Embedded newform 115.4.l.a.102.24

• Label: 115.4.l.a.102.24
• 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			102.24
Character	\(\chi\)	\(=\)	115.102
Dual form			115.4.l.a.53.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.147738 - 2.06565i) q^{2} +(5.23771 - 1.13939i) q^{3} +(3.67351 + 0.528171i) q^{4} +(4.29113 + 10.3241i) q^{5} +(-1.57978 - 10.9876i) q^{6} +(3.96906 - 2.16727i) q^{7} +(5.15538 - 23.6989i) q^{8} +(1.57536 - 0.719443i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 18 q^{2} - 26 q^{3} - 22 q^{5} - 76 q^{6} - 22 q^{7} - 50 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.147738	−	2.06565i	0.0522332	−	0.730316i	−0.902092	−	0.431545i	\(-0.857969\pi\)
							0.954325	−	0.298771i	\(-0.0965767\pi\)
\(3\)	5.23771	−	1.13939i	1.00800	−	0.219277i	0.321891	−	0.946777i	\(-0.395682\pi\)
							0.686108	+	0.727500i	\(0.259318\pi\)
\(5\)	4.29113	+	10.3241i	0.383811	+	0.923412i
							\(7\)	3.96906	−	2.16727i	0.214309	−	0.117022i	−0.368509	−	0.929624i	\(-0.620132\pi\)
0.582818	+	0.812602i	\(0.301950\pi\)
\(11\)	42.7289	+	37.0248i	1.17120	+	1.01485i	0.999559	+	0.0296944i	\(0.00945340\pi\)
							0.171644	+	0.985159i	\(0.445092\pi\)
\(13\)	−21.9447	+	40.1887i	−0.468182	+	0.857411i	0.531763	+	0.846893i	\(0.321530\pi\)
							−0.999945	+	0.0105180i	\(0.996652\pi\)
\(17\)	−90.8184	−	67.9858i	−1.29569	−	0.969940i	−0.999892	−	0.0146888i	\(-0.995324\pi\)
							−0.295796	−	0.955251i	\(-0.595585\pi\)
\(19\)	16.5939	−	115.413i	0.200364	−	1.39356i	−0.602843	−	0.797860i	\(-0.705966\pi\)
							0.803207	−	0.595700i	\(-0.203125\pi\)
\(23\)	20.3555	−	108.410i	0.184540	−	0.982825i
							\(29\)	−67.4393	+	9.69631i	−0.431834	+	0.0620883i	−0.354804	−	0.934941i	\(-0.615453\pi\)
−0.0770291	+	0.997029i	\(0.524543\pi\)
\(31\)	88.1264	−	56.6354i	0.510580	−	0.328130i	−0.259856	−	0.965647i	\(-0.583675\pi\)
							0.770436	+	0.637518i	\(0.220039\pi\)
\(37\)	−231.464	−	86.3316i	−1.02844	−	0.383590i	−0.222093	−	0.975026i	\(-0.571289\pi\)
							−0.806352	+	0.591435i	\(0.798561\pi\)
\(41\)	15.1490	−	33.1717i	0.0577044	−	0.126355i	−0.878583	−	0.477589i	\(-0.841511\pi\)
							0.936288	+	0.351234i	\(0.114238\pi\)
\(43\)	31.9600	+	146.918i	0.113346	+	0.521041i	0.998450	+	0.0556494i	\(0.0177229\pi\)
							−0.885105	+	0.465392i	\(0.845913\pi\)
\(47\)	−68.7986	−	68.7986i	−0.213517	−	0.213517i	0.592243	−	0.805760i	\(-0.298243\pi\)
							−0.805760	+	0.592243i	\(0.798243\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.l.a.102.24	yes	680
5.3	odd	4	inner	115.4.l.a.33.11	yes	680
23.7	odd	22	inner	115.4.l.a.7.11	✓	680
115.53	even	44	inner	115.4.l.a.53.24	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.l.a.7.11	✓	680	23.7	odd	22	inner
115.4.l.a.33.11	yes	680	5.3	odd	4	inner
115.4.l.a.53.24	yes	680	115.53	even	44	inner
115.4.l.a.102.24	yes	680	1.1	even	1	trivial