Properties

Label 720.2.f.d
Level $720$
Weight $2$
Character orbit 720.f
Analytic conductor $5.749$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} +O(q^{10})\) \( q + \beta q^{5} + 2 \beta q^{17} + 4 q^{19} + 4 \beta q^{23} -5 q^{25} -8 q^{31} + 4 \beta q^{47} + 7 q^{49} + 2 \beta q^{53} + 2 q^{61} + 16 q^{79} -8 \beta q^{83} -10 q^{85} + 4 \beta q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 8q^{19} - 10q^{25} - 16q^{31} + 14q^{49} + 4q^{61} + 32q^{79} - 20q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.23607i
2.23607i
0 0 0 2.23607i 0 0 0 0 0
289.2 0 0 0 2.23607i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.f.d 2
3.b odd 2 1 inner 720.2.f.d 2
4.b odd 2 1 45.2.b.a 2
5.b even 2 1 inner 720.2.f.d 2
5.c odd 4 2 3600.2.a.bs 2
8.b even 2 1 2880.2.f.j 2
8.d odd 2 1 2880.2.f.k 2
12.b even 2 1 45.2.b.a 2
15.d odd 2 1 CM 720.2.f.d 2
15.e even 4 2 3600.2.a.bs 2
20.d odd 2 1 45.2.b.a 2
20.e even 4 2 225.2.a.f 2
24.f even 2 1 2880.2.f.k 2
24.h odd 2 1 2880.2.f.j 2
28.d even 2 1 2205.2.d.a 2
36.f odd 6 2 405.2.j.c 4
36.h even 6 2 405.2.j.c 4
40.e odd 2 1 2880.2.f.k 2
40.f even 2 1 2880.2.f.j 2
60.h even 2 1 45.2.b.a 2
60.l odd 4 2 225.2.a.f 2
84.h odd 2 1 2205.2.d.a 2
120.i odd 2 1 2880.2.f.j 2
120.m even 2 1 2880.2.f.k 2
140.c even 2 1 2205.2.d.a 2
180.n even 6 2 405.2.j.c 4
180.p odd 6 2 405.2.j.c 4
420.o odd 2 1 2205.2.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 4.b odd 2 1
45.2.b.a 2 12.b even 2 1
45.2.b.a 2 20.d odd 2 1
45.2.b.a 2 60.h even 2 1
225.2.a.f 2 20.e even 4 2
225.2.a.f 2 60.l odd 4 2
405.2.j.c 4 36.f odd 6 2
405.2.j.c 4 36.h even 6 2
405.2.j.c 4 180.n even 6 2
405.2.j.c 4 180.p odd 6 2
720.2.f.d 2 1.a even 1 1 trivial
720.2.f.d 2 3.b odd 2 1 inner
720.2.f.d 2 5.b even 2 1 inner
720.2.f.d 2 15.d odd 2 1 CM
2205.2.d.a 2 28.d even 2 1
2205.2.d.a 2 84.h odd 2 1
2205.2.d.a 2 140.c even 2 1
2205.2.d.a 2 420.o odd 2 1
2880.2.f.j 2 8.b even 2 1
2880.2.f.j 2 24.h odd 2 1
2880.2.f.j 2 40.f even 2 1
2880.2.f.j 2 120.i odd 2 1
2880.2.f.k 2 8.d odd 2 1
2880.2.f.k 2 24.f even 2 1
2880.2.f.k 2 40.e odd 2 1
2880.2.f.k 2 120.m even 2 1
3600.2.a.bs 2 5.c odd 4 2
3600.2.a.bs 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7} \)
\( T_{11} \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 20 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 80 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 80 + T^{2} \)
$53$ \( 20 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -16 + T )^{2} \)
$83$ \( 320 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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