Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8025,2,Mod(1,8025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8025, 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("8025.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8025 = 3 \cdot 5^{2} \cdot 107 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8025.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: | \(64.0799476221\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{13} - x^{12} - 20 x^{11} + 20 x^{10} + 151 x^{9} - 152 x^{8} - 532 x^{7} + 540 x^{6} + 877 x^{5} + \cdots - 96 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{13} - x^{12} - 20 x^{11} + 20 x^{10} + 151 x^{9} - 152 x^{8} - 532 x^{7} + 540 x^{6} + 877 x^{5} + \cdots - 96 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{12} - 2 \nu^{11} + 19 \nu^{10} + 32 \nu^{9} - 135 \nu^{8} - 183 \nu^{7} + 438 \nu^{6} + \cdots - 2 ) / 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{12} + 2 \nu^{11} - 19 \nu^{10} - 32 \nu^{9} + 145 \nu^{8} + 183 \nu^{7} - 558 \nu^{6} - 444 \nu^{5} + \cdots + 122 ) / 10 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{12} + 2 \nu^{11} - 19 \nu^{10} - 32 \nu^{9} + 145 \nu^{8} + 183 \nu^{7} - 568 \nu^{6} - 444 \nu^{5} + \cdots + 212 ) / 10 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3 \nu^{12} + \nu^{11} - 52 \nu^{10} - 6 \nu^{9} + 335 \nu^{8} - 36 \nu^{7} - 984 \nu^{6} + 288 \nu^{5} + \cdots + 86 ) / 10 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - \nu^{12} - \nu^{11} + 18 \nu^{10} + 16 \nu^{9} - 123 \nu^{8} - 90 \nu^{7} + 400 \nu^{6} + 208 \nu^{5} + \cdots - 56 ) / 4 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( \nu^{12} - \nu^{11} - 20 \nu^{10} + 16 \nu^{9} + 147 \nu^{8} - 92 \nu^{7} - 484 \nu^{6} + 224 \nu^{5} + \cdots + 56 ) / 4 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 2 \nu^{12} - \nu^{11} - 43 \nu^{10} + 16 \nu^{9} + 340 \nu^{8} - 99 \nu^{7} - 1216 \nu^{6} + 292 \nu^{5} + \cdots + 174 ) / 10 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( \nu^{12} + 2 \nu^{11} - 19 \nu^{10} - 32 \nu^{9} + 140 \nu^{8} + 183 \nu^{7} - 503 \nu^{6} - 444 \nu^{5} + \cdots + 102 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 7\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{12} - \beta_{7} + \beta_{5} + 8\beta_{3} + 28\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + 9\beta_{4} + 43\beta_{2} + \beta _1 + 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 10 \beta_{7} + 10 \beta_{5} + 54 \beta_{3} + \cdots + 13 \)
|
| \(\nu^{8}\) | \(=\) |
\( -\beta_{12} - 11\beta_{7} + 13\beta_{6} + 64\beta_{4} + 259\beta_{2} + 15\beta _1 + 424 \)
|
| \(\nu^{9}\) | \(=\) |
\( 89 \beta_{12} - 12 \beta_{11} + 12 \beta_{10} + 14 \beta_{9} + \beta_{8} - 75 \beta_{7} - \beta_{6} + \cdots + 124 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 16 \beta_{12} - \beta_{11} - 2 \beta_{9} - 87 \beta_{7} + 116 \beta_{6} + 424 \beta_{4} + 1561 \beta_{2} + \cdots + 2460 \)
|
| \(\nu^{11}\) | \(=\) |
\( 643 \beta_{12} - 100 \beta_{11} + 99 \beta_{10} + 133 \beta_{9} + 16 \beta_{8} - 509 \beta_{7} + \cdots + 1032 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 166 \beta_{12} - 20 \beta_{11} + 3 \beta_{10} - 39 \beta_{9} - 612 \beta_{7} + 891 \beta_{6} + \cdots + 14492 \)
|
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 |
|
−2.47451 | 1.00000 | 4.12319 | 0 | −2.47451 | 1.50133 | −5.25384 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.34947 | 1.00000 | 3.52000 | 0 | −2.34947 | 0.312848 | −3.57119 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.22466 | 1.00000 | 2.94910 | 0 | −2.22466 | −0.678665 | −2.11142 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.30851 | 1.00000 | −0.287814 | 0 | −1.30851 | −1.21378 | 2.99362 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.21811 | 1.00000 | −0.516202 | 0 | −1.21811 | 4.09599 | 3.06502 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.473982 | 1.00000 | −1.77534 | 0 | −0.473982 | −3.11833 | 1.78944 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.472223 | 1.00000 | −1.77701 | 0 | 0.472223 | −4.19512 | −1.78359 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.966938 | 1.00000 | −1.06503 | 0 | 0.966938 | 4.46829 | −2.96370 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.18958 | 1.00000 | −0.584901 | 0 | 1.18958 | 2.16419 | −3.07494 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.53993 | 1.00000 | 0.371375 | 0 | 1.53993 | −0.653117 | −2.50796 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.87567 | 1.00000 | 1.51815 | 0 | 1.87567 | −4.26988 | −0.903796 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.49984 | 1.00000 | 4.24921 | 0 | 2.49984 | 3.77753 | 5.62266 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.50505 | 1.00000 | 4.27528 | 0 | 2.50505 | −0.191292 | 5.69970 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(107\) | \(-1\) |
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 | 8025.2.a.bh | yes | 13 |
| 5.b | even | 2 | 1 | 8025.2.a.bg | ✓ | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8025.2.a.bg | ✓ | 13 | 5.b | even | 2 | 1 | |
| 8025.2.a.bh | yes | 13 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):
|
\( T_{2}^{13} - T_{2}^{12} - 20 T_{2}^{11} + 20 T_{2}^{10} + 151 T_{2}^{9} - 152 T_{2}^{8} - 532 T_{2}^{7} + \cdots - 96 \)
|
|
\( T_{7}^{13} - 2 T_{7}^{12} - 51 T_{7}^{11} + 91 T_{7}^{10} + 914 T_{7}^{9} - 1365 T_{7}^{8} - 6731 T_{7}^{7} + \cdots + 404 \)
|
|
\( T_{11}^{13} + 2 T_{11}^{12} - 59 T_{11}^{11} - 136 T_{11}^{10} + 924 T_{11}^{9} + 1688 T_{11}^{8} + \cdots + 150 \)
|
|
\( T_{13}^{13} - T_{13}^{12} - 85 T_{13}^{11} + 85 T_{13}^{10} + 2498 T_{13}^{9} - 3915 T_{13}^{8} + \cdots + 547932 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} - T^{12} + \cdots - 96 \)
|
| $3$ |
\( (T - 1)^{13} \)
|
| $5$ |
\( T^{13} \)
|
| $7$ |
\( T^{13} - 2 T^{12} + \cdots + 404 \)
|
| $11$ |
\( T^{13} + 2 T^{12} + \cdots + 150 \)
|
| $13$ |
\( T^{13} - T^{12} + \cdots + 547932 \)
|
| $17$ |
\( T^{13} - 8 T^{12} + \cdots + 5265016 \)
|
| $19$ |
\( T^{13} - T^{12} + \cdots - 30413651 \)
|
| $23$ |
\( T^{13} - 28 T^{12} + \cdots + 619200 \)
|
| $29$ |
\( T^{13} - 23 T^{12} + \cdots - 8706452 \)
|
| $31$ |
\( T^{13} + 12 T^{12} + \cdots - 71494920 \)
|
| $37$ |
\( T^{13} - 19 T^{12} + \cdots - 3619000 \)
|
| $41$ |
\( T^{13} + \cdots + 915409800 \)
|
| $43$ |
\( T^{13} - 19 T^{12} + \cdots + 41865992 \)
|
| $47$ |
\( T^{13} + \cdots - 678818844 \)
|
| $53$ |
\( T^{13} - 19 T^{12} + \cdots - 87408 \)
|
| $59$ |
\( T^{13} + \cdots + 1446442528 \)
|
| $61$ |
\( T^{13} + \cdots - 2783207273 \)
|
| $67$ |
\( T^{13} - 37 T^{12} + \cdots + 17620800 \)
|
| $71$ |
\( T^{13} + \cdots + 849631000 \)
|
| $73$ |
\( T^{13} + \cdots - 9616057200 \)
|
| $79$ |
\( T^{13} + \cdots + 63286670300 \)
|
| $83$ |
\( T^{13} + \cdots + 2889025174 \)
|
| $89$ |
\( T^{13} + \cdots + 287697090 \)
|
| $97$ |
\( T^{13} + \cdots + 367564681032 \)
|
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