Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2299,4,Mod(1,2299)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2299, 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("2299.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2299 = 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2299.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: | \(135.645391103\) |
| 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: |
\( x^{10} - 4 x^{9} - 38 x^{8} + 128 x^{7} + 499 x^{6} - 1156 x^{5} - 2730 x^{4} + 2748 x^{3} + 4628 x^{2} + \cdots - 1952 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 209) |
| 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
\( x^{10} - 4 x^{9} - 38 x^{8} + 128 x^{7} + 499 x^{6} - 1156 x^{5} - 2730 x^{4} + 2748 x^{3} + 4628 x^{2} + \cdots - 1952 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5889 \nu^{9} + 22985 \nu^{8} - 430832 \nu^{7} - 827470 \nu^{6} + 9118029 \nu^{5} + 11143105 \nu^{4} + \cdots + 77877760 ) / 4171412 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 23661 \nu^{9} + 84558 \nu^{8} + 809814 \nu^{7} - 2105852 \nu^{6} - 9169883 \nu^{5} + \cdots - 101515464 ) / 8342824 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29615 \nu^{9} - 121174 \nu^{8} - 1171734 \nu^{7} + 3737104 \nu^{6} + 16979665 \nu^{5} + \cdots - 49128816 ) / 8342824 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 15468 \nu^{9} + 48535 \nu^{8} + 639678 \nu^{7} - 1551732 \nu^{6} - 8945646 \nu^{5} + \cdots + 28999224 ) / 4171412 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 8157 \nu^{9} + 35101 \nu^{8} + 304566 \nu^{7} - 1021370 \nu^{6} - 4013188 \nu^{5} + \cdots - 14978406 ) / 2085706 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 25792 \nu^{9} + 81199 \nu^{8} + 1021320 \nu^{7} - 2430470 \nu^{6} - 13994044 \nu^{5} + \cdots - 608584 ) / 4171412 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 70843 \nu^{9} - 365961 \nu^{8} - 2256990 \nu^{7} + 11610816 \nu^{6} + 21503271 \nu^{5} + \cdots - 89837780 ) / 4171412 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 16\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + 3\beta_{8} + 3\beta_{7} + \beta_{6} + 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 20\beta_{2} + 28\beta _1 + 148 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + 24 \beta_{8} + 34 \beta_{7} - 16 \beta_{6} + 35 \beta_{5} - 27 \beta_{4} + 3 \beta_{3} + \cdots + 229 \)
|
| \(\nu^{6}\) | \(=\) |
\( 34 \beta_{9} + 96 \beta_{8} + 128 \beta_{7} + 30 \beta_{6} + 86 \beta_{5} - 66 \beta_{4} + 96 \beta_{3} + \cdots + 2779 \)
|
| \(\nu^{7}\) | \(=\) |
\( 52 \beta_{9} + 543 \beta_{8} + 959 \beta_{7} - 229 \beta_{6} + 917 \beta_{5} - 681 \beta_{4} + \cdots + 5640 \)
|
| \(\nu^{8}\) | \(=\) |
\( 865 \beta_{9} + 2403 \beta_{8} + 3999 \beta_{7} + 681 \beta_{6} + 2630 \beta_{5} - 1982 \beta_{4} + \cdots + 55036 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1765 \beta_{9} + 12028 \beta_{8} + 25246 \beta_{7} - 3344 \beta_{6} + 21959 \beta_{5} - 16799 \beta_{4} + \cdots + 134205 \)
|
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 |
|
−3.83898 | −4.61833 | 6.73778 | −3.46804 | 17.7297 | −5.73385 | 4.84564 | −5.67104 | 13.3137 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −3.40746 | 3.18502 | 3.61081 | −19.4027 | −10.8528 | 8.97513 | 14.9560 | −16.8557 | 66.1141 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.64121 | −1.42142 | −1.02399 | 15.2418 | 3.75427 | 33.7413 | 23.8343 | −24.9796 | −40.2570 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.359183 | 5.32423 | −7.87099 | 1.62705 | −1.91237 | 14.9146 | 5.70058 | 1.34748 | −0.584408 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0787876 | −6.76160 | −7.99379 | 3.67867 | −0.532730 | −32.3926 | −1.26011 | 18.7192 | 0.289833 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.59567 | −8.62487 | −5.45385 | 16.8045 | −13.7624 | 28.5770 | −21.4678 | 47.3883 | 26.8144 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.36428 | 0.664333 | −2.41017 | −1.33164 | 1.57067 | −10.1376 | −24.6126 | −26.5587 | −3.14837 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.89060 | 9.67478 | 0.355548 | −16.5808 | 27.9659 | 8.25844 | −22.0970 | 66.6014 | −47.9284 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 4.09278 | −7.59208 | 8.75083 | −13.8757 | −31.0727 | 7.43945 | 3.07300 | 30.6397 | −56.7903 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 5.22473 | 1.16993 | 19.2978 | 7.30686 | 6.11255 | −0.641868 | 59.0281 | −25.6313 | 38.1764 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 2299.4.a.j | 10 | |
| 11.b | odd | 2 | 1 | 209.4.a.b | ✓ | 10 | |
| 33.d | even | 2 | 1 | 1881.4.a.e | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.4.a.b | ✓ | 10 | 11.b | odd | 2 | 1 | |
| 1881.4.a.e | 10 | 33.d | even | 2 | 1 | ||
| 2299.4.a.j | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2299))\):
|
\( T_{2}^{10} - 6 T_{2}^{9} - 29 T_{2}^{8} + 200 T_{2}^{7} + 205 T_{2}^{6} - 2146 T_{2}^{5} + 501 T_{2}^{4} + \cdots + 228 \)
|
|
\( T_{5}^{10} + 10 T_{5}^{9} - 671 T_{5}^{8} - 5094 T_{5}^{7} + 145960 T_{5}^{6} + 660422 T_{5}^{5} + \cdots - 230929132 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 6 T^{9} + \cdots + 228 \)
|
| $3$ |
\( T^{10} + 9 T^{9} + \cdots - 370616 \)
|
| $5$ |
\( T^{10} + \cdots - 230929132 \)
|
| $7$ |
\( T^{10} + \cdots + 9583861392 \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} + \cdots + 61\!\cdots\!12 \)
|
| $17$ |
\( T^{10} + \cdots + 18\!\cdots\!56 \)
|
| $19$ |
\( (T - 19)^{10} \)
|
| $23$ |
\( T^{10} + \cdots - 32\!\cdots\!04 \)
|
| $29$ |
\( T^{10} + \cdots + 26\!\cdots\!32 \)
|
| $31$ |
\( T^{10} + \cdots - 10\!\cdots\!36 \)
|
| $37$ |
\( T^{10} + \cdots - 27\!\cdots\!88 \)
|
| $41$ |
\( T^{10} + \cdots + 18\!\cdots\!08 \)
|
| $43$ |
\( T^{10} + \cdots - 94\!\cdots\!56 \)
|
| $47$ |
\( T^{10} + \cdots - 10\!\cdots\!48 \)
|
| $53$ |
\( T^{10} + \cdots + 19\!\cdots\!44 \)
|
| $59$ |
\( T^{10} + \cdots + 11\!\cdots\!96 \)
|
| $61$ |
\( T^{10} + \cdots + 17\!\cdots\!72 \)
|
| $67$ |
\( T^{10} + \cdots - 16\!\cdots\!56 \)
|
| $71$ |
\( T^{10} + \cdots - 15\!\cdots\!52 \)
|
| $73$ |
\( T^{10} + \cdots + 31\!\cdots\!88 \)
|
| $79$ |
\( T^{10} + \cdots + 13\!\cdots\!16 \)
|
| $83$ |
\( T^{10} + \cdots - 15\!\cdots\!76 \)
|
| $89$ |
\( T^{10} + \cdots - 47\!\cdots\!84 \)
|
| $97$ |
\( T^{10} + \cdots - 14\!\cdots\!04 \)
|
| show more | |
| show less |