Embedded newform 294.2.a.f.1.1

• Label: 294.2.a.f.1.1
• Level: $294$
• Weight: $2$
• Character: 294.1
• Self dual: yes
• Analytic conductor: $2.348$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.34760181943\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	294.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	1.00000	0.577350
\(5\)	−1.00000	−0.447214				−0.223607	−	0.974679i	\(-0.571783\pi\)
			−0.223607	+	0.974679i	\(0.571783\pi\)
\(7\)	0	0
			\(11\)	5.00000	1.50756	0.753778	−	0.657129i	\(-0.228229\pi\)
0.753778	+	0.657129i				\(0.228229\pi\)
\(13\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(17\)	4.00000	0.970143	0.485071	−	0.874475i	\(-0.338794\pi\)
			0.485071	+	0.874475i	\(0.338794\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	−4.00000	−0.834058	−0.417029	−	0.908893i	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	−5.00000	−0.928477	−0.464238	−	0.885710i	\(-0.653672\pi\)
			−0.464238	+	0.885710i	\(0.653672\pi\)
\(31\)	−3.00000	−0.538816	−0.269408	−	0.963026i	\(-0.586828\pi\)
			−0.269408	+	0.963026i	\(0.586828\pi\)
\(37\)	−4.00000	−0.657596	−0.328798	−	0.944400i	\(-0.606644\pi\)
			−0.328798	+	0.944400i	\(0.606644\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	2.00000	0.304997	0.152499	−	0.988304i	\(-0.451268\pi\)
			0.152499	+	0.988304i	\(0.451268\pi\)
\(47\)	6.00000	0.875190	0.437595	−	0.899172i	\(-0.355830\pi\)
			0.437595	+	0.899172i	\(0.355830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.2.a.f.1.1		1
3.2	odd	2		882.2.a.d.1.1		1
4.3	odd	2		2352.2.a.f.1.1		1
5.4	even	2		7350.2.a.q.1.1		1
7.2	even	3		294.2.e.b.67.1		2
7.3	odd	6		42.2.e.a.37.1	yes	2
7.4	even	3		294.2.e.b.79.1		2
7.5	odd	6		42.2.e.a.25.1	✓	2
7.6	odd	2		294.2.a.e.1.1		1
8.3	odd	2		9408.2.a.cr.1.1		1
8.5	even	2		9408.2.a.z.1.1		1
12.11	even	2		7056.2.a.bl.1.1		1
21.2	odd	6		882.2.g.i.361.1		2
21.5	even	6		126.2.g.c.109.1		2
21.11	odd	6		882.2.g.i.667.1		2
21.17	even	6		126.2.g.c.37.1		2
21.20	even	2		882.2.a.c.1.1		1
28.3	even	6		336.2.q.b.289.1		2
28.11	odd	6		2352.2.q.u.961.1		2
28.19	even	6		336.2.q.b.193.1		2
28.23	odd	6		2352.2.q.u.1537.1		2
28.27	even	2		2352.2.a.t.1.1		1
35.3	even	12		1050.2.o.a.499.2		4
35.12	even	12		1050.2.o.a.949.2		4
35.17	even	12		1050.2.o.a.499.1		4
35.19	odd	6		1050.2.i.l.151.1		2
35.24	odd	6		1050.2.i.l.751.1		2
35.33	even	12		1050.2.o.a.949.1		4
35.34	odd	2		7350.2.a.bl.1.1		1
56.3	even	6		1344.2.q.s.961.1		2
56.5	odd	6		1344.2.q.g.193.1		2
56.13	odd	2		9408.2.a.ce.1.1		1
56.19	even	6		1344.2.q.s.193.1		2
56.27	even	2		9408.2.a.q.1.1		1
56.45	odd	6		1344.2.q.g.961.1		2
63.5	even	6		1134.2.e.e.865.1		2
63.31	odd	6		1134.2.h.e.541.1		2
63.38	even	6		1134.2.e.e.919.1		2
63.40	odd	6		1134.2.e.l.865.1		2
63.47	even	6		1134.2.h.l.109.1		2
63.52	odd	6		1134.2.e.l.919.1		2
63.59	even	6		1134.2.h.l.541.1		2
63.61	odd	6		1134.2.h.e.109.1		2
84.47	odd	6		1008.2.s.k.865.1		2
84.59	odd	6		1008.2.s.k.289.1		2
84.83	odd	2		7056.2.a.w.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.e.a.25.1	✓	2	7.5	odd	6
42.2.e.a.37.1	yes	2	7.3	odd	6
126.2.g.c.37.1		2	21.17	even	6
126.2.g.c.109.1		2	21.5	even	6
294.2.a.e.1.1		1	7.6	odd	2
294.2.a.f.1.1		1	1.1	even	1	trivial
294.2.e.b.67.1		2	7.2	even	3
294.2.e.b.79.1		2	7.4	even	3
336.2.q.b.193.1		2	28.19	even	6
336.2.q.b.289.1		2	28.3	even	6
882.2.a.c.1.1		1	21.20	even	2
882.2.a.d.1.1		1	3.2	odd	2
882.2.g.i.361.1		2	21.2	odd	6
882.2.g.i.667.1		2	21.11	odd	6
1008.2.s.k.289.1		2	84.59	odd	6
1008.2.s.k.865.1		2	84.47	odd	6
1050.2.i.l.151.1		2	35.19	odd	6
1050.2.i.l.751.1		2	35.24	odd	6
1050.2.o.a.499.1		4	35.17	even	12
1050.2.o.a.499.2		4	35.3	even	12
1050.2.o.a.949.1		4	35.33	even	12
1050.2.o.a.949.2		4	35.12	even	12
1134.2.e.e.865.1		2	63.5	even	6
1134.2.e.e.919.1		2	63.38	even	6
1134.2.e.l.865.1		2	63.40	odd	6
1134.2.e.l.919.1		2	63.52	odd	6
1134.2.h.e.109.1		2	63.61	odd	6
1134.2.h.e.541.1		2	63.31	odd	6
1134.2.h.l.109.1		2	63.47	even	6
1134.2.h.l.541.1		2	63.59	even	6
1344.2.q.g.193.1		2	56.5	odd	6
1344.2.q.g.961.1		2	56.45	odd	6
1344.2.q.s.193.1		2	56.19	even	6
1344.2.q.s.961.1		2	56.3	even	6
2352.2.a.f.1.1		1	4.3	odd	2
2352.2.a.t.1.1		1	28.27	even	2
2352.2.q.u.961.1		2	28.11	odd	6
2352.2.q.u.1537.1		2	28.23	odd	6
7056.2.a.w.1.1		1	84.83	odd	2
7056.2.a.bl.1.1		1	12.11	even	2
7350.2.a.q.1.1		1	5.4	even	2
7350.2.a.bl.1.1		1	35.34	odd	2
9408.2.a.q.1.1		1	56.27	even	2
9408.2.a.z.1.1		1	8.5	even	2
9408.2.a.ce.1.1		1	56.13	odd	2
9408.2.a.cr.1.1		1	8.3	odd	2