Properties

Label 2160.4.a.x
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 540)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 5 q^{5} + (\beta + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + (\beta + 1) q^{7} + (\beta - 21) q^{11} + (5 \beta - 7) q^{13} + (7 \beta + 12) q^{17} + ( - \beta + 70) q^{19} + (8 \beta - 63) q^{23} + 25 q^{25} + ( - 11 \beta - 63) q^{29} + (17 \beta + 28) q^{31} + ( - 5 \beta - 5) q^{35} - 286 q^{37} + (21 \beta - 33) q^{41} + (7 \beta + 265) q^{43} + ( - 14 \beta + 198) q^{47} + (2 \beta - 153) q^{49} + ( - 25 \beta - 252) q^{53} + ( - 5 \beta + 105) q^{55} + ( - 7 \beta - 219) q^{59} + (42 \beta - 301) q^{61} + ( - 25 \beta + 35) q^{65} + ( - 42 \beta + 460) q^{67} + (15 \beta - 399) q^{71} + ( - 27 \beta - 385) q^{73} + ( - 20 \beta + 168) q^{77} + (21 \beta + 862) q^{79} + ( - 28 \beta + 243) q^{83} + ( - 35 \beta - 60) q^{85} + (63 \beta - 147) q^{89} + ( - 2 \beta + 938) q^{91} + (5 \beta - 350) q^{95} + ( - 80 \beta + 56) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{5} + 2 q^{7} - 42 q^{11} - 14 q^{13} + 24 q^{17} + 140 q^{19} - 126 q^{23} + 50 q^{25} - 126 q^{29} + 56 q^{31} - 10 q^{35} - 572 q^{37} - 66 q^{41} + 530 q^{43} + 396 q^{47} - 306 q^{49} - 504 q^{53} + 210 q^{55} - 438 q^{59} - 602 q^{61} + 70 q^{65} + 920 q^{67} - 798 q^{71} - 770 q^{73} + 336 q^{77} + 1724 q^{79} + 486 q^{83} - 120 q^{85} - 294 q^{89} + 1876 q^{91} - 700 q^{95} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.79129
2.79129
0 0 0 −5.00000 0 −12.7477 0 0 0
1.2 0 0 0 −5.00000 0 14.7477 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.4.a.x 2
3.b odd 2 1 2160.4.a.bc 2
4.b odd 2 1 540.4.a.e 2
12.b even 2 1 540.4.a.h yes 2
36.f odd 6 2 1620.4.i.r 4
36.h even 6 2 1620.4.i.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.e 2 4.b odd 2 1
540.4.a.h yes 2 12.b even 2 1
1620.4.i.o 4 36.h even 6 2
1620.4.i.r 4 36.f odd 6 2
2160.4.a.x 2 1.a even 1 1 trivial
2160.4.a.bc 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{2} - 2T_{7} - 188 \) Copy content Toggle raw display
\( T_{11}^{2} + 42T_{11} + 252 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 188 \) Copy content Toggle raw display
$11$ \( T^{2} + 42T + 252 \) Copy content Toggle raw display
$13$ \( T^{2} + 14T - 4676 \) Copy content Toggle raw display
$17$ \( T^{2} - 24T - 9117 \) Copy content Toggle raw display
$19$ \( T^{2} - 140T + 4711 \) Copy content Toggle raw display
$23$ \( T^{2} + 126T - 8127 \) Copy content Toggle raw display
$29$ \( T^{2} + 126T - 18900 \) Copy content Toggle raw display
$31$ \( T^{2} - 56T - 53837 \) Copy content Toggle raw display
$37$ \( (T + 286)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 66T - 82260 \) Copy content Toggle raw display
$43$ \( T^{2} - 530T + 60964 \) Copy content Toggle raw display
$47$ \( T^{2} - 396T + 2160 \) Copy content Toggle raw display
$53$ \( T^{2} + 504T - 54621 \) Copy content Toggle raw display
$59$ \( T^{2} + 438T + 38700 \) Copy content Toggle raw display
$61$ \( T^{2} + 602T - 242795 \) Copy content Toggle raw display
$67$ \( T^{2} - 920T - 121796 \) Copy content Toggle raw display
$71$ \( T^{2} + 798T + 116676 \) Copy content Toggle raw display
$73$ \( T^{2} + 770T + 10444 \) Copy content Toggle raw display
$79$ \( T^{2} - 1724 T + 659695 \) Copy content Toggle raw display
$83$ \( T^{2} - 486T - 89127 \) Copy content Toggle raw display
$89$ \( T^{2} + 294T - 728532 \) Copy content Toggle raw display
$97$ \( T^{2} - 112 T - 1206464 \) Copy content Toggle raw display
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