LMFDB - Newform orbit 9025.2.a.ba

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9025,2,Mod(1,9025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [3,1,1,7,0,-6,-2,-6,4,0,-5,-4,-15,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.361.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 6x + 7$ x^3 - x^2 - 6*x + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 95)
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.

$f(q)$	$=$	$q + ( - \beta_{2} - \beta_1 + 1) q^{2} + \beta_{2} q^{3} + (\beta_1 + 2) q^{4} + (\beta_{2} - \beta_1 - 2) q^{6} + (\beta_{2} - 1) q^{7} + ( - 2 \beta_{2} - \beta_1 - 1) q^{8} + ( - \beta_{2} - \beta_1 + 2) q^{9}+ \cdots + (4 \beta_{2} + \beta_1 - 6) q^{99}+O(q^{100})$ q + (-b2 - b1 + 1) * q^2 + b2 * q^3 + (b1 + 2) * q^4 + (b2 - b1 - 2) * q^6 + (b2 - 1) * q^7 + (-2b2 - b1 - 1) q^8 + (-b2 - b1 + 2) * q^9 + (b2 - 2) * q^11 + (3b2 + 2b1 - 3) * q^12 - 5 * q^13 + (2b2 - 3) q^14 + (b2 + 2b1) q^16 + (-b2 - 3b1 + 1) q^17 + (-b2 + 5) * q^18 + (-2b2 - b1 + 5) q^21 + (3b2 + b1 - 4) q^22 + (-b2 - 3b1) q^23 - 7 * q^24 + (5b2 + 5b1 - 5) * q^26 + (-b2 - b1 - 2) * q^27 + (3b2 + b1 - 5) q^28 + (-b2 + 1) * q^29 + (-b2 - b1 + 1) * q^31 + (b2 - b1 - 2) * q^32 + (-3b2 - b1 + 5) q^33 + (4b2 + 3b1 + 6) * q^34 + (-4b2 - 2b1 + 3) * q^36 + (3b2 - 2b1 - 1) * q^37 - 5b2 q^39 + (3b2 + 2b1 - 1) * q^41 + (-5b2 - 2b1 + 10) * q^42 + (-2b2 - 3b1 + 2) * q^43 + (3b2 - 7) q^44 + (5b2 + 4b1 + 5) * q^46 + (b2 + 2b1 - 3) q^47 + (b2 + 3b1 - 1) q^48 + (-3b2 - b1 - 1) q^49 + (-b2 - 5b1 + 4) q^51 + (-5b1 - 10) q^52 + (3b2 - b1 + 3) q^53 + (3b2 + 4b1 + 1) * q^54 + (2b2 + b1 - 6) q^56 + (-2b2 + 3) q^58 + (b2 + 2b1 - 3) q^59 + (-2b2 - 4b1 - 1) * q^61 + (b1 + 4) * q^62 + (3b2 - 4) q^63 + (3b2 - 2b1 - 3) * q^64 + (-6b2 - b1 + 12) q^66 + (-2b1 - 6) q^67 + (-6b2 - 7b1 - 7) * q^68 + (-2b2 - 5b1 + 4) * q^69 + (-2b2 + b1 + 10) q^71 + (-b2 + 3b1 + 3) q^72 + (b2 + 3b1 + 6) q^73 + (8b2 - 5) q^74 + (-4b2 - b1 + 7) q^77 + (-5b2 + 5b1 + 10) * q^78 + (-2b2 - 4b1 + 10) * q^79 + (b2 + 2b1 - 8) q^81 + (-4b1 - 9) q^82 + (3b2 + 3b1 - 1) * q^83 + (-7b2 - b1 + 12) q^84 + (2b2 + 3b1 + 9) * q^86 + (2b2 + b1 - 5) q^87 + (4b2 + 2b1 - 5) * q^88 + (-4b1 + 6) q^89 + (-5b2 + 5) q^91 + (-6b2 - 8b1 - 9) * q^92 + (b2 - b1 - 2) * q^93 - 7 * q^94 + (-4b2 - 3b1 + 8) * q^96 + (-b2 - 4b1 + 4) q^97 + (5b1 + 6) q^98 + (4b2 + b1 - 6) q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q + q^{2} + q^{3} + 7 q^{4} - 6 q^{6} - 2 q^{7} - 6 q^{8} + 4 q^{9} - 5 q^{11} - 4 q^{12} - 15 q^{13} - 7 q^{14} + 3 q^{16} - q^{17} + 14 q^{18} + 12 q^{21} - 8 q^{22} - 4 q^{23} - 21 q^{24} - 5 q^{26}+ \cdots - 13 q^{99}+O(q^{100})$ 3 * q + q^2 + q^3 + 7 * q^4 - 6 * q^6 - 2 * q^7 - 6 * q^8 + 4 * q^9 - 5 * q^11 - 4 * q^12 - 15 * q^13 - 7 * q^14 + 3 * q^16 - q^17 + 14 * q^18 + 12 * q^21 - 8 * q^22 - 4 * q^23 - 21 * q^24 - 5 * q^26 - 8 * q^27 - 11 * q^28 + 2 * q^29 + q^31 - 6 * q^32 + 11 * q^33 + 25 * q^34 + 3 * q^36 - 2 * q^37 - 5 * q^39 + 2 * q^41 + 23 * q^42 + q^43 - 18 * q^44 + 24 * q^46 - 6 * q^47 + q^48 - 7 * q^49 + 6 * q^51 - 35 * q^52 + 11 * q^53 + 10 * q^54 - 15 * q^56 + 7 * q^58 - 6 * q^59 - 9 * q^61 + 13 * q^62 - 9 * q^63 - 8 * q^64 + 29 * q^66 - 20 * q^67 - 34 * q^68 + 5 * q^69 + 29 * q^71 + 11 * q^72 + 22 * q^73 - 7 * q^74 + 16 * q^77 + 30 * q^78 + 24 * q^79 - 21 * q^81 - 31 * q^82 + 3 * q^83 + 28 * q^84 + 32 * q^86 - 12 * q^87 - 9 * q^88 + 14 * q^89 + 10 * q^91 - 41 * q^92 - 6 * q^93 - 21 * q^94 + 17 * q^96 + 7 * q^97 + 23 * q^98 - 13 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - x^{2} - 6x + 7$ x^3 - x^2 - 6*x + 7 :

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

2.28514
−2.50702
1.22188

−2.50702

1.22188

4.28514

−3.06327

0.221876

−5.72889

−1.50702

1.2

1.22188

2.28514

−0.507019

2.79216

1.28514

−3.06327

2.22188

1.3

2.28514

−2.50702

3.22188

−5.72889

−3.50702

2.79216

3.28514

Atkin-Lehner signs

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

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	9025.2.a.ba		3
5.b	even	2	1		1805.2.a.g		3
19.b	odd	2	1		9025.2.a.z		3
19.d	odd	6	2		475.2.e.d		6
95.d	odd	2	1		1805.2.a.h		3
95.h	odd	6	2		95.2.e.b	✓	6
95.l	even	12	4		475.2.j.b		12
285.q	even	6	2		855.2.k.g		6
380.s	even	6	2		1520.2.q.j		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.2.e.b	✓	6	95.h	odd	6	2
475.2.e.d		6	19.d	odd	6	2
475.2.j.b		12	95.l	even	12	4
855.2.k.g		6	285.q	even	6	2
1520.2.q.j		6	380.s	even	6	2
1805.2.a.g		3	5.b	even	2	1
1805.2.a.h		3	95.d	odd	2	1
9025.2.a.z		3	19.b	odd	2	1
9025.2.a.ba		3	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 $S_{2}^{\mathrm{new}}(\Gamma_0(9025))$ :

T_{2}^{3} - T_{2}^{2} - 6T_{2} + 7

T2^3 - T2^2 - 6*T2 + 7

T_{3}^{3} - T_{3}^{2} - 6T_{3} + 7

T3^3 - T3^2 - 6*T3 + 7

T_{7}^{3} + 2T_{7}^{2} - 5T_{7} + 1

T7^3 + 2*T7^2 - 5*T7 + 1

T_{11}^{3} + 5T_{11}^{2} + 2T_{11} - 1

T11^3 + 5*T11^2 + 2*T11 - 1

T_{29}^{3} - 2T_{29}^{2} - 5T_{29} - 1

T29^3 - 2*T29^2 - 5*T29 - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3} - T^{2} - 6T + 7$ T^3 - T^2 - 6*T + 7
$3$	$T^{3} - T^{2} - 6T + 7$ T^3 - T^2 - 6*T + 7
$5$	$T^{3}$ T^3
$7$	$T^{3} + 2 T^{2} + \cdots + 1$ T^3 + 2T^2 - 5T + 1
$11$	$T^{3} + 5 T^{2} + \cdots - 1$ T^3 + 5T^2 + 2T - 1
$13$	$(T + 5)^{3}$ (T + 5)^3
$17$	$T^{3} + T^{2} - 44T - 7$ T^3 + T^2 - 44*T - 7
$19$	$T^{3}$ T^3
$23$	$T^{3} + 4 T^{2} + \cdots - 49$ T^3 + 4T^2 - 39T - 49
$29$	$T^{3} - 2 T^{2} + \cdots - 1$ T^3 - 2T^2 - 5T - 1
$31$	$T^{3} - T^{2} - 6T + 7$ T^3 - T^2 - 6*T + 7
$37$	$T^{3} + 2 T^{2} + \cdots - 227$ T^3 + 2T^2 - 119T - 227
$41$	$T^{3} - 2 T^{2} + \cdots + 37$ T^3 - 2T^2 - 43T + 37
$43$	$T^{3} - T^{2} + \cdots + 121$ T^3 - T^2 - 44*T + 121
$47$	$T^{3} + 6 T^{2} + \cdots - 49$ T^3 + 6T^2 - 7T - 49
$53$	$T^{3} - 11 T^{2} + \cdots + 311$ T^3 - 11T^2 - 42T + 311
$59$	$T^{3} + 6 T^{2} + \cdots - 49$ T^3 + 6T^2 - 7T - 49
$61$	$T^{3} + 9 T^{2} + \cdots - 49$ T^3 + 9T^2 - 49T - 49
$67$	$T^{3} + 20 T^{2} + \cdots + 88$ T^3 + 20T^2 + 108T + 88
$71$	$T^{3} - 29 T^{2} + \cdots - 467$ T^3 - 29T^2 + 236T - 467
$73$	$T^{3} - 22 T^{2} + \cdots - 77$ T^3 - 22T^2 + 117T - 77
$79$	$T^{3} - 24 T^{2} + \cdots + 248$ T^3 - 24T^2 + 116T + 248
$83$	$T^{3} - 3 T^{2} + \cdots - 77$ T^3 - 3T^2 - 54T - 77
$89$	$T^{3} - 14 T^{2} + \cdots + 56$ T^3 - 14T^2 - 36T + 56
$97$	$T^{3} - 7 T^{2} + \cdots + 121$ T^3 - 7T^2 - 66T + 121
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