Embedded newform 896.2.bt.a.59.5

• Label: 896.2.bt.a.59.5
• 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			59.5
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40809 - 0.131506i) q^{2} +(-2.12607 - 2.26998i) q^{3} +(1.96541 + 0.370343i) q^{4} +(-1.49802 - 3.30620i) q^{5} +(2.69517 + 3.47592i) q^{6} +(-2.48135 - 0.918095i) q^{7} +(-2.71877 - 0.779939i) q^{8} +(-0.436434 + 6.65869i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40809	−	0.131506i	−0.995667	−	0.0929887i
							\(3\)	−2.12607	−	2.26998i	−1.22749	−	1.31057i	−0.935147	−	0.354259i	\(-0.884733\pi\)
−0.292339	−	0.956315i	\(-0.594434\pi\)
\(5\)	−1.49802	−	3.30620i	−0.669934	−	1.47858i	−0.867063	−	0.498198i	\(-0.833995\pi\)
							0.197129	−	0.980378i	\(-0.436838\pi\)
\(7\)	−2.48135	−	0.918095i	−0.937862	−	0.347007i
							\(11\)	0.178561	+	0.287150i	0.0538382	+	0.0865788i	0.875017	−	0.484092i	\(-0.160850\pi\)
−0.821179	+	0.570671i	\(0.806683\pi\)
\(13\)	−0.442531	−	4.49309i	−0.122736	−	1.24616i	−0.837361	−	0.546651i	\(-0.815903\pi\)
							0.714625	−	0.699508i	\(-0.246597\pi\)
\(17\)	−4.34426	−	0.571932i	−1.05364	−	0.138714i	−0.416244	−	0.909253i	\(-0.636654\pi\)
							−0.637393	+	0.770539i	\(0.719987\pi\)
\(19\)	−0.967350	+	5.85915i	−0.221925	+	1.34418i	0.611366	+	0.791348i	\(0.290620\pi\)
							−0.833292	+	0.552834i	\(0.813547\pi\)
\(23\)	3.21559	+	3.66668i	0.670498	+	0.764556i	0.982649	−	0.185477i	\(-0.0593829\pi\)
							−0.312151	+	0.950032i	\(0.601050\pi\)
\(29\)	−3.71258	−	6.94576i	−0.689410	−	1.28979i	−0.946280	−	0.323349i	\(-0.895191\pi\)
							0.256870	−	0.966446i	\(-0.417309\pi\)
\(31\)	−4.72203	+	1.26526i	−0.848102	+	0.227248i	−0.656595	−	0.754243i	\(-0.728004\pi\)
							−0.191507	+	0.981491i	\(0.561337\pi\)
\(37\)	1.35221	−	3.59272i	0.222301	−	0.590640i	−0.776941	−	0.629574i	\(-0.783229\pi\)
							0.999242	+	0.0389337i	\(0.0123961\pi\)
\(41\)	−1.48622	−	7.47173i	−0.232109	−	1.16689i	−0.904428	−	0.426627i	\(-0.859702\pi\)
							0.672319	−	0.740262i	\(-0.265298\pi\)
\(43\)	−4.16746	+	1.26419i	−0.635532	+	0.192787i	−0.591579	−	0.806247i	\(-0.701495\pi\)
							−0.0439532	+	0.999034i	\(0.513995\pi\)
\(47\)	2.94731	−	3.84101i	0.429910	−	0.560269i	−0.527295	−	0.849683i	\(-0.676794\pi\)
							0.957204	+	0.289413i	\(0.0934603\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.5	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.81	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.81	yes	4032
896.243	even	96	inner	896.2.bt.a.243.5	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.5	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.81	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.81	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.5	yes	4032	896.243	even	96	inner