Embedded newform 896.2.bt.a.59.30

• Label: 896.2.bt.a.59.30
• 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			59.30
Character	\(\chi\)	\(=\)	896.59
Dual form			896.2.bt.a.243.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.998861 + 1.00114i) q^{2} +(-1.89383 - 2.02202i) q^{3} +(-0.00455275 - 1.99999i) q^{4} +(-0.342931 - 0.756865i) q^{5} +(3.91599 + 0.123734i) q^{6} +(0.918387 + 2.48124i) q^{7} +(2.00682 + 1.99316i) q^{8} +(-0.305769 + 4.66513i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4032 q - 16 q^{2} - 48 q^{3} - 16 q^{4} - 48 q^{5} - 32 q^{7} - 64 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.998861	+	1.00114i	−0.706302	+	0.707911i
							\(3\)	−1.89383	−	2.02202i	−1.09340	−	1.16741i	−0.984865	−	0.173321i	\(-0.944550\pi\)
−0.108536	−	0.994092i	\(-0.534616\pi\)
\(5\)	−0.342931	−	0.756865i	−0.153363	−	0.338480i	0.818322	−	0.574761i	\(-0.194905\pi\)
							−0.971685	+	0.236281i	\(0.924072\pi\)
\(7\)	0.918387	+	2.48124i	0.347118	+	0.937822i
							\(11\)	−0.618931	−	0.995321i	−0.186615	−	0.300101i	0.742533	−	0.669809i	\(-0.233624\pi\)
−0.929148	+	0.369709i	\(0.879458\pi\)
\(13\)	−0.494583	−	5.02158i	−0.137173	−	1.39274i	−0.779546	−	0.626345i	\(-0.784550\pi\)
							0.642374	−	0.766391i	\(-0.277950\pi\)
\(17\)	−4.91903	−	0.647603i	−1.19304	−	0.157067i	−0.492263	−	0.870446i	\(-0.663830\pi\)
							−0.700778	+	0.713380i	\(0.747164\pi\)
\(19\)	0.591855	−	3.58481i	0.135781	−	0.822412i	−0.828979	−	0.559280i	\(-0.811078\pi\)
							0.964760	−	0.263132i	\(-0.0847555\pi\)
\(23\)	1.85714	+	2.11767i	0.387241	+	0.441564i	0.912313	−	0.409494i	\(-0.134295\pi\)
							−0.525071	+	0.851058i	\(0.675961\pi\)
\(29\)	1.34779	+	2.52153i	0.250278	+	0.468237i	0.975417	−	0.220365i	\(-0.0707249\pi\)
							−0.725140	+	0.688602i	\(0.758225\pi\)
\(31\)	−6.28478	+	1.68400i	−1.12878	+	0.302455i	−0.774431	−	0.632658i	\(-0.781964\pi\)
							−0.354348	+	0.935114i	\(0.615297\pi\)
\(37\)	−1.32607	+	3.52327i	−0.218004	+	0.579223i	−0.998971	−	0.0453493i	\(-0.985560\pi\)
							0.780967	+	0.624572i	\(0.214727\pi\)
\(41\)	−1.24339	−	6.25096i	−0.194185	−	0.976235i	−0.947787	−	0.318904i	\(-0.896685\pi\)
							0.753602	−	0.657331i	\(-0.228315\pi\)
\(43\)	−3.12693	+	0.948544i	−0.476852	+	0.144652i	−0.519556	−	0.854436i	\(-0.673903\pi\)
							0.0427041	+	0.999088i	\(0.486403\pi\)
\(47\)	−7.70932	+	10.0470i	−1.12452	+	1.46550i	−0.258148	+	0.966105i	\(0.583112\pi\)
							−0.866373	+	0.499398i	\(0.833555\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.bt.a.59.30	✓	4032
7.5	odd	6	inner	896.2.bt.a.187.117	yes	4032
128.115	odd	32	inner	896.2.bt.a.115.117	yes	4032
896.243	even	96	inner	896.2.bt.a.243.30	yes	4032

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.bt.a.59.30	✓	4032	1.1	even	1	trivial
896.2.bt.a.115.117	yes	4032	128.115	odd	32	inner
896.2.bt.a.187.117	yes	4032	7.5	odd	6	inner
896.2.bt.a.243.30	yes	4032	896.243	even	96	inner