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Properties

Label 72.19.b.b
Level $72$
Weight $19$
Character orbit 72.b
Analytic conductor $147.878$
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] = [72,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.19");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 19 \)
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: \(147.878019151\)
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: \( x^{16} + 596528635860 x^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{15}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 27) q^{2} + (\beta_{4} - 30 \beta_1 - 27760) q^{4} + (\beta_{4} + \beta_{3} + 375 \beta_1 - 141) q^{5} + (\beta_{7} + \beta_{5} + 13 \beta_{4} + \cdots - 2350) q^{7}+ \cdots + ( - \beta_{9} + 7 \beta_{5} + \cdots - 19045968) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 27) q^{2} + (\beta_{4} - 30 \beta_1 - 27760) q^{4} + (\beta_{4} + \beta_{3} + 375 \beta_1 - 141) q^{5} + (\beta_{7} + \beta_{5} + 13 \beta_{4} + \cdots - 2350) q^{7}+ \cdots + (66261920 \beta_{15} + \cdots + 51\!\cdots\!81) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 426 q^{2} - 444332 q^{4} - 304914744 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 426 q^{2} - 444332 q^{4} - 304914744 q^{8} - 1569837600 q^{10} + 2471571264 q^{11} - 26163923904 q^{14} - 192320075504 q^{16} + 176439301344 q^{17} - 833365634368 q^{19} - 1256486405760 q^{20} + 59927319356 q^{22} - 15320509140080 q^{25} - 4184514840864 q^{26} - 12301604294400 q^{28} - 141481742931936 q^{32} + 80653465357268 q^{34} - 20487495736320 q^{35} - 493456694265564 q^{38} - 519930573603840 q^{40} + 594931562445024 q^{41} - 25\!\cdots\!92 q^{43}+ \cdots + 81\!\cdots\!66 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 596528635860 x^{14} + \cdots + 10\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 62\!\cdots\!87 \nu^{15} + \cdots - 19\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 68\!\cdots\!57 \nu^{15} + \cdots - 37\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!83 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!01 \nu^{15} + \cdots + 66\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!69 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!59 \nu^{15} + \cdots - 16\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!81 \nu^{15} + \cdots - 55\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!39 \nu^{15} + \cdots - 16\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24\!\cdots\!49 \nu^{15} + \cdots + 53\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 97\!\cdots\!81 \nu^{15} + \cdots + 64\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15\!\cdots\!47 \nu^{15} + \cdots - 67\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{15} + \cdots - 37\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 71\!\cdots\!59 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 86\!\cdots\!73 \nu^{15} + \cdots - 12\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 24\!\cdots\!67 \nu^{15} + \cdots + 56\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} + 375\beta _1 - 141 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 320 \beta_{14} - 480 \beta_{13} - 448 \beta_{12} - 448 \beta_{11} - 1492 \beta_{10} + \cdots - 2385842968964 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 372504150 \beta_{15} + 593498955 \beta_{14} - 398643620 \beta_{13} - 233644000 \beta_{12} + \cdots + 17\!\cdots\!75 ) / 256 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 316870662303200 \beta_{14} + 710088435218000 \beta_{13} + 617464533772960 \beta_{12} + \cdots + 25\!\cdots\!80 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\!\cdots\!00 \beta_{15} + \cdots - 69\!\cdots\!50 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 49\!\cdots\!50 \beta_{14} + \cdots - 37\!\cdots\!50 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 65\!\cdots\!00 \beta_{15} + \cdots + 24\!\cdots\!50 ) / 1024 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 45\!\cdots\!00 \beta_{14} + \cdots + 30\!\cdots\!50 ) / 128 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15\!\cdots\!50 \beta_{15} + \cdots - 53\!\cdots\!25 ) / 128 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 87\!\cdots\!25 \beta_{14} + \cdots - 50\!\cdots\!50 ) / 128 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 23\!\cdots\!00 \beta_{15} + \cdots + 74\!\cdots\!50 ) / 1024 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 33\!\cdots\!00 \beta_{14} + \cdots + 17\!\cdots\!50 ) / 256 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 21\!\cdots\!00 \beta_{15} + \cdots - 65\!\cdots\!50 ) / 512 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 64\!\cdots\!50 \beta_{14} + \cdots - 31\!\cdots\!50 ) / 256 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 81\!\cdots\!00 \beta_{15} + \cdots + 23\!\cdots\!50 ) / 1024 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
− 335678.i
335678.i
− 14143.5i
14143.5i
237892.i
− 237892.i
112721.i
− 112721.i
− 375716.i
375716.i
285196.i
− 285196.i
− 54535.2i
54535.2i
− 434599.i
434599.i
−501.983 − 100.781i 0 241830. + 101181.i − 2.68543e6i 0 − 4.44900e7i −1.11198e8 − 7.51629e7i 0 −2.70640e8 + 1.34804e9i
19.2 −501.983 + 100.781i 0 241830. − 101181.i 2.68543e6i 0 4.44900e7i −1.11198e8 + 7.51629e7i 0 −2.70640e8 − 1.34804e9i
19.3 −431.385 − 275.773i 0 110042. + 237929.i − 113148.i 0 3.55480e7i 1.81440e7 − 1.32986e8i 0 −3.12032e7 + 4.88104e7i
19.4 −431.385 + 275.773i 0 110042. − 237929.i 113148.i 0 − 3.55480e7i 1.81440e7 + 1.32986e8i 0 −3.12032e7 − 4.88104e7i
19.5 −297.207 − 416.907i 0 −85479.4 + 247816.i 1.90314e6i 0 − 9.33592e6i 1.28721e8 − 3.80157e7i 0 7.93431e8 − 5.65626e8i
19.6 −297.207 + 416.907i 0 −85479.4 − 247816.i − 1.90314e6i 0 9.33592e6i 1.28721e8 + 3.80157e7i 0 7.93431e8 + 5.65626e8i
19.7 −74.4785 − 506.554i 0 −251050. + 75454.8i 901770.i 0 − 6.41362e7i 5.69197e7 + 1.21551e8i 0 4.56795e8 − 6.71625e7i
19.8 −74.4785 + 506.554i 0 −251050. − 75454.8i − 901770.i 0 6.41362e7i 5.69197e7 − 1.21551e8i 0 4.56795e8 + 6.71625e7i
19.9 0.0491315 − 512.000i 0 −262144. − 50.3107i − 3.00573e6i 0 5.44315e7i −38638.6 + 1.34218e8i 0 −1.53893e9 − 147676.i
19.10 0.0491315 + 512.000i 0 −262144. + 50.3107i 3.00573e6i 0 − 5.44315e7i −38638.6 − 1.34218e8i 0 −1.53893e9 + 147676.i
19.11 299.623 − 415.174i 0 −82595.7 − 248792.i 2.28157e6i 0 3.15838e7i −1.28040e8 − 4.02522e7i 0 9.47248e8 + 6.83611e8i
19.12 299.623 + 415.174i 0 −82595.7 + 248792.i − 2.28157e6i 0 − 3.15838e7i −1.28040e8 + 4.02522e7i 0 9.47248e8 − 6.83611e8i
19.13 365.984 − 358.050i 0 5744.83 − 262081.i − 436282.i 0 − 5.56713e7i −9.17355e7 − 9.79745e7i 0 −1.56211e8 − 1.59672e8i
19.14 365.984 + 358.050i 0 5744.83 + 262081.i 436282.i 0 5.56713e7i −9.17355e7 + 9.79745e7i 0 −1.56211e8 + 1.59672e8i
19.15 426.398 − 283.424i 0 101486. − 241703.i − 3.47680e6i 0 − 1.08296e7i −2.52310e7 − 1.31825e8i 0 −9.85407e8 − 1.48250e9i
19.16 426.398 + 283.424i 0 101486. + 241703.i 3.47680e6i 0 1.08296e7i −2.52310e7 + 1.31825e8i 0 −9.85407e8 + 1.48250e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
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 72.19.b.b 16
3.b odd 2 1 8.19.d.b ✓ 16
8.d odd 2 1 inner 72.19.b.b 16
12.b even 2 1 32.19.d.b 16
24.f even 2 1 8.19.d.b ✓ 16
24.h odd 2 1 32.19.d.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.19.d.b ✓ 16 3.b odd 2 1
8.19.d.b ✓ 16 24.f even 2 1
32.19.d.b 16 12.b even 2 1
32.19.d.b 16 24.h odd 2 1
72.19.b.b 16 1.a even 1 1 trivial
72.19.b.b 16 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 38177832695040 T_{5}^{14} + \cdots + 29\!\cdots\!00 \) acting on \(S_{19}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 43\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 19\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 60\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 80\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 83\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 45\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 87\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 55\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 14\!\cdots\!20)^{2} \) Copy content Toggle raw display
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