LMFDB - Newform orbit 7225.2.a.v

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.6224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 6x^{2} - 2x + 5$ x^4 - 6x^2 - 2x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 85)
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.

$f(q)$	$=$	$q + (\beta_{3} - 1) q^{2} + ( - \beta_1 + 1) q^{3} + ( - \beta_{2} - \beta_1 + 1) q^{4} + (2 \beta_{3} - \beta_{2} - 1) q^{6} + (\beta_{3} + 2) q^{7} + \beta_{2} q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+ \cdots + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{99}+O(q^{100})$ q + (b3 - 1) * q^2 + (-b1 + 1) * q^3 + (-b2 - b1 + 1) * q^4 + (2b3 - b2 - 1) q^6 + (b3 + 2) * q^7 + b2 * q^8 + (b2 - b1 + 1) * q^9 + (b3 + b2 + b1) * q^11 + (b3 + 2) * q^12 + (-b3 + b2 - 2b1 - 1) q^13 + (3b3 - b2 - b1) q^14 + (2b1 - 1) q^16 + (2b3 - 3b2) * q^18 + (2b3 + 2b1) * q^19 + (2b3 - b2 - 3b1 + 2) * q^21 + (-2b2 - b1 + 3) q^22 + (2b2 + b1 + 1) q^23 + (-b3 + b2 - b1 + 2) * q^24 + (-3b2 + b1) q^26 + (-b3 + 2b2 + b1 + 3) q^27 + (2b3 - 2b2 - 3b1 + 1) q^28 + (2b3 - 2b2) * q^29 + (-3b2 - 2b1 + 3) * q^31 + (-3b3 + 1) q^32 + (b3 - b2 - 2b1 - 1) q^33 + (2b3 + 2b2 - 1) * q^36 + (-2b3 + 2b2 + 4) * q^37 + (-2b1 + 4) q^38 + (-3b3 + 4b2 + b1 + 7) * q^39 + (-3b3 - b1 + 3) q^41 + (7b3 - 3b2 - 2b1 + 1) q^42 + (b3 + b2 + 2b1 - 5) q^43 + (2b3 + b2 - 2b1 - 5) * q^44 + (-3b2 + 1) q^46 + (-b3 + b2 - 2b1 - 1) q^47 + (-2b2 + b1 - 7) q^48 + (6b3 - b2 - b1 - 1) q^49 + (b3 + 5b2 + 4b1 - 1) * q^52 + (4b2 + 4b1 - 2) * q^53 + (b3 - 2b2 + b1 - 3) q^54 + (b2 + 1) * q^56 + (4b3 - 4b2 - 2b1 - 6) q^57 + (2b3 + 2b2 - 2b1 + 2) q^58 - 2 * q^59 + (-b3 + 4b2 + 3b1 - 1) * q^61 + (5b3 + 4b2 - 6) * q^62 + (2b3 - 3b1 + 3) * q^63 + (-2b3 + 3b2 - b1 - 5) * q^64 + (2b3 - b2 - b1 + 2) q^66 + (-b3 + 3b2 - 1) q^67 + (-2b3 + b2 - 3b1 + 2) * q^69 + (b3 - 3b2 - b1 + 2) q^71 + (-3b3 - 2b1 + 7) * q^72 + (-5b3 + 4b2 + b1 + 3) * q^73 + (2b3 - 2b2 + 2b1 - 6) q^74 + (2b3 - 2b2 - 4b1 - 4) q^76 + (3b3 + b2 + 2b1 + 3) * q^77 + (3b3 - 4b2 + 3b1 - 9) q^78 + (4b3 + b2 + 1) q^79 + (-4b3 - b2 - b1 + 1) q^81 + (b3 + 2b2 + 3b1 - 9) * q^82 + (-3b3 - b2 - 4b1 + 5) * q^83 + (6b3 - b2 - b1 + 6) q^84 + (-6b3 - b2 - b1 + 8) q^86 + (6b3 - 4b2 - 4) * q^87 + (-b3 - 2b2 + 4) q^88 + (-b2 + 3b1 + 6) q^89 + (-3b3 - 5b1 - 3) * q^91 + (b3 + 2b2 - 2b1 - 6) * q^92 + (3b3 - b2 + 3) q^93 + (-3b2 + b1) q^94 + (-6b3 + 3b2 + 2b1 + 1) q^96 + (2b3 + 2b2 - 2b1 - 2) q^97 + (6b3 - 5b2 - 6b1 + 12) q^98 + (-3b2 - 2b1 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{2} + 4 q^{3} + 4 q^{4} + 10 q^{7} + 4 q^{9} + 2 q^{11} + 10 q^{12} - 6 q^{13} + 6 q^{14} - 4 q^{16} + 4 q^{18} + 4 q^{19} + 12 q^{21} + 12 q^{22} + 4 q^{23} + 6 q^{24} + 10 q^{27} + 8 q^{28}+ \cdots + 12 q^{99}+O(q^{100})$ 4 * q - 2 * q^2 + 4 * q^3 + 4 * q^4 + 10 * q^7 + 4 * q^9 + 2 * q^11 + 10 * q^12 - 6 * q^13 + 6 * q^14 - 4 * q^16 + 4 * q^18 + 4 * q^19 + 12 * q^21 + 12 * q^22 + 4 * q^23 + 6 * q^24 + 10 * q^27 + 8 * q^28 + 4 * q^29 + 12 * q^31 - 2 * q^32 - 2 * q^33 + 12 * q^37 + 16 * q^38 + 22 * q^39 + 6 * q^41 + 18 * q^42 - 18 * q^43 - 16 * q^44 + 4 * q^46 - 6 * q^47 - 28 * q^48 + 8 * q^49 - 2 * q^52 - 8 * q^53 - 10 * q^54 + 4 * q^56 - 16 * q^57 + 12 * q^58 - 8 * q^59 - 6 * q^61 - 14 * q^62 + 16 * q^63 - 24 * q^64 + 12 * q^66 - 6 * q^67 + 4 * q^69 + 10 * q^71 + 22 * q^72 + 2 * q^73 - 20 * q^74 - 12 * q^76 + 18 * q^77 - 30 * q^78 + 12 * q^79 - 4 * q^81 - 34 * q^82 + 14 * q^83 + 36 * q^84 + 20 * q^86 - 4 * q^87 + 14 * q^88 + 24 * q^89 - 18 * q^91 - 22 * q^92 + 18 * q^93 - 8 * q^96 - 4 * q^97 + 60 * q^98 + 12 * q^99

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

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

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 3$ b2 + b1 + 3
$\nu^{3}$	$=$	$\beta_{3} + \beta_{2} + 5\beta _1 + 1$ b3 + b2 + 5*b1 + 1

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

0.796815
−1.87228
2.44579
−1.37033

−2.31627

0.203185

3.36509

−0.470630

0.683735

−3.16190

−2.95872

1.2

−1.57942

2.87228

0.494582

−4.53654

1.42058

2.37769

5.24997

1.3

−0.134632

−1.44579

−1.98187

0.194649

2.86537

0.536087

−0.909700

1.4

2.03032

2.37033

2.12221

4.81252

5.03032

0.248119

2.61845

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.v		4
5.b	even	2	1		7225.2.a.w		4
5.c	odd	4	2		1445.2.b.e		8
17.b	even	2	1		425.2.a.g		4
51.c	odd	2	1		3825.2.a.bj		4
68.d	odd	2	1		6800.2.a.bw		4
85.c	even	2	1		425.2.a.h		4
85.g	odd	4	2		85.2.b.a	✓	8
255.h	odd	2	1		3825.2.a.bh		4
255.o	even	4	2		765.2.b.c		8
340.d	odd	2	1		6800.2.a.bt		4
340.r	even	4	2		1360.2.e.d		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.2.b.a	✓	8	85.g	odd	4	2
425.2.a.g		4	17.b	even	2	1
425.2.a.h		4	85.c	even	2	1
765.2.b.c		8	255.o	even	4	2
1360.2.e.d		8	340.r	even	4	2
1445.2.b.e		8	5.c	odd	4	2
3825.2.a.bh		4	255.h	odd	2	1
3825.2.a.bj		4	51.c	odd	2	1
6800.2.a.bt		4	340.d	odd	2	1
6800.2.a.bw		4	68.d	odd	2	1
7225.2.a.v		4	1.a	even	1	1	trivial
7225.2.a.w		4	5.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}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 8T_{2} - 1

T_{3}^{4} - 4T_{3}^{3} + 10T_{3} - 2

T_{7}^{4} - 10T_{7}^{3} + 32T_{7}^{2} - 38T_{7} + 14

T_{11}^{4} - 2T_{11}^{3} - 14T_{11}^{2} + 26T_{11} + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 2 T^{3} + \cdots - 1$ T^4 + 2T^3 - 4T^2 - 8*T - 1
$3$	$T^{4} - 4 T^{3} + \cdots - 2$ T^4 - 4T^3 + 10T - 2
$5$	$T^{4}$ T^4
$7$	$T^{4} - 10 T^{3} + \cdots + 14$ T^4 - 10T^3 + 32T^2 - 38*T + 14
$11$	$T^{4} - 2 T^{3} + \cdots + 2$ T^4 - 2T^3 - 14T^2 + 26*T + 2
$13$	$T^{4} + 6 T^{3} + \cdots - 164$ T^4 + 6T^3 - 28T^2 - 192*T - 164
$17$	$T^{4}$ T^4
$19$	$T^{4} - 4 T^{3} + \cdots + 112$ T^4 - 4T^3 - 32T^2 + 80*T + 112
$23$	$T^{4} - 4 T^{3} + \cdots - 10$ T^4 - 4T^3 - 20T^2 + 82*T - 10
$29$	$T^{4} - 4 T^{3} + \cdots - 80$ T^4 - 4T^3 - 32T^2 + 144*T - 80
$31$	$T^{4} - 12 T^{3} + \cdots + 74$ T^4 - 12T^3 - 6T^2 + 190*T + 74
$37$	$T^{4} - 12 T^{3} + \cdots - 16$ T^4 - 12T^3 + 16T^2 + 48*T - 16
$41$	$T^{4} - 6 T^{3} + \cdots + 392$ T^4 - 6T^3 - 36T^2 + 152*T + 392
$43$	$T^{4} + 18 T^{3} + \cdots - 316$ T^4 + 18T^3 + 96T^2 + 88*T - 316
$47$	$T^{4} + 6 T^{3} + \cdots - 164$ T^4 + 6T^3 - 28T^2 - 192*T - 164
$53$	$T^{4} + 8 T^{3} + \cdots + 16$ T^4 + 8T^3 - 104T^2 - 736*T + 16
$59$	$(T + 2)^{4}$ (T + 2)^4
$61$	$T^{4} + 6 T^{3} + \cdots + 1880$ T^4 + 6T^3 - 92T^2 - 336*T + 1880
$67$	$T^{4} + 6 T^{3} + \cdots - 52$ T^4 + 6T^3 - 52T^2 - 216*T - 52
$71$	$T^{4} - 10 T^{3} + \cdots + 242$ T^4 - 10T^3 - 18T^2 + 198*T + 242
$73$	$T^{4} - 2 T^{3} + \cdots + 1256$ T^4 - 2T^3 - 176T^2 + 176*T + 1256
$79$	$T^{4} - 12 T^{3} + \cdots - 526$ T^4 - 12T^3 - 58T^2 + 570*T - 526
$83$	$T^{4} - 14 T^{3} + \cdots - 956$ T^4 - 14T^3 - 44T^2 + 904*T - 956
$89$	$T^{4} - 24 T^{3} + \cdots - 484$ T^4 - 24T^3 + 136T^2 + 132*T - 484
$97$	$T^{4} + 4 T^{3} + \cdots + 2000$ T^4 + 4T^3 - 120T^2 + 2000
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