LMFDB - Embedded newform 3459.1.bm.a.719.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3459,1,Mod(176,3459)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3459, base_ring=CyclotomicField(192))

chi = DirichletCharacter(H, H._module([96, 25]))

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

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

chi := DirichletCharacter("3459.176");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3459 = 3 \cdot 1153 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3459.bm (of order $192$, degree $64$, 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:	$1.72626587870$
Analytic rank:	$0$
Dimension:	$64$
Coefficient field:	$\Q(\zeta_{192})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{64} - x^{32} + 1 $ x^64 - x^32 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{192}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{192} - \cdots)$

Embedding invariants

Embedding label			719.1
Root			$0.999465 + 0.0327191i$ of defining polynomial
Character	$\chi$	$=$	3459.719
Dual form			3459.1.bm.a.3281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.162895 - 0.986643i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.463377 - 0.979728i) q^{7} +(-0.946930 + 0.321439i) q^{9} +O(q^{10})$	\(q+(-0.162895 - 0.986643i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.463377 - 0.979728i) q^{7} +(-0.946930 + 0.321439i) q^{9} +(-0.956940 + 0.290285i) q^{12} +(-0.114186 - 1.15935i) q^{13} +(-0.965926 + 0.258819i) q^{16} +(-0.199794 - 0.111033i) q^{19} +(-1.04212 - 0.297595i) q^{21} +(-0.812847 + 0.582478i) q^{25} +(0.471397 + 0.881921i) q^{27} +(-1.03183 - 0.331533i) q^{28} +(-0.157160 + 0.0448794i) q^{31} +(0.442289 + 0.896873i) q^{36} +(-0.800094 - 0.882768i) q^{37} +(-1.12526 + 0.301513i) q^{39} +(1.46658 - 0.783904i) q^{43} +(0.412707 + 0.910864i) q^{48} +(-0.110756 - 0.134957i) q^{49} +(-1.13452 + 0.264534i) q^{52} +(-0.0770042 + 0.215212i) q^{57} +(1.98450 + 0.0974923i) q^{61} +(-0.123862 + 1.07668i) q^{63} +(0.382683 + 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(-1.53127 + 0.330722i) q^{73} +(0.707107 + 0.707107i) q^{75} +(-0.0840046 + 0.212578i) q^{76} +(1.58047 - 1.17215i) q^{79} +(0.793353 - 0.608761i) q^{81} +(-0.159024 + 1.07205i) q^{84} +(-1.18875 - 0.425343i) q^{91} +(0.0698805 + 0.147750i) q^{93} +(-0.812847 - 1.40789i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q+O(q^{10}) $ 64 * q	$ 64 q - 64 q^{67}+O(q^{100}) $ 64 * q - 64 * q^67

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3459\mathbb{Z}\right)^\times$.

$n$	$1154$	$2311$
$\chi(n)$	$-1$	$e\left(\frac{77}{192}\right)$

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		0.659346	−	0.751840i	$-0.270833\pi$
$2$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$3$	−0.162895	−	0.986643i	−0.162895	−	0.986643i
$3$	−0.162895	−	0.986643i	−0.162895	−	0.986643i
$4$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$5$		0			0		−0.305903	−	0.952063i	$-0.598958\pi$
$5$		0			0		0.305903	+	0.952063i	$0.401042\pi$
$6$		0			0
$7$	0.463377	−	0.979728i	0.463377	−	0.979728i	−0.528068	−	0.849202i	$-0.677083\pi$
$7$	0.463377	−	0.979728i	0.463377	−	0.979728i	0.991445	−	0.130526i	$-0.0416667\pi$
$8$		0			0
$9$	−0.946930	+	0.321439i	−0.946930	+	0.321439i
$10$		0			0
$11$		0			0		−0.729864	−	0.683592i	$-0.760417\pi$
$11$		0			0		0.729864	+	0.683592i	$0.239583\pi$
$12$	−0.956940	+	0.290285i	−0.956940	+	0.290285i
$13$	−0.114186	−	1.15935i	−0.114186	−	1.15935i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	−0.114186	−	1.15935i	−0.114186	−	1.15935i	0.751840	−	0.659346i	$-0.229167\pi$
$14$		0			0
$15$		0			0
$16$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$17$		0			0		−0.0163617	−	0.999866i	$-0.505208\pi$
$17$		0			0		0.0163617	+	0.999866i	$0.494792\pi$
$18$		0			0
$19$	−0.199794	−	0.111033i	−0.199794	−	0.111033i	0.382683	−	0.923880i	$-0.375000\pi$
$19$	−0.199794	−	0.111033i	−0.199794	−	0.111033i	−0.582478	+	0.812847i	$0.697917\pi$
$20$		0			0
$21$	−1.04212	−	0.297595i	−1.04212	−	0.297595i
$22$		0			0
$23$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$23$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$	−0.812847	+	0.582478i	−0.812847	+	0.582478i
$26$		0			0
$27$	0.471397	+	0.881921i	0.471397	+	0.881921i
$28$	−1.03183	−	0.331533i	−1.03183	−	0.331533i
$29$		0			0		−0.986643	−	0.162895i	$-0.947917\pi$
$29$		0			0		0.986643	+	0.162895i	$0.0520833\pi$
$30$		0			0
$31$	−0.157160	+	0.0448794i	−0.157160	+	0.0448794i	−0.352250	−	0.935906i	$-0.614583\pi$
$31$	−0.157160	+	0.0448794i	−0.157160	+	0.0448794i	0.195090	+	0.980785i	$0.437500\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.442289	+	0.896873i	0.442289	+	0.896873i
$37$	−0.800094	−	0.882768i	−0.800094	−	0.882768i	0.195090	−	0.980785i	$-0.437500\pi$
$37$	−0.800094	−	0.882768i	−0.800094	−	0.882768i	−0.995185	+	0.0980171i	$0.968750\pi$
$38$		0			0
$39$	−1.12526	+	0.301513i	−1.12526	+	0.301513i
$40$		0			0
$41$		0			0		−0.582478	−	0.812847i	$-0.697917\pi$
$41$		0			0		0.582478	+	0.812847i	$0.302083\pi$
$42$		0			0
$43$	1.46658	−	0.783904i	1.46658	−	0.783904i	0.471397	−	0.881921i	$-0.343750\pi$
$43$	1.46658	−	0.783904i	1.46658	−	0.783904i	0.995185	+	0.0980171i	$0.0312500\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$	0.412707	+	0.910864i	0.412707	+	0.910864i
$49$	−0.110756	−	0.134957i	−0.110756	−	0.134957i
$50$		0			0
$51$		0			0
$52$	−1.13452	+	0.264534i	−1.13452	+	0.264534i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.0770042	+	0.215212i	−0.0770042	+	0.215212i
$58$		0			0
$59$		0			0		−0.0490677	−	0.998795i	$-0.515625\pi$
$59$		0			0		0.0490677	+	0.998795i	$0.484375\pi$
$60$		0			0
$61$	1.98450	+	0.0974923i	1.98450	+	0.0974923i	0.997859	−	0.0654031i	$-0.0208333\pi$
$61$	1.98450	+	0.0974923i	1.98450	+	0.0974923i	0.986643	+	0.162895i	$0.0520833\pi$
$62$		0			0
$63$	−0.123862	+	1.07668i	−0.123862	+	1.07668i
$64$	0.382683	+	0.923880i	0.382683	+	0.923880i
$65$		0			0
$66$		0			0
$67$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$67$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$71$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$72$		0			0
$73$	−1.53127	+	0.330722i	−1.53127	+	0.330722i	−0.896873	−	0.442289i	$-0.854167\pi$
$73$	−1.53127	+	0.330722i	−1.53127	+	0.330722i	−0.634393	+	0.773010i	$0.718750\pi$
$74$		0			0
$75$	0.707107	+	0.707107i	0.707107	+	0.707107i
$76$	−0.0840046	+	0.212578i	−0.0840046	+	0.212578i
$77$		0			0
$78$		0			0
$79$	1.58047	−	1.17215i	1.58047	−	1.17215i	0.683592	−	0.729864i	$-0.260417\pi$
$79$	1.58047	−	1.17215i	1.58047	−	1.17215i	0.896873	−	0.442289i	$-0.145833\pi$
$80$		0			0
$81$	0.793353	−	0.608761i	0.793353	−	0.608761i
$82$		0			0
$83$		0			0		0.0817211	−	0.996655i	$-0.473958\pi$
$83$		0			0		−0.0817211	+	0.996655i	$0.526042\pi$
$84$	−0.159024	+	1.07205i	−0.159024	+	1.07205i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.999465	−	0.0327191i	$-0.0104167\pi$
$89$		0			0		−0.999465	+	0.0327191i	$0.989583\pi$
$90$		0			0
$91$	−1.18875	−	0.425343i	−1.18875	−	0.425343i
$92$		0			0
$93$	0.0698805	+	0.147750i	0.0698805	+	0.147750i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.812847	−	1.40789i	−0.812847	−	1.40789i	−0.910864	−	0.412707i	$-0.864583\pi$
$97$	−0.812847	−	1.40789i	−0.812847	−	1.40789i	0.0980171	−	0.995185i	$-0.468750\pi$
$98$		0			0
$99$		0			0
$100$	0.683592	+	0.729864i	0.683592	+	0.729864i
$101$		0			0		0.242980	−	0.970031i	$-0.421875\pi$
$101$		0			0		−0.242980	+	0.970031i	$0.578125\pi$
$102$		0			0
$103$	−0.155174	+	0.392675i	−0.155174	+	0.392675i	−0.986643	−	0.162895i	$-0.947917\pi$
$103$	−0.155174	+	0.392675i	−0.155174	+	0.392675i	0.831470	+	0.555570i	$0.187500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$107$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$108$	0.812847	−	0.582478i	0.812847	−	0.582478i
$109$	−0.273678	−	0.902197i	−0.273678	−	0.902197i	−0.980785	−	0.195090i	$-0.937500\pi$
$109$	−0.273678	−	0.902197i	−0.273678	−	0.902197i	0.707107	−	0.707107i	$-0.250000\pi$
$110$		0			0
$111$	−0.740646	+	0.933207i	−0.740646	+	0.933207i
$112$	−0.194015	+	1.06628i	−0.194015	+	1.06628i
$113$		0			0		0.783287	−	0.621661i	$-0.213542\pi$
$113$		0			0		−0.783287	+	0.621661i	$0.786458\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.480785	+	1.06112i	0.480785	+	1.06112i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.0654031	+	0.997859i	0.0654031	+	0.997859i
$122$		0			0
$123$		0			0
$124$	0.0650089	+	0.149957i	0.0650089	+	0.149957i
$125$		0			0
$126$		0			0
$127$	0.761850	+	0.690501i	0.761850	+	0.690501i	0.956940	−	0.290285i	$-0.0937500\pi$
$127$	0.761850	+	0.690501i	0.761850	+	0.690501i	−0.195090	+	0.980785i	$0.562500\pi$
$128$		0			0
$129$	−1.01233	−	1.31930i	−1.01233	−	1.31930i
$130$		0			0
$131$		0			0		−0.582478	−	0.812847i	$-0.697917\pi$
$131$		0			0		0.582478	+	0.812847i	$0.302083\pi$
$132$		0			0
$133$	−0.201362	+	0.144294i	−0.201362	+	0.144294i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$137$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$138$		0			0
$139$	1.23417	+	0.464509i	1.23417	+	0.464509i	0.881921	−	0.471397i	$-0.156250\pi$
$139$	1.23417	+	0.464509i	1.23417	+	0.464509i	0.352250	+	0.935906i	$0.385417\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.831470	−	0.555570i	0.831470	−	0.555570i
$145$		0			0
$146$		0			0
$147$	−0.115113	+	0.131261i	−0.115113	+	0.131261i
$148$	−0.770783	+	0.908474i	−0.770783	+	0.908474i
$149$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$149$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$150$		0			0
$151$	0.246639	+	1.66270i	0.246639	+	1.66270i	0.659346	+	0.751840i	$0.270833\pi$
$151$	0.246639	+	1.66270i	0.246639	+	1.66270i	−0.412707	+	0.910864i	$0.635417\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.445809	+	1.07628i	0.445809	+	1.07628i
$157$	0.142243	+	0.658594i	0.142243	+	0.658594i	0.991445	+	0.130526i	$0.0416667\pi$
$157$	0.142243	+	0.658594i	0.142243	+	0.658594i	−0.849202	+	0.528068i	$0.822917\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.764369	−	0.843351i	−0.764369	−	0.843351i	0.227076	−	0.973877i	$-0.427083\pi$
$163$	−0.764369	−	0.843351i	−0.764369	−	0.843351i	−0.991445	+	0.130526i	$0.958333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.646956	−	0.762527i	$-0.723958\pi$
$167$		0			0		0.646956	+	0.762527i	$0.276042\pi$
$168$		0			0
$169$	−0.350259	+	0.0696709i	−0.350259	+	0.0696709i
$170$		0			0
$171$	0.224882	+	0.0409186i	0.224882	+	0.0409186i
$172$	−0.968625	−	1.35171i	−0.968625	−	1.35171i
$173$		0			0		−0.903989	−	0.427555i	$-0.859375\pi$
$173$		0			0		0.903989	+	0.427555i	$0.140625\pi$
$174$		0			0
$175$	0.194015	+	1.06628i	0.194015	+	1.06628i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.367516	−	0.930017i	$-0.380208\pi$
$179$		0			0		−0.367516	+	0.930017i	$0.619792\pi$
$180$		0			0
$181$	0.227272	−	0.317157i	0.227272	−	0.317157i	−0.683592	−	0.729864i	$-0.739583\pi$
$181$	0.227272	−	0.317157i	0.227272	−	0.317157i	0.910864	+	0.412707i	$0.135417\pi$
$182$		0			0
$183$	−0.227076	−	1.97388i	−0.227076	−	1.97388i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.08248	−	0.0531787i	1.08248	−	0.0531787i
$190$		0			0
$191$		0			0		−0.114287	−	0.993448i	$-0.536458\pi$
$191$		0			0		0.114287	+	0.993448i	$0.463542\pi$
$192$	0.849202	−	0.528068i	0.849202	−	0.528068i
$193$	−1.49068	−	1.26475i	−1.49068	−	1.26475i	−0.881921	−	0.471397i	$-0.843750\pi$
$193$	−1.49068	−	1.26475i	−1.49068	−	1.26475i	−0.608761	−	0.793353i	$-0.708333\pi$
$194$		0			0
$195$		0			0
$196$	−0.119346	+	0.127424i	−0.119346	+	0.127424i
$197$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$197$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$198$		0			0
$199$	−1.78480	−	0.739288i	−1.78480	−	0.739288i	−0.991445	−	0.130526i	$-0.958333\pi$
$199$	−1.78480	−	0.739288i	−1.78480	−	0.739288i	−0.793353	−	0.608761i	$-0.791667\pi$
$200$		0			0
$201$	0.975742	+	1.56912i	0.975742	+	1.56912i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.410356	+	1.09029i	0.410356	+	1.09029i
$209$		0			0
$210$		0			0
$211$	−0.541185	+	1.51251i	−0.541185	+	1.51251i	0.290285	+	0.956940i	$0.406250\pi$
$211$	−0.541185	+	1.51251i	−0.541185	+	1.51251i	−0.831470	+	0.555570i	$0.812500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.0288546	+	0.174770i	−0.0288546	+	0.174770i
$218$		0			0
$219$	0.575741	+	1.45694i	0.575741	+	1.45694i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.00854139	−	0.260913i	−0.00854139	−	0.260913i	−0.995185	−	0.0980171i	$-0.968750\pi$
$223$	−0.00854139	−	0.260913i	−0.00854139	−	0.260913i	0.986643	−	0.162895i	$-0.0520833\pi$
$224$		0			0
$225$	0.582478	−	0.812847i	0.582478	−	0.812847i
$226$		0			0
$227$		0			0		−0.762527	−	0.646956i	$-0.776042\pi$
$227$		0			0		0.762527	+	0.646956i	$0.223958\pi$
$228$	0.223422	+	0.0482546i	0.223422	+	0.0482546i
$229$	1.41617	−	1.12395i	1.41617	−	1.12395i	0.442289	−	0.896873i	$-0.354167\pi$
$229$	1.41617	−	1.12395i	1.41617	−	1.12395i	0.973877	−	0.227076i	$-0.0729167\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0163617	−	0.999866i	$-0.494792\pi$
$233$		0			0		−0.0163617	+	0.999866i	$0.505208\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.41395	−	1.36842i	−1.41395	−	1.36842i
$238$		0			0
$239$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$239$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$240$		0			0
$241$	0.113665	−	1.38623i	0.113665	−	1.38623i	−0.659346	−	0.751840i	$-0.729167\pi$
$241$	0.113665	−	1.38623i	0.113665	−	1.38623i	0.773010	−	0.634393i	$-0.218750\pi$
$242$		0			0
$243$	−0.729864	−	0.683592i	−0.729864	−	0.683592i
$244$	−0.162371	−	1.98025i	−0.162371	−	1.98025i
$245$		0			0
$246$		0			0
$247$	−0.105912	+	0.244309i	−0.105912	+	0.244309i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$251$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$252$	1.08364	−	0.0177326i	1.08364	−	0.0177326i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.866025	−	0.500000i	0.866025	−	0.500000i
$257$		0			0		0.695443	−	0.718582i	$-0.255208\pi$
$257$		0			0		−0.695443	+	0.718582i	$0.744792\pi$
$258$		0			0
$259$	−1.23562	+	0.374821i	−1.23562	+	0.374821i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.930017	−	0.367516i	$-0.880208\pi$
$263$		0			0		0.930017	+	0.367516i	$0.119792\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.923880	+	1.60021i	0.923880	+	1.60021i
$269$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$269$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$270$		0			0
$271$	−0.555570	+	0.168530i	−0.555570	+	0.168530i	−0.555570	−	0.831470i	$-0.687500\pi$
$271$	−0.555570	+	0.168530i	−0.555570	+	0.168530i			1.00000i	$0.5\pi$
$272$		0			0
$273$	−0.226019	+	1.24216i	−0.226019	+	1.24216i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.55503	−	0.864189i	1.55503	−	0.864189i	0.555570	−	0.831470i	$-0.312500\pi$
$277$	1.55503	−	0.864189i	1.55503	−	0.864189i	0.999465	−	0.0327191i	$-0.0104167\pi$
$278$		0			0
$279$	0.134393	−	0.0930150i	0.134393	−	0.0930150i
$280$		0			0
$281$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$281$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$282$		0			0
$283$	0.771657	−	1.78000i	0.771657	−	1.78000i	0.162895	−	0.986643i	$-0.447917\pi$
$283$	0.771657	−	1.78000i	0.771657	−	1.78000i	0.608761	−	0.793353i	$-0.291667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.999465	+	0.0327191i	−0.999465	+	0.0327191i
$290$		0			0
$291$	−1.25668	+	1.03133i	−1.25668	+	1.03133i
$292$	0.527763	+	1.47500i	0.527763	+	1.47500i
$293$		0			0		0.595699	−	0.803208i	$-0.296875\pi$
$293$		0			0		−0.595699	+	0.803208i	$0.703125\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.608761	−	0.793353i	0.608761	−	0.793353i
$301$	−0.0884330	−	1.80009i	−0.0884330	−	1.80009i
$302$		0			0
$303$		0			0
$304$	0.221724	+	0.0555389i	0.221724	+	0.0555389i
$305$		0			0
$306$		0			0
$307$	0.837889	−	0.642935i	0.837889	−	0.642935i	−0.0980171	−	0.995185i	$-0.531250\pi$
$307$	0.837889	−	0.642935i	0.837889	−	0.642935i	0.935906	+	0.352250i	$0.114583\pi$
$308$		0			0
$309$	0.412707	+	0.0891362i	0.412707	+	0.0891362i
$310$		0			0
$311$		0			0		0.456904	−	0.889516i	$-0.348958\pi$
$311$		0			0		−0.456904	+	0.889516i	$0.651042\pi$
$312$		0			0
$313$	1.04710	−	1.68387i	1.04710	−	1.68387i	0.412707	−	0.910864i	$-0.364583\pi$
$313$	1.04710	−	1.68387i	1.04710	−	1.68387i	0.634393	−	0.773010i	$-0.281250\pi$
$314$		0			0
$315$		0			0
$316$	−1.36842	−	1.41395i	−1.36842	−	1.41395i
$317$		0			0		0.903989	−	0.427555i	$-0.140625\pi$
$317$		0			0		−0.903989	+	0.427555i	$0.859375\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$325$	0.768108	+	0.875860i	0.768108	+	0.875860i
$326$		0			0
$327$	−0.845566	+	0.416987i	−0.845566	+	0.416987i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.380336	+	1.18372i	−0.380336	+	1.18372i	0.555570	+	0.831470i	$0.312500\pi$
$331$	−0.380336	+	1.18372i	−0.380336	+	1.18372i	−0.935906	+	0.352250i	$0.885417\pi$
$332$		0			0
$333$	1.04139	+	0.578738i	1.04139	+	0.578738i
$334$		0			0
$335$		0			0
$336$	1.08364	+	0.0177326i	1.08364	+	0.0177326i
$337$	0.439303	−	0.207775i	0.439303	−	0.207775i	−0.195090	−	0.980785i	$-0.562500\pi$
$337$	0.439303	−	0.207775i	0.439303	−	0.207775i	0.634393	+	0.773010i	$0.281250\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.867761	−	0.217363i	0.867761	−	0.217363i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$347$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$348$		0			0
$349$	1.16236	−	0.953924i	1.16236	−	0.953924i	0.162895	−	0.986643i	$-0.447917\pi$
$349$	1.16236	−	0.953924i	1.16236	−	0.953924i	0.999465	+	0.0327191i	$0.0104167\pi$
$350$		0			0
$351$	0.968625	−	0.647215i	0.968625	−	0.647215i
$352$		0			0
$353$		0			0		−0.162895	−	0.986643i	$-0.552083\pi$
$353$		0			0		0.162895	+	0.986643i	$0.447917\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$359$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$360$		0			0
$361$	−0.500478	−	0.804835i	−0.500478	−	0.804835i
$362$		0			0
$363$	0.973877	−	0.227076i	0.973877	−	0.227076i
$364$	−0.266541	+	1.23410i	−0.266541	+	1.23410i
$365$		0			0
$366$		0			0
$367$	0.0726721	+	1.10876i	0.0726721	+	1.10876i	0.866025	+	0.500000i	$0.166667\pi$
$367$	0.0726721	+	1.10876i	0.0726721	+	1.10876i	−0.793353	+	0.608761i	$0.791667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.137365	−	0.0885679i	0.137365	−	0.0885679i
$373$	1.45195	−	1.23189i	1.45195	−	1.23189i	0.528068	−	0.849202i	$-0.322917\pi$
$373$	1.45195	−	1.23189i	1.45195	−	1.23189i	0.923880	−	0.382683i	$-0.125000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.119338	−	1.21166i	−0.119338	−	1.21166i	−0.849202	−	0.528068i	$-0.822917\pi$
$379$	−0.119338	−	1.21166i	−0.119338	−	1.21166i	0.729864	−	0.683592i	$-0.239583\pi$
$380$		0			0
$381$	0.557176	−	0.864154i	0.557176	−	0.864154i
$382$		0			0
$383$		0			0		−0.999465	−	0.0327191i	$-0.989583\pi$
$383$		0			0		0.999465	+	0.0327191i	$0.0104167\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.13677	+	1.21372i	−1.13677	+	1.21372i
$388$	−1.28975	+	0.989659i	−1.28975	+	0.989659i
$389$		0			0		0.970031	−	0.242980i	$-0.0781250\pi$
$389$		0			0		−0.970031	+	0.242980i	$0.921875\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.257276	+	1.41395i	0.257276	+	1.41395i	0.812847	+	0.582478i	$0.197917\pi$
$397$	0.257276	+	1.41395i	0.257276	+	1.41395i	−0.555570	+	0.831470i	$0.687500\pi$
$398$		0			0
$399$	0.175168	+	0.175168i	0.175168	+	0.175168i
$400$	0.634393	−	0.773010i	0.634393	−	0.773010i
$401$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$401$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$402$		0			0
$403$	0.0699761	+	0.177078i	0.0699761	+	0.177078i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.189856	−	1.27991i	0.189856	−	1.27991i	−0.659346	−	0.751840i	$-0.729167\pi$
$409$	0.189856	−	1.27991i	0.189856	−	1.27991i	0.849202	−	0.528068i	$-0.177083\pi$
$410$		0			0
$411$		0			0
$412$	0.409570	+	0.102592i	0.409570	+	0.102592i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.257264	−	1.29335i	0.257264	−	1.29335i
$418$		0			0
$419$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$419$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$420$		0			0
$421$	−0.532719	+	0.133439i	−0.532719	+	0.133439i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−0.532719	+	0.133439i	−0.532719	+	0.133439i	−0.0327191	+	0.999465i	$0.510417\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.01509	−	1.89910i	1.01509	−	1.89910i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.986643	−	0.162895i	$-0.947917\pi$
$431$		0			0		0.986643	+	0.162895i	$0.0520833\pi$
$432$	−0.683592	−	0.729864i	−0.683592	−	0.729864i
$433$	−0.229864	−	0.182433i	−0.229864	−	0.182433i	0.500000	−	0.866025i	$-0.333333\pi$
$433$	−0.229864	−	0.182433i	−0.229864	−	0.182433i	−0.729864	+	0.683592i	$0.760417\pi$
$434$		0			0
$435$		0			0
$436$	−0.858756	+	0.389097i	−0.858756	+	0.389097i
$437$		0			0
$438$		0			0
$439$	1.01210	+	1.51471i	1.01210	+	1.51471i	0.849202	+	0.528068i	$0.177083\pi$
$439$	1.01210	+	1.51471i	1.01210	+	1.51471i	0.162895	+	0.986643i	$0.447917\pi$
$440$		0			0
$441$	0.148259	+	0.0921932i	0.148259	+	0.0921932i
$442$		0			0
$443$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$443$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$444$	1.02190	+	0.612501i	1.02190	+	0.612501i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.08248	+	0.0531787i	1.08248	+	0.0531787i
$449$		0			0		0.352250	−	0.935906i	$-0.385417\pi$
$449$		0			0		−0.352250	+	0.935906i	$0.614583\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.60032	−	0.514191i	1.60032	−	0.514191i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.06330	−	1.59133i	1.06330	−	1.59133i	0.290285	−	0.956940i	$-0.406250\pi$
$457$	1.06330	−	1.59133i	1.06330	−	1.59133i	0.773010	−	0.634393i	$-0.218750\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.305903	−	0.952063i	$-0.401042\pi$
$461$		0			0		−0.305903	+	0.952063i	$0.598958\pi$
$462$		0			0
$463$	−0.0326140	−	1.99304i	−0.0326140	−	1.99304i	−0.0980171	−	0.995185i	$-0.531250\pi$
$463$	−0.0326140	−	1.99304i	−0.0326140	−	1.99304i	0.0654031	−	0.997859i	$-0.479167\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.456904	−	0.889516i	$-0.348958\pi$
$467$		0			0		−0.456904	+	0.889516i	$0.651042\pi$
$468$	0.989283	−	0.615175i	0.989283	−	0.615175i
$469$	−0.0982615	+	2.00016i	−0.0982615	+	2.00016i
$470$		0			0
$471$	0.626627	−	0.247625i	0.626627	−	0.247625i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.227076	−	0.0261230i	0.227076	−	0.0261230i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.274589	−	0.961562i	$-0.411458\pi$
$479$		0			0		−0.274589	+	0.961562i	$0.588542\pi$
$480$		0			0
$481$	−0.932074	+	1.02839i	−0.932074	+	1.02839i
$482$		0			0
$483$		0			0
$484$	0.980785	−	0.195090i	0.980785	−	0.195090i
$485$		0			0
$486$		0			0
$487$	−0.805124	+	0.288078i	−0.805124	+	0.288078i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−0.805124	+	0.288078i	−0.805124	+	0.288078i	−0.0980171	+	0.995185i	$0.531250\pi$
$488$		0			0
$489$	−0.707574	+	0.891537i	−0.707574	+	0.891537i
$490$		0			0
$491$		0			0		−0.514103	−	0.857729i	$-0.671875\pi$
$491$		0			0		0.514103	+	0.857729i	$0.328125\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.140189	−	0.0840261i	0.140189	−	0.0840261i
$497$		0			0
$498$		0			0
$499$	−0.510724	+	1.90605i	−0.510724	+	1.90605i	−0.0980171	+	0.995185i	$0.531250\pi$
$499$	−0.510724	+	1.90605i	−0.510724	+	1.90605i	−0.412707	+	0.910864i	$0.635417\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.125796	+	0.334232i	0.125796	+	0.334232i
$508$	0.585152	−	0.845461i	0.585152	−	0.845461i
$509$		0			0		−0.998795	−	0.0490677i	$-0.984375\pi$
$509$		0			0		0.998795	+	0.0490677i	$0.0156250\pi$
$510$		0			0
$511$	−0.385536	+	1.65347i	−0.385536	+	1.65347i
$512$		0			0
$513$	0.00373987	−	0.228543i	0.00373987	−	0.228543i
$514$		0			0
$515$		0			0
$516$	−1.17588	+	1.17588i	−1.17588	+	1.17588i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$521$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$522$		0			0
$523$	0.729864	−	1.26416i	0.729864	−	1.26416i	−0.227076	−	0.973877i	$-0.572917\pi$
$523$	0.729864	−	1.26416i	0.729864	−	1.26416i	0.956940	−	0.290285i	$-0.0937500\pi$
$524$		0			0
$525$	1.02043	−	0.365116i	1.02043	−	0.365116i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	−	0.500000i	0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	0.169343	+	0.180805i	0.169343	+	0.180805i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.19472	+	0.613676i	−1.19472	+	0.613676i	−0.935906	−	0.352250i	$-0.885417\pi$
$541$	−1.19472	+	0.613676i	−1.19472	+	0.613676i	−0.258819	+	0.965926i	$0.583333\pi$
$542$		0			0
$543$	−0.349942	−	0.172572i	−0.349942	−	0.172572i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.38192	+	1.33742i	−1.38192	+	1.33742i	−0.500000	+	0.866025i	$0.666667\pi$
$547$	−1.38192	+	1.33742i	−1.38192	+	1.33742i	−0.881921	+	0.471397i	$0.843750\pi$
$548$		0			0
$549$	−1.91052	+	0.545579i	−1.91052	+	0.545579i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.416040	−	2.09158i	−0.416040	−	2.09158i
$554$		0			0
$555$		0			0
$556$	0.299444	−	1.28424i	0.299444	−	1.28424i
$557$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$557$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$558$		0			0
$559$	−1.07628	−	1.61076i	−1.07628	−	1.61076i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.352250	−	0.935906i	$-0.385417\pi$
$563$		0			0		−0.352250	+	0.935906i	$0.614583\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.228799	−	1.05936i	−0.228799	−	1.05936i
$568$		0			0
$569$		0			0		−0.930017	−	0.367516i	$-0.880208\pi$
$569$		0			0		0.930017	+	0.367516i	$0.119792\pi$
$570$		0			0
$571$	−0.0169366	+	0.517361i	−0.0169366	+	0.517361i	0.956940	+	0.290285i	$0.0937500\pi$
$571$	−0.0169366	+	0.517361i	−0.0169366	+	0.517361i	−0.973877	+	0.227076i	$0.927083\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.659346	−	0.751840i	−0.659346	−	0.751840i
$577$	1.31234	+	1.49644i	1.31234	+	1.49644i	0.729864	+	0.683592i	$0.239583\pi$
$577$	1.31234	+	1.49644i	1.31234	+	1.49644i	0.582478	+	0.812847i	$0.302083\pi$
$578$		0			0
$579$	−1.00503	+	1.67679i	−1.00503	+	1.67679i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$587$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$588$	0.145163	+	0.0969948i	0.145163	+	0.0969948i
$589$	0.0363827	+	0.00848325i	0.0363827	+	0.00848325i
$590$		0			0
$591$		0			0
$592$	1.00131	+	0.645609i	1.00131	+	0.645609i
$593$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$593$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.438678	+	1.88139i	−0.438678	+	1.88139i
$598$		0			0
$599$		0			0		0.211112	−	0.977462i	$-0.432292\pi$
$599$		0			0		−0.211112	+	0.977462i	$0.567708\pi$
$600$		0			0
$601$	−0.222708	+	1.34892i	−0.222708	+	1.34892i	0.608761	+	0.793353i	$0.291667\pi$
$601$	−0.222708	+	1.34892i	−0.222708	+	1.34892i	−0.831470	+	0.555570i	$0.812500\pi$
$602$		0			0
$603$	1.38922	−	1.21831i	1.38922	−	1.21831i
$604$	1.61629	−	0.461555i	1.61629	−	0.461555i
$605$		0			0
$606$		0			0
$607$	−0.568078	−	0.715774i	−0.568078	−	0.715774i	0.412707	−	0.910864i	$-0.364583\pi$
$607$	−0.568078	−	0.715774i	−0.568078	−	0.715774i	−0.980785	+	0.195090i	$0.937500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.638949	−	0.473877i	−0.638949	−	0.473877i	0.227076	−	0.973877i	$-0.427083\pi$
$613$	−0.638949	−	0.473877i	−0.638949	−	0.473877i	−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$617$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$618$		0			0
$619$	0.0566711	+	0.0327191i	0.0566711	+	0.0327191i	0.528068	−	0.849202i	$-0.322917\pi$
$619$	0.0566711	+	0.0327191i	0.0566711	+	0.0327191i	−0.471397	+	0.881921i	$0.656250\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.00888	−	0.582478i	1.00888	−	0.582478i
$625$	0.321439	−	0.946930i	0.321439	−	0.946930i
$626$		0			0
$627$		0			0
$628$	0.634393	−	0.226990i	0.634393	−	0.226990i
$629$		0			0
$630$		0			0
$631$	−0.172040	+	0.276662i	−0.172040	+	0.276662i	−0.923880	−	0.382683i	$-0.875000\pi$
$631$	−0.172040	+	0.276662i	−0.172040	+	0.276662i	0.751840	+	0.659346i	$0.229167\pi$
$632$		0			0
$633$	1.58047	+	0.287575i	1.58047	+	0.287575i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.143815	+	0.143815i	−0.143815	+	0.143815i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$641$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$642$		0			0
$643$	1.65404	+	0.385667i	1.65404	+	0.385667i	0.946930	−	0.321439i	$-0.104167\pi$
$643$	1.65404	+	0.385667i	1.65404	+	0.385667i	0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.935906	−	0.352250i	$-0.114583\pi$
$647$		0			0		−0.935906	+	0.352250i	$0.885417\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	0.177136			0.177136
$652$	−0.736366	+	0.867909i	−0.736366	+	0.867909i
$653$		0			0		0.485763	−	0.874090i	$-0.338542\pi$
$653$		0			0		−0.485763	+	0.874090i	$0.661458\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.34369	−	0.805380i	1.34369	−	0.805380i
$658$		0			0
$659$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$659$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$660$		0			0
$661$	0.155524	+	0.119338i	0.155524	+	0.119338i	0.683592	−	0.729864i	$-0.260417\pi$
$661$	0.155524	+	0.119338i	0.155524	+	0.119338i	−0.528068	+	0.849202i	$0.677083\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.256036	+	0.0509288i	−0.256036	+	0.0509288i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.0130157	+	0.795390i	0.0130157	+	0.795390i	0.923880	+	0.382683i	$0.125000\pi$
$673$	0.0130157	+	0.795390i	0.0130157	+	0.795390i	−0.910864	+	0.412707i	$0.864583\pi$
$674$		0			0
$675$	−0.896873	−	0.442289i	−0.896873	−	0.442289i
$676$	0.114793	+	0.338169i	0.114793	+	0.338169i
$677$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$677$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$678$		0			0
$679$	−1.75601	+	0.143984i	−1.75601	+	0.143984i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.595699	−	0.803208i	$-0.703125\pi$
$683$		0			0		0.595699	+	0.803208i	$0.296875\pi$
$684$	0.0112156	−	0.228299i	0.0112156	−	0.228299i
$685$		0			0
$686$		0			0
$687$	−1.33962	−	1.21416i	−1.33962	−	1.21416i
$688$	−1.21372	+	1.13677i	−1.21372	+	1.13677i
$689$		0			0
$690$		0			0
$691$	1.96743	+	0.357986i	1.96743	+	0.357986i	0.986643	+	0.162895i	$0.0520833\pi$
$691$	1.96743	+	0.357986i	1.96743	+	0.357986i	0.980785	+	0.195090i	$0.0625000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.03183	−	0.331533i	1.03183	−	0.331533i
$701$		0			0		−0.718582	−	0.695443i	$-0.755208\pi$
$701$		0			0		0.718582	+	0.695443i	$0.244792\pi$
$702$		0			0
$703$	0.0618380	+	0.265209i	0.0618380	+	0.265209i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.41057	−	0.845461i	−1.41057	−	0.845461i	−0.412707	−	0.910864i	$-0.635417\pi$
$709$	−1.41057	−	0.845461i	−1.41057	−	0.845461i	−0.997859	+	0.0654031i	$0.979167\pi$
$710$		0			0
$711$	−1.11981	+	1.61797i	−1.11981	+	1.61797i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$719$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$720$		0			0
$721$	0.312811	+	0.333985i	0.312811	+	0.333985i
$722$		0			0
$723$	−1.38623	+	0.113665i	−1.38623	+	0.113665i
$724$	−0.344109	−	0.183930i	−0.344109	−	0.183930i
$725$		0			0
$726$		0			0
$727$	−1.35315	+	1.30958i	−1.35315	+	1.30958i	−0.442289	+	0.896873i	$0.645833\pi$
$727$	−1.35315	+	1.30958i	−1.35315	+	1.30958i	−0.910864	+	0.412707i	$0.864583\pi$
$728$		0			0
$729$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$730$		0			0
$731$		0			0
$732$	−1.92735	+	0.482776i	−1.92735	+	0.482776i
$733$	0.510724	−	0.0843209i	0.510724	−	0.0843209i	0.0980171	−	0.995185i	$-0.468750\pi$
$733$	0.510724	−	0.0843209i	0.510724	−	0.0843209i	0.412707	+	0.910864i	$0.364583\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.184592	−	1.40211i	0.184592	−	1.40211i	−0.608761	−	0.793353i	$-0.708333\pi$
$739$	0.184592	−	1.40211i	0.184592	−	1.40211i	0.793353	−	0.608761i	$-0.208333\pi$
$740$		0			0
$741$	0.258298	+	0.0647004i	0.258298	+	0.0647004i
$742$		0			0
$743$		0			0		−0.211112	−	0.977462i	$-0.567708\pi$
$743$		0			0		0.211112	+	0.977462i	$0.432292\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.448087	−	0.0739794i	−0.448087	−	0.0739794i	−0.0654031	−	0.997859i	$-0.520833\pi$
$751$	−0.448087	−	0.0739794i	−0.448087	−	0.0739794i	−0.382683	+	0.923880i	$0.625000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.194015	−	1.06628i	−0.194015	−	1.06628i
$757$	0.290285	−	0.956940i	0.290285	−	0.956940i	−0.683592	−	0.729864i	$-0.739583\pi$
$757$	0.290285	−	0.956940i	0.290285	−	0.956940i	0.973877	−	0.227076i	$-0.0729167\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$761$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$762$		0			0
$763$	−1.01072	−	0.149927i	−1.01072	−	0.149927i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$769$	0.0213077	−	0.129059i	0.0213077	−	0.129059i	−0.973877	−	0.227076i	$-0.927083\pi$
$769$	0.0213077	−	0.129059i	0.0213077	−	0.129059i	0.995185	+	0.0980171i	$0.0312500\pi$
$770$		0			0
$771$		0			0
$772$	−1.05936	+	1.64301i	−1.05936	+	1.64301i
$773$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$773$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$774$		0			0
$775$	0.101606	−	0.128022i	0.101606	−	0.128022i
$776$		0			0
$777$	0.571091	+	1.15806i	0.571091	+	1.15806i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.141912	+	0.101692i	0.141912	+	0.101692i
$785$		0			0
$786$		0			0
$787$	1.52945	+	1.05855i	1.52945	+	1.05855i	0.973877	+	0.227076i	$0.0729167\pi$
$787$	1.52945	+	1.05855i	1.52945	+	1.05855i	0.555570	+	0.831470i	$0.312500\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.113574	−	2.31186i	−0.113574	−	2.31186i
$794$		0			0
$795$		0			0
$796$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$797$		0			0		−0.162895	−	0.986643i	$-0.552083\pi$
$797$		0			0		0.162895	+	0.986643i	$0.447917\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.42834	−	1.17221i	1.42834	−	1.17221i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$809$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$810$		0			0
$811$	0.280443	−	0.222575i	0.280443	−	0.222575i	−0.471397	−	0.881921i	$-0.656250\pi$
$811$	0.280443	−	0.222575i	0.280443	−	0.222575i	0.751840	+	0.659346i	$0.229167\pi$
$812$		0			0
$813$	0.256779	+	0.520697i	0.256779	+	0.520697i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.380054	−	0.00621917i	−0.380054	−	0.00621917i
$818$		0			0
$819$	1.26239	+	0.0206577i	1.26239	+	0.0206577i
$820$		0			0
$821$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$821$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$822$		0			0
$823$	0.514191	−	0.331533i	0.514191	−	0.331533i	−0.258819	−	0.965926i	$-0.583333\pi$
$823$	0.514191	−	0.331533i	0.514191	−	0.331533i	0.773010	+	0.634393i	$0.218750\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.999866	−	0.0163617i	$-0.00520833\pi$
$827$		0			0		−0.999866	+	0.0163617i	$0.994792\pi$
$828$		0			0
$829$	1.35332	+	1.35332i	1.35332	+	1.35332i	0.881921	+	0.471397i	$0.156250\pi$
$829$	1.35332	+	1.35332i	1.35332	+	1.35332i	0.471397	+	0.881921i	$0.343750\pi$
$830$		0			0
$831$	−1.10595	−	1.39349i	−1.10595	−	1.39349i
$832$	1.02740	−	0.549156i	1.02740	−	0.549156i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.113665	−	0.117447i	−0.113665	−	0.117447i
$838$		0			0
$839$		0			0		−0.874090	−	0.485763i	$-0.838542\pi$
$839$		0			0		0.874090	+	0.485763i	$0.161458\pi$
$840$		0			0
$841$	0.946930	+	0.321439i	0.946930	+	0.321439i
$842$		0			0
$843$		0			0
$844$	1.57021	+	0.339133i	1.57021	+	0.339133i
$845$		0			0
$846$		0			0
$847$	1.00794	+	0.398308i	1.00794	+	0.398308i
$848$		0			0
$849$	−1.88192	−	0.471397i	−1.88192	−	0.471397i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.05441	+	1.37413i	−1.05441	+	1.37413i	−0.130526	+	0.991445i	$0.541667\pi$
$853$	−1.05441	+	1.37413i	−1.05441	+	1.37413i	−0.923880	+	0.382683i	$0.875000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$857$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$858$		0			0
$859$	0.247198	+	1.87766i	0.247198	+	1.87766i	0.442289	+	0.896873i	$0.354167\pi$
$859$	0.247198	+	1.87766i	0.247198	+	1.87766i	−0.195090	+	0.980785i	$0.562500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.840448	−	0.541892i	$-0.817708\pi$
$863$		0			0		0.840448	+	0.541892i	$0.182292\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.195090	+	0.980785i	0.195090	+	0.980785i
$868$	0.177041	+	0.00579572i	0.177041	+	0.00579572i
$869$		0			0
$870$		0			0
$871$	1.01471	+	1.89839i	1.01471	+	1.89839i
$872$		0			0
$873$	1.22226	+	1.07189i	1.22226	+	1.07189i
$874$		0			0
$875$		0			0
$876$	1.36933	−	0.760984i	1.36933	−	0.760984i
$877$	−1.07746	+	0.670005i	−1.07746	+	0.670005i	−0.946930	−	0.321439i	$-0.895833\pi$
$877$	−1.07746	+	0.670005i	−1.07746	+	0.670005i	−0.130526	+	0.991445i	$0.541667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.274589	−	0.961562i	$-0.588542\pi$
$881$		0			0		0.274589	+	0.961562i	$0.411458\pi$
$882$		0			0
$883$	0.124629	+	1.51995i	0.124629	+	1.51995i	0.707107	+	0.707107i	$0.250000\pi$
$883$	0.124629	+	1.51995i	0.124629	+	1.51995i	−0.582478	+	0.812847i	$0.697917\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$887$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$888$		0			0
$889$	1.02953	−	0.426444i	1.02953	−	0.426444i
$890$		0			0
$891$		0			0
$892$	−0.257566	+	0.0425243i	−0.257566	+	0.0425243i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.881921	−	0.471397i	−0.881921	−	0.471397i
$901$		0			0
$902$		0			0
$903$	−1.76165	+	0.380479i	−1.76165	+	0.380479i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.0500574	+	0.763728i	0.0500574	+	0.763728i	0.946930	+	0.321439i	$0.104167\pi$
$907$	0.0500574	+	0.763728i	0.0500574	+	0.763728i	−0.896873	+	0.442289i	$0.854167\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$911$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$912$	0.0186793	−	0.227809i	0.0186793	−	0.227809i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.29918	−	1.25735i	−1.29918	−	1.25735i
$917$		0			0
$918$		0			0
$919$	−1.06330	−	0.322547i	−1.06330	−	0.322547i	−0.290285	−	0.956940i	$-0.593750\pi$
$919$	−1.06330	−	0.322547i	−1.06330	−	0.322547i	−0.773010	+	0.634393i	$0.781250\pi$
$920$		0			0
$921$	−0.770836	−	0.721966i	−0.770836	−	0.721966i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.16455	+	0.251518i	1.16455	+	0.251518i
$926$		0			0
$927$	0.0207175	−	0.421715i	0.0207175	−	0.421715i
$928$		0			0
$929$		0			0		−0.621661	−	0.783287i	$-0.713542\pi$
$929$		0			0		0.621661	+	0.783287i	$0.286458\pi$
$930$		0			0
$931$	0.00714381	+	0.0392612i	0.00714381	+	0.0392612i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.195090	+	1.98079i	−0.195090	+	1.98079i			1.00000i	$0.5\pi$
$937$	−0.195090	+	1.98079i	−0.195090	+	1.98079i	−0.195090	+	0.980785i	$0.562500\pi$
$938$		0			0
$939$	−1.83195	−	0.758819i	−1.83195	−	0.758819i
$940$		0			0
$941$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$941$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.456904	−	0.889516i	$-0.651042\pi$
$947$		0			0		0.456904	+	0.889516i	$0.348958\pi$
$948$	−1.17215	+	1.58047i	−1.17215	+	1.58047i
$949$	0.558269	+	1.73750i	0.558269	+	1.73750i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.999465	−	0.0327191i	$-0.0104167\pi$
$953$		0			0		−0.999465	+	0.0327191i	$0.989583\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.826517	+	0.513961i	−0.826517	+	0.513961i
$962$		0			0
$963$		0			0
$964$	−1.38921	+	0.0682475i	−1.38921	+	0.0682475i
$965$		0			0
$966$		0			0
$967$	1.81071	+	0.517075i	1.81071	+	0.517075i	0.997859	−	0.0654031i	$-0.0208333\pi$
$967$	1.81071	+	0.517075i	1.81071	+	0.517075i	0.812847	+	0.582478i	$0.197917\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.874090	−	0.485763i	$-0.161458\pi$
$971$		0			0		−0.874090	+	0.485763i	$0.838542\pi$
$972$	−0.582478	+	0.812847i	−0.582478	+	0.812847i
$973$	1.02698	−	0.993910i	1.02698	−	0.993910i
$974$		0			0
$975$	0.739040	−	0.900523i	0.739040	−	0.900523i
$976$	−1.94211	+	0.419457i	−1.94211	+	0.419457i
$977$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$977$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.549156	+	0.766347i	0.549156	+	0.766347i
$982$		0			0
$983$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$983$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.256043	+	0.0731170i	0.256043	+	0.0731170i
$989$		0			0
$990$		0			0
$991$	−0.0213077	+	0.325093i	−0.0213077	+	0.325093i	0.973877	+	0.227076i	$0.0729167\pi$
$991$	−0.0213077	+	0.325093i	−0.0213077	+	0.325093i	−0.995185	+	0.0980171i	$0.968750\pi$
$992$		0			0
$993$	1.22986	+	0.182433i	1.22986	+	0.182433i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0500574	−	0.120849i	−0.0500574	−	0.120849i	0.896873	−	0.442289i	$-0.145833\pi$
$997$	−0.0500574	−	0.120849i	−0.0500574	−	0.120849i	−0.946930	+	0.321439i	$0.895833\pi$
$998$		0			0
$999$	0.401370	−	1.12175i	0.401370	−	1.12175i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3459.1.bm.a.719.1	✓	64
3.2	odd	2	CM	3459.1.bm.a.719.1	✓	64
1153.975	even	192	inner	3459.1.bm.a.3281.1	yes	64
3459.3281	odd	192	inner	3459.1.bm.a.3281.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3459.1.bm.a.719.1	✓	64	1.1	even	1	trivial
3459.1.bm.a.719.1	✓	64	3.2	odd	2	CM
3459.1.bm.a.3281.1	yes	64	1153.975	even	192	inner
3459.1.bm.a.3281.1	yes	64	3459.3281	odd	192	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:	\( N \)	\(=\)	\( 3459 = 3 \cdot 1153 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3459.bm (of order \(192\), degree \(64\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.162895 - 0.986643i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.463377 - 0.979728i) q^{7} +(-0.946930 + 0.321439i) q^{9} +O(q^{10})\)	\(q+(-0.162895 - 0.986643i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.463377 - 0.979728i) q^{7} +(-0.946930 + 0.321439i) q^{9} +(-0.956940 + 0.290285i) q^{12} +(-0.114186 - 1.15935i) q^{13} +(-0.965926 + 0.258819i) q^{16} +(-0.199794 - 0.111033i) q^{19} +(-1.04212 - 0.297595i) q^{21} +(-0.812847 + 0.582478i) q^{25} +(0.471397 + 0.881921i) q^{27} +(-1.03183 - 0.331533i) q^{28} +(-0.157160 + 0.0448794i) q^{31} +(0.442289 + 0.896873i) q^{36} +(-0.800094 - 0.882768i) q^{37} +(-1.12526 + 0.301513i) q^{39} +(1.46658 - 0.783904i) q^{43} +(0.412707 + 0.910864i) q^{48} +(-0.110756 - 0.134957i) q^{49} +(-1.13452 + 0.264534i) q^{52} +(-0.0770042 + 0.215212i) q^{57} +(1.98450 + 0.0974923i) q^{61} +(-0.123862 + 1.07668i) q^{63} +(0.382683 + 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(-1.53127 + 0.330722i) q^{73} +(0.707107 + 0.707107i) q^{75} +(-0.0840046 + 0.212578i) q^{76} +(1.58047 - 1.17215i) q^{79} +(0.793353 - 0.608761i) q^{81} +(-0.159024 + 1.07205i) q^{84} +(-1.18875 - 0.425343i) q^{91} +(0.0698805 + 0.147750i) q^{93} +(-0.812847 - 1.40789i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 64 * q	\( 64 q - 64 q^{67}+O(q^{100}) \) 64 * q - 64 * q^67

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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