Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4,11,Mod(3,4)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4.3");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4 = 2^{2} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4.b (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: | \(2.54142901069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.26777625.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 59x^{2} - 58x + 336 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} + 59x^{2} - 58x + 336 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} + 4\nu^{2} + 78\nu + 161 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} + 8\nu^{2} + 492\nu + 154 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -32\nu^{3} + 160\nu^{2} - 1472\nu + 3920 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 24 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 2\beta_{2} + 44\beta _1 - 2736 ) / 96 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{3} - 41\beta_{2} + 202\beta _1 - 2064 ) / 48 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 |
|
−25.4722 | − | 19.3692i | 343.535i | 273.666 | + | 986.754i | −3266.44 | 6654.00 | − | 8750.58i | 10902.2i | 12141.8 | − | 30435.5i | −58967.0 | 83203.5 | + | 63268.4i | ||||||||||||||||||||
| 3.2 | −25.4722 | + | 19.3692i | − | 343.535i | 273.666 | − | 986.754i | −3266.44 | 6654.00 | + | 8750.58i | − | 10902.2i | 12141.8 | + | 30435.5i | −58967.0 | 83203.5 | − | 63268.4i | |||||||||||||||||||
| 3.3 | 19.4722 | − | 25.3936i | − | 120.267i | −265.666 | − | 988.937i | 2486.44 | −3054.00 | − | 2341.85i | 29129.9i | −30285.8 | − | 12510.6i | 44585.0 | 48416.5 | − | 63139.6i | ||||||||||||||||||||
| 3.4 | 19.4722 | + | 25.3936i | 120.267i | −265.666 | + | 988.937i | 2486.44 | −3054.00 | + | 2341.85i | − | 29129.9i | −30285.8 | + | 12510.6i | 44585.0 | 48416.5 | + | 63139.6i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4.11.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 36.11.d.c | 4 | ||
| 4.b | odd | 2 | 1 | inner | 4.11.b.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 100.11.b.d | 4 | ||
| 5.c | odd | 4 | 2 | 100.11.d.a | 8 | ||
| 8.b | even | 2 | 1 | 64.11.c.d | 4 | ||
| 8.d | odd | 2 | 1 | 64.11.c.d | 4 | ||
| 12.b | even | 2 | 1 | 36.11.d.c | 4 | ||
| 16.e | even | 4 | 2 | 256.11.d.f | 8 | ||
| 16.f | odd | 4 | 2 | 256.11.d.f | 8 | ||
| 20.d | odd | 2 | 1 | 100.11.b.d | 4 | ||
| 20.e | even | 4 | 2 | 100.11.d.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.11.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 4.11.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 36.11.d.c | 4 | 3.b | odd | 2 | 1 | ||
| 36.11.d.c | 4 | 12.b | even | 2 | 1 | ||
| 64.11.c.d | 4 | 8.b | even | 2 | 1 | ||
| 64.11.c.d | 4 | 8.d | odd | 2 | 1 | ||
| 100.11.b.d | 4 | 5.b | even | 2 | 1 | ||
| 100.11.b.d | 4 | 20.d | odd | 2 | 1 | ||
| 100.11.d.a | 8 | 5.c | odd | 4 | 2 | ||
| 100.11.d.a | 8 | 20.e | even | 4 | 2 | ||
| 256.11.d.f | 8 | 16.e | even | 4 | 2 | ||
| 256.11.d.f | 8 | 16.f | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(4, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 12 T^{3} + \cdots + 1048576 \)
|
| $3$ |
\( T^{4} + \cdots + 1706987520 \)
|
| $5$ |
\( (T^{2} + 780 T - 8121820)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 32\!\cdots\!20 \)
|
| $13$ |
\( (T^{2} - 106132 T - 122360135324)^{2} \)
|
| $17$ |
\( (T^{2} + 85692 T - 10111760764)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 33\!\cdots\!20 \)
|
| $23$ |
\( T^{4} + \cdots + 80\!\cdots\!20 \)
|
| $29$ |
\( (T^{2} + \cdots - 139887323593756)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 34\!\cdots\!20 \)
|
| $37$ |
\( (T^{2} + \cdots + 572093080702756)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots + 36\!\cdots\!24)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 33\!\cdots\!20 \)
|
| $53$ |
\( (T^{2} + \cdots + 86\!\cdots\!96)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 27\!\cdots\!20 \)
|
| $61$ |
\( (T^{2} + \cdots + 51\!\cdots\!44)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 90\!\cdots\!20 \)
|
| $71$ |
\( T^{4} + \cdots + 15\!\cdots\!20 \)
|
| $73$ |
\( (T^{2} + \cdots - 57\!\cdots\!24)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 85\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots + 30\!\cdots\!20 \)
|
| $89$ |
\( (T^{2} + \cdots - 34\!\cdots\!96)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 83\!\cdots\!96)^{2} \)
|
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