Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [30,10,Mod(19,30)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(30, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("30.19");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 30 = 2 \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 30.c (of order \(2\), degree \(1\), 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: | \(15.4510750849\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 33542x^{4} + 281266441x^{2} + 430585316100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2}\cdot 5^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 33542x^{4} + 281266441x^{2} + 430585316100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 16771\nu ) / 656190 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 213\nu^{4} + 27957\nu^{3} + 5540793\nu^{2} + 246001316\nu + 22009924980 ) / 10499040 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} - 89\nu^{4} - 83855\nu^{3} - 836429\nu^{2} - 422764412\nu + 7340141340 ) / 10499040 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{5} - 35\nu^{4} - 139753\nu^{3} - 3867935\nu^{2} - 599527508\nu - 36679708620 ) / 10499040 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{5} - 515\nu^{4} + 139753\nu^{3} - 11918015\nu^{2} + 914498708\nu - 36679708620 ) / 10499040 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + 3\beta_{2} - \beta _1 - 1 ) / 60 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{5} - 103\beta_{4} + 213\beta_{3} + 89\beta_{2} + 7\beta _1 - 670879 ) / 60 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -16771\beta_{5} - 16771\beta_{4} - 16771\beta_{3} - 50313\beta_{2} + 39388171\beta _1 + 16771 ) / 60 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -773587\beta_{5} + 2383603\beta_{4} - 5540793\beta_{3} - 836429\beta_{2} - 773587\beta _1 + 11251967899 ) / 60 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 348854011 \beta_{5} + 285859771 \beta_{4} + 222865531 \beta_{3} + 983567793 \beta_{2} + \cdots - 285859771 ) / 60 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(11\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
− | 16.0000i | 81.0000i | −256.000 | −1397.52 | + | 7.73699i | 1296.00 | − | 5223.99i | 4096.00i | −6561.00 | 123.792 | + | 22360.3i | ||||||||||||||||||||||||||||||
| 19.2 | − | 16.0000i | 81.0000i | −256.000 | 663.633 | − | 1229.93i | 1296.00 | 3848.97i | 4096.00i | −6561.00 | −19678.8 | − | 10618.1i | ||||||||||||||||||||||||||||||||
| 19.3 | − | 16.0000i | 81.0000i | −256.000 | 866.889 | + | 1096.19i | 1296.00 | − | 5048.98i | 4096.00i | −6561.00 | 17539.0 | − | 13870.2i | |||||||||||||||||||||||||||||||
| 19.4 | 16.0000i | − | 81.0000i | −256.000 | −1397.52 | − | 7.73699i | 1296.00 | 5223.99i | − | 4096.00i | −6561.00 | 123.792 | − | 22360.3i | |||||||||||||||||||||||||||||||
| 19.5 | 16.0000i | − | 81.0000i | −256.000 | 663.633 | + | 1229.93i | 1296.00 | − | 3848.97i | − | 4096.00i | −6561.00 | −19678.8 | + | 10618.1i | ||||||||||||||||||||||||||||||
| 19.6 | 16.0000i | − | 81.0000i | −256.000 | 866.889 | − | 1096.19i | 1296.00 | 5048.98i | − | 4096.00i | −6561.00 | 17539.0 | + | 13870.2i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 30.10.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.10.c.c | 6 | ||
| 4.b | odd | 2 | 1 | 240.10.f.b | 6 | ||
| 5.b | even | 2 | 1 | inner | 30.10.c.b | ✓ | 6 |
| 5.c | odd | 4 | 1 | 150.10.a.r | 3 | ||
| 5.c | odd | 4 | 1 | 150.10.a.s | 3 | ||
| 15.d | odd | 2 | 1 | 90.10.c.c | 6 | ||
| 20.d | odd | 2 | 1 | 240.10.f.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.10.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 30.10.c.b | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 90.10.c.c | 6 | 3.b | odd | 2 | 1 | ||
| 90.10.c.c | 6 | 15.d | odd | 2 | 1 | ||
| 150.10.a.r | 3 | 5.c | odd | 4 | 1 | ||
| 150.10.a.s | 3 | 5.c | odd | 4 | 1 | ||
| 240.10.f.b | 6 | 4.b | odd | 2 | 1 | ||
| 240.10.f.b | 6 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 67596792T_{7}^{4} + 1477628306089488T_{7}^{2} + 10306232572478448089344 \)
acting on \(S_{10}^{\mathrm{new}}(30, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 256)^{3} \)
|
| $3$ |
\( (T^{2} + 6561)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 74\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 10\!\cdots\!44 \)
|
| $11$ |
\( (T^{3} + \cdots + 258734071355312)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 45\!\cdots\!96 \)
|
| $17$ |
\( T^{6} + \cdots + 40\!\cdots\!04 \)
|
| $19$ |
\( (T^{3} + \cdots - 46\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 44\!\cdots\!16 \)
|
| $29$ |
\( (T^{3} + \cdots + 74\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 18\!\cdots\!52)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 81\!\cdots\!24 \)
|
| $41$ |
\( (T^{3} + \cdots - 48\!\cdots\!28)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 55\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots + 39\!\cdots\!84 \)
|
| $53$ |
\( T^{6} + \cdots + 56\!\cdots\!76 \)
|
| $59$ |
\( (T^{3} + \cdots - 78\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 16\!\cdots\!12)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 20\!\cdots\!04 \)
|
| $71$ |
\( (T^{3} + \cdots + 21\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 55\!\cdots\!16 \)
|
| $79$ |
\( (T^{3} + \cdots + 28\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 17\!\cdots\!36 \)
|
| $89$ |
\( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 60\!\cdots\!84 \)
|
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