Embedded newform 30.10.e.a.17.13

• Label: 30.10.e.a.17.13
• Level: $30$
• Weight: $10$
• Character: 30.17
• Analytic conductor: $15.451$
• Analytic rank: $0$
• Dimension: $36$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	30.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4510750849\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			17.13
Character	\(\chi\)	\(=\)	30.17
Dual form			30.10.e.a.23.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.3137 - 11.3137i) q^{2} +(-19.5470 + 138.928i) q^{3} -256.000i q^{4} +(-442.457 - 1325.65i) q^{5} +(1350.64 + 1792.94i) q^{6} +(4229.25 + 4229.25i) q^{7} +(-2896.31 - 2896.31i) q^{8} +(-18918.8 - 5431.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 296 q^{3} + 4544 q^{6} + 9756 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.3137	−	11.3137i	0.500000	−	0.500000i
							\(3\)	−19.5470	+	138.928i	−0.139326	+	0.990247i
\(5\)	−442.457	−	1325.65i	−0.316597	−	0.948560i
							\(7\)	4229.25	+	4229.25i	0.665767	+	0.665767i	0.956733	−	0.290966i	\(-0.0939768\pi\)
−0.290966	+	0.956733i	\(0.593977\pi\)
\(11\)		−	62494.5i		−	1.28699i	−0.765451	−	0.643495i	\(-0.777484\pi\)
							0.765451	−	0.643495i	\(-0.222516\pi\)
\(13\)	18069.4	−	18069.4i	0.175469	−	0.175469i	−0.613908	−	0.789377i	\(-0.710404\pi\)
							0.789377	+	0.613908i	\(0.210404\pi\)
\(17\)	363210.	−	363210.i	1.05472	−	1.05472i	0.0563082	−	0.998413i	\(-0.482067\pi\)
							0.998413	−	0.0563082i	\(-0.0179329\pi\)
\(19\)		−	979630.i		−	1.72453i	−0.506456	−	0.862266i	\(-0.669045\pi\)
							0.506456	−	0.862266i	\(-0.330955\pi\)
\(23\)	−260592.	−	260592.i	−0.194171	−	0.194171i	0.603325	−	0.797496i	\(-0.293842\pi\)
							−0.797496	+	0.603325i	\(0.793842\pi\)
\(29\)	1.27328e6			0.334297			0.167149	−	0.985932i	\(-0.446544\pi\)
							0.167149	+	0.985932i	\(0.446544\pi\)
\(31\)	−6.88480e6			−1.33895			−0.669474	−	0.742836i	\(-0.733480\pi\)
							−0.669474	+	0.742836i	\(0.733480\pi\)
\(37\)	1.46121e7	+	1.46121e7i	1.28176	+	1.28176i	0.939668	+	0.342087i	\(0.111134\pi\)
							0.342087	+	0.939668i	\(0.388866\pi\)
\(41\)			3.70774e6i			0.204919i	0.994737	+	0.102459i	\(0.0326711\pi\)
							−0.994737	+	0.102459i	\(0.967329\pi\)
\(43\)	−5.49298e6	+	5.49298e6i	−0.245019	+	0.245019i	−0.818923	−	0.573904i	\(-0.805428\pi\)
							0.573904	+	0.818923i	\(0.305428\pi\)
\(47\)	−1.61958e7	+	1.61958e7i	−0.484129	+	0.484129i	−0.906447	−	0.422319i	\(-0.861216\pi\)
							0.422319	+	0.906447i	\(0.361216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	30.10.e.a.17.13	yes	36
3.2	odd	2	inner	30.10.e.a.17.1	✓	36
5.3	odd	4	inner	30.10.e.a.23.1	yes	36
15.8	even	4	inner	30.10.e.a.23.13	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.10.e.a.17.1	✓	36	3.2	odd	2	inner
30.10.e.a.17.13	yes	36	1.1	even	1	trivial
30.10.e.a.23.1	yes	36	5.3	odd	4	inner
30.10.e.a.23.13	yes	36	15.8	even	4	inner