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: | \(0\) |
| 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: |
\( x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + \cdots - 45399525376 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{3} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4 x^{9} - 905 x^{8} + 4018 x^{7} + 291290 x^{6} - 1367036 x^{5} - 39566544 x^{4} + \cdots - 45399525376 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 183 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 513695351 \nu^{9} + 3795335454 \nu^{8} - 434332945127 \nu^{7} - 2833919119884 \nu^{6} + \cdots + 27\!\cdots\!48 ) / 108660048238336 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21853876391 \nu^{9} + 115977856880 \nu^{8} - 18386405748895 \nu^{7} - 83272556051534 \nu^{6} + \cdots + 80\!\cdots\!32 ) / 34\!\cdots\!52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 24484427321 \nu^{9} - 149368159280 \nu^{8} + 20587737251265 \nu^{7} + \cdots - 10\!\cdots\!20 ) / 34\!\cdots\!52 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 12618303639 \nu^{9} + 92581936416 \nu^{8} - 10541454758991 \nu^{7} - 68543900904286 \nu^{6} + \cdots + 59\!\cdots\!88 ) / 17\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 225128756 \nu^{9} - 1452004903 \nu^{8} + 189042392248 \nu^{7} + 1061357779511 \nu^{6} + \cdots - 95\!\cdots\!88 ) / 15522864034048 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 60967977303 \nu^{9} + 401317466608 \nu^{8} - 50996170201551 \nu^{7} - 294300157350190 \nu^{6} + \cdots + 26\!\cdots\!32 ) / 34\!\cdots\!52 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 99096631945 \nu^{9} + 644959110512 \nu^{8} - 82926543155601 \nu^{7} - 473909157578994 \nu^{6} + \cdots + 43\!\cdots\!68 ) / 34\!\cdots\!52 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 183 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{9} + \beta_{8} + 2\beta_{7} + 5\beta_{6} - \beta_{5} + 4\beta_{4} + \beta_{2} + 257\beta _1 - 217 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{9} + 2 \beta_{8} + 28 \beta_{7} + 14 \beta_{6} + 19 \beta_{5} + 34 \beta_{4} + 22 \beta_{3} + \cdots + 46897 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 1055 \beta_{9} + 668 \beta_{8} + 572 \beta_{7} + 1956 \beta_{6} - 777 \beta_{5} + 1302 \beta_{4} + \cdots - 112501 \)
|
| \(\nu^{6}\) | \(=\) |
\( 785 \beta_{9} + 2910 \beta_{8} + 12456 \beta_{7} + 4486 \beta_{6} + 10449 \beta_{5} + 15430 \beta_{4} + \cdots + 13549473 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 425293 \beta_{9} + 308574 \beta_{8} + 145520 \beta_{7} + 654390 \beta_{6} - 365545 \beta_{5} + \cdots - 41971261 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 66803 \beta_{9} + 1873846 \beta_{8} + 4082704 \beta_{7} + 902246 \beta_{6} + 4318669 \beta_{5} + \cdots + 4103690089 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 155327385 \beta_{9} + 123773242 \beta_{8} + 38634872 \beta_{7} + 211558658 \beta_{6} + \cdots - 14635553093 \)
|
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 |
|
−16.2140 | 0 | 134.894 | 220.871 | 0 | −808.413 | −111.778 | 0 | −3581.21 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −15.7200 | 0 | 119.119 | −316.724 | 0 | −1289.70 | 139.607 | 0 | 4978.91 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −12.2020 | 0 | 20.8894 | 415.456 | 0 | 362.670 | 1306.97 | 0 | −5069.41 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −3.89746 | 0 | −112.810 | 91.5007 | 0 | −1347.68 | 938.547 | 0 | −356.620 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.27803 | 0 | −117.254 | −243.887 | 0 | 767.318 | 803.953 | 0 | 799.468 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 11.5549 | 0 | 5.51464 | −118.339 | 0 | 925.681 | −1415.30 | 0 | −1367.39 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 12.4614 | 0 | 27.2868 | 16.1140 | 0 | −704.986 | −1255.03 | 0 | 200.803 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 12.5631 | 0 | 29.8308 | −318.311 | 0 | −75.8943 | −1233.31 | 0 | −3998.96 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 18.8983 | 0 | 229.145 | 439.542 | 0 | 380.249 | 1911.48 | 0 | 8306.59 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 19.8339 | 0 | 265.384 | 437.776 | 0 | 1289.75 | 2724.87 | 0 | 8682.81 | |||||||||||||||||||||||||||||||||||||||||||||||||
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.c | 10 | |
| 3.b | odd | 2 | 1 | 37.8.a.a | ✓ | 10 | |
| 12.b | even | 2 | 1 | 592.8.a.f | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.8.a.a | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 333.8.a.c | 10 | 1.a | even | 1 | 1 | trivial | |
| 592.8.a.f | 10 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} - 24 T_{2}^{9} - 653 T_{2}^{8} + 16962 T_{2}^{7} + 139726 T_{2}^{6} - 4135692 T_{2}^{5} + \cdots - 26941953024 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(333))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots - 26941953024 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} + \cdots + 75\!\cdots\!00 \)
|
| $7$ |
\( T^{10} + \cdots - 94\!\cdots\!24 \)
|
| $11$ |
\( T^{10} + \cdots - 11\!\cdots\!20 \)
|
| $13$ |
\( T^{10} + \cdots - 11\!\cdots\!12 \)
|
| $17$ |
\( T^{10} + \cdots + 53\!\cdots\!60 \)
|
| $19$ |
\( T^{10} + \cdots + 50\!\cdots\!60 \)
|
| $23$ |
\( T^{10} + \cdots - 83\!\cdots\!12 \)
|
| $29$ |
\( T^{10} + \cdots - 13\!\cdots\!32 \)
|
| $31$ |
\( T^{10} + \cdots + 26\!\cdots\!76 \)
|
| $37$ |
\( (T - 50653)^{10} \)
|
| $41$ |
\( T^{10} + \cdots - 28\!\cdots\!98 \)
|
| $43$ |
\( T^{10} + \cdots + 30\!\cdots\!68 \)
|
| $47$ |
\( T^{10} + \cdots - 32\!\cdots\!64 \)
|
| $53$ |
\( T^{10} + \cdots + 44\!\cdots\!96 \)
|
| $59$ |
\( T^{10} + \cdots - 91\!\cdots\!52 \)
|
| $61$ |
\( T^{10} + \cdots - 19\!\cdots\!16 \)
|
| $67$ |
\( T^{10} + \cdots + 31\!\cdots\!12 \)
|
| $71$ |
\( T^{10} + \cdots + 38\!\cdots\!72 \)
|
| $73$ |
\( T^{10} + \cdots + 32\!\cdots\!06 \)
|
| $79$ |
\( T^{10} + \cdots - 64\!\cdots\!08 \)
|
| $83$ |
\( T^{10} + \cdots - 30\!\cdots\!60 \)
|
| $89$ |
\( T^{10} + \cdots + 74\!\cdots\!12 \)
|
| $97$ |
\( T^{10} + \cdots + 48\!\cdots\!24 \)
|
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