Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,3,Mod(28,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.28");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.g (of order \(4\), 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: | \(3.67848356886\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 286x^{12} + 16269x^{8} + 85684x^{4} + 62500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{8}\cdot 5 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} + 286x^{12} + 16269x^{8} + 85684x^{4} + 62500 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{12} - 3932\nu^{8} - 311181\nu^{4} - 4873078 ) / 908888 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2321\nu^{14} + 667056\nu^{10} + 38743349\nu^{6} + 276667814\nu^{2} ) / 227222000 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -6963\nu^{14} - 2001168\nu^{10} - 116230047\nu^{6} - 716392442\nu^{2} ) / 113611000 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3212 \nu^{14} - 6535 \nu^{12} + 901592 \nu^{10} - 1801800 \nu^{8} + 47801228 \nu^{6} + \cdots - 119060250 ) / 45444400 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3212 \nu^{14} + 6535 \nu^{12} + 901592 \nu^{10} + 1801800 \nu^{8} + 47801228 \nu^{6} + \cdots + 119060250 ) / 45444400 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2321\nu^{15} + 667056\nu^{11} + 38743349\nu^{7} + 276667814\nu^{3} ) / 227222000 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 22304 \nu^{14} + 28525 \nu^{12} - 6344094 \nu^{10} + 8190750 \nu^{8} - 351886026 \nu^{6} + \cdots + 1230609250 ) / 227222000 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 22304 \nu^{14} - 28525 \nu^{12} - 6344094 \nu^{10} - 8190750 \nu^{8} - 351886026 \nu^{6} + \cdots - 1230609250 ) / 227222000 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 115811 \nu^{15} + 246250 \nu^{13} + 32721196 \nu^{11} + 70111500 \nu^{9} + \cdots + 14728508500 \nu ) / 2272220000 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 221887 \nu^{15} + 476250 \nu^{13} + 63721432 \nu^{11} + 135308000 \nu^{9} + \cdots + 17685119500 \nu ) / 2272220000 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 221887 \nu^{15} - 508750 \nu^{13} - 63721432 \nu^{11} - 145138000 \nu^{9} + \cdots - 41228914500 \nu ) / 2272220000 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 233253 \nu^{15} - 191000 \nu^{13} + 66425108 \nu^{11} - 53400500 \nu^{9} + \cdots - 5379027000 \nu ) / 2272220000 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 280369 \nu^{15} + 218625 \nu^{13} + 79941784 \nu^{11} + 61756000 \nu^{9} + \cdots + 10053767750 \nu ) / 1136110000 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 594161 \nu^{15} + 71500 \nu^{13} - 169538296 \nu^{11} + 21626000 \nu^{9} + \cdots + 18849159000 \nu ) / 2272220000 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 6\beta_{3} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} + \beta_{14} + \beta_{12} + 11\beta_{7} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{9} - 2\beta_{8} + \beta_{6} - \beta_{5} - 15\beta_{2} - 64 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{15} + \beta_{14} - 6\beta_{13} - 17\beta_{12} - 20\beta_{11} - 2\beta_{10} - 2\beta_{7} - 138\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 44\beta_{9} + 44\beta_{8} + 27\beta_{6} + 27\beta_{5} - 213\beta_{4} - 806\beta_{3} \)
|
| \(\nu^{7}\) | \(=\) |
\( - 284 \beta_{15} - 328 \beta_{14} - 44 \beta_{13} - 257 \beta_{12} - 27 \beta_{11} + 152 \beta_{10} + \cdots - 284 \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -754\beta_{9} + 754\beta_{8} - 507\beta_{6} + 507\beta_{5} + 3041\beta_{2} + 10794 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1261 \beta_{15} - 507 \beta_{14} + 2782 \beta_{13} + 3795 \beta_{12} + 5056 \beta_{11} + \cdots + 25232 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -11880\beta_{9} - 11880\beta_{8} - 8345\beta_{6} - 8345\beta_{5} + 43701\beta_{4} + 149366\beta_{3} \)
|
| \(\nu^{11}\) | \(=\) |
\( 63926 \beta_{15} + 75806 \beta_{14} + 11880 \beta_{13} + 55581 \beta_{12} + 8345 \beta_{11} + \cdots + 63926 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 180182\beta_{9} - 180182\beta_{8} + 129411\beta_{6} - 129411\beta_{5} - 630645\beta_{2} - 2107654 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 309593 \beta_{15} + 129411 \beta_{14} - 697826 \beta_{13} - 810827 \beta_{12} - 1120420 \beta_{11} + \cdots - 5052828 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 2679844\beta_{9} + 2679844\beta_{8} + 1947657\beta_{6} + 1947657\beta_{5} - 9123373\beta_{4} - 30090966\beta_{3} \)
|
| \(\nu^{15}\) | \(=\) |
\( - 13750874 \beta_{15} - 16430718 \beta_{14} - 2679844 \beta_{13} - 11803217 \beta_{12} + \cdots - 13750874 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−2.69407 | + | 2.69407i | 0 | − | 10.5160i | −4.29496 | − | 2.55995i | 0 | −1.13867 | + | 1.13867i | 17.5546 | + | 17.5546i | 0 | 18.4676 | − | 4.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.2 | −2.04178 | + | 2.04178i | 0 | − | 4.33774i | 4.12923 | + | 2.81948i | 0 | 8.42562 | − | 8.42562i | 0.689596 | + | 0.689596i | 0 | −14.1877 | + | 2.67423i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.3 | −1.05185 | + | 1.05185i | 0 | 1.78723i | −2.02789 | − | 4.57030i | 0 | −5.20088 | + | 5.20088i | −6.08729 | − | 6.08729i | 0 | 6.94029 | + | 2.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.4 | −0.683187 | + | 0.683187i | 0 | 3.06651i | −2.52793 | + | 4.31388i | 0 | 1.91393 | − | 1.91393i | −4.82775 | − | 4.82775i | 0 | −1.22013 | − | 4.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.5 | 0.683187 | − | 0.683187i | 0 | 3.06651i | 2.52793 | − | 4.31388i | 0 | 1.91393 | − | 1.91393i | 4.82775 | + | 4.82775i | 0 | −1.22013 | − | 4.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.6 | 1.05185 | − | 1.05185i | 0 | 1.78723i | 2.02789 | + | 4.57030i | 0 | −5.20088 | + | 5.20088i | 6.08729 | + | 6.08729i | 0 | 6.94029 | + | 2.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.7 | 2.04178 | − | 2.04178i | 0 | − | 4.33774i | −4.12923 | − | 2.81948i | 0 | 8.42562 | − | 8.42562i | −0.689596 | − | 0.689596i | 0 | −14.1877 | + | 2.67423i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.8 | 2.69407 | − | 2.69407i | 0 | − | 10.5160i | 4.29496 | + | 2.55995i | 0 | −1.13867 | + | 1.13867i | −17.5546 | − | 17.5546i | 0 | 18.4676 | − | 4.67423i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.1 | −2.69407 | − | 2.69407i | 0 | 10.5160i | −4.29496 | + | 2.55995i | 0 | −1.13867 | − | 1.13867i | 17.5546 | − | 17.5546i | 0 | 18.4676 | + | 4.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.2 | −2.04178 | − | 2.04178i | 0 | 4.33774i | 4.12923 | − | 2.81948i | 0 | 8.42562 | + | 8.42562i | 0.689596 | − | 0.689596i | 0 | −14.1877 | − | 2.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.3 | −1.05185 | − | 1.05185i | 0 | − | 1.78723i | −2.02789 | + | 4.57030i | 0 | −5.20088 | − | 5.20088i | −6.08729 | + | 6.08729i | 0 | 6.94029 | − | 2.67423i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.4 | −0.683187 | − | 0.683187i | 0 | − | 3.06651i | −2.52793 | − | 4.31388i | 0 | 1.91393 | + | 1.91393i | −4.82775 | + | 4.82775i | 0 | −1.22013 | + | 4.67423i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.5 | 0.683187 | + | 0.683187i | 0 | − | 3.06651i | 2.52793 | + | 4.31388i | 0 | 1.91393 | + | 1.91393i | 4.82775 | − | 4.82775i | 0 | −1.22013 | + | 4.67423i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.6 | 1.05185 | + | 1.05185i | 0 | − | 1.78723i | 2.02789 | − | 4.57030i | 0 | −5.20088 | − | 5.20088i | 6.08729 | − | 6.08729i | 0 | 6.94029 | − | 2.67423i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.7 | 2.04178 | + | 2.04178i | 0 | 4.33774i | −4.12923 | + | 2.81948i | 0 | 8.42562 | + | 8.42562i | −0.689596 | + | 0.689596i | 0 | −14.1877 | − | 2.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.8 | 2.69407 | + | 2.69407i | 0 | 10.5160i | 4.29496 | − | 2.55995i | 0 | −1.13867 | − | 1.13867i | −17.5546 | + | 17.5546i | 0 | 18.4676 | + | 4.67423i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.c | odd | 4 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 135.3.g.b | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 135.3.g.b | ✓ | 16 |
| 5.b | even | 2 | 1 | 675.3.g.j | 16 | ||
| 5.c | odd | 4 | 1 | inner | 135.3.g.b | ✓ | 16 |
| 5.c | odd | 4 | 1 | 675.3.g.j | 16 | ||
| 9.c | even | 3 | 2 | 405.3.l.n | 32 | ||
| 9.d | odd | 6 | 2 | 405.3.l.n | 32 | ||
| 15.d | odd | 2 | 1 | 675.3.g.j | 16 | ||
| 15.e | even | 4 | 1 | inner | 135.3.g.b | ✓ | 16 |
| 15.e | even | 4 | 1 | 675.3.g.j | 16 | ||
| 45.k | odd | 12 | 2 | 405.3.l.n | 32 | ||
| 45.l | even | 12 | 2 | 405.3.l.n | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.3.g.b | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 135.3.g.b | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 135.3.g.b | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 135.3.g.b | ✓ | 16 | 15.e | even | 4 | 1 | inner |
| 405.3.l.n | 32 | 9.c | even | 3 | 2 | ||
| 405.3.l.n | 32 | 9.d | odd | 6 | 2 | ||
| 405.3.l.n | 32 | 45.k | odd | 12 | 2 | ||
| 405.3.l.n | 32 | 45.l | even | 12 | 2 | ||
| 675.3.g.j | 16 | 5.b | even | 2 | 1 | ||
| 675.3.g.j | 16 | 5.c | odd | 4 | 1 | ||
| 675.3.g.j | 16 | 15.d | odd | 2 | 1 | ||
| 675.3.g.j | 16 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 286T_{2}^{12} + 16269T_{2}^{8} + 85684T_{2}^{4} + 62500 \)
acting on \(S_{3}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 286 T^{12} + \cdots + 62500 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + \cdots + 152587890625 \)
|
| $7$ |
\( (T^{8} - 8 T^{7} + \cdots + 145924)^{2} \)
|
| $11$ |
\( (T^{8} - 550 T^{6} + \cdots + 164430250)^{2} \)
|
| $13$ |
\( (T^{8} - 20 T^{7} + \cdots + 15625)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 62\!\cdots\!00 \)
|
| $19$ |
\( (T^{8} + 1638 T^{6} + \cdots + 3382352964)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 45\!\cdots\!00 \)
|
| $29$ |
\( (T^{8} + 3690 T^{6} + \cdots + 2609840250)^{2} \)
|
| $31$ |
\( (T^{4} - 50 T^{3} + \cdots - 1005650)^{4} \)
|
| $37$ |
\( (T^{8} - 8 T^{7} + \cdots + 2296326400)^{2} \)
|
| $41$ |
\( (T^{8} + \cdots + 1815186025000)^{2} \)
|
| $43$ |
\( (T^{8} - 68 T^{7} + \cdots + 45403938724)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 12\!\cdots\!00 \)
|
| $53$ |
\( T^{16} + \cdots + 11\!\cdots\!00 \)
|
| $59$ |
\( (T^{8} + \cdots + 1319432976000)^{2} \)
|
| $61$ |
\( (T^{4} + 70 T^{3} + \cdots + 3570142)^{4} \)
|
| $67$ |
\( (T^{8} + \cdots + 39225920563600)^{2} \)
|
| $71$ |
\( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{8} + \cdots + 102945374440000)^{2} \)
|
| $79$ |
\( (T^{8} + 24012 T^{6} + \cdots + 46753250625)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 21\!\cdots\!00 \)
|
| $89$ |
\( (T^{8} + \cdots + 490910422500000)^{2} \)
|
| $97$ |
\( (T^{8} + \cdots + 5463120878224)^{2} \)
|
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