Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [58,7,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, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58.17");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| 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: | \(13.3431368500\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 8 x^{14} + 20450 x^{13} + 8564487 x^{12} + 6394574 x^{11} + 115007058 x^{10} + \cdots + 15\!\cdots\!96 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 4 x^{15} + 8 x^{14} + 20450 x^{13} + 8564487 x^{12} + 6394574 x^{11} + 115007058 x^{10} + \cdots + 15\!\cdots\!96 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 56\!\cdots\!35 \nu^{15} + \cdots - 24\!\cdots\!12 ) / 40\!\cdots\!40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 21\!\cdots\!91 \nu^{15} + \cdots + 89\!\cdots\!60 ) / 41\!\cdots\!40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10\!\cdots\!49 \nu^{15} + \cdots + 26\!\cdots\!16 ) / 15\!\cdots\!60 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17\!\cdots\!14 \nu^{15} + \cdots + 61\!\cdots\!20 ) / 16\!\cdots\!60 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 32\!\cdots\!03 \nu^{15} + \cdots - 90\!\cdots\!60 ) / 15\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 34\!\cdots\!32 \nu^{15} + \cdots - 16\!\cdots\!56 ) / 16\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 39\!\cdots\!59 \nu^{15} + \cdots + 14\!\cdots\!60 ) / 15\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 24\!\cdots\!22 \nu^{15} + \cdots + 37\!\cdots\!60 ) / 80\!\cdots\!30 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 75\!\cdots\!37 \nu^{15} + \cdots - 22\!\cdots\!32 ) / 15\!\cdots\!60 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 12\!\cdots\!97 \nu^{15} + \cdots + 36\!\cdots\!76 ) / 15\!\cdots\!60 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 63\!\cdots\!62 \nu^{15} + \cdots + 25\!\cdots\!16 ) / 79\!\cdots\!80 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 15\!\cdots\!59 \nu^{15} + \cdots + 15\!\cdots\!96 ) / 15\!\cdots\!60 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 85\!\cdots\!27 \nu^{15} + \cdots + 24\!\cdots\!00 ) / 15\!\cdots\!60 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 87\!\cdots\!42 \nu^{15} + \cdots + 65\!\cdots\!00 ) / 79\!\cdots\!80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{13} - \beta_{11} - \beta_{8} + \beta_{4} - 4\beta_{3} + 1102\beta_{2} + 4\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{15} + 2 \beta_{13} + 14 \beta_{12} + 14 \beta_{9} + \beta_{8} - \beta_{7} - 7 \beta_{6} + \cdots - 3342 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2545 \beta_{13} + 2545 \beta_{11} - 66 \beta_{10} - 1443 \beta_{9} + 2632 \beta_{7} - 66 \beta_{6} + \cdots - 2156328 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 4033 \beta_{14} - 47645 \beta_{12} + 12803 \beta_{11} - 23887 \beta_{10} + 47645 \beta_{9} + \cdots - 11580664 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 18420 \beta_{15} + 18420 \beta_{14} - 6506793 \beta_{13} + 5197776 \beta_{12} + 6506793 \beta_{11} + \cdots + 18420 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11661195 \beta_{15} - 58191210 \beta_{13} - 129396057 \beta_{12} - 129396057 \beta_{9} + \cdots + 39938754531 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 66153711 \beta_{15} - 66153711 \beta_{14} - 16941831974 \beta_{13} - 16941831974 \beta_{11} + \cdots + 12818058802025 \)
|
| \(\nu^{9}\) | \(=\) |
\( 31151049467 \beta_{14} + 322332656038 \beta_{12} - 227393078245 \beta_{11} + 192091821515 \beta_{10} + \cdots + 134232721696241 \)
|
| \(\nu^{10}\) | \(=\) |
\( 158268334983 \beta_{15} - 158268334983 \beta_{14} + 44708933314802 \beta_{13} - 39254105324820 \beta_{12} + \cdots - 158268334983 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 81594495816311 \beta_{15} + 810944570178220 \beta_{13} + 769223946611950 \beta_{12} + \cdots - 43\!\cdots\!92 \)
|
| \(\nu^{12}\) | \(=\) |
\( 287043163818216 \beta_{15} + 287043163818216 \beta_{14} + \cdots - 89\!\cdots\!52 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 21\!\cdots\!73 \beta_{14} + \cdots - 13\!\cdots\!58 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 30\!\cdots\!78 \beta_{15} + \cdots + 30\!\cdots\!78 \)
|
| \(\nu^{15}\) | \(=\) |
\( 55\!\cdots\!39 \beta_{15} + \cdots + 42\!\cdots\!85 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 |
|
4.00000 | − | 4.00000i | −35.5579 | + | 35.5579i | − | 32.0000i | 230.586i | 284.463i | −308.085 | −128.000 | − | 128.000i | − | 1799.72i | 922.343 | + | 922.343i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | 4.00000 | − | 4.00000i | −22.1668 | + | 22.1668i | − | 32.0000i | − | 83.0798i | 177.334i | 178.568 | −128.000 | − | 128.000i | − | 253.731i | −332.319 | − | 332.319i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 4.00000 | − | 4.00000i | −17.4695 | + | 17.4695i | − | 32.0000i | − | 25.8464i | 139.756i | 189.045 | −128.000 | − | 128.000i | 118.632i | −103.385 | − | 103.385i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 4.00000 | − | 4.00000i | 0.521305 | − | 0.521305i | − | 32.0000i | 12.6283i | − | 4.17044i | −534.715 | −128.000 | − | 128.000i | 728.456i | 50.5134 | + | 50.5134i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 4.00000 | − | 4.00000i | 10.3630 | − | 10.3630i | − | 32.0000i | 239.375i | − | 82.9040i | 51.7768 | −128.000 | − | 128.000i | 514.217i | 957.502 | + | 957.502i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 4.00000 | − | 4.00000i | 15.7324 | − | 15.7324i | − | 32.0000i | − | 230.165i | − | 125.859i | 15.1401 | −128.000 | − | 128.000i | 233.983i | −920.660 | − | 920.660i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.7 | 4.00000 | − | 4.00000i | 25.5900 | − | 25.5900i | − | 32.0000i | 27.7967i | − | 204.720i | 596.779 | −128.000 | − | 128.000i | − | 580.698i | 111.187 | + | 111.187i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 4.00000 | − | 4.00000i | 36.9874 | − | 36.9874i | − | 32.0000i | 46.7049i | − | 295.899i | −576.510 | −128.000 | − | 128.000i | − | 2007.14i | 186.820 | + | 186.820i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.1 | 4.00000 | + | 4.00000i | −35.5579 | − | 35.5579i | 32.0000i | − | 230.586i | − | 284.463i | −308.085 | −128.000 | + | 128.000i | 1799.72i | 922.343 | − | 922.343i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.2 | 4.00000 | + | 4.00000i | −22.1668 | − | 22.1668i | 32.0000i | 83.0798i | − | 177.334i | 178.568 | −128.000 | + | 128.000i | 253.731i | −332.319 | + | 332.319i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.3 | 4.00000 | + | 4.00000i | −17.4695 | − | 17.4695i | 32.0000i | 25.8464i | − | 139.756i | 189.045 | −128.000 | + | 128.000i | − | 118.632i | −103.385 | + | 103.385i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.4 | 4.00000 | + | 4.00000i | 0.521305 | + | 0.521305i | 32.0000i | − | 12.6283i | 4.17044i | −534.715 | −128.000 | + | 128.000i | − | 728.456i | 50.5134 | − | 50.5134i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.5 | 4.00000 | + | 4.00000i | 10.3630 | + | 10.3630i | 32.0000i | − | 239.375i | 82.9040i | 51.7768 | −128.000 | + | 128.000i | − | 514.217i | 957.502 | − | 957.502i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.6 | 4.00000 | + | 4.00000i | 15.7324 | + | 15.7324i | 32.0000i | 230.165i | 125.859i | 15.1401 | −128.000 | + | 128.000i | − | 233.983i | −920.660 | + | 920.660i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.7 | 4.00000 | + | 4.00000i | 25.5900 | + | 25.5900i | 32.0000i | − | 27.7967i | 204.720i | 596.779 | −128.000 | + | 128.000i | 580.698i | 111.187 | − | 111.187i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 41.8 | 4.00000 | + | 4.00000i | 36.9874 | + | 36.9874i | 32.0000i | − | 46.7049i | 295.899i | −576.510 | −128.000 | + | 128.000i | 2007.14i | 186.820 | − | 186.820i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 58.7.c.b | ✓ | 16 |
| 29.c | odd | 4 | 1 | inner | 58.7.c.b | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.7.c.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 58.7.c.b | ✓ | 16 | 29.c | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 28 T_{3}^{15} + 392 T_{3}^{14} + 11254 T_{3}^{13} + 8220895 T_{3}^{12} + \cdots + 31\!\cdots\!00 \)
acting on \(S_{7}^{\mathrm{new}}(58, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 32)^{8} \)
|
| $3$ |
\( T^{16} + \cdots + 31\!\cdots\!00 \)
|
| $5$ |
\( T^{16} + \cdots + 20\!\cdots\!00 \)
|
| $7$ |
\( (T^{8} + \cdots - 14\!\cdots\!88)^{2} \)
|
| $11$ |
\( T^{16} + \cdots + 13\!\cdots\!00 \)
|
| $13$ |
\( T^{16} + \cdots + 20\!\cdots\!00 \)
|
| $17$ |
\( T^{16} + \cdots + 14\!\cdots\!00 \)
|
| $19$ |
\( T^{16} + \cdots + 33\!\cdots\!16 \)
|
| $23$ |
\( (T^{8} + \cdots + 39\!\cdots\!12)^{2} \)
|
| $29$ |
\( T^{16} + \cdots + 15\!\cdots\!61 \)
|
| $31$ |
\( T^{16} + \cdots + 12\!\cdots\!24 \)
|
| $37$ |
\( T^{16} + \cdots + 46\!\cdots\!76 \)
|
| $41$ |
\( T^{16} + \cdots + 17\!\cdots\!84 \)
|
| $43$ |
\( T^{16} + \cdots + 27\!\cdots\!56 \)
|
| $47$ |
\( T^{16} + \cdots + 11\!\cdots\!96 \)
|
| $53$ |
\( (T^{8} + \cdots + 79\!\cdots\!80)^{2} \)
|
| $59$ |
\( (T^{8} + \cdots + 31\!\cdots\!80)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 33\!\cdots\!16 \)
|
| $67$ |
\( T^{16} + \cdots + 47\!\cdots\!00 \)
|
| $71$ |
\( T^{16} + \cdots + 17\!\cdots\!00 \)
|
| $73$ |
\( T^{16} + \cdots + 34\!\cdots\!00 \)
|
| $79$ |
\( T^{16} + \cdots + 19\!\cdots\!24 \)
|
| $83$ |
\( (T^{8} + \cdots - 39\!\cdots\!60)^{2} \)
|
| $89$ |
\( T^{16} + \cdots + 18\!\cdots\!00 \)
|
| $97$ |
\( T^{16} + \cdots + 10\!\cdots\!00 \)
|
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