Embedded newform 35.5.l.a.2.7

• Label: 35.5.l.a.2.7
• Level: $35$
• Weight: $5$
• Character: 35.2
• Analytic conductor: $3.618$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	35.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.61794870793\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.495094 + 0.132660i) q^{2} +(15.4296 + 4.13434i) q^{3} +(-13.6289 + 7.86864i) q^{4} +(12.6360 + 21.5715i) q^{5} -8.18755 q^{6} +(-36.5078 - 32.6831i) q^{7} +(11.5027 - 11.5027i) q^{8} +(150.831 + 87.0822i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(22\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.495094	+	0.132660i	−0.123774	+	0.0331650i	−0.320174	−	0.947359i	\(-0.603741\pi\)
							0.196401	+	0.980524i	\(0.437075\pi\)
\(3\)	15.4296	+	4.13434i	1.71440	+	0.459371i	0.976495	−	0.215538i	\(-0.0691505\pi\)
							0.737901	+	0.674909i	\(0.235817\pi\)
\(5\)	12.6360	+	21.5715i	0.505442	+	0.862861i
							\(7\)	−36.5078	−	32.6831i	−0.745057	−	0.667001i
\(11\)	42.7420	+	74.0314i	0.353240	+	0.611830i								0.986815	−	0.161852i	\(-0.0517466\pi\)
							−0.633575	+	0.773681i	\(0.718413\pi\)
\(13\)	131.873	−	131.873i	0.780311	−	0.780311i	−0.199572	−	0.979883i	\(-0.563955\pi\)
							0.979883	+	0.199572i	\(0.0639553\pi\)
\(17\)	49.9889	−	186.561i	0.172972	−	0.645540i	−0.823916	−	0.566712i	\(-0.808215\pi\)
							0.996888	−	0.0788284i	\(-0.0251179\pi\)
\(19\)	−435.787	−	251.602i	−1.20717	−	0.696958i	−0.245028	−	0.969516i	\(-0.578797\pi\)
							−0.962139	+	0.272558i	\(0.912130\pi\)
\(23\)	−58.9699	−	220.079i	−0.111474	−	0.416028i	0.887525	−	0.460760i	\(-0.152423\pi\)
							−0.998999	+	0.0447323i	\(0.985757\pi\)
\(29\)		−	592.118i		−	0.704065i	−0.935988	−	0.352032i	\(-0.885491\pi\)
							0.935988	−	0.352032i	\(-0.114509\pi\)
\(31\)	244.720	+	423.868i	0.254652	+	0.441069i	0.964801	−	0.262982i	\(-0.0847059\pi\)
							−0.710149	+	0.704051i	\(0.751373\pi\)
\(37\)	75.9617	−	20.3539i	0.0554870	−	0.0148677i	−0.230969	−	0.972961i	\(-0.574190\pi\)
							0.286456	+	0.958093i	\(0.407523\pi\)
\(41\)	−1391.63			−0.827858			−0.413929	−	0.910309i	\(-0.635844\pi\)
							−0.413929	+	0.910309i	\(0.635844\pi\)
\(43\)	−838.944	+	838.944i	−0.453728	+	0.453728i	−0.896590	−	0.442862i	\(-0.853963\pi\)
							0.442862	+	0.896590i	\(0.353963\pi\)
\(47\)	1806.26	−	483.986i	0.817683	−	0.219097i	0.174350	−	0.984684i	\(-0.444218\pi\)
							0.643333	+	0.765586i	\(0.277551\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.7	✓	56
5.3	odd	4	inner	35.5.l.a.23.8	yes	56
7.4	even	3	inner	35.5.l.a.32.8	yes	56
35.18	odd	12	inner	35.5.l.a.18.7	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.7	✓	56	1.1	even	1	trivial
35.5.l.a.18.7	yes	56	35.18	odd	12	inner
35.5.l.a.23.8	yes	56	5.3	odd	4	inner
35.5.l.a.32.8	yes	56	7.4	even	3	inner