LMFDB - Newform orbit 7448.2.a.bj

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7448 = 2^{3} \cdot 7^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.25857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 8x^{2} + 9x + 3$ x^4 - x^3 - 8x^2 + 9x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1064)
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.

$f(q)$	$=$	$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{2} + 1) q^{9} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{11} + ( - \beta_{3} - \beta_1) q^{13} + ( - \beta_{3} + 2) q^{15} + (\beta_{3} - 1) q^{17}+ \cdots + ( - \beta_{2} + 3 \beta_1 - 1) q^{99}+O(q^{100})$ q + (-b2 - 1) * q^3 + (-b2 + b1 - 1) * q^5 + (b2 + 1) * q^9 + (b3 + b2 - b1 + 1) * q^11 + (-b3 - b1) * q^13 + (-b3 + 2) * q^15 + (b3 - 1) * q^17 - q^19 + (-b3 - b1 + 2) * q^23 + (-b3 - b2 - 1) * q^25 + (2b2 - 1) q^27 + (b2 + 2b1) q^29 + (b3 + 2b1 - 5) q^31 + (b2 - 3b1 + 1) q^33 + (2b3 + b2 - b1 + 1) q^37 + (2b3 + 3b1 - 1) * q^39 + (b2 + 2b1 - 2) q^41 + (b3 - 3b2 + 4b1 - 1) * q^43 + (b3 - 2) * q^45 + (-b3 + b2 - 2b1) q^47 + (-b3 + 2b2 - 3b1 + 4) * q^51 + (b2 - 2b1 - 3) q^53 + (4b2 - b1 - 2) q^55 + (b2 + 1) * q^57 + (-2b3 + b2 + b1 - 1) q^59 + (-b3 + b2 - 4b1 + 2) q^61 + (b3 - 2b2 + b1 - 4) q^65 + (-3b3 - 2b1 + 3) * q^67 + (2b3 - 2b2 + 3b1 - 3) q^69 + (-b3 - 3b2 - 2b1 + 6) * q^71 + (4b2 - 3b1 + 3) * q^73 + (b3 + 3b1 + 1) q^75 + (b3 + b2 + 4b1 + 3) q^79 + (-2b2 - 8) q^81 + (6b2 - 3b1 + 3) * q^83 + (-b3 + 4b2 - 2b1 + 3) * q^85 + (-2b3 - 2b2 - 7) * q^87 + (-3b2 + 3b1) * q^89 + (-3b3 + 4b2 - 3b1 + 4) q^93 + (b2 - b1 + 1) * q^95 + (-b3 + 5b2 + 2) q^97 + (-b2 + 3b1 - 1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{3} - q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 7 q^{15} - 3 q^{17} - 4 q^{19} + 6 q^{23} - 3 q^{25} - 8 q^{27} - 17 q^{31} - q^{33} + 3 q^{37} + q^{39} - 8 q^{41} + 7 q^{43} - 7 q^{45} - 5 q^{47}+ \cdots + q^{99}+O(q^{100})$ 4 * q - 2 * q^3 - q^5 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 7 * q^15 - 3 * q^17 - 4 * q^19 + 6 * q^23 - 3 * q^25 - 8 * q^27 - 17 * q^31 - q^33 + 3 * q^37 + q^39 - 8 * q^41 + 7 * q^43 - 7 * q^45 - 5 * q^47 + 8 * q^51 - 16 * q^53 - 17 * q^55 + 2 * q^57 - 7 * q^59 + q^61 - 10 * q^65 + 7 * q^67 - 3 * q^69 + 27 * q^71 + q^73 + 8 * q^75 + 15 * q^79 - 28 * q^81 - 3 * q^83 + q^85 - 26 * q^87 + 9 * q^89 + 2 * q^93 + q^95 - 3 * q^97 + q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - x^{3} - 8x^{2} + 9x + 3$ x^4 - x^3 - 8*x^2 + 9*x + 3 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{3} - 6\nu + 1 ) / 2$ (v^3 - 6*v + 1) / 2
$\beta_{3}$	$=$	$\nu^{2} - 4$ v^2 - 4

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} + 4$ b3 + 4
$\nu^{3}$	$=$	$2\beta_{2} + 6\beta _1 - 1$ 2b2 + 6b1 - 1

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

−0.270905
2.57368
−2.82557
1.52280

−2.30278

−2.57368

2.30278

1.2

−2.30278

0.270905

2.30278

1.3

1.30278

−1.52280

−1.30278

1.4

1.30278

2.82557

−1.30278

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$-1$
$19$	$+1$

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	7448.2.a.bj		4
7.b	odd	2	1		1064.2.a.h	✓	4
21.c	even	2	1		9576.2.a.ci		4
28.d	even	2	1		2128.2.a.t		4
56.e	even	2	1		8512.2.a.bu		4
56.h	odd	2	1		8512.2.a.bq		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1064.2.a.h	✓	4	7.b	odd	2	1
2128.2.a.t		4	28.d	even	2	1
7448.2.a.bj		4	1.a	even	1	1	trivial
8512.2.a.bq		4	56.h	odd	2	1
8512.2.a.bu		4	56.e	even	2	1
9576.2.a.ci		4	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7448))$ :

T_{3}^{2} + T_{3} - 3

T_{5}^{4} + T_{5}^{3} - 8T_{5}^{2} - 9T_{5} + 3

T_{11}^{4} - 2T_{11}^{3} - 27T_{11}^{2} + 28T_{11} + 79

T_{13}^{4} + 2T_{13}^{3} - 21T_{13}^{2} - 22T_{13} + 4

T_{17}^{4} + 3T_{17}^{3} - 17T_{17}^{2} - 24T_{17} + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$(T^{2} + T - 3)^{2}$ (T^2 + T - 3)^2
$5$	$T^{4} + T^{3} - 8 T^{2} + \cdots + 3$ T^4 + T^3 - 8T^2 - 9T + 3
$7$	$T^{4}$ T^4
$11$	$T^{4} - 2 T^{3} + \cdots + 79$ T^4 - 2T^3 - 27T^2 + 28*T + 79
$13$	$T^{4} + 2 T^{3} + \cdots + 4$ T^4 + 2T^3 - 21T^2 - 22*T + 4
$17$	$T^{4} + 3 T^{3} + \cdots + 64$ T^4 + 3T^3 - 17T^2 - 24*T + 64
$19$	$(T + 1)^{4}$ (T + 1)^4
$23$	$T^{4} - 6 T^{3} + \cdots - 36$ T^4 - 6T^3 - 9T^2 + 54*T - 36
$29$	$T^{4} - 53 T^{2} + \cdots - 29$ T^4 - 53T^2 + 78T - 29
$31$	$T^{4} + 17 T^{3} + \cdots - 636$ T^4 + 17T^3 + 67T^2 - 108*T - 636
$37$	$T^{4} - 3 T^{3} + \cdots + 1543$ T^4 - 3T^3 - 86T^2 + 69*T + 1543
$41$	$T^{4} + 8 T^{3} + \cdots - 69$ T^4 + 8T^3 - 29T^2 - 102*T - 69
$43$	$T^{4} - 7 T^{3} + \cdots + 1156$ T^4 - 7T^3 - 111T^2 + 680*T + 1156
$47$	$T^{4} + 5 T^{3} + \cdots - 87$ T^4 + 5T^3 - 32T^2 - 159*T - 87
$53$	$T^{4} + 16 T^{3} + \cdots - 23$ T^4 + 16T^3 + 69T^2 + 40*T - 23
$59$	$T^{4} + 7 T^{3} + \cdots + 417$ T^4 + 7T^3 - 110T^2 - 111*T + 417
$61$	$T^{4} - T^{3} + \cdots + 2643$ T^4 - T^3 - 116T^2 + 141T + 2643
$67$	$T^{4} - 7 T^{3} + \cdots + 2532$ T^4 - 7T^3 - 161T^2 + 348*T + 2532
$71$	$T^{4} - 27 T^{3} + \cdots - 6269$ T^4 - 27T^3 + 154T^2 + 825*T - 6269
$73$	$T^{4} - T^{3} + \cdots - 108$ T^4 - T^3 - 101T^2 + 270T - 108
$79$	$T^{4} - 15 T^{3} + \cdots + 412$ T^4 - 15T^3 - 71T^2 + 540*T + 412
$83$	$T^{4} + 3 T^{3} + \cdots + 1296$ T^4 + 3T^3 - 189T^2 + 108*T + 1296
$89$	$T^{4} - 9 T^{3} + \cdots + 324$ T^4 - 9T^3 - 45T^2 + 162*T + 324
$97$	$T^{4} + 3 T^{3} + \cdots + 7747$ T^4 + 3T^3 - 212T^2 - 375*T + 7747
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