Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5415,2,Mod(1,5415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("5415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5415.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: | \(43.2389926945\) |
| 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: |
\( x^{12} - 4 x^{11} - 9 x^{10} + 50 x^{9} + 9 x^{8} - 208 x^{7} + 82 x^{6} + 358 x^{5} - 204 x^{4} + \cdots - 19 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{12} - 4 x^{11} - 9 x^{10} + 50 x^{9} + 9 x^{8} - 208 x^{7} + 82 x^{6} + 358 x^{5} - 204 x^{4} + \cdots - 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 37 \nu^{11} + 546 \nu^{10} - 2819 \nu^{9} - 4972 \nu^{8} + 30484 \nu^{7} + 6921 \nu^{6} + \cdots + 14579 ) / 1361 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 140 \nu^{11} + 325 \nu^{10} + 2243 \nu^{9} - 4839 \nu^{8} - 13417 \nu^{7} + 25604 \nu^{6} + \cdots - 8853 ) / 1361 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 198 \nu^{11} - 1529 \nu^{10} + 658 \nu^{9} + 17129 \nu^{8} - 26968 \nu^{7} - 55732 \nu^{6} + \cdots - 11919 ) / 1361 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 243 \nu^{11} - 165 \nu^{10} + 5750 \nu^{9} + 321 \nu^{8} - 45346 \nu^{7} + 11855 \nu^{6} + \cdots - 20820 ) / 1361 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 379 \nu^{11} - 2095 \nu^{10} - 1435 \nu^{9} + 24367 \nu^{8} - 19314 \nu^{7} - 87240 \nu^{6} + \cdots - 9349 ) / 1361 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 498 \nu^{11} + 670 \nu^{10} + 8037 \nu^{9} - 9222 \nu^{8} - 47279 \nu^{7} + 44492 \nu^{6} + \cdots - 14440 ) / 1361 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 760 \nu^{11} + 2542 \nu^{10} + 7510 \nu^{9} - 29963 \nu^{8} - 16062 \nu^{7} + 110801 \nu^{6} + \cdots + 548 ) / 1361 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 760 \nu^{11} - 2542 \nu^{10} - 7510 \nu^{9} + 29963 \nu^{8} + 16062 \nu^{7} - 110801 \nu^{6} + \cdots - 4631 ) / 1361 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 1120 \nu^{11} - 2600 \nu^{10} - 15222 \nu^{9} + 33268 \nu^{8} + 73311 \nu^{7} - 143587 \nu^{6} + \cdots + 25911 ) / 1361 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1934 \nu^{11} + 5559 \nu^{10} + 22800 \nu^{9} - 68150 \nu^{8} - 85235 \nu^{7} + 271691 \nu^{6} + \cdots - 14565 ) / 1361 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{8} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} + 7\beta_{9} + 6\beta_{8} - \beta_{6} + 2\beta_{4} + \beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{11} + 8 \beta_{10} + 2 \beta_{9} - \beta_{8} + 8 \beta_{7} - 10 \beta_{6} + 9 \beta_{5} + \cdots - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 10 \beta_{11} + 10 \beta_{10} + 47 \beta_{9} + 35 \beta_{8} - 13 \beta_{6} + 4 \beta_{5} + 21 \beta_{4} + \cdots + 66 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{11} + 57 \beta_{10} + 27 \beta_{9} - 13 \beta_{8} + 56 \beta_{7} - 81 \beta_{6} + 70 \beta_{5} + \cdots - 23 \)
|
| \(\nu^{8}\) | \(=\) |
\( 84 \beta_{11} + 84 \beta_{10} + 321 \beta_{9} + 206 \beta_{8} + 2 \beta_{7} - 123 \beta_{6} + 60 \beta_{5} + \cdots + 362 \)
|
| \(\nu^{9}\) | \(=\) |
\( 144 \beta_{11} + 405 \beta_{10} + 271 \beta_{9} - 115 \beta_{8} + 377 \beta_{7} - 615 \beta_{6} + \cdots - 193 \)
|
| \(\nu^{10}\) | \(=\) |
\( 672 \beta_{11} + 676 \beta_{10} + 2231 \beta_{9} + 1220 \beta_{8} + 44 \beta_{7} - 1045 \beta_{6} + \cdots + 2065 \)
|
| \(\nu^{11}\) | \(=\) |
\( 1304 \beta_{11} + 2907 \beta_{10} + 2431 \beta_{9} - 873 \beta_{8} + 2506 \beta_{7} - 4562 \beta_{6} + \cdots - 1433 \)
|
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 |
|
−2.48065 | −1.00000 | 4.15364 | 1.00000 | 2.48065 | −4.92831 | −5.34244 | 1.00000 | −2.48065 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.78314 | −1.00000 | 1.17959 | 1.00000 | 1.78314 | −2.24903 | 1.46290 | 1.00000 | −1.78314 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.39592 | −1.00000 | −0.0514143 | 1.00000 | 1.39592 | −3.49575 | 2.86361 | 1.00000 | −1.39592 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.741114 | −1.00000 | −1.45075 | 1.00000 | 0.741114 | −1.35634 | 2.55740 | 1.00000 | −0.741114 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.703138 | −1.00000 | −1.50560 | 1.00000 | 0.703138 | 2.40347 | 2.46492 | 1.00000 | −0.703138 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.223198 | −1.00000 | −1.95018 | 1.00000 | −0.223198 | −2.94956 | −0.881674 | 1.00000 | 0.223198 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.04414 | −1.00000 | −0.909765 | 1.00000 | −1.04414 | 2.28851 | −3.03821 | 1.00000 | 1.04414 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.38437 | −1.00000 | −0.0835125 | 1.00000 | −1.38437 | 3.64105 | −2.88436 | 1.00000 | 1.38437 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.57066 | −1.00000 | 0.466982 | 1.00000 | −1.57066 | −1.75320 | −2.40785 | 1.00000 | 1.57066 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.87254 | −1.00000 | 1.50641 | 1.00000 | −1.87254 | −0.138560 | −0.924263 | 1.00000 | 1.87254 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.28168 | −1.00000 | 3.20606 | 1.00000 | −2.28168 | −1.99531 | 2.75183 | 1.00000 | 2.28168 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.72737 | −1.00000 | 5.43853 | 1.00000 | −2.72737 | 4.53303 | 9.37815 | 1.00000 | 2.72737 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(1\) |
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 | 5415.2.a.bq | yes | 12 |
| 19.b | odd | 2 | 1 | 5415.2.a.bn | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5415.2.a.bn | ✓ | 12 | 19.b | odd | 2 | 1 | |
| 5415.2.a.bq | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{12} - 4 T_{2}^{11} - 9 T_{2}^{10} + 50 T_{2}^{9} + 9 T_{2}^{8} - 208 T_{2}^{7} + 82 T_{2}^{6} + \cdots - 19 \)
|
|
\( T_{7}^{12} + 6 T_{7}^{11} - 34 T_{7}^{10} - 260 T_{7}^{9} + 184 T_{7}^{8} + 3682 T_{7}^{7} + 3407 T_{7}^{6} + \cdots + 6821 \)
|
|
\( T_{11}^{12} + 10 T_{11}^{11} - 14 T_{11}^{10} - 386 T_{11}^{9} - 559 T_{11}^{8} + 3908 T_{11}^{7} + \cdots + 81 \)
|
|
\( T_{13}^{12} - 12 T_{13}^{11} - 42 T_{13}^{10} + 940 T_{13}^{9} - 1041 T_{13}^{8} - 22464 T_{13}^{7} + \cdots + 79801 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} - 4 T^{11} + \cdots - 19 \)
|
| $3$ |
\( (T + 1)^{12} \)
|
| $5$ |
\( (T - 1)^{12} \)
|
| $7$ |
\( T^{12} + 6 T^{11} + \cdots + 6821 \)
|
| $11$ |
\( T^{12} + 10 T^{11} + \cdots + 81 \)
|
| $13$ |
\( T^{12} - 12 T^{11} + \cdots + 79801 \)
|
| $17$ |
\( T^{12} + 12 T^{11} + \cdots + 5449181 \)
|
| $19$ |
\( T^{12} \)
|
| $23$ |
\( T^{12} + 22 T^{11} + \cdots - 47975 \)
|
| $29$ |
\( T^{12} - 44 T^{11} + \cdots + 6205481 \)
|
| $31$ |
\( T^{12} + \cdots - 181035059 \)
|
| $37$ |
\( T^{12} - 10 T^{11} + \cdots + 7351241 \)
|
| $41$ |
\( T^{12} + \cdots - 324849895 \)
|
| $43$ |
\( T^{12} + 26 T^{11} + \cdots - 24816475 \)
|
| $47$ |
\( T^{12} + 14 T^{11} + \cdots + 13964305 \)
|
| $53$ |
\( T^{12} + \cdots + 1263881461 \)
|
| $59$ |
\( T^{12} + \cdots + 139315181 \)
|
| $61$ |
\( T^{12} + \cdots + 4450853741 \)
|
| $67$ |
\( T^{12} + \cdots + 199509701 \)
|
| $71$ |
\( T^{12} - 10 T^{11} + \cdots - 878975 \)
|
| $73$ |
\( T^{12} + \cdots - 70641129239 \)
|
| $79$ |
\( T^{12} + \cdots - 870629875 \)
|
| $83$ |
\( T^{12} + \cdots - 1725912899 \)
|
| $89$ |
\( T^{12} + \cdots + 1519246681 \)
|
| $97$ |
\( T^{12} + \cdots + 67821850781 \)
|
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