Embedded newform 315.2.r.b.184.5

• Label: 315.2.r.b.184.5
• Level: $315$
• Weight: $2$
• Character: 315.184
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			184.5
Character	\(\chi\)	\(=\)	315.184
Dual form			315.2.r.b.214.38

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.37032i q^{2} +(0.187779 + 1.72184i) q^{3} -3.61843 q^{4} +(2.03667 - 0.923024i) q^{5} +(4.08132 - 0.445097i) q^{6} +(0.987053 + 2.45474i) q^{7} +3.83621i q^{8} +(-2.92948 + 0.646652i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 88 q^{4} + 3 q^{5} + 6 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(136\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.37032i		−	1.67607i	−0.545615	−	0.838036i	\(-0.683704\pi\)
							0.545615	−	0.838036i	\(-0.316296\pi\)
\(3\)	0.187779	+	1.72184i	0.108414	+	0.994106i
							\(5\)	2.03667	−	0.923024i	0.910827	−	0.412789i
\(7\)	0.987053	+	2.45474i	0.373071	+	0.927803i
							\(11\)	2.27384	−	3.93841i	0.685589	−	1.18747i	−0.287663	−	0.957732i	\(-0.592878\pi\)
0.973251	−	0.229742i	\(-0.0737884\pi\)
\(13\)	3.86101	+	2.22916i	1.07085	+	0.618257i	0.928414	−	0.371547i	\(-0.121172\pi\)
							0.142438	+	0.989804i	\(0.454506\pi\)
\(17\)	4.70772	−	2.71800i	1.14179	−	0.659213i	0.194917	−	0.980820i	\(-0.437556\pi\)
							0.946873	+	0.321607i	\(0.104223\pi\)
\(19\)	1.13162	−	1.96003i	0.259612	−	0.449661i	−0.706526	−	0.707687i	\(-0.749739\pi\)
							0.966138	+	0.258026i	\(0.0830720\pi\)
\(23\)	−4.98344	+	2.87719i	−1.03912	+	0.599936i	−0.919584	−	0.392894i	\(-0.871474\pi\)
							−0.119535	+	0.992830i	\(0.538140\pi\)
\(29\)	0.995443	+	1.72416i	0.184849	+	0.320168i	0.943526	−	0.331299i	\(-0.107487\pi\)
							−0.758677	+	0.651467i	\(0.774154\pi\)
\(31\)	−8.97508			−1.61197			−0.805986	−	0.591934i	\(-0.798365\pi\)
							−0.805986	+	0.591934i	\(0.798365\pi\)
\(37\)	−0.398664	−	0.230169i	−0.0655399	−	0.0378395i	0.466872	−	0.884325i	\(-0.345381\pi\)
							−0.532412	+	0.846485i	\(0.678714\pi\)
\(41\)	−3.32066	+	5.75155i	−0.518600	+	0.898241i	0.481167	+	0.876629i	\(0.340213\pi\)
							−0.999766	+	0.0216119i	\(0.993120\pi\)
\(43\)	1.73012	−	0.998884i	0.263840	−	0.152328i	−0.362245	−	0.932083i	\(-0.617990\pi\)
							0.626085	+	0.779755i	\(0.284656\pi\)
\(47\)			0.885837i			0.129213i	0.997911	+	0.0646063i	\(0.0205792\pi\)
							−0.997911	+	0.0646063i	\(0.979421\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.r.b.184.5	✓	84
3.2	odd	2		945.2.r.b.604.38		84
5.4	even	2	inner	315.2.r.b.184.38	yes	84
7.4	even	3		315.2.bo.b.4.38	yes	84
9.2	odd	6		945.2.bo.b.289.38		84
9.7	even	3		315.2.bo.b.79.5	yes	84
15.14	odd	2		945.2.r.b.604.5		84
21.11	odd	6		945.2.bo.b.739.5		84
35.4	even	6		315.2.bo.b.4.5	yes	84
45.29	odd	6		945.2.bo.b.289.5		84
45.34	even	6		315.2.bo.b.79.38	yes	84
63.11	odd	6		945.2.r.b.424.38		84
63.25	even	3	inner	315.2.r.b.214.5	yes	84
105.74	odd	6		945.2.bo.b.739.38		84
315.74	odd	6		945.2.r.b.424.5		84
315.214	even	6	inner	315.2.r.b.214.38	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.r.b.184.5	✓	84	1.1	even	1	trivial
315.2.r.b.184.38	yes	84	5.4	even	2	inner
315.2.r.b.214.5	yes	84	63.25	even	3	inner
315.2.r.b.214.38	yes	84	315.214	even	6	inner
315.2.bo.b.4.5	yes	84	35.4	even	6
315.2.bo.b.4.38	yes	84	7.4	even	3
315.2.bo.b.79.5	yes	84	9.7	even	3
315.2.bo.b.79.38	yes	84	45.34	even	6
945.2.r.b.424.5		84	315.74	odd	6
945.2.r.b.424.38		84	63.11	odd	6
945.2.r.b.604.5		84	15.14	odd	2
945.2.r.b.604.38		84	3.2	odd	2
945.2.bo.b.289.5		84	45.29	odd	6
945.2.bo.b.289.38		84	9.2	odd	6
945.2.bo.b.739.5		84	21.11	odd	6
945.2.bo.b.739.38		84	105.74	odd	6