Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.4");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(5.29266605383\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{5})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 11^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - x^{19} + 78 x^{18} + 79 x^{17} + 10573 x^{16} - 33409 x^{15} + 1262953 x^{14} + \cdots + 25599187870096 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 24\!\cdots\!63 \nu^{19} + \cdots - 65\!\cdots\!64 ) / 34\!\cdots\!02 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 98\!\cdots\!84 \nu^{19} + \cdots + 78\!\cdots\!30 ) / 13\!\cdots\!11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 32\!\cdots\!82 \nu^{19} + \cdots - 50\!\cdots\!28 ) / 27\!\cdots\!22 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 68\!\cdots\!67 \nu^{19} + \cdots + 56\!\cdots\!00 ) / 54\!\cdots\!44 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 30\!\cdots\!65 \nu^{19} + \cdots + 20\!\cdots\!20 ) / 13\!\cdots\!08 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\!\cdots\!06 \nu^{19} + \cdots + 32\!\cdots\!16 ) / 69\!\cdots\!04 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 56\!\cdots\!13 \nu^{19} + \cdots - 29\!\cdots\!88 ) / 54\!\cdots\!44 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 16\!\cdots\!73 \nu^{19} + \cdots - 39\!\cdots\!64 ) / 34\!\cdots\!02 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 81\!\cdots\!62 \nu^{19} + \cdots + 18\!\cdots\!84 ) / 15\!\cdots\!76 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 24\!\cdots\!41 \nu^{19} + \cdots + 16\!\cdots\!84 ) / 31\!\cdots\!52 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 56\!\cdots\!84 \nu^{19} + \cdots + 10\!\cdots\!64 ) / 69\!\cdots\!04 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 40\!\cdots\!71 \nu^{19} + \cdots - 24\!\cdots\!12 ) / 40\!\cdots\!32 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 21\!\cdots\!25 \nu^{19} + \cdots + 40\!\cdots\!80 ) / 20\!\cdots\!16 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 23\!\cdots\!85 \nu^{19} + \cdots + 58\!\cdots\!24 ) / 20\!\cdots\!16 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 16\!\cdots\!35 \nu^{19} + \cdots - 10\!\cdots\!84 ) / 13\!\cdots\!08 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 61\!\cdots\!69 \nu^{19} + \cdots - 13\!\cdots\!12 ) / 40\!\cdots\!32 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 62\!\cdots\!69 \nu^{19} + \cdots - 11\!\cdots\!44 ) / 20\!\cdots\!16 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 80\!\cdots\!34 \nu^{19} + \cdots - 26\!\cdots\!28 ) / 20\!\cdots\!16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{12} + \beta_{9} - \beta_{8} - 49\beta_{7} - 6\beta_{6} + \beta_{4} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{19} + \beta_{16} + 2 \beta_{14} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots - 35 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6 \beta_{19} - 3 \beta_{18} + 5 \beta_{17} - 109 \beta_{16} + 3 \beta_{15} - 5 \beta_{14} + \cdots + 2 \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 17 \beta_{19} + 17 \beta_{18} - 249 \beta_{17} - 219 \beta_{16} + 249 \beta_{13} - 408 \beta_{11} + \cdots + 7332 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 391 \beta_{18} - 943 \beta_{17} - 470 \beta_{15} + 472 \beta_{14} + 2129 \beta_{12} - 943 \beta_{11} + \cdots - 129800 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3378 \beta_{19} + 10216 \beta_{18} - 18712 \beta_{17} + 18726 \beta_{16} + 3378 \beta_{15} + \cdots + 101768 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 103772 \beta_{19} + 1140633 \beta_{16} - 45416 \beta_{15} - 74628 \beta_{14} - 33804 \beta_{13} + \cdots - 24013831 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1450621 \beta_{19} - 960777 \beta_{18} + 2035773 \beta_{17} - 2777337 \beta_{16} + 960777 \beta_{15} + \cdots + 26404325 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 5314715 \beta_{19} + 5314715 \beta_{18} + 4888469 \beta_{17} + 37443407 \beta_{16} + \cdots + 2321917415 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 63162745 \beta_{18} + 291186469 \beta_{17} - 93206829 \beta_{15} + 215238939 \beta_{14} + \cdots - 5229729401 \)
|
| \(\nu^{12}\) | \(=\) |
\( 628941271 \beta_{19} + 732584482 \beta_{18} - 444888904 \beta_{17} - 4464778965 \beta_{16} + \cdots + 206326475808 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 17032821338 \beta_{19} + 43657317140 \beta_{16} - 7707637614 \beta_{15} + 30587060194 \beta_{14} + \cdots - 1324846024046 \)
|
| \(\nu^{14}\) | \(=\) |
\( 153953121956 \beta_{19} - 79410379408 \beta_{18} + 53176977104 \beta_{17} - 1277679603561 \beta_{16} + \cdots + 2160057360596 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 913604260816 \beta_{19} + 913604260816 \beta_{18} - 3235932238730 \beta_{17} + 863704227014 \beta_{16} + \cdots + 144241411965627 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 8796134072883 \beta_{18} + 3823389968623 \beta_{17} - 8576511792099 \beta_{15} + \cdots - 26\!\cdots\!65 \)
|
| \(\nu^{17}\) | \(=\) |
\( 106507268471761 \beta_{19} + 99981415905261 \beta_{18} - 246103968451527 \beta_{17} + \cdots + 14\!\cdots\!77 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 19\!\cdots\!09 \beta_{19} + \cdots - 25\!\cdots\!20 \)
|
| \(\nu^{19}\) | \(=\) |
\( 22\!\cdots\!26 \beta_{19} + \cdots + 55\!\cdots\!22 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(-1 - \beta_{2} + \beta_{6} + \beta_{7}\) | \(1\) |
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 |
|
−7.78409 | − | 5.65548i | −2.78115 | + | 8.55951i | 18.7192 | + | 57.6117i | 48.1408 | − | 34.9763i | 70.0568 | − | 50.8993i | 9.43666 | + | 29.0431i | 84.9654 | − | 261.497i | −65.5304 | − | 47.6106i | −572.540 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.2 | −3.08746 | − | 2.24317i | −2.78115 | + | 8.55951i | −5.38796 | − | 16.5824i | 3.77724 | − | 2.74432i | 27.7871 | − | 20.1885i | 49.4196 | + | 152.098i | −58.2998 | + | 179.428i | −65.5304 | − | 47.6106i | −17.8180 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.3 | 0.319007 | + | 0.231772i | −2.78115 | + | 8.55951i | −9.84050 | − | 30.2859i | −17.3054 | + | 12.5731i | −2.87106 | + | 2.08595i | −52.3175 | − | 161.017i | 7.77944 | − | 23.9427i | −65.5304 | − | 47.6106i | −8.43465 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.4 | 5.51306 | + | 4.00547i | −2.78115 | + | 8.55951i | 4.46148 | + | 13.7310i | −62.5986 | + | 45.4806i | −49.6176 | + | 36.0493i | 59.3828 | + | 182.761i | 36.9828 | − | 113.821i | −65.5304 | − | 47.6106i | −527.281 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 4.5 | 8.77555 | + | 6.37581i | −2.78115 | + | 8.55951i | 26.4708 | + | 81.4687i | 31.3852 | − | 22.8027i | −78.9800 | + | 57.3823i | −67.6896 | − | 208.327i | −179.871 | + | 553.584i | −65.5304 | − | 47.6106i | 420.808 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.1 | −2.65487 | − | 8.17084i | 7.28115 | + | 5.29007i | −33.8257 | + | 24.5758i | −21.1802 | + | 65.1858i | 23.8938 | − | 73.5376i | −196.384 | + | 142.681i | 68.1911 | + | 49.5438i | 25.0304 | + | 77.0356i | 588.853 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.2 | −1.95678 | − | 6.02234i | 7.28115 | + | 5.29007i | −6.55106 | + | 4.75963i | 8.65323 | − | 26.6319i | 17.6110 | − | 54.2011i | 121.539 | − | 88.3034i | −122.450 | − | 88.9651i | 25.0304 | + | 77.0356i | −177.319 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.3 | 0.218677 | + | 0.673018i | 7.28115 | + | 5.29007i | 25.4834 | − | 18.5148i | −21.0138 | + | 64.6739i | −1.96809 | + | 6.05716i | 98.3565 | − | 71.4602i | 36.3535 | + | 26.4124i | 25.0304 | + | 77.0356i | −48.1219 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.4 | 1.03535 | + | 3.18647i | 7.28115 | + | 5.29007i | 16.8069 | − | 12.2109i | 32.9083 | − | 101.281i | −9.31813 | + | 28.6783i | −32.7853 | + | 23.8199i | 143.049 | + | 103.931i | 25.0304 | + | 77.0356i | 356.801 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 16.5 | 2.62155 | + | 8.06830i | 7.28115 | + | 5.29007i | −32.3365 | + | 23.4938i | −8.26670 | + | 25.4423i | −23.5940 | + | 72.6147i | −58.4588 | + | 42.4728i | −54.7010 | − | 39.7426i | 25.0304 | + | 77.0356i | −226.948 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.1 | −7.78409 | + | 5.65548i | −2.78115 | − | 8.55951i | 18.7192 | − | 57.6117i | 48.1408 | + | 34.9763i | 70.0568 | + | 50.8993i | 9.43666 | − | 29.0431i | 84.9654 | + | 261.497i | −65.5304 | + | 47.6106i | −572.540 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | −3.08746 | + | 2.24317i | −2.78115 | − | 8.55951i | −5.38796 | + | 16.5824i | 3.77724 | + | 2.74432i | 27.7871 | + | 20.1885i | 49.4196 | − | 152.098i | −58.2998 | − | 179.428i | −65.5304 | + | 47.6106i | −17.8180 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.3 | 0.319007 | − | 0.231772i | −2.78115 | − | 8.55951i | −9.84050 | + | 30.2859i | −17.3054 | − | 12.5731i | −2.87106 | − | 2.08595i | −52.3175 | + | 161.017i | 7.77944 | + | 23.9427i | −65.5304 | + | 47.6106i | −8.43465 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.4 | 5.51306 | − | 4.00547i | −2.78115 | − | 8.55951i | 4.46148 | − | 13.7310i | −62.5986 | − | 45.4806i | −49.6176 | − | 36.0493i | 59.3828 | − | 182.761i | 36.9828 | + | 113.821i | −65.5304 | + | 47.6106i | −527.281 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 25.5 | 8.77555 | − | 6.37581i | −2.78115 | − | 8.55951i | 26.4708 | − | 81.4687i | 31.3852 | + | 22.8027i | −78.9800 | − | 57.3823i | −67.6896 | + | 208.327i | −179.871 | − | 553.584i | −65.5304 | + | 47.6106i | 420.808 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.1 | −2.65487 | + | 8.17084i | 7.28115 | − | 5.29007i | −33.8257 | − | 24.5758i | −21.1802 | − | 65.1858i | 23.8938 | + | 73.5376i | −196.384 | − | 142.681i | 68.1911 | − | 49.5438i | 25.0304 | − | 77.0356i | 588.853 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.2 | −1.95678 | + | 6.02234i | 7.28115 | − | 5.29007i | −6.55106 | − | 4.75963i | 8.65323 | + | 26.6319i | 17.6110 | + | 54.2011i | 121.539 | + | 88.3034i | −122.450 | + | 88.9651i | 25.0304 | − | 77.0356i | −177.319 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0.218677 | − | 0.673018i | 7.28115 | − | 5.29007i | 25.4834 | + | 18.5148i | −21.0138 | − | 64.6739i | −1.96809 | − | 6.05716i | 98.3565 | + | 71.4602i | 36.3535 | − | 26.4124i | 25.0304 | − | 77.0356i | −48.1219 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 1.03535 | − | 3.18647i | 7.28115 | − | 5.29007i | 16.8069 | + | 12.2109i | 32.9083 | + | 101.281i | −9.31813 | − | 28.6783i | −32.7853 | − | 23.8199i | 143.049 | − | 103.931i | 25.0304 | − | 77.0356i | 356.801 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 31.5 | 2.62155 | − | 8.06830i | 7.28115 | − | 5.29007i | −32.3365 | − | 23.4938i | −8.26670 | − | 25.4423i | −23.5940 | − | 72.6147i | −58.4588 | − | 42.4728i | −54.7010 | + | 39.7426i | 25.0304 | − | 77.0356i | −226.948 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 33.6.e.b | ✓ | 20 |
| 3.b | odd | 2 | 1 | 99.6.f.b | 20 | ||
| 11.c | even | 5 | 1 | inner | 33.6.e.b | ✓ | 20 |
| 11.c | even | 5 | 1 | 363.6.a.r | 10 | ||
| 11.d | odd | 10 | 1 | 363.6.a.t | 10 | ||
| 33.f | even | 10 | 1 | 1089.6.a.bi | 10 | ||
| 33.h | odd | 10 | 1 | 99.6.f.b | 20 | ||
| 33.h | odd | 10 | 1 | 1089.6.a.bk | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.6.e.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 33.6.e.b | ✓ | 20 | 11.c | even | 5 | 1 | inner |
| 99.6.f.b | 20 | 3.b | odd | 2 | 1 | ||
| 99.6.f.b | 20 | 33.h | odd | 10 | 1 | ||
| 363.6.a.r | 10 | 11.c | even | 5 | 1 | ||
| 363.6.a.t | 10 | 11.d | odd | 10 | 1 | ||
| 1089.6.a.bi | 10 | 33.f | even | 10 | 1 | ||
| 1089.6.a.bk | 10 | 33.h | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 6 T_{2}^{19} + 94 T_{2}^{18} - 488 T_{2}^{17} + 12401 T_{2}^{16} - 30756 T_{2}^{15} + \cdots + 1371559530496 \)
acting on \(S_{6}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + \cdots + 1371559530496 \)
|
| $3$ |
\( (T^{4} - 9 T^{3} + \cdots + 6561)^{5} \)
|
| $5$ |
\( T^{20} + \cdots + 43\!\cdots\!61 \)
|
| $7$ |
\( T^{20} + \cdots + 20\!\cdots\!61 \)
|
| $11$ |
\( T^{20} + \cdots + 11\!\cdots\!01 \)
|
| $13$ |
\( T^{20} + \cdots + 14\!\cdots\!76 \)
|
| $17$ |
\( T^{20} + \cdots + 96\!\cdots\!00 \)
|
| $19$ |
\( T^{20} + \cdots + 20\!\cdots\!00 \)
|
| $23$ |
\( (T^{10} + \cdots + 12\!\cdots\!56)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( T^{20} + \cdots + 91\!\cdots\!01 \)
|
| $37$ |
\( T^{20} + \cdots + 15\!\cdots\!16 \)
|
| $41$ |
\( T^{20} + \cdots + 54\!\cdots\!36 \)
|
| $43$ |
\( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 39\!\cdots\!16 \)
|
| $53$ |
\( T^{20} + \cdots + 16\!\cdots\!61 \)
|
| $59$ |
\( T^{20} + \cdots + 17\!\cdots\!25 \)
|
| $61$ |
\( T^{20} + \cdots + 12\!\cdots\!16 \)
|
| $67$ |
\( (T^{10} + \cdots + 33\!\cdots\!36)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 31\!\cdots\!36 \)
|
| $73$ |
\( T^{20} + \cdots + 77\!\cdots\!36 \)
|
| $79$ |
\( T^{20} + \cdots + 53\!\cdots\!25 \)
|
| $83$ |
\( T^{20} + \cdots + 75\!\cdots\!61 \)
|
| $89$ |
\( (T^{10} + \cdots - 11\!\cdots\!20)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 48\!\cdots\!25 \)
|
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