Properties

Label 1620.3.t.c
Level $1620$
Weight $3$
Character orbit 1620.t
Analytic conductor $44.142$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1485512441856.6
Defining polynomial: \(x^{8} - 24 x^{6} + 455 x^{4} - 2904 x^{2} + 14641\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 180)
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.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{5} + \beta_{6} ) q^{5} + \beta_{3} q^{7} -7 \beta_{4} q^{11} + 3 \beta_{7} q^{13} -4 \beta_{2} q^{17} + 12 q^{19} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{23} + ( -21 \beta_{1} + 2 \beta_{3} ) q^{25} + 6 \beta_{4} q^{29} + ( 38 - 38 \beta_{1} ) q^{31} + ( 2 \beta_{2} - 23 \beta_{4} - 23 \beta_{6} ) q^{35} + ( -\beta_{3} - \beta_{7} ) q^{37} -49 \beta_{6} q^{41} -10 \beta_{3} q^{43} + 16 \beta_{5} q^{47} + ( -3 + 3 \beta_{1} ) q^{49} + ( 14 + 7 \beta_{3} + 7 \beta_{7} ) q^{55} + 59 \beta_{6} q^{59} + 70 \beta_{1} q^{61} + ( 69 \beta_{4} + 6 \beta_{5} ) q^{65} -16 \beta_{7} q^{67} + ( 84 \beta_{4} + 84 \beta_{6} ) q^{71} + ( -2 \beta_{3} - 2 \beta_{7} ) q^{73} + ( 14 \beta_{2} + 14 \beta_{5} ) q^{77} -30 \beta_{1} q^{79} + 28 \beta_{5} q^{83} + ( -92 + 92 \beta_{1} + 4 \beta_{7} ) q^{85} + ( 23 \beta_{4} + 23 \beta_{6} ) q^{89} -138 q^{91} + ( 12 \beta_{2} + 12 \beta_{5} + 12 \beta_{6} ) q^{95} + 14 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 96 q^{19} - 84 q^{25} + 152 q^{31} - 12 q^{49} + 112 q^{55} + 280 q^{61} - 120 q^{79} - 368 q^{85} - 1104 q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 24 x^{6} + 455 x^{4} - 2904 x^{2} + 14641\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -24 \nu^{6} + 455 \nu^{4} - 10920 \nu^{2} + 69696 \)\()/55055\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - 2556 \)\()/455\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 13021 \nu \)\()/5005\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 3011 \nu \)\()/5005\)
\(\beta_{5}\)\(=\)\((\)\( -167 \nu^{6} + 5460 \nu^{4} - 75985 \nu^{2} + 484968 \)\()/55055\)
\(\beta_{6}\)\(=\)\((\)\( 191 \nu^{7} - 5915 \nu^{5} + 86905 \nu^{3} - 554664 \nu \)\()/605605\)
\(\beta_{7}\)\(=\)\((\)\( 719 \nu^{7} - 15925 \nu^{5} + 327145 \nu^{3} - 2087976 \nu \)\()/605605\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{2} - 12 \beta_{1} + 12\)
\(\nu^{3}\)\(=\)\((\)\(13 \beta_{7} - 35 \beta_{6} - 35 \beta_{4} + 13 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\(24 \beta_{5} - 167 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(191 \beta_{7} - 719 \beta_{6}\)\()/2\)
\(\nu^{6}\)\(=\)\(-455 \beta_{2} - 2556\)
\(\nu^{7}\)\(=\)\((\)\(13021 \beta_{4} - 3011 \beta_{3}\)\()/2\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
2.32446 + 1.34203i
−2.32446 1.34203i
−3.54921 2.04914i
3.54921 + 2.04914i
2.32446 1.34203i
−2.32446 + 1.34203i
−3.54921 + 2.04914i
3.54921 2.04914i
0 0 0 −3.62266 3.44621i 0 5.87367 + 3.39116i 0 0 0
269.2 0 0 0 −1.17317 4.86042i 0 −5.87367 3.39116i 0 0 0
269.3 0 0 0 1.17317 + 4.86042i 0 −5.87367 3.39116i 0 0 0
269.4 0 0 0 3.62266 + 3.44621i 0 5.87367 + 3.39116i 0 0 0
1349.1 0 0 0 −3.62266 + 3.44621i 0 5.87367 3.39116i 0 0 0
1349.2 0 0 0 −1.17317 + 4.86042i 0 −5.87367 + 3.39116i 0 0 0
1349.3 0 0 0 1.17317 4.86042i 0 −5.87367 + 3.39116i 0 0 0
1349.4 0 0 0 3.62266 3.44621i 0 5.87367 3.39116i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1349.4
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 1620.3.t.c 8
3.b odd 2 1 inner 1620.3.t.c 8
5.b even 2 1 inner 1620.3.t.c 8
9.c even 3 1 180.3.b.a 4
9.c even 3 1 inner 1620.3.t.c 8
9.d odd 6 1 180.3.b.a 4
9.d odd 6 1 inner 1620.3.t.c 8
15.d odd 2 1 inner 1620.3.t.c 8
36.f odd 6 1 720.3.c.b 4
36.h even 6 1 720.3.c.b 4
45.h odd 6 1 180.3.b.a 4
45.h odd 6 1 inner 1620.3.t.c 8
45.j even 6 1 180.3.b.a 4
45.j even 6 1 inner 1620.3.t.c 8
45.k odd 12 2 900.3.g.c 4
45.l even 12 2 900.3.g.c 4
72.j odd 6 1 2880.3.c.c 4
72.l even 6 1 2880.3.c.f 4
72.n even 6 1 2880.3.c.c 4
72.p odd 6 1 2880.3.c.f 4
180.n even 6 1 720.3.c.b 4
180.p odd 6 1 720.3.c.b 4
180.v odd 12 2 3600.3.l.r 4
180.x even 12 2 3600.3.l.r 4
360.z odd 6 1 2880.3.c.f 4
360.bd even 6 1 2880.3.c.f 4
360.bh odd 6 1 2880.3.c.c 4
360.bk even 6 1 2880.3.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.b.a 4 9.c even 3 1
180.3.b.a 4 9.d odd 6 1
180.3.b.a 4 45.h odd 6 1
180.3.b.a 4 45.j even 6 1
720.3.c.b 4 36.f odd 6 1
720.3.c.b 4 36.h even 6 1
720.3.c.b 4 180.n even 6 1
720.3.c.b 4 180.p odd 6 1
900.3.g.c 4 45.k odd 12 2
900.3.g.c 4 45.l even 12 2
1620.3.t.c 8 1.a even 1 1 trivial
1620.3.t.c 8 3.b odd 2 1 inner
1620.3.t.c 8 5.b even 2 1 inner
1620.3.t.c 8 9.c even 3 1 inner
1620.3.t.c 8 9.d odd 6 1 inner
1620.3.t.c 8 15.d odd 2 1 inner
1620.3.t.c 8 45.h odd 6 1 inner
1620.3.t.c 8 45.j even 6 1 inner
2880.3.c.c 4 72.j odd 6 1
2880.3.c.c 4 72.n even 6 1
2880.3.c.c 4 360.bh odd 6 1
2880.3.c.c 4 360.bk even 6 1
2880.3.c.f 4 72.l even 6 1
2880.3.c.f 4 72.p odd 6 1
2880.3.c.f 4 360.z odd 6 1
2880.3.c.f 4 360.bd even 6 1
3600.3.l.r 4 180.v odd 12 2
3600.3.l.r 4 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{4} - 46 T_{7}^{2} + 2116 \)
\( T_{17}^{2} - 368 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 390625 + 26250 T^{2} + 1139 T^{4} + 42 T^{6} + T^{8} \)
$7$ \( ( 2116 - 46 T^{2} + T^{4} )^{2} \)
$11$ \( ( 9604 - 98 T^{2} + T^{4} )^{2} \)
$13$ \( ( 171396 - 414 T^{2} + T^{4} )^{2} \)
$17$ \( ( -368 + T^{2} )^{4} \)
$19$ \( ( -12 + T )^{8} \)
$23$ \( ( 8464 + 92 T^{2} + T^{4} )^{2} \)
$29$ \( ( 5184 - 72 T^{2} + T^{4} )^{2} \)
$31$ \( ( 1444 - 38 T + T^{2} )^{4} \)
$37$ \( ( 46 + T^{2} )^{4} \)
$41$ \( ( 23059204 - 4802 T^{2} + T^{4} )^{2} \)
$43$ \( ( 21160000 - 4600 T^{2} + T^{4} )^{2} \)
$47$ \( ( 34668544 + 5888 T^{2} + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( ( 48469444 - 6962 T^{2} + T^{4} )^{2} \)
$61$ \( ( 4900 - 70 T + T^{2} )^{4} \)
$67$ \( ( 138674176 - 11776 T^{2} + T^{4} )^{2} \)
$71$ \( ( 14112 + T^{2} )^{4} \)
$73$ \( ( 184 + T^{2} )^{4} \)
$79$ \( ( 900 + 30 T + T^{2} )^{4} \)
$83$ \( ( 325153024 + 18032 T^{2} + T^{4} )^{2} \)
$89$ \( ( 1058 + T^{2} )^{4} \)
$97$ \( ( 81288256 - 9016 T^{2} + T^{4} )^{2} \)
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