Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,12,Mod(49,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.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: | \(57.6257385420\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 |
|
− | 78.0000i | − | 243.000i | −4036.00 | 0 | −18954.0 | 27760.0i | 155064.i | −59049.0 | 0 | ||||||||||||||||||||||
| 49.2 | 78.0000i | 243.000i | −4036.00 | 0 | −18954.0 | − | 27760.0i | − | 155064.i | −59049.0 | 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 | 75.12.b.a | 2 | |
| 3.b | odd | 2 | 1 | 225.12.b.a | 2 | ||
| 5.b | even | 2 | 1 | inner | 75.12.b.a | 2 | |
| 5.c | odd | 4 | 1 | 3.12.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 75.12.a.a | 1 | ||
| 15.d | odd | 2 | 1 | 225.12.b.a | 2 | ||
| 15.e | even | 4 | 1 | 9.12.a.a | 1 | ||
| 15.e | even | 4 | 1 | 225.12.a.f | 1 | ||
| 20.e | even | 4 | 1 | 48.12.a.f | 1 | ||
| 35.f | even | 4 | 1 | 147.12.a.c | 1 | ||
| 40.i | odd | 4 | 1 | 192.12.a.q | 1 | ||
| 40.k | even | 4 | 1 | 192.12.a.g | 1 | ||
| 45.k | odd | 12 | 2 | 81.12.c.a | 2 | ||
| 45.l | even | 12 | 2 | 81.12.c.e | 2 | ||
| 60.l | odd | 4 | 1 | 144.12.a.l | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.12.a.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 9.12.a.a | 1 | 15.e | even | 4 | 1 | ||
| 48.12.a.f | 1 | 20.e | even | 4 | 1 | ||
| 75.12.a.a | 1 | 5.c | odd | 4 | 1 | ||
| 75.12.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 75.12.b.a | 2 | 5.b | even | 2 | 1 | inner | |
| 81.12.c.a | 2 | 45.k | odd | 12 | 2 | ||
| 81.12.c.e | 2 | 45.l | even | 12 | 2 | ||
| 144.12.a.l | 1 | 60.l | odd | 4 | 1 | ||
| 147.12.a.c | 1 | 35.f | even | 4 | 1 | ||
| 192.12.a.g | 1 | 40.k | even | 4 | 1 | ||
| 192.12.a.q | 1 | 40.i | odd | 4 | 1 | ||
| 225.12.a.f | 1 | 15.e | even | 4 | 1 | ||
| 225.12.b.a | 2 | 3.b | odd | 2 | 1 | ||
| 225.12.b.a | 2 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 6084 \)
acting on \(S_{12}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 6084 \)
|
| $3$ |
\( T^{2} + 59049 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 770617600 \)
|
| $11$ |
\( (T - 637836)^{2} \)
|
| $13$ |
\( T^{2} + 587083893796 \)
|
| $17$ |
\( T^{2} + 9513239597316 \)
|
| $19$ |
\( (T - 19511404)^{2} \)
|
| $23$ |
\( T^{2} + 234468368769600 \)
|
| $29$ |
\( (T + 10751262)^{2} \)
|
| $31$ |
\( (T + 50937400)^{2} \)
|
| $37$ |
\( T^{2} + 44\!\cdots\!00 \)
|
| $41$ |
\( (T - 898833450)^{2} \)
|
| $43$ |
\( T^{2} + 91\!\cdots\!44 \)
|
| $47$ |
\( T^{2} + 24\!\cdots\!36 \)
|
| $53$ |
\( T^{2} + 14\!\cdots\!00 \)
|
| $59$ |
\( (T + 555306924)^{2} \)
|
| $61$ |
\( (T - 4950420998)^{2} \)
|
| $67$ |
\( T^{2} + 28\!\cdots\!56 \)
|
| $71$ |
\( (T + 14831086248)^{2} \)
|
| $73$ |
\( T^{2} + 19\!\cdots\!00 \)
|
| $79$ |
\( (T + 3720542360)^{2} \)
|
| $83$ |
\( T^{2} + 76\!\cdots\!96 \)
|
| $89$ |
\( (T - 25472769174)^{2} \)
|
| $97$ |
\( T^{2} + 15\!\cdots\!16 \)
|
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