Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [100,7,Mod(57,100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("100.57");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 100.f (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: | \(23.0054083620\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.691798081536.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{2}\cdot 5^{4} \) |
| 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
\( x^{8} - 4x^{7} + 26x^{6} - 64x^{5} + 229x^{4} - 356x^{3} + 164x^{2} + 4x + 985 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{6} - 6\nu^{5} + 55\nu^{4} - 100\nu^{3} + 443\nu^{2} - 394\nu + 140 ) / 975 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{6} + 24\nu^{5} - 220\nu^{4} + 400\nu^{3} - 3072\nu^{2} + 2876\nu - 7060 ) / 325 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 76\nu^{6} - 228\nu^{5} + 1440\nu^{4} - 2500\nu^{3} + 9684\nu^{2} - 8472\nu - 16780 ) / 975 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 58\nu^{7} - 203\nu^{6} + 447\nu^{5} - 610\nu^{4} - 5203\nu^{3} + 8313\nu^{2} - 108592\nu + 52895 ) / 19695 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 76\nu^{7} - 266\nu^{6} + 1944\nu^{5} - 4195\nu^{4} + 18084\nu^{3} - 23064\nu^{2} + 6891\nu + 265 ) / 6565 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 1824 \nu^{7} - 6384 \nu^{6} + 46656 \nu^{5} - 100680 \nu^{4} + 434016 \nu^{3} - 553536 \nu^{2} + \cdots - 387540 ) / 6565 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -32\nu^{7} + 112\nu^{6} - 776\nu^{5} + 1660\nu^{4} - 5488\nu^{3} + 6628\nu^{2} + 9176\nu - 5640 ) / 101 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} - 24\beta_{5} + 60 ) / 120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - 24\beta_{5} - 30\beta_{2} - 360\beta _1 - 540 ) / 120 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{7} - 4\beta_{6} + 324\beta_{5} + 72\beta_{4} - 45\beta_{2} - 540\beta _1 - 840 ) / 120 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6\beta_{7} - 3\beta_{6} + 224\beta_{5} + 48\beta_{4} - 60\beta_{3} + 80\beta_{2} + 3240\beta _1 + 260 ) / 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 120 \beta_{7} - 34 \beta_{6} - 1644 \beta_{5} - 3240 \beta_{4} - 450 \beta_{3} + 675 \beta_{2} + \cdots + 3360 ) / 120 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 405 \beta_{7} - 79 \beta_{6} - 6624 \beta_{5} - 10080 \beta_{4} + 3600 \beta_{3} - 180 \beta_{2} + \cdots + 69660 ) / 120 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 504 \beta_{7} + 1261 \beta_{6} - 15876 \beta_{5} + 38412 \beta_{4} + 14175 \beta_{3} - 3045 \beta_{2} + \cdots + 231060 ) / 120 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(77\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 57.1 |
|
0 | −27.4779 | + | 27.4779i | 0 | 0 | 0 | 67.2032 | + | 67.2032i | 0 | − | 781.068i | 0 | |||||||||||||||||||||||||||||||||||||
| 57.2 | 0 | −15.2304 | + | 15.2304i | 0 | 0 | 0 | 18.2134 | + | 18.2134i | 0 | 265.068i | 0 | |||||||||||||||||||||||||||||||||||||||
| 57.3 | 0 | 15.2304 | − | 15.2304i | 0 | 0 | 0 | −18.2134 | − | 18.2134i | 0 | 265.068i | 0 | |||||||||||||||||||||||||||||||||||||||
| 57.4 | 0 | 27.4779 | − | 27.4779i | 0 | 0 | 0 | −67.2032 | − | 67.2032i | 0 | − | 781.068i | 0 | ||||||||||||||||||||||||||||||||||||||
| 93.1 | 0 | −27.4779 | − | 27.4779i | 0 | 0 | 0 | 67.2032 | − | 67.2032i | 0 | 781.068i | 0 | |||||||||||||||||||||||||||||||||||||||
| 93.2 | 0 | −15.2304 | − | 15.2304i | 0 | 0 | 0 | 18.2134 | − | 18.2134i | 0 | − | 265.068i | 0 | ||||||||||||||||||||||||||||||||||||||
| 93.3 | 0 | 15.2304 | + | 15.2304i | 0 | 0 | 0 | −18.2134 | + | 18.2134i | 0 | − | 265.068i | 0 | ||||||||||||||||||||||||||||||||||||||
| 93.4 | 0 | 27.4779 | + | 27.4779i | 0 | 0 | 0 | −67.2032 | + | 67.2032i | 0 | 781.068i | 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 | 100.7.f.c | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 100.7.f.c | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 100.7.f.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 100.7.f.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 100.7.f.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 100.7.f.c | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 2495538T_{3}^{4} + 490796923761 \)
acting on \(S_{7}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 490796923761 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 35912501231616 \)
|
| $11$ |
\( (T^{2} + 1230 T - 2973375)^{4} \)
|
| $13$ |
\( T^{8} + \cdots + 42\!\cdots\!76 \)
|
| $17$ |
\( T^{8} + \cdots + 10\!\cdots\!21 \)
|
| $19$ |
\( (T^{4} + \cdots + 520531936498561)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 10\!\cdots\!56 \)
|
| $29$ |
\( (T^{4} + \cdots + 28\!\cdots\!56)^{2} \)
|
| $31$ |
\( (T^{2} - 1892 T - 1538105084)^{4} \)
|
| $37$ |
\( T^{8} + \cdots + 55\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + 54318 T - 842702319)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 70\!\cdots\!16 \)
|
| $53$ |
\( T^{8} + \cdots + 23\!\cdots\!36 \)
|
| $59$ |
\( (T^{4} + \cdots + 11\!\cdots\!16)^{2} \)
|
| $61$ |
\( (T^{2} - 63064 T - 4600579376)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 16\!\cdots\!21 \)
|
| $71$ |
\( (T^{2} - 63636 T - 179384133276)^{4} \)
|
| $73$ |
\( T^{8} + \cdots + 65\!\cdots\!01 \)
|
| $79$ |
\( (T^{4} + \cdots + 35\!\cdots\!16)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 30\!\cdots\!41 \)
|
| $89$ |
\( (T^{4} + \cdots + 21\!\cdots\!41)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 22\!\cdots\!76 \)
|
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