Embedded newform 26.3.f.a.7.1

• Label: 26.3.f.a.7.1
• Level: $26$
• Weight: $3$
• Character: 26.7
• Analytic conductor: $0.708$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 26 = 2 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	26.f (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.708448687337\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			7.1
Root			\(0.866025 - 0.500000i\) of defining polynomial
Character	\(\chi\)	\(=\)	26.7
Dual form			26.3.f.a.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(-0.866025 + 1.50000i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-1.73205 - 1.73205i) q^{5} +(-0.633975 + 2.36603i) q^{6} +(-8.96410 - 2.40192i) q^{7} +(2.00000 - 2.00000i) q^{8} +(3.00000 + 5.19615i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 6 q^{6} - 22 q^{7} + 8 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/26\mathbb{Z}\right)^\times\).

\(n\)	\(15\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36603	−	0.366025i	0.683013	−	0.183013i
							\(3\)	−0.866025	+	1.50000i	−0.288675	+	0.500000i	−0.973494	−	0.228714i	\(-0.926548\pi\)
0.684819	+	0.728714i	\(0.259881\pi\)
\(5\)	−1.73205	−	1.73205i	−0.346410	−	0.346410i	0.512360	−	0.858771i	\(-0.328771\pi\)
							−0.858771	+	0.512360i	\(0.828771\pi\)
\(7\)	−8.96410	−	2.40192i	−1.28059	−	0.343132i	−0.446509	−	0.894779i	\(-0.647333\pi\)
							−0.834077	+	0.551647i	\(0.813999\pi\)
\(11\)	1.96410	+	7.33013i	0.178555	+	0.666375i	0.995919	+	0.0902537i	\(0.0287678\pi\)
							−0.817364	+	0.576121i	\(0.804566\pi\)
\(13\)	3.92820	−	12.3923i	0.302169	−	0.953254i
							\(17\)	14.3038	−	8.25833i	0.841403	−	0.485784i	−0.0163380	−	0.999867i	\(-0.505201\pi\)
0.857741	+	0.514082i	\(0.171867\pi\)
\(19\)	−4.10770	+	15.3301i	−0.216194	+	0.806849i	0.769548	+	0.638589i	\(0.220481\pi\)
							−0.985743	+	0.168260i	\(0.946185\pi\)
\(23\)	21.1865	+	12.2321i	0.921154	+	0.531828i	0.884003	−	0.467481i	\(-0.154839\pi\)
							0.0371507	+	0.999310i	\(0.488172\pi\)
\(29\)	−27.3564	+	47.3827i	−0.943324	+	1.63389i	−0.184252	+	0.982879i	\(0.558986\pi\)
							−0.759072	+	0.651007i	\(0.774347\pi\)
\(31\)	−40.6603	−	40.6603i	−1.31162	−	1.31162i	−0.920220	−	0.391401i	\(-0.871991\pi\)
							−0.391401	−	0.920220i	\(-0.628009\pi\)
\(37\)	−1.25833	−	4.69615i	−0.0340089	−	0.126923i	0.946834	−	0.321721i	\(-0.104261\pi\)
							−0.980843	+	0.194798i	\(0.937595\pi\)
\(41\)	11.2583	−	3.01666i	0.274593	−	0.0735771i	−0.118894	−	0.992907i	\(-0.537935\pi\)
							0.393488	+	0.919330i	\(0.371268\pi\)
\(43\)	−3.77499	+	2.17949i	−0.0877905	+	0.0506859i	−0.543253	−	0.839569i	\(-0.682807\pi\)
							0.455462	+	0.890255i	\(0.349474\pi\)
\(47\)	33.0000	−	33.0000i	0.702128	−	0.702128i	−0.262739	−	0.964867i	\(-0.584626\pi\)
							0.964867	+	0.262739i	\(0.0846259\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	26.3.f.a.7.1	✓	4
3.2	odd	2		234.3.bb.b.163.1		4
4.3	odd	2		208.3.bd.c.33.1		4
13.2	odd	12	inner	26.3.f.a.15.1	yes	4
13.3	even	3		338.3.f.f.89.1		4
13.4	even	6		338.3.d.e.99.2		4
13.5	odd	4		338.3.f.f.19.1		4
13.6	odd	12		338.3.d.d.239.2		4
13.7	odd	12		338.3.d.e.239.2		4
13.8	odd	4		338.3.f.c.19.1		4
13.9	even	3		338.3.d.d.99.2		4
13.10	even	6		338.3.f.c.89.1		4
13.11	odd	12		338.3.f.d.249.1		4
13.12	even	2		338.3.f.d.319.1		4
39.2	even	12		234.3.bb.b.145.1		4
52.15	even	12		208.3.bd.c.145.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.3.f.a.7.1	✓	4	1.1	even	1	trivial
26.3.f.a.15.1	yes	4	13.2	odd	12	inner
208.3.bd.c.33.1		4	4.3	odd	2
208.3.bd.c.145.1		4	52.15	even	12
234.3.bb.b.145.1		4	39.2	even	12
234.3.bb.b.163.1		4	3.2	odd	2
338.3.d.d.99.2		4	13.9	even	3
338.3.d.d.239.2		4	13.6	odd	12
338.3.d.e.99.2		4	13.4	even	6
338.3.d.e.239.2		4	13.7	odd	12
338.3.f.c.19.1		4	13.8	odd	4
338.3.f.c.89.1		4	13.10	even	6
338.3.f.d.249.1		4	13.11	odd	12
338.3.f.d.319.1		4	13.12	even	2
338.3.f.f.19.1		4	13.5	odd	4
338.3.f.f.89.1		4	13.3	even	3