Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [93,4,Mod(25,93)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(93, base_ring=CyclotomicField(6))
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("93.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 93 = 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 93.e (of order \(3\), degree \(2\), 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: | \(5.48717763053\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Relative dimension: | \(7\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 3 x^{13} + 44 x^{12} - 81 x^{11} + 1206 x^{10} - 2049 x^{9} + 15925 x^{8} - 9870 x^{7} + \cdots + 1089936 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 3 x^{13} + 44 x^{12} - 81 x^{11} + 1206 x^{10} - 2049 x^{9} + 15925 x^{8} - 9870 x^{7} + \cdots + 1089936 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 55\!\cdots\!18 \nu^{13} + \cdots + 11\!\cdots\!80 ) / 91\!\cdots\!72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 52\!\cdots\!85 \nu^{13} + \cdots - 14\!\cdots\!96 ) / 47\!\cdots\!84 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 17\!\cdots\!11 \nu^{13} + \cdots + 94\!\cdots\!44 ) / 91\!\cdots\!72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 48\!\cdots\!93 \nu^{13} + \cdots - 32\!\cdots\!12 ) / 47\!\cdots\!84 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 53\!\cdots\!89 \nu^{13} + \cdots + 11\!\cdots\!52 ) / 47\!\cdots\!84 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 25\!\cdots\!49 \nu^{13} + \cdots - 41\!\cdots\!40 ) / 18\!\cdots\!44 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 20\!\cdots\!85 \nu^{13} + \cdots - 21\!\cdots\!88 ) / 91\!\cdots\!72 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11\!\cdots\!46 \nu^{13} + \cdots + 15\!\cdots\!20 ) / 47\!\cdots\!84 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 50\!\cdots\!93 \nu^{13} + \cdots - 59\!\cdots\!36 ) / 18\!\cdots\!44 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 61\!\cdots\!78 \nu^{13} + \cdots - 62\!\cdots\!92 ) / 16\!\cdots\!96 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 22\!\cdots\!02 \nu^{13} + \cdots + 30\!\cdots\!36 ) / 45\!\cdots\!36 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 13\!\cdots\!51 \nu^{13} + \cdots - 94\!\cdots\!72 ) / 15\!\cdots\!28 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 11\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{8} - \beta_{7} + \beta_{4} - 18\beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} - 2 \beta_{6} - 27 \beta_{5} + \cdots - 206 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{13} + 2 \beta_{12} - 33 \beta_{11} - 25 \beta_{9} - 37 \beta_{6} - 29 \beta_{5} + \cdots - 386 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 35\beta_{12} - 47\beta_{10} + 98\beta_{8} - 39\beta_{7} - 663\beta_{4} - 106\beta_{2} + 4540 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 82 \beta_{13} + 905 \beta_{11} - 905 \beta_{10} + 553 \beta_{9} + 1105 \beta_{8} + 553 \beta_{7} + \cdots + 904 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 987 \beta_{13} - 987 \beta_{12} + 1575 \beta_{11} - 1151 \beta_{9} + 3290 \beta_{6} + \cdots - 3426 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( -2562\beta_{12} + 23481\beta_{10} - 30273\beta_{8} - 12201\beta_{7} + 20505\beta_{4} - 201190\beta_{2} - 18936 \)
|
| \(\nu^{10}\) | \(=\) |
\( 26043 \beta_{13} - 46263 \beta_{11} + 46263 \beta_{10} + 30687 \beta_{9} - 95898 \beta_{8} + \cdots - 2454068 \)
|
| \(\nu^{11}\) | \(=\) |
\( 72306 \beta_{13} + 72306 \beta_{12} - 594001 \beta_{11} - 273169 \beta_{9} - 792985 \beta_{6} + \cdots - 4701222 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 666307 \beta_{12} - 1273471 \beta_{10} + 2613050 \beta_{8} - 777943 \beta_{7} - 9425439 \beta_{4} + \cdots + 58059812 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 1939778 \beta_{13} + 14815353 \beta_{11} - 14815353 \beta_{10} + 6205225 \beta_{9} + 20244145 \beta_{8} + \cdots + 14007576 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/93\mathbb{Z}\right)^\times\).
| \(n\) | \(32\) | \(34\) |
| \(\chi(n)\) | \(1\) | \(-1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
−4.96019 | −1.50000 | − | 2.59808i | 16.6035 | −4.64341 | + | 8.04262i | 7.44029 | + | 12.8870i | −16.3707 | − | 28.3548i | −42.6750 | −4.50000 | + | 7.79423i | 23.0322 | − | 39.8929i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 25.2 | −4.02856 | −1.50000 | − | 2.59808i | 8.22926 | 5.40824 | − | 9.36735i | 6.04283 | + | 10.4665i | 4.79760 | + | 8.30969i | −0.923582 | −4.50000 | + | 7.79423i | −21.7874 | + | 37.7369i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 25.3 | −2.00671 | −1.50000 | − | 2.59808i | −3.97311 | −4.33897 | + | 7.51532i | 3.01007 | + | 5.21359i | 3.49545 | + | 6.05430i | 24.0266 | −4.50000 | + | 7.79423i | 8.70707 | − | 15.0811i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 25.4 | −1.15961 | −1.50000 | − | 2.59808i | −6.65531 | 2.81505 | − | 4.87581i | 1.73941 | + | 3.01275i | 0.322107 | + | 0.557905i | 16.9944 | −4.50000 | + | 7.79423i | −3.26436 | + | 5.65403i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 25.5 | 2.08311 | −1.50000 | − | 2.59808i | −3.66064 | −7.77830 | + | 13.4724i | −3.12467 | − | 5.41208i | 9.43363 | + | 16.3395i | −24.2904 | −4.50000 | + | 7.79423i | −16.2031 | + | 28.0646i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 25.6 | 2.22258 | −1.50000 | − | 2.59808i | −3.06014 | 2.73591 | − | 4.73874i | −3.33387 | − | 5.77443i | −12.0756 | − | 20.9155i | −24.5820 | −4.50000 | + | 7.79423i | 6.08079 | − | 10.5322i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 25.7 | 4.84937 | −1.50000 | − | 2.59808i | 15.5164 | 3.80148 | − | 6.58436i | −7.27406 | − | 12.5990i | 1.89742 | + | 3.28643i | 36.4500 | −4.50000 | + | 7.79423i | 18.4348 | − | 31.9300i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 67.1 | −4.96019 | −1.50000 | + | 2.59808i | 16.6035 | −4.64341 | − | 8.04262i | 7.44029 | − | 12.8870i | −16.3707 | + | 28.3548i | −42.6750 | −4.50000 | − | 7.79423i | 23.0322 | + | 39.8929i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 67.2 | −4.02856 | −1.50000 | + | 2.59808i | 8.22926 | 5.40824 | + | 9.36735i | 6.04283 | − | 10.4665i | 4.79760 | − | 8.30969i | −0.923582 | −4.50000 | − | 7.79423i | −21.7874 | − | 37.7369i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 67.3 | −2.00671 | −1.50000 | + | 2.59808i | −3.97311 | −4.33897 | − | 7.51532i | 3.01007 | − | 5.21359i | 3.49545 | − | 6.05430i | 24.0266 | −4.50000 | − | 7.79423i | 8.70707 | + | 15.0811i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 67.4 | −1.15961 | −1.50000 | + | 2.59808i | −6.65531 | 2.81505 | + | 4.87581i | 1.73941 | − | 3.01275i | 0.322107 | − | 0.557905i | 16.9944 | −4.50000 | − | 7.79423i | −3.26436 | − | 5.65403i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 67.5 | 2.08311 | −1.50000 | + | 2.59808i | −3.66064 | −7.77830 | − | 13.4724i | −3.12467 | + | 5.41208i | 9.43363 | − | 16.3395i | −24.2904 | −4.50000 | − | 7.79423i | −16.2031 | − | 28.0646i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 67.6 | 2.22258 | −1.50000 | + | 2.59808i | −3.06014 | 2.73591 | + | 4.73874i | −3.33387 | + | 5.77443i | −12.0756 | + | 20.9155i | −24.5820 | −4.50000 | − | 7.79423i | 6.08079 | + | 10.5322i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 67.7 | 4.84937 | −1.50000 | + | 2.59808i | 15.5164 | 3.80148 | + | 6.58436i | −7.27406 | + | 12.5990i | 1.89742 | − | 3.28643i | 36.4500 | −4.50000 | − | 7.79423i | 18.4348 | + | 31.9300i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 93.4.e.a | ✓ | 14 |
| 3.b | odd | 2 | 1 | 279.4.h.b | 14 | ||
| 31.c | even | 3 | 1 | inner | 93.4.e.a | ✓ | 14 |
| 93.h | odd | 6 | 1 | 279.4.h.b | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 93.4.e.a | ✓ | 14 | 1.a | even | 1 | 1 | trivial |
| 93.4.e.a | ✓ | 14 | 31.c | even | 3 | 1 | inner |
| 279.4.h.b | 14 | 3.b | odd | 2 | 1 | ||
| 279.4.h.b | 14 | 93.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + 3T_{2}^{6} - 35T_{2}^{5} - 93T_{2}^{4} + 298T_{2}^{3} + 582T_{2}^{2} - 704T_{2} - 1044 \)
acting on \(S_{4}^{\mathrm{new}}(93, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{7} + 3 T^{6} + \cdots - 1044)^{2} \)
|
| $3$ |
\( (T^{2} + 3 T + 9)^{7} \)
|
| $5$ |
\( T^{14} + \cdots + 10088602177536 \)
|
| $7$ |
\( T^{14} + \cdots + 5985577543936 \)
|
| $11$ |
\( T^{14} + \cdots + 11\!\cdots\!96 \)
|
| $13$ |
\( T^{14} + \cdots + 80\!\cdots\!56 \)
|
| $17$ |
\( T^{14} + \cdots + 18\!\cdots\!16 \)
|
| $19$ |
\( T^{14} + \cdots + 45\!\cdots\!16 \)
|
| $23$ |
\( (T^{7} + \cdots - 12030272122752)^{2} \)
|
| $29$ |
\( (T^{7} + \cdots + 268942887928116)^{2} \)
|
| $31$ |
\( T^{14} + \cdots + 20\!\cdots\!31 \)
|
| $37$ |
\( T^{14} + \cdots + 13\!\cdots\!16 \)
|
| $41$ |
\( T^{14} + \cdots + 54\!\cdots\!24 \)
|
| $43$ |
\( T^{14} + \cdots + 15\!\cdots\!44 \)
|
| $47$ |
\( (T^{7} + \cdots - 601540697748816)^{2} \)
|
| $53$ |
\( T^{14} + \cdots + 24\!\cdots\!16 \)
|
| $59$ |
\( T^{14} + \cdots + 31\!\cdots\!16 \)
|
| $61$ |
\( (T^{7} + \cdots + 48\!\cdots\!12)^{2} \)
|
| $67$ |
\( T^{14} + \cdots + 35\!\cdots\!36 \)
|
| $71$ |
\( T^{14} + \cdots + 69\!\cdots\!16 \)
|
| $73$ |
\( T^{14} + \cdots + 40\!\cdots\!56 \)
|
| $79$ |
\( T^{14} + \cdots + 24\!\cdots\!76 \)
|
| $83$ |
\( T^{14} + \cdots + 65\!\cdots\!36 \)
|
| $89$ |
\( (T^{7} + \cdots - 54\!\cdots\!12)^{2} \)
|
| $97$ |
\( (T^{7} + \cdots + 22\!\cdots\!53)^{2} \)
|
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