Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [74,22,Mod(1,74)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(74, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("74.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 74 = 2 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 74.a (trivial) |
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: | yes |
| Analytic conductor: | \(206.813234772\) |
| Analytic rank: | \(1\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 4 x^{13} - 94555761504 x^{12} + 201235358550377 x^{11} + \cdots - 27\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{33}\cdot 3^{11}\cdot 5^{2}\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{14} - 4 x^{13} - 94555761504 x^{12} + 201235358550377 x^{11} + \cdots - 27\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 65\!\cdots\!09 \nu^{13} + \cdots - 42\!\cdots\!50 ) / 39\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 87\!\cdots\!31 \nu^{13} + \cdots - 18\!\cdots\!00 ) / 61\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 23\!\cdots\!07 \nu^{13} + \cdots - 15\!\cdots\!50 ) / 40\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 63\!\cdots\!39 \nu^{13} + \cdots - 78\!\cdots\!50 ) / 27\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 82\!\cdots\!81 \nu^{13} + \cdots - 59\!\cdots\!50 ) / 32\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\!\cdots\!63 \nu^{13} + \cdots + 13\!\cdots\!50 ) / 49\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 13\!\cdots\!09 \nu^{13} + \cdots - 96\!\cdots\!50 ) / 49\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 96\!\cdots\!13 \nu^{13} + \cdots - 72\!\cdots\!50 ) / 33\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 25\!\cdots\!15 \nu^{13} + \cdots - 15\!\cdots\!50 ) / 65\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 51\!\cdots\!63 \nu^{13} + \cdots - 27\!\cdots\!50 ) / 98\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 76\!\cdots\!11 \nu^{13} + \cdots + 53\!\cdots\!50 ) / 14\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 82\!\cdots\!87 \nu^{13} + \cdots - 53\!\cdots\!50 ) / 98\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - 36\beta_{2} - 3173\beta _1 + 13507966824 \)
|
| \(\nu^{3}\) | \(=\) |
\( 11751 \beta_{13} + 4735 \beta_{12} - 10737 \beta_{11} + 8906 \beta_{10} - 7914 \beta_{9} + \cdots - 43047306909495 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 775330353 \beta_{13} - 402198948 \beta_{12} + 470134611 \beta_{11} + 639609327 \beta_{10} + \cdots + 30\!\cdots\!96 \)
|
| \(\nu^{5}\) | \(=\) |
\( 416637983297430 \beta_{13} + 150516503271006 \beta_{12} - 405952646191479 \beta_{11} + \cdots - 38\!\cdots\!77 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 32\!\cdots\!00 \beta_{13} + \cdots + 77\!\cdots\!87 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12\!\cdots\!74 \beta_{13} + \cdots + 91\!\cdots\!12 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 11\!\cdots\!74 \beta_{13} + \cdots + 20\!\cdots\!05 \)
|
| \(\nu^{9}\) | \(=\) |
\( 35\!\cdots\!36 \beta_{13} + \cdots + 75\!\cdots\!78 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 35\!\cdots\!89 \beta_{13} + \cdots + 54\!\cdots\!62 \)
|
| \(\nu^{11}\) | \(=\) |
\( 99\!\cdots\!88 \beta_{13} + \cdots + 34\!\cdots\!16 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 10\!\cdots\!99 \beta_{13} + \cdots + 14\!\cdots\!28 \)
|
| \(\nu^{13}\) | \(=\) |
\( 28\!\cdots\!82 \beta_{13} + \cdots + 13\!\cdots\!38 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1024.00 | −179943. | 1.04858e6 | 2.49960e7 | −1.84262e8 | 8.73789e8 | 1.07374e9 | 2.19192e10 | 2.55959e10 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1024.00 | −169288. | 1.04858e6 | −3.28845e6 | −1.73351e8 | −6.41817e8 | 1.07374e9 | 1.81982e10 | −3.36737e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1024.00 | −157453. | 1.04858e6 | 1.18900e7 | −1.61232e8 | −1.12575e9 | 1.07374e9 | 1.43312e10 | 1.21753e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1024.00 | −95634.3 | 1.04858e6 | −4.12601e7 | −9.79295e7 | 9.94624e8 | 1.07374e9 | −1.31443e9 | −4.22504e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1024.00 | −68938.9 | 1.04858e6 | −3.10487e7 | −7.05934e7 | −6.49075e8 | 1.07374e9 | −5.70778e9 | −3.17939e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1024.00 | −52044.2 | 1.04858e6 | −8.38003e6 | −5.32933e7 | 8.52021e8 | 1.07374e9 | −7.75176e9 | −8.58115e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1024.00 | −50043.9 | 1.04858e6 | 2.21880e7 | −5.12449e7 | −1.26158e8 | 1.07374e9 | −7.95596e9 | 2.27205e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1024.00 | −3996.02 | 1.04858e6 | −7.50074e6 | −4.09193e6 | −5.37607e8 | 1.07374e9 | −1.04444e10 | −7.68075e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1024.00 | 44457.4 | 1.04858e6 | 3.74384e7 | 4.55244e7 | −5.01571e8 | 1.07374e9 | −8.48389e9 | 3.83369e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1024.00 | 62574.0 | 1.04858e6 | 1.55498e7 | 6.40758e7 | −1.30648e7 | 1.07374e9 | −6.54485e9 | 1.59230e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1024.00 | 95992.6 | 1.04858e6 | −3.65448e7 | 9.82964e7 | 8.15769e8 | 1.07374e9 | −1.24577e9 | −3.74219e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1024.00 | 147811. | 1.04858e6 | 9.77476e6 | 1.51359e8 | −5.66343e8 | 1.07374e9 | 1.13878e10 | 1.00094e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1024.00 | 150233. | 1.04858e6 | −2.65430e7 | 1.53839e8 | −1.37332e9 | 1.07374e9 | 1.21096e10 | −2.71801e10 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1024.00 | 159986. | 1.04858e6 | −3.54738e6 | 1.63826e8 | 5.74402e6 | 1.07374e9 | 1.51352e10 | −3.63251e9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(37\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 74.22.a.a | ✓ | 14 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 74.22.a.a | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{14} + 116288 T_{3}^{13} - 88277272716 T_{3}^{12} + \cdots + 31\!\cdots\!00 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(74))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1024)^{14} \)
|
| $3$ |
\( T^{14} + \cdots + 31\!\cdots\!00 \)
|
| $5$ |
\( T^{14} + \cdots + 34\!\cdots\!00 \)
|
| $7$ |
\( T^{14} + \cdots - 56\!\cdots\!00 \)
|
| $11$ |
\( T^{14} + \cdots - 54\!\cdots\!80 \)
|
| $13$ |
\( T^{14} + \cdots + 10\!\cdots\!00 \)
|
| $17$ |
\( T^{14} + \cdots - 15\!\cdots\!16 \)
|
| $19$ |
\( T^{14} + \cdots + 46\!\cdots\!00 \)
|
| $23$ |
\( T^{14} + \cdots - 90\!\cdots\!00 \)
|
| $29$ |
\( T^{14} + \cdots - 15\!\cdots\!00 \)
|
| $31$ |
\( T^{14} + \cdots + 13\!\cdots\!00 \)
|
| $37$ |
\( (T - 48\!\cdots\!49)^{14} \)
|
| $41$ |
\( T^{14} + \cdots + 70\!\cdots\!46 \)
|
| $43$ |
\( T^{14} + \cdots - 76\!\cdots\!92 \)
|
| $47$ |
\( T^{14} + \cdots + 22\!\cdots\!00 \)
|
| $53$ |
\( T^{14} + \cdots + 76\!\cdots\!60 \)
|
| $59$ |
\( T^{14} + \cdots + 54\!\cdots\!00 \)
|
| $61$ |
\( T^{14} + \cdots - 36\!\cdots\!00 \)
|
| $67$ |
\( T^{14} + \cdots - 34\!\cdots\!00 \)
|
| $71$ |
\( T^{14} + \cdots + 54\!\cdots\!00 \)
|
| $73$ |
\( T^{14} + \cdots - 22\!\cdots\!50 \)
|
| $79$ |
\( T^{14} + \cdots - 68\!\cdots\!00 \)
|
| $83$ |
\( T^{14} + \cdots - 30\!\cdots\!00 \)
|
| $89$ |
\( T^{14} + \cdots - 99\!\cdots\!00 \)
|
| $97$ |
\( T^{14} + \cdots + 25\!\cdots\!56 \)
|
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