LMFDB - Embedded newform 548.1.j.a.215.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [548,1,Mod(15,548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(548, base_ring=CyclotomicField(34))

chi = DirichletCharacter(H, H._module([17, 19]))

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

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

chi := DirichletCharacter("548.15");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 548 = 2^{2} \cdot 137 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	548.j (of order $34$, degree $16$, 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:	$0.273487626923$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 $ x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + 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:	$D_{34}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{34} - \cdots)$

Embedding invariants

Embedding label			215.1
Root			$-0.739009 + 0.673696i$ of defining polynomial
Character	$\chi$	$=$	548.215
Dual form			548.1.j.a.339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.445738 - 0.895163i) q^{2} +(-0.602635 - 0.798017i) q^{4} +(1.72198 - 0.857445i) q^{5} +(-0.982973 + 0.183750i) q^{8} +(-0.0922684 + 0.995734i) q^{9} +O(q^{10})$	\(q+(0.445738 - 0.895163i) q^{2} +(-0.602635 - 0.798017i) q^{4} +(1.72198 - 0.857445i) q^{5} +(-0.982973 + 0.183750i) q^{8} +(-0.0922684 + 0.995734i) q^{9} -1.92365i q^{10} +(-0.576554 + 1.48826i) q^{13} +(-0.273663 + 0.961826i) q^{16} +(-1.93247 - 0.361242i) q^{17} +(0.850217 + 0.526432i) q^{18} +(-1.72198 - 0.857445i) q^{20} +(1.62738 - 2.15499i) q^{25} +(1.07524 + 1.17948i) q^{26} +(-0.694903 - 0.197717i) q^{29} +(0.739009 + 0.673696i) q^{32} +(-1.18475 + 1.56886i) q^{34} +(0.850217 - 0.526432i) q^{36} -0.891477 q^{37} +(-1.53511 + 1.15926i) q^{40} -1.05286i q^{41} +(0.694903 + 1.79375i) q^{45} +(0.739009 + 0.673696i) q^{49} +(-1.20369 - 2.41733i) q^{50} +(1.53511 - 0.436776i) q^{52} +(0.193463 - 0.312454i) q^{53} +(-0.486734 + 0.533922i) q^{58} +(0.0505009 + 0.544991i) q^{61} +(0.932472 - 0.361242i) q^{64} +(0.283284 + 3.05712i) q^{65} +(0.876298 + 1.75984i) q^{68} +(-0.0922684 - 0.995734i) q^{72} +(0.172075 - 0.0666624i) q^{73} +(-0.397365 + 0.798017i) q^{74} +(0.353470 + 1.89090i) q^{80} +(-0.982973 - 0.183750i) q^{81} +(-0.942485 - 0.469302i) q^{82} +(-3.63743 + 1.03494i) q^{85} +(1.20614 - 0.600584i) q^{89} +(1.91545 + 0.177492i) q^{90} +(0.932472 - 0.361242i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} - q^{4} - q^{8} + q^{9}+O(q^{10}) $ 16 * q - q^2 - q^4 - q^8 + q^9	$ 16 q - q^{2} - q^{4} - q^{8} + q^{9} - q^{16} - 15 q^{17} + q^{18} - q^{25} - q^{32} + 2 q^{34} + q^{36} + 2 q^{37} - q^{49} - q^{50} + 2 q^{61} - q^{64} + 2 q^{68} + q^{72} - 2 q^{73} - 15 q^{74} - q^{81} - q^{98}+O(q^{100}) $ 16 * q - q^2 - q^4 - q^8 + q^9 - q^16 - 15 * q^17 + q^18 - q^25 - q^32 + 2 * q^34 + q^36 + 2 * q^37 - q^49 - q^50 + 2 * q^61 - q^64 + 2 * q^68 + q^72 - 2 * q^73 - 15 * q^74 - q^81 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/548\mathbb{Z}\right)^\times$.

$n$	$275$	$277$
$\chi(n)$	$-1$	$e\left(\frac{9}{34}\right)$

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 $

$2$	0.445738	−	0.895163i	0.445738	−	0.895163i
$2$	0.445738	−	0.895163i	0.445738	−	0.895163i
$3$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$3$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$4$	−0.602635	−	0.798017i	−0.602635	−	0.798017i
$5$	1.72198	−	0.857445i	1.72198	−	0.857445i	0.739009	−	0.673696i	$-0.235294\pi$
$5$	1.72198	−	0.857445i	1.72198	−	0.857445i	0.982973	−	0.183750i	$-0.0588235\pi$
$6$		0			0
$7$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$7$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$8$	−0.982973	+	0.183750i	−0.982973	+	0.183750i
$9$	−0.0922684	+	0.995734i	−0.0922684	+	0.995734i
$10$		−	1.92365i		−	1.92365i
$11$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$11$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$12$		0			0
$13$	−0.576554	+	1.48826i	−0.576554	+	1.48826i	0.273663	+	0.961826i	$0.411765\pi$
$13$	−0.576554	+	1.48826i	−0.576554	+	1.48826i	−0.850217	+	0.526432i	$0.823529\pi$
$14$		0			0
$15$		0			0
$16$	−0.273663	+	0.961826i	−0.273663	+	0.961826i
$17$	−1.93247	−	0.361242i	−1.93247	−	0.361242i	−0.932472	−	0.361242i	$-0.882353\pi$
$17$	−1.93247	−	0.361242i	−1.93247	−	0.361242i	−1.00000			$\pi$
$18$	0.850217	+	0.526432i	0.850217	+	0.526432i
$19$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$19$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$20$	−1.72198	−	0.857445i	−1.72198	−	0.857445i
$21$		0			0
$22$		0			0
$23$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$23$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$24$		0			0
$25$	1.62738	−	2.15499i	1.62738	−	2.15499i
$26$	1.07524	+	1.17948i	1.07524	+	1.17948i
$27$		0			0
$28$		0			0
$29$	−0.694903	−	0.197717i	−0.694903	−	0.197717i	−0.0922684	−	0.995734i	$-0.529412\pi$
$29$	−0.694903	−	0.197717i	−0.694903	−	0.197717i	−0.602635	+	0.798017i	$0.705882\pi$
$30$		0			0
$31$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$31$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$32$	0.739009	+	0.673696i	0.739009	+	0.673696i
$33$		0			0
$34$	−1.18475	+	1.56886i	−1.18475	+	1.56886i
$35$		0			0
$36$	0.850217	−	0.526432i	0.850217	−	0.526432i
$37$	−0.891477			−0.891477			−0.445738	−	0.895163i	$-0.647059\pi$
$37$	−0.891477			−0.891477			−0.445738	+	0.895163i	$0.647059\pi$
$38$		0			0
$39$		0			0
$40$	−1.53511	+	1.15926i	−1.53511	+	1.15926i
$41$		−	1.05286i		−	1.05286i	−0.850217	−	0.526432i	$-0.823529\pi$
$41$		−	1.05286i		−	1.05286i	0.850217	−	0.526432i	$-0.176471\pi$
$42$		0			0
$43$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$43$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$44$		0			0
$45$	0.694903	+	1.79375i	0.694903	+	1.79375i
$46$		0			0
$47$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$47$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$48$		0			0
$49$	0.739009	+	0.673696i	0.739009	+	0.673696i
$50$	−1.20369	−	2.41733i	−1.20369	−	2.41733i
$51$		0			0
$52$	1.53511	−	0.436776i	1.53511	−	0.436776i
$53$	0.193463	−	0.312454i	0.193463	−	0.312454i	−0.739009	−	0.673696i	$-0.764706\pi$
$53$	0.193463	−	0.312454i	0.193463	−	0.312454i	0.932472	+	0.361242i	$0.117647\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.486734	+	0.533922i	−0.486734	+	0.533922i
$59$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$59$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$60$		0			0
$61$	0.0505009	+	0.544991i	0.0505009	+	0.544991i	0.982973	+	0.183750i	$0.0588235\pi$
$61$	0.0505009	+	0.544991i	0.0505009	+	0.544991i	−0.932472	+	0.361242i	$0.882353\pi$
$62$		0			0
$63$		0			0
$64$	0.932472	−	0.361242i	0.932472	−	0.361242i
$65$	0.283284	+	3.05712i	0.283284	+	3.05712i
$66$		0			0
$67$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$67$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$68$	0.876298	+	1.75984i	0.876298	+	1.75984i
$69$		0			0
$70$		0			0
$71$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$71$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$72$	−0.0922684	−	0.995734i	−0.0922684	−	0.995734i
$73$	0.172075	−	0.0666624i	0.172075	−	0.0666624i	−0.273663	−	0.961826i	$-0.588235\pi$
$73$	0.172075	−	0.0666624i	0.172075	−	0.0666624i	0.445738	+	0.895163i	$0.352941\pi$
$74$	−0.397365	+	0.798017i	−0.397365	+	0.798017i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$79$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$80$	0.353470	+	1.89090i	0.353470	+	1.89090i
$81$	−0.982973	−	0.183750i	−0.982973	−	0.183750i
$82$	−0.942485	−	0.469302i	−0.942485	−	0.469302i
$83$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$83$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$84$		0			0
$85$	−3.63743	+	1.03494i	−3.63743	+	1.03494i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.20614	−	0.600584i	1.20614	−	0.600584i	0.273663	−	0.961826i	$-0.411765\pi$
$89$	1.20614	−	0.600584i	1.20614	−	0.600584i	0.932472	+	0.361242i	$0.117647\pi$
$90$	1.91545	+	0.177492i	1.91545	+	0.177492i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$97$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$98$	0.932472	−	0.361242i	0.932472	−	0.361242i
$99$		0			0
$100$	−2.70043			−2.70043
$101$	1.25664	−	0.778076i	1.25664	−	0.778076i	0.273663	−	0.961826i	$-0.411765\pi$
$101$	1.25664	−	0.778076i	1.25664	−	0.778076i	0.982973	+	0.183750i	$0.0588235\pi$
$102$		0			0
$103$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$103$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$104$	0.293271	−	1.56886i	0.293271	−	1.56886i
$105$		0			0
$106$	−0.193463	−	0.312454i	−0.193463	−	0.312454i
$107$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$107$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$108$		0			0
$109$	−0.404479	−	1.42160i	−0.404479	−	1.42160i	−0.850217	−	0.526432i	$-0.823529\pi$
$109$	−0.404479	−	1.42160i	−0.404479	−	1.42160i	0.445738	−	0.895163i	$-0.352941\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.58923	+	1.20013i	1.58923	+	1.20013i	0.850217	+	0.526432i	$0.176471\pi$
$113$	1.58923	+	1.20013i	1.58923	+	1.20013i	0.739009	+	0.673696i	$0.235294\pi$
$114$		0			0
$115$		0			0
$116$	0.260991	+	0.673696i	0.260991	+	0.673696i
$117$	−1.42871	−	0.711414i	−1.42871	−	0.711414i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.273663	+	0.961826i	−0.273663	+	0.961826i
$122$	0.510366	+	0.197717i	0.510366	+	0.197717i
$123$		0			0
$124$		0			0
$125$	0.601052	−	3.21535i	0.601052	−	3.21535i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$129$		0			0
$130$	2.86289	+	1.10909i	2.86289	+	1.10909i
$131$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$131$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.96595			1.96595
$137$	0.932472	−	0.361242i	0.932472	−	0.361242i
$137$	0.932472	−	0.361242i	0.932472	−	0.361242i
$138$		0			0
$139$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$139$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.932472	−	0.361242i	−0.932472	−	0.361242i
$145$	−1.36614	+	0.255376i	−1.36614	+	0.255376i
$146$	0.0170269	−	0.183750i	0.0170269	−	0.183750i
$147$		0			0
$148$	0.537235	+	0.711414i	0.537235	+	0.711414i
$149$	0.328972	−	1.75984i	0.328972	−	1.75984i	−0.273663	−	0.961826i	$-0.588235\pi$
$149$	0.328972	−	1.75984i	0.328972	−	1.75984i	0.602635	−	0.798017i	$-0.294118\pi$
$150$		0			0
$151$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$151$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$152$		0			0
$153$	0.538007	−	1.89090i	0.538007	−	1.89090i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.60263	−	0.798017i	−1.60263	−	0.798017i	−0.602635	−	0.798017i	$-0.705882\pi$
$157$	−1.60263	−	0.798017i	−1.60263	−	0.798017i	−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	1.85022	+	0.526432i	1.85022	+	0.526432i
$161$		0			0
$162$	−0.602635	+	0.798017i	−0.602635	+	0.798017i
$163$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$163$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$164$	−0.840204	+	0.634493i	−0.840204	+	0.634493i
$165$		0			0
$166$		0			0
$167$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$167$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$168$		0			0
$169$	−1.14349	−	1.04243i	−1.14349	−	1.04243i
$170$	−0.694903	+	3.71740i	−0.694903	+	3.71740i
$171$		0			0
$172$		0			0
$173$	−1.58561	+	0.981767i	−1.58561	+	0.981767i	−0.602635	+	0.798017i	$0.705882\pi$
$173$	−1.58561	+	0.981767i	−1.58561	+	0.981767i	−0.982973	+	0.183750i	$0.941176\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		−	1.34739i		−	1.34739i
$179$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$179$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$180$	1.01267	−	1.63552i	1.01267	−	1.63552i
$181$	−0.510366	+	1.79375i	−0.510366	+	1.79375i	0.0922684	+	0.995734i	$0.470588\pi$
$181$	−0.510366	+	1.79375i	−0.510366	+	1.79375i	−0.602635	+	0.798017i	$0.705882\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.53511	+	0.764392i	−1.53511	+	0.764392i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$191$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$192$		0			0
$193$	0.181395	+	0.0339085i	0.181395	+	0.0339085i	0.273663	−	0.961826i	$-0.411765\pi$
$193$	0.181395	+	0.0339085i	0.181395	+	0.0339085i	−0.0922684	+	0.995734i	$0.529412\pi$
$194$		0			0
$195$		0			0
$196$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$197$	0.404479	+	1.42160i	0.404479	+	1.42160i	0.850217	+	0.526432i	$0.176471\pi$
$197$	0.404479	+	1.42160i	0.404479	+	1.42160i	−0.445738	+	0.895163i	$0.647059\pi$
$198$		0			0
$199$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$199$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$200$	−1.20369	+	2.41733i	−1.20369	+	2.41733i
$201$		0			0
$202$	−0.136374	−	1.47171i	−0.136374	−	1.47171i
$203$		0			0
$204$		0			0
$205$	−0.902773	−	1.81301i	−0.902773	−	1.81301i
$206$		0			0
$207$		0			0
$208$	−1.27366	−	0.961826i	−1.27366	−	0.961826i
$209$		0			0
$210$		0			0
$211$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$211$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$212$	−0.365931	+	0.0339085i	−0.365931	+	0.0339085i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.45285	−	0.271585i	−1.45285	−	0.271585i
$219$		0			0
$220$		0			0
$221$	1.65180	−	2.66774i	1.65180	−	2.66774i
$222$		0			0
$223$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$223$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$224$		0			0
$225$	1.99565	+	1.81927i	1.99565	+	1.81927i
$226$	1.78269	−	0.887674i	1.78269	−	0.887674i
$227$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$227$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$228$		0			0
$229$	−0.380338	−	0.981767i	−0.380338	−	0.981767i	−0.982973	−	0.183750i	$-0.941176\pi$
$229$	−0.380338	−	0.981767i	−0.380338	−	0.981767i	0.602635	−	0.798017i	$-0.294118\pi$
$230$		0			0
$231$		0			0
$232$	0.719401	+	0.0666624i	0.719401	+	0.0666624i
$233$		−	0.367499i		−	0.367499i	−0.982973	−	0.183750i	$-0.941176\pi$
$233$		−	0.367499i		−	0.367499i	0.982973	−	0.183750i	$-0.0588235\pi$
$234$	−1.27366	+	0.961826i	−1.27366	+	0.961826i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$239$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$240$		0			0
$241$	−0.365931	+	1.95756i	−0.365931	+	1.95756i	−0.0922684	+	0.995734i	$0.529412\pi$
$241$	−0.365931	+	1.95756i	−0.365931	+	1.95756i	−0.273663	+	0.961826i	$0.588235\pi$
$242$	0.739009	+	0.673696i	0.739009	+	0.673696i
$243$		0			0
$244$	0.404479	−	0.368731i	0.404479	−	0.368731i
$245$	1.85022	+	0.526432i	1.85022	+	0.526432i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−2.61035	−	1.97124i	−2.61035	−	1.97124i
$251$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$251$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$257$	−1.18475	−	0.221468i	−1.18475	−	0.221468i	−0.445738	−	0.895163i	$-0.647059\pi$
$257$	−1.18475	−	0.221468i	−1.18475	−	0.221468i	−0.739009	+	0.673696i	$0.764706\pi$
$258$		0			0
$259$		0			0
$260$	2.26892	−	2.06839i	2.26892	−	2.06839i
$261$	0.260991	−	0.673696i	0.260991	−	0.673696i
$262$		0			0
$263$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$263$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$264$		0			0
$265$	0.0652282	−	0.703924i	0.0652282	−	0.703924i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.850217	+	0.526432i	$0.823529\pi$
$269$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.932472	+	0.361242i	$0.882353\pi$
$270$		0			0
$271$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$271$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$272$	0.876298	−	1.75984i	0.876298	−	1.75984i
$273$		0			0
$274$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$275$		0			0
$276$		0			0
$277$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$277$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.831277	+	0.322039i	0.831277	+	0.322039i	0.739009	−	0.673696i	$-0.235294\pi$
$281$	0.831277	+	0.322039i	0.831277	+	0.322039i	0.0922684	+	0.995734i	$0.470588\pi$
$282$		0			0
$283$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$283$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.739009	+	0.673696i	−0.739009	+	0.673696i
$289$	2.67148	+	1.03494i	2.67148	+	1.03494i
$290$	−0.380338	+	1.33675i	−0.380338	+	1.33675i
$291$		0			0
$292$	−0.156896	−	0.0971461i	−0.156896	−	0.0971461i
$293$	−1.02474	−	0.634493i	−1.02474	−	0.634493i	−0.0922684	−	0.995734i	$-0.529412\pi$
$293$	−1.02474	−	0.634493i	−1.02474	−	0.634493i	−0.932472	+	0.361242i	$0.882353\pi$
$294$		0			0
$295$		0			0
$296$	0.876298	−	0.163808i	0.876298	−	0.163808i
$297$		0			0
$298$	−1.42871	−	1.07891i	−1.42871	−	1.07891i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.554262	+	0.895163i	0.554262	+	0.895163i
$306$	−1.45285	−	1.32445i	−1.45285	−	1.32445i
$307$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$307$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	0.510366	−	0.197717i	0.510366	−	0.197717i	−0.0922684	−	0.995734i	$-0.529412\pi$
$313$	0.510366	−	0.197717i	0.510366	−	0.197717i	0.602635	+	0.798017i	$0.294118\pi$
$314$	−1.42871	+	1.07891i	−1.42871	+	1.07891i
$315$		0			0
$316$		0			0
$317$	−0.554262	+	0.895163i	−0.554262	+	0.895163i	0.445738	+	0.895163i	$0.352941\pi$
$317$	−0.554262	+	0.895163i	−0.554262	+	0.895163i	−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	1.29596	−	1.42160i	1.29596	−	1.42160i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.445738	+	0.895163i	0.445738	+	0.895163i
$325$	2.26892	+	3.66442i	2.26892	+	3.66442i
$326$		0			0
$327$		0			0
$328$	0.193463	+	1.03494i	0.193463	+	1.03494i
$329$		0			0
$330$		0			0
$331$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$331$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$332$		0			0
$333$	0.0822551	−	0.887674i	0.0822551	−	0.887674i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.537235	−	1.07891i	0.537235	−	1.07891i	−0.445738	−	0.895163i	$-0.647059\pi$
$337$	0.537235	−	1.07891i	0.537235	−	1.07891i	0.982973	−	0.183750i	$-0.0588235\pi$
$338$	−1.44284	+	0.558959i	−1.44284	+	0.558959i
$339$		0			0
$340$	3.01794	+	2.27904i	3.01794	+	2.27904i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.172075	+	1.85699i	0.172075	+	1.85699i
$347$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$347$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$348$		0			0
$349$	0.365931	−	0.0339085i	0.365931	−	0.0339085i	0.0922684	−	0.995734i	$-0.470588\pi$
$349$	0.365931	−	0.0339085i	0.365931	−	0.0339085i	0.273663	+	0.961826i	$0.411765\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.34164	−	1.47171i	1.34164	−	1.47171i	0.602635	−	0.798017i	$-0.294118\pi$
$353$	1.34164	−	1.47171i	1.34164	−	1.47171i	0.739009	−	0.673696i	$-0.235294\pi$
$354$		0			0
$355$		0			0
$356$	−1.20614	−	0.600584i	−1.20614	−	0.600584i
$357$		0			0
$358$		0			0
$359$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$359$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$360$	−1.01267	−	1.63552i	−1.01267	−	1.63552i
$361$	0.445738	+	0.895163i	0.445738	+	0.895163i
$362$	1.37821	+	1.25640i	1.37821	+	1.25640i
$363$		0			0
$364$		0			0
$365$	0.239151	−	0.262337i	0.239151	−	0.262337i
$366$		0			0
$367$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$367$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$368$		0			0
$369$	1.04837	+	0.0971461i	1.04837	+	0.0971461i
$370$			1.71489i			1.71489i
$371$		0			0
$372$		0			0
$373$	−1.02474	+	0.634493i	−1.02474	+	0.634493i	−0.932472	−	0.361242i	$-0.882353\pi$
$373$	−1.02474	+	0.634493i	−1.02474	+	0.634493i	−0.0922684	+	0.995734i	$0.529412\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.694903	−	0.920200i	0.694903	−	0.920200i
$378$		0			0
$379$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$379$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$383$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$384$		0			0
$385$		0			0
$386$	0.111208	−	0.147263i	0.111208	−	0.147263i
$387$		0			0
$388$		0			0
$389$	0.181395	−	0.0339085i	0.181395	−	0.0339085i	−0.0922684	−	0.995734i	$-0.529412\pi$
$389$	0.181395	−	0.0339085i	0.181395	−	0.0339085i	0.273663	+	0.961826i	$0.411765\pi$
$390$		0			0
$391$		0			0
$392$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$393$		0			0
$394$	1.45285	+	0.271585i	1.45285	+	0.271585i
$395$		0			0
$396$		0			0
$397$	−0.658809	+	0.600584i	−0.658809	+	0.600584i	−0.932472	−	0.361242i	$-0.882353\pi$
$397$	−0.658809	+	0.600584i	−0.658809	+	0.600584i	0.273663	+	0.961826i	$0.411765\pi$
$398$		0			0
$399$		0			0
$400$	1.62738	+	2.15499i	1.62738	+	2.15499i
$401$			0.722483i			0.722483i	0.932472	+	0.361242i	$0.117647\pi$
$401$			0.722483i			0.722483i	−0.932472	+	0.361242i	$0.882353\pi$
$402$		0			0
$403$		0			0
$404$	−1.37821	−	0.533922i	−1.37821	−	0.533922i
$405$	−1.85022	+	0.526432i	−1.85022	+	0.526432i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.757949	−	1.52217i	0.757949	−	1.52217i	−0.0922684	−	0.995734i	$-0.529412\pi$
$409$	0.757949	−	1.52217i	0.757949	−	1.52217i	0.850217	−	0.526432i	$-0.176471\pi$
$410$	−2.02534			−2.02534
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.42871	+	0.711414i	−1.42871	+	0.711414i
$417$		0			0
$418$		0			0
$419$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$419$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$420$		0			0
$421$			1.34739i			1.34739i	0.739009	+	0.673696i	$0.235294\pi$
$421$			1.34739i			1.34739i	−0.739009	+	0.673696i	$0.764706\pi$
$422$		0			0
$423$		0			0
$424$	−0.132756	+	0.342683i	−0.132756	+	0.342683i
$425$	−3.92333	+	3.57659i	−3.92333	+	3.57659i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$431$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$432$		0			0
$433$	−1.45285	+	0.271585i	−1.45285	+	0.271585i	−0.850217	−	0.526432i	$-0.823529\pi$
$433$	−1.45285	+	0.271585i	−1.45285	+	0.271585i	−0.602635	+	0.798017i	$0.705882\pi$
$434$		0			0
$435$		0			0
$436$	−0.890705	+	1.17948i	−0.890705	+	1.17948i
$437$		0			0
$438$		0			0
$439$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$439$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$440$		0			0
$441$	−0.739009	+	0.673696i	−0.739009	+	0.673696i
$442$	−1.65180	−	2.66774i	−1.65180	−	2.66774i
$443$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$443$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$444$		0			0
$445$	1.56198	−	2.06839i	1.56198	−	2.06839i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.58561	−	0.981767i	1.58561	−	0.981767i	0.602635	−	0.798017i	$-0.294118\pi$
$449$	1.58561	−	0.981767i	1.58561	−	0.981767i	0.982973	−	0.183750i	$-0.0588235\pi$
$450$	2.51808	−	0.975509i	2.51808	−	0.975509i
$451$		0			0
$452$		−	1.99147i		−	1.99147i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.907732	+	0.995734i	−0.907732	+	0.995734i	0.0922684	+	0.995734i	$0.470588\pi$
$457$	−0.907732	+	0.995734i	−0.907732	+	0.995734i	−1.00000			$\pi$
$458$	−1.04837	−	0.0971461i	−1.04837	−	0.0971461i
$459$		0			0
$460$		0			0
$461$	0.0822551	+	0.165190i	0.0822551	+	0.165190i	0.932472	−	0.361242i	$-0.117647\pi$
$461$	0.0822551	+	0.165190i	0.0822551	+	0.165190i	−0.850217	+	0.526432i	$0.823529\pi$
$462$		0			0
$463$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$463$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$464$	0.380338	−	0.614268i	0.380338	−	0.614268i
$465$		0			0
$466$	−0.328972	−	0.163808i	−0.328972	−	0.163808i
$467$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$467$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$468$	0.293271	+	1.56886i	0.293271	+	1.56886i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.293271	+	0.221468i	0.293271	+	0.221468i
$478$		0			0
$479$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$479$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$480$		0			0
$481$	0.513985	−	1.32675i	0.513985	−	1.32675i
$482$	1.58923	+	1.20013i	1.58923	+	1.20013i
$483$		0			0
$484$	0.932472	−	0.361242i	0.932472	−	0.361242i
$485$		0			0
$486$		0			0
$487$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$487$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$488$	−0.149783	−	0.526432i	−0.149783	−	0.526432i
$489$		0			0
$490$	1.29596	−	1.42160i	1.29596	−	1.42160i
$491$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$491$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$492$		0			0
$493$	1.27146	+	0.633110i	1.27146	+	0.633110i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$499$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$500$	−2.92812	+	1.45803i	−2.92812	+	1.45803i
$501$		0			0
$502$		0			0
$503$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$503$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$504$		0			0
$505$	1.49675	−	2.41733i	1.49675	−	2.41733i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.831277	+	0.322039i	−0.831277	+	0.322039i	−0.739009	−	0.673696i	$-0.764706\pi$
$509$	−0.831277	+	0.322039i	−0.831277	+	0.322039i	−0.0922684	+	0.995734i	$0.529412\pi$
$510$		0			0
$511$		0			0
$512$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$513$		0			0
$514$	−0.726337	+	0.961826i	−0.726337	+	0.961826i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.840204	−	2.95301i	−0.840204	−	2.95301i
$521$	−0.576554	+	0.435393i	−0.576554	+	0.435393i	−0.850217	−	0.526432i	$-0.823529\pi$
$521$	−0.576554	+	0.435393i	−0.576554	+	0.435393i	0.273663	+	0.961826i	$0.411765\pi$
$522$	−0.486734	−	0.533922i	−0.486734	−	0.533922i
$523$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$523$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.850217	+	0.526432i	0.850217	+	0.526432i
$530$	−0.601052	−	0.372156i	−0.601052	−	0.372156i
$531$		0			0
$532$		0			0
$533$	1.56693	+	0.607033i	1.56693	+	0.607033i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			1.99147i			1.99147i
$539$		0			0
$540$		0			0
$541$	1.58561	+	0.614268i	1.58561	+	0.614268i	0.982973	−	0.183750i	$-0.0588235\pi$
$541$	1.58561	+	0.614268i	1.58561	+	0.614268i	0.602635	+	0.798017i	$0.294118\pi$
$542$		0			0
$543$		0			0
$544$	−1.18475	−	1.56886i	−1.18475	−	1.56886i
$545$	−1.91545	−	2.10114i	−1.91545	−	2.10114i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$549$	−0.547326			−0.547326
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.181395	+	1.95756i	−0.181395	+	1.95756i	0.0922684	+	0.995734i	$0.470588\pi$
$557$	−0.181395	+	1.95756i	−0.181395	+	1.95756i	−0.273663	+	0.961826i	$0.588235\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.658809	−	0.600584i	0.658809	−	0.600584i
$563$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$563$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$564$		0			0
$565$	3.76566	+	0.703924i	3.76566	+	0.703924i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.380338	+	0.981767i	0.380338	+	0.981767i	0.982973	+	0.183750i	$0.0588235\pi$
$569$	0.380338	+	0.981767i	0.380338	+	0.981767i	−0.602635	+	0.798017i	$0.705882\pi$
$570$		0			0
$571$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$571$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.273663	+	0.961826i	0.273663	+	0.961826i
$577$	−1.72198	−	0.489946i	−1.72198	−	0.489946i	−0.739009	−	0.673696i	$-0.764706\pi$
$577$	−1.72198	−	0.489946i	−1.72198	−	0.489946i	−0.982973	+	0.183750i	$0.941176\pi$
$578$	2.11722	−	1.93010i	2.11722	−	1.93010i
$579$		0			0
$580$	1.02708	+	0.936306i	1.02708	+	0.936306i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.156896	+	0.0971461i	−0.156896	+	0.0971461i
$585$	−3.07021			−3.07021
$586$	−1.02474	+	0.634493i	−1.02474	+	0.634493i
$587$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$587$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.243964	−	0.857445i	0.243964	−	0.857445i
$593$	0.260991	+	0.673696i	0.260991	+	0.673696i	1.00000			$0$
$593$	0.260991	+	0.673696i	0.260991	+	0.673696i	−0.739009	+	0.673696i	$0.764706\pi$
$594$		0			0
$595$		0			0
$596$	−1.60263	+	0.798017i	−1.60263	+	0.798017i
$597$		0			0
$598$		0			0
$599$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$599$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$600$		0			0
$601$	−0.709310	+	1.14558i	−0.709310	+	1.14558i	0.273663	+	0.961826i	$0.411765\pi$
$601$	−0.709310	+	1.14558i	−0.709310	+	1.14558i	−0.982973	+	0.183750i	$0.941176\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.353470	+	1.89090i	0.353470	+	1.89090i
$606$		0			0
$607$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$607$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$608$		0			0
$609$		0			0
$610$	1.04837	−	0.0971461i	1.04837	−	0.0971461i
$611$		0			0
$612$	−1.83319	+	0.710182i	−1.83319	+	0.710182i
$613$	−0.156896	−	1.69318i	−0.156896	−	1.69318i	−0.602635	−	0.798017i	$-0.705882\pi$
$613$	−0.156896	−	1.69318i	−0.156896	−	1.69318i	0.445738	−	0.895163i	$-0.352941\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.876298	−	1.75984i	−0.876298	−	1.75984i	−0.602635	−	0.798017i	$-0.705882\pi$
$617$	−0.876298	−	1.75984i	−0.876298	−	1.75984i	−0.273663	−	0.961826i	$-0.588235\pi$
$618$		0			0
$619$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$619$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.982973	−	3.45479i	−0.982973	−	3.45479i
$626$	0.0505009	−	0.544991i	0.0505009	−	0.544991i
$627$		0			0
$628$	0.328972	+	1.75984i	0.328972	+	1.75984i
$629$	1.72275	+	0.322039i	1.72275	+	0.322039i
$630$		0			0
$631$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$631$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$632$		0			0
$633$		0			0
$634$	0.554262	+	0.895163i	0.554262	+	0.895163i
$635$		0			0
$636$		0			0
$637$	−1.42871	+	0.711414i	−1.42871	+	0.711414i
$638$		0			0
$639$		0			0
$640$	−0.694903	−	1.79375i	−0.694903	−	1.79375i
$641$	0.538007	−	1.89090i	0.538007	−	1.89090i	0.0922684	−	0.995734i	$-0.470588\pi$
$641$	0.538007	−	1.89090i	0.538007	−	1.89090i	0.445738	−	0.895163i	$-0.352941\pi$
$642$		0			0
$643$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$643$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$647$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$648$	1.00000			1.00000
$649$		0			0
$650$	4.29160	−	0.397675i	4.29160	−	0.397675i
$651$		0			0
$652$		0			0
$653$	−0.136374	−	0.124322i	−0.136374	−	0.124322i	0.602635	−	0.798017i	$-0.294118\pi$
$653$	−0.136374	−	0.124322i	−0.136374	−	0.124322i	−0.739009	+	0.673696i	$0.764706\pi$
$654$		0			0
$655$		0			0
$656$	1.01267	+	0.288130i	1.01267	+	0.288130i
$657$	0.0505009	+	0.177492i	0.0505009	+	0.177492i
$658$		0			0
$659$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$659$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$660$		0			0
$661$	−0.576554	−	0.435393i	−0.576554	−	0.435393i	0.273663	−	0.961826i	$-0.411765\pi$
$661$	−0.576554	−	0.435393i	−0.576554	−	0.435393i	−0.850217	+	0.526432i	$0.823529\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.757949	−	0.469302i	−0.757949	−	0.469302i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.0675278	+	0.361242i	−0.0675278	+	0.361242i	0.932472	+	0.361242i	$0.117647\pi$
$673$	−0.0675278	+	0.361242i	−0.0675278	+	0.361242i	−1.00000			$\pi$
$674$	−0.726337	−	0.961826i	−0.726337	−	0.961826i
$675$		0			0
$676$	−0.142769	+	1.54073i	−0.142769	+	1.54073i
$677$	−0.876298	+	0.163808i	−0.876298	+	0.163808i	−0.602635	−	0.798017i	$-0.705882\pi$
$677$	−0.876298	+	0.163808i	−0.876298	+	0.163808i	−0.273663	+	0.961826i	$0.588235\pi$
$678$		0			0
$679$		0			0
$680$	3.38532	−	1.68569i	3.38532	−	1.68569i
$681$		0			0
$682$		0			0
$683$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$683$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$684$		0			0
$685$	1.29596	−	1.42160i	1.29596	−	1.42160i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.353470	+	0.468070i	0.353470	+	0.468070i
$690$		0			0
$691$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$691$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$692$	1.73901	+	0.673696i	1.73901	+	0.673696i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.380338	+	2.03463i	−0.380338	+	2.03463i
$698$	0.132756	−	0.342683i	0.132756	−	0.342683i
$699$		0			0
$700$		0			0
$701$	0.329838	−	1.15926i	0.329838	−	1.15926i	−0.602635	−	0.798017i	$-0.705882\pi$
$701$	0.329838	−	1.15926i	0.329838	−	1.15926i	0.932472	−	0.361242i	$-0.117647\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.719401	−	1.85699i	−0.719401	−	1.85699i
$707$		0			0
$708$		0			0
$709$	1.58923	+	1.20013i	1.58923	+	1.20013i	0.850217	+	0.526432i	$0.176471\pi$
$709$	1.58923	+	1.20013i	1.58923	+	1.20013i	0.739009	+	0.673696i	$0.235294\pi$
$710$		0			0
$711$		0			0
$712$	−1.07524	+	0.811985i	−1.07524	+	0.811985i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$719$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$720$	−1.91545	+	0.177492i	−1.91545	+	0.177492i
$721$		0			0
$722$	1.00000			1.00000
$723$		0			0
$724$	1.73901	−	0.673696i	1.73901	−	0.673696i
$725$	−1.55695	+	1.17575i	−1.55695	+	1.17575i
$726$		0			0
$727$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$727$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$728$		0			0
$729$	0.273663	−	0.961826i	0.273663	−	0.961826i
$730$	−0.128235	−	0.331013i	−0.128235	−	0.331013i
$731$		0			0
$732$		0			0
$733$	1.42871	−	0.711414i	1.42871	−	0.711414i	0.445738	−	0.895163i	$-0.352941\pi$
$733$	1.42871	−	0.711414i	1.42871	−	0.711414i	0.982973	+	0.183750i	$0.0588235\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.554262	−	0.895163i	0.554262	−	0.895163i
$739$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$739$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$740$	1.53511	+	0.764392i	1.53511	+	0.764392i
$741$		0			0
$742$		0			0
$743$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$743$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$744$		0			0
$745$	−0.942485	−	3.31249i	−0.942485	−	3.31249i
$746$	0.111208	+	1.20013i	0.111208	+	1.20013i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$751$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$752$		0			0
$753$		0			0
$754$	−0.513985	−	1.03222i	−0.513985	−	1.03222i
$755$		0			0
$756$		0			0
$757$	−0.0822551	−	0.887674i	−0.0822551	−	0.887674i	−0.932472	−	0.361242i	$-0.882353\pi$
$757$	−0.0822551	−	0.887674i	−0.0822551	−	0.887674i	0.850217	−	0.526432i	$-0.176471\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.156896	−	1.69318i	−0.156896	−	1.69318i	−0.602635	−	0.798017i	$-0.705882\pi$
$761$	−0.156896	−	1.69318i	−0.156896	−	1.69318i	0.445738	−	0.895163i	$-0.352941\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.694903	−	3.71740i	−0.694903	−	3.71740i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.840204	−	1.35698i	0.840204	−	1.35698i	−0.0922684	−	0.995734i	$-0.529412\pi$
$769$	0.840204	−	1.35698i	0.840204	−	1.35698i	0.932472	−	0.361242i	$-0.117647\pi$
$770$		0			0
$771$		0			0
$772$	−0.0822551	−	0.165190i	−0.0822551	−	0.165190i
$773$	−1.25664	−	1.14558i	−1.25664	−	1.14558i	−0.982973	−	0.183750i	$-0.941176\pi$
$773$	−1.25664	−	1.14558i	−1.25664	−	1.14558i	−0.273663	−	0.961826i	$-0.588235\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.0505009	−	0.177492i	0.0505009	−	0.177492i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$785$	−3.44396			−3.44396
$786$		0			0
$787$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$787$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$788$	0.890705	−	1.17948i	0.890705	−	1.17948i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.840204	−	0.239059i	−0.840204	−	0.239059i
$794$	0.243964	+	0.857445i	0.243964	+	0.857445i
$795$		0			0
$796$		0			0
$797$	1.02474	−	1.35698i	1.02474	−	1.35698i	0.0922684	−	0.995734i	$-0.470588\pi$
$797$	1.02474	−	1.35698i	1.02474	−	1.35698i	0.932472	−	0.361242i	$-0.117647\pi$
$798$		0			0
$799$		0			0
$800$	2.65445	−	0.496203i	2.65445	−	0.496203i
$801$	0.486734	+	1.25640i	0.486734	+	1.25640i
$802$	0.646741	+	0.322039i	0.646741	+	0.322039i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.09227	+	0.995734i	−1.09227	+	0.995734i
$809$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	0.602635	+	0.798017i	$0.294118\pi$
$809$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	−0.982973	+	0.183750i	$0.941176\pi$
$810$	−0.353470	+	1.89090i	−0.353470	+	1.89090i
$811$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$811$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.02474	−	1.35698i	−1.02474	−	1.35698i
$819$		0			0
$820$	−0.902773	+	1.81301i	−0.902773	+	1.81301i
$821$	1.20527			1.20527			0.602635	−	0.798017i	$-0.294118\pi$
$821$	1.20527			1.20527			0.602635	+	0.798017i	$0.294118\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$827$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$828$		0			0
$829$	−0.510366	−	0.197717i	−0.510366	−	0.197717i	0.0922684	−	0.995734i	$-0.470588\pi$
$829$	−0.510366	−	0.197717i	−0.510366	−	0.197717i	−0.602635	+	0.798017i	$0.705882\pi$
$830$		0			0
$831$		0			0
$832$			1.59603i			1.59603i
$833$	−1.18475	−	1.56886i	−1.18475	−	1.56886i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$839$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$840$		0			0
$841$	−0.406419	−	0.251644i	−0.406419	−	0.251644i
$842$	1.20614	+	0.600584i	1.20614	+	0.600584i
$843$		0			0
$844$		0			0
$845$	−2.86289	−	0.814562i	−2.86289	−	0.814562i
$846$		0			0
$847$		0			0
$848$	0.247582	+	0.271585i	0.247582	+	0.271585i
$849$		0			0
$850$	1.45285	+	5.10624i	1.45285	+	5.10624i
$851$		0			0
$852$		0			0
$853$	0.709310	+	1.14558i	0.709310	+	1.14558i	0.982973	+	0.183750i	$0.0588235\pi$
$853$	0.709310	+	1.14558i	0.709310	+	1.14558i	−0.273663	+	0.961826i	$0.588235\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.719401	−	0.0666624i	0.719401	−	0.0666624i	0.273663	−	0.961826i	$-0.411765\pi$
$857$	0.719401	−	0.0666624i	0.719401	−	0.0666624i	0.445738	+	0.895163i	$0.352941\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	−1.88858	+	3.05016i	−1.88858	+	3.05016i
$866$	−0.404479	+	1.42160i	−0.404479	+	1.42160i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.658809	+	1.32307i	0.658809	+	1.32307i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.78269	+	0.887674i	1.78269	+	0.887674i	0.932472	+	0.361242i	$0.117647\pi$
$877$	1.78269	+	0.887674i	1.78269	+	0.887674i	0.850217	+	0.526432i	$0.176471\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.156896	+	1.69318i	−0.156896	+	1.69318i	0.445738	+	0.895163i	$0.352941\pi$
$881$	−0.156896	+	1.69318i	−0.156896	+	1.69318i	−0.602635	+	0.798017i	$0.705882\pi$
$882$	0.273663	+	0.961826i	0.273663	+	0.961826i
$883$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$883$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$884$	−3.12433	+	0.289512i	−3.12433	+	0.289512i
$885$		0			0
$886$		0			0
$887$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$887$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$888$		0			0
$889$		0			0
$890$	−1.15531	−	2.32018i	−1.15531	−	2.32018i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.172075	−	1.85699i	−0.172075	−	1.85699i
$899$		0			0
$900$	0.249165	−	2.68891i	0.249165	−	2.68891i
$901$	−0.486734	+	0.533922i	−0.486734	+	0.533922i
$902$		0			0
$903$		0			0
$904$	−1.78269	−	0.887674i	−1.78269	−	0.887674i
$905$	0.659202	+	3.52642i	0.659202	+	3.52642i
$906$		0			0
$907$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$907$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$908$		0			0
$909$	0.658809	+	1.32307i	0.658809	+	1.32307i
$910$		0			0
$911$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$911$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$912$		0			0
$913$		0			0
$914$	0.486734	+	1.25640i	0.486734	+	1.25640i
$915$		0			0
$916$	−0.554262	+	0.895163i	−0.554262	+	0.895163i
$917$		0			0
$918$		0			0
$919$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$919$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$920$		0			0
$921$		0			0
$922$	0.184537			0.184537
$923$		0			0
$924$		0			0
$925$	−1.45077	+	1.92113i	−1.45077	+	1.92113i
$926$		0			0
$927$		0			0
$928$	−0.380338	−	0.614268i	−0.380338	−	0.614268i
$929$	−0.890705	+	0.811985i	−0.890705	+	0.811985i	−0.982973	−	0.183750i	$-0.941176\pi$
$929$	−0.890705	+	0.811985i	−0.890705	+	0.811985i	0.0922684	+	0.995734i	$0.470588\pi$
$930$		0			0
$931$		0			0
$932$	−0.293271	+	0.221468i	−0.293271	+	0.221468i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	1.53511	+	0.436776i	1.53511	+	0.436776i
$937$	1.83319	−	0.342683i	1.83319	−	0.342683i	0.850217	−	0.526432i	$-0.176471\pi$
$937$	1.83319	−	0.342683i	1.83319	−	0.342683i	0.982973	+	0.183750i	$0.0588235\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.67148	−	1.03494i	−1.67148	−	1.03494i	−0.932472	−	0.361242i	$-0.882353\pi$
$941$	−1.67148	−	1.03494i	−1.67148	−	1.03494i	−0.739009	−	0.673696i	$-0.764706\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$947$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$948$		0			0
$949$			0.294527i			0.294527i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.72198	−	0.489946i	1.72198	−	0.489946i	0.739009	−	0.673696i	$-0.235294\pi$
$953$	1.72198	−	0.489946i	1.72198	−	0.489946i	0.982973	+	0.183750i	$0.0588235\pi$
$954$	0.328972	−	0.163808i	0.328972	−	0.163808i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.445738	+	0.895163i	−0.445738	+	0.895163i
$962$	−0.958553	−	1.05148i	−0.958553	−	1.05148i
$963$		0			0
$964$	1.78269	−	0.887674i	1.78269	−	0.887674i
$965$	0.341433	−	0.0971461i	0.341433	−	0.0971461i
$966$		0			0
$967$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$967$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$968$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$969$		0			0
$970$		0			0
$971$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$971$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.538007	−	0.100571i	−0.538007	−	0.100571i
$977$	1.58561	+	0.981767i	1.58561	+	0.981767i	0.982973	+	0.183750i	$0.0588235\pi$
$977$	1.58561	+	0.981767i	1.58561	+	0.981767i	0.602635	+	0.798017i	$0.294118\pi$
$978$		0			0
$979$		0			0
$980$	−0.694903	−	1.79375i	−0.694903	−	1.79375i
$981$	1.45285	−	0.271585i	1.45285	−	0.271585i
$982$		0			0
$983$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$983$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$984$		0			0
$985$	1.91545	+	2.10114i	1.91545	+	2.10114i
$986$	1.13347	−	0.855960i	1.13347	−	0.855960i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$991$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.67148	+	1.03494i	−1.67148	+	1.03494i	−0.739009	+	0.673696i	$0.764706\pi$
$997$	−1.67148	+	1.03494i	−1.67148	+	1.03494i	−0.932472	+	0.361242i	$0.882353\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	548.1.j.a.215.1	✓	16
4.3	odd	2	CM	548.1.j.a.215.1	✓	16
137.65	even	34	inner	548.1.j.a.339.1	yes	16
548.339	odd	34	inner	548.1.j.a.339.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
548.1.j.a.215.1	✓	16	1.1	even	1	trivial
548.1.j.a.215.1	✓	16	4.3	odd	2	CM
548.1.j.a.339.1	yes	16	137.65	even	34	inner
548.1.j.a.339.1	yes	16	548.339	odd	34	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 548 = 2^{2} \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	548.j (of order \(34\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.445738 - 0.895163i) q^{2} +(-0.602635 - 0.798017i) q^{4} +(1.72198 - 0.857445i) q^{5} +(-0.982973 + 0.183750i) q^{8} +(-0.0922684 + 0.995734i) q^{9} +O(q^{10})\)	\(q+(0.445738 - 0.895163i) q^{2} +(-0.602635 - 0.798017i) q^{4} +(1.72198 - 0.857445i) q^{5} +(-0.982973 + 0.183750i) q^{8} +(-0.0922684 + 0.995734i) q^{9} -1.92365i q^{10} +(-0.576554 + 1.48826i) q^{13} +(-0.273663 + 0.961826i) q^{16} +(-1.93247 - 0.361242i) q^{17} +(0.850217 + 0.526432i) q^{18} +(-1.72198 - 0.857445i) q^{20} +(1.62738 - 2.15499i) q^{25} +(1.07524 + 1.17948i) q^{26} +(-0.694903 - 0.197717i) q^{29} +(0.739009 + 0.673696i) q^{32} +(-1.18475 + 1.56886i) q^{34} +(0.850217 - 0.526432i) q^{36} -0.891477 q^{37} +(-1.53511 + 1.15926i) q^{40} -1.05286i q^{41} +(0.694903 + 1.79375i) q^{45} +(0.739009 + 0.673696i) q^{49} +(-1.20369 - 2.41733i) q^{50} +(1.53511 - 0.436776i) q^{52} +(0.193463 - 0.312454i) q^{53} +(-0.486734 + 0.533922i) q^{58} +(0.0505009 + 0.544991i) q^{61} +(0.932472 - 0.361242i) q^{64} +(0.283284 + 3.05712i) q^{65} +(0.876298 + 1.75984i) q^{68} +(-0.0922684 - 0.995734i) q^{72} +(0.172075 - 0.0666624i) q^{73} +(-0.397365 + 0.798017i) q^{74} +(0.353470 + 1.89090i) q^{80} +(-0.982973 - 0.183750i) q^{81} +(-0.942485 - 0.469302i) q^{82} +(-3.63743 + 1.03494i) q^{85} +(1.20614 - 0.600584i) q^{89} +(1.91545 + 0.177492i) q^{90} +(0.932472 - 0.361242i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - q^{4} - q^{8} + q^{9}+O(q^{10}) \) 16 * q - q^2 - q^4 - q^8 + q^9	\( 16 q - q^{2} - q^{4} - q^{8} + q^{9} - q^{16} - 15 q^{17} + q^{18} - q^{25} - q^{32} + 2 q^{34} + q^{36} + 2 q^{37} - q^{49} - q^{50} + 2 q^{61} - q^{64} + 2 q^{68} + q^{72} - 2 q^{73} - 15 q^{74} - q^{81} - q^{98}+O(q^{100}) \) 16 * q - q^2 - q^4 - q^8 + q^9 - q^16 - 15 * q^17 + q^18 - q^25 - q^32 + 2 * q^34 + q^36 + 2 * q^37 - q^49 - q^50 + 2 * q^61 - q^64 + 2 * q^68 + q^72 - 2 * q^73 - 15 * q^74 - q^81 - q^98

Introduction

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