Embedded newform 35.3.h.a.31.1

• Label: 35.3.h.a.31.1
• Level: $35$
• Weight: $3$
• Character: 35.31
• 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			31.1
Root			\(1.77870 - 3.08079i\) of defining polynomial
Character	\(\chi\)	\(=\)	35.31
Dual form			35.3.h.a.26.1

$q$-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 \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

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

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	−0.840648	−	0.541582i	\(-0.817826\pi\)
							−0.0486998	−	0.998813i	\(-0.515508\pi\)
\(3\)	−2.10717	−	1.21658i	−0.702390	−	0.405525i	0.105847	−	0.994382i	\(-0.466245\pi\)
							−0.808237	+	0.588857i	\(0.799578\pi\)
\(5\)	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								−0.355300	−	0.934752i	\(-0.615621\pi\)
							0.987169	−	0.159677i	\(-0.0510453\pi\)
\(13\)			0.702771i			0.0540593i	0.999635	+	0.0270296i	\(0.00860485\pi\)
							−0.999635	+	0.0270296i	\(0.991395\pi\)
\(17\)	23.6124	+	13.6326i	1.38896	+	0.801918i	0.993198	−	0.116435i	\(-0.0371466\pi\)
							0.395764	+	0.918352i	\(0.370480\pi\)
\(19\)	2.21652	−	1.27971i	0.116659	−	0.0673530i	−0.440535	−	0.897735i	\(-0.645211\pi\)
							0.557194	+	0.830382i	\(0.311878\pi\)
\(23\)	−16.4777	−	28.5402i	−0.716422	−	1.24088i	−0.962409	−	0.271605i	\(-0.912445\pi\)
							0.245987	−	0.969273i	\(-0.420888\pi\)
\(29\)	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	0.795869	−	0.605469i	\(-0.207014\pi\)
							−0.126417	+	0.991977i	\(0.540348\pi\)
\(37\)	11.9080	+	20.6253i	0.321838	+	0.557440i	0.980867	−	0.194677i	\(-0.0623659\pi\)
							−0.659029	+	0.752117i	\(0.729033\pi\)
\(41\)		−	25.1015i		−	0.612231i	−0.951994	−	0.306116i	\(-0.900971\pi\)
							0.951994	−	0.306116i	\(-0.0990295\pi\)
\(43\)	−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	−0.558364	−	0.829596i	\(-0.688571\pi\)
							0.439270	+	0.898355i	\(0.355237\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.h.a.26.1	✓	12	7.5	odd	6	inner
35.3.h.a.31.1	yes	12	1.1	even	1	trivial
175.3.i.d.26.6		12	35.19	odd	6
175.3.i.d.101.6		12	5.4	even	2
175.3.j.b.24.1		24	5.3	odd	4
175.3.j.b.24.12		24	5.2	odd	4
175.3.j.b.124.1		24	35.12	even	12
175.3.j.b.124.12		24	35.33	even	12
245.3.d.a.146.11		12	7.3	odd	6
245.3.d.a.146.12		12	7.4	even	3
245.3.h.c.31.1		12	7.6	odd	2
245.3.h.c.166.1		12	7.2	even	3
315.3.w.c.136.6		12	3.2	odd	2
315.3.w.c.271.6		12	21.5	even	6
560.3.bx.c.241.5		12	4.3	odd	2
560.3.bx.c.481.5		12	28.19	even	6