Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [261,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("261.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(200.537570126\) |
| Analytic rank: | \(0\) |
| Dimension: | \(14\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{14} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{14} - 23517 x^{12} - 42196 x^{11} + 214206700 x^{10} + 532863376 x^{9} - 951901011680 x^{8} + \cdots + 30\!\cdots\!04 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{20}\cdot 3^{7} \) |
| 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 \(1,\beta_1,\ldots,\beta_{13}\) 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
\( x^{14} - 23517 x^{12} - 42196 x^{11} + 214206700 x^{10} + 532863376 x^{9} - 951901011680 x^{8} + \cdots + 30\!\cdots\!04 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 3360 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 65\!\cdots\!65 \nu^{13} + \cdots + 36\!\cdots\!92 ) / 14\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 21\!\cdots\!97 \nu^{13} + \cdots - 13\!\cdots\!76 ) / 14\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 40\!\cdots\!27 \nu^{13} + \cdots - 18\!\cdots\!96 ) / 21\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 53\!\cdots\!03 \nu^{13} + \cdots - 30\!\cdots\!28 ) / 74\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 28\!\cdots\!77 \nu^{13} + \cdots + 61\!\cdots\!24 ) / 29\!\cdots\!20 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 14\!\cdots\!53 \nu^{13} + \cdots - 22\!\cdots\!08 ) / 14\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 22\!\cdots\!21 \nu^{13} + \cdots + 96\!\cdots\!64 ) / 14\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 12\!\cdots\!99 \nu^{13} + \cdots + 41\!\cdots\!00 ) / 74\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 27\!\cdots\!79 \nu^{13} + \cdots - 66\!\cdots\!96 ) / 14\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 22\!\cdots\!69 \nu^{13} + \cdots + 30\!\cdots\!12 ) / 74\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 62\!\cdots\!99 \nu^{13} + \cdots - 70\!\cdots\!08 ) / 17\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 3360 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{13} - \beta_{11} + \beta_{10} - 2 \beta_{9} + 2 \beta_{7} + \beta_{5} - \beta_{4} + \cdots + 9049 \)
|
| \(\nu^{4}\) | \(=\) |
\( 21 \beta_{13} - 66 \beta_{12} - 53 \beta_{11} + 15 \beta_{10} + 78 \beta_{9} + 74 \beta_{8} + \cdots + 17808190 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 13321 \beta_{13} - 2552 \beta_{12} - 10337 \beta_{11} + 10855 \beta_{10} - 20570 \beta_{9} + \cdots + 164163591 \)
|
| \(\nu^{6}\) | \(=\) |
\( 225101 \beta_{13} - 788730 \beta_{12} - 703893 \beta_{11} + 40983 \beta_{10} + 895822 \beta_{9} + \cdots + 107440644170 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 125395925 \beta_{13} - 35663488 \beta_{12} - 92565397 \beta_{11} + 92794059 \beta_{10} + \cdots + 1603648814167 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1571159353 \beta_{13} - 7278434650 \beta_{12} - 6964864137 \beta_{11} - 235952445 \beta_{10} + \cdots + 698085834836598 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1052023740033 \beta_{13} - 366122300872 \beta_{12} - 790114395049 \beta_{11} + 738470867527 \beta_{10} + \cdots + 13\!\cdots\!79 \)
|
| \(\nu^{10}\) | \(=\) |
\( 8830550613573 \beta_{13} - 61606633335418 \beta_{12} - 62288559148573 \beta_{11} - 4216420191633 \beta_{10} + \cdots + 47\!\cdots\!14 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 83\!\cdots\!17 \beta_{13} + \cdots + 10\!\cdots\!07 \)
|
| \(\nu^{12}\) | \(=\) |
\( 38\!\cdots\!89 \beta_{13} + \cdots + 33\!\cdots\!34 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 65\!\cdots\!45 \beta_{13} + \cdots + 82\!\cdots\!39 \)
|
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 |
|
−88.5004 | 0 | 5784.31 | −7590.53 | 0 | −38749.2 | −330665. | 0 | 671765. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −77.1568 | 0 | 3905.18 | 3176.79 | 0 | −18569.7 | −143294. | 0 | −245111. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −73.5071 | 0 | 3355.30 | −7553.66 | 0 | 46751.0 | −96095.7 | 0 | 555248. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −51.1787 | 0 | 571.255 | 4173.93 | 0 | −8856.29 | 75577.8 | 0 | −213616. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −35.2896 | 0 | −802.642 | 5887.06 | 0 | 66890.4 | 100598. | 0 | −207752. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −20.2914 | 0 | −1636.26 | −12035.6 | 0 | 8773.61 | 74758.8 | 0 | 244220. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −8.22600 | 0 | −1980.33 | −8469.53 | 0 | 2360.62 | 33137.1 | 0 | 69670.4 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −6.06439 | 0 | −2011.22 | 8656.32 | 0 | −42605.5 | 24616.7 | 0 | −52495.3 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 26.9644 | 0 | −1320.92 | 9762.32 | 0 | −73001.2 | −90841.0 | 0 | 263235. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 55.2092 | 0 | 1000.05 | −5132.07 | 0 | −31764.7 | −57856.2 | 0 | −283337. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 61.8148 | 0 | 1773.06 | 11752.2 | 0 | 18695.9 | −16995.1 | 0 | 726462. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 62.7554 | 0 | 1890.24 | −1400.24 | 0 | 77426.1 | −9900.16 | 0 | −87872.8 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 67.9786 | 0 | 2573.09 | −12131.4 | 0 | 74935.0 | 35694.6 | 0 | −824677. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 85.4920 | 0 | 5260.89 | 1144.42 | 0 | 2737.98 | 274676. | 0 | 97838.7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 261.12.a.e | 14 | |
| 3.b | odd | 2 | 1 | 29.12.a.b | ✓ | 14 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.12.a.b | ✓ | 14 | 3.b | odd | 2 | 1 | |
| 261.12.a.e | 14 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{14} - 23517 T_{2}^{12} + 42196 T_{2}^{11} + 214206700 T_{2}^{10} - 532863376 T_{2}^{9} + \cdots + 30\!\cdots\!04 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{14} + \cdots + 30\!\cdots\!04 \)
|
| $3$ |
\( T^{14} \)
|
| $5$ |
\( T^{14} + \cdots - 45\!\cdots\!00 \)
|
| $7$ |
\( T^{14} + \cdots + 12\!\cdots\!08 \)
|
| $11$ |
\( T^{14} + \cdots + 12\!\cdots\!04 \)
|
| $13$ |
\( T^{14} + \cdots + 17\!\cdots\!76 \)
|
| $17$ |
\( T^{14} + \cdots - 35\!\cdots\!36 \)
|
| $19$ |
\( T^{14} + \cdots + 15\!\cdots\!80 \)
|
| $23$ |
\( T^{14} + \cdots - 23\!\cdots\!24 \)
|
| $29$ |
\( (T - 20511149)^{14} \)
|
| $31$ |
\( T^{14} + \cdots + 97\!\cdots\!68 \)
|
| $37$ |
\( T^{14} + \cdots - 35\!\cdots\!80 \)
|
| $41$ |
\( T^{14} + \cdots + 42\!\cdots\!00 \)
|
| $43$ |
\( T^{14} + \cdots + 63\!\cdots\!08 \)
|
| $47$ |
\( T^{14} + \cdots - 13\!\cdots\!48 \)
|
| $53$ |
\( T^{14} + \cdots + 45\!\cdots\!68 \)
|
| $59$ |
\( T^{14} + \cdots - 76\!\cdots\!00 \)
|
| $61$ |
\( T^{14} + \cdots - 31\!\cdots\!36 \)
|
| $67$ |
\( T^{14} + \cdots + 49\!\cdots\!32 \)
|
| $71$ |
\( T^{14} + \cdots - 17\!\cdots\!36 \)
|
| $73$ |
\( T^{14} + \cdots + 32\!\cdots\!00 \)
|
| $79$ |
\( T^{14} + \cdots - 38\!\cdots\!00 \)
|
| $83$ |
\( T^{14} + \cdots + 62\!\cdots\!88 \)
|
| $89$ |
\( T^{14} + \cdots + 91\!\cdots\!40 \)
|
| $97$ |
\( T^{14} + \cdots - 35\!\cdots\!48 \)
|
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