Embedded newform 115.4.l.a.102.32

• Label: 115.4.l.a.102.32
• 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			102.32
Character	\(\chi\)	\(=\)	115.102
Dual form			115.4.l.a.53.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.371284 - 5.19124i) q^{2} +(-8.92216 + 1.94090i) q^{3} +(-18.8925 - 2.71633i) q^{4} +(-8.89093 - 6.77874i) q^{5} +(6.76299 + 47.0376i) q^{6} +(8.66048 - 4.72898i) q^{7} +(-12.2652 + 56.3824i) q^{8} +(51.2777 - 23.4178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 18 q^{2} - 26 q^{3} - 22 q^{5} - 76 q^{6} - 22 q^{7} - 50 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.371284	−	5.19124i	0.131269	−	1.83538i	−0.325207	−	0.945643i	\(-0.605434\pi\)
							0.456476	−	0.889736i	\(-0.349112\pi\)
\(3\)	−8.92216	+	1.94090i	−1.71707	+	0.373526i	−0.960042	−	0.279857i	\(-0.909713\pi\)
							−0.757028	+	0.653382i	\(0.773349\pi\)
\(5\)	−8.89093	−	6.77874i	−0.795229	−	0.606309i
							\(7\)	8.66048	−	4.72898i	0.467622	−	0.255341i	−0.228119	−	0.973633i	\(-0.573258\pi\)
0.695742	+	0.718292i	\(0.255076\pi\)
\(11\)	−37.0225	−	32.0802i	−1.01479	−	0.879321i	−0.0220694	−	0.999756i	\(-0.507025\pi\)
							−0.992721	+	0.120435i	\(0.961571\pi\)
\(13\)	−1.59878	+	2.92795i	−0.0341094	+	0.0624667i	−0.894178	−	0.447712i	\(-0.852239\pi\)
							0.860068	+	0.510179i	\(0.170421\pi\)
\(17\)	53.9561	+	40.3910i	0.769781	+	0.576251i	0.910359	−	0.413819i	\(-0.135805\pi\)
							−0.140578	+	0.990070i	\(0.544896\pi\)
\(19\)	9.07100	−	63.0902i	0.109528	−	0.761784i	−0.858838	−	0.512248i	\(-0.828813\pi\)
							0.968366	−	0.249536i	\(-0.0802780\pi\)
\(23\)	2.23227	+	110.282i	0.0202374	+	0.999795i
							\(29\)	−284.031	+	40.8374i	−1.81873	+	0.261494i	−0.965568	−	0.260152i	\(-0.916227\pi\)
−0.853162	+	0.521646i	\(0.825318\pi\)
\(31\)	58.5463	−	37.6254i	0.339201	−	0.217991i	−0.359939	−	0.932976i	\(-0.617203\pi\)
							0.699140	+	0.714985i	\(0.253566\pi\)
\(37\)	347.295	+	129.534i	1.54311	+	0.575549i	0.969823	−	0.243809i	\(-0.0783969\pi\)
							0.573283	+	0.819358i	\(0.305670\pi\)
\(41\)	20.3768	−	44.6189i	0.0776175	−	0.169959i	−0.866845	−	0.498578i	\(-0.833856\pi\)
							0.944463	+	0.328619i	\(0.106583\pi\)
\(43\)	11.1703	+	51.3489i	0.0396151	+	0.182108i	0.992703	−	0.120587i	\(-0.0384775\pi\)
							−0.953088	+	0.302694i	\(0.902114\pi\)
\(47\)	−446.522	−	446.522i	−1.38579	−	1.38579i	−0.833958	−	0.551827i	\(-0.813931\pi\)
							−0.551827	−	0.833958i	\(-0.686069\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.l.a.102.32	yes	680
5.3	odd	4	inner	115.4.l.a.33.3	yes	680
23.7	odd	22	inner	115.4.l.a.7.3	✓	680
115.53	even	44	inner	115.4.l.a.53.32	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.l.a.7.3	✓	680	23.7	odd	22	inner
115.4.l.a.33.3	yes	680	5.3	odd	4	inner
115.4.l.a.53.32	yes	680	115.53	even	44	inner
115.4.l.a.102.32	yes	680	1.1	even	1	trivial