Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,12,Mod(1,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 41 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 41.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: | \(31.5020704029\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 5 x^{19} - 33928 x^{18} + 70752 x^{17} + 486071480 x^{16} + 382640960 x^{15} + \cdots - 33\!\cdots\!56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | multiple of \( 2^{25}\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 5 x^{19} - 33928 x^{18} + 70752 x^{17} + 486071480 x^{16} + 382640960 x^{15} + \cdots - 33\!\cdots\!56 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 44\!\cdots\!65 \nu^{19} + \cdots - 97\!\cdots\!20 ) / 18\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 44\!\cdots\!65 \nu^{19} + \cdots - 35\!\cdots\!16 ) / 18\!\cdots\!28 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83\!\cdots\!43 \nu^{19} + \cdots - 27\!\cdots\!72 ) / 18\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13\!\cdots\!97 \nu^{19} + \cdots - 12\!\cdots\!72 ) / 18\!\cdots\!28 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 25\!\cdots\!93 \nu^{19} + \cdots - 14\!\cdots\!12 ) / 22\!\cdots\!16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 21\!\cdots\!73 \nu^{19} + \cdots + 58\!\cdots\!16 ) / 18\!\cdots\!28 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 27\!\cdots\!05 \nu^{19} + \cdots - 10\!\cdots\!44 ) / 18\!\cdots\!28 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 31\!\cdots\!15 \nu^{19} + \cdots + 36\!\cdots\!52 ) / 18\!\cdots\!28 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 10\!\cdots\!51 \nu^{19} + \cdots - 22\!\cdots\!84 ) / 22\!\cdots\!16 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 41\!\cdots\!01 \nu^{19} + \cdots + 31\!\cdots\!52 ) / 91\!\cdots\!64 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 42\!\cdots\!77 \nu^{19} + \cdots - 74\!\cdots\!32 ) / 91\!\cdots\!64 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 10\!\cdots\!27 \nu^{19} + \cdots - 10\!\cdots\!92 ) / 18\!\cdots\!28 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 10\!\cdots\!35 \nu^{19} + \cdots - 42\!\cdots\!16 ) / 18\!\cdots\!28 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 14\!\cdots\!03 \nu^{19} + \cdots - 13\!\cdots\!16 ) / 18\!\cdots\!28 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 18\!\cdots\!21 \nu^{19} + \cdots - 12\!\cdots\!08 ) / 18\!\cdots\!28 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 10\!\cdots\!63 \nu^{19} + \cdots + 33\!\cdots\!32 ) / 91\!\cdots\!64 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 26\!\cdots\!45 \nu^{19} + \cdots - 16\!\cdots\!80 ) / 18\!\cdots\!28 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 40\!\cdots\!63 \nu^{19} + \cdots + 20\!\cdots\!04 ) / 18\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 4\beta _1 + 3393 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} - \beta_{7} - \beta_{6} + 4\beta_{4} - 20\beta_{3} - \beta_{2} + 5301\beta _1 + 13519 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 39 \beta_{19} - 8 \beta_{18} + 6 \beta_{17} - 32 \beta_{16} - 6 \beta_{15} + 22 \beta_{14} + \cdots + 17986403 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 393 \beta_{19} - 766 \beta_{18} - 434 \beta_{17} - 1234 \beta_{16} + 1708 \beta_{15} + \cdots + 128175785 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 452140 \beta_{19} - 139212 \beta_{18} + 28208 \beta_{17} - 379284 \beta_{16} - 16172 \beta_{15} + \cdots + 109213243647 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 9891982 \beta_{19} - 12182060 \beta_{18} - 5591532 \beta_{17} - 20246564 \beta_{16} + \cdots + 1272032351983 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 4160052921 \beta_{19} - 1565835128 \beta_{18} + 18881818 \beta_{17} - 3624414704 \beta_{16} + \cdots + 719033961820155 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 136333112227 \beta_{19} - 138179530690 \beta_{18} - 56060062358 \beta_{17} - 232197550014 \beta_{16} + \cdots + 13\!\cdots\!85 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 35846515802198 \beta_{19} - 15396706207132 \beta_{18} - 1228000844620 \beta_{17} + \cdots + 50\!\cdots\!55 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 15\!\cdots\!52 \beta_{19} + \cdots + 13\!\cdots\!83 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 30\!\cdots\!11 \beta_{19} + \cdots + 37\!\cdots\!23 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 15\!\cdots\!57 \beta_{19} + \cdots + 13\!\cdots\!01 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 25\!\cdots\!92 \beta_{19} + \cdots + 28\!\cdots\!59 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 14\!\cdots\!62 \beta_{19} + \cdots + 12\!\cdots\!55 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 21\!\cdots\!01 \beta_{19} + \cdots + 22\!\cdots\!15 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 13\!\cdots\!51 \beta_{19} + \cdots + 11\!\cdots\!45 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 18\!\cdots\!98 \beta_{19} + \cdots + 18\!\cdots\!71 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 12\!\cdots\!52 \beta_{19} + \cdots + 10\!\cdots\!87 \)
|
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 |
|
−90.2840 | 726.607 | 6103.21 | −4746.70 | −65601.0 | 59034.5 | −366121. | 350811. | 428551. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −70.9995 | 320.953 | 2992.92 | −10237.6 | −22787.5 | −78746.0 | −67089.0 | −74136.3 | 726865. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −70.2468 | −615.245 | 2886.61 | −2325.09 | 43219.0 | 50712.6 | −58909.4 | 201379. | 163330. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −64.4865 | −772.386 | 2110.51 | 5581.17 | 49808.5 | −52609.7 | −4030.86 | 419434. | −359910. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −64.1966 | 26.5308 | 2073.20 | −1749.24 | −1703.19 | 39049.0 | −1617.73 | −176443. | 112295. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −58.7456 | 723.044 | 1403.05 | 6050.46 | −42475.7 | −54639.9 | 37888.0 | 345646. | −355438. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −39.6817 | −424.276 | −473.364 | 11414.2 | 16836.0 | 69720.0 | 100052. | 2863.25 | −452933. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −24.8604 | −201.814 | −1429.96 | −1700.87 | 5017.17 | −28637.2 | 86463.5 | −136418. | 42284.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −14.4054 | 601.543 | −1840.48 | 9988.48 | −8665.46 | 63461.7 | 56015.2 | 184707. | −143888. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 7.02697 | 311.756 | −1998.62 | −7817.39 | 2190.70 | −36914.7 | −28435.5 | −79955.5 | −54932.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 15.1229 | −70.4794 | −1819.30 | −8080.73 | −1065.85 | 77040.8 | −58484.8 | −172180. | −122204. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 25.0345 | −503.852 | −1421.27 | 12259.3 | −12613.7 | −28390.0 | −86851.6 | 76719.7 | 306906. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 31.7297 | −392.139 | −1041.23 | −5628.01 | −12442.5 | −72031.6 | −98020.2 | −23373.7 | −178575. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 40.3087 | 612.531 | −423.210 | 3610.67 | 24690.3 | 54490.2 | −99611.2 | 198047. | 145541. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 52.9604 | −613.134 | 756.800 | −11862.9 | −32471.8 | 8594.71 | −68382.4 | 198786. | −628264. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 60.3851 | −351.608 | 1598.36 | 6614.18 | −21231.9 | 9244.66 | −27151.4 | −53518.6 | 399398. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 71.0063 | 497.460 | 2993.89 | 10443.3 | 35322.8 | −63530.9 | 67164.1 | 70319.9 | 741543. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 79.7545 | 708.989 | 4312.78 | −3892.86 | 56545.0 | 8251.62 | 180626. | 325518. | −310473. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.19 | 83.2297 | 178.618 | 4879.19 | 10238.0 | 14866.3 | 63500.9 | 235639. | −145243. | 852103. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.20 | 86.3477 | −267.097 | 5407.92 | −8398.34 | −23063.2 | 46791.2 | 290122. | −105806. | −725177. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(41\) | \(1\) |
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 | 41.12.a.b | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 41.12.a.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 55 T_{2}^{19} - 32503 T_{2}^{18} + 1738275 T_{2}^{17} + 443222606 T_{2}^{16} + \cdots - 68\!\cdots\!64 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(41))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots - 68\!\cdots\!64 \)
|
| $3$ |
\( T^{20} + \cdots + 10\!\cdots\!64 \)
|
| $5$ |
\( T^{20} + \cdots - 55\!\cdots\!00 \)
|
| $7$ |
\( T^{20} + \cdots + 13\!\cdots\!96 \)
|
| $11$ |
\( T^{20} + \cdots - 44\!\cdots\!40 \)
|
| $13$ |
\( T^{20} + \cdots + 17\!\cdots\!00 \)
|
| $17$ |
\( T^{20} + \cdots + 10\!\cdots\!20 \)
|
| $19$ |
\( T^{20} + \cdots - 17\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 15\!\cdots\!32 \)
|
| $29$ |
\( T^{20} + \cdots - 65\!\cdots\!24 \)
|
| $31$ |
\( T^{20} + \cdots + 29\!\cdots\!20 \)
|
| $37$ |
\( T^{20} + \cdots - 54\!\cdots\!44 \)
|
| $41$ |
\( (T + 115856201)^{20} \)
|
| $43$ |
\( T^{20} + \cdots - 33\!\cdots\!52 \)
|
| $47$ |
\( T^{20} + \cdots + 27\!\cdots\!76 \)
|
| $53$ |
\( T^{20} + \cdots + 16\!\cdots\!72 \)
|
| $59$ |
\( T^{20} + \cdots - 95\!\cdots\!16 \)
|
| $61$ |
\( T^{20} + \cdots + 75\!\cdots\!68 \)
|
| $67$ |
\( T^{20} + \cdots - 29\!\cdots\!00 \)
|
| $71$ |
\( T^{20} + \cdots + 67\!\cdots\!96 \)
|
| $73$ |
\( T^{20} + \cdots + 25\!\cdots\!64 \)
|
| $79$ |
\( T^{20} + \cdots + 41\!\cdots\!68 \)
|
| $83$ |
\( T^{20} + \cdots - 18\!\cdots\!24 \)
|
| $89$ |
\( T^{20} + \cdots - 17\!\cdots\!60 \)
|
| $97$ |
\( T^{20} + \cdots + 16\!\cdots\!84 \)
|
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