Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [123,6,Mod(1,123)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(123, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("123.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 123 = 3 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 123.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: | \(19.7272098370\) |
| 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: |
\( x^{7} - x^{6} - 152x^{5} + 180x^{4} + 6087x^{3} - 2823x^{2} - 60512x - 42332 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{7} - x^{6} - 152x^{5} + 180x^{4} + 6087x^{3} - 2823x^{2} - 60512x - 42332 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -16\nu^{6} - 702\nu^{5} - 2683\nu^{4} + 65829\nu^{3} + 301519\nu^{2} - 1196629\nu - 3557894 ) / 140732 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -89\nu^{6} + 493\nu^{5} + 13662\nu^{4} - 27436\nu^{3} - 453571\nu^{2} - 1125921\nu + 2677958 ) / 281464 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -371\nu^{6} - 3084\nu^{5} + 41138\nu^{4} + 292806\nu^{3} - 966483\nu^{2} - 5438614\nu - 3165400 ) / 281464 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 371\nu^{6} + 3084\nu^{5} - 41138\nu^{4} - 292806\nu^{3} + 1247947\nu^{2} + 5438614\nu - 9219016 ) / 281464 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 194\nu^{6} - 284\nu^{5} - 24641\nu^{4} + 59409\nu^{3} + 675989\nu^{2} - 1547515\nu - 3205342 ) / 70366 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 44 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} - \beta_{5} - \beta_{4} + 8\beta_{3} + 2\beta_{2} + 71\beta _1 - 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 86\beta_{5} + 102\beta_{4} - 24\beta_{3} - 46\beta_{2} - 112\beta _1 + 3166 \)
|
| \(\nu^{5}\) | \(=\) |
\( 70\beta_{6} - 164\beta_{5} - 268\beta_{4} + 936\beta_{3} + 300\beta_{2} + 5825\beta _1 - 6518 \)
|
| \(\nu^{6}\) | \(=\) |
\( 540\beta_{6} + 7505\beta_{5} + 9385\beta_{4} - 4128\beta_{3} - 6016\beta_{2} - 19464\beta _1 + 263144 \)
|
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 |
|
−11.1568 | 9.00000 | 92.4739 | −19.1030 | −100.411 | 174.467 | −674.694 | 81.0000 | 213.128 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.05025 | 9.00000 | 17.7060 | −84.9899 | −63.4522 | 106.412 | 100.776 | 81.0000 | 599.200 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.95735 | 9.00000 | −16.3394 | 61.0736 | −35.6161 | −177.875 | 191.296 | 81.0000 | −241.689 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.77188 | 9.00000 | −28.8604 | −36.5536 | −15.9469 | 105.449 | 107.838 | 81.0000 | 64.7687 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 3.43813 | 9.00000 | −20.1793 | 22.2843 | 30.9431 | −65.4111 | −179.399 | 81.0000 | 76.6162 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.02406 | 9.00000 | 17.3374 | −61.3615 | 63.2165 | −64.7484 | −102.991 | 81.0000 | −431.006 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 7.47408 | 9.00000 | 23.8619 | −81.3500 | 67.2667 | −136.293 | −60.8249 | 81.0000 | −608.017 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 123.6.a.a | ✓ | 7 |
| 3.b | odd | 2 | 1 | 369.6.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 123.6.a.a | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 369.6.a.b | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 6T_{2}^{6} - 137T_{2}^{5} - 560T_{2}^{4} + 5302T_{2}^{3} + 15004T_{2}^{2} - 47936T_{2} - 99552 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(123))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + 6 T^{6} + \cdots - 99552 \)
|
| $3$ |
\( (T - 9)^{7} \)
|
| $5$ |
\( T^{7} + \cdots + 403184891584 \)
|
| $7$ |
\( T^{7} + \cdots - 201009160668032 \)
|
| $11$ |
\( T^{7} + \cdots + 38\!\cdots\!52 \)
|
| $13$ |
\( T^{7} + \cdots - 81\!\cdots\!56 \)
|
| $17$ |
\( T^{7} + \cdots + 17\!\cdots\!98 \)
|
| $19$ |
\( T^{7} + \cdots + 25\!\cdots\!24 \)
|
| $23$ |
\( T^{7} + \cdots - 11\!\cdots\!68 \)
|
| $29$ |
\( T^{7} + \cdots - 10\!\cdots\!20 \)
|
| $31$ |
\( T^{7} + \cdots + 66\!\cdots\!68 \)
|
| $37$ |
\( T^{7} + \cdots - 53\!\cdots\!20 \)
|
| $41$ |
\( (T - 1681)^{7} \)
|
| $43$ |
\( T^{7} + \cdots + 10\!\cdots\!56 \)
|
| $47$ |
\( T^{7} + \cdots + 30\!\cdots\!36 \)
|
| $53$ |
\( T^{7} + \cdots - 60\!\cdots\!52 \)
|
| $59$ |
\( T^{7} + \cdots - 12\!\cdots\!92 \)
|
| $61$ |
\( T^{7} + \cdots - 39\!\cdots\!04 \)
|
| $67$ |
\( T^{7} + \cdots + 96\!\cdots\!88 \)
|
| $71$ |
\( T^{7} + \cdots + 23\!\cdots\!16 \)
|
| $73$ |
\( T^{7} + \cdots + 40\!\cdots\!10 \)
|
| $79$ |
\( T^{7} + \cdots + 21\!\cdots\!44 \)
|
| $83$ |
\( T^{7} + \cdots + 25\!\cdots\!12 \)
|
| $89$ |
\( T^{7} + \cdots - 73\!\cdots\!28 \)
|
| $97$ |
\( T^{7} + \cdots + 35\!\cdots\!68 \)
|
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