Embedded newform 148.2.l.b.119.11

• Label: 148.2.l.b.119.11
• Level: $148$
• Weight: $2$
• Character: 148.119
• Analytic conductor: $1.182$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 148 = 2^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	148.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.18178594991\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			119.11
Character	\(\chi\)	\(=\)	148.119
Dual form			148.2.l.b.51.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.380601 + 1.36204i) q^{2} +(0.309523 - 0.536109i) q^{3} +(-1.71029 + 1.03678i) q^{4} +(1.97369 - 0.528848i) q^{5} +(0.848004 + 0.217538i) q^{6} +(1.80362 + 1.04132i) q^{7} +(-2.06307 - 1.93487i) q^{8} +(1.30839 + 2.26620i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 4 q^{2} - 4 q^{5} + 8 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/148\mathbb{Z}\right)^\times\).

\(n\)	\(75\)	\(113\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.380601	+	1.36204i	0.269125	+	0.963105i
							\(3\)	0.309523	−	0.536109i	0.178703	−	0.309523i	−0.762734	−	0.646713i	\(-0.776143\pi\)
0.941437	+	0.337190i	\(0.109477\pi\)
\(5\)	1.97369	−	0.528848i	0.882661	−	0.236508i	0.211106	−	0.977463i	\(-0.432293\pi\)
							0.671555	+	0.740955i	\(0.265627\pi\)
\(7\)	1.80362	+	1.04132i	0.681705	+	0.393583i	0.800497	−	0.599337i	\(-0.204569\pi\)
							−0.118792	+	0.992919i	\(0.537902\pi\)
\(11\)	−2.26068			−0.681622			−0.340811	−	0.940132i	\(-0.610702\pi\)
							−0.340811	+	0.940132i	\(0.610702\pi\)
\(13\)	−3.17107	+	0.849686i	−0.879497	+	0.235660i	−0.670190	−	0.742190i	\(-0.733787\pi\)
							−0.209307	+	0.977850i	\(0.567121\pi\)
\(17\)	0.734384	−	2.74076i	0.178114	−	0.664732i	−0.817886	−	0.575380i	\(-0.804854\pi\)
							0.996000	−	0.0893512i	\(-0.0284793\pi\)
\(19\)	−1.28294	−	4.78801i	−0.294327	−	1.09845i	−0.941750	−	0.336314i	\(-0.890820\pi\)
							0.647423	−	0.762131i	\(-0.275847\pi\)
\(23\)	3.38949	−	3.38949i	0.706758	−	0.706758i	−0.259094	−	0.965852i	\(-0.583424\pi\)
							0.965852	+	0.259094i	\(0.0834241\pi\)
\(29\)	−3.76318	+	3.76318i	−0.698805	+	0.698805i	−0.964153	−	0.265347i	\(-0.914513\pi\)
							0.265347	+	0.964153i	\(0.414513\pi\)
\(31\)	−3.58199	−	3.58199i	−0.643345	−	0.643345i	0.308031	−	0.951376i	\(-0.400330\pi\)
							−0.951376	+	0.308031i	\(0.900330\pi\)
\(37\)	0.759653	−	6.03514i	0.124886	−	0.992171i
							\(41\)	4.04772	+	2.33695i	0.632147	+	0.364970i	0.781583	−	0.623801i	\(-0.214412\pi\)
−0.149436	+	0.988771i	\(0.547746\pi\)
\(43\)	−7.40701	+	7.40701i	−1.12956	+	1.12956i	−0.139310	+	0.990249i	\(0.544488\pi\)
							−0.990249	+	0.139310i	\(0.955512\pi\)
\(47\)		−	0.911328i		−	0.132931i	−0.997789	−	0.0664654i	\(-0.978828\pi\)
							0.997789	−	0.0664654i	\(-0.0211722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	148.2.l.b.119.11	yes	64
4.3	odd	2	inner	148.2.l.b.119.13	yes	64
37.14	odd	12	inner	148.2.l.b.51.13	yes	64
148.51	even	12	inner	148.2.l.b.51.11	✓	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
148.2.l.b.51.11	✓	64	148.51	even	12	inner
148.2.l.b.51.13	yes	64	37.14	odd	12	inner
148.2.l.b.119.11	yes	64	1.1	even	1	trivial
148.2.l.b.119.13	yes	64	4.3	odd	2	inner