Properties

Label 1620.3.t.b
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: 8.0.12960000.1
Defining polynomial: \(x^{8} - 3 x^{6} + 8 x^{4} - 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 60)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{5} -2 \beta_{4} q^{7} +O(q^{10})\) \( q -\beta_{5} q^{5} -2 \beta_{4} q^{7} + ( \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{11} + 3 \beta_{1} q^{13} + 7 \beta_{2} q^{17} -6 q^{19} -\beta_{7} q^{23} + ( 15 \beta_{3} + 5 \beta_{4} ) q^{25} + ( 3 \beta_{2} + 6 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{29} + ( -34 + 34 \beta_{3} ) q^{31} + ( -6 \beta_{2} + 4 \beta_{6} ) q^{35} + ( 11 \beta_{1} + 11 \beta_{4} ) q^{37} + ( 4 \beta_{5} - 2 \beta_{7} ) q^{41} -7 \beta_{4} q^{43} + ( \beta_{2} + \beta_{7} ) q^{47} + ( 15 - 15 \beta_{3} ) q^{49} -9 \beta_{2} q^{53} + ( -40 - 5 \beta_{1} - 5 \beta_{4} ) q^{55} + ( 22 \beta_{5} - 11 \beta_{7} ) q^{59} -74 \beta_{3} q^{61} + ( -9 \beta_{2} + 6 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} ) q^{65} + 23 \beta_{1} q^{67} + ( 6 \beta_{2} + 12 \beta_{6} ) q^{71} + ( -14 \beta_{1} - 14 \beta_{4} ) q^{73} -16 \beta_{7} q^{77} + 78 \beta_{3} q^{79} + ( -23 \beta_{2} - 23 \beta_{7} ) q^{83} + ( 70 - 35 \beta_{1} - 70 \beta_{3} ) q^{85} + ( -2 \beta_{2} - 4 \beta_{6} ) q^{89} + 96 q^{91} + 6 \beta_{5} q^{95} + 8 \beta_{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 48q^{19} + 60q^{25} - 136q^{31} + 60q^{49} - 320q^{55} - 296q^{61} + 312q^{79} + 280q^{85} + 768q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 13 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} - 9 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( 3 \nu^{6} - 8 \nu^{4} + 24 \nu^{2} - 1 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{7} + 16 \nu^{5} - 40 \nu^{3} + 15 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{7} - 7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 58 \nu + 21 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -12 \nu^{7} + \nu^{6} + 32 \nu^{5} - 88 \nu^{3} + 4 \nu + 9 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{6} + 24 \nu^{4} - 56 \nu^{2} + 21 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{5} + \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 6 \beta_{3} + \beta_{2}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{6} + 2 \beta_{4} - \beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{7} + 14 \beta_{3} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + 11 \beta_{4} - 5 \beta_{2}\)\()/8\)
\(\nu^{6}\)\(=\)\(-2 \beta_{2} - 9\)
\(\nu^{7}\)\(=\)\((\)\(13 \beta_{7} - 26 \beta_{5} - 29 \beta_{1}\)\()/8\)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(1 - \beta_{3}\)

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
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
−1.40126 0.809017i
0 0 0 −4.99102 + 0.299576i 0 −6.92820 4.00000i 0 0 0
269.2 0 0 0 −2.75495 + 4.17256i 0 6.92820 + 4.00000i 0 0 0
269.3 0 0 0 2.75495 4.17256i 0 6.92820 + 4.00000i 0 0 0
269.4 0 0 0 4.99102 0.299576i 0 −6.92820 4.00000i 0 0 0
1349.1 0 0 0 −4.99102 0.299576i 0 −6.92820 + 4.00000i 0 0 0
1349.2 0 0 0 −2.75495 4.17256i 0 6.92820 4.00000i 0 0 0
1349.3 0 0 0 2.75495 + 4.17256i 0 6.92820 4.00000i 0 0 0
1349.4 0 0 0 4.99102 + 0.299576i 0 −6.92820 + 4.00000i 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.b 8
3.b odd 2 1 inner 1620.3.t.b 8
5.b even 2 1 inner 1620.3.t.b 8
9.c even 3 1 60.3.b.a 4
9.c even 3 1 inner 1620.3.t.b 8
9.d odd 6 1 60.3.b.a 4
9.d odd 6 1 inner 1620.3.t.b 8
15.d odd 2 1 inner 1620.3.t.b 8
36.f odd 6 1 240.3.c.d 4
36.h even 6 1 240.3.c.d 4
45.h odd 6 1 60.3.b.a 4
45.h odd 6 1 inner 1620.3.t.b 8
45.j even 6 1 60.3.b.a 4
45.j even 6 1 inner 1620.3.t.b 8
45.k odd 12 1 300.3.g.e 2
45.k odd 12 1 300.3.g.h 2
45.l even 12 1 300.3.g.e 2
45.l even 12 1 300.3.g.h 2
72.j odd 6 1 960.3.c.h 4
72.l even 6 1 960.3.c.g 4
72.n even 6 1 960.3.c.h 4
72.p odd 6 1 960.3.c.g 4
180.n even 6 1 240.3.c.d 4
180.p odd 6 1 240.3.c.d 4
180.v odd 12 1 1200.3.l.h 2
180.v odd 12 1 1200.3.l.q 2
180.x even 12 1 1200.3.l.h 2
180.x even 12 1 1200.3.l.q 2
360.z odd 6 1 960.3.c.g 4
360.bd even 6 1 960.3.c.g 4
360.bh odd 6 1 960.3.c.h 4
360.bk even 6 1 960.3.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a 4 9.c even 3 1
60.3.b.a 4 9.d odd 6 1
60.3.b.a 4 45.h odd 6 1
60.3.b.a 4 45.j even 6 1
240.3.c.d 4 36.f odd 6 1
240.3.c.d 4 36.h even 6 1
240.3.c.d 4 180.n even 6 1
240.3.c.d 4 180.p odd 6 1
300.3.g.e 2 45.k odd 12 1
300.3.g.e 2 45.l even 12 1
300.3.g.h 2 45.k odd 12 1
300.3.g.h 2 45.l even 12 1
960.3.c.g 4 72.l even 6 1
960.3.c.g 4 72.p odd 6 1
960.3.c.g 4 360.z odd 6 1
960.3.c.g 4 360.bd even 6 1
960.3.c.h 4 72.j odd 6 1
960.3.c.h 4 72.n even 6 1
960.3.c.h 4 360.bh odd 6 1
960.3.c.h 4 360.bk even 6 1
1200.3.l.h 2 180.v odd 12 1
1200.3.l.h 2 180.x even 12 1
1200.3.l.q 2 180.v odd 12 1
1200.3.l.q 2 180.x even 12 1
1620.3.t.b 8 1.a even 1 1 trivial
1620.3.t.b 8 3.b odd 2 1 inner
1620.3.t.b 8 5.b even 2 1 inner
1620.3.t.b 8 9.c even 3 1 inner
1620.3.t.b 8 9.d odd 6 1 inner
1620.3.t.b 8 15.d odd 2 1 inner
1620.3.t.b 8 45.h odd 6 1 inner
1620.3.t.b 8 45.j even 6 1 inner

Hecke kernels

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

\( T_{7}^{4} - 64 T_{7}^{2} + 4096 \)
\( T_{17}^{2} - 980 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 390625 - 18750 T^{2} + 275 T^{4} - 30 T^{6} + T^{8} \)
$7$ \( ( 4096 - 64 T^{2} + T^{4} )^{2} \)
$11$ \( ( 6400 - 80 T^{2} + T^{4} )^{2} \)
$13$ \( ( 20736 - 144 T^{2} + T^{4} )^{2} \)
$17$ \( ( -980 + T^{2} )^{4} \)
$19$ \( ( 6 + T )^{8} \)
$23$ \( ( 400 + 20 T^{2} + T^{4} )^{2} \)
$29$ \( ( 518400 - 720 T^{2} + T^{4} )^{2} \)
$31$ \( ( 1156 + 34 T + T^{2} )^{4} \)
$37$ \( ( 1936 + T^{2} )^{4} \)
$41$ \( ( 102400 - 320 T^{2} + T^{4} )^{2} \)
$43$ \( ( 614656 - 784 T^{2} + T^{4} )^{2} \)
$47$ \( ( 400 + 20 T^{2} + T^{4} )^{2} \)
$53$ \( ( -1620 + T^{2} )^{4} \)
$59$ \( ( 93702400 - 9680 T^{2} + T^{4} )^{2} \)
$61$ \( ( 5476 + 74 T + T^{2} )^{4} \)
$67$ \( ( 71639296 - 8464 T^{2} + T^{4} )^{2} \)
$71$ \( ( 2880 + T^{2} )^{4} \)
$73$ \( ( 3136 + T^{2} )^{4} \)
$79$ \( ( 6084 - 78 T + T^{2} )^{4} \)
$83$ \( ( 111936400 + 10580 T^{2} + T^{4} )^{2} \)
$89$ \( ( 320 + T^{2} )^{4} \)
$97$ \( ( 1048576 - 1024 T^{2} + T^{4} )^{2} \)
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