Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [273,12,Mod(1,273)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(273, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("273.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.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: | \(209.757688293\) |
| Analytic rank: | \(1\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 5 x^{12} - 16576 x^{11} + 33140 x^{10} + 102944048 x^{9} + 77145040 x^{8} + \cdots - 22\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{5}\cdot 7 \) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 5 x^{12} - 16576 x^{11} + 33140 x^{10} + 102944048 x^{9} + 77145040 x^{8} + \cdots - 22\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 2551 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu^{2} - 4164\nu + 322 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 23\!\cdots\!81 \nu^{12} + \cdots - 57\!\cdots\!00 ) / 35\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 50\!\cdots\!21 \nu^{12} + \cdots - 12\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 54\!\cdots\!51 \nu^{12} + \cdots + 18\!\cdots\!00 ) / 20\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 93\!\cdots\!91 \nu^{12} + \cdots + 20\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 36\!\cdots\!07 \nu^{12} + \cdots - 52\!\cdots\!00 ) / 56\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 37\!\cdots\!99 \nu^{12} + \cdots + 45\!\cdots\!00 ) / 20\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 15\!\cdots\!33 \nu^{12} + \cdots + 44\!\cdots\!00 ) / 60\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 76\!\cdots\!05 \nu^{12} + \cdots - 20\!\cdots\!00 ) / 30\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 66\!\cdots\!61 \nu^{12} + \cdots + 13\!\cdots\!00 ) / 11\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 2551 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 4180\beta _1 + 9882 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{11} + 16\beta_{10} - 16\beta_{4} - 7\beta_{3} + 5616\beta_{2} + 32976\beta _1 + 10662734 \)
|
| \(\nu^{5}\) | \(=\) |
\( 32 \beta_{12} - 368 \beta_{11} - 512 \beta_{10} + 32 \beta_{9} + 208 \beta_{8} - 64 \beta_{7} + \cdots + 82318058 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 11872 \beta_{12} + 128784 \beta_{11} + 134080 \beta_{10} + 19040 \beta_{9} + 25680 \beta_{8} + \cdots + 50673055578 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 262496 \beta_{12} - 4054000 \beta_{11} - 4407488 \beta_{10} + 561504 \beta_{9} + 1790416 \beta_{8} + \cdots + 491577831594 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 133465952 \beta_{12} + 820063248 \beta_{11} + 874502208 \beta_{10} + 214945120 \beta_{9} + \cdots + 252637444508042 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 4102575968 \beta_{12} - 30574392048 \beta_{11} - 28832441024 \beta_{10} + 5214028128 \beta_{9} + \cdots + 26\!\cdots\!66 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1047614612832 \beta_{12} + 4844942677520 \beta_{11} + 5255473732928 \beta_{10} + \cdots + 12\!\cdots\!66 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 33718436834144 \beta_{12} - 200643512541424 \beta_{11} - 173255330123456 \beta_{10} + \cdots + 13\!\cdots\!34 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 70\!\cdots\!76 \beta_{12} + \cdots + 66\!\cdots\!50 \)
|
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 |
|
−81.0323 | 243.000 | 4518.24 | −7853.03 | −19690.9 | 16807.0 | −200169. | 59049.0 | 636349. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −80.7092 | 243.000 | 4465.97 | 6134.38 | −19612.3 | 16807.0 | −195152. | 59049.0 | −495101. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −72.6196 | 243.000 | 3225.60 | 865.592 | −17646.6 | 16807.0 | −85516.8 | 59049.0 | −62858.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −44.9817 | 243.000 | −24.6469 | 1977.86 | −10930.6 | 16807.0 | 93231.2 | 59049.0 | −88967.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −36.8409 | 243.000 | −690.746 | −7657.49 | −8952.34 | 16807.0 | 100898. | 59049.0 | 282109. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −18.3613 | 243.000 | −1710.86 | −2003.70 | −4461.79 | 16807.0 | 69017.5 | 59049.0 | 36790.4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −15.7938 | 243.000 | −1798.55 | 9740.33 | −3837.90 | 16807.0 | 60751.9 | 59049.0 | −153837. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 12.6290 | 243.000 | −1888.51 | −754.865 | 3068.84 | 16807.0 | −49714.0 | 59049.0 | −9533.16 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 25.1031 | 243.000 | −1417.83 | −13212.2 | 6100.05 | 16807.0 | −87003.2 | 59049.0 | −331668. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 34.7414 | 243.000 | −841.037 | 10618.0 | 8442.16 | 16807.0 | −100369. | 59049.0 | 368883. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 43.6274 | 243.000 | −144.651 | 1576.14 | 10601.5 | 16807.0 | −95659.6 | 59049.0 | 68762.8 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 59.0297 | 243.000 | 1436.50 | −5804.86 | 14344.2 | 16807.0 | −36096.5 | 59049.0 | −342659. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 66.2083 | 243.000 | 2335.53 | 278.900 | 16088.6 | 16807.0 | 19037.1 | 59049.0 | 18465.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 273.12.a.a | ✓ | 13 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.12.a.a | ✓ | 13 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} + 109 T_{2}^{12} - 11104 T_{2}^{11} - 1324276 T_{2}^{10} + 45437168 T_{2}^{9} + \cdots + 42\!\cdots\!76 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + \cdots + 42\!\cdots\!76 \)
|
| $3$ |
\( (T - 243)^{13} \)
|
| $5$ |
\( T^{13} + \cdots - 33\!\cdots\!00 \)
|
| $7$ |
\( (T - 16807)^{13} \)
|
| $11$ |
\( T^{13} + \cdots + 48\!\cdots\!00 \)
|
| $13$ |
\( (T - 371293)^{13} \)
|
| $17$ |
\( T^{13} + \cdots + 47\!\cdots\!32 \)
|
| $19$ |
\( T^{13} + \cdots - 61\!\cdots\!84 \)
|
| $23$ |
\( T^{13} + \cdots - 35\!\cdots\!28 \)
|
| $29$ |
\( T^{13} + \cdots + 18\!\cdots\!04 \)
|
| $31$ |
\( T^{13} + \cdots + 49\!\cdots\!20 \)
|
| $37$ |
\( T^{13} + \cdots - 34\!\cdots\!40 \)
|
| $41$ |
\( T^{13} + \cdots + 46\!\cdots\!40 \)
|
| $43$ |
\( T^{13} + \cdots - 93\!\cdots\!32 \)
|
| $47$ |
\( T^{13} + \cdots - 47\!\cdots\!28 \)
|
| $53$ |
\( T^{13} + \cdots - 17\!\cdots\!00 \)
|
| $59$ |
\( T^{13} + \cdots - 16\!\cdots\!84 \)
|
| $61$ |
\( T^{13} + \cdots - 40\!\cdots\!00 \)
|
| $67$ |
\( T^{13} + \cdots + 47\!\cdots\!60 \)
|
| $71$ |
\( T^{13} + \cdots - 26\!\cdots\!48 \)
|
| $73$ |
\( T^{13} + \cdots - 19\!\cdots\!72 \)
|
| $79$ |
\( T^{13} + \cdots - 56\!\cdots\!16 \)
|
| $83$ |
\( T^{13} + \cdots + 26\!\cdots\!00 \)
|
| $89$ |
\( T^{13} + \cdots - 31\!\cdots\!80 \)
|
| $97$ |
\( T^{13} + \cdots - 86\!\cdots\!60 \)
|
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