LMFDB - Newform orbit 464.4.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,4,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$464 = 2^{4} \cdot 29$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	464.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:	$27.3768862427$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.19816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 42x - 54$ x^3 - x^2 - 42*x - 54
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 58)
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.

$f(q)$	$=$	$q + (\beta_1 - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + (4 \beta_{2} - 8) q^{7} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{2} - \beta_1 - 3) q^{11} + (11 \beta_{2} + 8 \beta_1 - 4) q^{13}+ \cdots + ( - 22 \beta_{2} - 38 \beta_1 + 114) q^{99}+O(q^{100})$ q + (b1 - 1) * q^3 + (b2 + 2b1 + 6) q^5 + (4b2 - 8) q^7 + (3b2 + 2b1 + 1) * q^9 + (2b2 - b1 - 3) q^11 + (11b2 + 8b1 - 4) * q^13 + (2b2 + 13b1 + 39) * q^15 + (-16b2 - 12b1 - 18) * q^17 + (-10b2 + 8b1 + 52) * q^19 + (-16b2 - 4b1 - 28) * q^21 + (-12b2 + 6b1 + 66) * q^23 + (11b2 + 40b1 + 13) * q^25 + (-6b2 - 17b1 + 53) * q^27 + 29 * q^29 + (36b2 + 11b1 + 25) * q^31 + (-11b2 - 4b1 - 42) * q^33 + (-12b2 - 24b1) * q^35 + (38b2 - 12b1 - 10) * q^37 + (-20b2 + 31b1 + 121) * q^39 + (-6b2 + 4b1 + 186) * q^41 + (-2b2 + 15b1 - 11) * q^43 + (4b2 + 26b1 + 132) * q^45 + (-2b2 - 33b1 - 207) * q^47 + (-80b2 - 64b1 + 201) * q^49 + (28b2 - 70b1 - 162) * q^51 + (-27b2 - 116b1 + 276) * q^53 + (-8b2 - 25b1 - 39) * q^55 + (64b2 + 66b1 + 254) * q^57 + (-4b2 + 86b1 - 90) * q^59 + (40b2 - 80b1 + 134) * q^61 + (-56b2 - 56b1 + 280) * q^63 + (9b2 + 90b1 + 468) * q^65 + (72b2 - 36b1 + 88) * q^67 + (66b2 + 72b1 + 204) * q^69 + (4b2 - 2b1 + 18) * q^71 + (30b2 + 40b1 - 178) * q^73 + (76b2 + 144b1 + 968) * q^75 + (-24b2 - 28b1 + 300) * q^77 + (38b2 + 37b1 + 691) * q^79 + (-108b2 - 58b1 - 485) * q^81 + (-12b2 - 162b1 + 150) * q^83 + (-38b2 - 184b1 - 840) * q^85 + (29b1 - 29) q^87 + (154b2 + 68b1 + 282) * q^89 + (-244b2 - 208b1 + 1064) * q^91 + (-111b2 + 94b1 - 52) * q^93 + (86b2 + 244b1 + 552) * q^95 + (-34b2 + 176b1 + 14) * q^97 + (-22b2 - 38b1 + 114) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9} - 10 q^{11} - 4 q^{13} + 130 q^{15} - 66 q^{17} + 164 q^{19} - 88 q^{21} + 204 q^{23} + 79 q^{25} + 142 q^{27} + 87 q^{29} + 86 q^{31} - 130 q^{33} - 24 q^{35}+ \cdots + 304 q^{99}+O(q^{100})$ 3 * q - 2 * q^3 + 20 * q^5 - 24 * q^7 + 5 * q^9 - 10 * q^11 - 4 * q^13 + 130 * q^15 - 66 * q^17 + 164 * q^19 - 88 * q^21 + 204 * q^23 + 79 * q^25 + 142 * q^27 + 87 * q^29 + 86 * q^31 - 130 * q^33 - 24 * q^35 - 42 * q^37 + 394 * q^39 + 562 * q^41 - 18 * q^43 + 422 * q^45 - 654 * q^47 + 539 * q^49 - 556 * q^51 + 712 * q^53 - 142 * q^55 + 828 * q^57 - 184 * q^59 + 322 * q^61 + 784 * q^63 + 1494 * q^65 + 228 * q^67 + 684 * q^69 + 52 * q^71 - 494 * q^73 + 3048 * q^75 + 872 * q^77 + 2110 * q^79 - 1513 * q^81 + 288 * q^83 - 2704 * q^85 - 58 * q^87 + 914 * q^89 + 2984 * q^91 - 62 * q^93 + 1900 * q^95 + 218 * q^97 + 304 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - x^{2} - 42x - 54$ x^3 - x^2 - 42*x - 54 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{2} - 4\nu - 27 ) / 3$ (v^2 - 4*v - 27) / 3

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$3\beta_{2} + 4\beta _1 + 27$ 3b2 + 4b1 + 27

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

−5.13291
−1.39712
7.53003

−6.13291

2.36031

18.5045

10.6126

1.2

−2.39712

−3.28077

−33.9461

−21.2538

1.3

6.53003

20.9205

−8.55839

15.6413

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$29$	$-1$

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	464.4.a.i		3
4.b	odd	2	1		58.4.a.d	✓	3
8.b	even	2	1		1856.4.a.s		3
8.d	odd	2	1		1856.4.a.r		3
12.b	even	2	1		522.4.a.k		3
20.d	odd	2	1		1450.4.a.h		3
116.d	odd	2	1		1682.4.a.d		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
58.4.a.d	✓	3	4.b	odd	2	1
464.4.a.i		3	1.a	even	1	1	trivial
522.4.a.k		3	12.b	even	2	1
1450.4.a.h		3	20.d	odd	2	1
1682.4.a.d		3	116.d	odd	2	1
1856.4.a.r		3	8.d	odd	2	1
1856.4.a.s		3	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{3} + 2T_{3}^{2} - 41T_{3} - 96$ T3^3 + 2*T3^2 - 41*T3 - 96 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(464))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} + 2 T^{2} + \cdots - 96$ T^3 + 2T^2 - 41T - 96
$5$	$T^{3} - 20 T^{2} + \cdots + 162$ T^3 - 20T^2 - 27T + 162
$7$	$T^{3} + 24 T^{2} + \cdots - 5376$ T^3 + 24T^2 - 496T - 5376
$11$	$T^{3} + 10 T^{2} + \cdots - 2424$ T^3 + 10T^2 - 233T - 2424
$13$	$T^{3} + 4 T^{2} + \cdots + 131706$ T^3 + 4T^2 - 5619T + 131706
$17$	$T^{3} + 66 T^{2} + \cdots - 679368$ T^3 + 66T^2 - 10660T - 679368
$19$	$T^{3} - 164 T^{2} + \cdots + 664448$ T^3 - 164T^2 - 124T + 664448
$23$	$T^{3} - 204 T^{2} + \cdots + 677376$ T^3 - 204T^2 + 4284T + 677376
$29$	$(T - 29)^{3}$ (T - 29)^3
$31$	$T^{3} - 86 T^{2} + \cdots + 4766172$ T^3 - 86T^2 - 48089T + 4766172
$37$	$T^{3} + 42 T^{2} + \cdots - 7684896$ T^3 + 42T^2 - 79456T - 7684896
$41$	$T^{3} - 562 T^{2} + \cdots - 5982048$ T^3 - 562T^2 + 102432T - 5982048
$43$	$T^{3} + 18 T^{2} + \cdots - 196488$ T^3 + 18T^2 - 10369T - 196488
$47$	$T^{3} + 654 T^{2} + \cdots + 3425124$ T^3 + 654T^2 + 98015T + 3425124
$53$	$T^{3} - 712 T^{2} + \cdots + 252120546$ T^3 - 712T^2 - 350571T + 252120546
$59$	$T^{3} + 184 T^{2} + \cdots - 57362928$ T^3 + 184T^2 - 311444T - 57362928
$61$	$T^{3} - 322 T^{2} + \cdots - 5254424$ T^3 - 322T^2 - 388372T - 5254424
$67$	$T^{3} - 228 T^{2} + \cdots - 47608192$ T^3 - 228T^2 - 327840T - 47608192
$71$	$T^{3} - 52 T^{2} + \cdots + 672$ T^3 - 52T^2 - 164T + 672
$73$	$T^{3} + 494 T^{2} + \cdots - 9410208$ T^3 + 494T^2 + 6112T - 9410208
$79$	$T^{3} - 2110 T^{2} + \cdots - 285187172$ T^3 - 2110T^2 + 1400543T - 285187172
$83$	$T^{3} - 288 T^{2} + \cdots + 437606064$ T^3 - 288T^2 - 1038996T + 437606064
$89$	$T^{3} - 914 T^{2} + \cdots + 598011552$ T^3 - 914T^2 - 664800T + 598011552
$97$	$T^{3} - 218 T^{2} + \cdots - 17006112$ T^3 - 218T^2 - 1500768T - 17006112
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