Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [58,9,Mod(17,58)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58.17");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.c (of order \(4\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(23.6279593835\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 8 x^{19} + 34 x^{18} + 520870 x^{17} + 612801803 x^{16} + 10955282558 x^{15} + \cdots + 14\!\cdots\!20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} - 8 x^{19} + 34 x^{18} + 520870 x^{17} + 612801803 x^{16} + 10955282558 x^{15} + \cdots + 14\!\cdots\!20 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 11\!\cdots\!91 \nu^{19} + \cdots - 19\!\cdots\!20 ) / 18\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 25\!\cdots\!61 \nu^{19} + \cdots + 45\!\cdots\!20 ) / 18\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 25\!\cdots\!61 \nu^{19} + \cdots + 45\!\cdots\!20 ) / 18\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 18\!\cdots\!43 \nu^{19} + \cdots + 35\!\cdots\!40 ) / 41\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 27\!\cdots\!68 \nu^{19} + \cdots + 48\!\cdots\!40 ) / 41\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14\!\cdots\!57 \nu^{19} + \cdots - 39\!\cdots\!40 ) / 83\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 67\!\cdots\!93 \nu^{19} + \cdots + 86\!\cdots\!60 ) / 29\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 29\!\cdots\!44 \nu^{19} + \cdots - 36\!\cdots\!80 ) / 73\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 20\!\cdots\!04 \nu^{19} + \cdots + 87\!\cdots\!20 ) / 13\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 10\!\cdots\!29 \nu^{19} + \cdots - 12\!\cdots\!80 ) / 48\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 28\!\cdots\!89 \nu^{19} + \cdots + 15\!\cdots\!80 ) / 73\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 49\!\cdots\!49 \nu^{19} + \cdots + 87\!\cdots\!80 ) / 73\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 10\!\cdots\!89 \nu^{19} + \cdots - 10\!\cdots\!80 ) / 10\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 30\!\cdots\!63 \nu^{19} + \cdots - 54\!\cdots\!60 ) / 25\!\cdots\!00 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 59\!\cdots\!17 \nu^{19} + \cdots - 14\!\cdots\!40 ) / 32\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 21\!\cdots\!03 \nu^{19} + \cdots - 22\!\cdots\!60 ) / 11\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 10\!\cdots\!33 \nu^{19} + \cdots + 25\!\cdots\!60 ) / 32\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 25\!\cdots\!48 \nu^{19} + \cdots - 28\!\cdots\!60 ) / 73\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 26\!\cdots\!92 \nu^{19} + \cdots + 66\!\cdots\!40 ) / 73\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( -\beta_{3} + \beta_{2} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{14} + \beta_{7} + 8481\beta_{3} + 12\beta_{2} - 10\beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 10 \beta_{18} - 4 \beta_{16} + 30 \beta_{14} - 12 \beta_{13} - 40 \beta_{12} + 3 \beta_{11} + \cdots - 72355 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 1437 \beta_{19} - 1078 \beta_{18} + 399 \beta_{17} - 917 \beta_{16} + 901 \beta_{15} + \cdots - 123049080 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 681799 \beta_{19} - 5290 \beta_{18} + 328371 \beta_{17} - 4545 \beta_{16} + 54380 \beta_{15} + \cdots - 3861943627 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 54672601 \beta_{19} + 34698953 \beta_{18} + 17852880 \beta_{17} + 25660868 \beta_{16} + \cdots - 21327372648 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 368440723 \beta_{19} + 9462992265 \beta_{18} + 118088334 \beta_{17} + 1227844362 \beta_{16} + \cdots + 107959589332121 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1425400031876 \beta_{19} + 1019002348917 \beta_{18} - 482121415287 \beta_{17} + 595012672326 \beta_{16} + \cdots + 46\!\cdots\!44 \)
|
| \(\nu^{9}\) | \(=\) |
\( 415247651346312 \beta_{19} + 8833268662317 \beta_{18} - 178422526990929 \beta_{17} + \cdots + 35\!\cdots\!55 \)
|
| \(\nu^{10}\) | \(=\) |
\( 41\!\cdots\!44 \beta_{19} + \cdots + 33\!\cdots\!10 \)
|
| \(\nu^{11}\) | \(=\) |
\( 43\!\cdots\!20 \beta_{19} + \cdots - 82\!\cdots\!11 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 94\!\cdots\!58 \beta_{19} + \cdots - 23\!\cdots\!44 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 24\!\cdots\!64 \beta_{19} + \cdots - 24\!\cdots\!47 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 26\!\cdots\!44 \beta_{19} + \cdots - 31\!\cdots\!78 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 37\!\cdots\!06 \beta_{19} + \cdots + 53\!\cdots\!93 \)
|
| \(\nu^{16}\) | \(=\) |
\( 58\!\cdots\!90 \beta_{19} + \cdots + 13\!\cdots\!48 \)
|
| \(\nu^{17}\) | \(=\) |
\( 14\!\cdots\!80 \beta_{19} + \cdots + 15\!\cdots\!37 \)
|
| \(\nu^{18}\) | \(=\) |
\( 16\!\cdots\!00 \beta_{19} + \cdots + 25\!\cdots\!26 \)
|
| \(\nu^{19}\) | \(=\) |
\( 29\!\cdots\!38 \beta_{19} + \cdots - 32\!\cdots\!59 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) |
| \(\chi(n)\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
8.00000 | − | 8.00000i | −109.436 | + | 109.436i | − | 128.000i | − | 214.288i | 1750.98i | −1333.30 | −1024.00 | − | 1024.00i | − | 17391.5i | −1714.30 | − | 1714.30i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | 8.00000 | − | 8.00000i | −64.1915 | + | 64.1915i | − | 128.000i | 811.656i | 1027.06i | 2217.69 | −1024.00 | − | 1024.00i | − | 1680.09i | 6493.25 | + | 6493.25i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 8.00000 | − | 8.00000i | −53.0142 | + | 53.0142i | − | 128.000i | 111.354i | 848.227i | −2383.30 | −1024.00 | − | 1024.00i | 939.996i | 890.831 | + | 890.831i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 8.00000 | − | 8.00000i | −52.9266 | + | 52.9266i | − | 128.000i | − | 1048.55i | 846.826i | 76.2136 | −1024.00 | − | 1024.00i | 958.550i | −8388.36 | − | 8388.36i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 8.00000 | − | 8.00000i | 1.27653 | − | 1.27653i | − | 128.000i | − | 148.377i | − | 20.4244i | 4685.03 | −1024.00 | − | 1024.00i | 6557.74i | −1187.01 | − | 1187.01i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 8.00000 | − | 8.00000i | 25.8038 | − | 25.8038i | − | 128.000i | − | 413.688i | − | 412.861i | −3714.12 | −1024.00 | − | 1024.00i | 5229.33i | −3309.50 | − | 3309.50i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.7 | 8.00000 | − | 8.00000i | 30.1373 | − | 30.1373i | − | 128.000i | 621.075i | − | 482.196i | −328.250 | −1024.00 | − | 1024.00i | 4744.49i | 4968.60 | + | 4968.60i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 8.00000 | − | 8.00000i | 67.2603 | − | 67.2603i | − | 128.000i | 928.868i | − | 1076.17i | −450.923 | −1024.00 | − | 1024.00i | − | 2486.91i | 7430.95 | + | 7430.95i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.9 | 8.00000 | − | 8.00000i | 81.5522 | − | 81.5522i | − | 128.000i | − | 734.196i | − | 1304.84i | 2005.78 | −1024.00 | − | 1024.00i | − | 6740.54i | −5873.57 | − | 5873.57i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.10 | 8.00000 | − | 8.00000i | 89.5380 | − | 89.5380i | − | 128.000i | − | 225.860i | − | 1432.61i | −1638.83 | −1024.00 | − | 1024.00i | − | 9473.12i | −1806.88 | − | 1806.88i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | 8.00000 | + | 8.00000i | −109.436 | − | 109.436i | 128.000i | 214.288i | − | 1750.98i | −1333.30 | −1024.00 | + | 1024.00i | 17391.5i | −1714.30 | + | 1714.30i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.2 | 8.00000 | + | 8.00000i | −64.1915 | − | 64.1915i | 128.000i | − | 811.656i | − | 1027.06i | 2217.69 | −1024.00 | + | 1024.00i | 1680.09i | 6493.25 | − | 6493.25i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.3 | 8.00000 | + | 8.00000i | −53.0142 | − | 53.0142i | 128.000i | − | 111.354i | − | 848.227i | −2383.30 | −1024.00 | + | 1024.00i | − | 939.996i | 890.831 | − | 890.831i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.4 | 8.00000 | + | 8.00000i | −52.9266 | − | 52.9266i | 128.000i | 1048.55i | − | 846.826i | 76.2136 | −1024.00 | + | 1024.00i | − | 958.550i | −8388.36 | + | 8388.36i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.5 | 8.00000 | + | 8.00000i | 1.27653 | + | 1.27653i | 128.000i | 148.377i | 20.4244i | 4685.03 | −1024.00 | + | 1024.00i | − | 6557.74i | −1187.01 | + | 1187.01i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.6 | 8.00000 | + | 8.00000i | 25.8038 | + | 25.8038i | 128.000i | 413.688i | 412.861i | −3714.12 | −1024.00 | + | 1024.00i | − | 5229.33i | −3309.50 | + | 3309.50i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.7 | 8.00000 | + | 8.00000i | 30.1373 | + | 30.1373i | 128.000i | − | 621.075i | 482.196i | −328.250 | −1024.00 | + | 1024.00i | − | 4744.49i | 4968.60 | − | 4968.60i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.8 | 8.00000 | + | 8.00000i | 67.2603 | + | 67.2603i | 128.000i | − | 928.868i | 1076.17i | −450.923 | −1024.00 | + | 1024.00i | 2486.91i | 7430.95 | − | 7430.95i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.9 | 8.00000 | + | 8.00000i | 81.5522 | + | 81.5522i | 128.000i | 734.196i | 1304.84i | 2005.78 | −1024.00 | + | 1024.00i | 6740.54i | −5873.57 | + | 5873.57i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.10 | 8.00000 | + | 8.00000i | 89.5380 | + | 89.5380i | 128.000i | 225.860i | 1432.61i | −1638.83 | −1024.00 | + | 1024.00i | 9473.12i | −1806.88 | + | 1806.88i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 58.9.c.b | ✓ | 20 |
| 29.c | odd | 4 | 1 | inner | 58.9.c.b | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.9.c.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 58.9.c.b | ✓ | 20 | 29.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} - 32 T_{3}^{19} + 512 T_{3}^{18} - 356574 T_{3}^{17} + 624069448 T_{3}^{16} + \cdots + 94\!\cdots\!36 \)
acting on \(S_{9}^{\mathrm{new}}(58, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 128)^{10} \)
|
| $3$ |
\( T^{20} + \cdots + 94\!\cdots\!36 \)
|
| $5$ |
\( T^{20} + \cdots + 14\!\cdots\!00 \)
|
| $7$ |
\( (T^{10} + \cdots + 45\!\cdots\!00)^{2} \)
|
| $11$ |
\( T^{20} + \cdots + 32\!\cdots\!56 \)
|
| $13$ |
\( T^{20} + \cdots + 16\!\cdots\!44 \)
|
| $17$ |
\( T^{20} + \cdots + 64\!\cdots\!76 \)
|
| $19$ |
\( T^{20} + \cdots + 71\!\cdots\!84 \)
|
| $23$ |
\( (T^{10} + \cdots - 25\!\cdots\!60)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 98\!\cdots\!01 \)
|
| $31$ |
\( T^{20} + \cdots + 11\!\cdots\!76 \)
|
| $37$ |
\( T^{20} + \cdots + 66\!\cdots\!76 \)
|
| $41$ |
\( T^{20} + \cdots + 73\!\cdots\!76 \)
|
| $43$ |
\( T^{20} + \cdots + 29\!\cdots\!00 \)
|
| $47$ |
\( T^{20} + \cdots + 96\!\cdots\!00 \)
|
| $53$ |
\( (T^{10} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{10} + \cdots - 41\!\cdots\!20)^{2} \)
|
| $61$ |
\( T^{20} + \cdots + 30\!\cdots\!64 \)
|
| $67$ |
\( T^{20} + \cdots + 21\!\cdots\!04 \)
|
| $71$ |
\( T^{20} + \cdots + 96\!\cdots\!00 \)
|
| $73$ |
\( T^{20} + \cdots + 89\!\cdots\!56 \)
|
| $79$ |
\( T^{20} + \cdots + 35\!\cdots\!64 \)
|
| $83$ |
\( (T^{10} + \cdots - 81\!\cdots\!00)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 28\!\cdots\!16 \)
|
| $97$ |
\( T^{20} + \cdots + 15\!\cdots\!44 \)
|
| show more | |
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