Embedded newform 43.7.h.a.3.15

• Label: 43.7.h.a.3.15
• Level: $43$
• Weight: $7$
• Character: 43.3
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.15
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.75934 - 5.72982i) q^{2} +(42.1286 + 3.15710i) q^{3} +(14.6864 + 18.4162i) q^{4} +(13.8848 - 45.0135i) q^{5} +(134.337 - 232.678i) q^{6} +(-37.7866 + 21.8161i) q^{7} +(542.857 - 123.904i) q^{8} +(1043.99 + 157.357i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.75934	−	5.72982i	0.344917	−	0.716228i	−0.654282	−	0.756251i	\(-0.727029\pi\)
							0.999199	+	0.0400233i	\(0.0127432\pi\)
\(3\)	42.1286	+	3.15710i	1.56032	+	0.116930i	0.826704	−	0.562637i	\(-0.190213\pi\)
							0.733614	+	0.679566i	\(0.237832\pi\)
\(5\)	13.8848	−	45.0135i	0.111079	−	0.360108i	−0.882917	−	0.469529i	\(-0.844424\pi\)
							0.993995	+	0.109421i	\(0.0348998\pi\)
\(7\)	−37.7866	+	21.8161i	−0.110165	+	0.0636038i	−0.554070	−	0.832470i	\(-0.686926\pi\)
							0.443905	+	0.896074i	\(0.353593\pi\)
\(11\)	−883.355	+	1107.69i	−0.663678	+	0.832226i	−0.993738	−	0.111734i	\(-0.964359\pi\)
							0.330060	+	0.943960i	\(0.392931\pi\)
\(13\)	490.092	−	454.739i	0.223073	−	0.206982i	−0.560717	−	0.828008i	\(-0.689474\pi\)
							0.783790	+	0.621026i	\(0.213284\pi\)
\(17\)	−547.203	+	168.790i	−0.111379	+	0.0343558i	−0.349944	−	0.936771i	\(-0.613799\pi\)
							0.238565	+	0.971127i	\(0.423323\pi\)
\(19\)	−898.477	−	5961.01i	−0.130992	−	0.869078i	−0.953732	−	0.300659i	\(-0.902793\pi\)
							0.822739	−	0.568419i	\(-0.192445\pi\)
\(23\)	−6644.37	−	16929.6i	−0.546097	−	1.39143i	−0.891518	−	0.452986i	\(-0.850359\pi\)
							0.345420	−	0.938448i	\(-0.387736\pi\)
\(29\)	−35612.8	+	2668.81i	−1.46020	+	0.109427i	−0.781002	−	0.624528i	\(-0.785291\pi\)
							−0.679198	+	0.733955i	\(0.737672\pi\)
\(31\)	−22638.9	+	15434.9i	−0.759923	+	0.518107i	−0.880182	−	0.474636i	\(-0.842580\pi\)
							0.120259	+	0.992743i	\(0.461627\pi\)
\(37\)	11931.5	+	6888.68i	0.235555	+	0.135998i	0.613132	−	0.789981i	\(-0.289909\pi\)
							−0.377577	+	0.925978i	\(0.623243\pi\)
\(41\)	29423.0	+	14169.3i	0.426908	+	0.205588i	0.634985	−	0.772524i	\(-0.281006\pi\)
							−0.208077	+	0.978112i	\(0.566720\pi\)
\(43\)	−26170.3	+	75076.5i	−0.329157	+	0.944275i
							\(47\)	−30817.4	−	38643.7i	−0.296826	−	0.372208i	0.610946	−	0.791672i	\(-0.290789\pi\)
−0.907772	+	0.419464i	\(0.862218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.15	✓	252
43.29	odd	42	inner	43.7.h.a.29.15	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.15	✓	252	1.1	even	1	trivial
43.7.h.a.29.15	yes	252	43.29	odd	42	inner