LMFDB - Embedded newform 3459.1.bm.a.266.1

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			266.1
Root			$-0.729864 + 0.683592i$ of defining polynomial
Character	$\chi$	$=$	3459.266
Dual form			3459.1.bm.a.3446.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.582478 - 0.812847i) q^{3} +(0.130526 + 0.991445i) q^{4} +(-1.21852 + 1.10440i) q^{7} +(-0.321439 - 0.946930i) q^{9} +O(q^{10})$	\(q+(0.582478 - 0.812847i) q^{3} +(0.130526 + 0.991445i) q^{4} +(-1.21852 + 1.10440i) q^{7} +(-0.321439 - 0.946930i) q^{9} +(0.881921 + 0.471397i) q^{12} +(-1.52537 - 1.25184i) q^{13} +(-0.965926 + 0.258819i) q^{16} +(-1.36933 + 1.08678i) q^{19} +(0.187949 + 1.63376i) q^{21} +(-0.162895 + 0.986643i) q^{25} +(-0.956940 - 0.290285i) q^{27} +(-1.25400 - 1.06394i) q^{28} +(0.0699215 - 0.607797i) q^{31} +(0.896873 - 0.442289i) q^{36} +(0.346392 + 0.968101i) q^{37} +(-1.90605 + 0.510724i) q^{39} +(-0.322547 + 1.06330i) q^{43} +(-0.352250 + 0.935906i) q^{48} +(0.167070 - 1.69629i) q^{49} +(1.04203 - 1.67572i) q^{52} +(0.0857792 + 1.74607i) q^{57} +(0.878250 + 0.415381i) q^{61} +(1.43747 + 0.798856i) q^{63} +(-0.382683 - 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(0.540306 + 0.0983120i) q^{73} +(0.707107 + 0.707107i) q^{75} +(-1.25621 - 1.21576i) q^{76} +(-1.44175 - 0.864154i) q^{79} +(-0.793353 + 0.608761i) q^{81} +(-1.59525 + 0.399590i) q^{84} +(3.24123 - 0.159232i) q^{91} +(-0.453318 - 0.410864i) q^{93} +(-0.162895 - 0.282143i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3459\mathbb{Z}\right)^\times$.

$n$	$1154$	$2311$
$\chi(n)$	$-1$	$e\left(\frac{53}{192}\right)$

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		−0.751840	−	0.659346i	$-0.770833\pi$
$2$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$3$	0.582478	−	0.812847i	0.582478	−	0.812847i
$3$	0.582478	−	0.812847i	0.582478	−	0.812847i
$4$	0.130526	+	0.991445i	0.130526	+	0.991445i
$5$		0			0		−0.646956	−	0.762527i	$-0.723958\pi$
$5$		0			0		0.646956	+	0.762527i	$0.276042\pi$
$6$		0			0
$7$	−1.21852	+	1.10440i	−1.21852	+	1.10440i	−0.227076	+	0.973877i	$0.572917\pi$
$7$	−1.21852	+	1.10440i	−1.21852	+	1.10440i	−0.991445	+	0.130526i	$0.958333\pi$
$8$		0			0
$9$	−0.321439	−	0.946930i	−0.321439	−	0.946930i
$10$		0			0
$11$		0			0		−0.0327191	−	0.999465i	$-0.510417\pi$
$11$		0			0		0.0327191	+	0.999465i	$0.489583\pi$
$12$	0.881921	+	0.471397i	0.881921	+	0.471397i
$13$	−1.52537	−	1.25184i	−1.52537	−	1.25184i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	−1.52537	−	1.25184i	−1.52537	−	1.25184i	−0.659346	−	0.751840i	$-0.729167\pi$
$14$		0			0
$15$		0			0
$16$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$17$		0			0		−0.930017	−	0.367516i	$-0.880208\pi$
$17$		0			0		0.930017	+	0.367516i	$0.119792\pi$
$18$		0			0
$19$	−1.36933	+	1.08678i	−1.36933	+	1.08678i	−0.382683	+	0.923880i	$0.625000\pi$
$19$	−1.36933	+	1.08678i	−1.36933	+	1.08678i	−0.986643	+	0.162895i	$0.947917\pi$
$20$		0			0
$21$	0.187949	+	1.63376i	0.187949	+	1.63376i
$22$		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
$25$	−0.162895	+	0.986643i	−0.162895	+	0.986643i
$26$		0			0
$27$	−0.956940	−	0.290285i	−0.956940	−	0.290285i
$28$	−1.25400	−	1.06394i	−1.25400	−	1.06394i
$29$		0			0		0.812847	−	0.582478i	$-0.197917\pi$
$29$		0			0		−0.812847	+	0.582478i	$0.802083\pi$
$30$		0			0
$31$	0.0699215	−	0.607797i	0.0699215	−	0.607797i	−0.910864	−	0.412707i	$-0.864583\pi$
$31$	0.0699215	−	0.607797i	0.0699215	−	0.607797i	0.980785	−	0.195090i	$-0.0625000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.896873	−	0.442289i	0.896873	−	0.442289i
$37$	0.346392	+	0.968101i	0.346392	+	0.968101i	0.980785	+	0.195090i	$0.0625000\pi$
$37$	0.346392	+	0.968101i	0.346392	+	0.968101i	−0.634393	+	0.773010i	$0.718750\pi$
$38$		0			0
$39$	−1.90605	+	0.510724i	−1.90605	+	0.510724i
$40$		0			0
$41$		0			0		−0.986643	−	0.162895i	$-0.947917\pi$
$41$		0			0		0.986643	+	0.162895i	$0.0520833\pi$
$42$		0			0
$43$	−0.322547	+	1.06330i	−0.322547	+	1.06330i	0.634393	+	0.773010i	$0.281250\pi$
$43$	−0.322547	+	1.06330i	−0.322547	+	1.06330i	−0.956940	+	0.290285i	$0.906250\pi$
$44$		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	$0.166667\pi$
$48$	−0.352250	+	0.935906i	−0.352250	+	0.935906i
$49$	0.167070	−	1.69629i	0.167070	−	1.69629i
$50$		0			0
$51$		0			0
$52$	1.04203	−	1.67572i	1.04203	−	1.67572i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.0857792	+	1.74607i	0.0857792	+	1.74607i
$58$		0			0
$59$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$59$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$60$		0			0
$61$	0.878250	+	0.415381i	0.878250	+	0.415381i	0.812847	−	0.582478i	$-0.197917\pi$
$61$	0.878250	+	0.415381i	0.878250	+	0.415381i	0.0654031	+	0.997859i	$0.479167\pi$
$62$		0			0
$63$	1.43747	+	0.798856i	1.43747	+	0.798856i
$64$	−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		−0.803208	−	0.595699i	$-0.796875\pi$
$71$		0			0		0.803208	+	0.595699i	$0.203125\pi$
$72$		0			0
$73$	0.540306	+	0.0983120i	0.540306	+	0.0983120i	0.442289	−	0.896873i	$-0.354167\pi$
$73$	0.540306	+	0.0983120i	0.540306	+	0.0983120i	0.0980171	+	0.995185i	$0.468750\pi$
$74$		0			0
$75$	0.707107	+	0.707107i	0.707107	+	0.707107i
$76$	−1.25621	−	1.21576i	−1.25621	−	1.21576i
$77$		0			0
$78$		0			0
$79$	−1.44175	−	0.864154i	−1.44175	−	0.864154i	−0.442289	−	0.896873i	$-0.645833\pi$
$79$	−1.44175	−	0.864154i	−1.44175	−	0.864154i	−0.999465	+	0.0327191i	$0.989583\pi$
$80$		0			0
$81$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$82$		0			0
$83$		0			0		−0.305903	−	0.952063i	$-0.598958\pi$
$83$		0			0		0.305903	+	0.952063i	$0.401042\pi$
$84$	−1.59525	+	0.399590i	−1.59525	+	0.399590i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.729864	−	0.683592i	$-0.760417\pi$
$89$		0			0		0.729864	+	0.683592i	$0.239583\pi$
$90$		0			0
$91$	3.24123	−	0.159232i	3.24123	−	0.159232i
$92$		0			0
$93$	−0.453318	−	0.410864i	−0.453318	−	0.410864i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.162895	−	0.282143i	−0.162895	−	0.282143i	0.773010	−	0.634393i	$-0.218750\pi$
$97$	−0.162895	−	0.282143i	−0.162895	−	0.282143i	−0.935906	+	0.352250i	$0.885417\pi$
$98$		0			0
$99$		0			0
$100$	−0.999465	−	0.0327191i	−0.999465	−	0.0327191i
$101$		0			0		−0.803208	−	0.595699i	$-0.796875\pi$
$101$		0			0		0.803208	+	0.595699i	$0.203125\pi$
$102$		0			0
$103$	−0.257276	−	0.248992i	−0.257276	−	0.248992i	0.555570	−	0.831470i	$-0.312500\pi$
$103$	−0.257276	−	0.248992i	−0.257276	−	0.248992i	−0.812847	+	0.582478i	$0.802083\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$107$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$108$	0.162895	−	0.986643i	0.162895	−	0.986643i
$109$	0.902197	−	1.68789i	0.902197	−	1.68789i	0.195090	−	0.980785i	$-0.437500\pi$
$109$	0.902197	−	1.68789i	0.902197	−	1.68789i	0.707107	−	0.707107i	$-0.250000\pi$
$110$		0			0
$111$	0.988683	+	0.282334i	0.988683	+	0.282334i
$112$	0.891160	−	1.38215i	0.891160	−	1.38215i
$113$		0			0		−0.274589	−	0.961562i	$-0.588542\pi$
$113$		0			0		0.274589	+	0.961562i	$0.411458\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.695090	+	1.84681i	−0.695090	+	1.84681i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.997859	+	0.0654031i	−0.997859	+	0.0654031i
$122$		0			0
$123$		0			0
$124$	0.611724	−	0.0100102i	0.611724	−	0.0100102i
$125$		0			0
$126$		0			0
$127$	−1.86271	−	0.666487i	−1.86271	−	0.666487i	−0.980785	−	0.195090i	$-0.937500\pi$
$127$	−1.86271	−	0.666487i	−1.86271	−	0.666487i	−0.881921	−	0.471397i	$-0.843750\pi$
$128$		0			0
$129$	0.676419	+	0.881527i	0.676419	+	0.881527i
$130$		0			0
$131$		0			0		−0.986643	−	0.162895i	$-0.947917\pi$
$131$		0			0		0.986643	+	0.162895i	$0.0520833\pi$
$132$		0			0
$133$	0.468316	−	2.83655i	0.468316	−	2.83655i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.803208	−	0.595699i	$-0.203125\pi$
$137$		0			0		−0.803208	+	0.595699i	$0.796875\pi$
$138$		0			0
$139$	0.620579	+	1.36965i	0.620579	+	1.36965i	0.910864	+	0.412707i	$0.135417\pi$
$139$	0.620579	+	1.36965i	0.620579	+	1.36965i	−0.290285	+	0.956940i	$0.593750\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.555570	+	0.831470i	0.555570	+	0.831470i
$145$		0			0
$146$		0			0
$147$	−1.28151	−	1.12386i	−1.28151	−	1.12386i
$148$	−0.914605	+	0.469791i	−0.914605	+	0.469791i
$149$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$149$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$150$		0			0
$151$	1.10409	+	0.276560i	1.10409	+	0.276560i	0.751840	−	0.659346i	$-0.229167\pi$
$151$	1.10409	+	0.276560i	1.10409	+	0.276560i	0.352250	+	0.935906i	$0.385417\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.755144	−	1.82308i	−0.755144	−	1.82308i
$157$	−0.0175679	+	0.0965501i	−0.0175679	+	0.0965501i	−0.991445	−	0.130526i	$-0.958333\pi$
$157$	−0.0175679	+	0.0965501i	−0.0175679	+	0.0965501i	0.973877	+	0.227076i	$0.0729167\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.142243	+	0.397542i	0.142243	+	0.397542i	0.991445	−	0.130526i	$-0.0416667\pi$
$163$	0.142243	+	0.397542i	0.142243	+	0.397542i	−0.849202	+	0.528068i	$0.822917\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.889516	−	0.456904i	$-0.848958\pi$
$167$		0			0		0.889516	+	0.456904i	$0.151042\pi$
$168$		0			0
$169$	0.564564	+	2.83826i	0.564564	+	2.83826i
$170$		0			0
$171$	1.46926	+	0.947324i	1.46926	+	0.947324i
$172$	−1.09630	−	0.181000i	−1.09630	−	0.181000i
$173$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$173$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$174$		0			0
$175$	−0.891160	−	1.38215i	−0.891160	−	1.38215i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.718582	−	0.695443i	$-0.755208\pi$
$179$		0			0		0.718582	+	0.695443i	$0.244792\pi$
$180$		0			0
$181$	1.93537	−	0.319531i	1.93537	−	0.319531i	0.935906	−	0.352250i	$-0.114583\pi$
$181$	1.93537	−	0.319531i	1.93537	−	0.319531i	0.999465	+	0.0327191i	$0.0104167\pi$
$182$		0			0
$183$	0.849202	−	0.471932i	0.849202	−	0.471932i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.48664	−	0.703130i	1.48664	−	0.703130i
$190$		0			0
$191$		0			0		0.874090	−	0.485763i	$-0.161458\pi$
$191$		0			0		−0.874090	+	0.485763i	$0.838542\pi$
$192$	−0.973877	−	0.227076i	−0.973877	−	0.227076i
$193$	0.899046	+	1.75029i	0.899046	+	1.75029i	0.608761	+	0.793353i	$0.291667\pi$
$193$	0.899046	+	1.75029i	0.899046	+	1.75029i	0.290285	+	0.956940i	$0.406250\pi$
$194$		0			0
$195$		0			0
$196$	1.70359	−	0.0557697i	1.70359	−	0.0557697i
$197$		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	$0.0416667\pi$
$199$	1.78480	+	0.739288i	1.78480	+	0.739288i	0.793353	+	0.608761i	$0.208333\pi$
$200$		0			0
$201$	−0.419582	+	1.79949i	−0.419582	+	1.79949i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.79740	+	0.814389i	1.79740	+	0.814389i
$209$		0			0
$210$		0			0
$211$	−0.0841735	−	1.71339i	−0.0841735	−	1.71339i	−0.555570	−	0.831470i	$-0.687500\pi$
$211$	−0.0841735	−	1.71339i	−0.0841735	−	1.71339i	0.471397	−	0.881921i	$-0.343750\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.586053	+	0.817835i	0.586053	+	0.817835i
$218$		0			0
$219$	0.394629	−	0.381921i	0.394629	−	0.381921i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.178453	−	0.190533i	0.178453	−	0.190533i	−0.634393	−	0.773010i	$-0.718750\pi$
$223$	0.178453	−	0.190533i	0.178453	−	0.190533i	0.812847	+	0.582478i	$0.197917\pi$
$224$		0			0
$225$	0.986643	−	0.162895i	0.986643	−	0.162895i
$226$		0			0
$227$		0			0		−0.456904	−	0.889516i	$-0.651042\pi$
$227$		0			0		0.456904	+	0.889516i	$0.348958\pi$
$228$	−1.71994	+	0.312954i	−1.71994	+	0.312954i
$229$	0.368805	+	1.29149i	0.368805	+	1.29149i	0.896873	+	0.442289i	$0.145833\pi$
$229$	0.368805	+	1.29149i	0.368805	+	1.29149i	−0.528068	+	0.849202i	$0.677083\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.930017	−	0.367516i	$-0.119792\pi$
$233$		0			0		−0.930017	+	0.367516i	$0.880208\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.54221	+	0.668574i	−1.54221	+	0.668574i
$238$		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
$241$	0.243345	+	0.757363i	0.243345	+	0.757363i	0.995185	+	0.0980171i	$0.0312500\pi$
$241$	0.243345	+	0.757363i	0.243345	+	0.757363i	−0.751840	+	0.659346i	$0.770833\pi$
$242$		0			0
$243$	0.0327191	+	0.999465i	0.0327191	+	0.999465i
$244$	−0.297193	+	0.924954i	−0.297193	+	0.924954i
$245$		0			0
$246$		0			0
$247$	3.44920	+	0.0564424i	3.44920	+	0.0564424i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$251$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$252$	−0.604393	+	1.52945i	−0.604393	+	1.52945i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.866025	−	0.500000i	0.866025	−	0.500000i
$257$		0			0		−0.397748	−	0.917494i	$-0.630208\pi$
$257$		0			0		0.397748	+	0.917494i	$0.369792\pi$
$258$		0			0
$259$	−1.49126	−	0.797095i	−1.49126	−	0.797095i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.695443	−	0.718582i	$-0.255208\pi$
$263$		0			0		−0.695443	+	0.718582i	$0.744792\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.923880	−	1.60021i	−0.923880	−	1.60021i
$269$		0			0		0.803208	−	0.595699i	$-0.203125\pi$
$269$		0			0		−0.803208	+	0.595699i	$0.796875\pi$
$270$		0			0
$271$	0.831470	+	0.444430i	0.831470	+	0.444430i	0.831470	−	0.555570i	$-0.187500\pi$
$271$	0.831470	+	0.444430i	0.831470	+	0.444430i			1.00000i	$0.5\pi$
$272$		0			0
$273$	1.75852	−	2.72737i	1.75852	−	2.72737i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.56133	−	1.23916i	−1.56133	−	1.23916i	−0.831470	−	0.555570i	$-0.812500\pi$
$277$	−1.56133	−	1.23916i	−1.56133	−	1.23916i	−0.729864	−	0.683592i	$-0.760417\pi$
$278$		0			0
$279$	−0.598017	+	0.129159i	−0.598017	+	0.129159i
$280$		0			0
$281$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$281$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$282$		0			0
$283$	−1.19124	−	0.0194933i	−1.19124	−	0.0194933i	−0.582478	−	0.812847i	$-0.697917\pi$
$283$	−1.19124	−	0.0194933i	−1.19124	−	0.0194933i	−0.608761	+	0.793353i	$0.708333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.729864	+	0.683592i	0.729864	+	0.683592i
$290$		0			0
$291$	−0.324222	−	0.0319331i	−0.324222	−	0.0319331i
$292$	−0.0269468	+	0.548516i	−0.0269468	+	0.548516i
$293$		0			0		−0.514103	−	0.857729i	$-0.671875\pi$
$293$		0			0		0.514103	+	0.857729i	$0.328125\pi$
$294$		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
$301$	−0.781276	−	1.65187i	−0.781276	−	1.65187i
$302$		0			0
$303$		0			0
$304$	1.04139	−	1.40415i	1.04139	−	1.40415i
$305$		0			0
$306$		0			0
$307$	−0.360303	+	0.276471i	−0.360303	+	0.276471i	−0.773010	−	0.634393i	$-0.781250\pi$
$307$	−0.360303	+	0.276471i	−0.360303	+	0.276471i	0.412707	+	0.910864i	$0.364583\pi$
$308$		0			0
$309$	−0.352250	+	0.0640941i	−0.352250	+	0.0640941i
$310$		0			0
$311$		0			0		0.0817211	−	0.996655i	$-0.473958\pi$
$311$		0			0		−0.0817211	+	0.996655i	$0.526042\pi$
$312$		0			0
$313$	−0.450267	−	1.93109i	−0.450267	−	1.93109i	−0.352250	−	0.935906i	$-0.614583\pi$
$313$	−0.450267	−	1.93109i	−0.450267	−	1.93109i	−0.0980171	−	0.995185i	$-0.531250\pi$
$314$		0			0
$315$		0			0
$316$	0.668574	−	1.54221i	0.668574	−	1.54221i
$317$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$317$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$318$		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
$325$	1.48360	−	1.30108i	1.48360	−	1.30108i
$326$		0			0
$327$	−0.846488	−	1.71651i	−0.846488	−	1.71651i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.24418	+	1.46643i	−1.24418	+	1.46643i	−0.412707	+	0.910864i	$0.635417\pi$
$331$	−1.24418	+	1.46643i	−1.24418	+	1.46643i	−0.831470	+	0.555570i	$0.812500\pi$
$332$		0			0
$333$	0.805380	−	0.639195i	0.805380	−	0.639195i
$334$		0			0
$335$		0			0
$336$	−0.604393	−	1.52945i	−0.604393	−	1.52945i
$337$	−1.07880	+	1.19028i	−1.07880	+	1.19028i	−0.0980171	+	0.995185i	$0.531250\pi$
$337$	−1.07880	+	1.19028i	−1.07880	+	1.19028i	−0.980785	+	0.195090i	$0.937500\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.690163	+	0.930577i	0.690163	+	0.930577i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.998795	−	0.0490677i	$-0.0156250\pi$
$347$		0			0		−0.998795	+	0.0490677i	$0.984375\pi$
$348$		0			0
$349$	−1.31234	−	0.129254i	−1.31234	−	0.129254i	−0.582478	−	0.812847i	$-0.697917\pi$
$349$	−1.31234	−	0.129254i	−1.31234	−	0.129254i	−0.729864	+	0.683592i	$0.760417\pi$
$350$		0			0
$351$	1.09630	+	1.64073i	1.09630	+	1.64073i
$352$		0			0
$353$		0			0		0.582478	−	0.812847i	$-0.302083\pi$
$353$		0			0		−0.582478	+	0.812847i	$0.697917\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$359$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$360$		0			0
$361$	0.466900	−	2.00242i	0.466900	−	2.00242i
$362$		0			0
$363$	−0.528068	+	0.849202i	−0.528068	+	0.849202i
$364$	0.580935	+	3.19272i	0.580935	+	3.19272i
$365$		0			0
$366$		0			0
$367$	1.65938	−	0.108761i	1.65938	−	0.108761i	0.793353	−	0.608761i	$-0.208333\pi$
$367$	1.65938	−	0.108761i	1.65938	−	0.108761i	0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.348179	−	0.503069i	0.348179	−	0.503069i
$373$	−0.696803	+	1.35656i	−0.696803	+	1.35656i	0.227076	+	0.973877i	$0.427083\pi$
$373$	−0.696803	+	1.35656i	−0.696803	+	1.35656i	−0.923880	+	0.382683i	$0.875000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.941158	+	0.772388i	0.941158	+	0.772388i	0.973877	−	0.227076i	$-0.0729167\pi$
$379$	0.941158	+	0.772388i	0.941158	+	0.772388i	−0.0327191	+	0.999465i	$0.510417\pi$
$380$		0			0
$381$	−1.62674	+	1.12588i	−1.62674	+	1.12588i
$382$		0			0
$383$		0			0		0.729864	−	0.683592i	$-0.239583\pi$
$383$		0			0		−0.729864	+	0.683592i	$0.760417\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.11055	−	0.0363555i	1.11055	−	0.0363555i
$388$	0.258467	−	0.198329i	0.258467	−	0.198329i
$389$		0			0		−0.595699	−	0.803208i	$-0.703125\pi$
$389$		0			0		0.595699	+	0.803208i	$0.296875\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.994365	+	1.54221i	0.994365	+	1.54221i	0.831470	+	0.555570i	$0.187500\pi$
$397$	0.994365	+	1.54221i	0.994365	+	1.54221i	0.162895	+	0.986643i	$0.447917\pi$
$398$		0			0
$399$	−2.03289	−	2.03289i	−2.03289	−	2.03289i
$400$	−0.0980171	−	0.995185i	−0.0980171	−	0.995185i
$401$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$401$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$402$		0			0
$403$	−0.867521	+	0.839586i	−0.867521	+	0.839586i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.72572	+	0.432270i	−1.72572	+	0.432270i	−0.973877	−	0.227076i	$-0.927083\pi$
$409$	−1.72572	+	0.432270i	−1.72572	+	0.432270i	−0.751840	+	0.659346i	$0.770833\pi$
$410$		0			0
$411$		0			0
$412$	0.213280	−	0.287575i	0.213280	−	0.287575i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.47479	+	0.293353i	1.47479	+	0.293353i
$418$		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
$421$	−1.18359	−	1.59589i	−1.18359	−	1.59589i	−0.683592	−	0.729864i	$-0.739583\pi$
$421$	−1.18359	−	1.59589i	−1.18359	−	1.59589i	−0.500000	−	0.866025i	$-0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.52891	+	0.463791i	−1.52891	+	0.463791i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.812847	−	0.582478i	$-0.197917\pi$
$431$		0			0		−0.812847	+	0.582478i	$0.802083\pi$
$432$	0.999465	+	0.0327191i	0.999465	+	0.0327191i
$433$	0.532719	−	1.86549i	0.532719	−	1.86549i	0.0327191	−	0.999465i	$-0.489583\pi$
$433$	0.532719	−	1.86549i	0.532719	−	1.86549i	0.500000	−	0.866025i	$-0.333333\pi$
$434$		0			0
$435$		0			0
$436$	1.79121	+	0.674165i	1.79121	+	0.674165i
$437$		0			0
$438$		0			0
$439$	−1.55635	+	1.03992i	−1.55635	+	1.03992i	−0.582478	+	0.812847i	$0.697917\pi$
$439$	−1.55635	+	1.03992i	−1.55635	+	1.03992i	−0.973877	+	0.227076i	$0.927083\pi$
$440$		0			0
$441$	−1.65997	+	0.387052i	−1.65997	+	0.387052i
$442$		0			0
$443$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$443$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$444$	−0.150869	+	1.01708i	−0.150869	+	1.01708i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.48664	+	0.703130i	1.48664	+	0.703130i
$449$		0			0		0.910864	−	0.412707i	$-0.135417\pi$
$449$		0			0		−0.910864	+	0.412707i	$0.864583\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.867909	−	0.736366i	0.867909	−	0.736366i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.46658	+	0.979938i	1.46658	+	0.979938i	0.995185	+	0.0980171i	$0.0312500\pi$
$457$	1.46658	+	0.979938i	1.46658	+	0.979938i	0.471397	+	0.881921i	$0.343750\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.646956	−	0.762527i	$-0.276042\pi$
$461$		0			0		−0.646956	+	0.762527i	$0.723958\pi$
$462$		0			0
$463$	−1.77087	−	0.699796i	−1.77087	−	0.699796i	−0.997859	−	0.0654031i	$-0.979167\pi$
$463$	−1.77087	−	0.699796i	−1.77087	−	0.699796i	−0.773010	−	0.634393i	$-0.781250\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.0817211	−	0.996655i	$-0.473958\pi$
$467$		0			0		−0.0817211	+	0.996655i	$0.526042\pi$
$468$	−1.92174	−	0.448087i	−1.92174	−	0.448087i
$469$	1.29921	−	2.74696i	1.29921	−	2.74696i
$470$		0			0
$471$	0.0682475	+	0.0705183i	0.0682475	+	0.0705183i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.849202	−	1.52807i	−0.849202	−	1.52807i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.993448	−	0.114287i	$-0.0364583\pi$
$479$		0			0		−0.993448	+	0.114287i	$0.963542\pi$
$480$		0			0
$481$	0.683531	−	1.91034i	0.683531	−	1.91034i
$482$		0			0
$483$		0			0
$484$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$485$		0			0
$486$		0			0
$487$	−1.48012	−	0.0727135i	−1.48012	−	0.0727135i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−1.48012	−	0.0727135i	−1.48012	−	0.0727135i	−0.773010	+	0.634393i	$0.781250\pi$
$488$		0			0
$489$	0.405994	+	0.115938i	0.405994	+	0.115938i
$490$		0			0
$491$		0			0		0.989177	−	0.146730i	$-0.0468750\pi$
$491$		0			0		−0.989177	+	0.146730i	$0.953125\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.0897706	+	0.605184i	0.0897706	+	0.605184i
$497$		0			0
$498$		0			0
$499$	−0.420760	+	1.57030i	−0.420760	+	1.57030i	0.352250	+	0.935906i	$0.385417\pi$
$499$	−0.420760	+	1.57030i	−0.420760	+	1.57030i	−0.773010	+	0.634393i	$0.781250\pi$
$500$		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
$507$	2.63591	+	1.19432i	2.63591	+	1.19432i
$508$	0.417653	−	1.93376i	0.417653	−	1.93376i
$509$		0			0		−0.903989	−	0.427555i	$-0.859375\pi$
$509$		0			0		0.903989	+	0.427555i	$0.140625\pi$
$510$		0			0
$511$	−0.766950	+	0.476920i	−0.766950	+	0.476920i
$512$		0			0
$513$	1.62584	−	0.642484i	1.62584	−	0.642484i
$514$		0			0
$515$		0			0
$516$	−0.785695	+	0.785695i	−0.785695	+	0.785695i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.903989	−	0.427555i	$-0.140625\pi$
$521$		0			0		−0.903989	+	0.427555i	$0.859375\pi$
$522$		0			0
$523$	−0.0327191	+	0.0566711i	−0.0327191	+	0.0566711i	−0.881921	−	0.471397i	$-0.843750\pi$
$523$	−0.0327191	+	0.0566711i	−0.0327191	+	0.0566711i	0.849202	+	0.528068i	$0.177083\pi$
$524$		0			0
$525$	−1.64256	−	0.0806936i	−1.64256	−	0.0806936i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.866025	−	0.500000i	0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	2.87341	+	0.0940656i	2.87341	+	0.0940656i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.671526	+	0.0550620i	−0.671526	+	0.0550620i	−0.412707	−	0.910864i	$-0.635417\pi$
$541$	−0.671526	+	0.0550620i	−0.671526	+	0.0550620i	−0.258819	+	0.965926i	$0.583333\pi$
$542$		0			0
$543$	0.867580	−	1.75928i	0.867580	−	1.75928i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.209715	−	0.0909149i	−0.209715	−	0.0909149i	0.290285	−	0.956940i	$-0.406250\pi$
$547$	−0.209715	−	0.0909149i	−0.209715	−	0.0909149i	−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$	0.111033	−	0.965161i	0.111033	−	0.965161i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.71118	−	0.539287i	2.71118	−	0.539287i
$554$		0			0
$555$		0			0
$556$	−1.27693	+	0.794045i	−1.27693	+	0.794045i
$557$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$557$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$558$		0			0
$559$	1.82308	−	1.21814i	1.82308	−	1.21814i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.910864	−	0.412707i	$-0.135417\pi$
$563$		0			0		−0.910864	+	0.412707i	$0.864583\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.294400	−	1.61797i	0.294400	−	1.61797i
$568$		0			0
$569$		0			0		0.695443	−	0.718582i	$-0.255208\pi$
$569$		0			0		−0.695443	+	0.718582i	$0.744792\pi$
$570$		0			0
$571$	−0.353853	−	0.377805i	−0.353853	−	0.377805i	0.528068	−	0.849202i	$-0.322917\pi$
$571$	−0.353853	−	0.377805i	−0.353853	−	0.377805i	−0.881921	+	0.471397i	$0.843750\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.751840	+	0.659346i	−0.751840	+	0.659346i
$577$	0.953924	−	0.836569i	0.953924	−	0.836569i	−0.0327191	−	0.999465i	$-0.510417\pi$
$577$	0.953924	−	0.836569i	0.953924	−	0.836569i	0.986643	+	0.162895i	$0.0520833\pi$
$578$		0			0
$579$	1.94639	+	0.288720i	1.94639	+	0.288720i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$587$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$588$	0.946970	−	1.41724i	0.946970	−	1.41724i
$589$	0.564794	+	0.908262i	0.564794	+	0.908262i
$590$		0			0
$591$		0			0
$592$	−0.585152	−	0.845461i	−0.585152	−	0.845461i
$593$		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
$597$	1.64053	−	1.02015i	1.64053	−	1.02015i
$598$		0			0
$599$		0			0		−0.179017	−	0.983846i	$-0.557292\pi$
$599$		0			0		0.179017	+	0.983846i	$0.442708\pi$
$600$		0			0
$601$	−1.16433	−	1.62482i	−1.16433	−	1.62482i	−0.608761	−	0.793353i	$-0.708333\pi$
$601$	−1.16433	−	1.62482i	−1.16433	−	1.62482i	−0.555570	−	0.831470i	$-0.687500\pi$
$602$		0			0
$603$	1.21831	+	1.38922i	1.21831	+	1.38922i
$604$	−0.130081	+	1.13074i	−0.130081	+	1.13074i
$605$		0			0
$606$		0			0
$607$	−0.157160	+	0.0448794i	−0.157160	+	0.0448794i	−0.352250	−	0.935906i	$-0.614583\pi$
$607$	−0.157160	+	0.0448794i	−0.157160	+	0.0448794i	0.195090	+	0.980785i	$0.437500\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.71523	+	1.02807i	−1.71523	+	1.02807i	−0.849202	+	0.528068i	$0.822917\pi$
$613$	−1.71523	+	1.02807i	−1.71523	+	1.02807i	−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$617$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$618$		0			0
$619$	1.18402	+	0.683592i	1.18402	+	0.683592i	0.956940	−	0.290285i	$-0.0937500\pi$
$619$	1.18402	+	0.683592i	1.18402	+	0.683592i	0.227076	+	0.973877i	$0.427083\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.70892	−	0.986643i	1.70892	−	0.986643i
$625$	−0.946930	−	0.321439i	−0.946930	−	0.321439i
$626$		0			0
$627$		0			0
$628$	−0.0980171	−	0.00481527i	−0.0980171	−	0.00481527i
$629$		0			0
$630$		0			0
$631$	0.264534	+	1.13452i	0.264534	+	1.13452i	0.923880	+	0.382683i	$0.125000\pi$
$631$	0.264534	+	1.13452i	0.264534	+	1.13452i	−0.659346	+	0.751840i	$0.729167\pi$
$632$		0			0
$633$	−1.44175	−	0.929592i	−1.44175	−	0.929592i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.37833	+	2.37833i	−2.37833	+	2.37833i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$641$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$642$		0			0
$643$	1.02855	+	1.65404i	1.02855	+	1.65404i	0.707107	+	0.707107i	$0.250000\pi$
$643$	1.02855	+	1.65404i	1.02855	+	1.65404i	0.321439	+	0.946930i	$0.395833\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.412707	−	0.910864i	$-0.364583\pi$
$647$		0			0		−0.412707	+	0.910864i	$0.635417\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.00614			1.00614
$652$	−0.375574	+	0.192915i	−0.375574	+	0.192915i
$653$		0			0		−0.621661	−	0.783287i	$-0.713542\pi$
$653$		0			0		0.621661	+	0.783287i	$0.286458\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.0805810	−	0.543233i	−0.0805810	−	0.543233i
$658$		0			0
$659$		0			0		−0.740951	−	0.671559i	$-0.765625\pi$
$659$		0			0		0.740951	+	0.671559i	$0.234375\pi$
$660$		0			0
$661$	−1.22654	−	0.941158i	−1.22654	−	0.941158i	−0.227076	−	0.973877i	$-0.572917\pi$
$661$	−1.22654	−	0.941158i	−1.22654	−	0.941158i	−0.999465	+	0.0327191i	$0.989583\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.0509288	−	0.256036i	−0.0509288	−	0.256036i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.85979	−	0.734933i	−1.85979	−	0.734933i	−0.935906	−	0.352250i	$-0.885417\pi$
$673$	−1.85979	−	0.734933i	−1.85979	−	0.734933i	−0.923880	−	0.382683i	$-0.875000\pi$
$674$		0			0
$675$	0.442289	−	0.896873i	0.442289	−	0.896873i
$676$	−2.74028	+	0.930201i	−2.74028	+	0.930201i
$677$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$677$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$678$		0			0
$679$	0.510091	+	0.163895i	0.510091	+	0.163895i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$683$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$684$	−0.747444	+	1.58034i	−0.747444	+	1.58034i
$685$		0			0
$686$		0			0
$687$	1.26460	+	0.452483i	1.26460	+	0.452483i
$688$	0.0363555	−	1.11055i	0.0363555	−	1.11055i
$689$		0			0
$690$		0			0
$691$	0.617756	+	0.398308i	0.617756	+	0.398308i	0.812847	−	0.582478i	$-0.197917\pi$
$691$	0.617756	+	0.398308i	0.617756	+	0.398308i	−0.195090	+	0.980785i	$0.562500\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.25400	−	1.06394i	1.25400	−	1.06394i
$701$		0			0		0.917494	−	0.397748i	$-0.130208\pi$
$701$		0			0		−0.917494	+	0.397748i	$0.869792\pi$
$702$		0			0
$703$	−1.52643	−	0.949196i	−1.52643	−	0.949196i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.286847	−	1.93376i	0.286847	−	1.93376i	−0.0654031	−	0.997859i	$-0.520833\pi$
$709$	0.286847	−	1.93376i	0.286847	−	1.93376i	0.352250	−	0.935906i	$-0.385417\pi$
$710$		0			0
$711$	−0.354857	+	1.64301i	−0.354857	+	1.64301i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$719$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$720$		0			0
$721$	0.588484	+	0.0192650i	0.588484	+	0.0192650i
$722$		0			0
$723$	0.757363	+	0.243345i	0.757363	+	0.243345i
$724$	0.569414	+	1.87711i	0.569414	+	1.87711i
$725$		0			0
$726$		0			0
$727$	−1.83278	−	0.794539i	−1.83278	−	0.794539i	−0.935906	−	0.352250i	$-0.885417\pi$
$727$	−1.83278	−	0.794539i	−1.83278	−	0.794539i	−0.896873	−	0.442289i	$-0.854167\pi$
$728$		0			0
$729$	0.831470	+	0.555570i	0.831470	+	0.555570i
$730$		0			0
$731$		0			0
$732$	0.578738	+	0.780338i	0.578738	+	0.780338i
$733$	0.420760	+	0.301513i	0.420760	+	0.301513i	0.773010	−	0.634393i	$-0.218750\pi$
$733$	0.420760	+	0.301513i	0.420760	+	0.301513i	−0.352250	+	0.935906i	$0.614583\pi$
$734$		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	$0.291667\pi$
$739$	−0.184592	+	1.40211i	−0.184592	+	1.40211i	−0.793353	+	0.608761i	$0.791667\pi$
$740$		0			0
$741$	2.05496	−	2.77079i	2.05496	−	2.77079i
$742$		0			0
$743$		0			0		0.179017	−	0.983846i	$-0.442708\pi$
$743$		0			0		−0.179017	+	0.983846i	$0.557292\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.38054	−	0.989283i	1.38054	−	0.989283i	0.382683	−	0.923880i	$-0.375000\pi$
$751$	1.38054	−	0.989283i	1.38054	−	0.989283i	0.997859	−	0.0654031i	$-0.0208333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.891160	+	1.38215i	0.891160	+	1.38215i
$757$	0.471397	+	0.881921i	0.471397	+	0.881921i	0.999465	+	0.0327191i	$0.0104167\pi$
$757$	0.471397	+	0.881921i	0.471397	+	0.881921i	−0.528068	+	0.849202i	$0.677083\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$761$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$762$		0			0
$763$	0.764767	+	3.05312i	0.764767	+	3.05312i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.0980171	−	0.995185i	0.0980171	−	0.995185i
$769$	1.16246	+	1.62221i	1.16246	+	1.62221i	0.634393	+	0.773010i	$0.281250\pi$
$769$	1.16246	+	1.62221i	1.16246	+	1.62221i	0.528068	+	0.849202i	$0.322917\pi$
$770$		0			0
$771$		0			0
$772$	−1.61797	+	1.11981i	−1.61797	+	1.11981i
$773$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$773$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$774$		0			0
$775$	0.588289	+	0.167995i	0.588289	+	0.167995i
$776$		0			0
$777$	−1.51654	+	0.747875i	−1.51654	+	0.747875i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.277655	+	1.68173i	0.277655	+	1.68173i
$785$		0			0
$786$		0			0
$787$	−1.35954	−	0.293632i	−1.35954	−	0.293632i	−0.528068	−	0.849202i	$-0.677083\pi$
$787$	−1.35954	−	0.293632i	−1.35954	−	0.293632i	−0.831470	+	0.555570i	$0.812500\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.819666	−	1.73304i	−0.819666	−	1.73304i
$794$		0			0
$795$		0			0
$796$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$797$		0			0		0.582478	−	0.812847i	$-0.302083\pi$
$797$		0			0		−0.582478	+	0.812847i	$0.697917\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.83886	−	0.181112i	−1.83886	−	0.181112i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$809$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$810$		0			0
$811$	0.297595	+	1.04212i	0.297595	+	1.04212i	0.956940	+	0.290285i	$0.0937500\pi$
$811$	0.297595	+	1.04212i	0.297595	+	1.04212i	−0.659346	+	0.751840i	$0.729167\pi$
$812$		0			0
$813$	0.845566	−	0.416987i	0.845566	−	0.416987i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.713890	−	1.80653i	−0.713890	−	1.80653i
$818$		0			0
$819$	−1.19264	−	3.01804i	−1.19264	−	3.01804i
$820$		0			0
$821$		0			0		0.903989	−	0.427555i	$-0.140625\pi$
$821$		0			0		−0.903989	+	0.427555i	$0.859375\pi$
$822$		0			0
$823$	0.736366	−	1.06394i	0.736366	−	1.06394i	−0.258819	−	0.965926i	$-0.583333\pi$
$823$	0.736366	−	1.06394i	0.736366	−	1.06394i	0.995185	−	0.0980171i	$-0.0312500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.367516	−	0.930017i	$-0.380208\pi$
$827$		0			0		−0.367516	+	0.930017i	$0.619792\pi$
$828$		0			0
$829$	−1.24723	−	1.24723i	−1.24723	−	1.24723i	−0.956940	−	0.290285i	$-0.906250\pi$
$829$	−1.24723	−	1.24723i	−1.24723	−	1.24723i	−0.290285	−	0.956940i	$-0.593750\pi$
$830$		0			0
$831$	−1.91669	+	0.547340i	−1.91669	+	0.547340i
$832$	−0.572815	+	1.88832i	−0.572815	+	1.88832i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.243345	+	0.561329i	−0.243345	+	0.561329i
$838$		0			0
$839$		0			0		0.783287	−	0.621661i	$-0.213542\pi$
$839$		0			0		−0.783287	+	0.621661i	$0.786458\pi$
$840$		0			0
$841$	0.321439	−	0.946930i	0.321439	−	0.946930i
$842$		0			0
$843$		0			0
$844$	1.68775	−	0.307096i	1.68775	−	0.307096i
$845$		0			0
$846$		0			0
$847$	1.14368	−	1.18173i	1.14368	−	1.18173i
$848$		0			0
$849$	−0.709715	+	0.956940i	−0.709715	+	0.956940i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.05441	−	1.37413i	1.05441	−	1.37413i	0.130526	−	0.991445i	$-0.458333\pi$
$853$	1.05441	−	1.37413i	1.05441	−	1.37413i	0.923880	−	0.382683i	$-0.125000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.970031	−	0.242980i	$-0.921875\pi$
$857$		0			0		0.970031	+	0.242980i	$0.0781250\pi$
$858$		0			0
$859$	−0.0839125	−	0.637379i	−0.0839125	−	0.637379i	−0.980785	−	0.195090i	$-0.937500\pi$
$859$	−0.0839125	−	0.637379i	−0.0839125	−	0.637379i	0.896873	−	0.442289i	$-0.145833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.569100	−	0.822268i	$-0.692708\pi$
$863$		0			0		0.569100	+	0.822268i	$0.307292\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.980785	−	0.195090i	0.980785	−	0.195090i
$868$	−0.734343	+	0.687788i	−0.734343	+	0.687788i
$869$		0			0
$870$		0			0
$871$	3.48916	+	1.05842i	3.48916	+	1.05842i
$872$		0			0
$873$	−0.214809	+	0.244943i	−0.214809	+	0.244943i
$874$		0			0
$875$		0			0
$876$	0.430163	+	0.341402i	0.430163	+	0.341402i
$877$	−0.190913	−	0.0445147i	−0.190913	−	0.0445147i	0.130526	−	0.991445i	$-0.458333\pi$
$877$	−0.190913	−	0.0445147i	−0.190913	−	0.0445147i	−0.321439	+	0.946930i	$0.604167\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.993448	−	0.114287i	$-0.963542\pi$
$881$		0			0		0.993448	+	0.114287i	$0.0364583\pi$
$882$		0			0
$883$	−0.279537	+	0.870002i	−0.279537	+	0.870002i	0.707107	+	0.707107i	$0.250000\pi$
$883$	−0.279537	+	0.870002i	−0.279537	+	0.870002i	−0.986643	+	0.162895i	$0.947917\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.773010	−	0.634393i	$-0.218750\pi$
$887$		0			0		−0.773010	+	0.634393i	$0.781250\pi$
$888$		0			0
$889$	3.00582	−	1.24505i	3.00582	−	1.24505i
$890$		0			0
$891$		0			0
$892$	0.212196	+	0.152057i	0.212196	+	0.152057i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.290285	+	0.956940i	0.290285	+	0.956940i
$901$		0			0
$902$		0			0
$903$	−1.79779	−	0.327119i	−1.79779	−	0.327119i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.763728	−	0.0500574i	0.763728	−	0.0500574i	0.321439	−	0.946930i	$-0.395833\pi$
$907$	0.763728	−	0.0500574i	0.763728	−	0.0500574i	0.442289	+	0.896873i	$0.354167\pi$
$908$		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	$0.208333\pi$
$912$	−0.534774	−	1.66438i	−0.534774	−	1.66438i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.23230	+	0.534223i	−1.23230	+	0.534223i
$917$		0			0
$918$		0			0
$919$	−1.46658	+	0.783904i	−1.46658	+	0.783904i	−0.995185	−	0.0980171i	$-0.968750\pi$
$919$	−1.46658	+	0.783904i	−1.46658	+	0.783904i	−0.471397	+	0.881921i	$0.656250\pi$
$920$		0			0
$921$	0.0148595	+	0.453909i	0.0148595	+	0.453909i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.01160	+	0.184066i	−1.01160	+	0.184066i
$926$		0			0
$927$	−0.153079	+	0.323659i	−0.153079	+	0.323659i
$928$		0			0
$929$		0			0		0.961562	−	0.274589i	$-0.0885417\pi$
$929$		0			0		−0.961562	+	0.274589i	$0.911458\pi$
$930$		0			0
$931$	1.61471	+	2.50435i	1.61471	+	2.50435i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.980785	+	0.804910i	−0.980785	+	0.804910i	−0.980785	−	0.195090i	$-0.937500\pi$
$937$	−0.980785	+	0.804910i	−0.980785	+	0.804910i			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		−0.0817211	−	0.996655i	$-0.526042\pi$
$947$		0			0		0.0817211	+	0.996655i	$0.473958\pi$
$948$	−0.864154	−	1.44175i	−0.864154	−	1.44175i
$949$	−0.701096	−	0.826339i	−0.701096	−	0.826339i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.729864	−	0.683592i	$-0.760417\pi$
$953$		0			0		0.729864	+	0.683592i	$0.239583\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.609348	+	0.142080i	0.609348	+	0.142080i
$962$		0			0
$963$		0			0
$964$	−0.719121	+	0.340119i	−0.719121	+	0.340119i
$965$		0			0
$966$		0			0
$967$	0.228299	+	1.98450i	0.228299	+	1.98450i	0.162895	+	0.986643i	$0.447917\pi$
$967$	0.228299	+	1.98450i	0.228299	+	1.98450i	0.0654031	+	0.997859i	$0.479167\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.783287	−	0.621661i	$-0.786458\pi$
$971$		0			0		0.783287	+	0.621661i	$0.213542\pi$
$972$	−0.986643	+	0.162895i	−0.986643	+	0.162895i
$973$	−2.26883	−	0.983575i	−2.26883	−	0.983575i
$974$		0			0
$975$	−0.193416	−	1.96378i	−0.193416	−	1.96378i
$976$	−0.955833	−	0.173920i	−0.955833	−	0.173920i
$977$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$977$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.88832	−	0.311762i	−1.88832	−	0.311762i
$982$		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
$988$	0.394251	+	3.42706i	0.394251	+	3.42706i
$989$		0			0
$990$		0			0
$991$	−1.16246	−	0.0761917i	−1.16246	−	0.0761917i	−0.528068	−	0.849202i	$-0.677083\pi$
$991$	−1.16246	−	0.0761917i	−1.16246	−	0.0761917i	−0.634393	+	0.773010i	$0.718750\pi$
$992$		0			0
$993$	0.467281	+	1.86549i	0.467281	+	1.86549i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.763728	−	1.84380i	−0.763728	−	1.84380i	−0.442289	−	0.896873i	$-0.645833\pi$
$997$	−0.763728	−	1.84380i	−0.763728	−	1.84380i	−0.321439	−	0.946930i	$-0.604167\pi$
$998$		0			0
$999$	−0.0504517	−	1.02697i	−0.0504517	−	1.02697i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3459.1.bm.a.266.1	✓	64
3.2	odd	2	CM	3459.1.bm.a.266.1	✓	64
1153.1140	even	192	inner	3459.1.bm.a.3446.1	yes	64
3459.3446	odd	192	inner	3459.1.bm.a.3446.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3459.1.bm.a.266.1	✓	64	1.1	even	1	trivial
3459.1.bm.a.266.1	✓	64	3.2	odd	2	CM
3459.1.bm.a.3446.1	yes	64	1153.1140	even	192	inner
3459.1.bm.a.3446.1	yes	64	3459.3446	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)

\(f(q)\)	\(=\)	\(q+(0.582478 - 0.812847i) q^{3} +(0.130526 + 0.991445i) q^{4} +(-1.21852 + 1.10440i) q^{7} +(-0.321439 - 0.946930i) q^{9} +O(q^{10})\)	\(q+(0.582478 - 0.812847i) q^{3} +(0.130526 + 0.991445i) q^{4} +(-1.21852 + 1.10440i) q^{7} +(-0.321439 - 0.946930i) q^{9} +(0.881921 + 0.471397i) q^{12} +(-1.52537 - 1.25184i) q^{13} +(-0.965926 + 0.258819i) q^{16} +(-1.36933 + 1.08678i) q^{19} +(0.187949 + 1.63376i) q^{21} +(-0.162895 + 0.986643i) q^{25} +(-0.956940 - 0.290285i) q^{27} +(-1.25400 - 1.06394i) q^{28} +(0.0699215 - 0.607797i) q^{31} +(0.896873 - 0.442289i) q^{36} +(0.346392 + 0.968101i) q^{37} +(-1.90605 + 0.510724i) q^{39} +(-0.322547 + 1.06330i) q^{43} +(-0.352250 + 0.935906i) q^{48} +(0.167070 - 1.69629i) q^{49} +(1.04203 - 1.67572i) q^{52} +(0.0857792 + 1.74607i) q^{57} +(0.878250 + 0.415381i) q^{61} +(1.43747 + 0.798856i) q^{63} +(-0.382683 - 0.923880i) q^{64} +(-1.70711 + 0.707107i) q^{67} +(0.540306 + 0.0983120i) q^{73} +(0.707107 + 0.707107i) q^{75} +(-1.25621 - 1.21576i) q^{76} +(-1.44175 - 0.864154i) q^{79} +(-0.793353 + 0.608761i) q^{81} +(-1.59525 + 0.399590i) q^{84} +(3.24123 - 0.159232i) q^{91} +(-0.453318 - 0.410864i) q^{93} +(-0.162895 - 0.282143i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 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

Related objects

Downloads

Learn more