Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5239,2,Mod(1,5239)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5239, 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("5239.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5239 = 13^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5239.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: | \(41.8336256189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} - 5 x^{17} - 15 x^{16} + 105 x^{15} + 50 x^{14} - 876 x^{13} + 294 x^{12} + 3702 x^{11} + \cdots + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 403) |
| 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 \(1,\beta_1,\ldots,\beta_{17}\) 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
\( x^{18} - 5 x^{17} - 15 x^{16} + 105 x^{15} + 50 x^{14} - 876 x^{13} + 294 x^{12} + 3702 x^{11} + \cdots + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 17 \nu^{17} + 162 \nu^{16} - 28 \nu^{15} - 3280 \nu^{14} + 5112 \nu^{13} + 26098 \nu^{12} + \cdots + 1974 ) / 474 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 45327 \nu^{17} + 193276 \nu^{16} + 803057 \nu^{15} - 4102782 \nu^{14} - 4895200 \nu^{13} + \cdots + 686697 ) / 35076 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 26517 \nu^{17} + 98344 \nu^{16} + 520055 \nu^{15} - 2091906 \nu^{14} - 3947152 \nu^{13} + \cdots - 99873 ) / 17538 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 77417 \nu^{17} + 334476 \nu^{16} + 1352945 \nu^{15} - 7083946 \nu^{14} - 7964604 \nu^{13} + \cdots + 1022697 ) / 35076 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 42983 \nu^{17} + 154837 \nu^{16} + 862069 \nu^{15} - 3310168 \nu^{14} - 6798940 \nu^{13} + \cdots - 4443 ) / 17538 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 90333 \nu^{17} - 293684 \nu^{16} - 1924627 \nu^{15} + 6313014 \nu^{14} + 16710296 \nu^{13} + \cdots + 647277 ) / 35076 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 105331 \nu^{17} + 482454 \nu^{16} + 1751581 \nu^{15} - 10212518 \nu^{14} - 8931252 \nu^{13} + \cdots + 2094033 ) / 35076 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 57118 \nu^{17} - 224021 \nu^{16} - 1080170 \nu^{15} + 4768856 \nu^{14} + 7631564 \nu^{13} + \cdots - 307770 ) / 17538 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 113809 \nu^{17} - 437876 \nu^{16} - 2187815 \nu^{15} + 9345386 \nu^{14} + 15965720 \nu^{13} + \cdots - 810723 ) / 35076 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 113809 \nu^{17} + 437876 \nu^{16} + 2187815 \nu^{15} - 9345386 \nu^{14} - 15965720 \nu^{13} + \cdots + 880875 ) / 35076 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 110847 \nu^{17} + 337100 \nu^{16} + 2453059 \nu^{15} - 7310238 \nu^{14} - 22437956 \nu^{13} + \cdots - 325857 ) / 35076 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 114463 \nu^{17} - 420780 \nu^{16} - 2261983 \nu^{15} + 8973758 \nu^{14} + 17390064 \nu^{13} + \cdots + 128421 ) / 35076 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 131169 \nu^{17} - 480680 \nu^{16} - 2604559 \nu^{15} + 10275270 \nu^{14} + 20187992 \nu^{13} + \cdots - 236199 ) / 35076 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 78955 \nu^{17} - 274928 \nu^{16} - 1618496 \nu^{15} + 5895062 \nu^{14} + 13231124 \nu^{13} + \cdots - 27174 ) / 17538 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 182945 \nu^{17} + 665380 \nu^{16} + 3646741 \nu^{15} - 14216362 \nu^{14} - 28460584 \nu^{13} + \cdots + 241137 ) / 35076 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} + \beta_{11} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{17} + \beta_{14} + \beta_{12} + 2\beta_{11} + \beta_{4} + 8\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{16} + \beta_{15} + \beta_{14} + 8 \beta_{12} + 10 \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8 \beta_{17} - 2 \beta_{16} + \beta_{15} + 11 \beta_{14} + 10 \beta_{12} + 23 \beta_{11} - 2 \beta_{10} + \cdots + 87 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 14 \beta_{16} + 13 \beta_{15} + 16 \beta_{14} - \beta_{13} + 55 \beta_{12} + 82 \beta_{11} + \cdots + 88 \)
|
| \(\nu^{8}\) | \(=\) |
\( 51 \beta_{17} - 30 \beta_{16} + 15 \beta_{15} + 98 \beta_{14} + 80 \beta_{12} + 208 \beta_{11} + \cdots + 550 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2 \beta_{17} - 138 \beta_{16} + 121 \beta_{15} + 176 \beta_{14} - 13 \beta_{13} + 370 \beta_{12} + \cdots + 741 \)
|
| \(\nu^{10}\) | \(=\) |
\( 300 \beta_{17} - 312 \beta_{16} + 158 \beta_{15} + 815 \beta_{14} + 607 \beta_{12} + 1729 \beta_{11} + \cdots + 3682 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 39 \beta_{17} - 1188 \beta_{16} + 989 \beta_{15} + 1671 \beta_{14} - 115 \beta_{13} + 2511 \beta_{12} + \cdots + 6084 \)
|
| \(\nu^{12}\) | \(=\) |
\( 1677 \beta_{17} - 2797 \beta_{16} + 1440 \beta_{15} + 6572 \beta_{14} + 6 \beta_{13} + 4561 \beta_{12} + \cdots + 25688 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 514 \beta_{17} - 9562 \beta_{16} + 7589 \beta_{15} + 14715 \beta_{14} - 844 \beta_{13} + 17345 \beta_{12} + \cdots + 49155 \)
|
| \(\nu^{14}\) | \(=\) |
\( 8869 \beta_{17} - 23212 \beta_{16} + 12147 \beta_{15} + 52146 \beta_{14} + 174 \beta_{13} + \cdots + 184786 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 5739 \beta_{17} - 74083 \beta_{16} + 56281 \beta_{15} + 124041 \beta_{14} - 5375 \beta_{13} + \cdots + 392987 \)
|
| \(\nu^{16}\) | \(=\) |
\( 42956 \beta_{17} - 184045 \beta_{16} + 97716 \beta_{15} + 409921 \beta_{14} + 3055 \beta_{13} + \cdots + 1359763 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 58464 \beta_{17} - 560585 \beta_{16} + 409267 \beta_{15} + 1017449 \beta_{14} - 28986 \beta_{13} + \cdots + 3119994 \)
|
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 |
|
−2.41340 | 1.76491 | 3.82450 | 3.22507 | −4.25943 | 2.94693 | −4.40325 | 0.114894 | −7.78339 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.25209 | −3.32589 | 3.07190 | −0.820579 | 7.49019 | −3.93743 | −2.41402 | 8.06151 | 1.84802 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.12512 | −1.58668 | 2.51612 | −1.24766 | 3.37187 | 3.24266 | −1.09681 | −0.482455 | 2.65143 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.61973 | 0.527034 | 0.623515 | −1.44443 | −0.853651 | −2.00144 | 2.22953 | −2.72224 | 2.33959 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.29375 | 2.99396 | −0.326220 | 2.87150 | −3.87343 | −1.15440 | 3.00954 | 5.96380 | −3.71499 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.994497 | −2.24109 | −1.01098 | 0.441402 | 2.22876 | −0.726525 | 2.99441 | 2.02250 | −0.438972 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.275968 | 1.46450 | −1.92384 | −1.37475 | −0.404155 | −0.479560 | 1.08286 | −0.855243 | 0.379388 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.210308 | −0.138682 | −1.95577 | 1.89472 | 0.0291659 | 4.57273 | 0.831932 | −2.98077 | −0.398475 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.0992449 | −1.63256 | −1.99015 | −0.352370 | −0.162023 | −4.11405 | −0.396002 | −0.334746 | −0.0349709 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.114709 | 3.14754 | −1.98684 | −1.85062 | 0.361050 | 3.91805 | −0.457325 | 6.90700 | −0.212282 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.36617 | −2.86498 | −0.133569 | −1.11791 | −3.91407 | 2.10992 | −2.91483 | 5.20813 | −1.52725 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.37173 | 1.54115 | −0.118364 | 3.15415 | 2.11404 | 3.42099 | −2.90582 | −0.624856 | 4.32663 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.49672 | −0.945774 | 0.240162 | 2.74524 | −1.41556 | −2.95814 | −2.63398 | −2.10551 | 4.10885 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.88972 | −1.98190 | 1.57104 | −1.89350 | −3.74523 | 0.913094 | −0.810613 | 0.927915 | −3.57819 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.00444 | 0.797964 | 2.01776 | −2.51167 | 1.59947 | −4.73648 | 0.0356005 | −2.36325 | −5.03448 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.39266 | −2.02648 | 3.72483 | 3.38808 | −4.84868 | 4.01703 | 4.12693 | 1.10663 | 8.10653 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.66314 | 2.24241 | 5.09232 | 4.27464 | 5.97185 | −2.88003 | 8.23530 | 2.02839 | 11.3840 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.78632 | 2.26457 | 5.76358 | −3.38130 | 6.30983 | 1.84667 | 10.4866 | 2.12830 | −9.42138 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(31\) | \(-1\) |
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 | 5239.2.a.p | 18 | |
| 13.b | even | 2 | 1 | 5239.2.a.o | 18 | ||
| 13.c | even | 3 | 2 | 403.2.f.c | ✓ | 36 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.f.c | ✓ | 36 | 13.c | even | 3 | 2 | |
| 5239.2.a.o | 18 | 13.b | even | 2 | 1 | ||
| 5239.2.a.p | 18 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\):
|
\( T_{2}^{18} - 5 T_{2}^{17} - 15 T_{2}^{16} + 105 T_{2}^{15} + 50 T_{2}^{14} - 876 T_{2}^{13} + 294 T_{2}^{12} + \cdots + 3 \)
|
|
\( T_{5}^{18} - 6 T_{5}^{17} - 33 T_{5}^{16} + 218 T_{5}^{15} + 494 T_{5}^{14} - 3250 T_{5}^{13} + \cdots + 23148 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} - 5 T^{17} + \cdots + 3 \)
|
| $3$ |
\( T^{18} - 38 T^{16} + \cdots - 2336 \)
|
| $5$ |
\( T^{18} - 6 T^{17} + \cdots + 23148 \)
|
| $7$ |
\( T^{18} - 4 T^{17} + \cdots - 4404442 \)
|
| $11$ |
\( T^{18} - 17 T^{17} + \cdots + 72927744 \)
|
| $13$ |
\( T^{18} \)
|
| $17$ |
\( T^{18} + 7 T^{17} + \cdots - 53973696 \)
|
| $19$ |
\( T^{18} + \cdots + 5838146228 \)
|
| $23$ |
\( T^{18} + 4 T^{17} + \cdots - 43716876 \)
|
| $29$ |
\( T^{18} + \cdots - 572747667 \)
|
| $31$ |
\( (T - 1)^{18} \)
|
| $37$ |
\( T^{18} + \cdots + 8815937470259 \)
|
| $41$ |
\( T^{18} + \cdots + 2505681124737 \)
|
| $43$ |
\( T^{18} + \cdots + 6203249882 \)
|
| $47$ |
\( T^{18} + \cdots + 122728339872 \)
|
| $53$ |
\( T^{18} + \cdots - 4476288182004 \)
|
| $59$ |
\( T^{18} + \cdots - 371876030574 \)
|
| $61$ |
\( T^{18} + \cdots - 115724517827123 \)
|
| $67$ |
\( T^{18} + \cdots + 11543955614 \)
|
| $71$ |
\( T^{18} + \cdots - 9174387926316 \)
|
| $73$ |
\( T^{18} + \cdots - 6266144237684 \)
|
| $79$ |
\( T^{18} + \cdots - 69106734445600 \)
|
| $83$ |
\( T^{18} + \cdots + 2010479436288 \)
|
| $89$ |
\( T^{18} + \cdots - 22\!\cdots\!22 \)
|
| $97$ |
\( T^{18} + \cdots - 16\!\cdots\!86 \)
|
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