Embedded newform 115.3.h.a.21.1

• Label: 115.3.h.a.21.1
• Level: $115$
• Weight: $3$
• Character: 115.21
• 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			21.1
Character	\(\chi\)	\(=\)	115.21
Dual form			115.3.h.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.20529 - 2.05992i) q^{2} +(-0.0243799 - 0.169566i) q^{3} +(4.36899 + 9.56674i) q^{4} +(0.629973 - 2.14549i) q^{5} +(-0.271147 + 0.593730i) q^{6} +(-1.96278 - 1.70076i) q^{7} +(3.53386 - 24.5785i) q^{8} +(8.60728 - 2.52732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q + 4 q^{2} - 16 q^{4} - 26 q^{6} - 22 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(51\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{13}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.20529	−	2.05992i	−1.60265	−	1.02996i	−0.965910	−	0.258876i	\(-0.916648\pi\)
							−0.636736	−	0.771082i	\(-0.719716\pi\)
\(3\)	−0.0243799	−	0.169566i	−0.00812665	−	0.0565221i	0.985355	−	0.170518i	\(-0.0545441\pi\)
							−0.993481	+	0.113996i	\(0.963635\pi\)
\(5\)	0.629973	−	2.14549i	0.125995	−	0.429098i
							\(7\)	−1.96278	−	1.70076i	−0.280398	−	0.242966i	0.503295	−	0.864115i	\(-0.332121\pi\)
−0.783693	+	0.621149i	\(0.786666\pi\)
\(11\)	−3.88947	−	6.05214i	−0.353588	−	0.550194i	0.618208	−	0.786014i	\(-0.287859\pi\)
							−0.971797	+	0.235820i	\(0.924222\pi\)
\(13\)	−6.79451	−	7.84128i	−0.522654	−	0.603175i	0.431639	−	0.902046i	\(-0.357935\pi\)
							−0.954294	+	0.298871i	\(0.903390\pi\)
\(17\)	−4.40189	−	2.01028i	−0.258935	−	0.118252i	0.281718	−	0.959497i	\(-0.409096\pi\)
							−0.540653	+	0.841246i	\(0.681823\pi\)
\(19\)	−25.8204	+	11.7918i	−1.35897	+	0.620621i	−0.955668	−	0.294446i	\(-0.904865\pi\)
							−0.403302	+	0.915067i	\(0.632138\pi\)
\(23\)	−2.84437	−	22.8234i	−0.123668	−	0.992324i
							\(29\)	20.2801	−	44.4072i	0.699313	−	1.53128i	−0.141489	−	0.989940i	\(-0.545189\pi\)
0.840802	−	0.541342i	\(-0.182084\pi\)
\(31\)	3.41938	−	23.7823i	0.110303	−	0.767171i	−0.857323	−	0.514779i	\(-0.827874\pi\)
							0.967625	−	0.252391i	\(-0.0812171\pi\)
\(37\)	5.31425	+	18.0987i	0.143628	+	0.489153i	0.999612	−	0.0278496i	\(-0.00886595\pi\)
							−0.855984	+	0.517003i	\(0.827048\pi\)
\(41\)	−40.8165	−	11.9848i	−0.995523	−	0.292312i	−0.256905	−	0.966437i	\(-0.582703\pi\)
							−0.738618	+	0.674125i	\(0.764521\pi\)
\(43\)	42.0521	−	6.04618i	0.977956	−	0.140609i	0.365241	−	0.930913i	\(-0.380987\pi\)
							0.612715	+	0.790304i	\(0.290077\pi\)
\(47\)	−78.7657			−1.67587			−0.837933	−	0.545774i	\(-0.816236\pi\)
							−0.837933	+	0.545774i	\(0.816236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.3.h.a.21.1	yes	160
23.11	odd	22	inner	115.3.h.a.11.1	✓	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.3.h.a.11.1	✓	160	23.11	odd	22	inner
115.3.h.a.21.1	yes	160	1.1	even	1	trivial