Embedded newform 403.2.bk.a.9.2

• Label: 403.2.bk.a.9.2
• Level: $403$
• Weight: $2$
• Character: 403.9
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	403.9
Dual form			403.2.bk.a.224.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65225 - 1.83501i) q^{2} +(2.14947 - 2.38723i) q^{3} +(-0.428272 + 4.07474i) q^{4} +(-1.22323 - 2.11870i) q^{5} -7.93205 q^{6} +(-1.37387 - 0.998173i) q^{7} +(4.18946 - 3.04382i) q^{8} +(-0.765054 - 7.27901i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65225	−	1.83501i	−1.16832	−	1.29755i	−0.946591	−	0.322438i	\(-0.895498\pi\)
							−0.221725	−	0.975109i	\(-0.571169\pi\)
\(3\)	2.14947	−	2.38723i	1.24100	−	1.37827i	0.342365	−	0.939567i	\(-0.388772\pi\)
							0.898633	−	0.438701i	\(-0.144561\pi\)
\(5\)	−1.22323	−	2.11870i	−0.547045	−	0.947510i	−0.998475	−	0.0552038i	\(-0.982419\pi\)
							0.451430	−	0.892307i	\(-0.350914\pi\)
\(7\)	−1.37387	−	0.998173i	−0.519273	−	0.377274i	0.297057	−	0.954860i	\(-0.403995\pi\)
							−0.816330	+	0.577586i	\(0.803995\pi\)
\(11\)	−0.826280	−	0.600327i	−0.249133	−	0.181005i	0.456210	−	0.889872i	\(-0.349207\pi\)
							−0.705342	+	0.708867i	\(0.749207\pi\)
\(13\)	3.54432	+	0.661685i	0.983016	+	0.183519i
							\(17\)	3.04780	−	2.21436i	0.739201	−	0.537061i	−0.153260	−	0.988186i	\(-0.548977\pi\)
0.892461	+	0.451125i	\(0.148977\pi\)
\(19\)	−2.30340	+	7.08915i	−0.528437	+	1.62636i	0.228980	+	0.973431i	\(0.426461\pi\)
							−0.757417	+	0.652931i	\(0.773539\pi\)
\(23\)	0.0153473	+	0.146019i	0.00320013	+	0.0304472i	0.996005	−	0.0892926i	\(-0.0284606\pi\)
							−0.992805	+	0.119740i	\(0.961794\pi\)
\(29\)	2.36732	+	2.62917i	0.439599	+	0.488225i	0.921707	−	0.387888i	\(-0.126795\pi\)
							−0.482107	+	0.876112i	\(0.660128\pi\)
\(31\)	−3.50615	−	4.32515i	−0.629722	−	0.776820i
							\(37\)	3.40632	+	5.89992i	0.559996	+	0.969941i	0.997496	+	0.0707231i	\(0.0225307\pi\)
−0.437500	+	0.899218i	\(0.644136\pi\)
\(41\)	2.36067	−	7.26539i	0.368674	−	1.13466i	−0.578973	−	0.815346i	\(-0.696547\pi\)
							0.947648	−	0.319317i	\(-0.103453\pi\)
\(43\)	−0.776602	+	2.39013i	−0.118431	+	0.364492i	−0.992647	−	0.121044i	\(-0.961376\pi\)
							0.874216	+	0.485536i	\(0.161376\pi\)
\(47\)	3.85676	+	11.8699i	0.562566	+	1.73140i	0.675075	+	0.737749i	\(0.264111\pi\)
							−0.112510	+	0.993651i	\(0.535889\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bk.a.9.2	yes	280
13.3	even	3		403.2.bj.a.133.34	yes	280
31.7	even	15		403.2.bj.a.100.34	✓	280
403.224	even	15	inner	403.2.bk.a.224.2	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.34	✓	280	31.7	even	15
403.2.bj.a.133.34	yes	280	13.3	even	3
403.2.bk.a.9.2	yes	280	1.1	even	1	trivial
403.2.bk.a.224.2	yes	280	403.224	even	15	inner