Embedded newform 403.2.bp.a.10.1

• Label: 403.2.bp.a.10.1
• Level: $403$
• Weight: $2$
• Character: 403.10
• 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.bp (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			10.1
Character	\(\chi\)	\(=\)	403.10
Dual form			403.2.bp.a.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.76032 - 0.290121i) q^{2} +(-0.0839252 + 0.0609752i) q^{3} +(5.57887 + 1.18583i) q^{4} +(-1.92798 + 1.11312i) q^{5} +(0.249350 - 0.143962i) q^{6} +(-0.0225283 - 0.0202846i) q^{7} +(-9.77606 - 3.17644i) q^{8} +(-0.923726 + 2.84293i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.76032	−	0.290121i	−1.95184	−	0.205146i	−0.955796	−	0.294032i	\(-0.905003\pi\)
							−0.996042	+	0.0888854i	\(0.971669\pi\)
\(3\)	−0.0839252	+	0.0609752i	−0.0484542	+	0.0352041i	−0.611749	−	0.791052i	\(-0.709534\pi\)
							0.563295	+	0.826256i	\(0.309534\pi\)
\(5\)	−1.92798	+	1.11312i	−0.862217	+	0.497801i	−0.864754	−	0.502196i	\(-0.832526\pi\)
							0.00253719	+	0.999997i	\(0.499192\pi\)
\(7\)	−0.0225283	−	0.0202846i	−0.00851490	−	0.00766685i	0.664862	−	0.746966i	\(-0.268490\pi\)
							−0.673377	+	0.739299i	\(0.735157\pi\)
\(11\)	0.954106	−	4.48872i	0.287674	−	1.35340i	−0.562456	−	0.826827i	\(-0.690144\pi\)
							0.850130	−	0.526572i	\(-0.176523\pi\)
\(13\)	−3.06801	−	1.89402i	−0.850914	−	0.525306i
							\(17\)	−1.91322	+	0.406668i	−0.464025	+	0.0986315i	−0.433991	−	0.900917i	\(-0.642895\pi\)
−0.0300340	+	0.999549i	\(0.509562\pi\)
\(19\)	2.23123	+	0.234512i	0.511880	+	0.0538008i	0.356948	−	0.934124i	\(-0.383817\pi\)
							0.154932	+	0.987925i	\(0.450484\pi\)
\(23\)	4.83240	−	1.02716i	1.00763	−	0.214177i	0.325602	−	0.945507i	\(-0.394433\pi\)
							0.682024	+	0.731330i	\(0.261100\pi\)
\(29\)	0.329865	−	3.13845i	0.0612543	−	0.582796i	−0.920247	−	0.391339i	\(-0.872012\pi\)
							0.981501	−	0.191457i	\(-0.0613213\pi\)
\(31\)	5.51004	+	0.799691i	0.989632	+	0.143629i
							\(37\)		−	6.27350i		−	1.03136i	−0.856782	−	0.515679i	\(-0.827540\pi\)
0.856782	−	0.515679i	\(-0.172460\pi\)
\(41\)	−3.07173	−	6.89922i	−0.479724	−	1.07748i	−0.977646	−	0.210256i	\(-0.932570\pi\)
							0.497922	−	0.867222i	\(-0.334096\pi\)
\(43\)	1.26116	−	11.9991i	0.192325	−	1.82985i	−0.293694	−	0.955900i	\(-0.594885\pi\)
							0.486018	−	0.873949i	\(-0.338449\pi\)
\(47\)	1.91807	+	2.64000i	0.279780	+	0.385084i	0.925661	−	0.378354i	\(-0.123510\pi\)
							−0.645881	+	0.763438i	\(0.723510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.10.1	✓	280
13.4	even	6		403.2.bt.a.134.1	yes	280
31.28	even	15		403.2.bt.a.400.1	yes	280
403.121	even	30	inner	403.2.bp.a.121.1	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.10.1	✓	280	1.1	even	1	trivial
403.2.bp.a.121.1	yes	280	403.121	even	30	inner
403.2.bt.a.134.1	yes	280	13.4	even	6
403.2.bt.a.400.1	yes	280	31.28	even	15