LMFDB - Newform orbit 2925.2.a.bi

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2925,2,Mod(1,2925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2925 = 3^{2} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,0,5,0,0,-5,3,0,0,1] 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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 5x + 3$ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 975)
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 $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_1 - 2) q^{7} + (\beta_{2} + \beta_1 + 1) q^{8} + (\beta_{2} - \beta_1 + 1) q^{11} - q^{13} + (\beta_{2} - 2 \beta_1 + 4) q^{14} + (2 \beta_1 + 1) q^{16}+ \cdots + ( - 3 \beta_{2} + 2 \beta_1 - 15) q^{98}+O(q^{100})$ q + b1 * q^2 + (b2 + 2) * q^4 + (b1 - 2) * q^7 + (b2 + b1 + 1) * q^8 + (b2 - b1 + 1) * q^11 - q^13 + (b2 - 2b1 + 4) q^14 + (2b1 + 1) q^16 + (-2b1 + 3) q^17 + (b2 - 2b1 + 3) q^19 + (2b1 - 3) q^22 + (b2 + 2b1 + 1) q^23 - b1 * q^26 + (-b2 + 3b1 - 3) q^28 + (2b2 + 5) q^29 + (-b2 + 3b1 + 1) q^31 + (-b1 + 6) * q^32 + (-2b2 + 3b1 - 8) * q^34 + (b2 + 2b1 - 1) q^37 + (-b2 + 4b1 - 7) q^38 + (-b2 + 2b1 - 1) q^41 + (3b2 + 1) q^43 + (-b1 + 6) * q^44 + (3b2 + 2b1 + 9) * q^46 + (-4b2 + b1 - 4) q^47 + (b2 - 4b1 + 1) q^49 + (-b2 - 2) * q^52 + (-b2 + 2b1 + 2) q^53 + 3 * q^56 + (2b2 + 7b1 + 2) * q^58 + (-2b2 + b1 - 2) q^59 + (-2b2 + 2b1 - 3) * q^61 + (2b2 + 11) q^62 + (-b2 + 2b1 - 6) q^64 + (-3b2 - 3b1 + 1) * q^67 + (b2 - 6b1 + 4) q^68 + (-b2 + 2b1 + 5) q^71 + (-b2 - 9) * q^73 + (3b2 + 9) q^74 + (b2 - 4b1 + 9) q^76 + (-2b2 + 4b1 - 5) * q^77 + (b2 - 4b1 + 3) q^79 + (b2 - 2b1 + 7) q^82 + (-5b1 + 6) q^83 + (3b2 + 4b1 + 3) * q^86 + (-b2 + 2b1 + 2) q^88 + (-2b1 + 6) q^89 + (-b1 + 2) * q^91 + (3b2 + 8b1 + 9) * q^92 + (-3b2 - 8b1) * q^94 + (-2b2 + 2b1 - 10) * q^97 + (-3b2 + 2b1 - 15) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$3 q + q^{2} + 5 q^{4} - 5 q^{7} + 3 q^{8} + q^{11} - 3 q^{13} + 9 q^{14} + 5 q^{16} + 7 q^{17} + 6 q^{19} - 7 q^{22} + 4 q^{23} - q^{26} - 5 q^{28} + 13 q^{29} + 7 q^{31} + 17 q^{32} - 19 q^{34} - 2 q^{37}+ \cdots - 40 q^{98}+O(q^{100})$ 3 * q + q^2 + 5 * q^4 - 5 * q^7 + 3 * q^8 + q^11 - 3 * q^13 + 9 * q^14 + 5 * q^16 + 7 * q^17 + 6 * q^19 - 7 * q^22 + 4 * q^23 - q^26 - 5 * q^28 + 13 * q^29 + 7 * q^31 + 17 * q^32 - 19 * q^34 - 2 * q^37 - 16 * q^38 + 17 * q^44 + 26 * q^46 - 7 * q^47 - 2 * q^49 - 5 * q^52 + 9 * q^53 + 9 * q^56 + 11 * q^58 - 3 * q^59 - 5 * q^61 + 31 * q^62 - 15 * q^64 + 3 * q^67 + 5 * q^68 + 18 * q^71 - 26 * q^73 + 24 * q^74 + 22 * q^76 - 9 * q^77 + 4 * q^79 + 18 * q^82 + 13 * q^83 + 10 * q^86 + 9 * q^88 + 16 * q^89 + 5 * q^91 + 32 * q^92 - 5 * q^94 - 26 * q^97 - 40 * q^98

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

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 4$ v^2 - 4

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 4$ b2 + 4

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

−2.08613
0.571993
2.51414

−2.08613

2.35194

−4.08613

−0.734191

1.2

0.571993

−1.67282

−1.42801

−2.10083

1.3

2.51414

4.32088

0.514137

5.83502

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$+1$
$13$	$+1$

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	2925.2.a.bi		3
3.b	odd	2	1		975.2.a.n	✓	3
5.b	even	2	1		2925.2.a.bg		3
5.c	odd	4	2		2925.2.c.x		6
15.d	odd	2	1		975.2.a.p	yes	3
15.e	even	4	2		975.2.c.j		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
975.2.a.n	✓	3	3.b	odd	2	1
975.2.a.p	yes	3	15.d	odd	2	1
975.2.c.j		6	15.e	even	4	2
2925.2.a.bg		3	5.b	even	2	1
2925.2.a.bi		3	1.a	even	1	1	trivial
2925.2.c.x		6	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2925))$ :

T_{2}^{3} - T_{2}^{2} - 5T_{2} + 3

T2^3 - T2^2 - 5*T2 + 3

T_{7}^{3} + 5T_{7}^{2} + 3T_{7} - 3

T7^3 + 5*T7^2 + 3*T7 - 3

T_{11}^{3} - T_{11}^{2} - 11T_{11} + 9

T11^3 - T11^2 - 11*T11 + 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3} - T^{2} - 5T + 3$ T^3 - T^2 - 5*T + 3
$3$	$T^{3}$ T^3
$5$	$T^{3}$ T^3
$7$	$T^{3} + 5 T^{2} + \cdots - 3$ T^3 + 5T^2 + 3T - 3
$11$	$T^{3} - T^{2} - 11T + 9$ T^3 - T^2 - 11*T + 9
$13$	$(T + 1)^{3}$ (T + 1)^3
$17$	$T^{3} - 7 T^{2} + \cdots + 27$ T^3 - 7T^2 - 5T + 27
$19$	$T^{3} - 6 T^{2} + \cdots + 4$ T^3 - 6T^2 - 12T + 4
$23$	$T^{3} - 4 T^{2} + \cdots - 36$ T^3 - 4T^2 - 32T - 36
$29$	$T^{3} - 13 T^{2} + \cdots + 129$ T^3 - 13T^2 + 19T + 129
$31$	$T^{3} - 7 T^{2} + \cdots + 223$ T^3 - 7T^2 - 31T + 223
$37$	$T^{3} + 2 T^{2} + \cdots - 108$ T^3 + 2T^2 - 36T - 108
$41$	$T^{3} - 24T + 36$ T^3 - 24*T + 36
$43$	$T^{3} - 84T + 164$ T^3 - 84*T + 164
$47$	$T^{3} + 7 T^{2} + \cdots - 909$ T^3 + 7T^2 - 125T - 909
$53$	$T^{3} - 9 T^{2} + \cdots + 81$ T^3 - 9T^2 + 3T + 81
$59$	$T^{3} + 3 T^{2} + \cdots - 117$ T^3 + 3T^2 - 33T - 117
$61$	$T^{3} + 5 T^{2} + \cdots - 113$ T^3 + 5T^2 - 37T - 113
$67$	$T^{3} - 3 T^{2} + \cdots + 863$ T^3 - 3T^2 - 159T + 863
$71$	$T^{3} - 18 T^{2} + \cdots - 36$ T^3 - 18T^2 + 84T - 36
$73$	$T^{3} + 26 T^{2} + \cdots + 564$ T^3 + 26T^2 + 216T + 564
$79$	$T^{3} - 4 T^{2} + \cdots - 164$ T^3 - 4T^2 - 76T - 164
$83$	$T^{3} - 13 T^{2} + \cdots + 339$ T^3 - 13T^2 - 77T + 339
$89$	$T^{3} - 16 T^{2} + \cdots - 48$ T^3 - 16T^2 + 64T - 48
$97$	$T^{3} + 26 T^{2} + \cdots + 216$ T^3 + 26T^2 + 180T + 216
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