Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [725,4,Mod(1,725)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(725, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("725.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 725 = 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 725.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: | \(42.7763847542\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 28x^{4} + 52x^{3} + 110x^{2} - 272x + 128 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 145) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 28x^{4} + 52x^{3} + 110x^{2} - 272x + 128 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 28\nu^{3} - 36\nu^{2} - 94\nu + 64 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 28\nu^{3} - 52\nu^{2} - 126\nu + 224 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} + 3\nu^{4} - 132\nu^{3} + 52\nu^{2} + 566\nu - 448 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -9\nu^{5} + \nu^{4} + 260\nu^{3} - 228\nu^{2} - 1310\nu + 1280 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - 2\beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 2\beta_{3} - 2\beta_{2} + 19\beta _1 - 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - 14\beta_{3} + 28\beta_{2} - 63\beta _1 + 192 \)
|
| \(\nu^{5}\) | \(=\) |
\( 27\beta_{5} + 29\beta_{4} - 34\beta_{3} - 80\beta_{2} + 447\beta _1 - 552 \)
|
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 |
|
−4.16002 | 7.93167 | 9.30577 | 0 | −32.9959 | 25.4521 | −5.43202 | 35.9114 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.43640 | 0.236828 | −5.93675 | 0 | −0.340181 | 1.74251 | 20.0188 | −26.9439 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.73392 | 4.69344 | −4.99351 | 0 | 8.13806 | −24.1441 | −22.5297 | −4.97161 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.39843 | −7.05006 | −2.24753 | 0 | −16.9091 | 34.7748 | −24.5780 | 22.7033 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.50685 | 10.0154 | 4.29800 | 0 | 35.1224 | 18.1119 | −12.9824 | 73.3075 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.95722 | −2.82724 | 16.5740 | 0 | −14.0153 | 23.0628 | 42.5033 | −19.0067 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(29\) | \( +1 \) |
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 | 725.4.a.e | 6 | |
| 5.b | even | 2 | 1 | 145.4.a.b | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 1305.4.a.k | 6 | ||
| 20.d | odd | 2 | 1 | 2320.4.a.q | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 145.4.a.b | ✓ | 6 | 5.b | even | 2 | 1 | |
| 725.4.a.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 1305.4.a.k | 6 | 15.d | odd | 2 | 1 | ||
| 2320.4.a.q | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 7T_{2}^{5} - 8T_{2}^{4} + 134T_{2}^{3} - 189T_{2}^{2} - 235T_{2} + 432 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(725))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 7 T^{5} + \cdots + 432 \)
|
| $3$ |
\( T^{6} - 13 T^{5} + \cdots + 1760 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 79 T^{5} + \cdots - 15554232 \)
|
| $11$ |
\( T^{6} + 44 T^{5} + \cdots - 825339712 \)
|
| $13$ |
\( T^{6} - 53 T^{5} + \cdots - 709906608 \)
|
| $17$ |
\( T^{6} + \cdots + 98710833232 \)
|
| $19$ |
\( T^{6} + \cdots + 4689647936 \)
|
| $23$ |
\( T^{6} + \cdots - 308235736584 \)
|
| $29$ |
\( (T + 29)^{6} \)
|
| $31$ |
\( T^{6} + \cdots - 263130127648 \)
|
| $37$ |
\( T^{6} + \cdots - 340583216640 \)
|
| $41$ |
\( T^{6} + \cdots + 42378124088000 \)
|
| $43$ |
\( T^{6} + \cdots - 12169261515536 \)
|
| $47$ |
\( T^{6} + \cdots - 411657204692736 \)
|
| $53$ |
\( T^{6} + \cdots + 201884510509264 \)
|
| $59$ |
\( T^{6} + \cdots - 27\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 54\!\cdots\!84 \)
|
| $67$ |
\( T^{6} + \cdots - 47\!\cdots\!12 \)
|
| $71$ |
\( T^{6} + \cdots - 44\!\cdots\!08 \)
|
| $73$ |
\( T^{6} + \cdots + 10\!\cdots\!72 \)
|
| $79$ |
\( T^{6} + \cdots + 660071840350080 \)
|
| $83$ |
\( T^{6} + \cdots + 308459236115232 \)
|
| $89$ |
\( T^{6} + \cdots - 836484908883776 \)
|
| $97$ |
\( T^{6} + \cdots + 11\!\cdots\!28 \)
|
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