Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7225,2,Mod(1,7225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7225, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7225.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7225 = 5^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7225.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: | \(57.6919154604\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.93924352.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 17x^{4} + 73x^{2} - 67 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 425) |
| 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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 17x^{4} + 73x^{2} - 67 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{4} - 21\nu^{2} + 18 ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 19\nu^{2} + 60 ) / 17 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 19\nu^{3} + 77\nu ) / 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 40\nu^{3} + 95\nu ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{3} + \beta_{2} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -21\beta_{3} + 19\beta_{2} + 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( 19\beta_{5} - 40\beta_{4} + 75\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.48119 | −3.29112 | 0.193937 | 0 | 4.87478 | 2.22194 | 2.67513 | 7.83146 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.48119 | 3.29112 | 0.193937 | 0 | −4.87478 | −2.22194 | 2.67513 | 7.83146 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.311108 | −1.12261 | −1.90321 | 0 | −0.349253 | −3.60843 | −1.21432 | −1.73975 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.311108 | 1.12261 | −1.90321 | 0 | 0.349253 | 3.60843 | −1.21432 | −1.73975 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.17009 | −2.21547 | 2.70928 | 0 | −4.80775 | −1.02091 | 1.53919 | 1.90829 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.17009 | 2.21547 | 2.70928 | 0 | 4.80775 | 1.02091 | 1.53919 | 1.90829 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(17\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7225.2.a.bg | 6 | |
| 5.b | even | 2 | 1 | 7225.2.a.ba | 6 | ||
| 17.b | even | 2 | 1 | inner | 7225.2.a.bg | 6 | |
| 17.c | even | 4 | 2 | 425.2.d.a | ✓ | 6 | |
| 85.c | even | 2 | 1 | 7225.2.a.ba | 6 | ||
| 85.f | odd | 4 | 2 | 425.2.c.c | 12 | ||
| 85.i | odd | 4 | 2 | 425.2.c.c | 12 | ||
| 85.j | even | 4 | 2 | 425.2.d.b | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 425.2.c.c | 12 | 85.f | odd | 4 | 2 | ||
| 425.2.c.c | 12 | 85.i | odd | 4 | 2 | ||
| 425.2.d.a | ✓ | 6 | 17.c | even | 4 | 2 | |
| 425.2.d.b | yes | 6 | 85.j | even | 4 | 2 | |
| 7225.2.a.ba | 6 | 5.b | even | 2 | 1 | ||
| 7225.2.a.ba | 6 | 85.c | even | 2 | 1 | ||
| 7225.2.a.bg | 6 | 1.a | even | 1 | 1 | trivial | |
| 7225.2.a.bg | 6 | 17.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):
|
\( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \)
|
|
\( T_{3}^{6} - 17T_{3}^{4} + 73T_{3}^{2} - 67 \)
|
|
\( T_{7}^{6} - 19T_{7}^{4} + 83T_{7}^{2} - 67 \)
|
|
\( T_{11}^{6} - 66T_{11}^{4} + 1292T_{11}^{2} - 6700 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - T^{2} - 3 T + 1)^{2} \)
|
| $3$ |
\( T^{6} - 17 T^{4} + \cdots - 67 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 19 T^{4} + \cdots - 67 \)
|
| $11$ |
\( T^{6} - 66 T^{4} + \cdots - 6700 \)
|
| $13$ |
\( (T^{3} - T^{2} - 13 T + 23)^{2} \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T^{3} + 6 T^{2} + \cdots - 100)^{2} \)
|
| $23$ |
\( T^{6} - 54 T^{4} + \cdots - 268 \)
|
| $29$ |
\( T^{6} - 116 T^{4} + \cdots - 1072 \)
|
| $31$ |
\( T^{6} - 89 T^{4} + \cdots - 1675 \)
|
| $37$ |
\( T^{6} - 144 T^{4} + \cdots - 1072 \)
|
| $41$ |
\( T^{6} - 100 T^{4} + \cdots - 26800 \)
|
| $43$ |
\( (T^{3} - 4 T^{2} + \cdots + 452)^{2} \)
|
| $47$ |
\( (T^{3} - 20 T^{2} + \cdots - 208)^{2} \)
|
| $53$ |
\( (T^{3} - 5 T^{2} - 69 T - 43)^{2} \)
|
| $59$ |
\( (T^{3} - 2 T^{2} - 44 T + 20)^{2} \)
|
| $61$ |
\( T^{6} - 228 T^{4} + \cdots - 107200 \)
|
| $67$ |
\( (T^{3} - 12 T^{2} + \cdots + 16)^{2} \)
|
| $71$ |
\( T^{6} - 91 T^{4} + \cdots - 1675 \)
|
| $73$ |
\( T^{6} - 32 T^{4} + \cdots - 1072 \)
|
| $79$ |
\( T^{6} - 135 T^{4} + \cdots - 67 \)
|
| $83$ |
\( (T^{3} - 10 T^{2} + \cdots - 184)^{2} \)
|
| $89$ |
\( (T^{3} + 4 T^{2} - 4 T - 20)^{2} \)
|
| $97$ |
\( T^{6} - 332 T^{4} + \cdots - 1239232 \)
|
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