Embedded newform 128.3.l.a.3.7

• Label: 128.3.l.a.3.7
• Level: $128$
• Weight: $3$
• Character: 128.3
• Analytic conductor: $3.488$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.48774738381\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(31\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57051 - 1.23834i) q^{2} +(-3.06977 + 2.51930i) q^{3} +(0.933028 + 3.88966i) q^{4} +(-2.31969 - 0.703669i) q^{5} +(7.94087 - 0.155169i) q^{6} +(0.668264 + 3.35959i) q^{7} +(3.35139 - 7.26417i) q^{8} +(1.32084 - 6.64029i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.57051	−	1.23834i	−0.785257	−	0.619170i
							\(3\)	−3.06977	+	2.51930i	−1.02326	+	0.839766i	−0.987185	−	0.159583i	\(-0.948985\pi\)
−0.0360737	+	0.999349i	\(0.511485\pi\)
\(5\)	−2.31969	−	0.703669i	−0.463937	−	0.140734i	0.0496583	−	0.998766i	\(-0.484187\pi\)
							−0.513595	+	0.858032i	\(0.671687\pi\)
\(7\)	0.668264	+	3.35959i	0.0954663	+	0.479941i	0.998708	+	0.0508100i	\(0.0161803\pi\)
							−0.903242	+	0.429131i	\(0.858820\pi\)
\(11\)	−0.756988	−	7.68583i	−0.0688171	−	0.698712i	−0.966258	−	0.257577i	\(-0.917076\pi\)
							0.897441	−	0.441135i	\(-0.145424\pi\)
\(13\)	−5.37205	−	17.7093i	−0.413235	−	1.36225i	−0.878779	−	0.477228i	\(-0.841642\pi\)
							0.465545	−	0.885024i	\(-0.345858\pi\)
\(17\)	7.14813	−	17.2571i	0.420479	−	1.01512i	−0.561728	−	0.827322i	\(-0.689863\pi\)
							0.982207	−	0.187803i	\(-0.0601367\pi\)
\(19\)	−5.16252	+	9.65839i	−0.271712	+	0.508337i	−0.980358	−	0.197227i	\(-0.936806\pi\)
							0.708646	+	0.705564i	\(0.249306\pi\)
\(23\)	9.35088	−	6.24806i	0.406560	−	0.271655i	−0.335431	−	0.942065i	\(-0.608882\pi\)
							0.741991	+	0.670410i	\(0.233882\pi\)
\(29\)	3.05110	−	30.9783i	0.105210	−	1.06822i	−0.787109	−	0.616814i	\(-0.788423\pi\)
							0.892320	−	0.451404i	\(-0.149077\pi\)
\(31\)	31.4307	+	31.4307i	1.01389	+	1.01389i	0.999902	+	0.0139910i	\(0.00445361\pi\)
							0.0139910	+	0.999902i	\(0.495546\pi\)
\(37\)	−1.11698	−	2.08972i	−0.0301886	−	0.0564788i	0.866375	−	0.499394i	\(-0.166444\pi\)
							−0.896563	+	0.442916i	\(0.853944\pi\)
\(41\)	−21.7739	+	14.5488i	−0.531070	+	0.354849i	−0.792013	−	0.610505i	\(-0.790967\pi\)
							0.260943	+	0.965354i	\(0.415967\pi\)
\(43\)	−24.7111	−	20.2799i	−0.574677	−	0.471625i	0.301599	−	0.953435i	\(-0.402480\pi\)
							−0.876276	+	0.481810i	\(0.839980\pi\)
\(47\)	10.7217	−	25.8845i	0.228122	−	0.550734i	−0.767827	−	0.640657i	\(-0.778662\pi\)
							0.995949	+	0.0899228i	\(0.0286620\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.7	✓	496
128.43	odd	32	inner	128.3.l.a.43.7	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.7	✓	496	1.1	even	1	trivial
128.3.l.a.43.7	yes	496	128.43	odd	32	inner