Embedded newform 15.11.f.a.13.2

• Label: 15.11.f.a.13.2
• Level: $15$
• Weight: $11$
• Character: 15.13
• 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			13.2
Root			\(28.5746 + 28.5746i\) of defining polynomial
Character	\(\chi\)	\(=\)	15.13
Dual form			15.11.f.a.7.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-32.7993 + 32.7993i) q^{2} +(99.2043 + 99.2043i) q^{3} -1127.59i q^{4} +(-1119.22 + 2917.70i) q^{5} -6507.67 q^{6} +(-16059.1 + 16059.1i) q^{7} +(3397.64 + 3397.64i) q^{8} +19683.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 64 q^{2} + 10676 q^{5} - 4860 q^{6} + 10604 q^{7} - 39948 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(7\)	\(11\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−32.7993	+	32.7993i	−1.02498	+	1.02498i	−0.0252984	+	0.999680i	\(0.508054\pi\)
							−0.999680	+	0.0252984i	\(0.991946\pi\)
\(3\)	99.2043	+	99.2043i	0.408248	+	0.408248i
							\(5\)	−1119.22	+	2917.70i	−0.358149	+	0.933664i
\(7\)	−16059.1	+	16059.1i	−0.955503	+	0.955503i								−0.999051	−	0.0435483i	\(-0.986134\pi\)
							0.0435483	+	0.999051i	\(0.486134\pi\)
\(11\)	246556.			1.53092			0.765461	−	0.643482i	\(-0.222511\pi\)
							0.765461	+	0.643482i	\(0.222511\pi\)
\(13\)	−463750.	−	463750.i	−1.24901	−	1.24901i	−0.956157	−	0.292856i	\(-0.905394\pi\)
							−0.292856	−	0.956157i	\(-0.594606\pi\)
\(17\)	221558.	−	221558.i	0.156042	−	0.156042i	−0.624768	−	0.780810i	\(-0.714807\pi\)
							0.780810	+	0.624768i	\(0.214807\pi\)
\(19\)			594565.i			0.240122i	0.992767	+	0.120061i	\(0.0383090\pi\)
							−0.992767	+	0.120061i	\(0.961691\pi\)
\(23\)	−664041.	−	664041.i	−0.103171	−	0.103171i	0.653637	−	0.756808i	\(-0.273242\pi\)
							−0.756808	+	0.653637i	\(0.773242\pi\)
\(29\)		−	3.97361e6i		−	0.193729i	−0.995298	−	0.0968646i	\(-0.969119\pi\)
							0.995298	−	0.0968646i	\(-0.0308814\pi\)
\(31\)	−2.75635e7			−0.962778			−0.481389	−	0.876507i	\(-0.659868\pi\)
							−0.481389	+	0.876507i	\(0.659868\pi\)
\(37\)	−4.26463e7	+	4.26463e7i	−0.614997	+	0.614997i	−0.944244	−	0.329247i	\(-0.893205\pi\)
							0.329247	+	0.944244i	\(0.393205\pi\)
\(41\)	−1.16388e7			−0.100459			−0.0502297	−	0.998738i	\(-0.515995\pi\)
							−0.0502297	+	0.998738i	\(0.515995\pi\)
\(43\)	7.66609e7	+	7.66609e7i	0.521473	+	0.521473i	0.918016	−	0.396543i	\(-0.129790\pi\)
							−0.396543	+	0.918016i	\(0.629790\pi\)
\(47\)	−1.67697e8	+	1.67697e8i	−0.731198	+	0.731198i	−0.970857	−	0.239659i	\(-0.922964\pi\)
							0.239659	+	0.970857i	\(0.422964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.11.f.a.13.2	yes	20
3.2	odd	2		45.11.g.c.28.9		20
5.2	odd	4	inner	15.11.f.a.7.2	✓	20
5.3	odd	4		75.11.f.d.7.9		20
5.4	even	2		75.11.f.d.43.9		20
15.2	even	4		45.11.g.c.37.9		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.11.f.a.7.2	✓	20	5.2	odd	4	inner
15.11.f.a.13.2	yes	20	1.1	even	1	trivial
45.11.g.c.28.9		20	3.2	odd	2
45.11.g.c.37.9		20	15.2	even	4
75.11.f.d.7.9		20	5.3	odd	4
75.11.f.d.43.9		20	5.4	even	2