Embedded newform 115.4.l.a.102.29

• Label: 115.4.l.a.102.29
• Level: $115$
• Weight: $4$
• Character: 115.102
• Analytic conductor: $6.785$
• Analytic rank: $0$
• Dimension: $680$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.l (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.78521965066\)
Analytic rank:	\(0\)
Dimension:	\(680\)
Relative dimension:	\(34\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			102.29
Character	\(\chi\)	\(=\)	115.102
Dual form			115.4.l.a.53.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.298608 - 4.17509i) q^{2} +(3.03712 - 0.660684i) q^{3} +(-9.42365 - 1.35492i) q^{4} +(-11.1630 - 0.622798i) q^{5} +(-1.85151 - 12.8775i) q^{6} +(-12.7837 + 6.98043i) q^{7} +(-1.35291 + 6.21924i) q^{8} +(-15.7725 + 7.20306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 18 q^{2} - 26 q^{3} - 22 q^{5} - 76 q^{6} - 22 q^{7} - 50 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.298608	−	4.17509i	0.105574	−	1.47612i	−0.618167	−	0.786047i	\(-0.712125\pi\)
							0.723741	−	0.690071i	\(-0.242421\pi\)
\(3\)	3.03712	−	0.660684i	0.584493	−	0.127149i	0.0894109	−	0.995995i	\(-0.471502\pi\)
							0.495082	+	0.868846i	\(0.335138\pi\)
\(5\)	−11.1630	−	0.622798i	−0.998447	−	0.0557047i
							\(7\)	−12.7837	+	6.98043i	−0.690255	+	0.376908i	−0.785769	−	0.618520i	\(-0.787732\pi\)
0.0955139	+	0.995428i	\(0.469551\pi\)
\(11\)	−20.6924	−	17.9301i	−0.567182	−	0.491466i	0.323416	−	0.946257i	\(-0.395169\pi\)
							−0.890598	+	0.454791i	\(0.849714\pi\)
\(13\)	4.74551	−	8.69076i	0.101244	−	0.185414i	−0.822147	−	0.569275i	\(-0.807224\pi\)
							0.923391	+	0.383861i	\(0.125406\pi\)
\(17\)	−24.1781	−	18.0995i	−0.344944	−	0.258222i	0.412732	−	0.910852i	\(-0.364574\pi\)
							−0.757676	+	0.652631i	\(0.773665\pi\)
\(19\)	16.8117	−	116.928i	0.202993	−	1.41184i	−0.592348	−	0.805682i	\(-0.701799\pi\)
							0.795341	−	0.606162i	\(-0.207292\pi\)
\(23\)	110.261	−	3.09784i	0.999606	−	0.0280845i
							\(29\)	251.814	−	36.2053i	1.61244	−	0.231833i	0.723653	−	0.690164i	\(-0.242461\pi\)
0.888782	+	0.458330i	\(0.151552\pi\)
\(31\)	−75.0715	+	48.2455i	−0.434943	+	0.279521i	−0.739735	−	0.672898i	\(-0.765049\pi\)
							0.304792	+	0.952419i	\(0.401413\pi\)
\(37\)	−300.203	−	111.970i	−1.33387	−	0.497506i	−0.421481	−	0.906837i	\(-0.638489\pi\)
							−0.912386	+	0.409331i	\(0.865762\pi\)
\(41\)	122.519	−	268.278i	0.466688	−	1.02190i	−0.519224	−	0.854638i	\(-0.673779\pi\)
							0.985912	−	0.167266i	\(-0.0534937\pi\)
\(43\)	4.25897	+	19.5782i	0.0151044	+	0.0694336i	0.984082	−	0.177714i	\(-0.0568703\pi\)
							−0.968978	+	0.247148i	\(0.920507\pi\)
\(47\)	−82.1440	−	82.1440i	−0.254935	−	0.254935i	0.568056	−	0.822990i	\(-0.307696\pi\)
							−0.822990	+	0.568056i	\(0.807696\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.l.a.102.29	yes	680
5.3	odd	4	inner	115.4.l.a.33.6	yes	680
23.7	odd	22	inner	115.4.l.a.7.6	✓	680
115.53	even	44	inner	115.4.l.a.53.29	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.l.a.7.6	✓	680	23.7	odd	22	inner
115.4.l.a.33.6	yes	680	5.3	odd	4	inner
115.4.l.a.53.29	yes	680	115.53	even	44	inner
115.4.l.a.102.29	yes	680	1.1	even	1	trivial