Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [50,26,Mod(49,50)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(50, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("50.49");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.b (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(197.998389976\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} + 5878907094 x^{6} + \cdots + 44\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{10}\cdot 5^{14} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 5878907094 x^{6} + \cdots + 44\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 30\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 15\nu^{4} + 44091803205\nu^{2} + 9981754453699758000 ) / 18641220777293 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} + 5878907094 \nu^{5} + \cdots + 33\!\cdots\!96 \nu ) / 24\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 500 \nu^{6} + 2259704459110 \nu^{4} + \cdots - 25\!\cdots\!50 ) / 18\!\cdots\!93 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5000 \nu^{6} + 25452780059335 \nu^{4} + \cdots + 75\!\cdots\!50 ) / 18\!\cdots\!93 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 135853707867217 \nu^{7} + \cdots - 45\!\cdots\!92 \nu ) / 16\!\cdots\!28 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 280566039606923 \nu^{7} + \cdots + 53\!\cdots\!48 \nu ) / 16\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 10\beta_{4} - 190249\beta_{2} - 440918032050 ) / 300 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1763867\beta_{7} - 476597\beta_{6} - 83885493497818500\beta_{3} - 248667639775\beta_1 ) / 3000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2939453547\beta_{5} + 29394535470\beta_{4} + 932052513409063\beta_{2} + 1096422984171637021350 ) / 300 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 89\!\cdots\!49 \beta_{7} + \cdots + 73\!\cdots\!65 \beta_1 ) / 3000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 48\!\cdots\!66 \beta_{5} + \cdots - 16\!\cdots\!50 ) / 1500 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 35\!\cdots\!03 \beta_{7} + \cdots - 23\!\cdots\!35 \beta_1 ) / 3000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 4096.00i | − | 1.71850e6i | −1.67772e7 | 0 | −7.03899e9 | 3.68156e8i | 6.87195e10i | −2.10596e12 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.2 | − | 4096.00i | − | 276233.i | −1.67772e7 | 0 | −1.13145e9 | 2.71222e10i | 6.87195e10i | 7.70984e11 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.3 | − | 4096.00i | 881731.i | −1.67772e7 | 0 | 3.61157e9 | 2.21890e10i | 6.87195e10i | 6.98383e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 49.4 | − | 4096.00i | 1.21936e6i | −1.67772e7 | 0 | 4.99450e9 | − | 3.85845e10i | 6.87195e10i | −6.39550e11 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.5 | 4096.00i | − | 1.21936e6i | −1.67772e7 | 0 | 4.99450e9 | 3.85845e10i | − | 6.87195e10i | −6.39550e11 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.6 | 4096.00i | − | 881731.i | −1.67772e7 | 0 | 3.61157e9 | − | 2.21890e10i | − | 6.87195e10i | 6.98383e10 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 49.7 | 4096.00i | 276233.i | −1.67772e7 | 0 | −1.13145e9 | − | 2.71222e10i | − | 6.87195e10i | 7.70984e11 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 49.8 | 4096.00i | 1.71850e6i | −1.67772e7 | 0 | −7.03899e9 | − | 3.68156e8i | − | 6.87195e10i | −2.10596e12 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.26.b.g | 8 | |
| 5.b | even | 2 | 1 | inner | 50.26.b.g | 8 | |
| 5.c | odd | 4 | 1 | 50.26.a.g | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 50.26.a.h | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.26.a.g | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 50.26.a.h | yes | 4 | 5.c | odd | 4 | 1 | |
| 50.26.b.g | 8 | 1.a | even | 1 | 1 | trivial | |
| 50.26.b.g | 8 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 5293844284284 T_{3}^{6} + \cdots + 26\!\cdots\!81 \)
acting on \(S_{26}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16777216)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 26\!\cdots\!81 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 73\!\cdots\!16 \)
|
| $11$ |
\( (T^{4} + \cdots - 11\!\cdots\!59)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 45\!\cdots\!96 \)
|
| $17$ |
\( T^{8} + \cdots + 10\!\cdots\!81 \)
|
| $19$ |
\( (T^{4} + \cdots - 17\!\cdots\!75)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 29\!\cdots\!36 \)
|
| $29$ |
\( (T^{4} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 31\!\cdots\!16)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 95\!\cdots\!36 \)
|
| $41$ |
\( (T^{4} + \cdots + 26\!\cdots\!41)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 98\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 31\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 16\!\cdots\!56 \)
|
| $59$ |
\( (T^{4} + \cdots - 13\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 73\!\cdots\!84)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 80\!\cdots\!81 \)
|
| $71$ |
\( (T^{4} + \cdots - 15\!\cdots\!84)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 22\!\cdots\!61 \)
|
| $79$ |
\( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 10\!\cdots\!01 \)
|
| $89$ |
\( (T^{4} + \cdots + 74\!\cdots\!25)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
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