LMFDB - Newform orbit 189.6.a.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [189,6,Mod(1,189)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(189, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("189.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$N$	$=$	$189 = 3^{3} \cdot 7$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	189.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,104] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.3125419447$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 5$ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	yes
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 $\beta = \sqrt{21}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 2 \beta q^{2} + 52 q^{4} + 19 \beta q^{5} + 49 q^{7} - 40 \beta q^{8} - 798 q^{10} - 7 \beta q^{11} - 940 q^{13} - 98 \beta q^{14} + 16 q^{16} + 2 \beta q^{17} - 1939 q^{19} + 988 \beta q^{20} + 294 q^{22} + \cdots - 4802 \beta q^{98} +O(q^{100})$ q - 2b q^2 + 52 * q^4 + 19b q^5 + 49 * q^7 - 40b q^8 - 798 * q^10 - 7b q^11 - 940 * q^13 - 98b q^14 + 16 * q^16 + 2b q^17 - 1939 * q^19 + 988b q^20 + 294 * q^22 - 449b q^23 + 4456 * q^25 + 1880b q^26 + 2548 * q^28 + 1272b q^29 - 8695 * q^31 + 1248b q^32 - 84 * q^34 + 931b q^35 - 2455 * q^37 + 3878b q^38 - 15960 * q^40 - 2535b q^41 + 6014 * q^43 - 364b q^44 + 18858 * q^46 - 5606b q^47 + 2401 * q^49 - 8912b q^50 - 48880 * q^52 + 4028b q^53 - 2793 * q^55 - 1960b q^56 - 53424 * q^58 - 3658b q^59 + 5792 * q^61 + 17390b q^62 - 52928 * q^64 - 17860b q^65 + 20996 * q^67 + 104b q^68 - 39102 * q^70 - 13713b q^71 - 79870 * q^73 + 4910b q^74 - 100828 * q^76 - 343b q^77 + 70502 * q^79 + 304b q^80 + 106470 * q^82 + 8976b q^83 + 798 * q^85 - 12028b q^86 + 5880 * q^88 + 17593b q^89 - 46060 * q^91 - 23348b q^92 + 235452 * q^94 - 36841b q^95 - 140488 * q^97 - 4802b q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 104 q^{4} + 98 q^{7} - 1596 q^{10} - 1880 q^{13} + 32 q^{16} - 3878 q^{19} + 588 q^{22} + 8912 q^{25} + 5096 q^{28} - 17390 q^{31} - 168 q^{34} - 4910 q^{37} - 31920 q^{40} + 12028 q^{43} + 37716 q^{46}+ \cdots - 280976 q^{97}+O(q^{100})$ 2 * q + 104 * q^4 + 98 * q^7 - 1596 * q^10 - 1880 * q^13 + 32 * q^16 - 3878 * q^19 + 588 * q^22 + 8912 * q^25 + 5096 * q^28 - 17390 * q^31 - 168 * q^34 - 4910 * q^37 - 31920 * q^40 + 12028 * q^43 + 37716 * q^46 + 4802 * q^49 - 97760 * q^52 - 5586 * q^55 - 106848 * q^58 + 11584 * q^61 - 105856 * q^64 + 41992 * q^67 - 78204 * q^70 - 159740 * q^73 - 201656 * q^76 + 141004 * q^79 + 212940 * q^82 + 1596 * q^85 + 11760 * q^88 - 92120 * q^91 + 470904 * q^94 - 280976 * q^97

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

2.79129
−1.79129

−9.16515

52.0000

87.0689

49.0000

−183.303

−798.000

1.2

9.16515

52.0000

−87.0689

49.0000

183.303

−798.000

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$7$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	189.6.a.g	✓	2
3.b	odd	2	1	inner	189.6.a.g	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.6.a.g	✓	2	1.a	even	1	1	trivial
189.6.a.g	✓	2	3.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{2} - 84$ T2^2 - 84 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(189))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 84$ T^2 - 84
$3$	$T^{2}$ T^2
$5$	$T^{2} - 7581$ T^2 - 7581
$7$	$(T - 49)^{2}$ (T - 49)^2
$11$	$T^{2} - 1029$ T^2 - 1029
$13$	$(T + 940)^{2}$ (T + 940)^2
$17$	$T^{2} - 84$ T^2 - 84
$19$	$(T + 1939)^{2}$ (T + 1939)^2
$23$	$T^{2} - 4233621$ T^2 - 4233621
$29$	$T^{2} - 33977664$ T^2 - 33977664
$31$	$(T + 8695)^{2}$ (T + 8695)^2
$37$	$(T + 2455)^{2}$ (T + 2455)^2
$41$	$T^{2} - 134950725$ T^2 - 134950725
$43$	$(T - 6014)^{2}$ (T - 6014)^2
$47$	$T^{2} - 659971956$ T^2 - 659971956
$53$	$T^{2} - 340720464$ T^2 - 340720464
$59$	$T^{2} - 281000244$ T^2 - 281000244
$61$	$(T - 5792)^{2}$ (T - 5792)^2
$67$	$(T - 20996)^{2}$ (T - 20996)^2
$71$	$T^{2} - 3948973749$ T^2 - 3948973749
$73$	$(T + 79870)^{2}$ (T + 79870)^2
$79$	$(T - 70502)^{2}$ (T - 70502)^2
$83$	$T^{2} - 1691940096$ T^2 - 1691940096
$89$	$T^{2} - 6499786629$ T^2 - 6499786629
$97$	$(T + 140488)^{2}$ (T + 140488)^2
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