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Properties

Label 72.4.f.a
Level $72$
Weight $4$
Character orbit 72.f
Analytic conductor $4.248$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,4,Mod(35,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.35");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 72.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.24813752041\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 11 x^{10} - 14 x^{9} + 47 x^{8} - 92 x^{7} + 412 x^{6} + 216 x^{5} - 3600 x^{4} + \cdots + 4752 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (\beta_{6} - 1) q^{4} - \beta_{5} q^{5} + (\beta_{6} + \beta_{2}) q^{7} + (\beta_{8} - \beta_{5} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (\beta_{6} - 1) q^{4} - \beta_{5} q^{5} + (\beta_{6} + \beta_{2}) q^{7} + (\beta_{8} - \beta_{5} - \beta_{3}) q^{8} + (\beta_{7} + \beta_{2} + 1) q^{10} + ( - \beta_{11} - \beta_{8} + \cdots - \beta_1) q^{11}+ \cdots + ( - 16 \beta_{11} + 16 \beta_{8} + \cdots + 32 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 12 q^{10} + 120 q^{16} + 48 q^{19} - 456 q^{22} + 300 q^{25} - 552 q^{28} + 804 q^{34} + 1632 q^{40} - 864 q^{43} - 1944 q^{46} + 132 q^{49} - 2136 q^{52} + 2676 q^{58} + 3936 q^{64} + 816 q^{67} - 4008 q^{70} - 432 q^{73} - 3408 q^{76} + 6084 q^{82} + 4320 q^{88} - 3600 q^{91} - 3624 q^{94} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 11 x^{10} - 14 x^{9} + 47 x^{8} - 92 x^{7} + 412 x^{6} + 216 x^{5} - 3600 x^{4} + \cdots + 4752 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4277057920 \nu^{11} - 59380220093 \nu^{10} + 114015248186 \nu^{9} - 593465861429 \nu^{8} + \cdots - 537322554029160 ) / 37934520016512 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 257038009 \nu^{11} - 3322938790 \nu^{10} + 10343321737 \nu^{9} - 36880573758 \nu^{8} + \cdots - 12703352038112 ) / 1806405715072 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6450074368 \nu^{11} + 10987186739 \nu^{10} - 64951877606 \nu^{9} + 117651492539 \nu^{8} + \cdots - 78720161644776 ) / 37934520016512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4080551143 \nu^{11} + 6820210750 \nu^{10} - 36090136719 \nu^{9} + 34756825278 \nu^{8} + \cdots + 22356767786592 ) / 6322420002752 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 302314799 \nu^{11} + 59308073 \nu^{10} + 3164181577 \nu^{9} + 2440099361 \nu^{8} + \cdots + 1589921453592 ) / 387086938944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33558357761 \nu^{11} - 22194050062 \nu^{10} + 315381078817 \nu^{9} - 107074576294 \nu^{8} + \cdots + 167479746043488 ) / 37934520016512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3905972177 \nu^{11} + 4714262576 \nu^{10} + 32111134909 \nu^{9} + 83497703744 \nu^{8} + \cdots + 50410663561968 ) / 2709608572608 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25232605974 \nu^{11} + 13361127351 \nu^{10} - 208778135020 \nu^{9} + \cdots - 38092119070568 ) / 12644840005504 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30731447147 \nu^{11} - 48562745558 \nu^{10} + 316071545139 \nu^{9} - 300336504846 \nu^{8} + \cdots - 135558548890496 ) / 12644840005504 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 87035551243 \nu^{11} + 119210097800 \nu^{10} - 836826710783 \nu^{9} + \cdots + 298139920856688 ) / 18967260008256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 78438633436 \nu^{11} + 159730339809 \nu^{10} - 892011382654 \nu^{9} + \cdots + 503755155345992 ) / 12644840005504 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -2\beta_{10} - \beta_{9} - \beta_{7} + 8\beta_{4} - \beta_{2} + 4 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{7} + 4\beta_{3} + \beta_{2} - 4\beta _1 - 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + 7 \beta_{10} + \beta_{9} - 9 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} - 6 \beta_{5} + \cdots - 16 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8 \beta_{11} + 2 \beta_{10} - 17 \beta_{9} + 8 \beta_{8} + 11 \beta_{7} + 12 \beta_{6} - 16 \beta_{5} + \cdots + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 80 \beta_{11} - 80 \beta_{10} + 95 \beta_{9} + 120 \beta_{8} + 17 \beta_{7} + 264 \beta_{6} + \cdots + 604 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 46 \beta_{11} - 35 \beta_{10} + 109 \beta_{9} - 22 \beta_{8} - 20 \beta_{7} - 166 \beta_{6} + 44 \beta_{5} + \cdots - 376 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 602 \beta_{11} + 486 \beta_{10} - 1789 \beta_{9} - 630 \beta_{8} - 117 \beta_{7} - 2700 \beta_{6} + \cdots - 17468 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 800 \beta_{11} + 1160 \beta_{10} - 2131 \beta_{9} - 416 \beta_{8} + 539 \beta_{7} + 1784 \beta_{6} + \cdots + 22132 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 891 \beta_{11} - 4079 \beta_{10} + 3490 \beta_{9} + 3969 \beta_{8} + 635 \beta_{7} + 11130 \beta_{6} + \cdots + 125228 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9356 \beta_{11} - 15470 \beta_{10} + 21545 \beta_{9} + 9908 \beta_{8} - 10463 \beta_{7} + 3964 \beta_{6} + \cdots - 211252 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5852 \beta_{11} + 103012 \beta_{10} + 92411 \beta_{9} - 147444 \beta_{8} - 42439 \beta_{7} + \cdots - 2603300 ) / 24 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
0.379272 + 0.549495i
0.379272 − 0.549495i
−1.75607 + 0.457021i
−1.75607 − 0.457021i
1.87680 − 0.723107i
1.87680 + 0.723107i
1.87680 − 2.10532i
1.87680 + 2.10532i
−1.75607 + 2.37141i
−1.75607 − 2.37141i
0.379272 − 3.37792i
0.379272 + 3.37792i
−2.77710 − 0.536371i 0 7.42461 + 2.97912i −5.96794 0 4.64674i −19.0210 − 12.2557i 0 16.5736 + 3.20103i
35.2 −2.77710 + 0.536371i 0 7.42461 − 2.97912i −5.96794 0 − 4.64674i −19.0210 + 12.2557i 0 16.5736 − 3.20103i
35.3 −1.35368 − 2.48346i 0 −4.33513 + 6.72359i −4.33351 0 28.4234i 22.5661 + 1.66455i 0 5.86617 + 10.7621i
35.4 −1.35368 + 2.48346i 0 −4.33513 − 6.72359i −4.33351 0 − 28.4234i 22.5661 − 1.66455i 0 5.86617 − 10.7621i
35.5 −0.977373 − 2.65419i 0 −6.08948 + 5.18827i 19.8898 0 − 12.9042i 19.7224 + 11.0918i 0 −19.4398 − 52.7914i
35.6 −0.977373 + 2.65419i 0 −6.08948 − 5.18827i 19.8898 0 12.9042i 19.7224 − 11.0918i 0 −19.4398 + 52.7914i
35.7 0.977373 − 2.65419i 0 −6.08948 − 5.18827i −19.8898 0 12.9042i −19.7224 + 11.0918i 0 −19.4398 + 52.7914i
35.8 0.977373 + 2.65419i 0 −6.08948 + 5.18827i −19.8898 0 − 12.9042i −19.7224 − 11.0918i 0 −19.4398 − 52.7914i
35.9 1.35368 − 2.48346i 0 −4.33513 − 6.72359i 4.33351 0 − 28.4234i −22.5661 + 1.66455i 0 5.86617 − 10.7621i
35.10 1.35368 + 2.48346i 0 −4.33513 + 6.72359i 4.33351 0 28.4234i −22.5661 − 1.66455i 0 5.86617 + 10.7621i
35.11 2.77710 − 0.536371i 0 7.42461 − 2.97912i 5.96794 0 − 4.64674i 19.0210 − 12.2557i 0 16.5736 − 3.20103i
35.12 2.77710 + 0.536371i 0 7.42461 + 2.97912i 5.96794 0 4.64674i 19.0210 + 12.2557i 0 16.5736 + 3.20103i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.12
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 72.4.f.a ✓ 12
3.b odd 2 1 inner 72.4.f.a ✓ 12
4.b odd 2 1 288.4.f.a 12
8.b even 2 1 288.4.f.a 12
8.d odd 2 1 inner 72.4.f.a ✓ 12
12.b even 2 1 288.4.f.a 12
16.e even 4 2 2304.4.c.n 24
16.f odd 4 2 2304.4.c.n 24
24.f even 2 1 inner 72.4.f.a ✓ 12
24.h odd 2 1 288.4.f.a 12
48.i odd 4 2 2304.4.c.n 24
48.k even 4 2 2304.4.c.n 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.f.a ✓ 12 1.a even 1 1 trivial
72.4.f.a ✓ 12 3.b odd 2 1 inner
72.4.f.a ✓ 12 8.d odd 2 1 inner
72.4.f.a ✓ 12 24.f even 2 1 inner
288.4.f.a 12 4.b odd 2 1
288.4.f.a 12 8.b even 2 1
288.4.f.a 12 12.b even 2 1
288.4.f.a 12 24.h odd 2 1
2304.4.c.n 24 16.e even 4 2
2304.4.c.n 24 16.f odd 4 2
2304.4.c.n 24 48.i odd 4 2
2304.4.c.n 24 48.k even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(72, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 450 T^{4} + \cdots - 264600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 996 T^{4} + \cdots + 2904768)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 4440 T^{4} + \cdots + 15323648)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 5700 T^{4} + \cdots + 2534148288)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 17094 T^{4} + \cdots + 42195125000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 12 T^{2} + \cdots - 338240)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 2940672038400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 2640040133400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 47901170923200)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 420084090667008)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 8295430241672)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 216 T^{2} + \cdots - 13422080)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 84658467686616)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 199401408561152)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 86704128000000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 204 T^{2} + \cdots + 36740800)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 108 T^{2} + \cdots - 45402560)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 218981353348800)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 24 T^{2} + \cdots + 327738880)^{4} \) Copy content Toggle raw display
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