Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,2,Mod(8,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([2, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.m (of order \(12\), degree \(4\), not 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: | \(1.07798042729\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 45) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + \cdots + 33432180594 ) / 3707507912227 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 94386116202 \nu^{15} - 619740656932 \nu^{14} + 2033008548312 \nu^{13} - 4441986378774 \nu^{12} + \cdots - 80029143512 ) / 3707507912227 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 140020985001 \nu^{15} - 843322184609 \nu^{14} + 2535236829090 \nu^{13} + \cdots - 5622211270526 ) / 3707507912227 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 140094553942 \nu^{15} + 840493754711 \nu^{14} - 2524530400854 \nu^{13} + \cdots - 1652709999986 ) / 3707507912227 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 190424165289 \nu^{15} - 1059314239720 \nu^{14} + 2941649532255 \nu^{13} + \cdots + 7166890770427 ) / 3707507912227 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 250381984552 \nu^{15} + 1889212496921 \nu^{14} - 6927613311171 \nu^{13} + \cdots + 5660844178894 ) / 3707507912227 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 254444403226 \nu^{15} + 1391318146672 \nu^{14} - 3674576400880 \nu^{13} + \cdots + 146893504700 ) / 3707507912227 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 196858530 \nu^{15} - 1212076232 \nu^{14} + 3734556994 \nu^{13} - 7676123862 \nu^{12} + \cdots + 785074438 ) / 1973128213 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 677106342206 \nu^{15} + 3890533925017 \nu^{14} - 11114551602021 \nu^{13} + \cdots - 11893459432082 ) / 3707507912227 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} + \cdots - 3343669588 ) / 1973128213 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} + \cdots - 10233974547 ) / 3810388399 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2029780580195 \nu^{15} + 12360646755715 \nu^{14} - 37626995630777 \nu^{13} + \cdots - 14351633608601 ) / 3707507912227 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 2802573231023 \nu^{15} - 17485025240258 \nu^{14} + 54572756591878 \nu^{13} + \cdots + 12411058785072 ) / 3707507912227 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2852772870096 \nu^{15} + 17521813816267 \nu^{14} - 53872302373806 \nu^{13} + \cdots - 17972736434118 ) / 3707507912227 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{3} - 2\beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 4\beta_{3} - \beta_{2} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + \beta_{12} + 8 \beta_{11} - \beta_{10} + \beta_{7} + \cdots + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 16 \beta_{15} + 8 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 3 \beta_{9} - 8 \beta_{8} - 8 \beta_{6} + \cdots + 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 35 \beta_{15} - 16 \beta_{14} + 8 \beta_{13} + 30 \beta_{12} - 8 \beta_{11} + 16 \beta_{10} + \cdots + 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{15} - \beta_{14} - 10 \beta_{12} - 10 \beta_{11} + 51 \beta_{10} - \beta_{8} + 51 \beta_{7} + \cdots + 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( 50 \beta_{15} + 148 \beta_{14} + 50 \beta_{13} - 50 \beta_{12} + 145 \beta_{11} + 50 \beta_{10} + \cdots + 50 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 288 \beta_{15} + 278 \beta_{14} + 298 \beta_{13} + 228 \beta_{12} + 456 \beta_{11} - 55 \beta_{9} + \cdots + 228 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 1385 \beta_{15} - 288 \beta_{14} + 576 \beta_{13} + 1032 \beta_{12} + 288 \beta_{11} + 288 \beta_{10} + \cdots + 288 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 1533 \beta_{15} - 1603 \beta_{14} + 817 \beta_{12} - 1245 \beta_{11} + 1673 \beta_{10} - 1603 \beta_{8} + \cdots - 817 \)
|
| \(\nu^{12}\) | \(=\) |
\( 3206 \beta_{15} - 1603 \beta_{13} - 3206 \beta_{12} - 1603 \beta_{11} + 3206 \beta_{10} + 428 \beta_{9} + \cdots - 3923 \)
|
| \(\nu^{13}\) | \(=\) |
\( 8782 \beta_{15} + 8782 \beta_{14} - 6751 \beta_{12} + 6751 \beta_{11} + 428 \beta_{8} + 9210 \beta_{6} + \cdots - 6751 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 8782 \beta_{15} + 8782 \beta_{14} + 8782 \beta_{13} + 8782 \beta_{12} + 17564 \beta_{11} + \cdots - 8782 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 47744 \beta_{15} - 45285 \beta_{14} + 36503 \beta_{12} - 22803 \beta_{11} - 45285 \beta_{8} + \cdots - 36503 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(1 + \beta_{11}\) | \(-\beta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8.1 |
|
−1.29724 | + | 0.347596i | 0 | −0.170031 | + | 0.0981673i | −0.561484 | − | 2.16443i | 0 | −0.530190 | − | 1.97869i | 2.08575 | − | 2.08575i | 0 | 1.48073 | + | 2.61262i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 8.2 | −0.186243 | + | 0.0499037i | 0 | −1.69985 | + | 0.981412i | 2.04963 | + | 0.893868i | 0 | 0.632007 | + | 2.35868i | 0.540289 | − | 0.540289i | 0 | −0.426337 | − | 0.0641924i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 8.3 | 1.60599 | − | 0.430324i | 0 | 0.661975 | − | 0.382191i | 1.24906 | − | 1.85468i | 0 | 0.465559 | + | 1.73749i | −1.45267 | + | 1.45267i | 0 | 1.20786 | − | 3.51610i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 8.4 | 2.24352 | − | 0.601150i | 0 | 2.93996 | − | 1.69739i | −2.10323 | + | 0.759216i | 0 | −0.201351 | − | 0.751454i | 2.29074 | − | 2.29074i | 0 | −4.26225 | + | 2.96768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.1 | −1.29724 | − | 0.347596i | 0 | −0.170031 | − | 0.0981673i | −0.561484 | + | 2.16443i | 0 | −0.530190 | + | 1.97869i | 2.08575 | + | 2.08575i | 0 | 1.48073 | − | 2.61262i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | −0.186243 | − | 0.0499037i | 0 | −1.69985 | − | 0.981412i | 2.04963 | − | 0.893868i | 0 | 0.632007 | − | 2.35868i | 0.540289 | + | 0.540289i | 0 | −0.426337 | + | 0.0641924i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 1.60599 | + | 0.430324i | 0 | 0.661975 | + | 0.382191i | 1.24906 | + | 1.85468i | 0 | 0.465559 | − | 1.73749i | −1.45267 | − | 1.45267i | 0 | 1.20786 | + | 3.51610i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 2.24352 | + | 0.601150i | 0 | 2.93996 | + | 1.69739i | −2.10323 | − | 0.759216i | 0 | −0.201351 | + | 0.751454i | 2.29074 | + | 2.29074i | 0 | −4.26225 | − | 2.96768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.1 | −0.347596 | − | 1.29724i | 0 | 0.170031 | − | 0.0981673i | 1.59371 | + | 1.56847i | 0 | 1.97869 | − | 0.530190i | −2.08575 | − | 2.08575i | 0 | 1.48073 | − | 2.61262i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.2 | −0.0499037 | − | 0.186243i | 0 | 1.69985 | − | 0.981412i | 0.250705 | − | 2.22197i | 0 | −2.35868 | + | 0.632007i | −0.540289 | − | 0.540289i | 0 | −0.426337 | + | 0.0641924i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.3 | 0.430324 | + | 1.60599i | 0 | −0.661975 | + | 0.382191i | 2.23073 | − | 0.154373i | 0 | −1.73749 | + | 0.465559i | 1.45267 | + | 1.45267i | 0 | 1.20786 | + | 3.51610i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 62.4 | 0.601150 | + | 2.24352i | 0 | −2.93996 | + | 1.69739i | −1.70912 | + | 1.44185i | 0 | 0.751454 | − | 0.201351i | −2.29074 | − | 2.29074i | 0 | −4.26225 | − | 2.96768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 98.1 | −0.347596 | + | 1.29724i | 0 | 0.170031 | + | 0.0981673i | 1.59371 | − | 1.56847i | 0 | 1.97869 | + | 0.530190i | −2.08575 | + | 2.08575i | 0 | 1.48073 | + | 2.61262i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 98.2 | −0.0499037 | + | 0.186243i | 0 | 1.69985 | + | 0.981412i | 0.250705 | + | 2.22197i | 0 | −2.35868 | − | 0.632007i | −0.540289 | + | 0.540289i | 0 | −0.426337 | − | 0.0641924i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 98.3 | 0.430324 | − | 1.60599i | 0 | −0.661975 | − | 0.382191i | 2.23073 | + | 0.154373i | 0 | −1.73749 | − | 0.465559i | 1.45267 | − | 1.45267i | 0 | 1.20786 | − | 3.51610i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 98.4 | 0.601150 | − | 2.24352i | 0 | −2.93996 | − | 1.69739i | −1.70912 | − | 1.44185i | 0 | 0.751454 | + | 0.201351i | −2.29074 | + | 2.29074i | 0 | −4.26225 | + | 2.96768i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
| 45.l | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 135.2.m.a | 16 | |
| 3.b | odd | 2 | 1 | 45.2.l.a | ✓ | 16 | |
| 5.b | even | 2 | 1 | 675.2.q.a | 16 | ||
| 5.c | odd | 4 | 1 | inner | 135.2.m.a | 16 | |
| 5.c | odd | 4 | 1 | 675.2.q.a | 16 | ||
| 9.c | even | 3 | 1 | 45.2.l.a | ✓ | 16 | |
| 9.c | even | 3 | 1 | 405.2.f.a | 16 | ||
| 9.d | odd | 6 | 1 | inner | 135.2.m.a | 16 | |
| 9.d | odd | 6 | 1 | 405.2.f.a | 16 | ||
| 12.b | even | 2 | 1 | 720.2.cu.c | 16 | ||
| 15.d | odd | 2 | 1 | 225.2.p.b | 16 | ||
| 15.e | even | 4 | 1 | 45.2.l.a | ✓ | 16 | |
| 15.e | even | 4 | 1 | 225.2.p.b | 16 | ||
| 36.f | odd | 6 | 1 | 720.2.cu.c | 16 | ||
| 45.h | odd | 6 | 1 | 675.2.q.a | 16 | ||
| 45.j | even | 6 | 1 | 225.2.p.b | 16 | ||
| 45.k | odd | 12 | 1 | 45.2.l.a | ✓ | 16 | |
| 45.k | odd | 12 | 1 | 225.2.p.b | 16 | ||
| 45.k | odd | 12 | 1 | 405.2.f.a | 16 | ||
| 45.l | even | 12 | 1 | inner | 135.2.m.a | 16 | |
| 45.l | even | 12 | 1 | 405.2.f.a | 16 | ||
| 45.l | even | 12 | 1 | 675.2.q.a | 16 | ||
| 60.l | odd | 4 | 1 | 720.2.cu.c | 16 | ||
| 180.x | even | 12 | 1 | 720.2.cu.c | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.2.l.a | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 45.2.l.a | ✓ | 16 | 9.c | even | 3 | 1 | |
| 45.2.l.a | ✓ | 16 | 15.e | even | 4 | 1 | |
| 45.2.l.a | ✓ | 16 | 45.k | odd | 12 | 1 | |
| 135.2.m.a | 16 | 1.a | even | 1 | 1 | trivial | |
| 135.2.m.a | 16 | 5.c | odd | 4 | 1 | inner | |
| 135.2.m.a | 16 | 9.d | odd | 6 | 1 | inner | |
| 135.2.m.a | 16 | 45.l | even | 12 | 1 | inner | |
| 225.2.p.b | 16 | 15.d | odd | 2 | 1 | ||
| 225.2.p.b | 16 | 15.e | even | 4 | 1 | ||
| 225.2.p.b | 16 | 45.j | even | 6 | 1 | ||
| 225.2.p.b | 16 | 45.k | odd | 12 | 1 | ||
| 405.2.f.a | 16 | 9.c | even | 3 | 1 | ||
| 405.2.f.a | 16 | 9.d | odd | 6 | 1 | ||
| 405.2.f.a | 16 | 45.k | odd | 12 | 1 | ||
| 405.2.f.a | 16 | 45.l | even | 12 | 1 | ||
| 675.2.q.a | 16 | 5.b | even | 2 | 1 | ||
| 675.2.q.a | 16 | 5.c | odd | 4 | 1 | ||
| 675.2.q.a | 16 | 45.h | odd | 6 | 1 | ||
| 675.2.q.a | 16 | 45.l | even | 12 | 1 | ||
| 720.2.cu.c | 16 | 12.b | even | 2 | 1 | ||
| 720.2.cu.c | 16 | 36.f | odd | 6 | 1 | ||
| 720.2.cu.c | 16 | 60.l | odd | 4 | 1 | ||
| 720.2.cu.c | 16 | 180.x | even | 12 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 6 T^{15} + \cdots + 1 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( T^{16} - 6 T^{15} + \cdots + 390625 \)
|
| $7$ |
\( T^{16} + 2 T^{15} + \cdots + 2401 \)
|
| $11$ |
\( (T^{8} - 10 T^{6} + 102 T^{4} + \cdots + 4)^{2} \)
|
| $13$ |
\( T^{16} + 2 T^{15} + \cdots + 4477456 \)
|
| $17$ |
\( T^{16} + 964 T^{12} + \cdots + 16 \)
|
| $19$ |
\( (T^{8} + 60 T^{6} + \cdots + 324)^{2} \)
|
| $23$ |
\( T^{16} + 18 T^{15} + \cdots + 62742241 \)
|
| $29$ |
\( T^{16} + \cdots + 981506241 \)
|
| $31$ |
\( (T^{8} + 2 T^{7} + \cdots + 676)^{2} \)
|
| $37$ |
\( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 4)^{2} \)
|
| $41$ |
\( (T^{8} - 12 T^{7} + \cdots + 32761)^{2} \)
|
| $43$ |
\( T^{16} + 2 T^{15} + \cdots + 11316496 \)
|
| $47$ |
\( T^{16} + \cdots + 33243864241 \)
|
| $53$ |
\( T^{16} + \cdots + 409600000000 \)
|
| $59$ |
\( T^{16} + \cdots + 592240896 \)
|
| $61$ |
\( (T^{8} - 4 T^{7} + \cdots + 11449)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 539415333601 \)
|
| $71$ |
\( (T^{8} + 116 T^{6} + \cdots + 128164)^{2} \)
|
| $73$ |
\( (T^{8} + 4 T^{7} + \cdots + 270400)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 17\!\cdots\!96 \)
|
| $83$ |
\( T^{16} + \cdots + 13841287201 \)
|
| $89$ |
\( (T^{8} - 300 T^{6} + \cdots + 3969)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 6146560000 \)
|
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