Embedded newform 15.11.f.a.7.1

• Label: 15.11.f.a.7.1
• Level: $15$
• Weight: $11$
• Character: 15.7
• Analytic conductor: $9.530$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	15.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.53035879011\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	x^20 - 4*x^19 + 8*x^18 - 49316*x^17 + 18276332*x^16 - 230627572*x^15 + 1992333560*x^14 - 394917674060*x^13 + 80687382741790*x^12 - 1533501926122980*x^11 + 15146257542506232*x^10 - 328741228253604588*x^9 + 82447502291888584636*x^8 - 1645271725800808466812*x^7 + 16207307909588298724904*x^6 + 13012195729526348972284*x^5 + 16940236408526758395395273*x^4 - 319136657167982267954012296*x^3 + 2999193261390255504529305632*x^2 + 9416677726299921146630526016*x + 14782945224391593673997322304
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{21}\cdot 3^{34}\cdot 5^{14} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.1
Root			\(39.4498 - 39.4498i\) of defining polynomial
Character	\(\chi\)	\(=\)	15.7
Dual form			15.11.f.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-41.2250 - 41.2250i) q^{2} +(-99.2043 + 99.2043i) q^{3} +2375.00i q^{4} +(953.956 - 2975.83i) q^{5} +8179.40 q^{6} +(16792.7 + 16792.7i) q^{7} +(55695.2 - 55695.2i) q^{8} -19683.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 64 q^{2} + 10676 q^{5} - 4860 q^{6} + 10604 q^{7} - 39948 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−41.2250	−	41.2250i	−1.28828	−	1.28828i	−0.935830	−	0.352452i	\(-0.885348\pi\)
							−0.352452	−	0.935830i	\(-0.614652\pi\)
\(3\)	−99.2043	+	99.2043i	−0.408248	+	0.408248i
							\(5\)	953.956	−	2975.83i	0.305266	−	0.952267i
\(7\)	16792.7	+	16792.7i	0.999152	+	0.999152i								1.00000		0.000847870i	\(-0.000269886\pi\)
							−0.000847870		1.00000i	\(0.500270\pi\)
\(11\)	−143131.			−0.888731			−0.444366	−	0.895846i	\(-0.646571\pi\)
							−0.444366	+	0.895846i	\(0.646571\pi\)
\(13\)	44078.6	−	44078.6i	0.118716	−	0.118716i	−0.645253	−	0.763969i	\(-0.723248\pi\)
							0.763969	+	0.645253i	\(0.223248\pi\)
\(17\)	534739.	+	534739.i	0.376615	+	0.376615i	0.869879	−	0.493265i	\(-0.164197\pi\)
							−0.493265	+	0.869879i	\(0.664197\pi\)
\(19\)		−	4.33349e6i		−	1.75013i	−0.484008	−	0.875063i	\(-0.660820\pi\)
							0.484008	−	0.875063i	\(-0.339180\pi\)
\(23\)	6.55189e6	−	6.55189e6i	1.01795	−	1.01795i	0.0181162	−	0.999836i	\(-0.494233\pi\)
							0.999836	−	0.0181162i	\(-0.00576687\pi\)
\(29\)		−	1.25631e7i		−	0.612503i	−0.951951	−	0.306252i	\(-0.900925\pi\)
							0.951951	−	0.306252i	\(-0.0990749\pi\)
\(31\)	1.29043e7			0.450739			0.225369	−	0.974273i	\(-0.427641\pi\)
							0.225369	+	0.974273i	\(0.427641\pi\)
\(37\)	1.59867e7	+	1.59867e7i	0.230542	+	0.230542i	0.812919	−	0.582377i	\(-0.197877\pi\)
							−0.582377	+	0.812919i	\(0.697877\pi\)
\(41\)	−9.99306e7			−0.862540			−0.431270	−	0.902223i	\(-0.641934\pi\)
							−0.431270	+	0.902223i	\(0.641934\pi\)
\(43\)	1.60483e8	−	1.60483e8i	1.09166	−	1.09166i	0.0963078	−	0.995352i	\(-0.469297\pi\)
							0.995352	−	0.0963078i	\(-0.0307033\pi\)
\(47\)	5.89584e7	+	5.89584e7i	0.257073	+	0.257073i	0.823863	−	0.566790i	\(-0.191815\pi\)
							−0.566790	+	0.823863i	\(0.691815\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.11.f.a.7.1	✓	20
3.2	odd	2		45.11.g.c.37.10		20
5.2	odd	4		75.11.f.d.43.10		20
5.3	odd	4	inner	15.11.f.a.13.1	yes	20
5.4	even	2		75.11.f.d.7.10		20
15.8	even	4		45.11.g.c.28.10		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.11.f.a.7.1	✓	20	1.1	even	1	trivial
15.11.f.a.13.1	yes	20	5.3	odd	4	inner
45.11.g.c.28.10		20	15.8	even	4
45.11.g.c.37.10		20	3.2	odd	2
75.11.f.d.7.10		20	5.4	even	2
75.11.f.d.43.10		20	5.2	odd	4