Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [401,2,Mod(1,401)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(401, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("401.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 401 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 401.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: | \(3.20200112105\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 10 x^{10} + 34 x^{9} + 29 x^{8} - 129 x^{7} - 24 x^{6} + 203 x^{5} + x^{4} + \cdots + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 10 x^{10} + 34 x^{9} + 29 x^{8} - 129 x^{7} - 24 x^{6} + 203 x^{5} + x^{4} + \cdots + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{11} - 3 \nu^{10} - 8 \nu^{9} + 32 \nu^{8} + 7 \nu^{7} - 111 \nu^{6} + 52 \nu^{5} + 157 \nu^{4} + \cdots + 14 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{11} + 3 \nu^{10} + 8 \nu^{9} - 32 \nu^{8} - 7 \nu^{7} + 111 \nu^{6} - 52 \nu^{5} - 157 \nu^{4} + \cdots - 18 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( - \nu^{11} + \nu^{10} + 13 \nu^{9} - 10 \nu^{8} - 60 \nu^{7} + 30 \nu^{6} + 123 \nu^{5} - 26 \nu^{4} + \cdots + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{11} - 14\nu^{9} - \nu^{8} + 69\nu^{7} + 8\nu^{6} - 146\nu^{5} - 23\nu^{4} + 123\nu^{3} + 27\nu^{2} - 25\nu - 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 3 \nu^{11} + 7 \nu^{10} + 30 \nu^{9} - 74 \nu^{8} - 85 \nu^{7} + 249 \nu^{6} + 56 \nu^{5} - 317 \nu^{4} + \cdots - 16 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 3 \nu^{11} + \nu^{10} + 40 \nu^{9} - 6 \nu^{8} - 189 \nu^{7} - 5 \nu^{6} + 396 \nu^{5} + 73 \nu^{4} + \cdots + 24 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{11} - 5 \nu^{10} - 6 \nu^{9} + 58 \nu^{8} - 15 \nu^{7} - 229 \nu^{6} + 132 \nu^{5} + 385 \nu^{4} + \cdots + 38 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( 2 \nu^{11} - 4 \nu^{10} - 21 \nu^{9} + 43 \nu^{8} + 66 \nu^{7} - 151 \nu^{6} - 63 \nu^{5} + 211 \nu^{4} + \cdots + 13 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 5 \nu^{11} + 5 \nu^{10} + 64 \nu^{9} - 48 \nu^{8} - 291 \nu^{7} + 129 \nu^{6} + 596 \nu^{5} + \cdots + 36 ) / 2 \)
|
| \(\beta_{11}\) | \(=\) |
\( 4 \nu^{11} - 8 \nu^{10} - 44 \nu^{9} + 86 \nu^{8} + 155 \nu^{7} - 298 \nu^{6} - 208 \nu^{5} + 394 \nu^{4} + \cdots + 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{8} + 2\beta_{6} - \beta_{5} - \beta_{4} + 5\beta_{3} + 6\beta_{2} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( 10 \beta_{11} + 9 \beta_{10} - 2 \beta_{9} - 9 \beta_{8} - 9 \beta_{7} + 10 \beta_{6} - 8 \beta_{5} + \cdots - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{11} + 10 \beta_{10} + 9 \beta_{9} - 20 \beta_{8} - \beta_{7} + 20 \beta_{6} - 10 \beta_{5} + \cdots + 39 \)
|
| \(\nu^{7}\) | \(=\) |
\( 74 \beta_{11} + 63 \beta_{10} - 18 \beta_{9} - 65 \beta_{8} - 63 \beta_{7} + 77 \beta_{6} - 55 \beta_{5} + \cdots - 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( 91 \beta_{11} + 78 \beta_{10} + 61 \beta_{9} - 152 \beta_{8} - 16 \beta_{7} + 156 \beta_{6} - 78 \beta_{5} + \cdots + 210 \)
|
| \(\nu^{9}\) | \(=\) |
\( 498 \beta_{11} + 411 \beta_{10} - 120 \beta_{9} - 440 \beta_{8} - 407 \beta_{7} + 537 \beta_{6} + \cdots - 17 \)
|
| \(\nu^{10}\) | \(=\) |
\( 678 \beta_{11} + 562 \beta_{10} + 374 \beta_{9} - 1056 \beta_{8} - 169 \beta_{7} + 1112 \beta_{6} + \cdots + 1192 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3229 \beta_{11} + 2619 \beta_{10} - 718 \beta_{9} - 2904 \beta_{8} - 2550 \beta_{7} + 3584 \beta_{6} + \cdots + 65 \)
|
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 |
|
−2.54250 | 0.462633 | 4.46431 | 2.03931 | −1.17625 | −1.72844 | −6.26551 | −2.78597 | −5.18494 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.85869 | −2.50269 | 1.45472 | 0.985636 | 4.65172 | −2.11223 | 1.01350 | 3.26346 | −1.83199 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.82098 | 2.71334 | 1.31596 | −3.63372 | −4.94094 | −2.90457 | 1.24563 | 4.36224 | 6.61692 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.53244 | 1.08207 | 0.348380 | 1.17495 | −1.65820 | −3.29109 | 2.53101 | −1.82913 | −1.80054 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.44790 | −2.35797 | 0.0964152 | −1.54450 | 3.41411 | 2.00193 | 2.75620 | 2.56004 | 2.23628 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.578957 | −0.601622 | −1.66481 | −0.0489583 | 0.348313 | 2.79065 | 2.12177 | −2.63805 | 0.0283447 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.257740 | 2.08548 | −1.93357 | −2.90849 | 0.537511 | −2.64864 | −1.01384 | 1.34921 | −0.749634 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.305046 | −0.792698 | −1.90695 | 3.43210 | −0.241809 | −4.26743 | −1.19180 | −2.37163 | 1.04695 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.960737 | 0.489694 | −1.07698 | −2.30095 | 0.470467 | −1.58751 | −2.95617 | −2.76020 | −2.21061 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.17976 | −0.735736 | −0.608160 | −1.18930 | −0.867993 | −0.108315 | −3.07701 | −2.45869 | −1.40309 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.72691 | −2.96155 | 0.982220 | 1.08846 | −5.11433 | −1.51444 | −1.75761 | 5.77079 | 1.87967 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.35127 | −1.88094 | 3.52847 | −4.09454 | −4.42260 | −4.62990 | 3.59384 | 0.537944 | −9.62736 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(401\) | \(1\) |
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 | 401.2.a.a | ✓ | 12 |
| 3.b | odd | 2 | 1 | 3609.2.a.b | 12 | ||
| 4.b | odd | 2 | 1 | 6416.2.a.k | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 401.2.a.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 3609.2.a.b | 12 | 3.b | odd | 2 | 1 | ||
| 6416.2.a.k | 12 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 3 T_{2}^{11} - 10 T_{2}^{10} - 34 T_{2}^{9} + 29 T_{2}^{8} + 129 T_{2}^{7} - 24 T_{2}^{6} + \cdots + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(401))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 3 T^{11} + \cdots + 4 \)
|
| $3$ |
\( T^{12} + 5 T^{11} + \cdots - 16 \)
|
| $5$ |
\( T^{12} + 7 T^{11} + \cdots - 79 \)
|
| $7$ |
\( T^{12} + 20 T^{11} + \cdots + 2657 \)
|
| $11$ |
\( T^{12} + 11 T^{11} + \cdots + 41849 \)
|
| $13$ |
\( T^{12} + 11 T^{11} + \cdots - 272 \)
|
| $17$ |
\( T^{12} - T^{11} + \cdots - 4208 \)
|
| $19$ |
\( T^{12} + 34 T^{11} + \cdots - 201104 \)
|
| $23$ |
\( T^{12} + 7 T^{11} + \cdots + 23888 \)
|
| $29$ |
\( T^{12} + \cdots - 522688447 \)
|
| $31$ |
\( T^{12} + 52 T^{11} + \cdots + 69618448 \)
|
| $37$ |
\( T^{12} - 3 T^{11} + \cdots - 74838512 \)
|
| $41$ |
\( T^{12} + 16 T^{11} + \cdots - 24965557 \)
|
| $43$ |
\( T^{12} + 2 T^{11} + \cdots - 32571323 \)
|
| $47$ |
\( T^{12} + \cdots - 120470771 \)
|
| $53$ |
\( T^{12} + \cdots + 3460081936 \)
|
| $59$ |
\( T^{12} + \cdots - 49673523056 \)
|
| $61$ |
\( T^{12} + 24 T^{11} + \cdots - 45635056 \)
|
| $67$ |
\( T^{12} + \cdots - 429818624 \)
|
| $71$ |
\( T^{12} + \cdots + 2314277776 \)
|
| $73$ |
\( T^{12} + 20 T^{11} + \cdots - 10554352 \)
|
| $79$ |
\( T^{12} + 53 T^{11} + \cdots - 19238224 \)
|
| $83$ |
\( T^{12} + \cdots - 3709520128 \)
|
| $89$ |
\( T^{12} + \cdots + 291613788629 \)
|
| $97$ |
\( T^{12} + \cdots - 9379281712 \)
|
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