Embedded newform 403.2.bf.a.37.8

• Label: 403.2.bf.a.37.8
• Level: $403$
• Weight: $2$
• Character: 403.37
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(35\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.8
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.504814 + 1.88399i) q^{2} +(-2.36532 - 1.36562i) q^{3} +(-1.56254 - 0.902134i) q^{4} +(-3.95125 + 1.05873i) q^{5} +(3.76686 - 3.76686i) q^{6} +(-1.17143 + 1.17143i) q^{7} +(-0.269949 + 0.269949i) q^{8} +(2.22981 + 3.86215i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.504814	+	1.88399i	−0.356958	+	1.33218i	0.521045	+	0.853529i	\(0.325542\pi\)
							−0.878003	+	0.478655i	\(0.841124\pi\)
\(3\)	−2.36532	−	1.36562i	−1.36562	−	0.788438i	−0.375251	−	0.926923i	\(-0.622443\pi\)
							−0.990365	+	0.138485i	\(0.955777\pi\)
\(5\)	−3.95125	+	1.05873i	−1.76705	+	0.473480i	−0.988127	−	0.153639i	\(-0.950901\pi\)
							−0.778924	+	0.627119i	\(0.784234\pi\)
\(7\)	−1.17143	+	1.17143i	−0.442759	+	0.442759i	−0.892938	−	0.450179i	\(-0.851360\pi\)
							0.450179	+	0.892938i	\(0.351360\pi\)
\(11\)	−2.48397	−	2.48397i	−0.748945	−	0.748945i	0.225336	−	0.974281i	\(-0.427652\pi\)
							−0.974281	+	0.225336i	\(0.927652\pi\)
\(13\)	3.46047	+	1.01248i	0.959763	+	0.280812i
							\(17\)	1.39093			0.337349			0.168675	−	0.985672i	\(-0.446051\pi\)
0.168675	+	0.985672i	\(0.446051\pi\)
\(19\)	−0.170067	−	0.170067i	−0.0390159	−	0.0390159i	0.687330	−	0.726346i	\(-0.258783\pi\)
							−0.726346	+	0.687330i	\(0.758783\pi\)
\(23\)	0.998214	+	1.72896i	0.208142	+	0.360512i	0.951129	−	0.308793i	\(-0.0999251\pi\)
							−0.742987	+	0.669306i	\(0.766592\pi\)
\(29\)	2.34447	−	1.35358i	0.435356	−	0.251353i	−0.266270	−	0.963899i	\(-0.585791\pi\)
							0.701626	+	0.712545i	\(0.252458\pi\)
\(31\)	−5.48389	+	0.962761i	−0.984936	+	0.172917i
							\(37\)	9.76687	−	2.61702i	1.60566	−	0.430236i	0.658916	−	0.752216i	\(-0.271015\pi\)
0.946746	+	0.321980i	\(0.104348\pi\)
\(41\)	−4.10192	−	4.10192i	−0.640612	−	0.640612i	0.310094	−	0.950706i	\(-0.399640\pi\)
							−0.950706	+	0.310094i	\(0.899640\pi\)
\(43\)	−1.38685			−0.211493			−0.105747	−	0.994393i	\(-0.533723\pi\)
							−0.105747	+	0.994393i	\(0.533723\pi\)
\(47\)	−4.90418	+	4.90418i	−0.715348	+	0.715348i	−0.967649	−	0.252301i	\(-0.918813\pi\)
							0.252301	+	0.967649i	\(0.418813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bf.a.37.8	yes	140
13.6	odd	12		403.2.ba.a.6.8	✓	140
31.26	odd	6		403.2.ba.a.336.8	yes	140
403.305	even	12	inner	403.2.bf.a.305.8	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.8	✓	140	13.6	odd	12
403.2.ba.a.336.8	yes	140	31.26	odd	6
403.2.bf.a.37.8	yes	140	1.1	even	1	trivial
403.2.bf.a.305.8	yes	140	403.305	even	12	inner