LMFDB - Embedded newform 3680.1.co.a.3649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3680,1,Mod(129,3680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3680, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 11, 21]))

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

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

chi := DirichletCharacter("3680.129");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3680 = 2^{5} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3680.co (of order $22$, degree $10$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.83655924649$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$4$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{88})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} - \cdots)$

Embedding invariants

Embedding label			3649.1
Root			$0.800541 - 0.599278i$ of defining polynomial
Character	$\chi$	$=$	3680.3649
Dual form			3680.1.co.a.2849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.45640 + 0.665114i) q^{3} +(0.755750 - 0.654861i) q^{5} +(-0.357643 - 0.229843i) q^{7} +(1.02385 - 1.18159i) q^{9} +(-0.665114 + 1.45640i) q^{15} +(0.673741 + 0.0968693i) q^{21} +(0.800541 - 0.599278i) q^{23} +(0.142315 - 0.989821i) q^{25} +(-0.254166 + 0.865611i) q^{27} +(-1.25667 + 0.368991i) q^{29} +(-0.420803 + 0.0605024i) q^{35} +(0.708089 + 0.817178i) q^{41} +(0.398174 + 0.871880i) q^{43} -1.56347i q^{45} -1.99490i q^{47} +(-0.340335 - 0.745229i) q^{49} +(-1.74557 - 0.797176i) q^{61} +(-0.637753 + 0.187261i) q^{63} +(0.266684 - 1.85483i) q^{67} +(-0.767317 + 1.40524i) q^{69} +(0.451077 + 1.53623i) q^{75} +(0.0169414 + 0.117830i) q^{81} +(-0.457701 + 0.528215i) q^{83} +(1.58479 - 1.37322i) q^{87} +(1.80075 - 0.822373i) q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 4 q^{9} + 4 q^{25} + 8 q^{29} - 4 q^{49} + 4 q^{69} + 4 q^{81}+O(q^{100}) $ 40 * q + 4 * q^9 + 4 * q^25 + 8 * q^29 - 4 * q^49 + 4 * q^69 + 4 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3680\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1381$	$3041$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{17}{22}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.45640	+	0.665114i	−1.45640	+	0.665114i	−0.977147	−	0.212565i	$-0.931818\pi$
$3$	−1.45640	+	0.665114i	−1.45640	+	0.665114i	−0.479249	+	0.877679i	$0.659091\pi$
$4$		0			0
$5$	0.755750	−	0.654861i	0.755750	−	0.654861i
$5$	0.755750	−	0.654861i	0.755750	−	0.654861i
$6$		0			0
$7$	−0.357643	−	0.229843i	−0.357643	−	0.229843i	0.349464	−	0.936950i	$-0.386364\pi$
$7$	−0.357643	−	0.229843i	−0.357643	−	0.229843i	−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$	1.02385	−	1.18159i	1.02385	−	1.18159i
$10$		0			0
$11$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$11$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$12$		0			0
$13$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$13$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$14$		0			0
$15$	−0.665114	+	1.45640i	−0.665114	+	1.45640i
$16$		0			0
$17$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$17$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$18$		0			0
$19$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$19$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$20$		0			0
$21$	0.673741	+	0.0968693i	0.673741	+	0.0968693i
$22$		0			0
$23$	0.800541	−	0.599278i	0.800541	−	0.599278i
$23$	0.800541	−	0.599278i	0.800541	−	0.599278i
$24$		0			0
$25$	0.142315	−	0.989821i	0.142315	−	0.989821i
$26$		0			0
$27$	−0.254166	+	0.865611i	−0.254166	+	0.865611i
$28$		0			0
$29$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$29$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.415415	+	0.909632i	$0.636364\pi$
$30$		0			0
$31$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$31$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.420803	+	0.0605024i	−0.420803	+	0.0605024i
$36$		0			0
$37$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$37$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.708089	+	0.817178i	0.708089	+	0.817178i	0.989821	−	0.142315i	$-0.0454545\pi$
$41$	0.708089	+	0.817178i	0.708089	+	0.817178i	−0.281733	+	0.959493i	$0.590909\pi$
$42$		0			0
$43$	0.398174	+	0.871880i	0.398174	+	0.871880i	0.997452	+	0.0713392i	$0.0227273\pi$
$43$	0.398174	+	0.871880i	0.398174	+	0.871880i	−0.599278	+	0.800541i	$0.704545\pi$
$44$		0			0
$45$		−	1.56347i		−	1.56347i
$46$		0			0
$47$		−	1.99490i		−	1.99490i	−0.0713392	−	0.997452i	$-0.522727\pi$
$47$		−	1.99490i		−	1.99490i	0.0713392	−	0.997452i	$-0.477273\pi$
$48$		0			0
$49$	−0.340335	−	0.745229i	−0.340335	−	0.745229i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$53$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$59$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$60$		0			0
$61$	−1.74557	−	0.797176i	−1.74557	−	0.797176i	−0.989821	−	0.142315i	$-0.954545\pi$
$61$	−1.74557	−	0.797176i	−1.74557	−	0.797176i	−0.755750	−	0.654861i	$-0.772727\pi$
$62$		0			0
$63$	−0.637753	+	0.187261i	−0.637753	+	0.187261i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.266684	−	1.85483i	0.266684	−	1.85483i	−0.212565	−	0.977147i	$-0.568182\pi$
$67$	0.266684	−	1.85483i	0.266684	−	1.85483i	0.479249	−	0.877679i	$-0.340909\pi$
$68$		0			0
$69$	−0.767317	+	1.40524i	−0.767317	+	1.40524i
$70$		0			0
$71$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$71$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$72$		0			0
$73$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$73$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$74$		0			0
$75$	0.451077	+	1.53623i	0.451077	+	1.53623i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$79$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$80$		0			0
$81$	0.0169414	+	0.117830i	0.0169414	+	0.117830i
$82$		0			0
$83$	−0.457701	+	0.528215i	−0.457701	+	0.528215i	−0.936950	−	0.349464i	$-0.886364\pi$
$83$	−0.457701	+	0.528215i	−0.457701	+	0.528215i	0.479249	+	0.877679i	$0.340909\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.58479	−	1.37322i	1.58479	−	1.37322i
$88$		0			0
$89$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$89$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$97$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$101$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$102$		0			0
$103$	−0.0203052	−	0.141226i	−0.0203052	−	0.141226i	0.977147	−	0.212565i	$-0.0681818\pi$
$103$	−0.0203052	−	0.141226i	−0.0203052	−	0.141226i	−0.997452	+	0.0713392i	$0.977273\pi$
$104$		0			0
$105$	0.572615	−	0.367998i	0.572615	−	0.367998i
$106$		0			0
$107$	0.0592707	−	0.129785i	0.0592707	−	0.129785i	−0.877679	−	0.479249i	$-0.840909\pi$
$107$	0.0592707	−	0.129785i	0.0592707	−	0.129785i	0.936950	+	0.349464i	$0.113636\pi$
$108$		0			0
$109$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	0.909632	−	0.415415i	$-0.136364\pi$
$109$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	−0.989821	+	0.142315i	$0.954545\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$113$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$114$		0			0
$115$	0.212565	−	0.977147i	0.212565	−	0.977147i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$122$		0			0
$123$	−1.57477	−	0.719175i	−1.57477	−	0.719175i
$124$		0			0
$125$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$126$		0			0
$127$	1.85483	−	0.266684i	1.85483	−	0.266684i	0.877679	−	0.479249i	$-0.159091\pi$
$127$	1.85483	−	0.266684i	1.85483	−	0.266684i	0.977147	+	0.212565i	$0.0681818\pi$
$128$		0			0
$129$	−1.15980	−	1.00497i	−1.15980	−	1.00497i
$130$		0			0
$131$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$131$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.374768	+	0.820628i	0.374768	+	0.820628i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	1.32684	+	2.90537i	1.32684	+	2.90537i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−0.708089	+	1.10181i	−0.708089	+	1.10181i
$146$		0			0
$147$	0.991324	+	0.858987i	0.991324	+	0.858987i
$148$		0			0
$149$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.540641	−	0.841254i	$-0.318182\pi$
$149$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.755750	+	0.654861i	$0.227273\pi$
$150$		0			0
$151$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$151$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$157$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.424047	+	0.0303285i	−0.424047	+	0.0303285i
$162$		0			0
$163$	1.73749	+	0.249813i	1.73749	+	0.249813i	0.936950	−	0.349464i	$-0.113636\pi$
$163$	1.73749	+	0.249813i	1.73749	+	0.249813i	0.800541	+	0.599278i	$0.204545\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.494541	−	1.68425i	−0.494541	−	1.68425i	−0.707107	−	0.707107i	$-0.750000\pi$
$167$	−0.494541	−	1.68425i	−0.494541	−	1.68425i	0.212565	−	0.977147i	$-0.431818\pi$
$168$		0			0
$169$	0.415415	−	0.909632i	0.415415	−	0.909632i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$173$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$174$		0			0
$175$	−0.278401	+	0.321292i	−0.278401	+	0.321292i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$179$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$180$		0			0
$181$	1.65486	−	0.755750i	1.65486	−	0.755750i	0.654861	−	0.755750i	$-0.272727\pi$
$181$	1.65486	−	0.755750i	1.65486	−	0.755750i	1.00000			$0$
$182$		0			0
$183$	3.07245			3.07245
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.289855	−	0.251161i	0.289855	−	0.251161i
$190$		0			0
$191$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$191$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$192$		0			0
$193$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$193$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$197$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$198$		0			0
$199$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$199$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$200$		0			0
$201$	0.845273	+	2.87874i	0.845273	+	2.87874i
$202$		0			0
$203$	0.534248	+	0.156869i	0.534248	+	0.156869i
$204$		0			0
$205$	1.07028	+	0.153882i	1.07028	+	0.153882i
$206$		0			0
$207$	0.111536	−	1.55948i	0.111536	−	1.55948i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$211$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.871880	+	0.398174i	0.871880	+	0.398174i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.518203	+	0.806340i	−0.518203	+	0.806340i	−0.997452	−	0.0713392i	$-0.977273\pi$
$223$	−0.518203	+	0.806340i	−0.518203	+	0.806340i	0.479249	+	0.877679i	$0.340909\pi$
$224$		0			0
$225$	−1.02385	−	1.18159i	−1.02385	−	1.18159i
$226$		0			0
$227$	0.828713	+	1.81463i	0.828713	+	1.81463i	0.479249	+	0.877679i	$0.340909\pi$
$227$	0.828713	+	1.81463i	0.828713	+	1.81463i	0.349464	+	0.936950i	$0.386364\pi$
$228$		0			0
$229$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$229$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$233$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$234$		0			0
$235$	−1.30638	−	1.50765i	−1.30638	−	1.50765i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$239$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$240$		0			0
$241$	−1.66538	+	0.239446i	−1.66538	+	0.239446i	−0.909632	−	0.415415i	$-0.863636\pi$
$241$	−1.66538	+	0.239446i	−1.66538	+	0.239446i	−0.755750	+	0.654861i	$0.772727\pi$
$242$		0			0
$243$	−0.590785	−	0.919280i	−0.590785	−	0.919280i
$244$		0			0
$245$	−0.745229	−	0.340335i	−0.745229	−	0.340335i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.315271	−	1.07371i	0.315271	−	1.07371i
$250$		0			0
$251$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$251$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$257$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.850647	+	1.86266i	−0.850647	+	1.86266i
$262$		0			0
$263$	0.806340	−	0.518203i	0.806340	−	0.518203i	−0.0713392	−	0.997452i	$-0.522727\pi$
$263$	0.806340	−	0.518203i	0.806340	−	0.518203i	0.877679	+	0.479249i	$0.159091\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.07563	+	2.39540i	−2.07563	+	2.39540i
$268$		0			0
$269$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.989821	+	0.142315i	$0.0454545\pi$
$269$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.540641	+	0.841254i	$0.318182\pi$
$270$		0			0
$271$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$271$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.45027	+	1.25667i	−1.45027	+	1.25667i	−0.540641	+	0.841254i	$0.681818\pi$
$281$	−1.45027	+	1.25667i	−1.45027	+	1.25667i	−0.909632	+	0.415415i	$0.863636\pi$
$282$		0			0
$283$	−1.00829	−	0.647988i	−1.00829	−	0.647988i	−0.0713392	−	0.997452i	$-0.522727\pi$
$283$	−1.00829	−	0.647988i	−1.00829	−	0.647988i	−0.936950	+	0.349464i	$0.886364\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.0654201	−	0.455007i	−0.0654201	−	0.455007i
$288$		0			0
$289$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$293$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	0.0579914	−	0.403339i	0.0579914	−	0.403339i
$302$		0			0
$303$	−0.374768	+	1.27634i	−0.374768	+	1.27634i
$304$		0			0
$305$	−1.84125	+	0.540641i	−1.84125	+	0.540641i
$306$		0			0
$307$	−1.77769	−	0.811843i	−1.77769	−	0.811843i	−0.977147	−	0.212565i	$-0.931818\pi$
$307$	−1.77769	−	0.811843i	−1.77769	−	0.811843i	−0.800541	−	0.599278i	$-0.795455\pi$
$308$		0			0
$309$	0.123504	+	0.192176i	0.123504	+	0.192176i
$310$		0			0
$311$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$311$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$312$		0			0
$313$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$313$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$314$		0			0
$315$	−0.359351	+	0.559162i	−0.359351	+	0.559162i
$316$		0			0
$317$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$317$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$			0.228440i			0.228440i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.298430	+	0.344407i	0.298430	+	0.344407i
$328$		0			0
$329$	−0.458515	+	0.713463i	−0.458515	+	0.713463i
$330$		0			0
$331$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$331$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.01311	−	1.57642i	−1.01311	−	1.57642i
$336$		0			0
$337$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$337$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.110070	+	0.765552i	−0.110070	+	0.765552i
$344$		0			0
$345$	0.340335	+	1.56449i	0.340335	+	1.56449i
$346$		0			0
$347$	−1.85483	−	0.266684i	−1.85483	−	0.266684i	−0.877679	−	0.479249i	$-0.840909\pi$
$347$	−1.85483	−	0.266684i	−1.85483	−	0.266684i	−0.977147	+	0.212565i	$0.931818\pi$
$348$		0			0
$349$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$349$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$353$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$359$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$360$		0			0
$361$	0.841254	+	0.540641i	0.841254	+	0.540641i
$362$		0			0
$363$	1.21002	−	1.04849i	1.21002	−	1.04849i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.698928			−0.698928			−0.349464	−	0.936950i	$-0.613636\pi$
$367$	−0.698928			−0.698928			−0.349464	+	0.936950i	$0.613636\pi$
$368$		0			0
$369$	1.69055			1.69055
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$373$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$374$		0			0
$375$	1.34692	+	0.865611i	1.34692	+	0.865611i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$379$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$380$		0			0
$381$	−2.52399	+	1.62207i	−2.52399	+	1.62207i
$382$		0			0
$383$	−0.811843	+	1.77769i	−0.811843	+	1.77769i	−0.212565	+	0.977147i	$0.568182\pi$
$383$	−0.811843	+	1.77769i	−0.811843	+	1.77769i	−0.599278	+	0.800541i	$0.704545\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.43788	+	0.422198i	1.43788	+	0.422198i
$388$		0			0
$389$	0.281733	+	0.0405070i	0.281733	+	0.0405070i	0.281733	−	0.959493i	$-0.409091\pi$
$389$	0.281733	+	0.0405070i	0.281733	+	0.0405070i			1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$397$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$401$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.0899659	+	0.0779559i	0.0899659	+	0.0779559i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.989821	−	1.14231i	−0.989821	−	1.14231i	−0.989821	−	0.142315i	$-0.954545\pi$
$409$	−0.989821	−	1.14231i	−0.989821	−	1.14231i		−	1.00000i	$-0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$			0.698928i			0.698928i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$419$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$420$		0			0
$421$	0.909632	−	1.41542i	0.909632	−	1.41542i		−	1.00000i	$-0.5\pi$
$421$	0.909632	−	1.41542i	0.909632	−	1.41542i	0.909632	−	0.415415i	$-0.136364\pi$
$422$		0			0
$423$	−2.35716	−	2.04249i	−2.35716	−	2.04249i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.441065	+	0.686311i	0.441065	+	0.686311i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$431$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$432$		0			0
$433$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$433$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$434$		0			0
$435$	0.298430	−	2.07563i	0.298430	−	2.07563i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$439$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$440$		0			0
$441$	−1.22901	−	0.360869i	−1.22901	−	0.360869i
$442$		0			0
$443$	0.270040	+	0.919672i	0.270040	+	0.919672i	0.977147	+	0.212565i	$0.0681818\pi$
$443$	0.270040	+	0.919672i	0.270040	+	0.919672i	−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	0.822373	−	1.80075i	0.822373	−	1.80075i
$446$		0			0
$447$	−1.76409	+	1.13371i	−1.76409	+	1.13371i
$448$		0			0
$449$	0.281733	+	1.95949i	0.281733	+	1.95949i	0.281733	+	0.959493i	$0.409091\pi$
$449$	0.281733	+	1.95949i	0.281733	+	1.95949i			1.00000i	$0.500000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$457$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$461$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$462$		0			0
$463$	−0.386712	+	0.176606i	−0.386712	+	0.176606i	−0.599278	−	0.800541i	$-0.704545\pi$
$463$	−0.386712	+	0.176606i	−0.386712	+	0.176606i	0.212565	+	0.977147i	$0.431818\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.18971	+	0.764582i	1.18971	+	0.764582i	0.977147	−	0.212565i	$-0.0681818\pi$
$467$	1.18971	+	0.764582i	1.18971	+	0.764582i	0.212565	+	0.977147i	$0.431818\pi$
$468$		0			0
$469$	−0.521696	+	0.602069i	−0.521696	+	0.602069i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$479$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	0.597409	−	0.326210i	0.597409	−	0.326210i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−0.550588	+	1.87513i	−0.550588	+	1.87513i	−0.0713392	+	0.997452i	$0.522727\pi$
$487$	−0.550588	+	1.87513i	−0.550588	+	1.87513i	−0.479249	+	0.877679i	$0.659091\pi$
$488$		0			0
$489$	−2.69663	+	0.791802i	−2.69663	+	0.791802i
$490$		0			0
$491$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$491$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$499$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$500$		0			0
$501$	1.84047	+	2.12401i	1.84047	+	2.12401i
$502$		0			0
$503$	−0.497898	−	1.09024i	−0.497898	−	1.09024i	−0.977147	−	0.212565i	$-0.931818\pi$
$503$	−0.497898	−	1.09024i	−0.497898	−	1.09024i	0.479249	−	0.877679i	$-0.340909\pi$
$504$		0			0
$505$		−	0.830830i		−	0.830830i
$506$		0			0
$507$			1.60108i			1.60108i
$508$		0			0
$509$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.654861	+	0.755750i	$0.272727\pi$
$509$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.107829	−	0.0934345i	−0.107829	−	0.0934345i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.37491	−	0.627899i	−1.37491	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$521$	−1.37491	−	0.627899i	−1.37491	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$522$		0			0
$523$	0.670617	−	0.196911i	0.670617	−	0.196911i	0.0713392	−	0.997452i	$-0.477273\pi$
$523$	0.670617	−	0.196911i	0.670617	−	0.196911i	0.599278	+	0.800541i	$0.295455\pi$
$524$		0			0
$525$	0.191767	−	0.653097i	0.191767	−	0.653097i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.281733	−	0.959493i	0.281733	−	0.959493i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.0401971	−	0.136899i	−0.0401971	−	0.136899i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.118239	+	0.822373i	0.118239	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$541$	0.118239	+	0.822373i	0.118239	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$542$		0			0
$543$	−1.90747	+	2.20134i	−1.90747	+	2.20134i
$544$		0			0
$545$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$546$		0			0
$547$	1.50765	−	1.30638i	1.50765	−	1.30638i	0.707107	−	0.707107i	$-0.250000\pi$
$547$	1.50765	−	1.30638i	1.50765	−	1.30638i	0.800541	−	0.599278i	$-0.204545\pi$
$548$		0			0
$549$	−2.72914	+	1.24636i	−2.72914	+	1.24636i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$557$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.278125	+	1.93440i	0.278125	+	1.93440i	0.349464	+	0.936950i	$0.386364\pi$
$563$	0.278125	+	1.93440i	0.278125	+	1.93440i	−0.0713392	+	0.997452i	$0.522727\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.0210235	−	0.0460350i	0.0210235	−	0.0460350i
$568$		0			0
$569$	0.234072	+	0.797176i	0.234072	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$569$	0.234072	+	0.797176i	0.234072	+	0.797176i	−0.755750	+	0.654861i	$0.772727\pi$
$570$		0			0
$571$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$571$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.479249	−	0.877679i	−0.479249	−	0.877679i
$576$		0			0
$577$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$577$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.285100	−	0.0837128i	0.285100	−	0.0837128i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.73749	+	0.249813i	−1.73749	+	0.249813i	−0.936950	−	0.349464i	$-0.886364\pi$
$587$	−1.73749	+	0.249813i	−1.73749	+	0.249813i	−0.800541	+	0.599278i	$0.795455\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$593$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	0.822373	+	1.80075i	0.822373	+	1.80075i	0.540641	+	0.841254i	$0.318182\pi$
$601$	0.822373	+	1.80075i	0.822373	+	1.80075i	0.281733	+	0.959493i	$0.409091\pi$
$602$		0			0
$603$	−1.91860	−	2.21418i	−1.91860	−	2.21418i
$604$		0			0
$605$	−0.540641	+	0.841254i	−0.540641	+	0.841254i
$606$		0			0
$607$	−1.32661	−	1.14952i	−1.32661	−	1.14952i	−0.977147	−	0.212565i	$-0.931818\pi$
$607$	−1.32661	−	1.14952i	−1.32661	−	1.14952i	−0.349464	−	0.936950i	$-0.613636\pi$
$608$		0			0
$609$	−0.882413	+	0.126872i	−0.882413	+	0.126872i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$613$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$614$		0			0
$615$	−1.66109	+	0.487741i	−1.66109	+	0.487741i
$616$		0			0
$617$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$617$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$618$		0			0
$619$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$619$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$620$		0			0
$621$	0.315271	+	0.845273i	0.315271	+	0.845273i
$622$		0			0
$623$	−0.833040	−	0.119773i	−0.833040	−	0.119773i
$624$		0			0
$625$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$631$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.22714	−	1.41620i	1.22714	−	1.41620i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.281733	−	0.959493i	$-0.409091\pi$
$641$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.909632	+	0.415415i	$0.136364\pi$
$642$		0			0
$643$	1.75536			1.75536			0.877679	−	0.479249i	$-0.159091\pi$
$643$	1.75536			1.75536			0.877679	+	0.479249i	$0.159091\pi$
$644$		0			0
$645$	−1.53463			−1.53463
$646$		0			0
$647$	−0.635768	+	0.290345i	−0.635768	+	0.290345i	−0.707107	−	0.707107i	$-0.750000\pi$
$647$	−0.635768	+	0.290345i	−0.635768	+	0.290345i	0.0713392	+	0.997452i	$0.477273\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$653$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$659$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$660$		0			0
$661$	−0.368991	−	1.25667i	−0.368991	−	1.25667i	−0.909632	−	0.415415i	$-0.863636\pi$
$661$	−0.368991	−	1.25667i	−0.368991	−	1.25667i	0.540641	−	0.841254i	$-0.318182\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.784887	+	1.04849i	−0.784887	+	1.04849i
$668$		0			0
$669$	0.218401	−	1.51901i	0.218401	−	1.51901i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$673$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$674$		0			0
$675$	0.820628	+	0.374768i	0.820628	+	0.374768i
$676$		0			0
$677$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$677$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−2.41387	−	2.09163i	−2.41387	−	2.09163i
$682$		0			0
$683$	1.05657	−	1.64406i	1.05657	−	1.64406i	0.349464	−	0.936950i	$-0.386364\pi$
$683$	1.05657	−	1.64406i	1.05657	−	1.64406i	0.707107	−	0.707107i	$-0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.31669	+	2.88314i	1.31669	+	2.88314i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.89945	+	0.273100i	−1.89945	+	0.273100i	−0.989821	−	0.142315i	$-0.954545\pi$
$701$	−1.89945	+	0.273100i	−1.89945	+	0.273100i	−0.909632	+	0.415415i	$0.863636\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	2.90537	+	1.32684i	2.90537	+	1.32684i
$706$		0			0
$707$	−0.338904	+	0.0995111i	−0.338904	+	0.0995111i
$708$		0			0
$709$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$709$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$719$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$720$		0			0
$721$	−0.0251978	+	0.0551755i	−0.0251978	+	0.0551755i
$722$		0			0
$723$	2.26620	−	1.45640i	2.26620	−	1.45640i
$724$		0			0
$725$	0.186393	+	1.29639i	0.186393	+	1.29639i
$726$		0			0
$727$	1.30638	−	1.50765i	1.30638	−	1.50765i	0.599278	−	0.800541i	$-0.295455\pi$
$727$	1.30638	−	1.50765i	1.30638	−	1.50765i	0.707107	−	0.707107i	$-0.250000\pi$
$728$		0			0
$729$	1.37170	+	0.881537i	1.37170	+	0.881537i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$733$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$734$		0			0
$735$	1.31171			1.31171
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$739$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.67822	+	1.07853i	1.67822	+	1.07853i	0.877679	+	0.479249i	$0.159091\pi$
$743$	1.67822	+	1.07853i	1.67822	+	1.07853i	0.800541	+	0.599278i	$0.204545\pi$
$744$		0			0
$745$	0.857685	−	0.989821i	0.857685	−	0.989821i
$746$		0			0
$747$	0.155515	+	1.08163i	0.155515	+	1.08163i
$748$		0			0
$749$	−0.0510279	+	0.0327936i	−0.0510279	+	0.0327936i
$750$		0			0
$751$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$751$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$757$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$761$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$762$		0			0
$763$	−0.0340910	+	0.116103i	−0.0340910	+	0.116103i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$769$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$773$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	1.18157i		−	1.18157i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1.04849	+	1.21002i	1.04849	+	1.21002i	0.977147	+	0.212565i	$0.0681818\pi$
$787$	1.04849	+	1.21002i	1.04849	+	1.21002i	0.0713392	+	0.997452i	$0.477273\pi$
$788$		0			0
$789$	−0.829686	+	1.29102i	−0.829686	+	1.29102i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$797$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.871991	−	2.96973i	0.871991	−	2.96973i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−0.300613	+	0.300613i	−0.300613	+	0.300613i
$806$		0			0
$807$	−2.88314	−	0.414533i	−2.88314	−	0.414533i
$808$		0			0
$809$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	0.142315	−	0.989821i	$-0.454545\pi$
$809$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	−0.415415	+	0.909632i	$0.636364\pi$
$810$		0			0
$811$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$811$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.47670	−	0.949018i	1.47670	−	0.949018i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.27155	−	0.817178i	−1.27155	−	0.817178i	−0.281733	−	0.959493i	$-0.590909\pi$
$821$	−1.27155	−	0.817178i	−1.27155	−	0.817178i	−0.989821	+	0.142315i	$0.954545\pi$
$822$		0			0
$823$	−0.321292	+	0.278401i	−0.321292	+	0.278401i	−0.800541	−	0.599278i	$-0.795455\pi$
$823$	−0.321292	+	0.278401i	−0.321292	+	0.278401i	0.479249	+	0.877679i	$0.340909\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$827$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−1.08128			−1.08128			−0.540641	−	0.841254i	$-0.681818\pi$
$829$	−1.08128			−1.08128			−0.540641	+	0.841254i	$0.681818\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1.47670	−	0.949018i	−1.47670	−	0.949018i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$839$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$840$		0			0
$841$	0.601808	−	0.386758i	0.601808	−	0.386758i
$842$		0			0
$843$	1.27634	−	2.79480i	1.27634	−	2.79480i
$844$		0			0
$845$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$846$		0			0
$847$	0.407910	+	0.119773i	0.407910	+	0.119773i
$848$		0			0
$849$	1.89945	+	0.273100i	1.89945	+	0.273100i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$853$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$857$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$858$		0			0
$859$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$859$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$860$		0			0
$861$	0.397909	+	0.619158i	0.397909	+	0.619158i
$862$		0			0
$863$	1.93440	−	0.278125i	1.93440	−	0.278125i	0.936950	−	0.349464i	$-0.113636\pi$
$863$	1.93440	−	0.278125i	1.93440	−	0.278125i	0.997452	+	0.0713392i	$0.0227273\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.865611	−	1.34692i	0.865611	−	1.34692i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			0.425131i			0.425131i
$876$		0			0
$877$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$877$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.449181	−	0.698939i	0.449181	−	0.698939i	−0.540641	−	0.841254i	$-0.681818\pi$
$881$	0.449181	−	0.698939i	0.449181	−	0.698939i	0.989821	+	0.142315i	$0.0454545\pi$
$882$		0			0
$883$	−0.905808	−	0.784887i	−0.905808	−	0.784887i	0.0713392	−	0.997452i	$-0.477273\pi$
$883$	−0.905808	−	0.784887i	−0.905808	−	0.784887i	−0.977147	+	0.212565i	$0.931818\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.647988	+	1.00829i	0.647988	+	1.00829i	0.997452	+	0.0713392i	$0.0227273\pi$
$887$	0.647988	+	1.00829i	0.647988	+	1.00829i	−0.349464	+	0.936950i	$0.613636\pi$
$888$		0			0
$889$	−0.724660	−	0.330941i	−0.724660	−	0.330941i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	0.183808	+	0.625992i	0.183808	+	0.625992i
$904$		0			0
$905$	0.755750	−	1.65486i	0.755750	−	1.65486i
$906$		0			0
$907$	−1.34692	+	0.865611i	−1.34692	+	0.865611i	−0.997452	−	0.0713392i	$-0.977273\pi$
$907$	−1.34692	+	0.865611i	−1.34692	+	0.865611i	−0.349464	+	0.936950i	$0.613636\pi$
$908$		0			0
$909$	−0.184863	−	1.28575i	−0.184863	−	1.28575i
$910$		0			0
$911$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$911$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	2.32201	−	2.01203i	2.32201	−	2.01203i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	3.12899			3.12899
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.187661	−	0.120602i	−0.187661	−	0.120602i
$928$		0			0
$929$	−0.989821	+	1.14231i	−0.989821	+	1.14231i			1.00000i	$0.5\pi$
$929$	−0.989821	+	1.14231i	−0.989821	+	1.14231i	−0.989821	+	0.142315i	$0.954545\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$937$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$941$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$942$		0			0
$943$	1.05657	+	0.229843i	1.05657	+	0.229843i
$944$		0			0
$945$	0.0545825	−	0.379629i	0.0545825	−	0.379629i
$946$		0			0
$947$	−0.337672	+	1.15001i	−0.337672	+	1.15001i	0.599278	+	0.800541i	$0.295455\pi$
$947$	−0.337672	+	1.15001i	−0.337672	+	1.15001i	−0.936950	+	0.349464i	$0.886364\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$953$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.654861	+	0.755750i	0.654861	+	0.755750i
$962$		0			0
$963$	−0.0926677	−	0.202914i	−0.0926677	−	0.202914i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1.19856i			1.19856i	0.800541	+	0.599278i	$0.204545\pi$
$967$			1.19856i			1.19856i	−0.800541	+	0.599278i	$0.795455\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$971$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$977$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.404794	−	0.184863i	−0.404794	−	0.184863i
$982$		0			0
$983$	−1.15001	+	0.337672i	−1.15001	+	0.337672i	−0.800541	−	0.599278i	$-0.795455\pi$
$983$	−1.15001	+	0.337672i	−1.15001	+	0.337672i	−0.349464	+	0.936950i	$0.613636\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	0.193245	−	1.34405i	0.193245	−	1.34405i
$988$		0			0
$989$	0.841254	+	0.459359i	0.841254	+	0.459359i
$990$		0			0
$991$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$991$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$997$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3680.1.co.a.3649.1	yes	40
4.3	odd	2	inner	3680.1.co.a.3649.4	yes	40
5.4	even	2	inner	3680.1.co.a.3649.4	yes	40
20.19	odd	2	CM	3680.1.co.a.3649.1	yes	40
23.20	odd	22	inner	3680.1.co.a.2849.4	yes	40
92.43	even	22	inner	3680.1.co.a.2849.1	✓	40
115.89	odd	22	inner	3680.1.co.a.2849.1	✓	40
460.319	even	22	inner	3680.1.co.a.2849.4	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.1.co.a.2849.1	✓	40	92.43	even	22	inner
3680.1.co.a.2849.1	✓	40	115.89	odd	22	inner
3680.1.co.a.2849.4	yes	40	23.20	odd	22	inner
3680.1.co.a.2849.4	yes	40	460.319	even	22	inner
3680.1.co.a.3649.1	yes	40	1.1	even	1	trivial
3680.1.co.a.3649.1	yes	40	20.19	odd	2	CM
3680.1.co.a.3649.4	yes	40	4.3	odd	2	inner
3680.1.co.a.3649.4	yes	40	5.4	even	2	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:	\( N \)	\(=\)	\( 3680 = 2^{5} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3680.co (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.45640 + 0.665114i) q^{3} +(0.755750 - 0.654861i) q^{5} +(-0.357643 - 0.229843i) q^{7} +(1.02385 - 1.18159i) q^{9} +(-0.665114 + 1.45640i) q^{15} +(0.673741 + 0.0968693i) q^{21} +(0.800541 - 0.599278i) q^{23} +(0.142315 - 0.989821i) q^{25} +(-0.254166 + 0.865611i) q^{27} +(-1.25667 + 0.368991i) q^{29} +(-0.420803 + 0.0605024i) q^{35} +(0.708089 + 0.817178i) q^{41} +(0.398174 + 0.871880i) q^{43} -1.56347i q^{45} -1.99490i q^{47} +(-0.340335 - 0.745229i) q^{49} +(-1.74557 - 0.797176i) q^{61} +(-0.637753 + 0.187261i) q^{63} +(0.266684 - 1.85483i) q^{67} +(-0.767317 + 1.40524i) q^{69} +(0.451077 + 1.53623i) q^{75} +(0.0169414 + 0.117830i) q^{81} +(-0.457701 + 0.528215i) q^{83} +(1.58479 - 1.37322i) q^{87} +(1.80075 - 0.822373i) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{9} + 4 q^{25} + 8 q^{29} - 4 q^{49} + 4 q^{69} + 4 q^{81}+O(q^{100}) \) 40 * q + 4 * q^9 + 4 * q^25 + 8 * q^29 - 4 * q^49 + 4 * q^69 + 4 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more