Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [100,20,Mod(49,100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(100, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 20, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("100.49");
S:= CuspForms(chi, 20);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 20 \) |
| Character orbit: | \([\chi]\) | \(=\) | 100.c (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: | \(228.816696556\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} + 2703441x^{4} + 1827426135900x^{2} + 19485019126562500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{6}\cdot 5^{8} \) |
| Twist minimal: | no (minimal twist has level 20) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 2703441x^{4} + 1827426135900x^{2} + 19485019126562500 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{5} - 1095082472\nu^{3} - 78680197834472800\nu ) / 1608027851784375 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 136885309\nu^{3} - 186857618602850\nu ) / 77185336885650 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 400\nu^{4} + 540688400\nu^{2} + 74267104000 ) / 4607901 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2752\nu^{4} - 6668992832\nu^{2} - 2658044501641600 ) / 23039505 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4438001684\nu^{5} + 9985463228339644\nu^{3} + 5294344608519167300600\nu ) / 4824083555353125 \)
|
| \(\nu\) | \(=\) |
\( ( 48\beta_{2} - 125\beta_1 ) / 6000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -125\beta_{4} - 172\beta_{3} - 14418352000 ) / 16000 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 375\beta_{5} - 27147011544\beta_{2} + 1351566250\beta_1 ) / 48000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 168965125\beta_{4} + 416812052\beta_{3} + 19486618499632000 ) / 16000 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 51331990875\beta_{5} + 60622432427687304\beta_{2} - 1848054837628750\beta_1 ) / 48000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(77\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 |
|
0 | − | 64511.3i | 0 | 0 | 0 | − | 1.51440e8i | 0 | −2.99945e9 | 0 | ||||||||||||||||||||||||||||||||||
| 49.2 | 0 | − | 46767.8i | 0 | 0 | 0 | 3.25847e7i | 0 | −1.02497e9 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.3 | 0 | − | 16342.5i | 0 | 0 | 0 | 6.88942e7i | 0 | 8.95184e8 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.4 | 0 | 16342.5i | 0 | 0 | 0 | − | 6.88942e7i | 0 | 8.95184e8 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.5 | 0 | 46767.8i | 0 | 0 | 0 | − | 3.25847e7i | 0 | −1.02497e9 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.6 | 0 | 64511.3i | 0 | 0 | 0 | 1.51440e8i | 0 | −2.99945e9 | 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 | 100.20.c.b | 6 | |
| 5.b | even | 2 | 1 | inner | 100.20.c.b | 6 | |
| 5.c | odd | 4 | 1 | 20.20.a.a | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 100.20.a.b | 3 | ||
| 20.e | even | 4 | 1 | 80.20.a.e | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.20.a.a | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 80.20.a.e | 3 | 20.e | even | 4 | 1 | ||
| 100.20.a.b | 3 | 5.c | odd | 4 | 1 | ||
| 100.20.c.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 100.20.c.b | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 6616012428T_{3}^{4} + 10798258892396282928T_{3}^{2} + 2431100481221246994526349376 \)
acting on \(S_{20}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 24\!\cdots\!76 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 11\!\cdots\!56 \)
|
| $11$ |
\( (T^{3} + \cdots - 50\!\cdots\!00)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 19\!\cdots\!36 \)
|
| $17$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $19$ |
\( (T^{3} + \cdots + 18\!\cdots\!44)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 77\!\cdots\!24 \)
|
| $29$ |
\( (T^{3} + \cdots + 15\!\cdots\!24)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots - 10\!\cdots\!96)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 11\!\cdots\!76 \)
|
| $41$ |
\( (T^{3} + \cdots - 95\!\cdots\!76)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 95\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 15\!\cdots\!64 \)
|
| $53$ |
\( T^{6} + \cdots + 61\!\cdots\!24 \)
|
| $59$ |
\( (T^{3} + \cdots + 74\!\cdots\!52)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 17\!\cdots\!72)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 11\!\cdots\!36 \)
|
| $71$ |
\( (T^{3} + \cdots - 20\!\cdots\!12)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 16\!\cdots\!16 \)
|
| $79$ |
\( (T^{3} + \cdots - 12\!\cdots\!72)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 40\!\cdots\!24 \)
|
| $89$ |
\( (T^{3} + \cdots - 60\!\cdots\!24)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 12\!\cdots\!56 \)
|
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