Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2450,4,Mod(1,2450)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2450, 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("2450.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2450.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: | \(144.554679514\) |
| 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: |
\( x^{8} - 146x^{6} + 4997x^{4} - 4646x^{2} + 676 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2\cdot 7^{2}\cdot 11^{2} \) |
| 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 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
\( x^{8} - 146x^{6} + 4997x^{4} - 4646x^{2} + 676 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{6} + 538\nu^{4} - 13801\nu^{2} - 82955 ) / 2541 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{6} + 269\nu^{4} - 8171\nu^{2} + 5531 ) / 363 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 115\nu^{7} - 16738\nu^{5} + 567661\nu^{3} - 387910\nu ) / 66066 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -113\nu^{6} + 16469\nu^{4} - 559490\nu^{2} + 288362 ) / 2541 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 115\nu^{7} - 16738\nu^{5} + 567661\nu^{3} - 321844\nu ) / 9438 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -251\nu^{7} + 36724\nu^{5} - 1264738\nu^{3} + 1473804\nu ) / 11011 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3743\nu^{7} - 545360\nu^{5} + 18520367\nu^{3} - 10278848\nu ) / 66066 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 7\beta_{3} ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{2} + 7\beta _1 + 259 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -14\beta_{7} - 7\beta_{6} + 128\beta_{5} - 532\beta_{3} ) / 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14\beta_{4} - 267\beta_{2} + 539\beta _1 + 20076 ) / 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1883\beta_{7} - 539\beta_{6} + 13941\beta_{5} - 43358\beta_{3} ) / 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1883\beta_{4} - 29011\beta_{2} + 43897\beta _1 + 1661436 ) / 7 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -204960\beta_{7} - 43897\beta_{6} + 1400624\beta_{5} - 3704204\beta_{3} ) / 7 \)
|
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 |
|
−2.00000 | −8.65990 | 4.00000 | 0 | 17.3198 | 0 | −8.00000 | 47.9939 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −8.20597 | 4.00000 | 0 | 16.4119 | 0 | −8.00000 | 40.3380 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −1.83880 | 4.00000 | 0 | 3.67759 | 0 | −8.00000 | −23.6188 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −0.535700 | 4.00000 | 0 | 1.07140 | 0 | −8.00000 | −26.7130 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 0.535700 | 4.00000 | 0 | −1.07140 | 0 | −8.00000 | −26.7130 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 1.83880 | 4.00000 | 0 | −3.67759 | 0 | −8.00000 | −23.6188 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.00000 | 8.20597 | 4.00000 | 0 | −16.4119 | 0 | −8.00000 | 40.3380 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.00000 | 8.65990 | 4.00000 | 0 | −17.3198 | 0 | −8.00000 | 47.9939 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2450.4.a.cz | ✓ | 8 |
| 5.b | even | 2 | 1 | 2450.4.a.da | yes | 8 | |
| 7.b | odd | 2 | 1 | inner | 2450.4.a.cz | ✓ | 8 |
| 35.c | odd | 2 | 1 | 2450.4.a.da | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2450.4.a.cz | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 2450.4.a.cz | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 2450.4.a.da | yes | 8 | 5.b | even | 2 | 1 | |
| 2450.4.a.da | yes | 8 | 35.c | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
|
\( T_{3}^{8} - 146T_{3}^{6} + 5573T_{3}^{4} - 18662T_{3}^{2} + 4900 \)
|
|
\( T_{11}^{4} - 18T_{11}^{3} - 4150T_{11}^{2} + 118654T_{11} - 636127 \)
|
|
\( T_{19}^{8} - 39264T_{19}^{6} + 420361613T_{19}^{4} - 999855668868T_{19}^{2} + 76252888644100 \)
|
|
\( T_{23}^{4} + 82T_{23}^{3} - 36231T_{23}^{2} - 2970884T_{23} - 47065480 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{8} \)
|
| $3$ |
\( T^{8} - 146 T^{6} + \cdots + 4900 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} - 18 T^{3} + \cdots - 636127)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 639936001600 \)
|
| $17$ |
\( T^{8} + \cdots + 13735444050496 \)
|
| $19$ |
\( T^{8} + \cdots + 76252888644100 \)
|
| $23$ |
\( (T^{4} + 82 T^{3} + \cdots - 47065480)^{2} \)
|
| $29$ |
\( (T^{4} - 196 T^{3} + \cdots + 759399292)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 65\!\cdots\!36 \)
|
| $37$ |
\( (T^{4} + 16 T^{3} + \cdots + 576478760)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 50\!\cdots\!76 \)
|
| $43$ |
\( (T^{4} + 376 T^{3} + \cdots - 4344960292)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 99\!\cdots\!00 \)
|
| $53$ |
\( (T^{4} + 350 T^{3} + \cdots + 1090048)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 39\!\cdots\!04 \)
|
| $61$ |
\( T^{8} + \cdots + 27\!\cdots\!00 \)
|
| $67$ |
\( (T^{4} + 776 T^{3} + \cdots - 18141327911)^{2} \)
|
| $71$ |
\( (T^{4} - 1324 T^{3} + \cdots - 233105755124)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 85\!\cdots\!44 \)
|
| $79$ |
\( (T^{4} + 958 T^{3} + \cdots + 177724623860)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 17\!\cdots\!24 \)
|
| $89$ |
\( T^{8} + \cdots + 14\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 16\!\cdots\!64 \)
|
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