Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{5} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} + ( 3 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{5} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} + ( 3 \beta_{2} + \beta_{3} ) q^{9} -\beta_{2} q^{11} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -6 + \beta_{1} + 3 \beta_{2} ) q^{15} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( 4 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + ( 3 - 2 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{21} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{23} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{25} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{29} + ( -4 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -\beta_{1} + \beta_{3} ) q^{33} + ( 7 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{37} + ( 6 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{39} + ( 7 - 2 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{41} + ( -2 + 4 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{45} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{47} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{49} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{51} + ( -6 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{55} + ( -3 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{57} + ( -7 + 7 \beta_{2} ) q^{59} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{61} + ( 3 - 4 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{63} + ( 3 - 6 \beta_{1} + 8 \beta_{2} + 3 \beta_{3} ) q^{65} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{67} + ( 6 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{69} -4 q^{71} + ( -2 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{73} + ( -3 - 3 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{75} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{77} + ( -1 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{79} + ( -6 + 5 \beta_{1} + 6 \beta_{2} ) q^{81} + ( -1 + 2 \beta_{1} - 12 \beta_{2} - \beta_{3} ) q^{83} + ( 8 - 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{85} + ( -3 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{87} + 6 q^{89} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{91} + ( -6 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{93} + ( 8 + 4 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} ) q^{95} + ( 2 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{3} + q^{5} - 3q^{7} + 5q^{9} + O(q^{10}) \) \( 4q + q^{3} + q^{5} - 3q^{7} + 5q^{9} - 2q^{11} - 5q^{13} - 17q^{15} - 10q^{17} + 14q^{19} - 3q^{21} - 5q^{23} - 7q^{25} + 16q^{27} + 3q^{29} - 7q^{31} - 2q^{33} + 30q^{35} + 12q^{37} + 19q^{39} + 12q^{41} - 8q^{43} + 5q^{45} - 3q^{47} - 7q^{49} - 19q^{51} - 20q^{53} - 2q^{55} - 13q^{57} - 14q^{59} + q^{61} - 9q^{63} + 19q^{65} - 4q^{67} + 19q^{69} - 16q^{71} - 14q^{73} - 7q^{75} - 3q^{77} + 7q^{79} - 7q^{81} - 25q^{83} + 14q^{85} + 3q^{87} + 24q^{89} - 18q^{91} - 13q^{93} + 20q^{95} + 8q^{97} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 −1.18614 + 1.26217i 0 1.68614 + 2.92048i 0 0.686141 1.18843i 0 −0.186141 2.99422i 0
25.2 0 1.68614 0.396143i 0 −1.18614 2.05446i 0 −2.18614 + 3.78651i 0 2.68614 1.33591i 0
49.1 0 −1.18614 1.26217i 0 1.68614 2.92048i 0 0.686141 + 1.18843i 0 −0.186141 + 2.99422i 0
49.2 0 1.68614 + 0.396143i 0 −1.18614 + 2.05446i 0 −2.18614 3.78651i 0 2.68614 + 1.33591i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.i.b 4
3.b odd 2 1 216.2.i.b 4
4.b odd 2 1 144.2.i.d 4
8.b even 2 1 576.2.i.j 4
8.d odd 2 1 576.2.i.l 4
9.c even 3 1 inner 72.2.i.b 4
9.c even 3 1 648.2.a.f 2
9.d odd 6 1 216.2.i.b 4
9.d odd 6 1 648.2.a.g 2
12.b even 2 1 432.2.i.d 4
24.f even 2 1 1728.2.i.j 4
24.h odd 2 1 1728.2.i.i 4
36.f odd 6 1 144.2.i.d 4
36.f odd 6 1 1296.2.a.n 2
36.h even 6 1 432.2.i.d 4
36.h even 6 1 1296.2.a.p 2
72.j odd 6 1 1728.2.i.i 4
72.j odd 6 1 5184.2.a.bp 2
72.l even 6 1 1728.2.i.j 4
72.l even 6 1 5184.2.a.bo 2
72.n even 6 1 576.2.i.j 4
72.n even 6 1 5184.2.a.bt 2
72.p odd 6 1 576.2.i.l 4
72.p odd 6 1 5184.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.b 4 1.a even 1 1 trivial
72.2.i.b 4 9.c even 3 1 inner
144.2.i.d 4 4.b odd 2 1
144.2.i.d 4 36.f odd 6 1
216.2.i.b 4 3.b odd 2 1
216.2.i.b 4 9.d odd 6 1
432.2.i.d 4 12.b even 2 1
432.2.i.d 4 36.h even 6 1
576.2.i.j 4 8.b even 2 1
576.2.i.j 4 72.n even 6 1
576.2.i.l 4 8.d odd 2 1
576.2.i.l 4 72.p odd 6 1
648.2.a.f 2 9.c even 3 1
648.2.a.g 2 9.d odd 6 1
1296.2.a.n 2 36.f odd 6 1
1296.2.a.p 2 36.h even 6 1
1728.2.i.i 4 24.h odd 2 1
1728.2.i.i 4 72.j odd 6 1
1728.2.i.j 4 24.f even 2 1
1728.2.i.j 4 72.l even 6 1
5184.2.a.bo 2 72.l even 6 1
5184.2.a.bp 2 72.j odd 6 1
5184.2.a.bs 2 72.p odd 6 1
5184.2.a.bt 2 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - T_{5}^{3} + 9 T_{5}^{2} + 8 T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(72, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \)
$5$ \( 1 - T - T^{2} + 8 T^{3} - 26 T^{4} + 40 T^{5} - 25 T^{6} - 125 T^{7} + 625 T^{8} \)
$7$ \( 1 + 3 T + T^{2} - 18 T^{3} - 48 T^{4} - 126 T^{5} + 49 T^{6} + 1029 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 + 5 T + T^{2} - 10 T^{3} + 82 T^{4} - 130 T^{5} + 169 T^{6} + 10985 T^{7} + 28561 T^{8} \)
$17$ \( ( 1 + 5 T + 32 T^{2} + 85 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 7 T + 42 T^{2} - 133 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 5 T - 19 T^{2} - 10 T^{3} + 832 T^{4} - 230 T^{5} - 10051 T^{6} + 60835 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 522 T^{5} - 36163 T^{6} - 73167 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 7 T - 17 T^{2} + 28 T^{3} + 1876 T^{4} + 868 T^{5} - 16337 T^{6} + 208537 T^{7} + 923521 T^{8} \)
$37$ \( ( 1 - 6 T + 50 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 1476 T^{5} + 99179 T^{6} - 827052 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 5848 T^{5} - 9245 T^{6} + 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 3 T - 79 T^{2} - 18 T^{3} + 5112 T^{4} - 846 T^{5} - 174511 T^{6} + 311469 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 + 10 T + 98 T^{2} + 530 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 488 T^{5} - 420473 T^{6} - 226981 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 4 T - 89 T^{2} - 116 T^{3} + 5464 T^{4} - 7772 T^{5} - 399521 T^{6} + 1203052 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 4 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 7 T + 84 T^{2} + 511 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 - 7 T - 113 T^{2} - 28 T^{3} + 16132 T^{4} - 2212 T^{5} - 705233 T^{6} - 3451273 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 25 T + 311 T^{2} + 3700 T^{3} + 39832 T^{4} + 307100 T^{5} + 2142479 T^{6} + 14294675 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{4} \)
$97$ \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 13192 T^{5} - 1063217 T^{6} - 7301384 T^{7} + 88529281 T^{8} \)
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