Properties

Label 2160.3.c.i
Level $2160$
Weight $3$
Character orbit 2160.c
Analytic conductor $58.856$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2160.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(58.8557371018\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \zeta_{8}^{3} q^{5} + 5 \zeta_{8}^{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \zeta_{8}^{3} q^{5} + 5 \zeta_{8}^{2} q^{7} + (\zeta_{8}^{3} + \zeta_{8}) q^{11} - 9 \zeta_{8}^{2} q^{13} + ( - 8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{17} - 21 q^{19} + (\zeta_{8}^{3} - \zeta_{8}) q^{23} - 25 \zeta_{8}^{2} q^{25} + ( - 27 \zeta_{8}^{3} - 27 \zeta_{8}) q^{29} - 40 q^{31} - 25 \zeta_{8} q^{35} + 25 \zeta_{8}^{2} q^{37} + ( - 37 \zeta_{8}^{3} - 37 \zeta_{8}) q^{41} + 64 \zeta_{8}^{2} q^{43} + (16 \zeta_{8}^{3} - 16 \zeta_{8}) q^{47} + 24 q^{49} + ( - 51 \zeta_{8}^{3} + 51 \zeta_{8}) q^{53} + ( - 5 \zeta_{8}^{2} - 5) q^{55} + (64 \zeta_{8}^{3} + 64 \zeta_{8}) q^{59} - 97 q^{61} + 45 \zeta_{8} q^{65} - 131 \zeta_{8}^{2} q^{67} + ( - 63 \zeta_{8}^{3} - 63 \zeta_{8}) q^{71} + 17 \zeta_{8}^{2} q^{73} + (5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{77} + 117 q^{79} + ( - 41 \zeta_{8}^{3} + 41 \zeta_{8}) q^{83} + (40 \zeta_{8}^{2} - 40) q^{85} + ( - 104 \zeta_{8}^{3} - 104 \zeta_{8}) q^{89} + 45 q^{91} - 105 \zeta_{8}^{3} q^{95} + 41 \zeta_{8}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 84 q^{19} - 160 q^{31} + 96 q^{49} - 20 q^{55} - 388 q^{61} + 468 q^{79} - 160 q^{85} + 180 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0 0 0 −3.53553 3.53553i 0 5.00000i 0 0 0
1889.2 0 0 0 −3.53553 + 3.53553i 0 5.00000i 0 0 0
1889.3 0 0 0 3.53553 3.53553i 0 5.00000i 0 0 0
1889.4 0 0 0 3.53553 + 3.53553i 0 5.00000i 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
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.3.c.i 4
3.b odd 2 1 inner 2160.3.c.i 4
4.b odd 2 1 270.3.b.c 4
5.b even 2 1 inner 2160.3.c.i 4
12.b even 2 1 270.3.b.c 4
15.d odd 2 1 inner 2160.3.c.i 4
20.d odd 2 1 270.3.b.c 4
20.e even 4 1 1350.3.d.f 2
20.e even 4 1 1350.3.d.g 2
36.f odd 6 2 810.3.j.e 8
36.h even 6 2 810.3.j.e 8
60.h even 2 1 270.3.b.c 4
60.l odd 4 1 1350.3.d.f 2
60.l odd 4 1 1350.3.d.g 2
180.n even 6 2 810.3.j.e 8
180.p odd 6 2 810.3.j.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.c 4 4.b odd 2 1
270.3.b.c 4 12.b even 2 1
270.3.b.c 4 20.d odd 2 1
270.3.b.c 4 60.h even 2 1
810.3.j.e 8 36.f odd 6 2
810.3.j.e 8 36.h even 6 2
810.3.j.e 8 180.n even 6 2
810.3.j.e 8 180.p odd 6 2
1350.3.d.f 2 20.e even 4 1
1350.3.d.f 2 60.l odd 4 1
1350.3.d.g 2 20.e even 4 1
1350.3.d.g 2 60.l odd 4 1
2160.3.c.i 4 1.a even 1 1 trivial
2160.3.c.i 4 3.b odd 2 1 inner
2160.3.c.i 4 5.b even 2 1 inner
2160.3.c.i 4 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{17}^{2} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$19$ \( (T + 21)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1458)^{2} \) Copy content Toggle raw display
$31$ \( (T + 40)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 625)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2738)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 5202)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8192)^{2} \) Copy content Toggle raw display
$61$ \( (T + 97)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 17161)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 7938)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 289)^{2} \) Copy content Toggle raw display
$79$ \( (T - 117)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 3362)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 21632)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1681)^{2} \) Copy content Toggle raw display
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