Properties

Label 480.2.f.d
Level $480$
Weight $2$
Character orbit 480.f
Analytic conductor $3.833$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{3} + ( 1 - 2 i ) q^{5} + 4 i q^{7} - q^{9} +O(q^{10})\) \( q + i q^{3} + ( 1 - 2 i ) q^{5} + 4 i q^{7} - q^{9} + 4 i q^{13} + ( 2 + i ) q^{15} + 8 q^{19} -4 q^{21} + 4 i q^{23} + ( -3 - 4 i ) q^{25} -i q^{27} + 6 q^{29} -8 q^{31} + ( 8 + 4 i ) q^{35} + 4 i q^{37} -4 q^{39} + 6 q^{41} + 4 i q^{43} + ( -1 + 2 i ) q^{45} + 4 i q^{47} -9 q^{49} -12 i q^{53} + 8 i q^{57} -6 q^{61} -4 i q^{63} + ( 8 + 4 i ) q^{65} -12 i q^{67} -4 q^{69} -16 q^{71} + ( 4 - 3 i ) q^{75} + 8 q^{79} + q^{81} -12 i q^{83} + 6 i q^{87} + 10 q^{89} -16 q^{91} -8 i q^{93} + ( 8 - 16 i ) q^{95} -8 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{9} + 4q^{15} + 16q^{19} - 8q^{21} - 6q^{25} + 12q^{29} - 16q^{31} + 16q^{35} - 8q^{39} + 12q^{41} - 2q^{45} - 18q^{49} - 12q^{61} + 16q^{65} - 8q^{69} - 32q^{71} + 8q^{75} + 16q^{79} + 2q^{81} + 20q^{89} - 32q^{91} + 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
1.00000i
1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 4.00000i 0 −1.00000 0
289.2 0 1.00000i 0 1.00000 2.00000i 0 4.00000i 0 −1.00000 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
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 480.2.f.d yes 2
3.b odd 2 1 1440.2.f.d 2
4.b odd 2 1 480.2.f.c 2
5.b even 2 1 inner 480.2.f.d yes 2
5.c odd 4 1 2400.2.a.o 1
5.c odd 4 1 2400.2.a.t 1
8.b even 2 1 960.2.f.e 2
8.d odd 2 1 960.2.f.d 2
12.b even 2 1 1440.2.f.b 2
15.d odd 2 1 1440.2.f.d 2
15.e even 4 1 7200.2.a.c 1
15.e even 4 1 7200.2.a.by 1
16.e even 4 1 3840.2.d.a 2
16.e even 4 1 3840.2.d.bf 2
16.f odd 4 1 3840.2.d.p 2
16.f odd 4 1 3840.2.d.q 2
20.d odd 2 1 480.2.f.c 2
20.e even 4 1 2400.2.a.p 1
20.e even 4 1 2400.2.a.s 1
24.f even 2 1 2880.2.f.o 2
24.h odd 2 1 2880.2.f.m 2
40.e odd 2 1 960.2.f.d 2
40.f even 2 1 960.2.f.e 2
40.i odd 4 1 4800.2.a.c 1
40.i odd 4 1 4800.2.a.cr 1
40.k even 4 1 4800.2.a.e 1
40.k even 4 1 4800.2.a.cp 1
60.h even 2 1 1440.2.f.b 2
60.l odd 4 1 7200.2.a.a 1
60.l odd 4 1 7200.2.a.ca 1
80.k odd 4 1 3840.2.d.p 2
80.k odd 4 1 3840.2.d.q 2
80.q even 4 1 3840.2.d.a 2
80.q even 4 1 3840.2.d.bf 2
120.i odd 2 1 2880.2.f.m 2
120.m even 2 1 2880.2.f.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.c 2 4.b odd 2 1
480.2.f.c 2 20.d odd 2 1
480.2.f.d yes 2 1.a even 1 1 trivial
480.2.f.d yes 2 5.b even 2 1 inner
960.2.f.d 2 8.d odd 2 1
960.2.f.d 2 40.e odd 2 1
960.2.f.e 2 8.b even 2 1
960.2.f.e 2 40.f even 2 1
1440.2.f.b 2 12.b even 2 1
1440.2.f.b 2 60.h even 2 1
1440.2.f.d 2 3.b odd 2 1
1440.2.f.d 2 15.d odd 2 1
2400.2.a.o 1 5.c odd 4 1
2400.2.a.p 1 20.e even 4 1
2400.2.a.s 1 20.e even 4 1
2400.2.a.t 1 5.c odd 4 1
2880.2.f.m 2 24.h odd 2 1
2880.2.f.m 2 120.i odd 2 1
2880.2.f.o 2 24.f even 2 1
2880.2.f.o 2 120.m even 2 1
3840.2.d.a 2 16.e even 4 1
3840.2.d.a 2 80.q even 4 1
3840.2.d.p 2 16.f odd 4 1
3840.2.d.p 2 80.k odd 4 1
3840.2.d.q 2 16.f odd 4 1
3840.2.d.q 2 80.k odd 4 1
3840.2.d.bf 2 16.e even 4 1
3840.2.d.bf 2 80.q even 4 1
4800.2.a.c 1 40.i odd 4 1
4800.2.a.e 1 40.k even 4 1
4800.2.a.cp 1 40.k even 4 1
4800.2.a.cr 1 40.i odd 4 1
7200.2.a.a 1 60.l odd 4 1
7200.2.a.c 1 15.e even 4 1
7200.2.a.by 1 15.e even 4 1
7200.2.a.ca 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\):

\( T_{7}^{2} + 16 \)
\( T_{11} \)
\( T_{19} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 2 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -8 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 8 + T )^{2} \)
$37$ \( 16 + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 6 + T )^{2} \)
$67$ \( 144 + T^{2} \)
$71$ \( ( 16 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 64 + T^{2} \)
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