Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [25,7,Mod(7,25)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25.7");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 25 = 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 25.c (of order \(4\), degree \(2\), 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: | \(5.75135209050\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.333061916000256.23 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 90x^{6} - 12x^{5} + 3011x^{4} + 528x^{3} + 41202x^{2} + 17580x + 243850 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3^{2}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 90x^{6} - 12x^{5} + 3011x^{4} + 528x^{3} + 41202x^{2} + 17580x + 243850 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 150 \nu^{7} - 462 \nu^{6} + 10174 \nu^{5} - 53319 \nu^{4} + 232394 \nu^{3} - 1499697 \nu^{2} + \cdots - 13362675 ) / 5194175 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 282286 \nu^{7} - 15339522 \nu^{6} + 35907577 \nu^{5} - 1002858672 \nu^{4} + 1947856606 \nu^{3} + \cdots - 99219003000 ) / 8378204275 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 121302 \nu^{7} - 349419 \nu^{6} - 14676598 \nu^{5} - 37861257 \nu^{4} - 662937606 \nu^{3} + \cdots - 13779356625 ) / 1675640855 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 963177 \nu^{7} - 1768660 \nu^{6} + 45747936 \nu^{5} - 88514646 \nu^{4} + 488036934 \nu^{3} + \cdots + 98741527500 ) / 8378204275 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 679482 \nu^{7} + 81620 \nu^{6} - 44765601 \nu^{5} + 16276001 \nu^{4} - 929477859 \nu^{3} + \cdots - 1500016825 ) / 1675640855 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3755130 \nu^{7} - 720363 \nu^{6} + 351434266 \nu^{5} - 120109521 \nu^{4} + 12942876266 \nu^{3} + \cdots + 34258391175 ) / 8378204275 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 144 \nu^{7} + 534 \nu^{6} + 12924 \nu^{5} + 34101 \nu^{4} + 351360 \nu^{3} + 846702 \nu^{2} + \cdots + 6566615 ) / 56455 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 3\beta_{3} - 8\beta_1 ) / 15 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 18\beta_{5} + 30\beta_{4} + 45\beta _1 - 338 ) / 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{7} - 23\beta_{6} - 9\beta_{5} - 225\beta_{3} - 45\beta_{2} + 514\beta _1 + 72 ) / 15 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -15\beta_{7} + 6\beta_{6} - 364\beta_{5} - 460\beta_{4} + 36\beta_{3} + 60\beta_{2} - 2028\beta _1 + 2605 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 690 \beta_{7} + 506 \beta_{6} + 2250 \beta_{5} + 450 \beta_{4} + 9588 \beta_{3} + 3375 \beta_{2} + \cdots - 15420 ) / 15 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1406 \beta_{7} - 2025 \beta_{6} + 47088 \beta_{5} + 48780 \beta_{4} - 16380 \beta_{3} - 20700 \beta_{2} + \cdots - 148528 ) / 15 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 34524 \beta_{7} - 6418 \beta_{6} - 201834 \beta_{5} - 70875 \beta_{4} - 337470 \beta_{3} + \cdots + 1211742 ) / 15 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−8.83897 | − | 8.83897i | −29.2322 | + | 29.2322i | 92.2549i | 0 | 516.765 | −106.298 | − | 106.298i | 249.744 | − | 249.744i | − | 980.039i | 0 | |||||||||||||||||||||||||||||||||
| 7.2 | −2.71525 | − | 2.71525i | −16.9847 | + | 16.9847i | − | 49.2549i | 0 | 92.2354 | 383.600 | + | 383.600i | −307.515 | + | 307.515i | 152.039i | 0 | ||||||||||||||||||||||||||||||||||
| 7.3 | 2.71525 | + | 2.71525i | 16.9847 | − | 16.9847i | − | 49.2549i | 0 | 92.2354 | −383.600 | − | 383.600i | 307.515 | − | 307.515i | 152.039i | 0 | ||||||||||||||||||||||||||||||||||
| 7.4 | 8.83897 | + | 8.83897i | 29.2322 | − | 29.2322i | 92.2549i | 0 | 516.765 | 106.298 | + | 106.298i | −249.744 | + | 249.744i | − | 980.039i | 0 | ||||||||||||||||||||||||||||||||||
| 18.1 | −8.83897 | + | 8.83897i | −29.2322 | − | 29.2322i | − | 92.2549i | 0 | 516.765 | −106.298 | + | 106.298i | 249.744 | + | 249.744i | 980.039i | 0 | ||||||||||||||||||||||||||||||||||
| 18.2 | −2.71525 | + | 2.71525i | −16.9847 | − | 16.9847i | 49.2549i | 0 | 92.2354 | 383.600 | − | 383.600i | −307.515 | − | 307.515i | − | 152.039i | 0 | ||||||||||||||||||||||||||||||||||
| 18.3 | 2.71525 | − | 2.71525i | 16.9847 | + | 16.9847i | 49.2549i | 0 | 92.2354 | −383.600 | + | 383.600i | 307.515 | + | 307.515i | − | 152.039i | 0 | ||||||||||||||||||||||||||||||||||
| 18.4 | 8.83897 | − | 8.83897i | 29.2322 | + | 29.2322i | − | 92.2549i | 0 | 516.765 | 106.298 | − | 106.298i | −249.744 | − | 249.744i | 980.039i | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 25.7.c.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 225.7.g.f | 8 | ||
| 5.b | even | 2 | 1 | inner | 25.7.c.d | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 25.7.c.d | ✓ | 8 |
| 15.d | odd | 2 | 1 | 225.7.g.f | 8 | ||
| 15.e | even | 4 | 2 | 225.7.g.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 25.7.c.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 25.7.c.d | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 25.7.c.d | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
| 225.7.g.f | 8 | 3.b | odd | 2 | 1 | ||
| 225.7.g.f | 8 | 15.d | odd | 2 | 1 | ||
| 225.7.g.f | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 24633T_{2}^{4} + 5308416 \)
acting on \(S_{7}^{\mathrm{new}}(25, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 24633 T^{4} + 5308416 \)
|
| $3$ |
\( T^{8} + \cdots + 972292630401 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 44\!\cdots\!16 \)
|
| $11$ |
\( (T^{2} - 1974 T - 1028331)^{4} \)
|
| $13$ |
\( T^{8} + \cdots + 12\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots + 24\!\cdots\!41 \)
|
| $19$ |
\( (T^{4} + \cdots + 685239255555625)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 86\!\cdots\!96 \)
|
| $29$ |
\( (T^{4} + \cdots + 220068324090000)^{2} \)
|
| $31$ |
\( (T^{2} - 27164 T + 176460724)^{4} \)
|
| $37$ |
\( T^{8} + \cdots + 56\!\cdots\!16 \)
|
| $41$ |
\( (T^{2} - 98034 T + 2274506289)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 16\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 78\!\cdots\!16 \)
|
| $53$ |
\( T^{8} + \cdots + 71\!\cdots\!76 \)
|
| $59$ |
\( (T^{4} + \cdots + 49\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} - 98224 T - 31430261456)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 14\!\cdots\!41 \)
|
| $71$ |
\( (T^{2} - 203844 T - 28860905916)^{4} \)
|
| $73$ |
\( T^{8} + \cdots + 20\!\cdots\!21 \)
|
| $79$ |
\( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 12\!\cdots\!81 \)
|
| $89$ |
\( (T^{4} + \cdots + 17\!\cdots\!25)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 38\!\cdots\!16 \)
|
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