Embedded newform 32.9.h.a.11.24

• Label: 32.9.h.a.11.24
• Level: $32$
• Weight: $9$
• Character: 32.11
• Analytic conductor: $13.036$
• Analytic rank: $0$
• Dimension: $124$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	32.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0361155220\)
Analytic rank:	\(0\)
Dimension:	\(124\)
Relative dimension:	\(31\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			11.24
Character	\(\chi\)	\(=\)	32.11
Dual form			32.9.h.a.3.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.6371 + 10.9808i) q^{2} +(8.35809 + 20.1782i) q^{3} +(14.8420 + 255.569i) q^{4} +(377.364 - 911.037i) q^{5} +(-124.310 + 326.594i) q^{6} +(2934.15 - 2934.15i) q^{7} +(-2633.65 + 3137.05i) q^{8} +(4302.02 - 4302.02i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 124 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.6371	+	10.9808i	0.727316	+	0.686303i
							\(3\)	8.35809	+	20.1782i	0.103186	+	0.249114i	0.967038	−	0.254633i	\(-0.0819546\pi\)
−0.863852	+	0.503746i	\(0.831955\pi\)
\(5\)	377.364	−	911.037i	0.603782	−	1.45766i	−0.265877	−	0.964007i	\(-0.585662\pi\)
							0.869660	−	0.493652i	\(-0.164338\pi\)
\(7\)	2934.15	−	2934.15i	1.22205	−	1.22205i	0.255153	−	0.966901i	\(-0.417874\pi\)
							0.966901	−	0.255153i	\(-0.0821258\pi\)
\(11\)	−5784.68	+	13965.5i	−0.395102	+	0.953860i	0.593708	+	0.804680i	\(0.297663\pi\)
							−0.988810	+	0.149180i	\(0.952337\pi\)
\(13\)	8908.24	+	21506.4i	0.311902	+	0.752999i	0.999635	+	0.0270344i	\(0.00860637\pi\)
							−0.687732	+	0.725965i	\(0.741394\pi\)
\(17\)		−	79306.8i		−	0.949544i	−0.880109	−	0.474772i	\(-0.842531\pi\)
							0.880109	−	0.474772i	\(-0.157469\pi\)
\(19\)	−87164.8	+	36104.8i	−0.668847	+	0.277045i	−0.691156	−	0.722706i	\(-0.742898\pi\)
							0.0223092	+	0.999751i	\(0.492898\pi\)
\(23\)	234451.	+	234451.i	0.837799	+	0.837799i	0.988569	−	0.150770i	\(-0.0481751\pi\)
							−0.150770	+	0.988569i	\(0.548175\pi\)
\(29\)	811062.	−	335953.i	1.14673	−	0.474992i	0.273296	−	0.961930i	\(-0.411886\pi\)
							0.873437	+	0.486938i	\(0.161886\pi\)
\(31\)		−	355528.i		−	0.384970i	−0.981300	−	0.192485i	\(-0.938345\pi\)
							0.981300	−	0.192485i	\(-0.0616547\pi\)
\(37\)	−353180.	+	852651.i	−0.188447	+	0.454951i	−0.989661	−	0.143427i	\(-0.954188\pi\)
							0.801214	+	0.598378i	\(0.204188\pi\)
\(41\)	−3.33542e6	+	3.33542e6i	−1.18036	+	1.18036i	−0.200710	+	0.979651i	\(0.564325\pi\)
							−0.979651	+	0.200710i	\(0.935675\pi\)
\(43\)	54215.5	−	130888.i	0.0158580	−	0.0382847i	−0.915752	−	0.401744i	\(-0.868404\pi\)
							0.931610	+	0.363459i	\(0.118404\pi\)
\(47\)	−4.22704e6			−0.866253			−0.433127	−	0.901333i	\(-0.642590\pi\)
							−0.433127	+	0.901333i	\(0.642590\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.h.a.11.24	yes	124
32.3	odd	8	inner	32.9.h.a.3.24	✓	124

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.9.h.a.3.24	✓	124	32.3	odd	8	inner
32.9.h.a.11.24	yes	124	1.1	even	1	trivial