Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,10,Mod(1,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.a (trivial) |
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: | yes |
| Analytic conductor: | \(180.262542657\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{457}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 114 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | no (minimal twist has level 70) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(-1 + 5\sqrt{457})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−16.0000 | −32.9439 | 256.000 | 0 | 527.102 | 2401.00 | −4096.00 | −18597.7 | 0 | ||||||||||||||||||||||||
| 1.2 | −16.0000 | 73.9439 | 256.000 | 0 | −1183.10 | 2401.00 | −4096.00 | −14215.3 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 350.10.a.e | 2 | |
| 5.b | even | 2 | 1 | 70.10.a.g | ✓ | 2 | |
| 5.c | odd | 4 | 2 | 350.10.c.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.10.a.g | ✓ | 2 | 5.b | even | 2 | 1 | |
| 350.10.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 350.10.c.i | 4 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 41T_{3} - 2436 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{2} \)
|
| $3$ |
\( T^{2} - 41T - 2436 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( (T - 2401)^{2} \)
|
| $11$ |
\( T^{2} + \cdots - 3553968100 \)
|
| $13$ |
\( T^{2} + \cdots + 4579896466 \)
|
| $17$ |
\( T^{2} + \cdots - 135613633614 \)
|
| $19$ |
\( T^{2} + \cdots - 782561931816 \)
|
| $23$ |
\( T^{2} + \cdots - 2900989310800 \)
|
| $29$ |
\( T^{2} + \cdots + 5249854835546 \)
|
| $31$ |
\( T^{2} + \cdots + 8061716729856 \)
|
| $37$ |
\( T^{2} + \cdots - 93112262314476 \)
|
| $41$ |
\( T^{2} + \cdots + 181677814279296 \)
|
| $43$ |
\( T^{2} + \cdots + 147491289239464 \)
|
| $47$ |
\( T^{2} + \cdots + 57908335950624 \)
|
| $53$ |
\( T^{2} + \cdots - 46\!\cdots\!84 \)
|
| $59$ |
\( T^{2} + \cdots - 92\!\cdots\!24 \)
|
| $61$ |
\( T^{2} + \cdots + 97650784628544 \)
|
| $67$ |
\( T^{2} + \cdots + 23\!\cdots\!16 \)
|
| $71$ |
\( T^{2} + \cdots - 54\!\cdots\!00 \)
|
| $73$ |
\( T^{2} + \cdots - 952313665523796 \)
|
| $79$ |
\( T^{2} + \cdots + 18\!\cdots\!64 \)
|
| $83$ |
\( T^{2} + \cdots - 33\!\cdots\!84 \)
|
| $89$ |
\( T^{2} + \cdots - 45\!\cdots\!44 \)
|
| $97$ |
\( T^{2} + \cdots + 57\!\cdots\!50 \)
|
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