Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,10,Mod(1,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, 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("23.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 23.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: | \(11.8458242318\) |
| Analytic rank: | \(1\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3199 \nu^{6} - 79358 \nu^{5} - 1488724 \nu^{4} + 30366663 \nu^{3} + 273707722 \nu^{2} + \cdots - 14876298328 ) / 11419280 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1391 \nu^{6} - 6842 \nu^{5} - 682136 \nu^{4} - 269473 \nu^{3} + 87308078 \nu^{2} + \cdots - 2521122692 ) / 2854820 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 7713 \nu^{6} + 105666 \nu^{5} + 3328828 \nu^{4} - 32964761 \nu^{3} - 360613014 \nu^{2} + \cdots + 6770868456 ) / 11419280 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 7713 \nu^{6} - 105666 \nu^{5} - 3328828 \nu^{4} + 32964761 \nu^{3} + 383451574 \nu^{2} + \cdots - 10950324936 ) / 5709640 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 29851 \nu^{6} + 497782 \nu^{5} + 11250276 \nu^{4} - 154091107 \nu^{3} - 1009430098 \nu^{2} + \cdots + 17652094632 ) / 11419280 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} + 6\beta _1 + 732 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{6} + 15\beta_{5} + 14\beta_{4} - 8\beta_{3} - 6\beta_{2} + 560\beta _1 + 2494 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4\beta_{6} + 243\beta_{5} + 456\beta_{4} - 32\beta_{3} - 54\beta_{2} + 2205\beta _1 + 103240 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 964\beta_{6} + 9063\beta_{5} + 10258\beta_{4} - 3564\beta_{3} - 3776\beta_{2} + 192494\beta _1 + 1742612 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1206 \beta_{6} + 223049 \beta_{5} + 374874 \beta_{4} - 42256 \beta_{3} - 72698 \beta_{2} + \cdots + 71615698 ) / 4 \)
|
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 |
|
−30.8448 | 107.347 | 439.399 | −1512.00 | −3311.09 | 8168.19 | 2239.35 | −8159.64 | 46637.2 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −30.1460 | −193.374 | 396.783 | 568.966 | 5829.45 | 3614.33 | 3473.33 | 17710.4 | −17152.1 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −20.7546 | 245.319 | −81.2472 | −680.714 | −5091.50 | −8213.17 | 12312.6 | 40498.5 | 14127.9 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.55091 | −23.1179 | −491.289 | 2146.10 | 105.207 | 615.663 | 4565.88 | −19148.6 | −9766.68 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 17.4059 | 86.8015 | −209.035 | −1221.93 | 1510.86 | −4757.43 | −12550.3 | −12148.5 | −21268.8 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 26.3074 | −105.922 | 180.082 | 92.1465 | −2786.54 | 2337.45 | −8731.92 | −8463.52 | 2424.14 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 42.5829 | −206.054 | 1301.31 | −1780.57 | −8774.39 | −11661.0 | 33611.0 | 22775.3 | −75821.7 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \(1\) |
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 | 23.10.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 207.10.a.b | 7 | ||
| 4.b | odd | 2 | 1 | 368.10.a.f | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.10.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 207.10.a.b | 7 | 3.b | odd | 2 | 1 | ||
| 368.10.a.f | 7 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 2560T_{2}^{5} - 11640T_{2}^{4} + 1832832T_{2}^{3} + 10289408T_{2}^{2} - 367442944T_{2} - 1712515072 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(23))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots - 1712515072 \)
|
| $3$ |
\( T^{7} + \cdots - 223029139672800 \)
|
| $5$ |
\( T^{7} + \cdots - 25\!\cdots\!80 \)
|
| $7$ |
\( T^{7} + \cdots + 19\!\cdots\!28 \)
|
| $11$ |
\( T^{7} + \cdots - 10\!\cdots\!48 \)
|
| $13$ |
\( T^{7} + \cdots - 72\!\cdots\!88 \)
|
| $17$ |
\( T^{7} + \cdots + 48\!\cdots\!00 \)
|
| $19$ |
\( T^{7} + \cdots - 58\!\cdots\!40 \)
|
| $23$ |
\( (T + 279841)^{7} \)
|
| $29$ |
\( T^{7} + \cdots - 15\!\cdots\!00 \)
|
| $31$ |
\( T^{7} + \cdots - 89\!\cdots\!28 \)
|
| $37$ |
\( T^{7} + \cdots + 12\!\cdots\!92 \)
|
| $41$ |
\( T^{7} + \cdots + 21\!\cdots\!20 \)
|
| $43$ |
\( T^{7} + \cdots - 77\!\cdots\!68 \)
|
| $47$ |
\( T^{7} + \cdots - 30\!\cdots\!40 \)
|
| $53$ |
\( T^{7} + \cdots - 18\!\cdots\!72 \)
|
| $59$ |
\( T^{7} + \cdots - 23\!\cdots\!60 \)
|
| $61$ |
\( T^{7} + \cdots + 16\!\cdots\!04 \)
|
| $67$ |
\( T^{7} + \cdots - 12\!\cdots\!56 \)
|
| $71$ |
\( T^{7} + \cdots + 17\!\cdots\!20 \)
|
| $73$ |
\( T^{7} + \cdots + 95\!\cdots\!60 \)
|
| $79$ |
\( T^{7} + \cdots + 26\!\cdots\!56 \)
|
| $83$ |
\( T^{7} + \cdots + 71\!\cdots\!08 \)
|
| $89$ |
\( T^{7} + \cdots + 63\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 62\!\cdots\!00 \)
|
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