Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [370,2,Mod(159,370)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(370, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("370.159");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 370 = 2 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 370.m (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(2.95446487479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 37x^{14} + 559x^{12} + 4431x^{10} + 19684x^{8} + 48248x^{6} + 58656x^{4} + 25392x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 37x^{14} + 559x^{12} + 4431x^{10} + 19684x^{8} + 48248x^{6} + 58656x^{4} + 25392x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{15} + 31\nu^{13} + 381\nu^{11} + 2361\nu^{9} + 7734\nu^{7} + 12700\nu^{5} + 8872\nu^{3} + 2272\nu - 128 ) / 256 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{15} + 2 \nu^{14} - 31 \nu^{13} + 54 \nu^{12} - 381 \nu^{11} + 546 \nu^{10} - 2361 \nu^{9} + \cdots - 384 ) / 256 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{15} - 2 \nu^{14} - 31 \nu^{13} - 54 \nu^{12} - 381 \nu^{11} - 546 \nu^{10} - 2361 \nu^{9} + \cdots + 384 ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 162 \nu^{12} - 273 \nu^{11} + 1670 \nu^{10} - 1237 \nu^{9} + \cdots + 5632 \nu ) / 256 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{15} - 6 \nu^{14} - 27 \nu^{13} - 162 \nu^{12} - 273 \nu^{11} - 1670 \nu^{10} - 1237 \nu^{9} + \cdots + 5632 \nu ) / 256 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{15} + 6 \nu^{14} + 27 \nu^{13} + 170 \nu^{12} + 289 \nu^{11} + 1870 \nu^{10} + 1637 \nu^{9} + \cdots + 768 ) / 256 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{15} + 6 \nu^{14} - 27 \nu^{13} + 170 \nu^{12} - 289 \nu^{11} + 1870 \nu^{10} - 1637 \nu^{9} + \cdots + 768 ) / 256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - \nu^{15} + 6 \nu^{14} - 31 \nu^{13} + 170 \nu^{12} - 381 \nu^{11} + 1870 \nu^{10} - 2361 \nu^{9} + \cdots - 384 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 170 \nu^{12} + 381 \nu^{11} + 1870 \nu^{10} + 2361 \nu^{9} + \cdots - 384 ) / 256 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 3 \nu^{14} + 89 \nu^{12} + 1035 \nu^{10} + 5975 \nu^{8} + 17774 \nu^{6} + 24892 \nu^{4} + 11560 \nu^{2} + \cdots + 128 ) / 128 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 3 \nu^{14} - 89 \nu^{12} - 1035 \nu^{10} - 5975 \nu^{8} - 17774 \nu^{6} - 24892 \nu^{4} + \cdots - 128 ) / 128 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 3 \nu^{15} + 6 \nu^{14} + 81 \nu^{13} + 170 \nu^{12} + 835 \nu^{11} + 1870 \nu^{10} + 4143 \nu^{9} + \cdots + 256 ) / 256 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( -3\nu^{14} - 85\nu^{12} - 935\nu^{10} - 5051\nu^{8} - 13938\nu^{6} - 18092\nu^{4} - 8056\nu^{2} - 128 ) / 64 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 3 \nu^{14} + 87 \nu^{12} + 985 \nu^{10} + 5513 \nu^{8} + 8 \nu^{7} + 15872 \nu^{6} + 112 \nu^{5} + \cdots + 320 ) / 64 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 3 \nu^{14} - 87 \nu^{12} - 985 \nu^{10} - 5513 \nu^{8} + 8 \nu^{7} - 15872 \nu^{6} + 112 \nu^{5} + \cdots - 320 ) / 64 \)
|
| \(\nu\) | \(=\) |
\( \beta_{11} + \beta_{10} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{13} + \beta_{7} + \beta_{6} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} + \beta_{14} - 6\beta_{11} - 6\beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} - \beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( -11\beta_{13} - 2\beta_{9} - 2\beta_{8} - 9\beta_{7} - 9\beta_{6} + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 11 \beta_{15} - 11 \beta_{14} + 42 \beta_{11} + 42 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} + \cdots + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{15} + 2 \beta_{14} + 101 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + 24 \beta_{9} + \cdots - 192 \)
|
| \(\nu^{7}\) | \(=\) |
\( 103 \beta_{15} + 103 \beta_{14} - 320 \beta_{11} - 320 \beta_{10} - 99 \beta_{9} + 99 \beta_{8} + \cdots - 28 \)
|
| \(\nu^{8}\) | \(=\) |
\( 28 \beta_{15} - 28 \beta_{14} - 871 \beta_{13} - 32 \beta_{11} + 32 \beta_{10} - 230 \beta_{9} + \cdots + 1510 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 919 \beta_{15} - 919 \beta_{14} + 4 \beta_{13} + 8 \beta_{12} + 2546 \beta_{11} + 2546 \beta_{10} + \cdots + 294 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 282 \beta_{15} + 282 \beta_{14} + 7329 \beta_{13} + 366 \beta_{11} - 366 \beta_{10} + 2056 \beta_{9} + \cdots - 12252 \)
|
| \(\nu^{11}\) | \(=\) |
\( 8039 \beta_{15} + 8039 \beta_{14} - 100 \beta_{13} - 200 \beta_{12} - 20692 \beta_{11} - 20692 \beta_{10} + \cdots - 2800 \)
|
| \(\nu^{12}\) | \(=\) |
\( 2500 \beta_{15} - 2500 \beta_{14} - 61043 \beta_{13} - 3692 \beta_{11} + 3692 \beta_{10} - 17886 \beta_{9} + \cdots + 100922 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 69671 \beta_{15} - 69671 \beta_{14} + 1576 \beta_{13} + 3152 \beta_{12} + 169910 \beta_{11} + \cdots + 25586 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 20794 \beta_{15} + 20794 \beta_{14} + 506205 \beta_{13} + 35122 \beta_{11} - 35122 \beta_{10} + \cdots - 837320 \)
|
| \(\nu^{15}\) | \(=\) |
\( 600927 \beta_{15} + 600927 \beta_{14} - 20200 \beta_{13} - 40400 \beta_{12} - 1402224 \beta_{11} + \cdots - 229220 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(297\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 159.1 |
|
−0.500000 | − | 0.866025i | −2.47681 | − | 1.42999i | −0.500000 | + | 0.866025i | 1.52446 | − | 1.63586i | 2.85998i | 4.15112 | + | 2.39665i | 1.00000 | 2.58973 | + | 4.48555i | −2.17892 | − | 0.502291i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 159.2 | −0.500000 | − | 0.866025i | −1.79089 | − | 1.03397i | −0.500000 | + | 0.866025i | 1.86778 | + | 1.22939i | 2.06794i | −1.25580 | − | 0.725037i | 1.00000 | 0.638185 | + | 1.10537i | 0.130790 | − | 2.23224i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 159.3 | −0.500000 | − | 0.866025i | −1.67492 | − | 0.967016i | −0.500000 | + | 0.866025i | −0.564048 | + | 2.16376i | 1.93403i | 0.358580 | + | 0.207027i | 1.00000 | 0.370239 | + | 0.641273i | 2.15589 | − | 0.593399i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 159.4 | −0.500000 | − | 0.866025i | 0.0880035 | + | 0.0508088i | −0.500000 | + | 0.866025i | −1.69259 | − | 1.46122i | − | 0.101618i | 1.94895 | + | 1.12523i | 1.00000 | −1.49484 | − | 2.58913i | −0.419156 | + | 2.19643i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 159.5 | −0.500000 | − | 0.866025i | 0.802594 | + | 0.463378i | −0.500000 | + | 0.866025i | 1.82024 | − | 1.29873i | − | 0.926756i | −3.13584 | − | 1.81048i | 1.00000 | −1.07056 | − | 1.85427i | −2.03486 | − | 0.927011i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 159.6 | −0.500000 | − | 0.866025i | 1.53027 | + | 0.883503i | −0.500000 | + | 0.866025i | 2.03272 | − | 0.931689i | − | 1.76701i | 3.45647 | + | 1.99560i | 1.00000 | 0.0611550 | + | 0.105924i | −1.82323 | − | 1.29454i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 159.7 | −0.500000 | − | 0.866025i | 2.50227 | + | 1.44468i | −0.500000 | + | 0.866025i | −0.791839 | + | 2.09117i | − | 2.88937i | 0.668346 | + | 0.385870i | 1.00000 | 2.67422 | + | 4.63189i | 2.20693 | − | 0.359832i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 159.8 | −0.500000 | − | 0.866025i | 2.51948 | + | 1.45462i | −0.500000 | + | 0.866025i | −1.19673 | − | 1.88887i | − | 2.90925i | −0.191824 | − | 0.110750i | 1.00000 | 2.73187 | + | 4.73173i | −1.03745 | + | 1.98083i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.1 | −0.500000 | + | 0.866025i | −2.47681 | + | 1.42999i | −0.500000 | − | 0.866025i | 1.52446 | + | 1.63586i | − | 2.85998i | 4.15112 | − | 2.39665i | 1.00000 | 2.58973 | − | 4.48555i | −2.17892 | + | 0.502291i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.2 | −0.500000 | + | 0.866025i | −1.79089 | + | 1.03397i | −0.500000 | − | 0.866025i | 1.86778 | − | 1.22939i | − | 2.06794i | −1.25580 | + | 0.725037i | 1.00000 | 0.638185 | − | 1.10537i | 0.130790 | + | 2.23224i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.3 | −0.500000 | + | 0.866025i | −1.67492 | + | 0.967016i | −0.500000 | − | 0.866025i | −0.564048 | − | 2.16376i | − | 1.93403i | 0.358580 | − | 0.207027i | 1.00000 | 0.370239 | − | 0.641273i | 2.15589 | + | 0.593399i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.4 | −0.500000 | + | 0.866025i | 0.0880035 | − | 0.0508088i | −0.500000 | − | 0.866025i | −1.69259 | + | 1.46122i | 0.101618i | 1.94895 | − | 1.12523i | 1.00000 | −1.49484 | + | 2.58913i | −0.419156 | − | 2.19643i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.5 | −0.500000 | + | 0.866025i | 0.802594 | − | 0.463378i | −0.500000 | − | 0.866025i | 1.82024 | + | 1.29873i | 0.926756i | −3.13584 | + | 1.81048i | 1.00000 | −1.07056 | + | 1.85427i | −2.03486 | + | 0.927011i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.6 | −0.500000 | + | 0.866025i | 1.53027 | − | 0.883503i | −0.500000 | − | 0.866025i | 2.03272 | + | 0.931689i | 1.76701i | 3.45647 | − | 1.99560i | 1.00000 | 0.0611550 | − | 0.105924i | −1.82323 | + | 1.29454i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.7 | −0.500000 | + | 0.866025i | 2.50227 | − | 1.44468i | −0.500000 | − | 0.866025i | −0.791839 | − | 2.09117i | 2.88937i | 0.668346 | − | 0.385870i | 1.00000 | 2.67422 | − | 4.63189i | 2.20693 | + | 0.359832i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 249.8 | −0.500000 | + | 0.866025i | 2.51948 | − | 1.45462i | −0.500000 | − | 0.866025i | −1.19673 | + | 1.88887i | 2.90925i | −0.191824 | + | 0.110750i | 1.00000 | 2.73187 | − | 4.73173i | −1.03745 | − | 1.98083i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 185.l | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 370.2.m.c | ✓ | 16 |
| 5.b | even | 2 | 1 | 370.2.m.d | yes | 16 | |
| 37.e | even | 6 | 1 | 370.2.m.d | yes | 16 | |
| 185.l | even | 6 | 1 | inner | 370.2.m.c | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 370.2.m.c | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 370.2.m.c | ✓ | 16 | 185.l | even | 6 | 1 | inner |
| 370.2.m.d | yes | 16 | 5.b | even | 2 | 1 | |
| 370.2.m.d | yes | 16 | 37.e | even | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 3 T_{3}^{15} - 14 T_{3}^{14} + 51 T_{3}^{13} + 154 T_{3}^{12} - 561 T_{3}^{11} - 771 T_{3}^{10} + \cdots + 256 \)
acting on \(S_{2}^{\mathrm{new}}(370, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{8} \)
|
| $3$ |
\( T^{16} - 3 T^{15} + \cdots + 256 \)
|
| $5$ |
\( T^{16} - 6 T^{15} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} - 12 T^{15} + \cdots + 256 \)
|
| $11$ |
\( (T^{8} + 3 T^{7} + \cdots - 11840)^{2} \)
|
| $13$ |
\( T^{16} - 6 T^{15} + \cdots + 1849600 \)
|
| $17$ |
\( T^{16} + 58 T^{14} + \cdots + 226576 \)
|
| $19$ |
\( T^{16} + \cdots + 1435500544 \)
|
| $23$ |
\( (T^{8} - 11 T^{7} + \cdots - 9224)^{2} \)
|
| $29$ |
\( T^{16} + 192 T^{14} + \cdots + 14500864 \)
|
| $31$ |
\( T^{16} + 121 T^{14} + \cdots + 16384 \)
|
| $37$ |
\( T^{16} + \cdots + 3512479453921 \)
|
| $41$ |
\( T^{16} + \cdots + 23754764740996 \)
|
| $43$ |
\( (T^{8} - 11 T^{7} + \cdots + 177208)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 35452370944 \)
|
| $53$ |
\( T^{16} + 3 T^{15} + \cdots + 802816 \)
|
| $59$ |
\( T^{16} + \cdots + 159386189824 \)
|
| $61$ |
\( T^{16} + \cdots + 899128754176 \)
|
| $67$ |
\( T^{16} + \cdots + 1893990400 \)
|
| $71$ |
\( T^{16} + \cdots + 451477504 \)
|
| $73$ |
\( T^{16} + \cdots + 10676686950400 \)
|
| $79$ |
\( T^{16} + \cdots + 5065884565504 \)
|
| $83$ |
\( T^{16} + \cdots + 1363677184 \)
|
| $89$ |
\( T^{16} + \cdots + 5611687210000 \)
|
| $97$ |
\( (T^{8} + 34 T^{7} + \cdots + 3678976)^{2} \)
|
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