Embedded newform 403.2.bz.a.38.12

• Label: 403.2.bz.a.38.12
• 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			38.12
Character	\(\chi\)	\(=\)	403.38
Dual form			403.2.bz.a.350.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08301 + 0.351891i) q^{2} +(-0.146913 + 0.0312274i) q^{3} +(-0.568950 + 0.413366i) q^{4} +(1.21706 + 0.702668i) q^{5} +(0.148120 - 0.0855171i) q^{6} +(1.11912 - 2.51358i) q^{7} +(1.80939 - 2.49042i) q^{8} +(-2.72003 + 1.21103i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08301	+	0.351891i	−0.765804	+	0.248825i	−0.665768	−	0.746159i	\(-0.731896\pi\)
							−0.100036	+	0.994984i	\(0.531896\pi\)
\(3\)	−0.146913	+	0.0312274i	−0.0848204	+	0.0180291i	−0.250126	−	0.968213i	\(-0.580472\pi\)
							0.165306	+	0.986242i	\(0.447139\pi\)
\(5\)	1.21706	+	0.702668i	0.544284	+	0.314243i	0.746814	−	0.665034i	\(-0.231583\pi\)
							−0.202529	+	0.979276i	\(0.564916\pi\)
\(7\)	1.11912	−	2.51358i	0.422988	−	0.950046i	−0.568839	−	0.822449i	\(-0.692607\pi\)
							0.991826	−	0.127596i	\(-0.0407262\pi\)
\(11\)	1.01360	−	0.106534i	0.305612	−	0.0321212i	0.0495189	−	0.998773i	\(-0.484231\pi\)
							0.256093	+	0.966652i	\(0.417565\pi\)
\(13\)	3.57005	+	0.504734i	0.990153	+	0.139988i
							\(17\)	0.511256	−	4.86428i	0.123998	−	1.17976i	−0.738698	−	0.674037i	\(-0.764559\pi\)
0.862696	−	0.505723i	\(-0.168774\pi\)
\(19\)	0.358914	+	0.323167i	0.0823404	+	0.0741397i	0.709273	−	0.704934i	\(-0.249023\pi\)
							−0.626933	+	0.779073i	\(0.715690\pi\)
\(23\)	6.02878	+	4.38016i	1.25709	+	0.913327i	0.998611	−	0.0526905i	\(-0.0167797\pi\)
							0.258476	+	0.966018i	\(0.416780\pi\)
\(29\)	2.53288	+	7.79541i	0.470344	+	1.44757i	0.852135	+	0.523322i	\(0.175307\pi\)
							−0.381791	+	0.924249i	\(0.624693\pi\)
\(31\)	1.89395	+	5.23574i	0.340163	+	0.940366i
							\(37\)	6.99133	−	4.03644i	1.14937	−	0.663587i	0.200634	−	0.979666i	\(-0.435700\pi\)
0.948733	+	0.316079i	\(0.102366\pi\)
\(41\)	1.63763	−	7.70443i	0.255754	−	1.20323i	−0.643376	−	0.765550i	\(-0.722467\pi\)
							0.899131	−	0.437680i	\(-0.144200\pi\)
\(43\)	0.590351	−	0.655651i	0.0900277	−	0.0999859i	−0.696448	−	0.717608i	\(-0.745237\pi\)
							0.786475	+	0.617622i	\(0.211904\pi\)
\(47\)	7.04552	+	2.28923i	1.02769	+	0.333918i	0.773878	−	0.633334i	\(-0.218314\pi\)
							0.253816	+	0.967252i	\(0.418314\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bz.a.38.12	✓	288
13.12	even	2	inner	403.2.bz.a.38.25	yes	288
31.9	even	15	inner	403.2.bz.a.350.25	yes	288
403.350	even	30	inner	403.2.bz.a.350.12	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bz.a.38.12	✓	288	1.1	even	1	trivial
403.2.bz.a.38.25	yes	288	13.12	even	2	inner
403.2.bz.a.350.12	yes	288	403.350	even	30	inner
403.2.bz.a.350.25	yes	288	31.9	even	15	inner