Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,4,Mod(1,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.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: | \(14.4554679514\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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 \(\beta = \sqrt{2}\). 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 |
|
−4.41421 | −3.24264 | 11.4853 | −5.00000 | 14.3137 | 0 | −15.3848 | −16.4853 | 22.0711 | ||||||||||||||||||||||||
| 1.2 | −1.58579 | 5.24264 | −5.48528 | −5.00000 | −8.31371 | 0 | 21.3848 | 0.485281 | 7.92893 | |||||||||||||||||||||||||
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 | 245.4.a.h | 2 | |
| 3.b | odd | 2 | 1 | 2205.4.a.bg | 2 | ||
| 5.b | even | 2 | 1 | 1225.4.a.v | 2 | ||
| 7.b | odd | 2 | 1 | 245.4.a.g | 2 | ||
| 7.c | even | 3 | 2 | 245.4.e.l | 4 | ||
| 7.d | odd | 6 | 2 | 35.4.e.b | ✓ | 4 | |
| 21.c | even | 2 | 1 | 2205.4.a.bf | 2 | ||
| 21.g | even | 6 | 2 | 315.4.j.c | 4 | ||
| 28.f | even | 6 | 2 | 560.4.q.i | 4 | ||
| 35.c | odd | 2 | 1 | 1225.4.a.x | 2 | ||
| 35.i | odd | 6 | 2 | 175.4.e.c | 4 | ||
| 35.k | even | 12 | 4 | 175.4.k.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.e.b | ✓ | 4 | 7.d | odd | 6 | 2 | |
| 175.4.e.c | 4 | 35.i | odd | 6 | 2 | ||
| 175.4.k.c | 8 | 35.k | even | 12 | 4 | ||
| 245.4.a.g | 2 | 7.b | odd | 2 | 1 | ||
| 245.4.a.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 245.4.e.l | 4 | 7.c | even | 3 | 2 | ||
| 315.4.j.c | 4 | 21.g | even | 6 | 2 | ||
| 560.4.q.i | 4 | 28.f | even | 6 | 2 | ||
| 1225.4.a.v | 2 | 5.b | even | 2 | 1 | ||
| 1225.4.a.x | 2 | 35.c | odd | 2 | 1 | ||
| 2205.4.a.bf | 2 | 21.c | even | 2 | 1 | ||
| 2205.4.a.bg | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(245))\):
|
\( T_{2}^{2} + 6T_{2} + 7 \)
|
|
\( T_{3}^{2} - 2T_{3} - 17 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 6T + 7 \)
|
| $3$ |
\( T^{2} - 2T - 17 \)
|
| $5$ |
\( (T + 5)^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 28T - 4 \)
|
| $13$ |
\( T^{2} - 36T + 124 \)
|
| $17$ |
\( T^{2} - 76T - 4388 \)
|
| $19$ |
\( T^{2} + 160T + 5048 \)
|
| $23$ |
\( T^{2} + 22T - 27257 \)
|
| $29$ |
\( T^{2} + 250T + 8425 \)
|
| $31$ |
\( T^{2} - 132T - 15644 \)
|
| $37$ |
\( T^{2} + 416T + 6272 \)
|
| $41$ |
\( T^{2} - 106T + 1009 \)
|
| $43$ |
\( T^{2} + 666T + 110839 \)
|
| $47$ |
\( T^{2} - 196T + 1412 \)
|
| $53$ |
\( T^{2} + 952T + 186248 \)
|
| $59$ |
\( T^{2} + 840T + 34888 \)
|
| $61$ |
\( T^{2} + 98T - 647399 \)
|
| $67$ |
\( T^{2} + 1286 T + 403927 \)
|
| $71$ |
\( T^{2} - 1064T + 38024 \)
|
| $73$ |
\( T^{2} - 172T - 20452 \)
|
| $79$ |
\( T^{2} + 1240 T + 292808 \)
|
| $83$ |
\( T^{2} - 1906 T + 908159 \)
|
| $89$ |
\( T^{2} + 650T - 104327 \)
|
| $97$ |
\( T^{2} - 628T - 401404 \)
|
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