Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7436,2,Mod(1,7436)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7436, 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("7436.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7436 = 2^{2} \cdot 11 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7436.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: | \(59.3767589430\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 2 x^{11} - 23 x^{10} + 46 x^{9} + 182 x^{8} - 372 x^{7} - 575 x^{6} + 1224 x^{5} + 624 x^{4} + \cdots + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 572) |
| 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
\( x^{12} - 2 x^{11} - 23 x^{10} + 46 x^{9} + 182 x^{8} - 372 x^{7} - 575 x^{6} + 1224 x^{5} + 624 x^{4} + \cdots + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 41 \nu^{11} - 125 \nu^{10} - 515 \nu^{9} + 3232 \nu^{8} - 975 \nu^{7} - 30347 \nu^{6} + 27551 \nu^{5} + \cdots + 32952 ) / 10434 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 43 \nu^{11} + 428 \nu^{10} + 1346 \nu^{9} - 8437 \nu^{8} - 15095 \nu^{7} + 51126 \nu^{6} + \cdots + 29826 ) / 10434 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17 \nu^{11} + 14 \nu^{10} + 413 \nu^{9} - 274 \nu^{8} - 3444 \nu^{7} + 1664 \nu^{6} + 11329 \nu^{5} + \cdots - 1200 ) / 564 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 362 \nu^{11} - 934 \nu^{10} - 7813 \nu^{9} + 20435 \nu^{8} + 56710 \nu^{7} - 157791 \nu^{6} + \cdots + 72846 ) / 10434 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1099 \nu^{11} + 424 \nu^{10} + 24493 \nu^{9} - 10796 \nu^{8} - 182418 \nu^{7} + 101518 \nu^{6} + \cdots - 34812 ) / 20868 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 31 \nu^{11} + 20 \nu^{10} + 731 \nu^{9} - 472 \nu^{8} - 5954 \nu^{7} + 4002 \nu^{6} + 19763 \nu^{5} + \cdots - 828 ) / 564 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 652 \nu^{11} - 345 \nu^{10} + 15273 \nu^{9} + 9129 \nu^{8} - 126160 \nu^{7} - 75967 \nu^{6} + \cdots + 56724 ) / 10434 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 237 \nu^{11} + 256 \nu^{10} + 5437 \nu^{9} - 5534 \nu^{8} - 43565 \nu^{7} + 42493 \nu^{6} + \cdots - 19378 ) / 3478 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 7 \nu^{11} + 3 \nu^{10} + 159 \nu^{9} - 52 \nu^{8} - 1255 \nu^{7} + 323 \nu^{6} + 4123 \nu^{5} + \cdots + 186 ) / 94 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{9} - \beta_{4} + 2\beta_{3} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{11} - 2\beta_{10} - \beta_{9} + 2\beta_{8} - 2\beta_{5} + \beta_{4} + \beta_{3} + 11\beta_{2} - \beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15 \beta_{11} - \beta_{10} - 14 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \beta_{6} - 13 \beta_{4} + \cdots + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 32 \beta_{11} - 29 \beta_{10} - 17 \beta_{9} + 30 \beta_{8} - 4 \beta_{7} - \beta_{6} - 26 \beta_{5} + \cdots + 216 \)
|
| \(\nu^{7}\) | \(=\) |
\( 176 \beta_{11} - 27 \beta_{10} - 156 \beta_{9} + 40 \beta_{8} - 32 \beta_{7} + 13 \beta_{6} - 4 \beta_{5} + \cdots + 59 \)
|
| \(\nu^{8}\) | \(=\) |
\( 407 \beta_{11} - 334 \beta_{10} - 228 \beta_{9} + 348 \beta_{8} - 76 \beta_{7} - 16 \beta_{6} + \cdots + 1903 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1923 \beta_{11} - 435 \beta_{10} - 1639 \beta_{9} + 582 \beta_{8} - 400 \beta_{7} + 133 \beta_{6} + \cdots + 1067 \)
|
| \(\nu^{10}\) | \(=\) |
\( 4793 \beta_{11} - 3584 \beta_{10} - 2817 \beta_{9} + 3744 \beta_{8} - 1044 \beta_{7} - 182 \beta_{6} + \cdots + 17740 \)
|
| \(\nu^{11}\) | \(=\) |
\( 20524 \beta_{11} - 5828 \beta_{10} - 16971 \beta_{9} + 7436 \beta_{8} - 4594 \beta_{7} + 1274 \beta_{6} + \cdots + 15749 \)
|
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 | −2.92122 | 0 | 2.68122 | 0 | 2.37957 | 0 | 5.53350 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.88955 | 0 | −0.307799 | 0 | 1.73849 | 0 | 5.34950 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.98897 | 0 | −2.64929 | 0 | −0.190541 | 0 | 0.955984 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.43274 | 0 | 2.80197 | 0 | −5.18181 | 0 | −0.947266 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.393335 | 0 | −4.15244 | 0 | −2.32766 | 0 | −2.84529 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.131989 | 0 | 0.505374 | 0 | −3.87828 | 0 | −2.98258 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.617973 | 0 | 3.59217 | 0 | 1.46848 | 0 | −2.61811 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.56042 | 0 | −0.661534 | 0 | 4.56146 | 0 | −0.565103 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.81802 | 0 | −1.10453 | 0 | −0.452124 | 0 | 0.305195 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.15388 | 0 | 3.60411 | 0 | −4.09242 | 0 | 1.63918 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.32788 | 0 | −3.47989 | 0 | −1.04773 | 0 | 2.41902 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.27963 | 0 | −0.829371 | 0 | −0.977430 | 0 | 7.75597 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
| \(13\) | \(-1\) |
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 | 7436.2.a.u | 12 | |
| 13.b | even | 2 | 1 | 7436.2.a.v | 12 | ||
| 13.f | odd | 12 | 2 | 572.2.p.a | ✓ | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 572.2.p.a | ✓ | 24 | 13.f | odd | 12 | 2 | |
| 7436.2.a.u | 12 | 1.a | even | 1 | 1 | trivial | |
| 7436.2.a.v | 12 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7436))\):
|
\( T_{3}^{12} - 2 T_{3}^{11} - 23 T_{3}^{10} + 46 T_{3}^{9} + 182 T_{3}^{8} - 372 T_{3}^{7} - 575 T_{3}^{6} + \cdots + 36 \)
|
|
\( T_{5}^{12} - 40 T_{5}^{10} + 564 T_{5}^{8} + 72 T_{5}^{7} - 3301 T_{5}^{6} - 1392 T_{5}^{5} + 6814 T_{5}^{4} + \cdots - 351 \)
|
|
\( T_{7}^{12} + 8 T_{7}^{11} - 17 T_{7}^{10} - 252 T_{7}^{9} - 194 T_{7}^{8} + 2042 T_{7}^{7} + 2797 T_{7}^{6} + \cdots + 468 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 2 T^{11} + \cdots + 36 \)
|
| $5$ |
\( T^{12} - 40 T^{10} + \cdots - 351 \)
|
| $7$ |
\( T^{12} + 8 T^{11} + \cdots + 468 \)
|
| $11$ |
\( (T + 1)^{12} \)
|
| $13$ |
\( T^{12} \)
|
| $17$ |
\( T^{12} - 108 T^{10} + \cdots + 140253 \)
|
| $19$ |
\( T^{12} + 22 T^{11} + \cdots - 236172 \)
|
| $23$ |
\( T^{12} - 10 T^{11} + \cdots - 937728 \)
|
| $29$ |
\( T^{12} - 8 T^{11} + \cdots - 1007667 \)
|
| $31$ |
\( T^{12} + 28 T^{11} + \cdots - 1139904 \)
|
| $37$ |
\( T^{12} - 16 T^{11} + \cdots - 7496211 \)
|
| $41$ |
\( T^{12} + 16 T^{11} + \cdots + 10652877 \)
|
| $43$ |
\( T^{12} - 10 T^{11} + \cdots - 91945152 \)
|
| $47$ |
\( T^{12} + \cdots + 62751861252 \)
|
| $53$ |
\( T^{12} - 8 T^{11} + \cdots + 37188801 \)
|
| $59$ |
\( T^{12} + 32 T^{11} + \cdots - 7569612 \)
|
| $61$ |
\( T^{12} + 6 T^{11} + \cdots + 70416421 \)
|
| $67$ |
\( T^{12} + \cdots - 32852206812 \)
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| $71$ |
\( T^{12} + \cdots - 6499756224 \)
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| $73$ |
\( T^{12} + \cdots + 2436392256 \)
|
| $79$ |
\( T^{12} + 16 T^{11} + \cdots + 1391872 \)
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| $83$ |
\( T^{12} + \cdots - 19048942044 \)
|
| $89$ |
\( T^{12} - 16 T^{11} + \cdots - 77323248 \)
|
| $97$ |
\( T^{12} + \cdots + 10448000784 \)
|
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