LMFDB - Newform orbit 8325.2.a.cf

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:	$5$
Coefficient field:	5.5.457904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 5x - 6$ x^5 - 2x^4 - 5x^3 + 8x^2 + 5x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1665)
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 $1,\beta_1,\beta_2,\beta_3,\beta_4$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{4} + \beta_{2}) q^{7} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{8} + ( - \beta_{4} + \beta_{3} - 2) q^{11} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{13}+ \cdots + (2 \beta_{4} - 4 \beta_{2} - 6 \beta_1 + 2) q^{98}+O(q^{100})$ q + b1 * q^2 + (b2 + b1) * q^4 + (-b4 + b2) * q^7 + (b3 + 2b2 + b1) q^8 + (-b4 + b3 - 2) * q^11 + (b4 + b3 - b2 - b1 + 1) * q^13 + (-b4 + b1 - 2) * q^14 + (b4 + b3 + b2 + 2b1 - 2) q^16 + (-2b3 - 2b2 - b1 + 2) * q^17 + (b1 - 2) * q^19 + (-2b3 - b2 - 4b1) * q^22 + (b3 - b1 + 4) * q^23 + (2b4 - b3 - b2 - 2b1) * q^26 + (b4 - b3 - 2b2 - 2b1 + 2) * q^28 + (2b4 - b3 - 2b2 - 4b1 - 1) q^29 + (-2b4 - b3 - 2b2 + b1 - 2) * q^31 + (2b4 - b3 + 2) q^32 + (-2b4 - 3b2 - b1 + 2) * q^34 - q^37 + (b2 - b1 + 2) * q^38 + (-b4 - 2b3 + 3b2 - 2) * q^41 + (b3 - 1) * q^43 + (-b3 - 5b2 - 4b1 - 2) * q^44 + (b4 - b3 - b2 + 2b1 - 2) q^46 + (2b1 + 3) q^47 + (3b4 - b3 - 4b2 - 3b1 + 1) q^49 + (-b4 + b2 + b1 - 4) * q^52 + (3b4 - b3 - 3b1 + 1) * q^53 + (2b4 - 3b2 - 4b1 + 4) q^56 + (b4 + b3 - 4b2 - 6b1 - 4) * q^58 + (b3 + b2 + 2b1 - 1) q^59 + (4b4 - 3b3 - 2b2 - 3b1) * q^61 + (-3b4 - 3b3 - 3b2 - 6b1 + 6) * q^62 + (-b4 + b3 + b1 + 4) * q^64 + (-2b4 + b3 + b2 - 2) q^67 + (-2b4 - b3 - 2b2 - 5b1) q^68 + (2b4 + b3 + b2 + 2b1 - 8) * q^71 + (b4 - 3b3 - 2b2 - 2) * q^73 - b1 * q^74 + (b3 + b1) * q^76 + (2b4 + 2b3 - 3b2 + b1 + 2) q^77 + (-3b3 - 2b2 - 4b1) q^79 + (-3b4 + 4b3 + 2b2 + 5b1 - 6) * q^82 + (-b4 - b3 + 3b2 + 3) q^83 + (b4 - b3 - 2b1) q^86 + (-b4 - 7b2 - 7b1 + 2) * q^88 + (3b4 - b3 + b2 + 3b1 - 5) * q^89 + (-4b4 + 3b3 + 6b2 + 3b1 - 8) * q^91 + (-b3 + 2b2 + 2b1 - 2) * q^92 + (2b2 + 5b1 + 4) * q^94 + (6b4 - 4b3 - 5b2 - 6b1 + 2) * q^97 + (2b4 - 4b2 - 6b1 + 2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$5 q + 2 q^{2} + 4 q^{4} + 6 q^{8} - 12 q^{11} + 3 q^{13} - 10 q^{14} - 2 q^{16} + 4 q^{17} - 8 q^{19} - 10 q^{22} + 18 q^{23} - 2 q^{26} + 4 q^{28} - 13 q^{29} - 16 q^{31} + 14 q^{32} - 2 q^{34} - 5 q^{37}+ \cdots - 6 q^{98}+O(q^{100})$ 5 * q + 2 * q^2 + 4 * q^4 + 6 * q^8 - 12 * q^11 + 3 * q^13 - 10 * q^14 - 2 * q^16 + 4 * q^17 - 8 * q^19 - 10 * q^22 + 18 * q^23 - 2 * q^26 + 4 * q^28 - 13 * q^29 - 16 * q^31 + 14 * q^32 - 2 * q^34 - 5 * q^37 + 10 * q^38 - 6 * q^41 - 5 * q^43 - 28 * q^44 - 6 * q^46 + 19 * q^47 - 3 * q^49 - 18 * q^52 + 5 * q^53 + 10 * q^56 - 38 * q^58 + q^59 - 2 * q^61 + 6 * q^62 + 20 * q^64 - 12 * q^67 - 18 * q^68 - 30 * q^71 - 12 * q^73 - 2 * q^74 + 2 * q^76 + 10 * q^77 - 12 * q^79 - 22 * q^82 + 19 * q^83 - 2 * q^86 - 20 * q^88 - 11 * q^89 - 30 * q^91 - 2 * q^92 + 34 * q^94 - 6 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 5x - 6$ x^5 - 2*x^4 - 5*x^3 + 8*x^2 + 5*x - 6 :

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

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 2$ b2 + b1 + 2
$\nu^{3}$	$=$	$\beta_{3} + 2\beta_{2} + 5\beta_1$ b3 + 2b2 + 5b1
$\nu^{4}$	$=$	$\beta_{4} + \beta_{3} + 7\beta_{2} + 8\beta _1 + 6$ b4 + b3 + 7b2 + 8b1 + 6

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

−1.80696
−1.07694
0.788997
1.51432
2.58059

−1.80696

1.26510

2.45070

1.32794

1.2

−1.07694

−0.840195

1.59547

3.05873

1.3

0.788997

−1.37748

−4.52826

−2.66482

1.4

1.51432

0.293161

1.43000

−2.58470

1.5

2.58059

4.65942

−0.947908

6.86286

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.cf		5
3.b	odd	2	1		8325.2.a.by		5
5.b	even	2	1		1665.2.a.o	✓	5
15.d	odd	2	1		1665.2.a.r	yes	5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1665.2.a.o	✓	5	5.b	even	2	1
1665.2.a.r	yes	5	15.d	odd	2	1
8325.2.a.by		5	3.b	odd	2	1
8325.2.a.cf		5	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 $S_{2}^{\mathrm{new}}(\Gamma_0(8325))$ :

T_{2}^{5} - 2T_{2}^{4} - 5T_{2}^{3} + 8T_{2}^{2} + 5T_{2} - 6

T_{7}^{5} - 16T_{7}^{3} + 24T_{7}^{2} + 11T_{7} - 24

T_{11}^{5} + 12T_{11}^{4} + 33T_{11}^{3} - 76T_{11}^{2} - 389T_{11} - 292

T_{13}^{5} - 3T_{13}^{4} - 30T_{13}^{3} + 38T_{13}^{2} + 249T_{13} + 113

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{5} - 2 T^{4} + \cdots - 6$ T^5 - 2T^4 - 5T^3 + 8T^2 + 5T - 6
$3$	$T^{5}$ T^5
$5$	$T^{5}$ T^5
$7$	$T^{5} - 16 T^{3} + \cdots - 24$ T^5 - 16T^3 + 24T^2 + 11*T - 24
$11$	$T^{5} + 12 T^{4} + \cdots - 292$ T^5 + 12T^4 + 33T^3 - 76T^2 - 389T - 292
$13$	$T^{5} - 3 T^{4} + \cdots + 113$ T^5 - 3T^4 - 30T^3 + 38T^2 + 249T + 113
$17$	$T^{5} - 4 T^{4} + \cdots - 2048$ T^5 - 4T^4 - 49T^3 + 178T^2 + 581T - 2048
$19$	$T^{5} + 8 T^{4} + \cdots - 4$ T^5 + 8T^4 + 19T^3 + 10T^2 - 7T - 4
$23$	$T^{5} - 18 T^{4} + \cdots - 128$ T^5 - 18T^4 + 110T^3 - 290T^2 + 329T - 128
$29$	$T^{5} + 13 T^{4} + \cdots + 4877$ T^5 + 13T^4 - 19T^3 - 535T^2 + 139T + 4877
$31$	$T^{5} + 16 T^{4} + \cdots - 7938$ T^5 + 16T^4 - 6T^3 - 1170T^2 - 5859T - 7938
$37$	$(T + 1)^{5}$ (T + 1)^5
$41$	$T^{5} + 6 T^{4} + \cdots - 334$ T^5 + 6T^4 - 168T^3 - 952T^2 + 1411T - 334
$43$	$T^{5} + 5 T^{4} + \cdots - 3$ T^5 + 5T^4 - 3T^3 - 35T^2 - 29T - 3
$47$	$T^{5} - 19 T^{4} + \cdots + 117$ T^5 - 19T^4 + 118T^3 - 242T^2 - 7T + 117
$53$	$T^{5} - 5 T^{4} + \cdots - 4176$ T^5 - 5T^4 - 120T^3 + 452T^2 + 1504T - 4176
$59$	$T^{5} - T^{4} + \cdots + 13$ T^5 - T^4 - 31T^3 - 23T^2 + 17*T + 13
$61$	$T^{5} + 2 T^{4} + \cdots + 1664$ T^5 + 2T^4 - 218T^3 - 78T^2 + 10001T + 1664
$67$	$T^{5} + 12 T^{4} + \cdots + 292$ T^5 + 12T^4 + 7T^3 - 276T^2 - 573T + 292
$71$	$T^{5} + 30 T^{4} + \cdots - 19678$ T^5 + 30T^4 + 251T^3 - 150T^2 - 8557T - 19678
$73$	$T^{5} + 12 T^{4} + \cdots - 444$ T^5 + 12T^4 - 51T^3 - 100T^2 + 515T - 444
$79$	$T^{5} + 12 T^{4} + \cdots + 2126$ T^5 + 12T^4 - 117T^3 - 1110T^2 + 117T + 2126
$83$	$T^{5} - 19 T^{4} + \cdots + 3071$ T^5 - 19T^4 + 25T^3 + 783T^2 - 3285T + 3071
$89$	$T^{5} + 11 T^{4} + \cdots + 6019$ T^5 + 11T^4 - 194T^3 - 1978T^2 - 1147T + 6019
$97$	$T^{5} - 456 T^{3} + \cdots + 245914$ T^5 - 456T^3 - 1286T^2 + 47695*T + 245914
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