Embedded newform 304.2.bg.a.3.9

• Label: 304.2.bg.a.3.9
• Level: $304$
• Weight: $2$
• Character: 304.3
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $456$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(456\)
Relative dimension:	\(38\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09186 - 0.898798i) q^{2} +(0.945637 + 0.440958i) q^{3} +(0.384323 + 1.96273i) q^{4} +(-1.51828 - 2.16833i) q^{5} +(-0.636173 - 1.33140i) q^{6} +(-0.308642 - 0.534584i) q^{7} +(1.34447 - 2.48845i) q^{8} +(-1.22858 - 1.46416i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09186	−	0.898798i	−0.772063	−	0.635546i
							\(3\)	0.945637	+	0.440958i	0.545964	+	0.254587i	0.675974	−	0.736925i	\(-0.263723\pi\)
−0.130010	+	0.991513i	\(0.541501\pi\)
\(5\)	−1.51828	−	2.16833i	−0.678995	−	0.969705i	−0.999746	−	0.0225417i	\(-0.992824\pi\)
							0.320751	−	0.947164i	\(-0.396065\pi\)
\(7\)	−0.308642	−	0.534584i	−0.116656	−	0.202054i	0.801785	−	0.597613i	\(-0.203884\pi\)
							−0.918440	+	0.395559i	\(0.870551\pi\)
\(11\)	−4.38023	+	1.17368i	−1.32069	+	0.353877i	−0.849236	−	0.528014i	\(-0.822937\pi\)
							−0.471453	+	0.881891i	\(0.656270\pi\)
\(13\)	4.83785	−	2.25593i	1.34178	−	0.625682i	0.386747	−	0.922186i	\(-0.373599\pi\)
							0.955032	+	0.296504i	\(0.0958207\pi\)
\(17\)	−2.70398	−	2.26891i	−0.655811	−	0.550291i	0.253017	−	0.967462i	\(-0.418577\pi\)
							−0.908828	+	0.417171i	\(0.863022\pi\)
\(19\)	0.727146	−	4.29782i	0.166819	−	0.985988i
							\(23\)	−0.160689	−	0.911310i	−0.0335059	−	0.190021i	0.963461	−	0.267849i	\(-0.0863128\pi\)
−0.996967	+	0.0778275i	\(0.975202\pi\)
\(29\)	0.268345	−	3.06720i	0.0498304	−	0.569564i	−0.929707	−	0.368301i	\(-0.879940\pi\)
							0.979537	−	0.201263i	\(-0.0645047\pi\)
\(31\)	2.64312	+	4.57802i	0.474718	+	0.822236i	0.999581	−	0.0289509i	\(-0.00921666\pi\)
							−0.524863	+	0.851187i	\(0.675883\pi\)
\(37\)	−1.03739	−	1.03739i	−0.170546	−	0.170546i	0.616673	−	0.787219i	\(-0.288480\pi\)
							−0.787219	+	0.616673i	\(0.788480\pi\)
\(41\)	−11.6696	+	4.24739i	−1.82249	+	0.663331i	−0.827722	+	0.561138i	\(0.810364\pi\)
							−0.994765	+	0.102193i	\(0.967414\pi\)
\(43\)	−3.16742	+	2.21785i	−0.483026	+	0.338219i	−0.789593	−	0.613630i	\(-0.789709\pi\)
							0.306567	+	0.951849i	\(0.400820\pi\)
\(47\)	5.75111	+	6.85390i	0.838885	+	0.999744i	0.999918	+	0.0127954i	\(0.00407302\pi\)
							−0.161033	+	0.986949i	\(0.551483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.9	✓	456
16.11	odd	4	inner	304.2.bg.a.155.29	yes	456
19.13	odd	18	inner	304.2.bg.a.51.29	yes	456
304.203	even	36	inner	304.2.bg.a.203.9	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.9	✓	456	1.1	even	1	trivial
304.2.bg.a.51.29	yes	456	19.13	odd	18	inner
304.2.bg.a.155.29	yes	456	16.11	odd	4	inner
304.2.bg.a.203.9	yes	456	304.203	even	36	inner