Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [310,8,Mod(1,310)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(310, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("310.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 310 = 2 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 310.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: | \(96.8393579001\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 14812 x^{8} - 76284 x^{7} + 65389315 x^{6} + 753120238 x^{5} - 93411415542 x^{4} + \cdots - 37\!\cdots\!08 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 5\cdot 7 \) |
| 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
\( x^{10} - 14812 x^{8} - 76284 x^{7} + 65389315 x^{6} + 753120238 x^{5} - 93411415542 x^{4} + \cdots - 37\!\cdots\!08 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 54\!\cdots\!79 \nu^{9} + \cdots + 47\!\cdots\!28 ) / 62\!\cdots\!12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 80\!\cdots\!25 \nu^{9} + \cdots + 28\!\cdots\!04 ) / 62\!\cdots\!12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 45\!\cdots\!71 \nu^{9} + \cdots + 12\!\cdots\!88 ) / 20\!\cdots\!04 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10\!\cdots\!09 \nu^{9} + \cdots + 17\!\cdots\!56 ) / 44\!\cdots\!08 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 25\!\cdots\!03 \nu^{9} + \cdots + 67\!\cdots\!76 ) / 10\!\cdots\!52 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 74\!\cdots\!81 \nu^{9} + \cdots - 19\!\cdots\!48 ) / 26\!\cdots\!88 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 16\!\cdots\!35 \nu^{9} + \cdots + 26\!\cdots\!96 ) / 31\!\cdots\!56 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!29 \nu^{9} + \cdots - 30\!\cdots\!24 ) / 20\!\cdots\!04 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{7} - \beta_{5} + 8\beta _1 + 2962 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{9} + 46 \beta_{8} - \beta_{7} - 7 \beta_{6} - 36 \beta_{5} - 59 \beta_{4} + 15 \beta_{3} + \cdots + 22865 \)
|
| \(\nu^{4}\) | \(=\) |
\( 246 \beta_{9} + 7789 \beta_{8} + 8231 \beta_{7} + 1444 \beta_{6} - 9407 \beta_{5} + 973 \beta_{4} + \cdots + 17719850 \)
|
| \(\nu^{5}\) | \(=\) |
\( 36066 \beta_{9} + 492856 \beta_{8} - 45019 \beta_{7} - 90062 \beta_{6} - 382121 \beta_{5} + \cdots + 188137478 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2639853 \beta_{9} + 62597754 \beta_{8} + 64762687 \beta_{7} + 14963458 \beta_{6} - 83748325 \beta_{5} + \cdots + 126569138845 \)
|
| \(\nu^{7}\) | \(=\) |
\( 476856961 \beta_{9} + 4389104642 \beta_{8} - 542716045 \beta_{7} - 869622517 \beta_{6} + \cdots + 1238396215337 \)
|
| \(\nu^{8}\) | \(=\) |
\( 25840801036 \beta_{9} + 512621157295 \beta_{8} + 510531283767 \beta_{7} + 126615714718 \beta_{6} + \cdots + 958579447406880 \)
|
| \(\nu^{9}\) | \(=\) |
\( 4662712777792 \beta_{9} + 37123324849614 \beta_{8} - 5299767600503 \beta_{7} - 7577345524116 \beta_{6} + \cdots + 76\!\cdots\!80 \)
|
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 |
|
−8.00000 | −89.7844 | 64.0000 | −125.000 | 718.275 | 78.5007 | −512.000 | 5874.23 | 1000.00 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | −53.1803 | 64.0000 | −125.000 | 425.443 | 1539.21 | −512.000 | 641.147 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | −45.4596 | 64.0000 | −125.000 | 363.677 | −1368.57 | −512.000 | −120.423 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | −23.1269 | 64.0000 | −125.000 | 185.015 | −1273.98 | −512.000 | −1652.15 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −8.00000 | −20.2671 | 64.0000 | −125.000 | 162.137 | 1660.61 | −512.000 | −1776.24 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −8.00000 | 6.47639 | 64.0000 | −125.000 | −51.8111 | −202.305 | −512.000 | −2145.06 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −8.00000 | 21.7969 | 64.0000 | −125.000 | −174.375 | 382.733 | −512.000 | −1711.90 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −8.00000 | 38.5839 | 64.0000 | −125.000 | −308.671 | −54.6838 | −512.000 | −698.286 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −8.00000 | 75.0454 | 64.0000 | −125.000 | −600.363 | 754.474 | −512.000 | 3444.81 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −8.00000 | 89.9159 | 64.0000 | −125.000 | −719.327 | −999.986 | −512.000 | 5897.86 | 1000.00 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(31\) | \( -1 \) |
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 | 310.8.a.f | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 310.8.a.f | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 14812 T_{3}^{8} - 76284 T_{3}^{7} + 65389315 T_{3}^{6} + 753120238 T_{3}^{5} + \cdots - 37\!\cdots\!08 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(310))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{10} \)
|
| $3$ |
\( T^{10} + \cdots - 37\!\cdots\!08 \)
|
| $5$ |
\( (T + 125)^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 11\!\cdots\!76 \)
|
| $11$ |
\( T^{10} + \cdots + 72\!\cdots\!80 \)
|
| $13$ |
\( T^{10} + \cdots - 40\!\cdots\!92 \)
|
| $17$ |
\( T^{10} + \cdots + 32\!\cdots\!00 \)
|
| $19$ |
\( T^{10} + \cdots + 92\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots - 71\!\cdots\!00 \)
|
| $29$ |
\( T^{10} + \cdots - 91\!\cdots\!00 \)
|
| $31$ |
\( (T - 29791)^{10} \)
|
| $37$ |
\( T^{10} + \cdots - 50\!\cdots\!24 \)
|
| $41$ |
\( T^{10} + \cdots - 93\!\cdots\!48 \)
|
| $43$ |
\( T^{10} + \cdots - 27\!\cdots\!40 \)
|
| $47$ |
\( T^{10} + \cdots + 41\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots - 13\!\cdots\!00 \)
|
| $59$ |
\( T^{10} + \cdots - 13\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 52\!\cdots\!00 \)
|
| $67$ |
\( T^{10} + \cdots + 11\!\cdots\!32 \)
|
| $71$ |
\( T^{10} + \cdots - 66\!\cdots\!40 \)
|
| $73$ |
\( T^{10} + \cdots - 11\!\cdots\!00 \)
|
| $79$ |
\( T^{10} + \cdots - 12\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots - 13\!\cdots\!28 \)
|
| $89$ |
\( T^{10} + \cdots + 80\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots + 54\!\cdots\!20 \)
|
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