LMFDB - Newform orbit 7225.2.a.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 4$ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1445)
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 $\beta = \frac{1}{2}(1 + \sqrt{17})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta q^{2} - \beta q^{3} + (\beta + 2) q^{4} + (\beta + 4) q^{6} + (\beta + 2) q^{7} + ( - \beta - 4) q^{8} + (\beta + 1) q^{9} + ( - \beta + 3) q^{11} + ( - 3 \beta - 4) q^{12} + (2 \beta - 4) q^{13} + \cdots + (\beta - 1) q^{99} +O(q^{100})$ q - b * q^2 - b * q^3 + (b + 2) * q^4 + (b + 4) * q^6 + (b + 2) * q^7 + (-b - 4) * q^8 + (b + 1) * q^9 + (-b + 3) * q^11 + (-3b - 4) q^12 + (2b - 4) q^13 + (-3b - 4) q^14 + 3b q^16 + (-2b - 4) q^18 + (b + 1) * q^19 + (-3b - 4) q^21 + (-2b + 4) q^22 - b * q^23 + (5b + 4) q^24 + (2b - 8) q^26 + (b - 4) * q^27 + (5b + 8) q^28 + (-b - 5) * q^29 + (-b - 4) * q^32 + (-2b + 4) q^33 + (4b + 6) q^36 + 10 * q^37 + (-2b - 4) q^38 + (2b - 8) q^39 + 7 * q^41 + (7b + 12) q^42 + (2b - 6) q^43 + 2 * q^44 + (b + 4) * q^46 + (2b - 2) q^47 + (-3b - 12) q^48 + (5b + 1) q^49 + 2b q^52 + (-2b - 2) q^53 + (3b - 4) q^54 + (-7b - 12) q^56 + (-2b - 4) q^57 + (6b + 4) q^58 + (-b - 1) * q^59 + (-b - 9) * q^61 + (4b + 6) q^63 + (-b + 4) * q^64 + (-2b + 8) q^66 + (b - 2) * q^67 + (b + 4) * q^69 + (3b - 9) q^71 + (-6b - 8) q^72 + (-2b - 10) q^73 - 10b q^74 + (4b + 6) q^76 + 2 * q^77 + (6b - 8) q^78 + (-b - 11) * q^79 - 7 * q^81 - 7b q^82 + (b - 6) * q^83 + (-13b - 20) q^84 + (4b - 8) q^86 + (6b + 4) q^87 + (2b - 8) q^88 + (-8b + 5) q^89 + 2b q^91 + (-3b - 4) q^92 - 8 * q^94 + (5b + 4) q^96 + (4b - 10) q^97 + (-6b - 20) q^98 + (b - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{2} - q^{3} + 5 q^{4} + 9 q^{6} + 5 q^{7} - 9 q^{8} + 3 q^{9} + 5 q^{11} - 11 q^{12} - 6 q^{13} - 11 q^{14} + 3 q^{16} - 10 q^{18} + 3 q^{19} - 11 q^{21} + 6 q^{22} - q^{23} + 13 q^{24} - 14 q^{26}+ \cdots - q^{99}+O(q^{100})$ 2 * q - q^2 - q^3 + 5 * q^4 + 9 * q^6 + 5 * q^7 - 9 * q^8 + 3 * q^9 + 5 * q^11 - 11 * q^12 - 6 * q^13 - 11 * q^14 + 3 * q^16 - 10 * q^18 + 3 * q^19 - 11 * q^21 + 6 * q^22 - q^23 + 13 * q^24 - 14 * q^26 - 7 * q^27 + 21 * q^28 - 11 * q^29 - 9 * q^32 + 6 * q^33 + 16 * q^36 + 20 * q^37 - 10 * q^38 - 14 * q^39 + 14 * q^41 + 31 * q^42 - 10 * q^43 + 4 * q^44 + 9 * q^46 - 2 * q^47 - 27 * q^48 + 7 * q^49 + 2 * q^52 - 6 * q^53 - 5 * q^54 - 31 * q^56 - 10 * q^57 + 14 * q^58 - 3 * q^59 - 19 * q^61 + 16 * q^63 + 7 * q^64 + 14 * q^66 - 3 * q^67 + 9 * q^69 - 15 * q^71 - 22 * q^72 - 22 * q^73 - 10 * q^74 + 16 * q^76 + 4 * q^77 - 10 * q^78 - 23 * q^79 - 14 * q^81 - 7 * q^82 - 11 * q^83 - 53 * q^84 - 12 * q^86 + 14 * q^87 - 14 * q^88 + 2 * q^89 + 2 * q^91 - 11 * q^92 - 16 * q^94 + 13 * q^96 - 16 * q^97 - 46 * q^98 - q^99

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

2.56155
−1.56155

−2.56155

4.56155

6.56155

4.56155

−6.56155

3.56155

1.2

1.56155

0.438447

2.43845

0.438447

−2.43845

−0.561553

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$17$	$+1$

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	7225.2.a.j		2
5.b	even	2	1		1445.2.a.i	yes	2
17.b	even	2	1		7225.2.a.k		2
85.c	even	2	1		1445.2.a.h	✓	2
85.j	even	4	2		1445.2.d.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1445.2.a.h	✓	2	85.c	even	2	1
1445.2.a.i	yes	2	5.b	even	2	1
1445.2.d.d		4	85.j	even	4	2
7225.2.a.j		2	1.a	even	1	1	trivial
7225.2.a.k		2	17.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7225))$ :

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

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

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

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + T - 4$ T^2 + T - 4
$3$	$T^{2} + T - 4$ T^2 + T - 4
$5$	$T^{2}$ T^2
$7$	$T^{2} - 5T + 2$ T^2 - 5*T + 2
$11$	$T^{2} - 5T + 2$ T^2 - 5*T + 2
$13$	$T^{2} + 6T - 8$ T^2 + 6*T - 8
$17$	$T^{2}$ T^2
$19$	$T^{2} - 3T - 2$ T^2 - 3*T - 2
$23$	$T^{2} + T - 4$ T^2 + T - 4
$29$	$T^{2} + 11T + 26$ T^2 + 11*T + 26
$31$	$T^{2}$ T^2
$37$	$(T - 10)^{2}$ (T - 10)^2
$41$	$(T - 7)^{2}$ (T - 7)^2
$43$	$T^{2} + 10T + 8$ T^2 + 10*T + 8
$47$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$53$	$T^{2} + 6T - 8$ T^2 + 6*T - 8
$59$	$T^{2} + 3T - 2$ T^2 + 3*T - 2
$61$	$T^{2} + 19T + 86$ T^2 + 19*T + 86
$67$	$T^{2} + 3T - 2$ T^2 + 3*T - 2
$71$	$T^{2} + 15T + 18$ T^2 + 15*T + 18
$73$	$T^{2} + 22T + 104$ T^2 + 22*T + 104
$79$	$T^{2} + 23T + 128$ T^2 + 23*T + 128
$83$	$T^{2} + 11T + 26$ T^2 + 11*T + 26
$89$	$T^{2} - 2T - 271$ T^2 - 2*T - 271
$97$	$T^{2} + 16T - 4$ T^2 + 16*T - 4
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