Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,22,Mod(1,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.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: | \(209.608008215\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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
\( x^{10} - 5 x^{9} - 15486550 x^{8} - 930225020 x^{7} + 80619202368510 x^{6} + \cdots + 47\!\cdots\!18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 97\nu - 3097264 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 26\!\cdots\!11 \nu^{9} + \cdots + 18\!\cdots\!30 ) / 94\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 26\!\cdots\!11 \nu^{9} + \cdots - 30\!\cdots\!82 ) / 94\!\cdots\!64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 42\!\cdots\!43 \nu^{9} + \cdots + 38\!\cdots\!10 ) / 11\!\cdots\!08 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 13\!\cdots\!05 \nu^{9} + \cdots + 29\!\cdots\!94 ) / 23\!\cdots\!16 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 80\!\cdots\!57 \nu^{9} + \cdots - 45\!\cdots\!06 ) / 59\!\cdots\!04 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 12\!\cdots\!93 \nu^{9} + \cdots - 27\!\cdots\!14 ) / 47\!\cdots\!32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 42\!\cdots\!51 \nu^{9} + \cdots + 89\!\cdots\!34 ) / 94\!\cdots\!64 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 97\beta _1 + 3097264 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} - 322\beta_{2} + 5075668\beta _1 + 299759510 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 128 \beta_{9} - 205 \beta_{8} + 85 \beta_{7} + 190 \beta_{6} + 171 \beta_{5} - 84 \beta_{4} + \cdots + 15720901830703 \)
|
| \(\nu^{5}\) | \(=\) |
\( 285232 \beta_{9} + 247348 \beta_{8} + 310540 \beta_{7} - 203224 \beta_{6} - 130188 \beta_{5} + \cdots + 483057304801666 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1393440912 \beta_{9} - 2236243116 \beta_{8} + 385164396 \beta_{7} + 846881256 \beta_{6} + \cdots + 89\!\cdots\!24 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3466587403696 \beta_{9} + 3256823931146 \beta_{8} + 3738315792070 \beta_{7} - 1249106068220 \beta_{6} + \cdots - 82\!\cdots\!68 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 11\!\cdots\!32 \beta_{9} + \cdots + 54\!\cdots\!39 \)
|
| \(\nu^{9}\) | \(=\) |
\( 31\!\cdots\!52 \beta_{9} + \cdots - 12\!\cdots\!48 \)
|
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 |
|
−2706.03 | 59049.0 | 5.22544e6 | 0 | −1.59788e8 | 9.90426e7 | −8.46523e9 | 3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2241.21 | 59049.0 | 2.92587e6 | 0 | −1.32341e8 | −1.14262e8 | −1.85733e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1468.87 | 59049.0 | 60422.4 | 0 | −8.67352e7 | −1.30137e9 | 2.99169e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1083.03 | 59049.0 | −924202. | 0 | −6.39517e7 | 7.89823e8 | 3.27221e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −359.921 | 59049.0 | −1.96761e6 | 0 | −2.12530e7 | 2.25801e8 | 1.46299e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −238.651 | 59049.0 | −2.04020e6 | 0 | −1.40921e7 | −6.18210e8 | 9.87381e8 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 998.710 | 59049.0 | −1.09973e6 | 0 | 5.89728e7 | 1.20991e9 | −3.19276e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1728.04 | 59049.0 | 888953. | 0 | 1.02039e8 | −3.15759e8 | −2.08781e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2338.38 | 59049.0 | 3.37088e6 | 0 | 1.38079e8 | −9.66595e8 | 2.97845e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2387.58 | 59049.0 | 3.60339e6 | 0 | 1.40984e8 | 8.76258e8 | 3.59626e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 75.22.a.m | 10 | |
| 5.b | even | 2 | 1 | 75.22.a.n | 10 | ||
| 5.c | odd | 4 | 2 | 15.22.b.a | ✓ | 20 | |
| 15.e | even | 4 | 2 | 45.22.b.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.22.b.a | ✓ | 20 | 5.c | odd | 4 | 2 | |
| 45.22.b.d | 20 | 15.e | even | 4 | 2 | ||
| 75.22.a.m | 10 | 1.a | even | 1 | 1 | trivial | |
| 75.22.a.n | 10 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 645 T_{2}^{9} - 15299350 T_{2}^{8} - 8951036520 T_{2}^{7} + 78367524408160 T_{2}^{6} + \cdots + 79\!\cdots\!68 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 79\!\cdots\!68 \)
|
| $3$ |
\( (T - 59049)^{10} \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + \cdots - 52\!\cdots\!00 \)
|
| $11$ |
\( T^{10} + \cdots - 61\!\cdots\!68 \)
|
| $13$ |
\( T^{10} + \cdots - 32\!\cdots\!32 \)
|
| $17$ |
\( T^{10} + \cdots - 11\!\cdots\!76 \)
|
| $19$ |
\( T^{10} + \cdots - 47\!\cdots\!00 \)
|
| $23$ |
\( T^{10} + \cdots - 16\!\cdots\!00 \)
|
| $29$ |
\( T^{10} + \cdots - 11\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots - 79\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots + 48\!\cdots\!00 \)
|
| $41$ |
\( T^{10} + \cdots + 82\!\cdots\!00 \)
|
| $43$ |
\( T^{10} + \cdots - 10\!\cdots\!76 \)
|
| $47$ |
\( T^{10} + \cdots + 11\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots + 71\!\cdots\!00 \)
|
| $59$ |
\( T^{10} + \cdots - 78\!\cdots\!00 \)
|
| $61$ |
\( T^{10} + \cdots + 17\!\cdots\!76 \)
|
| $67$ |
\( T^{10} + \cdots - 12\!\cdots\!76 \)
|
| $71$ |
\( T^{10} + \cdots - 32\!\cdots\!76 \)
|
| $73$ |
\( T^{10} + \cdots - 57\!\cdots\!00 \)
|
| $79$ |
\( T^{10} + \cdots + 21\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 31\!\cdots\!32 \)
|
| $89$ |
\( T^{10} + \cdots + 39\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots - 61\!\cdots\!76 \)
|
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