Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,32,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, 32, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.1");
S:= CuspForms(chi, 32);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 32 \) |
| 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: | \(152.192832048\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 5 x^{9} - 15839611600 x^{8} - 45590087880320 x^{7} + \cdots - 25\!\cdots\!32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{39}\cdot 3^{11}\cdot 5^{24} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 5 x^{9} - 15839611600 x^{8} - 45590087880320 x^{7} + \cdots - 25\!\cdots\!32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 53\!\cdots\!07 \nu^{9} + \cdots - 41\!\cdots\!56 ) / 37\!\cdots\!32 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 53\!\cdots\!07 \nu^{9} + \cdots + 33\!\cdots\!52 ) / 23\!\cdots\!52 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 89\!\cdots\!91 \nu^{9} + \cdots - 62\!\cdots\!24 ) / 18\!\cdots\!16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\!\cdots\!29 \nu^{9} + \cdots - 22\!\cdots\!56 ) / 37\!\cdots\!32 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 64\!\cdots\!29 \nu^{9} + \cdots - 47\!\cdots\!16 ) / 47\!\cdots\!04 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 76\!\cdots\!45 \nu^{9} + \cdots - 54\!\cdots\!20 ) / 15\!\cdots\!68 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 35\!\cdots\!17 \nu^{9} + \cdots - 23\!\cdots\!76 ) / 34\!\cdots\!12 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 84\!\cdots\!05 \nu^{9} + \cdots + 63\!\cdots\!96 ) / 37\!\cdots\!32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 16\beta_{2} + 4325\beta _1 + 3167920152 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - 21\beta_{4} - 3966\beta_{3} - 322486\beta_{2} + 5047086656\beta _1 + 13698262399413 \)
|
| \(\nu^{4}\) | \(=\) |
\( 35 \beta_{9} + 164 \beta_{8} - 979 \beta_{7} + 1137 \beta_{6} - 97127 \beta_{5} + 140999 \beta_{4} + \cdots + 15\!\cdots\!62 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 1513099 \beta_{9} + 4189980 \beta_{8} - 129564869 \beta_{7} + 7816414305 \beta_{6} + \cdots + 85\!\cdots\!76 \)
|
| \(\nu^{6}\) | \(=\) |
\( 342139142513 \beta_{9} + 1465860813164 \beta_{8} - 6025423461505 \beta_{7} + 21099238069205 \beta_{6} + \cdots + 92\!\cdots\!44 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 20\!\cdots\!83 \beta_{9} + \cdots + 45\!\cdots\!00 \)
|
| \(\nu^{8}\) | \(=\) |
\( 26\!\cdots\!29 \beta_{9} + \cdots + 56\!\cdots\!08 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 19\!\cdots\!07 \beta_{9} + \cdots + 22\!\cdots\!24 \)
|
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 |
|
−77381.3 | −1.11024e7 | 3.84039e9 | 0 | 8.59116e11 | 1.42766e13 | −1.30999e14 | −4.94411e14 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −74584.1 | 2.87502e7 | 3.41531e9 | 0 | −2.14431e12 | −2.26849e13 | −9.45597e13 | 2.08899e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −50468.3 | −3.54185e7 | 3.99569e8 | 0 | 1.78751e12 | −7.37897e12 | 8.82143e13 | 6.36795e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −32135.4 | 2.90388e7 | −1.11480e9 | 0 | −9.33176e11 | 1.18783e13 | 1.04835e14 | 2.25580e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −7493.91 | −1.11199e7 | −2.09132e9 | 0 | 8.33318e10 | −7.35420e12 | 3.17653e13 | −4.94020e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 32195.1 | −4.81231e7 | −1.11096e9 | 0 | −1.54933e12 | 1.83301e13 | −1.04906e14 | 1.69816e15 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 38499.8 | 3.74766e7 | −6.65246e8 | 0 | 1.44284e12 | −1.41937e13 | −1.08290e14 | 7.86820e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 39249.7 | 1.97176e6 | −6.06943e8 | 0 | 7.73911e10 | 1.05331e13 | −1.08110e14 | −6.13786e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 73139.2 | −2.03600e7 | 3.20186e9 | 0 | −1.48911e12 | −1.35439e13 | 7.71159e13 | −2.03144e14 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 84554.3 | 2.61258e7 | 5.00194e9 | 0 | 2.20905e12 | 2.00250e13 | 2.41357e14 | 6.48826e13 | 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 | 25.32.a.e | yes | 10 |
| 5.b | even | 2 | 1 | 25.32.a.d | ✓ | 10 | |
| 5.c | odd | 4 | 2 | 25.32.b.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.32.a.d | ✓ | 10 | 5.b | even | 2 | 1 | |
| 25.32.a.e | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 25.32.b.d | 20 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 25575 T_{2}^{9} - 15545275330 T_{2}^{8} + 367724526454800 T_{2}^{7} + \cdots - 21\!\cdots\!76 \)
acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(25))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 21\!\cdots\!76 \)
|
| $3$ |
\( T^{10} + \cdots - 69\!\cdots\!49 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 15\!\cdots\!76 \)
|
| $11$ |
\( T^{10} + \cdots - 50\!\cdots\!01 \)
|
| $13$ |
\( T^{10} + \cdots - 83\!\cdots\!24 \)
|
| $17$ |
\( T^{10} + \cdots + 80\!\cdots\!49 \)
|
| $19$ |
\( T^{10} + \cdots - 11\!\cdots\!25 \)
|
| $23$ |
\( T^{10} + \cdots - 56\!\cdots\!24 \)
|
| $29$ |
\( T^{10} + \cdots + 95\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots - 53\!\cdots\!76 \)
|
| $37$ |
\( T^{10} + \cdots + 37\!\cdots\!24 \)
|
| $41$ |
\( T^{10} + \cdots - 49\!\cdots\!51 \)
|
| $43$ |
\( T^{10} + \cdots + 80\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots - 75\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots + 13\!\cdots\!76 \)
|
| $59$ |
\( T^{10} + \cdots + 35\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 68\!\cdots\!24 \)
|
| $67$ |
\( T^{10} + \cdots + 91\!\cdots\!99 \)
|
| $71$ |
\( T^{10} + \cdots + 24\!\cdots\!24 \)
|
| $73$ |
\( T^{10} + \cdots + 18\!\cdots\!01 \)
|
| $79$ |
\( T^{10} + \cdots + 41\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 11\!\cdots\!51 \)
|
| $89$ |
\( T^{10} + \cdots + 15\!\cdots\!25 \)
|
| $97$ |
\( T^{10} + \cdots + 26\!\cdots\!24 \)
|
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