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: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 4 x^{10} - 12 x^{9} + 61 x^{8} + 27 x^{7} - 300 x^{6} + 90 x^{5} + 532 x^{4} - 301 x^{3} + \cdots + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 1605) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 4 x^{10} - 12 x^{9} + 61 x^{8} + 27 x^{7} - 300 x^{6} + 90 x^{5} + 532 x^{4} - 301 x^{3} + \cdots + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} + 2\nu^{8} + 14\nu^{7} - 24\nu^{6} - 62\nu^{5} + 73\nu^{4} + 107\nu^{3} - 37\nu^{2} - 59\nu - 9 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3 \nu^{10} - 3 \nu^{9} - 53 \nu^{8} + 40 \nu^{7} + 333 \nu^{6} - 153 \nu^{5} - 905 \nu^{4} + 165 \nu^{3} + \cdots - 208 ) / 20 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - \nu^{10} + \nu^{9} + 21 \nu^{8} - 20 \nu^{7} - 151 \nu^{6} + 131 \nu^{5} + 425 \nu^{4} - 305 \nu^{3} + \cdots + 86 ) / 10 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3 \nu^{10} - 7 \nu^{9} - 45 \nu^{8} + 106 \nu^{7} + 207 \nu^{6} - 511 \nu^{5} - 273 \nu^{4} + 843 \nu^{3} + \cdots - 14 ) / 10 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 3 \nu^{10} - 7 \nu^{9} - 45 \nu^{8} + 106 \nu^{7} + 207 \nu^{6} - 511 \nu^{5} - 263 \nu^{4} + 843 \nu^{3} + \cdots + 76 ) / 10 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 3 \nu^{10} + 9 \nu^{9} + 41 \nu^{8} - 134 \nu^{7} - 149 \nu^{6} + 625 \nu^{5} + 7 \nu^{4} + \cdots - 78 ) / 10 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 3 \nu^{10} + 3 \nu^{9} + 53 \nu^{8} - 40 \nu^{7} - 333 \nu^{6} + 163 \nu^{5} + 905 \nu^{4} + \cdots + 198 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - \beta_{7} + 8\beta_{2} + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + 2\beta_{5} + 10\beta_{3} + 38\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{10} + \beta_{9} + 12\beta_{8} - 11\beta_{7} + 2\beta_{5} + \beta_{4} + 61\beta_{2} - 3\beta _1 + 148 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 26 \beta_{5} + \beta_{4} + 85 \beta_{3} + \cdots + 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( 16 \beta_{10} + 18 \beta_{9} + 111 \beta_{8} - 91 \beta_{7} + 3 \beta_{6} + 30 \beta_{5} + 14 \beta_{4} + \cdots + 1012 \)
|
| \(\nu^{9}\) | \(=\) |
\( 142 \beta_{10} + 54 \beta_{9} + 35 \beta_{8} + 37 \beta_{7} + 6 \beta_{6} + 252 \beta_{5} + 13 \beta_{4} + \cdots + 72 \)
|
| \(\nu^{10}\) | \(=\) |
\( 178 \beta_{10} + 221 \beta_{9} + 939 \beta_{8} - 678 \beta_{7} + 59 \beta_{6} + 322 \beta_{5} + \cdots + 7184 \)
|
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 |
|
−2.73228 | −1.00000 | 5.46536 | 0 | 2.73228 | −0.227337 | −9.46835 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.97847 | −1.00000 | 1.91434 | 0 | 1.97847 | −4.41842 | 0.169484 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.82003 | −1.00000 | 1.31250 | 0 | 1.82003 | 3.55363 | 1.25126 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.473099 | −1.00000 | −1.77618 | 0 | 0.473099 | 3.23567 | 1.78650 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.315773 | −1.00000 | −1.90029 | 0 | 0.315773 | −1.07507 | 1.23160 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.612683 | −1.00000 | −1.62462 | 0 | −0.612683 | 4.75124 | −2.22074 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.09516 | −1.00000 | −0.800632 | 0 | −1.09516 | −1.57557 | −3.06713 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.03436 | −1.00000 | 2.13862 | 0 | −2.03436 | −0.991916 | 0.282009 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.28424 | −1.00000 | 3.21776 | 0 | −2.28424 | −4.79253 | 2.78166 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.49810 | −1.00000 | 4.24051 | 0 | −2.49810 | 4.74700 | 5.59703 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.79511 | −1.00000 | 5.81262 | 0 | −2.79511 | 0.793304 | 10.6567 | 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.bd | 11 | |
| 5.b | even | 2 | 1 | 1605.2.a.m | ✓ | 11 | |
| 15.d | odd | 2 | 1 | 4815.2.a.t | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1605.2.a.m | ✓ | 11 | 5.b | even | 2 | 1 | |
| 4815.2.a.t | 11 | 15.d | odd | 2 | 1 | ||
| 8025.2.a.bd | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8025))\):
|
\( T_{2}^{11} - 4 T_{2}^{10} - 12 T_{2}^{9} + 61 T_{2}^{8} + 27 T_{2}^{7} - 300 T_{2}^{6} + 90 T_{2}^{5} + \cdots + 32 \)
|
|
\( T_{7}^{11} - 4 T_{7}^{10} - 50 T_{7}^{9} + 191 T_{7}^{8} + 799 T_{7}^{7} - 2703 T_{7}^{6} - 4770 T_{7}^{5} + \cdots - 1664 \)
|
|
\( T_{11}^{11} - 13 T_{11}^{10} - 10 T_{11}^{9} + 724 T_{11}^{8} - 1900 T_{11}^{7} - 10639 T_{11}^{6} + \cdots - 15808 \)
|
|
\( T_{13}^{11} - T_{13}^{10} - 108 T_{13}^{9} + 26 T_{13}^{8} + 4139 T_{13}^{7} + 2154 T_{13}^{6} + \cdots - 68992 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 4 T^{10} + \cdots + 32 \)
|
| $3$ |
\( (T + 1)^{11} \)
|
| $5$ |
\( T^{11} \)
|
| $7$ |
\( T^{11} - 4 T^{10} + \cdots - 1664 \)
|
| $11$ |
\( T^{11} - 13 T^{10} + \cdots - 15808 \)
|
| $13$ |
\( T^{11} - T^{10} + \cdots - 68992 \)
|
| $17$ |
\( T^{11} - 7 T^{10} + \cdots + 822608 \)
|
| $19$ |
\( T^{11} - 33 T^{10} + \cdots - 1059712 \)
|
| $23$ |
\( T^{11} - 15 T^{10} + \cdots - 8385536 \)
|
| $29$ |
\( T^{11} - 14 T^{10} + \cdots - 734192 \)
|
| $31$ |
\( T^{11} + \cdots + 225880064 \)
|
| $37$ |
\( T^{11} + 2 T^{10} + \cdots - 48896 \)
|
| $41$ |
\( T^{11} - 32 T^{10} + \cdots + 34112 \)
|
| $43$ |
\( T^{11} - 4 T^{10} + \cdots + 282496 \)
|
| $47$ |
\( T^{11} - 11 T^{10} + \cdots - 1114112 \)
|
| $53$ |
\( T^{11} + \cdots - 889611008 \)
|
| $59$ |
\( T^{11} - 16 T^{10} + \cdots + 1179136 \)
|
| $61$ |
\( T^{11} + \cdots + 3280690424 \)
|
| $67$ |
\( T^{11} + \cdots - 896913664 \)
|
| $71$ |
\( T^{11} + \cdots + 4648315904 \)
|
| $73$ |
\( T^{11} + \cdots + 145852144 \)
|
| $79$ |
\( T^{11} + \cdots + 969932288 \)
|
| $83$ |
\( T^{11} + \cdots - 1800266752 \)
|
| $89$ |
\( T^{11} + \cdots - 5220748528 \)
|
| $97$ |
\( T^{11} + \cdots - 145854832 \)
|
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