Embedded newform 392.2.z.a.53.14

• Label: 392.2.z.a.53.14
• Level: $392$
• Weight: $2$
• Character: 392.53
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			53.14
Character	\(\chi\)	\(=\)	392.53
Dual form			392.2.z.a.37.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05324 - 0.943763i) q^{2} +(-1.08411 + 1.16839i) q^{3} +(0.218622 + 1.98802i) q^{4} +(0.529378 + 1.71620i) q^{5} +(2.24451 - 0.207452i) q^{6} +(-2.34752 - 1.22032i) q^{7} +(1.64596 - 2.30018i) q^{8} +(0.0343439 + 0.458288i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05324	−	0.943763i	−0.744752	−	0.667341i
							\(3\)	−1.08411	+	1.16839i	−0.625910	+	0.674571i	−0.963803	−	0.266614i	\(-0.914095\pi\)
0.337893	+	0.941184i	\(0.390286\pi\)
\(5\)	0.529378	+	1.71620i	0.236745	+	0.767509i	0.993502	+	0.113813i	\(0.0363063\pi\)
							−0.756757	+	0.653696i	\(0.773217\pi\)
\(7\)	−2.34752	−	1.22032i	−0.887277	−	0.461236i
							\(11\)	4.71137	+	0.353068i	1.42053	+	0.106454i	0.762720	−	0.646729i	\(-0.223863\pi\)
0.657811	+	0.753183i	\(0.271482\pi\)
\(13\)	0.558298	+	1.15932i	0.154844	+	0.321537i	0.963931	−	0.266152i	\(-0.0857523\pi\)
							−0.809087	+	0.587689i	\(0.800038\pi\)
\(17\)	−3.60762	+	0.543762i	−0.874977	+	0.131882i	−0.571150	−	0.820846i	\(-0.693503\pi\)
							−0.303827	+	0.952727i	\(0.598265\pi\)
\(19\)	−4.74129	+	2.73738i	−1.08773	+	0.627999i	−0.932970	−	0.359954i	\(-0.882793\pi\)
							−0.154756	+	0.987953i	\(0.549459\pi\)
\(23\)	−7.55176	−	1.13824i	−1.57465	−	0.237340i	−0.697154	−	0.716922i	\(-0.745550\pi\)
							−0.877497	+	0.479581i	\(0.840789\pi\)
\(29\)	−6.72404	+	5.36224i	−1.24862	+	0.995744i	−0.248994	+	0.968505i	\(0.580100\pi\)
							−0.999629	+	0.0272387i	\(0.991329\pi\)
\(31\)	−0.368485	+	0.638234i	−0.0661818	+	0.114630i	−0.897218	−	0.441589i	\(-0.854415\pi\)
							0.831036	+	0.556219i	\(0.187748\pi\)
\(37\)	6.57534	+	2.58063i	1.08098	+	0.424253i	0.837940	−	0.545763i	\(-0.183760\pi\)
							0.243040	+	0.970016i	\(0.421855\pi\)
\(41\)	−1.11229	+	4.87325i	−0.173710	+	0.761074i	0.810740	+	0.585407i	\(0.199065\pi\)
							−0.984450	+	0.175667i	\(0.943792\pi\)
\(43\)	0.167094	−	0.0381381i	0.0254816	−	0.00581601i	−0.209761	−	0.977753i	\(-0.567269\pi\)
							0.235243	+	0.971937i	\(0.424411\pi\)
\(47\)	−8.06065	−	5.49566i	−1.17577	−	0.801624i	−0.192056	−	0.981384i	\(-0.561516\pi\)
							−0.983711	+	0.179760i	\(0.942468\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.53.14	yes	648
8.5	even	2	inner	392.2.z.a.53.8	yes	648
49.37	even	21	inner	392.2.z.a.37.8	✓	648
392.37	even	42	inner	392.2.z.a.37.14	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.37.8	✓	648	49.37	even	21	inner
392.2.z.a.37.14	yes	648	392.37	even	42	inner
392.2.z.a.53.8	yes	648	8.5	even	2	inner
392.2.z.a.53.14	yes	648	1.1	even	1	trivial