Embedded newform 896.2.bt.a.59.63

• Label: 896.2.bt.a.59.63
• Level: $896$
• Weight: $2$
• Character: 896.59
• Analytic conductor: $7.155$
• Analytic rank: $0$
• Dimension: $4032$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.bt (of order \(96\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(4032\)
Relative dimension:	\(126\) over \(\Q(\zeta_{96})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{96}]$

Embedding invariants

Embedding label			59.63
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.63

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0715009 + 1.41240i) q^{2} +(-1.22748 - 1.31057i) q^{3} +(-1.98978 - 0.201977i) q^{4} +(-1.13546 - 2.50601i) q^{5} +(1.93882 - 1.64000i) q^{6} +(1.62416 - 2.08857i) q^{7} +(0.427543 - 2.79593i) q^{8} +(-0.0146708 + 0.223833i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{17}{32}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0715009	+	1.41240i	−0.0505588	+	0.998721i
							\(3\)	−1.22748	−	1.31057i	−0.708688	−	0.756659i	0.269810	−	0.962914i	\(-0.413039\pi\)
−0.978498	+	0.206255i	\(0.933872\pi\)
\(5\)	−1.13546	−	2.50601i	−0.507793	−	1.12072i	−0.971997	−	0.234994i	\(-0.924493\pi\)
							0.464204	−	0.885728i	\(-0.346341\pi\)
\(7\)	1.62416	−	2.08857i	0.613874	−	0.789404i
							\(11\)	−1.86382	−	2.99727i	−0.561963	−	0.903710i	−0.999989	−	0.00474742i	\(-0.998489\pi\)
0.438026	−	0.898962i	\(-0.355678\pi\)
\(13\)	−0.000677219	−	0.00687592i	−0.000187827	−	0.00190704i	0.995090	−	0.0989706i	\(-0.0315550\pi\)
							−0.995278	+	0.0970636i	\(0.969055\pi\)
\(17\)	−4.20471	−	0.553561i	−1.01979	−	0.134258i	−0.397955	−	0.917405i	\(-0.630280\pi\)
							−0.621837	+	0.783147i	\(0.713613\pi\)
\(19\)	−0.143465	+	0.868955i	−0.0329131	+	0.199352i	−0.998052	−	0.0623861i	\(-0.980129\pi\)
							0.965139	+	0.261738i	\(0.0842956\pi\)
\(23\)	4.17290	+	4.75828i	0.870109	+	0.992169i	1.00000		0.000641116i	\(0.000204073\pi\)
							−0.129891	+	0.991528i	\(0.541463\pi\)
\(29\)	0.703020	+	1.31526i	0.130548	+	0.244237i	0.938699	−	0.344737i	\(-0.112032\pi\)
							−0.808152	+	0.588974i	\(0.799532\pi\)
\(31\)	−4.59784	+	1.23199i	−0.825797	+	0.221272i	−0.646879	−	0.762592i	\(-0.723926\pi\)
							−0.178918	+	0.983864i	\(0.557260\pi\)
\(37\)	3.53312	−	9.38728i	0.580842	−	1.54326i	−0.239209	−	0.970968i	\(-0.576888\pi\)
							0.820051	−	0.572291i	\(-0.193945\pi\)
\(41\)	0.439299	+	2.20850i	0.0686069	+	0.344910i	0.999808	−	0.0195957i	\(-0.00623790\pi\)
							−0.931201	+	0.364506i	\(0.881238\pi\)
\(43\)	7.55960	−	2.29318i	1.15283	−	0.349707i	0.344700	−	0.938713i	\(-0.387981\pi\)
							0.808129	+	0.589006i	\(0.200481\pi\)
\(47\)	−4.97779	+	6.48718i	−0.726085	+	0.946252i	−0.999879	−	0.0155363i	\(-0.995054\pi\)
							0.273795	+	0.961788i	\(0.411721\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.63	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.106	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.106	yes	4032
896.243	even	96	inner	896.2.bt.a.243.63	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.63	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.106	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.106	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.63	yes	4032	896.243	even	96	inner