Properties

Label 720.2.x.a
Level $720$
Weight $2$
Character orbit 720.x
Analytic conductor $5.749$
Analytic rank $0$
Dimension $2$
CM discriminant -4
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.x (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - i ) q^{5} +O(q^{10})\) \( q + ( -2 - i ) q^{5} + ( -5 + 5 i ) q^{13} + ( 5 + 5 i ) q^{17} + ( 3 + 4 i ) q^{25} + 4 i q^{29} + ( 5 + 5 i ) q^{37} -8 q^{41} + 7 i q^{49} + ( -5 + 5 i ) q^{53} -12 q^{61} + ( 15 - 5 i ) q^{65} + ( 5 - 5 i ) q^{73} + ( -5 - 15 i ) q^{85} -16 i q^{89} + ( 5 + 5 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + O(q^{10}) \) \( 2q - 4q^{5} - 10q^{13} + 10q^{17} + 6q^{25} + 10q^{37} - 16q^{41} - 10q^{53} - 24q^{61} + 30q^{65} + 10q^{73} - 10q^{85} + 10q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 −2.00000 1.00000i 0 0 0 0 0
703.1 0 0 0 −2.00000 + 1.00000i 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 \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.x.a 2
3.b odd 2 1 80.2.n.a 2
4.b odd 2 1 CM 720.2.x.a 2
5.b even 2 1 3600.2.x.c 2
5.c odd 4 1 inner 720.2.x.a 2
5.c odd 4 1 3600.2.x.c 2
12.b even 2 1 80.2.n.a 2
15.d odd 2 1 400.2.n.a 2
15.e even 4 1 80.2.n.a 2
15.e even 4 1 400.2.n.a 2
20.d odd 2 1 3600.2.x.c 2
20.e even 4 1 inner 720.2.x.a 2
20.e even 4 1 3600.2.x.c 2
24.f even 2 1 320.2.n.d 2
24.h odd 2 1 320.2.n.d 2
48.i odd 4 1 1280.2.o.h 2
48.i odd 4 1 1280.2.o.i 2
48.k even 4 1 1280.2.o.h 2
48.k even 4 1 1280.2.o.i 2
60.h even 2 1 400.2.n.a 2
60.l odd 4 1 80.2.n.a 2
60.l odd 4 1 400.2.n.a 2
120.i odd 2 1 1600.2.n.g 2
120.m even 2 1 1600.2.n.g 2
120.q odd 4 1 320.2.n.d 2
120.q odd 4 1 1600.2.n.g 2
120.w even 4 1 320.2.n.d 2
120.w even 4 1 1600.2.n.g 2
240.z odd 4 1 1280.2.o.h 2
240.bb even 4 1 1280.2.o.h 2
240.bd odd 4 1 1280.2.o.i 2
240.bf even 4 1 1280.2.o.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.a 2 3.b odd 2 1
80.2.n.a 2 12.b even 2 1
80.2.n.a 2 15.e even 4 1
80.2.n.a 2 60.l odd 4 1
320.2.n.d 2 24.f even 2 1
320.2.n.d 2 24.h odd 2 1
320.2.n.d 2 120.q odd 4 1
320.2.n.d 2 120.w even 4 1
400.2.n.a 2 15.d odd 2 1
400.2.n.a 2 15.e even 4 1
400.2.n.a 2 60.h even 2 1
400.2.n.a 2 60.l odd 4 1
720.2.x.a 2 1.a even 1 1 trivial
720.2.x.a 2 4.b odd 2 1 CM
720.2.x.a 2 5.c odd 4 1 inner
720.2.x.a 2 20.e even 4 1 inner
1280.2.o.h 2 48.i odd 4 1
1280.2.o.h 2 48.k even 4 1
1280.2.o.h 2 240.z odd 4 1
1280.2.o.h 2 240.bb even 4 1
1280.2.o.i 2 48.i odd 4 1
1280.2.o.i 2 48.k even 4 1
1280.2.o.i 2 240.bd odd 4 1
1280.2.o.i 2 240.bf even 4 1
1600.2.n.g 2 120.i odd 2 1
1600.2.n.g 2 120.m even 2 1
1600.2.n.g 2 120.q odd 4 1
1600.2.n.g 2 120.w even 4 1
3600.2.x.c 2 5.b even 2 1
3600.2.x.c 2 5.c odd 4 1
3600.2.x.c 2 20.d odd 2 1
3600.2.x.c 2 20.e even 4 1

Hecke kernels

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

\( T_{7} \)
\( T_{11} \)
\( T_{13}^{2} + 10 T_{13} + 50 \)
\( T_{17}^{2} - 10 T_{17} + 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 50 + 10 T + T^{2} \)
$17$ \( 50 - 10 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 16 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 50 - 10 T + T^{2} \)
$41$ \( ( 8 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 50 + 10 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 12 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 50 - 10 T + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 256 + T^{2} \)
$97$ \( 50 - 10 T + T^{2} \)
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