Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2075,4,Mod(1,2075)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2075, 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("2075.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2075 = 5^{2} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2075.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: | \(122.428963262\) |
| Analytic rank: | \(1\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 5 x^{12} - 76 x^{11} + 374 x^{10} + 2129 x^{9} - 9943 x^{8} - 28670 x^{7} + 114750 x^{6} + \cdots - 6976 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 83) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 5 x^{12} - 76 x^{11} + 374 x^{10} + 2129 x^{9} - 9943 x^{8} - 28670 x^{7} + 114750 x^{6} + \cdots - 6976 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 279707708 \nu^{12} + 32799653612 \nu^{11} - 247350621751 \nu^{10} - 1625978100588 \nu^{9} + \cdots - 715170660049408 ) / 50909819752144 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3126316687 \nu^{12} - 921852208 \nu^{11} + 351747851852 \nu^{10} + \cdots + 149999827632064 ) / 50909819752144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 23874100150 \nu^{12} - 194702439969 \nu^{11} - 1182075881643 \nu^{10} + \cdots + 12\!\cdots\!24 ) / 50909819752144 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 31958242378 \nu^{12} + 225365018255 \nu^{11} + 2015105742751 \nu^{10} + \cdots - 252710825961488 ) / 50909819752144 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 35391627155 \nu^{12} + 247284352608 \nu^{11} + 2149166771102 \nu^{10} + \cdots + 50379320943328 ) / 50909819752144 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 38206728119 \nu^{12} + 271216957111 \nu^{11} + 2294159434744 \nu^{10} + \cdots - 331288340490208 ) / 50909819752144 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 27081172078 \nu^{12} - 227446639806 \nu^{11} - 1344183692221 \nu^{10} + \cdots - 259956897375520 ) / 25454909876072 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 67895566266 \nu^{12} - 473198431316 \nu^{11} - 4144413598717 \nu^{10} + \cdots - 11\!\cdots\!96 ) / 50909819752144 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 103680538656 \nu^{12} - 747204986625 \nu^{11} - 5997982781008 \nu^{10} + \cdots + 510318814633600 ) / 50909819752144 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 106872517630 \nu^{12} + 828186946772 \nu^{11} + 5958243168695 \nu^{10} + \cdots - 640456165136400 ) / 50909819752144 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{12} - \beta_{9} - \beta_{8} + 2\beta_{6} - \beta_{5} + \beta_{4} + 21\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{12} - 2 \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + \cdots + 304 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 28 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} - 26 \beta_{9} - 52 \beta_{8} + \beta_{7} + 63 \beta_{6} + \cdots - 34 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2 \beta_{12} - 30 \beta_{11} - 42 \beta_{10} + 8 \beta_{9} - 98 \beta_{8} + 74 \beta_{7} - 60 \beta_{6} + \cdots + 7422 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 679 \beta_{12} - 160 \beta_{11} - 232 \beta_{10} - 583 \beta_{9} - 1927 \beta_{8} + 114 \beta_{7} + \cdots - 3146 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1155 \beta_{12} - 1760 \beta_{11} - 306 \beta_{10} + 1403 \beta_{9} - 4677 \beta_{8} + 3239 \beta_{7} + \cdots + 188696 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 16314 \beta_{12} - 4652 \beta_{11} - 9600 \beta_{10} - 12908 \beta_{9} - 63418 \beta_{8} + \cdots - 133094 \)
|
| \(\nu^{10}\) | \(=\) |
\( 60606 \beta_{12} - 71802 \beta_{11} + 17882 \beta_{10} + 68108 \beta_{9} - 174958 \beta_{8} + \cdots + 4890654 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 400045 \beta_{12} - 114668 \beta_{11} - 349556 \beta_{10} - 292185 \beta_{9} - 1968821 \beta_{8} + \cdots - 4638122 \)
|
| \(\nu^{12}\) | \(=\) |
\( 2377915 \beta_{12} - 2530216 \beta_{11} + 1213538 \beta_{10} + 2572419 \beta_{9} - 5843261 \beta_{8} + \cdots + 128313476 \)
|
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 |
|
−5.26189 | 9.88379 | 19.6875 | 0 | −52.0074 | −11.0128 | −61.4985 | 70.6892 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.97334 | −1.07884 | 16.7341 | 0 | 5.36546 | −7.39925 | −43.4375 | −25.8361 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.61904 | −3.20706 | 13.3356 | 0 | 14.8135 | 22.2043 | −24.6451 | −16.7148 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.08231 | −9.18464 | 8.66522 | 0 | 37.4945 | −8.58868 | −2.71565 | 57.3577 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.90302 | −0.0552626 | 0.427512 | 0 | 0.160428 | −34.5834 | 21.9831 | −26.9969 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.16563 | −7.89819 | −3.31003 | 0 | 17.1046 | 4.90538 | 24.4934 | 35.3814 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.00424280 | 7.83724 | −7.99998 | 0 | −0.0332519 | −10.0055 | 0.0678848 | 34.4223 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.65981 | −7.37487 | −5.24503 | 0 | −12.2409 | −23.5648 | −21.9842 | 27.3887 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.74457 | 4.85948 | −4.95647 | 0 | 8.47771 | −29.0420 | −22.6035 | −3.38542 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.59240 | −1.71328 | −1.27948 | 0 | −4.44149 | 24.8560 | −24.0561 | −24.0647 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.61289 | 8.27665 | −1.17281 | 0 | 21.6260 | 31.6072 | −23.9675 | 41.5030 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 5.06588 | 0.527559 | 17.6632 | 0 | 2.67255 | −1.40455 | 48.9526 | −26.7217 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 5.33392 | −7.87257 | 20.4507 | 0 | −41.9917 | −9.97200 | 66.4112 | 34.9774 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(83\) | \( -1 \) |
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 | 2075.4.a.d | 13 | |
| 5.b | even | 2 | 1 | 83.4.a.b | ✓ | 13 | |
| 15.d | odd | 2 | 1 | 747.4.a.f | 13 | ||
| 20.d | odd | 2 | 1 | 1328.4.a.j | 13 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 83.4.a.b | ✓ | 13 | 5.b | even | 2 | 1 | |
| 747.4.a.f | 13 | 15.d | odd | 2 | 1 | ||
| 1328.4.a.j | 13 | 20.d | odd | 2 | 1 | ||
| 2075.4.a.d | 13 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{13} + 5 T_{2}^{12} - 76 T_{2}^{11} - 374 T_{2}^{10} + 2129 T_{2}^{9} + 9943 T_{2}^{8} + \cdots + 6976 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2075))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} + 5 T^{12} + \cdots + 6976 \)
|
| $3$ |
\( T^{13} + 7 T^{12} + \cdots - 2267708 \)
|
| $5$ |
\( T^{13} \)
|
| $7$ |
\( T^{13} + \cdots + 198634180601741 \)
|
| $11$ |
\( T^{13} + \cdots + 18\!\cdots\!16 \)
|
| $13$ |
\( T^{13} + \cdots - 40\!\cdots\!64 \)
|
| $17$ |
\( T^{13} + \cdots - 15\!\cdots\!21 \)
|
| $19$ |
\( T^{13} + \cdots + 12\!\cdots\!56 \)
|
| $23$ |
\( T^{13} + \cdots - 11\!\cdots\!48 \)
|
| $29$ |
\( T^{13} + \cdots + 31\!\cdots\!04 \)
|
| $31$ |
\( T^{13} + \cdots + 52\!\cdots\!69 \)
|
| $37$ |
\( T^{13} + \cdots - 65\!\cdots\!04 \)
|
| $41$ |
\( T^{13} + \cdots + 32\!\cdots\!84 \)
|
| $43$ |
\( T^{13} + \cdots + 12\!\cdots\!48 \)
|
| $47$ |
\( T^{13} + \cdots - 56\!\cdots\!08 \)
|
| $53$ |
\( T^{13} + \cdots - 67\!\cdots\!96 \)
|
| $59$ |
\( T^{13} + \cdots + 15\!\cdots\!64 \)
|
| $61$ |
\( T^{13} + \cdots - 51\!\cdots\!28 \)
|
| $67$ |
\( T^{13} + \cdots + 75\!\cdots\!32 \)
|
| $71$ |
\( T^{13} + \cdots - 12\!\cdots\!08 \)
|
| $73$ |
\( T^{13} + \cdots - 49\!\cdots\!68 \)
|
| $79$ |
\( T^{13} + \cdots + 48\!\cdots\!96 \)
|
| $83$ |
\( (T - 83)^{13} \)
|
| $89$ |
\( T^{13} + \cdots + 20\!\cdots\!76 \)
|
| $97$ |
\( T^{13} + \cdots + 32\!\cdots\!04 \)
|
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