Embedded newform 95.11.m.a.7.19

• Label: 95.11.m.a.7.19
• Level: $95$
• Weight: $11$
• Character: 95.7
• Analytic conductor: $60.359$
• Analytic rank: $0$
• Dimension: $392$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	95.m (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(60.3589390040\)
Analytic rank:	\(0\)
Dimension:	\(392\)
Relative dimension:	\(98\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			7.19
Character	\(\chi\)	\(=\)	95.7
Dual form			95.11.m.a.68.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.5704 + 43.1812i) q^{2} +(28.4153 + 7.61384i) q^{3} +(-843.934 - 487.246i) q^{4} +(81.7868 + 3123.93i) q^{5} +(-657.550 + 1138.91i) q^{6} +(-7638.87 - 7638.87i) q^{7} +(-1565.02 + 1565.02i) q^{8} +(-50388.5 - 29091.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.5704	+	43.1812i	−0.361574	+	1.34941i	0.510432	+	0.859918i	\(0.329486\pi\)
							−0.872006	+	0.489495i	\(0.837181\pi\)
\(3\)	28.4153	+	7.61384i	0.116935	+	0.0313327i	0.316812	−	0.948488i	\(-0.397388\pi\)
							−0.199877	+	0.979821i	\(0.564054\pi\)
\(5\)	81.7868	+	3123.93i	0.0261718	+	0.999657i
							\(7\)	−7638.87	−	7638.87i	−0.454505	−	0.454505i	0.442341	−	0.896847i	\(-0.354148\pi\)
−0.896847	+	0.442341i	\(0.854148\pi\)
\(11\)	−189178.			−1.17464			−0.587322	−	0.809353i	\(-0.699818\pi\)
							−0.587322	+	0.809353i	\(0.699818\pi\)
\(13\)	178832.	+	667409.i	0.481646	+	1.79753i	0.594710	+	0.803940i	\(0.297267\pi\)
							−0.113064	+	0.993588i	\(0.536067\pi\)
\(17\)	−25365.9	+	94666.9i	−0.0178651	+	0.0666736i	−0.974283	−	0.225328i	\(-0.927655\pi\)
							0.956418	+	0.292002i	\(0.0943213\pi\)
\(19\)	−2.41134e6	+	562579.i	−0.973847	+	0.227204i
							\(23\)	2.61776e6	+	9.76961e6i	0.406715	+	1.51788i	0.800870	+	0.598838i	\(0.204371\pi\)
−0.394155	+	0.919044i	\(0.628963\pi\)
\(29\)	−5.72165e6	−	3.30340e6i	−0.278953	−	0.161054i	0.353996	−	0.935247i	\(-0.384823\pi\)
							−0.632949	+	0.774193i	\(0.718156\pi\)
\(31\)	3.80661e7			1.32963			0.664813	−	0.747010i	\(-0.268511\pi\)
							0.664813	+	0.747010i	\(0.268511\pi\)
\(37\)	−5.94190e7	−	5.94190e7i	−0.856873	−	0.856873i	0.134095	−	0.990968i	\(-0.457187\pi\)
							−0.990968	+	0.134095i	\(0.957187\pi\)
\(41\)	−9.06135e7	−	1.56947e8i	−0.782120	−	1.35467i	−0.930704	−	0.365772i	\(-0.880805\pi\)
							0.148584	−	0.988900i	\(-0.452528\pi\)
\(43\)	1.08603e8	+	2.91000e7i	0.738752	+	0.197948i	0.608523	−	0.793536i	\(-0.291762\pi\)
							0.130229	+	0.991484i	\(0.458429\pi\)
\(47\)	1.14438e8	−	3.06636e7i	0.498977	−	0.133701i	−0.000547927	−	1.00000i	\(-0.500174\pi\)
							0.499525	+	0.866299i	\(0.333508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.m.a.7.19	✓	392
5.3	odd	4	inner	95.11.m.a.83.80	yes	392
19.11	even	3	inner	95.11.m.a.87.80	yes	392
95.68	odd	12	inner	95.11.m.a.68.19	yes	392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.m.a.7.19	✓	392	1.1	even	1	trivial
95.11.m.a.68.19	yes	392	95.68	odd	12	inner
95.11.m.a.83.80	yes	392	5.3	odd	4	inner
95.11.m.a.87.80	yes	392	19.11	even	3	inner