Embedded newform 35.5.l.a.2.13

• Label: 35.5.l.a.2.13
• 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			2.13
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.17220 - 1.65384i) q^{2} +(2.01200 + 0.539113i) q^{3} +(21.5045 - 12.4156i) q^{4} +(3.38702 + 24.7695i) q^{5} +13.3101 q^{6} +(2.59467 - 48.9313i) q^{7} +(39.9028 - 39.9028i) q^{8} +(-66.3906 - 38.3306i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.17220	−	1.65384i	1.54305	−	0.413459i	0.615800	−	0.787902i	\(-0.288833\pi\)
							0.927250	+	0.374443i	\(0.122166\pi\)
\(3\)	2.01200	+	0.539113i	0.223555	+	0.0599015i	0.368858	−	0.929486i	\(-0.379749\pi\)
							−0.145303	+	0.989387i	\(0.546416\pi\)
\(5\)	3.38702	+	24.7695i	0.135481	+	0.990780i
							\(7\)	2.59467	−	48.9313i	0.0529525	−	0.998597i
\(11\)	−2.47446	−	4.28589i	−0.0204501	−	0.0354206i								0.855619	−	0.517606i	\(-0.173177\pi\)
							−0.876069	+	0.482185i	\(0.839843\pi\)
\(13\)	−145.178	+	145.178i	−0.859042	+	0.859042i	−0.991225	−	0.132183i	\(-0.957801\pi\)
							0.132183	+	0.991225i	\(0.457801\pi\)
\(17\)	34.4477	−	128.561i	0.119196	−	0.444846i	−0.880370	−	0.474287i	\(-0.842706\pi\)
							0.999567	+	0.0294407i	\(0.00937264\pi\)
\(19\)	516.304	+	298.088i	1.43020	+	0.825729i	0.997136	−	0.0756326i	\(-0.0240976\pi\)
							0.433068	+	0.901361i	\(0.357431\pi\)
\(23\)	−70.2844	−	262.305i	−0.132863	−	0.495851i	0.867135	−	0.498074i	\(-0.165959\pi\)
							−0.999998	+	0.00222290i	\(0.999292\pi\)
\(29\)		−	870.666i		−	1.03528i	−0.855600	−	0.517638i	\(-0.826812\pi\)
							0.855600	−	0.517638i	\(-0.173188\pi\)
\(31\)	−16.7638	−	29.0357i	−0.0174441	−	0.0302141i	0.857172	−	0.515031i	\(-0.172220\pi\)
							−0.874616	+	0.484817i	\(0.838886\pi\)
\(37\)	2173.69	−	582.437i	1.58779	−	0.425447i	0.646464	−	0.762944i	\(-0.276247\pi\)
							0.941327	+	0.337497i	\(0.109580\pi\)
\(41\)	441.857			0.262854			0.131427	−	0.991326i	\(-0.458044\pi\)
							0.131427	+	0.991326i	\(0.458044\pi\)
\(43\)	631.163	−	631.163i	0.341353	−	0.341353i	−0.515523	−	0.856876i	\(-0.672402\pi\)
							0.856876	+	0.515523i	\(0.172402\pi\)
\(47\)	748.705	−	200.615i	0.338934	−	0.0908171i	−0.0853376	−	0.996352i	\(-0.527197\pi\)
							0.424272	+	0.905535i	\(0.360530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.13	✓	56
5.3	odd	4	inner	35.5.l.a.23.2	yes	56
7.4	even	3	inner	35.5.l.a.32.2	yes	56
35.18	odd	12	inner	35.5.l.a.18.13	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.13	✓	56	1.1	even	1	trivial
35.5.l.a.18.13	yes	56	35.18	odd	12	inner
35.5.l.a.23.2	yes	56	5.3	odd	4	inner
35.5.l.a.32.2	yes	56	7.4	even	3	inner