Embedded newform 35.3.h.a.31.5

• Label: 35.3.h.a.31.5
• 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			31.5
Root			\(-0.925400 + 1.60284i\) of defining polynomial
Character	\(\chi\)	\(=\)	35.31
Dual form			35.3.h.a.26.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.925400 + 1.60284i) q^{2} +(0.731043 + 0.422068i) q^{3} +(0.287270 - 0.497567i) q^{4} +(-1.93649 + 1.11803i) q^{5} +1.56233i q^{6} +(-6.88972 + 1.23763i) q^{7} +8.46656 q^{8} +(-4.14372 - 7.17713i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 6 q^{3} - 10 q^{4} - 2 q^{7} - 4 q^{8} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.925400	+	1.60284i	0.462700	+	0.801420i	0.999094	−	0.0425476i	\(-0.0135474\pi\)
							−0.536395	+	0.843967i	\(0.680214\pi\)
\(3\)	0.731043	+	0.422068i	0.243681	+	0.140689i	0.616867	−	0.787067i	\(-0.288401\pi\)
							−0.373186	+	0.927756i	\(0.621735\pi\)
\(5\)	−1.93649	+	1.11803i	−0.387298	+	0.223607i
							\(7\)	−6.88972	+	1.23763i	−0.984246	+	0.176804i
\(11\)	3.93873	−	6.82208i	0.358066	−	0.620189i								−0.629572	−	0.776942i	\(-0.716770\pi\)
							0.987638	+	0.156754i	\(0.0501029\pi\)
\(13\)			2.73368i			0.210283i	0.994457	+	0.105142i	\(0.0335296\pi\)
							−0.994457	+	0.105142i	\(0.966470\pi\)
\(17\)	4.29344	+	2.47882i	0.252555	+	0.145813i	0.620934	−	0.783863i	\(-0.286754\pi\)
							−0.368379	+	0.929676i	\(0.620087\pi\)
\(19\)	−25.3152	+	14.6158i	−1.33238	+	0.769250i	−0.985664	−	0.168720i	\(-0.946037\pi\)
							−0.346716	+	0.937970i	\(0.612703\pi\)
\(23\)	18.0840	+	31.3224i	0.786261	+	1.36184i	0.928243	+	0.371975i	\(0.121319\pi\)
							−0.141982	+	0.989869i	\(0.545347\pi\)
\(29\)	0.399005			0.0137588			0.00687940	−	0.999976i	\(-0.497810\pi\)
							0.00687940	+	0.999976i	\(0.497810\pi\)
\(31\)	17.4396	+	10.0688i	0.562568	+	0.324799i	0.754176	−	0.656673i	\(-0.228037\pi\)
							−0.191607	+	0.981472i	\(0.561370\pi\)
\(37\)	3.61299	+	6.25789i	0.0976485	+	0.169132i	0.910711	−	0.413044i	\(-0.135535\pi\)
							−0.813062	+	0.582177i	\(0.802201\pi\)
\(41\)		−	71.1139i		−	1.73448i	−0.497887	−	0.867242i	\(-0.665890\pi\)
							0.497887	−	0.867242i	\(-0.334110\pi\)
\(43\)	−51.7627			−1.20378			−0.601892	−	0.798577i	\(-0.705586\pi\)
							−0.601892	+	0.798577i	\(0.705586\pi\)
\(47\)	−19.9766	+	11.5335i	−0.425034	+	0.245393i	−0.697229	−	0.716849i	\(-0.745584\pi\)
							0.272195	+	0.962242i	\(0.412250\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.3.h.a.31.5	yes	12
3.2	odd	2		315.3.w.c.136.2		12
4.3	odd	2		560.3.bx.c.241.2		12
5.2	odd	4		175.3.j.b.24.4		24
5.3	odd	4		175.3.j.b.24.9		24
5.4	even	2		175.3.i.d.101.2		12
7.2	even	3		245.3.h.c.166.5		12
7.3	odd	6		245.3.d.a.146.4		12
7.4	even	3		245.3.d.a.146.3		12
7.5	odd	6	inner	35.3.h.a.26.5	✓	12
7.6	odd	2		245.3.h.c.31.5		12
21.5	even	6		315.3.w.c.271.2		12
28.19	even	6		560.3.bx.c.481.2		12
35.12	even	12		175.3.j.b.124.9		24
35.19	odd	6		175.3.i.d.26.2		12
35.33	even	12		175.3.j.b.124.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.h.a.26.5	✓	12	7.5	odd	6	inner
35.3.h.a.31.5	yes	12	1.1	even	1	trivial
175.3.i.d.26.2		12	35.19	odd	6
175.3.i.d.101.2		12	5.4	even	2
175.3.j.b.24.4		24	5.2	odd	4
175.3.j.b.24.9		24	5.3	odd	4
175.3.j.b.124.4		24	35.33	even	12
175.3.j.b.124.9		24	35.12	even	12
245.3.d.a.146.3		12	7.4	even	3
245.3.d.a.146.4		12	7.3	odd	6
245.3.h.c.31.5		12	7.6	odd	2
245.3.h.c.166.5		12	7.2	even	3
315.3.w.c.136.2		12	3.2	odd	2
315.3.w.c.271.2		12	21.5	even	6
560.3.bx.c.241.2		12	4.3	odd	2
560.3.bx.c.481.2		12	28.19	even	6