Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2025,4,Mod(1,2025)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2025, 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("2025.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2025 = 3^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2025.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: | \(119.478867762\) |
| 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: |
\( x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{12}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 45) |
| 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
\( x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 19\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{14} - 86\nu^{12} + 2754\nu^{10} - 40102\nu^{8} + 252737\nu^{6} - 416664\nu^{4} - 951920\nu^{2} + 182272 ) / 10368 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13 \nu^{14} - 1022 \nu^{12} + 29466 \nu^{10} - 370126 \nu^{8} + 1713581 \nu^{6} + 1226376 \nu^{4} + \cdots + 15782656 ) / 176256 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{14} + 94 \nu^{12} - 11574 \nu^{10} + 387218 \nu^{8} - 5621959 \nu^{6} + 36266220 \nu^{4} + \cdots + 17199616 ) / 352512 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 25 \nu^{14} - 2138 \nu^{12} + 70506 \nu^{10} - 1127470 \nu^{8} + 9049121 \nu^{6} - 34165980 \nu^{4} + \cdots - 17007872 ) / 352512 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11 \nu^{14} - 938 \nu^{12} + 30582 \nu^{10} - 479522 \nu^{8} + 3772331 \nu^{6} - 14336208 \nu^{4} + \cdots - 6685568 ) / 58752 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 73 \nu^{14} + 6194 \nu^{12} - 200394 \nu^{10} + 3084622 \nu^{8} - 22981937 \nu^{6} + \cdots + 8294528 ) / 176256 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 101 \nu^{15} - 8407 \nu^{13} + 263820 \nu^{11} - 3850424 \nu^{9} + 25802659 \nu^{7} + \cdots - 100002400 \nu ) / 2878848 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 895 \nu^{15} + 75908 \nu^{13} - 2465730 \nu^{11} + 38436238 \nu^{9} - 295408283 \nu^{7} + \cdots - 79369792 \nu ) / 17273088 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 283 \nu^{15} - 25214 \nu^{13} + 875550 \nu^{11} - 15014794 \nu^{9} + 133274387 \nu^{7} + \cdots - 288310016 \nu ) / 5757696 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 365 \nu^{15} + 31990 \nu^{13} - 1088058 \nu^{11} + 18206894 \nu^{9} - 157923085 \nu^{7} + \cdots + 640762624 \nu ) / 5757696 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 1831 \nu^{15} - 155204 \nu^{13} + 5037714 \nu^{11} - 78348574 \nu^{9} + 598222883 \nu^{7} + \cdots - 1433400128 \nu ) / 17273088 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 137 \nu^{15} - 12712 \nu^{13} + 468198 \nu^{11} - 8743514 \nu^{9} + 87674005 \nu^{7} + \cdots - 475010752 \nu ) / 1016064 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 19\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - \beta_{6} + \beta_{5} + 27\beta_{2} + 208 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{15} + \beta_{14} - 3\beta_{13} + \beta_{12} - 6\beta_{10} + 33\beta_{3} + 418\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5\beta_{9} + \beta_{8} - 20\beta_{7} - 38\beta_{6} + 41\beta_{5} + 4\beta_{4} + 687\beta_{2} + 4583 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 45 \beta_{15} + 57 \beta_{14} - 143 \beta_{13} + 39 \beta_{12} + 30 \beta_{11} - 268 \beta_{10} + \cdots + 9830 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 241\beta_{9} + 29\beta_{8} - 235\beta_{7} - 1141\beta_{6} + 1288\beta_{5} + 200\beta_{4} + 17333\beta_{2} + 108106 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1476 \beta_{15} + 2112 \beta_{14} - 4732 \beta_{13} + 1254 \beta_{12} + 1518 \beta_{11} + \cdots + 238799 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 8180 \beta_{9} + 508 \beta_{8} + 212 \beta_{7} - 31636 \beta_{6} + 36676 \beta_{5} + 6868 \beta_{4} + \cdots + 2633567 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 42740 \beta_{15} + 66188 \beta_{14} - 136604 \beta_{13} + 36404 \beta_{12} + 52824 \beta_{11} + \cdots + 5902367 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 242180 \beta_{9} + 4228 \beta_{8} + 125815 \beta_{7} - 843953 \beta_{6} + 996029 \beta_{5} + \cdots + 65230888 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1162845 \beta_{15} + 1911261 \beta_{14} - 3697319 \beta_{13} + 983541 \beta_{12} + \cdots + 147362278 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 6700657 \beta_{9} - 125203 \beta_{8} + 5450348 \beta_{7} - 22023454 \beta_{6} + 26358613 \beta_{5} + \cdots + 1630940951 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 30553497 \beta_{15} + 52782405 \beta_{14} - 96685123 \beta_{13} + 25248315 \beta_{12} + \cdots + 3702009370 \beta_1 \)
|
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 |
|
−5.05435 | 0 | 17.5465 | 0 | 0 | −21.0117 | −48.2513 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −5.02371 | 0 | 17.2377 | 0 | 0 | 5.38197 | −46.4074 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −4.07626 | 0 | 8.61587 | 0 | 0 | −13.3430 | −2.51043 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.08740 | 0 | 1.53204 | 0 | 0 | 31.3204 | 19.9692 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −2.67506 | 0 | −0.844033 | 0 | 0 | −15.4153 | 23.6584 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.46385 | 0 | −1.92944 | 0 | 0 | 19.2401 | 24.4647 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.785333 | 0 | −7.38325 | 0 | 0 | −20.9136 | 12.0810 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.473990 | 0 | −7.77533 | 0 | 0 | −8.20657 | 7.47735 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.473990 | 0 | −7.77533 | 0 | 0 | 8.20657 | −7.47735 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.785333 | 0 | −7.38325 | 0 | 0 | 20.9136 | −12.0810 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.46385 | 0 | −1.92944 | 0 | 0 | −19.2401 | −24.4647 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.67506 | 0 | −0.844033 | 0 | 0 | 15.4153 | −23.6584 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 3.08740 | 0 | 1.53204 | 0 | 0 | −31.3204 | −19.9692 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 4.07626 | 0 | 8.61587 | 0 | 0 | 13.3430 | 2.51043 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 5.02371 | 0 | 17.2377 | 0 | 0 | −5.38197 | 46.4074 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 5.05435 | 0 | 17.5465 | 0 | 0 | 21.0117 | 48.2513 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2025.4.a.bl | 16 | |
| 3.b | odd | 2 | 1 | 2025.4.a.bk | 16 | ||
| 5.b | even | 2 | 1 | inner | 2025.4.a.bl | 16 | |
| 5.c | odd | 4 | 2 | 405.4.b.f | 16 | ||
| 9.d | odd | 6 | 2 | 225.4.e.g | 32 | ||
| 15.d | odd | 2 | 1 | 2025.4.a.bk | 16 | ||
| 15.e | even | 4 | 2 | 405.4.b.e | 16 | ||
| 45.h | odd | 6 | 2 | 225.4.e.g | 32 | ||
| 45.k | odd | 12 | 4 | 135.4.j.a | 32 | ||
| 45.l | even | 12 | 4 | 45.4.j.a | ✓ | 32 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.4.j.a | ✓ | 32 | 45.l | even | 12 | 4 | |
| 135.4.j.a | 32 | 45.k | odd | 12 | 4 | ||
| 225.4.e.g | 32 | 9.d | odd | 6 | 2 | ||
| 225.4.e.g | 32 | 45.h | odd | 6 | 2 | ||
| 405.4.b.e | 16 | 15.e | even | 4 | 2 | ||
| 405.4.b.f | 16 | 5.c | odd | 4 | 2 | ||
| 2025.4.a.bk | 16 | 3.b | odd | 2 | 1 | ||
| 2025.4.a.bk | 16 | 15.d | odd | 2 | 1 | ||
| 2025.4.a.bl | 16 | 1.a | even | 1 | 1 | trivial | |
| 2025.4.a.bl | 16 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2025))\):
|
\( T_{2}^{16} - 91 T_{2}^{14} + 3268 T_{2}^{12} - 59128 T_{2}^{10} + 571975 T_{2}^{8} - 2881141 T_{2}^{6} + \cdots + 614656 \)
|
|
\( T_{7}^{16} - 2742 T_{7}^{14} + 2969739 T_{7}^{12} - 1665664668 T_{7}^{10} + 525057427827 T_{7}^{8} + \cdots + 57\!\cdots\!16 \)
|
|
\( T_{11}^{8} - 45 T_{11}^{7} - 3975 T_{11}^{6} + 160551 T_{11}^{5} + 4235328 T_{11}^{4} + \cdots + 137637673416 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 91 T^{14} + \cdots + 614656 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 57\!\cdots\!16 \)
|
| $11$ |
\( (T^{8} - 45 T^{7} + \cdots + 137637673416)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 53\!\cdots\!96 \)
|
| $17$ |
\( T^{16} + \cdots + 18\!\cdots\!56 \)
|
| $19$ |
\( (T^{8} + \cdots + 11815095359200)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 30\!\cdots\!64 \)
|
| $29$ |
\( (T^{8} + \cdots - 16\!\cdots\!16)^{2} \)
|
| $31$ |
\( (T^{8} + \cdots - 12\!\cdots\!60)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 12\!\cdots\!24 \)
|
| $41$ |
\( (T^{8} + \cdots - 50\!\cdots\!35)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 22\!\cdots\!56 \)
|
| $47$ |
\( T^{16} + \cdots + 89\!\cdots\!44 \)
|
| $53$ |
\( T^{16} + \cdots + 45\!\cdots\!44 \)
|
| $59$ |
\( (T^{8} + \cdots - 79\!\cdots\!32)^{2} \)
|
| $61$ |
\( (T^{8} + \cdots - 49\!\cdots\!22)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 26\!\cdots\!89 \)
|
| $71$ |
\( (T^{8} + \cdots - 15\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 43\!\cdots\!96 \)
|
| $79$ |
\( (T^{8} + \cdots + 78\!\cdots\!32)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 15\!\cdots\!76 \)
|
| $89$ |
\( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 17\!\cdots\!96 \)
|
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