Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,10,Mod(1,261)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 261.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: | \(134.424353239\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{12} - 4 x^{11} - 4803 x^{10} + 14952 x^{9} + 8248476 x^{8} - 14809944 x^{7} - 6122244486 x^{6} + \cdots + 40\!\cdots\!38 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 29) |
| 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
\( x^{12} - 4 x^{11} - 4803 x^{10} + 14952 x^{9} + 8248476 x^{8} - 14809944 x^{7} - 6122244486 x^{6} + \cdots + 40\!\cdots\!38 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 802 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 40\!\cdots\!49 \nu^{11} + \cdots + 33\!\cdots\!98 ) / 76\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13\!\cdots\!41 \nu^{11} + \cdots + 11\!\cdots\!26 ) / 63\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 17\!\cdots\!85 \nu^{11} + \cdots + 18\!\cdots\!74 ) / 76\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 37\!\cdots\!47 \nu^{11} + \cdots - 19\!\cdots\!06 ) / 76\!\cdots\!60 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 16\!\cdots\!79 \nu^{11} + \cdots - 41\!\cdots\!50 ) / 19\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 23\!\cdots\!11 \nu^{11} + \cdots + 28\!\cdots\!78 ) / 25\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 19\!\cdots\!63 \nu^{11} + \cdots - 15\!\cdots\!42 ) / 19\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 10\!\cdots\!15 \nu^{11} + \cdots - 28\!\cdots\!42 ) / 76\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 38\!\cdots\!67 \nu^{11} + \cdots - 22\!\cdots\!50 ) / 19\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 802 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + 2 \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + \cdots + 615 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 22 \beta_{11} - 30 \beta_{10} + 10 \beta_{9} + 13 \beta_{8} - 15 \beta_{7} + 17 \beta_{6} + \cdots + 1101721 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2465 \beta_{11} + 4576 \beta_{10} - 282 \beta_{9} + 2176 \beta_{8} - 3889 \beta_{7} - 2358 \beta_{6} + \cdots + 264774 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 42992 \beta_{11} - 64952 \beta_{10} - 144 \beta_{9} + 14284 \beta_{8} - 32284 \beta_{7} + \cdots + 1729835436 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4998295 \beta_{11} + 8628182 \beta_{10} - 1436000 \beta_{9} + 3624579 \beta_{8} - 6506170 \beta_{7} + \cdots - 348834123 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 64582010 \beta_{11} - 115136266 \beta_{10} - 45622794 \beta_{9} - 4277177 \beta_{8} + \cdots + 2869629595335 \)
|
| \(\nu^{9}\) | \(=\) |
\( 9607745489 \beta_{11} + 15447922780 \beta_{10} - 4471116534 \beta_{9} + 5456505878 \beta_{8} + \cdots - 396878366204 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 84569969644 \beta_{11} - 194888768844 \beta_{10} - 160024203100 \beta_{9} - 62985305038 \beta_{8} + \cdots + 48\!\cdots\!14 \)
|
| \(\nu^{11}\) | \(=\) |
\( 18065382227179 \beta_{11} + 27225396210594 \beta_{10} - 11292791223524 \beta_{9} + 7759824849265 \beta_{8} + \cdots + 18\!\cdots\!75 \)
|
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 |
|
−44.2474 | 0 | 1445.83 | −350.462 | 0 | −3350.56 | −41319.6 | 0 | 15507.0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −39.0522 | 0 | 1013.08 | 303.923 | 0 | 12556.4 | −19568.2 | 0 | −11868.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −30.2132 | 0 | 400.835 | 389.202 | 0 | 3096.10 | 3358.64 | 0 | −11759.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −27.3150 | 0 | 234.111 | −2383.64 | 0 | 3032.73 | 7590.56 | 0 | 65109.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −10.2072 | 0 | −407.813 | 1619.94 | 0 | −263.065 | 9388.73 | 0 | −16535.1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −5.88142 | 0 | −477.409 | −959.601 | 0 | −2615.28 | 5819.13 | 0 | 5643.82 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 6.09515 | 0 | −474.849 | 546.084 | 0 | 6563.37 | −6015.00 | 0 | 3328.46 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.2767 | 0 | −406.390 | 2693.22 | 0 | 4178.53 | −9437.99 | 0 | 27677.4 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 20.0654 | 0 | −109.378 | 1017.86 | 0 | −11164.8 | −12468.2 | 0 | 20423.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 23.9566 | 0 | 61.9199 | −2644.17 | 0 | 2059.91 | −10782.4 | 0 | −63345.5 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 40.0148 | 0 | 1089.19 | 146.392 | 0 | 8248.48 | 23096.1 | 0 | 5857.84 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 40.5077 | 0 | 1128.88 | −2140.75 | 0 | −10261.8 | 24988.3 | 0 | −86716.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(29\) | \(1\) |
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 | 261.10.a.e | 12 | |
| 3.b | odd | 2 | 1 | 29.10.a.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.10.a.b | ✓ | 12 | 3.b | odd | 2 | 1 | |
| 261.10.a.e | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 16 T_{2}^{11} - 4693 T_{2}^{10} - 62542 T_{2}^{9} + 7898928 T_{2}^{8} + \cdots + 41\!\cdots\!60 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 41\!\cdots\!60 \)
|
| $3$ |
\( T^{12} \)
|
| $5$ |
\( T^{12} + \cdots - 19\!\cdots\!00 \)
|
| $7$ |
\( T^{12} + \cdots - 14\!\cdots\!08 \)
|
| $11$ |
\( T^{12} + \cdots - 13\!\cdots\!72 \)
|
| $13$ |
\( T^{12} + \cdots + 20\!\cdots\!56 \)
|
| $17$ |
\( T^{12} + \cdots + 31\!\cdots\!32 \)
|
| $19$ |
\( T^{12} + \cdots + 78\!\cdots\!60 \)
|
| $23$ |
\( T^{12} + \cdots + 47\!\cdots\!76 \)
|
| $29$ |
\( (T + 707281)^{12} \)
|
| $31$ |
\( T^{12} + \cdots - 22\!\cdots\!04 \)
|
| $37$ |
\( T^{12} + \cdots - 18\!\cdots\!12 \)
|
| $41$ |
\( T^{12} + \cdots + 35\!\cdots\!52 \)
|
| $43$ |
\( T^{12} + \cdots - 16\!\cdots\!40 \)
|
| $47$ |
\( T^{12} + \cdots + 45\!\cdots\!92 \)
|
| $53$ |
\( T^{12} + \cdots - 21\!\cdots\!08 \)
|
| $59$ |
\( T^{12} + \cdots - 25\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots + 22\!\cdots\!20 \)
|
| $67$ |
\( T^{12} + \cdots + 96\!\cdots\!20 \)
|
| $71$ |
\( T^{12} + \cdots + 31\!\cdots\!76 \)
|
| $73$ |
\( T^{12} + \cdots - 53\!\cdots\!16 \)
|
| $79$ |
\( T^{12} + \cdots - 64\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots + 21\!\cdots\!76 \)
|
| $89$ |
\( T^{12} + \cdots - 32\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots + 95\!\cdots\!04 \)
|
| show more | |
| show less |