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: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 6 x^{15} - 7 x^{14} + 98 x^{13} - 50 x^{12} - 620 x^{11} + 642 x^{10} + 1958 x^{9} - 2445 x^{8} + \cdots - 143 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 6 x^{15} - 7 x^{14} + 98 x^{13} - 50 x^{12} - 620 x^{11} + 642 x^{10} + 1958 x^{9} - 2445 x^{8} + \cdots - 143 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 50 \nu^{15} + 253 \nu^{14} + 699 \nu^{13} - 4825 \nu^{12} - 2853 \nu^{11} + 36709 \nu^{10} + \cdots - 14863 ) / 981 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 252 \nu^{15} + 1310 \nu^{14} + 2821 \nu^{13} - 22574 \nu^{12} - 5149 \nu^{11} + 153944 \nu^{10} + \cdots - 51880 ) / 981 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 598 \nu^{15} + 2882 \nu^{14} + 7536 \nu^{13} - 50186 \nu^{12} - 25803 \nu^{11} + 345740 \nu^{10} + \cdots - 116861 ) / 981 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 571 \nu^{15} + 2983 \nu^{14} + 6280 \nu^{13} - 50905 \nu^{12} - 10657 \nu^{11} + 343198 \nu^{10} + \cdots - 119493 ) / 981 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 539 \nu^{15} + 2917 \nu^{14} + 5619 \nu^{13} - 49888 \nu^{12} - 4785 \nu^{11} + 337759 \nu^{10} + \cdots - 114829 ) / 981 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 676 \nu^{15} - 3656 \nu^{14} - 7083 \nu^{13} + 62618 \nu^{12} + 6435 \nu^{11} - 424418 \nu^{10} + \cdots + 143108 ) / 981 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 803 \nu^{15} - 4279 \nu^{14} - 8538 \nu^{13} + 73075 \nu^{12} + 9738 \nu^{11} - 493075 \nu^{10} + \cdots + 161809 ) / 981 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 912 \nu^{15} - 4715 \nu^{14} - 9955 \nu^{13} + 79724 \nu^{12} + 16060 \nu^{11} - 530462 \nu^{10} + \cdots + 163444 ) / 981 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 884 \nu^{15} - 4739 \nu^{14} - 9254 \nu^{13} + 80837 \nu^{12} + 8306 \nu^{11} - 545282 \nu^{10} + \cdots + 193044 ) / 981 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1164 \nu^{15} + 6025 \nu^{14} + 12776 \nu^{13} - 102298 \nu^{12} - 21209 \nu^{11} + 684406 \nu^{10} + \cdots - 213362 ) / 981 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 1232 \nu^{15} + 6356 \nu^{14} + 13622 \nu^{13} - 108206 \nu^{12} - 23441 \nu^{11} + 726116 \nu^{10} + \cdots - 233301 ) / 981 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 1404 \nu^{15} + 7283 \nu^{14} + 15499 \nu^{13} - 124259 \nu^{12} - 26554 \nu^{11} + 836978 \nu^{10} + \cdots - 269161 ) / 981 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 1610 \nu^{15} + 8648 \nu^{14} + 17036 \nu^{13} - 148280 \nu^{12} - 17594 \nu^{11} + 1006409 \nu^{10} + \cdots - 345129 ) / 981 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} + \beta_{10} - \beta_{4} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + \beta_{11} + \beta_{8} - \beta_{3} + 7\beta_{2} + 8\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} - \beta_{14} + 8 \beta_{12} + \beta_{11} + 7 \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{15} - \beta_{14} + 2 \beta_{12} + 10 \beta_{11} + 2 \beta_{10} - \beta_{9} + 11 \beta_{8} + \cdots + 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13 \beta_{15} - 12 \beta_{14} + \beta_{13} + 55 \beta_{12} + 14 \beta_{11} + 45 \beta_{10} + 11 \beta_{9} + \cdots + 93 \)
|
| \(\nu^{8}\) | \(=\) |
\( 80 \beta_{15} - 14 \beta_{14} + \beta_{13} + 29 \beta_{12} + 82 \beta_{11} + 29 \beta_{10} - 10 \beta_{9} + \cdots + 540 \)
|
| \(\nu^{9}\) | \(=\) |
\( 124 \beta_{15} - 107 \beta_{14} + 14 \beta_{13} + 369 \beta_{12} + 139 \beta_{11} + 293 \beta_{10} + \cdots + 728 \)
|
| \(\nu^{10}\) | \(=\) |
\( 601 \beta_{15} - 143 \beta_{14} + 18 \beta_{13} + 297 \beta_{12} + 633 \beta_{11} + 293 \beta_{10} + \cdots + 3492 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1058 \beta_{15} - 855 \beta_{14} + 140 \beta_{13} + 2480 \beta_{12} + 1213 \beta_{11} + 1954 \beta_{10} + \cdots + 5527 \)
|
| \(\nu^{12}\) | \(=\) |
\( 4417 \beta_{15} - 1292 \beta_{14} + 218 \beta_{13} + 2655 \beta_{12} + 4766 \beta_{11} + 2574 \beta_{10} + \cdots + 23225 \)
|
| \(\nu^{13}\) | \(=\) |
\( 8561 \beta_{15} - 6496 \beta_{14} + 1236 \beta_{13} + 16814 \beta_{12} + 9949 \beta_{11} + 13313 \beta_{10} + \cdots + 41361 \)
|
| \(\nu^{14}\) | \(=\) |
\( 32218 \beta_{15} - 10942 \beta_{14} + 2231 \beta_{13} + 22144 \beta_{12} + 35490 \beta_{11} + \cdots + 157701 \)
|
| \(\nu^{15}\) | \(=\) |
\( 67260 \beta_{15} - 48176 \beta_{14} + 10312 \beta_{13} + 115172 \beta_{12} + 78888 \beta_{11} + \cdots + 307238 \)
|
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 |
|
−2.39803 | −2.07445 | 3.75053 | 1.72048 | 4.97458 | 4.03440 | −4.19782 | 1.30333 | −4.12575 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.98020 | −0.356066 | 1.92119 | −0.216560 | 0.705081 | −0.588879 | 0.156056 | −2.87322 | 0.428832 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.62343 | 1.83615 | 0.635531 | 0.228119 | −2.98086 | 5.09130 | 2.21512 | 0.371437 | −0.370336 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.35230 | 2.07433 | −0.171290 | −1.78146 | −2.80512 | −0.741185 | 2.93623 | 1.30286 | 2.40906 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.14863 | −1.93162 | −0.680658 | 0.935257 | 2.21870 | −2.40530 | 3.07907 | 0.731137 | −1.07426 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.822168 | −3.18869 | −1.32404 | 3.52839 | 2.62164 | −1.36337 | 2.73292 | 7.16777 | −2.90093 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.408195 | −1.70305 | −1.83338 | −1.20169 | 0.695175 | 1.38189 | 1.56477 | −0.0996374 | 0.490526 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.514549 | 0.898143 | −1.73524 | −4.15188 | 0.462139 | −0.640143 | −1.92196 | −2.19334 | −2.13635 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.701596 | 1.14830 | −1.50776 | 3.76385 | 0.805645 | 1.01624 | −2.46103 | −1.68140 | 2.64070 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.713407 | 0.0834579 | −1.49105 | 0.324859 | 0.0595394 | −4.45265 | −2.49054 | −2.99303 | 0.231756 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.67822 | 2.97280 | 0.816413 | −0.831548 | 4.98901 | 4.40715 | −1.98632 | 5.83757 | −1.39552 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.13091 | 1.11226 | 2.54077 | 3.03030 | 2.37013 | 2.53524 | 1.15233 | −1.76287 | 6.45728 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.26843 | −1.51173 | 3.14579 | −3.73652 | −3.42925 | 2.29899 | 2.59914 | −0.714686 | −8.47605 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.45204 | −3.22909 | 4.01251 | −2.63485 | −7.91787 | −4.98078 | 4.93476 | 7.42704 | −6.46076 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.55255 | −0.858433 | 4.51550 | 2.98015 | −2.19119 | 1.30961 | 6.42092 | −2.26309 | 7.60698 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.72125 | 2.72766 | 5.40519 | −1.95689 | 7.42265 | −2.90252 | 9.26636 | 4.44015 | −5.32518 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.w | yes | 16 |
| 11.b | odd | 2 | 1 | 2299.2.a.v | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2299.2.a.v | ✓ | 16 | 11.b | odd | 2 | 1 | |
| 2299.2.a.w | yes | 16 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\):
|
\( T_{2}^{16} - 6 T_{2}^{15} - 7 T_{2}^{14} + 98 T_{2}^{13} - 50 T_{2}^{12} - 620 T_{2}^{11} + 642 T_{2}^{10} + \cdots - 143 \)
|
|
\( T_{3}^{16} + 2 T_{3}^{15} - 29 T_{3}^{14} - 52 T_{3}^{13} + 330 T_{3}^{12} + 520 T_{3}^{11} - 1892 T_{3}^{10} + \cdots + 96 \)
|
|
\( T_{7}^{16} - 4 T_{7}^{15} - 62 T_{7}^{14} + 248 T_{7}^{13} + 1359 T_{7}^{12} - 5476 T_{7}^{11} + \cdots + 57232 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 6 T^{15} + \cdots - 143 \)
|
| $3$ |
\( T^{16} + 2 T^{15} + \cdots + 96 \)
|
| $5$ |
\( T^{16} - 48 T^{14} + \cdots + 441 \)
|
| $7$ |
\( T^{16} - 4 T^{15} + \cdots + 57232 \)
|
| $11$ |
\( T^{16} \)
|
| $13$ |
\( T^{16} - 8 T^{15} + \cdots + 361117 \)
|
| $17$ |
\( T^{16} + \cdots - 209397747 \)
|
| $19$ |
\( (T - 1)^{16} \)
|
| $23$ |
\( T^{16} - 10 T^{15} + \cdots + 4167636 \)
|
| $29$ |
\( T^{16} + \cdots + 184284523609 \)
|
| $31$ |
\( T^{16} + 2 T^{15} + \cdots + 3017988 \)
|
| $37$ |
\( T^{16} + \cdots - 2748908871 \)
|
| $41$ |
\( T^{16} - 40 T^{15} + \cdots + 6637261 \)
|
| $43$ |
\( T^{16} + \cdots + 8592299892 \)
|
| $47$ |
\( T^{16} + \cdots + 22180838724 \)
|
| $53$ |
\( T^{16} + \cdots - 6988479303 \)
|
| $59$ |
\( T^{16} + \cdots - 18263902448 \)
|
| $61$ |
\( T^{16} + \cdots - 942785235392 \)
|
| $67$ |
\( T^{16} + \cdots + 30119784372 \)
|
| $71$ |
\( T^{16} + \cdots - 517194176 \)
|
| $73$ |
\( T^{16} + \cdots - 85040723810736 \)
|
| $79$ |
\( T^{16} + \cdots - 152249159215868 \)
|
| $83$ |
\( T^{16} + \cdots - 712822823856 \)
|
| $89$ |
\( T^{16} + \cdots + 59310074133 \)
|
| $97$ |
\( T^{16} + \cdots - 5421805914447 \)
|
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