Embedded newform 310.3.i.a.99.15

• Label: 310.3.i.a.99.15
• Level: $310$
• Weight: $3$
• Character: 310.99
• Analytic conductor: $8.447$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	310.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.44688819517\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			99.15
Character	\(\chi\)	\(=\)	310.99
Dual form			310.3.i.a.119.31

$q$-expansion

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +(2.55575 + 4.42669i) q^{3} -2.00000 q^{4} +(-4.97323 + 0.516724i) q^{5} +(6.26029 - 3.61438i) q^{6} +(-1.46156 + 0.843833i) q^{7} +2.82843i q^{8} +(-8.56373 + 14.8328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 128 q^{4} + 6 q^{5} - 56 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.41421i		−	0.707107i
							\(3\)	2.55575	+	4.42669i	0.851917	+	1.47556i	0.879476	+	0.475944i	\(0.157893\pi\)
−0.0275586	+	0.999620i	\(0.508773\pi\)
\(5\)	−4.97323	+	0.516724i	−0.994646	+	0.103345i
							\(7\)	−1.46156	+	0.843833i	−0.208795	+	0.120548i	−0.600751	−	0.799436i	\(-0.705132\pi\)
0.391957	+	0.919984i	\(0.371798\pi\)
\(11\)	−15.9723	−	9.22159i	−1.45202	−	0.838326i	−0.453428	−	0.891293i	\(-0.649799\pi\)
							−0.998596	+	0.0529665i	\(0.983132\pi\)
\(13\)	−0.219739	+	0.380599i	−0.0169030	+	0.0292769i	−0.874353	−	0.485290i	\(-0.838714\pi\)
							0.857450	+	0.514567i	\(0.172047\pi\)
\(17\)	−13.5274	−	23.4302i	−0.795730	−	1.37825i	−0.922375	−	0.386297i	\(-0.873754\pi\)
							0.126644	−	0.991948i	\(-0.459579\pi\)
\(19\)	13.1230	+	22.7297i	0.690684	+	1.19630i	0.971614	+	0.236572i	\(0.0760238\pi\)
							−0.280930	+	0.959728i	\(0.590643\pi\)
\(23\)	−13.3027			−0.578378			−0.289189	−	0.957272i	\(-0.593386\pi\)
							−0.289189	+	0.957272i	\(0.593386\pi\)
\(29\)			6.74872i			0.232714i	0.993207	+	0.116357i	\(0.0371217\pi\)
							−0.993207	+	0.116357i	\(0.962878\pi\)
\(31\)	−16.6877	+	26.1251i	−0.538312	+	0.842745i
							\(37\)	−20.4903	−	35.4902i	−0.553792	−	0.959196i	−0.997996	−	0.0632705i	\(-0.979847\pi\)
0.444204	−	0.895925i	\(-0.353486\pi\)
\(41\)	−23.6999	+	41.0495i	−0.578047	+	1.00121i	0.417656	+	0.908605i	\(0.362852\pi\)
							−0.995703	+	0.0926019i	\(0.970482\pi\)
\(43\)	−5.79153	−	10.0312i	−0.134687	−	0.233284i	0.790791	−	0.612086i	\(-0.209669\pi\)
							−0.925478	+	0.378802i	\(0.876336\pi\)
\(47\)			17.4760i			0.371830i	0.982566	+	0.185915i	\(0.0595249\pi\)
							−0.982566	+	0.185915i	\(0.940475\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.3.i.a.99.15	✓	64
5.4	even	2	inner	310.3.i.a.99.18	yes	64
31.26	odd	6	inner	310.3.i.a.119.2	yes	64
155.119	odd	6	inner	310.3.i.a.119.31	yes	64

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.3.i.a.99.15	✓	64	1.1	even	1	trivial
310.3.i.a.99.18	yes	64	5.4	even	2	inner
310.3.i.a.119.2	yes	64	31.26	odd	6	inner
310.3.i.a.119.31	yes	64	155.119	odd	6	inner