Embedded newform 88.2.o.a.37.7

• Label: 88.2.o.a.37.7
• Level: $88$
• Weight: $2$
• Character: 88.37
• Analytic conductor: $0.703$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	88.o (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.702683537787\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			37.7
Character	\(\chi\)	\(=\)	88.37
Dual form			88.2.o.a.69.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.210425 - 1.39847i) q^{2} +(-2.32812 - 0.756451i) q^{3} +(-1.91144 - 0.588548i) q^{4} +(-0.117836 - 0.162187i) q^{5} +(-1.54777 + 3.09663i) q^{6} +(-0.725001 - 2.23132i) q^{7} +(-1.22528 + 2.54925i) q^{8} +(2.42086 + 1.75886i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 5 q^{2} - q^{4} - 7 q^{6} - 10 q^{7} - 5 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/88\mathbb{Z}\right)^\times\).

\(n\)	\(23\)	\(45\)	\(57\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.210425	−	1.39847i	0.148793	−	0.988868i
							\(3\)	−2.32812	−	0.756451i	−1.34414	−	0.436737i	−0.453422	−	0.891296i	\(-0.649797\pi\)
−0.890717	+	0.454559i	\(0.849797\pi\)
\(5\)	−0.117836	−	0.162187i	−0.0526977	−	0.0725322i	0.781855	−	0.623460i	\(-0.214274\pi\)
							−0.834553	+	0.550928i	\(0.814274\pi\)
\(7\)	−0.725001	−	2.23132i	−0.274025	−	0.843361i	−0.989476	−	0.144699i	\(-0.953779\pi\)
							0.715451	−	0.698663i	\(-0.246221\pi\)
\(11\)	1.47592	−	2.97013i	0.445005	−	0.895528i
							\(13\)	0.959026	−	1.31999i	0.265986	−	0.366098i	−0.655044	−	0.755591i	\(-0.727350\pi\)
0.921029	+	0.389493i	\(0.127350\pi\)
\(17\)	2.69920	−	1.96108i	0.654651	−	0.475632i	−0.210201	−	0.977658i	\(-0.567412\pi\)
							0.864852	+	0.502026i	\(0.167412\pi\)
\(19\)	4.68224	+	1.52135i	1.07418	+	0.349022i	0.792114	−	0.610373i	\(-0.208981\pi\)
							0.282065	+	0.959395i	\(0.408981\pi\)
\(23\)	−9.25109			−1.92899			−0.964493	−	0.264108i	\(-0.914922\pi\)
							−0.964493	+	0.264108i	\(0.914922\pi\)
\(29\)	−5.88863	+	1.91333i	−1.09349	+	0.355297i	−0.799595	−	0.600540i	\(-0.794952\pi\)
							−0.293897	+	0.955837i	\(0.594952\pi\)
\(31\)	6.68182	+	4.85463i	1.20009	+	0.871917i	0.994294	−	0.106678i	\(-0.0340214\pi\)
							0.205797	+	0.978595i	\(0.434021\pi\)
\(37\)	1.21740	−	0.395559i	0.200140	−	0.0650295i	−0.207232	−	0.978292i	\(-0.566445\pi\)
							0.407372	+	0.913262i	\(0.366445\pi\)
\(41\)	−1.86354	+	5.73540i	−0.291037	+	0.895718i	0.693487	+	0.720469i	\(0.256073\pi\)
							−0.984524	+	0.175250i	\(0.943927\pi\)
\(43\)			2.47184i			0.376952i	0.982078	+	0.188476i	\(0.0603549\pi\)
							−0.982078	+	0.188476i	\(0.939645\pi\)
\(47\)	1.47060	−	4.52605i	0.214509	−	0.660192i	−0.784679	−	0.619903i	\(-0.787172\pi\)
							0.999188	−	0.0402894i	\(-0.0128280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.2.o.a.37.7	yes	40
3.2	odd	2		792.2.br.b.37.4		40
4.3	odd	2		352.2.w.a.81.9		40
8.3	odd	2		352.2.w.a.81.2		40
8.5	even	2	inner	88.2.o.a.37.2	✓	40
11.2	odd	10		968.2.o.d.269.9		40
11.3	even	5	inner	88.2.o.a.69.2	yes	40
11.4	even	5		968.2.o.j.493.10		40
11.5	even	5		968.2.c.h.485.14		20
11.6	odd	10		968.2.c.i.485.7		20
11.7	odd	10		968.2.o.d.493.1		40
11.8	odd	10		968.2.o.i.245.9		40
11.9	even	5		968.2.o.j.269.2		40
11.10	odd	2		968.2.o.i.565.4		40
24.5	odd	2		792.2.br.b.37.9		40
33.14	odd	10		792.2.br.b.685.9		40
44.3	odd	10		352.2.w.a.113.2		40
44.27	odd	10		3872.2.c.h.1937.17		20
44.39	even	10		3872.2.c.i.1937.17		20
88.3	odd	10		352.2.w.a.113.9		40
88.5	even	10		968.2.c.h.485.13		20
88.13	odd	10		968.2.o.d.269.1		40
88.21	odd	2		968.2.o.i.565.9		40
88.27	odd	10		3872.2.c.h.1937.4		20
88.29	odd	10		968.2.o.d.493.9		40
88.37	even	10		968.2.o.j.493.2		40
88.53	even	10		968.2.o.j.269.10		40
88.61	odd	10		968.2.c.i.485.8		20
88.69	even	10	inner	88.2.o.a.69.7	yes	40
88.83	even	10		3872.2.c.i.1937.4		20
88.85	odd	10		968.2.o.i.245.4		40
264.245	odd	10		792.2.br.b.685.4		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.o.a.37.2	✓	40	8.5	even	2	inner
88.2.o.a.37.7	yes	40	1.1	even	1	trivial
88.2.o.a.69.2	yes	40	11.3	even	5	inner
88.2.o.a.69.7	yes	40	88.69	even	10	inner
352.2.w.a.81.2		40	8.3	odd	2
352.2.w.a.81.9		40	4.3	odd	2
352.2.w.a.113.2		40	44.3	odd	10
352.2.w.a.113.9		40	88.3	odd	10
792.2.br.b.37.4		40	3.2	odd	2
792.2.br.b.37.9		40	24.5	odd	2
792.2.br.b.685.4		40	264.245	odd	10
792.2.br.b.685.9		40	33.14	odd	10
968.2.c.h.485.13		20	88.5	even	10
968.2.c.h.485.14		20	11.5	even	5
968.2.c.i.485.7		20	11.6	odd	10
968.2.c.i.485.8		20	88.61	odd	10
968.2.o.d.269.1		40	88.13	odd	10
968.2.o.d.269.9		40	11.2	odd	10
968.2.o.d.493.1		40	11.7	odd	10
968.2.o.d.493.9		40	88.29	odd	10
968.2.o.i.245.4		40	88.85	odd	10
968.2.o.i.245.9		40	11.8	odd	10
968.2.o.i.565.4		40	11.10	odd	2
968.2.o.i.565.9		40	88.21	odd	2
968.2.o.j.269.2		40	11.9	even	5
968.2.o.j.269.10		40	88.53	even	10
968.2.o.j.493.2		40	88.37	even	10
968.2.o.j.493.10		40	11.4	even	5
3872.2.c.h.1937.4		20	88.27	odd	10
3872.2.c.h.1937.17		20	44.27	odd	10
3872.2.c.i.1937.4		20	88.83	even	10
3872.2.c.i.1937.17		20	44.39	even	10