Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [110,4,Mod(89,110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, 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("110.89");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 110 = 2 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 110.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: | \(6.49021010063\) |
| 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: | \( 2^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{7} + 1065\nu^{5} + 43460\nu^{3} + 522203\nu ) / 31460 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 140\nu^{4} + 5955\nu^{2} + 72996 ) / 220 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 140\nu^{4} + 6175\nu^{2} + 81356 ) / 220 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 80 \nu^{7} + 1001 \nu^{6} - 10650 \nu^{5} + 124410 \nu^{4} - 450330 \nu^{3} + 4686825 \nu^{2} + \cdots + 52336856 ) / 188760 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 80 \nu^{7} - 1001 \nu^{6} - 10650 \nu^{5} - 124410 \nu^{4} - 450330 \nu^{3} - 4686825 \nu^{2} + \cdots - 52336856 ) / 188760 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 323 \nu^{7} - 429 \nu^{6} - 40050 \nu^{5} - 60060 \nu^{4} - 1499085 \nu^{3} - 2554695 \nu^{2} + \cdots - 31409664 ) / 188760 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{6} - 6\beta_{5} - 20\beta_{2} - 45\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6\beta_{6} - 6\beta_{5} - 81\beta_{4} + 95\beta_{3} + 1760 \)
|
| \(\nu^{5}\) | \(=\) |
\( 64\beta_{7} + 520\beta_{6} + 520\beta_{5} + 32\beta_{3} + 2164\beta_{2} + 2329\beta _1 + 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( -840\beta_{6} + 840\beta_{5} + 5385\beta_{4} - 7125\beta_{3} - 93106 \)
|
| \(\nu^{7}\) | \(=\) |
\( -8520\beta_{7} - 36630\beta_{6} - 36630\beta_{5} - 4260\beta_{3} - 175500\beta_{2} - 130861\beta _1 - 4260 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 89.1 |
|
− | 2.00000i | − | 5.97576i | −4.00000 | 1.78092 | − | 11.0376i | −11.9515 | − | 8.09941i | 8.00000i | −8.70973 | −22.0752 | − | 3.56184i | |||||||||||||||||||||||||||||||||||
| 89.2 | − | 2.00000i | − | 2.48251i | −4.00000 | 3.44238 | + | 10.6372i | −4.96501 | 31.7569i | 8.00000i | 20.8372 | 21.2744 | − | 6.88475i | |||||||||||||||||||||||||||||||||||||
| 89.3 | − | 2.00000i | 6.82127i | −4.00000 | −8.37442 | − | 7.40737i | 13.6425 | − | 13.6360i | 8.00000i | −19.5297 | −14.8147 | + | 16.7488i | |||||||||||||||||||||||||||||||||||||
| 89.4 | − | 2.00000i | 8.63700i | −4.00000 | 11.1511 | + | 0.807757i | 17.2740 | 0.978517i | 8.00000i | −47.5977 | 1.61551 | − | 22.3022i | ||||||||||||||||||||||||||||||||||||||
| 89.5 | 2.00000i | − | 8.63700i | −4.00000 | 11.1511 | − | 0.807757i | 17.2740 | − | 0.978517i | − | 8.00000i | −47.5977 | 1.61551 | + | 22.3022i | ||||||||||||||||||||||||||||||||||||
| 89.6 | 2.00000i | − | 6.82127i | −4.00000 | −8.37442 | + | 7.40737i | 13.6425 | 13.6360i | − | 8.00000i | −19.5297 | −14.8147 | − | 16.7488i | |||||||||||||||||||||||||||||||||||||
| 89.7 | 2.00000i | 2.48251i | −4.00000 | 3.44238 | − | 10.6372i | −4.96501 | − | 31.7569i | − | 8.00000i | 20.8372 | 21.2744 | + | 6.88475i | |||||||||||||||||||||||||||||||||||||
| 89.8 | 2.00000i | 5.97576i | −4.00000 | 1.78092 | + | 11.0376i | −11.9515 | 8.09941i | − | 8.00000i | −8.70973 | −22.0752 | + | 3.56184i | ||||||||||||||||||||||||||||||||||||||
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 | 110.4.b.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 990.4.c.i | 8 | ||
| 4.b | odd | 2 | 1 | 880.4.b.h | 8 | ||
| 5.b | even | 2 | 1 | inner | 110.4.b.c | ✓ | 8 |
| 5.c | odd | 4 | 1 | 550.4.a.ba | 4 | ||
| 5.c | odd | 4 | 1 | 550.4.a.bb | 4 | ||
| 15.d | odd | 2 | 1 | 990.4.c.i | 8 | ||
| 20.d | odd | 2 | 1 | 880.4.b.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.b.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 110.4.b.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 550.4.a.ba | 4 | 5.c | odd | 4 | 1 | ||
| 550.4.a.bb | 4 | 5.c | odd | 4 | 1 | ||
| 880.4.b.h | 8 | 4.b | odd | 2 | 1 | ||
| 880.4.b.h | 8 | 20.d | odd | 2 | 1 | ||
| 990.4.c.i | 8 | 3.b | odd | 2 | 1 | ||
| 990.4.c.i | 8 | 15.d | odd | 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 \(S_{4}^{\mathrm{new}}(110, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{4} \)
|
| $3$ |
\( T^{8} + 163 T^{6} + \cdots + 763876 \)
|
| $5$ |
\( T^{8} - 16 T^{7} + \cdots + 244140625 \)
|
| $7$ |
\( T^{8} + 1261 T^{6} + \cdots + 11778624 \)
|
| $11$ |
\( (T + 11)^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 5498837401600 \)
|
| $17$ |
\( T^{8} + \cdots + 136200103430400 \)
|
| $19$ |
\( (T^{4} + 151 T^{3} + \cdots - 390272)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 73\!\cdots\!84 \)
|
| $29$ |
\( (T^{4} + 29 T^{3} + \cdots + 15788512)^{2} \)
|
| $31$ |
\( (T^{4} - 511 T^{3} + \cdots - 256907000)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 49077590691600 \)
|
| $41$ |
\( (T^{4} - 226 T^{3} + \cdots + 3452698624)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 75\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 15\!\cdots\!76 \)
|
| $59$ |
\( (T^{4} - 134 T^{3} + \cdots + 1066184592)^{2} \)
|
| $61$ |
\( (T^{4} + 245 T^{3} + \cdots + 22378892000)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 80\!\cdots\!44 \)
|
| $71$ |
\( (T^{4} - 1017 T^{3} + \cdots - 50573201976)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 16\!\cdots\!00 \)
|
| $79$ |
\( (T^{4} - 136 T^{3} + \cdots + 145164269184)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 40\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} + \cdots - 1272726658650)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 77\!\cdots\!96 \)
|
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