Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,10,Mod(1,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 387.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: | \(199.318868595\) |
| Analytic rank: | \(0\) |
| Dimension: | \(15\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{15} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} + \cdots + 52\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{10}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 43) |
| 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 \(1,\beta_1,\ldots,\beta_{14}\) 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
\( x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} + \cdots + 52\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 724 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 19\!\cdots\!31 \nu^{14} + \cdots - 11\!\cdots\!12 ) / 25\!\cdots\!24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 55\!\cdots\!29 \nu^{14} + \cdots - 11\!\cdots\!00 ) / 48\!\cdots\!20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 30\!\cdots\!41 \nu^{14} + \cdots + 54\!\cdots\!28 ) / 25\!\cdots\!24 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 14\!\cdots\!33 \nu^{14} + \cdots + 15\!\cdots\!20 ) / 48\!\cdots\!20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 99\!\cdots\!51 \nu^{14} + \cdots - 28\!\cdots\!00 ) / 32\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 18\!\cdots\!43 \nu^{14} + \cdots + 51\!\cdots\!00 ) / 38\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 32\!\cdots\!87 \nu^{14} + \cdots + 16\!\cdots\!00 ) / 48\!\cdots\!20 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 15\!\cdots\!89 \nu^{14} + \cdots - 66\!\cdots\!80 ) / 12\!\cdots\!20 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 40\!\cdots\!87 \nu^{14} + \cdots - 50\!\cdots\!00 ) / 19\!\cdots\!80 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 87\!\cdots\!79 \nu^{14} + \cdots - 26\!\cdots\!76 ) / 40\!\cdots\!16 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 86\!\cdots\!09 \nu^{14} + \cdots - 15\!\cdots\!80 ) / 38\!\cdots\!60 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 34\!\cdots\!11 \nu^{14} + \cdots - 60\!\cdots\!84 ) / 12\!\cdots\!12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 724 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + 2\beta_{9} - \beta_{8} + 3\beta_{5} + 12\beta_{3} - \beta_{2} + 1260\beta _1 - 969 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 25 \beta_{14} - 51 \beta_{13} + 3 \beta_{12} + 5 \beta_{11} - 65 \beta_{10} - 45 \beta_{9} + \cdots + 912593 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 451 \beta_{14} - 129 \beta_{13} + 361 \beta_{12} + 2211 \beta_{11} + 17 \beta_{10} + 4281 \beta_{9} + \cdots - 2541141 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 56261 \beta_{14} - 137047 \beta_{13} + 29647 \beta_{12} + 13355 \beta_{11} - 166145 \beta_{10} + \cdots + 1382468255 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 1476377 \beta_{14} - 101683 \beta_{13} + 624443 \beta_{12} + 3891795 \beta_{11} + 499011 \beta_{10} + \cdots - 7989903641 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 99010825 \beta_{14} - 289235299 \beta_{13} + 92490507 \beta_{12} + 24459623 \beta_{11} + \cdots + 2293879739123 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 3505734621 \beta_{14} + 504553057 \beta_{13} + 400597703 \beta_{12} + 6498434995 \beta_{11} + \cdots - 22522119936385 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 159606282881 \beta_{14} - 567035896587 \beta_{13} + 222487096275 \beta_{12} + 36914590119 \beta_{11} + \cdots + 40\!\cdots\!63 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 7366410638861 \beta_{14} + 2712863734673 \beta_{13} - 1119132304553 \beta_{12} + \cdots - 57\!\cdots\!21 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 245248710611449 \beta_{14} + \cdots + 72\!\cdots\!59 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 14\!\cdots\!65 \beta_{14} + \cdots - 13\!\cdots\!37 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 36\!\cdots\!09 \beta_{14} + \cdots + 13\!\cdots\!79 \)
|
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 |
|
−43.0936 | 0 | 1345.05 | 1330.99 | 0 | −743.545 | −35899.3 | 0 | −57356.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −33.4490 | 0 | 606.833 | 752.925 | 0 | −7164.62 | −3172.07 | 0 | −25184.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −26.3075 | 0 | 180.084 | −879.044 | 0 | 8431.97 | 8731.88 | 0 | 23125.4 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −23.8680 | 0 | 57.6838 | 578.837 | 0 | −413.035 | 10843.6 | 0 | −13815.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −20.4329 | 0 | −94.4979 | 2583.58 | 0 | −1769.99 | 12392.5 | 0 | −52790.0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −15.0644 | 0 | −285.063 | −1876.98 | 0 | −7041.18 | 12007.3 | 0 | 28275.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.77990 | 0 | −508.832 | −1139.29 | 0 | 4324.96 | −1816.98 | 0 | −2027.82 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.38595 | 0 | −492.763 | 1237.15 | 0 | 12549.3 | −4406.84 | 0 | 5426.06 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 9.62988 | 0 | −419.265 | −636.384 | 0 | −3288.51 | −8967.97 | 0 | −6128.30 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 17.9205 | 0 | −190.855 | −238.484 | 0 | −4524.82 | −12595.5 | 0 | −4273.75 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 18.4876 | 0 | −170.210 | 2363.43 | 0 | 7025.62 | −12612.4 | 0 | 43694.1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 23.7142 | 0 | 50.3616 | 764.097 | 0 | 4079.63 | −10947.4 | 0 | 18119.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 34.6728 | 0 | 690.205 | −1855.36 | 0 | −11539.9 | 6178.86 | 0 | −64330.7 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 41.2075 | 0 | 1186.05 | 1998.59 | 0 | −10690.1 | 27776.1 | 0 | 82356.9 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 42.4171 | 0 | 1287.21 | −267.050 | 0 | 1084.23 | 32882.2 | 0 | −11327.5 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(43\) | \(1\) |
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 | 387.10.a.c | 15 | |
| 3.b | odd | 2 | 1 | 43.10.a.a | ✓ | 15 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.10.a.a | ✓ | 15 | 3.b | odd | 2 | 1 | |
| 387.10.a.c | 15 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{15} - 32 T_{2}^{14} - 4949 T_{2}^{13} + 151570 T_{2}^{12} + 9265782 T_{2}^{11} + \cdots - 99\!\cdots\!40 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(387))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{15} + \cdots - 99\!\cdots\!40 \)
|
| $3$ |
\( T^{15} \)
|
| $5$ |
\( T^{15} + \cdots + 94\!\cdots\!00 \)
|
| $7$ |
\( T^{15} + \cdots + 71\!\cdots\!00 \)
|
| $11$ |
\( T^{15} + \cdots + 24\!\cdots\!72 \)
|
| $13$ |
\( T^{15} + \cdots + 49\!\cdots\!00 \)
|
| $17$ |
\( T^{15} + \cdots - 24\!\cdots\!14 \)
|
| $19$ |
\( T^{15} + \cdots - 49\!\cdots\!04 \)
|
| $23$ |
\( T^{15} + \cdots - 25\!\cdots\!72 \)
|
| $29$ |
\( T^{15} + \cdots - 13\!\cdots\!92 \)
|
| $31$ |
\( T^{15} + \cdots + 22\!\cdots\!88 \)
|
| $37$ |
\( T^{15} + \cdots + 41\!\cdots\!00 \)
|
| $41$ |
\( T^{15} + \cdots + 40\!\cdots\!50 \)
|
| $43$ |
\( (T + 3418801)^{15} \)
|
| $47$ |
\( T^{15} + \cdots + 15\!\cdots\!00 \)
|
| $53$ |
\( T^{15} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( T^{15} + \cdots - 82\!\cdots\!32 \)
|
| $61$ |
\( T^{15} + \cdots + 23\!\cdots\!44 \)
|
| $67$ |
\( T^{15} + \cdots - 22\!\cdots\!04 \)
|
| $71$ |
\( T^{15} + \cdots + 26\!\cdots\!16 \)
|
| $73$ |
\( T^{15} + \cdots - 23\!\cdots\!52 \)
|
| $79$ |
\( T^{15} + \cdots - 15\!\cdots\!80 \)
|
| $83$ |
\( T^{15} + \cdots + 48\!\cdots\!00 \)
|
| $89$ |
\( T^{15} + \cdots + 52\!\cdots\!32 \)
|
| $97$ |
\( T^{15} + \cdots + 17\!\cdots\!82 \)
|
| show more | |
| show less |