Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6045,2,Mod(1,6045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("6045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6045.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: | \(48.2695680219\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 4x^{9} - 12x^{8} + 59x^{7} + 38x^{6} - 302x^{5} + 13x^{4} + 626x^{3} - 167x^{2} - 457x + 135 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4x^{9} - 12x^{8} + 59x^{7} + 38x^{6} - 302x^{5} + 13x^{4} + 626x^{3} - 167x^{2} - 457x + 135 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{9} - 7\nu^{8} - 25\nu^{7} + 93\nu^{6} + 95\nu^{5} - 384\nu^{4} - 106\nu^{3} + 474\nu^{2} + 33\nu - 40 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{9} - 7\nu^{8} - 25\nu^{7} + 93\nu^{6} + 100\nu^{5} - 389\nu^{4} - 146\nu^{3} + 499\nu^{2} + 98\nu - 50 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{9} - 7\nu^{8} - 25\nu^{7} + 93\nu^{6} + 100\nu^{5} - 389\nu^{4} - 151\nu^{3} + 504\nu^{2} + 123\nu - 65 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{9} - 6\nu^{8} + 69\nu^{6} - 115\nu^{5} - 212\nu^{4} + 602\nu^{3} + 42\nu^{2} - 751\nu + 225 ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3\nu^{9} - 13\nu^{8} - 25\nu^{7} + 162\nu^{6} - 15\nu^{5} - 606\nu^{4} + 456\nu^{3} + 581\nu^{2} - 653\nu + 125 ) / 5 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -2\nu^{9} + 2\nu^{8} + 40\nu^{7} - 33\nu^{6} - 290\nu^{5} + 194\nu^{4} + 881\nu^{3} - 459\nu^{2} - 903\nu + 305 ) / 5 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} + 4\nu^{8} - 35\nu^{7} - 41\nu^{6} + 330\nu^{5} + 83\nu^{4} - 1143\nu^{3} + 182\nu^{2} + 1224\nu - 355 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{4} + 8\beta_{2} + 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - 8\beta_{5} + 10\beta_{4} - \beta_{3} + 11\beta_{2} + 27\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{9} + 3 \beta_{8} - 8 \beta_{7} + 8 \beta_{6} - 3 \beta_{5} + 15 \beta_{4} - 2 \beta_{3} + \cdots + 134 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} + 11 \beta_{6} - 55 \beta_{5} + 84 \beta_{4} - 17 \beta_{3} + \cdots + 113 \)
|
| \(\nu^{8}\) | \(=\) |
\( 33 \beta_{9} + 47 \beta_{8} - 49 \beta_{7} + 52 \beta_{6} - 44 \beta_{5} + 159 \beta_{4} - 37 \beta_{3} + \cdots + 863 \)
|
| \(\nu^{9}\) | \(=\) |
\( 60 \beta_{9} + 75 \beta_{8} - 69 \beta_{7} + 92 \beta_{6} - 375 \beta_{5} + 679 \beta_{4} - 199 \beta_{3} + \cdots + 981 \)
|
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 |
|
−2.40552 | 1.00000 | 3.78653 | −1.00000 | −2.40552 | −2.80645 | −4.29753 | 1.00000 | 2.40552 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.24384 | 1.00000 | 3.03480 | −1.00000 | −2.24384 | 3.47187 | −2.32191 | 1.00000 | 2.24384 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.35516 | 1.00000 | −0.163543 | −1.00000 | −1.35516 | −3.94543 | 2.93195 | 1.00000 | 1.35516 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.33219 | 1.00000 | −0.225281 | −1.00000 | −1.33219 | −1.08911 | 2.96449 | 1.00000 | 1.33219 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.297952 | 1.00000 | −1.91122 | −1.00000 | 0.297952 | 2.52653 | −1.16536 | 1.00000 | −0.297952 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.60945 | 1.00000 | 0.590323 | −1.00000 | 1.60945 | −4.62845 | −2.26880 | 1.00000 | −1.60945 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.70000 | 1.00000 | 0.889984 | −1.00000 | 1.70000 | 2.93970 | −1.88702 | 1.00000 | −1.70000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.37998 | 1.00000 | 3.66431 | −1.00000 | 2.37998 | 2.09247 | 3.96103 | 1.00000 | −2.37998 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.55963 | 1.00000 | 4.55168 | −1.00000 | 2.55963 | −3.20072 | 6.53135 | 1.00000 | −2.55963 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.78970 | 1.00000 | 5.78242 | −1.00000 | 2.78970 | 3.63959 | 10.5518 | 1.00000 | −2.78970 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
| \(31\) | \(-1\) |
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 | 6045.2.a.v | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6045.2.a.v | ✓ | 10 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):
|
\( T_{2}^{10} - 4 T_{2}^{9} - 12 T_{2}^{8} + 59 T_{2}^{7} + 38 T_{2}^{6} - 302 T_{2}^{5} + 13 T_{2}^{4} + \cdots + 135 \)
|
|
\( T_{7}^{10} + T_{7}^{9} - 50 T_{7}^{8} - 25 T_{7}^{7} + 944 T_{7}^{6} + 86 T_{7}^{5} - 8176 T_{7}^{4} + \cdots - 35084 \)
|
|
\( T_{11}^{10} - 6 T_{11}^{9} - 16 T_{11}^{8} + 108 T_{11}^{7} + 77 T_{11}^{6} - 500 T_{11}^{5} - 32 T_{11}^{4} + \cdots - 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 4 T^{9} + \cdots + 135 \)
|
| $3$ |
\( (T - 1)^{10} \)
|
| $5$ |
\( (T + 1)^{10} \)
|
| $7$ |
\( T^{10} + T^{9} + \cdots - 35084 \)
|
| $11$ |
\( T^{10} - 6 T^{9} + \cdots - 3 \)
|
| $13$ |
\( (T - 1)^{10} \)
|
| $17$ |
\( T^{10} + T^{9} + \cdots - 7044 \)
|
| $19$ |
\( T^{10} - 2 T^{9} + \cdots + 108587 \)
|
| $23$ |
\( T^{10} - T^{9} + \cdots - 53697 \)
|
| $29$ |
\( T^{10} - 15 T^{9} + \cdots + 706725 \)
|
| $31$ |
\( (T - 1)^{10} \)
|
| $37$ |
\( T^{10} + 3 T^{9} + \cdots - 142292 \)
|
| $41$ |
\( T^{10} - 43 T^{9} + \cdots - 823869 \)
|
| $43$ |
\( T^{10} + 8 T^{9} + \cdots + 255700 \)
|
| $47$ |
\( T^{10} + 11 T^{9} + \cdots - 833343 \)
|
| $53$ |
\( T^{10} - 10 T^{9} + \cdots + 20411772 \)
|
| $59$ |
\( T^{10} - 48 T^{9} + \cdots - 8197500 \)
|
| $61$ |
\( T^{10} - 6 T^{9} + \cdots - 34531880 \)
|
| $67$ |
\( T^{10} + \cdots - 6426253268 \)
|
| $71$ |
\( T^{10} - 53 T^{9} + \cdots + 3019212 \)
|
| $73$ |
\( T^{10} - 7 T^{9} + \cdots + 407224 \)
|
| $79$ |
\( T^{10} + T^{9} + \cdots + 21042572 \)
|
| $83$ |
\( T^{10} - 41 T^{9} + \cdots - 30556260 \)
|
| $89$ |
\( T^{10} + \cdots - 202561500 \)
|
| $97$ |
\( T^{10} + \cdots - 1855747904 \)
|
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