LMFDB - Embedded newform 3459.1.bm.a.1502.1

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			1502.1
Root			$-0.0327191 + 0.999465i$ of defining polynomial
Character	$\chi$	$=$	3459.1502
Dual form			3459.1.bm.a.707.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.986643 + 0.162895i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.142243 + 0.397542i) q^{7} +(0.946930 - 0.321439i) q^{9} +O(q^{10})$	\(q+(-0.986643 + 0.162895i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.142243 + 0.397542i) q^{7} +(0.946930 - 0.321439i) q^{9} +(0.290285 + 0.956940i) q^{12} +(-1.61787 + 0.159346i) q^{13} +(-0.965926 + 0.258819i) q^{16} +(1.19553 - 0.341402i) q^{19} +(-0.205101 - 0.369061i) q^{21} +(-0.582478 - 0.812847i) q^{25} +(-0.881921 + 0.471397i) q^{27} +(0.375574 - 0.192915i) q^{28} +(0.740816 - 1.33304i) q^{31} +(-0.442289 - 0.896873i) q^{36} +(-0.293107 - 0.0143994i) q^{37} +(1.57030 - 0.420760i) q^{39} +(-0.783904 - 1.46658i) q^{43} +(0.910864 - 0.412707i) q^{48} +(0.635204 - 0.521299i) q^{49} +(0.369156 + 1.58323i) q^{52} +(-1.12395 + 0.531588i) q^{57} +(-1.16075 + 1.05205i) q^{61} +(0.262479 + 0.330722i) q^{63} +(0.382683 + 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(0.123862 - 0.192105i) q^{73} +(0.707107 + 0.707107i) q^{75} +(-0.494529 - 1.14074i) q^{76} +(-1.62674 - 0.241304i) q^{79} +(0.793353 - 0.608761i) q^{81} +(-0.339133 + 0.251518i) q^{84} +(-0.293476 - 0.620503i) q^{91} +(-0.513775 + 1.43591i) q^{93} +(-0.582478 - 1.00888i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3459\mathbb{Z}\right)^\times$.

$n$	$1154$	$2311$
$\chi(n)$	$-1$	$e\left(\frac{125}{192}\right)$

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	$0.729167\pi$
$3$	−0.986643	+	0.162895i	−0.986643	+	0.162895i
$3$	−0.986643	+	0.162895i	−0.986643	+	0.162895i
$4$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$5$		0			0		0.456904	−	0.889516i	$-0.348958\pi$
$5$		0			0		−0.456904	+	0.889516i	$0.651042\pi$
$6$		0			0
$7$	0.142243	+	0.397542i	0.142243	+	0.397542i	0.991445	−	0.130526i	$-0.0416667\pi$
$7$	0.142243	+	0.397542i	0.142243	+	0.397542i	−0.849202	+	0.528068i	$0.822917\pi$
$8$		0			0
$9$	0.946930	−	0.321439i	0.946930	−	0.321439i
$10$		0			0
$11$		0			0		0.683592	−	0.729864i	$-0.260417\pi$
$11$		0			0		−0.683592	+	0.729864i	$0.739583\pi$
$12$	0.290285	+	0.956940i	0.290285	+	0.956940i
$13$	−1.61787	+	0.159346i	−1.61787	+	0.159346i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	−1.61787	+	0.159346i	−1.61787	+	0.159346i	−0.751840	+	0.659346i	$0.770833\pi$
$14$		0			0
$15$		0			0
$16$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$17$		0			0		−0.718582	−	0.695443i	$-0.755208\pi$
$17$		0			0		0.718582	+	0.695443i	$0.244792\pi$
$18$		0			0
$19$	1.19553	−	0.341402i	1.19553	−	0.341402i	0.382683	−	0.923880i	$-0.375000\pi$
$19$	1.19553	−	0.341402i	1.19553	−	0.341402i	0.812847	+	0.582478i	$0.197917\pi$
$20$		0			0
$21$	−0.205101	−	0.369061i	−0.205101	−	0.369061i
$22$		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
$25$	−0.582478	−	0.812847i	−0.582478	−	0.812847i
$26$		0			0
$27$	−0.881921	+	0.471397i	−0.881921	+	0.471397i
$28$	0.375574	−	0.192915i	0.375574	−	0.192915i
$29$		0			0		0.162895	−	0.986643i	$-0.447917\pi$
$29$		0			0		−0.162895	+	0.986643i	$0.552083\pi$
$30$		0			0
$31$	0.740816	−	1.33304i	0.740816	−	1.33304i	−0.195090	−	0.980785i	$-0.562500\pi$
$31$	0.740816	−	1.33304i	0.740816	−	1.33304i	0.935906	−	0.352250i	$-0.114583\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.442289	−	0.896873i	−0.442289	−	0.896873i
$37$	−0.293107	−	0.0143994i	−0.293107	−	0.0143994i	−0.0980171	−	0.995185i	$-0.531250\pi$
$37$	−0.293107	−	0.0143994i	−0.293107	−	0.0143994i	−0.195090	+	0.980785i	$0.562500\pi$
$38$		0			0
$39$	1.57030	−	0.420760i	1.57030	−	0.420760i
$40$		0			0
$41$		0			0		0.812847	−	0.582478i	$-0.197917\pi$
$41$		0			0		−0.812847	+	0.582478i	$0.802083\pi$
$42$		0			0
$43$	−0.783904	−	1.46658i	−0.783904	−	1.46658i	−0.881921	−	0.471397i	$-0.843750\pi$
$43$	−0.783904	−	1.46658i	−0.783904	−	1.46658i	0.0980171	−	0.995185i	$-0.468750\pi$
$44$		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	$0.166667\pi$
$48$	0.910864	−	0.412707i	0.910864	−	0.412707i
$49$	0.635204	−	0.521299i	0.635204	−	0.521299i
$50$		0			0
$51$		0			0
$52$	0.369156	+	1.58323i	0.369156	+	1.58323i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.12395	+	0.531588i	−1.12395	+	0.531588i
$58$		0			0
$59$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$59$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$60$		0			0
$61$	−1.16075	+	1.05205i	−1.16075	+	1.05205i	−0.162895	+	0.986643i	$0.552083\pi$
$61$	−1.16075	+	1.05205i	−1.16075	+	1.05205i	−0.997859	+	0.0654031i	$0.979167\pi$
$62$		0			0
$63$	0.262479	+	0.330722i	0.262479	+	0.330722i
$64$	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		−0.514103	−	0.857729i	$-0.671875\pi$
$71$		0			0		0.514103	+	0.857729i	$0.328125\pi$
$72$		0			0
$73$	0.123862	−	0.192105i	0.123862	−	0.192105i	−0.773010	−	0.634393i	$-0.781250\pi$
$73$	0.123862	−	0.192105i	0.123862	−	0.192105i	0.896873	+	0.442289i	$0.145833\pi$
$74$		0			0
$75$	0.707107	+	0.707107i	0.707107	+	0.707107i
$76$	−0.494529	−	1.14074i	−0.494529	−	1.14074i
$77$		0			0
$78$		0			0
$79$	−1.62674	−	0.241304i	−1.62674	−	0.241304i	−0.729864	−	0.683592i	$-0.760417\pi$
$79$	−1.62674	−	0.241304i	−1.62674	−	0.241304i	−0.896873	+	0.442289i	$0.854167\pi$
$80$		0			0
$81$	0.793353	−	0.608761i	0.793353	−	0.608761i
$82$		0			0
$83$		0			0		0.762527	−	0.646956i	$-0.223958\pi$
$83$		0			0		−0.762527	+	0.646956i	$0.776042\pi$
$84$	−0.339133	+	0.251518i	−0.339133	+	0.251518i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.0327191	−	0.999465i	$-0.510417\pi$
$89$		0			0		0.0327191	+	0.999465i	$0.489583\pi$
$90$		0			0
$91$	−0.293476	−	0.620503i	−0.293476	−	0.620503i
$92$		0			0
$93$	−0.513775	+	1.43591i	−0.513775	+	1.43591i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.582478	−	1.00888i	−0.582478	−	1.00888i	−0.995185	−	0.0980171i	$-0.968750\pi$
$97$	−0.582478	−	1.00888i	−0.582478	−	1.00888i	0.412707	−	0.910864i	$-0.364583\pi$
$98$		0			0
$99$		0			0
$100$	−0.729864	+	0.683592i	−0.729864	+	0.683592i
$101$		0			0		−0.514103	−	0.857729i	$-0.671875\pi$
$101$		0			0		0.514103	+	0.857729i	$0.328125\pi$
$102$		0			0
$103$	−0.668574	−	1.54221i	−0.668574	−	1.54221i	−0.831470	−	0.555570i	$-0.812500\pi$
$103$	−0.668574	−	1.54221i	−0.668574	−	1.54221i	0.162895	−	0.986643i	$-0.447917\pi$
$104$		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	$0.395833\pi$
$108$	0.582478	+	0.812847i	0.582478	+	0.812847i
$109$	1.68789	−	0.512016i	1.68789	−	0.512016i	0.707107	−	0.707107i	$-0.250000\pi$
$109$	1.68789	−	0.512016i	1.68789	−	0.512016i	0.980785	+	0.195090i	$0.0625000\pi$
$110$		0			0
$111$	0.291538	−	0.0335388i	0.291538	−	0.0335388i
$112$	−0.240287	−	0.347181i	−0.240287	−	0.347181i
$113$		0			0		0.114287	−	0.993448i	$-0.463542\pi$
$113$		0			0		−0.114287	+	0.993448i	$0.536458\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.48079	+	0.670935i	−1.48079	+	0.670935i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.0654031	−	0.997859i	−0.0654031	−	0.997859i
$122$		0			0
$123$		0			0
$124$	−1.41833	−	0.560482i	−1.41833	−	0.560482i
$125$		0			0
$126$		0			0
$127$	−0.0951944	−	1.93773i	−0.0951944	−	1.93773i	−0.290285	−	0.956940i	$-0.593750\pi$
$127$	−0.0951944	−	1.93773i	−0.0951944	−	1.93773i	0.195090	−	0.980785i	$-0.437500\pi$
$128$		0			0
$129$	1.01233	+	1.31930i	1.01233	+	1.31930i
$130$		0			0
$131$		0			0		0.812847	−	0.582478i	$-0.197917\pi$
$131$		0			0		−0.812847	+	0.582478i	$0.802083\pi$
$132$		0			0
$133$	0.305777	+	0.426711i	0.305777	+	0.426711i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$137$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$138$		0			0
$139$	−0.464509	+	1.23417i	−0.464509	+	1.23417i	0.471397	+	0.881921i	$0.343750\pi$
$139$	−0.464509	+	1.23417i	−0.464509	+	1.23417i	−0.935906	+	0.352250i	$0.885417\pi$
$140$		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
$147$	−0.541803	+	0.617808i	−0.541803	+	0.617808i
$148$	0.0239819	+	0.292479i	0.0239819	+	0.292479i
$149$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$149$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$150$		0			0
$151$	−1.57021	−	1.16455i	−1.57021	−	1.16455i	−0.910864	−	0.412707i	$-0.864583\pi$
$151$	−1.57021	−	1.16455i	−1.57021	−	1.16455i	−0.659346	−	0.751840i	$-0.729167\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.622126	−	1.50194i	−0.622126	−	1.50194i
$157$	1.51951	+	0.979728i	1.51951	+	0.979728i	0.991445	+	0.130526i	$0.0416667\pi$
$157$	1.51951	+	0.979728i	1.51951	+	0.979728i	0.528068	+	0.849202i	$0.322917\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.96532	−	0.0965501i	−1.96532	−	0.0965501i	−0.973877	−	0.227076i	$-0.927083\pi$
$163$	−1.96532	−	0.0965501i	−1.96532	−	0.0965501i	−0.991445	+	0.130526i	$0.958333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.0817211	−	0.996655i	$-0.473958\pi$
$167$		0			0		−0.0817211	+	0.996655i	$0.526042\pi$
$168$		0			0
$169$	1.61131	−	0.320510i	1.61131	−	0.320510i
$170$		0			0
$171$	1.02234	−	0.707574i	1.02234	−	0.707574i
$172$	−1.35171	+	0.968625i	−1.35171	+	0.968625i
$173$		0			0		0.941544	−	0.336890i	$-0.109375\pi$
$173$		0			0		−0.941544	+	0.336890i	$0.890625\pi$
$174$		0			0
$175$	0.240287	−	0.347181i	0.240287	−	0.347181i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.397748	−	0.917494i	$-0.630208\pi$
$179$		0			0		0.397748	+	0.917494i	$0.369792\pi$
$180$		0			0
$181$	0.317157	+	0.227272i	0.317157	+	0.227272i	0.729864	−	0.683592i	$-0.239583\pi$
$181$	0.317157	+	0.227272i	0.317157	+	0.227272i	−0.412707	+	0.910864i	$0.635417\pi$
$182$		0			0
$183$	0.973877	−	1.22708i	0.973877	−	1.22708i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.312847	−	0.283548i	−0.312847	−	0.283548i
$190$		0			0
$191$		0			0		0.621661	−	0.783287i	$-0.286458\pi$
$191$		0			0		−0.621661	+	0.783287i	$0.713542\pi$
$192$	−0.528068	−	0.849202i	−0.528068	−	0.849202i
$193$	−1.08016	+	0.0885679i	−1.08016	+	0.0885679i	−0.608761	−	0.793353i	$-0.708333\pi$
$193$	−1.08016	+	0.0885679i	−1.08016	+	0.0885679i	−0.471397	+	0.881921i	$0.656250\pi$
$194$		0			0
$195$		0			0
$196$	−0.599749	−	0.561727i	−0.599749	−	0.561727i
$197$		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
$201$	1.56912	−	0.975742i	1.56912	−	0.975742i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.52150	−	0.572651i	1.52150	−	0.572651i
$209$		0			0
$210$		0			0
$211$	1.78841	−	0.845855i	1.78841	−	0.845855i	0.831470	−	0.555570i	$-0.187500\pi$
$211$	1.78841	−	0.845855i	1.78841	−	0.845855i	0.956940	−	0.290285i	$-0.0937500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.635313	+	0.104891i	0.635313	+	0.104891i
$218$		0			0
$219$	−0.0909149	+	0.209715i	−0.0909149	+	0.209715i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.260913	+	0.00854139i	−0.260913	+	0.00854139i	−0.162895	−	0.986643i	$-0.552083\pi$
$223$	−0.260913	+	0.00854139i	−0.260913	+	0.00854139i	−0.0980171	+	0.995185i	$0.531250\pi$
$224$		0			0
$225$	−0.812847	−	0.582478i	−0.812847	−	0.582478i
$226$		0			0
$227$		0			0		0.996655	−	0.0817211i	$-0.0260417\pi$
$227$		0			0		−0.996655	+	0.0817211i	$0.973958\pi$
$228$	0.673745	+	1.04495i	0.673745	+	1.04495i
$229$	−0.215212	+	1.87075i	−0.215212	+	1.87075i	0.227076	+	0.973877i	$0.427083\pi$
$229$	−0.215212	+	1.87075i	−0.215212	+	1.87075i	−0.442289	+	0.896873i	$0.645833\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.718582	−	0.695443i	$-0.244792\pi$
$233$		0			0		−0.718582	+	0.695443i	$0.755208\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.64432	−	0.0269075i	1.64432	−	0.0269075i
$238$		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
$241$	0.0249525	−	0.0211706i	0.0249525	−	0.0211706i	−0.634393	−	0.773010i	$-0.718750\pi$
$241$	0.0249525	−	0.0211706i	0.0249525	−	0.0211706i	0.659346	+	0.751840i	$0.270833\pi$
$242$		0			0
$243$	−0.683592	+	0.729864i	−0.683592	+	0.729864i
$244$	1.19455	+	1.01350i	1.19455	+	1.01350i
$245$		0			0
$246$		0			0
$247$	−1.87981	+	0.742845i	−1.87981	+	0.742845i
$248$		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	$0.270833\pi$
$252$	0.293632	−	0.303402i	0.293632	−	0.303402i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.866025	−	0.500000i	0.866025	−	0.500000i
$257$		0			0		−0.0163617	−	0.999866i	$-0.505208\pi$
$257$		0			0		0.0163617	+	0.999866i	$0.494792\pi$
$258$		0			0
$259$	−0.0359680	−	0.118571i	−0.0359680	−	0.118571i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.917494	−	0.397748i	$-0.130208\pi$
$263$		0			0		−0.917494	+	0.397748i	$0.869792\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.923880	+	1.60021i	0.923880	+	1.60021i
$269$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$269$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$270$		0			0
$271$	0.555570	+	1.83147i	0.555570	+	1.83147i	0.555570	+	0.831470i	$0.312500\pi$
$271$	0.555570	+	1.83147i	0.555570	+	1.83147i			1.00000i	$0.5\pi$
$272$		0			0
$273$	0.390633	+	0.564409i	0.390633	+	0.564409i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.588289	−	0.167995i	−0.588289	−	0.167995i	−0.0327191	−	0.999465i	$-0.510417\pi$
$277$	−0.588289	−	0.167995i	−0.588289	−	0.167995i	−0.555570	+	0.831470i	$0.687500\pi$
$278$		0			0
$279$	0.273010	−	1.50042i	0.273010	−	1.50042i
$280$		0			0
$281$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$281$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$282$		0			0
$283$	1.59540	−	0.630458i	1.59540	−	0.630458i	0.608761	−	0.793353i	$-0.291667\pi$
$283$	1.59540	−	0.630458i	1.59540	−	0.630458i	0.986643	+	0.162895i	$0.0520833\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.0327191	+	0.999465i	0.0327191	+	0.999465i
$290$		0			0
$291$	0.739040	+	0.900523i	0.739040	+	0.900523i
$292$	−0.206628	−	0.0977279i	−0.206628	−	0.0977279i
$293$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$293$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$294$		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
$301$	0.471523	−	0.520245i	0.471523	−	0.520245i
$302$		0			0
$303$		0			0
$304$	−1.06643	+	0.639195i	−1.06643	+	0.639195i
$305$		0			0
$306$		0			0
$307$	1.34743	−	1.03392i	1.34743	−	1.03392i	0.352250	−	0.935906i	$-0.385417\pi$
$307$	1.34743	−	1.03392i	1.34743	−	1.03392i	0.995185	−	0.0980171i	$-0.0312500\pi$
$308$		0			0
$309$	0.910864	+	1.41271i	0.910864	+	1.41271i
$310$		0			0
$311$		0			0		0.952063	−	0.305903i	$-0.0989583\pi$
$311$		0			0		−0.952063	+	0.305903i	$0.901042\pi$
$312$		0			0
$313$	1.68387	+	1.04710i	1.68387	+	1.04710i	0.910864	+	0.412707i	$0.135417\pi$
$313$	1.68387	+	1.04710i	1.68387	+	1.04710i	0.773010	+	0.634393i	$0.218750\pi$
$314$		0			0
$315$		0			0
$316$	−0.0269075	+	1.64432i	−0.0269075	+	1.64432i
$317$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$317$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$318$		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
$325$	1.07189	+	1.22226i	1.07189	+	1.22226i
$326$		0			0
$327$	−1.58194	+	0.780128i	−1.58194	+	0.780128i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.907820	−	1.76738i	−0.907820	−	1.76738i	−0.555570	−	0.831470i	$-0.687500\pi$
$331$	−0.907820	−	1.76738i	−0.907820	−	1.76738i	−0.352250	−	0.935906i	$-0.614583\pi$
$332$		0			0
$333$	−0.282181	+	0.0805810i	−0.282181	+	0.0805810i
$334$		0			0
$335$		0			0
$336$	0.293632	+	0.303402i	0.293632	+	0.303402i
$337$	0.968101	+	0.346392i	0.968101	+	0.346392i	0.773010	−	0.634393i	$-0.218750\pi$
$337$	0.968101	+	0.346392i	0.968101	+	0.346392i	0.195090	+	0.980785i	$0.437500\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.659744	+	0.395435i	0.659744	+	0.395435i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$347$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$348$		0			0
$349$	0.953924	+	1.16236i	0.953924	+	1.16236i	0.986643	+	0.162895i	$0.0520833\pi$
$349$	0.953924	+	1.16236i	0.953924	+	1.16236i	−0.0327191	+	0.999465i	$0.510417\pi$
$350$		0			0
$351$	1.35171	−	0.903187i	1.35171	−	0.903187i
$352$		0			0
$353$		0			0		0.986643	−	0.162895i	$-0.0520833\pi$
$353$		0			0		−0.986643	+	0.162895i	$0.947917\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.242980	−	0.970031i	$-0.578125\pi$
$359$		0			0		0.242980	+	0.970031i	$0.421875\pi$
$360$		0			0
$361$	0.463535	−	0.288245i	0.463535	−	0.288245i
$362$		0			0
$363$	0.227076	+	0.973877i	0.227076	+	0.973877i
$364$	−0.576888	+	0.371957i	−0.576888	+	0.371957i
$365$		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
$372$	1.49068	+	0.321957i	1.49068	+	0.321957i
$373$	1.77308	+	0.145384i	1.77308	+	0.145384i	0.923880	−	0.382683i	$-0.125000\pi$
$373$	1.77308	+	0.145384i	1.77308	+	0.145384i	0.849202	+	0.528068i	$0.177083\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.21166	−	0.119338i	1.21166	−	0.119338i	0.528068	−	0.849202i	$-0.322917\pi$
$379$	1.21166	−	0.119338i	1.21166	−	0.119338i	0.683592	+	0.729864i	$0.260417\pi$
$380$		0			0
$381$	0.409570	+	1.89634i	0.409570	+	1.89634i
$382$		0			0
$383$		0			0		0.0327191	−	0.999465i	$-0.489583\pi$
$383$		0			0		−0.0327191	+	0.999465i	$0.510417\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.21372	−	1.13677i	−1.21372	−	1.13677i
$388$	−0.924221	+	0.709180i	−0.924221	+	0.709180i
$389$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$389$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.13805	−	1.64432i	1.13805	−	1.64432i	0.555570	−	0.831470i	$-0.312500\pi$
$397$	1.13805	−	1.64432i	1.13805	−	1.64432i	0.582478	−	0.812847i	$-0.302083\pi$
$398$		0			0
$399$	−0.371202	−	0.371202i	−0.371202	−	0.371202i
$400$	0.773010	+	0.634393i	0.773010	+	0.634393i
$401$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$401$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$402$		0			0
$403$	−0.986126	+	2.27472i	−0.986126	+	2.27472i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.131278	−	0.0973624i	0.131278	−	0.0973624i	−0.528068	−	0.849202i	$-0.677083\pi$
$409$	0.131278	−	0.0973624i	0.131278	−	0.0973624i	0.659346	+	0.751840i	$0.270833\pi$
$410$		0			0
$411$		0			0
$412$	−1.44175	+	0.864154i	−1.44175	+	0.864154i
$413$		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
$421$	−1.49946	−	0.898744i	−1.49946	−	0.898744i	−0.999465	−	0.0327191i	$-0.989583\pi$
$421$	−1.49946	−	0.898744i	−1.49946	−	0.898744i	−0.500000	−	0.866025i	$-0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.583341	−	0.311802i	−0.583341	−	0.311802i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.162895	−	0.986643i	$-0.447917\pi$
$431$		0			0		−0.162895	+	0.986643i	$0.552083\pi$
$432$	0.729864	−	0.683592i	0.729864	−	0.683592i
$433$	−0.183592	−	1.59589i	−0.183592	−	1.59589i	−0.683592	−	0.729864i	$-0.739583\pi$
$433$	−0.183592	−	1.59589i	−0.183592	−	1.59589i	0.500000	−	0.866025i	$-0.333333\pi$
$434$		0			0
$435$		0			0
$436$	−0.727950	−	1.60662i	−0.727950	−	1.60662i
$437$		0			0
$438$		0			0
$439$	0.458575	+	0.686307i	0.458575	+	0.686307i	0.986643	−	0.162895i	$-0.0520833\pi$
$439$	0.458575	+	0.686307i	0.458575	+	0.686307i	−0.528068	+	0.849202i	$0.677083\pi$
$440$		0			0
$441$	0.433928	−	0.697813i	0.433928	−	0.697813i
$442$		0			0
$443$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$443$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$444$	−0.0713052	−	0.284666i	−0.0713052	−	0.284666i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.312847	+	0.283548i	−0.312847	+	0.283548i
$449$		0			0		−0.935906	−	0.352250i	$-0.885417\pi$
$449$		0			0		0.935906	+	0.352250i	$0.114583\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.73894	+	0.893212i	1.73894	+	0.893212i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.322547	−	0.482726i	0.322547	−	0.482726i	−0.634393	−	0.773010i	$-0.718750\pi$
$457$	0.322547	−	0.482726i	0.322547	−	0.482726i	0.956940	+	0.290285i	$0.0937500\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.456904	−	0.889516i	$-0.651042\pi$
$461$		0			0		0.456904	+	0.889516i	$0.348958\pi$
$462$		0			0
$463$	0.929782	+	0.899842i	0.929782	+	0.899842i	0.995185	−	0.0980171i	$-0.0312500\pi$
$463$	0.929782	+	0.899842i	0.929782	+	0.899842i	−0.0654031	+	0.997859i	$0.520833\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.952063	−	0.305903i	$-0.0989583\pi$
$467$		0			0		−0.952063	+	0.305903i	$0.901042\pi$
$468$	0.858476	+	1.38054i	0.858476	+	1.38054i
$469$	−0.523928	−	0.578065i	−0.523928	−	0.578065i
$470$		0			0
$471$	−1.65881	−	0.719121i	−1.65881	−	0.719121i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.973877	−	0.772924i	−0.973877	−	0.772924i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.874090	−	0.485763i	$-0.161458\pi$
$479$		0			0		−0.874090	+	0.485763i	$0.838542\pi$
$480$		0			0
$481$	0.476503	−	0.0234091i	0.476503	−	0.0234091i
$482$		0			0
$483$		0			0
$484$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$485$		0			0
$486$		0			0
$487$	0.288078	−	0.609090i	0.288078	−	0.609090i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	0.288078	−	0.609090i	0.288078	−	0.609090i	0.995185	+	0.0980171i	$0.0312500\pi$
$488$		0			0
$489$	1.95480	−	0.224882i	1.95480	−	0.224882i
$490$		0			0
$491$		0			0		−0.970031	−	0.242980i	$-0.921875\pi$
$491$		0			0		0.970031	+	0.242980i	$0.0781250\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.370558	+	1.47935i	−0.370558	+	1.47935i
$497$		0			0
$498$		0			0
$499$	0.0843209	−	0.314690i	0.0843209	−	0.314690i	−0.910864	−	0.412707i	$-0.864583\pi$
$499$	0.0843209	−	0.314690i	0.0843209	−	0.314690i	0.995185	+	0.0980171i	$0.0312500\pi$
$500$		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
$507$	−1.53758	+	0.578704i	−1.53758	+	0.578704i
$508$	−1.90872	+	0.347304i	−1.90872	+	0.347304i
$509$		0			0		0.740951	−	0.671559i	$-0.234375\pi$
$509$		0			0		−0.740951	+	0.671559i	$0.765625\pi$
$510$		0			0
$511$	0.0939881	+	0.0219149i	0.0939881	+	0.0219149i
$512$		0			0
$513$	−0.893428	+	0.864659i	−0.893428	+	0.864659i
$514$		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		−0.740951	−	0.671559i	$-0.765625\pi$
$521$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$522$		0			0
$523$	0.683592	−	1.18402i	0.683592	−	1.18402i	−0.290285	−	0.956940i	$-0.593750\pi$
$523$	0.683592	−	1.18402i	0.683592	−	1.18402i	0.973877	−	0.227076i	$-0.0729167\pi$
$524$		0			0
$525$	−0.180524	+	0.381685i	−0.180524	+	0.381685i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	−	0.500000i	0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	0.383149	−	0.358858i	0.383149	−	0.358858i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.611069	+	1.90183i	−0.611069	+	1.90183i	−0.258819	+	0.965926i	$0.583333\pi$
$541$	−0.611069	+	1.90183i	−0.611069	+	1.90183i	−0.352250	+	0.935906i	$0.614583\pi$
$542$		0			0
$543$	−0.349942	−	0.172572i	−0.349942	−	0.172572i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.971397	−	0.0158959i	−0.971397	−	0.0158959i	−0.471397	−	0.881921i	$-0.656250\pi$
$547$	−0.971397	−	0.0158959i	−0.971397	−	0.0158959i	−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$	−0.760984	+	1.36933i	−0.760984	+	1.36933i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.135463	−	0.681019i	−0.135463	−	0.681019i
$554$		0			0
$555$		0			0
$556$	1.28424	+	0.299444i	1.28424	+	0.299444i
$557$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$557$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$558$		0			0
$559$	1.50194	+	2.24782i	1.50194	+	2.24782i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.935906	−	0.352250i	$-0.885417\pi$
$563$		0			0		0.935906	+	0.352250i	$0.114583\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.354857	+	0.228799i	0.354857	+	0.228799i
$568$		0			0
$569$		0			0		0.917494	−	0.397748i	$-0.130208\pi$
$569$		0			0		−0.917494	+	0.397748i	$0.869792\pi$
$570$		0			0
$571$	−0.517361	−	0.0169366i	−0.517361	−	0.0169366i	−0.227076	−	0.973877i	$-0.572917\pi$
$571$	−0.517361	−	0.0169366i	−0.517361	−	0.0169366i	−0.290285	+	0.956940i	$0.593750\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.659346	+	0.751840i	0.659346	+	0.751840i
$577$	−0.129254	−	0.147386i	−0.129254	−	0.147386i	0.683592	−	0.729864i	$-0.260417\pi$
$577$	−0.129254	−	0.147386i	−0.129254	−	0.147386i	−0.812847	+	0.582478i	$0.802083\pi$
$578$		0			0
$579$	1.05130	−	0.263338i	1.05130	−	0.263338i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$587$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$588$	0.683242	+	0.456527i	0.683242	+	0.456527i
$589$	0.430567	−	1.84660i	0.430567	−	1.84660i
$590$		0			0
$591$		0			0
$592$	0.286847	−	0.0619530i	0.286847	−	0.0619530i
$593$		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
$597$	1.88139	+	0.438678i	1.88139	+	0.438678i
$598$		0			0
$599$		0			0		0.840448	−	0.541892i	$-0.182292\pi$
$599$		0			0		−0.840448	+	0.541892i	$0.817708\pi$
$600$		0			0
$601$	1.44023	+	0.237783i	1.44023	+	0.237783i	0.831470	−	0.555570i	$-0.187500\pi$
$601$	1.44023	+	0.237783i	1.44023	+	0.237783i	0.608761	+	0.793353i	$0.291667\pi$
$602$		0			0
$603$	−1.38922	+	1.21831i	−1.38922	+	1.21831i
$604$	−0.949630	+	1.70878i	−0.949630	+	1.70878i
$605$		0			0
$606$		0			0
$607$	1.89165	+	0.217617i	1.89165	+	0.217617i	0.980785	−	0.195090i	$-0.0625000\pi$
$607$	1.89165	+	0.217617i	1.89165	+	0.217617i	0.910864	+	0.412707i	$0.135417\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.83990	+	0.272924i	−1.83990	+	0.272924i	−0.973877	−	0.227076i	$-0.927083\pi$
$613$	−1.83990	+	0.272924i	−1.83990	+	0.272924i	−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
$619$	1.73112	+	0.999465i	1.73112	+	0.999465i	0.881921	+	0.471397i	$0.156250\pi$
$619$	1.73112	+	0.999465i	1.73112	+	0.999465i	0.849202	+	0.528068i	$0.177083\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.40789	+	0.812847i	−1.40789	+	0.812847i
$625$	−0.321439	+	0.946930i	−0.321439	+	0.946930i
$626$		0			0
$627$		0			0
$628$	0.773010	−	1.63439i	0.773010	−	1.63439i
$629$		0			0
$630$		0			0
$631$	−1.67572	−	1.04203i	−1.67572	−	1.04203i	−0.923880	−	0.382683i	$-0.875000\pi$
$631$	−1.67572	−	1.04203i	−1.67572	−	1.04203i	−0.751840	−	0.659346i	$-0.770833\pi$
$632$		0			0
$633$	−1.62674	+	1.12588i	−1.62674	+	1.12588i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.944608	+	0.944608i	−0.944608	+	0.944608i
$638$		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
$643$	−0.239823	+	1.02855i	−0.239823	+	1.02855i	0.707107	+	0.707107i	$0.250000\pi$
$643$	−0.239823	+	1.02855i	−0.239823	+	1.02855i	−0.946930	+	0.321439i	$0.895833\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.352250	−	0.935906i	$-0.614583\pi$
$647$		0			0		0.352250	+	0.935906i	$0.385417\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.643913			−0.643913
$652$	0.160802	+	1.96111i	0.160802	+	1.96111i
$653$		0			0		−0.274589	−	0.961562i	$-0.588542\pi$
$653$		0			0		0.274589	+	0.961562i	$0.411458\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.0555389	−	0.221724i	0.0555389	−	0.221724i
$658$		0			0
$659$		0			0		0.336890	−	0.941544i	$-0.390625\pi$
$659$		0			0		−0.336890	+	0.941544i	$0.609375\pi$
$660$		0			0
$661$	−1.57907	−	1.21166i	−1.57907	−	1.21166i	−0.849202	−	0.528068i	$-0.822917\pi$
$661$	−1.57907	−	1.21166i	−1.57907	−	1.21166i	−0.729864	−	0.683592i	$-0.760417\pi$
$662$		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
$673$	1.33659	+	1.29355i	1.33659	+	1.29355i	0.923880	+	0.382683i	$0.125000\pi$
$673$	1.33659	+	1.29355i	1.33659	+	1.29355i	0.412707	+	0.910864i	$0.364583\pi$
$674$		0			0
$675$	0.896873	+	0.442289i	0.896873	+	0.442289i
$676$	−0.528086	−	1.55569i	−0.528086	−	1.55569i
$677$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$677$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$678$		0			0
$679$	0.318219	−	0.375065i	0.318219	−	0.375065i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$683$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$684$	−0.834963	−	0.921240i	−0.834963	−	0.921240i
$685$		0			0
$686$		0			0
$687$	−0.0923988	−	1.88082i	−0.0923988	−	1.88082i
$688$	1.13677	+	1.21372i	1.13677	+	1.21372i
$689$		0			0
$690$		0			0
$691$	−1.14368	+	0.791553i	−1.14368	+	0.791553i	−0.980785	−	0.195090i	$-0.937500\pi$
$691$	−1.14368	+	0.791553i	−1.14368	+	0.791553i	−0.162895	+	0.986643i	$0.552083\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.375574	−	0.192915i	−0.375574	−	0.192915i
$701$		0			0		0.999866	−	0.0163617i	$-0.00520833\pi$
$701$		0			0		−0.999866	+	0.0163617i	$0.994792\pi$
$702$		0			0
$703$	−0.355335	+	0.0828525i	−0.355335	+	0.0828525i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.0869951	+	0.347304i	0.0869951	+	0.347304i	0.997859	−	0.0654031i	$-0.0208333\pi$
$709$	0.0869951	+	0.347304i	0.0869951	+	0.347304i	−0.910864	+	0.412707i	$0.864583\pi$
$710$		0			0
$711$	−1.61797	+	0.294400i	−1.61797	+	0.294400i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.773010	−	0.634393i	$-0.218750\pi$
$719$		0			0		−0.773010	+	0.634393i	$0.781250\pi$
$720$		0			0
$721$	0.517994	−	0.485155i	0.517994	−	0.485155i
$722$		0			0
$723$	−0.0211706	+	0.0249525i	−0.0211706	+	0.0249525i
$724$	0.183930	−	0.344109i	0.183930	−	0.344109i
$725$		0			0
$726$		0			0
$727$	0.854996	+	0.0139911i	0.854996	+	0.0139911i	0.442289	−	0.896873i	$-0.354167\pi$
$727$	0.854996	+	0.0139911i	0.854996	+	0.0139911i	0.412707	+	0.910864i	$0.364583\pi$
$728$		0			0
$729$	0.555570	−	0.831470i	0.555570	−	0.831470i
$730$		0			0
$731$		0			0
$732$	−1.34369	−	0.805380i	−1.34369	−	0.805380i
$733$	−0.0843209	−	0.510724i	−0.0843209	−	0.510724i	−0.995185	−	0.0980171i	$-0.968750\pi$
$733$	−0.0843209	−	0.510724i	−0.0843209	−	0.510724i	0.910864	−	0.412707i	$-0.135417\pi$
$734$		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
$741$	1.73369	−	1.03913i	1.73369	−	1.03913i
$742$		0			0
$743$		0			0		−0.840448	−	0.541892i	$-0.817708\pi$
$743$		0			0		0.840448	+	0.541892i	$0.182292\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.317280	+	1.92174i	−0.317280	+	1.92174i	0.0654031	+	0.997859i	$0.479167\pi$
$751$	−0.317280	+	1.92174i	−0.317280	+	1.92174i	−0.382683	+	0.923880i	$0.625000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.240287	+	0.347181i	−0.240287	+	0.347181i
$757$	0.956940	+	0.290285i	0.956940	+	0.290285i	0.729864	−	0.683592i	$-0.239583\pi$
$757$	0.956940	+	0.290285i	0.956940	+	0.290285i	0.227076	+	0.973877i	$0.427083\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$761$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$762$		0			0
$763$	0.443638	+	0.598177i	0.443638	+	0.598177i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.773010	+	0.634393i	−0.773010	+	0.634393i
$769$	−0.129059	−	0.0213077i	−0.129059	−	0.0213077i	0.0980171	−	0.995185i	$-0.468750\pi$
$769$	−0.129059	−	0.0213077i	−0.129059	−	0.0213077i	−0.227076	+	0.973877i	$0.572917\pi$
$770$		0			0
$771$		0			0
$772$	0.228799	+	1.05936i	0.228799	+	1.05936i
$773$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$773$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$774$		0			0
$775$	−1.51506	+	0.174294i	−1.51506	+	0.174294i
$776$		0			0
$777$	0.0548022	+	0.111128i	0.0548022	+	0.111128i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.478638	+	0.667939i	−0.478638	+	0.667939i
$785$		0			0
$786$		0			0
$787$	−0.328494	−	1.80535i	−0.328494	−	1.80535i	−0.555570	−	0.831470i	$-0.687500\pi$
$787$	−0.328494	−	1.80535i	−0.328494	−	1.80535i	0.227076	−	0.973877i	$-0.427083\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.71030	−	1.88703i	1.71030	−	1.88703i
$794$		0			0
$795$		0			0
$796$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$797$		0			0		0.986643	−	0.162895i	$-0.0520833\pi$
$797$		0			0		−0.986643	+	0.162895i	$0.947917\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.17221	−	1.42834i	−1.17221	−	1.42834i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$809$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$810$		0			0
$811$	0.130081	−	1.13074i	0.130081	−	1.13074i	−0.751840	−	0.659346i	$-0.770833\pi$
$811$	0.130081	−	1.13074i	0.130081	−	1.13074i	0.881921	−	0.471397i	$-0.156250\pi$
$812$		0			0
$813$	−0.846488	−	1.71651i	−0.846488	−	1.71651i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.43787	−	1.48572i	−1.43787	−	1.48572i
$818$		0			0
$819$	−0.477356	−	0.493238i	−0.477356	−	0.493238i
$820$		0			0
$821$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$821$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$822$		0			0
$823$	−0.893212	−	0.192915i	−0.893212	−	0.192915i	−0.258819	−	0.965926i	$-0.583333\pi$
$823$	−0.893212	−	0.192915i	−0.893212	−	0.192915i	−0.634393	+	0.773010i	$0.718750\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.695443	−	0.718582i	$-0.255208\pi$
$827$		0			0		−0.695443	+	0.718582i	$0.744792\pi$
$828$		0			0
$829$	−0.410525	−	0.410525i	−0.410525	−	0.410525i	0.471397	−	0.881921i	$-0.343750\pi$
$829$	−0.410525	−	0.410525i	−0.410525	−	0.410525i	−0.881921	+	0.471397i	$0.843750\pi$
$830$		0			0
$831$	0.607797	+	0.0699215i	0.607797	+	0.0699215i
$832$	−0.766347	−	1.43373i	−0.766347	−	1.43373i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.0249525	+	1.52485i	−0.0249525	+	1.52485i
$838$		0			0
$839$		0			0		0.961562	−	0.274589i	$-0.0885417\pi$
$839$		0			0		−0.961562	+	0.274589i	$0.911458\pi$
$840$		0			0
$841$	−0.946930	−	0.321439i	−0.946930	−	0.321439i
$842$		0			0
$843$		0			0
$844$	−1.07205	−	1.66270i	−1.07205	−	1.66270i
$845$		0			0
$846$		0			0
$847$	0.387387	−	0.167939i	0.387387	−	0.167939i
$848$		0			0
$849$	−1.47140	+	0.881921i	−1.47140	+	0.881921i
$850$		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		−0.803208	−	0.595699i	$-0.796875\pi$
$857$		0			0		0.803208	+	0.595699i	$0.203125\pi$
$858$		0			0
$859$	−0.247198	−	1.87766i	−0.247198	−	1.87766i	−0.442289	−	0.896873i	$-0.645833\pi$
$859$	−0.247198	−	1.87766i	−0.247198	−	1.87766i	0.195090	−	0.980785i	$-0.437500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.977462	−	0.211112i	$-0.0677083\pi$
$863$		0			0		−0.977462	+	0.211112i	$0.932292\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$868$	0.0210682	−	0.643568i	0.0210682	−	0.643568i
$869$		0			0
$870$		0			0
$871$	2.64919	−	1.41602i	2.64919	−	1.41602i
$872$		0			0
$873$	−0.875860	−	0.768108i	−0.875860	−	0.768108i
$874$		0			0
$875$		0			0
$876$	0.219788	+	0.0627638i	0.219788	+	0.0627638i
$877$	0.816404	+	1.31288i	0.816404	+	1.31288i	0.946930	+	0.321439i	$0.104167\pi$
$877$	0.816404	+	1.31288i	0.816404	+	1.31288i	−0.130526	+	0.991445i	$0.541667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.874090	−	0.485763i	$-0.838542\pi$
$881$		0			0		0.874090	+	0.485763i	$0.161458\pi$
$882$		0			0
$883$	1.51995	+	1.28958i	1.51995	+	1.28958i	0.812847	+	0.582478i	$0.197917\pi$
$883$	1.51995	+	1.28958i	1.51995	+	1.28958i	0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$887$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$888$		0			0
$889$	0.756786	−	0.313471i	0.756786	−	0.313471i
$890$		0			0
$891$		0			0
$892$	0.0425243	+	0.257566i	0.0425243	+	0.257566i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.471397	+	0.881921i	−0.471397	+	0.881921i
$901$		0			0
$902$		0			0
$903$	−0.380479	+	0.590105i	−0.380479	+	0.590105i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.0500574	−	0.763728i	−0.0500574	−	0.763728i	−0.946930	−	0.321439i	$-0.895833\pi$
$907$	−0.0500574	−	0.763728i	−0.0500574	−	0.763728i	0.896873	−	0.442289i	$-0.145833\pi$
$908$		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	$0.208333\pi$
$912$	0.948066	−	0.804374i	0.948066	−	0.804374i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.88284	−	0.0308106i	1.88284	−	0.0308106i
$917$		0			0
$918$		0			0
$919$	−0.322547	+	1.06330i	−0.322547	+	1.06330i	0.634393	+	0.773010i	$0.281250\pi$
$919$	−0.322547	+	1.06330i	−0.322547	+	1.06330i	−0.956940	+	0.290285i	$0.906250\pi$
$920$		0			0
$921$	−1.16102	+	1.23960i	−1.16102	+	1.23960i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.159024	+	0.246639i	0.159024	+	0.246639i
$926$		0			0
$927$	−1.12882	−	1.24546i	−1.12882	−	1.24546i
$928$		0			0
$929$		0			0		−0.993448	−	0.114287i	$-0.963542\pi$
$929$		0			0		0.993448	+	0.114287i	$0.0364583\pi$
$930$		0			0
$931$	0.581433	−	0.840088i	0.581433	−	0.840088i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.195090	+	0.0192147i	0.195090	+	0.0192147i	0.195090	−	0.980785i	$-0.437500\pi$
$937$	0.195090	+	0.0192147i	0.195090	+	0.0192147i			1.00000i	$0.5\pi$
$938$		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		−0.952063	−	0.305903i	$-0.901042\pi$
$947$		0			0		0.952063	+	0.305903i	$0.0989583\pi$
$948$	−0.241304	−	1.62674i	−0.241304	−	1.62674i
$949$	−0.169781	+	0.330536i	−0.169781	+	0.330536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.0327191	−	0.999465i	$-0.510417\pi$
$953$		0			0		0.0327191	+	0.999465i	$0.489583\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.700108	−	1.12586i	−0.700108	−	1.12586i
$962$		0			0
$963$		0			0
$964$	−0.0242465	−	0.0219757i	−0.0242465	−	0.0219757i
$965$		0			0
$966$		0			0
$967$	−0.415381	−	0.747444i	−0.415381	−	0.747444i	0.582478	−	0.812847i	$-0.302083\pi$
$967$	−0.415381	−	0.747444i	−0.415381	−	0.747444i	−0.997859	+	0.0654031i	$0.979167\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.961562	−	0.274589i	$-0.911458\pi$
$971$		0			0		0.961562	+	0.274589i	$0.0885417\pi$
$972$	0.812847	+	0.582478i	0.812847	+	0.582478i
$973$	−0.556708	−	0.00910992i	−0.556708	−	0.00910992i
$974$		0			0
$975$	−1.25668	−	1.03133i	−1.25668	−	1.03133i
$976$	0.848913	−	1.31662i	0.848913	−	1.31662i
$977$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$977$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.43373	−	1.02740i	1.43373	−	1.02740i
$982$		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
$988$	0.981854	+	1.76676i	0.981854	+	1.76676i
$989$		0			0
$990$		0			0
$991$	0.129059	−	1.96906i	0.129059	−	1.96906i	−0.0980171	−	0.995185i	$-0.531250\pi$
$991$	0.129059	−	1.96906i	0.129059	−	1.96906i	0.227076	−	0.973877i	$-0.427083\pi$
$992$		0			0
$993$	1.18359	+	1.59589i	1.18359	+	1.59589i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.0500574	+	0.120849i	0.0500574	+	0.120849i	0.946930	−	0.321439i	$-0.104167\pi$
$997$	0.0500574	+	0.120849i	0.0500574	+	0.120849i	−0.896873	+	0.442289i	$0.854167\pi$
$998$		0			0
$999$	0.265286	−	0.125471i	0.265286	−	0.125471i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3459.1.bm.a.1502.1	yes	64
3.2	odd	2	CM	3459.1.bm.a.1502.1	yes	64
1153.707	even	192	inner	3459.1.bm.a.707.1	✓	64
3459.707	odd	192	inner	3459.1.bm.a.707.1	✓	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3459.1.bm.a.707.1	✓	64	1153.707	even	192	inner
3459.1.bm.a.707.1	✓	64	3459.707	odd	192	inner
3459.1.bm.a.1502.1	yes	64	1.1	even	1	trivial
3459.1.bm.a.1502.1	yes	64	3.2	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(-0.986643 + 0.162895i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.142243 + 0.397542i) q^{7} +(0.946930 - 0.321439i) q^{9} +O(q^{10})\)	\(q+(-0.986643 + 0.162895i) q^{3} +(-0.130526 - 0.991445i) q^{4} +(0.142243 + 0.397542i) q^{7} +(0.946930 - 0.321439i) q^{9} +(0.290285 + 0.956940i) q^{12} +(-1.61787 + 0.159346i) q^{13} +(-0.965926 + 0.258819i) q^{16} +(1.19553 - 0.341402i) q^{19} +(-0.205101 - 0.369061i) q^{21} +(-0.582478 - 0.812847i) q^{25} +(-0.881921 + 0.471397i) q^{27} +(0.375574 - 0.192915i) q^{28} +(0.740816 - 1.33304i) q^{31} +(-0.442289 - 0.896873i) q^{36} +(-0.293107 - 0.0143994i) q^{37} +(1.57030 - 0.420760i) q^{39} +(-0.783904 - 1.46658i) q^{43} +(0.910864 - 0.412707i) q^{48} +(0.635204 - 0.521299i) q^{49} +(0.369156 + 1.58323i) q^{52} +(-1.12395 + 0.531588i) q^{57} +(-1.16075 + 1.05205i) q^{61} +(0.262479 + 0.330722i) q^{63} +(0.382683 + 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(0.123862 - 0.192105i) q^{73} +(0.707107 + 0.707107i) q^{75} +(-0.494529 - 1.14074i) q^{76} +(-1.62674 - 0.241304i) q^{79} +(0.793353 - 0.608761i) q^{81} +(-0.339133 + 0.251518i) q^{84} +(-0.293476 - 0.620503i) q^{91} +(-0.513775 + 1.43591i) q^{93} +(-0.582478 - 1.00888i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 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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