Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,4,Mod(4,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.e (of order \(5\), degree \(4\), 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: | \(1.94706303019\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | 8.0.682515625.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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
\( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + \cdots + 440220 ) / 1168519 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + \cdots + 528473 ) / 1168519 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + \cdots - 701074 ) / 1168519 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + \cdots + 2809884 ) / 1168519 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} + \cdots - 2305776 ) / 1168519 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} + \cdots + 665808 ) / 1168519 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} - 5\beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 6\beta_{6} + 6\beta_{5} + 2\beta_{4} + 4\beta_{3} - 10\beta_{2} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 14\beta_{3} - 13\beta_{2} - 10\beta _1 - 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( 43\beta_{7} + 25\beta_{5} + 49\beta_{4} + 18\beta_{2} - 25\beta _1 - 62 \)
|
| \(\nu^{6}\) | \(=\) |
\( 97\beta_{7} - 92\beta_{6} + 92\beta_{4} - 97\beta_{3} + 221\beta_{2} - 44\beta _1 - 221 \)
|
| \(\nu^{7}\) | \(=\) |
\( -449\beta_{6} - 260\beta_{5} - 412\beta_{3} + 896\beta_{2} - 412 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(-1 + \beta_{2} - \beta_{3} + \beta_{7}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4.1 |
|
−2.02339 | − | 1.47008i | 0.927051 | − | 2.85317i | −0.539165 | − | 1.65938i | −8.44146 | + | 6.13308i | −6.07016 | + | 4.41023i | −10.1220 | − | 31.1524i | −7.53140 | + | 23.1793i | −7.28115 | − | 5.29007i | 26.0964 | ||||||||||||||||||||||||||
| 4.2 | 0.523388 | + | 0.380264i | 0.927051 | − | 2.85317i | −2.34280 | − | 7.21040i | 9.01441 | − | 6.54935i | 1.57016 | − | 1.14079i | 8.07696 | + | 24.8583i | 3.11499 | − | 9.58696i | −7.28115 | − | 5.29007i | 7.20851 | |||||||||||||||||||||||||||
| 16.1 | −1.45957 | − | 4.49208i | −2.42705 | − | 1.76336i | −11.5763 | + | 8.41069i | 1.86000 | − | 5.72450i | −4.37870 | + | 13.4762i | −8.05785 | + | 5.85437i | 24.1083 | + | 17.5157i | 2.78115 | + | 8.55951i | −28.4297 | |||||||||||||||||||||||||||
| 16.2 | −0.0404346 | − | 0.124445i | −2.42705 | − | 1.76336i | 6.45828 | − | 4.69222i | 2.06705 | − | 6.36172i | −0.121304 | + | 0.373335i | 11.6029 | − | 8.43002i | −1.69194 | − | 1.22926i | 2.78115 | + | 8.55951i | −0.875265 | |||||||||||||||||||||||||||
| 25.1 | −2.02339 | + | 1.47008i | 0.927051 | + | 2.85317i | −0.539165 | + | 1.65938i | −8.44146 | − | 6.13308i | −6.07016 | − | 4.41023i | −10.1220 | + | 31.1524i | −7.53140 | − | 23.1793i | −7.28115 | + | 5.29007i | 26.0964 | |||||||||||||||||||||||||||
| 25.2 | 0.523388 | − | 0.380264i | 0.927051 | + | 2.85317i | −2.34280 | + | 7.21040i | 9.01441 | + | 6.54935i | 1.57016 | + | 1.14079i | 8.07696 | − | 24.8583i | 3.11499 | + | 9.58696i | −7.28115 | + | 5.29007i | 7.20851 | |||||||||||||||||||||||||||
| 31.1 | −1.45957 | + | 4.49208i | −2.42705 | + | 1.76336i | −11.5763 | − | 8.41069i | 1.86000 | + | 5.72450i | −4.37870 | − | 13.4762i | −8.05785 | − | 5.85437i | 24.1083 | − | 17.5157i | 2.78115 | − | 8.55951i | −28.4297 | |||||||||||||||||||||||||||
| 31.2 | −0.0404346 | + | 0.124445i | −2.42705 | + | 1.76336i | 6.45828 | + | 4.69222i | 2.06705 | + | 6.36172i | −0.121304 | − | 0.373335i | 11.6029 | + | 8.43002i | −1.69194 | + | 1.22926i | 2.78115 | − | 8.55951i | −0.875265 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 33.4.e.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 99.4.f.b | 8 | ||
| 11.c | even | 5 | 1 | inner | 33.4.e.b | ✓ | 8 |
| 11.c | even | 5 | 1 | 363.4.a.p | 4 | ||
| 11.d | odd | 10 | 1 | 363.4.a.t | 4 | ||
| 33.f | even | 10 | 1 | 1089.4.a.z | 4 | ||
| 33.h | odd | 10 | 1 | 99.4.f.b | 8 | ||
| 33.h | odd | 10 | 1 | 1089.4.a.bg | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.4.e.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 33.4.e.b | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 99.4.f.b | 8 | 3.b | odd | 2 | 1 | ||
| 99.4.f.b | 8 | 33.h | odd | 10 | 1 | ||
| 363.4.a.p | 4 | 11.c | even | 5 | 1 | ||
| 363.4.a.t | 4 | 11.d | odd | 10 | 1 | ||
| 1089.4.a.z | 4 | 33.f | even | 10 | 1 | ||
| 1089.4.a.bg | 4 | 33.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 6T_{2}^{7} + 34T_{2}^{6} + 72T_{2}^{5} + 49T_{2}^{4} - 96T_{2}^{3} + 51T_{2}^{2} + 3T_{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 6 T^{7} + \cdots + 1 \)
|
| $3$ |
\( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \)
|
| $5$ |
\( T^{8} - 9 T^{7} + \cdots + 21911761 \)
|
| $7$ |
\( T^{8} + \cdots + 14957045401 \)
|
| $11$ |
\( T^{8} + \cdots + 3138428376721 \)
|
| $13$ |
\( T^{8} + \cdots + 153370490192656 \)
|
| $17$ |
\( T^{8} + \cdots + 16499324068096 \)
|
| $19$ |
\( T^{8} + \cdots + 23\!\cdots\!96 \)
|
| $23$ |
\( (T^{4} + 42 T^{3} + \cdots - 46471644)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 26\!\cdots\!16 \)
|
| $31$ |
\( T^{8} + \cdots + 2860289355121 \)
|
| $37$ |
\( T^{8} + \cdots + 19\!\cdots\!16 \)
|
| $41$ |
\( T^{8} + \cdots + 11\!\cdots\!36 \)
|
| $43$ |
\( (T^{4} + 322 T^{3} + \cdots + 5520039844)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 20\!\cdots\!16 \)
|
| $53$ |
\( T^{8} + \cdots + 27\!\cdots\!21 \)
|
| $59$ |
\( T^{8} + \cdots + 36\!\cdots\!61 \)
|
| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!16 \)
|
| $67$ |
\( (T^{4} + 259 T^{3} + \cdots + 1798706704)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 12\!\cdots\!76 \)
|
| $73$ |
\( T^{8} + \cdots + 27\!\cdots\!56 \)
|
| $79$ |
\( T^{8} + \cdots + 11\!\cdots\!81 \)
|
| $83$ |
\( T^{8} + \cdots + 82\!\cdots\!21 \)
|
| $89$ |
\( (T^{4} - 894 T^{3} + \cdots - 245710544796)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 98\!\cdots\!81 \)
|
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