Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [799,2,Mod(1,799)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(799, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("799.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 799 = 17 \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 799.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: | \(6.38004712150\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{12} - 4 x^{11} - 8 x^{10} + 44 x^{9} + 11 x^{8} - 168 x^{7} + 41 x^{6} + 272 x^{5} - 111 x^{4} + \cdots - 17 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{12} - 4 x^{11} - 8 x^{10} + 44 x^{9} + 11 x^{8} - 168 x^{7} + 41 x^{6} + 272 x^{5} - 111 x^{4} + \cdots - 17 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23 \nu^{11} + 42 \nu^{10} + 314 \nu^{9} - 430 \nu^{8} - 1656 \nu^{7} + 1421 \nu^{6} + 4108 \nu^{5} + \cdots - 470 ) / 89 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 43 \nu^{11} - 183 \nu^{10} - 351 \nu^{9} + 2077 \nu^{8} + 604 \nu^{7} - 8194 \nu^{6} + 829 \nu^{5} + \cdots + 1107 ) / 178 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 26 \nu^{11} - 121 \nu^{10} - 146 \nu^{9} + 1260 \nu^{8} - 264 \nu^{7} - 4435 \nu^{6} + 2422 \nu^{5} + \cdots + 392 ) / 89 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 49 \nu^{11} + 163 \nu^{10} + 549 \nu^{9} - 1957 \nu^{8} - 2282 \nu^{7} + 8526 \nu^{6} + 4801 \nu^{5} + \cdots - 3443 ) / 178 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 53 \nu^{11} + 209 \nu^{10} + 414 \nu^{9} - 2233 \nu^{8} - 523 \nu^{7} + 8065 \nu^{6} - 2007 \nu^{5} + \cdots - 847 ) / 89 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 125 \nu^{11} + 503 \nu^{10} + 921 \nu^{9} - 5243 \nu^{8} - 812 \nu^{7} + 18190 \nu^{6} - 5469 \nu^{5} + \cdots - 755 ) / 178 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 137 \nu^{11} + 463 \nu^{10} + 1317 \nu^{9} - 5003 \nu^{8} - 3990 \nu^{7} + 18498 \nu^{6} + \cdots - 3113 ) / 178 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 207 \nu^{11} + 645 \nu^{10} + 2203 \nu^{9} - 7163 \nu^{8} - 8140 \nu^{7} + 27474 \nu^{6} + \cdots - 5209 ) / 178 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 259 \nu^{11} - 887 \nu^{10} - 2495 \nu^{9} + 9683 \nu^{8} + 7612 \nu^{7} - 36344 \nu^{6} - 8009 \nu^{5} + \cdots + 6349 ) / 178 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{5} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + 2\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 7\beta_{2} + 8\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9 \beta_{11} + 9 \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} - 7 \beta_{5} + \beta_{4} + \cdots + 31 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 13 \beta_{11} + 20 \beta_{10} - 5 \beta_{9} + 10 \beta_{8} - 10 \beta_{7} + \beta_{6} - 10 \beta_{5} + \cdots + 30 \)
|
| \(\nu^{7}\) | \(=\) |
\( 69 \beta_{11} + 67 \beta_{10} + 7 \beta_{9} + 23 \beta_{8} - 14 \beta_{7} + 10 \beta_{6} - 43 \beta_{5} + \cdots + 1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 125 \beta_{11} + 157 \beta_{10} - \beta_{9} + 84 \beta_{8} - 83 \beta_{7} + 14 \beta_{6} - 73 \beta_{5} + \cdots + 139 \)
|
| \(\nu^{9}\) | \(=\) |
\( 512 \beta_{11} + 474 \beta_{10} + 121 \beta_{9} + 208 \beta_{8} - 150 \beta_{7} + 79 \beta_{6} + \cdots + 10 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1074 \beta_{11} + 1150 \beta_{10} + 264 \beta_{9} + 673 \beta_{8} - 663 \beta_{7} + 139 \beta_{6} + \cdots + 655 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3785 \beta_{11} + 3304 \beta_{10} + 1419 \beta_{9} + 1738 \beta_{8} - 1416 \beta_{7} + 591 \beta_{6} + \cdots + 34 \)
|
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 |
|
−2.71308 | 1.21731 | 5.36082 | −2.50173 | −3.30266 | 3.12035 | −9.11818 | −1.51816 | 6.78741 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.50912 | −1.79229 | 4.29566 | −0.903725 | 4.49706 | 0.667112 | −5.76007 | 0.212306 | 2.26755 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.79209 | −1.51007 | 1.21159 | 3.47759 | 2.70619 | 0.267552 | 1.41290 | −0.719681 | −6.23215 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.49145 | 2.89346 | 0.224430 | −2.71401 | −4.31546 | 0.661301 | 2.64818 | 5.37212 | 4.04782 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.26302 | 0.309492 | −0.404793 | −2.50842 | −0.390894 | 1.11284 | 3.03729 | −2.90421 | 3.16818 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.551268 | 0.225258 | −1.69610 | 1.20101 | −0.124178 | −2.44081 | 2.03754 | −2.94926 | −0.662077 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.465230 | −2.94303 | −1.78356 | −0.522650 | 1.36919 | −0.353063 | 1.76023 | 5.66144 | 0.243152 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.676150 | 1.57724 | −1.54282 | 0.592656 | 1.06645 | −2.73886 | −2.39548 | −0.512301 | 0.400724 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.829033 | −0.269629 | −1.31270 | −1.24530 | −0.223531 | 4.95448 | −2.74634 | −2.92730 | −1.03240 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.35580 | 1.74298 | −0.161810 | −2.78238 | 2.36313 | −3.70739 | −2.93098 | 0.0379915 | −3.77234 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.72832 | −1.57467 | 0.987078 | 0.906069 | −2.72153 | −0.995688 | −1.75065 | −0.520416 | 1.56597 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 2.19596 | −0.876055 | 2.82222 | −3.99910 | −1.92378 | 0.452168 | 1.80556 | −2.23253 | −8.78184 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \( +1 \) |
| \(47\) | \( +1 \) |
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 | 799.2.a.e | ✓ | 12 |
| 3.b | odd | 2 | 1 | 7191.2.a.v | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 799.2.a.e | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 7191.2.a.v | 12 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(799))\):
|
\( T_{2}^{12} + 4 T_{2}^{11} - 8 T_{2}^{10} - 44 T_{2}^{9} + 11 T_{2}^{8} + 168 T_{2}^{7} + 41 T_{2}^{6} + \cdots - 17 \)
|
|
\( T_{5}^{12} + 11 T_{5}^{11} + 30 T_{5}^{10} - 82 T_{5}^{9} - 572 T_{5}^{8} - 805 T_{5}^{7} + 713 T_{5}^{6} + \cdots + 250 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + 4 T^{11} + \cdots - 17 \)
|
| $3$ |
\( T^{12} + T^{11} + \cdots + 2 \)
|
| $5$ |
\( T^{12} + 11 T^{11} + \cdots + 250 \)
|
| $7$ |
\( T^{12} - T^{11} + \cdots - 8 \)
|
| $11$ |
\( T^{12} + 6 T^{11} + \cdots + 40472 \)
|
| $13$ |
\( T^{12} + 13 T^{11} + \cdots + 7444 \)
|
| $17$ |
\( (T + 1)^{12} \)
|
| $19$ |
\( T^{12} + 9 T^{11} + \cdots + 76540 \)
|
| $23$ |
\( T^{12} + 2 T^{11} + \cdots + 1494920 \)
|
| $29$ |
\( T^{12} + 14 T^{11} + \cdots - 21512 \)
|
| $31$ |
\( T^{12} + 18 T^{11} + \cdots - 8416 \)
|
| $37$ |
\( T^{12} - 8 T^{11} + \cdots + 14253196 \)
|
| $41$ |
\( T^{12} + 43 T^{11} + \cdots + 7535336 \)
|
| $43$ |
\( T^{12} + 7 T^{11} + \cdots - 17612 \)
|
| $47$ |
\( (T + 1)^{12} \)
|
| $53$ |
\( T^{12} + \cdots - 151695260 \)
|
| $59$ |
\( T^{12} + 25 T^{11} + \cdots - 93639488 \)
|
| $61$ |
\( T^{12} + \cdots + 126046784 \)
|
| $67$ |
\( T^{12} + \cdots + 1146468764 \)
|
| $71$ |
\( T^{12} + \cdots + 10302753706 \)
|
| $73$ |
\( T^{12} + \cdots + 4216197410 \)
|
| $79$ |
\( T^{12} + \cdots - 1851768206 \)
|
| $83$ |
\( T^{12} + 45 T^{11} + \cdots - 3542984 \)
|
| $89$ |
\( T^{12} + 65 T^{11} + \cdots - 2646016 \)
|
| $97$ |
\( T^{12} + 21 T^{11} + \cdots - 33823804 \)
|
| show more | |
| show less |