LMFDB - Newform orbit 168.4.q.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [168,4,Mod(25,168)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("168.25"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$168 = 2^{3} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	168.q (of order $3$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$9.91232088096$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 3 \zeta_{6} q^{3} + (2 \zeta_{6} - 2) q^{5} + ( - 7 \zeta_{6} + 21) q^{7} + (9 \zeta_{6} - 9) q^{9} + 18 \zeta_{6} q^{11} + 33 q^{13} - 6 q^{15} + 68 \zeta_{6} q^{17} + (25 \zeta_{6} - 25) q^{19} + (42 \zeta_{6} + 21) q^{21}+ \cdots - 162 q^{99}+O(q^{100})$ q + 3z q^3 + (2z - 2) q^5 + (-7z + 21) q^7 + (9z - 9) q^9 + 18z q^11 + 33 * q^13 - 6 * q^15 + 68z q^17 + (25z - 25) q^19 + (42z + 21) q^21 + (92z - 92) q^23 + 121z q^25 - 27 * q^27 + 92 * q^29 - 25z q^31 + (54z - 54) q^33 + (42z - 28) q^35 + (-213z + 213) q^37 + 99z q^39 + 94 * q^41 - 67 * q^43 - 18z q^45 + (278z - 278) q^47 + (-245z + 392) q^49 + (204z - 204) q^51 + 400z q^53 - 36 * q^55 - 75 * q^57 - 744z q^59 + (-734z + 734) q^61 + (189z - 126) q^63 + (66z - 66) q^65 - 555z q^67 - 276 * q^69 - 642 * q^71 - 973z q^73 + (363z - 363) q^75 + (252z + 126) q^77 + (-785z + 785) q^79 - 81z q^81 - 822 * q^83 - 136 * q^85 + 276z q^87 + (424z - 424) q^89 + (-231z + 693) q^91 + (-75z + 75) q^93 - 50z q^95 - 734 * q^97 - 162 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{3} - 2 q^{5} + 35 q^{7} - 9 q^{9} + 18 q^{11} + 66 q^{13} - 12 q^{15} + 68 q^{17} - 25 q^{19} + 84 q^{21} - 92 q^{23} + 121 q^{25} - 54 q^{27} + 184 q^{29} - 25 q^{31} - 54 q^{33} - 14 q^{35}+ \cdots - 324 q^{99}+O(q^{100})$ 2 * q + 3 * q^3 - 2 * q^5 + 35 * q^7 - 9 * q^9 + 18 * q^11 + 66 * q^13 - 12 * q^15 + 68 * q^17 - 25 * q^19 + 84 * q^21 - 92 * q^23 + 121 * q^25 - 54 * q^27 + 184 * q^29 - 25 * q^31 - 54 * q^33 - 14 * q^35 + 213 * q^37 + 99 * q^39 + 188 * q^41 - 134 * q^43 - 18 * q^45 - 278 * q^47 + 539 * q^49 - 204 * q^51 + 400 * q^53 - 72 * q^55 - 150 * q^57 - 744 * q^59 + 734 * q^61 - 63 * q^63 - 66 * q^65 - 555 * q^67 - 552 * q^69 - 1284 * q^71 - 973 * q^73 - 363 * q^75 + 504 * q^77 + 785 * q^79 - 81 * q^81 - 1644 * q^83 - 272 * q^85 + 276 * q^87 - 424 * q^89 + 1155 * q^91 + 75 * q^93 - 50 * q^95 - 1468 * q^97 - 324 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$ .

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$-1 + \zeta_{6}$	$1$	$1$	$1$

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}

25.1

0.500000	−	0.866025i
0.500000	+	0.866025i

1.50000

−

2.59808i

−1.00000

−

1.73205i

17.5000

6.06218i

−4.50000

−

7.79423i

121.1

1.50000

2.59808i

−1.00000

1.73205i

17.5000

−

6.06218i

−4.50000

7.79423i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.4.q.b	✓	2
3.b	odd	2	1		504.4.s.d		2
4.b	odd	2	1		336.4.q.b		2
7.c	even	3	1	inner	168.4.q.b	✓	2
7.c	even	3	1		1176.4.a.d		1
7.d	odd	6	1		1176.4.a.k		1
21.h	odd	6	1		504.4.s.d		2
28.f	even	6	1		2352.4.a.j		1
28.g	odd	6	1		336.4.q.b		2
28.g	odd	6	1		2352.4.a.bc		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.q.b	✓	2	1.a	even	1	1	trivial
168.4.q.b	✓	2	7.c	even	3	1	inner
336.4.q.b		2	4.b	odd	2	1
336.4.q.b		2	28.g	odd	6	1
504.4.s.d		2	3.b	odd	2	1
504.4.s.d		2	21.h	odd	6	1
1176.4.a.d		1	7.c	even	3	1
1176.4.a.k		1	7.d	odd	6	1
2352.4.a.j		1	28.f	even	6	1
2352.4.a.bc		1	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} + 2T_{5} + 4$ T5^2 + 2*T5 + 4 acting on $S_{4}^{\mathrm{new}}(168, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$5$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$7$	$T^{2} - 35T + 343$ T^2 - 35*T + 343
$11$	$T^{2} - 18T + 324$ T^2 - 18*T + 324
$13$	$(T - 33)^{2}$ (T - 33)^2
$17$	$T^{2} - 68T + 4624$ T^2 - 68*T + 4624
$19$	$T^{2} + 25T + 625$ T^2 + 25*T + 625
$23$	$T^{2} + 92T + 8464$ T^2 + 92*T + 8464
$29$	$(T - 92)^{2}$ (T - 92)^2
$31$	$T^{2} + 25T + 625$ T^2 + 25*T + 625
$37$	$T^{2} - 213T + 45369$ T^2 - 213*T + 45369
$41$	$(T - 94)^{2}$ (T - 94)^2
$43$	$(T + 67)^{2}$ (T + 67)^2
$47$	$T^{2} + 278T + 77284$ T^2 + 278*T + 77284
$53$	$T^{2} - 400T + 160000$ T^2 - 400*T + 160000
$59$	$T^{2} + 744T + 553536$ T^2 + 744*T + 553536
$61$	$T^{2} - 734T + 538756$ T^2 - 734*T + 538756
$67$	$T^{2} + 555T + 308025$ T^2 + 555*T + 308025
$71$	$(T + 642)^{2}$ (T + 642)^2
$73$	$T^{2} + 973T + 946729$ T^2 + 973*T + 946729
$79$	$T^{2} - 785T + 616225$ T^2 - 785*T + 616225
$83$	$(T + 822)^{2}$ (T + 822)^2
$89$	$T^{2} + 424T + 179776$ T^2 + 424*T + 179776
$97$	$(T + 734)^{2}$ (T + 734)^2
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