Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [460,2,Mod(91,460)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(460, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("460.91");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 460 = 2^{2} \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 460.e (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(3.67311849298\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | 16.0.7465802011608416256.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{15} + 5\nu^{13} + 3\nu^{11} + 4\nu^{9} + 172\nu^{7} - 96\nu^{5} - 112\nu^{3} - 256\nu ) / 1152 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{14} - \nu^{12} - 15\nu^{10} - 2\nu^{8} + 28\nu^{6} - 96\nu^{4} + 80\nu^{2} - 160 ) / 288 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{15} + 7\nu^{13} + 9\nu^{11} - 16\nu^{9} + 44\nu^{7} + 144\nu^{5} - 80\nu^{3} + 640\nu ) / 1152 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2 \nu^{14} - 3 \nu^{13} + 4 \nu^{12} + 15 \nu^{11} - 12 \nu^{10} - 27 \nu^{9} - 22 \nu^{8} + \cdots + 64 ) / 576 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{15} + 6 \nu^{14} + 11 \nu^{13} - 6 \nu^{12} - 15 \nu^{11} + 30 \nu^{10} + 22 \nu^{9} + \cdots + 768 ) / 1152 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 15 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 24 \nu^{7} + \cdots - 64 ) / 576 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{15} - 9\nu^{13} + 9\nu^{11} - 16\nu^{9} - 36\nu^{7} + 144\nu^{5} - 80\nu^{3} ) / 384 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 3 \nu^{15} - 4 \nu^{14} - 3 \nu^{13} + 8 \nu^{12} + 3 \nu^{11} - 24 \nu^{10} - 54 \nu^{9} + \cdots + 128 ) / 1152 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2 \nu^{14} + 9 \nu^{13} - 4 \nu^{12} - 9 \nu^{11} + 12 \nu^{10} - 27 \nu^{9} + 22 \nu^{8} + 36 \nu^{7} + \cdots - 64 ) / 576 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 3 \nu^{15} + 8 \nu^{14} + 3 \nu^{13} - 4 \nu^{12} - 39 \nu^{11} + 36 \nu^{10} - 12 \nu^{9} + \cdots + 896 ) / 1152 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - \nu^{15} - 6 \nu^{14} + 11 \nu^{13} + 30 \nu^{12} - 15 \nu^{11} - 6 \nu^{10} + 22 \nu^{9} + \cdots + 1152 ) / 1152 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( \nu^{15} - 4\nu^{12} + 8\nu^{10} + 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} - 8\nu^{3} - 128\nu^{2} + 64 ) / 192 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( \nu^{15} + 4\nu^{12} + 5\nu^{9} + 20\nu^{6} - 96\nu^{4} - 8\nu^{3} + 96\nu^{2} + 64 ) / 192 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 3 \nu^{15} + 8 \nu^{14} - 3 \nu^{13} - 4 \nu^{12} + 39 \nu^{11} + 36 \nu^{10} + 12 \nu^{9} + \cdots + 896 ) / 1152 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( -\nu^{15} - 4\nu^{12} + 8\nu^{10} - 5\nu^{9} - 8\nu^{8} - 12\nu^{6} + 32\nu^{4} + 8\nu^{3} - 128\nu^{2} + 64 ) / 192 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{14} + \beta_{10} - \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 2\beta_{4} + \beta_{3} - 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} - \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + \cdots + 1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{15} - \beta_{12} + \beta_{9} - 3\beta_{8} + \beta_{7} - 2\beta_{6} + 2\beta_{4} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 5 \beta_{15} + \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} + \beta_{10} + \beta_{7} + \cdots - 1 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{15} - \beta_{14} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 5 \beta_{7} - 3 \beta_{6} + \cdots + 2 \beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{15} + \beta_{12} + 5\beta_{11} + 5\beta_{7} + 3\beta_{6} + 5\beta_{5} - 3\beta_{4} + 6\beta_{2} + 5\beta _1 - 5 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -5\beta_{14} + 5\beta_{10} - \beta_{9} - 10\beta_{8} + 7\beta_{7} - 3\beta_{6} + 6\beta_{4} + 5\beta_{3} + 26\beta_1 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 9 \beta_{15} - 25 \beta_{14} - \beta_{13} - 8 \beta_{12} - 18 \beta_{11} - 25 \beta_{10} + 9 \beta_{7} + \cdots + 23 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -\beta_{15} + \beta_{12} - 21\beta_{9} - 9\beta_{8} - 13\beta_{7} + 6\beta_{6} - 6\beta_{4} ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 49 \beta_{15} - 13 \beta_{14} + 43 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 13 \beta_{10} + 3 \beta_{7} + \cdots - 35 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 22 \beta_{15} + 61 \beta_{14} - 22 \beta_{12} - 61 \beta_{10} + 7 \beta_{9} - 17 \beta_{7} + \cdots + 22 \beta_1 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 29 \beta_{15} - 29 \beta_{12} - \beta_{11} - \beta_{7} - 39 \beta_{6} - \beta_{5} + 39 \beta_{4} + \cdots - 31 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 65 \beta_{14} - 65 \beta_{10} + 45 \beta_{9} + 130 \beta_{8} - 91 \beta_{7} - 25 \beta_{6} + \cdots - 50 \beta_1 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 43 \beta_{15} + 229 \beta_{14} + 109 \beta_{13} - 152 \beta_{12} - 86 \beta_{11} + 229 \beta_{10} + \cdots - 203 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( -179\beta_{15} + 179\beta_{12} + 113\beta_{9} + 21\beta_{8} + 73\beta_{7} - 46\beta_{6} + 46\beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).
| \(n\) | \(231\) | \(277\) | \(281\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−1.35760 | − | 0.396143i | 1.47175i | 1.68614 | + | 1.07561i | − | 1.00000i | 0.583024 | − | 1.99804i | −3.53986 | −1.86301 | − | 2.12819i | 0.833952 | −0.396143 | + | 1.35760i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.2 | −1.35760 | − | 0.396143i | 1.47175i | 1.68614 | + | 1.07561i | 1.00000i | 0.583024 | − | 1.99804i | 3.53986 | −1.86301 | − | 2.12819i | 0.833952 | 0.396143 | − | 1.35760i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.3 | −1.35760 | + | 0.396143i | − | 1.47175i | 1.68614 | − | 1.07561i | − | 1.00000i | 0.583024 | + | 1.99804i | 3.53986 | −1.86301 | + | 2.12819i | 0.833952 | 0.396143 | + | 1.35760i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.4 | −1.35760 | + | 0.396143i | − | 1.47175i | 1.68614 | − | 1.07561i | 1.00000i | 0.583024 | + | 1.99804i | −3.53986 | −1.86301 | + | 2.12819i | 0.833952 | −0.396143 | − | 1.35760i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.5 | −0.637910 | − | 1.26217i | 2.87247i | −1.18614 | + | 1.61030i | − | 1.00000i | 3.62554 | − | 1.83238i | −2.68161 | 2.78912 | + | 0.469882i | −5.25109 | −1.26217 | + | 0.637910i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.6 | −0.637910 | − | 1.26217i | 2.87247i | −1.18614 | + | 1.61030i | 1.00000i | 3.62554 | − | 1.83238i | 2.68161 | 2.78912 | + | 0.469882i | −5.25109 | 1.26217 | − | 0.637910i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.7 | −0.637910 | + | 1.26217i | − | 2.87247i | −1.18614 | − | 1.61030i | − | 1.00000i | 3.62554 | + | 1.83238i | 2.68161 | 2.78912 | − | 0.469882i | −5.25109 | 1.26217 | + | 0.637910i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.8 | −0.637910 | + | 1.26217i | − | 2.87247i | −1.18614 | − | 1.61030i | 1.00000i | 3.62554 | + | 1.83238i | −2.68161 | 2.78912 | − | 0.469882i | −5.25109 | −1.26217 | − | 0.637910i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.9 | 0.637910 | − | 1.26217i | − | 0.348132i | −1.18614 | − | 1.61030i | − | 1.00000i | −0.439402 | − | 0.222077i | −4.89140 | −2.78912 | + | 0.469882i | 2.87880 | −1.26217 | − | 0.637910i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.10 | 0.637910 | − | 1.26217i | − | 0.348132i | −1.18614 | − | 1.61030i | 1.00000i | −0.439402 | − | 0.222077i | 4.89140 | −2.78912 | + | 0.469882i | 2.87880 | 1.26217 | + | 0.637910i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.11 | 0.637910 | + | 1.26217i | 0.348132i | −1.18614 | + | 1.61030i | − | 1.00000i | −0.439402 | + | 0.222077i | 4.89140 | −2.78912 | − | 0.469882i | 2.87880 | 1.26217 | − | 0.637910i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.12 | 0.637910 | + | 1.26217i | 0.348132i | −1.18614 | + | 1.61030i | 1.00000i | −0.439402 | + | 0.222077i | −4.89140 | −2.78912 | − | 0.469882i | 2.87880 | −1.26217 | + | 0.637910i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.13 | 1.35760 | − | 0.396143i | − | 0.679463i | 1.68614 | − | 1.07561i | − | 1.00000i | −0.269165 | − | 0.922437i | 1.16300 | 1.86301 | − | 2.12819i | 2.53833 | −0.396143 | − | 1.35760i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.14 | 1.35760 | − | 0.396143i | − | 0.679463i | 1.68614 | − | 1.07561i | 1.00000i | −0.269165 | − | 0.922437i | −1.16300 | 1.86301 | − | 2.12819i | 2.53833 | 0.396143 | + | 1.35760i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.15 | 1.35760 | + | 0.396143i | 0.679463i | 1.68614 | + | 1.07561i | − | 1.00000i | −0.269165 | + | 0.922437i | −1.16300 | 1.86301 | + | 2.12819i | 2.53833 | 0.396143 | − | 1.35760i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 91.16 | 1.35760 | + | 0.396143i | 0.679463i | 1.68614 | + | 1.07561i | 1.00000i | −0.269165 | + | 0.922437i | 1.16300 | 1.86301 | + | 2.12819i | 2.53833 | −0.396143 | + | 1.35760i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 23.b | odd | 2 | 1 | inner |
| 92.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 460.2.e.a | ✓ | 16 |
| 4.b | odd | 2 | 1 | inner | 460.2.e.a | ✓ | 16 |
| 23.b | odd | 2 | 1 | inner | 460.2.e.a | ✓ | 16 |
| 92.b | even | 2 | 1 | inner | 460.2.e.a | ✓ | 16 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 460.2.e.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 460.2.e.a | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
| 460.2.e.a | ✓ | 16 | 23.b | odd | 2 | 1 | inner |
| 460.2.e.a | ✓ | 16 | 92.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 11T_{3}^{6} + 24T_{3}^{4} + 11T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(460, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} - T^{6} - 4 T^{2} + 16)^{2} \)
|
| $3$ |
\( (T^{8} + 11 T^{6} + 24 T^{4} + \cdots + 1)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{8} \)
|
| $7$ |
\( (T^{8} - 45 T^{6} + \cdots + 2916)^{2} \)
|
| $11$ |
\( (T^{8} - 45 T^{6} + \cdots + 2916)^{2} \)
|
| $13$ |
\( (T^{4} - T^{3} - 24 T^{2} + \cdots + 31)^{4} \)
|
| $17$ |
\( (T^{8} + 99 T^{6} + \cdots + 11664)^{2} \)
|
| $19$ |
\( (T^{8} - 45 T^{6} + \cdots + 2916)^{2} \)
|
| $23$ |
\( (T^{8} - 16 T^{6} + \cdots + 279841)^{2} \)
|
| $29$ |
\( (T^{4} + 12 T^{3} + \cdots + 12)^{4} \)
|
| $31$ |
\( (T^{8} + 207 T^{6} + \cdots + 4124961)^{2} \)
|
| $37$ |
\( (T^{8} + 108 T^{6} + \cdots + 46656)^{2} \)
|
| $41$ |
\( (T^{4} + 9 T^{3} + \cdots - 681)^{4} \)
|
| $43$ |
\( (T^{8} - 180 T^{6} + \cdots + 46656)^{2} \)
|
| $47$ |
\( (T^{8} + 362 T^{6} + \cdots + 163216)^{2} \)
|
| $53$ |
\( (T^{4} + 180 T^{2} + 6912)^{4} \)
|
| $59$ |
\( (T^{4} + 76 T^{2} + 256)^{4} \)
|
| $61$ |
\( (T^{8} + 243 T^{6} + \cdots + 13089924)^{2} \)
|
| $67$ |
\( (T^{8} - 288 T^{6} + \cdots + 6718464)^{2} \)
|
| $71$ |
\( (T^{8} + 215 T^{6} + \cdots + 6889)^{2} \)
|
| $73$ |
\( (T^{4} - 2 T^{3} + \cdots - 908)^{4} \)
|
| $79$ |
\( (T^{8} - 324 T^{6} + \cdots + 3779136)^{2} \)
|
| $83$ |
\( (T^{4} - 252 T^{2} + 5184)^{4} \)
|
| $89$ |
\( (T^{8} + 396 T^{6} + \cdots + 13483584)^{2} \)
|
| $97$ |
\( (T^{8} + 531 T^{6} + \cdots + 29746116)^{2} \)
|
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