Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,10,Mod(1,147)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(147, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 147.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: | \(75.7102679161\) |
| 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: |
\( x^{10} - 4 x^{9} - 3802 x^{8} + 19928 x^{7} + 4922742 x^{6} - 34016432 x^{5} - 2500007760 x^{4} + \cdots + 19340135623186 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{2}\cdot 7^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 4 x^{9} - 3802 x^{8} + 19928 x^{7} + 4922742 x^{6} - 34016432 x^{5} - 2500007760 x^{4} + \cdots + 19340135623186 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 351070661325529 \nu^{9} + \cdots + 93\!\cdots\!18 ) / 76\!\cdots\!88 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 17\!\cdots\!21 \nu^{9} + \cdots + 45\!\cdots\!82 ) / 30\!\cdots\!52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 17\!\cdots\!51 \nu^{9} + \cdots + 35\!\cdots\!10 ) / 30\!\cdots\!52 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 34\!\cdots\!41 \nu^{9} + \cdots - 91\!\cdots\!06 ) / 35\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 10\!\cdots\!23 \nu^{9} + \cdots + 26\!\cdots\!02 ) / 35\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 89\!\cdots\!73 \nu^{9} + \cdots + 13\!\cdots\!10 ) / 15\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 29\!\cdots\!55 \nu^{9} + \cdots - 10\!\cdots\!58 ) / 53\!\cdots\!60 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 56\!\cdots\!27 \nu^{9} + \cdots + 14\!\cdots\!22 ) / 10\!\cdots\!84 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 51\!\cdots\!03 \nu^{9} + \cdots - 98\!\cdots\!14 ) / 53\!\cdots\!60 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{2} - 49\beta_1 ) / 49 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{6} - 41\beta_{3} - 49\beta_{2} + 97\beta _1 + 37367 ) / 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -35\beta_{8} + 13\beta_{6} + 147\beta_{5} + 49\beta_{4} - 218\beta_{3} + 7533\beta_{2} - 58820\beta _1 - 92583 ) / 49 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 637 \beta_{9} + 497 \beta_{8} + 7 \beta_{7} + 17296 \beta_{6} - 84 \beta_{5} - 924 \beta_{4} + \cdots + 44838674 ) / 49 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 5341 \beta_{9} - 97034 \beta_{8} + 2009 \beta_{7} - 8913 \beta_{6} + 316393 \beta_{5} + \cdots - 122819944 ) / 49 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 222740 \beta_{9} + 176760 \beta_{8} - 5700 \beta_{7} + 4535468 \beta_{6} - 331580 \beta_{5} + \cdots + 8826692045 ) / 7 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 19866560 \beta_{9} - 206443321 \beta_{8} + 3339840 \beta_{7} - 77000665 \beta_{6} + \cdots - 204407365421 ) / 49 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3010622279 \beta_{9} + 2449504715 \beta_{8} - 218522171 \beta_{7} + 55315375032 \beta_{6} + \cdots + 90599878927944 ) / 49 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 47979734495 \beta_{9} - 394523979584 \beta_{8} + 3400039475 \beta_{7} - 228501521277 \beta_{6} + \cdots - 389698009981274 ) / 49 \)
|
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 |
|
−36.9053 | 81.0000 | 850.004 | 1542.64 | −2989.33 | 0 | −12474.2 | 6561.00 | −56931.8 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −32.2484 | 81.0000 | 527.959 | −1223.57 | −2612.12 | 0 | −514.666 | 6561.00 | 39458.1 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −25.1604 | 81.0000 | 121.045 | −1284.25 | −2037.99 | 0 | 9836.57 | 6561.00 | 32312.3 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −16.3248 | 81.0000 | −245.501 | 2098.09 | −1322.31 | 0 | 12366.1 | 6561.00 | −34250.8 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.01742 | 81.0000 | −495.860 | 2145.02 | 325.411 | 0 | −4049.00 | 6561.00 | 8617.44 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 11.1866 | 81.0000 | −386.860 | −1768.41 | 906.115 | 0 | −10055.2 | 6561.00 | −19782.5 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 16.9299 | 81.0000 | −225.378 | 851.551 | 1371.32 | 0 | −12483.7 | 6561.00 | 14416.7 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 31.7120 | 81.0000 | 493.652 | −2528.82 | 2568.67 | 0 | −581.840 | 6561.00 | −80193.9 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 36.9916 | 81.0000 | 856.382 | 528.735 | 2996.32 | 0 | 12739.3 | 6561.00 | 19558.8 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 43.8013 | 81.0000 | 1406.56 | 2139.02 | 3547.91 | 0 | 39182.7 | 6561.00 | 93691.7 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
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 | 147.10.a.n | yes | 10 |
| 7.b | odd | 2 | 1 | 147.10.a.m | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.10.a.m | ✓ | 10 | 7.b | odd | 2 | 1 | |
| 147.10.a.n | yes | 10 | 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 \(S_{10}^{\mathrm{new}}(\Gamma_0(147))\):
|
\( T_{2}^{10} - 34 T_{2}^{9} - 3433 T_{2}^{8} + 106896 T_{2}^{7} + 4043144 T_{2}^{6} + \cdots + 19110761689088 \)
|
|
\( T_{5}^{10} - 2500 T_{5}^{9} - 11690758 T_{5}^{8} + 31876380768 T_{5}^{7} + 40632537385832 T_{5}^{6} + \cdots + 46\!\cdots\!00 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 19110761689088 \)
|
| $3$ |
\( (T - 81)^{10} \)
|
| $5$ |
\( T^{10} + \cdots + 46\!\cdots\!00 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( T^{10} + \cdots - 28\!\cdots\!92 \)
|
| $13$ |
\( T^{10} + \cdots + 58\!\cdots\!76 \)
|
| $17$ |
\( T^{10} + \cdots - 15\!\cdots\!32 \)
|
| $19$ |
\( T^{10} + \cdots - 12\!\cdots\!08 \)
|
| $23$ |
\( T^{10} + \cdots + 94\!\cdots\!48 \)
|
| $29$ |
\( T^{10} + \cdots + 15\!\cdots\!48 \)
|
| $31$ |
\( T^{10} + \cdots - 26\!\cdots\!08 \)
|
| $37$ |
\( T^{10} + \cdots - 20\!\cdots\!52 \)
|
| $41$ |
\( T^{10} + \cdots - 55\!\cdots\!52 \)
|
| $43$ |
\( T^{10} + \cdots - 29\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots + 13\!\cdots\!76 \)
|
| $53$ |
\( T^{10} + \cdots - 11\!\cdots\!32 \)
|
| $59$ |
\( T^{10} + \cdots + 37\!\cdots\!88 \)
|
| $61$ |
\( T^{10} + \cdots + 42\!\cdots\!84 \)
|
| $67$ |
\( T^{10} + \cdots + 23\!\cdots\!92 \)
|
| $71$ |
\( T^{10} + \cdots - 21\!\cdots\!04 \)
|
| $73$ |
\( T^{10} + \cdots + 37\!\cdots\!16 \)
|
| $79$ |
\( T^{10} + \cdots - 17\!\cdots\!52 \)
|
| $83$ |
\( T^{10} + \cdots + 21\!\cdots\!36 \)
|
| $89$ |
\( T^{10} + \cdots - 84\!\cdots\!84 \)
|
| $97$ |
\( T^{10} + \cdots - 54\!\cdots\!12 \)
|
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