LMFDB - Newform orbit 9025.2.a.bl

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9025 = 5^{2} \cdot 19^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 5x^{2} + 5$ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1805)
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.

$f(q)$	$=$	$q + \beta_1 q^{2} + \beta_1 q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{2} + 3) q^{6} + (2 \beta_{2} + 3) q^{7} + (\beta_{3} - \beta_1) q^{8} + \beta_{2} q^{9} + ( - 3 \beta_{2} - 4) q^{11} + (\beta_{3} + \beta_1) q^{12}+ \cdots + ( - \beta_{2} - 3) q^{99}+O(q^{100})$ q + b1 * q^2 + b1 * q^3 + (b2 + 1) * q^4 + (b2 + 3) * q^6 + (2b2 + 3) q^7 + (b3 - b1) * q^8 + b2 * q^9 + (-3b2 - 4) q^11 + (b3 + b1) * q^12 + (-b3 - b1) * q^13 + (2b3 + 3b1) * q^14 + (-b2 - 4) * q^16 + (-2b2 - 4) q^17 + b3 * q^18 + (2b3 + 3b1) * q^21 + (-3b3 - 4b1) * q^22 + (-5b2 - 1) q^23 + (b2 - 2) * q^24 + (-3b2 - 4) q^26 + (b3 - 3b1) q^27 + (3b2 + 5) q^28 + (-4b3 + b1) q^29 + b1 * q^31 + (-3b3 - 2b1) * q^32 + (-3b3 - 4b1) * q^33 + (-2b3 - 4b1) * q^34 + q^36 + (4b3 - b1) q^37 + (-3b2 - 4) q^39 + (-b3 - 3b1) q^41 + (7b2 + 11) q^42 + (b2 - 1) * q^43 + (-4b2 - 7) q^44 + (-5b3 - b1) q^46 + (4b2 - 1) q^47 + (-b3 - 4b1) q^48 + (8b2 + 6) q^49 + (-2b3 - 4b1) * q^51 + (-b3 - 2b1) q^52 + (-3b3 - 4b1) * q^53 + (-b2 - 8) * q^54 + (-b3 - b1) * q^56 + (-7b2 - 1) q^58 + (-b3 + 8b1) q^59 + (8b2 - 1) q^61 + (b2 + 3) * q^62 + (b2 + 2) * q^63 + (-6b2 - 1) q^64 + (-10b2 - 15) q^66 + (3b3 - 5b1) * q^67 + (-4b2 - 6) q^68 + (-5b3 - b1) q^69 + (5b3 - 3b1) * q^71 + (-2b3 + b1) q^72 + q^73 + (7b2 + 1) q^74 + (-11b2 - 18) q^77 + (-3b3 - 4b1) * q^78 + (2b3 - 4b1) * q^79 + (-4b2 - 8) q^81 + (-5b2 - 10) q^82 + (-2b2 + 10) q^83 + (3b3 + 5b1) * q^84 + (b3 - b1) * q^86 + (-7b2 - 1) q^87 + (2b3 + b1) q^88 + (5b3 + b1) q^89 + (-3b3 - 5b1) * q^91 + (-b2 - 6) * q^92 + (b2 + 3) * q^93 + (4b3 - b1) q^94 + (-8b2 - 9) q^96 + (-b3 + 2b1) q^97 + (8b3 + 6b1) * q^98 + (-b2 - 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49}+ \cdots - 10 q^{99}+O(q^{100})$ 4 * q + 2 * q^4 + 10 * q^6 + 8 * q^7 - 2 * q^9 - 10 * q^11 - 14 * q^16 - 12 * q^17 + 6 * q^23 - 10 * q^24 - 10 * q^26 + 14 * q^28 + 4 * q^36 - 10 * q^39 + 30 * q^42 - 6 * q^43 - 20 * q^44 - 12 * q^47 + 8 * q^49 - 30 * q^54 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 6 * q^63 + 8 * q^64 - 40 * q^66 - 16 * q^68 + 4 * q^73 - 10 * q^74 - 50 * q^77 - 24 * q^81 - 30 * q^82 + 44 * q^83 + 10 * q^87 - 22 * q^92 + 10 * q^93 - 20 * q^96 - 10 * q^99

Basis of coefficient ring in terms of $\nu = \zeta_{20} + \zeta_{20}^{-1}$ :

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

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 3$ b2 + 3
$\nu^{3}$	$=$	$\beta_{3} + 3\beta_1$ b3 + 3*b1

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

−1.90211
−1.17557
1.17557
1.90211

−1.90211

1.61803

3.61803

4.23607

0.726543

0.618034

1.2

−1.17557

−0.618034

1.38197

−0.236068

3.07768

−1.61803

1.3

1.17557

−0.618034

1.38197

−0.236068

−3.07768

−1.61803

1.4

1.90211

1.61803

3.61803

4.23607

−0.726543

0.618034

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$19$	$+1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9025.2.a.bl		4
5.b	even	2	1		1805.2.a.l	✓	4
19.b	odd	2	1	inner	9025.2.a.bl		4
95.d	odd	2	1		1805.2.a.l	✓	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1805.2.a.l	✓	4	5.b	even	2	1
1805.2.a.l	✓	4	95.d	odd	2	1
9025.2.a.bl		4	1.a	even	1	1	trivial
9025.2.a.bl		4	19.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9025))$ :

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

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

T_{7}^{2} - 4T_{7} - 1

T_{11}^{2} + 5T_{11} - 5

T_{29}^{4} - 85T_{29}^{2} + 605

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} - 5T^{2} + 5$ T^4 - 5*T^2 + 5
$3$	$T^{4} - 5T^{2} + 5$ T^4 - 5*T^2 + 5
$5$	$T^{4}$ T^4
$7$	$(T^{2} - 4 T - 1)^{2}$ (T^2 - 4*T - 1)^2
$11$	$(T^{2} + 5 T - 5)^{2}$ (T^2 + 5*T - 5)^2
$13$	$T^{4} - 10T^{2} + 5$ T^4 - 10*T^2 + 5
$17$	$(T^{2} + 6 T + 4)^{2}$ (T^2 + 6*T + 4)^2
$19$	$T^{4}$ T^4
$23$	$(T^{2} - 3 T - 29)^{2}$ (T^2 - 3*T - 29)^2
$29$	$T^{4} - 85T^{2} + 605$ T^4 - 85*T^2 + 605
$31$	$T^{4} - 5T^{2} + 5$ T^4 - 5*T^2 + 5
$37$	$T^{4} - 85T^{2} + 605$ T^4 - 85*T^2 + 605
$41$	$T^{4} - 50T^{2} + 125$ T^4 - 50*T^2 + 125
$43$	$(T^{2} + 3 T + 1)^{2}$ (T^2 + 3*T + 1)^2
$47$	$(T^{2} + 6 T - 11)^{2}$ (T^2 + 6*T - 11)^2
$53$	$T^{4} - 125T^{2} + 125$ T^4 - 125*T^2 + 125
$59$	$T^{4} - 325 T^{2} + 25205$ T^4 - 325*T^2 + 25205
$61$	$(T^{2} + 10 T - 55)^{2}$ (T^2 + 10*T - 55)^2
$67$	$T^{4} - 170T^{2} + 4805$ T^4 - 170*T^2 + 4805
$71$	$T^{4} - 170T^{2} + 5$ T^4 - 170*T^2 + 5
$73$	$(T - 1)^{4}$ (T - 1)^4
$79$	$T^{4} - 100T^{2} + 2000$ T^4 - 100*T^2 + 2000
$83$	$(T^{2} - 22 T + 116)^{2}$ (T^2 - 22*T + 116)^2
$89$	$T^{4} - 130T^{2} + 4205$ T^4 - 130*T^2 + 4205
$97$	$T^{4} - 25T^{2} + 125$ T^4 - 25*T^2 + 125
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