Embedded newform 35.3.h.a.26.1

• Label: 35.3.h.a.26.1
• Level: $35$
• Weight: $3$
• Character: 35.26
• Analytic conductor: $0.954$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	35.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.953680925261\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - 2*x^11 + 19*x^10 - 26*x^9 + 244*x^8 - 338*x^7 + 1249*x^6 - 986*x^5 + 3532*x^4 - 3618*x^3 + 3579*x^2 - 1386*x + 441
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			26.1
Root			\(1.77870 + 3.08079i\) of defining polynomial
Character	\(\chi\)	\(=\)	35.26
Dual form			35.3.h.a.31.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.77870 + 3.08079i) q^{2} +(-2.10717 + 1.21658i) q^{3} +(-4.32752 - 7.49548i) q^{4} +(1.93649 + 1.11803i) q^{5} -8.65567i q^{6} +(-5.39402 + 4.46146i) q^{7} +16.5598 q^{8} +(-1.53989 + 2.66717i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 6 q^{3} - 10 q^{4} - 2 q^{7} - 4 q^{8} + 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\)	\(22\)	\(31\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.77870	+	3.08079i	−0.889348	+	1.54040i	−0.0486998	+	0.998813i	\(0.515508\pi\)
							−0.840648	+	0.541582i	\(0.817826\pi\)
\(3\)	−2.10717	+	1.21658i	−0.702390	+	0.405525i	−0.808237	−	0.588857i	\(-0.799578\pi\)
							0.105847	+	0.994382i	\(0.466245\pi\)
\(5\)	1.93649	+	1.11803i	0.387298	+	0.223607i
							\(7\)	−5.39402	+	4.46146i	−0.770574	+	0.637351i
\(11\)	6.95056	+	12.0387i	0.631869	+	1.09443i								0.987169	+	0.159677i	\(0.0510453\pi\)
							−0.355300	+	0.934752i	\(0.615621\pi\)
\(13\)		−	0.702771i		−	0.0540593i	−0.999635	−	0.0270296i	\(-0.991395\pi\)
							0.999635	−	0.0270296i	\(-0.00860485\pi\)
\(17\)	23.6124	−	13.6326i	1.38896	−	0.801918i	0.395764	−	0.918352i	\(-0.370480\pi\)
							0.993198	+	0.116435i	\(0.0371466\pi\)
\(19\)	2.21652	+	1.27971i	0.116659	+	0.0673530i	0.557194	−	0.830382i	\(-0.311878\pi\)
							−0.440535	+	0.897735i	\(0.645211\pi\)
\(23\)	−16.4777	+	28.5402i	−0.716422	+	1.24088i	0.245987	+	0.969273i	\(0.420888\pi\)
							−0.962409	+	0.271605i	\(0.912445\pi\)
\(29\)	3.39850			0.117190			0.0585949	−	0.998282i	\(-0.481338\pi\)
							0.0585949	+	0.998282i	\(0.481338\pi\)
\(31\)	20.7530	−	11.9818i	0.669452	−	0.386508i	−0.126417	−	0.991977i	\(-0.540348\pi\)
							0.795869	+	0.605469i	\(0.207014\pi\)
\(37\)	11.9080	−	20.6253i	0.321838	−	0.557440i	−0.659029	−	0.752117i	\(-0.729033\pi\)
							0.980867	+	0.194677i	\(0.0623659\pi\)
\(41\)			25.1015i			0.612231i	0.951994	+	0.306116i	\(0.0990295\pi\)
							−0.951994	+	0.306116i	\(0.900971\pi\)
\(43\)	−25.1201			−0.584189			−0.292094	−	0.956390i	\(-0.594352\pi\)
							−0.292094	+	0.956390i	\(0.594352\pi\)
\(47\)	−5.59742	−	3.23167i	−0.119094	−	0.0687590i	0.439270	−	0.898355i	\(-0.355237\pi\)
							−0.558364	+	0.829596i	\(0.688571\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.3.h.a.26.1	✓	12
3.2	odd	2		315.3.w.c.271.6		12
4.3	odd	2		560.3.bx.c.481.5		12
5.2	odd	4		175.3.j.b.124.1		24
5.3	odd	4		175.3.j.b.124.12		24
5.4	even	2		175.3.i.d.26.6		12
7.2	even	3		245.3.d.a.146.11		12
7.3	odd	6	inner	35.3.h.a.31.1	yes	12
7.4	even	3		245.3.h.c.31.1		12
7.5	odd	6		245.3.d.a.146.12		12
7.6	odd	2		245.3.h.c.166.1		12
21.17	even	6		315.3.w.c.136.6		12
28.3	even	6		560.3.bx.c.241.5		12
35.3	even	12		175.3.j.b.24.1		24
35.17	even	12		175.3.j.b.24.12		24
35.24	odd	6		175.3.i.d.101.6		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.h.a.26.1	✓	12	1.1	even	1	trivial
35.3.h.a.31.1	yes	12	7.3	odd	6	inner
175.3.i.d.26.6		12	5.4	even	2
175.3.i.d.101.6		12	35.24	odd	6
175.3.j.b.24.1		24	35.3	even	12
175.3.j.b.24.12		24	35.17	even	12
175.3.j.b.124.1		24	5.2	odd	4
175.3.j.b.124.12		24	5.3	odd	4
245.3.d.a.146.11		12	7.2	even	3
245.3.d.a.146.12		12	7.5	odd	6
245.3.h.c.31.1		12	7.4	even	3
245.3.h.c.166.1		12	7.6	odd	2
315.3.w.c.136.6		12	21.17	even	6
315.3.w.c.271.6		12	3.2	odd	2
560.3.bx.c.241.5		12	28.3	even	6
560.3.bx.c.481.5		12	4.3	odd	2