Embedded newform 35.6.e.a.11.6

• Label: 35.6.e.a.11.6
• Level: $35$
• Weight: $6$
• Character: 35.11
• Analytic conductor: $5.613$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.61343369345\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - x^11 + 110*x^10 + 27*x^9 + 9714*x^8 - 3257*x^7 + 240269*x^6 - 608504*x^5 + 5067668*x^4 - 7120352*x^3 + 9029136*x^2 - 1794240*x + 313600
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.6
Root			\(-4.08626 - 7.07762i\) of defining polynomial
Character	\(\chi\)	\(=\)	35.11
Dual form			35.6.e.a.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.58626 + 7.94364i) q^{2} +(-7.16289 + 12.4065i) q^{3} +(-26.0676 + 45.1505i) q^{4} +(12.5000 + 21.6506i) q^{5} -131.404 q^{6} +(-5.98242 - 129.504i) q^{7} -184.691 q^{8} +(18.8861 + 32.7118i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{2} - 20 q^{3} - 31 q^{4} + 150 q^{5} + 192 q^{6} - 20 q^{7} - 270 q^{8} - 378 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.58626	+	7.94364i	0.810745	+	1.40425i	0.912344	+	0.409425i	\(0.134271\pi\)
							−0.101599	+	0.994825i	\(0.532396\pi\)
\(3\)	−7.16289	+	12.4065i	−0.459499	+	0.795876i	−0.998934	−	0.0461510i	\(-0.985304\pi\)
							0.539435	+	0.842027i	\(0.318638\pi\)
\(5\)	12.5000	+	21.6506i	0.223607	+	0.387298i
							\(7\)	−5.98242	−	129.504i	−0.0461457	−	0.998935i
\(11\)	−107.693	+	186.529i	−0.268352	+	0.464799i								−0.968436	−	0.249261i	\(-0.919812\pi\)
							0.700084	+	0.714060i	\(0.253146\pi\)
\(13\)	132.696			0.217771			0.108886	−	0.994054i	\(-0.465272\pi\)
							0.108886	+	0.994054i	\(0.465272\pi\)
\(17\)	−220.819	+	382.470i	−0.185317	+	0.320978i	−0.943683	−	0.330851i	\(-0.892664\pi\)
							0.758367	+	0.651828i	\(0.225998\pi\)
\(19\)	−259.501	−	449.469i	−0.164913	−	0.285638i	0.771711	−	0.635973i	\(-0.219401\pi\)
							−0.936624	+	0.350335i	\(0.886068\pi\)
\(23\)	2194.83	+	3801.55i	0.865129	+	1.49845i	0.866919	+	0.498450i	\(0.166097\pi\)
							−0.00178940	+	0.999998i	\(0.500570\pi\)
\(29\)	7114.31			1.57086			0.785430	−	0.618950i	\(-0.212442\pi\)
							0.785430	+	0.618950i	\(0.212442\pi\)
\(31\)	5152.36	−	8924.15i	0.962946	−	1.66787i	0.247912	−	0.968783i	\(-0.420256\pi\)
							0.715034	−	0.699090i	\(-0.246411\pi\)
\(37\)	−2784.04	−	4822.10i	−0.334327	−	0.579071i	0.649028	−	0.760764i	\(-0.275176\pi\)
							−0.983355	+	0.181693i	\(0.941842\pi\)
\(41\)	7246.54			0.673242			0.336621	−	0.941640i	\(-0.390716\pi\)
							0.336621	+	0.941640i	\(0.390716\pi\)
\(43\)	−184.308			−0.0152010			−0.00760050	−	0.999971i	\(-0.502419\pi\)
							−0.00760050	+	0.999971i	\(0.502419\pi\)
\(47\)	−8087.09	−	14007.2i	−0.534008	−	0.924928i	−0.999211	−	0.0397243i	\(-0.987352\pi\)
							0.465203	−	0.885204i	\(-0.345981\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.6.e.a.11.6	✓	12
7.2	even	3	inner	35.6.e.a.16.6	yes	12
7.3	odd	6		245.6.a.h.1.1		6
7.4	even	3		245.6.a.i.1.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.e.a.11.6	✓	12	1.1	even	1	trivial
35.6.e.a.16.6	yes	12	7.2	even	3	inner
245.6.a.h.1.1		6	7.3	odd	6
245.6.a.i.1.1		6	7.4	even	3