Embedded newform 35.3.h.a.31.3

• Label: 35.3.h.a.31.3
• 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			31.3
Root			\(0.410701 - 0.711354i\) of defining polynomial
Character	\(\chi\)	\(=\)	35.31
Dual form			35.3.h.a.26.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.410701 - 0.711354i) q^{2} +(0.507487 + 0.292998i) q^{3} +(1.66265 - 2.87979i) q^{4} +(1.93649 - 1.11803i) q^{5} -0.481337i q^{6} +(1.91172 + 6.73389i) q^{7} -6.01701 q^{8} +(-4.32830 - 7.49684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 6 q^{3} - 10 q^{4} - 2 q^{7} - 4 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.410701	−	0.711354i	−0.205350	−	0.355677i	0.744894	−	0.667183i	\(-0.232500\pi\)
							−0.950244	+	0.311506i	\(0.899167\pi\)
\(3\)	0.507487	+	0.292998i	0.169162	+	0.0976659i	0.582191	−	0.813052i	\(-0.302196\pi\)
							−0.413028	+	0.910718i	\(0.635529\pi\)
\(5\)	1.93649	−	1.11803i	0.387298	−	0.223607i
							\(7\)	1.91172	+	6.73389i	0.273103	+	0.961985i
\(11\)	−6.95248	+	12.0421i	−0.632044	+	1.09473i								0.355089	+	0.934832i	\(0.384450\pi\)
							−0.987133	+	0.159900i	\(0.948883\pi\)
\(13\)			18.3546i			1.41189i	0.708266	+	0.705946i	\(0.249478\pi\)
							−0.708266	+	0.705946i	\(0.750522\pi\)
\(17\)	−8.98966	−	5.19018i	−0.528804	−	0.305305i	0.211726	−	0.977329i	\(-0.432092\pi\)
							−0.740529	+	0.672024i	\(0.765425\pi\)
\(19\)	24.4251	−	14.1019i	1.28553	−	0.742203i	0.307679	−	0.951490i	\(-0.400448\pi\)
							0.977854	+	0.209287i	\(0.0671143\pi\)
\(23\)	−4.84179	−	8.38623i	−0.210513	−	0.364619i	0.741362	−	0.671105i	\(-0.234180\pi\)
							−0.951875	+	0.306486i	\(0.900847\pi\)
\(29\)	−17.7531			−0.612176			−0.306088	−	0.952003i	\(-0.599020\pi\)
							−0.306088	+	0.952003i	\(0.599020\pi\)
\(31\)	25.1484	+	14.5194i	0.811239	+	0.468369i	0.847386	−	0.530978i	\(-0.178175\pi\)
							−0.0361471	+	0.999346i	\(0.511508\pi\)
\(37\)	−17.2496	−	29.8772i	−0.466205	−	0.807491i	0.533050	−	0.846084i	\(-0.321046\pi\)
							−0.999255	+	0.0385930i	\(0.987712\pi\)
\(41\)		−	13.3513i		−	0.325641i	−0.986656	−	0.162821i	\(-0.947941\pi\)
							0.986656	−	0.162821i	\(-0.0520592\pi\)
\(43\)	0.607058			0.0141176			0.00705881	−	0.999975i	\(-0.497753\pi\)
							0.00705881	+	0.999975i	\(0.497753\pi\)
\(47\)	−5.51096	+	3.18175i	−0.117254	+	0.0676969i	−0.557480	−	0.830190i	\(-0.688232\pi\)
							0.440226	+	0.897887i	\(0.354898\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.3.h.a.31.3	yes	12
3.2	odd	2		315.3.w.c.136.4		12
4.3	odd	2		560.3.bx.c.241.3		12
5.2	odd	4		175.3.j.b.24.8		24
5.3	odd	4		175.3.j.b.24.5		24
5.4	even	2		175.3.i.d.101.4		12
7.2	even	3		245.3.h.c.166.3		12
7.3	odd	6		245.3.d.a.146.8		12
7.4	even	3		245.3.d.a.146.7		12
7.5	odd	6	inner	35.3.h.a.26.3	✓	12
7.6	odd	2		245.3.h.c.31.3		12
21.5	even	6		315.3.w.c.271.4		12
28.19	even	6		560.3.bx.c.481.3		12
35.12	even	12		175.3.j.b.124.5		24
35.19	odd	6		175.3.i.d.26.4		12
35.33	even	12		175.3.j.b.124.8		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.h.a.26.3	✓	12	7.5	odd	6	inner
35.3.h.a.31.3	yes	12	1.1	even	1	trivial
175.3.i.d.26.4		12	35.19	odd	6
175.3.i.d.101.4		12	5.4	even	2
175.3.j.b.24.5		24	5.3	odd	4
175.3.j.b.24.8		24	5.2	odd	4
175.3.j.b.124.5		24	35.12	even	12
175.3.j.b.124.8		24	35.33	even	12
245.3.d.a.146.7		12	7.4	even	3
245.3.d.a.146.8		12	7.3	odd	6
245.3.h.c.31.3		12	7.6	odd	2
245.3.h.c.166.3		12	7.2	even	3
315.3.w.c.136.4		12	3.2	odd	2
315.3.w.c.271.4		12	21.5	even	6
560.3.bx.c.241.3		12	4.3	odd	2
560.3.bx.c.481.3		12	28.19	even	6