Embedded newform 930.2.bt.b.13.9

• Label: 930.2.bt.b.13.9
• Level: $930$
• Weight: $2$
• Character: 930.13
• Analytic conductor: $7.426$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bt (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			13.9
Character	\(\chi\)	\(=\)	930.13
Dual form			930.2.bt.b.787.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(0.358368 - 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(-2.22514 + 0.220752i) q^{5} +(-0.866025 - 0.500000i) q^{6} +(-2.35743 - 0.123548i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 16 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{11}{30}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.156434	−	0.987688i	0.110616	−	0.698401i
							\(3\)	0.358368	−	0.933580i	0.206904	−	0.539003i
\(5\)	−2.22514	+	0.220752i	−0.995115	+	0.0987234i
							\(7\)	−2.35743	−	0.123548i	−0.891024	−	0.0466966i	−0.398673	−	0.917093i	\(-0.630529\pi\)
−0.492351	+	0.870397i	\(0.663862\pi\)
\(11\)	0.827344	+	3.89235i	0.249454	+	1.17359i	0.907316	+	0.420449i	\(0.138128\pi\)
							−0.657862	+	0.753138i	\(0.728539\pi\)
\(13\)	1.94803	+	2.40562i	0.540287	+	0.667199i	0.972046	−	0.234790i	\(-0.0754403\pi\)
							−0.431759	+	0.901989i	\(0.642107\pi\)
\(17\)	−0.0414150	−	0.0637735i	−0.0100446	−	0.0154673i	0.833613	−	0.552350i	\(-0.186269\pi\)
							−0.843657	+	0.536882i	\(0.819602\pi\)
\(19\)	2.01930	−	0.212237i	0.463260	−	0.0486906i	0.129976	−	0.991517i	\(-0.458510\pi\)
							0.333284	+	0.942827i	\(0.391843\pi\)
\(23\)	6.06220	+	3.08884i	1.26406	+	0.644069i	0.952030	−	0.306005i	\(-0.0989924\pi\)
							0.312026	+	0.950074i	\(0.398992\pi\)
\(29\)	−1.68738	+	1.22595i	−0.313338	+	0.227654i	−0.733328	−	0.679875i	\(-0.762034\pi\)
							0.419989	+	0.907529i	\(0.362034\pi\)
\(31\)	−4.86544	−	2.70694i	−0.873858	−	0.486181i
							\(37\)	−0.219035	+	0.0586904i	−0.0360092	+	0.00964863i	−0.276779	−	0.960934i	\(-0.589267\pi\)
0.240769	+	0.970582i	\(0.422600\pi\)
\(41\)	0.145347	+	0.0647128i	0.0226994	+	0.0101064i	0.418055	−	0.908422i	\(-0.362712\pi\)
							−0.395356	+	0.918528i	\(0.629379\pi\)
\(43\)	1.15869	+	0.938287i	0.176698	+	0.143087i	0.713590	−	0.700564i	\(-0.247068\pi\)
							−0.536891	+	0.843651i	\(0.680401\pi\)
\(47\)	1.88075	−	0.297882i	0.274336	−	0.0434506i	−0.0177505	−	0.999842i	\(-0.505650\pi\)
							0.292087	+	0.956392i	\(0.405650\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.b.13.9	yes	256
5.2	odd	4		930.2.bt.a.757.14	yes	256
31.12	odd	30		930.2.bt.a.43.14	✓	256
155.12	even	60	inner	930.2.bt.b.787.9	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.43.14	✓	256	31.12	odd	30
930.2.bt.a.757.14	yes	256	5.2	odd	4
930.2.bt.b.13.9	yes	256	1.1	even	1	trivial
930.2.bt.b.787.9	yes	256	155.12	even	60	inner