Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [880,4,Mod(529,880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("880.529");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 880.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: | \(51.9216808051\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 110) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{7} + 1065\nu^{5} + 43460\nu^{3} + 490743\nu ) / 31460 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{7} + 1065\nu^{5} + 43460\nu^{3} + 522203\nu ) / 15730 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 32 \nu^{7} - 1001 \nu^{6} + 4260 \nu^{5} - 124410 \nu^{4} + 189570 \nu^{3} - 4686825 \nu^{2} + \cdots - 52336856 ) / 188760 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 32 \nu^{7} - 1001 \nu^{6} - 4260 \nu^{5} - 124410 \nu^{4} - 189570 \nu^{3} - 4686825 \nu^{2} + \cdots - 52336856 ) / 188760 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8\nu^{7} + 143\nu^{6} + 1065\nu^{5} + 20020\nu^{4} + 43460\nu^{3} + 851565\nu^{2} + 522203\nu + 10438428 ) / 31460 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 8\nu^{7} + 143\nu^{6} + 1065\nu^{5} + 20020\nu^{4} + 43460\nu^{3} + 883025\nu^{2} + 522203\nu + 11633908 ) / 31460 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -379\nu^{7} - 47505\nu^{5} - 1819035\nu^{3} - 20813749\nu ) / 94380 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12\beta_{4} + 12\beta_{3} - 53\beta_{2} + 90\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -81\beta_{6} + 95\beta_{5} + 6\beta_{4} + 6\beta_{3} - 7\beta_{2} + 1760 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 64\beta_{7} + 976\beta_{4} - 976\beta_{3} + 3485\beta_{2} - 4658\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5385\beta_{6} - 7125\beta_{5} - 840\beta_{4} - 840\beta_{3} + 870\beta_{2} - 93106 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -8520\beta_{7} - 64740\beta_{4} + 64740\beta_{3} - 237361\beta_{2} + 261722\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).
| \(n\) | \(111\) | \(177\) | \(321\) | \(661\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 529.1 |
|
0 | − | 8.63700i | 0 | 11.1511 | + | 0.807757i | 0 | − | 0.978517i | 0 | −47.5977 | 0 | ||||||||||||||||||||||||||||||||||||||
| 529.2 | 0 | − | 6.82127i | 0 | −8.37442 | − | 7.40737i | 0 | 13.6360i | 0 | −19.5297 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 529.3 | 0 | − | 5.97576i | 0 | 1.78092 | + | 11.0376i | 0 | − | 8.09941i | 0 | −8.70973 | 0 | |||||||||||||||||||||||||||||||||||||||
| 529.4 | 0 | − | 2.48251i | 0 | 3.44238 | − | 10.6372i | 0 | 31.7569i | 0 | 20.8372 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 529.5 | 0 | 2.48251i | 0 | 3.44238 | + | 10.6372i | 0 | − | 31.7569i | 0 | 20.8372 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 529.6 | 0 | 5.97576i | 0 | 1.78092 | − | 11.0376i | 0 | 8.09941i | 0 | −8.70973 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 529.7 | 0 | 6.82127i | 0 | −8.37442 | + | 7.40737i | 0 | − | 13.6360i | 0 | −19.5297 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 529.8 | 0 | 8.63700i | 0 | 11.1511 | − | 0.807757i | 0 | 0.978517i | 0 | −47.5977 | 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 | 880.4.b.h | 8 | |
| 4.b | odd | 2 | 1 | 110.4.b.c | ✓ | 8 | |
| 5.b | even | 2 | 1 | inner | 880.4.b.h | 8 | |
| 12.b | even | 2 | 1 | 990.4.c.i | 8 | ||
| 20.d | odd | 2 | 1 | 110.4.b.c | ✓ | 8 | |
| 20.e | even | 4 | 1 | 550.4.a.ba | 4 | ||
| 20.e | even | 4 | 1 | 550.4.a.bb | 4 | ||
| 60.h | even | 2 | 1 | 990.4.c.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.b.c | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 110.4.b.c | ✓ | 8 | 20.d | odd | 2 | 1 | |
| 550.4.a.ba | 4 | 20.e | even | 4 | 1 | ||
| 550.4.a.bb | 4 | 20.e | even | 4 | 1 | ||
| 880.4.b.h | 8 | 1.a | even | 1 | 1 | trivial | |
| 880.4.b.h | 8 | 5.b | even | 2 | 1 | inner | |
| 990.4.c.i | 8 | 12.b | even | 2 | 1 | ||
| 990.4.c.i | 8 | 60.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 163T_{3}^{6} + 8763T_{3}^{4} + 171997T_{3}^{2} + 763876 \)
acting on \(S_{4}^{\mathrm{new}}(880, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 163 T^{6} + \cdots + 763876 \)
|
| $5$ |
\( T^{8} - 16 T^{7} + \cdots + 244140625 \)
|
| $7$ |
\( T^{8} + 1261 T^{6} + \cdots + 11778624 \)
|
| $11$ |
\( (T - 11)^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 5498837401600 \)
|
| $17$ |
\( T^{8} + \cdots + 136200103430400 \)
|
| $19$ |
\( (T^{4} - 151 T^{3} + \cdots - 390272)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 73\!\cdots\!84 \)
|
| $29$ |
\( (T^{4} + 29 T^{3} + \cdots + 15788512)^{2} \)
|
| $31$ |
\( (T^{4} + 511 T^{3} + \cdots - 256907000)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 49077590691600 \)
|
| $41$ |
\( (T^{4} - 226 T^{3} + \cdots + 3452698624)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 75\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 15\!\cdots\!76 \)
|
| $59$ |
\( (T^{4} + 134 T^{3} + \cdots + 1066184592)^{2} \)
|
| $61$ |
\( (T^{4} + 245 T^{3} + \cdots + 22378892000)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 80\!\cdots\!44 \)
|
| $71$ |
\( (T^{4} + 1017 T^{3} + \cdots - 50573201976)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $79$ |
\( (T^{4} + 136 T^{3} + \cdots + 145164269184)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 40\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} + \cdots - 1272726658650)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 77\!\cdots\!96 \)
|
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