Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,12,Mod(1,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(185.939049695\) |
| 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: |
\( x^{10} - 2 x^{9} - 1081758 x^{8} + 886790 x^{7} + 382181877876 x^{6} - 10654635891750 x^{5} + \cdots + 10\!\cdots\!80 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{2}\cdot 5\cdot 11^{9} \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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
\( x^{10} - 2 x^{9} - 1081758 x^{8} + 886790 x^{7} + 382181877876 x^{6} - 10654635891750 x^{5} + \cdots + 10\!\cdots\!80 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 58\!\cdots\!17 \nu^{9} + \cdots + 16\!\cdots\!20 ) / 19\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 83\!\cdots\!31 \nu^{9} + \cdots + 22\!\cdots\!60 ) / 16\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 12\!\cdots\!13 \nu^{9} + \cdots + 15\!\cdots\!80 ) / 12\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 49\!\cdots\!27 \nu^{9} + \cdots - 15\!\cdots\!80 ) / 48\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 41\!\cdots\!87 \nu^{9} + \cdots + 11\!\cdots\!80 ) / 69\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 35\!\cdots\!53 \nu^{9} + \cdots - 12\!\cdots\!20 ) / 31\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\!\cdots\!93 \nu^{9} + \cdots + 76\!\cdots\!80 ) / 87\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 24\!\cdots\!69 \nu^{9} + \cdots - 10\!\cdots\!60 ) / 69\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 16\!\cdots\!87 \nu^{9} + \cdots + 34\!\cdots\!80 ) / 31\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( 7\beta_{2} - 121\beta _1 + 64 ) / 121 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 121 \beta_{8} - 242 \beta_{7} + 121 \beta_{6} - 451 \beta_{4} - 121 \beta_{3} - 32114 \beta_{2} + \cdots + 26162766 ) / 121 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 43659 \beta_{9} + 70994 \beta_{8} - 10780 \beta_{7} + 30404 \beta_{6} + 29931 \beta_{5} + \cdots + 56981355 ) / 121 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 415701 \beta_{9} + 57182477 \beta_{8} - 119682827 \beta_{7} + 82350884 \beta_{6} + \cdots + 9812866199302 ) / 121 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 25817226138 \beta_{9} + 41181931318 \beta_{8} - 9678293460 \beta_{7} + 28918703440 \beta_{6} + \cdots + 682762461473150 ) / 121 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 1255337574282 \beta_{9} + 27265110469003 \beta_{8} - 55056198618828 \beta_{7} + 50351910146395 \beta_{6} + \cdots + 41\!\cdots\!70 ) / 121 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 13\!\cdots\!33 \beta_{9} + \cdots + 64\!\cdots\!29 ) / 121 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 16\!\cdots\!45 \beta_{9} + \cdots + 19\!\cdots\!66 ) / 121 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 66\!\cdots\!16 \beta_{9} + \cdots + 46\!\cdots\!64 ) / 121 \)
|
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 |
|
−32.0000 | −740.392 | 1024.00 | 7333.96 | 23692.6 | −72663.6 | −32768.0 | 371034. | −234687. | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −32.0000 | −571.788 | 1024.00 | −3923.81 | 18297.2 | 59339.0 | −32768.0 | 149794. | 125562. | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −32.0000 | −286.091 | 1024.00 | 438.563 | 9154.92 | −39816.5 | −32768.0 | −95298.7 | −14034.0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −32.0000 | −254.401 | 1024.00 | −9628.64 | 8140.82 | 116.735 | −32768.0 | −112427. | 308117. | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −32.0000 | −131.306 | 1024.00 | 4814.08 | 4201.80 | 11852.7 | −32768.0 | −159906. | −154051. | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −32.0000 | 8.09373 | 1024.00 | 12375.0 | −258.999 | 28798.8 | −32768.0 | −177081. | −396000. | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −32.0000 | 261.388 | 1024.00 | −9107.97 | −8364.41 | −36571.1 | −32768.0 | −108823. | 291455. | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −32.0000 | 487.884 | 1024.00 | 811.242 | −15612.3 | 46405.7 | −32768.0 | 60884.1 | −25959.8 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −32.0000 | 627.820 | 1024.00 | −7505.66 | −20090.2 | −64324.8 | −32768.0 | 217010. | 240181. | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −32.0000 | 651.793 | 1024.00 | −1152.78 | −20857.4 | 52937.0 | −32768.0 | 247687. | 36889.0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
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 | 242.12.a.m | 10 | |
| 11.b | odd | 2 | 1 | 242.12.a.n | 10 | ||
| 11.d | odd | 10 | 2 | 22.12.c.a | ✓ | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.12.c.a | ✓ | 20 | 11.d | odd | 10 | 2 | |
| 242.12.a.m | 10 | 1.a | even | 1 | 1 | trivial | |
| 242.12.a.n | 10 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{10} - 53 T_{3}^{9} - 1080767 T_{3}^{8} + 49530330 T_{3}^{7} + 380504223009 T_{3}^{6} + \cdots - 17\!\cdots\!01 \)
|
|
\( T_{7}^{10} + 13926 T_{7}^{9} - 10796670602 T_{7}^{8} - 58226882683410 T_{7}^{7} + \cdots + 39\!\cdots\!80 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 32)^{10} \)
|
| $3$ |
\( T^{10} + \cdots - 17\!\cdots\!01 \)
|
| $5$ |
\( T^{10} + \cdots - 46\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots + 39\!\cdots\!80 \)
|
| $11$ |
\( T^{10} \)
|
| $13$ |
\( T^{10} + \cdots + 51\!\cdots\!20 \)
|
| $17$ |
\( T^{10} + \cdots + 23\!\cdots\!55 \)
|
| $19$ |
\( T^{10} + \cdots + 82\!\cdots\!25 \)
|
| $23$ |
\( T^{10} + \cdots - 17\!\cdots\!16 \)
|
| $29$ |
\( T^{10} + \cdots - 53\!\cdots\!00 \)
|
| $31$ |
\( T^{10} + \cdots + 26\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots - 24\!\cdots\!80 \)
|
| $41$ |
\( T^{10} + \cdots - 83\!\cdots\!09 \)
|
| $43$ |
\( T^{10} + \cdots + 11\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots - 63\!\cdots\!20 \)
|
| $53$ |
\( T^{10} + \cdots - 14\!\cdots\!80 \)
|
| $59$ |
\( T^{10} + \cdots + 38\!\cdots\!25 \)
|
| $61$ |
\( T^{10} + \cdots - 24\!\cdots\!00 \)
|
| $67$ |
\( T^{10} + \cdots + 17\!\cdots\!56 \)
|
| $71$ |
\( T^{10} + \cdots + 27\!\cdots\!20 \)
|
| $73$ |
\( T^{10} + \cdots - 56\!\cdots\!45 \)
|
| $79$ |
\( T^{10} + \cdots - 13\!\cdots\!00 \)
|
| $83$ |
\( T^{10} + \cdots + 84\!\cdots\!55 \)
|
| $89$ |
\( T^{10} + \cdots + 11\!\cdots\!00 \)
|
| $97$ |
\( T^{10} + \cdots + 19\!\cdots\!31 \)
|
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