Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [13,14,Mod(1,13)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 13 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 13.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: | \(13.9400207637\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{7} - x^{6} - 35596x^{5} + 594856x^{4} + 263632368x^{3} + 252644208x^{2} - 410033371968x - 5442114981888 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| 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
\( x^{7} - x^{6} - 35596x^{5} + 594856x^{4} + 263632368x^{3} + 252644208x^{2} - 410033371968x - 5442114981888 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 969 \nu^{6} - 153125 \nu^{5} - 37273116 \nu^{4} + 5141065076 \nu^{3} + 205885624704 \nu^{2} + \cdots - 372250932068352 ) / 215581424640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 969 \nu^{6} + 153125 \nu^{5} + 37273116 \nu^{4} - 5141065076 \nu^{3} + \cdots - 724411775075328 ) / 107790712320 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2191 \nu^{6} + 13092 \nu^{5} + 104657371 \nu^{4} - 1116041580 \nu^{3} + \cdots + 22\!\cdots\!04 ) / 26947678080 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 90703 \nu^{6} - 3354157 \nu^{5} - 3281844634 \nu^{4} + 139887596428 \nu^{3} + \cdots - 19\!\cdots\!80 ) / 215581424640 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 51142 \nu^{6} + 709161 \nu^{5} + 1685935189 \nu^{4} - 60895203816 \nu^{3} + \cdots + 92\!\cdots\!84 ) / 107790712320 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} - 24\beta _1 + 10174 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{6} - 10\beta_{5} - 8\beta_{4} - 77\beta_{3} + 4\beta_{2} + 20778\beta _1 - 242636 \)
|
| \(\nu^{4}\) | \(=\) |
\( 34\beta_{6} + 302\beta_{5} + 1056\beta_{4} + 28003\beta_{3} + 50428\beta_{2} - 1445466\beta _1 + 211264092 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 187182 \beta_{6} - 296962 \beta_{5} - 268640 \beta_{4} - 3095465 \beta_{3} - 3011468 \beta_{2} + \cdots - 14663379932 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3561850 \beta_{6} + 17744918 \beta_{5} + 40612416 \beta_{4} + 784047859 \beta_{3} + 1240168804 \beta_{2} + \cdots + 5319008263956 \)
|
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 |
|
−157.134 | 286.918 | 16499.3 | 17641.6 | −45084.7 | −100427. | −1.30536e6 | −1.51200e6 | −2.77210e6 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −52.2059 | −945.382 | −5466.54 | −30724.8 | 49354.5 | −524401. | 713057. | −700576. | 1.60402e6 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −19.9195 | 1765.94 | −7795.21 | 53134.3 | −35176.8 | 17946.1 | 318458. | 1.52424e6 | −1.05841e6 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.73404 | −42.7679 | −8184.53 | −45417.0 | −116.929 | 591080. | −44774.1 | −1.59249e6 | −124172. | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 66.8519 | −2327.76 | −3722.83 | 7472.85 | −155615. | −60830.3 | −796528. | 3.82413e6 | 499574. | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 135.726 | 2027.33 | 10229.5 | −15679.1 | 275162. | 295835. | 276541. | 2.51576e6 | −2.12805e6 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 150.948 | −36.2889 | 14593.4 | 48458.2 | −5477.75 | −87338.4 | 966274. | −1.59301e6 | 7.31468e6 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(-1\) |
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 | 13.14.a.b | ✓ | 7 |
| 3.b | odd | 2 | 1 | 117.14.a.d | 7 | ||
| 13.b | even | 2 | 1 | 169.14.a.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 13.14.a.b | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 117.14.a.d | 7 | 3.b | odd | 2 | 1 | ||
| 169.14.a.b | 7 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 127 T_{2}^{6} - 28684 T_{2}^{5} + 3589516 T_{2}^{4} + 109262496 T_{2}^{3} + \cdots + 611898900480 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(13))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + \cdots + 611898900480 \)
|
| $3$ |
\( T^{7} + \cdots - 35\!\cdots\!72 \)
|
| $5$ |
\( T^{7} + \cdots + 74\!\cdots\!00 \)
|
| $7$ |
\( T^{7} + \cdots - 87\!\cdots\!32 \)
|
| $11$ |
\( T^{7} + \cdots + 25\!\cdots\!84 \)
|
| $13$ |
\( (T - 4826809)^{7} \)
|
| $17$ |
\( T^{7} + \cdots + 10\!\cdots\!12 \)
|
| $19$ |
\( T^{7} + \cdots - 59\!\cdots\!40 \)
|
| $23$ |
\( T^{7} + \cdots - 57\!\cdots\!12 \)
|
| $29$ |
\( T^{7} + \cdots + 17\!\cdots\!40 \)
|
| $31$ |
\( T^{7} + \cdots - 20\!\cdots\!00 \)
|
| $37$ |
\( T^{7} + \cdots + 12\!\cdots\!56 \)
|
| $41$ |
\( T^{7} + \cdots + 14\!\cdots\!68 \)
|
| $43$ |
\( T^{7} + \cdots - 24\!\cdots\!00 \)
|
| $47$ |
\( T^{7} + \cdots - 69\!\cdots\!20 \)
|
| $53$ |
\( T^{7} + \cdots - 44\!\cdots\!28 \)
|
| $59$ |
\( T^{7} + \cdots - 26\!\cdots\!20 \)
|
| $61$ |
\( T^{7} + \cdots - 14\!\cdots\!84 \)
|
| $67$ |
\( T^{7} + \cdots - 21\!\cdots\!68 \)
|
| $71$ |
\( T^{7} + \cdots - 39\!\cdots\!28 \)
|
| $73$ |
\( T^{7} + \cdots - 10\!\cdots\!28 \)
|
| $79$ |
\( T^{7} + \cdots + 45\!\cdots\!20 \)
|
| $83$ |
\( T^{7} + \cdots - 22\!\cdots\!92 \)
|
| $89$ |
\( T^{7} + \cdots + 65\!\cdots\!00 \)
|
| $97$ |
\( T^{7} + \cdots - 13\!\cdots\!20 \)
|
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