Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,4,Mod(99,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.99");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.b (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: | \(10.3253342510\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + \zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 5.41421i | − | 4.65685i | −21.3137 | 0 | −25.2132 | 7.00000i | 72.0833i | 5.31371 | 0 | ||||||||||||||||||||||||||||
| 99.2 | − | 2.58579i | 6.65685i | 1.31371 | 0 | 17.2132 | 7.00000i | − | 24.0833i | −17.3137 | 0 | |||||||||||||||||||||||||||||
| 99.3 | 2.58579i | − | 6.65685i | 1.31371 | 0 | 17.2132 | − | 7.00000i | 24.0833i | −17.3137 | 0 | |||||||||||||||||||||||||||||
| 99.4 | 5.41421i | 4.65685i | −21.3137 | 0 | −25.2132 | − | 7.00000i | − | 72.0833i | 5.31371 | 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 | 175.4.b.c | 4 | |
| 5.b | even | 2 | 1 | inner | 175.4.b.c | 4 | |
| 5.c | odd | 4 | 1 | 35.4.a.b | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 175.4.a.c | 2 | ||
| 15.e | even | 4 | 1 | 315.4.a.f | 2 | ||
| 15.e | even | 4 | 1 | 1575.4.a.z | 2 | ||
| 20.e | even | 4 | 1 | 560.4.a.r | 2 | ||
| 35.f | even | 4 | 1 | 245.4.a.k | 2 | ||
| 35.f | even | 4 | 1 | 1225.4.a.m | 2 | ||
| 35.k | even | 12 | 2 | 245.4.e.i | 4 | ||
| 35.l | odd | 12 | 2 | 245.4.e.h | 4 | ||
| 40.i | odd | 4 | 1 | 2240.4.a.bn | 2 | ||
| 40.k | even | 4 | 1 | 2240.4.a.bo | 2 | ||
| 105.k | odd | 4 | 1 | 2205.4.a.u | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.4.a.b | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 175.4.a.c | 2 | 5.c | odd | 4 | 1 | ||
| 175.4.b.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 175.4.b.c | 4 | 5.b | even | 2 | 1 | inner | |
| 245.4.a.k | 2 | 35.f | even | 4 | 1 | ||
| 245.4.e.h | 4 | 35.l | odd | 12 | 2 | ||
| 245.4.e.i | 4 | 35.k | even | 12 | 2 | ||
| 315.4.a.f | 2 | 15.e | even | 4 | 1 | ||
| 560.4.a.r | 2 | 20.e | even | 4 | 1 | ||
| 1225.4.a.m | 2 | 35.f | even | 4 | 1 | ||
| 1575.4.a.z | 2 | 15.e | even | 4 | 1 | ||
| 2205.4.a.u | 2 | 105.k | odd | 4 | 1 | ||
| 2240.4.a.bn | 2 | 40.i | odd | 4 | 1 | ||
| 2240.4.a.bo | 2 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\):
|
\( T_{2}^{4} + 36T_{2}^{2} + 196 \)
|
|
\( T_{3}^{4} + 66T_{3}^{2} + 961 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 36T^{2} + 196 \)
|
| $3$ |
\( T^{4} + 66T^{2} + 961 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 49)^{2} \)
|
| $11$ |
\( (T^{2} + 14 T - 1999)^{2} \)
|
| $13$ |
\( T^{4} + 1314 T^{2} + 351649 \)
|
| $17$ |
\( T^{4} + 8994 T^{2} + 10543009 \)
|
| $19$ |
\( (T^{2} + 36 T - 3548)^{2} \)
|
| $23$ |
\( T^{4} + 48264 T^{2} + 31764496 \)
|
| $29$ |
\( (T^{2} - 26 T - 983)^{2} \)
|
| $31$ |
\( (T^{2} + 120 T - 61200)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 5230760976 \)
|
| $41$ |
\( (T^{2} + 328 T - 3856)^{2} \)
|
| $43$ |
\( T^{4} + 52296 T^{2} + 58553104 \)
|
| $47$ |
\( T^{4} + 130946 T^{2} + 17833729 \)
|
| $53$ |
\( T^{4} + \cdots + 1022976256 \)
|
| $59$ |
\( (T - 616)^{4} \)
|
| $61$ |
\( (T^{2} - 336 T + 4896)^{2} \)
|
| $67$ |
\( T^{4} + 27936 T^{2} + 5837056 \)
|
| $71$ |
\( (T + 952)^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 14988615184 \)
|
| $79$ |
\( (T^{2} + 1014 T + 134041)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 468753838336 \)
|
| $89$ |
\( (T^{2} - 216 T + 7792)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 3178522031281 \)
|
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