Properties

Label 720.3.l.c
Level $720$
Weight $3$
Character orbit 720.l
Analytic conductor $19.619$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(19.6185790339\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - \beta_{3} + 4) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + ( - \beta_{3} + 4) q^{7} + (6 \beta_{2} - \beta_1) q^{11} + (\beta_{3} + 2) q^{13} - 4 \beta_1 q^{17} + ( - 2 \beta_{3} - 8) q^{19} + (6 \beta_{2} - 2 \beta_1) q^{23} - 5 q^{25} + ( - 6 \beta_{2} + 8 \beta_1) q^{29} + ( - 2 \beta_{3} - 2) q^{31} + ( - 4 \beta_{2} + 5 \beta_1) q^{35} + ( - 3 \beta_{3} - 34) q^{37} + (24 \beta_{2} - 3 \beta_1) q^{41} + (2 \beta_{3} - 20) q^{43} + (12 \beta_{2} + 14 \beta_1) q^{47} + ( - 8 \beta_{3} + 57) q^{49} + (24 \beta_{2} + 10 \beta_1) q^{53} + ( - \beta_{3} + 30) q^{55} + ( - 18 \beta_{2} - 17 \beta_1) q^{59} + ( - 6 \beta_{3} - 10) q^{61} + ( - 2 \beta_{2} - 5 \beta_1) q^{65} + 76 q^{67} + (12 \beta_{2} - 12 \beta_1) q^{71} + ( - 6 \beta_{3} + 38) q^{73} + (42 \beta_{2} - 34 \beta_1) q^{77} + ( - 6 \beta_{3} - 50) q^{79} + (24 \beta_{2} - 2 \beta_1) q^{83} - 4 \beta_{3} q^{85} + (12 \beta_{2} + 21 \beta_1) q^{89} + (2 \beta_{3} - 82) q^{91} + (8 \beta_{2} + 10 \beta_1) q^{95} + (2 \beta_{3} - 106) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{7} + 8 q^{13} - 32 q^{19} - 20 q^{25} - 8 q^{31} - 136 q^{37} - 80 q^{43} + 228 q^{49} + 120 q^{55} - 40 q^{61} + 304 q^{67} + 152 q^{73} - 200 q^{79} - 328 q^{91} - 424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 6 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.58114 + 0.707107i
−1.58114 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
0 0 0 2.23607i 0 −5.48683 0 0 0
161.2 0 0 0 2.23607i 0 13.4868 0 0 0
161.3 0 0 0 2.23607i 0 −5.48683 0 0 0
161.4 0 0 0 2.23607i 0 13.4868 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.3.l.c 4
3.b odd 2 1 inner 720.3.l.c 4
4.b odd 2 1 180.3.g.a 4
5.b even 2 1 3600.3.l.n 4
5.c odd 4 2 3600.3.c.k 8
8.b even 2 1 2880.3.l.f 4
8.d odd 2 1 2880.3.l.b 4
12.b even 2 1 180.3.g.a 4
15.d odd 2 1 3600.3.l.n 4
15.e even 4 2 3600.3.c.k 8
20.d odd 2 1 900.3.g.d 4
20.e even 4 2 900.3.b.b 8
24.f even 2 1 2880.3.l.b 4
24.h odd 2 1 2880.3.l.f 4
36.f odd 6 2 1620.3.o.f 8
36.h even 6 2 1620.3.o.f 8
60.h even 2 1 900.3.g.d 4
60.l odd 4 2 900.3.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 4.b odd 2 1
180.3.g.a 4 12.b even 2 1
720.3.l.c 4 1.a even 1 1 trivial
720.3.l.c 4 3.b odd 2 1 inner
900.3.b.b 8 20.e even 4 2
900.3.b.b 8 60.l odd 4 2
900.3.g.d 4 20.d odd 2 1
900.3.g.d 4 60.h even 2 1
1620.3.o.f 8 36.f odd 6 2
1620.3.o.f 8 36.h even 6 2
2880.3.l.b 4 8.d odd 2 1
2880.3.l.b 4 24.f even 2 1
2880.3.l.f 4 8.b even 2 1
2880.3.l.f 4 24.h odd 2 1
3600.3.c.k 8 5.c odd 4 2
3600.3.c.k 8 15.e even 4 2
3600.3.l.n 4 5.b even 2 1
3600.3.l.n 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 8T_{7} - 74 \) acting on \(S_{3}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8 T - 74)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 396 T^{2} + 26244 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 86)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 288)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16 T - 296)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 504 T^{2} + 11664 \) Copy content Toggle raw display
$29$ \( T^{4} + 2664 T^{2} + 944784 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 356)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 68 T + 346)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 6084 T^{2} + \cdots + 7387524 \) Copy content Toggle raw display
$43$ \( (T^{2} + 40 T + 40)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8496 T^{2} + \cdots + 7884864 \) Copy content Toggle raw display
$53$ \( T^{4} + 9360 T^{2} + \cdots + 1166400 \) Copy content Toggle raw display
$59$ \( T^{4} + 13644 T^{2} + \cdots + 12830724 \) Copy content Toggle raw display
$61$ \( (T^{2} + 20 T - 3140)^{2} \) Copy content Toggle raw display
$67$ \( (T - 76)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 6624 T^{2} + \cdots + 3504384 \) Copy content Toggle raw display
$73$ \( (T^{2} - 76 T - 1796)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 100 T - 740)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 5904 T^{2} + \cdots + 7884864 \) Copy content Toggle raw display
$89$ \( T^{4} + 17316 T^{2} + \cdots + 52099524 \) Copy content Toggle raw display
$97$ \( (T^{2} + 212 T + 10876)^{2} \) Copy content Toggle raw display
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