Embedded newform 403.2.bz.a.38.17

• Label: 403.2.bz.a.38.17
• Level: $403$
• Weight: $2$
• Character: 403.38
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bz (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			38.17
Character	\(\chi\)	\(=\)	403.38
Dual form			403.2.bz.a.350.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.181578 + 0.0589982i) q^{2} +(2.72617 - 0.579466i) q^{3} +(-1.58854 + 1.15415i) q^{4} +(-2.54353 - 1.46851i) q^{5} +(-0.460825 + 0.266057i) q^{6} +(0.826174 - 1.85562i) q^{7} +(0.444794 - 0.612207i) q^{8} +(4.35561 - 1.93924i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.181578	+	0.0589982i	−0.128395	+	0.0417180i	−0.372510	−	0.928028i	\(-0.621503\pi\)
							0.244115	+	0.969746i	\(0.421503\pi\)
\(3\)	2.72617	−	0.579466i	1.57396	−	0.334555i	0.663506	−	0.748171i	\(-0.269068\pi\)
							0.910451	+	0.413616i	\(0.135734\pi\)
\(5\)	−2.54353	−	1.46851i	−1.13750	−	0.656737i	−0.191692	−	0.981455i	\(-0.561397\pi\)
							−0.945811	+	0.324718i	\(0.894731\pi\)
\(7\)	0.826174	−	1.85562i	0.312264	−	0.701357i	−0.687426	−	0.726255i	\(-0.741259\pi\)
							0.999690	+	0.0248975i	\(0.00792594\pi\)
\(11\)	2.62206	−	0.275589i	0.790580	−	0.0830933i	0.299373	−	0.954136i	\(-0.403222\pi\)
							0.491207	+	0.871043i	\(0.336556\pi\)
\(13\)	3.60520	−	0.0505023i	0.999902	−	0.0140068i
							\(17\)	0.623691	−	5.93402i	0.151267	−	1.43921i	−0.610836	−	0.791757i	\(-0.709167\pi\)
0.762103	−	0.647455i	\(-0.224167\pi\)
\(19\)	1.75638	+	1.58145i	0.402942	+	0.362810i	0.845542	−	0.533909i	\(-0.179277\pi\)
							−0.442601	+	0.896719i	\(0.645944\pi\)
\(23\)	0.691154	+	0.502153i	0.144116	+	0.104706i	0.657507	−	0.753448i	\(-0.271611\pi\)
							−0.513391	+	0.858155i	\(0.671611\pi\)
\(29\)	−1.29318	−	3.98001i	−0.240138	−	0.739069i	−0.996398	−	0.0847979i	\(-0.972976\pi\)
							0.756260	−	0.654271i	\(-0.227024\pi\)
\(31\)	−4.60611	+	3.12790i	−0.827282	+	0.561787i
							\(37\)	−7.80326	+	4.50522i	−1.28285	+	0.740653i	−0.977368	−	0.211546i	\(-0.932150\pi\)
−0.305480	+	0.952198i	\(0.598817\pi\)
\(41\)	−2.04040	+	9.59931i	−0.318656	+	1.49916i	0.469084	+	0.883153i	\(0.344584\pi\)
							−0.787741	+	0.616007i	\(0.788749\pi\)
\(43\)	−3.06929	+	3.40879i	−0.468062	+	0.519836i	−0.930240	−	0.366951i	\(-0.880402\pi\)
							0.462178	+	0.886787i	\(0.347068\pi\)
\(47\)	1.83600	+	0.596551i	0.267808	+	0.0870160i	0.439843	−	0.898075i	\(-0.355034\pi\)
							−0.172035	+	0.985091i	\(0.555034\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bz.a.38.17	✓	288
13.12	even	2	inner	403.2.bz.a.38.20	yes	288
31.9	even	15	inner	403.2.bz.a.350.20	yes	288
403.350	even	30	inner	403.2.bz.a.350.17	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bz.a.38.17	✓	288	1.1	even	1	trivial
403.2.bz.a.38.20	yes	288	13.12	even	2	inner
403.2.bz.a.350.17	yes	288	403.350	even	30	inner
403.2.bz.a.350.20	yes	288	31.9	even	15	inner