LMFDB - Newform orbit 4275.2.a.bo

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4275,2,Mod(1,4275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$4275 = 3^{2} \cdot 5^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-2,0,8,0,0,-4,-12,0,0,-4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$ x^4 - 2x^3 - 4x^2 + 4*x + 3
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{2} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (2 \beta_1 - 2) q^{7} + (\beta_{3} + \beta_{2} - 2) q^{8} - 2 \beta_1 q^{11} + ( - \beta_{3} - 2 \beta_1) q^{13} + (2 \beta_{3} + 2 \beta_1 + 2) q^{14}+ \cdots + ( - 4 \beta_{3} + \beta_{2} + 8) q^{98}+O(q^{100})$ q + b2 * q^2 + (-b2 + b1 + 1) * q^4 + (2b1 - 2) q^7 + (b3 + b2 - 2) * q^8 - 2b1 q^11 + (-b3 - 2b1) q^13 + (2b3 + 2b1 + 2) * q^14 + (-2b3 - 2b2 - 1) * q^16 + (2b3 + 2) q^17 + q^19 + (-2b3 - 2b2 - 2b1 - 2) q^22 + (-2b3 - 2b1 - 2) * q^23 + (-b2 - 3b1) q^26 + (-2b3 + 2b2 + 2) * q^28 + (-2b3 - 2) q^29 + (2b3 + 2b2 - 2b1 + 4) q^31 + (2b3 + b2 - 4b1 + 2) * q^32 + (-4b3 + 2b1 - 4) * q^34 + (-b3 - 2b2) q^37 + b2 * q^38 + (-2b2 + 2b1 - 6) * q^41 + (2b1 - 2) q^43 + (2b3 - 2b1 - 4) * q^44 + (2b3 - 2b2 - 4b1 + 2) q^46 + (-4b2 + 2b1 - 6) * q^47 + (4b2 - 4b1 + 9) * q^49 + (-b3 - 2b2 - 6) q^52 + (-b3 - 2b2 - 4) q^53 + (2b2 - 4b1 + 6) * q^56 + (4b3 - 2b1 + 4) * q^58 + (-2b2 - 2b1) * q^59 + (-2b2 + 4b1 + 2) * q^61 + (-6b3 - 2b2 + 2b1) q^62 + (-4b3 - b2 - b1 - 3) q^64 + (3b3 - 2b2 + 2b1 + 4) q^67 + (6b3 + 2b2 - 2b1 + 6) q^68 + (-2b3 + 2b2 + 2b1 + 4) q^71 + (2b3 - 6) q^73 + (2b3 + 3b2 - 3b1 - 4) q^74 + (-b2 + b1 + 1) * q^76 + (-4b2 - 12) q^77 + (2b2 + 2b1 - 4) * q^79 + (2b3 - 2b2 - 4) * q^82 + (2b3 - 2b1 + 2) * q^83 + (2b3 + 2b1 + 2) * q^86 + (-2b3 - 4b2 + 4b1 - 2) q^88 + (2b3 + 4b2 + 2) * q^89 + (2b3 - 6b2 + 2b1 - 12) q^91 + (-4b3 - 2b2 - 10) * q^92 + (2b3 - 2b1 - 10) * q^94 + (b3 + 2b2 - 4b1 - 4) * q^97 + (-4b3 + b2 + 8) q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 12 q^{8} - 4 q^{11} - 2 q^{13} + 8 q^{14} + 4 q^{16} + 4 q^{17} + 4 q^{19} - 4 q^{22} - 8 q^{23} - 4 q^{26} + 8 q^{28} - 4 q^{29} + 4 q^{31} - 6 q^{32} - 4 q^{34} + 6 q^{37}+ \cdots + 38 q^{98}+O(q^{100})$ 4 * q - 2 * q^2 + 8 * q^4 - 4 * q^7 - 12 * q^8 - 4 * q^11 - 2 * q^13 + 8 * q^14 + 4 * q^16 + 4 * q^17 + 4 * q^19 - 4 * q^22 - 8 * q^23 - 4 * q^26 + 8 * q^28 - 4 * q^29 + 4 * q^31 - 6 * q^32 - 4 * q^34 + 6 * q^37 - 2 * q^38 - 16 * q^41 - 4 * q^43 - 24 * q^44 - 12 * q^47 + 20 * q^49 - 18 * q^52 - 10 * q^53 + 12 * q^56 + 4 * q^58 + 20 * q^61 + 20 * q^62 - 4 * q^64 + 18 * q^67 + 4 * q^68 + 20 * q^71 - 28 * q^73 - 32 * q^74 + 8 * q^76 - 40 * q^77 - 16 * q^79 - 16 * q^82 + 8 * q^86 + 12 * q^88 - 4 * q^89 - 36 * q^91 - 28 * q^92 - 48 * q^94 - 30 * q^97 + 38 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - 2x^{3} - 4x^{2} + 4x + 3$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 :

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

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 3$ b2 + b1 + 3
$\nu^{3}$	$=$	$\beta_{3} + 2\beta_{2} + 5\beta _1 + 4$ b3 + 2b2 + 5b1 + 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

1.28734
−0.552409
−1.51658
2.78165

−2.63010

4.91744

0.574672

−7.67316

1.2

−2.14243

2.59002

−3.10482

−1.26409

1.3

0.816594

−1.33317

−5.03316

−2.72185

1.4

1.95594

1.82571

3.56331

−0.340899

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$5$	$+1$
$19$	$-1$

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	4275.2.a.bo		4
3.b	odd	2	1		475.2.a.i		4
5.b	even	2	1		855.2.a.m		4
12.b	even	2	1		7600.2.a.cf		4
15.d	odd	2	1		95.2.a.b	✓	4
15.e	even	4	2		475.2.b.e		8
57.d	even	2	1		9025.2.a.bf		4
60.h	even	2	1		1520.2.a.t		4
105.g	even	2	1		4655.2.a.y		4
120.i	odd	2	1		6080.2.a.cc		4
120.m	even	2	1		6080.2.a.ch		4
285.b	even	2	1		1805.2.a.p		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
95.2.a.b	✓	4	15.d	odd	2	1
475.2.a.i		4	3.b	odd	2	1
475.2.b.e		8	15.e	even	4	2
855.2.a.m		4	5.b	even	2	1
1520.2.a.t		4	60.h	even	2	1
1805.2.a.p		4	285.b	even	2	1
4275.2.a.bo		4	1.a	even	1	1	trivial
4655.2.a.y		4	105.g	even	2	1
6080.2.a.cc		4	120.i	odd	2	1
6080.2.a.ch		4	120.m	even	2	1
7600.2.a.cf		4	12.b	even	2	1
9025.2.a.bf		4	57.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4275))$ :

T_{2}^{4} + 2T_{2}^{3} - 6T_{2}^{2} - 8T_{2} + 9

T2^4 + 2*T2^3 - 6*T2^2 - 8*T2 + 9

T_{7}^{4} + 4T_{7}^{3} - 16T_{7}^{2} - 48T_{7} + 32

T7^4 + 4*T7^3 - 16*T7^2 - 48*T7 + 32

T_{11}^{4} + 4T_{11}^{3} - 16T_{11}^{2} - 32T_{11} + 48

T11^4 + 4*T11^3 - 16*T11^2 - 32*T11 + 48

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 2 T^{3} + \cdots + 9$ T^4 + 2T^3 - 6T^2 - 8*T + 9
$3$	$T^{4}$ T^4
$5$	$T^{4}$ T^4
$7$	$T^{4} + 4 T^{3} + \cdots + 32$ T^4 + 4T^3 - 16T^2 - 48*T + 32
$11$	$T^{4} + 4 T^{3} + \cdots + 48$ T^4 + 4T^3 - 16T^2 - 32*T + 48
$13$	$T^{4} + 2 T^{3} + \cdots + 20$ T^4 + 2T^3 - 24T^2 - 32*T + 20
$17$	$T^{4} - 4 T^{3} + \cdots + 48$ T^4 - 4T^3 - 32T^2 + 16*T + 48
$19$	$(T - 1)^{4}$ (T - 1)^4
$23$	$T^{4} + 8 T^{3} + \cdots + 288$ T^4 + 8T^3 - 24T^2 - 176*T + 288
$29$	$T^{4} + 4 T^{3} + \cdots + 48$ T^4 + 4T^3 - 32T^2 - 16*T + 48
$31$	$T^{4} - 4 T^{3} + \cdots - 640$ T^4 - 4T^3 - 80T^2 + 512*T - 640
$37$	$T^{4} - 6 T^{3} + \cdots + 4$ T^4 - 6T^3 - 24T^2 + 40*T + 4
$41$	$T^{4} + 16 T^{3} + \cdots - 240$ T^4 + 16T^3 + 56T^2 - 32*T - 240
$43$	$T^{4} + 4 T^{3} + \cdots + 32$ T^4 + 4T^3 - 16T^2 - 48*T + 32
$47$	$T^{4} + 12 T^{3} + \cdots + 1056$ T^4 + 12T^3 - 64T^2 - 656*T + 1056
$53$	$T^{4} + 10 T^{3} + \cdots - 348$ T^4 + 10T^3 - 184T - 348
$59$	$T^{4} - 64 T^{2} + \cdots - 192$ T^4 - 64T^2 + 224T - 192
$61$	$T^{4} - 20 T^{3} + \cdots - 2656$ T^4 - 20T^3 + 56T^2 + 688*T - 2656
$67$	$T^{4} - 18 T^{3} + \cdots - 1076$ T^4 - 18T^3 + 8T^2 + 488*T - 1076
$71$	$T^{4} - 20 T^{3} + \cdots - 4224$ T^4 - 20T^3 + 32T^2 + 1024*T - 4224
$73$	$T^{4} + 28 T^{3} + \cdots + 176$ T^4 + 28T^3 + 256T^2 + 784*T + 176
$79$	$T^{4} + 16 T^{3} + \cdots - 1856$ T^4 + 16T^3 + 32T^2 - 480*T - 1856
$83$	$T^{4} - 72 T^{2} + \cdots + 480$ T^4 - 72T^2 - 112T + 480
$89$	$T^{4} + 4 T^{3} + \cdots + 240$ T^4 + 4T^3 - 144T^2 + 176*T + 240
$97$	$T^{4} + 30 T^{3} + \cdots - 1388$ T^4 + 30T^3 + 224T^2 + 8*T - 1388
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