LMFDB - Embedded newform 3040.1.cn.a.1709.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,1,Mod(189,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 3, 4, 4]))

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

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

chi := DirichletCharacter("3040.189");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3040.cn (of order $8$, degree $4$, 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:	$1.51715763840$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$8$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{64})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + 1 $ x^32 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{32}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{32} + \cdots)$

Embedding invariants

Embedding label			1709.5
Root			$0.956940 + 0.290285i$ of defining polynomial
Character	$\chi$	$=$	3040.1709
Dual form			3040.1.cn.a.2469.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0980171 - 0.995185i) q^{2} +(0.485544 - 1.17221i) q^{3} +(-0.980785 - 0.195090i) q^{4} +(0.923880 - 0.382683i) q^{5} +(-1.11897 - 0.598102i) q^{6} +(-0.290285 + 0.956940i) q^{8} +(-0.431207 - 0.431207i) q^{9} +O(q^{10})$	\(q+(0.0980171 - 0.995185i) q^{2} +(0.485544 - 1.17221i) q^{3} +(-0.980785 - 0.195090i) q^{4} +(0.923880 - 0.382683i) q^{5} +(-1.11897 - 0.598102i) q^{6} +(-0.290285 + 0.956940i) q^{8} +(-0.431207 - 0.431207i) q^{9} +(-0.290285 - 0.956940i) q^{10} +(-0.636379 - 1.53636i) q^{11} +(-0.704900 + 1.05496i) q^{12} +(1.62958 + 0.674993i) q^{13} -1.26879i q^{15} +(0.923880 + 0.382683i) q^{16} +(-0.471397 + 0.386865i) q^{18} +(-0.923880 - 0.382683i) q^{19} +(-0.980785 + 0.195090i) q^{20} +(-1.59133 + 0.482726i) q^{22} +(0.980785 + 0.804910i) q^{24} +(0.707107 - 0.707107i) q^{25} +(0.831470 - 1.55557i) q^{26} +(0.457372 - 0.189450i) q^{27} +(-1.26268 - 0.124363i) q^{30} +(0.471397 - 0.881921i) q^{32} -2.10991 q^{33} +(0.338797 + 0.507046i) q^{36} +(-1.83886 + 0.761681i) q^{37} +(-0.471397 + 0.881921i) q^{38} +(1.58246 - 1.58246i) q^{39} +(0.0980171 + 0.995185i) q^{40} +(0.324423 + 1.63099i) q^{44} +(-0.563400 - 0.233368i) q^{45} +(0.897168 - 0.897168i) q^{48} -1.00000i q^{49} +(-0.634393 - 0.773010i) q^{50} +(-1.46658 - 0.979938i) q^{52} +(0.591637 + 1.42834i) q^{53} +(-0.143707 - 0.473739i) q^{54} +(-1.17588 - 1.17588i) q^{55} +(-0.897168 + 0.897168i) q^{57} +(-0.247528 + 1.24441i) q^{60} +(-0.425215 + 1.02656i) q^{61} +(-0.831470 - 0.555570i) q^{64} +1.76384 q^{65} +(-0.206808 + 2.09976i) q^{66} +(-0.732410 + 1.76820i) q^{67} +(0.537813 - 0.287467i) q^{72} +(0.577774 + 1.90466i) q^{74} +(-0.485544 - 1.17221i) q^{75} +(0.831470 + 0.555570i) q^{76} +(-1.41973 - 1.72995i) q^{78} +1.00000 q^{80} -1.23794i q^{81} +(1.65493 - 0.162997i) q^{88} +(-0.287467 + 0.537813i) q^{90} -1.00000 q^{95} +(-0.804910 - 0.980785i) q^{96} -0.942793 q^{97} +(-0.995185 - 0.0980171i) q^{98} +(-0.388076 + 0.936899i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) $ 32 * q - 32 * q^66 + 32 * q^80 - 32 * q^95 - 32 * q^96 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$.

$n$	$191$	$1217$	$1921$	$2661$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{7}{8}\right)$

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 $

$2$	0.0980171	−	0.995185i	0.0980171	−	0.995185i
$2$	0.0980171	−	0.995185i	0.0980171	−	0.995185i
$3$	0.485544	−	1.17221i	0.485544	−	1.17221i	−0.471397	−	0.881921i	$-0.656250\pi$
$3$	0.485544	−	1.17221i	0.485544	−	1.17221i	0.956940	−	0.290285i	$-0.0937500\pi$
$4$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$5$	0.923880	−	0.382683i	0.923880	−	0.382683i
$5$	0.923880	−	0.382683i	0.923880	−	0.382683i
$6$	−1.11897	−	0.598102i	−1.11897	−	0.598102i
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−0.290285	+	0.956940i	−0.290285	+	0.956940i
$9$	−0.431207	−	0.431207i	−0.431207	−	0.431207i
$10$	−0.290285	−	0.956940i	−0.290285	−	0.956940i
$11$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	−0.831470	−	0.555570i	$-0.812500\pi$
$11$	−0.636379	−	1.53636i	−0.636379	−	1.53636i	0.195090	−	0.980785i	$-0.437500\pi$
$12$	−0.704900	+	1.05496i	−0.704900	+	1.05496i
$13$	1.62958	+	0.674993i	1.62958	+	0.674993i	0.995185	−	0.0980171i	$-0.0312500\pi$
$13$	1.62958	+	0.674993i	1.62958	+	0.674993i	0.634393	+	0.773010i	$0.281250\pi$
$14$		0			0
$15$		−	1.26879i		−	1.26879i
$16$	0.923880	+	0.382683i	0.923880	+	0.382683i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.471397	+	0.386865i	−0.471397	+	0.386865i
$19$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$19$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$20$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$21$		0			0
$22$	−1.59133	+	0.482726i	−1.59133	+	0.482726i
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$	0.980785	+	0.804910i	0.980785	+	0.804910i
$25$	0.707107	−	0.707107i	0.707107	−	0.707107i
$26$	0.831470	−	1.55557i	0.831470	−	1.55557i
$27$	0.457372	−	0.189450i	0.457372	−	0.189450i
$28$		0			0
$29$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$29$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$30$	−1.26268	−	0.124363i	−1.26268	−	0.124363i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.471397	−	0.881921i	0.471397	−	0.881921i
$33$	−2.10991			−2.10991
$34$		0			0
$35$		0			0
$36$	0.338797	+	0.507046i	0.338797	+	0.507046i
$37$	−1.83886	+	0.761681i	−1.83886	+	0.761681i	−0.881921	+	0.471397i	$0.843750\pi$
$37$	−1.83886	+	0.761681i	−1.83886	+	0.761681i	−0.956940	+	0.290285i	$0.906250\pi$
$38$	−0.471397	+	0.881921i	−0.471397	+	0.881921i
$39$	1.58246	−	1.58246i	1.58246	−	1.58246i
$40$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$43$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$44$	0.324423	+	1.63099i	0.324423	+	1.63099i
$45$	−0.563400	−	0.233368i	−0.563400	−	0.233368i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	0.897168	−	0.897168i	0.897168	−	0.897168i
$49$		−	1.00000i		−	1.00000i
$50$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$51$		0			0
$52$	−1.46658	−	0.979938i	−1.46658	−	0.979938i
$53$	0.591637	+	1.42834i	0.591637	+	1.42834i	0.881921	+	0.471397i	$0.156250\pi$
$53$	0.591637	+	1.42834i	0.591637	+	1.42834i	−0.290285	+	0.956940i	$0.593750\pi$
$54$	−0.143707	−	0.473739i	−0.143707	−	0.473739i
$55$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$56$		0			0
$57$	−0.897168	+	0.897168i	−0.897168	+	0.897168i
$58$		0			0
$59$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$59$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$60$	−0.247528	+	1.24441i	−0.247528	+	1.24441i
$61$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$61$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$62$		0			0
$63$		0			0
$64$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$65$	1.76384			1.76384
$66$	−0.206808	+	2.09976i	−0.206808	+	2.09976i
$67$	−0.732410	+	1.76820i	−0.732410	+	1.76820i	−0.0980171	+	0.995185i	$0.531250\pi$
$67$	−0.732410	+	1.76820i	−0.732410	+	1.76820i	−0.634393	+	0.773010i	$0.718750\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$71$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$72$	0.537813	−	0.287467i	0.537813	−	0.287467i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$	0.577774	+	1.90466i	0.577774	+	1.90466i
$75$	−0.485544	−	1.17221i	−0.485544	−	1.17221i
$76$	0.831470	+	0.555570i	0.831470	+	0.555570i
$77$		0			0
$78$	−1.41973	−	1.72995i	−1.41973	−	1.72995i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	1.00000			1.00000
$81$		−	1.23794i		−	1.23794i
$82$		0			0
$83$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$83$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.65493	−	0.162997i	1.65493	−	0.162997i
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$	−0.287467	+	0.537813i	−0.287467	+	0.537813i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000			−1.00000
$96$	−0.804910	−	0.980785i	−0.804910	−	0.980785i
$97$	−0.942793			−0.942793			−0.471397	−	0.881921i	$-0.656250\pi$
$97$	−0.942793			−0.942793			−0.471397	+	0.881921i	$0.656250\pi$
$98$	−0.995185	−	0.0980171i	−0.995185	−	0.0980171i
$99$	−0.388076	+	0.936899i	−0.388076	+	0.936899i
$100$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$101$	1.53636	−	0.636379i	1.53636	−	0.636379i	0.555570	−	0.831470i	$-0.312500\pi$
$101$	1.53636	−	0.636379i	1.53636	−	0.636379i	0.980785	+	0.195090i	$0.0625000\pi$
$102$		0			0
$103$	0.138617	−	0.138617i	0.138617	−	0.138617i	−0.634393	−	0.773010i	$-0.718750\pi$
$103$	0.138617	−	0.138617i	0.138617	−	0.138617i	0.773010	+	0.634393i	$0.218750\pi$
$104$	−1.11897	+	1.36347i	−1.11897	+	1.36347i
$105$		0			0
$106$	1.47945	−	0.448786i	1.47945	−	0.448786i
$107$	0.222174	+	0.536376i	0.222174	+	0.536376i	0.995185	−	0.0980171i	$-0.0312500\pi$
$107$	0.222174	+	0.536376i	0.222174	+	0.536376i	−0.773010	+	0.634393i	$0.781250\pi$
$108$	−0.485544	+	0.0965806i	−0.485544	+	0.0965806i
$109$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$109$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$110$	−1.28547	+	1.05496i	−1.28547	+	1.05496i
$111$			2.52535i			2.52535i
$112$		0			0
$113$		−	1.54602i		−	1.54602i	−0.634393	−	0.773010i	$-0.718750\pi$
$113$		−	1.54602i		−	1.54602i	0.634393	−	0.773010i	$-0.281250\pi$
$114$	0.804910	+	0.980785i	0.804910	+	0.980785i
$115$		0			0
$116$		0			0
$117$	−0.411624	−	0.993748i	−0.411624	−	0.993748i
$118$		0			0
$119$		0			0
$120$	1.21415	+	0.368309i	1.21415	+	0.368309i
$121$	−1.24830	+	1.24830i	−1.24830	+	1.24830i
$122$	0.979938	+	0.523788i	0.979938	+	0.523788i
$123$		0			0
$124$		0			0
$125$	0.382683	−	0.923880i	0.382683	−	0.923880i
$126$		0			0
$127$	1.91388			1.91388			0.956940	−	0.290285i	$-0.0937500\pi$
$127$	1.91388			1.91388			0.956940	+	0.290285i	$0.0937500\pi$
$128$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$129$		0			0
$130$	0.172887	−	1.75535i	0.172887	−	1.75535i
$131$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$131$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$132$	2.06937	+	0.411624i	2.06937	+	0.411624i
$133$		0			0
$134$	1.68789	+	0.902197i	1.68789	+	0.902197i
$135$	0.350057	−	0.350057i	0.350057	−	0.350057i
$136$		0			0
$137$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$137$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$138$		0			0
$139$	0.750661	+	1.81225i	0.750661	+	1.81225i	0.555570	+	0.831470i	$0.312500\pi$
$139$	0.750661	+	1.81225i	0.750661	+	1.81225i	0.195090	+	0.980785i	$0.437500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.93316i		−	2.93316i
$144$	−0.233368	−	0.563400i	−0.233368	−	0.563400i
$145$		0			0
$146$		0			0
$147$	−1.17221	−	0.485544i	−1.17221	−	0.485544i
$148$	1.95213	−	0.388302i	1.95213	−	0.388302i
$149$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.555570	−	0.831470i	$-0.687500\pi$
$149$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.195090	−	0.980785i	$-0.562500\pi$
$150$	−1.21415	+	0.368309i	−1.21415	+	0.368309i
$151$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$151$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$152$	0.634393	−	0.773010i	0.634393	−	0.773010i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.86078	+	1.24333i	−1.86078	+	1.24333i
$157$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$157$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$158$		0			0
$159$	1.96157			1.96157
$160$	0.0980171	−	0.995185i	0.0980171	−	0.995185i
$161$		0			0
$162$	−1.23198	−	0.121339i	−1.23198	−	0.121339i
$163$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$163$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$164$		0			0
$165$	−1.94931	+	0.807429i	−1.94931	+	0.807429i
$166$		0			0
$167$	−0.410525	+	0.410525i	−0.410525	+	0.410525i	−0.881921	−	0.471397i	$-0.843750\pi$
$167$	−0.410525	+	0.410525i	−0.410525	+	0.410525i	0.471397	+	0.881921i	$0.343750\pi$
$168$		0			0
$169$	1.49280	+	1.49280i	1.49280	+	1.49280i
$170$		0			0
$171$	0.233368	+	0.563400i	0.233368	+	0.563400i
$172$		0			0
$173$	1.76820	+	0.732410i	1.76820	+	0.732410i	0.995185	+	0.0980171i	$0.0312500\pi$
$173$	1.76820	+	0.732410i	1.76820	+	0.732410i	0.773010	+	0.634393i	$0.218750\pi$
$174$		0			0
$175$		0			0
$176$		−	1.66294i		−	1.66294i
$177$		0			0
$178$		0			0
$179$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$179$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$180$	0.507046	+	0.338797i	0.507046	+	0.338797i
$181$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$181$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$182$		0			0
$183$	0.996879	+	0.996879i	0.996879	+	0.996879i
$184$		0			0
$185$	−1.40740	+	1.40740i	−1.40740	+	1.40740i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$191$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$191$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$192$	−1.05496	+	0.704900i	−1.05496	+	0.704900i
$193$	0.580569			0.580569			0.290285	−	0.956940i	$-0.406250\pi$
$193$	0.580569			0.580569			0.290285	+	0.956940i	$0.406250\pi$
$194$	−0.0924099	+	0.938254i	−0.0924099	+	0.938254i
$195$	0.856422	−	2.06759i	0.856422	−	2.06759i
$196$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$197$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$197$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$198$	0.894350	+	0.478040i	0.894350	+	0.478040i
$199$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$199$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$200$	0.471397	+	0.881921i	0.471397	+	0.881921i
$201$	1.71707	+	1.71707i	1.71707	+	1.71707i
$202$	−0.482726	−	1.59133i	−0.482726	−	1.59133i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.124363	−	0.151537i	−0.124363	−	0.151537i
$207$		0			0
$208$	1.24723	+	1.24723i	1.24723	+	1.24723i
$209$			1.66294i			1.66294i
$210$		0			0
$211$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$211$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$212$	−0.301614	−	1.51631i	−0.301614	−	1.51631i
$213$		0			0
$214$	0.555570	−	0.168530i	0.555570	−	0.168530i
$215$		0			0
$216$	0.0485240	+	0.492672i	0.0485240	+	0.492672i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.923880	+	1.38268i	0.923880	+	1.38268i
$221$		0			0
$222$	2.51319	+	0.247528i	2.51319	+	0.247528i
$223$	0.942793			0.942793			0.471397	−	0.881921i	$-0.343750\pi$
$223$	0.942793			0.942793			0.471397	+	0.881921i	$0.343750\pi$
$224$		0			0
$225$	−0.609819			−0.609819
$226$	−1.53858	−	0.151537i	−1.53858	−	0.151537i
$227$	−0.360791	+	0.871028i	−0.360791	+	0.871028i	0.634393	+	0.773010i	$0.281250\pi$
$227$	−0.360791	+	0.871028i	−0.360791	+	0.871028i	−0.995185	+	0.0980171i	$0.968750\pi$
$228$	1.05496	−	0.704900i	1.05496	−	0.704900i
$229$	−1.81225	+	0.750661i	−1.81225	+	0.750661i	−0.831470	+	0.555570i	$0.812500\pi$
$229$	−1.81225	+	0.750661i	−1.81225	+	0.750661i	−0.980785	+	0.195090i	$0.937500\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$	−1.02931	+	0.312238i	−1.02931	+	0.312238i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$239$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$240$	0.485544	−	1.17221i	0.485544	−	1.17221i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	1.11994	+	1.36465i	1.11994	+	1.36465i
$243$	−0.993748	−	0.411624i	−0.993748	−	0.411624i
$244$	0.617317	−	0.923880i	0.617317	−	0.923880i
$245$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$246$		0			0
$247$	−1.24723	−	1.24723i	−1.24723	−	1.24723i
$248$		0			0
$249$		0			0
$250$	−0.881921	−	0.471397i	−0.881921	−	0.471397i
$251$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$251$	1.30656	−	0.541196i	1.30656	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$252$		0			0
$253$		0			0
$254$	0.187593	−	1.90466i	0.187593	−	1.90466i
$255$		0			0
$256$	0.707107	+	0.707107i	0.707107	+	0.707107i
$257$	−0.580569			−0.580569			−0.290285	−	0.956940i	$-0.593750\pi$
$257$	−0.580569			−0.580569			−0.290285	+	0.956940i	$0.593750\pi$
$258$		0			0
$259$		0			0
$260$	−1.72995	−	0.344109i	−1.72995	−	0.344109i
$261$		0			0
$262$	1.62958	+	0.871028i	1.62958	+	0.871028i
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$	0.612476	−	2.01906i	0.612476	−	2.01906i
$265$	1.09320	+	1.09320i	1.09320	+	1.09320i
$266$		0			0
$267$		0			0
$268$	1.06330	−	1.59133i	1.06330	−	1.59133i
$269$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$269$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$270$	−0.314060	−	0.382683i	−0.314060	−	0.382683i
$271$		−	1.96157i		−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$271$		−	1.96157i		−	1.96157i	0.195090	−	0.980785i	$-0.437500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.53636	−	0.636379i	−1.53636	−	0.636379i
$276$		0			0
$277$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$277$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$278$	1.87711	−	0.569414i	1.87711	−	0.569414i
$279$		0			0
$280$		0			0
$281$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$281$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$283$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$284$		0			0
$285$	−0.485544	+	1.17221i	−0.485544	+	1.17221i
$286$	−2.91904	−	0.287500i	−2.91904	−	0.287500i
$287$		0			0
$288$	−0.583561	+	0.177021i	−0.583561	+	0.177021i
$289$	−1.00000			−1.00000
$290$		0			0
$291$	−0.457767	+	1.10515i	−0.457767	+	1.10515i
$292$		0			0
$293$	0.181112	−	0.0750191i	0.181112	−	0.0750191i	−0.290285	−	0.956940i	$-0.593750\pi$
$293$	0.181112	−	0.0750191i	0.181112	−	0.0750191i	0.471397	+	0.881921i	$0.343750\pi$
$294$	−0.598102	+	1.11897i	−0.598102	+	1.11897i
$295$		0			0
$296$	−0.195090	−	1.98079i	−0.195090	−	1.98079i
$297$	−0.582124	−	0.582124i	−0.582124	−	0.582124i
$298$	−1.87711	+	0.569414i	−1.87711	+	0.569414i
$299$		0			0
$300$	0.247528	+	1.24441i	0.247528	+	1.24441i
$301$		0			0
$302$		0			0
$303$		−	2.10991i		−	2.10991i
$304$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$305$			1.11114i			1.11114i
$306$		0			0
$307$	−0.871028	−	0.360791i	−0.871028	−	0.360791i	−0.0980171	−	0.995185i	$-0.531250\pi$
$307$	−0.871028	−	0.360791i	−0.871028	−	0.360791i	−0.773010	+	0.634393i	$0.781250\pi$
$308$		0			0
$309$	−0.0951832	−	0.229793i	−0.0951832	−	0.229793i
$310$		0			0
$311$	0.785695	+	0.785695i	0.785695	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$311$	0.785695	+	0.785695i	0.785695	+	0.785695i	−0.195090	+	0.980785i	$0.562500\pi$
$312$	1.05496	+	1.97369i	1.05496	+	1.97369i
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.222174	+	0.536376i	−0.222174	+	0.536376i	−0.995185	−	0.0980171i	$-0.968750\pi$
$317$	−0.222174	+	0.536376i	−0.222174	+	0.536376i	0.773010	+	0.634393i	$0.218750\pi$
$318$	0.192268	−	1.95213i	0.192268	−	1.95213i
$319$		0			0
$320$	−0.980785	−	0.195090i	−0.980785	−	0.195090i
$321$	0.736619			0.736619
$322$		0			0
$323$		0			0
$324$	−0.241510	+	1.21415i	−0.241510	+	1.21415i
$325$	1.62958	−	0.674993i	1.62958	−	0.674993i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	0.612476	+	2.01906i	0.612476	+	2.01906i
$331$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$331$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$332$		0			0
$333$	1.12137	+	0.464488i	1.12137	+	0.464488i
$334$	0.368309	+	0.448786i	0.368309	+	0.448786i
$335$			1.91388i			1.91388i
$336$		0			0
$337$		−	1.99037i		−	1.99037i	−0.0980171	−	0.995185i	$-0.531250\pi$
$337$		−	1.99037i		−	1.99037i	0.0980171	−	0.995185i	$-0.468750\pi$
$338$	1.63193	−	1.33929i	1.63193	−	1.33929i
$339$	−1.81225	−	0.750661i	−1.81225	−	0.750661i
$340$		0			0
$341$		0			0
$342$	0.583561	−	0.177021i	0.583561	−	0.177021i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.902197	−	1.68789i	0.902197	−	1.68789i
$347$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$347$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$348$		0			0
$349$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$349$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$350$		0			0
$351$	0.873201			0.873201
$352$	−1.65493	−	0.162997i	−1.65493	−	0.162997i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.555570	−	0.831470i	$-0.312500\pi$
$359$	1.38704	−	1.38704i	1.38704	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$360$	0.386865	−	0.471397i	0.386865	−	0.471397i
$361$	0.707107	+	0.707107i	0.707107	+	0.707107i
$362$		0			0
$363$	0.857163	+	2.06937i	0.857163	+	2.06937i
$364$		0			0
$365$		0			0
$366$	1.08979	−	0.894368i	1.08979	−	0.894368i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	1.26268	+	1.53858i	1.26268	+	1.53858i
$371$		0			0
$372$		0			0
$373$	−0.485544	−	1.17221i	−0.485544	−	1.17221i	−0.956940	−	0.290285i	$-0.906250\pi$
$373$	−0.485544	−	1.17221i	−0.485544	−	1.17221i	0.471397	−	0.881921i	$-0.343750\pi$
$374$		0			0
$375$	−0.897168	−	0.897168i	−0.897168	−	0.897168i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$379$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$380$	0.980785	+	0.195090i	0.980785	+	0.195090i
$381$	0.929273	−	2.24346i	0.929273	−	2.24346i
$382$	−0.0750191	+	0.761681i	−0.0750191	+	0.761681i
$383$	−0.196034			−0.196034			−0.0980171	−	0.995185i	$-0.531250\pi$
$383$	−0.196034			−0.196034			−0.0980171	+	0.995185i	$0.531250\pi$
$384$	0.598102	+	1.11897i	0.598102	+	1.11897i
$385$		0			0
$386$	0.0569057	−	0.577774i	0.0569057	−	0.577774i
$387$		0			0
$388$	0.924678	+	0.183930i	0.924678	+	0.183930i
$389$	1.70711	−	0.707107i	1.70711	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$389$	1.70711	−	0.707107i	1.70711	−	0.707107i	1.00000			$0$
$390$	−1.97369	−	1.05496i	−1.97369	−	1.05496i
$391$		0			0
$392$	0.956940	+	0.290285i	0.956940	+	0.290285i
$393$	1.65775	+	1.65775i	1.65775	+	1.65775i
$394$		0			0
$395$		0			0
$396$	0.563400	−	0.843187i	0.563400	−	0.843187i
$397$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$397$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$398$	0.485544	+	0.591637i	0.485544	+	0.591637i
$399$		0			0
$400$	0.923880	−	0.382683i	0.923880	−	0.382683i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$	1.87711	−	1.54050i	1.87711	−	1.54050i
$403$		0			0
$404$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$405$	−0.473739	−	1.14371i	−0.473739	−	1.14371i
$406$		0			0
$407$	2.34043	+	2.34043i	2.34043	+	2.34043i
$408$		0			0
$409$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$409$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$410$		0			0
$411$		0			0
$412$	−0.162997	+	0.108911i	−0.162997	+	0.108911i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.36347	−	1.11897i	1.36347	−	1.11897i
$417$	2.48881			2.48881
$418$	1.65493	+	0.162997i	1.65493	+	0.162997i
$419$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$419$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$420$		0			0
$421$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$421$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$422$		0			0
$423$		0			0
$424$	−1.53858	+	0.151537i	−1.53858	+	0.151537i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.113263	−	0.569414i	−0.113263	−	0.569414i
$429$	−3.43827	−	1.42418i	−3.43827	−	1.42418i
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	0.495056			0.495056
$433$			1.26879i			1.26879i	0.773010	+	0.634393i	$0.218750\pi$
$433$			1.26879i			1.26879i	−0.773010	+	0.634393i	$0.781250\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$439$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$440$	1.46658	−	0.783904i	1.46658	−	0.783904i
$441$	−0.431207	+	0.431207i	−0.431207	+	0.431207i
$442$		0			0
$443$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$443$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$444$	0.492672	−	2.47683i	0.492672	−	2.47683i
$445$		0			0
$446$	0.0924099	−	0.938254i	0.0924099	−	0.938254i
$447$	−2.48881			−2.48881
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.0597727	+	0.606883i	−0.0597727	+	0.606883i
$451$		0			0
$452$	−0.301614	+	1.51631i	−0.301614	+	1.51631i
$453$		0			0
$454$	0.831470	+	0.444430i	0.831470	+	0.444430i
$455$		0			0
$456$	−0.598102	−	1.11897i	−0.598102	−	1.11897i
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$	0.569414	+	1.87711i	0.569414	+	1.87711i
$459$		0			0
$460$		0			0
$461$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$461$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$467$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$468$	0.209844	+	1.05496i	0.209844	+	1.05496i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$476$		0			0
$477$	0.360791	−	0.871028i	0.360791	−	0.871028i
$478$	−1.40740	−	0.138617i	−1.40740	−	0.138617i
$479$	1.66294			1.66294			0.831470	−	0.555570i	$-0.187500\pi$
$479$	1.66294			1.66294			0.831470	+	0.555570i	$0.187500\pi$
$480$	−1.11897	−	0.598102i	−1.11897	−	0.598102i
$481$	−3.51070			−3.51070
$482$		0			0
$483$		0			0
$484$	1.46785	−	0.980785i	1.46785	−	0.980785i
$485$	−0.871028	+	0.360791i	−0.871028	+	0.360791i
$486$	−0.507046	+	0.948617i	−0.507046	+	0.948617i
$487$	−1.09320	+	1.09320i	−1.09320	+	1.09320i	−0.0980171	+	0.995185i	$0.531250\pi$
$487$	−1.09320	+	1.09320i	−1.09320	+	1.09320i	−0.995185	+	0.0980171i	$0.968750\pi$
$488$	−0.858923	−	0.704900i	−0.858923	−	0.704900i
$489$		0			0
$490$	−0.956940	+	0.290285i	−0.956940	+	0.290285i
$491$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$491$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$492$		0			0
$493$		0			0
$494$	−1.36347	+	1.11897i	−1.36347	+	1.11897i
$495$			1.01409i			1.01409i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.360480	+	0.149316i	0.360480	+	0.149316i	0.555570	−	0.831470i	$-0.312500\pi$
$499$	0.360480	+	0.149316i	0.360480	+	0.149316i	−0.195090	+	0.980785i	$0.562500\pi$
$500$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$501$	0.281892	+	0.680547i	0.281892	+	0.680547i
$502$	−0.410525	−	1.35332i	−0.410525	−	1.35332i
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$	1.17588	−	1.17588i	1.17588	−	1.17588i
$506$		0			0
$507$	2.47469	−	1.02505i	2.47469	−	1.02505i
$508$	−1.87711	−	0.373380i	−1.87711	−	0.373380i
$509$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$509$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$510$		0			0
$511$		0			0
$512$	0.773010	−	0.634393i	0.773010	−	0.634393i
$513$	−0.495056			−0.495056
$514$	−0.0569057	+	0.577774i	−0.0569057	+	0.577774i
$515$	0.0750191	−	0.181112i	0.0750191	−	0.181112i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.71707	−	1.71707i	1.71707	−	1.71707i
$520$	−0.512016	+	1.68789i	−0.512016	+	1.68789i
$521$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$521$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$522$		0			0
$523$	0.0750191	+	0.181112i	0.0750191	+	0.181112i	0.956940	−	0.290285i	$-0.0937500\pi$
$523$	0.0750191	+	0.181112i	0.0750191	+	0.181112i	−0.881921	+	0.471397i	$0.843750\pi$
$524$	1.02656	−	1.53636i	1.02656	−	1.53636i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.94931	−	0.807429i	−1.94931	−	0.807429i
$529$			1.00000i			1.00000i
$530$	1.19509	−	0.980785i	1.19509	−	0.980785i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.410525	+	0.410525i	0.410525	+	0.410525i
$536$	−1.47945	−	1.21415i	−1.47945	−	1.21415i
$537$		0			0
$538$		0			0
$539$	−1.53636	+	0.636379i	−1.53636	+	0.636379i
$540$	−0.411624	+	0.275038i	−0.411624	+	0.275038i
$541$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	−0.980785	−	0.195090i	$-0.937500\pi$
$541$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	0.831470	+	0.555570i	$0.187500\pi$
$542$	−1.95213	−	0.192268i	−1.95213	−	0.192268i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.761681	−	1.83886i	0.761681	−	1.83886i	0.290285	−	0.956940i	$-0.406250\pi$
$547$	0.761681	−	1.83886i	0.761681	−	1.83886i	0.471397	−	0.881921i	$-0.343750\pi$
$548$		0			0
$549$	0.626016	−	0.259304i	0.626016	−	0.259304i
$550$	−0.783904	+	1.46658i	−0.783904	+	1.46658i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.966411	+	2.33312i	0.966411	+	2.33312i
$556$	−0.382683	−	1.92388i	−0.382683	−	1.92388i
$557$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$557$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.83886	−	0.761681i	−1.83886	−	0.761681i	−0.956940	−	0.290285i	$-0.906250\pi$
$563$	−1.83886	−	0.761681i	−1.83886	−	0.761681i	−0.881921	−	0.471397i	$-0.843750\pi$
$564$		0			0
$565$	−0.591637	−	1.42834i	−0.591637	−	1.42834i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$569$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$570$	1.11897	+	0.598102i	1.11897	+	0.598102i
$571$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	−0.555570	−	0.831470i	$-0.687500\pi$
$571$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	0.195090	+	0.980785i	$0.437500\pi$
$572$	−0.572232	+	2.87680i	−0.572232	+	2.87680i
$573$	−0.371619	+	0.897168i	−0.371619	+	0.897168i
$574$		0			0
$575$		0			0
$576$	0.118970	+	0.598102i	0.118970	+	0.598102i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.0980171	+	0.995185i	−0.0980171	+	0.995185i
$579$	0.281892	−	0.680547i	0.281892	−	0.680547i
$580$		0			0
$581$		0			0
$582$	1.05496	+	0.563887i	1.05496	+	0.563887i
$583$	1.81793	−	1.81793i	1.81793	−	1.81793i
$584$		0			0
$585$	−0.760582	−	0.760582i	−0.760582	−	0.760582i
$586$	−0.0569057	−	0.187593i	−0.0569057	−	0.187593i
$587$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$587$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$588$	1.05496	+	0.704900i	1.05496	+	0.704900i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.99037			−1.99037
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	−0.636379	+	0.522263i	−0.636379	+	0.522263i
$595$		0			0
$596$	0.382683	+	1.92388i	0.382683	+	1.92388i
$597$	0.371619	+	0.897168i	0.371619	+	0.897168i
$598$		0			0
$599$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$599$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$600$	1.26268	−	0.124363i	1.26268	−	0.124363i
$601$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$601$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$	1.07828	−	0.446638i	1.07828	−	0.446638i
$604$		0			0
$605$	−0.675577	+	1.63099i	−0.675577	+	1.63099i
$606$	−2.09976	−	0.206808i	−2.09976	−	0.206808i
$607$	−1.54602			−1.54602			−0.773010	−	0.634393i	$-0.781250\pi$
$607$	−1.54602			−1.54602			−0.773010	+	0.634393i	$0.781250\pi$
$608$	−0.773010	+	0.634393i	−0.773010	+	0.634393i
$609$		0			0
$610$	1.10579	+	0.108911i	1.10579	+	0.108911i
$611$		0			0
$612$		0			0
$613$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$613$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$614$	−0.444430	+	0.831470i	−0.444430	+	0.831470i
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$	−0.238016	+	0.0722012i	−0.238016	+	0.0722012i
$619$	−0.425215	−	1.02656i	−0.425215	−	1.02656i	−0.980785	−	0.195090i	$-0.937500\pi$
$619$	−0.425215	−	1.02656i	−0.425215	−	1.02656i	0.555570	−	0.831470i	$-0.312500\pi$
$620$		0			0
$621$		0			0
$622$	0.858923	−	0.704900i	0.858923	−	0.704900i
$623$		0			0
$624$	2.06759	−	0.856422i	2.06759	−	0.856422i
$625$		−	1.00000i		−	1.00000i
$626$		0			0
$627$	1.94931	+	0.807429i	1.94931	+	0.807429i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.275899	+	0.275899i	0.275899	+	0.275899i	0.831470	−	0.555570i	$-0.187500\pi$
$631$	0.275899	+	0.275899i	0.275899	+	0.275899i	−0.555570	+	0.831470i	$0.687500\pi$
$632$		0			0
$633$		0			0
$634$	0.512016	+	0.273678i	0.512016	+	0.273678i
$635$	1.76820	−	0.732410i	1.76820	−	0.732410i
$636$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$637$	0.674993	−	1.62958i	0.674993	−	1.62958i
$638$		0			0
$639$		0			0
$640$	−0.290285	+	0.956940i	−0.290285	+	0.956940i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	0.0722012	−	0.733072i	0.0722012	−	0.733072i
$643$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$643$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	1.18463	+	0.359355i	1.18463	+	0.359355i
$649$		0			0
$650$	−0.512016	−	1.68789i	−0.512016	−	1.68789i
$651$		0			0
$652$		0			0
$653$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$653$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$654$		0			0
$655$			1.84776i			1.84776i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$659$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$660$	2.06937	−	0.411624i	2.06937	−	0.411624i
$661$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$661$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.572165	−	1.07045i	0.572165	−	1.07045i
$667$		0			0
$668$	0.482726	−	0.322547i	0.482726	−	0.322547i
$669$	0.457767	−	1.10515i	0.457767	−	1.10515i
$670$	1.90466	+	0.187593i	1.90466	+	0.187593i
$671$	1.84776			1.84776
$672$		0			0
$673$	−1.99037			−1.99037			−0.995185	−	0.0980171i	$-0.968750\pi$
$673$	−1.99037			−1.99037			−0.995185	+	0.0980171i	$0.968750\pi$
$674$	−1.98079	−	0.195090i	−1.98079	−	0.195090i
$675$	0.189450	−	0.457372i	0.189450	−	0.457372i
$676$	−1.17289	−	1.75535i	−1.17289	−	1.75535i
$677$	1.83886	−	0.761681i	1.83886	−	0.761681i	0.881921	−	0.471397i	$-0.156250\pi$
$677$	1.83886	−	0.761681i	1.83886	−	0.761681i	0.956940	−	0.290285i	$-0.0937500\pi$
$678$	−0.924678	+	1.72995i	−0.924678	+	1.72995i
$679$		0			0
$680$		0			0
$681$	0.845844	+	0.845844i	0.845844	+	0.845844i
$682$		0			0
$683$	0.761681	+	1.83886i	0.761681	+	1.83886i	0.471397	+	0.881921i	$0.343750\pi$
$683$	0.761681	+	1.83886i	0.761681	+	1.83886i	0.290285	+	0.956940i	$0.406250\pi$
$684$	−0.118970	−	0.598102i	−0.118970	−	0.598102i
$685$		0			0
$686$		0			0
$687$			2.48881i			2.48881i
$688$		0			0
$689$			2.72694i			2.72694i
$690$		0			0
$691$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.831470	−	0.555570i	$-0.187500\pi$
$691$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.195090	+	0.980785i	$0.437500\pi$
$692$	−1.59133	−	1.06330i	−1.59133	−	1.06330i
$693$		0			0
$694$		0			0
$695$	1.38704	+	1.38704i	1.38704	+	1.38704i
$696$		0			0
$697$		0			0
$698$	1.24723	+	0.666656i	1.24723	+	0.666656i
$699$		0			0
$700$		0			0
$701$	0.149316	−	0.360480i	0.149316	−	0.360480i	−0.831470	−	0.555570i	$-0.812500\pi$
$701$	0.149316	−	0.360480i	0.149316	−	0.360480i	0.980785	+	0.195090i	$0.0625000\pi$
$702$	0.0855886	−	0.868996i	0.0855886	−	0.868996i
$703$	1.99037			1.99037
$704$	−0.324423	+	1.63099i	−0.324423	+	1.63099i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$709$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−1.12247	−	2.70989i	−1.12247	−	2.70989i
$716$		0			0
$717$	−1.65775	−	0.686662i	−1.65775	−	0.686662i
$718$	−1.24441	−	1.51631i	−1.24441	−	1.51631i
$719$			0.390181i			0.390181i	0.980785	+	0.195090i	$0.0625000\pi$
$719$			0.390181i			0.390181i	−0.980785	+	0.195090i	$0.937500\pi$
$720$	−0.431207	−	0.431207i	−0.431207	−	0.431207i
$721$		0			0
$722$	0.773010	−	0.634393i	0.773010	−	0.634393i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	2.14343	−	0.650201i	2.14343	−	0.650201i
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	−0.0896606	+	0.0896606i	−0.0896606	+	0.0896606i
$730$		0			0
$731$		0			0
$732$	−0.783243	−	1.17221i	−0.783243	−	1.17221i
$733$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$733$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$734$		0			0
$735$	−1.26879			−1.26879
$736$		0			0
$737$	3.18267			3.18267
$738$		0			0
$739$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$739$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$740$	1.65493	−	1.10579i	1.65493	−	1.10579i
$741$	−2.06759	+	0.856422i	−2.06759	+	0.856422i
$742$		0			0
$743$	−1.24723	+	1.24723i	−1.24723	+	1.24723i	−0.290285	+	0.956940i	$0.593750\pi$
$743$	−1.24723	+	1.24723i	−1.24723	+	1.24723i	−0.956940	+	0.290285i	$0.906250\pi$
$744$		0			0
$745$	−1.38704	−	1.38704i	−1.38704	−	1.38704i
$746$	−1.21415	+	0.368309i	−1.21415	+	0.368309i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−0.980785	+	0.804910i	−0.980785	+	0.804910i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		−	1.79434i		−	1.79434i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$757$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$758$		0			0
$759$		0			0
$760$	0.290285	−	0.956940i	0.290285	−	0.956940i
$761$	−0.785695	+	0.785695i	−0.785695	+	0.785695i	−0.980785	−	0.195090i	$-0.937500\pi$
$761$	−0.785695	+	0.785695i	−0.785695	+	0.785695i	0.195090	+	0.980785i	$0.437500\pi$
$762$	−2.14157	−	1.14470i	−2.14157	−	1.14470i
$763$		0			0
$764$	0.750661	+	0.149316i	0.750661	+	0.149316i
$765$		0			0
$766$	−0.0192147	+	0.195090i	−0.0192147	+	0.195090i
$767$		0			0
$768$	1.17221	−	0.485544i	1.17221	−	0.485544i
$769$	1.96157			1.96157			0.980785	−	0.195090i	$-0.0625000\pi$
$769$	1.96157			1.96157			0.980785	+	0.195090i	$0.0625000\pi$
$770$		0			0
$771$	−0.281892	+	0.680547i	−0.281892	+	0.680547i
$772$	−0.569414	−	0.113263i	−0.569414	−	0.113263i
$773$	−0.871028	+	0.360791i	−0.871028	+	0.360791i	−0.773010	−	0.634393i	$-0.781250\pi$
$773$	−0.871028	+	0.360791i	−0.871028	+	0.360791i	−0.0980171	+	0.995185i	$0.531250\pi$
$774$		0			0
$775$		0			0
$776$	0.273678	−	0.902197i	0.273678	−	0.902197i
$777$		0			0
$778$	−0.536376	−	1.76820i	−0.536376	−	1.76820i
$779$		0			0
$780$	−1.24333	+	1.86078i	−1.24333	+	1.86078i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.382683	−	0.923880i	0.382683	−	0.923880i
$785$		0			0
$786$	1.81225	−	1.48728i	1.81225	−	1.48728i
$787$	−0.181112	−	0.0750191i	−0.181112	−	0.0750191i	0.290285	−	0.956940i	$-0.406250\pi$
$787$	−0.181112	−	0.0750191i	−0.181112	−	0.0750191i	−0.471397	+	0.881921i	$0.656250\pi$
$788$		0			0
$789$		0			0
$790$		0

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:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3040.cn (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0980171 - 0.995185i) q^{2} +(0.485544 - 1.17221i) q^{3} +(-0.980785 - 0.195090i) q^{4} +(0.923880 - 0.382683i) q^{5} +(-1.11897 - 0.598102i) q^{6} +(-0.290285 + 0.956940i) q^{8} +(-0.431207 - 0.431207i) q^{9} +O(q^{10})\)	\(q+(0.0980171 - 0.995185i) q^{2} +(0.485544 - 1.17221i) q^{3} +(-0.980785 - 0.195090i) q^{4} +(0.923880 - 0.382683i) q^{5} +(-1.11897 - 0.598102i) q^{6} +(-0.290285 + 0.956940i) q^{8} +(-0.431207 - 0.431207i) q^{9} +(-0.290285 - 0.956940i) q^{10} +(-0.636379 - 1.53636i) q^{11} +(-0.704900 + 1.05496i) q^{12} +(1.62958 + 0.674993i) q^{13} -1.26879i q^{15} +(0.923880 + 0.382683i) q^{16} +(-0.471397 + 0.386865i) q^{18} +(-0.923880 - 0.382683i) q^{19} +(-0.980785 + 0.195090i) q^{20} +(-1.59133 + 0.482726i) q^{22} +(0.980785 + 0.804910i) q^{24} +(0.707107 - 0.707107i) q^{25} +(0.831470 - 1.55557i) q^{26} +(0.457372 - 0.189450i) q^{27} +(-1.26268 - 0.124363i) q^{30} +(0.471397 - 0.881921i) q^{32} -2.10991 q^{33} +(0.338797 + 0.507046i) q^{36} +(-1.83886 + 0.761681i) q^{37} +(-0.471397 + 0.881921i) q^{38} +(1.58246 - 1.58246i) q^{39} +(0.0980171 + 0.995185i) q^{40} +(0.324423 + 1.63099i) q^{44} +(-0.563400 - 0.233368i) q^{45} +(0.897168 - 0.897168i) q^{48} -1.00000i q^{49} +(-0.634393 - 0.773010i) q^{50} +(-1.46658 - 0.979938i) q^{52} +(0.591637 + 1.42834i) q^{53} +(-0.143707 - 0.473739i) q^{54} +(-1.17588 - 1.17588i) q^{55} +(-0.897168 + 0.897168i) q^{57} +(-0.247528 + 1.24441i) q^{60} +(-0.425215 + 1.02656i) q^{61} +(-0.831470 - 0.555570i) q^{64} +1.76384 q^{65} +(-0.206808 + 2.09976i) q^{66} +(-0.732410 + 1.76820i) q^{67} +(0.537813 - 0.287467i) q^{72} +(0.577774 + 1.90466i) q^{74} +(-0.485544 - 1.17221i) q^{75} +(0.831470 + 0.555570i) q^{76} +(-1.41973 - 1.72995i) q^{78} +1.00000 q^{80} -1.23794i q^{81} +(1.65493 - 0.162997i) q^{88} +(-0.287467 + 0.537813i) q^{90} -1.00000 q^{95} +(-0.804910 - 0.980785i) q^{96} -0.942793 q^{97} +(-0.995185 - 0.0980171i) q^{98} +(-0.388076 + 0.936899i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 32 q^{66} + 32 q^{80} - 32 q^{95} - 32 q^{96} - 32 q^{99}+O(q^{100}) \) 32 * q - 32 * q^66 + 32 * q^80 - 32 * q^95 - 32 * q^96 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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