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: | \(1\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 6 x^{17} - 90 x^{16} + 574 x^{15} + 3140 x^{14} - 22274 x^{13} - 51450 x^{12} + 448726 x^{11} + \cdots + 2334720 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 6 x^{17} - 90 x^{16} + 574 x^{15} + 3140 x^{14} - 22274 x^{13} - 51450 x^{12} + 448726 x^{11} + \cdots + 2334720 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\!\cdots\!66 \nu^{17} + \cdots - 26\!\cdots\!16 ) / 91\!\cdots\!04 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 21\!\cdots\!86 \nu^{17} + \cdots + 48\!\cdots\!08 ) / 91\!\cdots\!04 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25\!\cdots\!41 \nu^{17} + \cdots + 37\!\cdots\!88 ) / 91\!\cdots\!04 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 69\!\cdots\!73 \nu^{17} + \cdots + 39\!\cdots\!84 ) / 22\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 46\!\cdots\!07 \nu^{17} + \cdots - 96\!\cdots\!96 ) / 91\!\cdots\!04 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 49\!\cdots\!78 \nu^{17} + \cdots - 98\!\cdots\!96 ) / 91\!\cdots\!04 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 55\!\cdots\!07 \nu^{17} + \cdots + 50\!\cdots\!20 ) / 91\!\cdots\!04 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 68\!\cdots\!18 \nu^{17} + \cdots + 10\!\cdots\!12 ) / 91\!\cdots\!04 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 87\!\cdots\!55 \nu^{17} + \cdots - 48\!\cdots\!44 ) / 91\!\cdots\!04 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 88\!\cdots\!11 \nu^{17} + \cdots + 21\!\cdots\!00 ) / 91\!\cdots\!04 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 13\!\cdots\!89 \nu^{17} + \cdots - 33\!\cdots\!28 ) / 91\!\cdots\!04 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 18\!\cdots\!27 \nu^{17} + \cdots - 33\!\cdots\!12 ) / 91\!\cdots\!04 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 23\!\cdots\!23 \nu^{17} + \cdots - 41\!\cdots\!08 ) / 91\!\cdots\!04 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 29\!\cdots\!52 \nu^{17} + \cdots + 31\!\cdots\!56 ) / 91\!\cdots\!04 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 34\!\cdots\!81 \nu^{17} + \cdots - 24\!\cdots\!40 ) / 91\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{6} + \beta_{4} + \beta_{2} + 19\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{16} + \beta_{13} + \beta_{12} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} + 3 \beta_{6} + \cdots + 225 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{17} + 3 \beta_{16} + 3 \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + \cdots + 53 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 4 \beta_{17} + 43 \beta_{16} + 4 \beta_{15} + 8 \beta_{14} + 47 \beta_{13} + 27 \beta_{12} + \cdots + 4841 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 82 \beta_{17} + 147 \beta_{16} + 117 \beta_{15} + 57 \beta_{14} + 91 \beta_{13} - 118 \beta_{12} + \cdots + 2959 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 168 \beta_{17} + 1439 \beta_{16} + 174 \beta_{15} + 476 \beta_{14} + 1729 \beta_{13} + 455 \beta_{12} + \cdots + 112731 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2406 \beta_{17} + 5403 \beta_{16} + 3447 \beta_{15} + 2533 \beta_{14} + 4633 \beta_{13} + \cdots + 119933 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 4892 \beta_{17} + 44511 \beta_{16} + 5436 \beta_{15} + 19804 \beta_{14} + 58111 \beta_{13} + \cdots + 2769809 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 61802 \beta_{17} + 179867 \beta_{16} + 92161 \beta_{15} + 98657 \beta_{14} + 189655 \beta_{13} + \cdots + 4316243 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 120944 \beta_{17} + 1337343 \beta_{16} + 151538 \beta_{15} + 715540 \beta_{14} + 1868285 \beta_{13} + \cdots + 70854567 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1476142 \beta_{17} + 5734875 \beta_{16} + 2366667 \beta_{15} + 3541421 \beta_{14} + 6968437 \beta_{13} + \cdots + 146471353 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 2668948 \beta_{17} + 39776007 \beta_{16} + 4058400 \beta_{15} + 24115796 \beta_{14} + \cdots + 1871928965 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 33469218 \beta_{17} + 179069539 \beta_{16} + 59826957 \beta_{15} + 120699649 \beta_{14} + \cdots + 4805230199 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 52122376 \beta_{17} + 1180733799 \beta_{16} + 107987766 \beta_{15} + 782413628 \beta_{14} + \cdots + 50789070427 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 721930726 \beta_{17} + 5532118931 \beta_{16} + 1507761927 \beta_{15} + 3977375933 \beta_{14} + \cdots + 154319134221 \)
|
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 |
|
−5.52634 | −6.93606 | 22.5404 | −20.1287 | 38.3310 | 19.4784 | −80.3550 | 21.1090 | 111.238 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.88721 | 9.51146 | 15.8848 | −20.7883 | −46.4845 | −19.9656 | −38.5349 | 63.4679 | 101.597 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.38401 | −3.40421 | 11.2195 | 10.5795 | 14.9241 | 33.0336 | −14.1145 | −15.4114 | −46.3806 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.93998 | −2.40498 | 7.52342 | −10.1845 | 9.47556 | −32.6275 | 1.87773 | −21.2161 | 40.1266 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.49487 | 7.26127 | 4.21412 | 10.8334 | −25.3772 | −13.8581 | 13.2312 | 25.7260 | −37.8612 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.81206 | 4.39643 | −0.0923432 | −16.8494 | −12.3630 | 12.9158 | 22.7561 | −7.67144 | 47.3813 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −2.21271 | 2.70882 | −3.10390 | 2.73676 | −5.99385 | 11.5141 | 24.5697 | −19.6623 | −6.05566 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −2.08014 | −6.72517 | −3.67300 | 1.31825 | 13.9893 | 7.10424 | 24.2815 | 18.2280 | −2.74215 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.15150 | −7.57076 | −6.67404 | 18.3685 | 8.71774 | −25.2111 | 16.8972 | 30.3164 | −21.1513 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −0.105296 | −0.343644 | −7.98891 | −16.0139 | 0.0361843 | −22.6655 | 1.68357 | −26.8819 | 1.68620 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.358037 | −9.51080 | −7.87181 | −14.0403 | −3.40522 | 35.1451 | −5.68270 | 63.4552 | −5.02694 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.837228 | 1.70494 | −7.29905 | 11.3759 | 1.42743 | −27.5404 | −12.8088 | −24.0932 | 9.52418 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.61431 | 3.64054 | −1.16538 | 0.302892 | 9.51751 | 27.0597 | −23.9611 | −13.7464 | 0.791853 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 3.15480 | −5.38807 | 1.95277 | −14.4007 | −16.9983 | −5.91572 | −19.0778 | 2.03128 | −45.4314 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 3.74414 | 9.42877 | 6.01859 | 4.00968 | 35.3026 | −31.4657 | −7.41866 | 61.9016 | 15.0128 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 4.22467 | −9.90185 | 9.84786 | 6.64131 | −41.8321 | −0.361822 | 7.80659 | 71.0466 | 28.0573 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 4.82054 | 2.55220 | 15.2376 | −8.04245 | 12.3030 | −14.9441 | 34.8890 | −20.4863 | −38.7689 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 4.84039 | 2.98110 | 15.4294 | −3.71799 | 14.4297 | 4.30459 | 35.9609 | −18.1130 | −17.9965 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.m | ✓ | 18 |
| 11.b | odd | 2 | 1 | 2299.4.a.n | yes | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2299.4.a.m | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 2299.4.a.n | yes | 18 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2299))\):
|
\( T_{2}^{18} + 6 T_{2}^{17} - 90 T_{2}^{16} - 574 T_{2}^{15} + 3140 T_{2}^{14} + 22274 T_{2}^{13} + \cdots + 2334720 \)
|
|
\( T_{5}^{18} + 58 T_{5}^{17} + 317 T_{5}^{16} - 39016 T_{5}^{15} - 638304 T_{5}^{14} + \cdots - 48\!\cdots\!72 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} + 6 T^{17} + \cdots + 2334720 \)
|
| $3$ |
\( T^{18} + \cdots - 184638336588 \)
|
| $5$ |
\( T^{18} + \cdots - 48\!\cdots\!72 \)
|
| $7$ |
\( T^{18} + \cdots + 39\!\cdots\!60 \)
|
| $11$ |
\( T^{18} \)
|
| $13$ |
\( T^{18} + \cdots - 32\!\cdots\!00 \)
|
| $17$ |
\( T^{18} + \cdots - 21\!\cdots\!60 \)
|
| $19$ |
\( (T - 19)^{18} \)
|
| $23$ |
\( T^{18} + \cdots - 30\!\cdots\!40 \)
|
| $29$ |
\( T^{18} + \cdots - 19\!\cdots\!20 \)
|
| $31$ |
\( T^{18} + \cdots - 12\!\cdots\!00 \)
|
| $37$ |
\( T^{18} + \cdots + 16\!\cdots\!00 \)
|
| $41$ |
\( T^{18} + \cdots + 38\!\cdots\!00 \)
|
| $43$ |
\( T^{18} + \cdots - 66\!\cdots\!80 \)
|
| $47$ |
\( T^{18} + \cdots + 53\!\cdots\!96 \)
|
| $53$ |
\( T^{18} + \cdots - 11\!\cdots\!00 \)
|
| $59$ |
\( T^{18} + \cdots + 62\!\cdots\!80 \)
|
| $61$ |
\( T^{18} + \cdots + 13\!\cdots\!60 \)
|
| $67$ |
\( T^{18} + \cdots - 20\!\cdots\!84 \)
|
| $71$ |
\( T^{18} + \cdots - 96\!\cdots\!60 \)
|
| $73$ |
\( T^{18} + \cdots - 45\!\cdots\!80 \)
|
| $79$ |
\( T^{18} + \cdots - 12\!\cdots\!00 \)
|
| $83$ |
\( T^{18} + \cdots - 93\!\cdots\!40 \)
|
| $89$ |
\( T^{18} + \cdots - 46\!\cdots\!16 \)
|
| $97$ |
\( T^{18} + \cdots - 78\!\cdots\!84 \)
|
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