LMFDB - Newform orbit 1710.4.a.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,4,Mod(1,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, 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("1710.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1710.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:	$100.893266110$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{106})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 106$ x^2 - 106
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 570)
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 $\beta = \sqrt{106}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 2 q^{2} + 4 q^{4} + 5 q^{5} + (\beta + 12) q^{7} - 8 q^{8} - 10 q^{10} + ( - 5 \beta + 10) q^{11} + ( - 3 \beta - 10) q^{13} + ( - 2 \beta - 24) q^{14} + 16 q^{16} + 6 \beta q^{17} + 19 q^{19} + 20 q^{20}+ \cdots + ( - 48 \beta + 186) q^{98}+O(q^{100})$ q - 2 * q^2 + 4 * q^4 + 5 * q^5 + (b + 12) * q^7 - 8 * q^8 - 10 * q^10 + (-5b + 10) q^11 + (-3b - 10) q^13 + (-2b - 24) q^14 + 16 * q^16 + 6b q^17 + 19 * q^19 + 20 * q^20 + (10b - 20) q^22 + (2b - 106) q^23 + 25 * q^25 + (6b + 20) q^26 + (4b + 48) q^28 + (15b - 108) q^29 + (22b - 30) q^31 - 32 * q^32 - 12b q^34 + (5b + 60) q^35 + (-23b - 42) q^37 - 38 * q^38 - 40 * q^40 + (-9b - 252) q^41 + (17b + 184) q^43 + (-20b + 40) q^44 + (-4b + 212) q^46 + (-18b - 222) q^47 + (24b - 93) q^49 - 50 * q^50 + (-12b - 40) q^52 + (-2b - 152) q^53 + (-25b + 50) q^55 + (-8b - 96) q^56 + (-30b + 216) q^58 + (6b - 588) q^59 + (-4b + 172) q^61 + (-44b + 60) q^62 + 64 * q^64 + (-15b - 50) q^65 + (-40b + 16) q^67 + 24b q^68 + (-10b - 120) q^70 + (-30b - 852) q^71 + (-12b + 938) q^73 + (46b + 84) q^74 + 76 * q^76 + (-50b - 410) q^77 + (-20b - 24) q^79 + 80 * q^80 + (18b + 504) q^82 + (4b - 362) q^83 + 30b q^85 + (-34b - 368) q^86 + (40b - 80) q^88 + (133b - 92) q^89 + (-46b - 438) q^91 + (8b - 424) q^92 + (36b + 444) q^94 + 95 * q^95 + (-47b - 978) q^97 + (-48b + 186) q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{2} + 8 q^{4} + 10 q^{5} + 24 q^{7} - 16 q^{8} - 20 q^{10} + 20 q^{11} - 20 q^{13} - 48 q^{14} + 32 q^{16} + 38 q^{19} + 40 q^{20} - 40 q^{22} - 212 q^{23} + 50 q^{25} + 40 q^{26} + 96 q^{28}+ \cdots + 372 q^{98}+O(q^{100})$ 2 * q - 4 * q^2 + 8 * q^4 + 10 * q^5 + 24 * q^7 - 16 * q^8 - 20 * q^10 + 20 * q^11 - 20 * q^13 - 48 * q^14 + 32 * q^16 + 38 * q^19 + 40 * q^20 - 40 * q^22 - 212 * q^23 + 50 * q^25 + 40 * q^26 + 96 * q^28 - 216 * q^29 - 60 * q^31 - 64 * q^32 + 120 * q^35 - 84 * q^37 - 76 * q^38 - 80 * q^40 - 504 * q^41 + 368 * q^43 + 80 * q^44 + 424 * q^46 - 444 * q^47 - 186 * q^49 - 100 * q^50 - 80 * q^52 - 304 * q^53 + 100 * q^55 - 192 * q^56 + 432 * q^58 - 1176 * q^59 + 344 * q^61 + 120 * q^62 + 128 * q^64 - 100 * q^65 + 32 * q^67 - 240 * q^70 - 1704 * q^71 + 1876 * q^73 + 168 * q^74 + 152 * q^76 - 820 * q^77 - 48 * q^79 + 160 * q^80 + 1008 * q^82 - 724 * q^83 - 736 * q^86 - 160 * q^88 - 184 * q^89 - 876 * q^91 - 848 * q^92 + 888 * q^94 + 190 * q^95 - 1956 * q^97 + 372 * q^98

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

−10.2956
10.2956

−2.00000

4.00000

5.00000

1.70437

−8.00000

−10.0000

1.2

−2.00000

4.00000

5.00000

22.2956

−8.00000

−10.0000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$5$	$-1$
$19$	$-1$

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	1710.4.a.m		2
3.b	odd	2	1		570.4.a.n	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
570.4.a.n	✓	2	3.b	odd	2	1
1710.4.a.m		2	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 $S_{4}^{\mathrm{new}}(\Gamma_0(1710))$ :

T_{7}^{2} - 24T_{7} + 38

T_{11}^{2} - 20T_{11} - 2550

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T + 2)^{2}$ (T + 2)^2
$3$	$T^{2}$ T^2
$5$	$(T - 5)^{2}$ (T - 5)^2
$7$	$T^{2} - 24T + 38$ T^2 - 24*T + 38
$11$	$T^{2} - 20T - 2550$ T^2 - 20*T - 2550
$13$	$T^{2} + 20T - 854$ T^2 + 20*T - 854
$17$	$T^{2} - 3816$ T^2 - 3816
$19$	$(T - 19)^{2}$ (T - 19)^2
$23$	$T^{2} + 212T + 10812$ T^2 + 212*T + 10812
$29$	$T^{2} + 216T - 12186$ T^2 + 216*T - 12186
$31$	$T^{2} + 60T - 50404$ T^2 + 60*T - 50404
$37$	$T^{2} + 84T - 54310$ T^2 + 84*T - 54310
$41$	$T^{2} + 504T + 54918$ T^2 + 504*T + 54918
$43$	$T^{2} - 368T + 3222$ T^2 - 368*T + 3222
$47$	$T^{2} + 444T + 14940$ T^2 + 444*T + 14940
$53$	$T^{2} + 304T + 22680$ T^2 + 304*T + 22680
$59$	$T^{2} + 1176 T + 341928$ T^2 + 1176*T + 341928
$61$	$T^{2} - 344T + 27888$ T^2 - 344*T + 27888
$67$	$T^{2} - 32T - 169344$ T^2 - 32*T - 169344
$71$	$T^{2} + 1704 T + 630504$ T^2 + 1704*T + 630504
$73$	$T^{2} - 1876 T + 864580$ T^2 - 1876*T + 864580
$79$	$T^{2} + 48T - 41824$ T^2 + 48*T - 41824
$83$	$T^{2} + 724T + 129348$ T^2 + 724*T + 129348
$89$	$T^{2} + 184 T - 1866570$ T^2 + 184*T - 1866570
$97$	$T^{2} + 1956 T + 722330$ T^2 + 1956*T + 722330
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