Embedded newform 315.2.z.b.104.8

• Label: 315.2.z.b.104.8
• Level: $315$
• Weight: $2$
• Character: 315.104
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.z (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			104.8
Character	\(\chi\)	\(=\)	315.104
Dual form			315.2.z.b.209.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.05869 - 1.83370i) q^{2} +(1.03202 - 1.39102i) q^{3} +(-1.24164 + 2.15058i) q^{4} +(1.87597 + 1.21685i) q^{5} +(-3.64330 - 0.419758i) q^{6} +(2.55499 - 0.687047i) q^{7} +1.02327 q^{8} +(-0.869874 - 2.87112i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.05869	−	1.83370i	−0.748605	−	1.29662i	−0.948491	−	0.316803i	\(-0.897391\pi\)
							0.199887	−	0.979819i	\(-0.435943\pi\)
\(3\)	1.03202	−	1.39102i	0.595836	−	0.803106i
							\(5\)	1.87597	+	1.21685i	0.838961	+	0.544192i
\(7\)	2.55499	−	0.687047i	0.965695	−	0.259679i
							\(11\)	−1.72589	+	0.996444i	−0.520376	+	0.300439i	−0.737088	−	0.675796i	\(-0.763800\pi\)
0.216713	+	0.976235i	\(0.430467\pi\)
\(13\)	1.33105	−	2.30545i	0.369167	−	0.639416i	−0.620268	−	0.784390i	\(-0.712976\pi\)
							0.989436	+	0.144973i	\(0.0463096\pi\)
\(17\)		−	5.46375i		−	1.32515i	−0.748994	−	0.662577i	\(-0.769463\pi\)
							0.748994	−	0.662577i	\(-0.230537\pi\)
\(19\)		−	3.05175i		−	0.700121i	−0.936727	−	0.350060i	\(-0.886161\pi\)
							0.936727	−	0.350060i	\(-0.113839\pi\)
\(23\)	−0.801634	+	1.38847i	−0.167152	+	0.289516i	−0.937417	−	0.348207i	\(-0.886790\pi\)
							0.770265	+	0.637724i	\(0.220124\pi\)
\(29\)	−6.66642	+	3.84886i	−1.23792	+	0.714715i	−0.968669	−	0.248355i	\(-0.920110\pi\)
							−0.269253	+	0.963069i	\(0.586777\pi\)
\(31\)	6.51499	+	3.76143i	1.17013	+	0.675573i	0.953710	−	0.300729i	\(-0.0972299\pi\)
							0.216416	+	0.976301i	\(0.430563\pi\)
\(37\)			4.99246i			0.820756i	0.911916	+	0.410378i	\(0.134603\pi\)
							−0.911916	+	0.410378i	\(0.865397\pi\)
\(41\)	−4.47806	+	7.75624i	−0.699356	+	1.21132i	0.269334	+	0.963047i	\(0.413197\pi\)
							−0.968690	+	0.248274i	\(0.920137\pi\)
\(43\)	−5.97669	+	3.45064i	−0.911437	+	0.526218i	−0.880893	−	0.473315i	\(-0.843057\pi\)
							−0.0305436	+	0.999533i	\(0.509724\pi\)
\(47\)	1.30305	−	0.752317i	0.190070	−	0.109737i	−0.401946	−	0.915664i	\(-0.631666\pi\)
							0.592015	+	0.805927i	\(0.298333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.b.104.8	yes	80
3.2	odd	2		945.2.z.b.314.33		80
5.4	even	2	inner	315.2.z.b.104.33	yes	80
7.6	odd	2	inner	315.2.z.b.104.7	✓	80
9.2	odd	6	inner	315.2.z.b.209.34	yes	80
9.7	even	3		945.2.z.b.629.7		80
15.14	odd	2		945.2.z.b.314.8		80
21.20	even	2		945.2.z.b.314.34		80
35.34	odd	2	inner	315.2.z.b.104.34	yes	80
45.29	odd	6	inner	315.2.z.b.209.7	yes	80
45.34	even	6		945.2.z.b.629.34		80
63.20	even	6	inner	315.2.z.b.209.33	yes	80
63.34	odd	6		945.2.z.b.629.8		80
105.104	even	2		945.2.z.b.314.7		80
315.34	odd	6		945.2.z.b.629.33		80
315.209	even	6	inner	315.2.z.b.209.8	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.b.104.7	✓	80	7.6	odd	2	inner
315.2.z.b.104.8	yes	80	1.1	even	1	trivial
315.2.z.b.104.33	yes	80	5.4	even	2	inner
315.2.z.b.104.34	yes	80	35.34	odd	2	inner
315.2.z.b.209.7	yes	80	45.29	odd	6	inner
315.2.z.b.209.8	yes	80	315.209	even	6	inner
315.2.z.b.209.33	yes	80	63.20	even	6	inner
315.2.z.b.209.34	yes	80	9.2	odd	6	inner
945.2.z.b.314.7		80	105.104	even	2
945.2.z.b.314.8		80	15.14	odd	2
945.2.z.b.314.33		80	3.2	odd	2
945.2.z.b.314.34		80	21.20	even	2
945.2.z.b.629.7		80	9.7	even	3
945.2.z.b.629.8		80	63.34	odd	6
945.2.z.b.629.33		80	315.34	odd	6
945.2.z.b.629.34		80	45.34	even	6