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Properties

Label 72.18.d.b
Level $72$
Weight $18$
Character orbit 72.d
Analytic conductor $131.920$
Analytic rank $0$
Dimension $16$
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] = [72,18,Mod(37,72)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.37");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 72.d (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: \(131.919902888\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 109505575668 x^{14} - 766539029536 x^{13} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{20}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 17) q^{2} + (\beta_{4} + 17 \beta_1 - 1713) q^{4} + ( - \beta_{4} - \beta_{3} + 63 \beta_1 + 8) q^{5} + (\beta_{5} + 11 \beta_{4} + \cdots + 721562) q^{7}+ \cdots + (\beta_{7} - \beta_{6} + \cdots - 1520694) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 17) q^{2} + (\beta_{4} + 17 \beta_1 - 1713) q^{4} + ( - \beta_{4} - \beta_{3} + 63 \beta_1 + 8) q^{5} + (\beta_{5} + 11 \beta_{4} + \cdots + 721562) q^{7}+ \cdots + (7959952 \beta_{15} + \cdots - 21\!\cdots\!81) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 270 q^{2} - 27436 q^{4} + 11529600 q^{7} - 24334920 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 270 q^{2} - 27436 q^{4} + 11529600 q^{7} - 24334920 q^{8} + 131002712 q^{10} - 16363788528 q^{14} + 26500434192 q^{16} + 7489125600 q^{17} + 209445719856 q^{20} + 223126527100 q^{22} - 746845345920 q^{23} - 1809682431664 q^{25} - 2467726531080 q^{26} + 3220542267040 q^{28} - 318979758592 q^{31} - 1455647316000 q^{32} - 4461251980292 q^{34} - 24076283913900 q^{38} + 60626292962592 q^{40} - 7482251536032 q^{41} - 193654716236040 q^{44} - 195097141003568 q^{46} + 376698804821760 q^{47} + 127691292101520 q^{49} - 474997408872102 q^{50} - 272251877663120 q^{52} + 22\!\cdots\!52 q^{55}+ \cdots - 33\!\cdots\!90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 109505575668 x^{14} - 766539029536 x^{13} + \cdots + 23\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 81\!\cdots\!73 \nu^{15} + \cdots - 11\!\cdots\!56 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 54\!\cdots\!61 \nu^{15} + \cdots + 29\!\cdots\!92 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!31 \nu^{15} + \cdots - 29\!\cdots\!04 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 64\!\cdots\!49 \nu^{15} + \cdots + 74\!\cdots\!72 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69\!\cdots\!99 \nu^{15} + \cdots + 11\!\cdots\!28 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64\!\cdots\!79 \nu^{15} + \cdots - 79\!\cdots\!68 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!41 \nu^{15} + \cdots + 10\!\cdots\!52 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 35\!\cdots\!99 \nu^{15} + \cdots - 28\!\cdots\!72 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66\!\cdots\!17 \nu^{15} + \cdots - 10\!\cdots\!24 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 33\!\cdots\!57 \nu^{15} + \cdots + 21\!\cdots\!24 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!13 \nu^{15} + \cdots + 91\!\cdots\!64 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 22\!\cdots\!93 \nu^{15} + \cdots - 98\!\cdots\!04 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 97\!\cdots\!61 \nu^{15} + \cdots - 34\!\cdots\!08 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!09 \nu^{15} + \cdots + 26\!\cdots\!52 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 30\!\cdots\!73 \nu^{15} + \cdots - 71\!\cdots\!56 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - 63\beta _1 - 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 28 \beta_{15} - 50 \beta_{14} - 100 \beta_{13} - 128 \beta_{12} - 472 \beta_{11} + \cdots - 438007585166 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 42474661 \beta_{15} + 20432520 \beta_{14} - 184624480 \beta_{13} - 63610411 \beta_{12} + \cdots + 49133479861965 ) / 256 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6048673051210 \beta_{15} + 4859492388390 \beta_{14} + 35378369834860 \beta_{13} + \cdots + 91\!\cdots\!71 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\!\cdots\!90 \beta_{15} + \cdots - 44\!\cdots\!56 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 18\!\cdots\!04 \beta_{15} + \cdots - 29\!\cdots\!58 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 27\!\cdots\!94 \beta_{15} + \cdots + 36\!\cdots\!20 ) / 1024 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 15\!\cdots\!80 \beta_{15} + \cdots + 25\!\cdots\!59 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 25\!\cdots\!55 \beta_{15} + \cdots - 18\!\cdots\!39 ) / 128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 20\!\cdots\!84 \beta_{15} + \cdots - 36\!\cdots\!38 ) / 256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15\!\cdots\!06 \beta_{15} + \cdots + 59\!\cdots\!80 ) / 1024 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 34\!\cdots\!90 \beta_{15} + \cdots + 68\!\cdots\!38 ) / 128 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 40\!\cdots\!30 \beta_{15} + \cdots - 11\!\cdots\!56 ) / 512 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 24\!\cdots\!64 \beta_{15} + \cdots - 52\!\cdots\!18 ) / 256 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 36\!\cdots\!06 \beta_{15} + \cdots + 94\!\cdots\!40 ) / 1024 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 58156.1i
0.500000 − 58156.1i
0.500000 + 82996.3i
0.500000 − 82996.3i
0.500000 − 151568.i
0.500000 + 151568.i
0.500000 + 159067.i
0.500000 − 159067.i
0.500000 + 12044.5i
0.500000 − 12044.5i
0.500000 − 198996.i
0.500000 + 198996.i
0.500000 + 65608.9i
0.500000 − 65608.9i
0.500000 + 83132.4i
0.500000 − 83132.4i
−351.815 − 85.4288i 0 116476. + 60110.3i − 465248.i 0 2.24795e7 −3.58428e7 − 3.10981e7i 0 −3.97456e7 + 1.63681e8i
37.2 −351.815 + 85.4288i 0 116476. − 60110.3i 465248.i 0 2.24795e7 −3.58428e7 + 3.10981e7i 0 −3.97456e7 − 1.63681e8i
37.3 −328.641 − 151.878i 0 84938.1 + 99826.8i − 663971.i 0 −1.66742e7 −1.27527e7 − 4.57074e7i 0 −1.00843e8 + 2.18208e8i
37.4 −328.641 + 151.878i 0 84938.1 − 99826.8i 663971.i 0 −1.66742e7 −1.27527e7 + 4.57074e7i 0 −1.00843e8 − 2.18208e8i
37.5 −214.125 − 291.929i 0 −39373.4 + 125018.i 1.21254e6i 0 1.76580e7 4.49273e7 − 1.52753e7i 0 3.53976e8 − 2.59635e8i
37.6 −214.125 + 291.929i 0 −39373.4 − 125018.i − 1.21254e6i 0 1.76580e7 4.49273e7 + 1.52753e7i 0 3.53976e8 + 2.59635e8i
37.7 −42.2945 − 359.560i 0 −127494. + 30414.8i − 1.27253e6i 0 5.50569e6 1.63282e7 + 4.45555e7i 0 −4.57551e8 + 5.38211e7i
37.8 −42.2945 + 359.560i 0 −127494. − 30414.8i 1.27253e6i 0 5.50569e6 1.63282e7 − 4.45555e7i 0 −4.57551e8 − 5.38211e7i
37.9 −18.3339 − 361.574i 0 −130400. + 13258.2i − 96356.3i 0 −1.47728e7 7.18455e6 + 4.69061e7i 0 −3.48399e7 + 1.76659e6i
37.10 −18.3339 + 361.574i 0 −130400. − 13258.2i 96356.3i 0 −1.47728e7 7.18455e6 − 4.69061e7i 0 −3.48399e7 − 1.76659e6i
37.11 200.394 − 301.520i 0 −50756.5 − 120846.i 1.59197e6i 0 −1.66055e7 −4.66086e7 − 8.91263e6i 0 4.80011e8 + 3.19021e8i
37.12 200.394 + 301.520i 0 −50756.5 + 120846.i − 1.59197e6i 0 −1.66055e7 −4.66086e7 + 8.91263e6i 0 4.80011e8 − 3.19021e8i
37.13 257.790 − 254.197i 0 1839.67 − 131059.i − 524871.i 0 1.57495e7 −3.28406e7 − 3.42534e7i 0 −1.33421e8 − 1.35307e8i
37.14 257.790 + 254.197i 0 1839.67 + 131059.i 524871.i 0 1.57495e7 −3.28406e7 + 3.42534e7i 0 −1.33421e8 + 1.35307e8i
37.15 362.025 − 3.13586i 0 131052. − 2270.52i − 665059.i 0 −7.57536e6 4.74371e7 − 1.23295e6i 0 −2.08553e6 − 2.40768e8i
37.16 362.025 + 3.13586i 0 131052. + 2270.52i 665059.i 0 −7.57536e6 4.74371e7 + 1.23295e6i 0 −2.08553e6 + 2.40768e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.18.d.b 16
3.b odd 2 1 8.18.b.a ✓ 16
8.b even 2 1 inner 72.18.d.b 16
12.b even 2 1 32.18.b.a 16
24.f even 2 1 32.18.b.a 16
24.h odd 2 1 8.18.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.18.b.a ✓ 16 3.b odd 2 1
8.18.b.a ✓ 16 24.h odd 2 1
32.18.b.a 16 12.b even 2 1
32.18.b.a 16 24.f even 2 1
72.18.d.b 16 1.a even 1 1 trivial
72.18.d.b 16 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 7008356840832 T_{5}^{14} + \cdots + 65\!\cdots\!00 \) acting on \(S_{18}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 21\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
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