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Properties

Label 72.21.b.b
Level $72$
Weight $21$
Character orbit 72.b
Analytic conductor $182.530$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,21,Mod(19,72)]
 
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, 0]))
 
N = Newforms(chi, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.19");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 72.b (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: \(182.529910874\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 16502665988595 x^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{153}\cdot 3^{19}\cdot 5^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 8)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 22) q^{2} + ( - \beta_{3} - \beta_{2} + \cdots + 40275) q^{4}+ \cdots + ( - \beta_{8} + \beta_{6} + \cdots + 172512128) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 22) q^{2} + ( - \beta_{3} - \beta_{2} + \cdots + 40275) q^{4}+ \cdots + (48972896 \beta_{17} + \cdots - 86\!\cdots\!58) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 398 q^{2} + 724980 q^{4} + 3105291368 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 398 q^{2} + 724980 q^{4} + 3105291368 q^{8} + 742754160 q^{10} - 27862395020 q^{11} + 431248136928 q^{14} - 958534870512 q^{16} - 4343925139172 q^{17} - 360681653556 q^{19} + 32520172742880 q^{20} - 66275482922148 q^{22} - 395727477008910 q^{25} - 154289718058128 q^{26} - 394619539621440 q^{28} + 28\!\cdots\!48 q^{32}+ \cdots - 15\!\cdots\!02 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 16502665988595 x^{16} + \cdots + 20\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 60\!\cdots\!31 \nu^{17} + \cdots + 39\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30\!\cdots\!55 \nu^{17} + \cdots + 26\!\cdots\!00 ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 54\!\cdots\!77 \nu^{17} + \cdots + 22\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!17 \nu^{17} + \cdots - 86\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 52\!\cdots\!21 \nu^{17} + \cdots + 27\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!43 \nu^{17} + \cdots - 51\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49\!\cdots\!27 \nu^{17} + \cdots - 19\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 56\!\cdots\!61 \nu^{17} + \cdots + 15\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36\!\cdots\!77 \nu^{17} + \cdots + 16\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!77 \nu^{17} + \cdots + 55\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 54\!\cdots\!53 \nu^{17} + \cdots - 12\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 68\!\cdots\!43 \nu^{17} + \cdots + 53\!\cdots\!00 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 76\!\cdots\!31 \nu^{17} + \cdots - 98\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!53 \nu^{17} + \cdots + 27\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 22\!\cdots\!43 \nu^{17} + \cdots + 17\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 90\!\cdots\!23 \nu^{17} + \cdots + 11\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 48\!\cdots\!47 \nu^{17} + \cdots + 77\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{3} - 39\beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2278 \beta_{15} - 980 \beta_{14} + 868 \beta_{13} + 1414 \beta_{12} - 58 \beta_{11} + \cdots - 117352117406015 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 878571 \beta_{17} + 404130005 \beta_{16} - 4901229363 \beta_{15} - 4551249152 \beta_{14} + \cdots + 73\!\cdots\!33 ) / 256 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 85\!\cdots\!50 \beta_{15} + \cdots + 30\!\cdots\!05 ) / 512 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 47\!\cdots\!60 \beta_{17} + \cdots - 70\!\cdots\!05 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 37\!\cdots\!40 \beta_{15} + \cdots - 11\!\cdots\!75 ) / 512 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 69\!\cdots\!40 \beta_{17} + \cdots + 69\!\cdots\!70 ) / 1024 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 79\!\cdots\!00 \beta_{15} + \cdots + 23\!\cdots\!25 ) / 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 95\!\cdots\!50 \beta_{17} + \cdots - 84\!\cdots\!75 ) / 256 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16\!\cdots\!75 \beta_{15} + \cdots - 48\!\cdots\!00 ) / 128 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 19\!\cdots\!00 \beta_{17} + \cdots + 15\!\cdots\!00 ) / 1024 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 29\!\cdots\!00 \beta_{15} + \cdots + 82\!\cdots\!75 ) / 512 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 45\!\cdots\!00 \beta_{17} + \cdots - 35\!\cdots\!75 ) / 512 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 12\!\cdots\!00 \beta_{15} + \cdots - 35\!\cdots\!25 ) / 512 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 42\!\cdots\!00 \beta_{17} + \cdots + 31\!\cdots\!50 ) / 1024 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 68\!\cdots\!75 \beta_{15} + \cdots + 19\!\cdots\!75 ) / 64 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 48\!\cdots\!50 \beta_{17} + \cdots - 34\!\cdots\!50 ) / 256 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
607700.i
− 607700.i
− 1.30366e6i
1.30366e6i
1.74596e6i
− 1.74596e6i
− 2.07165e6i
2.07165e6i
357294.i
− 357294.i
608151.i
− 608151.i
− 1.05610e6i
1.05610e6i
2.10254e6i
− 2.10254e6i
1.02967e6i
− 1.02967e6i
−979.944 − 297.130i 0 872004. + 582341.i 4.86160e6i 0 − 3.96260e8i −6.81484e8 − 8.29760e8i 0 1.44453e9 − 4.76409e9i
19.2 −979.944 + 297.130i 0 872004. − 582341.i − 4.86160e6i 0 3.96260e8i −6.81484e8 + 8.29760e8i 0 1.44453e9 + 4.76409e9i
19.3 −829.658 − 600.204i 0 328087. + 995927.i − 1.04292e7i 0 3.68005e8i 3.25559e8 − 1.02320e9i 0 −6.25967e9 + 8.65271e9i
19.4 −829.658 + 600.204i 0 328087. − 995927.i 1.04292e7i 0 − 3.68005e8i 3.25559e8 + 1.02320e9i 0 −6.25967e9 − 8.65271e9i
19.5 −607.571 − 824.278i 0 −310291. + 1.00161e6i 1.39677e7i 0 2.84328e8i 1.01413e9 − 3.52785e8i 0 1.15133e10 − 8.48636e9i
19.6 −607.571 + 824.278i 0 −310291. − 1.00161e6i − 1.39677e7i 0 − 2.84328e8i 1.01413e9 + 3.52785e8i 0 1.15133e10 + 8.48636e9i
19.7 −492.862 − 897.587i 0 −562749. + 884774.i − 1.65732e7i 0 − 1.30215e8i 1.07152e9 + 6.90447e7i 0 −1.48759e10 + 8.16831e9i
19.8 −492.862 + 897.587i 0 −562749. − 884774.i 1.65732e7i 0 1.30215e8i 1.07152e9 − 6.90447e7i 0 −1.48759e10 − 8.16831e9i
19.9 39.1296 − 1023.25i 0 −1.04551e6 − 80078.8i 2.85835e6i 0 − 1.54680e8i −1.22851e8 + 1.06669e9i 0 2.92482e9 + 1.11846e8i
19.10 39.1296 + 1023.25i 0 −1.04551e6 + 80078.8i − 2.85835e6i 0 1.54680e8i −1.22851e8 − 1.06669e9i 0 2.92482e9 − 1.11846e8i
19.11 340.525 − 965.722i 0 −816662. − 657705.i 4.86521e6i 0 1.26224e8i −9.13253e8 + 5.64703e8i 0 4.69844e9 + 1.65673e9i
19.12 340.525 + 965.722i 0 −816662. + 657705.i − 4.86521e6i 0 − 1.26224e8i −9.13253e8 − 5.64703e8i 0 4.69844e9 − 1.65673e9i
19.13 749.926 − 697.272i 0 76200.7 − 1.04580e6i − 8.44883e6i 0 6.17349e7i −6.72064e8 − 8.37407e8i 0 −5.89113e9 − 6.33599e9i
19.14 749.926 + 697.272i 0 76200.7 + 1.04580e6i 8.44883e6i 0 − 6.17349e7i −6.72064e8 + 8.37407e8i 0 −5.89113e9 + 6.33599e9i
19.15 980.704 − 294.612i 0 874983. − 577854.i 1.68203e7i 0 3.64653e8i 6.87856e8 − 8.24485e8i 0 4.95546e9 + 1.64957e10i
19.16 980.704 + 294.612i 0 874983. + 577854.i − 1.68203e7i 0 − 3.64653e8i 6.87856e8 + 8.24485e8i 0 4.95546e9 − 1.64957e10i
19.17 998.751 − 225.992i 0 946431. − 451420.i 8.23733e6i 0 − 5.26995e8i 8.43231e8 − 6.64742e8i 0 1.86157e9 + 8.22704e9i
19.18 998.751 + 225.992i 0 946431. + 451420.i − 8.23733e6i 0 5.26995e8i 8.43231e8 + 6.64742e8i 0 1.86157e9 − 8.22704e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.21.b.b 18
3.b odd 2 1 8.21.d.b ✓ 18
8.d odd 2 1 inner 72.21.b.b 18
12.b even 2 1 32.21.d.b 18
24.f even 2 1 8.21.d.b ✓ 18
24.h odd 2 1 32.21.d.b 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.21.d.b ✓ 18 3.b odd 2 1
8.21.d.b ✓ 18 24.f even 2 1
32.21.d.b 18 12.b even 2 1
32.21.d.b 18 24.h odd 2 1
72.21.b.b 18 1.a even 1 1 trivial
72.21.b.b 18 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + \cdots + 36\!\cdots\!00 \) acting on \(S_{21}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots - 80\!\cdots\!92)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 75\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots - 31\!\cdots\!08)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 83\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots + 87\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 61\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 17\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
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