LMFDB - Embedded newform 3459.1.bm.a.1412.1

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			1412.1
Root			$-0.162895 + 0.986643i$ of defining polynomial
Character	$\chi$	$=$	3459.1412
Dual form			3459.1.bm.a.2168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.683592 + 0.729864i) q^{3} +(-0.608761 - 0.793353i) q^{4} +(1.72926 - 0.256511i) q^{7} +(-0.0654031 - 0.997859i) q^{9} +O(q^{10})$	\(q+(-0.683592 + 0.729864i) q^{3} +(-0.608761 - 0.793353i) q^{4} +(1.72926 - 0.256511i) q^{7} +(-0.0654031 - 0.997859i) q^{9} +(0.995185 + 0.0980171i) q^{12} +(1.76290 - 0.942289i) q^{13} +(-0.258819 + 0.965926i) q^{16} +(-0.0755851 + 0.415403i) q^{19} +(-0.994890 + 1.43747i) q^{21} +(-0.0327191 + 0.999465i) q^{25} +(0.773010 + 0.634393i) q^{27} +(-1.25621 - 1.21576i) q^{28} +(-1.05855 - 1.52945i) q^{31} +(-0.751840 + 0.659346i) q^{36} +(-1.30287 - 0.326351i) q^{37} +(-0.517361 + 1.93082i) q^{39} +(1.24441 - 1.51631i) q^{43} +(-0.528068 - 0.849202i) q^{48} +(1.96760 - 0.596865i) q^{49} +(-1.82075 - 0.824972i) q^{52} +(-0.251518 - 0.339133i) q^{57} +(-1.67679 + 1.00503i) q^{61} +(-0.369061 - 1.70878i) q^{63} +(0.923880 - 0.382683i) q^{64} +(-0.292893 - 0.707107i) q^{67} +(0.297595 + 1.04212i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(0.375574 - 0.192915i) q^{76} +(1.47219 + 1.33432i) q^{79} +(-0.991445 + 0.130526i) q^{81} +(1.74607 - 0.0857792i) q^{84} +(2.80680 - 2.08166i) q^{91} +(1.83990 + 0.272924i) q^{93} +(-0.0327191 + 0.0566711i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3459\mathbb{Z}\right)^\times$.

$n$	$1154$	$2311$
$\chi(n)$	$-1$	$e\left(\frac{49}{192}\right)$

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		0.442289	−	0.896873i	$-0.354167\pi$
$2$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$3$	−0.683592	+	0.729864i	−0.683592	+	0.729864i
$3$	−0.683592	+	0.729864i	−0.683592	+	0.729864i
$4$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$5$		0			0		−0.695443	−	0.718582i	$-0.744792\pi$
$5$		0			0		0.695443	+	0.718582i	$0.255208\pi$
$6$		0			0
$7$	1.72926	−	0.256511i	1.72926	−	0.256511i	0.793353	−	0.608761i	$-0.208333\pi$
$7$	1.72926	−	0.256511i	1.72926	−	0.256511i	0.935906	+	0.352250i	$0.114583\pi$
$8$		0			0
$9$	−0.0654031	−	0.997859i	−0.0654031	−	0.997859i
$10$		0			0
$11$		0			0		0.582478	−	0.812847i	$-0.302083\pi$
$11$		0			0		−0.582478	+	0.812847i	$0.697917\pi$
$12$	0.995185	+	0.0980171i	0.995185	+	0.0980171i
$13$	1.76290	−	0.942289i	1.76290	−	0.942289i	0.866025	−	0.500000i	$-0.166667\pi$
$13$	1.76290	−	0.942289i	1.76290	−	0.942289i	0.896873	−	0.442289i	$-0.145833\pi$
$14$		0			0
$15$		0			0
$16$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$17$		0			0		−0.762527	−	0.646956i	$-0.776042\pi$
$17$		0			0		0.762527	+	0.646956i	$0.223958\pi$
$18$		0			0
$19$	−0.0755851	+	0.415403i	−0.0755851	+	0.415403i	0.923880	+	0.382683i	$0.125000\pi$
$19$	−0.0755851	+	0.415403i	−0.0755851	+	0.415403i	−0.999465	+	0.0327191i	$0.989583\pi$
$20$		0			0
$21$	−0.994890	+	1.43747i	−0.994890	+	1.43747i
$22$		0			0
$23$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$23$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$24$		0			0
$25$	−0.0327191	+	0.999465i	−0.0327191	+	0.999465i
$26$		0			0
$27$	0.773010	+	0.634393i	0.773010	+	0.634393i
$28$	−1.25621	−	1.21576i	−1.25621	−	1.21576i
$29$		0			0		0.729864	−	0.683592i	$-0.239583\pi$
$29$		0			0		−0.729864	+	0.683592i	$0.760417\pi$
$30$		0			0
$31$	−1.05855	−	1.52945i	−1.05855	−	1.52945i	−0.831470	−	0.555570i	$-0.812500\pi$
$31$	−1.05855	−	1.52945i	−1.05855	−	1.52945i	−0.227076	−	0.973877i	$-0.572917\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.751840	+	0.659346i	−0.751840	+	0.659346i
$37$	−1.30287	−	0.326351i	−1.30287	−	0.326351i	−0.471397	−	0.881921i	$-0.656250\pi$
$37$	−1.30287	−	0.326351i	−1.30287	−	0.326351i	−0.831470	+	0.555570i	$0.812500\pi$
$38$		0			0
$39$	−0.517361	+	1.93082i	−0.517361	+	1.93082i
$40$		0			0
$41$		0			0		−0.999465	−	0.0327191i	$-0.989583\pi$
$41$		0			0		0.999465	+	0.0327191i	$0.0104167\pi$
$42$		0			0
$43$	1.24441	−	1.51631i	1.24441	−	1.51631i	0.471397	−	0.881921i	$-0.343750\pi$
$43$	1.24441	−	1.51631i	1.24441	−	1.51631i	0.773010	−	0.634393i	$-0.218750\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	−0.528068	−	0.849202i	−0.528068	−	0.849202i
$49$	1.96760	−	0.596865i	1.96760	−	0.596865i
$50$		0			0
$51$		0			0
$52$	−1.82075	−	0.824972i	−1.82075	−	0.824972i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.251518	−	0.339133i	−0.251518	−	0.339133i
$58$		0			0
$59$		0			0		0.514103	−	0.857729i	$-0.328125\pi$
$59$		0			0		−0.514103	+	0.857729i	$0.671875\pi$
$60$		0			0
$61$	−1.67679	+	1.00503i	−1.67679	+	1.00503i	−0.729864	+	0.683592i	$0.760417\pi$
$61$	−1.67679	+	1.00503i	−1.67679	+	1.00503i	−0.946930	+	0.321439i	$0.895833\pi$
$62$		0			0
$63$	−0.369061	−	1.70878i	−0.369061	−	1.70878i
$64$	0.923880	−	0.382683i	0.923880	−	0.382683i
$65$		0			0
$66$		0			0
$67$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$67$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.427555	−	0.903989i	$-0.359375\pi$
$71$		0			0		−0.427555	+	0.903989i	$0.640625\pi$
$72$		0			0
$73$	0.297595	+	1.04212i	0.297595	+	1.04212i	0.956940	+	0.290285i	$0.0937500\pi$
$73$	0.297595	+	1.04212i	0.297595	+	1.04212i	−0.659346	+	0.751840i	$0.729167\pi$
$74$		0			0
$75$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$76$	0.375574	−	0.192915i	0.375574	−	0.192915i
$77$		0			0
$78$		0			0
$79$	1.47219	+	1.33432i	1.47219	+	1.33432i	0.812847	+	0.582478i	$0.197917\pi$
$79$	1.47219	+	1.33432i	1.47219	+	1.33432i	0.659346	+	0.751840i	$0.270833\pi$
$80$		0			0
$81$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$82$		0			0
$83$		0			0		0.930017	−	0.367516i	$-0.119792\pi$
$83$		0			0		−0.930017	+	0.367516i	$0.880208\pi$
$84$	1.74607	−	0.0857792i	1.74607	−	0.0857792i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.162895	−	0.986643i	$-0.552083\pi$
$89$		0			0		0.162895	+	0.986643i	$0.447917\pi$
$90$		0			0
$91$	2.80680	−	2.08166i	2.80680	−	2.08166i
$92$		0			0
$93$	1.83990	+	0.272924i	1.83990	+	0.272924i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.0327191	+	0.0566711i	−0.0327191	+	0.0566711i	−0.881921	−	0.471397i	$-0.843750\pi$
$97$	−0.0327191	+	0.0566711i	−0.0327191	+	0.0566711i	0.849202	+	0.528068i	$0.177083\pi$
$98$		0			0
$99$		0			0
$100$	0.812847	−	0.582478i	0.812847	−	0.582478i
$101$		0			0		0.427555	−	0.903989i	$-0.359375\pi$
$101$		0			0		−0.427555	+	0.903989i	$0.640625\pi$
$102$		0			0
$103$	1.71065	−	0.878683i	1.71065	−	0.878683i	0.729864	−	0.683592i	$-0.239583\pi$
$103$	1.71065	−	0.878683i	1.71065	−	0.878683i	0.980785	−	0.195090i	$-0.0625000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$107$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$108$	0.0327191	−	0.999465i	0.0327191	−	0.999465i
$109$	−0.151537	+	1.53858i	−0.151537	+	1.53858i	0.555570	+	0.831470i	$0.312500\pi$
$109$	−0.151537	+	1.53858i	−0.151537	+	1.53858i	−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$	1.12882	−	0.727824i	1.12882	−	0.727824i
$112$	−0.199794	+	1.73673i	−0.199794	+	1.73673i
$113$		0			0		0.541892	−	0.840448i	$-0.317708\pi$
$113$		0			0		−0.541892	+	0.840448i	$0.682292\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.05557	−	1.69750i	−1.05557	−	1.69750i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.321439	−	0.946930i	−0.321439	−	0.946930i
$122$		0			0
$123$		0			0
$124$	−0.568990	+	1.77087i	−0.568990	+	1.77087i
$125$		0			0
$126$		0			0
$127$	−0.163715	−	0.653587i	−0.163715	−	0.653587i	−0.995185	−	0.0980171i	$-0.968750\pi$
$127$	−0.163715	−	0.653587i	−0.163715	−	0.653587i	0.831470	−	0.555570i	$-0.187500\pi$
$128$		0			0
$129$	0.256036	+	1.94479i	0.256036	+	1.94479i
$130$		0			0
$131$		0			0		−0.999465	−	0.0327191i	$-0.989583\pi$
$131$		0			0		0.999465	+	0.0327191i	$0.0104167\pi$
$132$		0			0
$133$	−0.0241507	+	0.737727i	−0.0241507	+	0.737727i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$137$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$138$		0			0
$139$	0.861470	+	0.200867i	0.861470	+	0.200867i	0.634393	−	0.773010i	$-0.281250\pi$
$139$	0.861470	+	0.200867i	0.861470	+	0.200867i	0.227076	+	0.973877i	$0.427083\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.980785	+	0.195090i	0.980785	+	0.195090i
$145$		0			0
$146$		0			0
$147$	−0.909406	+	1.84409i	−0.909406	+	1.84409i
$148$	0.534223	+	1.23230i	0.534223	+	1.23230i
$149$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$149$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$150$		0			0
$151$	0.970357	+	0.0476706i	0.970357	+	0.0476706i	0.528068	−	0.849202i	$-0.322917\pi$
$151$	0.970357	+	0.0476706i	0.970357	+	0.0476706i	0.442289	+	0.896873i	$0.354167\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.84677	−	0.764957i	1.84677	−	0.764957i
$157$	1.14560	−	0.327144i	1.14560	−	0.327144i	0.352250	−	0.935906i	$-0.385417\pi$
$157$	1.14560	−	0.327144i	1.14560	−	0.327144i	0.793353	+	0.608761i	$0.208333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.20606	−	0.302102i	−1.20606	−	0.302102i	−0.412707	−	0.910864i	$-0.635417\pi$
$163$	−1.20606	−	0.302102i	−1.20606	−	0.302102i	−0.793353	+	0.608761i	$0.791667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.397748	−	0.917494i	$-0.369792\pi$
$167$		0			0		−0.397748	+	0.917494i	$0.630208\pi$
$168$		0			0
$169$	1.66433	−	2.49085i	1.66433	−	2.49085i
$170$		0			0
$171$	0.419457	+	0.0482546i	0.419457	+	0.0482546i
$172$	−1.96052	−	0.0641808i	−1.96052	−	0.0641808i
$173$		0			0		−0.146730	−	0.989177i	$-0.546875\pi$
$173$		0			0		0.146730	+	0.989177i	$0.453125\pi$
$174$		0			0
$175$	0.199794	+	1.73673i	0.199794	+	1.73673i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.889516	−	0.456904i	$-0.151042\pi$
$179$		0			0		−0.889516	+	0.456904i	$0.848958\pi$
$180$		0			0
$181$	−1.66205	+	0.0544098i	−1.66205	+	0.0544098i	−0.849202	−	0.528068i	$-0.822917\pi$
$181$	−1.66205	+	0.0544098i	−1.66205	+	0.0544098i	−0.812847	+	0.582478i	$0.802083\pi$
$182$		0			0
$183$	0.412707	−	1.91086i	0.412707	−	1.91086i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.49946	+	0.898744i	1.49946	+	0.898744i
$190$		0			0
$191$		0			0		0.211112	−	0.977462i	$-0.432292\pi$
$191$		0			0		−0.211112	+	0.977462i	$0.567708\pi$
$192$	−0.352250	+	0.935906i	−0.352250	+	0.935906i
$193$	−0.503867	+	0.218434i	−0.503867	+	0.218434i	−0.634393	−	0.773010i	$-0.718750\pi$
$193$	−0.503867	+	0.218434i	−0.503867	+	0.218434i	0.130526	+	0.991445i	$0.458333\pi$
$194$		0			0
$195$		0			0
$196$	−1.67132	−	1.19765i	−1.67132	−	1.19765i
$197$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$197$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$198$		0			0
$199$	0.198092	−	0.478235i	0.198092	−	0.478235i	−0.793353	−	0.608761i	$-0.791667\pi$
$199$	0.198092	−	0.478235i	0.198092	−	0.478235i	0.991445	+	0.130526i	$0.0416667\pi$
$200$		0			0
$201$	0.716311	+	0.269601i	0.716311	+	0.269601i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.453909	+	1.94671i	0.453909	+	1.94671i
$209$		0			0
$210$		0			0
$211$	−0.882768	−	1.19028i	−0.882768	−	1.19028i	−0.980785	−	0.195090i	$-0.937500\pi$
$211$	−0.882768	−	1.19028i	−0.882768	−	1.19028i	0.0980171	−	0.995185i	$-0.468750\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.22282	−	2.37328i	−2.22282	−	2.37328i
$218$		0			0
$219$	−0.964043	−	0.495185i	−0.964043	−	0.495185i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.20126	+	0.198329i	−1.20126	+	0.198329i	−0.729864	−	0.683592i	$-0.760417\pi$
$223$	−1.20126	+	0.198329i	−1.20126	+	0.198329i	−0.471397	+	0.881921i	$0.656250\pi$
$224$		0			0
$225$	0.999465	−	0.0327191i	0.999465	−	0.0327191i
$226$		0			0
$227$		0			0		0.917494	−	0.397748i	$-0.130208\pi$
$227$		0			0		−0.917494	+	0.397748i	$0.869792\pi$
$228$	−0.115938	+	0.405994i	−0.115938	+	0.405994i
$229$	0.159024	−	0.246639i	0.159024	−	0.246639i	−0.751840	−	0.659346i	$-0.770833\pi$
$229$	0.159024	−	0.246639i	0.159024	−	0.246639i	0.910864	+	0.412707i	$0.135417\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.762527	−	0.646956i	$-0.223958\pi$
$233$		0			0		−0.762527	+	0.646956i	$0.776042\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.98025	+	0.162371i	−1.98025	+	0.162371i
$238$		0			0
$239$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$239$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$240$		0			0
$241$	−0.152004	+	0.0600676i	−0.152004	+	0.0600676i	−0.442289	−	0.896873i	$-0.645833\pi$
$241$	−0.152004	+	0.0600676i	−0.152004	+	0.0600676i	0.290285	+	0.956940i	$0.406250\pi$
$242$		0			0
$243$	0.582478	−	0.812847i	0.582478	−	0.812847i
$244$	1.81811	+	0.718466i	1.81811	+	0.718466i
$245$		0			0
$246$		0			0
$247$	0.258180	+	0.803535i	0.258180	+	0.803535i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$251$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$252$	−1.13100	+	1.33304i	−1.13100	+	1.33304i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$257$		0			0		−0.0817211	−	0.996655i	$-0.526042\pi$
$257$		0			0		0.0817211	+	0.996655i	$0.473958\pi$
$258$		0			0
$259$	−2.33671	−	0.230145i	−2.33671	−	0.230145i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.456904	−	0.889516i	$-0.651042\pi$
$263$		0			0		0.456904	+	0.889516i	$0.348958\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$269$		0			0		−0.427555	−	0.903989i	$-0.640625\pi$
$269$		0			0		0.427555	+	0.903989i	$0.359375\pi$
$270$		0			0
$271$	0.195090	+	0.0192147i	0.195090	+	0.0192147i	0.195090	−	0.980785i	$-0.437500\pi$
$271$	0.195090	+	0.0192147i	0.195090	+	0.0192147i			1.00000i	$0.5\pi$
$272$		0			0
$273$	−0.399375	+	3.47159i	−0.399375	+	3.47159i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.357986	−	1.96743i	−0.357986	−	1.96743i	−0.195090	−	0.980785i	$-0.562500\pi$
$277$	−0.357986	−	1.96743i	−0.357986	−	1.96743i	−0.162895	−	0.986643i	$-0.552083\pi$
$278$		0			0
$279$	−1.45694	+	1.15631i	−1.45694	+	1.15631i
$280$		0			0
$281$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$281$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$282$		0			0
$283$	0.553066	+	1.72131i	0.553066	+	1.72131i	0.683592	+	0.729864i	$0.260417\pi$
$283$	0.553066	+	1.72131i	0.553066	+	1.72131i	−0.130526	+	0.991445i	$0.541667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.162895	+	0.986643i	0.162895	+	0.986643i
$290$		0			0
$291$	−0.0189957	−	0.0626204i	−0.0189957	−	0.0626204i
$292$	0.645609	−	0.870503i	0.645609	−	0.870503i
$293$		0			0		−0.671559	−	0.740951i	$-0.734375\pi$
$293$		0			0		0.671559	+	0.740951i	$0.265625\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$301$	1.76295	−	2.94131i	1.76295	−	2.94131i
$302$		0			0
$303$		0			0
$304$	−0.381685	−	0.180524i	−0.381685	−	0.180524i
$305$		0			0
$306$		0			0
$307$	1.85580	−	0.244320i	1.85580	−	0.244320i	0.881921	−	0.471397i	$-0.156250\pi$
$307$	1.85580	−	0.244320i	1.85580	−	0.244320i	0.973877	+	0.227076i	$0.0729167\pi$
$308$		0			0
$309$	−0.528068	+	1.84920i	−0.528068	+	1.84920i
$310$		0			0
$311$		0			0		0.0163617	−	0.999866i	$-0.494792\pi$
$311$		0			0		−0.0163617	+	0.999866i	$0.505208\pi$
$312$		0			0
$313$	−1.48501	+	0.558918i	−1.48501	+	0.558918i	−0.956940	−	0.290285i	$-0.906250\pi$
$313$	−1.48501	+	0.558918i	−1.48501	+	0.558918i	−0.528068	+	0.849202i	$0.677083\pi$
$314$		0			0
$315$		0			0
$316$	0.162371	−	1.98025i	0.162371	−	1.98025i
$317$		0			0		0.146730	−	0.989177i	$-0.453125\pi$
$317$		0			0		−0.146730	+	0.989177i	$0.546875\pi$
$318$		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
$325$	0.884104	+	1.79279i	0.884104	+	1.79279i
$326$		0			0
$327$	−1.01936	−	1.16236i	−1.01936	−	1.16236i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.16897	+	1.20786i	−1.16897	+	1.20786i	−0.195090	+	0.980785i	$0.562500\pi$
$331$	−1.16897	+	1.20786i	−1.16897	+	1.20786i	−0.973877	+	0.227076i	$0.927083\pi$
$332$		0			0
$333$	−0.240441	+	1.32142i	−0.240441	+	1.32142i
$334$		0			0
$335$		0			0
$336$	−1.13100	−	1.33304i	−1.13100	−	1.33304i
$337$	−0.125471	+	0.845855i	−0.125471	+	0.845855i	0.831470	+	0.555570i	$0.187500\pi$
$337$	−0.125471	+	0.845855i	−0.125471	+	0.845855i	−0.956940	+	0.290285i	$0.906250\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.66905	−	0.789402i	1.66905	−	0.789402i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.803208	−	0.595699i	$-0.203125\pi$
$347$		0			0		−0.803208	+	0.595699i	$0.796875\pi$
$348$		0			0
$349$	0.520697	+	1.71651i	0.520697	+	1.71651i	0.683592	+	0.729864i	$0.260417\pi$
$349$	0.520697	+	1.71651i	0.520697	+	1.71651i	−0.162895	+	0.986643i	$0.552083\pi$
$350$		0			0
$351$	1.96052	+	0.389972i	1.96052	+	0.389972i
$352$		0			0
$353$		0			0		0.683592	−	0.729864i	$-0.260417\pi$
$353$		0			0		−0.683592	+	0.729864i	$0.739583\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.941544	−	0.336890i	$-0.890625\pi$
$359$		0			0		0.941544	+	0.336890i	$0.109375\pi$
$360$		0			0
$361$	0.769060	+	0.289454i	0.769060	+	0.289454i
$362$		0			0
$363$	0.910864	+	0.412707i	0.910864	+	0.412707i
$364$	−3.36017	−	0.959547i	−3.36017	−	0.959547i
$365$		0			0
$366$		0			0
$367$	0.125419	+	0.369474i	0.125419	+	0.369474i	0.991445	−	0.130526i	$-0.0416667\pi$
$367$	0.125419	+	0.369474i	0.125419	+	0.369474i	−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.903537	−	1.62584i	−0.903537	−	1.62584i
$373$	−1.31859	−	0.571629i	−1.31859	−	0.571629i	−0.382683	−	0.923880i	$-0.625000\pi$
$373$	−1.31859	−	0.571629i	−1.31859	−	0.571629i	−0.935906	+	0.352250i	$0.885417\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.230228	+	0.123059i	−0.230228	+	0.123059i	−0.582478	−	0.812847i	$-0.697917\pi$
$379$	−0.230228	+	0.123059i	−0.230228	+	0.123059i	0.352250	+	0.935906i	$0.385417\pi$
$380$		0			0
$381$	0.588944	+	0.327298i	0.588944	+	0.327298i
$382$		0			0
$383$		0			0		0.162895	−	0.986643i	$-0.447917\pi$
$383$		0			0		−0.162895	+	0.986643i	$0.552083\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.59446	−	1.14257i	−1.59446	−	1.14257i
$388$	0.0648783	−	0.00854139i	0.0648783	−	0.00854139i
$389$		0			0		0.903989	−	0.427555i	$-0.140625\pi$
$389$		0			0		−0.903989	+	0.427555i	$0.859375\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.227809	+	1.98025i	0.227809	+	1.98025i	0.195090	+	0.980785i	$0.437500\pi$
$397$	0.227809	+	1.98025i	0.227809	+	1.98025i	0.0327191	+	0.999465i	$0.489583\pi$
$398$		0			0
$399$	−0.521931	−	0.521931i	−0.521931	−	0.521931i
$400$	−0.956940	−	0.290285i	−0.956940	−	0.290285i
$401$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$401$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$402$		0			0
$403$	−3.30729	−	1.69880i	−3.30729	−	1.69880i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.794539	+	0.0390332i	−0.794539	+	0.0390332i	−0.442289	−	0.896873i	$-0.645833\pi$
$409$	−0.794539	+	0.0390332i	−0.794539	+	0.0390332i	−0.352250	+	0.935906i	$0.614583\pi$
$410$		0			0
$411$		0			0
$412$	−1.73848	−	0.822241i	−1.73848	−	0.822241i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.735499	+	0.491445i	−0.735499	+	0.491445i
$418$		0			0
$419$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$419$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$420$		0			0
$421$	−1.48664	+	0.703130i	−1.48664	+	0.703130i	−0.986643	−	0.162895i	$-0.947917\pi$
$421$	−1.48664	+	0.703130i	−1.48664	+	0.703130i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.64181	+	2.16808i	−2.64181	+	2.16808i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.729864	−	0.683592i	$-0.239583\pi$
$431$		0			0		−0.729864	+	0.683592i	$0.760417\pi$
$432$	−0.812847	+	0.582478i	−0.812847	+	0.582478i
$433$	1.08248	+	1.67887i	1.08248	+	1.67887i	0.582478	+	0.812847i	$0.302083\pi$
$433$	1.08248	+	1.67887i	1.08248	+	1.67887i	0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$	1.31288	−	0.816404i	1.31288	−	0.816404i
$437$		0			0
$438$		0			0
$439$	0.331342	−	1.66577i	0.331342	−	1.66577i	−0.352250	−	0.935906i	$-0.614583\pi$
$439$	0.331342	−	1.66577i	0.331342	−	1.66577i	0.683592	−	0.729864i	$-0.260417\pi$
$440$		0			0
$441$	−0.724274	−	1.92435i	−0.724274	−	1.92435i
$442$		0			0
$443$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$443$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$444$	−1.26460	−	0.452483i	−1.26460	−	0.452483i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.49946	−	0.898744i	1.49946	−	0.898744i
$449$		0			0		0.227076	−	0.973877i	$-0.427083\pi$
$449$		0			0		−0.227076	+	0.973877i	$0.572917\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.698121	+	0.675641i	−0.698121	+	0.675641i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.388302	+	1.95213i	0.388302	+	1.95213i	0.290285	+	0.956940i	$0.406250\pi$
$457$	0.388302	+	1.95213i	0.388302	+	1.95213i	0.0980171	+	0.995185i	$0.468750\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.695443	−	0.718582i	$-0.255208\pi$
$461$		0			0		−0.695443	+	0.718582i	$0.744792\pi$
$462$		0			0
$463$	0.560482	+	0.475533i	0.560482	+	0.475533i	0.881921	−	0.471397i	$-0.156250\pi$
$463$	0.560482	+	0.475533i	0.560482	+	0.475533i	−0.321439	+	0.946930i	$0.604167\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.0163617	−	0.999866i	$-0.494792\pi$
$467$		0			0		−0.0163617	+	0.999866i	$0.505208\pi$
$468$	−0.704123	+	1.87081i	−0.704123	+	1.87081i
$469$	−0.687869	−	1.14764i	−0.687869	−	1.14764i
$470$		0			0
$471$	−0.544355	+	1.05977i	−0.544355	+	1.05977i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.412707	−	0.0891362i	−0.412707	−	0.0891362i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.822268	−	0.569100i	$-0.807292\pi$
$479$		0			0		0.822268	+	0.569100i	$0.192708\pi$
$480$		0			0
$481$	−2.60434	+	0.652353i	−2.60434	+	0.652353i
$482$		0			0
$483$		0			0
$484$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$485$		0			0
$486$		0			0
$487$	1.58903	+	1.17850i	1.58903	+	1.17850i	0.881921	+	0.471397i	$0.156250\pi$
$487$	1.58903	+	1.17850i	1.58903	+	1.17850i	0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$	1.04495	−	0.673745i	1.04495	−	0.673745i
$490$		0			0
$491$		0			0		−0.336890	−	0.941544i	$-0.609375\pi$
$491$		0			0		0.336890	+	0.941544i	$0.390625\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.75130	−	0.626627i	1.75130	−	0.626627i
$497$		0			0
$498$		0			0
$499$	1.40999	−	0.377805i	1.40999	−	0.377805i	0.528068	−	0.849202i	$-0.322917\pi$
$499$	1.40999	−	0.377805i	1.40999	−	0.377805i	0.881921	+	0.471397i	$0.156250\pi$
$500$		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
$507$	0.680256	+	2.91746i	0.680256	+	2.91746i
$508$	−0.418862	+	0.527763i	−0.418862	+	0.527763i
$509$		0			0		0.857729	−	0.514103i	$-0.171875\pi$
$509$		0			0		−0.857729	+	0.514103i	$0.828125\pi$
$510$		0			0
$511$	0.781935	+	1.72577i	0.781935	+	1.72577i
$512$		0			0
$513$	−0.321957	+	0.273160i	−0.321957	+	0.273160i
$514$		0			0
$515$		0			0
$516$	1.38704	−	1.38704i	1.38704	−	1.38704i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$521$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$522$		0			0
$523$	−0.582478	−	1.00888i	−0.582478	−	1.00888i	−0.995185	−	0.0980171i	$-0.968750\pi$
$523$	−0.582478	−	1.00888i	−0.582478	−	1.00888i	0.412707	−	0.910864i	$-0.364583\pi$
$524$		0			0
$525$	−1.40415	−	1.04139i	−1.40415	−	1.04139i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$	0.599980	−	0.429940i	0.599980	−	0.429940i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.93980	+	0.0317428i	−1.93980	+	0.0317428i	−0.973877	−	0.227076i	$-0.927083\pi$
$541$	−1.93980	+	0.0317428i	−1.93980	+	0.0317428i	−0.965926	+	0.258819i	$0.916667\pi$
$542$		0			0
$543$	1.09645	−	1.25026i	1.09645	−	1.25026i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.13439	−	0.0930150i	−1.13439	−	0.0930150i	−0.500000	−	0.866025i	$-0.666667\pi$
$547$	−1.13439	−	0.0930150i	−1.13439	−	0.0930150i	−0.634393	+	0.773010i	$0.718750\pi$
$548$		0			0
$549$	1.11255	+	1.60747i	1.11255	+	1.60747i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.88807	+	1.92975i	2.88807	+	1.92975i
$554$		0			0
$555$		0			0
$556$	−0.365071	−	0.805730i	−0.365071	−	0.805730i
$557$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$557$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$558$		0			0
$559$	0.764957	−	3.84570i	0.764957	−	3.84570i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.227076	−	0.973877i	$-0.427083\pi$
$563$		0			0		−0.227076	+	0.973877i	$0.572917\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.68098	+	0.480031i	−1.68098	+	0.480031i
$568$		0			0
$569$		0			0		−0.456904	−	0.889516i	$-0.651042\pi$
$569$		0			0		0.456904	+	0.889516i	$0.348958\pi$
$570$		0			0
$571$	−1.90605	−	0.314690i	−1.90605	−	0.314690i	−0.910864	−	0.412707i	$-0.864583\pi$
$571$	−1.90605	−	0.314690i	−1.90605	−	0.314690i	−0.995185	+	0.0980171i	$0.968750\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.442289	−	0.896873i	−0.442289	−	0.896873i
$577$	0.416987	+	0.845566i	0.416987	+	0.845566i	0.999465	+	0.0327191i	$0.0104167\pi$
$577$	0.416987	+	0.845566i	0.416987	+	0.845566i	−0.582478	+	0.812847i	$0.697917\pi$
$578$		0			0
$579$	0.185012	−	0.517075i	0.185012	−	0.517075i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.989177	−	0.146730i	$-0.953125\pi$
$587$		0			0		0.989177	+	0.146730i	$0.0468750\pi$
$588$	2.01663	−	0.401132i	2.01663	−	0.401132i
$589$	0.715346	−	0.324119i	0.715346	−	0.324119i
$590$		0			0
$591$		0			0
$592$	0.652438	−	1.17401i	0.652438	−	1.17401i
$593$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$593$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.213633	+	0.471498i	0.213633	+	0.471498i
$598$		0			0
$599$		0			0		−0.961562	−	0.274589i	$-0.911458\pi$
$599$		0			0		0.961562	+	0.274589i	$0.0885417\pi$
$600$		0			0
$601$	−1.11131	−	1.18654i	−1.11131	−	1.18654i	−0.980785	−	0.195090i	$-0.937500\pi$
$601$	−1.11131	−	1.18654i	−1.11131	−	1.18654i	−0.130526	−	0.991445i	$-0.541667\pi$
$602$		0			0
$603$	−0.686437	+	0.338513i	−0.686437	+	0.338513i
$604$	−0.552896	−	0.798856i	−0.552896	−	0.798856i
$605$		0			0
$606$		0			0
$607$	0.0275024	+	0.0177326i	0.0275024	+	0.0177326i	0.555570	−	0.831470i	$-0.312500\pi$
$607$	0.0275024	+	0.0177326i	0.0275024	+	0.0177326i	−0.528068	+	0.849202i	$0.677083\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.453318	−	0.410864i	0.453318	−	0.410864i	−0.412707	−	0.910864i	$-0.635417\pi$
$613$	0.453318	−	0.410864i	0.453318	−	0.410864i	0.866025	+	0.500000i	$0.166667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$617$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$618$		0			0
$619$	−1.70892	+	0.986643i	−1.70892	+	0.986643i	−0.773010	+	0.634393i	$0.781250\pi$
$619$	−1.70892	+	0.986643i	−1.70892	+	0.986643i	−0.935906	+	0.352250i	$0.885417\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.73112	−	0.999465i	−1.73112	−	0.999465i
$625$	−0.997859	−	0.0654031i	−0.997859	−	0.0654031i
$626$		0			0
$627$		0			0
$628$	−0.956940	−	0.709715i	−0.956940	−	0.709715i
$629$		0			0
$630$		0			0
$631$	1.27956	−	0.481591i	1.27956	−	0.481591i	0.382683	−	0.923880i	$-0.375000\pi$
$631$	1.27956	−	0.481591i	1.27956	−	0.481591i	0.896873	+	0.442289i	$0.145833\pi$
$632$		0			0
$633$	1.47219	+	0.169362i	1.47219	+	0.169362i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.90626	−	2.90626i	2.90626	−	2.90626i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$641$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$642$		0			0
$643$	−0.641704	+	0.290752i	−0.641704	+	0.290752i	−0.707107	−	0.707107i	$-0.750000\pi$
$643$	−0.641704	+	0.290752i	−0.641704	+	0.290752i	0.0654031	+	0.997859i	$0.479167\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.973877	−	0.227076i	$-0.0729167\pi$
$647$		0			0		−0.973877	+	0.227076i	$0.927083\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	3.25168			3.25168
$652$	0.494529	+	1.14074i	0.494529	+	1.14074i
$653$		0			0		−0.983846	−	0.179017i	$-0.942708\pi$
$653$		0			0		0.983846	+	0.179017i	$0.0572917\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.02043	−	0.365116i	1.02043	−	0.365116i
$658$		0			0
$659$		0			0		−0.989177	−	0.146730i	$-0.953125\pi$
$659$		0			0		0.989177	+	0.146730i	$0.0468750\pi$
$660$		0			0
$661$	1.74875	+	0.230228i	1.74875	+	0.230228i	0.935906	−	0.352250i	$-0.114583\pi$
$661$	1.74875	+	0.230228i	1.74875	+	0.230228i	0.812847	+	0.582478i	$0.197917\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.676419	−	1.01233i	0.676419	−	1.01233i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.466519	+	0.395812i	0.466519	+	0.395812i	0.849202	−	0.528068i	$-0.177083\pi$
$673$	0.466519	+	0.395812i	0.466519	+	0.395812i	−0.382683	+	0.923880i	$0.625000\pi$
$674$		0			0
$675$	−0.659346	+	0.751840i	−0.659346	+	0.751840i
$676$	−2.98930	+	0.195929i	−2.98930	+	0.195929i
$677$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$677$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$678$		0			0
$679$	−0.0420430	+	0.106392i	−0.0420430	+	0.106392i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.671559	−	0.740951i	$-0.265625\pi$
$683$		0			0		−0.671559	+	0.740951i	$0.734375\pi$
$684$	−0.217066	−	0.362153i	−0.217066	−	0.362153i
$685$		0			0
$686$		0			0
$687$	0.0713052	+	0.284666i	0.0713052	+	0.284666i
$688$	1.14257	+	1.59446i	1.14257	+	1.59446i
$689$		0			0
$690$		0			0
$691$	−1.28543	−	0.147877i	−1.28543	−	0.147877i	−0.555570	−	0.831470i	$-0.687500\pi$
$691$	−1.28543	−	0.147877i	−1.28543	−	0.147877i	−0.729864	+	0.683592i	$0.760417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.25621	−	1.21576i	1.25621	−	1.21576i
$701$		0			0		0.996655	−	0.0817211i	$-0.0260417\pi$
$701$		0			0		−0.996655	+	0.0817211i	$0.973958\pi$
$702$		0			0
$703$	0.234044	−	0.516547i	0.234044	−	0.516547i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.47500	+	0.527763i	1.47500	+	0.527763i	0.946930	−	0.321439i	$-0.104167\pi$
$709$	1.47500	+	0.527763i	1.47500	+	0.527763i	0.528068	+	0.849202i	$0.322917\pi$
$710$		0			0
$711$	1.23517	−	1.55631i	1.23517	−	1.55631i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$719$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$720$		0			0
$721$	2.73276	−	1.95827i	2.73276	−	1.95827i
$722$		0			0
$723$	0.0600676	−	0.152004i	0.0600676	−	0.152004i
$724$	1.05496	+	1.28547i	1.05496	+	1.28547i
$725$		0			0
$726$		0			0
$727$	1.60104	+	0.131278i	1.60104	+	0.131278i	0.849202	−	0.528068i	$-0.177083\pi$
$727$	1.60104	+	0.131278i	1.60104	+	0.131278i	0.751840	+	0.659346i	$0.229167\pi$
$728$		0			0
$729$	0.195090	+	0.980785i	0.195090	+	0.980785i
$730$		0			0
$731$		0			0
$732$	−1.76723	+	0.835838i	−1.76723	+	0.835838i
$733$	−1.40999	−	1.32060i	−1.40999	−	1.32060i	−0.881921	−	0.471397i	$-0.843750\pi$
$733$	−1.40999	−	1.32060i	−1.40999	−	1.32060i	−0.528068	−	0.849202i	$-0.677083\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.860919	+	1.12197i	−0.860919	+	1.12197i	0.130526	+	0.991445i	$0.458333\pi$
$739$	−0.860919	+	1.12197i	−0.860919	+	1.12197i	−0.991445	+	0.130526i	$0.958333\pi$
$740$		0			0
$741$	−0.762962	−	0.360854i	−0.762962	−	0.360854i
$742$		0			0
$743$		0			0		0.961562	−	0.274589i	$-0.0885417\pi$
$743$		0			0		−0.961562	+	0.274589i	$0.911458\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.602440	+	0.564247i	−0.602440	+	0.564247i	−0.923880	−	0.382683i	$-0.875000\pi$
$751$	−0.602440	+	0.564247i	−0.602440	+	0.564247i	0.321439	+	0.946930i	$0.395833\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.199794	−	1.73673i	−0.199794	−	1.73673i
$757$	0.0980171	+	0.995185i	0.0980171	+	0.995185i	0.910864	+	0.412707i	$0.135417\pi$
$757$	0.0980171	+	0.995185i	0.0980171	+	0.995185i	−0.812847	+	0.582478i	$0.802083\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$761$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$762$		0			0
$763$	0.132616	+	2.69947i	0.132616	+	2.69947i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.956940	−	0.290285i	0.956940	−	0.290285i
$769$	−0.439467	−	0.469214i	−0.439467	−	0.469214i	0.471397	−	0.881921i	$-0.343750\pi$
$769$	−0.439467	−	0.469214i	−0.439467	−	0.469214i	−0.910864	+	0.412707i	$0.864583\pi$
$770$		0			0
$771$		0			0
$772$	0.480031	+	0.266770i	0.480031	+	0.266770i
$773$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$773$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$774$		0			0
$775$	1.56326	−	1.00794i	1.56326	−	1.00794i
$776$		0			0
$777$	1.76533	−	1.54815i	1.76533	−	1.54815i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.0672749	+	2.05503i	0.0672749	+	2.05503i
$785$		0			0
$786$		0			0
$787$	0.715774	+	0.568078i	0.715774	+	0.568078i	0.910864	−	0.412707i	$-0.135417\pi$
$787$	0.715774	+	0.568078i	0.715774	+	0.568078i	−0.195090	+	0.980785i	$0.562500\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.00899	+	3.35179i	−2.00899	+	3.35179i
$794$		0			0
$795$		0			0
$796$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$797$		0			0		0.683592	−	0.729864i	$-0.260417\pi$
$797$		0			0		−0.683592	+	0.729864i	$0.739583\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−0.222174	−	0.732410i	−0.222174	−	0.732410i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$809$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$810$		0			0
$811$	0.123862	−	0.192105i	0.123862	−	0.192105i	−0.773010	−	0.634393i	$-0.781250\pi$
$811$	0.123862	−	0.192105i	0.123862	−	0.192105i	0.896873	+	0.442289i	$0.145833\pi$
$812$		0			0
$813$	−0.147386	+	0.129254i	−0.147386	+	0.129254i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.535822	+	0.631541i	0.535822	+	0.631541i
$818$		0			0
$819$	−2.26078	−	2.66464i	−2.26078	−	2.66464i
$820$		0			0
$821$		0			0		−0.857729	−	0.514103i	$-0.828125\pi$
$821$		0			0		0.857729	+	0.514103i	$0.171875\pi$
$822$		0			0
$823$	−0.675641	−	1.21576i	−0.675641	−	1.21576i	−0.965926	−	0.258819i	$-0.916667\pi$
$823$	−0.675641	−	1.21576i	−0.675641	−	1.21576i	0.290285	−	0.956940i	$-0.406250\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.646956	−	0.762527i	$-0.276042\pi$
$827$		0			0		−0.646956	+	0.762527i	$0.723958\pi$
$828$		0			0
$829$	1.40740	+	1.40740i	1.40740	+	1.40740i	0.773010	+	0.634393i	$0.218750\pi$
$829$	1.40740	+	1.40740i	1.40740	+	1.40740i	0.634393	+	0.773010i	$0.281250\pi$
$830$		0			0
$831$	1.68067	+	1.08364i	1.68067	+	1.08364i
$832$	1.26811	−	1.54519i	1.26811	−	1.54519i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.152004	−	1.85381i	0.152004	−	1.85381i
$838$		0			0
$839$		0			0		0.179017	−	0.983846i	$-0.442708\pi$
$839$		0			0		−0.179017	+	0.983846i	$0.557292\pi$
$840$		0			0
$841$	0.0654031	−	0.997859i	0.0654031	−	0.997859i
$842$		0			0
$843$		0			0
$844$	−0.406913	+	1.42494i	−0.406913	+	1.42494i
$845$		0			0
$846$		0			0
$847$	−0.798751	−	1.55503i	−0.798751	−	1.55503i
$848$		0			0
$849$	−1.63439	−	0.773010i	−1.63439	−	0.773010i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.226078	+	1.71723i	−0.226078	+	1.71723i	0.382683	+	0.923880i	$0.375000\pi$
$853$	−0.226078	+	1.71723i	−0.226078	+	1.71723i	−0.608761	+	0.793353i	$0.708333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.998795	−	0.0490677i	$-0.984375\pi$
$857$		0			0		0.998795	+	0.0490677i	$0.0156250\pi$
$858$		0			0
$859$	0.0796298	+	0.103776i	0.0796298	+	0.103776i	0.831470	−	0.555570i	$-0.187500\pi$
$859$	0.0796298	+	0.103776i	0.0796298	+	0.103776i	−0.751840	+	0.659346i	$0.770833\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.485763	−	0.874090i	$-0.338542\pi$
$863$		0			0		−0.485763	+	0.874090i	$0.661458\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$868$	−0.529683	+	3.20824i	−0.529683	+	3.20824i
$869$		0			0
$870$		0			0
$871$	−1.18264	−	0.970567i	−1.18264	−	0.970567i
$872$		0			0
$873$	0.0586897	+	0.0289426i	0.0586897	+	0.0289426i
$874$		0			0
$875$		0			0
$876$	0.194015	+	1.06628i	0.194015	+	1.06628i
$877$	−0.674165	+	1.79121i	−0.674165	+	1.79121i	−0.0654031	+	0.997859i	$0.520833\pi$
$877$	−0.674165	+	1.79121i	−0.674165	+	1.79121i	−0.608761	+	0.793353i	$0.708333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.822268	−	0.569100i	$-0.192708\pi$
$881$		0			0		−0.822268	+	0.569100i	$0.807292\pi$
$882$		0			0
$883$	−1.70657	−	0.674388i	−1.70657	−	0.674388i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−1.70657	−	0.674388i	−1.70657	−	0.674388i	−0.999465	+	0.0327191i	$0.989583\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$887$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$888$		0			0
$889$	−0.450758	−	1.08823i	−0.450758	−	1.08823i
$890$		0			0
$891$		0			0
$892$	0.888626	+	0.832289i	0.888626	+	0.832289i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.634393	−	0.773010i	−0.634393	−	0.773010i
$901$		0			0
$902$		0			0
$903$	0.941614	+	3.29737i	0.941614	+	3.29737i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.593943	−	1.74970i	−0.593943	−	1.74970i	−0.659346	−	0.751840i	$-0.729167\pi$
$907$	−0.593943	−	1.74970i	−0.593943	−	1.74970i	0.0654031	−	0.997859i	$-0.479167\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$911$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$912$	0.392675	−	0.155174i	0.392675	−	0.155174i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.292479	+	0.0239819i	−0.292479	+	0.0239819i
$917$		0			0
$918$		0			0
$919$	−0.388302	+	0.0382444i	−0.388302	+	0.0382444i	−0.290285	−	0.956940i	$-0.593750\pi$
$919$	−0.388302	+	0.0382444i	−0.388302	+	0.0382444i	−0.0980171	+	0.995185i	$0.531250\pi$
$920$		0			0
$921$	−1.09029	+	1.52150i	−1.09029	+	1.52150i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.368805	−	1.29149i	0.368805	−	1.29149i
$926$		0			0
$927$	−0.988683	−	1.64952i	−0.988683	−	1.64952i
$928$		0			0
$929$		0			0		−0.840448	−	0.541892i	$-0.817708\pi$
$929$		0			0		0.840448	+	0.541892i	$0.182292\pi$
$930$		0			0
$931$	0.0992180	+	0.862460i	0.0992180	+	0.862460i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.831470	+	0.444430i	0.831470	+	0.444430i	0.831470	−	0.555570i	$-0.187500\pi$
$937$	0.831470	+	0.444430i	0.831470	+	0.444430i			1.00000i	$0.5\pi$
$938$		0			0
$939$	0.607206	−	1.46593i	0.607206	−	1.46593i
$940$		0			0
$941$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$941$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.0163617	−	0.999866i	$-0.505208\pi$
$947$		0			0		0.0163617	+	0.999866i	$0.494792\pi$
$948$	1.33432	+	1.47219i	1.33432	+	1.47219i
$949$	1.50661	+	1.55674i	1.50661	+	1.55674i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.162895	−	0.986643i	$-0.552083\pi$
$953$		0			0		0.162895	+	0.986643i	$0.447917\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.866439	+	2.30207i	−0.866439	+	2.30207i
$962$		0			0
$963$		0			0
$964$	0.140189	+	0.0840261i	0.140189	+	0.0840261i
$965$		0			0
$966$		0			0
$967$	−0.914211	+	1.32090i	−0.914211	+	1.32090i	0.0327191	+	0.999465i	$0.489583\pi$
$967$	−0.914211	+	1.32090i	−0.914211	+	1.32090i	−0.946930	+	0.321439i	$0.895833\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.179017	−	0.983846i	$-0.557292\pi$
$971$		0			0		0.179017	+	0.983846i	$0.442708\pi$
$972$	−0.999465	+	0.0327191i	−0.999465	+	0.0327191i
$973$	1.54123	+	0.126374i	1.54123	+	0.126374i
$974$		0			0
$975$	−1.91286	−	0.580259i	−1.91286	−	0.580259i
$976$	−0.536800	−	1.87978i	−0.536800	−	1.87978i
$977$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$977$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.54519	+	0.0505844i	1.54519	+	0.0505844i
$982$		0			0
$983$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$983$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.480317	−	0.693990i	0.480317	−	0.693990i
$989$		0			0
$990$		0			0
$991$	0.439467	−	1.29463i	0.439467	−	1.29463i	−0.471397	−	0.881921i	$-0.656250\pi$
$991$	0.439467	−	1.29463i	0.439467	−	1.29463i	0.910864	−	0.412707i	$-0.135417\pi$
$992$		0			0
$993$	−0.0824777	−	1.67887i	−0.0824777	−	1.67887i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.593943	−	0.246019i	0.593943	−	0.246019i	−0.0654031	−	0.997859i	$-0.520833\pi$
$997$	0.593943	−	0.246019i	0.593943	−	0.246019i	0.659346	+	0.751840i	$0.270833\pi$
$998$		0			0
$999$	−0.800094	−	1.07880i	−0.800094	−	1.07880i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3459.1.bm.a.1412.1	✓	64
3.2	odd	2	CM	3459.1.bm.a.1412.1	✓	64
1153.1015	even	192	inner	3459.1.bm.a.2168.1	yes	64
3459.2168	odd	192	inner	3459.1.bm.a.2168.1	yes	64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3459.1.bm.a.1412.1	✓	64	1.1	even	1	trivial
3459.1.bm.a.1412.1	✓	64	3.2	odd	2	CM
3459.1.bm.a.2168.1	yes	64	1153.1015	even	192	inner
3459.1.bm.a.2168.1	yes	64	3459.2168	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)

\(f(q)\)	\(=\)	\(q+(-0.683592 + 0.729864i) q^{3} +(-0.608761 - 0.793353i) q^{4} +(1.72926 - 0.256511i) q^{7} +(-0.0654031 - 0.997859i) q^{9} +O(q^{10})\)	\(q+(-0.683592 + 0.729864i) q^{3} +(-0.608761 - 0.793353i) q^{4} +(1.72926 - 0.256511i) q^{7} +(-0.0654031 - 0.997859i) q^{9} +(0.995185 + 0.0980171i) q^{12} +(1.76290 - 0.942289i) q^{13} +(-0.258819 + 0.965926i) q^{16} +(-0.0755851 + 0.415403i) q^{19} +(-0.994890 + 1.43747i) q^{21} +(-0.0327191 + 0.999465i) q^{25} +(0.773010 + 0.634393i) q^{27} +(-1.25621 - 1.21576i) q^{28} +(-1.05855 - 1.52945i) q^{31} +(-0.751840 + 0.659346i) q^{36} +(-1.30287 - 0.326351i) q^{37} +(-0.517361 + 1.93082i) q^{39} +(1.24441 - 1.51631i) q^{43} +(-0.528068 - 0.849202i) q^{48} +(1.96760 - 0.596865i) q^{49} +(-1.82075 - 0.824972i) q^{52} +(-0.251518 - 0.339133i) q^{57} +(-1.67679 + 1.00503i) q^{61} +(-0.369061 - 1.70878i) q^{63} +(0.923880 - 0.382683i) q^{64} +(-0.292893 - 0.707107i) q^{67} +(0.297595 + 1.04212i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(0.375574 - 0.192915i) q^{76} +(1.47219 + 1.33432i) q^{79} +(-0.991445 + 0.130526i) q^{81} +(1.74607 - 0.0857792i) q^{84} +(2.80680 - 2.08166i) q^{91} +(1.83990 + 0.272924i) q^{93} +(-0.0327191 + 0.0566711i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q+O(q^{10}) \) 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