Properties

Label 1620.3.o.b
Level $1620$
Weight $3$
Character orbit 1620.o
Analytic conductor $44.142$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -2 + 2 \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{3} ) q^{5} + ( -2 + 2 \beta_{2} ) q^{7} -6 \beta_{1} q^{11} -8 \beta_{2} q^{13} + 6 \beta_{3} q^{17} -34 q^{19} + ( 18 \beta_{1} - 18 \beta_{3} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} + 18 \beta_{1} q^{29} -14 \beta_{2} q^{31} -2 \beta_{3} q^{35} + 56 q^{37} + ( -12 \beta_{1} + 12 \beta_{3} ) q^{41} + ( -8 + 8 \beta_{2} ) q^{43} + 18 \beta_{1} q^{47} + 45 \beta_{2} q^{49} -18 \beta_{3} q^{53} + 30 q^{55} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{59} + ( 46 - 46 \beta_{2} ) q^{61} + 8 \beta_{1} q^{65} -32 \beta_{2} q^{67} + 24 \beta_{3} q^{71} -106 q^{73} + ( 12 \beta_{1} - 12 \beta_{3} ) q^{77} + ( 22 - 22 \beta_{2} ) q^{79} + 54 \beta_{1} q^{83} -30 \beta_{2} q^{85} + 48 \beta_{3} q^{89} + 16 q^{91} + ( 34 \beta_{1} - 34 \beta_{3} ) q^{95} + ( -122 + 122 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} - 16q^{13} - 136q^{19} + 10q^{25} - 28q^{31} + 224q^{37} - 16q^{43} + 90q^{49} + 120q^{55} + 92q^{61} - 64q^{67} - 424q^{73} + 44q^{79} - 60q^{85} + 64q^{91} - 244q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

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
1.93649 1.11803i
−1.93649 + 1.11803i
1.93649 + 1.11803i
−1.93649 1.11803i
0 0 0 −1.93649 1.11803i 0 −1.00000 1.73205i 0 0 0
701.2 0 0 0 1.93649 + 1.11803i 0 −1.00000 1.73205i 0 0 0
1241.1 0 0 0 −1.93649 + 1.11803i 0 −1.00000 + 1.73205i 0 0 0
1241.2 0 0 0 1.93649 1.11803i 0 −1.00000 + 1.73205i 0 0 0
\(n\): e.g. 2-40 or 990-1000
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.b 4
3.b odd 2 1 inner 1620.3.o.b 4
9.c even 3 1 60.3.g.a 2
9.c even 3 1 inner 1620.3.o.b 4
9.d odd 6 1 60.3.g.a 2
9.d odd 6 1 inner 1620.3.o.b 4
36.f odd 6 1 240.3.l.a 2
36.h even 6 1 240.3.l.a 2
45.h odd 6 1 300.3.g.d 2
45.j even 6 1 300.3.g.d 2
45.k odd 12 2 300.3.b.c 4
45.l even 12 2 300.3.b.c 4
72.j odd 6 1 960.3.l.a 2
72.l even 6 1 960.3.l.d 2
72.n even 6 1 960.3.l.a 2
72.p odd 6 1 960.3.l.d 2
180.n even 6 1 1200.3.l.r 2
180.p odd 6 1 1200.3.l.r 2
180.v odd 12 2 1200.3.c.e 4
180.x even 12 2 1200.3.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 9.c even 3 1
60.3.g.a 2 9.d odd 6 1
240.3.l.a 2 36.f odd 6 1
240.3.l.a 2 36.h even 6 1
300.3.b.c 4 45.k odd 12 2
300.3.b.c 4 45.l even 12 2
300.3.g.d 2 45.h odd 6 1
300.3.g.d 2 45.j even 6 1
960.3.l.a 2 72.j odd 6 1
960.3.l.a 2 72.n even 6 1
960.3.l.d 2 72.l even 6 1
960.3.l.d 2 72.p odd 6 1
1200.3.c.e 4 180.v odd 12 2
1200.3.c.e 4 180.x even 12 2
1200.3.l.r 2 180.n even 6 1
1200.3.l.r 2 180.p odd 6 1
1620.3.o.b 4 1.a even 1 1 trivial
1620.3.o.b 4 3.b odd 2 1 inner
1620.3.o.b 4 9.c even 3 1 inner
1620.3.o.b 4 9.d odd 6 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 - 5 T^{2} + T^{4} \)
$7$ \( ( 4 + 2 T + T^{2} )^{2} \)
$11$ \( 32400 - 180 T^{2} + T^{4} \)
$13$ \( ( 64 + 8 T + T^{2} )^{2} \)
$17$ \( ( 180 + T^{2} )^{2} \)
$19$ \( ( 34 + T )^{4} \)
$23$ \( 2624400 - 1620 T^{2} + T^{4} \)
$29$ \( 2624400 - 1620 T^{2} + T^{4} \)
$31$ \( ( 196 + 14 T + T^{2} )^{2} \)
$37$ \( ( -56 + T )^{4} \)
$41$ \( 518400 - 720 T^{2} + T^{4} \)
$43$ \( ( 64 + 8 T + T^{2} )^{2} \)
$47$ \( 2624400 - 1620 T^{2} + T^{4} \)
$53$ \( ( 1620 + T^{2} )^{2} \)
$59$ \( 32400 - 180 T^{2} + T^{4} \)
$61$ \( ( 2116 - 46 T + T^{2} )^{2} \)
$67$ \( ( 1024 + 32 T + T^{2} )^{2} \)
$71$ \( ( 2880 + T^{2} )^{2} \)
$73$ \( ( 106 + T )^{4} \)
$79$ \( ( 484 - 22 T + T^{2} )^{2} \)
$83$ \( 212576400 - 14580 T^{2} + T^{4} \)
$89$ \( ( 11520 + T^{2} )^{2} \)
$97$ \( ( 14884 + 122 T + T^{2} )^{2} \)
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