Embedded newform 403.2.be.c.57.7

• Label: 403.2.be.c.57.7
• Level: $403$
• Weight: $2$
• Character: 403.57
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			57.7
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31740 + 1.31740i) q^{2} +(-1.87836 - 1.08447i) q^{3} -1.47108i q^{4} +(1.82236 + 0.488300i) q^{5} +(3.90323 - 1.04587i) q^{6} +(-2.35057 + 0.629834i) q^{7} +(-0.696799 - 0.696799i) q^{8} +(0.852156 + 1.47598i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31740	+	1.31740i	−0.931542	+	0.931542i	−0.997802	−	0.0662606i	\(-0.978893\pi\)
							0.0662606	+	0.997802i	\(0.478893\pi\)
\(3\)	−1.87836	−	1.08447i	−1.08447	−	0.626120i	−0.152372	−	0.988323i	\(-0.548691\pi\)
							−0.932099	+	0.362203i	\(0.882025\pi\)
\(5\)	1.82236	+	0.488300i	0.814984	+	0.218374i	0.642152	−	0.766577i	\(-0.278042\pi\)
							0.172832	+	0.984951i	\(0.444708\pi\)
\(7\)	−2.35057	+	0.629834i	−0.888433	+	0.238055i	−0.674042	−	0.738693i	\(-0.735443\pi\)
							−0.214391	+	0.976748i	\(0.568777\pi\)
\(11\)	4.63048	+	1.24073i	1.39614	+	0.374095i	0.876957	−	0.480568i	\(-0.159569\pi\)
							0.519183	+	0.854663i	\(0.326236\pi\)
\(13\)	−0.00574928	−	3.60555i	−0.00159456	−	0.999999i
							\(17\)	−2.20150	+	3.81311i	−0.533943	+	0.924816i	0.465271	+	0.885168i	\(0.345957\pi\)
−0.999214	+	0.0396474i	\(0.987377\pi\)
\(19\)	1.85539	+	6.92442i	0.425656	+	1.58857i	0.762485	+	0.647006i	\(0.223979\pi\)
							−0.336829	+	0.941566i	\(0.609354\pi\)
\(23\)	−3.30950			−0.690079			−0.345039	−	0.938588i	\(-0.612134\pi\)
							−0.345039	+	0.938588i	\(0.612134\pi\)
\(29\)		−	1.17176i		−	0.217590i	−0.994064	−	0.108795i	\(-0.965301\pi\)
							0.994064	−	0.108795i	\(-0.0346992\pi\)
\(31\)	−3.73105	+	4.13271i	−0.670116	+	0.742256i
							\(37\)	0.886205	+	3.30736i	0.145691	+	0.543727i	0.999724	+	0.0235072i	\(0.00748328\pi\)
−0.854032	+	0.520220i	\(0.825850\pi\)
\(41\)	−3.69603	−	0.990349i	−0.577224	−	0.154667i	−0.0416161	−	0.999134i	\(-0.513251\pi\)
							−0.535607	+	0.844467i	\(0.679917\pi\)
\(43\)	−6.09655	+	10.5595i	−0.929716	+	1.61031i	−0.145919	+	0.989297i	\(0.546614\pi\)
							−0.783796	+	0.621018i	\(0.786719\pi\)
\(47\)	2.08119	+	2.08119i	0.303573	+	0.303573i	0.842410	−	0.538837i	\(-0.181136\pi\)
							−0.538837	+	0.842410i	\(0.681136\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.7	✓	136
13.8	odd	4	inner	403.2.be.c.398.7	yes	136
31.6	odd	6	inner	403.2.be.c.161.7	yes	136
403.99	even	12	inner	403.2.be.c.99.7	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.7	✓	136	1.1	even	1	trivial
403.2.be.c.99.7	yes	136	403.99	even	12	inner
403.2.be.c.161.7	yes	136	31.6	odd	6	inner
403.2.be.c.398.7	yes	136	13.8	odd	4	inner