Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,6,Mod(1,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 37 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 37.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: | \(5.93420133308\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 189x^{6} + 402x^{5} + 11742x^{4} - 7676x^{3} - 246400x^{2} + 52288x + 1478656 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 189x^{6} + 402x^{5} + 11742x^{4} - 7676x^{3} - 246400x^{2} + 52288x + 1478656 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 48 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23 \nu^{7} + 1808 \nu^{6} - 25019 \nu^{5} - 197854 \nu^{4} + 2214298 \nu^{3} + 6254116 \nu^{2} + \cdots - 29115904 ) / 977664 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 85 \nu^{7} - 40 \nu^{6} + 14089 \nu^{5} + 37802 \nu^{4} - 678110 \nu^{3} - 2698988 \nu^{2} + \cdots + 26158592 ) / 488832 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 47 \nu^{7} + 640 \nu^{6} + 5123 \nu^{5} - 84242 \nu^{4} - 141322 \nu^{3} + 3208124 \nu^{2} + \cdots - 28758272 ) / 257280 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 65 \nu^{7} + 868 \nu^{6} + 6281 \nu^{5} - 107678 \nu^{4} - 53086 \nu^{3} + 3496532 \nu^{2} + \cdots - 21273920 ) / 244416 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5533 \nu^{7} - 45920 \nu^{6} - 747817 \nu^{5} + 4817158 \nu^{4} + 29637758 \nu^{3} + \cdots + 497767168 ) / 4888320 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 48 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{6} - 5\beta_{5} - \beta_{4} - \beta_{3} + 6\beta_{2} + 74\beta _1 + 101 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{7} - 33\beta_{5} - 17\beta_{4} + 3\beta_{3} + 114\beta_{2} + 348\beta _1 + 3597 \)
|
| \(\nu^{5}\) | \(=\) |
\( -56\beta_{7} + 400\beta_{6} - 681\beta_{5} - 281\beta_{4} - 149\beta_{3} + 880\beta_{2} + 6756\beta _1 + 17237 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1496 \beta_{7} + 200 \beta_{6} - 5831 \beta_{5} - 3887 \beta_{4} + 229 \beta_{3} + 12432 \beta_{2} + \cdots + 329523 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 12136 \beta_{7} + 34296 \beta_{6} - 84921 \beta_{5} - 50081 \beta_{4} - 15493 \beta_{3} + \cdots + 2279565 \)
|
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 |
|
−8.96299 | 22.7902 | 48.3352 | 17.3605 | −204.269 | −81.2526 | −146.413 | 276.394 | −155.602 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −7.25785 | −12.1192 | 20.6764 | −80.7145 | 87.9597 | −97.9320 | 82.1850 | −96.1240 | 585.814 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.42642 | −23.5874 | −26.1125 | 2.85205 | 57.2330 | 64.2015 | 141.005 | 313.366 | −6.92029 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.28512 | 12.2979 | −30.3485 | 43.6844 | −15.8042 | 173.605 | 80.1253 | −91.7626 | −56.1397 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 5.09632 | 26.6348 | −6.02756 | 54.7193 | 135.739 | −103.209 | −193.800 | 466.411 | 278.867 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.78419 | −18.6587 | 28.5936 | 107.310 | −145.243 | 185.149 | −26.5159 | 105.148 | 835.321 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.69044 | 14.0145 | 43.5237 | −41.3568 | 121.792 | 185.978 | 100.146 | −46.5941 | −359.408 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.3614 | 3.62805 | 75.3596 | 32.1450 | 37.5918 | −227.539 | 449.268 | −229.837 | 333.068 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(37\) | \(-1\) |
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 | 37.6.a.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 333.6.a.d | 8 | ||
| 4.b | odd | 2 | 1 | 592.6.a.h | 8 | ||
| 5.b | even | 2 | 1 | 925.6.a.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.6.a.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 333.6.a.d | 8 | 3.b | odd | 2 | 1 | ||
| 592.6.a.h | 8 | 4.b | odd | 2 | 1 | ||
| 925.6.a.b | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 12T_{2}^{7} - 133T_{2}^{6} + 1754T_{2}^{5} + 4422T_{2}^{4} - 71652T_{2}^{3} - 24744T_{2}^{2} + 654576T_{2} + 724608 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(37))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 12 T^{7} + \cdots + 724608 \)
|
| $3$ |
\( T^{8} + \cdots - 2024485812 \)
|
| $5$ |
\( T^{8} + \cdots + 1362821073408 \)
|
| $7$ |
\( T^{8} + \cdots + 71\!\cdots\!80 \)
|
| $11$ |
\( T^{8} + \cdots - 15\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots - 17\!\cdots\!64 \)
|
| $17$ |
\( T^{8} + \cdots - 39\!\cdots\!20 \)
|
| $19$ |
\( T^{8} + \cdots - 40\!\cdots\!40 \)
|
| $23$ |
\( T^{8} + \cdots + 55\!\cdots\!72 \)
|
| $29$ |
\( T^{8} + \cdots - 13\!\cdots\!48 \)
|
| $31$ |
\( T^{8} + \cdots - 39\!\cdots\!60 \)
|
| $37$ |
\( (T - 1369)^{8} \)
|
| $41$ |
\( T^{8} + \cdots - 20\!\cdots\!10 \)
|
| $43$ |
\( T^{8} + \cdots + 65\!\cdots\!20 \)
|
| $47$ |
\( T^{8} + \cdots - 11\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots - 36\!\cdots\!32 \)
|
| $59$ |
\( T^{8} + \cdots - 86\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots - 25\!\cdots\!32 \)
|
| $67$ |
\( T^{8} + \cdots - 43\!\cdots\!76 \)
|
| $71$ |
\( T^{8} + \cdots - 84\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots + 45\!\cdots\!62 \)
|
| $79$ |
\( T^{8} + \cdots - 12\!\cdots\!36 \)
|
| $83$ |
\( T^{8} + \cdots - 10\!\cdots\!80 \)
|
| $89$ |
\( T^{8} + \cdots + 16\!\cdots\!52 \)
|
| $97$ |
\( T^{8} + \cdots + 50\!\cdots\!20 \)
|
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