Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,34,Mod(1,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 34, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 34);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 34 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.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: | \(172.457072203\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - x^{5} - 9680197848 x^{4} + 8052423720422 x^{3} + \cdots - 10\!\cdots\!50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{5}\cdot 5^{6}\cdot 7\cdot 11^{2} \) |
| Twist minimal: | no (minimal twist has level 5) |
| 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
\( x^{6} - x^{5} - 9680197848 x^{4} + 8052423720422 x^{3} + \cdots - 10\!\cdots\!50 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 64759 \nu^{5} + 178753080 \nu^{4} - 443366379644016 \nu^{3} + \cdots + 31\!\cdots\!22 ) / 11\!\cdots\!72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 64759 \nu^{5} - 178753080 \nu^{4} + 443366379644016 \nu^{3} + \cdots - 15\!\cdots\!34 ) / 11\!\cdots\!72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 693353137 \nu^{5} + 22711573265160 \nu^{4} + \cdots + 90\!\cdots\!78 ) / 50\!\cdots\!44 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2773802021 \nu^{5} - 549722510060376 \nu^{4} + \cdots - 30\!\cdots\!22 ) / 40\!\cdots\!52 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2493\beta _1 + 12906931296 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -20\beta_{5} - 67\beta_{4} + 5000\beta_{3} + 1889408\beta_{2} + 9344083493\beta _1 - 16088601109849 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 2708988 \beta_{5} + 716451 \beta_{4} + 6365565999 \beta_{3} + 21984654807 \beta_{2} + \cdots + 60\!\cdots\!41 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 64725270960 \beta_{5} - 230343390564 \beta_{4} + 25466060786969 \beta_{3} + \cdots - 14\!\cdots\!00 ) / 2 \)
|
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 |
|
−178863. | −8.55112e7 | 2.34019e10 | 0 | 1.52948e13 | 8.09549e13 | −2.64931e15 | 1.75311e15 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −132922. | 1.17526e8 | 9.07819e9 | 0 | −1.56217e13 | 1.34853e13 | −6.49000e13 | 8.25322e15 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −76409.2 | −7.43191e7 | −2.75157e9 | 0 | 5.67866e12 | −1.36987e14 | 8.66595e14 | −3.57381e13 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 18258.4 | 3.67420e7 | −8.25656e9 | 0 | 6.70852e11 | 1.51681e13 | −3.07591e14 | −4.20908e15 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 96184.0 | −1.07272e8 | 6.61431e8 | 0 | −1.03179e13 | 8.61302e13 | −7.62595e14 | 5.94832e15 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 126401. | 8.63211e7 | 7.38725e9 | 0 | 1.09111e13 | −7.78650e13 | −1.52021e14 | 1.89228e15 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
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 | 25.34.a.c | 6 | |
| 5.b | even | 2 | 1 | 5.34.a.b | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 25.34.b.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.34.a.b | ✓ | 6 | 5.b | even | 2 | 1 | |
| 25.34.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 25.34.b.c | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 147350 T_{2}^{5} - 29674115352 T_{2}^{4} + \cdots - 40\!\cdots\!04 \)
acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 40\!\cdots\!04 \)
|
| $3$ |
\( T^{6} + \cdots - 25\!\cdots\!36 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 15\!\cdots\!56 \)
|
| $11$ |
\( T^{6} + \cdots + 24\!\cdots\!24 \)
|
| $13$ |
\( T^{6} + \cdots + 16\!\cdots\!84 \)
|
| $17$ |
\( T^{6} + \cdots - 40\!\cdots\!24 \)
|
| $19$ |
\( T^{6} + \cdots - 28\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 66\!\cdots\!96 \)
|
| $29$ |
\( T^{6} + \cdots - 16\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 11\!\cdots\!44 \)
|
| $37$ |
\( T^{6} + \cdots + 71\!\cdots\!16 \)
|
| $41$ |
\( T^{6} + \cdots + 96\!\cdots\!04 \)
|
| $43$ |
\( T^{6} + \cdots + 72\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots + 21\!\cdots\!36 \)
|
| $53$ |
\( T^{6} + \cdots + 68\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots + 95\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 77\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots + 17\!\cdots\!76 \)
|
| $71$ |
\( T^{6} + \cdots + 14\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots + 40\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots - 48\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 12\!\cdots\!76 \)
|
| $89$ |
\( T^{6} + \cdots + 81\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 94\!\cdots\!64 \)
|
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