Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5635,2,Mod(1,5635)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("5635.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5635 = 5 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5635.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: | \(44.9957015390\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{13} - 22 x^{11} - 2 x^{10} + 183 x^{9} + 30 x^{8} - 716 x^{7} - 150 x^{6} + 1324 x^{5} + 300 x^{4} + \cdots + 18 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{12}\) 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
\( x^{13} - 22 x^{11} - 2 x^{10} + 183 x^{9} + 30 x^{8} - 716 x^{7} - 150 x^{6} + 1324 x^{5} + 300 x^{4} + \cdots + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 237 \nu^{12} + 1331 \nu^{11} - 6788 \nu^{10} - 24087 \nu^{9} + 68552 \nu^{8} + 150711 \nu^{7} + \cdots + 56814 ) / 23962 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 462 \nu^{12} + 1533 \nu^{11} - 6711 \nu^{10} - 30272 \nu^{9} + 20041 \nu^{8} + 205071 \nu^{7} + \cdots - 32869 ) / 11981 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 980 \nu^{12} - 4341 \nu^{11} + 17503 \nu^{10} + 80188 \nu^{9} - 95155 \nu^{8} - 500350 \nu^{7} + \cdots - 2890 ) / 23962 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1331 \nu^{12} + 1574 \nu^{11} + 23613 \nu^{10} - 25181 \nu^{9} - 143601 \nu^{8} + 140897 \nu^{7} + \cdots + 4266 ) / 23962 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 993 \nu^{12} - 572 \nu^{11} + 19948 \nu^{10} + 13718 \nu^{9} - 147092 \nu^{8} - 110514 \nu^{7} + \cdots - 36793 ) / 11981 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2205 \nu^{12} + 6772 \nu^{11} - 42377 \nu^{10} - 132499 \nu^{9} + 276999 \nu^{8} + 904139 \nu^{7} + \cdots + 24474 ) / 23962 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 1325 \nu^{12} - 3852 \nu^{11} + 26171 \nu^{10} + 73697 \nu^{9} - 174927 \nu^{8} - 487854 \nu^{7} + \cdots + 47562 ) / 11981 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1365 \nu^{12} + 372 \nu^{11} + 29086 \nu^{10} - 6408 \nu^{9} - 229669 \nu^{8} + 41627 \nu^{7} + \cdots - 73344 ) / 11981 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 3118 \nu^{12} + 3811 \nu^{11} - 56495 \nu^{10} - 78788 \nu^{9} + 338569 \nu^{8} + 552028 \nu^{7} + \cdots - 144602 ) / 23962 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{12} + \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} + 7\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} + 9\beta_{3} + 30\beta _1 + 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 9 \beta_{12} + \beta_{10} + 11 \beta_{9} - 9 \beta_{7} - 11 \beta_{6} - 12 \beta_{5} - 22 \beta_{4} + \cdots + 97 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 12 \beta_{12} - 2 \beta_{11} - 11 \beta_{10} - 10 \beta_{9} - 10 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 13 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 66 \beta_{12} - \beta_{11} + 17 \beta_{10} + 96 \beta_{9} - 65 \beta_{7} - 95 \beta_{6} - 107 \beta_{5} + \cdots + 630 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 105 \beta_{12} - 35 \beta_{11} - 89 \beta_{10} - 78 \beta_{9} - 74 \beta_{8} + 37 \beta_{7} + \cdots + 120 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 458 \beta_{12} - 22 \beta_{11} + 196 \beta_{10} + 777 \beta_{9} + 4 \beta_{8} - 436 \beta_{7} + \cdots + 4262 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 818 \beta_{12} - 411 \beta_{11} - 637 \beta_{10} - 555 \beta_{9} - 489 \beta_{8} + 458 \beta_{7} + \cdots + 963 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 3123 \beta_{12} - 318 \beta_{11} + 1924 \beta_{10} + 6085 \beta_{9} + 89 \beta_{8} - 2809 \beta_{7} + \cdots + 29538 \)
|
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 |
|
−2.73163 | 2.35856 | 5.46180 | 1.00000 | −6.44271 | 0 | −9.45635 | 2.56280 | −2.73163 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.63750 | −0.690651 | 4.95643 | 1.00000 | 1.82159 | 0 | −7.79760 | −2.52300 | −2.63750 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.85803 | −1.59237 | 1.45228 | 1.00000 | 2.95868 | 0 | 1.01769 | −0.464342 | −1.85803 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.82080 | −0.472997 | 1.31530 | 1.00000 | 0.861231 | 0 | 1.24670 | −2.77627 | −1.82080 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.14950 | 3.36237 | −0.678643 | 1.00000 | −3.86506 | 0 | 3.07911 | 8.30555 | −1.14950 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.398199 | 1.76556 | −1.84144 | 1.00000 | −0.703044 | 0 | 1.52966 | 0.117208 | −0.398199 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.119411 | −2.64217 | −1.98574 | 1.00000 | −0.315505 | 0 | −0.475942 | 3.98107 | 0.119411 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.501340 | 0.597983 | −1.74866 | 1.00000 | 0.299793 | 0 | −1.87935 | −2.64242 | 0.501340 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.25752 | −1.68113 | −0.418633 | 1.00000 | −2.11406 | 0 | −3.04149 | −0.173808 | 1.25752 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.72511 | −3.26688 | 0.976017 | 1.00000 | −5.63574 | 0 | −1.76649 | 7.67249 | 1.72511 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.07450 | 2.90142 | 2.30354 | 1.00000 | 6.01900 | 0 | 0.629686 | 5.41827 | 2.07450 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.21858 | −0.298156 | 2.92209 | 1.00000 | −0.661484 | 0 | 2.04574 | −2.91110 | 2.21858 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 2.69920 | 0.658454 | 5.28566 | 1.00000 | 1.77730 | 0 | 8.86865 | −2.56644 | 2.69920 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 5635.2.a.bh | yes | 13 |
| 7.b | odd | 2 | 1 | 5635.2.a.bg | ✓ | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5635.2.a.bg | ✓ | 13 | 7.b | odd | 2 | 1 | |
| 5635.2.a.bh | yes | 13 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5635))\):
|
\( T_{2}^{13} - 22 T_{2}^{11} + 2 T_{2}^{10} + 183 T_{2}^{9} - 30 T_{2}^{8} - 716 T_{2}^{7} + 150 T_{2}^{6} + \cdots - 18 \)
|
|
\( T_{3}^{13} - T_{3}^{12} - 26 T_{3}^{11} + 20 T_{3}^{10} + 240 T_{3}^{9} - 118 T_{3}^{8} - 963 T_{3}^{7} + \cdots + 36 \)
|
|
\( T_{11}^{13} - 3 T_{11}^{12} - 97 T_{11}^{11} + 267 T_{11}^{10} + 3442 T_{11}^{9} - 8516 T_{11}^{8} + \cdots - 171840 \)
|
|
\( T_{17}^{13} - 19 T_{17}^{12} + 45 T_{17}^{11} + 1285 T_{17}^{10} - 9640 T_{17}^{9} - 600 T_{17}^{8} + \cdots - 13103936 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} - 22 T^{11} + \cdots - 18 \)
|
| $3$ |
\( T^{13} - T^{12} + \cdots + 36 \)
|
| $5$ |
\( (T - 1)^{13} \)
|
| $7$ |
\( T^{13} \)
|
| $11$ |
\( T^{13} - 3 T^{12} + \cdots - 171840 \)
|
| $13$ |
\( T^{13} - 15 T^{12} + \cdots + 140732 \)
|
| $17$ |
\( T^{13} - 19 T^{12} + \cdots - 13103936 \)
|
| $19$ |
\( T^{13} - 8 T^{12} + \cdots - 4708864 \)
|
| $23$ |
\( (T - 1)^{13} \)
|
| $29$ |
\( T^{13} + \cdots + 135560448 \)
|
| $31$ |
\( T^{13} + \cdots + 358269784 \)
|
| $37$ |
\( T^{13} + \cdots + 221715200 \)
|
| $41$ |
\( T^{13} + \cdots + 633150432 \)
|
| $43$ |
\( T^{13} + \cdots - 310169088 \)
|
| $47$ |
\( T^{13} + \cdots + 138407508 \)
|
| $53$ |
\( T^{13} + \cdots + 604531584 \)
|
| $59$ |
\( T^{13} + \cdots + 6342646656 \)
|
| $61$ |
\( T^{13} + \cdots - 244767488 \)
|
| $67$ |
\( T^{13} + \cdots - 2023497728 \)
|
| $71$ |
\( T^{13} + \cdots + 20223180288 \)
|
| $73$ |
\( T^{13} + \cdots - 53992275624 \)
|
| $79$ |
\( T^{13} + \cdots - 112221504 \)
|
| $83$ |
\( T^{13} + \cdots - 14887259136 \)
|
| $89$ |
\( T^{13} - 16 T^{12} + \cdots + 88915968 \)
|
| $97$ |
\( T^{13} - 39 T^{12} + \cdots - 11474368 \)
|
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