Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2299,2,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, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2299.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2299 = 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| 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: | \(18.3576074247\) |
| Analytic rank: | \(1\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - x^{13} - 17 x^{12} + 15 x^{11} + 107 x^{10} - 80 x^{9} - 310 x^{8} + 179 x^{7} + 415 x^{6} + \cdots - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - x^{13} - 17 x^{12} + 15 x^{11} + 107 x^{10} - 80 x^{9} - 310 x^{8} + 179 x^{7} + 415 x^{6} + \cdots - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{13} + 16 \nu^{11} - 91 \nu^{9} + 5 \nu^{8} + 224 \nu^{7} - 40 \nu^{6} - 228 \nu^{5} + 96 \nu^{4} + \cdots + 5 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{13} - 16 \nu^{11} + 92 \nu^{9} - 5 \nu^{8} - 237 \nu^{7} + 39 \nu^{6} + 281 \nu^{5} - 89 \nu^{4} + \cdots + 27 \nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{13} - 2 \nu^{12} + 15 \nu^{11} + 31 \nu^{10} - 75 \nu^{9} - 163 \nu^{8} + 143 \nu^{7} + \cdots - 10 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{13} - 2 \nu^{12} + 15 \nu^{11} + 31 \nu^{10} - 75 \nu^{9} - 163 \nu^{8} + 143 \nu^{7} + 347 \nu^{6} + \cdots - 6 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2 \nu^{13} + \nu^{12} - 31 \nu^{11} - 16 \nu^{10} + 167 \nu^{9} + 81 \nu^{8} - 376 \nu^{7} - 148 \nu^{6} + \cdots - 1 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2 \nu^{13} - 4 \nu^{12} + 31 \nu^{11} + 63 \nu^{10} - 165 \nu^{9} - 340 \nu^{8} + 360 \nu^{7} + \cdots - 6 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2 \nu^{13} - 6 \nu^{12} + 28 \nu^{11} + 93 \nu^{10} - 119 \nu^{9} - 491 \nu^{8} + 126 \nu^{7} + \cdots - 7 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 4 \nu^{13} - 4 \nu^{12} + 60 \nu^{11} + 63 \nu^{10} - 302 \nu^{9} - 329 \nu^{8} + 579 \nu^{7} + \cdots + 2 ) / 2 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 4 \nu^{13} - \nu^{12} + 63 \nu^{11} + 17 \nu^{10} - 349 \nu^{9} - 88 \nu^{8} + 823 \nu^{7} + 167 \nu^{6} + \cdots - 1 ) / 2 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 5 \nu^{13} + 5 \nu^{12} - 77 \nu^{11} - 79 \nu^{10} + 408 \nu^{9} + 417 \nu^{8} - 883 \nu^{7} - 877 \nu^{6} + \cdots + 4 ) / 2 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 7 \nu^{13} + 2 \nu^{12} - 110 \nu^{11} - 32 \nu^{10} + 607 \nu^{9} + 151 \nu^{8} - 1424 \nu^{7} + \cdots - 5 ) / 2 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 8 \nu^{13} - 9 \nu^{12} + 121 \nu^{11} + 142 \nu^{10} - 619 \nu^{9} - 751 \nu^{8} + 1238 \nu^{7} + \cdots - 9 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{13} + \beta_{11} - \beta_{9} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} - \beta_{12} + \beta_{11} - 2\beta_{10} - \beta_{8} + \beta_{7} - 6\beta_{5} + 5\beta_{4} - \beta_{2} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8 \beta_{13} - \beta_{12} + 10 \beta_{11} - \beta_{10} - 6 \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} - 21 \beta_{10} - 12 \beta_{8} + 9 \beta_{7} + \beta_{6} + \cdots + 39 \)
|
| \(\nu^{7}\) | \(=\) |
\( 57 \beta_{13} - 12 \beta_{12} + 80 \beta_{11} - 12 \beta_{10} - 35 \beta_{9} - 11 \beta_{8} + 23 \beta_{7} + \cdots + 47 \)
|
| \(\nu^{8}\) | \(=\) |
\( 93 \beta_{13} - 67 \beta_{12} + 71 \beta_{11} - 169 \beta_{10} - 2 \beta_{9} - 102 \beta_{8} + \cdots + 213 \)
|
| \(\nu^{9}\) | \(=\) |
\( 396 \beta_{13} - 105 \beta_{12} + 587 \beta_{11} - 110 \beta_{10} - 212 \beta_{9} - 95 \beta_{8} + \cdots + 323 \)
|
| \(\nu^{10}\) | \(=\) |
\( 717 \beta_{13} - 472 \beta_{12} + 546 \beta_{11} - 1237 \beta_{10} - 34 \beta_{9} - 769 \beta_{8} + \cdots + 1265 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2731 \beta_{13} - 823 \beta_{12} + 4137 \beta_{11} - 921 \beta_{10} - 1328 \beta_{9} - 760 \beta_{8} + \cdots + 2256 \)
|
| \(\nu^{12}\) | \(=\) |
\( 5304 \beta_{13} - 3263 \beta_{12} + 4157 \beta_{11} - 8682 \beta_{10} - 388 \beta_{9} - 5499 \beta_{8} + \cdots + 7977 \)
|
| \(\nu^{13}\) | \(=\) |
\( 18806 \beta_{13} - 6144 \beta_{12} + 28587 \beta_{11} - 7383 \beta_{10} - 8519 \beta_{9} - 5877 \beta_{8} + \cdots + 15868 \)
|
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 |
|
−2.48307 | −1.78784 | 4.16566 | −0.159606 | 4.43935 | 1.23706 | −5.37748 | 0.196384 | 0.396313 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.02502 | −0.0624840 | 2.10070 | −3.35497 | 0.126531 | −4.75396 | −0.203923 | −2.99610 | 6.79389 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.93293 | 2.02543 | 1.73621 | −2.10790 | −3.91500 | 3.28904 | 0.509892 | 1.10235 | 4.07443 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.04769 | 2.88767 | −0.902343 | −3.92698 | −3.02538 | −0.194286 | 3.04076 | 5.33862 | 4.11427 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.905029 | −0.561091 | −1.18092 | 2.75301 | 0.507804 | −0.551138 | 2.87883 | −2.68518 | −2.49156 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.366340 | −1.87701 | −1.86579 | −2.24160 | 0.687624 | −4.43568 | 1.41620 | 0.523161 | 0.821188 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.200027 | 1.22622 | −1.95999 | 0.0816230 | −0.245277 | 0.895749 | 0.792103 | −1.49638 | −0.0163268 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.154596 | 1.75951 | −1.97610 | −0.465506 | 0.272013 | 2.01956 | −0.614689 | 0.0958795 | −0.0719653 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.626946 | −1.68011 | −1.60694 | −4.03200 | −1.05334 | 3.53548 | −2.26136 | −0.177220 | −2.52785 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.849796 | −1.98168 | −1.27785 | 1.57133 | −1.68402 | 0.302738 | −2.78550 | 0.927060 | 1.33531 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.86285 | 0.912707 | 1.47022 | −0.733992 | 1.70024 | −0.296160 | −0.986904 | −2.16697 | −1.36732 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.86740 | 0.278036 | 1.48720 | 1.45924 | 0.519205 | −3.87596 | −0.957603 | −2.92270 | 2.72498 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.95517 | −2.86067 | 1.82269 | 0.394240 | −5.59309 | 0.353121 | −0.346674 | 5.18342 | 0.770806 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.64334 | −0.278680 | 4.98726 | −3.23687 | −0.736647 | −1.52557 | 7.89636 | −2.92234 | −8.55617 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 2299.2.a.u | 14 | |
| 11.b | odd | 2 | 1 | 2299.2.a.t | 14 | ||
| 11.d | odd | 10 | 2 | 209.2.f.b | ✓ | 28 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.f.b | ✓ | 28 | 11.d | odd | 10 | 2 | |
| 2299.2.a.t | 14 | 11.b | odd | 2 | 1 | ||
| 2299.2.a.u | 14 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\):
|
\( T_{2}^{14} - T_{2}^{13} - 17 T_{2}^{12} + 15 T_{2}^{11} + 107 T_{2}^{10} - 80 T_{2}^{9} - 310 T_{2}^{8} + \cdots - 1 \)
|
|
\( T_{3}^{14} + 2 T_{3}^{13} - 18 T_{3}^{12} - 36 T_{3}^{11} + 113 T_{3}^{10} + 224 T_{3}^{9} - 315 T_{3}^{8} + \cdots + 1 \)
|
|
\( T_{7}^{14} + 4 T_{7}^{13} - 37 T_{7}^{12} - 114 T_{7}^{11} + 529 T_{7}^{10} + 920 T_{7}^{9} - 3313 T_{7}^{8} + \cdots - 11 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} - T^{13} + \cdots - 1 \)
|
| $3$ |
\( T^{14} + 2 T^{13} + \cdots + 1 \)
|
| $5$ |
\( T^{14} + 14 T^{13} + \cdots - 9 \)
|
| $7$ |
\( T^{14} + 4 T^{13} + \cdots - 11 \)
|
| $11$ |
\( T^{14} \)
|
| $13$ |
\( T^{14} - 5 T^{13} + \cdots + 12419 \)
|
| $17$ |
\( T^{14} + 10 T^{13} + \cdots + 2737744 \)
|
| $19$ |
\( (T - 1)^{14} \)
|
| $23$ |
\( T^{14} + 21 T^{13} + \cdots - 2496119 \)
|
| $29$ |
\( T^{14} - 3 T^{13} + \cdots + 1961455 \)
|
| $31$ |
\( T^{14} + 13 T^{13} + \cdots - 98865679 \)
|
| $37$ |
\( T^{14} + 31 T^{13} + \cdots - 34143901 \)
|
| $41$ |
\( T^{14} + \cdots + 532426169 \)
|
| $43$ |
\( T^{14} - 6 T^{13} + \cdots + 11826491 \)
|
| $47$ |
\( T^{14} + \cdots + 30365724429 \)
|
| $53$ |
\( T^{14} + 42 T^{13} + \cdots - 1220029 \)
|
| $59$ |
\( T^{14} + \cdots + 179630369395 \)
|
| $61$ |
\( T^{14} + \cdots + 724582251 \)
|
| $67$ |
\( T^{14} + \cdots + 19444726741 \)
|
| $71$ |
\( T^{14} + \cdots + 3465674591 \)
|
| $73$ |
\( T^{14} + \cdots - 137430711 \)
|
| $79$ |
\( T^{14} + \cdots + 1933875295 \)
|
| $83$ |
\( T^{14} + \cdots - 1241650112679 \)
|
| $89$ |
\( T^{14} + \cdots - 9538214501275 \)
|
| $97$ |
\( T^{14} + \cdots + 254469301 \)
|
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