Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [58,10,Mod(1,58)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(58, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("58.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.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: | \(29.8720784975\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 24277x^{4} - 438236x^{3} + 114209352x^{2} + 1666208856x - 67206942720 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 24277x^{4} - 438236x^{3} + 114209352x^{2} + 1666208856x - 67206942720 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 7014823 \nu^{5} - 86671145 \nu^{4} + 244229864971 \nu^{3} + 1747969777484 \nu^{2} + \cdots - 46\!\cdots\!80 ) / 232964528276760 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14587789 \nu^{5} - 503340735 \nu^{4} - 302984494153 \nu^{3} - 2986144580412 \nu^{2} + \cdots + 39\!\cdots\!30 ) / 58241132069190 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 27079359 \nu^{5} - 943432985 \nu^{4} + 653722617903 \nu^{3} + 25245990225092 \nu^{2} + \cdots - 28\!\cdots\!30 ) / 58241132069190 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 76319473 \nu^{5} - 793513355 \nu^{4} + 1669184359141 \nu^{3} + 54716299833524 \nu^{2} + \cdots - 13\!\cdots\!20 ) / 116482264138380 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 217060127 \nu^{5} - 25281219935 \nu^{4} - 3890967228779 \nu^{3} + 384623894622044 \nu^{2} + \cdots - 87\!\cdots\!20 ) / 232964528276760 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - \beta_{2} - 2\beta _1 + 4 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{5} + 9\beta_{4} - 29\beta_{3} - 55\beta_{2} + 42\beta _1 + 64720 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 456\beta_{5} - 7775\beta_{4} + 5153\beta_{3} - 10103\beta_{2} + 19682\beta _1 + 2769230 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 98720\beta_{5} - 54377\beta_{4} - 488583\beta_{3} - 1261953\beta_{2} + 1284974\beta _1 + 964540524 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 34073616 \beta_{5} - 281320405 \beta_{4} + 103918537 \beta_{3} - 446507533 \beta_{2} + \cdots + 188697419824 ) / 24 \)
|
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 |
|
16.0000 | −174.926 | 256.000 | −2146.80 | −2798.82 | 3870.64 | 4096.00 | 10916.2 | −34348.9 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | −113.340 | 256.000 | −1830.37 | −1813.44 | −10939.9 | 4096.00 | −6836.99 | −29286.0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | −60.3167 | 256.000 | 2278.24 | −965.068 | 7604.78 | 4096.00 | −16044.9 | 36451.8 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | 153.396 | 256.000 | −572.092 | 2454.33 | 4491.54 | 4096.00 | 3847.21 | −9153.48 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 16.0000 | 189.356 | 256.000 | 1951.82 | 3029.70 | −11047.8 | 4096.00 | 16172.7 | 31229.1 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 16.0000 | 252.832 | 256.000 | 1682.21 | 4045.30 | 7850.76 | 4096.00 | 44240.8 | 26915.4 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(29\) | \(1\) |
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 | 58.10.a.d | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.10.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 247T_{3}^{5} - 54692T_{3}^{4} + 12021042T_{3}^{3} + 1033609329T_{3}^{2} - 134924198715T_{3} - 8782137281358 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(58))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 8782137281358 \)
|
| $5$ |
\( T^{6} + \cdots - 16\!\cdots\!50 \)
|
| $7$ |
\( T^{6} + \cdots + 12\!\cdots\!96 \)
|
| $11$ |
\( T^{6} + \cdots + 36\!\cdots\!02 \)
|
| $13$ |
\( T^{6} + \cdots - 34\!\cdots\!98 \)
|
| $17$ |
\( T^{6} + \cdots - 45\!\cdots\!16 \)
|
| $19$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 24\!\cdots\!08 \)
|
| $29$ |
\( (T + 707281)^{6} \)
|
| $31$ |
\( T^{6} + \cdots + 30\!\cdots\!06 \)
|
| $37$ |
\( T^{6} + \cdots - 37\!\cdots\!20 \)
|
| $41$ |
\( T^{6} + \cdots - 23\!\cdots\!56 \)
|
| $43$ |
\( T^{6} + \cdots - 94\!\cdots\!38 \)
|
| $47$ |
\( T^{6} + \cdots + 10\!\cdots\!18 \)
|
| $53$ |
\( T^{6} + \cdots - 14\!\cdots\!02 \)
|
| $59$ |
\( T^{6} + \cdots + 86\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 58\!\cdots\!60 \)
|
| $67$ |
\( T^{6} + \cdots - 47\!\cdots\!20 \)
|
| $71$ |
\( T^{6} + \cdots - 32\!\cdots\!52 \)
|
| $73$ |
\( T^{6} + \cdots - 51\!\cdots\!64 \)
|
| $79$ |
\( T^{6} + \cdots - 87\!\cdots\!50 \)
|
| $83$ |
\( T^{6} + \cdots - 14\!\cdots\!28 \)
|
| $89$ |
\( T^{6} + \cdots + 45\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 17\!\cdots\!44 \)
|
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