Properties

Label 1620.2.r.g
Level $1620$
Weight $2$
Character orbit 1620.r
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.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.2
Defining polynomial: \(x^{8} - 25 x^{4} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{3} + \beta_{7} ) q^{5} + ( \beta_{2} - \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{3} + \beta_{7} ) q^{5} + ( \beta_{2} - \beta_{6} ) q^{7} + ( -\beta_{1} - \beta_{7} ) q^{11} + 3 \beta_{2} q^{13} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{17} -3 q^{19} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{23} + ( 5 \beta_{2} - 5 \beta_{6} ) q^{25} + ( -3 \beta_{1} - 3 \beta_{7} ) q^{29} + ( 2 - 2 \beta_{4} ) q^{31} + ( -\beta_{1} + \beta_{5} ) q^{35} + \beta_{6} q^{37} + ( -\beta_{3} + \beta_{5} + \beta_{7} ) q^{41} + ( -10 \beta_{2} + 10 \beta_{6} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{7} ) q^{47} + ( -6 + 6 \beta_{4} ) q^{49} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} ) q^{53} + ( 5 + 5 \beta_{6} ) q^{55} + ( 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} ) q^{59} + \beta_{4} q^{61} -3 \beta_{1} q^{65} -11 \beta_{2} q^{67} + ( -3 \beta_{1} - 3 \beta_{3} + 3 \beta_{5} ) q^{71} -13 \beta_{6} q^{73} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{77} -3 \beta_{4} q^{79} + ( -5 \beta_{1} + 5 \beta_{7} ) q^{83} + ( 10 - 10 \beta_{2} - 10 \beta_{4} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{3} + 4 \beta_{5} ) q^{89} + 3 q^{91} + ( 3 \beta_{3} - 3 \beta_{7} ) q^{95} + ( -\beta_{2} + \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 24 q^{19} + 8 q^{31} - 24 q^{49} + 40 q^{55} + 4 q^{61} - 12 q^{79} + 40 q^{85} + 24 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 25 x^{4} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/25\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/25\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/125\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/125\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)
\(\nu^{4}\)\(=\)\(25 \beta_{4}\)
\(\nu^{5}\)\(=\)\(25 \beta_{5}\)
\(\nu^{6}\)\(=\)\(125 \beta_{6}\)
\(\nu^{7}\)\(=\)\(125 \beta_{7}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
2.15988 + 0.578737i
0.578737 2.15988i
−0.578737 + 2.15988i
−2.15988 0.578737i
2.15988 0.578737i
0.578737 + 2.15988i
−0.578737 2.15988i
−2.15988 + 0.578737i
0 0 0 −2.15988 + 0.578737i 0 0.866025 0.500000i 0 0 0
109.2 0 0 0 −0.578737 2.15988i 0 −0.866025 + 0.500000i 0 0 0
109.3 0 0 0 0.578737 + 2.15988i 0 −0.866025 + 0.500000i 0 0 0
109.4 0 0 0 2.15988 0.578737i 0 0.866025 0.500000i 0 0 0
1189.1 0 0 0 −2.15988 0.578737i 0 0.866025 + 0.500000i 0 0 0
1189.2 0 0 0 −0.578737 + 2.15988i 0 −0.866025 0.500000i 0 0 0
1189.3 0 0 0 0.578737 2.15988i 0 −0.866025 0.500000i 0 0 0
1189.4 0 0 0 2.15988 + 0.578737i 0 0.866025 + 0.500000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1189.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.d odd 2 1 inner
45.h odd 6 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.r.g 8
3.b odd 2 1 inner 1620.2.r.g 8
5.b even 2 1 inner 1620.2.r.g 8
9.c even 3 1 540.2.d.c 4
9.c even 3 1 inner 1620.2.r.g 8
9.d odd 6 1 540.2.d.c 4
9.d odd 6 1 inner 1620.2.r.g 8
15.d odd 2 1 inner 1620.2.r.g 8
36.f odd 6 1 2160.2.f.l 4
36.h even 6 1 2160.2.f.l 4
45.h odd 6 1 540.2.d.c 4
45.h odd 6 1 inner 1620.2.r.g 8
45.j even 6 1 540.2.d.c 4
45.j even 6 1 inner 1620.2.r.g 8
45.k odd 12 1 2700.2.a.v 2
45.k odd 12 1 2700.2.a.w 2
45.l even 12 1 2700.2.a.v 2
45.l even 12 1 2700.2.a.w 2
180.n even 6 1 2160.2.f.l 4
180.p odd 6 1 2160.2.f.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.d.c 4 9.c even 3 1
540.2.d.c 4 9.d odd 6 1
540.2.d.c 4 45.h odd 6 1
540.2.d.c 4 45.j even 6 1
1620.2.r.g 8 1.a even 1 1 trivial
1620.2.r.g 8 3.b odd 2 1 inner
1620.2.r.g 8 5.b even 2 1 inner
1620.2.r.g 8 9.c even 3 1 inner
1620.2.r.g 8 9.d odd 6 1 inner
1620.2.r.g 8 15.d odd 2 1 inner
1620.2.r.g 8 45.h odd 6 1 inner
1620.2.r.g 8 45.j even 6 1 inner
2160.2.f.l 4 36.f odd 6 1
2160.2.f.l 4 36.h even 6 1
2160.2.f.l 4 180.n even 6 1
2160.2.f.l 4 180.p odd 6 1
2700.2.a.v 2 45.k odd 12 1
2700.2.a.v 2 45.l even 12 1
2700.2.a.w 2 45.k odd 12 1
2700.2.a.w 2 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{4} - T_{7}^{2} + 1 \)
\( T_{11}^{4} + 10 T_{11}^{2} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 625 - 25 T^{4} + T^{8} \)
$7$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$11$ \( ( 100 + 10 T^{2} + T^{4} )^{2} \)
$13$ \( ( 81 - 9 T^{2} + T^{4} )^{2} \)
$17$ \( ( 40 + T^{2} )^{4} \)
$19$ \( ( 3 + T )^{8} \)
$23$ \( ( 100 - 10 T^{2} + T^{4} )^{2} \)
$29$ \( ( 8100 + 90 T^{2} + T^{4} )^{2} \)
$31$ \( ( 4 - 2 T + T^{2} )^{4} \)
$37$ \( ( 1 + T^{2} )^{4} \)
$41$ \( ( 100 + 10 T^{2} + T^{4} )^{2} \)
$43$ \( ( 10000 - 100 T^{2} + T^{4} )^{2} \)
$47$ \( ( 1600 - 40 T^{2} + T^{4} )^{2} \)
$53$ \( ( 90 + T^{2} )^{4} \)
$59$ \( ( 1600 + 40 T^{2} + T^{4} )^{2} \)
$61$ \( ( 1 - T + T^{2} )^{4} \)
$67$ \( ( 14641 - 121 T^{2} + T^{4} )^{2} \)
$71$ \( ( -90 + T^{2} )^{4} \)
$73$ \( ( 169 + T^{2} )^{4} \)
$79$ \( ( 9 + 3 T + T^{2} )^{4} \)
$83$ \( ( 62500 - 250 T^{2} + T^{4} )^{2} \)
$89$ \( ( -160 + T^{2} )^{4} \)
$97$ \( ( 1 - T^{2} + T^{4} )^{2} \)
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