Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1305,4,Mod(1,1305)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1305.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1305 = 3^{2} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1305.a (trivial) |
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: | yes |
| Analytic conductor: | \(76.9974925575\) |
| 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: |
\( x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 435) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - x^{5} - 49x^{4} + 27x^{3} + 692x^{2} - 82x - 2588 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 21\nu + 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 35\nu^{3} + 62\nu^{2} + 242\nu - 372 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 33\nu^{2} - 4\nu + 192 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 22\beta _1 + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 33\beta_{2} + 37\beta _1 + 336 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 8\beta_{4} + 35\beta_{3} + 39\beta_{2} + 540\beta _1 + 297 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.42717 | 0 | 21.4542 | 5.00000 | 0 | −10.2400 | −73.0180 | 0 | −27.1358 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −4.39070 | 0 | 11.2783 | 5.00000 | 0 | 31.5336 | −14.3939 | 0 | −21.9535 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.35472 | 0 | −2.45528 | 5.00000 | 0 | −3.94392 | 24.6193 | 0 | −11.7736 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.57352 | 0 | −1.37700 | 5.00000 | 0 | 31.5281 | −24.1319 | 0 | 12.8676 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.54861 | 0 | 4.59262 | 5.00000 | 0 | −21.2667 | −12.0915 | 0 | 17.7430 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.05047 | 0 | 17.5072 | 5.00000 | 0 | 19.3889 | 48.0160 | 0 | 25.2523 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(29\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1305.4.a.h | 6 | |
| 3.b | odd | 2 | 1 | 435.4.a.h | ✓ | 6 | |
| 15.d | odd | 2 | 1 | 2175.4.a.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 435.4.a.h | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 1305.4.a.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 2175.4.a.k | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} - 49T_{2}^{4} - 27T_{2}^{3} + 692T_{2}^{2} + 82T_{2} - 2588 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1305))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots - 2588 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} - 47 T^{5} + \cdots - 16555808 \)
|
| $11$ |
\( T^{6} + 81 T^{5} + \cdots + 51862040 \)
|
| $13$ |
\( T^{6} - 169 T^{5} + \cdots + 158715904 \)
|
| $17$ |
\( T^{6} - T^{5} + \cdots - 178244224 \)
|
| $19$ |
\( T^{6} + \cdots + 105108863360 \)
|
| $23$ |
\( T^{6} + \cdots - 670601037824 \)
|
| $29$ |
\( (T + 29)^{6} \)
|
| $31$ |
\( T^{6} + \cdots + 1225416704 \)
|
| $37$ |
\( T^{6} + \cdots - 128139819712 \)
|
| $41$ |
\( T^{6} + \cdots - 4772989110016 \)
|
| $43$ |
\( T^{6} + \cdots + 142035778816 \)
|
| $47$ |
\( T^{6} + \cdots - 11309241395200 \)
|
| $53$ |
\( T^{6} + \cdots - 89482466320 \)
|
| $59$ |
\( T^{6} + \cdots + 452631511470080 \)
|
| $61$ |
\( T^{6} + \cdots - 205098218864640 \)
|
| $67$ |
\( T^{6} + \cdots + 22\!\cdots\!92 \)
|
| $71$ |
\( T^{6} + \cdots + 756565599059968 \)
|
| $73$ |
\( T^{6} + \cdots + 43\!\cdots\!56 \)
|
| $79$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 11\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots + 226457216857600 \)
|
| $97$ |
\( T^{6} + \cdots + 72\!\cdots\!32 \)
|
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