Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [407,2,Mod(1,407)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(407, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("407.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 407 = 11 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 407.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: | \(3.24991136227\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 2 x^{10} - 16 x^{9} + 32 x^{8} + 89 x^{7} - 179 x^{6} - 201 x^{5} + 407 x^{4} + 168 x^{3} + \cdots + 75 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 2 x^{10} - 16 x^{9} + 32 x^{8} + 89 x^{7} - 179 x^{6} - 201 x^{5} + 407 x^{4} + 168 x^{3} + \cdots + 75 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10 \nu^{10} - \nu^{9} - 156 \nu^{8} + 831 \nu^{6} + 78 \nu^{5} - 1732 \nu^{4} - 330 \nu^{3} + \cdots - 196 ) / 59 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 29 \nu^{10} + 3 \nu^{9} - 476 \nu^{8} - 59 \nu^{7} + 2758 \nu^{6} + 356 \nu^{5} - 6722 \nu^{4} + \cdots - 1536 ) / 59 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29 \nu^{10} + 3 \nu^{9} - 476 \nu^{8} - 59 \nu^{7} + 2758 \nu^{6} + 356 \nu^{5} - 6722 \nu^{4} + \cdots - 1536 ) / 59 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 31 \nu^{10} + 9 \nu^{9} + 519 \nu^{8} - 118 \nu^{7} - 3054 \nu^{6} + 478 \nu^{5} + 7446 \nu^{4} + \cdots + 1587 ) / 59 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 48 \nu^{10} + 7 \nu^{9} - 796 \nu^{8} - 118 \nu^{7} + 4626 \nu^{6} + 634 \nu^{5} - 11122 \nu^{4} + \cdots - 2345 ) / 59 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 71 \nu^{10} - 13 \nu^{9} - 1143 \nu^{8} + 177 \nu^{7} + 6378 \nu^{6} - 815 \nu^{5} - 14374 \nu^{4} + \cdots - 2076 ) / 59 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 83 \nu^{10} + 26 \nu^{9} + 1342 \nu^{8} - 354 \nu^{7} - 7564 \nu^{6} + 1630 \nu^{5} + 17420 \nu^{4} + \cdots + 3090 ) / 59 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 100 \nu^{10} + 10 \nu^{9} + 1619 \nu^{8} - 118 \nu^{7} - 9136 \nu^{6} + 459 \nu^{5} + 21155 \nu^{4} + \cdots + 4025 ) / 59 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{8} + \beta_{4} + 7\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - 8\beta_{5} + 9\beta_{4} + \beta_{2} + 19\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10\beta_{10} + 10\beta_{8} - \beta_{7} + 12\beta_{4} - \beta_{3} + 45\beta_{2} + \beta _1 + 74 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11 \beta_{9} + 12 \beta_{8} - 11 \beta_{7} - 10 \beta_{6} - 55 \beta_{5} + 66 \beta_{4} - 4 \beta_{3} + \cdots - 7 \)
|
| \(\nu^{8}\) | \(=\) |
\( 76 \beta_{10} + \beta_{9} + 78 \beta_{8} - 15 \beta_{7} + \beta_{6} - 2 \beta_{5} + 106 \beta_{4} + \cdots + 416 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{10} + 92 \beta_{9} + 108 \beta_{8} - 89 \beta_{7} - 74 \beta_{6} - 362 \beta_{5} + 456 \beta_{4} + \cdots - 21 \)
|
| \(\nu^{10}\) | \(=\) |
\( 528 \beta_{10} + 17 \beta_{9} + 562 \beta_{8} - 152 \beta_{7} + 16 \beta_{6} - 38 \beta_{5} + \cdots + 2439 \)
|
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 |
|
−2.47114 | 1.05832 | 4.10655 | −4.21457 | −2.61527 | −3.78215 | −5.20559 | −1.87995 | 10.4148 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.33155 | −2.43538 | 3.43614 | 3.89694 | 5.67823 | 4.53659 | −3.34845 | 2.93109 | −9.08593 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.77363 | 2.13379 | 1.14575 | 0.365878 | −3.78454 | 3.16809 | 1.51512 | 1.55305 | −0.648930 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.937517 | −1.66676 | −1.12106 | 2.41098 | 1.56262 | −2.77842 | 2.92605 | −0.221897 | −2.26033 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.656152 | −1.15507 | −1.56946 | −3.87537 | 0.757901 | 1.46037 | 2.34211 | −1.66581 | 2.54283 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.532912 | 2.21777 | −1.71601 | 2.52510 | 1.18187 | −0.0960799 | −1.98030 | 1.91849 | 1.34565 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.15166 | −3.23521 | −0.673682 | −2.60808 | −3.72586 | 3.62737 | −3.07917 | 7.46660 | −3.00362 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.84206 | 0.484221 | 1.39318 | 3.85955 | 0.891963 | 1.31125 | −1.11779 | −2.76553 | 7.10952 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.88758 | 1.67145 | 1.56297 | −1.60937 | 3.15501 | 4.68001 | −0.824935 | −0.206245 | −3.03782 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.12601 | 3.05580 | 2.51992 | −1.92378 | 6.49666 | −2.08676 | 1.10536 | 6.33790 | −4.08997 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.62977 | −2.12892 | 4.91569 | 2.17273 | −5.59857 | −1.04026 | 7.66761 | 1.53230 | 5.71378 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(37\) | \(-1\) |
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 | 407.2.a.c | ✓ | 11 |
| 3.b | odd | 2 | 1 | 3663.2.a.u | 11 | ||
| 4.b | odd | 2 | 1 | 6512.2.a.bb | 11 | ||
| 11.b | odd | 2 | 1 | 4477.2.a.k | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 407.2.a.c | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 3663.2.a.u | 11 | 3.b | odd | 2 | 1 | ||
| 4477.2.a.k | 11 | 11.b | odd | 2 | 1 | ||
| 6512.2.a.bb | 11 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 2 T_{2}^{10} - 16 T_{2}^{9} + 32 T_{2}^{8} + 89 T_{2}^{7} - 179 T_{2}^{6} - 201 T_{2}^{5} + \cdots + 75 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(407))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 2 T^{10} + \cdots + 75 \)
|
| $3$ |
\( T^{11} - 24 T^{9} + \cdots + 400 \)
|
| $5$ |
\( T^{11} - T^{10} + \cdots + 9600 \)
|
| $7$ |
\( T^{11} - 9 T^{10} + \cdots + 1024 \)
|
| $11$ |
\( (T + 1)^{11} \)
|
| $13$ |
\( T^{11} - 22 T^{10} + \cdots - 10228 \)
|
| $17$ |
\( T^{11} - 19 T^{10} + \cdots + 343956 \)
|
| $19$ |
\( T^{11} - 14 T^{10} + \cdots - 4852 \)
|
| $23$ |
\( T^{11} + 14 T^{10} + \cdots + 311040 \)
|
| $29$ |
\( T^{11} - 13 T^{10} + \cdots + 4860 \)
|
| $31$ |
\( T^{11} - 12 T^{10} + \cdots + 43424000 \)
|
| $37$ |
\( (T - 1)^{11} \)
|
| $41$ |
\( T^{11} - 8 T^{10} + \cdots - 248448 \)
|
| $43$ |
\( T^{11} + \cdots + 1018403500 \)
|
| $47$ |
\( T^{11} + 14 T^{10} + \cdots - 366336 \)
|
| $53$ |
\( T^{11} + \cdots + 116464080 \)
|
| $59$ |
\( T^{11} + \cdots + 256770816 \)
|
| $61$ |
\( T^{11} + \cdots + 4605256000 \)
|
| $67$ |
\( T^{11} + \cdots + 2503722752 \)
|
| $71$ |
\( T^{11} + 15 T^{10} + \cdots - 8674800 \)
|
| $73$ |
\( T^{11} - 47 T^{10} + \cdots - 1136768 \)
|
| $79$ |
\( T^{11} + \cdots - 167023700 \)
|
| $83$ |
\( T^{11} + \cdots - 1510296576 \)
|
| $89$ |
\( T^{11} - 4 T^{10} + \cdots + 9600 \)
|
| $97$ |
\( T^{11} + \cdots + 4933959040 \)
|
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