LMFDB - Embedded newform 3040.1.cn.a.2469.3

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			2469.3
Root			$-0.471397 + 0.881921i$ of defining polynomial
Character	$\chi$	$=$	3040.2469
Dual form			3040.1.cn.a.1709.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.634393 + 0.773010i) q^{2} +(-0.761681 - 1.83886i) q^{3} +(-0.195090 - 0.980785i) q^{4} +(-0.923880 - 0.382683i) q^{5} +(1.90466 + 0.577774i) q^{6} +(0.881921 + 0.471397i) q^{8} +(-2.09415 + 2.09415i) q^{9} +O(q^{10})$	\(q+(-0.634393 + 0.773010i) q^{2} +(-0.761681 - 1.83886i) q^{3} +(-0.195090 - 0.980785i) q^{4} +(-0.923880 - 0.382683i) q^{5} +(1.90466 + 0.577774i) q^{6} +(0.881921 + 0.471397i) q^{8} +(-2.09415 + 2.09415i) q^{9} +(0.881921 - 0.471397i) q^{10} +(-0.425215 + 1.02656i) q^{11} +(-1.65493 + 1.10579i) q^{12} +(1.76820 - 0.732410i) q^{13} +1.99037i q^{15} +(-0.923880 + 0.382683i) q^{16} +(-0.290285 - 2.94731i) q^{18} +(0.923880 - 0.382683i) q^{19} +(-0.195090 + 0.980785i) q^{20} +(-0.523788 - 0.979938i) q^{22} +(0.195090 - 1.98079i) q^{24} +(0.707107 + 0.707107i) q^{25} +(-0.555570 + 1.83147i) q^{26} +(3.60706 + 1.49409i) q^{27} +(-1.53858 - 1.26268i) q^{30} +(0.290285 - 0.956940i) q^{32} +2.21158 q^{33} +(2.46246 + 1.64536i) q^{36} +(1.42834 + 0.591637i) q^{37} +(-0.290285 + 0.956940i) q^{38} +(-2.69360 - 2.69360i) q^{39} +(-0.634393 - 0.773010i) q^{40} +(1.08979 + 0.216773i) q^{44} +(2.73613 - 1.13334i) q^{45} +(1.40740 + 1.40740i) q^{48} +1.00000i q^{49} +(-0.995185 + 0.0980171i) q^{50} +(-1.06330 - 1.59133i) q^{52} +(-0.0750191 + 0.181112i) q^{53} +(-3.44324 + 1.84045i) q^{54} +(0.785695 - 0.785695i) q^{55} +(-1.40740 - 1.40740i) q^{57} +(1.95213 - 0.388302i) q^{60} +(0.636379 + 1.53636i) q^{61} +(0.555570 + 0.831470i) q^{64} -1.91388 q^{65} +(-1.40301 + 1.70957i) q^{66} +(-0.360791 - 0.871028i) q^{67} +(-2.83405 + 0.859699i) q^{72} +(-1.36347 + 0.728789i) q^{74} +(0.761681 - 1.83886i) q^{75} +(-0.555570 - 0.831470i) q^{76} +(3.79099 - 0.373380i) q^{78} +1.00000 q^{80} -4.80933i q^{81} +(-0.858923 + 0.704900i) q^{88} +(-0.859699 + 2.83405i) q^{90} -1.00000 q^{95} +(-1.98079 + 0.195090i) q^{96} -0.580569 q^{97} +(-0.773010 - 0.634393i) q^{98} +(-1.25930 - 3.04023i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3040\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$2$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$3$	−0.761681	−	1.83886i	−0.761681	−	1.83886i	−0.471397	−	0.881921i	$-0.656250\pi$
$3$	−0.761681	−	1.83886i	−0.761681	−	1.83886i	−0.290285	−	0.956940i	$-0.593750\pi$
$4$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$5$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$5$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$6$	1.90466	+	0.577774i	1.90466	+	0.577774i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	0.881921	+	0.471397i	0.881921	+	0.471397i
$9$	−2.09415	+	2.09415i	−2.09415	+	2.09415i
$10$	0.881921	−	0.471397i	0.881921	−	0.471397i
$11$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$11$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$12$	−1.65493	+	1.10579i	−1.65493	+	1.10579i
$13$	1.76820	−	0.732410i	1.76820	−	0.732410i	0.773010	−	0.634393i	$-0.218750\pi$
$13$	1.76820	−	0.732410i	1.76820	−	0.732410i	0.995185	−	0.0980171i	$-0.0312500\pi$
$14$		0			0
$15$			1.99037i			1.99037i
$16$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.290285	−	2.94731i	−0.290285	−	2.94731i
$19$	0.923880	−	0.382683i	0.923880	−	0.382683i
$19$	0.923880	−	0.382683i	0.923880	−	0.382683i
$20$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$21$		0			0
$22$	−0.523788	−	0.979938i	−0.523788	−	0.979938i
$23$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$23$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$24$	0.195090	−	1.98079i	0.195090	−	1.98079i
$25$	0.707107	+	0.707107i	0.707107	+	0.707107i
$26$	−0.555570	+	1.83147i	−0.555570	+	1.83147i
$27$	3.60706	+	1.49409i	3.60706	+	1.49409i
$28$		0			0
$29$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$29$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$30$	−1.53858	−	1.26268i	−1.53858	−	1.26268i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.290285	−	0.956940i	0.290285	−	0.956940i
$33$	2.21158			2.21158
$34$		0			0
$35$		0			0
$36$	2.46246	+	1.64536i	2.46246	+	1.64536i
$37$	1.42834	+	0.591637i	1.42834	+	0.591637i	0.956940	−	0.290285i	$-0.0937500\pi$
$37$	1.42834	+	0.591637i	1.42834	+	0.591637i	0.471397	+	0.881921i	$0.343750\pi$
$38$	−0.290285	+	0.956940i	−0.290285	+	0.956940i
$39$	−2.69360	−	2.69360i	−2.69360	−	2.69360i
$40$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$43$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$44$	1.08979	+	0.216773i	1.08979	+	0.216773i
$45$	2.73613	−	1.13334i	2.73613	−	1.13334i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	1.40740	+	1.40740i	1.40740	+	1.40740i
$49$			1.00000i			1.00000i
$50$	−0.995185	+	0.0980171i	−0.995185	+	0.0980171i
$51$		0			0
$52$	−1.06330	−	1.59133i	−1.06330	−	1.59133i
$53$	−0.0750191	+	0.181112i	−0.0750191	+	0.181112i	−0.956940	−	0.290285i	$-0.906250\pi$
$53$	−0.0750191	+	0.181112i	−0.0750191	+	0.181112i	0.881921	+	0.471397i	$0.156250\pi$
$54$	−3.44324	+	1.84045i	−3.44324	+	1.84045i
$55$	0.785695	−	0.785695i	0.785695	−	0.785695i
$56$		0			0
$57$	−1.40740	−	1.40740i	−1.40740	−	1.40740i
$58$		0			0
$59$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$59$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$60$	1.95213	−	0.388302i	1.95213	−	0.388302i
$61$	0.636379	+	1.53636i	0.636379	+	1.53636i	0.831470	+	0.555570i	$0.187500\pi$
$61$	0.636379	+	1.53636i	0.636379	+	1.53636i	−0.195090	+	0.980785i	$0.562500\pi$
$62$		0			0
$63$		0			0
$64$	0.555570	+	0.831470i	0.555570	+	0.831470i
$65$	−1.91388			−1.91388
$66$	−1.40301	+	1.70957i	−1.40301	+	1.70957i
$67$	−0.360791	−	0.871028i	−0.360791	−	0.871028i	−0.995185	−	0.0980171i	$-0.968750\pi$
$67$	−0.360791	−	0.871028i	−0.360791	−	0.871028i	0.634393	−	0.773010i	$-0.281250\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$	−2.83405	+	0.859699i	−2.83405	+	0.859699i
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$	−1.36347	+	0.728789i	−1.36347	+	0.728789i
$75$	0.761681	−	1.83886i	0.761681	−	1.83886i
$76$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$77$		0			0
$78$	3.79099	−	0.373380i	3.79099	−	0.373380i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	1.00000			1.00000
$81$		−	4.80933i		−	4.80933i
$82$		0			0
$83$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$83$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.858923	+	0.704900i	−0.858923	+	0.704900i
$89$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$89$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$90$	−0.859699	+	2.83405i	−0.859699	+	2.83405i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000			−1.00000
$96$	−1.98079	+	0.195090i	−1.98079	+	0.195090i
$97$	−0.580569			−0.580569			−0.290285	−	0.956940i	$-0.593750\pi$
$97$	−0.580569			−0.580569			−0.290285	+	0.956940i	$0.593750\pi$
$98$	−0.773010	−	0.634393i	−0.773010	−	0.634393i
$99$	−1.25930	−	3.04023i	−1.25930	−	3.04023i
$100$	0.555570	−	0.831470i	0.555570	−	0.831470i
$101$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.831470	−	0.555570i	$-0.187500\pi$
$101$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.195090	+	0.980785i	$0.437500\pi$
$102$		0			0
$103$	−0.897168	−	0.897168i	−0.897168	−	0.897168i	0.0980171	−	0.995185i	$-0.468750\pi$
$103$	−0.897168	−	0.897168i	−0.897168	−	0.897168i	−0.995185	+	0.0980171i	$0.968750\pi$
$104$	1.90466	+	0.187593i	1.90466	+	0.187593i
$105$		0			0
$106$	−0.0924099	−	0.172887i	−0.0924099	−	0.172887i
$107$	0.674993	−	1.62958i	0.674993	−	1.62958i	−0.0980171	−	0.995185i	$-0.531250\pi$
$107$	0.674993	−	1.62958i	0.674993	−	1.62958i	0.773010	−	0.634393i	$-0.218750\pi$
$108$	0.761681	−	3.82923i	0.761681	−	3.82923i
$109$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$109$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$110$	0.108911	+	1.10579i	0.108911	+	1.10579i
$111$		−	3.07715i		−	3.07715i
$112$		0			0
$113$			0.196034i			0.196034i	0.995185	+	0.0980171i	$0.0312500\pi$
$113$			0.196034i			0.196034i	−0.995185	+	0.0980171i	$0.968750\pi$
$114$	1.98079	−	0.195090i	1.98079	−	0.195090i
$115$		0			0
$116$		0			0
$117$	−2.16909	+	5.23663i	−2.16909	+	5.23663i
$118$		0			0
$119$		0			0
$120$	−0.938254	+	1.75535i	−0.938254	+	1.75535i
$121$	−0.165911	−	0.165911i	−0.165911	−	0.165911i
$122$	−1.59133	−	0.482726i	−1.59133	−	0.482726i
$123$		0			0
$124$		0			0
$125$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$126$		0			0
$127$	−0.942793			−0.942793			−0.471397	−	0.881921i	$-0.656250\pi$
$127$	−0.942793			−0.942793			−0.471397	+	0.881921i	$0.656250\pi$
$128$	−0.995185	−	0.0980171i	−0.995185	−	0.0980171i
$129$		0			0
$130$	1.21415	−	1.47945i	1.21415	−	1.47945i
$131$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$131$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$132$	−0.431458	−	2.16909i	−0.431458	−	2.16909i
$133$		0			0
$134$	0.902197	+	0.273678i	0.902197	+	0.273678i
$135$	−2.76072	−	2.76072i	−2.76072	−	2.76072i
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	−0.980785	−	0.195090i	$-0.937500\pi$
$139$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	0.831470	+	0.555570i	$0.187500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.12659i			2.12659i
$144$	1.13334	−	2.73613i	1.13334	−	2.73613i
$145$		0			0
$146$		0			0
$147$	1.83886	−	0.761681i	1.83886	−	0.761681i
$148$	0.301614	−	1.51631i	0.301614	−	1.51631i
$149$	0.149316	−	0.360480i	0.149316	−	0.360480i	−0.831470	−	0.555570i	$-0.812500\pi$
$149$	0.149316	−	0.360480i	0.149316	−	0.360480i	0.980785	+	0.195090i	$0.0625000\pi$
$150$	0.938254	+	1.75535i	0.938254	+	1.75535i
$151$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$151$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$152$	0.995185	+	0.0980171i	0.995185	+	0.0980171i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2.11635	+	3.16734i	−2.11635	+	3.16734i
$157$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$157$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$158$		0			0
$159$	0.390181			0.390181
$160$	−0.634393	+	0.773010i	−0.634393	+	0.773010i
$161$		0			0
$162$	3.71766	+	3.05101i	3.71766	+	3.05101i
$163$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$163$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$164$		0			0
$165$	−2.04323	−	0.846335i	−2.04323	−	0.846335i
$166$		0			0
$167$	1.24723	+	1.24723i	1.24723	+	1.24723i	0.956940	+	0.290285i	$0.0937500\pi$
$167$	1.24723	+	1.24723i	1.24723	+	1.24723i	0.290285	+	0.956940i	$0.406250\pi$
$168$		0			0
$169$	1.88298	−	1.88298i	1.88298	−	1.88298i
$170$		0			0
$171$	−1.13334	+	2.73613i	−1.13334	+	2.73613i
$172$		0			0
$173$	0.871028	−	0.360791i	0.871028	−	0.360791i	0.0980171	−	0.995185i	$-0.468750\pi$
$173$	0.871028	−	0.360791i	0.871028	−	0.360791i	0.773010	+	0.634393i	$0.218750\pi$
$174$		0			0
$175$		0			0
$176$		−	1.11114i		−	1.11114i
$177$		0			0
$178$		0			0
$179$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$179$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$180$	−1.64536	−	2.46246i	−1.64536	−	2.46246i
$181$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$181$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$182$		0			0
$183$	2.34043	−	2.34043i	2.34043	−	2.34043i
$184$		0			0
$185$	−1.09320	−	1.09320i	−1.09320	−	1.09320i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.634393	−	0.773010i	0.634393	−	0.773010i
$191$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$191$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$192$	1.10579	−	1.65493i	1.10579	−	1.65493i
$193$	−1.76384			−1.76384			−0.881921	−	0.471397i	$-0.843750\pi$
$193$	−1.76384			−1.76384			−0.881921	+	0.471397i	$0.843750\pi$
$194$	0.368309	−	0.448786i	0.368309	−	0.448786i
$195$	1.45777	+	3.51936i	1.45777	+	3.51936i
$196$	0.980785	−	0.195090i	0.980785	−	0.195090i
$197$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$197$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$198$	3.14902	+	0.955246i	3.14902	+	0.955246i
$199$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$199$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$200$	0.290285	+	0.956940i	0.290285	+	0.956940i
$201$	−1.32689	+	1.32689i	−1.32689	+	1.32689i
$202$	−0.979938	+	0.523788i	−0.979938	+	0.523788i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	1.26268	−	0.124363i	1.26268	−	0.124363i
$207$		0			0
$208$	−1.35332	+	1.35332i	−1.35332	+	1.35332i
$209$			1.11114i			1.11114i
$210$		0			0
$211$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$211$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$212$	0.192268	+	0.0382444i	0.192268	+	0.0382444i
$213$		0			0
$214$	0.831470	+	1.55557i	0.831470	+	1.55557i
$215$		0			0
$216$	2.47683	+	3.01803i	2.47683	+	3.01803i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$221$		0			0
$222$	2.37867	+	1.95213i	2.37867	+	1.95213i
$223$	0.580569			0.580569			0.290285	−	0.956940i	$-0.406250\pi$
$223$	0.580569			0.580569			0.290285	+	0.956940i	$0.406250\pi$
$224$		0			0
$225$	−2.96157			−2.96157
$226$	−0.151537	−	0.124363i	−0.151537	−	0.124363i
$227$	0.222174	+	0.536376i	0.222174	+	0.536376i	0.995185	−	0.0980171i	$-0.0312500\pi$
$227$	0.222174	+	0.536376i	0.222174	+	0.536376i	−0.773010	+	0.634393i	$0.781250\pi$
$228$	−1.10579	+	1.65493i	−1.10579	+	1.65493i
$229$	0.360480	+	0.149316i	0.360480	+	0.149316i	0.555570	−	0.831470i	$-0.312500\pi$
$229$	0.360480	+	0.149316i	0.360480	+	0.149316i	−0.195090	+	0.980785i	$0.562500\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$233$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$234$	−2.67192	−	4.99881i	−2.67192	−	4.99881i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$239$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$240$	−0.761681	−	1.83886i	−0.761681	−	1.83886i
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	0.233503	−	0.0229981i	0.233503	−	0.0229981i
$243$	−5.23663	+	2.16909i	−5.23663	+	2.16909i
$244$	1.38268	−	0.923880i	1.38268	−	0.923880i
$245$	0.382683	−	0.923880i	0.382683	−	0.923880i
$246$		0			0
$247$	1.35332	−	1.35332i	1.35332	−	1.35332i
$248$		0			0
$249$		0			0
$250$	0.956940	+	0.290285i	0.956940	+	0.290285i
$251$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$251$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$252$		0			0
$253$		0			0
$254$	0.598102	−	0.728789i	0.598102	−	0.728789i
$255$		0			0
$256$	0.707107	−	0.707107i	0.707107	−	0.707107i
$257$	1.76384			1.76384			0.881921	−	0.471397i	$-0.156250\pi$
$257$	1.76384			1.76384			0.881921	+	0.471397i	$0.156250\pi$
$258$		0			0
$259$		0			0
$260$	0.373380	+	1.87711i	0.373380	+	1.87711i
$261$		0			0
$262$	1.76820	+	0.536376i	1.76820	+	0.536376i
$263$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$263$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$264$	1.95044	+	1.04253i	1.95044	+	1.04253i
$265$	0.138617	−	0.138617i	0.138617	−	0.138617i
$266$		0			0
$267$		0			0
$268$	−0.783904	+	0.523788i	−0.783904	+	0.523788i
$269$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$269$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$270$	3.88545	−	0.382683i	3.88545	−	0.382683i
$271$			0.390181i			0.390181i	0.980785	+	0.195090i	$0.0625000\pi$
$271$			0.390181i			0.390181i	−0.980785	+	0.195090i	$0.937500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.02656	+	0.425215i	−1.02656	+	0.425215i
$276$		0			0
$277$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$277$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$278$	−0.183930	−	0.344109i	−0.183930	−	0.344109i
$279$		0			0
$280$		0			0
$281$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$281$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$282$		0			0
$283$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$283$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$284$		0			0
$285$	0.761681	+	1.83886i	0.761681	+	1.83886i
$286$	−1.64388	−	1.34909i	−1.64388	−	1.34909i
$287$		0			0
$288$	1.39607	+	2.61187i	1.39607	+	2.61187i
$289$	−1.00000			−1.00000
$290$		0			0
$291$	0.442209	+	1.06759i	0.442209	+	1.06759i
$292$		0			0
$293$	1.17221	+	0.485544i	1.17221	+	0.485544i	0.881921	−	0.471397i	$-0.156250\pi$
$293$	1.17221	+	0.485544i	1.17221	+	0.485544i	0.290285	+	0.956940i	$0.406250\pi$
$294$	−0.577774	+	1.90466i	−0.577774	+	1.90466i
$295$		0			0
$296$	0.980785	+	1.19509i	0.980785	+	1.19509i
$297$	−3.06755	+	3.06755i	−3.06755	+	3.06755i
$298$	0.183930	+	0.344109i	0.183930	+	0.344109i
$299$		0			0
$300$	−1.95213	−	0.388302i	−1.95213	−	0.388302i
$301$		0			0
$302$		0			0
$303$		−	2.21158i		−	2.21158i
$304$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$305$		−	1.66294i		−	1.66294i
$306$		0			0
$307$	0.536376	−	0.222174i	0.536376	−	0.222174i	−0.0980171	−	0.995185i	$-0.531250\pi$
$307$	0.536376	−	0.222174i	0.536376	−	0.222174i	0.634393	+	0.773010i	$0.281250\pi$
$308$		0			0
$309$	−0.966411	+	2.33312i	−0.966411	+	2.33312i
$310$		0			0
$311$	1.17588	−	1.17588i	1.17588	−	1.17588i	0.195090	−	0.980785i	$-0.437500\pi$
$311$	1.17588	−	1.17588i	1.17588	−	1.17588i	0.980785	−	0.195090i	$-0.0625000\pi$
$312$	−1.10579	−	3.64530i	−1.10579	−	3.64530i
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.674993	−	1.62958i	−0.674993	−	1.62958i	−0.773010	−	0.634393i	$-0.781250\pi$
$317$	−0.674993	−	1.62958i	−0.674993	−	1.62958i	0.0980171	−	0.995185i	$-0.468750\pi$
$318$	−0.247528	+	0.301614i	−0.247528	+	0.301614i
$319$		0			0
$320$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$321$	−3.51070			−3.51070
$322$		0			0
$323$		0			0
$324$	−4.71692	+	0.938254i	−4.71692	+	0.938254i
$325$	1.76820	+	0.732410i	1.76820	+	0.732410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	1.95044	−	1.04253i	1.95044	−	1.04253i
$331$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$331$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$332$		0			0
$333$	−4.23012	+	1.75217i	−4.23012	+	1.75217i
$334$	−1.75535	+	0.172887i	−1.75535	+	0.172887i
$335$			0.942793i			0.942793i
$336$		0			0
$337$			1.54602i			1.54602i	0.634393	+	0.773010i	$0.281250\pi$
$337$			1.54602i			1.54602i	−0.634393	+	0.773010i	$0.718750\pi$
$338$	0.261014	+	2.65012i	0.261014	+	2.65012i
$339$	0.360480	−	0.149316i	0.360480	−	0.149316i
$340$		0			0
$341$		0			0
$342$	−1.39607	−	2.61187i	−1.39607	−	2.61187i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.273678	+	0.902197i	−0.273678	+	0.902197i
$347$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$347$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$348$		0			0
$349$	0.541196	+	1.30656i	0.541196	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$349$	0.541196	+	1.30656i	0.541196	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$350$		0			0
$351$	7.47227			7.47227
$352$	0.858923	+	0.704900i	0.858923	+	0.704900i
$353$		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
$359$	0.275899	+	0.275899i	0.275899	+	0.275899i	0.831470	−	0.555570i	$-0.187500\pi$
$359$	0.275899	+	0.275899i	0.275899	+	0.275899i	−0.555570	+	0.831470i	$0.687500\pi$
$360$	2.94731	+	0.290285i	2.94731	+	0.290285i
$361$	0.707107	−	0.707107i	0.707107	−	0.707107i
$362$		0			0
$363$	−0.178716	+	0.431458i	−0.178716	+	0.431458i
$364$		0			0
$365$		0			0
$366$	0.324423	+	3.29393i	0.324423	+	3.29393i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	1.53858	−	0.151537i	1.53858	−	0.151537i
$371$		0			0
$372$		0			0
$373$	0.761681	−	1.83886i	0.761681	−	1.83886i	0.290285	−	0.956940i	$-0.406250\pi$
$373$	0.761681	−	1.83886i	0.761681	−	1.83886i	0.471397	−	0.881921i	$-0.343750\pi$
$374$		0			0
$375$	−1.40740	+	1.40740i	−1.40740	+	1.40740i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$379$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$380$	0.195090	+	0.980785i	0.195090	+	0.980785i
$381$	0.718108	+	1.73367i	0.718108	+	1.73367i
$382$	−0.485544	+	0.591637i	−0.485544	+	0.591637i
$383$	1.26879			1.26879			0.634393	−	0.773010i	$-0.281250\pi$
$383$	1.26879			1.26879			0.634393	+	0.773010i	$0.281250\pi$
$384$	0.577774	+	1.90466i	0.577774	+	1.90466i
$385$		0			0
$386$	1.11897	−	1.36347i	1.11897	−	1.36347i
$387$		0			0
$388$	0.113263	+	0.569414i	0.113263	+	0.569414i
$389$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$389$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$390$	−3.64530	−	1.10579i	−3.64530	−	1.10579i
$391$		0			0
$392$	−0.471397	+	0.881921i	−0.471397	+	0.881921i
$393$	−2.60054	+	2.60054i	−2.60054	+	2.60054i
$394$		0			0
$395$		0			0
$396$	−2.73613	+	1.82823i	−2.73613	+	1.82823i
$397$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$397$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$398$	−0.761681	+	0.0750191i	−0.761681	+	0.0750191i
$399$		0			0
$400$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$	−0.183930	−	1.86747i	−0.183930	−	1.86747i
$403$		0			0
$404$	0.216773	−	1.08979i	0.216773	−	1.08979i
$405$	−1.84045	+	4.44324i	−1.84045	+	4.44324i
$406$		0			0
$407$	−1.21470	+	1.21470i	−1.21470	+	1.21470i
$408$		0			0
$409$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$409$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$410$		0			0
$411$		0			0
$412$	−0.704900	+	1.05496i	−0.704900	+	1.05496i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.187593	−	1.90466i	−0.187593	−	1.90466i
$417$	0.776604			0.776604
$418$	−0.858923	−	0.704900i	−0.858923	−	0.704900i
$419$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$419$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$421$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$422$		0			0
$423$		0			0
$424$	−0.151537	+	0.124363i	−0.151537	+	0.124363i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.72995	−	0.344109i	−1.72995	−	0.344109i
$429$	3.91051	−	1.61978i	3.91051	−	1.61978i
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−3.90425			−3.90425
$433$		−	1.99037i		−	1.99037i	−0.0980171	−	0.995185i	$-0.531250\pi$
$433$		−	1.99037i		−	1.99037i	0.0980171	−	0.995185i	$-0.468750\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$439$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$440$	1.06330	−	0.322547i	1.06330	−	0.322547i
$441$	−2.09415	−	2.09415i	−2.09415	−	2.09415i
$442$		0			0
$443$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$443$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$444$	−3.01803	+	0.600323i	−3.01803	+	0.600323i
$445$		0			0
$446$	−0.368309	+	0.448786i	−0.368309	+	0.448786i
$447$	−0.776604			−0.776604
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	1.87880	−	2.28933i	1.87880	−	2.28933i
$451$		0			0
$452$	0.192268	−	0.0382444i	0.192268	−	0.0382444i
$453$		0			0
$454$	−0.555570	−	0.168530i	−0.555570	−	0.168530i
$455$		0			0
$456$	−0.577774	−	1.90466i	−0.577774	−	1.90466i
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$	−0.344109	+	0.183930i	−0.344109	+	0.183930i
$459$		0			0
$460$		0			0
$461$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$461$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$462$		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	$-0.125000\pi$
$467$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$468$	5.55918	+	1.10579i	5.55918	+	1.10579i
$469$		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
$477$	−0.222174	−	0.536376i	−0.222174	−	0.536376i
$478$	−1.09320	−	0.897168i	−1.09320	−	0.897168i
$479$	−1.11114			−1.11114			−0.555570	−	0.831470i	$-0.687500\pi$
$479$	−1.11114			−1.11114			−0.555570	+	0.831470i	$0.687500\pi$
$480$	1.90466	+	0.577774i	1.90466	+	0.577774i
$481$	2.95890			2.95890
$482$		0			0
$483$		0			0
$484$	−0.130355	+	0.195090i	−0.130355	+	0.195090i
$485$	0.536376	+	0.222174i	0.536376	+	0.222174i
$486$	1.64536	−	5.42403i	1.64536	−	5.42403i
$487$	−0.138617	−	0.138617i	−0.138617	−	0.138617i	0.634393	−	0.773010i	$-0.281250\pi$
$487$	−0.138617	−	0.138617i	−0.138617	−	0.138617i	−0.773010	+	0.634393i	$0.781250\pi$
$488$	−0.162997	+	1.65493i	−0.162997	+	1.65493i
$489$		0			0
$490$	0.471397	+	0.881921i	0.471397	+	0.881921i
$491$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$491$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$492$		0			0
$493$		0			0
$494$	0.187593	+	1.90466i	0.187593	+	1.90466i
$495$			3.29072i			3.29072i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.831470	−	0.555570i	$-0.187500\pi$
$499$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.980785	−	0.195090i	$-0.0625000\pi$
$500$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$501$	1.34349	−	3.24346i	1.34349	−	3.24346i
$502$	1.24723	−	0.666656i	1.24723	−	0.666656i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$506$		0			0
$507$	−4.89678	−	2.02831i	−4.89678	−	2.02831i
$508$	0.183930	+	0.924678i	0.183930	+	0.924678i
$509$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$509$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$510$		0			0
$511$		0			0
$512$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$513$	3.90425			3.90425
$514$	−1.11897	+	1.36347i	−1.11897	+	1.36347i
$515$	0.485544	+	1.17221i	0.485544	+	1.17221i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.32689	−	1.32689i	−1.32689	−	1.32689i
$520$	−1.68789	−	0.902197i	−1.68789	−	0.902197i
$521$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$521$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$522$		0			0
$523$	0.485544	−	1.17221i	0.485544	−	1.17221i	−0.471397	−	0.881921i	$-0.656250\pi$
$523$	0.485544	−	1.17221i	0.485544	−	1.17221i	0.956940	−	0.290285i	$-0.0937500\pi$
$524$	−1.53636	+	1.02656i	−1.53636	+	1.02656i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−2.04323	+	0.846335i	−2.04323	+	0.846335i
$529$		−	1.00000i		−	1.00000i
$530$	0.0192147	+	0.195090i	0.0192147	+	0.195090i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.24723	+	1.24723i	−1.24723	+	1.24723i
$536$	0.0924099	−	0.938254i	0.0924099	−	0.938254i
$537$		0			0
$538$		0			0
$539$	−1.02656	−	0.425215i	−1.02656	−	0.425215i
$540$	−2.16909	+	3.24627i	−2.16909	+	3.24627i
$541$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.555570	−	0.831470i	$-0.687500\pi$
$541$	−0.750661	−	1.81225i	−0.750661	−	1.81225i	−0.195090	−	0.980785i	$-0.562500\pi$
$542$	−0.301614	−	0.247528i	−0.301614	−	0.247528i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.591637	−	1.42834i	−0.591637	−	1.42834i	−0.881921	−	0.471397i	$-0.843750\pi$
$547$	−0.591637	−	1.42834i	−0.591637	−	1.42834i	0.290285	−	0.956940i	$-0.406250\pi$
$548$		0			0
$549$	−4.55003	−	1.88468i	−4.55003	−	1.88468i
$550$	0.322547	−	1.06330i	0.322547	−	1.06330i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.17758	+	2.84292i	−1.17758	+	2.84292i
$556$	0.382683	+	0.0761205i	0.382683	+	0.0761205i
$557$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$557$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.42834	−	0.591637i	1.42834	−	0.591637i	0.471397	−	0.881921i	$-0.343750\pi$
$563$	1.42834	−	0.591637i	1.42834	−	0.591637i	0.956940	+	0.290285i	$0.0937500\pi$
$564$		0			0
$565$	0.0750191	−	0.181112i	0.0750191	−	0.181112i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$	−1.90466	−	0.577774i	−1.90466	−	0.577774i
$571$	−1.81225	−	0.750661i	−1.81225	−	0.750661i	−0.980785	−	0.195090i	$-0.937500\pi$
$571$	−1.81225	−	0.750661i	−1.81225	−	0.750661i	−0.831470	−	0.555570i	$-0.812500\pi$
$572$	2.08573	−	0.414877i	2.08573	−	0.414877i
$573$	−0.582966	−	1.40740i	−0.582966	−	1.40740i
$574$		0			0
$575$		0			0
$576$	−2.90466	−	0.577774i	−2.90466	−	0.577774i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.634393	−	0.773010i	0.634393	−	0.773010i
$579$	1.34349	+	3.24346i	1.34349	+	3.24346i
$580$		0			0
$581$		0			0
$582$	−1.10579	−	0.335438i	−1.10579	−	0.335438i
$583$	−0.154023	−	0.154023i	−0.154023	−	0.154023i
$584$		0			0
$585$	4.00795	−	4.00795i	4.00795	−	4.00795i
$586$	−1.11897	+	0.598102i	−1.11897	+	0.598102i
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$	−1.10579	−	1.65493i	−1.10579	−	1.65493i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.54602			−1.54602
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	−0.425215	−	4.31728i	−0.425215	−	4.31728i
$595$		0			0
$596$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$597$	0.582966	−	1.40740i	0.582966	−	1.40740i
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$	1.53858	−	1.26268i	1.53858	−	1.26268i
$601$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$601$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$602$		0			0
$603$	2.57961	+	1.06851i	2.57961	+	1.06851i
$604$		0			0
$605$	0.0897902	+	0.216773i	0.0897902	+	0.216773i
$606$	1.70957	+	1.40301i	1.70957	+	1.40301i
$607$	−0.196034			−0.196034			−0.0980171	−	0.995185i	$-0.531250\pi$
$607$	−0.196034			−0.196034			−0.0980171	+	0.995185i	$0.531250\pi$
$608$	−0.0980171	−	0.995185i	−0.0980171	−	0.995185i
$609$		0			0
$610$	1.28547	+	1.05496i	1.28547	+	1.05496i
$611$		0			0
$612$		0			0
$613$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$613$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$614$	−0.168530	+	0.555570i	−0.168530	+	0.555570i
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$	−1.19044	−	2.22716i	−1.19044	−	2.22716i
$619$	0.636379	−	1.53636i	0.636379	−	1.53636i	−0.195090	−	0.980785i	$-0.562500\pi$
$619$	0.636379	−	1.53636i	0.636379	−	1.53636i	0.831470	−	0.555570i	$-0.187500\pi$
$620$		0			0
$621$		0			0
$622$	0.162997	+	1.65493i	0.162997	+	1.65493i
$623$		0			0
$624$	3.51936	+	1.45777i	3.51936	+	1.45777i
$625$			1.00000i			1.00000i
$626$		0			0
$627$	2.04323	−	0.846335i	2.04323	−	0.846335i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.38704	+	1.38704i	−1.38704	+	1.38704i	−0.555570	+	0.831470i	$0.687500\pi$
$631$	−1.38704	+	1.38704i	−1.38704	+	1.38704i	−0.831470	+	0.555570i	$0.812500\pi$
$632$		0			0
$633$		0			0
$634$	1.68789	+	0.512016i	1.68789	+	0.512016i
$635$	0.871028	+	0.360791i	0.871028	+	0.360791i
$636$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$637$	0.732410	+	1.76820i	0.732410	+	1.76820i
$638$		0			0
$639$		0			0
$640$	0.881921	+	0.471397i	0.881921	+	0.471397i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	2.22716	−	2.71381i	2.22716	−	2.71381i
$643$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$643$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	2.26710	−	4.24145i	2.26710	−	4.24145i
$649$		0			0
$650$	−1.68789	+	0.902197i	−1.68789	+	0.902197i
$651$		0			0
$652$		0			0
$653$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$653$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$654$		0			0
$655$			1.84776i			1.84776i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$659$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$660$	−0.431458	+	2.16909i	−0.431458	+	2.16909i
$661$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$661$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.32911	−	4.38149i	1.32911	−	4.38149i
$667$		0			0
$668$	0.979938	−	1.46658i	0.979938	−	1.46658i
$669$	−0.442209	−	1.06759i	−0.442209	−	1.06759i
$670$	−0.728789	−	0.598102i	−0.728789	−	0.598102i
$671$	−1.84776			−1.84776
$672$		0			0
$673$	−1.54602			−1.54602			−0.773010	−	0.634393i	$-0.781250\pi$
$673$	−1.54602			−1.54602			−0.773010	+	0.634393i	$0.781250\pi$
$674$	−1.19509	−	0.980785i	−1.19509	−	0.980785i
$675$	1.49409	+	3.60706i	1.49409	+	3.60706i
$676$	−2.21415	−	1.47945i	−2.21415	−	1.47945i
$677$	−1.42834	−	0.591637i	−1.42834	−	0.591637i	−0.471397	−	0.881921i	$-0.656250\pi$
$677$	−1.42834	−	0.591637i	−1.42834	−	0.591637i	−0.956940	+	0.290285i	$0.906250\pi$
$678$	−0.113263	+	0.373380i	−0.113263	+	0.373380i
$679$		0			0
$680$		0			0
$681$	0.817095	−	0.817095i	0.817095	−	0.817095i
$682$		0			0
$683$	−0.591637	+	1.42834i	−0.591637	+	1.42834i	0.290285	+	0.956940i	$0.406250\pi$
$683$	−0.591637	+	1.42834i	−0.591637	+	1.42834i	−0.881921	+	0.471397i	$0.843750\pi$
$684$	2.90466	+	0.577774i	2.90466	+	0.577774i
$685$		0			0
$686$		0			0
$687$		−	0.776604i		−	0.776604i
$688$		0			0
$689$			0.375186i			0.375186i
$690$		0			0
$691$	−1.53636	+	0.636379i	−1.53636	+	0.636379i	−0.980785	−	0.195090i	$-0.937500\pi$
$691$	−1.53636	+	0.636379i	−1.53636	+	0.636379i	−0.555570	+	0.831470i	$0.687500\pi$
$692$	−0.523788	−	0.783904i	−0.523788	−	0.783904i
$693$		0			0
$694$		0			0
$695$	0.275899	−	0.275899i	0.275899	−	0.275899i
$696$		0			0
$697$		0			0
$698$	−1.35332	−	0.410525i	−1.35332	−	0.410525i
$699$		0			0
$700$		0			0
$701$	0.750661	+	1.81225i	0.750661	+	1.81225i	0.555570	+	0.831470i	$0.312500\pi$
$701$	0.750661	+	1.81225i	0.750661	+	1.81225i	0.195090	+	0.980785i	$0.437500\pi$
$702$	−4.74036	+	5.77614i	−4.74036	+	5.77614i
$703$	1.54602			1.54602
$704$	−1.08979	+	0.216773i	−1.08979	+	0.216773i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$709$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	0.813811	−	1.96471i	0.813811	−	1.96471i
$716$		0			0
$717$	2.60054	−	1.07718i	2.60054	−	1.07718i
$718$	−0.388302	+	0.0382444i	−0.388302	+	0.0382444i
$719$			1.96157i			1.96157i	0.195090	+	0.980785i	$0.437500\pi$
$719$			1.96157i			1.96157i	−0.195090	+	0.980785i	$0.562500\pi$
$720$	−2.09415	+	2.09415i	−2.09415	+	2.09415i
$721$		0			0
$722$	0.0980171	+	0.995185i	0.0980171	+	0.995185i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−0.220145	−	0.411863i	−0.220145	−	0.411863i
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	4.57659	+	4.57659i	4.57659	+	4.57659i
$730$		0			0
$731$		0			0
$732$	−2.75205	−	1.83886i	−2.75205	−	1.83886i
$733$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$733$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$734$		0			0
$735$	−1.99037			−1.99037
$736$		0			0
$737$	1.04758			1.04758
$738$		0			0
$739$	0.541196	+	1.30656i	0.541196	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$739$	0.541196	+	1.30656i	0.541196	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$740$	−0.858923	+	1.28547i	−0.858923	+	1.28547i
$741$	−3.51936	−	1.45777i	−3.51936	−	1.45777i
$742$		0			0
$743$	1.35332	+	1.35332i	1.35332	+	1.35332i	0.881921	+	0.471397i	$0.156250\pi$
$743$	1.35332	+	1.35332i	1.35332	+	1.35332i	0.471397	+	0.881921i	$0.343750\pi$
$744$		0			0
$745$	−0.275899	+	0.275899i	−0.275899	+	0.275899i
$746$	0.938254	+	1.75535i	0.938254	+	1.75535i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−0.195090	−	1.98079i	−0.195090	−	1.98079i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$			2.81481i			2.81481i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$757$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$758$		0			0
$759$		0			0
$760$	−0.881921	−	0.471397i	−0.881921	−	0.471397i
$761$	−1.17588	−	1.17588i	−1.17588	−	1.17588i	−0.980785	−	0.195090i	$-0.937500\pi$
$761$	−1.17588	−	1.17588i	−1.17588	−	1.17588i	−0.195090	−	0.980785i	$-0.562500\pi$
$762$	−1.79571	−	0.544721i	−1.79571	−	0.544721i
$763$		0			0
$764$	−0.149316	−	0.750661i	−0.149316	−	0.750661i
$765$		0			0
$766$	−0.804910	+	0.980785i	−0.804910	+	0.980785i
$767$		0			0
$768$	−1.83886	−	0.761681i	−1.83886	−	0.761681i
$769$	0.390181			0.390181			0.195090	−	0.980785i	$-0.437500\pi$
$769$	0.390181			0.390181			0.195090	+	0.980785i	$0.437500\pi$
$770$		0			0
$771$	−1.34349	−	3.24346i	−1.34349	−	3.24346i
$772$	0.344109	+	1.72995i	0.344109	+	1.72995i
$773$	0.536376	+	0.222174i	0.536376	+	0.222174i	0.634393	−	0.773010i	$-0.281250\pi$
$773$	0.536376	+	0.222174i	0.536376	+	0.222174i	−0.0980171	+	0.995185i	$0.531250\pi$
$774$		0			0
$775$		0			0
$776$	−0.512016	−	0.273678i	−0.512016	−	0.273678i
$777$		0			0
$778$	−1.62958	+	0.871028i	−1.62958	+	0.871028i
$779$		0			0
$780$	3.16734	−	2.11635i	3.16734	−	2.11635i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$785$		0			0
$786$	−0.360480	−	3.66001i	−0.360480	−	3.66001i
$787$	−1.17221	+	0.485544i	−1.17221	+	0.485544i	−0.881921	−	0.471397i	$-0.843750\pi$
$787$	−1.17221	+	0.485544i	−1.17221	+	0.485544i	−0.290285	+	0.956940i	$0.593750\pi$
$788$		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)

\(f(q)\)	\(=\)	\(q+(-0.634393 + 0.773010i) q^{2} +(-0.761681 - 1.83886i) q^{3} +(-0.195090 - 0.980785i) q^{4} +(-0.923880 - 0.382683i) q^{5} +(1.90466 + 0.577774i) q^{6} +(0.881921 + 0.471397i) q^{8} +(-2.09415 + 2.09415i) q^{9} +O(q^{10})\)	\(q+(-0.634393 + 0.773010i) q^{2} +(-0.761681 - 1.83886i) q^{3} +(-0.195090 - 0.980785i) q^{4} +(-0.923880 - 0.382683i) q^{5} +(1.90466 + 0.577774i) q^{6} +(0.881921 + 0.471397i) q^{8} +(-2.09415 + 2.09415i) q^{9} +(0.881921 - 0.471397i) q^{10} +(-0.425215 + 1.02656i) q^{11} +(-1.65493 + 1.10579i) q^{12} +(1.76820 - 0.732410i) q^{13} +1.99037i q^{15} +(-0.923880 + 0.382683i) q^{16} +(-0.290285 - 2.94731i) q^{18} +(0.923880 - 0.382683i) q^{19} +(-0.195090 + 0.980785i) q^{20} +(-0.523788 - 0.979938i) q^{22} +(0.195090 - 1.98079i) q^{24} +(0.707107 + 0.707107i) q^{25} +(-0.555570 + 1.83147i) q^{26} +(3.60706 + 1.49409i) q^{27} +(-1.53858 - 1.26268i) q^{30} +(0.290285 - 0.956940i) q^{32} +2.21158 q^{33} +(2.46246 + 1.64536i) q^{36} +(1.42834 + 0.591637i) q^{37} +(-0.290285 + 0.956940i) q^{38} +(-2.69360 - 2.69360i) q^{39} +(-0.634393 - 0.773010i) q^{40} +(1.08979 + 0.216773i) q^{44} +(2.73613 - 1.13334i) q^{45} +(1.40740 + 1.40740i) q^{48} +1.00000i q^{49} +(-0.995185 + 0.0980171i) q^{50} +(-1.06330 - 1.59133i) q^{52} +(-0.0750191 + 0.181112i) q^{53} +(-3.44324 + 1.84045i) q^{54} +(0.785695 - 0.785695i) q^{55} +(-1.40740 - 1.40740i) q^{57} +(1.95213 - 0.388302i) q^{60} +(0.636379 + 1.53636i) q^{61} +(0.555570 + 0.831470i) q^{64} -1.91388 q^{65} +(-1.40301 + 1.70957i) q^{66} +(-0.360791 - 0.871028i) q^{67} +(-2.83405 + 0.859699i) q^{72} +(-1.36347 + 0.728789i) q^{74} +(0.761681 - 1.83886i) q^{75} +(-0.555570 - 0.831470i) q^{76} +(3.79099 - 0.373380i) q^{78} +1.00000 q^{80} -4.80933i q^{81} +(-0.858923 + 0.704900i) q^{88} +(-0.859699 + 2.83405i) q^{90} -1.00000 q^{95} +(-1.98079 + 0.195090i) q^{96} -0.580569 q^{97} +(-0.773010 - 0.634393i) q^{98} +(-1.25930 - 3.04023i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 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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