Properties

Label 720.2.o.a
Level $720$
Weight $2$
Character orbit 720.o
Analytic conductor $5.749$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} +O(q^{10})\) \( q + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{5} + 6 \zeta_{8}^{2} q^{13} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{29} + 12 \zeta_{8}^{2} q^{37} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{41} -7 q^{49} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{53} + 10 q^{61} + ( -6 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{65} -6 \zeta_{8}^{2} q^{73} + ( 9 + 3 \zeta_{8}^{2} ) q^{85} + ( 13 \zeta_{8} + 13 \zeta_{8}^{3} ) q^{89} -18 \zeta_{8}^{2} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{25} - 28q^{49} + 40q^{61} + 36q^{85} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 0 0 −2.12132 0.707107i 0 0 0 0 0
719.2 0 0 0 −2.12132 + 0.707107i 0 0 0 0 0
719.3 0 0 0 2.12132 0.707107i 0 0 0 0 0
719.4 0 0 0 2.12132 + 0.707107i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h 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.o.a 4
3.b odd 2 1 inner 720.2.o.a 4
4.b odd 2 1 CM 720.2.o.a 4
5.b even 2 1 inner 720.2.o.a 4
5.c odd 4 1 3600.2.h.a 2
5.c odd 4 1 3600.2.h.c 2
8.b even 2 1 2880.2.o.b 4
8.d odd 2 1 2880.2.o.b 4
12.b even 2 1 inner 720.2.o.a 4
15.d odd 2 1 inner 720.2.o.a 4
15.e even 4 1 3600.2.h.a 2
15.e even 4 1 3600.2.h.c 2
20.d odd 2 1 inner 720.2.o.a 4
20.e even 4 1 3600.2.h.a 2
20.e even 4 1 3600.2.h.c 2
24.f even 2 1 2880.2.o.b 4
24.h odd 2 1 2880.2.o.b 4
40.e odd 2 1 2880.2.o.b 4
40.f even 2 1 2880.2.o.b 4
60.h even 2 1 inner 720.2.o.a 4
60.l odd 4 1 3600.2.h.a 2
60.l odd 4 1 3600.2.h.c 2
120.i odd 2 1 2880.2.o.b 4
120.m even 2 1 2880.2.o.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.o.a 4 1.a even 1 1 trivial
720.2.o.a 4 3.b odd 2 1 inner
720.2.o.a 4 4.b odd 2 1 CM
720.2.o.a 4 5.b even 2 1 inner
720.2.o.a 4 12.b even 2 1 inner
720.2.o.a 4 15.d odd 2 1 inner
720.2.o.a 4 20.d odd 2 1 inner
720.2.o.a 4 60.h even 2 1 inner
2880.2.o.b 4 8.b even 2 1
2880.2.o.b 4 8.d odd 2 1
2880.2.o.b 4 24.f even 2 1
2880.2.o.b 4 24.h odd 2 1
2880.2.o.b 4 40.e odd 2 1
2880.2.o.b 4 40.f even 2 1
2880.2.o.b 4 120.i odd 2 1
2880.2.o.b 4 120.m even 2 1
3600.2.h.a 2 5.c odd 4 1
3600.2.h.a 2 15.e even 4 1
3600.2.h.a 2 20.e even 4 1
3600.2.h.a 2 60.l odd 4 1
3600.2.h.c 2 5.c odd 4 1
3600.2.h.c 2 15.e even 4 1
3600.2.h.c 2 20.e even 4 1
3600.2.h.c 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 8 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( ( 36 + T^{2} )^{2} \)
$17$ \( ( -18 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( 98 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( ( 144 + T^{2} )^{2} \)
$41$ \( ( 2 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -162 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( -10 + T )^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 36 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 338 + T^{2} )^{2} \)
$97$ \( ( 324 + T^{2} )^{2} \)
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