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: |
\( x^{16} + 217x^{12} + 9264x^{8} + 59497x^{4} + 28561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{8} \) |
| 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
\( x^{16} + 217x^{12} + 9264x^{8} + 59497x^{4} + 28561 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 23\nu^{12} + 5326\nu^{8} + 334140\nu^{4} + 5452367 ) / 1000354 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10883\nu^{13} + 2302656\nu^{9} + 89995272\nu^{5} + 364351187\nu ) / 52018408 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3253\nu^{14} + 709788\nu^{10} + 31035886\nu^{6} + 250013401\nu^{2} ) / 169059826 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 102443 \nu^{14} - 202124 \nu^{12} + 21417072 \nu^{10} - 46804888 \nu^{8} + 797507904 \nu^{6} + \cdots - 9369760868 ) / 1352478608 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 102443 \nu^{14} + 202124 \nu^{12} + 21417072 \nu^{10} + 46804888 \nu^{8} + 797507904 \nu^{6} + \cdots + 9369760868 ) / 1352478608 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -9759\nu^{14} - 2129364\nu^{10} - 93107658\nu^{6} - 665510290\nu^{2} ) / 84529913 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 360147 \nu^{14} + 433147 \nu^{12} - 77438888 \nu^{10} + 92951352 \nu^{8} + \cdots + 11565687211 ) / 1352478608 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 360147 \nu^{14} - 433147 \nu^{12} - 77438888 \nu^{10} - 92951352 \nu^{8} + \cdots - 10213208603 ) / 1352478608 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -3253\nu^{15} - 709788\nu^{11} - 31035886\nu^{7} - 250013401\nu^{3} ) / 169059826 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1740189 \nu^{15} + 1996904 \nu^{13} - 380220064 \nu^{11} + 433011800 \nu^{9} + \cdots + 77138103120 \nu ) / 17582221904 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1740189 \nu^{15} + 1996904 \nu^{13} + 380220064 \nu^{11} + 433011800 \nu^{9} + \cdots + 77138103120 \nu ) / 17582221904 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2140700 \nu^{15} + 359801 \nu^{13} - 462308536 \nu^{11} + 90667824 \nu^{9} + \cdots + 81060375617 \nu ) / 17582221904 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 2140700 \nu^{15} - 359801 \nu^{13} - 462308536 \nu^{11} - 90667824 \nu^{9} + \cdots - 81060375617 \nu ) / 17582221904 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1753759\nu^{15} + 377360280\nu^{11} + 15600114456\nu^{7} + 82428904231\nu^{3} ) / 8791110952 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 6\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - 8\beta_{10} \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} + 13\beta_{2} - 57 \)
|
| \(\nu^{5}\) | \(=\) |
\( -13\beta_{14} + 13\beta_{13} - 18\beta_{12} - 18\beta_{11} + 17\beta_{3} - 81\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 18\beta_{9} + 18\beta_{8} - 160\beta_{7} + 22\beta_{6} + 22\beta_{5} - 635\beta_{4} - 18 \)
|
| \(\nu^{7}\) | \(=\) |
\( 248\beta_{15} + 258\beta_{14} + 258\beta_{13} + 156\beta_{12} - 156\beta_{11} + 911\beta_{10} \)
|
| \(\nu^{8}\) | \(=\) |
\( 92\beta_{9} - 92\beta_{8} + 350\beta_{6} - 350\beta_{5} - 1987\beta_{2} + 7462 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1895\beta_{14} - 1895\beta_{13} + 3461\beta_{12} + 3461\beta_{11} - 3387\beta_{3} + 10828\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -3461\beta_{9} - 3461\beta_{8} + 24927\beta_{7} - 4953\beta_{6} - 4953\beta_{5} + 92763\beta_{4} + 3461 \)
|
| \(\nu^{11}\) | \(=\) |
\( -44739\beta_{15} - 45216\beta_{14} - 45216\beta_{13} - 23435\beta_{12} + 23435\beta_{11} - 132711\beta_{10} \)
|
| \(\nu^{12}\) | \(=\) |
\( -21304\beta_{9} + 21304\beta_{8} - 66520\beta_{6} + 66520\beta_{5} + 314752\beta_{2} - 1136913 \)
|
| \(\nu^{13}\) | \(=\) |
\( -293448\beta_{14} + 293448\beta_{13} - 583440\beta_{12} - 583440\beta_{11} + 580832\beta_{3} - 1654681\beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 583440 \beta_{9} + 583440 \beta_{8} - 3989289 \beta_{7} + 870824 \beta_{6} + 870824 \beta_{5} + \cdots - 583440 \)
|
| \(\nu^{15}\) | \(=\) |
\( 7472585 \beta_{15} + 7481257 \beta_{14} + 7481257 \beta_{13} + 3701905 \beta_{12} + \cdots + 20828168 \beta_{10} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(1\) | \(\beta_{4}\) |
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 |
|
−2.52312 | + | 2.52312i | 0 | − | 8.73230i | 2.81400 | + | 4.13296i | 0 | −4.39136 | + | 4.39136i | 11.9402 | + | 11.9402i | 0 | −17.5280 | − | 3.32790i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.2 | −1.85358 | + | 1.85358i | 0 | − | 2.87153i | −4.93699 | + | 0.791263i | 0 | 1.80972 | − | 1.80972i | −2.09171 | − | 2.09171i | 0 | 7.68445 | − | 10.6178i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.3 | −1.15623 | + | 1.15623i | 0 | 1.32628i | 0.910983 | − | 4.91631i | 0 | 7.71617 | − | 7.71617i | −6.15839 | − | 6.15839i | 0 | 4.63107 | + | 6.73767i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.4 | −0.601022 | + | 0.601022i | 0 | 3.27754i | 4.98774 | − | 0.349865i | 0 | −7.13454 | + | 7.13454i | −4.37397 | − | 4.37397i | 0 | −2.78747 | + | 3.20802i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.5 | 0.601022 | − | 0.601022i | 0 | 3.27754i | −4.98774 | + | 0.349865i | 0 | −7.13454 | + | 7.13454i | 4.37397 | + | 4.37397i | 0 | −2.78747 | + | 3.20802i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.6 | 1.15623 | − | 1.15623i | 0 | 1.32628i | −0.910983 | + | 4.91631i | 0 | 7.71617 | − | 7.71617i | 6.15839 | + | 6.15839i | 0 | 4.63107 | + | 6.73767i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.7 | 1.85358 | − | 1.85358i | 0 | − | 2.87153i | 4.93699 | − | 0.791263i | 0 | 1.80972 | − | 1.80972i | 2.09171 | + | 2.09171i | 0 | 7.68445 | − | 10.6178i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 28.8 | 2.52312 | − | 2.52312i | 0 | − | 8.73230i | −2.81400 | − | 4.13296i | 0 | −4.39136 | + | 4.39136i | −11.9402 | − | 11.9402i | 0 | −17.5280 | − | 3.32790i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.1 | −2.52312 | − | 2.52312i | 0 | 8.73230i | 2.81400 | − | 4.13296i | 0 | −4.39136 | − | 4.39136i | 11.9402 | − | 11.9402i | 0 | −17.5280 | + | 3.32790i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.2 | −1.85358 | − | 1.85358i | 0 | 2.87153i | −4.93699 | − | 0.791263i | 0 | 1.80972 | + | 1.80972i | −2.09171 | + | 2.09171i | 0 | 7.68445 | + | 10.6178i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.3 | −1.15623 | − | 1.15623i | 0 | − | 1.32628i | 0.910983 | + | 4.91631i | 0 | 7.71617 | + | 7.71617i | −6.15839 | + | 6.15839i | 0 | 4.63107 | − | 6.73767i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.4 | −0.601022 | − | 0.601022i | 0 | − | 3.27754i | 4.98774 | + | 0.349865i | 0 | −7.13454 | − | 7.13454i | −4.37397 | + | 4.37397i | 0 | −2.78747 | − | 3.20802i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.5 | 0.601022 | + | 0.601022i | 0 | − | 3.27754i | −4.98774 | − | 0.349865i | 0 | −7.13454 | − | 7.13454i | 4.37397 | − | 4.37397i | 0 | −2.78747 | − | 3.20802i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.6 | 1.15623 | + | 1.15623i | 0 | − | 1.32628i | −0.910983 | − | 4.91631i | 0 | 7.71617 | + | 7.71617i | 6.15839 | − | 6.15839i | 0 | 4.63107 | − | 6.73767i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.7 | 1.85358 | + | 1.85358i | 0 | 2.87153i | 4.93699 | + | 0.791263i | 0 | 1.80972 | + | 1.80972i | 2.09171 | − | 2.09171i | 0 | 7.68445 | + | 10.6178i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 82.8 | 2.52312 | + | 2.52312i | 0 | 8.73230i | −2.81400 | + | 4.13296i | 0 | −4.39136 | − | 4.39136i | −11.9402 | + | 11.9402i | 0 | −17.5280 | + | 3.32790i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | inner | 135.3.g.a | ✓ | 16 |
| 5.b | even | 2 | 1 | 675.3.g.k | 16 | ||
| 5.c | odd | 4 | 1 | inner | 135.3.g.a | ✓ | 16 |
| 5.c | odd | 4 | 1 | 675.3.g.k | 16 | ||
| 9.c | even | 3 | 2 | 405.3.l.o | 32 | ||
| 9.d | odd | 6 | 2 | 405.3.l.o | 32 | ||
| 15.d | odd | 2 | 1 | 675.3.g.k | 16 | ||
| 15.e | even | 4 | 1 | inner | 135.3.g.a | ✓ | 16 |
| 15.e | even | 4 | 1 | 675.3.g.k | 16 | ||
| 45.k | odd | 12 | 2 | 405.3.l.o | 32 | ||
| 45.l | even | 12 | 2 | 405.3.l.o | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.3.g.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 135.3.g.a | ✓ | 16 | 3.b | odd | 2 | 1 | inner |
| 135.3.g.a | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 135.3.g.a | ✓ | 16 | 15.e | even | 4 | 1 | inner |
| 405.3.l.o | 32 | 9.c | even | 3 | 2 | ||
| 405.3.l.o | 32 | 9.d | odd | 6 | 2 | ||
| 405.3.l.o | 32 | 45.k | odd | 12 | 2 | ||
| 405.3.l.o | 32 | 45.l | even | 12 | 2 | ||
| 675.3.g.k | 16 | 5.b | even | 2 | 1 | ||
| 675.3.g.k | 16 | 5.c | odd | 4 | 1 | ||
| 675.3.g.k | 16 | 15.d | odd | 2 | 1 | ||
| 675.3.g.k | 16 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 217T_{2}^{12} + 9264T_{2}^{8} + 59497T_{2}^{4} + 28561 \)
acting on \(S_{3}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 217 T^{12} + \cdots + 28561 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} + \cdots + 152587890625 \)
|
| $7$ |
\( (T^{8} + 4 T^{7} + \cdots + 3062500)^{2} \)
|
| $11$ |
\( (T^{8} - 478 T^{6} + \cdots + 2500)^{2} \)
|
| $13$ |
\( (T^{8} - 14 T^{7} + \cdots + 376360000)^{2} \)
|
| $17$ |
\( T^{16} + \cdots + 83\!\cdots\!96 \)
|
| $19$ |
\( (T^{8} + 1386 T^{6} + \cdots + 4283440704)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 16\!\cdots\!16 \)
|
| $29$ |
\( (T^{8} + 3174 T^{6} + \cdots + 8593290000)^{2} \)
|
| $31$ |
\( (T^{4} + 52 T^{3} + \cdots - 255749)^{4} \)
|
| $37$ |
\( (T^{8} + 88 T^{7} + \cdots + 3648160000)^{2} \)
|
| $41$ |
\( (T^{8} - 7498 T^{6} + \cdots + 154291840000)^{2} \)
|
| $43$ |
\( (T^{8} + \cdots + 4471533160000)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 15\!\cdots\!00 \)
|
| $53$ |
\( T^{16} + \cdots + 10\!\cdots\!61 \)
|
| $59$ |
\( (T^{8} + \cdots + 63541623690000)^{2} \)
|
| $61$ |
\( (T^{4} - 74 T^{3} + \cdots + 488272)^{4} \)
|
| $67$ |
\( (T^{8} + \cdots + 6996025000000)^{2} \)
|
| $71$ |
\( (T^{8} + \cdots + 33777019240000)^{2} \)
|
| $73$ |
\( (T^{8} + 28 T^{7} + \cdots + 24790502500)^{2} \)
|
| $79$ |
\( (T^{8} + \cdots + 616589288038656)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 13\!\cdots\!21 \)
|
| $89$ |
\( (T^{8} + \cdots + 14493249000000)^{2} \)
|
| $97$ |
\( (T^{8} + \cdots + 147923973760000)^{2} \)
|
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