Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [35,6,Mod(1,35)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35, 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("35.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 35 = 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 35.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: | \(5.61343369345\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{65}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). 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 |
|
−3.53113 | 13.5934 | −19.5311 | −25.0000 | −48.0000 | −49.0000 | 181.963 | −58.2198 | 88.2782 | ||||||||||||||||||||||||
| 1.2 | 4.53113 | −10.5934 | −11.4689 | −25.0000 | −48.0000 | −49.0000 | −196.963 | −130.780 | −113.278 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 35.6.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 315.6.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 560.6.a.l | 2 | ||
| 5.b | even | 2 | 1 | 175.6.a.d | 2 | ||
| 5.c | odd | 4 | 2 | 175.6.b.d | 4 | ||
| 7.b | odd | 2 | 1 | 245.6.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.6.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 175.6.a.d | 2 | 5.b | even | 2 | 1 | ||
| 175.6.b.d | 4 | 5.c | odd | 4 | 2 | ||
| 245.6.a.c | 2 | 7.b | odd | 2 | 1 | ||
| 315.6.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 560.6.a.l | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 16 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(35))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T - 16 \)
|
| $3$ |
\( T^{2} - 3T - 144 \)
|
| $5$ |
\( (T + 25)^{2} \)
|
| $7$ |
\( (T + 49)^{2} \)
|
| $11$ |
\( T^{2} + 601T - 62596 \)
|
| $13$ |
\( T^{2} + 577T + 37586 \)
|
| $17$ |
\( T^{2} - 41T - 1023346 \)
|
| $19$ |
\( T^{2} - 630T - 20960 \)
|
| $23$ |
\( T^{2} + 442 T - 13172224 \)
|
| $29$ |
\( T^{2} - 5885 T - 5853350 \)
|
| $31$ |
\( T^{2} + 396 T - 13834656 \)
|
| $37$ |
\( T^{2} + 8904 T - 5978196 \)
|
| $41$ |
\( T^{2} - 1774 T - 236091496 \)
|
| $43$ |
\( T^{2} + 27122 T + 170086856 \)
|
| $47$ |
\( T^{2} + 21289 T + 72995224 \)
|
| $53$ |
\( T^{2} + 55582 T + 768990296 \)
|
| $59$ |
\( T^{2} - 59600 T + 451339840 \)
|
| $61$ |
\( T^{2} + 51846 T + 142927344 \)
|
| $67$ |
\( T^{2} + 45344 T + 174936944 \)
|
| $71$ |
\( T^{2} + \cdots + 1172746624 \)
|
| $73$ |
\( T^{2} + 13532 T - 239780284 \)
|
| $79$ |
\( T^{2} + \cdots - 4382959400 \)
|
| $83$ |
\( T^{2} + \cdots + 2765333536 \)
|
| $89$ |
\( T^{2} + 37650 T - 177665240 \)
|
| $97$ |
\( T^{2} + 96339 T - 838817066 \)
|
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