Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5780,2,Mod(1,5780)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5780, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("5780.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5780.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: | \(46.1535323683\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 24x^{10} + 206x^{8} - 16x^{7} - 776x^{6} + 152x^{5} + 1226x^{4} - 384x^{3} - 588x^{2} + 200x + 34 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 340) |
| 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
\( x^{12} - 24x^{10} + 206x^{8} - 16x^{7} - 776x^{6} + 152x^{5} + 1226x^{4} - 384x^{3} - 588x^{2} + 200x + 34 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13053 \nu^{11} - 19736 \nu^{10} - 329634 \nu^{9} + 631844 \nu^{8} + 2960146 \nu^{7} + \cdots + 10567812 ) / 5841353 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 19736 \nu^{11} + 16362 \nu^{10} - 631844 \nu^{9} - 271228 \nu^{8} + 6896805 \nu^{7} + \cdots - 5397551 ) / 5841353 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 135425 \nu^{11} - 271796 \nu^{10} + 3134166 \nu^{9} + 5961993 \nu^{8} - 24896785 \nu^{7} + \cdots + 24875675 ) / 5841353 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 140626 \nu^{11} + 112715 \nu^{10} - 3297561 \nu^{9} - 2605786 \nu^{8} + 26952296 \nu^{7} + \cdots - 11744552 ) / 5841353 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 155784 \nu^{11} - 18945 \nu^{10} - 3565979 \nu^{9} + 283535 \nu^{8} + 28463791 \nu^{7} + \cdots + 19717433 ) / 5841353 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 684 \nu^{11} + 724 \nu^{10} - 16054 \nu^{9} - 16049 \nu^{8} + 132402 \nu^{7} + 111029 \nu^{6} + \cdots - 7548 ) / 22729 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 724 \nu^{11} - 362 \nu^{10} + 16049 \nu^{9} + 8502 \nu^{8} - 121973 \nu^{7} - 57520 \nu^{6} + \cdots - 44931 ) / 22729 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 724 \nu^{11} + 362 \nu^{10} - 16049 \nu^{9} - 8502 \nu^{8} + 121973 \nu^{7} + 57520 \nu^{6} + \cdots - 45985 ) / 22729 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 186258 \nu^{11} + 180989 \nu^{10} - 4443205 \nu^{9} - 4289273 \nu^{8} + 37600762 \nu^{7} + \cdots - 44070528 ) / 5841353 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 313440 \nu^{11} + 120812 \nu^{10} - 7526903 \nu^{9} - 3056274 \nu^{8} + 64266723 \nu^{7} + \cdots + 6853805 ) / 5841353 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{8} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{11} + \beta_{8} + 2\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{10} + 10\beta_{9} + 10\beta_{8} + 3\beta_{7} + \beta_{4} + \beta_{3} - 3\beta_{2} + \beta _1 + 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 12 \beta_{11} + 3 \beta_{9} + 13 \beta_{8} + 24 \beta_{7} + 13 \beta_{6} + 7 \beta_{5} + 11 \beta_{4} + \cdots + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 3 \beta_{11} - 25 \beta_{10} + 97 \beta_{9} + 92 \beta_{8} + 43 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots + 236 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 122 \beta_{11} - 2 \beta_{10} + 69 \beta_{9} + 155 \beta_{8} + 263 \beta_{7} + 135 \beta_{6} + \cdots + 163 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 71 \beta_{11} - 257 \beta_{10} + 951 \beta_{9} + 864 \beta_{8} + 520 \beta_{7} - 5 \beta_{6} + \cdots + 2170 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1208 \beta_{11} - 66 \beta_{10} + 1078 \beta_{9} + 1789 \beta_{8} + 2809 \beta_{7} + 1329 \beta_{6} + \cdots + 2398 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1144 \beta_{11} - 2537 \beta_{10} + 9447 \beta_{9} + 8404 \beta_{8} + 6037 \beta_{7} + 127 \beta_{6} + \cdots + 20914 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 11984 \beta_{11} - 1271 \beta_{10} + 14399 \beta_{9} + 20209 \beta_{8} + 29676 \beta_{7} + \cdots + 31050 \)
|
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 | −2.89366 | 0 | −1.00000 | 0 | 3.07222 | 0 | 5.37326 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.68860 | 0 | −1.00000 | 0 | 1.08148 | 0 | 4.22856 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.27331 | 0 | −1.00000 | 0 | −2.37337 | 0 | 2.16796 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.70227 | 0 | −1.00000 | 0 | −4.39180 | 0 | −0.102270 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.833172 | 0 | −1.00000 | 0 | −1.87711 | 0 | −2.30582 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.127664 | 0 | −1.00000 | 0 | 3.09319 | 0 | −2.98370 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.567193 | 0 | −1.00000 | 0 | −2.93861 | 0 | −2.67829 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.815249 | 0 | −1.00000 | 0 | 2.85898 | 0 | −2.33537 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.47256 | 0 | −1.00000 | 0 | 1.27134 | 0 | −0.831572 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.96743 | 0 | −1.00000 | 0 | −4.13403 | 0 | 0.870795 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.41635 | 0 | −1.00000 | 0 | 0.733789 | 0 | 2.83875 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.27990 | 0 | −1.00000 | 0 | −4.39608 | 0 | 7.75771 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(17\) | \(-1\) |
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 | 5780.2.a.q | 12 | |
| 17.b | even | 2 | 1 | 5780.2.a.r | 12 | ||
| 17.c | even | 4 | 2 | 5780.2.c.j | 24 | ||
| 17.e | odd | 16 | 2 | 340.2.u.a | ✓ | 24 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 340.2.u.a | ✓ | 24 | 17.e | odd | 16 | 2 | |
| 5780.2.a.q | 12 | 1.a | even | 1 | 1 | trivial | |
| 5780.2.a.r | 12 | 17.b | even | 2 | 1 | ||
| 5780.2.c.j | 24 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5780))\):
|
\( T_{3}^{12} - 24 T_{3}^{10} + 206 T_{3}^{8} - 16 T_{3}^{7} - 776 T_{3}^{6} + 152 T_{3}^{5} + 1226 T_{3}^{4} + \cdots + 34 \)
|
|
\( T_{7}^{12} + 8 T_{7}^{11} - 20 T_{7}^{10} - 264 T_{7}^{9} + 40 T_{7}^{8} + 3312 T_{7}^{7} + 1260 T_{7}^{6} + \cdots + 28642 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} - 24 T^{10} + \cdots + 34 \)
|
| $5$ |
\( (T + 1)^{12} \)
|
| $7$ |
\( T^{12} + 8 T^{11} + \cdots + 28642 \)
|
| $11$ |
\( T^{12} - 60 T^{10} + \cdots - 9886 \)
|
| $13$ |
\( T^{12} - 8 T^{11} + \cdots + 68 \)
|
| $17$ |
\( T^{12} \)
|
| $19$ |
\( T^{12} - 108 T^{10} + \cdots + 3804224 \)
|
| $23$ |
\( T^{12} + 8 T^{11} + \cdots - 9214 \)
|
| $29$ |
\( T^{12} + 16 T^{11} + \cdots + 7172488 \)
|
| $31$ |
\( T^{12} + 24 T^{11} + \cdots + 578 \)
|
| $37$ |
\( T^{12} + 24 T^{11} + \cdots + 47950472 \)
|
| $41$ |
\( T^{12} + 24 T^{11} + \cdots + 1591424 \)
|
| $43$ |
\( T^{12} + \cdots + 105998276 \)
|
| $47$ |
\( T^{12} + \cdots - 527676668 \)
|
| $53$ |
\( T^{12} - 16 T^{11} + \cdots + 464912 \)
|
| $59$ |
\( T^{12} - 8 T^{11} + \cdots + 70258688 \)
|
| $61$ |
\( T^{12} + \cdots + 553702432 \)
|
| $67$ |
\( T^{12} - 16 T^{11} + \cdots + 31396292 \)
|
| $71$ |
\( T^{12} + \cdots - 725790526 \)
|
| $73$ |
\( T^{12} + \cdots + 305848864 \)
|
| $79$ |
\( T^{12} + \cdots - 108160311998 \)
|
| $83$ |
\( T^{12} + \cdots - 41340702076 \)
|
| $89$ |
\( T^{12} + \cdots - 35522326844 \)
|
| $97$ |
\( T^{12} + \cdots + 100254773768 \)
|
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