Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,6,Mod(1,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.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: | \(132.316651346\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 5 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 44 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2\nu^{2} - 70\nu - 44 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 14\nu^{5} + 164\nu^{4} + 1600\nu^{3} - 7927\nu^{2} - 36334\nu + 55536 ) / 160 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 160\nu^{5} + 176\nu^{4} + 7407\nu^{3} - 13084\nu^{2} - 85572\nu + 99104 ) / 320 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} - 188\nu^{5} + 344\nu^{4} + 10287\nu^{3} - 15018\nu^{2} - 146720\nu + 89856 ) / 320 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} - 4\nu^{6} + 504\nu^{5} + 208\nu^{4} - 24861\nu^{3} + 6664\nu^{2} + 314100\nu - 129568 ) / 320 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 44 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 2\beta_{2} + 72\beta _1 - 44 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 91\beta_{2} - 159\beta _1 + 3164 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{7} + 2\beta_{6} + 10\beta_{5} - 10\beta_{4} + 115\beta_{3} - 306\beta_{2} + 5750\beta _1 - 6972 \)
|
| \(\nu^{6}\) | \(=\) |
\( -56\beta_{7} - 356\beta_{6} + 188\beta_{5} + 308\beta_{4} - 338\beta_{3} + 8081\beta_{2} - 19783\beta _1 + 252852 \)
|
| \(\nu^{7}\) | \(=\) |
\( 640 \beta_{7} + 672 \beta_{6} + 1568 \beta_{5} - 1952 \beta_{4} + 11345 \beta_{3} - 37078 \beta_{2} + \cdots - 869884 \)
|
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 |
|
−9.03274 | −9.00000 | 49.5905 | 0 | 81.2947 | −36.7509 | −158.890 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.09383 | −9.00000 | 18.3224 | 0 | 63.8444 | 4.62438 | 97.0268 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.15857 | −9.00000 | −22.0234 | 0 | 28.4272 | −35.7175 | 170.637 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.29244 | −9.00000 | −30.3296 | 0 | −11.6320 | 91.7671 | −80.5574 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.89416 | −9.00000 | −28.4122 | 0 | −17.0474 | −191.987 | −114.430 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6.43413 | −9.00000 | 9.39797 | 0 | −57.9071 | −9.96708 | −145.424 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.96197 | −9.00000 | 48.3168 | 0 | −80.6577 | 191.454 | 146.231 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 9.70245 | −9.00000 | 62.1375 | 0 | −87.3220 | −79.4231 | 292.408 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 825.6.a.q | yes | 8 |
| 5.b | even | 2 | 1 | 825.6.a.p | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 825.6.a.p | ✓ | 8 | 5.b | even | 2 | 1 | |
| 825.6.a.q | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 9T_{2}^{7} - 141T_{2}^{6} + 1251T_{2}^{5} + 5160T_{2}^{4} - 45922T_{2}^{3} - 22268T_{2}^{2} + 305880T_{2} - 277200 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 9 T^{7} + \cdots - 277200 \)
|
| $3$ |
\( (T + 9)^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots - 16208419192764 \)
|
| $11$ |
\( (T + 121)^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 29\!\cdots\!92 \)
|
| $17$ |
\( T^{8} + \cdots - 29\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots - 27\!\cdots\!23 \)
|
| $23$ |
\( T^{8} + \cdots - 31\!\cdots\!28 \)
|
| $29$ |
\( T^{8} + \cdots + 72\!\cdots\!84 \)
|
| $31$ |
\( T^{8} + \cdots - 95\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots - 29\!\cdots\!64 \)
|
| $41$ |
\( T^{8} + \cdots - 33\!\cdots\!96 \)
|
| $43$ |
\( T^{8} + \cdots + 68\!\cdots\!72 \)
|
| $47$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots - 55\!\cdots\!44 \)
|
| $59$ |
\( T^{8} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots - 59\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 39\!\cdots\!88 \)
|
| $73$ |
\( T^{8} + \cdots + 59\!\cdots\!56 \)
|
| $79$ |
\( T^{8} + \cdots + 21\!\cdots\!12 \)
|
| $83$ |
\( T^{8} + \cdots + 27\!\cdots\!00 \)
|
| $89$ |
\( T^{8} + \cdots + 21\!\cdots\!32 \)
|
| $97$ |
\( T^{8} + \cdots - 13\!\cdots\!75 \)
|
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