Embedded newform 76.7.l.a.23.8

• Label: 76.7.l.a.23.8
• Level: $76$
• Weight: $7$
• Character: 76.23
• Analytic conductor: $17.484$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	76.l (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.4841103551\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(58\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			23.8
Character	\(\chi\)	\(=\)	76.23
Dual form			76.7.l.a.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.25479 - 3.37166i) q^{2} +(-16.9114 + 2.98194i) q^{3} +(41.2638 + 48.9213i) q^{4} +(113.326 - 95.0915i) q^{5} +(132.743 + 35.3862i) q^{6} +(331.831 + 191.583i) q^{7} +(-134.414 - 494.041i) q^{8} +(-407.932 + 148.475i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 6 q^{2} + 192 q^{4} - 12 q^{5} - 492 q^{6} - 3 q^{8} + 1368 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.25479	−	3.37166i	−0.906848	−	0.421457i
							\(3\)	−16.9114	+	2.98194i	−0.626348	+	0.110442i	−0.477808	−	0.878464i	\(-0.658569\pi\)
−0.148540	+	0.988906i	\(0.547457\pi\)
\(5\)	113.326	−	95.0915i	0.906606	−	0.760732i	−0.0648647	−	0.997894i	\(-0.520662\pi\)
							0.971470	+	0.237162i	\(0.0762171\pi\)
\(7\)	331.831	+	191.583i	0.967438	+	0.558550i	0.898454	−	0.439067i	\(-0.144691\pi\)
							0.0689836	+	0.997618i	\(0.478024\pi\)
\(11\)	−681.944	+	393.721i	−0.512355	+	0.295808i	−0.733801	−	0.679364i	\(-0.762256\pi\)
							0.221446	+	0.975173i	\(0.428922\pi\)
\(13\)	120.952	−	685.953i	0.0550533	−	0.312223i	−0.944829	−	0.327564i	\(-0.893772\pi\)
							0.999882	+	0.0153413i	\(0.00488347\pi\)
\(17\)	−378.553	−	137.782i	−0.0770512	−	0.0280444i	0.303207	−	0.952925i	\(-0.401943\pi\)
							−0.380258	+	0.924880i	\(0.624165\pi\)
\(19\)	2302.39	+	6461.03i	0.335675	+	0.941978i
							\(23\)	4994.51	−	5952.22i	0.410496	−	0.489210i	−0.520694	−	0.853743i	\(-0.674327\pi\)
0.931191	+	0.364533i	\(0.118771\pi\)
\(29\)	121.599	−	44.2584i	0.00498581	−	0.00181469i	−0.339526	−	0.940597i	\(-0.610267\pi\)
							0.344512	+	0.938782i	\(0.388044\pi\)
\(31\)	33066.8	+	19091.1i	1.10996	+	0.640835i	0.938819	−	0.344412i	\(-0.111922\pi\)
							0.171140	+	0.985247i	\(0.445255\pi\)
\(37\)	26780.5			0.528705			0.264353	−	0.964426i	\(-0.414842\pi\)
							0.264353	+	0.964426i	\(0.414842\pi\)
\(41\)	−5385.57	−	30543.1i	−0.0781412	−	0.443161i	−0.998627	−	0.0523846i	\(-0.983318\pi\)
							0.920486	−	0.390776i	\(-0.127793\pi\)
\(43\)	75606.2	+	90104.0i	0.950938	+	1.13328i	0.990970	+	0.134085i	\(0.0428095\pi\)
							−0.0400322	+	0.999198i	\(0.512746\pi\)
\(47\)	51926.4	+	142667.i	0.500144	+	1.37413i	0.891135	+	0.453739i	\(0.149910\pi\)
							−0.390991	+	0.920394i	\(0.627868\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.7.l.a.23.8	yes	348
4.3	odd	2	inner	76.7.l.a.23.5	✓	348
19.5	even	9	inner	76.7.l.a.43.5	yes	348
76.43	odd	18	inner	76.7.l.a.43.8	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.7.l.a.23.5	✓	348	4.3	odd	2	inner
76.7.l.a.23.8	yes	348	1.1	even	1	trivial
76.7.l.a.43.5	yes	348	19.5	even	9	inner
76.7.l.a.43.8	yes	348	76.43	odd	18	inner