Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,6,Mod(1,75)]
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, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
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: | yes |
| Analytic conductor: | \(12.0287864860\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{409}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 102 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 15) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{409})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−9.61187 | 9.00000 | 60.3881 | 0 | −86.5069 | 217.790 | −272.863 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 10.6119 | 9.00000 | 80.6119 | 0 | 95.5069 | −105.790 | 515.863 | 81.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.6.a.h | 2 | |
| 3.b | odd | 2 | 1 | 225.6.a.m | 2 | ||
| 5.b | even | 2 | 1 | 15.6.a.c | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 75.6.b.e | 4 | ||
| 15.d | odd | 2 | 1 | 45.6.a.e | 2 | ||
| 15.e | even | 4 | 2 | 225.6.b.g | 4 | ||
| 20.d | odd | 2 | 1 | 240.6.a.q | 2 | ||
| 35.c | odd | 2 | 1 | 735.6.a.g | 2 | ||
| 40.e | odd | 2 | 1 | 960.6.a.bf | 2 | ||
| 40.f | even | 2 | 1 | 960.6.a.bj | 2 | ||
| 60.h | even | 2 | 1 | 720.6.a.bd | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.6.a.c | ✓ | 2 | 5.b | even | 2 | 1 | |
| 45.6.a.e | 2 | 15.d | odd | 2 | 1 | ||
| 75.6.a.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 75.6.b.e | 4 | 5.c | odd | 4 | 2 | ||
| 225.6.a.m | 2 | 3.b | odd | 2 | 1 | ||
| 225.6.b.g | 4 | 15.e | even | 4 | 2 | ||
| 240.6.a.q | 2 | 20.d | odd | 2 | 1 | ||
| 720.6.a.bd | 2 | 60.h | even | 2 | 1 | ||
| 735.6.a.g | 2 | 35.c | odd | 2 | 1 | ||
| 960.6.a.bf | 2 | 40.e | odd | 2 | 1 | ||
| 960.6.a.bj | 2 | 40.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 102 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T - 102 \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 112T - 23040 \)
|
| $11$ |
\( T^{2} - 248T - 89328 \)
|
| $13$ |
\( T^{2} + 876T + 165668 \)
|
| $17$ |
\( T^{2} + 2036 T + 381924 \)
|
| $19$ |
\( T^{2} - 1464 T - 3887920 \)
|
| $23$ |
\( T^{2} - 3216 T + 2350080 \)
|
| $29$ |
\( T^{2} - 1948 T + 529860 \)
|
| $31$ |
\( T^{2} - 2672 T - 1382400 \)
|
| $37$ |
\( T^{2} + 8668 T - 49300220 \)
|
| $41$ |
\( T^{2} + 7628 T - 15712860 \)
|
| $43$ |
\( T^{2} - 16440 T + 67149584 \)
|
| $47$ |
\( T^{2} - 19360 T - 61495104 \)
|
| $53$ |
\( T^{2} - 14356 T - 476289180 \)
|
| $59$ |
\( T^{2} + \cdots - 1067881200 \)
|
| $61$ |
\( T^{2} - 20220 T - 149182204 \)
|
| $67$ |
\( T^{2} - 12904 T - 125898096 \)
|
| $71$ |
\( T^{2} + 40976 T - 627281856 \)
|
| $73$ |
\( T^{2} + 59124 T + 836113700 \)
|
| $79$ |
\( T^{2} + \cdots + 1448870400 \)
|
| $83$ |
\( T^{2} + \cdots + 2064089232 \)
|
| $89$ |
\( T^{2} + \cdots - 2895368220 \)
|
| $97$ |
\( T^{2} + \cdots - 15772164476 \)
|
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