Embedded newform 930.2.bt.a.13.15

• Label: 930.2.bt.a.13.15
• 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			13.15
Character	\(\chi\)	\(=\)	930.13
Dual form			930.2.bt.a.787.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(-0.358368 + 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(1.45393 - 1.69884i) q^{5} +(0.866025 + 0.500000i) q^{6} +(3.84003 + 0.201247i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 8 q^{7}+O(q^{10}) \)

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
\(5\)	1.45393	−	1.69884i	0.650219	−	0.759746i
							\(7\)	3.84003	+	0.201247i	1.45140	+	0.0760644i	0.761650	−	0.647988i	\(-0.224389\pi\)
0.689745	+	0.724053i	\(0.257723\pi\)
\(11\)	−0.346890	−	1.63199i	−0.104591	−	0.492064i	−0.998996	−	0.0448046i	\(-0.985733\pi\)
							0.894404	−	0.447259i	\(-0.147600\pi\)
\(13\)	−3.27565	−	4.04509i	−0.908502	−	1.12191i	−0.992105	−	0.125408i	\(-0.959976\pi\)
							0.0836028	−	0.996499i	\(-0.473357\pi\)
\(17\)	1.55582	+	2.39575i	0.377341	+	0.581055i	0.975412	−	0.220388i	\(-0.0707325\pi\)
							−0.598071	+	0.801443i	\(0.704066\pi\)
\(19\)	4.31184	−	0.453193i	0.989204	−	0.103969i	0.403917	−	0.914796i	\(-0.367649\pi\)
							0.585287	+	0.810826i	\(0.300982\pi\)
\(23\)	−0.726128	−	0.369981i	−0.151408	−	0.0771464i	0.376645	−	0.926358i	\(-0.377078\pi\)
							−0.528053	+	0.849211i	\(0.677078\pi\)
\(29\)	−5.91927	+	4.30060i	−1.09918	+	0.798602i	−0.980926	−	0.194381i	\(-0.937730\pi\)
							−0.118256	+	0.992983i	\(0.537730\pi\)
\(31\)	5.45718	−	1.10419i	0.980138	−	0.198318i
							\(37\)	4.29500	−	1.15084i	0.706094	−	0.189197i	0.112135	−	0.993693i	\(-0.464231\pi\)
0.593959	+	0.804496i	\(0.297564\pi\)
\(41\)	4.12794	+	1.83788i	0.644676	+	0.287028i	0.702914	−	0.711274i	\(-0.251882\pi\)
							−0.0582384	+	0.998303i	\(0.518548\pi\)
\(43\)	−6.80139	−	5.50766i	−1.03720	−	0.839910i	−0.0498987	−	0.998754i	\(-0.515890\pi\)
							−0.987304	+	0.158844i	\(0.949223\pi\)
\(47\)	1.05187	−	0.166600i	0.153431	−	0.0243012i	−0.0792461	−	0.996855i	\(-0.525251\pi\)
							0.232678	+	0.972554i	\(0.425251\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.a.13.15	✓	256
5.2	odd	4		930.2.bt.b.757.10	yes	256
31.12	odd	30		930.2.bt.b.43.10	yes	256
155.12	even	60	inner	930.2.bt.a.787.15	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.13.15	✓	256	1.1	even	1	trivial
930.2.bt.a.787.15	yes	256	155.12	even	60	inner
930.2.bt.b.43.10	yes	256	31.12	odd	30
930.2.bt.b.757.10	yes	256	5.2	odd	4