Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6040,2,Mod(1,6040)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6040, 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("6040.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6040 = 2^{3} \cdot 5 \cdot 151 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6040.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: | \(48.2296428209\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} + \cdots - 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} + \cdots - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2008 \nu^{11} + 1424 \nu^{10} + 2835 \nu^{9} - 76496 \nu^{8} - 561574 \nu^{7} + 560150 \nu^{6} + \cdots + 4169518 ) / 1691597 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 10958 \nu^{11} + 39405 \nu^{10} + 302967 \nu^{9} - 957392 \nu^{8} - 2950336 \nu^{7} + \cdots - 4607858 ) / 1691597 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 87088 \nu^{11} - 120205 \nu^{10} - 1528205 \nu^{9} + 1743641 \nu^{8} + 8148517 \nu^{7} + \cdots + 1976415 ) / 1691597 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 179015 \nu^{11} + 513295 \nu^{10} + 3434568 \nu^{9} - 9422340 \nu^{8} - 23531438 \nu^{7} + \cdots - 25948112 ) / 3383194 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 141470 \nu^{11} - 108597 \nu^{10} - 2902088 \nu^{9} + 1424177 \nu^{8} + 20904860 \nu^{7} + \cdots + 2912349 ) / 1691597 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 321409 \nu^{11} + 547103 \nu^{10} + 6257848 \nu^{9} - 9375818 \nu^{8} - 42338020 \nu^{7} + \cdots - 10358630 ) / 3383194 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 334279 \nu^{11} - 834511 \nu^{10} - 6126792 \nu^{9} + 14627522 \nu^{8} + 38485960 \nu^{7} + \cdots + 28264038 ) / 3383194 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 358363 \nu^{11} + 1060051 \nu^{10} + 6884672 \nu^{9} - 19330712 \nu^{8} - 47460040 \nu^{7} + \cdots - 25308154 ) / 3383194 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 486873 \nu^{11} - 1009353 \nu^{10} - 9090678 \nu^{9} + 17063422 \nu^{8} + 58119654 \nu^{7} + \cdots + 3648252 ) / 3383194 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 508789 \nu^{11} + 1088163 \nu^{10} + 9696612 \nu^{9} - 18978206 \nu^{8} - 64020326 \nu^{7} + \cdots - 26396744 ) / 3383194 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{3} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} + \beta_{9} + \beta_{7} - \beta_{5} - 2\beta_{2} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7 \beta_{11} + 6 \beta_{10} + 3 \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + \cdots + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 13 \beta_{11} + 12 \beta_{9} + 4 \beta_{8} + 15 \beta_{7} + \beta_{6} - 6 \beta_{5} - \beta_{4} + \cdots + 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 43 \beta_{11} + 36 \beta_{10} + 4 \beta_{9} + 42 \beta_{8} + 32 \beta_{7} + 13 \beta_{6} + 29 \beta_{5} + \cdots + 205 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 129 \beta_{11} + 3 \beta_{10} + 123 \beta_{9} + 74 \beta_{8} + 170 \beta_{7} + 18 \beta_{6} + \cdots + 101 \)
|
| \(\nu^{8}\) | \(=\) |
\( 243 \beta_{11} + 231 \beta_{10} + 93 \beta_{9} + 470 \beta_{8} + 390 \beta_{7} + 134 \beta_{6} + \cdots + 1674 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1186 \beta_{11} + 69 \beta_{10} + 1214 \beta_{9} + 990 \beta_{8} + 1773 \beta_{7} + 231 \beta_{6} + \cdots + 1463 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1160 \beta_{11} + 1613 \beta_{10} + 1436 \beta_{9} + 4923 \beta_{8} + 4350 \beta_{7} + 1309 \beta_{6} + \cdots + 14457 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 10689 \beta_{11} + 1056 \beta_{10} + 11872 \beta_{9} + 11702 \beta_{8} + 17976 \beta_{7} + \cdots + 18469 \)
|
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 |
|
0 | −2.73116 | 0 | −1.00000 | 0 | −0.736166 | 0 | 4.45925 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.38247 | 0 | −1.00000 | 0 | 3.11118 | 0 | 2.67614 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.89604 | 0 | −1.00000 | 0 | −1.82001 | 0 | 0.594974 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.08777 | 0 | −1.00000 | 0 | 2.80815 | 0 | −1.81675 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.467276 | 0 | −1.00000 | 0 | −0.199223 | 0 | −2.78165 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.168382 | 0 | −1.00000 | 0 | −3.12001 | 0 | −2.97165 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.416740 | 0 | −1.00000 | 0 | 5.24669 | 0 | −2.82633 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.983156 | 0 | −1.00000 | 0 | −2.43489 | 0 | −2.03340 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 2.09968 | 0 | −1.00000 | 0 | 1.50711 | 0 | 1.40866 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.24403 | 0 | −1.00000 | 0 | 3.06714 | 0 | 2.03565 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.45327 | 0 | −1.00000 | 0 | −3.17952 | 0 | 3.01854 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.19946 | 0 | −1.00000 | 0 | 0.749556 | 0 | 7.23656 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(151\) | \(-1\) |
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 | 6040.2.a.m | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6040.2.a.m | ✓ | 12 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6040))\):
|
\( T_{3}^{12} - 3 T_{3}^{11} - 18 T_{3}^{10} + 54 T_{3}^{9} + 110 T_{3}^{8} - 335 T_{3}^{7} - 258 T_{3}^{6} + \cdots - 16 \)
|
|
\( T_{7}^{12} - 5 T_{7}^{11} - 31 T_{7}^{10} + 148 T_{7}^{9} + 359 T_{7}^{8} - 1577 T_{7}^{7} - 1917 T_{7}^{6} + \cdots + 1024 \)
|
|
\( T_{11}^{12} - 10 T_{11}^{11} + 9 T_{11}^{10} + 206 T_{11}^{9} - 662 T_{11}^{8} - 697 T_{11}^{7} + \cdots + 6652 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 3 T^{11} + \cdots - 16 \)
|
| $5$ |
\( (T + 1)^{12} \)
|
| $7$ |
\( T^{12} - 5 T^{11} + \cdots + 1024 \)
|
| $11$ |
\( T^{12} - 10 T^{11} + \cdots + 6652 \)
|
| $13$ |
\( T^{12} - 11 T^{11} + \cdots + 2956 \)
|
| $17$ |
\( T^{12} + 4 T^{11} + \cdots - 76288 \)
|
| $19$ |
\( T^{12} - 5 T^{11} + \cdots - 94048 \)
|
| $23$ |
\( T^{12} - 18 T^{11} + \cdots - 12 \)
|
| $29$ |
\( T^{12} - 16 T^{11} + \cdots + 35058 \)
|
| $31$ |
\( T^{12} + T^{11} + \cdots - 6780096 \)
|
| $37$ |
\( T^{12} - 2 T^{11} + \cdots - 3530624 \)
|
| $41$ |
\( T^{12} - 4 T^{11} + \cdots - 4089856 \)
|
| $43$ |
\( T^{12} - 7 T^{11} + \cdots + 55296 \)
|
| $47$ |
\( T^{12} - 191 T^{10} + \cdots - 466944 \)
|
| $53$ |
\( T^{12} + \cdots - 205285376 \)
|
| $59$ |
\( T^{12} + 4 T^{11} + \cdots - 3832788 \)
|
| $61$ |
\( T^{12} + 32 T^{11} + \cdots + 235264 \)
|
| $67$ |
\( T^{12} - 4 T^{11} + \cdots - 16157562 \)
|
| $71$ |
\( T^{12} + \cdots - 56525407616 \)
|
| $73$ |
\( T^{12} + 10 T^{11} + \cdots - 41830738 \)
|
| $79$ |
\( T^{12} - 32 T^{11} + \cdots + 1183744 \)
|
| $83$ |
\( T^{12} - 9 T^{11} + \cdots - 81975886 \)
|
| $89$ |
\( T^{12} + \cdots - 2419536128 \)
|
| $97$ |
\( T^{12} - 15 T^{11} + \cdots - 57394368 \)
|
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