Embedded newform 35.6.e.a.16.4

• Label: 35.6.e.a.16.4
• Level: $35$
• Weight: $6$
• Character: 35.16
• 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			16.4
Root			\(0.101659 - 0.176079i\) of defining polynomial
Character	\(\chi\)	\(=\)	35.16
Dual form			35.6.e.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.398341 - 0.689946i) q^{2} +(-5.28741 - 9.15807i) q^{3} +(15.6826 + 27.1631i) q^{4} +(12.5000 - 21.6506i) q^{5} -8.42477 q^{6} +(103.426 - 78.1675i) q^{7} +50.4819 q^{8} +(65.5865 - 113.599i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{2} - 20 q^{3} - 31 q^{4} + 150 q^{5} + 192 q^{6} - 20 q^{7} - 270 q^{8} - 378 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.398341	−	0.689946i	0.0704173	−	0.121966i	−0.828667	−	0.559742i	\(-0.810900\pi\)
							0.899084	+	0.437776i	\(0.144234\pi\)
\(3\)	−5.28741	−	9.15807i	−0.339188	−	0.587490i	0.645092	−	0.764105i	\(-0.276819\pi\)
							−0.984280	+	0.176614i	\(0.943486\pi\)
\(5\)	12.5000	−	21.6506i	0.223607	−	0.387298i
							\(7\)	103.426	−	78.1675i	0.797779	−	0.602950i
\(11\)	−262.646	−	454.915i	−0.654468	−	1.13357i								−0.982027	−	0.188741i	\(-0.939559\pi\)
							0.327559	−	0.944831i	\(-0.393774\pi\)
\(13\)	819.361			1.34467			0.672337	−	0.740245i	\(-0.265291\pi\)
							0.672337	+	0.740245i	\(0.265291\pi\)
\(17\)	460.684	+	797.927i	0.386616	+	0.669639i	0.991992	−	0.126301i	\(-0.0403104\pi\)
							−0.605376	+	0.795940i	\(0.706977\pi\)
\(19\)	348.088	−	602.906i	0.221210	−	0.383147i	−0.733966	−	0.679187i	\(-0.762333\pi\)
							0.955176	+	0.296040i	\(0.0956661\pi\)
\(23\)	−1190.94	+	2062.76i	−0.469428	+	0.813073i	−0.999389	−	0.0349486i	\(-0.988873\pi\)
							0.529961	+	0.848022i	\(0.322207\pi\)
\(29\)	−6200.04			−1.36899			−0.684494	−	0.729019i	\(-0.739977\pi\)
							−0.684494	+	0.729019i	\(0.739977\pi\)
\(31\)	4005.67	+	6938.03i	0.748637	+	1.29668i	0.948476	+	0.316849i	\(0.102625\pi\)
							−0.199839	+	0.979829i	\(0.564042\pi\)
\(37\)	4068.83	−	7047.42i	0.488614	−	0.846304i	−0.511301	−	0.859402i	\(-0.670836\pi\)
							0.999914	+	0.0130984i	\(0.00416947\pi\)
\(41\)	1703.92			0.158304			0.0791518	−	0.996863i	\(-0.474779\pi\)
							0.0791518	+	0.996863i	\(0.474779\pi\)
\(43\)	−8132.54			−0.670741			−0.335370	−	0.942086i	\(-0.608861\pi\)
							−0.335370	+	0.942086i	\(0.608861\pi\)
\(47\)	1663.01	−	2880.41i	0.109812	−	0.190200i	−0.805882	−	0.592076i	\(-0.798308\pi\)
							0.915694	+	0.401876i	\(0.131642\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.6.e.a.16.4	yes	12
7.2	even	3		245.6.a.i.1.3		6
7.4	even	3	inner	35.6.e.a.11.4	✓	12
7.5	odd	6		245.6.a.h.1.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.e.a.11.4	✓	12	7.4	even	3	inner
35.6.e.a.16.4	yes	12	1.1	even	1	trivial
245.6.a.h.1.3		6	7.5	odd	6
245.6.a.i.1.3		6	7.2	even	3