LMFDB - Embedded newform 3204.1.cd.b.719.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3204,1,Mod(35,3204)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3204, base_ring=CyclotomicField(88))

chi = DirichletCharacter(H, H._module([44, 44, 63]))

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

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

chi := DirichletCharacter("3204.35");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3204 = 2^{2} \cdot 3^{2} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3204.cd (of order $88$, degree $40$, 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:	$1.59900430048$
Analytic rank:	$0$
Dimension:	$40$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{88}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{88} - \cdots)$

Embedding invariants

Embedding label			719.1
Root			$-0.349464 - 0.936950i$ of defining polynomial
Character	$\chi$	$=$	3204.719
Dual form			3204.1.cd.b.1475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.755750 + 0.654861i) q^{2} +(0.142315 + 0.989821i) q^{4} +(0.550588 - 0.119773i) q^{5} +(-0.540641 + 0.841254i) q^{8} +O(q^{10})$	\(q+(0.755750 + 0.654861i) q^{2} +(0.142315 + 0.989821i) q^{4} +(0.550588 - 0.119773i) q^{5} +(-0.540641 + 0.841254i) q^{8} +(0.494541 + 0.270040i) q^{10} +(0.183863 + 0.109091i) q^{13} +(-0.959493 + 0.281733i) q^{16} +(1.86912 + 0.133682i) q^{17} +(0.196911 + 0.527938i) q^{20} +(-0.620830 + 0.283524i) q^{25} +(0.0675149 + 0.202850i) q^{26} +(1.11946 - 0.777256i) q^{29} +(-0.909632 - 0.415415i) q^{32} +(1.32505 + 1.32505i) q^{34} +(-0.583522 - 0.241703i) q^{37} +(-0.196911 + 0.527938i) q^{40} +(-1.47925 + 0.877679i) q^{41} +(-0.936950 - 0.349464i) q^{49} +(-0.654861 - 0.192284i) q^{50} +(-0.0818140 + 0.197516i) q^{52} +(1.19550 + 1.59700i) q^{53} +(1.35503 + 0.145681i) q^{58} +(-0.0126746 + 0.354880i) q^{61} +(-0.415415 - 0.909632i) q^{64} +(0.114299 + 0.0380423i) q^{65} +(0.133682 + 1.86912i) q^{68} +(0.0801894 + 0.273100i) q^{73} +(-0.282715 - 0.564792i) q^{74} +(-0.494541 + 0.270040i) q^{80} +(-1.69270 - 0.305397i) q^{82} +(1.04513 - 0.150267i) q^{85} +(0.212565 - 0.977147i) q^{89} +(-1.34692 - 0.865611i) q^{97} +(-0.479249 - 0.877679i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 4 q^{4}+O(q^{10}) $ 40 * q + 4 * q^4	$ 40 q + 4 q^{4} + 4 q^{13} - 4 q^{16} + 4 q^{17} + 4 q^{29} - 4 q^{34} - 4 q^{37} - 40 q^{41} - 4 q^{50} - 4 q^{52} + 4 q^{53} + 4 q^{64} - 4 q^{68}+O(q^{100}) $ 40 * q + 4 * q^4 + 4 * q^13 - 4 * q^16 + 4 * q^17 + 4 * q^29 - 4 * q^34 - 4 * q^37 - 40 * q^41 - 4 * q^50 - 4 * q^52 + 4 * q^53 + 4 * q^64 - 4 * q^68

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times$.

$n$	$181$	$713$	$1603$
$\chi(n)$	$e\left(\frac{81}{88}\right)$	$-1$	$-1$

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 $

$2$	0.755750	+	0.654861i	0.755750	+	0.654861i
$2$	0.755750	+	0.654861i	0.755750	+	0.654861i
$3$		0			0
$3$		0			0
$4$	0.142315	+	0.989821i	0.142315	+	0.989821i
$5$	0.550588	−	0.119773i	0.550588	−	0.119773i	0.0713392	−	0.997452i	$-0.477273\pi$
$5$	0.550588	−	0.119773i	0.550588	−	0.119773i	0.479249	+	0.877679i	$0.340909\pi$
$6$		0			0
$7$		0			0		0.177553	−	0.984111i	$-0.443182\pi$
$7$		0			0		−0.177553	+	0.984111i	$0.556818\pi$
$8$	−0.540641	+	0.841254i	−0.540641	+	0.841254i
$9$		0			0
$10$	0.494541	+	0.270040i	0.494541	+	0.270040i
$11$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$11$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$12$		0			0
$13$	0.183863	+	0.109091i	0.183863	+	0.109091i	0.599278	−	0.800541i	$-0.295455\pi$
$13$	0.183863	+	0.109091i	0.183863	+	0.109091i	−0.415415	+	0.909632i	$0.636364\pi$
$14$		0			0
$15$		0			0
$16$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$17$	1.86912	+	0.133682i	1.86912	+	0.133682i	0.959493	−	0.281733i	$-0.0909091\pi$
$17$	1.86912	+	0.133682i	1.86912	+	0.133682i	0.909632	+	0.415415i	$0.136364\pi$
$18$		0			0
$19$		0			0		−0.778642	−	0.627469i	$-0.784091\pi$
$19$		0			0		0.778642	+	0.627469i	$0.215909\pi$
$20$	0.196911	+	0.527938i	0.196911	+	0.527938i
$21$		0			0
$22$		0			0
$23$		0			0		0.994270	−	0.106895i	$-0.0340909\pi$
$23$		0			0		−0.994270	+	0.106895i	$0.965909\pi$
$24$		0			0
$25$	−0.620830	+	0.283524i	−0.620830	+	0.283524i
$26$	0.0675149	+	0.202850i	0.0675149	+	0.202850i
$27$		0			0
$28$		0			0
$29$	1.11946	−	0.777256i	1.11946	−	0.777256i	0.142315	−	0.989821i	$-0.454545\pi$
$29$	1.11946	−	0.777256i	1.11946	−	0.777256i	0.977147	+	0.212565i	$0.0681818\pi$
$30$		0			0
$31$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$31$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$32$	−0.909632	−	0.415415i	−0.909632	−	0.415415i
$33$		0			0
$34$	1.32505	+	1.32505i	1.32505	+	1.32505i
$35$		0			0
$36$		0			0
$37$	−0.583522	−	0.241703i	−0.583522	−	0.241703i	0.0713392	−	0.997452i	$-0.477273\pi$
$37$	−0.583522	−	0.241703i	−0.583522	−	0.241703i	−0.654861	+	0.755750i	$0.727273\pi$
$38$		0			0
$39$		0			0
$40$	−0.196911	+	0.527938i	−0.196911	+	0.527938i
$41$	−1.47925	+	0.877679i	−1.47925	+	0.877679i	−0.479249	+	0.877679i	$0.659091\pi$
$41$	−1.47925	+	0.877679i	−1.47925	+	0.877679i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.570323	−	0.821421i	$-0.306818\pi$
$43$		0			0		−0.570323	+	0.821421i	$0.693182\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$47$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$48$		0			0
$49$	−0.936950	−	0.349464i	−0.936950	−	0.349464i
$50$	−0.654861	−	0.192284i	−0.654861	−	0.192284i
$51$		0			0
$52$	−0.0818140	+	0.197516i	−0.0818140	+	0.197516i
$53$	1.19550	+	1.59700i	1.19550	+	1.59700i	0.654861	+	0.755750i	$0.272727\pi$
$53$	1.19550	+	1.59700i	1.19550	+	1.59700i	0.540641	+	0.841254i	$0.318182\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.35503	+	0.145681i	1.35503	+	0.145681i
$59$		0			0		−0.968937	−	0.247307i	$-0.920455\pi$
$59$		0			0		0.968937	+	0.247307i	$0.0795455\pi$
$60$		0			0
$61$	−0.0126746	+	0.354880i	−0.0126746	+	0.354880i	0.977147	+	0.212565i	$0.0681818\pi$
$61$	−0.0126746	+	0.354880i	−0.0126746	+	0.354880i	−0.989821	+	0.142315i	$0.954545\pi$
$62$		0			0
$63$		0			0
$64$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$65$	0.114299	+	0.0380423i	0.114299	+	0.0380423i
$66$		0			0
$67$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$67$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$68$	0.133682	+	1.86912i	0.133682	+	1.86912i
$69$		0			0
$70$		0			0
$71$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$71$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$72$		0			0
$73$	0.0801894	+	0.273100i	0.0801894	+	0.273100i	0.989821	−	0.142315i	$-0.0454545\pi$
$73$	0.0801894	+	0.273100i	0.0801894	+	0.273100i	−0.909632	+	0.415415i	$0.863636\pi$
$74$	−0.282715	−	0.564792i	−0.282715	−	0.564792i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$79$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$80$	−0.494541	+	0.270040i	−0.494541	+	0.270040i
$81$		0			0
$82$	−1.69270	−	0.305397i	−1.69270	−	0.305397i
$83$		0			0		0.447617	−	0.894225i	$-0.352273\pi$
$83$		0			0		−0.447617	+	0.894225i	$0.647727\pi$
$84$		0			0
$85$	1.04513	−	0.150267i	1.04513	−	0.150267i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.212565	−	0.977147i	0.212565	−	0.977147i
$89$	0.212565	−	0.977147i	0.212565	−	0.977147i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.34692	−	0.865611i	−1.34692	−	0.865611i	−0.349464	−	0.936950i	$-0.613636\pi$
$97$	−1.34692	−	0.865611i	−1.34692	−	0.865611i	−0.997452	+	0.0713392i	$0.977273\pi$
$98$	−0.479249	−	0.877679i	−0.479249	−	0.877679i
$99$		0			0
$100$	−0.368991	−	0.574161i	−0.368991	−	0.574161i
$101$	1.79036	−	0.741593i	1.79036	−	0.741593i	0.800541	−	0.599278i	$-0.204545\pi$
$101$	1.79036	−	0.741593i	1.79036	−	0.741593i	0.989821	−	0.142315i	$-0.0454545\pi$
$102$		0			0
$103$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$103$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$104$	−0.191177	+	0.0956962i	−0.191177	+	0.0956962i
$105$		0			0
$106$	−0.142315	+	1.98982i	−0.142315	+	1.98982i
$107$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$107$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$108$		0			0
$109$	−1.85483	+	0.691814i	−1.85483	+	0.691814i	−0.877679	+	0.479249i	$0.840909\pi$
$109$	−1.85483	+	0.691814i	−1.85483	+	0.691814i	−0.977147	+	0.212565i	$0.931818\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.599278	−	1.80054i	0.599278	−	1.80054i		−	1.00000i	$-0.5\pi$
$113$	0.599278	−	1.80054i	0.599278	−	1.80054i	0.599278	−	0.800541i	$-0.295455\pi$
$114$		0			0
$115$		0			0
$116$	0.928661	+	0.997452i	0.928661	+	0.997452i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.415415	−	0.909632i	0.415415	−	0.909632i
$122$	−0.241976	+	0.259900i	−0.241976	+	0.259900i
$123$		0			0
$124$		0			0
$125$	−0.758940	+	0.568136i	−0.758940	+	0.568136i
$126$		0			0
$127$		0			0		0.984111	−	0.177553i	$-0.0568182\pi$
$127$		0			0		−0.984111	+	0.177553i	$0.943182\pi$
$128$	0.281733	−	0.959493i	0.281733	−	0.959493i
$129$		0			0
$130$	0.0614688	+	0.103600i	0.0614688	+	0.103600i
$131$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$131$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.12299	+	1.50013i	−1.12299	+	1.50013i
$137$	−0.0364255	−	0.0613918i	−0.0364255	−	0.0613918i	0.841254	−	0.540641i	$-0.181818\pi$
$137$	−0.0364255	−	0.0613918i	−0.0364255	−	0.0613918i	−0.877679	+	0.479249i	$0.840909\pi$
$138$		0			0
$139$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$139$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.523268	−	0.562029i	0.523268	−	0.562029i
$146$	−0.118239	+	0.258908i	−0.118239	+	0.258908i
$147$		0			0
$148$	0.156199	−	0.611980i	0.156199	−	0.611980i
$149$	−0.436506	−	0.628688i	−0.436506	−	0.628688i	0.540641	−	0.841254i	$-0.318182\pi$
$149$	−0.436506	−	0.628688i	−0.436506	−	0.628688i	−0.977147	+	0.212565i	$0.931818\pi$
$150$		0			0
$151$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$151$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.30638	+	0.0934345i	−1.30638	+	0.0934345i	−0.707107	−	0.707107i	$-0.750000\pi$
$157$	−1.30638	+	0.0934345i	−1.30638	+	0.0934345i	−0.599278	+	0.800541i	$0.704545\pi$
$158$		0			0
$159$		0			0
$160$	−0.550588	−	0.119773i	−0.550588	−	0.119773i
$161$		0			0
$162$		0			0
$163$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$163$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$164$	−1.07926	−	1.33929i	−1.07926	−	1.33929i
$165$		0			0
$166$		0			0
$167$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$167$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$168$		0			0
$169$	−0.457344	−	0.837564i	−0.457344	−	0.837564i
$170$	0.888260	+	0.570850i	0.888260	+	0.570850i
$171$		0			0
$172$		0			0
$173$	0.229843	+	1.05657i	0.229843	+	1.05657i	0.936950	+	0.349464i	$0.113636\pi$
$173$	0.229843	+	1.05657i	0.229843	+	1.05657i	−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.800541	−	0.599278i	0.800541	−	0.599278i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	1.10521	−	0.282089i	1.10521	−	0.282089i	0.349464	−	0.936950i	$-0.386364\pi$
$181$	1.10521	−	0.282089i	1.10521	−	0.282089i	0.755750	+	0.654861i	$0.227273\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.350230	−	0.0631884i	−0.350230	−	0.0631884i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.510270	−	0.860014i	$-0.329545\pi$
$191$		0			0		−0.510270	+	0.860014i	$0.670455\pi$
$192$		0			0
$193$	−0.735364	−	1.46907i	−0.735364	−	1.46907i	−0.877679	−	0.479249i	$-0.840909\pi$
$193$	−0.735364	−	1.46907i	−0.735364	−	1.46907i	0.142315	−	0.989821i	$-0.454545\pi$
$194$	−0.451077	−	1.53623i	−0.451077	−	1.53623i
$195$		0			0
$196$	0.212565	−	0.977147i	0.212565	−	0.977147i
$197$	0.715720	−	0.888154i	0.715720	−	0.888154i	−0.281733	−	0.959493i	$-0.590909\pi$
$197$	0.715720	−	0.888154i	0.715720	−	0.888154i	0.997452	+	0.0713392i	$0.0227273\pi$
$198$		0			0
$199$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$199$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$200$	0.0971309	−	0.675560i	0.0971309	−	0.675560i
$201$		0			0
$202$	1.83871	+	0.611980i	1.83871	+	0.611980i
$203$		0			0
$204$		0			0
$205$	−0.709335	+	0.660414i	−0.709335	+	0.660414i
$206$		0			0
$207$		0			0
$208$	−0.207149	−	0.0528717i	−0.207149	−	0.0528717i
$209$		0			0
$210$		0			0
$211$		0			0		−0.731895	−	0.681418i	$-0.761364\pi$
$211$		0			0		0.731895	+	0.681418i	$0.238636\pi$
$212$	−1.41061	+	1.41061i	−1.41061	+	1.41061i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.85483	−	0.691814i	−1.85483	−	0.691814i
$219$		0			0
$220$		0			0
$221$	0.329079	+	0.228483i	0.329079	+	0.228483i
$222$		0			0
$223$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$223$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$224$		0			0
$225$		0			0
$226$	1.63201	−	0.968315i	1.63201	−	0.968315i
$227$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$227$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$228$		0			0
$229$	0.158952	+	0.881010i	0.158952	+	0.881010i	0.959493	+	0.281733i	$0.0909091\pi$
$229$	0.158952	+	0.881010i	0.158952	+	0.881010i	−0.800541	+	0.599278i	$0.795455\pi$
$230$		0			0
$231$		0			0
$232$	0.0486428	+	1.36197i	0.0486428	+	1.36197i
$233$	0.587486	+	0.587486i	0.587486	+	0.587486i	0.936950	−	0.349464i	$-0.113636\pi$
$233$	0.587486	+	0.587486i	0.587486	+	0.587486i	−0.349464	+	0.936950i	$0.613636\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$239$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$240$		0			0
$241$	0.241703	+	0.726200i	0.241703	+	0.726200i	0.997452	+	0.0713392i	$0.0227273\pi$
$241$	0.241703	+	0.726200i	0.241703	+	0.726200i	−0.755750	+	0.654861i	$0.772727\pi$
$242$	0.909632	−	0.415415i	0.909632	−	0.415415i
$243$		0			0
$244$	−0.353072	+	0.0379591i	−0.353072	+	0.0379591i
$245$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.945619	−	0.0676320i	−0.945619	−	0.0676320i
$251$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$251$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.841254	−	0.540641i	0.841254	−	0.540641i
$257$	−1.14952	−	0.627683i	−1.14952	−	0.627683i	−0.212565	−	0.977147i	$-0.568182\pi$
$257$	−1.14952	−	0.627683i	−1.14952	−	0.627683i	−0.936950	+	0.349464i	$0.886364\pi$
$258$		0			0
$259$		0			0
$260$	−0.0213887	+	0.118549i	−0.0213887	+	0.118549i
$261$		0			0
$262$		0			0
$263$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$263$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$264$		0			0
$265$	0.849507	+	0.736102i	0.849507	+	0.736102i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.37491	−	1.19136i	−1.37491	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$269$	−1.37491	−	1.19136i	−1.37491	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$270$		0			0
$271$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$271$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$272$	−1.83107	+	0.398326i	−1.83107	+	0.398326i
$273$		0			0
$274$	0.0126746	−	0.0702505i	0.0126746	−	0.0702505i
$275$		0			0
$276$		0			0
$277$	0.373128	+	0.203743i	0.373128	+	0.203743i	0.654861	−	0.755750i	$-0.272727\pi$
$277$	0.373128	+	0.203743i	0.373128	+	0.203743i	−0.281733	+	0.959493i	$0.590909\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.794635	+	0.640357i	−0.794635	+	0.640357i	−0.936950	−	0.349464i	$-0.886364\pi$
$281$	−0.794635	+	0.640357i	−0.794635	+	0.640357i	0.142315	+	0.989821i	$0.454545\pi$
$282$		0			0
$283$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$283$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.48594	+	0.357424i	2.48594	+	0.357424i
$290$	0.763511	−	0.0820859i	0.763511	−	0.0820859i
$291$		0			0
$292$	−0.258908	+	0.118239i	−0.258908	+	0.118239i
$293$	−0.462264	−	1.38888i	−0.462264	−	1.38888i	−0.877679	−	0.479249i	$-0.840909\pi$
$293$	−0.462264	−	1.38888i	−0.462264	−	1.38888i	0.415415	−	0.909632i	$-0.363636\pi$
$294$		0			0
$295$		0			0
$296$	0.518809	−	0.360215i	0.518809	−	0.360215i
$297$		0			0
$298$	0.0818140	−	0.760982i	0.0818140	−	0.760982i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.0355266	+	0.196911i	0.0355266	+	0.196911i
$306$		0			0
$307$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$307$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$311$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$312$		0			0
$313$	−1.64179	−	1.13992i	−1.64179	−	1.13992i	−0.841254	−	0.540641i	$-0.818182\pi$
$313$	−1.64179	−	1.13992i	−1.64179	−	1.13992i	−0.800541	−	0.599278i	$-0.795455\pi$
$314$	−1.04849	−	0.784887i	−1.04849	−	0.784887i
$315$		0			0
$316$		0			0
$317$	0.407910	+	0.119773i	0.407910	+	0.119773i	0.479249	−	0.877679i	$-0.340909\pi$
$317$	0.407910	+	0.119773i	0.407910	+	0.119773i	−0.0713392	+	0.997452i	$0.522727\pi$
$318$		0			0
$319$		0			0
$320$	−0.337672	−	0.451077i	−0.337672	−	0.451077i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.145077	−	0.0155974i	−0.145077	−	0.0155974i
$326$		0			0
$327$		0			0
$328$	0.0613918	−	1.71893i	0.0613918	−	1.71893i
$329$		0			0
$330$		0			0
$331$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$331$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.977147	−	1.21257i	0.977147	−	1.21257i		−	1.00000i	$-0.5\pi$
$337$	0.977147	−	1.21257i	0.977147	−	1.21257i	0.977147	−	0.212565i	$-0.0681818\pi$
$338$	0.202850	−	0.932485i	0.202850	−	0.932485i
$339$		0			0
$340$	0.297475	+	1.01311i	0.297475	+	1.01311i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.518203	+	0.949018i	−0.518203	+	0.949018i
$347$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$347$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$348$		0			0
$349$	−1.44053	−	0.259900i	−1.44053	−	0.259900i	−0.599278	−	0.800541i	$-0.704545\pi$
$349$	−1.44053	−	0.259900i	−1.44053	−	0.259900i	−0.841254	+	0.540641i	$0.818182\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.479249	+	0.122321i	−0.479249	+	0.122321i	−0.479249	−	0.877679i	$-0.659091\pi$
$353$	−0.479249	+	0.122321i	−0.479249	+	0.122321i			1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$	0.997452	+	0.0713392i	0.997452	+	0.0713392i
$357$		0			0
$358$		0			0
$359$		0			0		−0.247307	−	0.968937i	$-0.579545\pi$
$359$		0			0		0.247307	+	0.968937i	$0.420455\pi$
$360$		0			0
$361$	0.212565	+	0.977147i	0.212565	+	0.977147i
$362$	1.01999	+	0.510572i	1.01999	+	0.510572i
$363$		0			0
$364$		0			0
$365$	0.0768614	+	0.140761i	0.0768614	+	0.140761i
$366$		0			0
$367$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$367$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$368$		0			0
$369$		0			0
$370$	−0.223306	−	0.277106i	−0.223306	−	0.277106i
$371$		0			0
$372$		0			0
$373$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	−0.909632	−	0.415415i	$-0.863636\pi$
$373$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	0.959493	−	0.281733i	$-0.0909091\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.290619	−	0.0207855i	0.290619	−	0.0207855i
$378$		0			0
$379$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$379$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$383$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$384$		0			0
$385$		0			0
$386$	0.406285	−	1.59181i	0.406285	−	1.59181i
$387$		0			0
$388$	0.665114	−	1.45640i	0.665114	−	1.45640i
$389$	1.36197	−	1.46286i	1.36197	−	1.46286i	0.654861	−	0.755750i	$-0.272727\pi$
$389$	1.36197	−	1.46286i	1.36197	−	1.46286i	0.707107	−	0.707107i	$-0.250000\pi$
$390$		0			0
$391$		0			0
$392$	0.800541	−	0.599278i	0.800541	−	0.599278i
$393$		0			0
$394$	1.12252	−	0.202525i	1.12252	−	0.202525i
$395$		0			0
$396$		0			0
$397$	0.794635	+	1.33929i	0.794635	+	1.33929i	0.936950	+	0.349464i	$0.113636\pi$
$397$	0.794635	+	1.33929i	0.794635	+	1.33929i	−0.142315	+	0.989821i	$0.545455\pi$
$398$		0			0
$399$		0			0
$400$	0.515804	−	0.446947i	0.515804	−	0.446947i
$401$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$401$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$402$		0			0
$403$		0			0
$404$	0.988839	+	1.66660i	0.988839	+	1.66660i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.865611	−	0.647988i	0.865611	−	0.647988i	−0.0713392	−	0.997452i	$-0.522727\pi$
$409$	0.865611	−	0.647988i	0.865611	−	0.647988i	0.936950	+	0.349464i	$0.113636\pi$
$410$	−0.968559	+	0.0345922i	−0.968559	+	0.0345922i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.121929	−	0.175612i	−0.121929	−	0.175612i
$417$		0			0
$418$		0			0
$419$		0			0		0.948826	−	0.315800i	$-0.102273\pi$
$419$		0			0		−0.948826	+	0.315800i	$0.897727\pi$
$420$		0			0
$421$	−0.396309	+	1.19072i	−0.396309	+	1.19072i	0.540641	+	0.841254i	$0.318182\pi$
$421$	−0.396309	+	1.19072i	−0.396309	+	1.19072i	−0.936950	+	0.349464i	$0.886364\pi$
$422$		0			0
$423$		0			0
$424$	−1.98982	+	0.142315i	−1.98982	+	0.142315i
$425$	−1.19831	+	0.446947i	−1.19831	+	0.446947i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$431$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$432$		0			0
$433$	1.75320	−	0.726200i	1.75320	−	0.726200i	0.755750	−	0.654861i	$-0.227273\pi$
$433$	1.75320	−	0.726200i	1.75320	−	0.726200i	0.997452	−	0.0713392i	$-0.0227273\pi$
$434$		0			0
$435$		0			0
$436$	−0.948742	−	1.73749i	−0.948742	−	1.73749i
$437$		0			0
$438$		0			0
$439$		0			0		−0.894225	−	0.447617i	$-0.852273\pi$
$439$		0			0		0.894225	+	0.447617i	$0.147727\pi$
$440$		0			0
$441$		0			0
$442$	0.0990763	+	0.388177i	0.0990763	+	0.388177i
$443$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$443$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$444$		0			0
$445$		−	0.563465i		−	0.563465i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.141226	−	0.0203052i	0.141226	−	0.0203052i	−0.0713392	−	0.997452i	$-0.522727\pi$
$449$	0.141226	−	0.0203052i	0.141226	−	0.0203052i	0.212565	+	0.977147i	$0.431818\pi$
$450$		0			0
$451$		0			0
$452$	1.86750	+	0.336934i	1.86750	+	0.336934i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.741593	−	1.79036i	−0.741593	−	1.79036i	−0.599278	−	0.800541i	$-0.704545\pi$
$457$	−0.741593	−	1.79036i	−0.741593	−	1.79036i	−0.142315	−	0.989821i	$-0.545455\pi$
$458$	−0.456811	+	0.769914i	−0.456811	+	0.769914i
$459$		0			0
$460$		0			0
$461$	0.304632	+	1.03748i	0.304632	+	1.03748i	0.959493	+	0.281733i	$0.0909091\pi$
$461$	0.304632	+	1.03748i	0.304632	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$462$		0			0
$463$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$463$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$464$	−0.855137	+	1.06116i	−0.855137	+	1.06116i
$465$		0			0
$466$	0.0592707	+	0.828713i	0.0592707	+	0.828713i
$467$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$467$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$	−0.0809203	−	0.108097i	−0.0809203	−	0.108097i
$482$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$483$		0			0
$484$	0.959493	+	0.281733i	0.959493	+	0.281733i
$485$	−0.845273	−	0.315271i	−0.845273	−	0.315271i
$486$		0			0
$487$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$487$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$488$	−0.291692	−	0.202525i	−0.291692	−	0.202525i
$489$		0			0
$490$	−0.368991	−	0.425839i	−0.368991	−	0.425839i
$491$		0			0		0.570323	−	0.821421i	$-0.306818\pi$
$491$		0			0		−0.570323	+	0.821421i	$0.693182\pi$
$492$		0			0
$493$	2.19632	−	1.30314i	2.19632	−	1.30314i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.0356923	−	0.999363i	$-0.511364\pi$
$499$		0			0		0.0356923	+	0.999363i	$0.488636\pi$
$500$	−0.670361	−	0.670361i	−0.670361	−	0.670361i
$501$		0			0
$502$		0			0
$503$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$503$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$504$		0			0
$505$	0.896930	−	0.622749i	0.896930	−	0.622749i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.258908	+	0.118239i	−0.258908	+	0.118239i	−0.540641	−	0.841254i	$-0.681818\pi$
$509$	−0.258908	+	0.118239i	−0.258908	+	0.118239i	0.281733	+	0.959493i	$0.409091\pi$
$510$		0			0
$511$		0			0
$512$	0.989821	+	0.142315i	0.989821	+	0.142315i
$513$		0			0
$514$	−0.457701	−	1.22714i	−0.457701	−	1.22714i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.0937978	+	0.0755870i	−0.0937978	+	0.0755870i
$521$	−1.71017	−	1.01469i	−1.71017	−	1.01469i	−0.909632	−	0.415415i	$-0.863636\pi$
$521$	−1.71017	−	1.01469i	−1.71017	−	1.01469i	−0.800541	−	0.599278i	$-0.795455\pi$
$522$		0			0
$523$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$523$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.977147	−	0.212565i	0.977147	−	0.212565i
$530$	0.159970	+	1.11262i	0.159970	+	1.11262i
$531$		0			0
$532$		0			0
$533$	−0.367725			−0.367725
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.258908	−	1.80075i	−0.258908	−	1.80075i
$539$		0			0
$540$		0			0
$541$	−0.291692	+	1.61674i	−0.291692	+	1.61674i	0.415415	+	0.909632i	$0.363636\pi$
$541$	−0.291692	+	1.61674i	−0.291692	+	1.61674i	−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$		0			0
$544$	−1.64468	−	0.898064i	−1.64468	−	0.898064i
$545$	−0.938384	+	0.603063i	−0.938384	+	0.603063i
$546$		0			0
$547$		0			0		−0.860014	−	0.510270i	$-0.829545\pi$
$547$		0			0		0.860014	+	0.510270i	$0.170455\pi$
$548$	0.0555831	−	0.0447917i	0.0555831	−	0.0447917i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.148568	+	0.398326i	0.148568	+	0.398326i
$555$		0			0
$556$		0			0
$557$	1.97715	−	0.212565i	1.97715	−	0.212565i	0.977147	−	0.212565i	$-0.0681818\pi$
$557$	1.97715	−	0.212565i	1.97715	−	0.212565i	1.00000			$0$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.01989	−	0.0364255i	−1.01989	−	0.0364255i
$563$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$563$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$564$		0			0
$565$	0.114299	−	1.06313i	0.114299	−	1.06313i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.0407123	+	1.13992i	0.0407123	+	1.13992i	0.841254	+	0.540641i	$0.181818\pi$
$569$	0.0407123	+	1.13992i	0.0407123	+	1.13992i	−0.800541	+	0.599278i	$0.795455\pi$
$570$		0			0
$571$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$571$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.05382	−	1.51779i	1.05382	−	1.51779i	0.212565	−	0.977147i	$-0.431818\pi$
$577$	1.05382	−	1.51779i	1.05382	−	1.51779i	0.841254	−	0.540641i	$-0.181818\pi$
$578$	1.64468	+	1.89806i	1.64468	+	1.89806i
$579$		0			0
$580$	0.630778	+	0.437957i	0.630778	+	0.437957i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$585$		0			0
$586$	0.560168	−	1.35236i	0.560168	−	1.35236i
$587$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$587$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.627980	+	0.0675149i	0.627980	+	0.0675149i
$593$	−1.59181	−	0.406285i	−1.59181	−	0.406285i	−0.654861	−	0.755750i	$-0.727273\pi$
$593$	−1.59181	−	0.406285i	−1.59181	−	0.406285i	−0.936950	+	0.349464i	$0.886364\pi$
$594$		0			0
$595$		0			0
$596$	0.560168	−	0.521535i	0.560168	−	0.521535i
$597$		0			0
$598$		0			0
$599$		0			0		−0.948826	−	0.315800i	$-0.897727\pi$
$599$		0			0		0.948826	+	0.315800i	$0.102273\pi$
$600$		0			0
$601$	−0.215109	+	1.49611i	−0.215109	+	1.49611i	0.540641	+	0.841254i	$0.318182\pi$
$601$	−0.215109	+	1.49611i	−0.215109	+	1.49611i	−0.755750	+	0.654861i	$0.772727\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.119773	−	0.550588i	0.119773	−	0.550588i
$606$		0			0
$607$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$607$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$608$		0			0
$609$		0			0
$610$	−0.102100	+	0.172080i	−0.102100	+	0.172080i
$611$		0			0
$612$		0			0
$613$	−0.398174	+	0.729202i	−0.398174	+	0.729202i	−0.997452	−	0.0713392i	$-0.977273\pi$
$613$	−0.398174	+	0.729202i	−0.398174	+	0.729202i	0.599278	+	0.800541i	$0.295455\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.697067	+	1.39256i	−0.697067	+	1.39256i	0.212565	+	0.977147i	$0.431818\pi$
$617$	−0.697067	+	1.39256i	−0.697067	+	1.39256i	−0.909632	+	0.415415i	$0.863636\pi$
$618$		0			0
$619$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$619$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.0971309	−	0.112095i	0.0971309	−	0.112095i
$626$	−0.494298	−	1.93664i	−0.494298	−	1.93664i
$627$		0			0
$628$	−0.278401	−	1.27979i	−0.278401	−	1.27979i
$629$	−1.05836	−	0.529779i	−1.05836	−	0.529779i
$630$		0			0
$631$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$631$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$632$		0			0
$633$		0			0
$634$	0.229843	+	0.357643i	0.229843	+	0.357643i
$635$		0			0
$636$		0			0
$637$	−0.134147	−	0.166466i	−0.134147	−	0.166466i
$638$		0			0
$639$		0			0
$640$	0.0401971	−	0.562029i	0.0401971	−	0.562029i
$641$	1.47696	+	0.321292i	1.47696	+	0.321292i	0.877679	−	0.479249i	$-0.159091\pi$
$641$	1.47696	+	0.321292i	1.47696	+	0.321292i	0.599278	+	0.800541i	$0.295455\pi$
$642$		0			0
$643$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$643$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.315800	−	0.948826i	$-0.397727\pi$
$647$		0			0		−0.315800	+	0.948826i	$0.602273\pi$
$648$		0			0
$649$		0			0
$650$	−0.0994280	−	0.106793i	−0.0994280	−	0.106793i
$651$		0			0
$652$		0			0
$653$	−0.469302	+	1.83871i	−0.469302	+	1.83871i	0.0713392	+	0.997452i	$0.477273\pi$
$653$	−0.469302	+	1.83871i	−0.469302	+	1.83871i	−0.540641	+	0.841254i	$0.681818\pi$
$654$		0			0
$655$		0			0
$656$	1.17206	−	1.25888i	1.17206	−	1.25888i
$657$		0			0
$658$		0			0
$659$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$659$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$660$		0			0
$661$	−1.93695	+	0.349464i	−1.93695	+	0.349464i	−0.936950	+	0.349464i	$0.886364\pi$
$661$	−1.93695	+	0.349464i	−1.93695	+	0.349464i	−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.562029	+	1.91410i	−0.562029	+	1.91410i	−0.212565	+	0.977147i	$0.568182\pi$
$673$	−0.562029	+	1.91410i	−0.562029	+	1.91410i	−0.349464	+	0.936950i	$0.613636\pi$
$674$	1.53254	−	0.276501i	1.53254	−	0.276501i
$675$		0			0
$676$	0.763951	−	0.571887i	0.763951	−	0.571887i
$677$	−0.764879	+	0.0273177i	−0.764879	+	0.0273177i	−0.415415	−	0.909632i	$-0.636364\pi$
$677$	−0.764879	+	0.0273177i	−0.764879	+	0.0273177i	−0.349464	+	0.936950i	$0.613636\pi$
$678$		0			0
$679$		0			0
$680$	−0.438627	+	0.960459i	−0.438627	+	0.960459i
$681$		0			0
$682$		0			0
$683$		0			0		−0.570323	−	0.821421i	$-0.693182\pi$
$683$		0			0		0.570323	+	0.821421i	$0.306818\pi$
$684$		0			0
$685$	−0.0274085	−	0.0294388i	−0.0274085	−	0.0294388i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.0455898	+	0.424047i	0.0455898	+	0.424047i
$690$		0			0
$691$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$691$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$692$	−1.01311	+	0.377869i	−1.01311	+	0.377869i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.88223	+	1.44274i	−2.88223	+	1.44274i
$698$	−0.918482	−	1.13977i	−0.918482	−	1.13977i
$699$		0			0
$700$		0			0
$701$	1.05657	+	1.64406i	1.05657	+	1.64406i	0.707107	+	0.707107i	$0.250000\pi$
$701$	1.05657	+	1.64406i	1.05657	+	1.64406i	0.349464	+	0.936950i	$0.386364\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.442295	−	0.221397i	−0.442295	−	0.221397i
$707$		0			0
$708$		0			0
$709$	0.491779	+	1.92677i	0.491779	+	1.92677i	0.349464	+	0.936950i	$0.386364\pi$
$709$	0.491779	+	1.92677i	0.491779	+	1.92677i	0.142315	+	0.989821i	$0.454545\pi$
$710$		0			0
$711$		0			0
$712$	0.707107	+	0.707107i	0.707107	+	0.707107i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.984111	−	0.177553i	$-0.943182\pi$
$719$		0			0		0.984111	+	0.177553i	$0.0568182\pi$
$720$		0			0
$721$		0			0
$722$	−0.479249	+	0.877679i	−0.479249	+	0.877679i
$723$		0			0
$724$	0.436506	+	1.05382i	0.436506	+	1.05382i
$725$	−0.474625	+	0.799938i	−0.474625	+	0.799938i
$726$		0			0
$727$		0			0		−0.447617	−	0.894225i	$-0.647727\pi$
$727$		0			0		0.447617	+	0.894225i	$0.352273\pi$
$728$		0			0
$729$		0			0
$730$	−0.0340910	+	0.156714i	−0.0340910	+	0.156714i
$731$		0			0
$732$		0			0
$733$	−0.0771377	−	1.07853i	−0.0771377	−	1.07853i	−0.877679	−	0.479249i	$-0.840909\pi$
$733$	−0.0771377	−	1.07853i	−0.0771377	−	1.07853i	0.800541	−	0.599278i	$-0.204545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.731895	−	0.681418i	$-0.238636\pi$
$739$		0			0		−0.731895	+	0.681418i	$0.761364\pi$
$740$	0.0127023	−	0.355657i	0.0127023	−	0.355657i
$741$		0			0
$742$		0			0
$743$		0			0		−0.994270	−	0.106895i	$-0.965909\pi$
$743$		0			0		0.994270	+	0.106895i	$0.0340909\pi$
$744$		0			0
$745$	−0.315635	−	0.293867i	−0.315635	−	0.293867i
$746$	0.494217	−	0.494217i	0.494217	−	0.494217i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$751$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$752$		0			0
$753$		0			0
$754$	0.233247	+	0.174606i	0.233247	+	0.174606i
$755$		0			0
$756$		0			0
$757$	1.27979	+	1.47696i	1.27979	+	1.47696i	0.800541	+	0.599278i	$0.204545\pi$
$757$	1.27979	+	1.47696i	1.27979	+	1.47696i	0.479249	+	0.877679i	$0.340909\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.0498610	−	0.133682i	0.0498610	−	0.133682i	−0.909632	−	0.415415i	$-0.863636\pi$
$761$	0.0498610	−	0.133682i	0.0498610	−	0.133682i	0.959493	+	0.281733i	$0.0909091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.635768	+	0.290345i	0.635768	+	0.290345i	0.707107	−	0.707107i	$-0.250000\pi$
$769$	0.635768	+	0.290345i	0.635768	+	0.290345i	−0.0713392	+	0.997452i	$0.522727\pi$
$770$		0			0
$771$		0			0
$772$	1.34946	−	0.936950i	1.34946	−	0.936950i
$773$	−1.01989	−	0.0364255i	−1.01989	−	0.0364255i	−0.479249	−	0.877679i	$-0.659091\pi$
$773$	−1.01989	−	0.0364255i	−1.01989	−	0.0364255i	−0.540641	+	0.841254i	$0.681818\pi$
$774$		0			0
$775$		0			0
$776$	1.45640	−	0.665114i	1.45640	−	0.665114i
$777$		0			0
$778$	1.98727	−	0.213654i	1.98727	−	0.213654i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.997452	+	0.0713392i	0.997452	+	0.0713392i
$785$	−0.708089	+	0.207914i	−0.708089	+	0.207914i
$786$		0			0
$787$		0			0		0.778642	−	0.627469i	$-0.215909\pi$
$787$		0			0		−0.778642	+	0.627469i	$0.784091\pi$
$788$	0.980971	+	0.582037i	0.980971	+	0.582037i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.0410445	+	0.0638665i	−0.0410445	+	0.0638665i
$794$	−0.276501	+	1.53254i	−0.276501	+	1.53254i
$795$		0			0
$796$		0			0
$797$	0.136408	+	0.948742i	0.136408	+	0.948742i	0.936950	+	0.349464i	$0.113636\pi$
$797$	0.136408	+	0.948742i	0.136408	+	0.948742i	−0.800541	+	0.599278i	$0.795455\pi$
$798$		0			0
$799$		0			0
$800$	0.682507			0.682507
$801$		0			0
$802$	−1.97964			−1.97964
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.344076	+	1.90708i	−0.344076	+	1.90708i
$809$	−0.949018	+	1.47670i	−0.949018	+	1.47670i	−0.0713392	+	0.997452i	$0.522727\pi$
$809$	−0.949018	+	1.47670i	−0.949018	+	1.47670i	−0.877679	+	0.479249i	$0.840909\pi$
$810$		0			0
$811$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$811$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.07853	+	0.0771377i	1.07853	+	0.0771377i
$819$		0			0
$820$	−0.754641	−	0.608128i	−0.754641	−	0.608128i
$821$	0.613435	+	1.64468i	0.613435	+	1.64468i	0.755750	+	0.654861i	$0.227273\pi$
$821$	0.613435	+	1.64468i	0.613435	+	1.64468i	−0.142315	+	0.989821i	$0.545455\pi$
$822$		0			0
$823$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$823$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.315800	−	0.948826i	$-0.602273\pi$
$827$		0			0		0.315800	+	0.948826i	$0.397727\pi$
$828$		0			0
$829$	1.84658	+	0.0659508i	1.84658	+	0.0659508i	0.936950	−	0.349464i	$-0.113636\pi$
$829$	1.84658	+	0.0659508i	1.84658	+	0.0659508i	0.909632	+	0.415415i	$0.136364\pi$
$830$		0			0
$831$		0			0
$832$	0.0228531	−	0.212565i	0.0228531	−	0.212565i
$833$	−1.70456	−	0.778446i	−1.70456	−	0.778446i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.177553	−	0.984111i	$-0.556818\pi$
$839$		0			0		0.177553	+	0.984111i	$0.443182\pi$
$840$		0			0
$841$	0.299603	−	0.803267i	0.299603	−	0.803267i
$842$	−1.07926	+	0.640357i	−1.07926	+	0.640357i
$843$		0			0
$844$		0			0
$845$	−0.352126	−	0.406375i	−0.352126	−	0.406375i
$846$		0			0
$847$		0			0
$848$	−1.59700	−	1.19550i	−1.59700	−	1.19550i
$849$		0			0
$850$	−1.19831	−	0.446947i	−1.19831	−	0.446947i
$851$		0			0
$852$		0			0
$853$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$853$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.30896	−	1.21868i	−1.30896	−	1.21868i	−0.959493	−	0.281733i	$-0.909091\pi$
$857$	−1.30896	−	1.21868i	−1.30896	−	1.21868i	−0.349464	−	0.936950i	$-0.613636\pi$
$858$		0			0
$859$		0			0		−0.994270	−	0.106895i	$-0.965909\pi$
$859$		0			0		0.994270	+	0.106895i	$0.0340909\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.731895	−	0.681418i	$-0.238636\pi$
$863$		0			0		−0.731895	+	0.681418i	$0.761364\pi$
$864$		0			0
$865$	0.253098	+	0.554206i	0.253098	+	0.554206i
$866$	1.80054	+	0.599278i	1.80054	+	0.599278i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.420803	−	1.93440i	0.420803	−	1.93440i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.877679	+	1.47925i	−0.877679	+	1.47925i			1.00000i	$0.5\pi$
$877$	−0.877679	+	1.47925i	−0.877679	+	1.47925i	−0.877679	+	0.479249i	$0.840909\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.613435	−	0.334961i	0.613435	−	0.334961i	−0.142315	−	0.989821i	$-0.545455\pi$
$881$	0.613435	−	0.334961i	0.613435	−	0.334961i	0.755750	+	0.654861i	$0.227273\pi$
$882$		0			0
$883$		0			0		−0.984111	−	0.177553i	$-0.943182\pi$
$883$		0			0		0.984111	+	0.177553i	$0.0568182\pi$
$884$	−0.179325	+	0.358246i	−0.179325	+	0.358246i
$885$		0			0
$886$		0			0
$887$		0			0		0.968937	−	0.247307i	$-0.0795455\pi$
$887$		0			0		−0.968937	+	0.247307i	$0.920455\pi$
$888$		0			0
$889$		0			0
$890$	0.368991	−	0.425839i	0.368991	−	0.425839i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.120029	+	0.0771377i	0.120029	+	0.0771377i
$899$		0			0
$900$		0			0
$901$	2.02105	+	3.14482i	2.02105	+	3.14482i
$902$		0			0
$903$		0			0
$904$	1.19072	+	1.47759i	1.19072	+	1.47759i
$905$	0.574731	−	0.287690i	0.574731	−	0.287690i
$906$		0			0
$907$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$907$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$911$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$912$		0			0
$913$		0			0
$914$	0.611980	−	1.83871i	0.611980	−	1.83871i
$915$		0			0
$916$	−0.849422	+	0.282715i	−0.849422	+	0.282715i
$917$		0			0
$918$		0			0
$919$		0			0		−0.570323	−	0.821421i	$-0.693182\pi$
$919$		0			0		0.570323	+	0.821421i	$0.306818\pi$
$920$		0			0
$921$		0			0
$922$	−0.449181	+	0.983568i	−0.449181	+	0.983568i
$923$		0			0
$924$		0			0
$925$	0.430796	−	0.0153859i	0.430796	−	0.0153859i
$926$		0			0
$927$		0			0
$928$	−1.34118	+	0.241976i	−1.34118	+	0.241976i
$929$	0.270040	−	0.919672i	0.270040	−	0.919672i	−0.707107	−	0.707107i	$-0.750000\pi$
$929$	0.270040	−	0.919672i	0.270040	−	0.919672i	0.977147	−	0.212565i	$-0.0681818\pi$
$930$		0			0
$931$		0			0
$932$	−0.497898	+	0.665114i	−0.497898	+	0.665114i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.959493	+	1.28173i	−0.959493	+	1.28173i			1.00000i	$0.5\pi$
$937$	−0.959493	+	1.28173i	−0.959493	+	1.28173i	−0.959493	+	0.281733i	$0.909091\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.76003	−	0.317545i	1.76003	−	0.317545i	0.800541	−	0.599278i	$-0.204545\pi$
$941$	1.76003	−	0.317545i	1.76003	−	0.317545i	0.959493	+	0.281733i	$0.0909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$947$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$948$		0			0
$949$	−0.0150489	+	0.0589609i	−0.0150489	+	0.0589609i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.83871	−	0.611980i	1.83871	−	0.611980i	0.841254	−	0.540641i	$-0.181818\pi$
$953$	1.83871	−	0.611980i	1.83871	−	0.611980i	0.997452	−	0.0713392i	$-0.0227273\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.977147	−	0.212565i	−0.977147	−	0.212565i
$962$	0.00963291	−	0.134686i	0.00963291	−	0.134686i
$963$		0			0
$964$	−0.684410	+	0.342591i	−0.684410	+	0.342591i
$965$	−0.580838	−	0.720776i	−0.580838	−	0.720776i
$966$		0			0
$967$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$967$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$968$	0.540641	+	0.841254i	0.540641	+	0.841254i
$969$		0			0
$970$	−0.432356	−	0.791802i	−0.432356	−	0.791802i
$971$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$971$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.0878201	−	0.344076i	−0.0878201	−	0.344076i
$977$	1.14952	−	1.32661i	1.14952	−	1.32661i	0.212565	−	0.977147i	$-0.431818\pi$
$977$	1.14952	−	1.32661i	1.14952	−	1.32661i	0.936950	−	0.349464i	$-0.113636\pi$
$978$		0			0
$979$		0			0
$980$		−	0.563465i		−	0.563465i
$981$		0			0
$982$		0			0
$983$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$983$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$984$		0			0
$985$	0.287690	−	0.574731i	0.287690	−	0.574731i
$986$	2.51324	+	0.453438i	2.51324	+	0.453438i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$991$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.254771	+	1.17116i	−0.254771	+	1.17116i	0.654861	+	0.755750i	$0.272727\pi$
$997$	−0.254771	+	1.17116i	−0.254771	+	1.17116i	−0.909632	+	0.415415i	$0.863636\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3204.1.cd.b.719.1	yes	40
3.2	odd	2		3204.1.cd.a.719.1	✓	40
4.3	odd	2	CM	3204.1.cd.b.719.1	yes	40
12.11	even	2		3204.1.cd.a.719.1	✓	40
89.51	odd	88		3204.1.cd.a.1475.1	yes	40
267.140	even	88	inner	3204.1.cd.b.1475.1	yes	40
356.51	even	88		3204.1.cd.a.1475.1	yes	40
1068.407	odd	88	inner	3204.1.cd.b.1475.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3204.1.cd.a.719.1	✓	40	3.2	odd	2
3204.1.cd.a.719.1	✓	40	12.11	even	2
3204.1.cd.a.1475.1	yes	40	89.51	odd	88
3204.1.cd.a.1475.1	yes	40	356.51	even	88
3204.1.cd.b.719.1	yes	40	1.1	even	1	trivial
3204.1.cd.b.719.1	yes	40	4.3	odd	2	CM
3204.1.cd.b.1475.1	yes	40	267.140	even	88	inner
3204.1.cd.b.1475.1	yes	40	1068.407	odd	88	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:	\( N \)	\(=\)	\( 3204 = 2^{2} \cdot 3^{2} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3204.cd (of order \(88\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\(q+(0.755750 + 0.654861i) q^{2} +(0.142315 + 0.989821i) q^{4} +(0.550588 - 0.119773i) q^{5} +(-0.540641 + 0.841254i) q^{8} +O(q^{10})\)	\(q+(0.755750 + 0.654861i) q^{2} +(0.142315 + 0.989821i) q^{4} +(0.550588 - 0.119773i) q^{5} +(-0.540641 + 0.841254i) q^{8} +(0.494541 + 0.270040i) q^{10} +(0.183863 + 0.109091i) q^{13} +(-0.959493 + 0.281733i) q^{16} +(1.86912 + 0.133682i) q^{17} +(0.196911 + 0.527938i) q^{20} +(-0.620830 + 0.283524i) q^{25} +(0.0675149 + 0.202850i) q^{26} +(1.11946 - 0.777256i) q^{29} +(-0.909632 - 0.415415i) q^{32} +(1.32505 + 1.32505i) q^{34} +(-0.583522 - 0.241703i) q^{37} +(-0.196911 + 0.527938i) q^{40} +(-1.47925 + 0.877679i) q^{41} +(-0.936950 - 0.349464i) q^{49} +(-0.654861 - 0.192284i) q^{50} +(-0.0818140 + 0.197516i) q^{52} +(1.19550 + 1.59700i) q^{53} +(1.35503 + 0.145681i) q^{58} +(-0.0126746 + 0.354880i) q^{61} +(-0.415415 - 0.909632i) q^{64} +(0.114299 + 0.0380423i) q^{65} +(0.133682 + 1.86912i) q^{68} +(0.0801894 + 0.273100i) q^{73} +(-0.282715 - 0.564792i) q^{74} +(-0.494541 + 0.270040i) q^{80} +(-1.69270 - 0.305397i) q^{82} +(1.04513 - 0.150267i) q^{85} +(0.212565 - 0.977147i) q^{89} +(-1.34692 - 0.865611i) q^{97} +(-0.479249 - 0.877679i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{4}+O(q^{10}) \) 40 * q + 4 * q^4	\( 40 q + 4 q^{4} + 4 q^{13} - 4 q^{16} + 4 q^{17} + 4 q^{29} - 4 q^{34} - 4 q^{37} - 40 q^{41} - 4 q^{50} - 4 q^{52} + 4 q^{53} + 4 q^{64} - 4 q^{68}+O(q^{100}) \) 40 * q + 4 * q^4 + 4 * q^13 - 4 * q^16 + 4 * q^17 + 4 * q^29 - 4 * q^34 - 4 * q^37 - 40 * q^41 - 4 * q^50 - 4 * q^52 + 4 * q^53 + 4 * q^64 - 4 * q^68

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more