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: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} - 2 \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - 2 \beta_{4} q^{7} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{2}) q^{11} + 3 \beta_1 q^{13} + 7 \beta_{2} q^{17} - 6 q^{19} - \beta_{7} q^{23} + (5 \beta_{4} + 15 \beta_{3}) q^{25} + ( - 3 \beta_{7} + 6 \beta_{6} + 6 \beta_{5} + 3 \beta_{2}) q^{29} + (34 \beta_{3} - 34) q^{31} + (4 \beta_{6} - 6 \beta_{2}) q^{35} + (11 \beta_{4} + 11 \beta_1) q^{37} + ( - 2 \beta_{7} + 4 \beta_{5}) q^{41} - 7 \beta_{4} q^{43} + (\beta_{7} + \beta_{2}) q^{47} + ( - 15 \beta_{3} + 15) q^{49} - 9 \beta_{2} q^{53} + ( - 5 \beta_{4} - 5 \beta_1 - 40) q^{55} + ( - 11 \beta_{7} + 22 \beta_{5}) q^{59} - 74 \beta_{3} q^{61} + ( - 15 \beta_{7} + 6 \beta_{6} + 6 \beta_{5} - 9 \beta_{2}) q^{65} + 23 \beta_1 q^{67} + (12 \beta_{6} + 6 \beta_{2}) q^{71} + ( - 14 \beta_{4} - 14 \beta_1) q^{73} - 16 \beta_{7} q^{77} + 78 \beta_{3} q^{79} + ( - 23 \beta_{7} - 23 \beta_{2}) q^{83} + ( - 70 \beta_{3} - 35 \beta_1 + 70) q^{85} + ( - 4 \beta_{6} - 2 \beta_{2}) q^{89} + 96 q^{91} + 6 \beta_{5} q^{95} + 8 \beta_{4} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{19} + 60 q^{25} - 136 q^{31} + 60 q^{49} - 320 q^{55} - 296 q^{61} + 312 q^{79} + 280 q^{85} + 768 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 13\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} - 8\nu^{4} + 24\nu^{2} - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 15\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} + 24\nu^{4} - 56\nu^{2} + 58\nu + 21 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -12\nu^{7} + \nu^{6} + 32\nu^{5} - 88\nu^{3} + 4\nu + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{6} + 24\nu^{4} - 56\nu^{2} + 21 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 6\beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{6} + 2\beta_{4} - \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 14\beta_{3} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 10\beta_{6} - 10\beta_{5} + 11\beta_{4} - 5\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{7} - 26\beta_{5} - 29\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

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

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

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

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