Properties

Label 1620.3.o.e
Level $1620$
Weight $3$
Character orbit 1620.o
Analytic conductor $44.142$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.o (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: \( 2^{4}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 540)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{4} ) q^{5} + ( -5 + 5 \beta_{2} + \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{4} ) q^{5} + ( -5 + 5 \beta_{2} + \beta_{6} ) q^{7} + ( -3 \beta_{1} - \beta_{3} ) q^{11} + ( 7 \beta_{2} - \beta_{5} ) q^{13} + ( 5 \beta_{3} + 5 \beta_{7} ) q^{17} + ( -1 + 2 \beta_{5} - 2 \beta_{6} ) q^{19} + ( -6 \beta_{1} + 6 \beta_{4} + 7 \beta_{7} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} + ( -3 \beta_{1} + \beta_{3} ) q^{29} + ( 7 \beta_{2} - 2 \beta_{5} ) q^{31} + ( -5 \beta_{3} - 5 \beta_{4} - 5 \beta_{7} ) q^{35} + ( -16 + 6 \beta_{5} - 6 \beta_{6} ) q^{37} + ( 15 \beta_{1} - 15 \beta_{4} + 3 \beta_{7} ) q^{41} + ( -29 + 29 \beta_{2} + 7 \beta_{6} ) q^{43} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -21 \beta_{2} - 10 \beta_{5} ) q^{49} + ( \beta_{3} - 30 \beta_{4} + \beta_{7} ) q^{53} + ( 15 + \beta_{5} - \beta_{6} ) q^{55} + ( 9 \beta_{1} - 9 \beta_{4} - 17 \beta_{7} ) q^{59} + ( 1 - \beta_{2} + 12 \beta_{6} ) q^{61} + ( -7 \beta_{1} + 5 \beta_{3} ) q^{65} -14 \beta_{2} q^{67} + ( -15 \beta_{3} - 21 \beta_{4} - 15 \beta_{7} ) q^{71} + ( -7 + 3 \beta_{5} - 3 \beta_{6} ) q^{73} + ( 24 \beta_{1} - 24 \beta_{4} - 20 \beta_{7} ) q^{77} + ( 67 - 67 \beta_{2} - 12 \beta_{6} ) q^{79} + ( -48 \beta_{1} - 11 \beta_{3} ) q^{83} -5 \beta_{5} q^{85} + ( 3 \beta_{3} - 51 \beta_{4} + 3 \beta_{7} ) q^{89} + ( 10 - 2 \beta_{5} + 2 \beta_{6} ) q^{91} + ( \beta_{1} - \beta_{4} + 10 \beta_{7} ) q^{95} + ( -2 + 2 \beta_{2} + 2 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{7} + O(q^{10}) \) \( 8 q - 20 q^{7} + 28 q^{13} - 8 q^{19} + 20 q^{25} + 28 q^{31} - 128 q^{37} - 116 q^{43} - 84 q^{49} + 120 q^{55} + 4 q^{61} - 56 q^{67} - 56 q^{73} + 268 q^{79} + 80 q^{91} - 8 q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 29 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{6} - 8 \nu^{4} + 24 \nu^{2} - 1 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} - 39 \nu \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} - 8 \nu^{5} + 22 \nu^{3} - \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -27 \nu^{6} + 72 \nu^{4} - 168 \nu^{2} + 9 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -21 \nu^{6} + 72 \nu^{4} - 168 \nu^{2} + 63 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -15 \nu^{7} + 48 \nu^{5} - 120 \nu^{3} + 45 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 9 \beta_{2}\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + 3 \beta_{4} + 2 \beta_{3}\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{6} + 7 \beta_{2} - 7\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(11 \beta_{7} + 15 \beta_{4} - 15 \beta_{1}\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(4 \beta_{6} - 4 \beta_{5} - 27\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-29 \beta_{3} - 39 \beta_{1}\)\()/6\)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
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
−0.535233 0.309017i
0 0 0 −1.93649 1.11803i 0 −5.85410 10.1396i 0 0 0
701.2 0 0 0 −1.93649 1.11803i 0 0.854102 + 1.47935i 0 0 0
701.3 0 0 0 1.93649 + 1.11803i 0 −5.85410 10.1396i 0 0 0
701.4 0 0 0 1.93649 + 1.11803i 0 0.854102 + 1.47935i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −5.85410 + 10.1396i 0 0 0
1241.2 0 0 0 −1.93649 + 1.11803i 0 0.854102 1.47935i 0 0 0
1241.3 0 0 0 1.93649 1.11803i 0 −5.85410 + 10.1396i 0 0 0
1241.4 0 0 0 1.93649 1.11803i 0 0.854102 1.47935i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.3.o.e 8
3.b odd 2 1 inner 1620.3.o.e 8
9.c even 3 1 540.3.g.d 4
9.c even 3 1 inner 1620.3.o.e 8
9.d odd 6 1 540.3.g.d 4
9.d odd 6 1 inner 1620.3.o.e 8
36.f odd 6 1 2160.3.l.e 4
36.h even 6 1 2160.3.l.e 4
45.h odd 6 1 2700.3.g.n 4
45.j even 6 1 2700.3.g.n 4
45.k odd 12 1 2700.3.b.g 4
45.k odd 12 1 2700.3.b.l 4
45.l even 12 1 2700.3.b.g 4
45.l even 12 1 2700.3.b.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.d 4 9.c even 3 1
540.3.g.d 4 9.d odd 6 1
1620.3.o.e 8 1.a even 1 1 trivial
1620.3.o.e 8 3.b odd 2 1 inner
1620.3.o.e 8 9.c even 3 1 inner
1620.3.o.e 8 9.d odd 6 1 inner
2160.3.l.e 4 36.f odd 6 1
2160.3.l.e 4 36.h even 6 1
2700.3.b.g 4 45.k odd 12 1
2700.3.b.g 4 45.l even 12 1
2700.3.b.l 4 45.k odd 12 1
2700.3.b.l 4 45.l even 12 1
2700.3.g.n 4 45.h odd 6 1
2700.3.g.n 4 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 10 T_{7}^{3} + 120 T_{7}^{2} - 200 T_{7} + 400 \) acting on \(S_{3}^{\mathrm{new}}(1620, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 25 - 5 T^{2} + T^{4} )^{2} \)
$7$ \( ( 400 - 200 T + 120 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$11$ \( 1679616 - 139968 T^{2} + 10368 T^{4} - 108 T^{6} + T^{8} \)
$13$ \( ( 16 - 56 T + 192 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$17$ \( ( 225 + T^{2} )^{4} \)
$19$ \( ( -179 + 2 T + T^{2} )^{4} \)
$23$ \( 4640470641 - 84606282 T^{2} + 1474443 T^{4} - 1242 T^{6} + T^{8} \)
$29$ \( 1679616 - 139968 T^{2} + 10368 T^{4} - 108 T^{6} + T^{8} \)
$31$ \( ( 17161 + 1834 T + 327 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$37$ \( ( -1364 + 32 T + T^{2} )^{4} \)
$41$ \( 1187960484096 - 2628925632 T^{2} + 4727808 T^{4} - 2412 T^{6} + T^{8} \)
$43$ \( ( 1860496 - 79112 T + 4728 T^{2} + 58 T^{3} + T^{4} )^{2} \)
$47$ \( 1679616 - 839808 T^{2} + 418608 T^{4} - 648 T^{6} + T^{8} \)
$53$ \( ( 20169081 + 9018 T^{2} + T^{4} )^{2} \)
$59$ \( 23255696077056 - 28992364992 T^{2} + 31321728 T^{4} - 6012 T^{6} + T^{8} \)
$61$ \( ( 41977441 + 12958 T + 6483 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$67$ \( ( 196 + 14 T + T^{2} )^{4} \)
$71$ \( ( 32400 + 8460 T^{2} + T^{4} )^{2} \)
$73$ \( ( -356 + 14 T + T^{2} )^{4} \)
$79$ \( ( 3964081 + 266794 T + 19947 T^{2} - 134 T^{3} + T^{4} )^{2} \)
$83$ \( 11838693626789121 - 2743863680898 T^{2} + 527141763 T^{4} - 25218 T^{6} + T^{8} \)
$89$ \( ( 167029776 + 26172 T^{2} + T^{4} )^{2} \)
$97$ \( ( 30976 - 704 T + 192 T^{2} + 4 T^{3} + T^{4} )^{2} \)
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