Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [23,10,Mod(1,23)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(23, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("23.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 23.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: | \(11.8458242318\) |
| 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: |
\( x^{10} - 2 x^{9} - 4043 x^{8} + 23717 x^{7} + 5199055 x^{6} - 39386579 x^{5} - 2188157037 x^{4} + \cdots - 6733347025500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2} \) |
| 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
\( x^{10} - 2 x^{9} - 4043 x^{8} + 23717 x^{7} + 5199055 x^{6} - 39386579 x^{5} - 2188157037 x^{4} + \cdots - 6733347025500 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 84\!\cdots\!99 \nu^{9} + \cdots - 16\!\cdots\!00 ) / 17\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 84\!\cdots\!99 \nu^{9} + \cdots + 31\!\cdots\!00 ) / 17\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 93\!\cdots\!19 \nu^{9} + \cdots - 26\!\cdots\!80 ) / 28\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 53\!\cdots\!73 \nu^{9} + \cdots + 66\!\cdots\!00 ) / 14\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 82\!\cdots\!25 \nu^{9} + \cdots + 33\!\cdots\!60 ) / 17\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 85\!\cdots\!61 \nu^{9} + \cdots + 64\!\cdots\!00 ) / 17\!\cdots\!80 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 46\!\cdots\!15 \nu^{9} + \cdots - 51\!\cdots\!80 ) / 53\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 88\!\cdots\!77 \nu^{9} + \cdots + 76\!\cdots\!20 ) / 85\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - 6\beta _1 + 810 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - 3 \beta_{8} + \beta_{7} + 4 \beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + \cdots - 4983 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 29 \beta_{9} - 25 \beta_{8} - 47 \beta_{7} + 29 \beta_{6} + 95 \beta_{5} - 99 \beta_{4} + \cdots + 1181110 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1531 \beta_{9} - 5355 \beta_{8} + 1534 \beta_{7} + 5719 \beta_{6} - 9769 \beta_{5} + 10900 \beta_{4} + \cdots - 17627431 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 103208 \beta_{9} - 78304 \beta_{8} - 104666 \beta_{7} + 61603 \beta_{6} + 246926 \beta_{5} + \cdots + 1961498525 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3391076 \beta_{9} - 7920866 \beta_{8} + 1701131 \beta_{7} + 7301044 \beta_{6} - 17792060 \beta_{5} + \cdots - 45620121291 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 266176680 \beta_{9} - 183788614 \beta_{8} - 172745997 \beta_{7} + 129797040 \beta_{6} + \cdots + 3487962158423 \)
|
| \(\nu^{9}\) | \(=\) |
\( 8632716785 \beta_{9} - 10114241089 \beta_{8} + 1392689540 \beta_{7} + 8071181340 \beta_{6} + \cdots - 105294324517858 \)
|
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 |
|
−43.2938 | −58.8465 | 1362.36 | −1789.97 | 2547.69 | −6037.52 | −36815.2 | −16220.1 | 77494.5 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −37.3742 | 119.020 | 884.831 | 2050.18 | −4448.26 | −874.748 | −13934.2 | −5517.33 | −76624.0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −16.3111 | −77.3257 | −245.949 | −458.096 | 1261.27 | −11680.1 | 12363.0 | −13703.7 | 7472.04 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −8.46671 | 96.1325 | −440.315 | −280.538 | −813.926 | 6902.25 | 8062.98 | −10441.6 | 2375.24 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.09725 | −171.465 | −510.796 | −2559.07 | 188.140 | 5875.08 | 1122.26 | 9717.35 | 2807.94 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 11.7924 | 238.231 | −372.940 | 1373.99 | 2809.31 | 4110.33 | −10435.5 | 37071.0 | 16202.6 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 13.9147 | −239.919 | −318.382 | 1176.77 | −3338.39 | −2360.52 | −11554.5 | 37878.0 | 16374.4 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 36.3798 | −32.9195 | 811.492 | 589.007 | −1197.60 | 9311.52 | 10895.5 | −18599.3 | 21428.0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 37.0181 | 129.534 | 858.343 | 2322.43 | 4795.10 | −10588.7 | 12821.0 | −2904.01 | 85972.0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 39.4380 | 232.559 | 1043.36 | −2312.71 | 9171.67 | 6622.46 | 20955.8 | 34400.7 | −91208.6 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(23\) | \(-1\) |
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 | 23.10.a.b | ✓ | 10 |
| 3.b | odd | 2 | 1 | 207.10.a.f | 10 | ||
| 4.b | odd | 2 | 1 | 368.10.a.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.10.a.b | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 207.10.a.f | 10 | 3.b | odd | 2 | 1 | ||
| 368.10.a.i | 10 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 32 T_{2}^{9} - 3584 T_{2}^{8} + 116861 T_{2}^{7} + 3703708 T_{2}^{6} - 122455688 T_{2}^{5} + \cdots - 2136806320128 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(23))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 2136806320128 \)
|
| $3$ |
\( T^{10} + \cdots - 50\!\cdots\!00 \)
|
| $5$ |
\( T^{10} + \cdots - 61\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 15\!\cdots\!48 \)
|
| $11$ |
\( T^{10} + \cdots - 22\!\cdots\!00 \)
|
| $13$ |
\( T^{10} + \cdots + 46\!\cdots\!56 \)
|
| $17$ |
\( T^{10} + \cdots + 78\!\cdots\!60 \)
|
| $19$ |
\( T^{10} + \cdots + 14\!\cdots\!00 \)
|
| $23$ |
\( (T - 279841)^{10} \)
|
| $29$ |
\( T^{10} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots + 31\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots - 22\!\cdots\!84 \)
|
| $41$ |
\( T^{10} + \cdots - 21\!\cdots\!12 \)
|
| $43$ |
\( T^{10} + \cdots - 91\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots + 15\!\cdots\!00 \)
|
| $53$ |
\( T^{10} + \cdots + 90\!\cdots\!24 \)
|
| $59$ |
\( T^{10} + \cdots - 54\!\cdots\!20 \)
|
| $61$ |
\( T^{10} + \cdots - 27\!\cdots\!48 \)
|
| $67$ |
\( T^{10} + \cdots + 61\!\cdots\!00 \)
|
| $71$ |
\( T^{10} + \cdots - 31\!\cdots\!00 \)
|
| $73$ |
\( T^{10} + \cdots + 31\!\cdots\!04 \)
|
| $79$ |
\( T^{10} + \cdots + 41\!\cdots\!60 \)
|
| $83$ |
\( T^{10} + \cdots + 82\!\cdots\!08 \)
|
| $89$ |
\( T^{10} + \cdots - 23\!\cdots\!80 \)
|
| $97$ |
\( T^{10} + \cdots - 25\!\cdots\!56 \)
|
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