Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1502,2,Mod(1,1502)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1502, 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("1502.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1502 = 2 \cdot 751 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1502.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: | \(11.9935303836\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - 20x^{9} - 7x^{8} + 134x^{7} + 70x^{6} - 354x^{5} - 193x^{4} + 341x^{3} + 163x^{2} - 72x - 16 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 20x^{9} - 7x^{8} + 134x^{7} + 70x^{6} - 354x^{5} - 193x^{4} + 341x^{3} + 163x^{2} - 72x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3377 \nu^{10} + 5788 \nu^{9} + 56308 \nu^{8} - 66273 \nu^{7} - 324370 \nu^{6} + 232522 \nu^{5} + \cdots + 65684 ) / 26056 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 7719 \nu^{10} + 19580 \nu^{9} + 106732 \nu^{8} - 224575 \nu^{7} - 488494 \nu^{6} + 800590 \nu^{5} + \cdots - 11908 ) / 52112 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 10331 \nu^{10} + 22668 \nu^{9} + 154204 \nu^{8} - 260419 \nu^{7} - 773302 \nu^{6} + 917910 \nu^{5} + \cdots + 128476 ) / 52112 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3309 \nu^{10} + 5548 \nu^{9} + 53496 \nu^{8} - 58357 \nu^{7} - 297982 \nu^{6} + 174802 \nu^{5} + \cdots + 59296 ) / 13028 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 944 \nu^{10} - 2757 \nu^{9} - 12598 \nu^{8} + 33449 \nu^{7} + 55763 \nu^{6} - 130539 \nu^{5} + \cdots - 1891 ) / 3257 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16813 \nu^{10} - 33284 \nu^{9} - 255572 \nu^{8} + 361301 \nu^{7} + 1316490 \nu^{6} - 1152074 \nu^{5} + \cdots - 286404 ) / 52112 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 8321 \nu^{10} - 20172 \nu^{9} - 119748 \nu^{8} + 234113 \nu^{7} + 582246 \nu^{6} - 838742 \nu^{5} + \cdots - 54224 ) / 13028 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 17407 \nu^{10} + 43044 \nu^{9} + 248332 \nu^{8} - 498655 \nu^{7} - 1204054 \nu^{6} + \cdots + 137972 ) / 26056 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 34863 \nu^{10} + 75532 \nu^{9} + 516508 \nu^{8} - 846295 \nu^{7} - 2578286 \nu^{6} + \cdots + 194524 ) / 52112 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + \beta_{7} - \beta_{4} - \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} - \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{10} + \beta_{9} + \beta_{8} + 6\beta_{7} - 3\beta_{5} - 10\beta_{4} - 11\beta_{3} + 2\beta_{2} + 14\beta _1 + 29 \)
|
| \(\nu^{5}\) | \(=\) |
\( 18 \beta_{10} + 5 \beta_{9} + 6 \beta_{8} + 3 \beta_{7} + \beta_{6} - 17 \beta_{5} - 25 \beta_{4} + \cdots + 45 \)
|
| \(\nu^{6}\) | \(=\) |
\( 91 \beta_{10} + 23 \beta_{9} + 23 \beta_{8} + 43 \beta_{7} + 7 \beta_{6} - 58 \beta_{5} - 99 \beta_{4} + \cdots + 251 \)
|
| \(\nu^{7}\) | \(=\) |
\( 241 \beta_{10} + 98 \beta_{9} + 104 \beta_{8} + 55 \beta_{7} + 34 \beta_{6} - 236 \beta_{5} - 286 \beta_{4} + \cdots + 563 \)
|
| \(\nu^{8}\) | \(=\) |
\( 977 \beta_{10} + 358 \beta_{9} + 350 \beta_{8} + 366 \beta_{7} + 143 \beta_{6} - 812 \beta_{5} + \cdots + 2431 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2939 \beta_{10} + 1380 \beta_{9} + 1379 \beta_{8} + 750 \beta_{7} + 565 \beta_{6} - 2977 \beta_{5} + \cdots + 6654 \)
|
| \(\nu^{10}\) | \(=\) |
\( 10815 \beta_{10} + 4764 \beta_{9} + 4580 \beta_{8} + 3524 \beta_{7} + 2082 \beta_{6} - 10189 \beta_{5} + \cdots + 25322 \)
|
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 |
|
1.00000 | −3.37027 | 1.00000 | −4.42460 | −3.37027 | 1.01197 | 1.00000 | 8.35874 | −4.42460 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −3.36248 | 1.00000 | 2.63222 | −3.36248 | −0.528777 | 1.00000 | 8.30627 | 2.63222 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −3.14010 | 1.00000 | −0.286978 | −3.14010 | −4.72107 | 1.00000 | 6.86022 | −0.286978 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −2.29048 | 1.00000 | −2.28669 | −2.29048 | 2.59902 | 1.00000 | 2.24628 | −2.28669 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.79935 | 1.00000 | −0.966216 | −1.79935 | −0.438762 | 1.00000 | 0.237673 | −0.966216 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.17875 | 1.00000 | 3.48258 | −1.17875 | −3.41427 | 1.00000 | −1.61054 | 3.48258 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | −0.565115 | 1.00000 | −0.615969 | −0.565115 | 3.05802 | 1.00000 | −2.68064 | −0.615969 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 0.213882 | 1.00000 | −2.70660 | 0.213882 | 1.35942 | 1.00000 | −2.95425 | −2.70660 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.00000 | 0.642822 | 1.00000 | −0.305614 | 0.642822 | −3.12232 | 1.00000 | −2.58678 | −0.305614 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.00000 | 1.47084 | 1.00000 | −3.42444 | 1.47084 | −2.00214 | 1.00000 | −0.836618 | −3.42444 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.00000 | 2.37900 | 1.00000 | −3.09771 | 2.37900 | −2.80109 | 1.00000 | 2.65965 | −3.09771 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(751\) | \(-1\) |
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 | 1502.2.a.f | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1502.2.a.f | ✓ | 11 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{11} + 11 T_{3}^{10} + 35 T_{3}^{9} - 22 T_{3}^{8} - 312 T_{3}^{7} - 406 T_{3}^{6} + 430 T_{3}^{5} + \cdots + 47 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1502))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{11} \)
|
| $3$ |
\( T^{11} + 11 T^{10} + \cdots + 47 \)
|
| $5$ |
\( T^{11} + 12 T^{10} + \cdots + 139 \)
|
| $7$ |
\( T^{11} + 9 T^{10} + \cdots + 716 \)
|
| $11$ |
\( T^{11} + 7 T^{10} + \cdots + 58033 \)
|
| $13$ |
\( T^{11} + 21 T^{10} + \cdots - 1522733 \)
|
| $17$ |
\( T^{11} + 16 T^{10} + \cdots - 1907 \)
|
| $19$ |
\( T^{11} + 22 T^{10} + \cdots + 282847 \)
|
| $23$ |
\( T^{11} + 2 T^{10} + \cdots - 2340 \)
|
| $29$ |
\( T^{11} - 4 T^{10} + \cdots + 451739 \)
|
| $31$ |
\( T^{11} + 28 T^{10} + \cdots - 8892109 \)
|
| $37$ |
\( T^{11} + 22 T^{10} + \cdots + 11443076 \)
|
| $41$ |
\( T^{11} + 5 T^{10} + \cdots - 58363456 \)
|
| $43$ |
\( T^{11} + 7 T^{10} + \cdots + 80426041 \)
|
| $47$ |
\( T^{11} + 31 T^{10} + \cdots + 51477127 \)
|
| $53$ |
\( T^{11} + 17 T^{10} + \cdots - 36646727 \)
|
| $59$ |
\( T^{11} + 18 T^{10} + \cdots + 969156 \)
|
| $61$ |
\( T^{11} + \cdots + 529426183 \)
|
| $67$ |
\( T^{11} + \cdots - 954410769 \)
|
| $71$ |
\( T^{11} + \cdots + 185244353 \)
|
| $73$ |
\( T^{11} + 33 T^{10} + \cdots + 1345940 \)
|
| $79$ |
\( T^{11} - 9 T^{10} + \cdots - 70579841 \)
|
| $83$ |
\( T^{11} + \cdots - 185607811 \)
|
| $89$ |
\( T^{11} - 274 T^{9} + \cdots + 86060 \)
|
| $97$ |
\( T^{11} + \cdots + 140486732 \)
|
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