Properties

Label 1620.2.x.b
Level $1620$
Weight $2$
Character orbit 1620.x
Analytic conductor $12.936$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{3} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{4} ) q^{7} -2 \beta_{2} q^{11} + ( 3 - 3 \beta_{1} + 3 \beta_{4} - 3 \beta_{5} ) q^{13} + ( -\beta_{3} - \beta_{7} ) q^{17} -2 \beta_{5} q^{19} + ( -\beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{23} + 5 \beta_{4} q^{25} -2 \beta_{6} q^{29} + ( -4 - 4 \beta_{4} ) q^{31} + ( \beta_{3} - \beta_{7} ) q^{35} + ( -3 - 3 \beta_{5} ) q^{37} + ( -4 \beta_{2} + 4 \beta_{7} ) q^{41} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{43} + ( -3 \beta_{2} - 3 \beta_{6} ) q^{47} + ( 5 \beta_{1} + 5 \beta_{5} ) q^{49} + ( -\beta_{3} + \beta_{7} ) q^{53} + 10 \beta_{5} q^{55} + ( 4 \beta_{3} - 4 \beta_{6} ) q^{59} -6 \beta_{4} q^{61} + ( -3 \beta_{2} - 3 \beta_{6} ) q^{65} + ( 1 + \beta_{1} + \beta_{4} + \beta_{5} ) q^{67} -2 \beta_{7} q^{71} + ( 1 - \beta_{5} ) q^{73} + ( -2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{77} -6 \beta_{1} q^{79} + ( -3 \beta_{2} + 3 \beta_{6} ) q^{83} + ( -5 + 5 \beta_{1} - 5 \beta_{4} + 5 \beta_{5} ) q^{85} -2 \beta_{3} q^{89} + 6 q^{91} + ( -2 \beta_{2} + 2 \beta_{7} ) q^{95} + ( 9 \beta_{1} + 9 \beta_{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{7} + O(q^{10}) \) \( 8q + 4q^{7} + 12q^{13} - 20q^{25} - 16q^{31} - 24q^{37} + 12q^{43} + 24q^{61} + 4q^{67} + 8q^{73} - 20q^{85} + 48q^{91} - 36q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 13 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 29 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 9 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} - 8 \nu^{4} + 24 \nu^{2} - 9 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 8 \nu^{5} - 20 \nu^{3} + \nu \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 21 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} - 8 \nu^{5} + 22 \nu^{3} - \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + 3 \beta_{4} - \beta_{3} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{5}\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{6} + 7 \beta_{4}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} + 11 \beta_{5} - 5 \beta_{2} + 11 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{3} - 9\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{2} + 29 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(\beta_{5}\) \(1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
1.40126 0.809017i
−0.535233 + 0.309017i
−1.40126 + 0.809017i
0.535233 0.309017i
−1.40126 0.809017i
0.535233 + 0.309017i
1.40126 + 0.809017i
−0.535233 0.309017i
0 0 0 −1.11803 + 1.93649i 0 1.36603 + 0.366025i 0 0 0
53.2 0 0 0 1.11803 1.93649i 0 1.36603 + 0.366025i 0 0 0
377.1 0 0 0 −1.11803 + 1.93649i 0 −0.366025 + 1.36603i 0 0 0
377.2 0 0 0 1.11803 1.93649i 0 −0.366025 + 1.36603i 0 0 0
593.1 0 0 0 −1.11803 1.93649i 0 −0.366025 1.36603i 0 0 0
593.2 0 0 0 1.11803 + 1.93649i 0 −0.366025 1.36603i 0 0 0
917.1 0 0 0 −1.11803 1.93649i 0 1.36603 0.366025i 0 0 0
917.2 0 0 0 1.11803 + 1.93649i 0 1.36603 0.366025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 917.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.x.b 8
3.b odd 2 1 inner 1620.2.x.b 8
5.c odd 4 1 inner 1620.2.x.b 8
9.c even 3 1 60.2.i.a 4
9.c even 3 1 inner 1620.2.x.b 8
9.d odd 6 1 60.2.i.a 4
9.d odd 6 1 inner 1620.2.x.b 8
15.e even 4 1 inner 1620.2.x.b 8
36.f odd 6 1 240.2.v.b 4
36.h even 6 1 240.2.v.b 4
45.h odd 6 1 300.2.i.a 4
45.j even 6 1 300.2.i.a 4
45.k odd 12 1 60.2.i.a 4
45.k odd 12 1 300.2.i.a 4
45.k odd 12 1 inner 1620.2.x.b 8
45.l even 12 1 60.2.i.a 4
45.l even 12 1 300.2.i.a 4
45.l even 12 1 inner 1620.2.x.b 8
72.j odd 6 1 960.2.v.e 4
72.l even 6 1 960.2.v.h 4
72.n even 6 1 960.2.v.e 4
72.p odd 6 1 960.2.v.h 4
180.n even 6 1 1200.2.v.i 4
180.p odd 6 1 1200.2.v.i 4
180.v odd 12 1 240.2.v.b 4
180.v odd 12 1 1200.2.v.i 4
180.x even 12 1 240.2.v.b 4
180.x even 12 1 1200.2.v.i 4
360.bo even 12 1 960.2.v.h 4
360.br even 12 1 960.2.v.e 4
360.bt odd 12 1 960.2.v.h 4
360.bu odd 12 1 960.2.v.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.i.a 4 9.c even 3 1
60.2.i.a 4 9.d odd 6 1
60.2.i.a 4 45.k odd 12 1
60.2.i.a 4 45.l even 12 1
240.2.v.b 4 36.f odd 6 1
240.2.v.b 4 36.h even 6 1
240.2.v.b 4 180.v odd 12 1
240.2.v.b 4 180.x even 12 1
300.2.i.a 4 45.h odd 6 1
300.2.i.a 4 45.j even 6 1
300.2.i.a 4 45.k odd 12 1
300.2.i.a 4 45.l even 12 1
960.2.v.e 4 72.j odd 6 1
960.2.v.e 4 72.n even 6 1
960.2.v.e 4 360.br even 12 1
960.2.v.e 4 360.bu odd 12 1
960.2.v.h 4 72.l even 6 1
960.2.v.h 4 72.p odd 6 1
960.2.v.h 4 360.bo even 12 1
960.2.v.h 4 360.bt odd 12 1
1200.2.v.i 4 180.n even 6 1
1200.2.v.i 4 180.p odd 6 1
1200.2.v.i 4 180.v odd 12 1
1200.2.v.i 4 180.x even 12 1
1620.2.x.b 8 1.a even 1 1 trivial
1620.2.x.b 8 3.b odd 2 1 inner
1620.2.x.b 8 5.c odd 4 1 inner
1620.2.x.b 8 9.c even 3 1 inner
1620.2.x.b 8 9.d odd 6 1 inner
1620.2.x.b 8 15.e even 4 1 inner
1620.2.x.b 8 45.k odd 12 1 inner
1620.2.x.b 8 45.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 2 T_{7}^{3} + 2 T_{7}^{2} - 4 T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 25 + 5 T^{2} + T^{4} )^{2} \)
$7$ \( ( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 400 - 20 T^{2} + T^{4} )^{2} \)
$13$ \( ( 324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$17$ \( ( 100 + T^{4} )^{2} \)
$19$ \( ( 4 + T^{2} )^{4} \)
$23$ \( 10000 - 100 T^{4} + T^{8} \)
$29$ \( ( 400 + 20 T^{2} + T^{4} )^{2} \)
$31$ \( ( 16 + 4 T + T^{2} )^{4} \)
$37$ \( ( 18 + 6 T + T^{2} )^{4} \)
$41$ \( ( 6400 - 80 T^{2} + T^{4} )^{2} \)
$43$ \( ( 324 - 108 T + 18 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$47$ \( 65610000 - 8100 T^{4} + T^{8} \)
$53$ \( ( 100 + T^{4} )^{2} \)
$59$ \( ( 6400 + 80 T^{2} + T^{4} )^{2} \)
$61$ \( ( 36 - 6 T + T^{2} )^{4} \)
$67$ \( ( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$71$ \( ( 20 + T^{2} )^{4} \)
$73$ \( ( 2 - 2 T + T^{2} )^{4} \)
$79$ \( ( 1296 - 36 T^{2} + T^{4} )^{2} \)
$83$ \( 65610000 - 8100 T^{4} + T^{8} \)
$89$ \( ( -20 + T^{2} )^{4} \)
$97$ \( ( 26244 + 2916 T + 162 T^{2} + 18 T^{3} + T^{4} )^{2} \)
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