Properties

Label 72.2.l.a
Level $72$
Weight $2$
Character orbit 72.l
Analytic conductor $0.575$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 - \beta_{1} - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{6} + 2 \beta_{3} q^{8} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{9} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{11} + ( 2 - 2 \beta_{3} ) q^{12} + ( -4 + 4 \beta_{2} ) q^{16} + ( -3 + 6 \beta_{2} - 2 \beta_{3} ) q^{17} + ( -4 + \beta_{3} ) q^{18} + ( -1 + 6 \beta_{1} - 3 \beta_{3} ) q^{19} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{22} + ( 4 + 2 \beta_{1} - 4 \beta_{2} ) q^{24} + ( 5 - 5 \beta_{2} ) q^{25} + ( 5 + \beta_{3} ) q^{27} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + ( -6 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 4 - 3 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{34} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{36} + ( 6 - \beta_{1} + 6 \beta_{2} ) q^{38} + ( 6 + 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{41} + ( -5 - 3 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} ) q^{43} + ( 6 - 12 \beta_{2} - 2 \beta_{3} ) q^{44} + ( 4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{48} -7 \beta_{2} q^{49} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{50} + ( -1 + \beta_{1} + 7 \beta_{2} - 6 \beta_{3} ) q^{51} + ( -2 + 5 \beta_{1} + 2 \beta_{2} ) q^{54} + ( -7 + 4 \beta_{1} - 5 \beta_{2} - 6 \beta_{3} ) q^{57} + ( 6 - 5 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} ) q^{59} -8 q^{64} + ( -8 - 6 \beta_{1} + 12 \beta_{2} + 5 \beta_{3} ) q^{66} + ( -3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{67} + ( -12 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -2 \beta_{1} - 8 \beta_{2} + 2 \beta_{3} ) q^{72} + ( -1 - 12 \beta_{1} + 6 \beta_{3} ) q^{73} + ( -5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{75} + ( 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{76} + ( 7 - 4 \beta_{1} - 7 \beta_{2} ) q^{81} + ( 8 + 6 \beta_{1} - 3 \beta_{3} ) q^{82} + 2 \beta_{1} q^{83} + ( -12 - 5 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} ) q^{86} + ( 4 + 6 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} ) q^{88} + 4 \beta_{3} q^{89} + ( 8 + 4 \beta_{3} ) q^{96} + ( -5 + 6 \beta_{1} + 5 \beta_{2} - 12 \beta_{3} ) q^{97} -7 \beta_{3} q^{98} + ( 7 + 12 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 4q^{4} - 4q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 4q^{4} - 4q^{6} + 2q^{9} - 18q^{11} + 8q^{12} - 8q^{16} - 16q^{18} - 4q^{19} - 4q^{22} + 8q^{24} + 10q^{25} + 20q^{27} - 14q^{33} + 8q^{34} - 4q^{36} + 36q^{38} + 18q^{41} - 10q^{43} + 8q^{48} - 14q^{49} + 10q^{51} - 4q^{54} - 38q^{57} + 18q^{59} - 32q^{64} - 8q^{66} + 14q^{67} - 36q^{68} - 16q^{72} - 4q^{73} - 10q^{75} - 4q^{76} + 14q^{81} + 32q^{82} - 36q^{86} + 8q^{88} + 32q^{96} - 10q^{97} + 16q^{99} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 1.72474 0.158919i 1.00000 + 1.73205i 0 −2.22474 1.02494i 0 2.82843i 2.94949 0.548188i 0
11.2 1.22474 + 0.707107i −0.724745 1.57313i 1.00000 + 1.73205i 0 0.224745 2.43916i 0 2.82843i −1.94949 + 2.28024i 0
59.1 −1.22474 + 0.707107i 1.72474 + 0.158919i 1.00000 1.73205i 0 −2.22474 + 1.02494i 0 2.82843i 2.94949 + 0.548188i 0
59.2 1.22474 0.707107i −0.724745 + 1.57313i 1.00000 1.73205i 0 0.224745 + 2.43916i 0 2.82843i −1.94949 2.28024i 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
9.d odd 6 1 inner
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.l.a 4
3.b odd 2 1 216.2.l.a 4
4.b odd 2 1 288.2.p.a 4
8.b even 2 1 288.2.p.a 4
8.d odd 2 1 CM 72.2.l.a 4
9.c even 3 1 216.2.l.a 4
9.c even 3 1 648.2.f.a 4
9.d odd 6 1 inner 72.2.l.a 4
9.d odd 6 1 648.2.f.a 4
12.b even 2 1 864.2.p.a 4
24.f even 2 1 216.2.l.a 4
24.h odd 2 1 864.2.p.a 4
36.f odd 6 1 864.2.p.a 4
36.f odd 6 1 2592.2.f.a 4
36.h even 6 1 288.2.p.a 4
36.h even 6 1 2592.2.f.a 4
72.j odd 6 1 288.2.p.a 4
72.j odd 6 1 2592.2.f.a 4
72.l even 6 1 inner 72.2.l.a 4
72.l even 6 1 648.2.f.a 4
72.n even 6 1 864.2.p.a 4
72.n even 6 1 2592.2.f.a 4
72.p odd 6 1 216.2.l.a 4
72.p odd 6 1 648.2.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.l.a 4 1.a even 1 1 trivial
72.2.l.a 4 8.d odd 2 1 CM
72.2.l.a 4 9.d odd 6 1 inner
72.2.l.a 4 72.l even 6 1 inner
216.2.l.a 4 3.b odd 2 1
216.2.l.a 4 9.c even 3 1
216.2.l.a 4 24.f even 2 1
216.2.l.a 4 72.p odd 6 1
288.2.p.a 4 4.b odd 2 1
288.2.p.a 4 8.b even 2 1
288.2.p.a 4 36.h even 6 1
288.2.p.a 4 72.j odd 6 1
648.2.f.a 4 9.c even 3 1
648.2.f.a 4 9.d odd 6 1
648.2.f.a 4 72.l even 6 1
648.2.f.a 4 72.p odd 6 1
864.2.p.a 4 12.b even 2 1
864.2.p.a 4 24.h odd 2 1
864.2.p.a 4 36.f odd 6 1
864.2.p.a 4 72.n even 6 1
2592.2.f.a 4 36.f odd 6 1
2592.2.f.a 4 36.h even 6 1
2592.2.f.a 4 72.j odd 6 1
2592.2.f.a 4 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( 9 - 6 T + T^{2} - 2 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 625 + 450 T + 133 T^{2} + 18 T^{3} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( 361 + 70 T^{2} + T^{4} \)
$19$ \( ( -53 + 2 T + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( 25 + 90 T + 103 T^{2} - 18 T^{3} + T^{4} \)
$43$ \( 841 - 290 T + 129 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( 529 + 414 T + 85 T^{2} - 18 T^{3} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( 25 + 70 T + 201 T^{2} - 14 T^{3} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -215 + 2 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 64 - 8 T^{2} + T^{4} \)
$89$ \( ( 32 + T^{2} )^{2} \)
$97$ \( 36481 - 1910 T + 291 T^{2} + 10 T^{3} + T^{4} \)
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