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: \( x^{8} - 24x^{6} + 455x^{4} - 2904x^{2} + 14641 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} + \beta_{2}) q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5} + \beta_{2}) q^{5} + \beta_{3} q^{7} - 7 \beta_{4} q^{11} + 3 \beta_{7} q^{13} - 4 \beta_{2} q^{17} + 12 q^{19} + (2 \beta_{5} + 2 \beta_{2}) q^{23} + (2 \beta_{3} - 21 \beta_1) q^{25} + 6 \beta_{4} q^{29} + ( - 38 \beta_1 + 38) q^{31} + ( - 23 \beta_{6} - 23 \beta_{4} + 2 \beta_{2}) q^{35} + ( - \beta_{7} - \beta_{3}) q^{37} - 49 \beta_{6} q^{41} - 10 \beta_{3} q^{43} + 16 \beta_{5} q^{47} + (3 \beta_1 - 3) q^{49} + (7 \beta_{7} + 7 \beta_{3} + 14) q^{55} + 59 \beta_{6} q^{59} + 70 \beta_1 q^{61} + (6 \beta_{5} + 69 \beta_{4}) q^{65} - 16 \beta_{7} q^{67} + (84 \beta_{6} + 84 \beta_{4}) q^{71} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{73} + (14 \beta_{5} + 14 \beta_{2}) q^{77} - 30 \beta_1 q^{79} + 28 \beta_{5} q^{83} + (4 \beta_{7} + 92 \beta_1 - 92) q^{85} + (23 \beta_{6} + 23 \beta_{4}) q^{89} - 138 q^{91} + (12 \beta_{6} + 12 \beta_{5} + 12 \beta_{2}) q^{95} + 14 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 96 q^{19} - 84 q^{25} + 152 q^{31} - 12 q^{49} + 112 q^{55} + 280 q^{61} - 120 q^{79} - 368 q^{85} - 1104 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -24\nu^{6} + 455\nu^{4} - 10920\nu^{2} + 69696 ) / 55055 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 2556 ) / 455 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 13021\nu ) / 5005 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 3011\nu ) / 5005 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -167\nu^{6} + 5460\nu^{4} - 75985\nu^{2} + 484968 ) / 55055 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 191\nu^{7} - 5915\nu^{5} + 86905\nu^{3} - 554664\nu ) / 605605 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 719\nu^{7} - 15925\nu^{5} + 327145\nu^{3} - 2087976\nu ) / 605605 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{2} - 12\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{7} - 35\beta_{6} - 35\beta_{4} + 13\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 24\beta_{5} - 167\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 191\beta_{7} - 719\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -455\beta_{2} - 2556 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13021\beta_{4} - 3011\beta_{3} ) / 2 \) Copy content Toggle raw display

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 \(S_{3}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{4} - 46T_{7}^{2} + 2116 \) Copy content Toggle raw display
\( T_{17}^{2} - 368 \) Copy content Toggle raw display

Hecke characteristic polynomials

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