Embedded newform 315.2.bq.a.194.41

• Label: 315.2.bq.a.194.41
• Level: $315$
• Weight: $2$
• Character: 315.194
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(44\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			194.41
Character	\(\chi\)	\(=\)	315.194
Dual form			315.2.bq.a.164.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.46032 q^{2} +(-0.957408 - 1.44339i) q^{3} +4.05317 q^{4} +(-1.64136 - 1.51853i) q^{5} +(-2.35553 - 3.55119i) q^{6} +(0.227160 - 2.63598i) q^{7} +5.05144 q^{8} +(-1.16674 + 2.76382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 76 q^{4} - 3 q^{5} - 12 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.46032			1.73971			0.869854	−	0.493310i	\(-0.164213\pi\)
							0.869854	+	0.493310i	\(0.164213\pi\)
\(3\)	−0.957408	−	1.44339i	−0.552760	−	0.833341i
							\(5\)	−1.64136	−	1.51853i	−0.734038	−	0.679108i
\(7\)	0.227160	−	2.63598i	0.0858585	−	0.996307i
							\(11\)	2.30650	+	1.33166i	0.695435	+	0.401510i	0.805645	−	0.592399i	\(-0.201819\pi\)
−0.110210	+	0.993908i	\(0.535152\pi\)
\(13\)	−1.60728	+	2.78389i	−0.445779	+	0.772113i	−0.998106	−	0.0615151i	\(-0.980407\pi\)
							0.552327	+	0.833628i	\(0.313740\pi\)
\(17\)	2.53088	−	1.46120i	0.613828	−	0.354394i	−0.160634	−	0.987014i	\(-0.551354\pi\)
							0.774462	+	0.632620i	\(0.218021\pi\)
\(19\)	5.06365	+	2.92350i	1.16168	+	0.670697i	0.951706	−	0.307009i	\(-0.0993283\pi\)
							0.209975	+	0.977707i	\(0.432662\pi\)
\(23\)	−1.27648	−	2.21092i	−0.266164	−	0.461009i	0.701704	−	0.712468i	\(-0.252423\pi\)
							−0.967868	+	0.251459i	\(0.919090\pi\)
\(29\)	4.19625	−	2.42271i	0.779224	−	0.449885i	−0.0569312	−	0.998378i	\(-0.518132\pi\)
							0.836155	+	0.548493i	\(0.184798\pi\)
\(31\)			7.07814i			1.27127i	0.771989	+	0.635636i	\(0.219262\pi\)
							−0.771989	+	0.635636i	\(0.780738\pi\)
\(37\)	−1.59927	−	0.923338i	−0.262918	−	0.151796i	0.362747	−	0.931888i	\(-0.381839\pi\)
							−0.625665	+	0.780092i	\(0.715172\pi\)
\(41\)	−5.20058	+	9.00767i	−0.812194	+	1.40676i	0.0991306	+	0.995074i	\(0.468394\pi\)
							−0.911325	+	0.411688i	\(0.864939\pi\)
\(43\)	−2.40794	+	1.39022i	−0.367207	+	0.212007i	−0.672237	−	0.740336i	\(-0.734667\pi\)
							0.305031	+	0.952343i	\(0.401333\pi\)
\(47\)		−	6.04398i		−	0.881605i	−0.897604	−	0.440803i	\(-0.854694\pi\)
							0.897604	−	0.440803i	\(-0.145306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bq.a.194.41	yes	88
3.2	odd	2		945.2.bq.a.719.4		88
5.4	even	2	inner	315.2.bq.a.194.4	yes	88
7.3	odd	6		315.2.u.a.59.4	✓	88
9.2	odd	6		315.2.u.a.299.41	yes	88
9.7	even	3		945.2.u.a.89.4		88
15.14	odd	2		945.2.bq.a.719.41		88
21.17	even	6		945.2.u.a.584.41		88
35.24	odd	6		315.2.u.a.59.41	yes	88
45.29	odd	6		315.2.u.a.299.4	yes	88
45.34	even	6		945.2.u.a.89.41		88
63.38	even	6	inner	315.2.bq.a.164.4	yes	88
63.52	odd	6		945.2.bq.a.899.41		88
105.59	even	6		945.2.u.a.584.4		88
315.164	even	6	inner	315.2.bq.a.164.41	yes	88
315.304	odd	6		945.2.bq.a.899.4		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.u.a.59.4	✓	88	7.3	odd	6
315.2.u.a.59.41	yes	88	35.24	odd	6
315.2.u.a.299.4	yes	88	45.29	odd	6
315.2.u.a.299.41	yes	88	9.2	odd	6
315.2.bq.a.164.4	yes	88	63.38	even	6	inner
315.2.bq.a.164.41	yes	88	315.164	even	6	inner
315.2.bq.a.194.4	yes	88	5.4	even	2	inner
315.2.bq.a.194.41	yes	88	1.1	even	1	trivial
945.2.u.a.89.4		88	9.7	even	3
945.2.u.a.89.41		88	45.34	even	6
945.2.u.a.584.4		88	105.59	even	6
945.2.u.a.584.41		88	21.17	even	6
945.2.bq.a.719.4		88	3.2	odd	2
945.2.bq.a.719.41		88	15.14	odd	2
945.2.bq.a.899.4		88	315.304	odd	6
945.2.bq.a.899.41		88	63.52	odd	6