Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,21,Mod(5,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 21, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12.5");
S:= CuspForms(chi, 21);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 21 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.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: | \(30.4216518123\) |
| 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: |
\( x^{6} + 24769850x^{4} + 131733035896000x^{2} + 250851218720256000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{22}\cdot 5^{2} \) |
| 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
\( x^{6} + 24769850x^{4} + 131733035896000x^{2} + 250851218720256000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 967 \nu^{5} - 6552 \nu^{4} + 23795771510 \nu^{3} + 304111800720 \nu^{2} + \cdots + 32\!\cdots\!00 ) / 97219198876800 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 50121 \nu^{5} - 58968 \nu^{4} + 1267838842410 \nu^{3} + 2737006206480 \nu^{2} + \cdots + 29\!\cdots\!00 ) / 32406399625600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 207905 \nu^{5} - 90145944 \nu^{4} - 5116090874650 \nu^{3} + \cdots - 66\!\cdots\!00 ) / 97219198876800 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 22241 \nu^{5} - 10400922 \nu^{4} + 547302744730 \nu^{3} - 209526224698980 \nu^{2} + \cdots - 51\!\cdots\!00 ) / 12152399859600 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5702877 \nu^{5} + 471744 \nu^{4} - 140438836537650 \nu^{3} - 21896049651840 \nu^{2} + \cdots - 23\!\cdots\!00 ) / 16203199812800 \)
|
| \(\nu\) | \(=\) |
\( ( -9\beta_{5} - 211\beta_{4} + 211\beta_{3} - 54\beta_{2} - 225879\beta_1 ) / 8398080 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -23548\beta_{4} + 21091\beta_{3} + 8867397\beta _1 - 1155662121600 ) / 139968 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 14135373\beta_{5} + 261813727\beta_{4} - 261813727\beta_{3} + 874348398\beta_{2} + 259759811283\beta_1 ) / 839808 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 55245686020\beta_{4} - 74252673565\beta_{3} - 26129531044155\beta _1 + 2722217392864368000 ) / 23328 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 116169420430665 \beta_{5} + \cdots - 17\!\cdots\!15 \beta_1 ) / 419904 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | −53241.6 | − | 25536.6i | 0 | 9.13793e6i | 0 | 9.41859e7 | 0 | 2.18255e9 | + | 2.71922e9i | 0 | ||||||||||||||||||||||||||||||||
| 5.2 | 0 | −53241.6 | + | 25536.6i | 0 | − | 9.13793e6i | 0 | 9.41859e7 | 0 | 2.18255e9 | − | 2.71922e9i | 0 | ||||||||||||||||||||||||||||||||
| 5.3 | 0 | −8570.43 | − | 58423.7i | 0 | − | 1.60830e7i | 0 | −4.81200e8 | 0 | −3.33988e9 | + | 1.00143e9i | 0 | ||||||||||||||||||||||||||||||||
| 5.4 | 0 | −8570.43 | + | 58423.7i | 0 | 1.60830e7i | 0 | −4.81200e8 | 0 | −3.33988e9 | − | 1.00143e9i | 0 | |||||||||||||||||||||||||||||||||
| 5.5 | 0 | 19623.0 | − | 55693.1i | 0 | 7.88530e6i | 0 | 3.14494e8 | 0 | −2.71666e9 | − | 2.18573e9i | 0 | |||||||||||||||||||||||||||||||||
| 5.6 | 0 | 19623.0 | + | 55693.1i | 0 | − | 7.88530e6i | 0 | 3.14494e8 | 0 | −2.71666e9 | + | 2.18573e9i | 0 | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.21.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 12.21.c.b | ✓ | 6 |
| 4.b | odd | 2 | 1 | 48.21.e.d | 6 | ||
| 12.b | even | 2 | 1 | 48.21.e.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.21.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 12.21.c.b | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 48.21.e.d | 6 | 4.b | odd | 2 | 1 | ||
| 48.21.e.d | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 404343642432960 T_{5}^{4} + \cdots + 13\!\cdots\!00 \)
acting on \(S_{21}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 42\!\cdots\!01 \)
|
| $5$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots + 14\!\cdots\!24)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 11\!\cdots\!56)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 82\!\cdots\!00 \)
|
| $19$ |
\( (T^{3} + \cdots + 57\!\cdots\!96)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 57\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots + 22\!\cdots\!96)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 49\!\cdots\!28)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 59\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots - 71\!\cdots\!56)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 28\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots - 18\!\cdots\!64)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 22\!\cdots\!16)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 59\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 39\!\cdots\!52)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 15\!\cdots\!68)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 56\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots + 34\!\cdots\!44)^{2} \)
|
| show more | |
| show less |