Embedded newform 35.5.l.a.2.14

• Label: 35.5.l.a.2.14
• 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			2.14
Character	\(\chi\)	\(=\)	35.2
Dual form			35.5.l.a.18.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.18170 - 1.92433i) q^{2} +(-11.0263 - 2.95448i) q^{3} +(34.0173 - 19.6399i) q^{4} +(13.9161 - 20.7688i) q^{5} -84.8726 q^{6} +(-4.26291 + 48.8142i) q^{7} +(122.391 - 122.391i) q^{8} +(42.7013 + 24.6536i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.18170	−	1.92433i	1.79542	−	0.481083i	0.802176	−	0.597088i	\(-0.203676\pi\)
							0.993249	+	0.116005i	\(0.0370089\pi\)
\(3\)	−11.0263	−	2.95448i	−1.22514	−	0.328275i	−0.412454	−	0.910978i	\(-0.635328\pi\)
							−0.812685	+	0.582703i	\(0.801995\pi\)
\(5\)	13.9161	−	20.7688i	0.556643	−	0.830752i
							\(7\)	−4.26291	+	48.8142i	−0.0869981	+	0.996208i
\(11\)	68.1224	+	117.991i	0.562995	+	0.975136i								0.997233	+	0.0743380i	\(0.0236844\pi\)
							−0.434238	+	0.900798i	\(0.642982\pi\)
\(13\)	−117.781	+	117.781i	−0.696927	+	0.696927i	−0.963747	−	0.266819i	\(-0.914027\pi\)
							0.266819	+	0.963747i	\(0.414027\pi\)
\(17\)	1.86004	−	6.94178i	0.00643614	−	0.0240200i	−0.962633	−	0.270809i	\(-0.912709\pi\)
							0.969069	+	0.246789i	\(0.0793755\pi\)
\(19\)	42.2095	+	24.3697i	0.116924	+	0.0675061i	0.557321	−	0.830297i	\(-0.311829\pi\)
							−0.440397	+	0.897803i	\(0.645162\pi\)
\(23\)	−0.793256	−	2.96047i	−0.00149954	−	0.00559635i	0.965172	−	0.261616i	\(-0.0842554\pi\)
							−0.966672	+	0.256020i	\(0.917589\pi\)
\(29\)		−	867.840i		−	1.03191i	−0.856614	−	0.515957i	\(-0.827436\pi\)
							0.856614	−	0.515957i	\(-0.172564\pi\)
\(31\)	−348.746	−	604.045i	−0.362899	−	0.628559i	0.625538	−	0.780194i	\(-0.284880\pi\)
							−0.988437	+	0.151635i	\(0.951546\pi\)
\(37\)	−2114.00	+	566.445i	−1.54419	+	0.413765i	−0.927618	−	0.373531i	\(-0.878147\pi\)
							−0.616575	+	0.787296i	\(0.711480\pi\)
\(41\)	1811.76			1.07778			0.538892	−	0.842375i	\(-0.318843\pi\)
							0.538892	+	0.842375i	\(0.318843\pi\)
\(43\)	−345.292	+	345.292i	−0.186745	+	0.186745i	−0.794287	−	0.607542i	\(-0.792156\pi\)
							0.607542	+	0.794287i	\(0.292156\pi\)
\(47\)	−372.321	+	99.7630i	−0.168547	+	0.0451621i	−0.342106	−	0.939661i	\(-0.611140\pi\)
							0.173559	+	0.984824i	\(0.444473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.5.l.a.2.14	✓	56
5.3	odd	4	inner	35.5.l.a.23.1	yes	56
7.4	even	3	inner	35.5.l.a.32.1	yes	56
35.18	odd	12	inner	35.5.l.a.18.14	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.5.l.a.2.14	✓	56	1.1	even	1	trivial
35.5.l.a.18.14	yes	56	35.18	odd	12	inner
35.5.l.a.23.1	yes	56	5.3	odd	4	inner
35.5.l.a.32.1	yes	56	7.4	even	3	inner