Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2672,2,Mod(1,2672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2672, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2672.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2672 = 2^{4} \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2672.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: | \(21.3360274201\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} + \cdots + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 1336) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} + \cdots + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 513 \nu^{11} + 769 \nu^{10} + 12409 \nu^{9} - 16400 \nu^{8} - 105696 \nu^{7} + 120839 \nu^{6} + \cdots - 47568 ) / 4136 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 53519 \nu^{11} - 83931 \nu^{10} - 1289331 \nu^{9} + 1796400 \nu^{8} + 10951328 \nu^{7} + \cdots + 5721888 ) / 57904 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 107109 \nu^{11} + 169593 \nu^{10} + 2577481 \nu^{9} - 3638784 \nu^{8} - 21858800 \nu^{7} + \cdots - 12293136 ) / 57904 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 125227 \nu^{11} - 198647 \nu^{10} - 3012143 \nu^{9} + 4262080 \nu^{8} + 25523536 \nu^{7} + \cdots + 14118640 ) / 57904 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 32139 \nu^{11} + 50481 \nu^{10} + 774057 \nu^{9} - 1084624 \nu^{8} - 6567776 \nu^{7} + \cdots - 3617364 ) / 14476 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 367753 \nu^{11} + 578405 \nu^{10} + 8865733 \nu^{9} - 12442928 \nu^{8} - 75314336 \nu^{7} + \cdots - 41904752 ) / 115808 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 56675 \nu^{11} - 88801 \nu^{10} - 1366609 \nu^{9} + 1910016 \nu^{8} + 11611044 \nu^{7} + \cdots + 6392900 ) / 14476 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 474791 \nu^{11} + 746267 \nu^{10} + 11444395 \nu^{9} - 16035728 \nu^{8} - 97216992 \nu^{7} + \cdots - 52769488 ) / 115808 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 540349 \nu^{11} - 847241 \nu^{10} - 13023209 \nu^{9} + 18193152 \nu^{8} + 110627008 \nu^{7} + \cdots + 60098064 ) / 115808 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 56631 \nu^{11} - 89035 \nu^{10} - 1365163 \nu^{9} + 1913696 \nu^{8} + 11597760 \nu^{7} + \cdots + 6377904 ) / 10528 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{9} + \beta_{7} - \beta_{3} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} + \beta_{10} + \beta_{8} + 2\beta_{6} + \beta_{5} - \beta_{3} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{11} - \beta_{10} - 10 \beta_{9} + 2 \beta_{8} + 10 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 38 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 13 \beta_{11} + 11 \beta_{10} - 2 \beta_{9} + 16 \beta_{8} + 4 \beta_{7} + 26 \beta_{6} + 12 \beta_{5} + \cdots + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 22 \beta_{11} - 16 \beta_{10} - 103 \beta_{9} + 35 \beta_{8} + 97 \beta_{7} + 19 \beta_{6} + \cdots + 322 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 156 \beta_{11} + 110 \beta_{10} - 50 \beta_{9} + 202 \beta_{8} + 74 \beta_{7} + 294 \beta_{6} + \cdots + 96 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 335 \beta_{11} - 187 \beta_{10} - 1076 \beta_{9} + 477 \beta_{8} + 958 \beta_{7} + 282 \beta_{6} + \cdots + 2910 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1827 \beta_{11} + 1073 \beta_{10} - 849 \beta_{9} + 2372 \beta_{8} + 1049 \beta_{7} + 3189 \beta_{6} + \cdots + 1597 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 4456 \beta_{11} - 1910 \beta_{10} - 11292 \beta_{9} + 5969 \beta_{8} + 9670 \beta_{7} + 3806 \beta_{6} + \cdots + 27578 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 21047 \beta_{11} + 10383 \beta_{10} - 12259 \beta_{9} + 27045 \beta_{8} + 13533 \beta_{7} + \cdots + 22864 \)
|
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 |
|
0 | −3.02141 | 0 | 2.82320 | 0 | 1.06132 | 0 | 6.12894 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.49668 | 0 | 0.494819 | 0 | −2.18883 | 0 | 3.23342 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.42555 | 0 | −0.247339 | 0 | −2.30877 | 0 | 2.88329 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.17540 | 0 | 3.77550 | 0 | 4.01031 | 0 | −1.61844 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.571891 | 0 | −2.42234 | 0 | 3.89487 | 0 | −2.67294 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.569286 | 0 | 3.51919 | 0 | −3.94871 | 0 | −2.67591 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.295455 | 0 | −3.76247 | 0 | −1.37107 | 0 | −2.91271 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.830621 | 0 | −1.03423 | 0 | 1.08177 | 0 | −2.31007 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.79442 | 0 | 3.50243 | 0 | 0.885520 | 0 | 0.219933 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.31477 | 0 | −1.78615 | 0 | 4.41953 | 0 | 2.35817 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.68363 | 0 | 0.670957 | 0 | −4.03610 | 0 | 4.20190 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.34132 | 0 | 2.46644 | 0 | 2.50015 | 0 | 8.16441 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(167\) | \(-1\) |
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 | 2672.2.a.p | 12 | |
| 4.b | odd | 2 | 1 | 1336.2.a.d | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1336.2.a.d | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 2672.2.a.p | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2672))\):
|
\( T_{3}^{12} - T_{3}^{11} - 25 T_{3}^{10} + 20 T_{3}^{9} + 224 T_{3}^{8} - 135 T_{3}^{7} - 865 T_{3}^{6} + \cdots + 64 \)
|
|
\( T_{7}^{12} - 4 T_{7}^{11} - 44 T_{7}^{10} + 172 T_{7}^{9} + 679 T_{7}^{8} - 2585 T_{7}^{7} + \cdots - 19376 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - T^{11} + \cdots + 64 \)
|
| $5$ |
\( T^{12} - 8 T^{11} + \cdots - 448 \)
|
| $7$ |
\( T^{12} - 4 T^{11} + \cdots - 19376 \)
|
| $11$ |
\( T^{12} + 6 T^{11} + \cdots - 29696 \)
|
| $13$ |
\( T^{12} - 13 T^{11} + \cdots - 256 \)
|
| $17$ |
\( T^{12} - 10 T^{11} + \cdots + 340736 \)
|
| $19$ |
\( T^{12} - T^{11} + \cdots + 5824 \)
|
| $23$ |
\( T^{12} + 3 T^{11} + \cdots - 567296 \)
|
| $29$ |
\( T^{12} + \cdots + 302326208 \)
|
| $31$ |
\( T^{12} - 3 T^{11} + \cdots + 4303232 \)
|
| $37$ |
\( T^{12} + \cdots - 520493056 \)
|
| $41$ |
\( T^{12} - 20 T^{11} + \cdots - 97679104 \)
|
| $43$ |
\( T^{12} - T^{11} + \cdots + 35037184 \)
|
| $47$ |
\( T^{12} + \cdots + 117468224 \)
|
| $53$ |
\( T^{12} + \cdots + 105953024 \)
|
| $59$ |
\( T^{12} + \cdots + 2202297344 \)
|
| $61$ |
\( T^{12} + \cdots - 662178112 \)
|
| $67$ |
\( T^{12} - 9 T^{11} + \cdots + 16152832 \)
|
| $71$ |
\( T^{12} + 29 T^{11} + \cdots + 18153472 \)
|
| $73$ |
\( T^{12} - 12 T^{11} + \cdots + 398336 \)
|
| $79$ |
\( T^{12} + \cdots - 13407404032 \)
|
| $83$ |
\( T^{12} + \cdots - 2627101184 \)
|
| $89$ |
\( T^{12} + \cdots + 366901024 \)
|
| $97$ |
\( T^{12} + \cdots + 105555332936 \)
|
| show more | |
| show less |