Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,2,Mod(481,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("930.481");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.n (of order \(5\), 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: | \(7.42608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.1816890625.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 8x^{6} - 15x^{5} + 41x^{4} + 15x^{3} + 62x^{2} + 133x + 361 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - x^{7} + 8x^{6} - 15x^{5} + 41x^{4} + 15x^{3} + 62x^{2} + 133x + 361 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 128795 \nu^{7} + 470503 \nu^{6} - 1652991 \nu^{5} + 5286232 \nu^{4} - 16599235 \nu^{3} + \cdots + 22451179 ) / 303029309 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 214762 \nu^{7} + 1363340 \nu^{6} + 2203394 \nu^{5} - 2198831 \nu^{4} + 4048801 \nu^{3} + \cdots - 89195880 ) / 303029309 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 17568 \nu^{7} - 45727 \nu^{6} - 524081 \nu^{5} - 229321 \nu^{4} - 419871 \nu^{3} + \cdots - 20195518 ) / 15948911 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 31542 \nu^{7} + 141229 \nu^{6} - 379903 \nu^{5} + 1151570 \nu^{4} - 5411059 \nu^{3} + \cdots + 2447105 ) / 15948911 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 719365 \nu^{7} + 3230557 \nu^{6} - 7084160 \nu^{5} + 28988770 \nu^{4} - 51773137 \nu^{3} + \cdots + 100589097 ) / 303029309 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 132168 \nu^{7} - 69960 \nu^{6} + 957805 \nu^{5} - 1172588 \nu^{4} + 4400707 \nu^{3} + \cdots + 13667935 ) / 15948911 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 5\beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{6} - 5\beta_{5} + \beta_{4} + \beta_{3} - \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 11\beta_{7} + 21\beta_{6} + 2\beta_{5} + 2\beta_{4} - 31\beta_{3} - 31\beta_{2} - 21 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{7} - 19\beta_{6} - 42\beta_{4} + 19\beta_{2} + 12\beta _1 - 38 \)
|
| \(\nu^{6}\) | \(=\) |
\( -31\beta_{7} - 31\beta_{5} + 222\beta_{3} + 90\beta_{2} - 68\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( 321\beta_{7} + 223\beta_{6} + 200\beta_{5} + 321\beta_{4} - 223\beta_{3} - 526\beta_{2} - 200\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(871\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 481.1 |
|
0.809017 | − | 0.587785i | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | −1.00000 | 1.00000 | −0.785038 | + | 2.41610i | −0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | ||||||||||||||||||||||||||||
| 481.2 | 0.809017 | − | 0.587785i | 0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | −1.00000 | 1.00000 | 0.712089 | − | 2.19158i | −0.309017 | − | 0.951057i | 0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | |||||||||||||||||||||||||||||
| 721.1 | −0.309017 | + | 0.951057i | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −1.00000 | 1.00000 | −3.25533 | − | 2.36513i | 0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | |||||||||||||||||||||||||||||
| 721.2 | −0.309017 | + | 0.951057i | −0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | −1.00000 | 1.00000 | −0.171724 | − | 0.124765i | 0.809017 | − | 0.587785i | −0.809017 | + | 0.587785i | 0.309017 | − | 0.951057i | |||||||||||||||||||||||||||||
| 841.1 | −0.309017 | − | 0.951057i | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −1.00000 | 1.00000 | −3.25533 | + | 2.36513i | 0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | |||||||||||||||||||||||||||||
| 841.2 | −0.309017 | − | 0.951057i | −0.309017 | + | 0.951057i | −0.809017 | + | 0.587785i | −1.00000 | 1.00000 | −0.171724 | + | 0.124765i | 0.809017 | + | 0.587785i | −0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | |||||||||||||||||||||||||||||
| 901.1 | 0.809017 | + | 0.587785i | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | −1.00000 | 1.00000 | −0.785038 | − | 2.41610i | −0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||
| 901.2 | 0.809017 | + | 0.587785i | 0.809017 | − | 0.587785i | 0.309017 | + | 0.951057i | −1.00000 | 1.00000 | 0.712089 | + | 2.19158i | −0.309017 | + | 0.951057i | 0.309017 | − | 0.951057i | −0.809017 | − | 0.587785i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 930.2.n.b | ✓ | 8 |
| 31.d | even | 5 | 1 | inner | 930.2.n.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.2.n.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 930.2.n.b | ✓ | 8 | 31.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 7T_{7}^{7} + 29T_{7}^{6} + 73T_{7}^{5} + 206T_{7}^{4} + 275T_{7}^{3} + 635T_{7}^{2} + 200T_{7} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} + 7 T^{7} + \cdots + 25 \)
|
| $11$ |
\( T^{8} - T^{7} + \cdots + 361 \)
|
| $13$ |
\( T^{8} - 11 T^{7} + \cdots + 1 \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots + 121 \)
|
| $19$ |
\( T^{8} - 12 T^{7} + \cdots + 28561 \)
|
| $23$ |
\( T^{8} + 6 T^{7} + \cdots + 25 \)
|
| $29$ |
\( T^{8} - 9 T^{7} + \cdots + 14641 \)
|
| $31$ |
\( T^{8} + 15 T^{7} + \cdots + 923521 \)
|
| $37$ |
\( (T^{4} - 16 T^{3} + \cdots - 155)^{2} \)
|
| $41$ |
\( T^{8} + 13 T^{7} + \cdots + 361 \)
|
| $43$ |
\( T^{8} - 13 T^{7} + \cdots + 3568321 \)
|
| $47$ |
\( T^{8} + 15 T^{7} + \cdots + 1890625 \)
|
| $53$ |
\( T^{8} - 25 T^{7} + \cdots + 130321 \)
|
| $59$ |
\( T^{8} - 3 T^{7} + \cdots + 185761 \)
|
| $61$ |
\( (T^{4} - 14 T^{3} + \cdots - 761)^{2} \)
|
| $67$ |
\( (T^{4} + 4 T^{3} + \cdots + 919)^{2} \)
|
| $71$ |
\( T^{8} + 15 T^{7} + \cdots + 18931201 \)
|
| $73$ |
\( T^{8} + 24 T^{7} + \cdots + 3025 \)
|
| $79$ |
\( T^{8} - 2 T^{7} + \cdots + 3568321 \)
|
| $83$ |
\( T^{8} + 20 T^{7} + \cdots + 502681 \)
|
| $89$ |
\( T^{8} - 37 T^{7} + \cdots + 31798321 \)
|
| $97$ |
\( T^{8} + 16 T^{7} + \cdots + 3025 \)
|
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