Properties

Label 2160.4.a.ba
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{69}) \)
Defining polynomial: \( x^{2} - x - 17 \) 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 \(\beta = 3\sqrt{69}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{5} + ( - \beta - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + ( - \beta - 5) q^{7} + (\beta - 9) q^{11} + (\beta + 17) q^{13} + (\beta + 12) q^{17} + (\beta - 14) q^{19} + ( - 4 \beta - 15) q^{23} + 25 q^{25} + ( - 5 \beta - 21) q^{29} + (7 \beta - 128) q^{31} + ( - 5 \beta - 25) q^{35} + 74 q^{37} + ( - 9 \beta - 111) q^{41} + (5 \beta - 209) q^{43} + (10 \beta + 6) q^{47} + (10 \beta + 303) q^{49} + (17 \beta - 24) q^{53} + (5 \beta - 45) q^{55} + ( - 7 \beta + 177) q^{59} + (6 \beta + 419) q^{61} + (5 \beta + 85) q^{65} + ( - 6 \beta - 620) q^{67} + (27 \beta - 231) q^{71} + (21 \beta + 443) q^{73} + (4 \beta - 576) q^{77} + (3 \beta - 578) q^{79} + ( - 28 \beta - 573) q^{83} + (5 \beta + 60) q^{85} + ( - 3 \beta + 75) q^{89} + ( - 22 \beta - 706) q^{91} + (5 \beta - 70) q^{95} + ( - 16 \beta + 392) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} - 10 q^{7} - 18 q^{11} + 34 q^{13} + 24 q^{17} - 28 q^{19} - 30 q^{23} + 50 q^{25} - 42 q^{29} - 256 q^{31} - 50 q^{35} + 148 q^{37} - 222 q^{41} - 418 q^{43} + 12 q^{47} + 606 q^{49} - 48 q^{53} - 90 q^{55} + 354 q^{59} + 838 q^{61} + 170 q^{65} - 1240 q^{67} - 462 q^{71} + 886 q^{73} - 1152 q^{77} - 1156 q^{79} - 1146 q^{83} + 120 q^{85} + 150 q^{89} - 1412 q^{91} - 140 q^{95} + 784 q^{97}+O(q^{100}) \) 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
4.65331
−3.65331
0 0 0 5.00000 0 −29.9199 0 0 0
1.2 0 0 0 5.00000 0 19.9199 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.ba 2
3.b odd 2 1 2160.4.a.v 2
4.b odd 2 1 540.4.a.i yes 2
12.b even 2 1 540.4.a.f 2
36.f odd 6 2 1620.4.i.n 4
36.h even 6 2 1620.4.i.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.4.a.f 2 12.b even 2 1
540.4.a.i yes 2 4.b odd 2 1
1620.4.i.n 4 36.f odd 6 2
1620.4.i.q 4 36.h even 6 2
2160.4.a.v 2 3.b odd 2 1
2160.4.a.ba 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7}^{2} + 10T_{7} - 596 \) Copy content Toggle raw display
\( T_{11}^{2} + 18T_{11} - 540 \) 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
$7$ \( T^{2} + 10T - 596 \) Copy content Toggle raw display
$11$ \( T^{2} + 18T - 540 \) Copy content Toggle raw display
$13$ \( T^{2} - 34T - 332 \) Copy content Toggle raw display
$17$ \( T^{2} - 24T - 477 \) Copy content Toggle raw display
$19$ \( T^{2} + 28T - 425 \) Copy content Toggle raw display
$23$ \( T^{2} + 30T - 9711 \) Copy content Toggle raw display
$29$ \( T^{2} + 42T - 15084 \) Copy content Toggle raw display
$31$ \( T^{2} + 256T - 14045 \) Copy content Toggle raw display
$37$ \( (T - 74)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 222T - 37980 \) Copy content Toggle raw display
$43$ \( T^{2} + 418T + 28156 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T - 62064 \) Copy content Toggle raw display
$53$ \( T^{2} + 48T - 178893 \) Copy content Toggle raw display
$59$ \( T^{2} - 354T + 900 \) Copy content Toggle raw display
$61$ \( T^{2} - 838T + 153205 \) Copy content Toggle raw display
$67$ \( T^{2} + 1240 T + 362044 \) Copy content Toggle raw display
$71$ \( T^{2} + 462T - 399348 \) Copy content Toggle raw display
$73$ \( T^{2} - 886T - 77612 \) Copy content Toggle raw display
$79$ \( T^{2} + 1156 T + 328495 \) Copy content Toggle raw display
$83$ \( T^{2} + 1146 T - 158535 \) Copy content Toggle raw display
$89$ \( T^{2} - 150T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 784T - 5312 \) Copy content Toggle raw display
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