Embedded newform 72.7.p.b.43.13

• Label: 72.7.p.b.43.13
• Level: $72$
• Weight: $7$
• Character: 72.43
• Analytic conductor: $16.564$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	72.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.5638940206\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(68\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			43.13
Character	\(\chi\)	\(=\)	72.43
Dual form			72.7.p.b.67.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.67611 - 4.40790i) q^{2} +(-5.67293 + 26.3973i) q^{3} +(25.1408 + 58.8553i) q^{4} +(99.0560 - 57.1900i) q^{5} +(154.230 - 151.225i) q^{6} +(231.955 + 133.919i) q^{7} +(91.5860 - 503.742i) q^{8} +(-664.636 - 299.500i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 17 q^{2} + 42 q^{3} + 127 q^{4} - 561 q^{6} + 1774 q^{8} + 654 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(55\)	\(65\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.67611	−	4.40790i	−0.834513	−	0.550988i
							\(3\)	−5.67293	+	26.3973i	−0.210108	+	0.977678i
\(5\)	99.0560	−	57.1900i	0.792448	−	0.457520i								−0.0483757	−	0.998829i	\(-0.515404\pi\)
							0.840824	+	0.541309i	\(0.182071\pi\)
\(7\)	231.955	+	133.919i	0.676254	+	0.390436i	0.798442	−	0.602071i	\(-0.205658\pi\)
							−0.122188	+	0.992507i	\(0.538991\pi\)
\(11\)	447.272	−	774.697i	0.336042	−	0.582042i	−0.647643	−	0.761944i	\(-0.724245\pi\)
							0.983684	+	0.179903i	\(0.0575783\pi\)
\(13\)	200.351	−	115.673i	0.0911929	−	0.0526502i	−0.453710	−	0.891149i	\(-0.649900\pi\)
							0.544903	+	0.838499i	\(0.316566\pi\)
\(17\)	1128.95			0.229787			0.114894	−	0.993378i	\(-0.463347\pi\)
							0.114894	+	0.993378i	\(0.463347\pi\)
\(19\)	2848.33			0.415268			0.207634	−	0.978207i	\(-0.433424\pi\)
							0.207634	+	0.978207i	\(0.433424\pi\)
\(23\)	8667.92	−	5004.43i	0.712413	−	0.411312i	−0.0995412	−	0.995033i	\(-0.531737\pi\)
							0.811954	+	0.583722i	\(0.198404\pi\)
\(29\)	16045.5	+	9263.88i	0.657900	+	0.379839i	0.791476	−	0.611200i	\(-0.209313\pi\)
							−0.133577	+	0.991038i	\(0.542646\pi\)
\(31\)	39271.3	−	22673.3i	1.31823	−	0.761078i	0.334783	−	0.942295i	\(-0.391337\pi\)
							0.983443	+	0.181217i	\(0.0580037\pi\)
\(37\)			13689.1i			0.270253i	0.990828	+	0.135127i	\(0.0431442\pi\)
							−0.990828	+	0.135127i	\(0.956856\pi\)
\(41\)	51581.1	+	89341.1i	0.748409	+	1.29628i	0.948585	+	0.316523i	\(0.102515\pi\)
							−0.200176	+	0.979760i	\(0.564151\pi\)
\(43\)	−68346.1	+	118379.i	−0.859624	+	1.48891i	0.0126640	+	0.999920i	\(0.495969\pi\)
							−0.872288	+	0.488993i	\(0.837365\pi\)
\(47\)	80846.5	+	46676.8i	0.778696	+	0.449580i	0.835968	−	0.548778i	\(-0.184907\pi\)
							−0.0572721	+	0.998359i	\(0.518240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.p.b.43.13	✓	136
8.3	odd	2	inner	72.7.p.b.43.59	yes	136
9.4	even	3	inner	72.7.p.b.67.59	yes	136
72.67	odd	6	inner	72.7.p.b.67.13	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.p.b.43.13	✓	136	1.1	even	1	trivial
72.7.p.b.43.59	yes	136	8.3	odd	2	inner
72.7.p.b.67.13	yes	136	72.67	odd	6	inner
72.7.p.b.67.59	yes	136	9.4	even	3	inner