Embedded newform 403.2.bl.b.295.19

• Label: 403.2.bl.b.295.19
• Level: $403$
• Weight: $2$
• Character: 403.295
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bl (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			295.19
Character	\(\chi\)	\(=\)	403.295
Dual form			403.2.bl.b.250.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.00453535 + 0.0431510i) q^{2} +(1.83864 - 0.818617i) q^{3} +(1.95445 - 0.415432i) q^{4} +0.511564 q^{5} +(0.0436630 + 0.0756265i) q^{6} +(-2.40559 + 0.511325i) q^{7} +(0.0536061 + 0.164983i) q^{8} +(0.703083 - 0.780853i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.00453535	+	0.0431510i	0.00320698	+	0.0305123i	0.996008	−	0.0892600i	\(-0.0284502\pi\)
							−0.992801	+	0.119772i	\(0.961784\pi\)
\(3\)	1.83864	−	0.818617i	1.06154	−	0.472628i	0.199726	−	0.979852i	\(-0.435995\pi\)
							0.861815	+	0.507223i	\(0.169328\pi\)
\(5\)	0.511564			0.228778			0.114389	−	0.993436i	\(-0.463509\pi\)
							0.114389	+	0.993436i	\(0.463509\pi\)
\(7\)	−2.40559	+	0.511325i	−0.909229	+	0.193263i	−0.638706	−	0.769451i	\(-0.720530\pi\)
							−0.270523	+	0.962713i	\(0.587197\pi\)
\(11\)	1.54275	+	1.71340i	0.465157	+	0.516609i	0.929388	−	0.369105i	\(-0.120336\pi\)
							−0.464231	+	0.885714i	\(0.653669\pi\)
\(13\)	3.36984	+	1.28226i	0.934625	+	0.355634i
							\(17\)	2.04233	−	2.26824i	0.495338	−	0.550129i	−0.442696	−	0.896672i	\(-0.645978\pi\)
0.938034	+	0.346543i	\(0.112645\pi\)
\(19\)	−4.61308	−	2.05388i	−1.05831	−	0.471192i	−0.197601	−	0.980283i	\(-0.563315\pi\)
							−0.860713	+	0.509091i	\(0.829982\pi\)
\(23\)	−2.55152	−	0.542343i	−0.532029	−	0.113086i	−0.0659392	−	0.997824i	\(-0.521004\pi\)
							−0.466090	+	0.884737i	\(0.654338\pi\)
\(29\)	−1.01958	−	9.70061i	−0.189330	−	1.80136i	−0.516389	−	0.856354i	\(-0.672724\pi\)
							0.327058	−	0.945004i	\(-0.393943\pi\)
\(31\)	−4.35284	+	3.47171i	−0.781793	+	0.623538i
							\(37\)	−0.852503	+	1.47658i	−0.140151	+	0.242748i	−0.927553	−	0.373691i	\(-0.878092\pi\)
0.787403	+	0.616439i	\(0.211425\pi\)
\(41\)	0.0533698	+	0.507780i	0.00833497	+	0.0793019i	0.997899	−	0.0647944i	\(-0.0206392\pi\)
							−0.989564	+	0.144096i	\(0.953972\pi\)
\(43\)	−0.759999	−	0.338373i	−0.115899	−	0.0516014i	0.347968	−	0.937506i	\(-0.386872\pi\)
							−0.463867	+	0.885905i	\(0.653538\pi\)
\(47\)	0.0827363	+	0.0601114i	0.0120683	+	0.00876815i	0.593803	−	0.804610i	\(-0.297626\pi\)
							−0.581735	+	0.813379i	\(0.697626\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.295.19	yes	280
13.3	even	3	inner	403.2.bl.b.16.17	✓	280
31.2	even	5	inner	403.2.bl.b.126.17	yes	280
403.250	even	15	inner	403.2.bl.b.250.19	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.17	✓	280	13.3	even	3	inner
403.2.bl.b.126.17	yes	280	31.2	even	5	inner
403.2.bl.b.250.19	yes	280	403.250	even	15	inner
403.2.bl.b.295.19	yes	280	1.1	even	1	trivial