Embedded newform 315.2.z.b.104.13

• Label: 315.2.z.b.104.13
• 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			104.13
Character	\(\chi\)	\(=\)	315.104
Dual form			315.2.z.b.209.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.639446 - 1.10755i) q^{2} +(-0.624436 - 1.61557i) q^{3} +(0.182218 - 0.315610i) q^{4} +(2.10896 - 0.743174i) q^{5} +(-1.39004 + 1.72467i) q^{6} +(-2.47149 - 0.944330i) q^{7} -3.02386 q^{8} +(-2.22016 + 2.01765i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 52 q^{4} - 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.639446	−	1.10755i	−0.452157	−	0.783158i	0.546363	−	0.837548i	\(-0.316012\pi\)
							−0.998520	+	0.0543901i	\(0.982679\pi\)
\(3\)	−0.624436	−	1.61557i	−0.360518	−	0.932752i
							\(5\)	2.10896	−	0.743174i	0.943153	−	0.332357i
\(7\)	−2.47149	−	0.944330i	−0.934134	−	0.356923i
							\(11\)	0.756854	−	0.436970i	0.228200	−	0.131751i	−0.381541	−	0.924352i	\(-0.624607\pi\)
0.609741	+	0.792600i	\(0.291273\pi\)
\(13\)	2.77311	−	4.80317i	0.769123	−	1.33216i	−0.168915	−	0.985631i	\(-0.554026\pi\)
							0.938039	−	0.346530i	\(-0.112640\pi\)
\(17\)			2.70681i			0.656497i	0.944591	+	0.328249i	\(0.106458\pi\)
							−0.944591	+	0.328249i	\(0.893542\pi\)
\(19\)		−	0.623463i		−	0.143032i	−0.997439	−	0.0715161i	\(-0.977216\pi\)
							0.997439	−	0.0715161i	\(-0.0227837\pi\)
\(23\)	−2.10353	+	3.64342i	−0.438616	+	0.759706i	−0.997583	−	0.0694844i	\(-0.977865\pi\)
							0.558967	+	0.829190i	\(0.311198\pi\)
\(29\)	−2.54943	+	1.47191i	−0.473417	+	0.273328i	−0.717669	−	0.696384i	\(-0.754791\pi\)
							0.244252	+	0.969712i	\(0.421458\pi\)
\(31\)	−5.50830	−	3.18022i	−0.989320	−	0.571184i	−0.0842489	−	0.996445i	\(-0.526849\pi\)
							−0.905071	+	0.425261i	\(0.860182\pi\)
\(37\)			1.16158i			0.190963i	0.995431	+	0.0954813i	\(0.0304390\pi\)
							−0.995431	+	0.0954813i	\(0.969561\pi\)
\(41\)	2.97054	−	5.14513i	0.463921	−	0.803535i	−0.535231	−	0.844706i	\(-0.679775\pi\)
							0.999152	+	0.0411708i	\(0.0131088\pi\)
\(43\)	10.2412	−	5.91274i	1.56176	−	0.901684i	0.564684	−	0.825307i	\(-0.308998\pi\)
							0.997079	−	0.0763773i	\(-0.0243354\pi\)
\(47\)	−4.94324	+	2.85398i	−0.721045	+	0.416296i	−0.815137	−	0.579268i	\(-0.803339\pi\)
							0.0940920	+	0.995564i	\(0.470005\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.z.b.104.13	✓	80
3.2	odd	2		945.2.z.b.314.27		80
5.4	even	2	inner	315.2.z.b.104.28	yes	80
7.6	odd	2	inner	315.2.z.b.104.14	yes	80
9.2	odd	6	inner	315.2.z.b.209.27	yes	80
9.7	even	3		945.2.z.b.629.13		80
15.14	odd	2		945.2.z.b.314.14		80
21.20	even	2		945.2.z.b.314.28		80
35.34	odd	2	inner	315.2.z.b.104.27	yes	80
45.29	odd	6	inner	315.2.z.b.209.14	yes	80
45.34	even	6		945.2.z.b.629.28		80
63.20	even	6	inner	315.2.z.b.209.28	yes	80
63.34	odd	6		945.2.z.b.629.14		80
105.104	even	2		945.2.z.b.314.13		80
315.34	odd	6		945.2.z.b.629.27		80
315.209	even	6	inner	315.2.z.b.209.13	yes	80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.z.b.104.13	✓	80	1.1	even	1	trivial
315.2.z.b.104.14	yes	80	7.6	odd	2	inner
315.2.z.b.104.27	yes	80	35.34	odd	2	inner
315.2.z.b.104.28	yes	80	5.4	even	2	inner
315.2.z.b.209.13	yes	80	315.209	even	6	inner
315.2.z.b.209.14	yes	80	45.29	odd	6	inner
315.2.z.b.209.27	yes	80	9.2	odd	6	inner
315.2.z.b.209.28	yes	80	63.20	even	6	inner
945.2.z.b.314.13		80	105.104	even	2
945.2.z.b.314.14		80	15.14	odd	2
945.2.z.b.314.27		80	3.2	odd	2
945.2.z.b.314.28		80	21.20	even	2
945.2.z.b.629.13		80	9.7	even	3
945.2.z.b.629.14		80	63.34	odd	6
945.2.z.b.629.27		80	315.34	odd	6
945.2.z.b.629.28		80	45.34	even	6