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Properties

Label 480.2.d.b
Level $480$
Weight $2$
Character orbit 480.d
Analytic conductor $3.833$
Analytic rank $0$
Dimension $6$
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.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.839056.1
Defining polynomial: \( x^{6} + 6x^{4} + 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{4} q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_{4} q^{5} + \beta_1 q^{7} + q^{9} + (\beta_{5} + \beta_1) q^{11} + (\beta_{3} + 1) q^{13} + \beta_{4} q^{15} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{17} + (\beta_{4} - \beta_{2} + \beta_1) q^{19} + \beta_1 q^{21} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{23} + ( - \beta_{5} + \beta_{3}) q^{25} + q^{27} + (2 \beta_{5} + \beta_{4} - \beta_{2}) q^{29} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{31} + (\beta_{5} + \beta_1) q^{33} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{35} + ( - \beta_{3} + 3) q^{37} + (\beta_{3} + 1) q^{39} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}) q^{41} + (2 \beta_{4} + 2 \beta_{2}) q^{43} + \beta_{4} q^{45} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{47} + (2 \beta_{4} + 2 \beta_{2} - 1) q^{49} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{51} + ( - \beta_{4} - \beta_{2} - 4) q^{53} + (2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 1) q^{55} + (\beta_{4} - \beta_{2} + \beta_1) q^{57} + ( - \beta_{5} - 3 \beta_1) q^{59} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{61} + \beta_1 q^{63} + ( - \beta_{5} + \beta_{4} + 3 \beta_{2} - 2 \beta_1 - 2) q^{65} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{67} + ( - 2 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{69} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{71} + (2 \beta_{5} + 4 \beta_{4} - 4 \beta_{2}) q^{73} + ( - \beta_{5} + \beta_{3}) q^{75} - 4 q^{77} + (\beta_{4} + \beta_{3} + \beta_{2} - 3) q^{79} + q^{81} + ( - 2 \beta_{3} - 2) q^{83} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 3) q^{85} + (2 \beta_{5} + \beta_{4} - \beta_{2}) q^{87} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 4) q^{89} + ( - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{2} + 2 \beta_1) q^{91} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 3) q^{93} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 - 6) q^{95} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{97} + (\beta_{5} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 6 q^{9} + 8 q^{13} + 2 q^{25} + 6 q^{27} + 16 q^{31} - 4 q^{35} + 16 q^{37} + 8 q^{39} - 4 q^{41} - 6 q^{49} - 24 q^{53} + 8 q^{55} - 12 q^{65} - 16 q^{71} + 2 q^{75} - 24 q^{77} - 16 q^{79} + 6 q^{81} - 16 q^{83} + 16 q^{85} - 20 q^{89} + 16 q^{93} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 6x^{4} + 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{5} + 5\nu^{3} + \nu^{2} + 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{4} + 8\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} - 5\nu^{3} + \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} + 12\nu^{3} + 14\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{2} - 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{4} + \beta_{3} - 4\beta_{2} + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{5} - 6\beta_{4} + 6\beta_{2} + 11\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
2.02852i
− 2.02852i
− 1.32132i
1.32132i
0.373087i
− 0.373087i
0 1.00000 0 −2.11491 − 0.726062i 0 4.05705i 0 1.00000 0
49.2 0 1.00000 0 −2.11491 + 0.726062i 0 − 4.05705i 0 1.00000 0
49.3 0 1.00000 0 0.254102 − 2.22158i 0 − 2.64265i 0 1.00000 0
49.4 0 1.00000 0 0.254102 + 2.22158i 0 2.64265i 0 1.00000 0
49.5 0 1.00000 0 1.86081 − 1.23992i 0 0.746175i 0 1.00000 0
49.6 0 1.00000 0 1.86081 + 1.23992i 0 − 0.746175i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f 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.d.b 6
3.b odd 2 1 1440.2.d.f 6
4.b odd 2 1 120.2.d.b yes 6
5.b even 2 1 480.2.d.a 6
5.c odd 4 2 2400.2.k.f 12
8.b even 2 1 480.2.d.a 6
8.d odd 2 1 120.2.d.a ✓ 6
12.b even 2 1 360.2.d.e 6
15.d odd 2 1 1440.2.d.e 6
15.e even 4 2 7200.2.k.u 12
16.e even 4 2 3840.2.f.m 12
16.f odd 4 2 3840.2.f.l 12
20.d odd 2 1 120.2.d.a ✓ 6
20.e even 4 2 600.2.k.f 12
24.f even 2 1 360.2.d.f 6
24.h odd 2 1 1440.2.d.e 6
40.e odd 2 1 120.2.d.b yes 6
40.f even 2 1 inner 480.2.d.b 6
40.i odd 4 2 2400.2.k.f 12
40.k even 4 2 600.2.k.f 12
60.h even 2 1 360.2.d.f 6
60.l odd 4 2 1800.2.k.u 12
80.k odd 4 2 3840.2.f.l 12
80.q even 4 2 3840.2.f.m 12
120.i odd 2 1 1440.2.d.f 6
120.m even 2 1 360.2.d.e 6
120.q odd 4 2 1800.2.k.u 12
120.w even 4 2 7200.2.k.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a ✓ 6 8.d odd 2 1
120.2.d.a ✓ 6 20.d odd 2 1
120.2.d.b yes 6 4.b odd 2 1
120.2.d.b yes 6 40.e odd 2 1
360.2.d.e 6 12.b even 2 1
360.2.d.e 6 120.m even 2 1
360.2.d.f 6 24.f even 2 1
360.2.d.f 6 60.h even 2 1
480.2.d.a 6 5.b even 2 1
480.2.d.a 6 8.b even 2 1
480.2.d.b 6 1.a even 1 1 trivial
480.2.d.b 6 40.f even 2 1 inner
600.2.k.f 12 20.e even 4 2
600.2.k.f 12 40.k even 4 2
1440.2.d.e 6 15.d odd 2 1
1440.2.d.e 6 24.h odd 2 1
1440.2.d.f 6 3.b odd 2 1
1440.2.d.f 6 120.i odd 2 1
1800.2.k.u 12 60.l odd 4 2
1800.2.k.u 12 120.q odd 4 2
2400.2.k.f 12 5.c odd 4 2
2400.2.k.f 12 40.i odd 4 2
3840.2.f.l 12 16.f odd 4 2
3840.2.f.l 12 80.k odd 4 2
3840.2.f.m 12 16.e even 4 2
3840.2.f.m 12 80.q even 4 2
7200.2.k.u 12 15.e even 4 2
7200.2.k.u 12 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{3} - 4T_{13}^{2} - 16T_{13} + 56 \) acting on \(S_{2}^{\mathrm{new}}(480, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + 8 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + 128 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 32 T^{4} + 96 T^{2} + 64 \) Copy content Toggle raw display
$13$ \( (T^{3} - 4 T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 36 T^{4} + 368 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{6} + 92 T^{4} + 2304 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544 \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} - 4 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} - 100 T + 56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 64 T + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( (T^{3} + 12 T^{2} + 32 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776 \) Copy content Toggle raw display
$61$ \( T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$67$ \( (T^{3} - 64 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} - 80 T - 128)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 384 T^{4} + 34560 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$79$ \( (T^{3} + 8 T^{2} - 4 T - 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 8 T^{2} - 64 T - 448)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 10 T^{2} - 164 T - 1384)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 336 T^{4} + 28416 T^{2} + \cdots + 262144 \) Copy content Toggle raw display
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