Properties

Label 2160.4.a.bd
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: \(\Q(\sqrt{241}) \)
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{241})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 q^{5} + ( - \beta + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + ( - \beta + 3) q^{7} + (\beta + 2) q^{11} + ( - 6 \beta - 9) q^{13} + (7 \beta - 24) q^{17} + ( - 11 \beta - 29) q^{19} + (17 \beta + 42) q^{23} + 25 q^{25} + (13 \beta - 58) q^{29} + (19 \beta + 44) q^{31} + ( - 5 \beta + 15) q^{35} + (5 \beta - 131) q^{37} + (10 \beta + 54) q^{41} + (25 \beta + 48) q^{43} + ( - 7 \beta - 4) q^{47} + ( - 5 \beta - 274) q^{49} + ( - 50 \beta - 258) q^{53} + (5 \beta + 10) q^{55} + ( - 24 \beta - 252) q^{59} + ( - 5 \beta - 85) q^{61} + ( - 30 \beta - 45) q^{65} + ( - 73 \beta + 137) q^{67} + (46 \beta - 10) q^{71} + (79 \beta - 501) q^{73} - 54 q^{77} + ( - 30 \beta + 203) q^{79} + ( - 36 \beta - 558) q^{83} + (35 \beta - 120) q^{85} + (132 \beta + 8) q^{89} + ( - 3 \beta + 333) q^{91} + ( - 55 \beta - 145) q^{95} + ( - 69 \beta - 637) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{5} + 5 q^{7} + 5 q^{11} - 24 q^{13} - 41 q^{17} - 69 q^{19} + 101 q^{23} + 50 q^{25} - 103 q^{29} + 107 q^{31} + 25 q^{35} - 257 q^{37} + 118 q^{41} + 121 q^{43} - 15 q^{47} - 553 q^{49} - 566 q^{53} + 25 q^{55} - 528 q^{59} - 175 q^{61} - 120 q^{65} + 201 q^{67} + 26 q^{71} - 923 q^{73} - 108 q^{77} + 376 q^{79} - 1152 q^{83} - 205 q^{85} + 148 q^{89} + 663 q^{91} - 345 q^{95} - 1343 q^{97}+O(q^{100}) \) 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
8.26209
−7.26209
0 0 0 5.00000 0 −5.26209 0 0 0
1.2 0 0 0 5.00000 0 10.2621 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.bd 2
3.b odd 2 1 2160.4.a.y 2
4.b odd 2 1 1080.4.a.b yes 2
12.b even 2 1 1080.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.a 2 12.b even 2 1
1080.4.a.b yes 2 4.b odd 2 1
2160.4.a.y 2 3.b odd 2 1
2160.4.a.bd 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{2} - 5T_{7} - 54 \) Copy content Toggle raw display
\( T_{11}^{2} - 5T_{11} - 54 \) 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
$7$ \( T^{2} - 5T - 54 \) Copy content Toggle raw display
$11$ \( T^{2} - 5T - 54 \) Copy content Toggle raw display
$13$ \( T^{2} + 24T - 2025 \) Copy content Toggle raw display
$17$ \( T^{2} + 41T - 2532 \) Copy content Toggle raw display
$19$ \( T^{2} + 69T - 6100 \) Copy content Toggle raw display
$23$ \( T^{2} - 101T - 14862 \) Copy content Toggle raw display
$29$ \( T^{2} + 103T - 7530 \) Copy content Toggle raw display
$31$ \( T^{2} - 107T - 18888 \) Copy content Toggle raw display
$37$ \( T^{2} + 257T + 15006 \) Copy content Toggle raw display
$41$ \( T^{2} - 118T - 2544 \) Copy content Toggle raw display
$43$ \( T^{2} - 121T - 33996 \) Copy content Toggle raw display
$47$ \( T^{2} + 15T - 2896 \) Copy content Toggle raw display
$53$ \( T^{2} + 566T - 70536 \) Copy content Toggle raw display
$59$ \( T^{2} + 528T + 34992 \) Copy content Toggle raw display
$61$ \( T^{2} + 175T + 6150 \) Copy content Toggle raw display
$67$ \( T^{2} - 201T - 310972 \) Copy content Toggle raw display
$71$ \( T^{2} - 26T - 127320 \) Copy content Toggle raw display
$73$ \( T^{2} + 923T - 163038 \) Copy content Toggle raw display
$79$ \( T^{2} - 376T - 18881 \) Copy content Toggle raw display
$83$ \( T^{2} + 1152 T + 253692 \) Copy content Toggle raw display
$89$ \( T^{2} - 148 T - 1044320 \) Copy content Toggle raw display
$97$ \( T^{2} + 1343 T + 164062 \) Copy content Toggle raw display
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