Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [130,3,Mod(3,130)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(130, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([9, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("130.3");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 130 = 2 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 130.q (of order \(12\), degree \(4\), 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: | \(3.54224343668\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.5802782976.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 11x^{6} + 85x^{4} + 396x^{2} + 1296 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 11x^{6} + 85x^{4} + 396x^{2} + 1296 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -11\nu^{6} - 85\nu^{4} - 935\nu^{2} - 4356 ) / 3060 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 19\nu^{7} + 425\nu^{5} + 1615\nu^{3} + 7524\nu ) / 18360 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 17\nu^{4} - 85\nu^{2} + 102\nu - 396 ) / 102 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + 5\nu^{6} + 85\nu^{4} + 425\nu^{2} + 539\nu + 1980 ) / 510 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11\nu^{7} + 85\nu^{5} + 323\nu^{3} + 1296\nu ) / 3672 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -11\nu^{7} + 36\nu^{6} - 85\nu^{5} - 935\nu^{3} - 1296\nu - 1044 ) / 3060 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -121\nu^{7} - 216\nu^{6} - 935\nu^{5} - 7225\nu^{3} - 14256\nu + 6264 ) / 18360 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 12\beta _1 - 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{7} - 5\beta_{6} - 17\beta_{5} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 11\beta_{5} + 11\beta_{4} - 11\beta_{3} - 11\beta_{2} + 60\beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19\beta_{7} + 19\beta_{6} - 19\beta_{4} - 19\beta_{3} + 151\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -85\beta_{7} + 85\beta_{6} - 85\beta_{5} + 58 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1049\beta_{5} + 29\beta_{4} + 29\beta_{3} - 1049\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(-\beta_{5}\) | \(-1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−1.36603 | − | 0.366025i | −1.06071 | + | 3.95863i | 1.73205 | + | 1.00000i | 3.00000 | + | 4.00000i | 2.89792 | − | 5.01934i | 0.403391 | + | 1.50548i | −2.00000 | − | 2.00000i | −6.75139 | − | 3.89792i | −2.63397 | − | 6.56218i | ||||||||||||||||||||||||
| 3.2 | −1.36603 | − | 0.366025i | 0.694685 | − | 2.59260i | 1.73205 | + | 1.00000i | 3.00000 | + | 4.00000i | −1.89792 | + | 3.28729i | 2.15879 | + | 8.05670i | −2.00000 | − | 2.00000i | 1.55524 | + | 0.897916i | −2.63397 | − | 6.56218i | |||||||||||||||||||||||||
| 87.1 | −1.36603 | + | 0.366025i | −1.06071 | − | 3.95863i | 1.73205 | − | 1.00000i | 3.00000 | − | 4.00000i | 2.89792 | + | 5.01934i | 0.403391 | − | 1.50548i | −2.00000 | + | 2.00000i | −6.75139 | + | 3.89792i | −2.63397 | + | 6.56218i | |||||||||||||||||||||||||
| 87.2 | −1.36603 | + | 0.366025i | 0.694685 | + | 2.59260i | 1.73205 | − | 1.00000i | 3.00000 | − | 4.00000i | −1.89792 | − | 3.28729i | 2.15879 | − | 8.05670i | −2.00000 | + | 2.00000i | 1.55524 | − | 0.897916i | −2.63397 | + | 6.56218i | |||||||||||||||||||||||||
| 107.1 | 0.366025 | − | 1.36603i | −2.59260 | − | 0.694685i | −1.73205 | − | 1.00000i | 3.00000 | − | 4.00000i | −1.89792 | + | 3.28729i | −8.05670 | + | 2.15879i | −2.00000 | + | 2.00000i | −1.55524 | − | 0.897916i | −4.36603 | − | 5.56218i | |||||||||||||||||||||||||
| 107.2 | 0.366025 | − | 1.36603i | 3.95863 | + | 1.06071i | −1.73205 | − | 1.00000i | 3.00000 | − | 4.00000i | 2.89792 | − | 5.01934i | −1.50548 | + | 0.403391i | −2.00000 | + | 2.00000i | 6.75139 | + | 3.89792i | −4.36603 | − | 5.56218i | |||||||||||||||||||||||||
| 113.1 | 0.366025 | + | 1.36603i | −2.59260 | + | 0.694685i | −1.73205 | + | 1.00000i | 3.00000 | + | 4.00000i | −1.89792 | − | 3.28729i | −8.05670 | − | 2.15879i | −2.00000 | − | 2.00000i | −1.55524 | + | 0.897916i | −4.36603 | + | 5.56218i | |||||||||||||||||||||||||
| 113.2 | 0.366025 | + | 1.36603i | 3.95863 | − | 1.06071i | −1.73205 | + | 1.00000i | 3.00000 | + | 4.00000i | 2.89792 | + | 5.01934i | −1.50548 | − | 0.403391i | −2.00000 | − | 2.00000i | 6.75139 | − | 3.89792i | −4.36603 | + | 5.56218i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 13.c | even | 3 | 1 | inner |
| 65.q | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 130.3.q.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | inner | 130.3.q.a | ✓ | 8 |
| 13.c | even | 3 | 1 | inner | 130.3.q.a | ✓ | 8 |
| 65.q | odd | 12 | 1 | inner | 130.3.q.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.3.q.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 130.3.q.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 130.3.q.a | ✓ | 8 | 13.c | even | 3 | 1 | inner |
| 130.3.q.a | ✓ | 8 | 65.q | odd | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} - 48T_{3}^{5} - 73T_{3}^{4} + 528T_{3}^{3} + 242T_{3}^{2} + 2662T_{3} + 14641 \)
acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 14641 \)
|
| $5$ |
\( (T^{2} - 6 T + 25)^{4} \)
|
| $7$ |
\( T^{8} + 14 T^{7} + \cdots + 28561 \)
|
| $11$ |
\( (T^{4} + 23 T^{2} + 529)^{2} \)
|
| $13$ |
\( T^{8} + 24 T^{7} + \cdots + 815730721 \)
|
| $17$ |
\( T^{8} + 18 T^{7} + \cdots + 15752961 \)
|
| $19$ |
\( T^{8} + \cdots + 18141126721 \)
|
| $23$ |
\( T^{8} + \cdots + 1908029761 \)
|
| $29$ |
\( T^{8} + \cdots + 8731794511521 \)
|
| $31$ |
\( (T^{2} - 40 T + 32)^{4} \)
|
| $37$ |
\( T^{8} + \cdots + 2918114646001 \)
|
| $41$ |
\( (T^{4} + 30 T^{3} + \cdots + 20449)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 738446330241 \)
|
| $47$ |
\( (T^{4} - 12 T^{3} + \cdots + 8561476)^{2} \)
|
| $53$ |
\( (T^{4} - 40 T^{3} + \cdots + 4218916)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 203942419667281 \)
|
| $61$ |
\( (T^{4} + 190 T^{3} + \cdots + 74943649)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 5380214781841 \)
|
| $71$ |
\( (T^{4} - 96 T^{3} + \cdots + 1385329)^{2} \)
|
| $73$ |
\( (T^{4} - 212 T^{3} + \cdots + 29528356)^{2} \)
|
| $79$ |
\( (T^{4} + 24704 T^{2} + 6512704)^{2} \)
|
| $83$ |
\( (T^{4} - 44 T^{3} + \cdots + 18992164)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 268776384148801 \)
|
| $97$ |
\( T^{8} + \cdots + 86\!\cdots\!61 \)
|
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