Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [36,15,Mod(19,36)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(36, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("36.19");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 36.d (of order \(2\), degree \(1\), 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: | \(44.7584285347\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 88x^{4} - 1824x^{3} + 325632x^{2} + 21572352x + 982333440 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{30}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 4) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 88x^{4} - 1824x^{3} + 325632x^{2} + 21572352x + 982333440 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 3\nu^{4} - 100\nu^{3} + 1424\nu^{2} - 319936\nu - 18919936 ) / 262144 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{5} - 503\nu^{4} + 29484\nu^{3} - 233648\nu^{2} + 1295680\nu - 84863488 ) / 131072 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} - 21\nu^{4} - 700\nu^{3} + 9968\nu^{2} + 266195904\nu - 176479744 ) / 262144 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -19\nu^{5} + 967\nu^{4} + 5268\nu^{3} + 158128\nu^{2} - 7012672\nu - 141571584 ) / 65536 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 47\nu^{5} - 1907\nu^{4} - 9636\nu^{3} + 8059536\nu^{2} - 58592704\nu + 709930496 ) / 131072 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 7\beta _1 + 168 ) / 1024 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 16\beta_{5} + 16\beta_{4} + 9\beta_{3} + 225\beta _1 - 29800 ) / 1024 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 80\beta_{5} + 1104\beta_{4} + 49\beta_{3} + 4096\beta_{2} - 52151\beta _1 + 872600 ) / 1024 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2608\beta_{5} + 55760\beta_{4} - 583\beta_{3} - 28672\beta_{2} - 4650863\beta _1 - 220047784 ) / 1024 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 22608\beta_{5} - 254896\beta_{4} - 310271\beta_{3} - 323584\beta_{2} - 246707815\beta _1 - 18897315560 ) / 1024 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).
| \(n\) | \(19\) | \(29\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−100.593 | − | 79.1526i | 0 | 3853.74 | + | 15924.3i | 27633.0 | 0 | 784634.i | 872793. | − | 1.90690e6i | 0 | −2.77967e6 | − | 2.18722e6i | ||||||||||||||||||||||||||||
| 19.2 | −100.593 | + | 79.1526i | 0 | 3853.74 | − | 15924.3i | 27633.0 | 0 | − | 784634.i | 872793. | + | 1.90690e6i | 0 | −2.77967e6 | + | 2.18722e6i | ||||||||||||||||||||||||||||
| 19.3 | 32.2815 | − | 123.862i | 0 | −14299.8 | − | 7996.93i | −113343. | 0 | − | 697667.i | −1.45214e6 | + | 1.51306e6i | 0 | −3.65890e6 | + | 1.40390e7i | ||||||||||||||||||||||||||||
| 19.4 | 32.2815 | + | 123.862i | 0 | −14299.8 | + | 7996.93i | −113343. | 0 | 697667.i | −1.45214e6 | − | 1.51306e6i | 0 | −3.65890e6 | − | 1.40390e7i | |||||||||||||||||||||||||||||
| 19.5 | 114.311 | − | 57.5931i | 0 | 9750.07 | − | 13167.1i | 81680.5 | 0 | − | 1.09921e6i | 356209. | − | 2.06668e6i | 0 | 9.33699e6 | − | 4.70423e6i | ||||||||||||||||||||||||||||
| 19.6 | 114.311 | + | 57.5931i | 0 | 9750.07 | + | 13167.1i | 81680.5 | 0 | 1.09921e6i | 356209. | + | 2.06668e6i | 0 | 9.33699e6 | + | 4.70423e6i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 36.15.d.c | 6 | |
| 3.b | odd | 2 | 1 | 4.15.b.a | ✓ | 6 | |
| 4.b | odd | 2 | 1 | inner | 36.15.d.c | 6 | |
| 12.b | even | 2 | 1 | 4.15.b.a | ✓ | 6 | |
| 24.f | even | 2 | 1 | 64.15.c.d | 6 | ||
| 24.h | odd | 2 | 1 | 64.15.c.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.15.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 4.15.b.a | ✓ | 6 | 12.b | even | 2 | 1 | |
| 36.15.d.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 36.15.d.c | 6 | 4.b | odd | 2 | 1 | inner | |
| 64.15.c.d | 6 | 24.f | even | 2 | 1 | ||
| 64.15.c.d | 6 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 4030T_{5}^{2} - 10132896340T_{5} + 255824948369000 \)
acting on \(S_{15}^{\mathrm{new}}(36, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots + 4398046511104 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{3} + \cdots + 255824948369000)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $11$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 33\!\cdots\!40 \)
|
| $29$ |
\( (T^{3} + \cdots + 36\!\cdots\!08)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots + 22\!\cdots\!88)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 65\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 72\!\cdots\!40 \)
|
| $53$ |
\( (T^{3} + \cdots + 40\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 85\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 24\!\cdots\!32)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 62\!\cdots\!40 \)
|
| $71$ |
\( T^{6} + \cdots + 50\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 34\!\cdots\!00)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!40 \)
|
| $89$ |
\( (T^{3} + \cdots + 12\!\cdots\!08)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 12\!\cdots\!00)^{2} \)
|
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