Embedded newform 43.7.h.a.19.15

• Label: 43.7.h.a.19.15
• Level: $43$
• Weight: $7$
• Character: 43.19
• 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			19.15
Character	\(\chi\)	\(=\)	43.19
Dual form			43.7.h.a.34.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.48941 + 3.58019i) q^{2} +(7.30942 + 48.4948i) q^{3} +(-6.90425 - 30.2495i) q^{4} +(-128.302 - 188.184i) q^{5} +(-140.806 + 243.882i) q^{6} +(-377.012 + 217.668i) q^{7} +(236.755 - 491.626i) q^{8} +(-1601.71 + 494.061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{19}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.48941	+	3.58019i	0.561177	+	0.447523i	0.862542	−	0.505985i	\(-0.168871\pi\)
							−0.301366	+	0.953509i	\(0.597442\pi\)
\(3\)	7.30942	+	48.4948i	0.270719	+	1.79610i	0.549230	+	0.835671i	\(0.314921\pi\)
							−0.278510	+	0.960433i	\(0.589841\pi\)
\(5\)	−128.302	−	188.184i	−1.02641	−	1.50547i	−0.853879	−	0.520471i	\(-0.825756\pi\)
							−0.172536	−	0.985003i	\(-0.555196\pi\)
\(7\)	−377.012	+	217.668i	−1.09916	+	0.634601i	−0.936001	−	0.351998i	\(-0.885502\pi\)
							−0.163161	+	0.986600i	\(0.552169\pi\)
\(11\)	−293.120	+	1284.24i	−0.220226	+	0.964871i	0.737083	+	0.675802i	\(0.236203\pi\)
							−0.957308	+	0.289069i	\(0.906654\pi\)
\(13\)	−108.497	−	1447.79i	−0.0493840	−	0.658984i	−0.965888	−	0.258959i	\(-0.916620\pi\)
							0.916504	−	0.400025i	\(-0.130999\pi\)
\(17\)	571.269	+	389.485i	0.116277	+	0.0792764i	0.620062	−	0.784553i	\(-0.287107\pi\)
							−0.503785	+	0.863829i	\(0.668060\pi\)
\(19\)	−2911.09	+	9437.53i	−0.424419	+	1.37593i	0.450981	+	0.892534i	\(0.351074\pi\)
							−0.875400	+	0.483400i	\(0.839402\pi\)
\(23\)	−48.1778	−	44.7025i	−0.00395971	−	0.00367407i	0.678190	−	0.734886i	\(-0.262764\pi\)
							−0.682150	+	0.731212i	\(0.738955\pi\)
\(29\)	4126.41	−	27376.9i	0.169191	−	1.12251i	−0.729517	−	0.683963i	\(-0.760255\pi\)
							0.898708	−	0.438548i	\(-0.144507\pi\)
\(31\)	−603.176	−	1536.87i	−0.0202469	−	0.0515883i	0.920393	−	0.390995i	\(-0.127869\pi\)
							−0.940640	+	0.339407i	\(0.889774\pi\)
\(37\)	15274.9	+	8818.97i	0.301560	+	0.174106i	0.643143	−	0.765746i	\(-0.277630\pi\)
							−0.341584	+	0.939851i	\(0.610963\pi\)
\(41\)	−37162.8	+	46600.7i	−0.539209	+	0.676147i	−0.974563	−	0.224112i	\(-0.928052\pi\)
							0.435354	+	0.900259i	\(0.356623\pi\)
\(43\)	−73914.7	−	29291.2i	−0.929663	−	0.368410i
							\(47\)	−20520.2	−	89904.7i	−0.197646	−	0.865942i	−0.972333	−	0.233598i	\(-0.924950\pi\)
0.774688	−	0.632344i	\(-0.217907\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.19.15	✓	252
43.34	odd	42	inner	43.7.h.a.34.15	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.19.15	✓	252	1.1	even	1	trivial
43.7.h.a.34.15	yes	252	43.34	odd	42	inner