Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2175,2,Mod(349,2175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2175.349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2175 = 3 \cdot 5^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2175.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: | \(17.3674624396\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.3356224.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 8x^{4} + 16x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 87) |
| 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
\( x^{6} + 8x^{4} + 16x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 4\nu^{2} \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 7\nu^{3} + 12\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 4\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 7\beta_{3} + 16\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).
| \(n\) | \(901\) | \(1451\) | \(2002\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 349.1 |
|
− | 2.47283i | − | 1.00000i | −4.11491 | 0 | −2.47283 | − | 1.64207i | 5.22982i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 349.2 | − | 1.93543i | 1.00000i | −1.74590 | 0 | 1.93543 | 3.68133i | − | 0.491797i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 349.3 | − | 1.46260i | − | 1.00000i | −0.139194 | 0 | −1.46260 | 1.32340i | − | 2.72161i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 349.4 | 1.46260i | 1.00000i | −0.139194 | 0 | −1.46260 | − | 1.32340i | 2.72161i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 349.5 | 1.93543i | − | 1.00000i | −1.74590 | 0 | 1.93543 | − | 3.68133i | 0.491797i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
| 349.6 | 2.47283i | 1.00000i | −4.11491 | 0 | −2.47283 | 1.64207i | − | 5.22982i | −1.00000 | 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 | 2175.2.c.l | 6 | |
| 5.b | even | 2 | 1 | inner | 2175.2.c.l | 6 | |
| 5.c | odd | 4 | 1 | 87.2.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 2175.2.a.t | 3 | ||
| 15.e | even | 4 | 1 | 261.2.a.e | 3 | ||
| 15.e | even | 4 | 1 | 6525.2.a.bg | 3 | ||
| 20.e | even | 4 | 1 | 1392.2.a.u | 3 | ||
| 35.f | even | 4 | 1 | 4263.2.a.m | 3 | ||
| 40.i | odd | 4 | 1 | 5568.2.a.cb | 3 | ||
| 40.k | even | 4 | 1 | 5568.2.a.bx | 3 | ||
| 60.l | odd | 4 | 1 | 4176.2.a.bx | 3 | ||
| 145.h | odd | 4 | 1 | 2523.2.a.h | 3 | ||
| 435.p | even | 4 | 1 | 7569.2.a.t | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.2.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 261.2.a.e | 3 | 15.e | even | 4 | 1 | ||
| 1392.2.a.u | 3 | 20.e | even | 4 | 1 | ||
| 2175.2.a.t | 3 | 5.c | odd | 4 | 1 | ||
| 2175.2.c.l | 6 | 1.a | even | 1 | 1 | trivial | |
| 2175.2.c.l | 6 | 5.b | even | 2 | 1 | inner | |
| 2523.2.a.h | 3 | 145.h | odd | 4 | 1 | ||
| 4176.2.a.bx | 3 | 60.l | odd | 4 | 1 | ||
| 4263.2.a.m | 3 | 35.f | even | 4 | 1 | ||
| 5568.2.a.bx | 3 | 40.k | even | 4 | 1 | ||
| 5568.2.a.cb | 3 | 40.i | odd | 4 | 1 | ||
| 6525.2.a.bg | 3 | 15.e | even | 4 | 1 | ||
| 7569.2.a.t | 3 | 435.p | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2175, [\chi])\):
|
\( T_{2}^{6} + 12T_{2}^{4} + 44T_{2}^{2} + 49 \)
|
|
\( T_{7}^{6} + 18T_{7}^{4} + 65T_{7}^{2} + 64 \)
|
|
\( T_{11}^{3} + 8T_{11}^{2} + 15T_{11} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 12 T^{4} + \cdots + 49 \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + 18 T^{4} + \cdots + 64 \)
|
| $11$ |
\( (T^{3} + 8 T^{2} + 15 T + 4)^{2} \)
|
| $13$ |
\( T^{6} + 30 T^{4} + \cdots + 676 \)
|
| $17$ |
\( T^{6} + 70 T^{4} + \cdots + 8836 \)
|
| $19$ |
\( (T^{3} - 2 T^{2} - 20 T - 16)^{2} \)
|
| $23$ |
\( T^{6} + 44 T^{4} + \cdots + 1024 \)
|
| $29$ |
\( (T + 1)^{6} \)
|
| $31$ |
\( (T^{3} - 6 T^{2} - 4 T + 32)^{2} \)
|
| $37$ |
\( T^{6} + 64 T^{4} + \cdots + 64 \)
|
| $41$ |
\( (T^{3} + 2 T^{2} - 100 T + 56)^{2} \)
|
| $43$ |
\( T^{6} + 208 T^{4} + \cdots + 65536 \)
|
| $47$ |
\( T^{6} + 162 T^{4} + \cdots + 46656 \)
|
| $53$ |
\( T^{6} + 272 T^{4} + \cdots + 61504 \)
|
| $59$ |
\( (T^{3} - 20 T^{2} + \cdots - 112)^{2} \)
|
| $61$ |
\( (T^{3} - 4 T^{2} - 16 T + 56)^{2} \)
|
| $67$ |
\( T^{6} + 114 T^{4} + \cdots + 2704 \)
|
| $71$ |
\( (T^{3} + 14 T^{2} + \cdots - 416)^{2} \)
|
| $73$ |
\( T^{6} + 64 T^{4} + \cdots + 64 \)
|
| $79$ |
\( (T^{3} - 2 T^{2} + \cdots + 224)^{2} \)
|
| $83$ |
\( T^{6} + 120 T^{4} + \cdots + 43264 \)
|
| $89$ |
\( (T^{3} - 8 T^{2} - 131 T + 74)^{2} \)
|
| $97$ |
\( T^{6} + 160 T^{4} + \cdots + 10816 \)
|
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