LMFDB - Newform orbit 2368.2.a.bg

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2368,2,Mod(1,2368)] 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("2368.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$2368 = 2^{6} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,-5,0,-1,0,8,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.9085751986$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.48389.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 8x^{2} + 3x + 10$ x^4 - 2x^3 - 8x^2 + 3*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 296)
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_1 - 1) q^{3} + (\beta_{3} - 1) q^{5} + (\beta_{3} - \beta_{2}) q^{7} + (\beta_{2} + 2) q^{9} + ( - \beta_{2} - 1) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{13} + ( - 2 \beta_{3} + \beta_{2} - \beta_1) q^{15}+ \cdots + (2 \beta_{3} - \beta_{2} - \beta_1 - 10) q^{99}+O(q^{100})$ q + (b1 - 1) * q^3 + (b3 - 1) * q^5 + (b3 - b2) * q^7 + (b2 + 2) * q^9 + (-b2 - 1) * q^11 + (-b3 + b2 - b1 - 1) * q^13 + (-2b3 + b2 - b1) q^15 + 2 * q^17 + (-2b3 - 2b1) * q^19 + (-3b3 + b2 - 2b1) * q^21 + (-b3 - b2 + b1 - 3) * q^23 + (-b3 - b2 - b1 + 2) * q^25 + (b3 + b1) * q^27 + (b3 - b2 - b1 - 1) * q^29 + (-3b3 + 2b2 - 1) * q^31 + (-b3 - 3b1 + 2) q^33 + (2b2 - 2b1 + 4) * q^35 - q^37 + (3b3 - 2b2 - 3) * q^39 + (2b2 - b1 + 1) q^41 + (-2b3 - 2) q^43 + (2b3 - 3b2 + b1) * q^45 + (b3 + b2 + 2b1 - 8) q^47 + (-b3 + b2 - 2b1 + 3) q^49 + (2b1 - 2) q^51 + (-b3 + b2 + 2b1) q^53 + (-b3 + 3b2 - b1 - 1) q^55 + (4b3 - 4b2 - 2b1 - 6) q^57 + (2b3 + 2b2 + 2b1 + 2) q^59 + (3b3 + 1) q^61 + (4b3 - 2b2 - 6) * q^63 + (-3b2 + 3b1 - 2) * q^65 + (-b3 - 2b1 + 1) q^67 + (b3 - 4b1 + 9) q^69 + (b3 + b2 - 4b1 - 2) q^71 + (4b3 + b2 + 3) q^73 + (b3 - 2b2 - b1 - 4) q^75 + (-3b3 + b2 + 6) q^77 + (-b3 - b2 + 5b1 + 1) q^79 + (-2b3 - b2 + b1 - 3) q^81 + (3b3 - 3b2 - 2b1 - 2) q^83 + (2b3 - 2) q^85 + (-3b3 - 4b1 - 3) * q^87 + (2b2 + 2b1 - 6) * q^89 + (2b3 + 4b1 - 10) * q^91 + (8b3 - 3b2 + 3b1 + 2) q^93 + (2b3 + 4b1 - 10) * q^95 + (-2b3 + 2b2 + 4b1 - 2) q^97 + (2b3 - b2 - b1 - 10) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{3} - 5 q^{5} - q^{7} + 8 q^{9} - 4 q^{11} - 5 q^{13} + 8 q^{17} - 2 q^{19} - q^{21} - 9 q^{23} + 7 q^{25} + q^{27} - 7 q^{29} - q^{31} + 3 q^{33} + 12 q^{35} - 4 q^{37} - 15 q^{39} + 2 q^{41}+ \cdots - 44 q^{99}+O(q^{100})$ 4 * q - 2 * q^3 - 5 * q^5 - q^7 + 8 * q^9 - 4 * q^11 - 5 * q^13 + 8 * q^17 - 2 * q^19 - q^21 - 9 * q^23 + 7 * q^25 + q^27 - 7 * q^29 - q^31 + 3 * q^33 + 12 * q^35 - 4 * q^37 - 15 * q^39 + 2 * q^41 - 6 * q^43 - 29 * q^47 + 9 * q^49 - 4 * q^51 + 5 * q^53 - 5 * q^55 - 32 * q^57 + 10 * q^59 + q^61 - 28 * q^63 - 2 * q^65 + q^67 + 27 * q^69 - 17 * q^71 + 8 * q^73 - 19 * q^75 + 27 * q^77 + 15 * q^79 - 8 * q^81 - 15 * q^83 - 10 * q^85 - 17 * q^87 - 20 * q^89 - 34 * q^91 + 6 * q^93 - 34 * q^95 + 2 * q^97 - 44 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - 2x^{3} - 8x^{2} + 3x + 10$ x^4 - 2*x^3 - 8*x^2 + 3*x + 10 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 2\nu - 4$ v^2 - 2*v - 4
$\beta_{3}$	$=$	$\nu^{3} - 3\nu^{2} - 4\nu + 5$ v^3 - 3v^2 - 4v + 5

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 2\beta _1 + 4$ b2 + 2*b1 + 4
$\nu^{3}$	$=$	$\beta_{3} + 3\beta_{2} + 10\beta _1 + 7$ b3 + 3b2 + 10b1 + 7

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

−1.70606
−1.26726
1.23838
3.73494

−2.70606

−2.87345

−4.19623

4.32278

1.2

−2.26726

2.21602

3.07555

2.14048

1.3

0.238381

−3.65512

2.28805

−2.94317

1.4

2.73494

−0.687447

−2.16737

4.47992

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$37$	$+1$

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	2368.2.a.bg		4
4.b	odd	2	1		2368.2.a.bh		4
8.b	even	2	1		296.2.a.d	✓	4
8.d	odd	2	1		592.2.a.j		4
24.f	even	2	1		5328.2.a.bp		4
24.h	odd	2	1		2664.2.a.r		4
40.f	even	2	1		7400.2.a.n		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
296.2.a.d	✓	4	8.b	even	2	1
592.2.a.j		4	8.d	odd	2	1
2368.2.a.bg		4	1.a	even	1	1	trivial
2368.2.a.bh		4	4.b	odd	2	1
2664.2.a.r		4	24.h	odd	2	1
5328.2.a.bp		4	24.f	even	2	1
7400.2.a.n		4	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2368))$ :

T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 15T_{3} + 4

T3^4 + 2*T3^3 - 8*T3^2 - 15*T3 + 4

T_{5}^{4} + 5T_{5}^{3} - T_{5}^{2} - 26T_{5} - 16

T5^4 + 5*T5^3 - T5^2 - 26*T5 - 16

T_{7}^{4} + T_{7}^{3} - 18T_{7}^{2} - 4T_{7} + 64

T7^4 + T7^3 - 18*T7^2 - 4*T7 + 64

T_{11}^{4} + 4T_{11}^{3} - 12T_{11}^{2} - 63T_{11} - 52

T11^4 + 4*T11^3 - 12*T11^2 - 63*T11 - 52

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 2 T^{3} + \cdots + 4$ T^4 + 2T^3 - 8T^2 - 15*T + 4
$5$	$T^{4} + 5 T^{3} + \cdots - 16$ T^4 + 5T^3 - T^2 - 26T - 16
$7$	$T^{4} + T^{3} + \cdots + 64$ T^4 + T^3 - 18T^2 - 4T + 64
$11$	$T^{4} + 4 T^{3} + \cdots - 52$ T^4 + 4T^3 - 12T^2 - 63*T - 52
$13$	$T^{4} + 5 T^{3} + \cdots - 160$ T^4 + 5T^3 - 17T^2 - 122*T - 160
$17$	$(T - 2)^{4}$ (T - 2)^4
$19$	$T^{4} + 2 T^{3} + \cdots + 640$ T^4 + 2T^3 - 68T^2 - 72*T + 640
$23$	$T^{4} + 9 T^{3} + \cdots - 472$ T^4 + 9T^3 - 21T^2 - 302*T - 472
$29$	$T^{4} + 7 T^{3} + \cdots + 4$ T^4 + 7T^3 - 11T^2 - 80*T + 4
$31$	$T^{4} + T^{3} + \cdots + 848$ T^4 + T^3 - 105*T^2 + 848
$37$	$(T + 1)^{4}$ (T + 1)^4
$41$	$T^{4} - 2 T^{3} + \cdots - 422$ T^4 - 2T^3 - 82T^2 + 371*T - 422
$43$	$T^{4} + 6 T^{3} + \cdots + 128$ T^4 + 6T^3 - 28T^2 - 48*T + 128
$47$	$T^{4} + 29 T^{3} + \cdots - 2336$ T^4 + 29T^3 + 246T^2 + 316*T - 2336
$53$	$T^{4} - 5 T^{3} + \cdots - 8$ T^4 - 5T^3 - 50T^2 + 52*T - 8
$59$	$T^{4} - 10 T^{3} + \cdots + 512$ T^4 - 10T^3 - 140T^2 + 928*T + 512
$61$	$T^{4} - T^{3} + \cdots + 664$ T^4 - T^3 - 93T^2 - 166T + 664
$67$	$T^{4} - T^{3} + \cdots - 16$ T^4 - T^3 - 43T^2 + 64T - 16
$71$	$T^{4} + 17 T^{3} + \cdots + 6976$ T^4 + 17T^3 - 96T^2 - 1436*T + 6976
$73$	$T^{4} - 8 T^{3} + \cdots + 2938$ T^4 - 8T^3 - 200T^2 + 967*T + 2938
$79$	$T^{4} - 15 T^{3} + \cdots + 17320$ T^4 - 15T^3 - 209T^2 + 1966*T + 17320
$83$	$T^{4} + 15 T^{3} + \cdots + 4160$ T^4 + 15T^3 - 128T^2 - 1536*T + 4160
$89$	$T^{4} + 20 T^{3} + \cdots - 3392$ T^4 + 20T^3 + 44T^2 - 848*T - 3392
$97$	$T^{4} - 2 T^{3} + \cdots - 160$ T^4 - 2T^3 - 236T^2 - 472*T - 160
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