Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9898,2,Mod(1,9898)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9898, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9898.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9898 = 2 \cdot 7^{2} \cdot 101 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9898.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: | \(79.0359279207\) |
| 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: |
\( x^{12} - 3 x^{11} - 19 x^{10} + 57 x^{9} + 119 x^{8} - 385 x^{7} - 223 x^{6} + 1087 x^{5} - 249 x^{4} + \cdots + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{12} - 3 x^{11} - 19 x^{10} + 57 x^{9} + 119 x^{8} - 385 x^{7} - 223 x^{6} + 1087 x^{5} - 249 x^{4} + \cdots + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 367 \nu^{11} - 576 \nu^{10} + 10492 \nu^{9} + 10922 \nu^{8} - 99426 \nu^{7} - 66142 \nu^{6} + \cdots - 143098 ) / 20066 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16201 \nu^{11} - 39883 \nu^{10} - 329563 \nu^{9} + 747649 \nu^{8} + 2325175 \nu^{7} - 4979233 \nu^{6} + \cdots - 305076 ) / 40132 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21295 \nu^{11} - 49603 \nu^{10} - 438451 \nu^{9} + 924433 \nu^{8} + 3153103 \nu^{7} - 6153069 \nu^{6} + \cdots - 123816 ) / 40132 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 13440 \nu^{11} + 32133 \nu^{10} + 272719 \nu^{9} - 597281 \nu^{8} - 1916987 \nu^{7} + \cdots + 226272 ) / 20066 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 6866 \nu^{11} + 17710 \nu^{10} + 137950 \nu^{9} - 333375 \nu^{8} - 958913 \nu^{7} + 2242588 \nu^{6} + \cdots + 166189 ) / 10033 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7715 \nu^{11} - 19330 \nu^{10} - 156098 \nu^{9} + 362839 \nu^{8} + 1096901 \nu^{7} - 2434883 \nu^{6} + \cdots - 95847 ) / 10033 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 32847 \nu^{11} + 79587 \nu^{10} + 670399 \nu^{9} - 1491869 \nu^{8} - 4756391 \nu^{7} + \cdots + 463924 ) / 40132 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 21240 \nu^{11} + 51767 \nu^{10} + 431083 \nu^{9} - 965327 \nu^{8} - 3040005 \nu^{7} + \cdots + 254394 ) / 20066 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 21240 \nu^{11} - 51767 \nu^{10} - 431083 \nu^{9} + 965327 \nu^{8} + 3040005 \nu^{7} - 6430127 \nu^{6} + \cdots - 334658 ) / 20066 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 48025 \nu^{11} - 121595 \nu^{10} - 969267 \nu^{9} + 2279777 \nu^{8} + 6789903 \nu^{7} + \cdots - 954372 ) / 40132 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{10} + \beta_{9} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{6} - \beta_{4} - \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{11} + 9\beta_{10} + 9\beta_{9} + 2\beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} - 3\beta_{2} + 4\beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14 \beta_{11} + 13 \beta_{10} + 16 \beta_{9} + 2 \beta_{8} + 12 \beta_{6} - 2 \beta_{5} - 10 \beta_{4} + \cdots + 24 \)
|
| \(\nu^{6}\) | \(=\) |
\( 32 \beta_{11} + 79 \beta_{10} + 88 \beta_{9} + 26 \beta_{8} + 3 \beta_{7} + 19 \beta_{6} - 6 \beta_{5} + \cdots + 220 \)
|
| \(\nu^{7}\) | \(=\) |
\( 158 \beta_{11} + 145 \beta_{10} + 207 \beta_{9} + 41 \beta_{8} + 12 \beta_{7} + 127 \beta_{6} + \cdots + 339 \)
|
| \(\nu^{8}\) | \(=\) |
\( 408 \beta_{11} + 705 \beta_{10} + 908 \beta_{9} + 285 \beta_{8} + 74 \beta_{7} + 265 \beta_{6} + \cdots + 1991 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1714 \beta_{11} + 1531 \beta_{10} + 2472 \beta_{9} + 606 \beta_{8} + 277 \beta_{7} + 1332 \beta_{6} + \cdots + 4122 \)
|
| \(\nu^{10}\) | \(=\) |
\( 4839 \beta_{11} + 6448 \beta_{10} + 9648 \beta_{9} + 3064 \beta_{8} + 1218 \beta_{7} + 3301 \beta_{6} + \cdots + 19160 \)
|
| \(\nu^{11}\) | \(=\) |
\( 18542 \beta_{11} + 15792 \beta_{10} + 28464 \beta_{9} + 7856 \beta_{8} + 4418 \beta_{7} + 14138 \beta_{6} + \cdots + 47162 \)
|
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 |
|
−1.00000 | −2.61792 | 1.00000 | 1.11081 | 2.61792 | 0 | −1.00000 | 3.85352 | −1.11081 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.55843 | 1.00000 | 0.229065 | 2.55843 | 0 | −1.00000 | 3.54554 | −0.229065 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.97001 | 1.00000 | −2.16598 | 1.97001 | 0 | −1.00000 | 0.880924 | 2.16598 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.61421 | 1.00000 | 2.25632 | 1.61421 | 0 | −1.00000 | −0.394316 | −2.25632 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.0525366 | 1.00000 | −0.894828 | −0.0525366 | 0 | −1.00000 | −2.99724 | 0.894828 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.458484 | 1.00000 | 3.26386 | −0.458484 | 0 | −1.00000 | −2.78979 | −3.26386 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 0.518361 | 1.00000 | −0.990803 | −0.518361 | 0 | −1.00000 | −2.73130 | 0.990803 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 1.07626 | 1.00000 | 1.38510 | −1.07626 | 0 | −1.00000 | −1.84166 | −1.38510 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 1.59664 | 1.00000 | −4.18293 | −1.59664 | 0 | −1.00000 | −0.450726 | 4.18293 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | 1.78468 | 1.00000 | 4.01959 | −1.78468 | 0 | −1.00000 | 0.185094 | −4.01959 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | 2.96221 | 1.00000 | −2.10866 | −2.96221 | 0 | −1.00000 | 5.77468 | 2.10866 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | −1.00000 | 3.31139 | 1.00000 | 3.07846 | −3.31139 | 0 | −1.00000 | 7.96528 | −3.07846 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(101\) | \( +1 \) |
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 | 9898.2.a.bb | yes | 12 |
| 7.b | odd | 2 | 1 | 9898.2.a.ba | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9898.2.a.ba | ✓ | 12 | 7.b | odd | 2 | 1 | |
| 9898.2.a.bb | yes | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):
|
\( T_{3}^{12} - 3 T_{3}^{11} - 19 T_{3}^{10} + 57 T_{3}^{9} + 119 T_{3}^{8} - 385 T_{3}^{7} - 223 T_{3}^{6} + \cdots + 8 \)
|
|
\( T_{5}^{12} - 5 T_{5}^{11} - 24 T_{5}^{10} + 145 T_{5}^{9} + 108 T_{5}^{8} - 1218 T_{5}^{7} + 203 T_{5}^{6} + \cdots - 544 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{12} \)
|
| $3$ |
\( T^{12} - 3 T^{11} + \cdots + 8 \)
|
| $5$ |
\( T^{12} - 5 T^{11} + \cdots - 544 \)
|
| $7$ |
\( T^{12} \)
|
| $11$ |
\( T^{12} - 5 T^{11} + \cdots + 2458 \)
|
| $13$ |
\( T^{12} - 3 T^{11} + \cdots + 20264 \)
|
| $17$ |
\( T^{12} - 14 T^{11} + \cdots - 121984 \)
|
| $19$ |
\( T^{12} - 7 T^{11} + \cdots + 57838 \)
|
| $23$ |
\( T^{12} + 6 T^{11} + \cdots - 26951 \)
|
| $29$ |
\( T^{12} - 11 T^{11} + \cdots + 139366 \)
|
| $31$ |
\( T^{12} + 4 T^{11} + \cdots + 1407232 \)
|
| $37$ |
\( T^{12} + T^{11} + \cdots - 1661528 \)
|
| $41$ |
\( T^{12} + T^{11} + \cdots - 4028942 \)
|
| $43$ |
\( T^{12} + 3 T^{11} + \cdots + 257272 \)
|
| $47$ |
\( T^{12} - 36 T^{11} + \cdots - 7202816 \)
|
| $53$ |
\( T^{12} + T^{11} + \cdots - 28035668 \)
|
| $59$ |
\( T^{12} - 11 T^{11} + \cdots + 69673928 \)
|
| $61$ |
\( T^{12} + \cdots + 23257397912 \)
|
| $67$ |
\( T^{12} - 13 T^{11} + \cdots - 3899876 \)
|
| $71$ |
\( T^{12} + \cdots + 6322537487 \)
|
| $73$ |
\( T^{12} + \cdots + 33648721391 \)
|
| $79$ |
\( T^{12} + \cdots + 4852556131 \)
|
| $83$ |
\( T^{12} + \cdots - 2992779416 \)
|
| $89$ |
\( T^{12} + \cdots + 21487869191 \)
|
| $97$ |
\( T^{12} + \cdots - 26245372672 \)
|
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