Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [335,2,Mod(1,335)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(335, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("335.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 335 = 5 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 335.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: | \(2.67498846771\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 18x^{9} - 2x^{8} + 114x^{7} + 24x^{6} - 306x^{5} - 86x^{4} + 332x^{3} + 109x^{2} - 114x - 46 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 18x^{9} - 2x^{8} + 114x^{7} + 24x^{6} - 306x^{5} - 86x^{4} + 332x^{3} + 109x^{2} - 114x - 46 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 43 \nu^{10} + 84 \nu^{9} - 1344 \nu^{8} - 1488 \nu^{7} + 13496 \nu^{6} + 9411 \nu^{5} - 54847 \nu^{4} + \cdots - 28424 ) / 5261 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 228 \nu^{10} + 44 \nu^{9} + 4557 \nu^{8} - 1531 \nu^{7} - 33265 \nu^{6} + 14700 \nu^{5} + \cdots + 45738 ) / 5261 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 312 \nu^{10} - 614 \nu^{9} - 5959 \nu^{8} + 10125 \nu^{7} + 41644 \nu^{6} - 56389 \nu^{5} + \cdots - 49021 ) / 5261 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 581 \nu^{10} + 1312 \nu^{9} + 10574 \nu^{8} - 21738 \nu^{7} - 69792 \nu^{6} + 122189 \nu^{5} + \cdots + 106445 ) / 5261 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 697 \nu^{10} - 596 \nu^{9} - 11508 \nu^{8} + 8303 \nu^{7} + 65580 \nu^{6} - 37462 \nu^{5} + \cdots - 51354 ) / 5261 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 781 \nu^{10} + 1166 \nu^{9} + 12910 \nu^{8} - 16897 \nu^{7} - 73959 \nu^{6} + 79151 \nu^{5} + \cdots + 65159 ) / 5261 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 926 \nu^{10} + 271 \nu^{9} + 16708 \nu^{8} - 4049 \nu^{7} - 105521 \nu^{6} + 24537 \nu^{5} + \cdots + 79156 ) / 5261 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1624 \nu^{10} - 498 \nu^{9} - 28859 \nu^{8} + 6567 \nu^{7} + 177777 \nu^{6} - 29113 \nu^{5} + \cdots - 112574 ) / 5261 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 7\beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + 2\beta_{9} + 9\beta_{8} + 9\beta_{7} + 9\beta_{5} + 8\beta_{4} + 9\beta_{2} + 30\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{10} - 12 \beta_{8} - 10 \beta_{7} + 12 \beta_{6} + 11 \beta_{5} - 14 \beta_{4} - 10 \beta_{3} + \cdots + 101 \)
|
| \(\nu^{7}\) | \(=\) |
\( 13 \beta_{10} + 26 \beta_{9} + 71 \beta_{8} + 70 \beta_{7} - \beta_{6} + 69 \beta_{5} + 54 \beta_{4} + \cdots + 70 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 14 \beta_{10} + 2 \beta_{9} - 107 \beta_{8} - 76 \beta_{7} + 109 \beta_{6} + 94 \beta_{5} + \cdots + 682 \)
|
| \(\nu^{9}\) | \(=\) |
\( 123 \beta_{10} + 247 \beta_{9} + 541 \beta_{8} + 528 \beta_{7} - 10 \beta_{6} + 510 \beta_{5} + \cdots + 524 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 133 \beta_{10} + 42 \beta_{9} - 860 \beta_{8} - 528 \beta_{7} + 901 \beta_{6} + 746 \beta_{5} + \cdots + 4771 \)
|
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 |
|
−2.77487 | −2.82525 | 5.69989 | 1.00000 | 7.83971 | 3.48753 | −10.2667 | 4.98206 | −2.77487 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.41031 | 1.39579 | 3.80961 | 1.00000 | −3.36429 | 2.87116 | −4.36172 | −1.05177 | −2.41031 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.79815 | −2.00677 | 1.23333 | 1.00000 | 3.60846 | −4.23694 | 1.37858 | 1.02711 | −1.79815 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.15059 | 2.47793 | −0.676135 | 1.00000 | −2.85109 | −3.62172 | 3.07914 | 3.14014 | −1.15059 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.848725 | 2.32514 | −1.27967 | 1.00000 | −1.97341 | 4.41520 | 2.78354 | 2.40629 | −0.848725 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.480320 | −3.31336 | −1.76929 | 1.00000 | −1.59147 | −3.07009 | −1.81047 | 7.97835 | 0.480320 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.635615 | −1.50352 | −1.59599 | 1.00000 | −0.955659 | 4.02117 | −2.28567 | −0.739431 | 0.635615 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.23991 | 3.31819 | −0.462632 | 1.00000 | 4.11424 | −1.46927 | −3.05343 | 8.01037 | 1.23991 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.79850 | 0.651292 | 1.23462 | 1.00000 | 1.17135 | 3.51418 | −1.37654 | −2.57582 | 1.79850 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.14026 | 1.09818 | 2.58073 | 1.00000 | 2.35039 | −0.213589 | 1.24291 | −1.79401 | 2.14026 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.68804 | −1.61762 | 5.22554 | 1.00000 | −4.34822 | −1.69764 | 8.67036 | −0.383302 | 2.68804 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(67\) | \(1\) |
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 | 335.2.a.e | ✓ | 11 |
| 3.b | odd | 2 | 1 | 3015.2.a.q | 11 | ||
| 4.b | odd | 2 | 1 | 5360.2.a.bo | 11 | ||
| 5.b | even | 2 | 1 | 1675.2.a.l | 11 | ||
| 5.c | odd | 4 | 2 | 1675.2.c.j | 22 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 335.2.a.e | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
| 1675.2.a.l | 11 | 5.b | even | 2 | 1 | ||
| 1675.2.c.j | 22 | 5.c | odd | 4 | 2 | ||
| 3015.2.a.q | 11 | 3.b | odd | 2 | 1 | ||
| 5360.2.a.bo | 11 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} - 18 T_{2}^{9} + 2 T_{2}^{8} + 114 T_{2}^{7} - 24 T_{2}^{6} - 306 T_{2}^{5} + 86 T_{2}^{4} + \cdots + 46 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(335))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 18 T^{9} + \cdots + 46 \)
|
| $3$ |
\( T^{11} - 27 T^{9} + \cdots + 872 \)
|
| $5$ |
\( (T - 1)^{11} \)
|
| $7$ |
\( T^{11} - 4 T^{10} + \cdots - 15680 \)
|
| $11$ |
\( T^{11} - 6 T^{10} + \cdots + 542720 \)
|
| $13$ |
\( T^{11} - 4 T^{10} + \cdots - 18144 \)
|
| $17$ |
\( T^{11} - 2 T^{10} + \cdots - 1017344 \)
|
| $19$ |
\( T^{11} - 10 T^{10} + \cdots + 54976 \)
|
| $23$ |
\( T^{11} + 10 T^{10} + \cdots + 1765376 \)
|
| $29$ |
\( T^{11} - 26 T^{10} + \cdots - 3378814 \)
|
| $31$ |
\( T^{11} + 8 T^{10} + \cdots - 229376 \)
|
| $37$ |
\( T^{11} - 12 T^{10} + \cdots + 90113536 \)
|
| $41$ |
\( T^{11} - 14 T^{10} + \cdots + 9554944 \)
|
| $43$ |
\( T^{11} - 14 T^{10} + \cdots - 78711120 \)
|
| $47$ |
\( T^{11} + 16 T^{10} + \cdots + 89513984 \)
|
| $53$ |
\( T^{11} + \cdots + 1069913124 \)
|
| $59$ |
\( T^{11} - 2 T^{10} + \cdots - 3843756 \)
|
| $61$ |
\( T^{11} - 20 T^{10} + \cdots + 90128384 \)
|
| $67$ |
\( (T + 1)^{11} \)
|
| $71$ |
\( T^{11} + 24 T^{10} + \cdots + 1050112 \)
|
| $73$ |
\( T^{11} - 16 T^{10} + \cdots + 953856 \)
|
| $79$ |
\( T^{11} + \cdots + 2169896960 \)
|
| $83$ |
\( T^{11} + 36 T^{10} + \cdots - 2424832 \)
|
| $89$ |
\( T^{11} + \cdots + 64518103126 \)
|
| $97$ |
\( T^{11} - 20 T^{10} + \cdots - 28025252 \)
|
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