Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1225,4,Mod(1,1225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, 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("1225.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1225 = 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1225.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: | \(72.2773397570\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 202x^{6} + 12253x^{4} - 210844x^{2} + 592900 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 245) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 202x^{6} + 12253x^{4} - 210844x^{2} + 592900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -9\nu^{6} - 1168\nu^{4} + 246211\nu^{2} - 4412990 ) / 82160 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{6} - 311\nu^{4} + 12612\nu^{2} - 97115 ) / 5135 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 31\nu^{6} - 7388\nu^{4} + 395751\nu^{2} - 1117590 ) / 41080 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -211\nu^{7} + 53864\nu^{5} - 3938119\nu^{3} + 68723526\nu ) / 12652640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -16\nu^{7} + 1461\nu^{5} - 118355\nu^{3} + 9602958\nu ) / 790790 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -53\nu^{7} + 9782\nu^{5} - 505727\nu^{3} + 6138778\nu ) / 1581580 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1909\nu^{7} - 337416\nu^{5} + 16290961\nu^{3} - 113564394\nu ) / 12652640 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 5\beta_{6} - \beta_{4} ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{3} + 5\beta_{2} + 4\beta _1 + 255 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 77\beta_{7} + 420\beta_{6} - 25\beta_{5} - 117\beta_{4} ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -236\beta_{3} + 545\beta_{2} + 312\beta _1 + 20645 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1393\beta_{7} + 7446\beta_{6} - 645\beta_{5} - 1577\beta_{4} \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -24086\beta_{3} + 66055\beta_{2} + 23292\beta _1 + 1845055 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 666593\beta_{7} + 3293670\beta_{6} - 356675\beta_{5} - 454713\beta_{4} ) / 5 \)
|
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 |
|
−5.30045 | −8.20271 | 20.0948 | 0 | 43.4781 | 0 | −64.1080 | 40.2844 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.30045 | 8.20271 | 20.0948 | 0 | −43.4781 | 0 | −64.1080 | 40.2844 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.62777 | −1.92758 | −1.09481 | 0 | 5.06525 | 0 | 23.8991 | −23.2844 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.62777 | 1.92758 | −1.09481 | 0 | −5.06525 | 0 | 23.8991 | −23.2844 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.62777 | −1.92758 | −1.09481 | 0 | −5.06525 | 0 | −23.8991 | −23.2844 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.62777 | 1.92758 | −1.09481 | 0 | 5.06525 | 0 | −23.8991 | −23.2844 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.30045 | −8.20271 | 20.0948 | 0 | −43.4781 | 0 | 64.1080 | 40.2844 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.30045 | 8.20271 | 20.0948 | 0 | 43.4781 | 0 | 64.1080 | 40.2844 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1225.4.a.bm | 8 | |
| 5.b | even | 2 | 1 | inner | 1225.4.a.bm | 8 | |
| 5.c | odd | 4 | 2 | 245.4.b.c | ✓ | 8 | |
| 7.b | odd | 2 | 1 | inner | 1225.4.a.bm | 8 | |
| 35.c | odd | 2 | 1 | inner | 1225.4.a.bm | 8 | |
| 35.f | even | 4 | 2 | 245.4.b.c | ✓ | 8 | |
| 35.k | even | 12 | 4 | 245.4.j.c | 16 | ||
| 35.l | odd | 12 | 4 | 245.4.j.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 245.4.b.c | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 245.4.b.c | ✓ | 8 | 35.f | even | 4 | 2 | |
| 245.4.j.c | 16 | 35.k | even | 12 | 4 | ||
| 245.4.j.c | 16 | 35.l | odd | 12 | 4 | ||
| 1225.4.a.bm | 8 | 1.a | even | 1 | 1 | trivial | |
| 1225.4.a.bm | 8 | 5.b | even | 2 | 1 | inner | |
| 1225.4.a.bm | 8 | 7.b | odd | 2 | 1 | inner | |
| 1225.4.a.bm | 8 | 35.c | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):
|
\( T_{2}^{4} - 35T_{2}^{2} + 194 \)
|
|
\( T_{3}^{4} - 71T_{3}^{2} + 250 \)
|
|
\( T_{19}^{4} - 20546T_{19}^{2} + 93896000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 35 T^{2} + 194)^{2} \)
|
| $3$ |
\( (T^{4} - 71 T^{2} + 250)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{2} + 15 T - 56)^{4} \)
|
| $13$ |
\( (T^{4} - 6391 T^{2} + 3906250)^{2} \)
|
| $17$ |
\( (T^{4} - 8701 T^{2} + 17956000)^{2} \)
|
| $19$ |
\( (T^{4} - 20546 T^{2} + 93896000)^{2} \)
|
| $23$ |
\( (T^{4} - 12100 T^{2} + 34876544)^{2} \)
|
| $29$ |
\( (T^{2} - 229 T + 4018)^{4} \)
|
| $31$ |
\( (T^{4} - 45736 T^{2} + 3104000)^{2} \)
|
| $37$ |
\( (T^{4} - 154084 T^{2} + 5825897600)^{2} \)
|
| $41$ |
\( (T^{4} - 156776 T^{2} + 3802400000)^{2} \)
|
| $43$ |
\( (T^{4} - 27440 T^{2} + 119243264)^{2} \)
|
| $47$ |
\( (T^{4} - 379749 T^{2} + 35988001000)^{2} \)
|
| $53$ |
\( (T^{4} - 261700 T^{2} + 16420160000)^{2} \)
|
| $59$ |
\( (T^{4} - 316226 T^{2} + 24586784000)^{2} \)
|
| $61$ |
\( (T^{4} - 645250 T^{2} + 23765000000)^{2} \)
|
| $67$ |
\( (T^{4} - 1025728 T^{2} + 40946180096)^{2} \)
|
| $71$ |
\( (T^{2} - 658 T - 53848)^{4} \)
|
| $73$ |
\( (T^{4} - 610044 T^{2} + 91508356000)^{2} \)
|
| $79$ |
\( (T^{2} + 367 T - 119998)^{4} \)
|
| $83$ |
\( (T^{4} - 1018386 T^{2} + 135536164000)^{2} \)
|
| $89$ |
\( (T^{4} - 1818144 T^{2} + 486756864000)^{2} \)
|
| $97$ |
\( (T^{4} - 1264725 T^{2} + 392832400000)^{2} \)
|
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