Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [650,6,Mod(1,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, 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("650.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.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: | \(104.249482878\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{10}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 130) |
| 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 \(\beta = 5\sqrt{10}\). 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 |
|
4.00000 | −13.8114 | 16.0000 | 0 | −55.2456 | −213.246 | 64.0000 | −52.2456 | 0 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 17.8114 | 16.0000 | 0 | 71.2456 | −86.7544 | 64.0000 | 74.2456 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 650.6.a.h | 2 | |
| 5.b | even | 2 | 1 | 130.6.a.a | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 650.6.b.g | 4 | ||
| 20.d | odd | 2 | 1 | 1040.6.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.6.a.a | ✓ | 2 | 5.b | even | 2 | 1 | |
| 650.6.a.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 650.6.b.g | 4 | 5.c | odd | 4 | 2 | ||
| 1040.6.a.e | 2 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 4T_{3} - 246 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(650))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( T^{2} - 4T - 246 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 300T + 18500 \)
|
| $11$ |
\( T^{2} - 384T - 119386 \)
|
| $13$ |
\( (T - 169)^{2} \)
|
| $17$ |
\( T^{2} - 244T - 66116 \)
|
| $19$ |
\( T^{2} + 2408 T + 189366 \)
|
| $23$ |
\( T^{2} + 332 T - 9674694 \)
|
| $29$ |
\( T^{2} - 3528 T - 17052304 \)
|
| $31$ |
\( T^{2} - 880 T - 19828650 \)
|
| $37$ |
\( T^{2} + 10744 T + 23817384 \)
|
| $41$ |
\( T^{2} - 16020 T + 14431100 \)
|
| $43$ |
\( T^{2} + 2964 T + 474074 \)
|
| $47$ |
\( T^{2} + 38292 T + 360485316 \)
|
| $53$ |
\( T^{2} + 42052 T + 427931676 \)
|
| $59$ |
\( T^{2} - 42872 T + 110811846 \)
|
| $61$ |
\( T^{2} + 33192 T + 275363216 \)
|
| $67$ |
\( T^{2} + 37420 T + 349943100 \)
|
| $71$ |
\( T^{2} + 69520 T - 648148650 \)
|
| $73$ |
\( T^{2} + \cdots - 3075333144 \)
|
| $79$ |
\( T^{2} + \cdots - 5805828184 \)
|
| $83$ |
\( T^{2} + 31500 T - 90661500 \)
|
| $89$ |
\( T^{2} + 18452 T - 219584924 \)
|
| $97$ |
\( T^{2} + \cdots - 1343944636 \)
|
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