Embedded newform 896.2.bt.a.59.75

• Label: 896.2.bt.a.59.75
• 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			59.75
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.75

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.440874 - 1.34374i) q^{2} +(1.56195 + 1.66768i) q^{3} +(-1.61126 - 1.18484i) q^{4} +(-0.189884 - 0.419084i) q^{5} +(2.92955 - 1.36362i) q^{6} +(2.14623 - 1.54716i) q^{7} +(-2.30247 + 1.64275i) q^{8} +(-0.145251 + 2.21611i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.440874	−	1.34374i	0.311745	−	0.950166i
							\(3\)	1.56195	+	1.66768i	0.901795	+	0.962836i	0.999476	−	0.0323813i	\(-0.0103091\pi\)
−0.0976808	+	0.995218i	\(0.531142\pi\)
\(5\)	−0.189884	−	0.419084i	−0.0849189	−	0.187420i	0.863571	−	0.504227i	\(-0.168223\pi\)
							−0.948490	+	0.316807i	\(0.897389\pi\)
\(7\)	2.14623	−	1.54716i	0.811198	−	0.584772i
							\(11\)	−2.63978	−	4.24511i	−0.795923	−	1.27995i	−0.956153	−	0.292868i	\(-0.905390\pi\)
0.160230	−	0.987080i	\(-0.448776\pi\)
\(13\)	−0.572090	−	5.80852i	−0.158669	−	1.61099i	−0.663772	−	0.747935i	\(-0.731045\pi\)
							0.505103	−	0.863059i	\(-0.331455\pi\)
\(17\)	−4.42229	−	0.582206i	−1.07256	−	0.141206i	−0.426498	−	0.904488i	\(-0.640253\pi\)
							−0.646065	+	0.763283i	\(0.723586\pi\)
\(19\)	−0.189886	+	1.15012i	−0.0435628	+	0.263856i	−0.999565	−	0.0295047i	\(-0.990607\pi\)
							0.956002	+	0.293361i	\(0.0947737\pi\)
\(23\)	2.29920	+	2.62174i	0.479417	+	0.546670i	0.940102	−	0.340893i	\(-0.110730\pi\)
							−0.460685	+	0.887564i	\(0.652396\pi\)
\(29\)	1.78620	+	3.34175i	0.331690	+	0.620548i	0.991384	−	0.130986i	\(-0.0418143\pi\)
							−0.659694	+	0.751534i	\(0.729314\pi\)
\(31\)	8.65056	−	2.31791i	1.55369	−	0.416309i	0.623028	−	0.782200i	\(-0.285902\pi\)
							0.930658	+	0.365891i	\(0.119236\pi\)
\(37\)	−0.220595	+	0.586108i	−0.0362657	+	0.0963555i	−0.952798	−	0.303605i	\(-0.901810\pi\)
							0.916532	+	0.399961i	\(0.130976\pi\)
\(41\)	0.667701	+	3.35676i	0.104277	+	0.524238i	0.997249	+	0.0741248i	\(0.0236163\pi\)
							−0.892972	+	0.450113i	\(0.851384\pi\)
\(43\)	4.08078	−	1.23789i	0.622314	−	0.188777i	0.0366463	−	0.999328i	\(-0.488333\pi\)
							0.585667	+	0.810552i	\(0.300833\pi\)
\(47\)	−7.49148	+	9.76309i	−1.09275	+	1.42409i	−0.196516	+	0.980501i	\(0.562963\pi\)
							−0.896229	+	0.443593i	\(0.853704\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.75	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.10	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.10	yes	4032
896.243	even	96	inner	896.2.bt.a.243.75	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.75	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.10	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.10	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.75	yes	4032	896.243	even	96	inner