^67 - 9 * q^68 + 10 * q^72 + 2 * q^73 - q^81 + 2 * q^88 - q^89 - 2 * q^97 + q^98 - 9 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/712\mathbb{Z}\right)^\times$.

$n$	$357$	$535$	$537$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{9}{22}\right)$

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 $

$2$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$2$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$3$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$3$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$4$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$5$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$5$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$6$	1.49611	−	0.215109i	1.49611	−	0.215109i
$7$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$7$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$8$	0.415415	−	0.909632i	0.415415	−	0.909632i
$9$	−1.08070	+	0.694523i	−1.08070	+	0.694523i
$10$		0			0
$11$	−0.345139	−	0.755750i	−0.345139	−	0.755750i	0.654861	−	0.755750i	$-0.272727\pi$
$11$	−0.345139	−	0.755750i	−0.345139	−	0.755750i	−1.00000			$\pi$
$12$			1.51150i			1.51150i
$13$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$13$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$14$		0			0
$15$		0			0
$16$	0.841254	+	0.540641i	0.841254	+	0.540641i
$17$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i	0.959493	−	0.281733i	$-0.0909091\pi$
$17$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i	−1.00000			$\pi$
$18$	−0.533654	−	1.16854i	−0.533654	−	1.16854i
$19$	−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			0
$21$		0			0
$22$	0.797176	−	0.234072i	0.797176	−	0.234072i
$23$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$23$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$24$	−1.49611	−	0.215109i	−1.49611	−	0.215109i
$25$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$26$		0			0
$27$	0.325137	+	0.281733i	0.325137	+	0.281733i
$28$		0			0
$29$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$29$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$30$		0			0
$31$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$31$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$32$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$33$	−0.949069	+	0.822373i	−0.949069	+	0.822373i
$34$	0.284630			0.284630
$35$		0			0
$36$	1.23259	−	0.361922i	1.23259	−	0.361922i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	1.65486	−	0.755750i	1.65486	−	0.755750i
$39$		0			0
$40$		0			0
$41$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$41$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$42$		0			0
$43$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$43$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$44$	0.118239	+	0.822373i	0.118239	+	0.822373i
$45$		0			0
$46$		0			0
$47$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$47$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$48$	0.425839	−	1.45027i	0.425839	−	1.45027i
$49$	0.654861	+	0.755750i	0.654861	+	0.755750i
$50$	0.841254	−	0.540641i	0.841254	−	0.540641i
$51$	−0.391340	+	0.178719i	−0.391340	+	0.178719i
$52$		0			0
$53$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$53$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$54$	−0.325137	+	0.281733i	−0.325137	+	0.281733i
$55$		0			0
$56$		0			0
$57$	−1.80075	+	2.07817i	−1.80075	+	2.07817i
$58$		0			0
$59$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$59$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$60$		0			0
$61$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$61$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$62$		0			0
$63$		0			0
$64$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$65$		0			0
$66$	−0.678936	−	1.05645i	−0.678936	−	1.05645i
$67$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$67$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.959493	+	0.281733i	$0.0909091\pi$
$68$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i
$69$		0			0
$70$		0			0
$71$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$71$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$72$	0.182822	+	1.27155i	0.182822	+	1.27155i
$73$	1.10181	+	0.708089i	1.10181	+	0.708089i	0.959493	−	0.281733i	$-0.0909091\pi$
$73$	1.10181	+	0.708089i	1.10181	+	0.708089i	0.142315	+	0.989821i	$0.454545\pi$
$74$		0			0
$75$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$76$	0.512546	+	1.74557i	0.512546	+	1.74557i
$77$		0			0
$78$		0			0
$79$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$79$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$80$		0			0
$81$	−0.263521	+	0.577031i	−0.263521	+	0.577031i
$82$		0			0
$83$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$83$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$84$		0			0
$85$		0			0
$86$	0.557730	+	1.89945i	0.557730	+	1.89945i
$87$		0			0
$88$	−0.830830			−0.830830
$89$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$89$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.37491	+	0.627899i	1.37491	+	0.627899i
$97$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$97$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$98$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$99$	0.897877	+	0.577031i	0.897877	+	0.577031i
$100$	0.415415	+	0.909632i	0.415415	+	0.909632i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−0.121206	−	0.412791i	−0.121206	−	0.412791i
$103$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$103$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.830830	+	1.81926i	0.830830	+	1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$107$	0.830830	+	1.81926i	0.830830	+	1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$108$	−0.232593	−	0.361922i	−0.232593	−	0.361922i
$109$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$109$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$113$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$114$	−1.80075	−	2.07817i	−1.80075	−	2.07817i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.80075	−	0.822373i	−1.80075	−	0.822373i
$119$		0			0
$120$		0			0
$121$	0.202824	−	0.234072i	0.202824	−	0.234072i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$127$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$128$	0.841254	−	0.540641i	0.841254	−	0.540641i
$129$	−1.95949	−	2.26138i	−1.95949	−	2.26138i
$130$		0			0
$131$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.415415	−	0.909632i	$-0.636364\pi$
$131$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.841254	+	0.540641i	$0.818182\pi$
$132$	1.14231	−	0.521678i	1.14231	−	0.521678i
$133$		0			0
$134$	0.239446	+	1.66538i	0.239446	+	1.66538i
$135$		0			0
$136$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$137$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.142315	−	0.989821i	$-0.454545\pi$
$137$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$138$		0			0
$139$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$139$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.28463			−1.28463
$145$		0			0
$146$	−0.857685	+	0.989821i	−0.857685	+	0.989821i
$147$	0.817178	−	1.27155i	0.817178	−	1.27155i
$148$		0			0
$149$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$149$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$150$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$151$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$151$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$152$	−1.80075	+	0.258908i	−1.80075	+	0.258908i
$153$	0.239446	+	0.276335i	0.239446	+	0.276335i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$157$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.533654	−	0.342959i	−0.533654	−	0.342959i
$163$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$163$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$164$		0			0
$165$		0			0
$166$		−	0.563465i		−	0.563465i
$167$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$167$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$168$		0			0
$169$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$170$		0			0
$171$	2.12588	+	0.970858i	2.12588	+	0.970858i
$172$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$173$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$173$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$174$		0			0
$175$		0			0
$176$	0.118239	−	0.822373i	0.118239	−	0.822373i
$177$	2.99223			2.99223
$178$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$179$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$179$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$180$		0			0
$181$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$181$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.198939	+	0.127850i	−0.198939	+	0.127850i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$191$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$192$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$193$	1.07028	+	0.153882i	1.07028	+	0.153882i	0.654861	−	0.755750i	$-0.272727\pi$
$193$	1.07028	+	0.153882i	1.07028	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$194$	1.41542	+	0.909632i	1.41542	+	0.909632i
$195$		0			0
$196$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$197$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$197$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$198$	−0.698939	+	0.806618i	−0.698939	+	0.806618i
$199$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$199$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$200$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$201$	−1.37491	−	2.13940i	−1.37491	−	2.13940i
$202$		0			0
$203$		0			0
$204$	0.425839	−	0.0612263i	0.425839	−	0.0612263i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$210$		0			0
$211$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$211$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$212$		0			0
$213$		0			0
$214$	−1.91899	+	0.563465i	−1.91899	+	0.563465i
$215$		0			0
$216$	0.391340	−	0.178719i	0.391340	−	0.178719i
$217$		0			0
$218$		0			0
$219$	0.557730	−	1.89945i	0.557730	−	1.89945i
$220$		0			0
$221$		0			0
$222$		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
$225$	1.23259	+	0.361922i	1.23259	+	0.361922i
$226$	−0.158746	+	0.540641i	−0.158746	+	0.540641i
$227$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$227$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$228$	2.31329	−	1.48666i	2.31329	−	1.48666i
$229$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$229$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.91899			1.91899			0.959493	−	0.281733i	$-0.0909091\pi$
$233$	1.91899			1.91899			0.959493	+	0.281733i	$0.0909091\pi$
$234$		0			0
$235$		0			0
$236$	1.07028	−	1.66538i	1.07028	−	1.66538i
$237$		0			0
$238$		0			0
$239$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$239$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$240$		0			0
$241$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$241$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$242$	0.202824	+	0.234072i	0.202824	+	0.234072i
$243$	1.37491	+	0.197682i	1.37491	+	0.197682i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.353799	+	0.774713i	0.353799	+	0.774713i
$250$		0			0
$251$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$251$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.415415	+	0.909632i	0.415415	+	0.909632i
$257$	−0.698939	−	0.449181i	−0.698939	−	0.449181i	0.142315	−	0.989821i	$-0.454545\pi$
$257$	−0.698939	−	0.449181i	−0.698939	−	0.449181i	−0.841254	+	0.540641i	$0.818182\pi$
$258$	2.51722	−	1.61772i	2.51722	−	1.61772i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.544078	−	1.19136i	0.544078	−	1.19136i
$263$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$263$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$264$	0.353799	+	1.20493i	0.353799	+	1.20493i
$265$		0			0
$266$		0			0
$267$	1.37491	+	0.627899i	1.37491	+	0.627899i
$268$	−1.68251			−1.68251
$269$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$269$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$270$		0			0
$271$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$271$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$272$	0.118239	−	0.258908i	0.118239	−	0.258908i
$273$		0			0
$274$	1.80075	+	0.822373i	1.80075	+	0.822373i
$275$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$276$		0			0
$277$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$277$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$278$	0.345139	+	0.755750i	0.345139	+	0.755750i
$279$		0			0
$280$		0			0
$281$	0.983568	−	1.53046i	0.983568	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$281$	0.983568	−	1.53046i	0.983568	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$282$		0			0
$283$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.959493	+	0.281733i	$0.0909091\pi$
$283$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.182822	−	1.27155i	0.182822	−	1.27155i
$289$	0.881761	−	0.258908i	0.881761	−	0.258908i
$290$		0			0
$291$	−2.51722	−	0.361922i	−2.51722	−	0.361922i
$292$	−0.857685	−	0.989821i	−0.857685	−	0.989821i
$293$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$293$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$294$	1.14231	+	0.989821i	1.14231	+	0.989821i
$295$		0			0
$296$		0			0
$297$	0.100702	−	0.342959i	0.100702	−	0.342959i
$298$		0			0
$299$		0			0
$300$	1.14231	−	0.989821i	1.14231	−	0.989821i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	1.81926i		−	1.81926i
$305$		0			0
$306$	−0.307599	+	0.197682i	−0.307599	+	0.197682i
$307$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$307$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.415415	−	0.909632i	$-0.636364\pi$
$308$		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
$313$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$313$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$317$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	2.28463	−	1.97964i	2.28463	−	1.97964i
$322$		0			0
$323$	−0.391340	+	0.339098i	−0.391340	+	0.339098i
$324$	0.415415	−	0.479414i	0.415415	−	0.479414i
$325$		0			0
$326$	0.425839	−	1.45027i	0.425839	−	1.45027i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$331$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$332$	0.557730	+	0.0801894i	0.557730	+	0.0801894i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$337$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$338$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$339$	−0.121206	−	0.843008i	−0.121206	−	0.843008i
$340$		0			0
$341$		0			0
$342$	−1.26352	+	1.96608i	−1.26352	+	1.96608i
$343$		0			0
$344$		−	1.97964i		−	1.97964i
$345$		0			0
$346$		0			0
$347$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$347$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$348$		0			0
$349$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$349$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$350$		0			0
$351$		0			0
$352$	0.797176	+	0.234072i	0.797176	+	0.234072i
$353$	−0.304632	−	1.03748i	−0.304632	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$353$	−0.304632	−	1.03748i	−0.304632	−	1.03748i	0.654861	−	0.755750i	$-0.272727\pi$
$354$	−0.425839	+	2.96177i	−0.425839	+	2.96177i
$355$		0			0
$356$	0.841254	−	0.540641i	0.841254	−	0.540641i
$357$		0			0
$358$	−0.239446	+	1.66538i	−0.239446	+	1.66538i
$359$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$359$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$360$		0			0
$361$	−0.959493	+	2.10100i	−0.959493	+	2.10100i
$362$		0			0
$363$	−0.425839	−	0.194474i	−0.425839	−	0.194474i
$364$		0			0
$365$		0			0
$366$		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			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$373$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$374$	−0.0982369	−	0.215109i	−0.0982369	−	0.215109i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$379$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$383$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$384$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$385$		0			0
$386$	−0.304632	+	1.03748i	−0.304632	+	1.03748i
$387$	−1.37491	+	2.13940i	−1.37491	+	2.13940i
$388$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$389$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$389$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$390$		0			0
$391$		0			0
$392$	0.959493	−	0.281733i	0.959493	−	0.281733i
$393$			1.97964i			1.97964i
$394$		0			0
$395$		0			0
$396$	−0.698939	−	0.806618i	−0.698939	−	0.806618i
$397$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$397$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$398$		0			0
$399$		0			0
$400$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$401$	0.239446	+	1.66538i	0.239446	+	1.66538i	0.654861	+	0.755750i	$0.272727\pi$
$401$	0.239446	+	1.66538i	0.239446	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$402$	2.31329	−	1.05645i	2.31329	−	1.05645i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$			0.430218i			0.430218i
$409$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$409$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.959493	+	0.281733i	$0.0909091\pi$
$410$		0			0
$411$	−2.99223			−2.99223
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.949069	−	0.822373i	−0.949069	−	0.822373i
$418$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$419$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$419$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$420$		0			0
$421$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$421$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$422$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$423$		0			0
$424$		0			0
$425$	−0.186393	+	0.215109i	−0.186393	+	0.215109i
$426$		0			0
$427$		0			0
$428$	−0.284630	−	1.97964i	−0.284630	−	1.97964i
$429$		0			0
$430$		0			0
$431$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$431$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$432$	0.121206	+	0.412791i	0.121206	+	0.412791i
$433$		−	1.51150i		−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$433$		−	1.51150i		−	1.51150i	0.654861	−	0.755750i	$-0.272727\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.80075	+	0.822373i	1.80075	+	0.822373i
$439$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$439$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$440$		0			0
$441$	−1.23259	−	0.361922i	−1.23259	−	0.361922i
$442$		0			0
$443$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$443$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.84125	−	0.540641i	−1.84125	−	0.540641i	−0.841254	−	0.540641i	$-0.818182\pi$
$449$	−1.84125	−	0.540641i	−1.84125	−	0.540641i	−1.00000			$\pi$
$450$	−0.533654	+	1.16854i	−0.533654	+	1.16854i
$451$		0			0
$452$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$453$		0			0
$454$	−1.10181	+	0.708089i	−1.10181	+	0.708089i
$455$		0			0
$456$	1.14231	+	2.50132i	1.14231	+	2.50132i
$457$		−	1.08128i		−	1.08128i	−0.841254	−	0.540641i	$-0.818182\pi$
$457$		−	1.08128i		−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$458$		0			0
$459$	0.0662029	−	0.103014i	0.0662029	−	0.103014i
$460$		0			0
$461$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$461$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$462$		0			0
$463$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$463$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$464$		0			0
$465$		0			0
$466$	−0.273100	+	1.89945i	−0.273100	+	1.89945i
$467$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$467$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.49611	+	1.29639i	1.49611	+	1.29639i
$473$	−1.24302	−	1.07708i	−1.24302	−	1.07708i
$474$		0			0
$475$	−0.512546	+	1.74557i	−0.512546	+	1.74557i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		−	0.563465i		−	0.563465i
$483$		0			0
$484$	−0.260554	+	0.167448i	−0.260554	+	0.167448i
$485$		0			0
$486$	−0.391340	+	1.33278i	−0.391340	+	1.33278i
$487$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$487$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$488$		0			0
$489$	0.325137	+	2.26138i	0.325137	+	2.26138i
$490$		0			0
$491$	1.65486	−	0.755750i	1.65486	−	0.755750i	0.654861	−	0.755750i	$-0.272727\pi$
$491$	1.65486	−	0.755750i	1.65486	−	0.755750i	1.00000			$0$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.817178	+	0.239945i	−0.817178	+	0.239945i
$499$	−0.425839	+	0.368991i	−0.425839	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$499$	−0.425839	+	0.368991i	−0.425839	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$500$		0			0
$501$		0			0
$502$	1.25667	−	1.45027i	1.25667	−	1.45027i
$503$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$503$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.14231	+	0.989821i	1.14231	+	0.989821i
$508$		0			0
$509$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$509$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$510$		0			0
$511$		0			0
$512$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$513$	0.111387	−	0.774713i	0.111387	−	0.774713i
$514$	0.544078	−	0.627899i	0.544078	−	0.627899i
$515$		0			0
$516$	1.24302	+	2.72183i	1.24302	+	2.72183i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.557730	−	1.89945i	−0.557730	−	1.89945i	−0.415415	−	0.909632i	$-0.636364\pi$
$521$	−0.557730	−	1.89945i	−0.557730	−	1.89945i	−0.142315	−	0.989821i	$-0.545455\pi$
$522$		0			0
$523$	0.544078	+	1.19136i	0.544078	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$523$	0.544078	+	1.19136i	0.544078	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$524$	1.10181	+	0.708089i	1.10181	+	0.708089i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.24302	+	0.178719i	−1.24302	+	0.178719i
$529$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$530$		0			0
$531$	−0.716476	−	2.44009i	−0.716476	−	2.44009i
$532$		0			0
$533$		0			0
$534$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$535$		0			0
$536$	0.239446	−	1.66538i	0.239446	−	1.66538i
$537$	−0.716476	−	2.44009i	−0.716476	−	2.44009i
$538$		0			0
$539$	0.345139	−	0.755750i	0.345139	−	0.755750i
$540$		0			0
$541$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$541$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$542$		0			0
$543$		0			0
$544$	0.239446	+	0.153882i	0.239446	+	0.153882i
$545$		0			0
$546$		0			0
$547$	−0.512546	−	1.74557i	−0.512546	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$547$	−0.512546	−	1.74557i	−0.512546	−	1.74557i	0.142315	−	0.989821i	$-0.454545\pi$
$548$	−1.07028	+	1.66538i	−1.07028	+	1.66538i
$549$		0			0
$550$	−0.698939	−	0.449181i	−0.698939	−	0.449181i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$557$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$557$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	0.270133	+	0.234072i	0.270133	+	0.234072i
$562$	1.37491	+	1.19136i	1.37491	+	1.19136i
$563$	1.65486	+	0.755750i	1.65486	+	0.755750i	1.00000			$0$
$563$	1.65486	+	0.755750i	1.65486	+	0.755750i	0.654861	+	0.755750i	$0.272727\pi$
$564$		0			0
$565$		0			0
$566$	−1.25667	+	1.45027i	−1.25667	+	1.45027i
$567$		0			0
$568$		0			0
$569$	0.817178	−	0.708089i	0.817178	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$569$	0.817178	−	0.708089i	0.817178	−	0.708089i	0.959493	+	0.281733i	$0.0909091\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.23259	+	0.361922i	1.23259	+	0.361922i
$577$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$577$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−1.00000			$\pi$
$578$	0.130785	+	0.909632i	0.130785	+	0.909632i
$579$	−0.232593	−	1.61772i	−0.232593	−	1.61772i
$580$		0			0
$581$		0			0
$582$	0.716476	−	2.44009i	0.716476	−	2.44009i
$583$		0			0
$584$	1.10181	−	0.708089i	1.10181	−	0.708089i
$585$		0			0
$586$		0			0
$587$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$587$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$588$	−1.14231	+	0.989821i	−1.14231	+	0.989821i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.304632	−	1.03748i	0.304632	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$593$	0.304632	−	1.03748i	0.304632	−	1.03748i	0.959493	−	0.281733i	$-0.0909091\pi$
$594$	0.325137	+	0.148485i	0.325137	+	0.148485i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$599$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$600$	0.817178	+	1.27155i	0.817178	+	1.27155i
$601$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$601$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$602$		0			0
$603$	−1.41542	+	1.63348i	−1.41542	+	1.63348i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$607$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$608$	1.80075	+	0.258908i	1.80075	+	0.258908i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.151894	−	0.332601i	−0.151894	−	0.332601i
$613$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$613$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$614$	1.61435	−	1.03748i	1.61435	−	1.03748i
$615$		0			0
$616$		0			0
$617$	−1.49611	+	0.215109i	−1.49611	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$617$	−1.49611	+	0.215109i	−1.49611	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$618$		0			0
$619$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$619$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$626$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$627$	2.19209	+	0.643655i	2.19209	+	0.643655i
$628$		0			0
$629$		0			0
$630$		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
$633$	2.51722	+	1.61772i	2.51722	+	1.61772i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.544078	+	1.19136i	0.544078	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$641$	0.544078	+	1.19136i	0.544078	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$642$	1.63436	+	2.54311i	1.63436	+	2.54311i
$643$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$643$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$644$		0			0
$645$		0			0
$646$	−0.279953	−	0.435615i	−0.279953	−	0.435615i
$647$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$647$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$648$	0.415415	+	0.479414i	0.415415	+	0.479414i
$649$	1.62801	−	0.234072i	1.62801	−	0.234072i
$650$		0			0
$651$		0			0
$652$	1.37491	+	0.627899i	1.37491	+	0.627899i
$653$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$653$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.68251			−1.68251
$658$		0			0
$659$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$659$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.415415	+	0.909632i	$0.636364\pi$
$660$		0			0
$661$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$661$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$662$	0.698939	−	0.449181i	0.698939	−	0.449181i
$663$		0			0
$664$	−0.158746	+	0.540641i	−0.158746	+	0.540641i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.68251	+	1.08128i	−1.68251	+	1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$673$	−1.68251	+	1.08128i	−1.68251	+	1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$674$	−1.65486	+	0.755750i	−1.65486	+	0.755750i
$675$		−	0.430218i		−	0.430218i
$676$	0.959493	−	0.281733i	0.959493	−	0.281733i
$677$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$677$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$678$	0.851677			0.851677
$679$		0			0
$680$		0			0
$681$	1.07028	−	1.66538i	1.07028	−	1.66538i
$682$		0			0
$683$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$683$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$684$	−1.76625	−	1.53046i	−1.76625	−	1.53046i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1.95949	+	0.281733i	1.95949	+	0.281733i
$689$		0			0
$690$		0			0
$691$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$691$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$692$		0			0
$693$		0			0
$694$	−0.797176	−	1.74557i	−0.797176	−	1.74557i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−0.817178	−	2.78305i	−0.817178	−	2.78305i
$700$		0			0
$701$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$701$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$702$		0			0
$703$		0			0
$704$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$705$		0			0
$706$	1.07028	−	0.153882i	1.07028	−	0.153882i
$707$		0			0
$708$	−2.87102	−	0.843008i	−2.87102	−	0.843008i
$709$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$709$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$710$		0			0
$711$		0			0
$712$	0.415415	+	0.909632i	0.415415	+	0.909632i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.61435	−	0.474017i	−1.61435	−	0.474017i
$717$		0			0
$718$		0			0
$719$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$719$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$720$		0			0
$721$		0			0
$722$	−1.94306	−	1.24873i	−1.94306	−	1.24873i
$723$	0.353799	+	0.774713i	0.353799	+	0.774713i
$724$		0			0
$725$		0			0
$726$	0.253098	−	0.393828i	0.253098	−	0.393828i
$727$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$727$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$728$		0			0
$729$	−0.208518	−	1.45027i	−0.208518	−	1.45027i
$730$		0			0
$731$	−0.304632	−	0.474017i	−0.304632	−	0.474017i
$732$		0			0
$733$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$733$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.915415	−	1.05645i	−0.915415	−	1.05645i
$738$		0			0
$739$	0.817178	+	0.708089i	0.817178	+	0.708089i	0.959493	−	0.281733i	$-0.0909091\pi$
$739$	0.817178	+	0.708089i	0.817178	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$743$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.547045	−	0.474017i	0.547045	−	0.474017i
$748$	0.226900	−	0.0666238i	0.226900	−	0.0666238i
$749$		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
$753$	−0.817178	+	2.78305i	−0.817178	+	2.78305i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$757$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$758$	0.983568	−	0.449181i	0.983568	−	0.449181i
$759$		0			0
$760$		0			0
$761$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$761$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.14231	−	0.989821i	1.14231	−	0.989821i
$769$	−1.25667	+	1.45027i	−1.25667	+	1.45027i	−0.415415	+	0.909632i	$0.636364\pi$
$769$	−1.25667	+	1.45027i	−1.25667	+	1.45027i	−0.841254	+	0.540641i	$0.818182\pi$
$770$		0			0
$771$	−0.353799	+	1.20493i	−0.353799	+	1.20493i
$772$	−0.983568	−	0.449181i	−0.983568	−	0.449181i
$773$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$773$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$774$	−1.92195	−	1.66538i	−1.92195	−	1.66538i
$775$		0			0
$776$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.142315	+	0.989821i	0.142315	+	0.989821i
$785$		0			0
$786$	−1.95949	−	0.281733i	−1.95949	−	0.281733i
$787$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$787$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.897877	−	0.577031i	0.897877	−	0.577031i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$797$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			1.00000
$801$	0.182822	−	1.27155i	0.182822	−	1.27155i
$802$	−1.68251			−1.68251
$803$	0.154861	−	1.07708i	0.154861	−	1.07708i
$804$	0.716476	+	2.44009i	0.716476	+	2.44009i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$809$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$810$		0			0
$811$	1.68251	+	1.08128i	1.68251	+	1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$811$	1.68251	+	1.08128i	1.68251	+	1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.425839	−	0.0612263i	−0.425839	−	0.0612263i
$817$	−3.02977	−	1.94711i	−3.02977	−	1.94711i
$818$	0.239446	+	1.66538i	0.239446	+	1.66538i
$819$		0			0
$820$		0			0
$821$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$821$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$822$	0.425839	−	2.96177i	0.425839	−	2.96177i
$823$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$823$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$824$		0			0
$825$	1.24302	+	0.178719i	1.24302	+	0.178719i
$826$		0			0
$827$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$827$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$828$		0			0
$829$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$829$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.186393	−	0.215109i	0.186393	−	0.215109i
$834$	0.949069	−	0.822373i	0.949069	−	0.822373i
$835$		0			0
$836$	1.14231	−	0.989821i	1.14231	−	0.989821i
$837$		0			0
$838$		−	1.08128i		−	1.08128i
$839$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$839$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$840$		0			0
$841$	0.654861	+	0.755750i	0.654861	+	0.755750i
$842$		0			0
$843$	−2.63843	−	0.774713i	−2.63843	−	0.774713i
$844$	1.80075	−	0.822373i	1.80075	−	0.822373i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.817178	−	2.78305i	0.817178	−	2.78305i
$850$	−0.186393	−	0.215109i	−0.186393	−	0.215109i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	2.00000			2.00000
$857$	1.37491	−	1.19136i	1.37491	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$857$	1.37491	−	1.19136i	1.37491	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$858$		0			0
$859$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$859$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$863$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$864$	−0.425839	+	0.0612263i	−0.425839	+	0.0612263i
$865$		0			0
$866$	1.49611	+	0.215109i	1.49611	+	0.215109i
$867$	−0.750975	−	1.16854i	−0.750975	−	1.16854i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.307599	+	2.13940i	0.307599	+	2.13940i
$874$		0			0
$875$		0			0
$876$	−1.07028	+	1.66538i	−1.07028	+	1.66538i
$877$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$877$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$881$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$882$	0.533654	−	1.16854i	0.533654	−	1.16854i
$883$	0.512546	+	0.234072i	0.512546	+	0.234072i	0.654861	−	0.755750i	$-0.272727\pi$
$883$	0.512546	+	0.234072i	0.512546	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$884$		0			0
$885$		0			0
$886$	−1.84125	−	0.540641i	−1.84125	−	0.540641i
$887$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$887$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.527042			0.527042
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.797176	−	1.74557i	0.797176	−	1.74557i
$899$		0			0
$900$	−1.08070	−	0.694523i	−1.08070	−	0.694523i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.304632	−	0.474017i	0.304632	−	0.474017i
$905$		0			0
$906$		0			0
$907$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$907$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$908$	−0.544078	−	1.19136i	−0.544078	−	1.19136i
$909$		0			0
$910$		0			0
$911$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$911$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$912$	−2.63843	+	0.774713i	−2.63843	+	0.774713i
$913$	0.253098	+	0.393828i	0.253098	+	0.393828i
$914$	1.07028	+	0.153882i	1.07028	+	0.153882i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.0925436	+	0.0801894i	0.0925436	+	0.0801894i
$919$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$919$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$920$		0			0
$921$	−1.56815	+	2.44009i	−1.56815	+	2.44009i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$929$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$930$		0			0
$931$	0.512546	−	1.74557i	0.512546	−	1.74557i
$932$	−1.84125	−	0.540641i	−1.84125	−	0.540641i
$933$		0			0
$934$	0.0405070	+	0.281733i	0.0405070	+	0.281733i
$935$		0			0
$936$		0			0
$937$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$937$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$938$		0			0
$939$	1.07028	+	1.23516i	1.07028	+	1.23516i
$940$		0			0
$941$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$941$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$942$		0			0
$943$		0			0
$944$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$945$		0			0
$946$	1.24302	−	1.07708i	1.24302	−	1.07708i
$947$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$947$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$948$		0			0
$949$		0			0
$950$	−1.65486	−	0.755750i	−1.65486	−	0.755750i
$951$		0			0
$952$		0			0
$953$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$953$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$962$		0			0
$963$	−2.16140	−	1.38905i	−2.16140	−	1.38905i
$964$	0.557730	+	0.0801894i	0.557730	+	0.0801894i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−0.128663	−	0.281733i	−0.128663	−	0.281733i
$969$	0.658432	+	0.423148i	0.658432	+	0.423148i
$970$		0			0
$971$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$971$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$972$	−1.26352	−	0.577031i	−1.26352	−	0.577031i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	0.654861	+	0.755750i	$0.272727\pi$
$977$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$978$	−2.28463			−2.28463
$979$	0.797176	+	0.234072i	0.797176	+	0.234072i
$980$		0			0
$981$		0			0
$982$	0.512546	+	1.74557i	0.512546	+	1.74557i
$983$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$983$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	−0.678936	+	1.05645i	−0.678936	+	1.05645i
$994$		0			0
$995$		0			0
$996$	−0.121206	−	0.843008i	−0.121206	−	0.843008i
$997$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$997$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$998$	−0.304632	−	0.474017i	−0.304632	−	0.474017i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.1.w.a.235.1	yes	10
4.3	odd	2		2848.1.by.a.591.1		10
8.3	odd	2	CM	712.1.w.a.235.1	yes	10
8.5	even	2		2848.1.by.a.591.1		10
89.25	even	22	inner	712.1.w.a.203.1	✓	10
356.203	odd	22		2848.1.by.a.559.1		10
712.203	odd	22	inner	712.1.w.a.203.1	✓	10
712.381	even	22		2848.1.by.a.559.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.1.w.a.203.1	✓	10	89.25	even	22	inner
712.1.w.a.203.1	✓	10	712.203	odd	22	inner
712.1.w.a.235.1	yes	10	1.1	even	1	trivial
712.1.w.a.235.1	yes	10	8.3	odd	2	CM
2848.1.by.a.559.1		10	356.203	odd	22
2848.1.by.a.559.1		10	712.381	even	22
2848.1.by.a.591.1		10	4.3	odd	2
2848.1.by.a.591.1		10	8.5	even	2

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:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	712.w (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.142315 + 0.989821i) q^{2} +(-0.425839 - 1.45027i) q^{3} +(-0.959493 - 0.281733i) q^{4} +(1.49611 - 0.215109i) q^{6} +(0.415415 - 0.909632i) q^{8} +(-1.08070 + 0.694523i) q^{9} +O(q^{10})\)	\(q+(-0.142315 + 0.989821i) q^{2} +(-0.425839 - 1.45027i) q^{3} +(-0.959493 - 0.281733i) q^{4} +(1.49611 - 0.215109i) q^{6} +(0.415415 - 0.909632i) q^{8} +(-1.08070 + 0.694523i) q^{9} +(-0.345139 - 0.755750i) q^{11} +1.51150i q^{12} +(0.841254 + 0.540641i) q^{16} +(-0.0405070 - 0.281733i) q^{17} +(-0.533654 - 1.16854i) q^{18} +(-0.983568 - 1.53046i) q^{19} +(0.797176 - 0.234072i) q^{22} +(-1.49611 - 0.215109i) q^{24} +(-0.654861 - 0.755750i) q^{25} +(0.325137 + 0.281733i) q^{27} +(-0.654861 + 0.755750i) q^{32} +(-0.949069 + 0.822373i) q^{33} +0.284630 q^{34} +(1.23259 - 0.361922i) q^{36} +(1.65486 - 0.755750i) q^{38} +(1.80075 - 0.822373i) q^{43} +(0.118239 + 0.822373i) q^{44} +(0.425839 - 1.45027i) q^{48} +(0.654861 + 0.755750i) q^{49} +(0.841254 - 0.540641i) q^{50} +(-0.391340 + 0.178719i) q^{51} +(-0.325137 + 0.281733i) q^{54} +(-1.80075 + 2.07817i) q^{57} +(-0.557730 + 1.89945i) q^{59} +(-0.654861 - 0.755750i) q^{64} +(-0.678936 - 1.05645i) q^{66} +(1.61435 - 0.474017i) q^{67} +(-0.0405070 + 0.281733i) q^{68} +(0.182822 + 1.27155i) q^{72} +(1.10181 + 0.708089i) q^{73} +(-0.817178 + 1.27155i) q^{75} +(0.512546 + 1.74557i) q^{76} +(-0.263521 + 0.577031i) q^{81} +(-0.557730 + 0.0801894i) q^{83} +(0.557730 + 1.89945i) q^{86} -0.830830 q^{88} +(-0.654861 + 0.755750i) q^{89} +(1.37491 + 0.627899i) q^{96} +(0.698939 - 1.53046i) q^{97} +(-0.841254 + 0.540641i) q^{98} +(0.897877 + 0.577031i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} - q^{4} - q^{8} - q^{9}+O(q^{10}) \) 10 * q - q^2 - q^4 - q^8 - q^9	\( 10 q - q^{2} - q^{4} - q^{8} - q^{9} - 9 q^{11} - q^{16} - 9 q^{17} - q^{18} + 2 q^{22} - q^{25} + 11 q^{27} - q^{32} + 2 q^{34} - q^{36} + 11 q^{38} + 2 q^{44} + q^{49} - q^{50} - 11 q^{54} - q^{64} + 2 q^{67} - 9 q^{68} + 10 q^{72} + 2 q^{73} - q^{81} + 2 q^{88} - q^{89} - 2 q^{97} + q^{98} - 9 q^{99}+O(q^{100}) \) 10 * q - q^2 - q^4 - q^8 - q^9 - 9 * q^11 - q^16 - 9 * q^17 - q^18 + 2 * q^22 - q^25 + 11 * q^27 - q^32 + 2 * q^34 - q^36 + 11 * q^38 + 2 * q^44 + q^49 - q^50 - 11 * q^54 - q^64 + 2 * q^67 - 9 * q^68 + 10 * q^72 + 2 * q^73 - q^81 + 2 * q^88 - q^89 - 2 * q^97 + q^98 - 9 * q^99

Introduction

L-functions

Modular forms

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