LMFDB - Newform orbit 5004.2.a.l

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5004,2,Mod(1,5004)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5004.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5004, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5004 = 2^{2} \cdot 3^{2} \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5004.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.9571411714$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.62452821.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 2x^{3} + 18x^{2} - 5x - 4 $ x^6 - x^5 - 10x^4 + 2x^3 + 18x^2 - 5x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1668)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{2} + 1) q^{5} + (\beta_{3} + 1) q^{7} + (\beta_{3} - \beta_1 + 1) q^{11} + ( - \beta_{4} + \beta_{2} - 1) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{17} + ( - 2 \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{19}+ \cdots + (3 \beta_{5} - 3 \beta_{4} - \beta_{3} + \cdots - 7) q^{97}+O(q^{100}) $ q + (b2 + 1) * q^5 + (b3 + 1) * q^7 + (b3 - b1 + 1) * q^11 + (-b4 + b2 - 1) * q^13 + (-b5 - b4 + b3 - 2b2 + 2b1 + 1) * q^17 + (-2b5 + b4 - b3 + b2 - b1) q^19 + (-2b5 - b4 + b3 - b2 + b1) q^23 + (-b5 + b3 + b2 + b1 + 1) * q^25 + (b5 - b4 - 2b1 + 2) q^29 + (b5 - 3b4 + b3 - b2 + 2) q^31 + (2b4 + 2b2 + b1 + 3) * q^35 + (2b4 + b2 + b1 - 2) q^37 + (b5 + 3b4 - b3 + b2 - 5b1 + 3) * q^41 + (3b5 - b4 - b3 - 1) q^43 + (2b5 + b4 - b3 + 2b2 + 2b1 + 2) q^47 + (b5 + b4 + 2b3 - b1 + 1) q^49 + (2b5 + 4) q^53 + (b5 + 2b4 - b3 + 2b2 + b1 + 2) * q^55 + (3b5 - 2b4 - 2b3 + 3b2 - b1 + 3) * q^59 + (b5 - b2 - 3b1 - 3) q^61 + (-3b5 - b4 - b2 + 2b1 + 4) * q^65 + (3b5 + 2b4 - b1) * q^67 + (2b5 - b4 + 2b3 - b2 + 3) * q^71 + (-2b5 + 3b4 + b3 - b2 - b1 - 2) * q^73 + (b5 + 3b3 - 2b2 + 8) * q^77 + (-2b5 - 2b4 - 2b3 + b2 + 1) q^79 + (-3b5 + 2b4 - 3b3 + 3b2 + b1 + 4) * q^83 + (-3b5 - b4 - b3 + 2b2 + 2b1 - 5) q^85 + (-2b5 - b3 + 2b1 + 5) * q^89 + (-2b5 + b4 - 2b3 + 2b1 - 1) q^91 + (-5b4 + b3 - b2 + 3b1 + 2) * q^95 + (3b5 - 3b4 - b3 - 4b1 - 7) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} + 4 q^{7} + 3 q^{11} - 9 q^{13} + 13 q^{17} + 4 q^{23} + 3 q^{25} + 9 q^{29} + 12 q^{31} + 13 q^{35} - 14 q^{37} + 11 q^{41} - 7 q^{43} + 8 q^{47} + 22 q^{53} + 8 q^{55} + 9 q^{59} - 19 q^{61}+ \cdots - 47 q^{97}+O(q^{100}) $ 6 * q + 3 * q^5 + 4 * q^7 + 3 * q^11 - 9 * q^13 + 13 * q^17 + 4 * q^23 + 3 * q^25 + 9 * q^29 + 12 * q^31 + 13 * q^35 - 14 * q^37 + 11 * q^41 - 7 * q^43 + 8 * q^47 + 22 * q^53 + 8 * q^55 + 9 * q^59 - 19 * q^61 + 32 * q^65 - 4 * q^67 + 15 * q^71 - 10 * q^73 + 47 * q^77 + 9 * q^79 + 25 * q^83 - 29 * q^85 + 36 * q^89 + 2 * q^91 + 16 * q^95 - 47 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 10x^{4} + 2x^{3} + 18x^{2} - 5x - 4 $ x^6 - x^5 - 10*x^4 + 2*x^3 + 18*x^2 - 5*x - 4 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 7\nu^{2} + 14\nu - 4 ) / 3 $ (v^5 - 2v^4 - 8v^3 + 7v^2 + 14v - 4) / 3
$\beta_{3}$	$=$	$ ( \nu^{5} + \nu^{4} - 14\nu^{3} - 11\nu^{2} + 26\nu + 2 ) / 3 $ (v^5 + v^4 - 14v^3 - 11v^2 + 26*v + 2) / 3
$\beta_{4}$	$=$	$ ( \nu^{5} + \nu^{4} - 11\nu^{3} - 17\nu^{2} + 11\nu + 14 ) / 3 $ (v^5 + v^4 - 11v^3 - 17v^2 + 11*v + 14) / 3
$\beta_{5}$	$=$	$ ( 2\nu^{5} - \nu^{4} - 19\nu^{3} - 7\nu^{2} + 22\nu + 1 ) / 3 $ (2v^5 - v^4 - 19v^3 - 7v^2 + 22v + 1) / 3

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} - \beta_{4} - \beta_{2} + \beta _1 + 3 $ b5 - b4 - b2 + b1 + 3
$\nu^{3}$	$=$	$ 2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 7\beta _1 + 2 $ 2b5 - b4 - b3 - 2b2 + 7*b1 + 2
$\nu^{4}$	$=$	$ 10\beta_{5} - 8\beta_{4} - \beta_{3} - 11\beta_{2} + 16\beta _1 + 20 $ 10b5 - 8b4 - b3 - 11b2 + 16b1 + 20
$\nu^{5}$	$=$	$ 29\beta_{5} - 17\beta_{4} - 10\beta_{3} - 28\beta_{2} + 67\beta _1 + 39 $ 29b5 - 17b4 - 10b3 - 28b2 + 67*b1 + 39

Display $\beta_i$ in terms of $\nu^j$

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

−2.11822
−0.367756
3.36923
−1.77670
0.750556
1.14288

−2.04046

3.70568

1.2

−1.61576

−1.78049

1.3

−1.29989

−1.56475

1.4

1.15247

−1.71296

1.5

3.22404

4.31797

1.6

3.57960

1.03455

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$139$	$ -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	5004.2.a.l		6
3.b	odd	2	1		1668.2.a.g	✓	6
12.b	even	2	1		6672.2.a.bl		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1668.2.a.g	✓	6	3.b	odd	2	1
5004.2.a.l		6	1.a	even	1	1	trivial
6672.2.a.bl		6	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5004))$:

$ T_{5}^{6} - 3T_{5}^{5} - 12T_{5}^{4} + 23T_{5}^{3} + 56T_{5}^{2} - 24T_{5} - 57 $

T5^6 - 3*T5^5 - 12*T5^4 + 23*T5^3 + 56*T5^2 - 24*T5 - 57

$ T_{7}^{6} - 4T_{7}^{5} - 13T_{7}^{4} + 34T_{7}^{3} + 80T_{7}^{2} - 25T_{7} - 79 $

T7^6 - 4*T7^5 - 13*T7^4 + 34*T7^3 + 80*T7^2 - 25*T7 - 79

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 3 T^{5} + \cdots - 57 $ T^6 - 3T^5 - 12T^4 + 23T^3 + 56T^2 - 24*T - 57
$7$	$ T^{6} - 4 T^{5} + \cdots - 79 $ T^6 - 4T^5 - 13T^4 + 34T^3 + 80T^2 - 25*T - 79
$11$	$ T^{6} - 3 T^{5} + \cdots - 1 $ T^6 - 3T^5 - 32T^4 + 65T^3 + 148T^2 + 6*T - 1
$13$	$ T^{6} + 9 T^{5} + \cdots + 404 $ T^6 + 9T^5 - T^4 - 172T^3 - 379T^2 + 30T + 404
$17$	$ T^{6} - 13 T^{5} + \cdots + 6672 $ T^6 - 13T^5 + 9T^4 + 388T^3 - 836T^2 - 2808*T + 6672
$19$	$ T^{6} - 93 T^{4} + \cdots - 7184 $ T^6 - 93T^4 + 24T^3 + 1736T^2 + 224T - 7184
$23$	$ T^{6} - 4 T^{5} + \cdots - 3056 $ T^6 - 4T^5 - 61T^4 + 108T^3 + 1152T^2 + 336*T - 3056
$29$	$ T^{6} - 9 T^{5} + \cdots + 64 $ T^6 - 9T^5 - 17T^4 + 150T^3 + 413T^2 + 296*T + 64
$31$	$ T^{6} - 12 T^{5} + \cdots + 31573 $ T^6 - 12T^5 - 61T^4 + 1046T^3 - 1808T^2 - 11431*T + 31573
$37$	$ T^{6} + 14 T^{5} + \cdots + 578 $ T^6 + 14T^5 - 3T^4 - 629T^3 - 1868T^2 - 101*T + 578
$41$	$ T^{6} - 11 T^{5} + \cdots + 408866 $ T^6 - 11T^5 - 212T^4 + 2698T^3 + 7425T^2 - 162991*T + 408866
$43$	$ T^{6} + 7 T^{5} + \cdots + 17152 $ T^6 + 7T^5 - 105T^4 - 860T^3 + 848T^2 + 14336*T + 17152
$47$	$ T^{6} - 8 T^{5} + \cdots + 1152 $ T^6 - 8T^5 - 205T^4 + 1505T^3 + 4844T^2 - 5073*T + 1152
$53$	$ T^{6} - 22 T^{5} + \cdots + 1536 $ T^6 - 22T^5 + 148T^4 - 200T^3 - 928T^2 + 1536*T + 1536
$59$	$ T^{6} - 9 T^{5} + \cdots + 126304 $ T^6 - 9T^5 - 219T^4 + 1648T^3 + 8676T^2 - 81240*T + 126304
$61$	$ T^{6} + 19 T^{5} + \cdots + 12928 $ T^6 + 19T^5 + 39T^4 - 800T^3 - 2788T^2 + 5432*T + 12928
$67$	$ T^{6} + 4 T^{5} + \cdots + 1753 $ T^6 + 4T^5 - 193T^4 + 98T^3 + 2700T^2 + 4253*T + 1753
$71$	$ T^{6} - 15 T^{5} + \cdots - 1788 $ T^6 - 15T^5 - 35T^4 + 1046T^3 - 1019T^2 - 11508*T - 1788
$73$	$ T^{6} + 10 T^{5} + \cdots - 97024 $ T^6 + 10T^5 - 155T^4 - 1532T^3 + 4596T^2 + 35112*T - 97024
$79$	$ T^{6} - 9 T^{5} + \cdots - 47479 $ T^6 - 9T^5 - 218T^4 + 1483T^3 + 7252T^2 - 16062*T - 47479
$83$	$ T^{6} - 25 T^{5} + \cdots + 669 $ T^6 - 25T^5 - 108T^4 + 6139T^3 - 36170T^2 + 40422*T + 669
$89$	$ T^{6} - 36 T^{5} + \cdots + 2075 $ T^6 - 36T^5 + 445T^4 - 2042T^3 + 1444T^2 + 5145*T + 2075
$97$	$ T^{6} + 47 T^{5} + \cdots + 149808 $ T^6 + 47T^5 + 643T^4 - 292T^3 - 61412T^2 - 296616*T + 149808
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Level:	\( N \)	\(=\)	\( 5004 = 2^{2} \cdot 3^{2} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5004.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta_{2} + 1) q^{5} + (\beta_{3} + 1) q^{7} + (\beta_{3} - \beta_1 + 1) q^{11} + ( - \beta_{4} + \beta_{2} - 1) q^{13} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 1) q^{17} + ( - 2 \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{19}+ \cdots + (3 \beta_{5} - 3 \beta_{4} - \beta_{3} + \cdots - 7) q^{97}+O(q^{100}) \) q + (b2 + 1) * q^5 + (b3 + 1) * q^7 + (b3 - b1 + 1) * q^11 + (-b4 + b2 - 1) * q^13 + (-b5 - b4 + b3 - 2b2 + 2b1 + 1) * q^17 + (-2b5 + b4 - b3 + b2 - b1) q^19 + (-2b5 - b4 + b3 - b2 + b1) q^23 + (-b5 + b3 + b2 + b1 + 1) * q^25 + (b5 - b4 - 2b1 + 2) q^29 + (b5 - 3b4 + b3 - b2 + 2) q^31 + (2b4 + 2b2 + b1 + 3) * q^35 + (2b4 + b2 + b1 - 2) q^37 + (b5 + 3b4 - b3 + b2 - 5b1 + 3) * q^41 + (3b5 - b4 - b3 - 1) q^43 + (2b5 + b4 - b3 + 2b2 + 2b1 + 2) q^47 + (b5 + b4 + 2b3 - b1 + 1) q^49 + (2b5 + 4) q^53 + (b5 + 2b4 - b3 + 2b2 + b1 + 2) * q^55 + (3b5 - 2b4 - 2b3 + 3b2 - b1 + 3) * q^59 + (b5 - b2 - 3b1 - 3) q^61 + (-3b5 - b4 - b2 + 2b1 + 4) * q^65 + (3b5 + 2b4 - b1) * q^67 + (2b5 - b4 + 2b3 - b2 + 3) * q^71 + (-2b5 + 3b4 + b3 - b2 - b1 - 2) * q^73 + (b5 + 3b3 - 2b2 + 8) * q^77 + (-2b5 - 2b4 - 2b3 + b2 + 1) q^79 + (-3b5 + 2b4 - 3b3 + 3b2 + b1 + 4) * q^83 + (-3b5 - b4 - b3 + 2b2 + 2b1 - 5) q^85 + (-2b5 - b3 + 2b1 + 5) * q^89 + (-2b5 + b4 - 2b3 + 2b1 - 1) q^91 + (-5b4 + b3 - b2 + 3b1 + 2) * q^95 + (3b5 - 3b4 - b3 - 4b1 - 7) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} + 4 q^{7} + 3 q^{11} - 9 q^{13} + 13 q^{17} + 4 q^{23} + 3 q^{25} + 9 q^{29} + 12 q^{31} + 13 q^{35} - 14 q^{37} + 11 q^{41} - 7 q^{43} + 8 q^{47} + 22 q^{53} + 8 q^{55} + 9 q^{59} - 19 q^{61}+ \cdots - 47 q^{97}+O(q^{100}) \) 6 * q + 3 * q^5 + 4 * q^7 + 3 * q^11 - 9 * q^13 + 13 * q^17 + 4 * q^23 + 3 * q^25 + 9 * q^29 + 12 * q^31 + 13 * q^35 - 14 * q^37 + 11 * q^41 - 7 * q^43 + 8 * q^47 + 22 * q^53 + 8 * q^55 + 9 * q^59 - 19 * q^61 + 32 * q^65 - 4 * q^67 + 15 * q^71 - 10 * q^73 + 47 * q^77 + 9 * q^79 + 25 * q^83 - 29 * q^85 + 36 * q^89 + 2 * q^91 + 16 * q^95 - 47 * q^97

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