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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 92x^{4} + 26x^{3} + 2116x^{2} + 80x - 7800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 70) |
| 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
\( x^{6} - x^{5} - 92x^{4} + 26x^{3} + 2116x^{2} + 80x - 7800 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{5} + 33\nu^{4} + 806\nu^{3} - 948\nu^{2} - 7828\nu - 1500 ) / 1260 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 13\nu^{4} + 256\nu^{3} - 628\nu^{2} - 5188\nu + 2820 ) / 420 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 16\nu^{4} - 27\nu^{3} + 886\nu^{2} - 394\nu - 6330 ) / 210 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 13\nu^{4} - 256\nu^{3} + 1048\nu^{2} + 4768\nu - 15420 ) / 420 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{3} + \beta _1 + 30 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 6\beta_{3} - 3\beta_{2} + 46\beta _1 + 23 \)
|
| \(\nu^{4}\) | \(=\) |
\( 63\beta_{5} - 13\beta_{4} + 76\beta_{3} - 15\beta_{2} + 106\beta _1 + 1393 \)
|
| \(\nu^{5}\) | \(=\) |
\( 149\beta_{5} + 29\beta_{4} + 492\beta_{3} - 321\beta_{2} + 2446\beta _1 + 2659 \)
|
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 | −6.00242 | 4.00000 | 0 | −12.0048 | 0 | 8.00000 | 9.02909 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −4.49484 | 4.00000 | 0 | −8.98967 | 0 | 8.00000 | −6.79645 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −1.20031 | 4.00000 | 0 | −2.40061 | 0 | 8.00000 | −25.5593 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.06759 | 4.00000 | 0 | 6.13518 | 0 | 8.00000 | −17.5899 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 6.44232 | 4.00000 | 0 | 12.8846 | 0 | 8.00000 | 14.5035 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 9.18765 | 4.00000 | 0 | 18.3753 | 0 | 8.00000 | 57.4130 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
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 | 2450.4.a.cy | 6 | |
| 5.b | even | 2 | 1 | 2450.4.a.cv | 6 | ||
| 5.c | odd | 4 | 2 | 490.4.c.f | 12 | ||
| 7.b | odd | 2 | 1 | 2450.4.a.cx | 6 | ||
| 7.c | even | 3 | 2 | 350.4.e.n | 12 | ||
| 35.c | odd | 2 | 1 | 2450.4.a.cw | 6 | ||
| 35.f | even | 4 | 2 | 490.4.c.e | 12 | ||
| 35.j | even | 6 | 2 | 350.4.e.o | 12 | ||
| 35.l | odd | 12 | 4 | 70.4.i.a | ✓ | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.4.i.a | ✓ | 24 | 35.l | odd | 12 | 4 | |
| 350.4.e.n | 12 | 7.c | even | 3 | 2 | ||
| 350.4.e.o | 12 | 35.j | even | 6 | 2 | ||
| 490.4.c.e | 12 | 35.f | even | 4 | 2 | ||
| 490.4.c.f | 12 | 5.c | odd | 4 | 2 | ||
| 2450.4.a.cv | 6 | 5.b | even | 2 | 1 | ||
| 2450.4.a.cw | 6 | 35.c | odd | 2 | 1 | ||
| 2450.4.a.cx | 6 | 7.b | odd | 2 | 1 | ||
| 2450.4.a.cy | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
|
\( T_{3}^{6} - 7T_{3}^{5} - 72T_{3}^{4} + 364T_{3}^{3} + 1511T_{3}^{2} - 3717T_{3} - 5880 \)
|
|
\( T_{11}^{6} - 31T_{11}^{5} - 3033T_{11}^{4} + 96751T_{11}^{3} + 1345208T_{11}^{2} - 49232064T_{11} + 286379520 \)
|
|
\( T_{19}^{6} - 93T_{19}^{5} - 10611T_{19}^{4} + 120401T_{19}^{3} + 17245338T_{19}^{2} + 170214204T_{19} - 1453282440 \)
|
|
\( T_{23}^{6} + 94T_{23}^{5} - 31643T_{23}^{4} - 987212T_{23}^{3} + 264194947T_{23}^{2} - 4601928506T_{23} - 5244328929 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( T^{6} - 7 T^{5} + \cdots - 5880 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 31 T^{5} + \cdots + 286379520 \)
|
| $13$ |
\( T^{6} + \cdots - 26675500800 \)
|
| $17$ |
\( T^{6} + \cdots - 42613097680 \)
|
| $19$ |
\( T^{6} + \cdots - 1453282440 \)
|
| $23$ |
\( T^{6} + \cdots - 5244328929 \)
|
| $29$ |
\( T^{6} + \cdots - 1796409740592 \)
|
| $31$ |
\( T^{6} + \cdots - 495991036080 \)
|
| $37$ |
\( T^{6} + \cdots + 13994158315904 \)
|
| $41$ |
\( T^{6} + \cdots - 1146896772372 \)
|
| $43$ |
\( T^{6} + \cdots - 75964400923000 \)
|
| $47$ |
\( T^{6} + \cdots + 28595013492080 \)
|
| $53$ |
\( T^{6} + \cdots + 91905926717016 \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 12462758786850 \)
|
| $67$ |
\( T^{6} + \cdots + 16\!\cdots\!10 \)
|
| $71$ |
\( T^{6} + \cdots + 16\!\cdots\!40 \)
|
| $73$ |
\( T^{6} + \cdots + 31\!\cdots\!20 \)
|
| $79$ |
\( T^{6} + \cdots - 14\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 76\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots - 106054121644030 \)
|
| $97$ |
\( T^{6} + \cdots - 81\!\cdots\!00 \)
|
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