Embedded newform 403.2.bg.a.123.13

• Label: 403.2.bg.a.123.13
• Level: $403$
• Weight: $2$
• Character: 403.123
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			123.13
Character	\(\chi\)	\(=\)	403.123
Dual form			403.2.bg.a.154.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.248585 - 0.927732i) q^{2} +(-0.327048 - 0.188821i) q^{3} +(0.933159 - 0.538760i) q^{4} +(1.52097 - 1.52097i) q^{5} +(-0.0938763 + 0.350351i) q^{6} +(-0.204081 + 0.761641i) q^{7} +(-2.09009 - 2.09009i) q^{8} +(-1.42869 - 2.47457i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 8 q^{2} - 12 q^{4} - 8 q^{5} - 16 q^{7} - 28 q^{8} + 64 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.248585	−	0.927732i	−0.175776	−	0.656005i	−0.996418	−	0.0845628i	\(-0.973051\pi\)
							0.820642	−	0.571443i	\(-0.193616\pi\)
\(3\)	−0.327048	−	0.188821i	−0.188821	−	0.109016i	0.402609	−	0.915372i	\(-0.368103\pi\)
							−0.591431	+	0.806356i	\(0.701437\pi\)
\(5\)	1.52097	−	1.52097i	0.680198	−	0.680198i	−0.279846	−	0.960045i	\(-0.590284\pi\)
							0.960045	+	0.279846i	\(0.0902835\pi\)
\(7\)	−0.204081	+	0.761641i	−0.0771354	+	0.287873i	−0.993709	−	0.111993i	\(-0.964277\pi\)
							0.916574	+	0.399866i	\(0.130943\pi\)
\(11\)	0.412635	+	1.53998i	0.124414	+	0.464320i	0.999818	−	0.0190720i	\(-0.00607117\pi\)
							−0.875404	+	0.483392i	\(0.839405\pi\)
\(13\)	−1.50765	−	3.27521i	−0.418146	−	0.908380i
							\(17\)	2.13726	+	3.70183i	0.518361	+	0.897827i	0.999772	+	0.0213323i	\(0.00679081\pi\)
−0.481412	+	0.876495i	\(0.659876\pi\)
\(19\)	3.47654	+	0.931535i	0.797572	+	0.213709i	0.634518	−	0.772908i	\(-0.281199\pi\)
							0.163054	+	0.986617i	\(0.447865\pi\)
\(23\)	2.14965	−	3.72330i	0.448233	−	0.776362i	−0.550038	−	0.835139i	\(-0.685387\pi\)
							0.998271	+	0.0587775i	\(0.0187203\pi\)
\(29\)	−6.10960	−	3.52738i	−1.13452	−	0.655018i	−0.189455	−	0.981889i	\(-0.560672\pi\)
							−0.945069	+	0.326872i	\(0.894005\pi\)
\(31\)	−3.15031	−	4.59081i	−0.565813	−	0.824534i
							\(37\)	2.08282	+	7.77320i	0.342414	+	1.27791i	0.895604	+	0.444852i	\(0.146744\pi\)
−0.553190	+	0.833055i	\(0.686590\pi\)
\(41\)	−2.13610	−	7.97202i	−0.333602	−	1.24502i	−0.905377	−	0.424610i	\(-0.860411\pi\)
							0.571774	−	0.820411i	\(-0.306255\pi\)
\(43\)	3.48830	+	6.04191i	0.531960	+	0.921382i	0.999304	+	0.0373066i	\(0.0118778\pi\)
							−0.467343	+	0.884076i	\(0.654789\pi\)
\(47\)	1.55895	+	1.55895i	0.227396	+	0.227396i	0.811604	−	0.584208i	\(-0.198595\pi\)
							−0.584208	+	0.811604i	\(0.698595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bg.a.123.13	✓	144
13.11	odd	12	inner	403.2.bg.a.154.14	yes	144
31.30	odd	2	inner	403.2.bg.a.123.14	yes	144
403.154	even	12	inner	403.2.bg.a.154.13	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bg.a.123.13	✓	144	1.1	even	1	trivial
403.2.bg.a.123.14	yes	144	31.30	odd	2	inner
403.2.bg.a.154.13	yes	144	403.154	even	12	inner
403.2.bg.a.154.14	yes	144	13.11	odd	12	inner