Properties

Label 1620.2.d.e
Level $1620$
Weight $2$
Character orbit 1620.d
Analytic conductor $12.936$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.28356903014400.1
Defining polynomial: \(x^{8} - 3 x^{6} + 20 x^{4} - 75 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} -\beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} -\beta_{2} q^{7} + ( -\beta_{1} + \beta_{4} + \beta_{6} ) q^{11} + ( -\beta_{3} + \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{17} + ( -1 + \beta_{3} + \beta_{5} ) q^{19} + ( 2 \beta_{1} + 2 \beta_{6} ) q^{23} + ( 1 + \beta_{2} + \beta_{3} ) q^{25} -3 \beta_{4} q^{29} + ( -1 - \beta_{3} - \beta_{5} ) q^{31} + ( 2 \beta_{6} + \beta_{7} ) q^{35} + ( -\beta_{2} + \beta_{3} - \beta_{5} ) q^{37} + ( -\beta_{1} - \beta_{4} + \beta_{6} ) q^{41} + \beta_{2} q^{43} + ( 2 \beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{47} + ( -7 - 2 \beta_{3} - 2 \beta_{5} ) q^{49} + \beta_{7} q^{53} + ( 5 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{55} + ( -\beta_{1} + 3 \beta_{4} + \beta_{6} ) q^{59} + ( 4 + \beta_{3} + \beta_{5} ) q^{61} + ( 5 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{65} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( -3 \beta_{1} - \beta_{4} + 3 \beta_{6} ) q^{71} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{6} + \beta_{7} ) q^{77} -6 q^{79} + ( 2 \beta_{1} + 2 \beta_{6} - \beta_{7} ) q^{83} + ( 2 + 2 \beta_{2} - 3 \beta_{3} ) q^{85} + ( -2 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} ) q^{89} + ( 4 + 2 \beta_{3} + 2 \beta_{5} ) q^{91} + ( 3 \beta_{1} - 5 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{95} + ( \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q - 12 q^{19} + 6 q^{25} - 4 q^{31} - 48 q^{49} + 40 q^{55} + 28 q^{61} - 48 q^{79} + 22 q^{85} + 24 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{6} + 20 x^{4} - 75 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 2 \nu^{4} + 30 \nu^{2} - 25 \)\()/50\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 2 \nu^{4} + 20 \nu^{2} - 25 \)\()/50\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{5} + 20 \nu^{3} + 25 \nu \)\()/250\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 8 \nu^{4} - 10 \nu^{2} + 75 \)\()/50\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{5} + 20 \nu^{3} - 75 \nu \)\()/125\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 22 \nu^{5} + 70 \nu^{3} + 175 \nu \)\()/250\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6} + 5 \beta_{4}\)
\(\nu^{4}\)\(=\)\(5 \beta_{5} - 2 \beta_{3} + 3 \beta_{2} - 7\)
\(\nu^{5}\)\(=\)\(8 \beta_{7} - 9 \beta_{6} - 10 \beta_{4} - 10 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-10 \beta_{5} - 26 \beta_{3} + 14 \beta_{2} + 9\)
\(\nu^{7}\)\(=\)\(4 \beta_{7} + 58 \beta_{6} - 130 \beta_{4} + 45 \beta_{1}\)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
2.07237 + 0.839805i
2.07237 0.839805i
1.20635 + 1.88274i
1.20635 1.88274i
−1.20635 + 1.88274i
−1.20635 1.88274i
−2.07237 + 0.839805i
−2.07237 0.839805i
0 0 0 −2.07237 0.839805i 0 4.93536i 0 0 0
649.2 0 0 0 −2.07237 + 0.839805i 0 4.93536i 0 0 0
649.3 0 0 0 −1.20635 1.88274i 0 1.28148i 0 0 0
649.4 0 0 0 −1.20635 + 1.88274i 0 1.28148i 0 0 0
649.5 0 0 0 1.20635 1.88274i 0 1.28148i 0 0 0
649.6 0 0 0 1.20635 + 1.88274i 0 1.28148i 0 0 0
649.7 0 0 0 2.07237 0.839805i 0 4.93536i 0 0 0
649.8 0 0 0 2.07237 + 0.839805i 0 4.93536i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.8
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
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.d.e 8
3.b odd 2 1 inner 1620.2.d.e 8
5.b even 2 1 inner 1620.2.d.e 8
5.c odd 4 2 8100.2.a.be 8
9.c even 3 2 1620.2.r.h 16
9.d odd 6 2 1620.2.r.h 16
15.d odd 2 1 inner 1620.2.d.e 8
15.e even 4 2 8100.2.a.be 8
45.h odd 6 2 1620.2.r.h 16
45.j even 6 2 1620.2.r.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.d.e 8 1.a even 1 1 trivial
1620.2.d.e 8 3.b odd 2 1 inner
1620.2.d.e 8 5.b even 2 1 inner
1620.2.d.e 8 15.d odd 2 1 inner
1620.2.r.h 16 9.c even 3 2
1620.2.r.h 16 9.d odd 6 2
1620.2.r.h 16 45.h odd 6 2
1620.2.r.h 16 45.j even 6 2
8100.2.a.be 8 5.c odd 4 2
8100.2.a.be 8 15.e even 4 2

Hecke kernels

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

\( T_{7}^{4} + 26 T_{7}^{2} + 40 \)
\( T_{11}^{4} - 23 T_{11}^{2} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 625 - 75 T^{2} + 20 T^{4} - 3 T^{6} + T^{8} \)
$7$ \( ( 40 + 26 T^{2} + T^{4} )^{2} \)
$11$ \( ( 100 - 23 T^{2} + T^{4} )^{2} \)
$13$ \( ( 360 + 51 T^{2} + T^{4} )^{2} \)
$17$ \( ( 1690 + 83 T^{2} + T^{4} )^{2} \)
$19$ \( ( -30 + 3 T + T^{2} )^{4} \)
$23$ \( ( 640 + 68 T^{2} + T^{4} )^{2} \)
$29$ \( ( -27 + T^{2} )^{4} \)
$31$ \( ( -32 + T + T^{2} )^{4} \)
$37$ \( ( 1690 + 89 T^{2} + T^{4} )^{2} \)
$41$ \( ( 16 - 35 T^{2} + T^{4} )^{2} \)
$43$ \( ( 40 + 26 T^{2} + T^{4} )^{2} \)
$47$ \( ( 4000 + 170 T^{2} + T^{4} )^{2} \)
$53$ \( ( 360 + 78 T^{2} + T^{4} )^{2} \)
$59$ \( ( 64 - 59 T^{2} + T^{4} )^{2} \)
$61$ \( ( -20 - 7 T + T^{2} )^{4} \)
$67$ \( ( 4000 + 260 T^{2} + T^{4} )^{2} \)
$71$ \( ( 6084 - 231 T^{2} + T^{4} )^{2} \)
$73$ \( ( 250 + 65 T^{2} + T^{4} )^{2} \)
$79$ \( ( 6 + T )^{8} \)
$83$ \( ( 2560 + 122 T^{2} + T^{4} )^{2} \)
$89$ \( ( 961 - 110 T^{2} + T^{4} )^{2} \)
$97$ \( ( 160 + 206 T^{2} + T^{4} )^{2} \)
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