LMFDB - Newform orbit 336.4.q.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(193,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, 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("336.193"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.q (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-3,0,11] 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:	$19.8246417619$
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:	no (minimal twist has level 168)
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.

$f(q)$	$=$	$q + (3 \zeta_{6} - 3) q^{3} + 11 \zeta_{6} q^{5} + ( - 21 \zeta_{6} + 7) q^{7} - 9 \zeta_{6} q^{9} + ( - 39 \zeta_{6} + 39) q^{11} - 32 q^{13} - 33 q^{15} + (12 \zeta_{6} - 12) q^{17} - 88 \zeta_{6} q^{19} + \cdots - 351 q^{99} +O(q^{100})$ q + (3z - 3) q^3 + 11z q^5 + (-21z + 7) q^7 - 9z q^9 + (-39z + 39) q^11 - 32 * q^13 - 33 * q^15 + (12z - 12) q^17 - 88z q^19 + (21z + 42) q^21 - 92z q^23 + (-4z + 4) q^25 + 27 * q^27 + 255 * q^29 + (35z - 35) q^31 + 117z q^33 + (-154z + 231) q^35 + 4z q^37 + (-96z + 96) q^39 + 16 * q^41 + 330 * q^43 + (-99z + 99) q^45 - 298z q^47 + (147z - 392) q^49 - 36z q^51 + (-717z + 717) q^53 + 429 * q^55 + 264 * q^57 + (217z - 217) q^59 - 386z q^61 + (126z - 189) q^63 - 352z q^65 + (-906z + 906) q^67 + 276 * q^69 + 34 * q^71 + (-838z + 838) q^73 + 12z q^75 + (-273z - 546) q^77 + 1325z q^79 + (81z - 81) q^81 - 1163 * q^83 - 132 * q^85 + (765z - 765) q^87 + 54z q^89 + (672z - 224) q^91 - 105z q^93 + (-968z + 968) q^95 + 7 * q^97 - 351 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{3} + 11 q^{5} - 7 q^{7} - 9 q^{9} + 39 q^{11} - 64 q^{13} - 66 q^{15} - 12 q^{17} - 88 q^{19} + 105 q^{21} - 92 q^{23} + 4 q^{25} + 54 q^{27} + 510 q^{29} - 35 q^{31} + 117 q^{33} + 308 q^{35}+ \cdots - 702 q^{99}+O(q^{100})$ 2 * q - 3 * q^3 + 11 * q^5 - 7 * q^7 - 9 * q^9 + 39 * q^11 - 64 * q^13 - 66 * q^15 - 12 * q^17 - 88 * q^19 + 105 * q^21 - 92 * q^23 + 4 * q^25 + 54 * q^27 + 510 * q^29 - 35 * q^31 + 117 * q^33 + 308 * q^35 + 4 * q^37 + 96 * q^39 + 32 * q^41 + 660 * q^43 + 99 * q^45 - 298 * q^47 - 637 * q^49 - 36 * q^51 + 717 * q^53 + 858 * q^55 + 528 * q^57 - 217 * q^59 - 386 * q^61 - 252 * q^63 - 352 * q^65 + 906 * q^67 + 552 * q^69 + 68 * q^71 + 838 * q^73 + 12 * q^75 - 1365 * q^77 + 1325 * q^79 - 81 * q^81 - 2326 * q^83 - 264 * q^85 - 765 * q^87 + 54 * q^89 + 224 * q^91 - 105 * q^93 + 968 * q^95 + 14 * q^97 - 702 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$ .

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$-\zeta_{6}$

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}

193.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.50000

2.59808i

5.50000

9.52628i

−3.50000

−

18.1865i

−4.50000

−

7.79423i

289.1

−1.50000

−

2.59808i

5.50000

−

9.52628i

−3.50000

18.1865i

−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	336.4.q.c		2
4.b	odd	2	1		168.4.q.c	✓	2
7.c	even	3	1	inner	336.4.q.c		2
7.c	even	3	1		2352.4.a.y		1
7.d	odd	6	1		2352.4.a.o		1
12.b	even	2	1		504.4.s.a		2
28.f	even	6	1		1176.4.a.n		1
28.g	odd	6	1		168.4.q.c	✓	2
28.g	odd	6	1		1176.4.a.c		1
84.n	even	6	1		504.4.s.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.q.c	✓	2	4.b	odd	2	1
168.4.q.c	✓	2	28.g	odd	6	1
336.4.q.c		2	1.a	even	1	1	trivial
336.4.q.c		2	7.c	even	3	1	inner
504.4.s.a		2	12.b	even	2	1
504.4.s.a		2	84.n	even	6	1
1176.4.a.c		1	28.g	odd	6	1
1176.4.a.n		1	28.f	even	6	1
2352.4.a.o		1	7.d	odd	6	1
2352.4.a.y		1	7.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{2} - 11T_{5} + 121$ T5^2 - 11*T5 + 121 acting on $S_{4}^{\mathrm{new}}(336, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$5$	$T^{2} - 11T + 121$ T^2 - 11*T + 121
$7$	$T^{2} + 7T + 343$ T^2 + 7*T + 343
$11$	$T^{2} - 39T + 1521$ T^2 - 39*T + 1521
$13$	$(T + 32)^{2}$ (T + 32)^2
$17$	$T^{2} + 12T + 144$ T^2 + 12*T + 144
$19$	$T^{2} + 88T + 7744$ T^2 + 88*T + 7744
$23$	$T^{2} + 92T + 8464$ T^2 + 92*T + 8464
$29$	$(T - 255)^{2}$ (T - 255)^2
$31$	$T^{2} + 35T + 1225$ T^2 + 35*T + 1225
$37$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$41$	$(T - 16)^{2}$ (T - 16)^2
$43$	$(T - 330)^{2}$ (T - 330)^2
$47$	$T^{2} + 298T + 88804$ T^2 + 298*T + 88804
$53$	$T^{2} - 717T + 514089$ T^2 - 717*T + 514089
$59$	$T^{2} + 217T + 47089$ T^2 + 217*T + 47089
$61$	$T^{2} + 386T + 148996$ T^2 + 386*T + 148996
$67$	$T^{2} - 906T + 820836$ T^2 - 906*T + 820836
$71$	$(T - 34)^{2}$ (T - 34)^2
$73$	$T^{2} - 838T + 702244$ T^2 - 838*T + 702244
$79$	$T^{2} - 1325 T + 1755625$ T^2 - 1325*T + 1755625
$83$	$(T + 1163)^{2}$ (T + 1163)^2
$89$	$T^{2} - 54T + 2916$ T^2 - 54*T + 2916
$97$	$(T - 7)^{2}$ (T - 7)^2
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