LMFDB - Newform orbit 8325.2.a.bq

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8325,2,Mod(1,8325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8325 = 3^{2} \cdot 5^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8325.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:	$66.4754596827$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 5x + 4$ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 555)
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.

$f(q)$	$=$	$q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + ( - \beta_{2} + \beta_1 - 2) q^{7} + (\beta_{2} + \beta_1) q^{8} + \beta_{2} q^{11} - 5 q^{13} + ( - 3 \beta_1 + 4) q^{14} + \beta_1 q^{16} + ( - 2 \beta_{2} - \beta_1 + 2) q^{17}+ \cdots + ( - 4 \beta_{2} + 6 \beta_1 - 20) q^{98}+O(q^{100})$ q + b1 * q^2 + (b2 + 2) * q^4 + (-b2 + b1 - 2) * q^7 + (b2 + b1) * q^8 + b2 * q^11 - 5 * q^13 + (-3b1 + 4) q^14 + b1 * q^16 + (-2b2 - b1 + 2) q^17 + (b1 - 2) * q^19 + (b2 + b1) * q^22 + (-b2 + b1 - 2) * q^23 - 5b1 q^26 + (-b2 + 2b1 - 8) q^28 + (-b2 - 1) * q^29 + (b2 - b1 + 2) * q^31 + (-b2 - 2b1 + 4) q^32 + (-3b2 - 4) q^34 + q^37 + (b2 - 2b1 + 4) q^38 + (b2 - 3b1 + 4) q^41 + (b2 - 1) * q^43 + (b1 + 4) * q^44 + (-3b1 + 4) q^46 + (-2b1 - 1) q^47 + (b2 - 5b1 + 5) q^49 + (-5b2 - 10) q^52 + (-b2 - 3b1 + 5) q^53 + (b2 - 3b1) q^56 + (-b2 - 2b1) q^58 + (-2b2 + 5b1 + 3) * q^59 + (-3b2 - b1 + 2) q^61 + (3b1 - 4) q^62 + (-3b2 + b1 - 8) q^64 + (4b2 + 3b1 - 4) * q^67 + (b2 - 5b1 - 4) q^68 + (2b2 - 3b1 + 4) * q^71 + (-b2 - 6b1 + 2) q^73 + b1 * q^74 + (-b2 + 3b1 - 4) q^76 + (b2 - 4) * q^77 + (-3b2 - 2b1 - 4) * q^79 + (-2b2 + 5b1 - 12) * q^82 + (4b2 + b1 - 3) q^83 + b2 * q^86 + (-b2 + 2b1 + 4) q^88 + (-2b2 - 7) q^89 + (5b2 - 5b1 + 10) * q^91 + (-b2 + 2b1 - 8) q^92 + (-2b2 - b1 - 8) q^94 + (-3b2 + b1 - 10) q^97 + (-4b2 + 6b1 - 20) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$3 q + q^{2} + 5 q^{4} - 4 q^{7} - q^{11} - 15 q^{13} + 9 q^{14} + q^{16} + 7 q^{17} - 5 q^{19} - 4 q^{23} - 5 q^{26} - 21 q^{28} - 2 q^{29} + 4 q^{31} + 11 q^{32} - 9 q^{34} + 3 q^{37} + 9 q^{38} + 8 q^{41}+ \cdots - 50 q^{98}+O(q^{100})$ 3 * q + q^2 + 5 * q^4 - 4 * q^7 - q^11 - 15 * q^13 + 9 * q^14 + q^16 + 7 * q^17 - 5 * q^19 - 4 * q^23 - 5 * q^26 - 21 * q^28 - 2 * q^29 + 4 * q^31 + 11 * q^32 - 9 * q^34 + 3 * q^37 + 9 * q^38 + 8 * q^41 - 4 * q^43 + 13 * q^44 + 9 * q^46 - 5 * q^47 + 9 * q^49 - 25 * q^52 + 13 * q^53 - 4 * q^56 - q^58 + 16 * q^59 + 8 * q^61 - 9 * q^62 - 20 * q^64 - 13 * q^67 - 18 * q^68 + 7 * q^71 + q^73 + q^74 - 8 * q^76 - 13 * q^77 - 11 * q^79 - 29 * q^82 - 12 * q^83 - q^86 + 15 * q^88 - 19 * q^89 + 20 * q^91 - 21 * q^92 - 23 * q^94 - 26 * q^97 - 50 * q^98

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

$\beta_{1}$	$=$	$\nu$ 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

−2.16425
0.772866
2.39138

−2.16425

2.68397

−4.84822

−1.48028

1.2

0.772866

−1.40268

2.17554

−2.62981

1.3

2.39138

3.71871

−1.32733

4.11009

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$+1$
$37$	$-1$

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	8325.2.a.bq		3
3.b	odd	2	1		2775.2.a.t		3
5.b	even	2	1		1665.2.a.n		3
15.d	odd	2	1		555.2.a.h	✓	3
60.h	even	2	1		8880.2.a.ca		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
555.2.a.h	✓	3	15.d	odd	2	1
1665.2.a.n		3	5.b	even	2	1
2775.2.a.t		3	3.b	odd	2	1
8325.2.a.bq		3	1.a	even	1	1	trivial
8880.2.a.ca		3	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8325))$ :

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

T_{7}^{3} + 4T_{7}^{2} - 7T_{7} - 14

T_{11}^{3} + T_{11}^{2} - 7T_{11} + 4

T_{13} + 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3} - T^{2} - 5T + 4$ T^3 - T^2 - 5*T + 4
$3$	$T^{3}$ T^3
$5$	$T^{3}$ T^3
$7$	$T^{3} + 4 T^{2} + \cdots - 14$ T^3 + 4T^2 - 7T - 14
$11$	$T^{3} + T^{2} - 7T + 4$ T^3 + T^2 - 7*T + 4
$13$	$(T + 5)^{3}$ (T + 5)^3
$17$	$T^{3} - 7 T^{2} + \cdots + 86$ T^3 - 7T^2 - 19T + 86
$19$	$T^{3} + 5 T^{2} + \cdots - 2$ T^3 + 5T^2 + 3T - 2
$23$	$T^{3} + 4 T^{2} + \cdots - 14$ T^3 + 4T^2 - 7T - 14
$29$	$T^{3} + 2 T^{2} + \cdots - 11$ T^3 + 2T^2 - 6T - 11
$31$	$T^{3} - 4 T^{2} + \cdots + 14$ T^3 - 4T^2 - 7T + 14
$37$	$(T - 1)^{3}$ (T - 1)^3
$41$	$T^{3} - 8 T^{2} + \cdots - 28$ T^3 - 8T^2 - 33T - 28
$43$	$T^{3} + 4 T^{2} + \cdots - 1$ T^3 + 4T^2 - 2T - 1
$47$	$T^{3} + 5 T^{2} + \cdots - 49$ T^3 + 5T^2 - 13T - 49
$53$	$T^{3} - 13T^{2} + 256$ T^3 - 13*T^2 + 256
$59$	$T^{3} - 16 T^{2} + \cdots + 1447$ T^3 - 16T^2 - 74T + 1447
$61$	$T^{3} - 8 T^{2} + \cdots + 134$ T^3 - 8T^2 - 51T + 134
$67$	$T^{3} + 13 T^{2} + \cdots - 1192$ T^3 + 13T^2 - 113T - 1192
$71$	$T^{3} - 7 T^{2} + \cdots + 16$ T^3 - 7T^2 - 59T + 16
$73$	$T^{3} - T^{2} + \cdots + 154$ T^3 - T^2 - 201*T + 154
$79$	$T^{3} + 11 T^{2} + \cdots - 112$ T^3 + 11T^2 - 49T - 112
$83$	$T^{3} + 12 T^{2} + \cdots - 241$ T^3 + 12T^2 - 76T - 241
$89$	$T^{3} + 19 T^{2} + \cdots + 17$ T^3 + 19T^2 + 91T + 17
$97$	$T^{3} + 26 T^{2} + \cdots - 178$ T^3 + 26T^2 + 155T - 178
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