Embedded newform 403.2.bi.a.40.3

• Label: 403.2.bi.a.40.3
• Level: $403$
• Weight: $2$
• Character: 403.40
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bi (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			40.3
Character	\(\chi\)	\(=\)	403.40
Dual form			403.2.bi.a.131.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.608805 + 1.87371i) q^{2} +(-1.89076 - 0.401893i) q^{3} +(-1.52211 - 1.10588i) q^{4} +(0.912174 + 1.57993i) q^{5} +(1.90413 - 3.29806i) q^{6} +(-0.756252 + 0.336705i) q^{7} +(-0.188985 + 0.137305i) q^{8} +(0.672810 + 0.299554i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.608805	+	1.87371i	−0.430490	+	1.32491i	0.467148	+	0.884179i	\(0.345282\pi\)
							−0.897638	+	0.440733i	\(0.854718\pi\)
\(3\)	−1.89076	−	0.401893i	−1.09163	−	0.232033i	−0.373282	−	0.927718i	\(-0.621768\pi\)
							−0.718347	+	0.695685i	\(0.755101\pi\)
\(5\)	0.912174	+	1.57993i	0.407937	+	0.706567i	0.994658	−	0.103222i	\(-0.0329151\pi\)
							−0.586722	+	0.809789i	\(0.699582\pi\)
\(7\)	−0.756252	+	0.336705i	−0.285836	+	0.127263i	−0.544647	−	0.838666i	\(-0.683336\pi\)
							0.258810	+	0.965928i	\(0.416670\pi\)
\(11\)	−0.313423	+	2.98202i	−0.0945005	+	0.899112i	0.839865	+	0.542796i	\(0.182634\pi\)
							−0.934365	+	0.356317i	\(0.884032\pi\)
\(13\)	−0.669131	+	0.743145i	−0.185583	+	0.206111i
							\(17\)	−0.326993	−	3.11113i	−0.0793075	−	0.754561i	−0.959835	−	0.280565i	\(-0.909478\pi\)
0.880527	−	0.473995i	\(-0.157189\pi\)
\(19\)	−2.54997	−	2.83203i	−0.585003	−	0.649711i	0.375879	−	0.926669i	\(-0.377341\pi\)
							−0.960882	+	0.276957i	\(0.910674\pi\)
\(23\)	−6.69190	+	4.86195i	−1.39536	+	1.01379i	−0.400104	+	0.916470i	\(0.631026\pi\)
							−0.995253	+	0.0973173i	\(0.968974\pi\)
\(29\)	2.00453	−	6.16932i	0.372233	−	1.14561i	−0.573094	−	0.819490i	\(-0.694257\pi\)
							0.945327	−	0.326125i	\(-0.105743\pi\)
\(31\)	−4.00978	−	3.86286i	−0.720177	−	0.693790i
							\(37\)	1.34944	−	2.33730i	0.221847	−	0.384250i	−0.733522	−	0.679666i	\(-0.762125\pi\)
0.955369	+	0.295416i	\(0.0954581\pi\)
\(41\)	3.58703	−	0.762446i	0.560199	−	0.119074i	0.0808948	−	0.996723i	\(-0.474222\pi\)
							0.479305	+	0.877649i	\(0.340889\pi\)
\(43\)	−2.12836	−	2.36379i	−0.324572	−	0.360474i	0.558671	−	0.829390i	\(-0.311312\pi\)
							−0.883243	+	0.468915i	\(0.844645\pi\)
\(47\)	3.06569	+	9.43522i	0.447176	+	1.37627i	0.880079	+	0.474827i	\(0.157489\pi\)
							−0.432903	+	0.901441i	\(0.642511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.40.3	✓	120
31.7	even	15	inner	403.2.bi.a.131.3	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.40.3	✓	120	1.1	even	1	trivial
403.2.bi.a.131.3	yes	120	31.7	even	15	inner