Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7230,2,Mod(1,7230)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7230, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7230.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7230 = 2 \cdot 3 \cdot 5 \cdot 241 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7230.a (trivial) |
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: | yes |
| Analytic conductor: | \(57.7318406614\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3x^{9} - 13x^{8} + 38x^{7} + 53x^{6} - 158x^{5} - 54x^{4} + 230x^{3} - 51x^{2} - 56x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3x^{9} - 13x^{8} + 38x^{7} + 53x^{6} - 158x^{5} - 54x^{4} + 230x^{3} - 51x^{2} - 56x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 53 \nu^{9} - 13 \nu^{8} + 1005 \nu^{7} + 1094 \nu^{6} - 7497 \nu^{5} - 9662 \nu^{4} + 21806 \nu^{3} + \cdots - 15380 ) / 2712 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 329 \nu^{9} - 431 \nu^{8} - 4857 \nu^{7} + 2266 \nu^{6} + 25149 \nu^{5} + 11366 \nu^{4} + \cdots + 18308 ) / 8136 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 181 \nu^{9} + 979 \nu^{8} + 669 \nu^{7} - 10310 \nu^{6} + 4971 \nu^{5} + 35750 \nu^{4} + \cdots + 8240 ) / 4068 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 121 \nu^{9} + 175 \nu^{8} + 2013 \nu^{7} - 2210 \nu^{6} - 11001 \nu^{5} + 9590 \nu^{4} + \cdots + 5516 ) / 1356 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 913 \nu^{9} + 1567 \nu^{8} + 16545 \nu^{7} - 25058 \nu^{6} - 94965 \nu^{5} + 125738 \nu^{4} + \cdots + 70220 ) / 8136 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 143 \nu^{9} - 289 \nu^{8} - 1927 \nu^{7} + 2694 \nu^{6} + 8851 \nu^{5} - 6526 \nu^{4} - 14522 \nu^{3} + \cdots + 220 ) / 904 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 75 \nu^{9} + 101 \nu^{8} + 1371 \nu^{7} - 1650 \nu^{6} - 7607 \nu^{5} + 8066 \nu^{4} + 13366 \nu^{3} + \cdots + 1484 ) / 452 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 1961 \nu^{9} + 4943 \nu^{8} + 29049 \nu^{7} - 65290 \nu^{6} - 141789 \nu^{5} + 279826 \nu^{4} + \cdots + 68260 ) / 8136 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 406 \nu^{9} - 943 \nu^{8} - 5799 \nu^{7} + 11192 \nu^{6} + 27681 \nu^{5} - 42320 \nu^{4} + \cdots - 9302 ) / 1017 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{9} + \beta_{8} - \beta_{6} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{9} + 3\beta_{8} + \beta_{6} - 4\beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 16 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11 \beta_{9} + 15 \beta_{8} - 2 \beta_{7} - 7 \beta_{6} - 6 \beta_{5} + 7 \beta_{4} - 3 \beta_{3} + \cdots + 18 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5 \beta_{9} + 16 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 19 \beta_{5} + 17 \beta_{4} - 8 \beta_{3} + \cdots + 54 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 35 \beta_{9} + 65 \beta_{8} - 14 \beta_{7} - 13 \beta_{6} - 44 \beta_{5} + 43 \beta_{4} - 27 \beta_{3} + \cdots + 100 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 50 \beta_{9} + 168 \beta_{8} - 48 \beta_{7} + 30 \beta_{6} - 182 \beta_{5} + 167 \beta_{4} - 103 \beta_{3} + \cdots + 439 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 509 \beta_{9} + 1227 \beta_{8} - 348 \beta_{7} - 59 \beta_{6} - 1004 \beta_{5} + 927 \beta_{4} + \cdots + 2124 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 1001 \beta_{9} + 3537 \beta_{8} - 1202 \beta_{7} + 515 \beta_{6} - 3598 \beta_{5} + 3269 \beta_{4} + \cdots + 7930 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 2078 \beta_{9} + 6083 \beta_{8} - 2027 \beta_{7} + 223 \beta_{6} - 5379 \beta_{5} + 4842 \beta_{4} + \cdots + 11044 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −3.94458 | −1.00000 | 1.00000 | −1.00000 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −3.17988 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −2.25390 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −0.455668 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | −0.274536 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.29040 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.65647 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 2.25029 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 3.18187 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | −1.00000 | 1.00000 | 1.00000 | 1.00000 | 4.72953 | −1.00000 | 1.00000 | −1.00000 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(241\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7230.2.a.bq | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7230.2.a.bq | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7230))\):
|
\( T_{7}^{10} - 3 T_{7}^{9} - 32 T_{7}^{8} + 88 T_{7}^{7} + 304 T_{7}^{6} - 821 T_{7}^{5} - 827 T_{7}^{4} + \cdots - 256 \)
|
|
\( T_{11}^{10} + 5 T_{11}^{9} - 53 T_{11}^{8} - 251 T_{11}^{7} + 875 T_{11}^{6} + 3672 T_{11}^{5} + \cdots - 1600 \)
|
|
\( T_{13}^{10} - 4 T_{13}^{9} - 43 T_{13}^{8} + 196 T_{13}^{7} + 309 T_{13}^{6} - 2156 T_{13}^{5} + \cdots + 3776 \)
|
|
\( T_{17}^{10} - 15 T_{17}^{9} + 26 T_{17}^{8} + 656 T_{17}^{7} - 3881 T_{17}^{6} + 1290 T_{17}^{5} + \cdots + 13568 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{10} \)
|
| $3$ |
\( (T + 1)^{10} \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( T^{10} - 3 T^{9} + \cdots - 256 \)
|
| $11$ |
\( T^{10} + 5 T^{9} + \cdots - 1600 \)
|
| $13$ |
\( T^{10} - 4 T^{9} + \cdots + 3776 \)
|
| $17$ |
\( T^{10} - 15 T^{9} + \cdots + 13568 \)
|
| $19$ |
\( T^{10} + 14 T^{9} + \cdots + 242496 \)
|
| $23$ |
\( T^{10} - 9 T^{9} + \cdots - 125824 \)
|
| $29$ |
\( T^{10} - T^{9} + \cdots - 176944 \)
|
| $31$ |
\( T^{10} + 11 T^{9} + \cdots + 320256 \)
|
| $37$ |
\( T^{10} - 6 T^{9} + \cdots - 3113792 \)
|
| $41$ |
\( T^{10} - 8 T^{9} + \cdots - 42624 \)
|
| $43$ |
\( T^{10} - 2 T^{9} + \cdots + 32823424 \)
|
| $47$ |
\( T^{10} - 24 T^{9} + \cdots + 768 \)
|
| $53$ |
\( T^{10} - 27 T^{9} + \cdots + 2764032 \)
|
| $59$ |
\( T^{10} + \cdots + 123819776 \)
|
| $61$ |
\( T^{10} + 7 T^{9} + \cdots - 96122304 \)
|
| $67$ |
\( T^{10} - 5 T^{9} + \cdots - 13675776 \)
|
| $71$ |
\( T^{10} - T^{9} + \cdots - 325296 \)
|
| $73$ |
\( T^{10} - 12 T^{9} + \cdots + 3016528 \)
|
| $79$ |
\( T^{10} + 21 T^{9} + \cdots + 302336 \)
|
| $83$ |
\( T^{10} - 36 T^{9} + \cdots - 11114496 \)
|
| $89$ |
\( T^{10} - 9 T^{9} + \cdots - 88077568 \)
|
| $97$ |
\( T^{10} + \cdots + 1116606272 \)
|
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