Properties

Label 72.2.f.a
Level 72
Weight 2
Character orbit 72.f
Analytic conductor 0.575
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 72.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{5} -2 \beta_{3} q^{7} + 2 \beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{3} ) q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{5} -2 \beta_{3} q^{7} + 2 \beta_{2} q^{8} + ( -3 - \beta_{3} ) q^{10} -2 \beta_{2} q^{11} + 2 \beta_{3} q^{13} + ( 2 \beta_{1} - 4 \beta_{2} ) q^{14} + ( -2 + 2 \beta_{3} ) q^{16} + \beta_{2} q^{17} -4 q^{19} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{20} + ( 2 - 2 \beta_{3} ) q^{22} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{23} + q^{25} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{26} + ( 6 - 2 \beta_{3} ) q^{28} + ( 2 \beta_{1} - \beta_{2} ) q^{29} + 2 \beta_{3} q^{31} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{32} + ( -1 + \beta_{3} ) q^{34} + 6 \beta_{2} q^{35} -4 \beta_{1} q^{38} -4 \beta_{3} q^{40} + \beta_{2} q^{41} + 8 q^{43} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{44} + ( 6 + 2 \beta_{3} ) q^{46} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{47} -5 q^{49} + \beta_{1} q^{50} + ( -6 + 2 \beta_{3} ) q^{52} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{53} + 4 \beta_{3} q^{55} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{56} + ( 3 + \beta_{3} ) q^{58} -8 \beta_{2} q^{59} -8 \beta_{3} q^{61} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{62} -8 q^{64} -6 \beta_{2} q^{65} -4 q^{67} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -6 + 6 \beta_{3} ) q^{70} + ( -12 \beta_{1} + 6 \beta_{2} ) q^{71} -4 q^{73} + ( -4 - 4 \beta_{3} ) q^{76} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{77} -2 \beta_{3} q^{79} + ( 4 \beta_{1} - 8 \beta_{2} ) q^{80} + ( -1 + \beta_{3} ) q^{82} + 10 \beta_{2} q^{83} -2 \beta_{3} q^{85} + 8 \beta_{1} q^{86} + 8 q^{88} -5 \beta_{2} q^{89} + 12 q^{91} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -6 - 2 \beta_{3} ) q^{94} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{95} + 8 q^{97} -5 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + O(q^{10}) \) \( 4q + 4q^{4} - 12q^{10} - 8q^{16} - 16q^{19} + 8q^{22} + 4q^{25} + 24q^{28} - 4q^{34} + 32q^{43} + 24q^{46} - 20q^{49} - 24q^{52} + 12q^{58} - 32q^{64} - 16q^{67} - 24q^{70} - 16q^{73} - 16q^{76} - 4q^{82} + 32q^{88} + 48q^{91} - 24q^{94} + 32q^{97} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
−1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 2.44949 0 3.46410i 2.82843i 0 −3.00000 1.73205i
35.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 2.44949 0 3.46410i 2.82843i 0 −3.00000 + 1.73205i
35.3 1.22474 0.707107i 0 1.00000 1.73205i −2.44949 0 3.46410i 2.82843i 0 −3.00000 + 1.73205i
35.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i −2.44949 0 3.46410i 2.82843i 0 −3.00000 1.73205i
\(n\): 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
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.f.a 4
3.b odd 2 1 inner 72.2.f.a 4
4.b odd 2 1 288.2.f.a 4
5.b even 2 1 1800.2.b.c 4
5.c odd 4 2 1800.2.m.c 8
8.b even 2 1 288.2.f.a 4
8.d odd 2 1 inner 72.2.f.a 4
9.c even 3 1 648.2.l.a 4
9.c even 3 1 648.2.l.c 4
9.d odd 6 1 648.2.l.a 4
9.d odd 6 1 648.2.l.c 4
12.b even 2 1 288.2.f.a 4
15.d odd 2 1 1800.2.b.c 4
15.e even 4 2 1800.2.m.c 8
16.e even 4 2 2304.2.c.i 8
16.f odd 4 2 2304.2.c.i 8
20.d odd 2 1 7200.2.b.c 4
20.e even 4 2 7200.2.m.c 8
24.f even 2 1 inner 72.2.f.a 4
24.h odd 2 1 288.2.f.a 4
36.f odd 6 1 2592.2.p.a 4
36.f odd 6 1 2592.2.p.c 4
36.h even 6 1 2592.2.p.a 4
36.h even 6 1 2592.2.p.c 4
40.e odd 2 1 1800.2.b.c 4
40.f even 2 1 7200.2.b.c 4
40.i odd 4 2 7200.2.m.c 8
40.k even 4 2 1800.2.m.c 8
48.i odd 4 2 2304.2.c.i 8
48.k even 4 2 2304.2.c.i 8
60.h even 2 1 7200.2.b.c 4
60.l odd 4 2 7200.2.m.c 8
72.j odd 6 1 2592.2.p.a 4
72.j odd 6 1 2592.2.p.c 4
72.l even 6 1 648.2.l.a 4
72.l even 6 1 648.2.l.c 4
72.n even 6 1 2592.2.p.a 4
72.n even 6 1 2592.2.p.c 4
72.p odd 6 1 648.2.l.a 4
72.p odd 6 1 648.2.l.c 4
120.i odd 2 1 7200.2.b.c 4
120.m even 2 1 1800.2.b.c 4
120.q odd 4 2 1800.2.m.c 8
120.w even 4 2 7200.2.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.f.a 4 1.a even 1 1 trivial
72.2.f.a 4 3.b odd 2 1 inner
72.2.f.a 4 8.d odd 2 1 inner
72.2.f.a 4 24.f even 2 1 inner
288.2.f.a 4 4.b odd 2 1
288.2.f.a 4 8.b even 2 1
288.2.f.a 4 12.b even 2 1
288.2.f.a 4 24.h odd 2 1
648.2.l.a 4 9.c even 3 1
648.2.l.a 4 9.d odd 6 1
648.2.l.a 4 72.l even 6 1
648.2.l.a 4 72.p odd 6 1
648.2.l.c 4 9.c even 3 1
648.2.l.c 4 9.d odd 6 1
648.2.l.c 4 72.l even 6 1
648.2.l.c 4 72.p odd 6 1
1800.2.b.c 4 5.b even 2 1
1800.2.b.c 4 15.d odd 2 1
1800.2.b.c 4 40.e odd 2 1
1800.2.b.c 4 120.m even 2 1
1800.2.m.c 8 5.c odd 4 2
1800.2.m.c 8 15.e even 4 2
1800.2.m.c 8 40.k even 4 2
1800.2.m.c 8 120.q odd 4 2
2304.2.c.i 8 16.e even 4 2
2304.2.c.i 8 16.f odd 4 2
2304.2.c.i 8 48.i odd 4 2
2304.2.c.i 8 48.k even 4 2
2592.2.p.a 4 36.f odd 6 1
2592.2.p.a 4 36.h even 6 1
2592.2.p.a 4 72.j odd 6 1
2592.2.p.a 4 72.n even 6 1
2592.2.p.c 4 36.f odd 6 1
2592.2.p.c 4 36.h even 6 1
2592.2.p.c 4 72.j odd 6 1
2592.2.p.c 4 72.n even 6 1
7200.2.b.c 4 20.d odd 2 1
7200.2.b.c 4 40.f even 2 1
7200.2.b.c 4 60.h even 2 1
7200.2.b.c 4 120.i odd 2 1
7200.2.m.c 8 20.e even 4 2
7200.2.m.c 8 40.i odd 4 2
7200.2.m.c 8 60.l odd 4 2
7200.2.m.c 8 120.w even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(72, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 4 T + 7 T^{2} )^{2}( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 32 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 22 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 52 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 50 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 52 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 70 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 4 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 74 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 4 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 - 146 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 34 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 128 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 8 T + 97 T^{2} )^{4} \)
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