Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [495,6,Mod(1,495)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(495, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("495.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 495.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: | \(79.3899908074\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 3 x^{9} - 246 x^{8} + 640 x^{7} + 20433 x^{6} - 44595 x^{5} - 667026 x^{4} + 1173648 x^{3} + \cdots - 30445728 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{6} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 3 x^{9} - 246 x^{8} + 640 x^{7} + 20433 x^{6} - 44595 x^{5} - 667026 x^{4} + 1173648 x^{3} + \cdots - 30445728 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 50 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 81\nu + 42 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 36261 \nu^{9} - 1606053 \nu^{8} + 35087194 \nu^{7} + 397191392 \nu^{6} + \cdots + 5972691017616 ) / 15052677456 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 990403 \nu^{9} - 2004389 \nu^{8} - 226762278 \nu^{7} + 392139288 \nu^{6} + \cdots - 3962509873776 ) / 15052677456 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 609313 \nu^{9} + 5945958 \nu^{8} - 159878853 \nu^{7} - 1390758642 \nu^{6} + \cdots + 15026040504024 ) / 3763169364 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1358011 \nu^{9} - 277646 \nu^{8} - 326211473 \nu^{7} - 91884424 \nu^{6} + 25729327247 \nu^{5} + \cdots + 9252689315160 ) / 7526338728 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1503735 \nu^{9} - 2203557 \nu^{8} - 357687014 \nu^{7} + 389613236 \nu^{6} + \cdots + 3646401695016 ) / 7526338728 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 3429903 \nu^{9} + 9514463 \nu^{8} + 856846848 \nu^{7} - 2101119672 \nu^{6} + \cdots + 18290207994864 ) / 15052677456 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 50 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 81\beta _1 + 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - 3\beta_{5} + \beta_{4} + \beta_{3} + 115\beta_{2} + 21\beta _1 + 4037 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{9} - 11\beta_{8} + 7\beta_{7} + 4\beta_{5} + 5\beta_{4} + 123\beta_{3} + 153\beta_{2} + 7565\beta _1 + 1922 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 26 \beta_{9} + 166 \beta_{8} - 72 \beta_{7} + 17 \beta_{6} - 431 \beta_{5} + 205 \beta_{4} + \cdots + 376878 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 410 \beta_{9} - 1958 \beta_{8} + 1136 \beta_{7} + 17 \beta_{6} + 1407 \beta_{5} + 1047 \beta_{4} + \cdots + 303848 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 5574 \beta_{9} + 21221 \beta_{8} - 16644 \beta_{7} + 4437 \beta_{6} - 47898 \beta_{5} + \cdots + 37211083 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 44547 \beta_{9} - 260943 \beta_{8} + 137525 \beta_{7} + 6141 \beta_{6} + 266999 \beta_{5} + \cdots + 39807038 \)
|
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 |
|
−10.5165 | 0 | 78.5975 | −25.0000 | 0 | −37.1308 | −490.044 | 0 | 262.913 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −9.47270 | 0 | 57.7320 | −25.0000 | 0 | 154.743 | −243.752 | 0 | 236.817 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −6.10779 | 0 | 5.30512 | −25.0000 | 0 | −241.797 | 163.047 | 0 | 152.695 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −4.18500 | 0 | −14.4858 | −25.0000 | 0 | 86.0787 | 194.543 | 0 | 104.625 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.25634 | 0 | −21.3963 | −25.0000 | 0 | −3.69124 | 173.876 | 0 | 81.4085 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.97152 | 0 | −28.1131 | −25.0000 | 0 | 213.116 | −118.514 | 0 | −49.2880 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.29463 | 0 | −21.1454 | −25.0000 | 0 | −180.725 | −175.095 | 0 | −82.3658 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 6.41346 | 0 | 9.13253 | −25.0000 | 0 | 14.3353 | −146.660 | 0 | −160.337 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.54982 | 0 | 41.0994 | −25.0000 | 0 | −123.368 | 77.7982 | 0 | −213.745 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 10.3089 | 0 | 74.2740 | −25.0000 | 0 | 234.439 | 435.800 | 0 | −257.723 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 495.6.a.o | ✓ | 10 |
| 3.b | odd | 2 | 1 | 495.6.a.p | yes | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 495.6.a.o | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 495.6.a.p | yes | 10 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 3 T_{2}^{9} - 246 T_{2}^{8} - 640 T_{2}^{7} + 20433 T_{2}^{6} + 44595 T_{2}^{5} + \cdots - 30445728 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 3 T^{9} + \cdots - 30445728 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( (T + 25)^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 70\!\cdots\!96 \)
|
| $11$ |
\( (T - 121)^{10} \)
|
| $13$ |
\( T^{10} + \cdots - 25\!\cdots\!48 \)
|
| $17$ |
\( T^{10} + \cdots - 78\!\cdots\!96 \)
|
| $19$ |
\( T^{10} + \cdots + 86\!\cdots\!48 \)
|
| $23$ |
\( T^{10} + \cdots - 13\!\cdots\!28 \)
|
| $29$ |
\( T^{10} + \cdots - 87\!\cdots\!24 \)
|
| $31$ |
\( T^{10} + \cdots + 71\!\cdots\!24 \)
|
| $37$ |
\( T^{10} + \cdots - 70\!\cdots\!36 \)
|
| $41$ |
\( T^{10} + \cdots + 50\!\cdots\!00 \)
|
| $43$ |
\( T^{10} + \cdots - 21\!\cdots\!44 \)
|
| $47$ |
\( T^{10} + \cdots - 28\!\cdots\!24 \)
|
| $53$ |
\( T^{10} + \cdots - 16\!\cdots\!00 \)
|
| $59$ |
\( T^{10} + \cdots - 14\!\cdots\!04 \)
|
| $61$ |
\( T^{10} + \cdots - 93\!\cdots\!84 \)
|
| $67$ |
\( T^{10} + \cdots + 18\!\cdots\!92 \)
|
| $71$ |
\( T^{10} + \cdots + 38\!\cdots\!00 \)
|
| $73$ |
\( T^{10} + \cdots + 59\!\cdots\!48 \)
|
| $79$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots - 27\!\cdots\!00 \)
|
| $89$ |
\( T^{10} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots - 42\!\cdots\!76 \)
|
| show more | |
| show less |