Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,6,Mod(49,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.49");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 400.c (of order \(2\), degree \(1\), not 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: | \(64.1535279252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{409})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 205x^{2} + 10404 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 100) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 205x^{2} + 10404 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 307\nu ) / 102 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{3} - 515\nu ) / 102 \)
|
| \(\beta_{3}\) | \(=\) |
\( 10\nu^{2} + 1025 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 1025 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -307\beta_{2} - 515\beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(177\) | \(351\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 30.2237i | 0 | 0 | 0 | − | 141.342i | 0 | −670.475 | 0 | ||||||||||||||||||||||||||||
| 49.2 | 0 | − | 10.2237i | 0 | 0 | 0 | − | 101.342i | 0 | 138.475 | 0 | |||||||||||||||||||||||||||||
| 49.3 | 0 | 10.2237i | 0 | 0 | 0 | 101.342i | 0 | 138.475 | 0 | |||||||||||||||||||||||||||||||
| 49.4 | 0 | 30.2237i | 0 | 0 | 0 | 141.342i | 0 | −670.475 | 0 | |||||||||||||||||||||||||||||||
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 | 400.6.c.m | 4 | |
| 4.b | odd | 2 | 1 | 100.6.c.c | 4 | ||
| 5.b | even | 2 | 1 | inner | 400.6.c.m | 4 | |
| 5.c | odd | 4 | 1 | 400.6.a.p | 2 | ||
| 5.c | odd | 4 | 1 | 400.6.a.v | 2 | ||
| 12.b | even | 2 | 1 | 900.6.d.m | 4 | ||
| 20.d | odd | 2 | 1 | 100.6.c.c | 4 | ||
| 20.e | even | 4 | 1 | 100.6.a.c | ✓ | 2 | |
| 20.e | even | 4 | 1 | 100.6.a.e | yes | 2 | |
| 60.h | even | 2 | 1 | 900.6.d.m | 4 | ||
| 60.l | odd | 4 | 1 | 900.6.a.m | 2 | ||
| 60.l | odd | 4 | 1 | 900.6.a.s | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 100.6.a.c | ✓ | 2 | 20.e | even | 4 | 1 | |
| 100.6.a.e | yes | 2 | 20.e | even | 4 | 1 | |
| 100.6.c.c | 4 | 4.b | odd | 2 | 1 | ||
| 100.6.c.c | 4 | 20.d | odd | 2 | 1 | ||
| 400.6.a.p | 2 | 5.c | odd | 4 | 1 | ||
| 400.6.a.v | 2 | 5.c | odd | 4 | 1 | ||
| 400.6.c.m | 4 | 1.a | even | 1 | 1 | trivial | |
| 400.6.c.m | 4 | 5.b | even | 2 | 1 | inner | |
| 900.6.a.m | 2 | 60.l | odd | 4 | 1 | ||
| 900.6.a.s | 2 | 60.l | odd | 4 | 1 | ||
| 900.6.d.m | 4 | 12.b | even | 2 | 1 | ||
| 900.6.d.m | 4 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 1018T_{3}^{2} + 95481 \)
acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 1018 T^{2} + 95481 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 30248 T^{2} + 205176976 \)
|
| $11$ |
\( (T^{2} - 60 T - 91125)^{2} \)
|
| $13$ |
\( T^{4} + 894368 T^{2} + 575232256 \)
|
| $17$ |
\( T^{4} + \cdots + 4235894980641 \)
|
| $19$ |
\( (T^{2} - 2092 T + 265891)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 60612390876816 \)
|
| $29$ |
\( (T^{2} + 3552 T - 20404224)^{2} \)
|
| $31$ |
\( (T^{2} - 8888 T + 10546636)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 23\!\cdots\!76 \)
|
| $41$ |
\( (T^{2} + 12438 T - 14330439)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 539261284000000 \)
|
| $47$ |
\( T^{4} + \cdots + 45784385485056 \)
|
| $53$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $59$ |
\( (T^{2} + 36696 T - 194887296)^{2} \)
|
| $61$ |
\( (T^{2} - 19204 T - 1062163196)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 35\!\cdots\!21 \)
|
| $71$ |
\( (T^{2} + 2736 T - 1601572176)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 14\!\cdots\!41 \)
|
| $79$ |
\( (T^{2} + 16184 T + 3271564)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 14\!\cdots\!61 \)
|
| $89$ |
\( (T^{2} - 47322 T - 12174944679)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 93\!\cdots\!96 \)
|
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