LMFDB - Newform orbit 9200.2.a.ch

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9200 = 2^{4} \cdot 5^{2} \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 2x + 1$ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 4600)
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.

$f(q)$	$=$	$q + ( - \beta_1 + 1) q^{3} + ( - 2 \beta_1 + 2) q^{7} + (\beta_{2} - 2 \beta_1) q^{9} + (2 \beta_{2} + 2 \beta_1) q^{11} + ( - \beta_{2} + 2 \beta_1 - 1) q^{13} + (2 \beta_1 - 2) q^{17} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{19}+ \cdots + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{99}+O(q^{100})$ q + (-b1 + 1) * q^3 + (-2b1 + 2) q^7 + (b2 - 2b1) q^9 + (2b2 + 2b1) * q^11 + (-b2 + 2b1 - 1) q^13 + (2b1 - 2) q^17 + (-4b2 + 4b1 - 4) * q^19 + (2b2 - 4b1 + 6) * q^21 - q^23 + (2b2 + b1) q^27 + (-5b2 + 3b1 - 4) * q^29 + (b2 - 3b1 - 2) q^31 + (-2b2 + 2b1 - 6) * q^33 + (6b2 - 2b1 + 6) * q^37 + (-2b2 + 3b1 - 4) * q^39 + (-2b2 - 3b1 - 1) * q^41 + (-4b2 + 2b1 - 10) * q^43 + (b2 + 2b1 + 5) q^47 + (4b2 - 8b1 + 5) * q^49 + (-2b2 + 4b1 - 6) * q^51 + (-4b1 - 2) q^53 + (-4b2 + 8b1 - 8) * q^57 + (6b2 - 4b1 + 7) * q^59 + (-6b2 + 8b1 - 6) * q^61 + (4b2 - 4b1 + 6) * q^63 + (-4b2 + 8b1) * q^67 + (b1 - 1) * q^69 + (-9b2 + 2b1 - 5) * q^71 + (9b2 - 7b1 + 4) * q^73 + (-4b2 + 4b1 - 12) * q^77 + (8b2 - 6b1 + 6) * q^79 + (-4b2 + 7b1 - 4) * q^81 + (4b2 + 8b1 - 4) * q^83 + (-3b2 + 7b1 - 5) * q^87 + (-2b2 - 4b1 - 6) * q^89 + (-4b2 + 6b1 - 8) * q^91 + (3b2 - b1 + 3) q^93 + (-2b2 + 2b1 - 14) * q^97 + (-8b2 + 2b1 - 8) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 2 q^{3} + 4 q^{7} - 3 q^{9} - 4 q^{17} - 4 q^{19} + 12 q^{21} - 3 q^{23} - q^{27} - 4 q^{29} - 10 q^{31} - 14 q^{33} + 10 q^{37} - 7 q^{39} - 4 q^{41} - 24 q^{43} + 16 q^{47} + 3 q^{49} - 12 q^{51}+ \cdots - 14 q^{99}+O(q^{100})$ 3 * q + 2 * q^3 + 4 * q^7 - 3 * q^9 - 4 * q^17 - 4 * q^19 + 12 * q^21 - 3 * q^23 - q^27 - 4 * q^29 - 10 * q^31 - 14 * q^33 + 10 * q^37 - 7 * q^39 - 4 * q^41 - 24 * q^43 + 16 * q^47 + 3 * q^49 - 12 * q^51 - 10 * q^53 - 12 * q^57 + 11 * q^59 - 4 * q^61 + 10 * q^63 + 12 * q^67 - 2 * q^69 - 4 * q^71 - 4 * q^73 - 28 * q^77 + 4 * q^79 - q^81 - 8 * q^83 - 5 * q^87 - 20 * q^89 - 14 * q^91 + 5 * q^93 - 38 * q^97 - 14 * q^99

Basis of coefficient ring in terms of $\nu = \zeta_{14} + \zeta_{14}^{-1}$ :

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

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

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

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

1.80194
0.445042
−1.24698

−0.801938

−1.60388

−2.35690

1.2

0.554958

1.10992

−2.69202

1.3

2.24698

4.49396

2.04892

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$23$	$+1$

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	9200.2.a.ch		3
4.b	odd	2	1		4600.2.a.w	✓	3
5.b	even	2	1		9200.2.a.cb		3
20.d	odd	2	1		4600.2.a.z	yes	3
20.e	even	4	2		4600.2.e.s		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4600.2.a.w	✓	3	4.b	odd	2	1
4600.2.a.z	yes	3	20.d	odd	2	1
4600.2.e.s		6	20.e	even	4	2
9200.2.a.cb		3	5.b	even	2	1
9200.2.a.ch		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 $S_{2}^{\mathrm{new}}(\Gamma_0(9200))$ :

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

T_{7}^{3} - 4T_{7}^{2} - 4T_{7} + 8

T_{11}^{3} - 28T_{11} - 56

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} - 2T^{2} - T + 1$ T^3 - 2*T^2 - T + 1
$5$	$T^{3}$ T^3
$7$	$T^{3} - 4 T^{2} + \cdots + 8$ T^3 - 4T^2 - 4T + 8
$11$	$T^{3} - 28T - 56$ T^3 - 28*T - 56
$13$	$T^{3} - 7T + 7$ T^3 - 7*T + 7
$17$	$T^{3} + 4 T^{2} + \cdots - 8$ T^3 + 4T^2 - 4T - 8
$19$	$T^{3} + 4 T^{2} + \cdots - 64$ T^3 + 4T^2 - 32T - 64
$23$	$(T + 1)^{3}$ (T + 1)^3
$29$	$T^{3} + 4 T^{2} + \cdots - 169$ T^3 + 4T^2 - 39T - 169
$31$	$T^{3} + 10 T^{2} + \cdots - 41$ T^3 + 10T^2 + 17T - 41
$37$	$T^{3} - 10 T^{2} + \cdots + 328$ T^3 - 10T^2 - 32T + 328
$41$	$T^{3} + 4 T^{2} + \cdots + 41$ T^3 + 4T^2 - 39T + 41
$43$	$T^{3} + 24 T^{2} + \cdots + 232$ T^3 + 24T^2 + 164T + 232
$47$	$T^{3} - 16 T^{2} + \cdots - 83$ T^3 - 16T^2 + 69T - 83
$53$	$T^{3} + 10 T^{2} + \cdots - 104$ T^3 + 10T^2 - 4T - 104
$59$	$T^{3} - 11 T^{2} + \cdots + 379$ T^3 - 11T^2 - 25T + 379
$61$	$T^{3} + 4 T^{2} + \cdots + 104$ T^3 + 4T^2 - 116T + 104
$67$	$T^{3} - 12 T^{2} + \cdots + 832$ T^3 - 12T^2 - 64T + 832
$71$	$T^{3} + 4 T^{2} + \cdots - 533$ T^3 + 4T^2 - 151T - 533
$73$	$T^{3} + 4 T^{2} + \cdots + 349$ T^3 + 4T^2 - 151T + 349
$79$	$T^{3} - 4 T^{2} + \cdots + 568$ T^3 - 4T^2 - 116T + 568
$83$	$T^{3} + 8 T^{2} + \cdots - 1856$ T^3 + 8T^2 - 240T - 1856
$89$	$T^{3} + 20 T^{2} + \cdots + 8$ T^3 + 20T^2 + 68T + 8
$97$	$T^{3} + 38 T^{2} + \cdots + 1912$ T^3 + 38T^2 + 472T + 1912
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