LMFDB - Newform orbit 3888.2.a.bm

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3888,2,Mod(1,3888)] 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("3888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3888 = 2^{4} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,-3,0,0,0,3,0,3,0,0,0,9,0,-9,0,0,0,6,0,18,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$31.0458363059$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.9926793.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} - x^{3} + 36x^{2} + 9x - 9 $ x^6 - 12x^4 - x^3 + 36x^2 + 9*x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 1944)
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} q^{5} + (\beta_{4} - 1) q^{7} + (\beta_{5} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{17} + (\beta_{5} - \beta_{2} - 2) q^{19}+ \cdots + (\beta_{5} - \beta_{3} + \beta_{2} + \cdots + 4) q^{97}+O(q^{100}) $ q - b2 * q^5 + (b4 - 1) * q^7 + (b5 + b3 + b1) * q^11 + (-b4 - b2 + b1 + 1) * q^13 + (b4 - b3 - b1 + 1) * q^17 + (b5 - b2 - 2) * q^19 + (-b5 + b4 - b3 + b2 - b1 + 1) * q^23 + (b5 + b4 - 2b1 + 2) q^25 + (-b4 + 2b3 - b2 + 3b1) * q^29 + (-b5 - b4 + b3 - b2 - 3) * q^31 + (-b5 - 3b3 + b2 - b1 + 3) q^35 + (b5 + 3b3 - 2b2 + 2b1 + 2) q^37 + (b4 + b2 - b1 + 3) * q^41 + (b5 - b4 + b3 - b2 + 3b1 - 3) q^43 + (-b5 + b4 + 2b3 + b1 + 1) q^47 + (-2b5 + b4 - 4b3 + 2b2 - b1 + 4) q^49 + (-2b3 + b2 + b1 + 2) q^53 + (-b5 - b4 - 4b3 - b2 - 3b1 - 2) * q^55 + (b5 - b4 + 2b3 - 2b2 - b1 + 2) * q^59 + (-2b5 - b3 - b2 - 2b1 + 4) * q^61 + (b5 + 2b4 + 2b3 + b2 - 2b1 + 4) q^65 + (3b3 + b2 - 4) q^67 + (2b5 + b4 + 2b3 - b2 + 3b1 + 1) q^71 + (-2b4 + 4b3 - b2 + 5b1 + 2) q^73 + (-b5 - b3 + 2b2 - 5b1 + 1) * q^77 + (-b5 - b3 + 2b2 - 3) q^79 + (-b5 - 3b4 - b2 + 5) q^83 + (b5 + b4 - 2b3 - 2b2 - b1 + 3) * q^85 + (-b5 - 3b4 + 2b3 + 3b1 + 5) q^89 + (-b5 + b4 - 2b3 - 3b1 - 5) * q^91 + (2b5 + b4 - 3b3 - 5b1 + 5) q^95 + (b5 - b3 + b2 - 2b1 + 4) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{7} + 3 q^{11} + 3 q^{13} + 9 q^{17} - 9 q^{19} + 6 q^{23} + 18 q^{25} - 3 q^{29} - 24 q^{31} + 15 q^{35} + 15 q^{37} + 21 q^{41} - 18 q^{43} + 6 q^{47} + 21 q^{49} + 12 q^{53} - 18 q^{55} + 12 q^{59}+ \cdots + 27 q^{97}+O(q^{100}) $ 6 * q - 3 * q^7 + 3 * q^11 + 3 * q^13 + 9 * q^17 - 9 * q^19 + 6 * q^23 + 18 * q^25 - 3 * q^29 - 24 * q^31 + 15 * q^35 + 15 * q^37 + 21 * q^41 - 18 * q^43 + 6 * q^47 + 21 * q^49 + 12 * q^53 - 18 * q^55 + 12 * q^59 + 18 * q^61 + 33 * q^65 - 24 * q^67 + 15 * q^71 + 6 * q^73 + 3 * q^77 - 21 * q^79 + 18 * q^83 + 24 * q^85 + 18 * q^89 - 30 * q^91 + 39 * q^95 + 27 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 12x^{4} - x^{3} + 36x^{2} + 9x - 9 $ x^6 - 12*x^4 - x^3 + 36*x^2 + 9*x - 9 :

$\beta_{1}$	$=$	$ ( \nu^{4} + 2\nu^{3} - 7\nu^{2} - 11\nu + 3 ) / 2 $ (v^4 + 2v^3 - 7v^2 - 11*v + 3) / 2
$\beta_{2}$	$=$	$ ( \nu^{5} + \nu^{4} - 7\nu^{3} - 4\nu^{2} - 7 ) / 4 $ (v^5 + v^4 - 7v^3 - 4v^2 - 7) / 4
$\beta_{3}$	$=$	$ ( \nu^{5} - \nu^{4} - 11\nu^{3} + 6\nu^{2} + 26\nu + 3 ) / 4 $ (v^5 - v^4 - 11v^3 + 6v^2 + 26*v + 3) / 4
$\beta_{4}$	$=$	$ ( -\nu^{5} - \nu^{4} + 11\nu^{3} + 4\nu^{2} - 28\nu + 7 ) / 4 $ (-v^5 - v^4 + 11v^3 + 4v^2 - 28*v + 7) / 4
$\beta_{5}$	$=$	$ ( -\nu^{4} + 9\nu^{2} + \nu - 11 ) / 2 $ (-v^4 + 9*v^2 + v - 11) / 2

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

$\nu$	$=$	$ ( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta_1 ) / 3 $ (b5 - b4 + b3 - 2b2 + 2b1) / 3
$\nu^{2}$	$=$	$ ( \beta_{5} - \beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 12 ) / 3 $ (b5 - b4 - 2*b3 + b2 - b1 + 12) / 3
$\nu^{3}$	$=$	$ ( 7\beta_{5} - 4\beta_{4} + 7\beta_{3} - 11\beta_{2} + 14\beta_1 ) / 3 $ (7b5 - 4b4 + 7b3 - 11b2 + 14*b1) / 3
$\nu^{4}$	$=$	$ ( 4\beta_{5} - 10\beta_{4} - 17\beta_{3} + 7\beta_{2} - 7\beta _1 + 75 ) / 3 $ (4b5 - 10b4 - 17b3 + 7b2 - 7*b1 + 75) / 3
$\nu^{5}$	$=$	$ ( 49\beta_{5} - 22\beta_{4} + 58\beta_{3} - 68\beta_{2} + 101\beta _1 - 6 ) / 3 $ (49b5 - 22b4 + 58b3 - 68b2 + 101*b1 - 6) / 3

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

−1.93230
−2.71168
2.58232
−0.702936
0.400207
2.36439

−3.89275

1.41858

1.2

−2.65344

−4.61130

1.3

−1.27086

−3.12719

1.4

1.61815

5.19137

1.5

2.01336

−1.72399

1.6

4.18553

−0.147474

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -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	3888.2.a.bm		6
3.b	odd	2	1		3888.2.a.bl		6
4.b	odd	2	1		1944.2.a.q	✓	6
12.b	even	2	1		1944.2.a.r	yes	6
36.f	odd	6	2		1944.2.i.r		12
36.h	even	6	2		1944.2.i.q		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1944.2.a.q	✓	6	4.b	odd	2	1
1944.2.a.r	yes	6	12.b	even	2	1
1944.2.i.q		12	36.h	even	6	2
1944.2.i.r		12	36.f	odd	6	2
3888.2.a.bl		6	3.b	odd	2	1
3888.2.a.bm		6	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 $S_{2}^{\mathrm{new}}(\Gamma_0(3888))$:

$ T_{5}^{6} - 24T_{5}^{4} - 2T_{5}^{3} + 135T_{5}^{2} - 12T_{5} - 179 $

T5^6 - 24*T5^4 - 2*T5^3 + 135*T5^2 - 12*T5 - 179

$ T_{7}^{6} + 3T_{7}^{5} - 27T_{7}^{4} - 93T_{7}^{3} + 27T_{7}^{2} + 189T_{7} + 27 $

T7^6 + 3*T7^5 - 27*T7^4 - 93*T7^3 + 27*T7^2 + 189*T7 + 27

$ T_{11}^{6} - 3T_{11}^{5} - 39T_{11}^{4} + 65T_{11}^{3} + 465T_{11}^{2} - 129T_{11} - 1223 $

T11^6 - 3*T11^5 - 39*T11^4 + 65*T11^3 + 465*T11^2 - 129*T11 - 1223

$ T_{13}^{6} - 3T_{13}^{5} - 36T_{13}^{4} + 151T_{13}^{3} - 114T_{13}^{2} - 72T_{13} - 8 $

T13^6 - 3*T13^5 - 36*T13^4 + 151*T13^3 - 114*T13^2 - 72*T13 - 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} - 24 T^{4} + \cdots - 179 $ T^6 - 24T^4 - 2T^3 + 135T^2 - 12T - 179
$7$	$ T^{6} + 3 T^{5} + \cdots + 27 $ T^6 + 3T^5 - 27T^4 - 93T^3 + 27T^2 + 189*T + 27
$11$	$ T^{6} - 3 T^{5} + \cdots - 1223 $ T^6 - 3T^5 - 39T^4 + 65T^3 + 465T^2 - 129*T - 1223
$13$	$ T^{6} - 3 T^{5} + \cdots - 8 $ T^6 - 3T^5 - 36T^4 + 151T^3 - 114T^2 - 72*T - 8
$17$	$ T^{6} - 9 T^{5} + \cdots + 136 $ T^6 - 9T^5 + 3T^4 + 101T^3 - 36T^2 - 204*T + 136
$19$	$ T^{6} + 9 T^{5} + \cdots + 568 $ T^6 + 9T^5 - 24T^4 - 479T^3 - 1602T^2 - 1416*T + 568
$23$	$ T^{6} - 6 T^{5} + \cdots + 712 $ T^6 - 6T^5 - 45T^4 + 191T^3 + 444T^2 - 1476*T + 712
$29$	$ T^{6} + 3 T^{5} + \cdots - 6777 $ T^6 + 3T^5 - 90T^4 - 201T^3 + 2106T^2 + 3510*T - 6777
$31$	$ T^{6} + 24 T^{5} + \cdots - 50651 $ T^6 + 24T^5 + 144T^4 - 722T^3 - 11373T^2 - 42912*T - 50651
$37$	$ T^{6} - 15 T^{5} + \cdots - 23976 $ T^6 - 15T^5 - 54T^4 + 2049T^3 - 12798T^2 + 30456*T - 23976
$41$	$ T^{6} - 21 T^{5} + \cdots - 648 $ T^6 - 21T^5 + 144T^4 - 375T^3 + 162T^2 + 648*T - 648
$43$	$ T^{6} + 18 T^{5} + \cdots - 23624 $ T^6 + 18T^5 + 21T^4 - 1303T^3 - 9594T^2 - 25620*T - 23624
$47$	$ T^{6} - 6 T^{5} + \cdots - 173016 $ T^6 - 6T^5 - 171T^4 + 645T^3 + 9828T^2 - 14580*T - 173016
$53$	$ T^{6} - 12 T^{5} + \cdots - 15263 $ T^6 - 12T^5 - 72T^4 + 934T^3 + 759T^2 - 10188*T - 15263
$59$	$ T^{6} - 12 T^{5} + \cdots - 10259 $ T^6 - 12T^5 - 174T^4 + 1937T^3 + 1860T^2 - 15681*T - 10259
$61$	$ T^{6} - 18 T^{5} + \cdots + 242776 $ T^6 - 18T^5 - 87T^4 + 2527T^3 - 2520T^2 - 86700*T + 242776
$67$	$ T^{6} + 24 T^{5} + \cdots + 284608 $ T^6 + 24T^5 + 27T^4 - 2855T^3 - 16044T^2 + 40176*T + 284608
$71$	$ T^{6} - 15 T^{5} + \cdots + 303832 $ T^6 - 15T^5 - 174T^4 + 4165T^3 - 18120T^2 - 41748*T + 303832
$73$	$ T^{6} - 6 T^{5} + \cdots - 11177 $ T^6 - 6T^5 - 264T^4 + 898T^3 + 18213T^2 - 7422*T - 11177
$79$	$ T^{6} + 21 T^{5} + \cdots - 359 $ T^6 + 21T^5 + 72T^4 - 313T^3 - 1554T^2 - 1764*T - 359
$83$	$ T^{6} - 18 T^{5} + \cdots + 139303 $ T^6 - 18T^5 - 132T^4 + 2914T^3 + 3069T^2 - 108174*T + 139303
$89$	$ T^{6} - 18 T^{5} + \cdots - 174312 $ T^6 - 18T^5 - 153T^4 + 3159T^3 + 3564T^2 - 84564*T - 174312
$97$	$ T^{6} - 27 T^{5} + \cdots + 115471 $ T^6 - 27T^5 + 156T^4 + 1145T^3 - 10368T^2 - 8394*T + 115471
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3888 = 2^{4} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3888.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{5} + (\beta_{4} - 1) q^{7} + (\beta_{5} + \beta_{3} + \beta_1) q^{11} + ( - \beta_{4} - \beta_{2} + \beta_1 + 1) q^{13} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{17} + (\beta_{5} - \beta_{2} - 2) q^{19}+ \cdots + (\beta_{5} - \beta_{3} + \beta_{2} + \cdots + 4) q^{97}+O(q^{100}) \) q - b2 * q^5 + (b4 - 1) * q^7 + (b5 + b3 + b1) * q^11 + (-b4 - b2 + b1 + 1) * q^13 + (b4 - b3 - b1 + 1) * q^17 + (b5 - b2 - 2) * q^19 + (-b5 + b4 - b3 + b2 - b1 + 1) * q^23 + (b5 + b4 - 2b1 + 2) q^25 + (-b4 + 2b3 - b2 + 3b1) * q^29 + (-b5 - b4 + b3 - b2 - 3) * q^31 + (-b5 - 3b3 + b2 - b1 + 3) q^35 + (b5 + 3b3 - 2b2 + 2b1 + 2) q^37 + (b4 + b2 - b1 + 3) * q^41 + (b5 - b4 + b3 - b2 + 3b1 - 3) q^43 + (-b5 + b4 + 2b3 + b1 + 1) q^47 + (-2b5 + b4 - 4b3 + 2b2 - b1 + 4) q^49 + (-2b3 + b2 + b1 + 2) q^53 + (-b5 - b4 - 4b3 - b2 - 3b1 - 2) * q^55 + (b5 - b4 + 2b3 - 2b2 - b1 + 2) * q^59 + (-2b5 - b3 - b2 - 2b1 + 4) * q^61 + (b5 + 2b4 + 2b3 + b2 - 2b1 + 4) q^65 + (3b3 + b2 - 4) q^67 + (2b5 + b4 + 2b3 - b2 + 3b1 + 1) q^71 + (-2b4 + 4b3 - b2 + 5b1 + 2) q^73 + (-b5 - b3 + 2b2 - 5b1 + 1) * q^77 + (-b5 - b3 + 2b2 - 3) q^79 + (-b5 - 3b4 - b2 + 5) q^83 + (b5 + b4 - 2b3 - 2b2 - b1 + 3) * q^85 + (-b5 - 3b4 + 2b3 + 3b1 + 5) q^89 + (-b5 + b4 - 2b3 - 3b1 - 5) * q^91 + (2b5 + b4 - 3b3 - 5b1 + 5) q^95 + (b5 - b3 + b2 - 2b1 + 4) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{7} + 3 q^{11} + 3 q^{13} + 9 q^{17} - 9 q^{19} + 6 q^{23} + 18 q^{25} - 3 q^{29} - 24 q^{31} + 15 q^{35} + 15 q^{37} + 21 q^{41} - 18 q^{43} + 6 q^{47} + 21 q^{49} + 12 q^{53} - 18 q^{55} + 12 q^{59}+ \cdots + 27 q^{97}+O(q^{100}) \) 6 * q - 3 * q^7 + 3 * q^11 + 3 * q^13 + 9 * q^17 - 9 * q^19 + 6 * q^23 + 18 * q^25 - 3 * q^29 - 24 * q^31 + 15 * q^35 + 15 * q^37 + 21 * q^41 - 18 * q^43 + 6 * q^47 + 21 * q^49 + 12 * q^53 - 18 * q^55 + 12 * q^59 + 18 * q^61 + 33 * q^65 - 24 * q^67 + 15 * q^71 + 6 * q^73 + 3 * q^77 - 21 * q^79 + 18 * q^83 + 24 * q^85 + 18 * q^89 - 30 * q^91 + 39 * q^95 + 27 * q^97

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