Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [19,14,Mod(1,19)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(19, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("19.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 19 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 19.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: | \(20.3738765009\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3 x^{8} - 52410 x^{7} + 406204 x^{6} + 897751800 x^{5} - 12830529264 x^{4} + \cdots - 12\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{11}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3 x^{8} - 52410 x^{7} + 406204 x^{6} + 897751800 x^{5} - 12830529264 x^{4} + \cdots - 12\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2836605030931 \nu^{8} - 464393667479871 \nu^{7} + \cdots + 49\!\cdots\!88 ) / 18\!\cdots\!36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2836605030931 \nu^{8} - 464393667479871 \nu^{7} + \cdots - 61\!\cdots\!12 ) / 94\!\cdots\!68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 181258093914655 \nu^{8} + \cdots - 10\!\cdots\!00 ) / 18\!\cdots\!36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 46012593085903 \nu^{8} + \cdots - 41\!\cdots\!76 ) / 47\!\cdots\!84 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 393673805612291 \nu^{8} + \cdots - 15\!\cdots\!60 ) / 18\!\cdots\!36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 447888235317953 \nu^{8} + \cdots + 15\!\cdots\!60 ) / 18\!\cdots\!36 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 16\!\cdots\!33 \nu^{8} + \cdots + 57\!\cdots\!28 ) / 18\!\cdots\!36 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 2\beta_{2} - 7\beta _1 + 11650 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + 8\beta_{7} - 7\beta_{6} + 5\beta_{5} - 4\beta_{4} + 4\beta_{3} - 234\beta_{2} + 18197\beta _1 - 89070 \)
|
| \(\nu^{4}\) | \(=\) |
\( 283 \beta_{8} + 752 \beta_{7} + 587 \beta_{6} - 349 \beta_{5} - 316 \beta_{4} + 24030 \beta_{3} + \cdots + 211112042 \)
|
| \(\nu^{5}\) | \(=\) |
\( 35139 \beta_{8} + 260784 \beta_{7} - 211197 \beta_{6} + 232803 \beta_{5} - 200124 \beta_{4} + \cdots - 1741775062 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11102315 \beta_{8} + 27390064 \beta_{7} + 22933675 \beta_{6} - 13127093 \beta_{5} - 9132764 \beta_{4} + \cdots + 4262057976090 \)
|
| \(\nu^{7}\) | \(=\) |
\( 878783027 \beta_{8} + 6993847408 \beta_{7} - 5599325837 \beta_{6} + 7152406675 \beta_{5} + \cdots - 22196113910294 \)
|
| \(\nu^{8}\) | \(=\) |
\( 342597684075 \beta_{8} + 771993888624 \beta_{7} + 677712761643 \beta_{6} - 377842770549 \beta_{5} + \cdots + 91\!\cdots\!10 \)
|
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 |
|
−170.554 | 1590.00 | 20896.8 | −14341.9 | −271182. | −43865.1 | −2.16686e6 | 933789. | 2.44608e6 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −129.538 | −1976.19 | 8588.14 | −51907.1 | 255992. | −208548. | −51315.2 | 2.31100e6 | 6.72395e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −117.199 | 895.295 | 5543.72 | 10358.6 | −104928. | −343975. | 310377. | −792770. | −1.21403e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −51.8583 | −448.573 | −5502.72 | −180.798 | 23262.2 | 432134. | 710185. | −1.39311e6 | 9375.88 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −35.8847 | −2287.91 | −6904.29 | 26790.5 | 82101.0 | 137823. | 541725. | 3.64022e6 | −961367. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 22.4951 | 1462.94 | −7685.97 | 13997.4 | 32909.0 | −234474. | −357177. | 545861. | 314874. | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 96.1158 | 537.125 | 1046.25 | −45995.8 | 51626.2 | 526355. | −686820. | −1.30582e6 | −4.42092e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 118.284 | −1376.59 | 5799.05 | 36643.3 | −162829. | −130590. | −283048. | 300685. | 4.33431e6 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 139.140 | 275.905 | 11168.0 | −37306.2 | 38389.5 | −570821. | 414085. | −1.51820e6 | −5.19079e6 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \(1\) |
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 | 19.14.a.a | ✓ | 9 |
| 3.b | odd | 2 | 1 | 171.14.a.c | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.14.a.a | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 171.14.a.c | 9 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + 129 T_{2}^{8} - 45018 T_{2}^{7} - 5295424 T_{2}^{6} + 653212512 T_{2}^{5} + \cdots + 17\!\cdots\!64 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(19))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + \cdots + 17\!\cdots\!64 \)
|
| $3$ |
\( T^{9} + \cdots - 86\!\cdots\!44 \)
|
| $5$ |
\( T^{9} + \cdots + 32\!\cdots\!00 \)
|
| $7$ |
\( T^{9} + \cdots - 17\!\cdots\!84 \)
|
| $11$ |
\( T^{9} + \cdots + 22\!\cdots\!96 \)
|
| $13$ |
\( T^{9} + \cdots + 50\!\cdots\!44 \)
|
| $17$ |
\( T^{9} + \cdots + 11\!\cdots\!54 \)
|
| $19$ |
\( (T + 47045881)^{9} \)
|
| $23$ |
\( T^{9} + \cdots - 26\!\cdots\!36 \)
|
| $29$ |
\( T^{9} + \cdots + 68\!\cdots\!60 \)
|
| $31$ |
\( T^{9} + \cdots - 79\!\cdots\!44 \)
|
| $37$ |
\( T^{9} + \cdots - 38\!\cdots\!56 \)
|
| $41$ |
\( T^{9} + \cdots + 14\!\cdots\!36 \)
|
| $43$ |
\( T^{9} + \cdots - 12\!\cdots\!52 \)
|
| $47$ |
\( T^{9} + \cdots - 10\!\cdots\!84 \)
|
| $53$ |
\( T^{9} + \cdots - 16\!\cdots\!52 \)
|
| $59$ |
\( T^{9} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{9} + \cdots - 19\!\cdots\!68 \)
|
| $67$ |
\( T^{9} + \cdots + 12\!\cdots\!56 \)
|
| $71$ |
\( T^{9} + \cdots - 10\!\cdots\!96 \)
|
| $73$ |
\( T^{9} + \cdots + 34\!\cdots\!78 \)
|
| $79$ |
\( T^{9} + \cdots - 20\!\cdots\!40 \)
|
| $83$ |
\( T^{9} + \cdots - 21\!\cdots\!92 \)
|
| $89$ |
\( T^{9} + \cdots + 99\!\cdots\!40 \)
|
| $97$ |
\( T^{9} + \cdots - 13\!\cdots\!08 \)
|
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