Embedded newform 115.3.h.a.11.14

• Label: 115.3.h.a.11.14
• Level: $115$
• Weight: $3$
• Character: 115.11
• Analytic conductor: $3.134$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	115.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13352304014\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.14
Character	\(\chi\)	\(=\)	115.11
Dual form			115.3.h.a.21.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.28753 - 1.47011i) q^{2} +(0.630718 - 4.38674i) q^{3} +(1.40992 - 3.08729i) q^{4} +(0.629973 + 2.14549i) q^{5} +(-5.00618 - 10.9620i) q^{6} +(2.08243 - 1.80444i) q^{7} +(0.234513 + 1.63107i) q^{8} +(-10.2102 - 2.99799i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.28753	−	1.47011i	1.14376	−	0.735053i	0.175376	−	0.984501i	\(-0.443886\pi\)
							0.968388	+	0.249449i	\(0.0802495\pi\)
\(3\)	0.630718	−	4.38674i	0.210239	−	1.46225i	−0.562118	−	0.827057i	\(-0.690013\pi\)
							0.772357	−	0.635189i	\(-0.219078\pi\)
\(5\)	0.629973	+	2.14549i	0.125995	+	0.429098i
							\(7\)	2.08243	−	1.80444i	0.297490	−	0.257777i	−0.493307	−	0.869856i	\(-0.664212\pi\)
0.790797	+	0.612079i	\(0.209666\pi\)
\(11\)	−10.9198	+	16.9915i	−0.992708	+	1.54468i	−0.162893	+	0.986644i	\(0.552083\pi\)
							−0.829814	+	0.558040i	\(0.811554\pi\)
\(13\)	14.7680	−	17.0432i	1.13600	−	1.31101i	0.191876	−	0.981419i	\(-0.438543\pi\)
							0.944123	−	0.329594i	\(-0.106912\pi\)
\(17\)	−14.8421	+	6.77817i	−0.873066	+	0.398716i	−0.800986	−	0.598683i	\(-0.795691\pi\)
							−0.0720798	+	0.997399i	\(0.522964\pi\)
\(19\)	−9.43934	−	4.31080i	−0.496807	−	0.226884i	0.151222	−	0.988500i	\(-0.451679\pi\)
							−0.648029	+	0.761616i	\(0.724407\pi\)
\(23\)	22.9992	+	0.192953i	0.999965	+	0.00838926i
							\(29\)	8.26493	+	18.0977i	0.284998	+	0.624058i	0.996939	−	0.0781809i	\(-0.0249112\pi\)
−0.711942	+	0.702239i	\(0.752184\pi\)
\(31\)	−4.62897	−	32.1952i	−0.149322	−	1.03855i	−0.917334	−	0.398119i	\(-0.869663\pi\)
							0.768012	−	0.640435i	\(-0.221246\pi\)
\(37\)	−6.81806	+	23.2202i	−0.184272	+	0.627572i	0.814597	+	0.580027i	\(0.196958\pi\)
							−0.998869	+	0.0475453i	\(0.984860\pi\)
\(41\)	15.6218	−	4.58696i	0.381018	−	0.111877i	−0.0856142	−	0.996328i	\(-0.527285\pi\)
							0.466633	+	0.884451i	\(0.345467\pi\)
\(43\)	−20.8261	−	2.99434i	−0.484328	−	0.0696359i	−0.104174	−	0.994559i	\(-0.533220\pi\)
							−0.380154	+	0.924923i	\(0.624129\pi\)
\(47\)	−76.2451			−1.62224			−0.811118	−	0.584883i	\(-0.801141\pi\)
							−0.811118	+	0.584883i	\(0.801141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.11.14	✓	160
23.21	odd	22	inner	115.3.h.a.21.14	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.14	✓	160	1.1	even	1	trivial
115.3.h.a.21.14	yes	160	23.21	odd	22	inner