Embedded newform 15.11.f.a.13.5

• Label: 15.11.f.a.13.5
• 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			13.5
Root			\(10.7802 + 10.7802i\) of defining polynomial
Character	\(\chi\)	\(=\)	15.13
Dual form			15.11.f.a.7.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.5555 + 12.5555i) q^{2} +(-99.2043 - 99.2043i) q^{3} +708.721i q^{4} +(3105.25 - 350.813i) q^{5} +2491.11 q^{6} +(-8580.16 + 8580.16i) q^{7} +(-21755.1 - 21755.1i) q^{8} +19683.0i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 64 q^{2} + 10676 q^{5} - 4860 q^{6} + 10604 q^{7} - 39948 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.5555	+	12.5555i	−0.392358	+	0.392358i	−0.875527	−	0.483169i	\(-0.839486\pi\)
							0.483169	+	0.875527i	\(0.339486\pi\)
\(3\)	−99.2043	−	99.2043i	−0.408248	−	0.408248i
							\(5\)	3105.25	−	350.813i	0.993679	−	0.112260i
\(7\)	−8580.16	+	8580.16i	−0.510511	+	0.510511i								−0.914683	−	0.404172i	\(-0.867560\pi\)
							0.404172	+	0.914683i	\(0.367560\pi\)
\(11\)	−117349.			−0.728642			−0.364321	−	0.931273i	\(-0.618699\pi\)
							−0.364321	+	0.931273i	\(0.618699\pi\)
\(13\)	−468858.	−	468858.i	−1.26277	−	1.26277i	−0.949744	−	0.313026i	\(-0.898657\pi\)
							−0.313026	−	0.949744i	\(-0.601343\pi\)
\(17\)	−1.23683e6	+	1.23683e6i	−0.871094	+	0.871094i	−0.992592	−	0.121498i	\(-0.961230\pi\)
							0.121498	+	0.992592i	\(0.461230\pi\)
\(19\)		−	733454.i		−	0.296214i	−0.988971	−	0.148107i	\(-0.952682\pi\)
							0.988971	−	0.148107i	\(-0.0473179\pi\)
\(23\)	−1.35683e6	−	1.35683e6i	−0.210807	−	0.210807i	0.593803	−	0.804610i	\(-0.297626\pi\)
							−0.804610	+	0.593803i	\(0.797626\pi\)
\(29\)			2.76954e7i			1.35026i	0.737697	+	0.675132i	\(0.235913\pi\)
							−0.737697	+	0.675132i	\(0.764087\pi\)
\(31\)	−2.91035e7			−1.01657			−0.508285	−	0.861189i	\(-0.669720\pi\)
							−0.508285	+	0.861189i	\(0.669720\pi\)
\(37\)	2.07001e7	−	2.07001e7i	0.298513	−	0.298513i	−0.541918	−	0.840431i	\(-0.682302\pi\)
							0.840431	+	0.541918i	\(0.182302\pi\)
\(41\)	−5.61462e7			−0.484620			−0.242310	−	0.970199i	\(-0.577905\pi\)
							−0.242310	+	0.970199i	\(0.577905\pi\)
\(43\)	−1.82942e8	−	1.82942e8i	−1.24443	−	1.24443i	−0.958142	−	0.286293i	\(-0.907577\pi\)
							−0.286293	−	0.958142i	\(-0.592423\pi\)
\(47\)	5.88764e7	−	5.88764e7i	0.256715	−	0.256715i	−0.567001	−	0.823717i	\(-0.691897\pi\)
							0.823717	+	0.567001i	\(0.191897\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.11.f.a.13.5	yes	20
3.2	odd	2		45.11.g.c.28.6		20
5.2	odd	4	inner	15.11.f.a.7.5	✓	20
5.3	odd	4		75.11.f.d.7.6		20
5.4	even	2		75.11.f.d.43.6		20
15.2	even	4		45.11.g.c.37.6		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.11.f.a.7.5	✓	20	5.2	odd	4	inner
15.11.f.a.13.5	yes	20	1.1	even	1	trivial
45.11.g.c.28.6		20	3.2	odd	2
45.11.g.c.37.6		20	15.2	even	4
75.11.f.d.7.6		20	5.3	odd	4
75.11.f.d.43.6		20	5.4	even	2