Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [333,8,Mod(1,333)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(333, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("333.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 333 = 3^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 333.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: | \(104.024213486\) |
| Analytic rank: | \(1\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - 5 x^{10} - 1078 x^{9} + 4966 x^{8} + 379692 x^{7} - 1385588 x^{6} - 48765978 x^{5} + \cdots + 6680404080 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 37) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 5 x^{10} - 1078 x^{9} + 4966 x^{8} + 379692 x^{7} - 1385588 x^{6} - 48765978 x^{5} + \cdots + 6680404080 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 198 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15\!\cdots\!17 \nu^{10} + \cdots - 11\!\cdots\!56 ) / 31\!\cdots\!72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\!\cdots\!37 \nu^{10} + \cdots - 93\!\cdots\!44 ) / 15\!\cdots\!36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 20\!\cdots\!65 \nu^{10} + \cdots - 17\!\cdots\!64 ) / 15\!\cdots\!36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\!\cdots\!61 \nu^{10} + \cdots - 13\!\cdots\!52 ) / 31\!\cdots\!72 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 65\!\cdots\!69 \nu^{10} + \cdots - 30\!\cdots\!36 ) / 15\!\cdots\!36 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 82\!\cdots\!29 \nu^{10} + \cdots + 10\!\cdots\!52 ) / 15\!\cdots\!36 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 69\!\cdots\!63 \nu^{10} + \cdots + 80\!\cdots\!36 ) / 79\!\cdots\!68 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 44\!\cdots\!95 \nu^{10} + \cdots + 20\!\cdots\!20 ) / 39\!\cdots\!84 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 198 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + 3\beta_{8} + 3\beta_{7} + \beta_{6} + 3\beta_{4} + \beta_{3} - \beta_{2} + 373\beta _1 - 45 \)
|
| \(\nu^{4}\) | \(=\) |
\( 6 \beta_{10} - 3 \beta_{9} + 17 \beta_{8} - 24 \beta_{7} - 21 \beta_{6} - 16 \beta_{5} + 9 \beta_{4} + \cdots + 74060 \)
|
| \(\nu^{5}\) | \(=\) |
\( 594 \beta_{10} - 78 \beta_{9} + 1672 \beta_{8} + 1398 \beta_{7} + 384 \beta_{6} + 492 \beta_{5} + \cdots - 92818 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2598 \beta_{10} - 2782 \beta_{9} + 9956 \beta_{8} - 15934 \beta_{7} - 13572 \beta_{6} - 14532 \beta_{5} + \cdots + 31110828 \)
|
| \(\nu^{7}\) | \(=\) |
\( 305411 \beta_{10} - 33482 \beta_{9} + 798175 \beta_{8} + 625297 \beta_{7} + 123893 \beta_{6} + \cdots - 73009987 \)
|
| \(\nu^{8}\) | \(=\) |
\( 850092 \beta_{10} - 1519241 \beta_{9} + 4550125 \beta_{8} - 8591574 \beta_{7} - 7141489 \beta_{6} + \cdots + 13533654002 \)
|
| \(\nu^{9}\) | \(=\) |
\( 151917708 \beta_{10} - 5774168 \beta_{9} + 365393820 \beta_{8} + 286873436 \beta_{7} + \cdots - 45807258608 \)
|
| \(\nu^{10}\) | \(=\) |
\( 191429772 \beta_{10} - 693474316 \beta_{9} + 1862539160 \beta_{8} - 4372551324 \beta_{7} + \cdots + 5992816281474 \)
|
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 |
|
−21.9474 | 0 | 353.689 | −19.5004 | 0 | 1140.68 | −4953.28 | 0 | 427.983 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −21.2283 | 0 | 322.641 | 351.409 | 0 | −895.269 | −4131.89 | 0 | −7459.83 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −15.8861 | 0 | 124.370 | −303.592 | 0 | 157.297 | 57.6738 | 0 | 4822.91 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −8.16645 | 0 | −61.3091 | 56.3977 | 0 | 1198.36 | 1545.98 | 0 | −460.569 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −5.43681 | 0 | −98.4411 | 143.945 | 0 | 1592.07 | 1231.12 | 0 | −782.599 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.25926 | 0 | −126.414 | −409.245 | 0 | −873.009 | 320.374 | 0 | 515.347 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 3.12221 | 0 | −118.252 | 342.676 | 0 | −975.157 | −768.850 | 0 | 1069.91 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 6.85832 | 0 | −80.9635 | 335.413 | 0 | 735.210 | −1433.14 | 0 | 2300.37 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 8.08402 | 0 | −62.6486 | −414.761 | 0 | 49.3599 | −1541.21 | 0 | −3352.94 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 18.7203 | 0 | 222.450 | 4.96717 | 0 | −1164.11 | 1768.14 | 0 | 92.9870 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 21.1395 | 0 | 318.879 | −463.709 | 0 | 1277.57 | 4035.09 | 0 | −9802.57 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(37\) | \(1\) |
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 | 333.8.a.d | 11 | |
| 3.b | odd | 2 | 1 | 37.8.a.b | ✓ | 11 | |
| 12.b | even | 2 | 1 | 592.8.a.g | 11 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.8.a.b | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 333.8.a.d | 11 | 1.a | even | 1 | 1 | trivial | |
| 592.8.a.g | 11 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{11} + 16 T_{2}^{10} - 973 T_{2}^{9} - 14278 T_{2}^{8} + 302086 T_{2}^{7} + 3815344 T_{2}^{6} + \cdots - 28348180480 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(333))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + \cdots - 28348180480 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} + \cdots + 75\!\cdots\!00 \)
|
| $7$ |
\( T^{11} + \cdots - 14\!\cdots\!64 \)
|
| $11$ |
\( T^{11} + \cdots - 11\!\cdots\!52 \)
|
| $13$ |
\( T^{11} + \cdots + 16\!\cdots\!72 \)
|
| $17$ |
\( T^{11} + \cdots - 27\!\cdots\!72 \)
|
| $19$ |
\( T^{11} + \cdots - 66\!\cdots\!60 \)
|
| $23$ |
\( T^{11} + \cdots - 10\!\cdots\!48 \)
|
| $29$ |
\( T^{11} + \cdots - 51\!\cdots\!60 \)
|
| $31$ |
\( T^{11} + \cdots - 68\!\cdots\!92 \)
|
| $37$ |
\( (T + 50653)^{11} \)
|
| $41$ |
\( T^{11} + \cdots + 36\!\cdots\!24 \)
|
| $43$ |
\( T^{11} + \cdots + 32\!\cdots\!04 \)
|
| $47$ |
\( T^{11} + \cdots + 12\!\cdots\!12 \)
|
| $53$ |
\( T^{11} + \cdots - 14\!\cdots\!64 \)
|
| $59$ |
\( T^{11} + \cdots + 44\!\cdots\!40 \)
|
| $61$ |
\( T^{11} + \cdots - 16\!\cdots\!72 \)
|
| $67$ |
\( T^{11} + \cdots + 19\!\cdots\!84 \)
|
| $71$ |
\( T^{11} + \cdots - 32\!\cdots\!80 \)
|
| $73$ |
\( T^{11} + \cdots + 83\!\cdots\!68 \)
|
| $79$ |
\( T^{11} + \cdots + 35\!\cdots\!00 \)
|
| $83$ |
\( T^{11} + \cdots + 64\!\cdots\!04 \)
|
| $89$ |
\( T^{11} + \cdots - 13\!\cdots\!80 \)
|
| $97$ |
\( T^{11} + \cdots + 31\!\cdots\!28 \)
|
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