LMFDB - Newform orbit 294.4.e.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,4,Mod(67,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([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("294.67"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	294.e (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,-2,3,-4,-18,-12,0,16,-9,-36,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$17.3465615417$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
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.

$f(q)$	$=$	$q - 2 \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 18 \zeta_{6} q^{5} - 6 q^{6} + 8 q^{8} - 9 \zeta_{6} q^{9} + (36 \zeta_{6} - 36) q^{10} + ( - 72 \zeta_{6} + 72) q^{11} + 12 \zeta_{6} q^{12} + \cdots - 648 q^{99} +O(q^{100})$ q - 2z q^2 + (-3z + 3) q^3 + (4z - 4) q^4 - 18z q^5 - 6 * q^6 + 8 * q^8 - 9z q^9 + (36z - 36) q^10 + (-72z + 72) q^11 + 12z q^12 - 34 * q^13 - 54 * q^15 - 16z q^16 + (6z - 6) q^17 + (18z - 18) q^18 - 92z q^19 + 72 * q^20 - 144 * q^22 + 180z q^23 + (-24z + 24) q^24 + (199z - 199) q^25 + 68z q^26 - 27 * q^27 - 114 * q^29 + 108z q^30 + (56z - 56) q^31 + (32z - 32) q^32 - 216z q^33 + 12 * q^34 + 36 * q^36 + 34z q^37 + (184z - 184) q^38 + (102z - 102) q^39 - 144z q^40 + 6 * q^41 + 164 * q^43 + 288z q^44 + (162z - 162) q^45 + (-360z + 360) q^46 - 168z q^47 - 48 * q^48 + 398 * q^50 + 18z q^51 + (-136z + 136) q^52 + (654z - 654) q^53 + 54z q^54 - 1296 * q^55 - 276 * q^57 + 228z q^58 + (-492z + 492) q^59 + (-216z + 216) q^60 + 250z q^61 + 112 * q^62 + 64 * q^64 + 612z q^65 + (432z - 432) q^66 + (-124z + 124) q^67 - 24z q^68 + 540 * q^69 + 36 * q^71 - 72z q^72 + (1010z - 1010) q^73 + (-68z + 68) q^74 + 597z q^75 + 368 * q^76 + 204 * q^78 - 56z q^79 + (288z - 288) q^80 + (81z - 81) q^81 - 12z q^82 + 228 * q^83 + 108 * q^85 - 328z q^86 + (342z - 342) q^87 + (-576z + 576) q^88 - 390z q^89 + 324 * q^90 - 720 * q^92 + 168z q^93 + (336z - 336) q^94 + (1656z - 1656) q^95 + 96z q^96 - 70 * q^97 - 648 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} - 18 q^{5} - 12 q^{6} + 16 q^{8} - 9 q^{9} - 36 q^{10} + 72 q^{11} + 12 q^{12} - 68 q^{13} - 108 q^{15} - 16 q^{16} - 6 q^{17} - 18 q^{18} - 92 q^{19} + 144 q^{20} - 288 q^{22}+ \cdots - 1296 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 + 3 * q^3 - 4 * q^4 - 18 * q^5 - 12 * q^6 + 16 * q^8 - 9 * q^9 - 36 * q^10 + 72 * q^11 + 12 * q^12 - 68 * q^13 - 108 * q^15 - 16 * q^16 - 6 * q^17 - 18 * q^18 - 92 * q^19 + 144 * q^20 - 288 * q^22 + 180 * q^23 + 24 * q^24 - 199 * q^25 + 68 * q^26 - 54 * q^27 - 228 * q^29 + 108 * q^30 - 56 * q^31 - 32 * q^32 - 216 * q^33 + 24 * q^34 + 72 * q^36 + 34 * q^37 - 184 * q^38 - 102 * q^39 - 144 * q^40 + 12 * q^41 + 328 * q^43 + 288 * q^44 - 162 * q^45 + 360 * q^46 - 168 * q^47 - 96 * q^48 + 796 * q^50 + 18 * q^51 + 136 * q^52 - 654 * q^53 + 54 * q^54 - 2592 * q^55 - 552 * q^57 + 228 * q^58 + 492 * q^59 + 216 * q^60 + 250 * q^61 + 224 * q^62 + 128 * q^64 + 612 * q^65 - 432 * q^66 + 124 * q^67 - 24 * q^68 + 1080 * q^69 + 72 * q^71 - 72 * q^72 - 1010 * q^73 + 68 * q^74 + 597 * q^75 + 736 * q^76 + 408 * q^78 - 56 * q^79 - 288 * q^80 - 81 * q^81 - 12 * q^82 + 456 * q^83 + 216 * q^85 - 328 * q^86 - 342 * q^87 + 576 * q^88 - 390 * q^89 + 648 * q^90 - 1440 * q^92 + 168 * q^93 - 336 * q^94 - 1656 * q^95 + 96 * q^96 - 140 * q^97 - 1296 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$ .

$n$	$197$	$199$
$\chi(n)$	$1$	$-\zeta_{6}$

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}

67.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

−

1.73205i

1.50000

−

2.59808i

−2.00000

3.46410i

−9.00000

−

15.5885i

−6.00000

8.00000

−4.50000

−

7.79423i

−18.0000

31.1769i

79.1

−1.00000

1.73205i

1.50000

2.59808i

−2.00000

−

3.46410i

−9.00000

15.5885i

−6.00000

8.00000

−4.50000

7.79423i

−18.0000

−

31.1769i

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	294.4.e.c		2
3.b	odd	2	1		882.4.g.w		2
7.b	odd	2	1		294.4.e.b		2
7.c	even	3	1		42.4.a.a	✓	1
7.c	even	3	1	inner	294.4.e.c		2
7.d	odd	6	1		294.4.a.i		1
7.d	odd	6	1		294.4.e.b		2
21.c	even	2	1		882.4.g.o		2
21.g	even	6	1		882.4.a.g		1
21.g	even	6	1		882.4.g.o		2
21.h	odd	6	1		126.4.a.a		1
21.h	odd	6	1		882.4.g.w		2
28.f	even	6	1		2352.4.a.a		1
28.g	odd	6	1		336.4.a.l		1
35.j	even	6	1		1050.4.a.g		1
35.l	odd	12	2		1050.4.g.a		2
56.k	odd	6	1		1344.4.a.a		1
56.p	even	6	1		1344.4.a.o		1
84.n	even	6	1		1008.4.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.a.a	✓	1	7.c	even	3	1
126.4.a.a		1	21.h	odd	6	1
294.4.a.i		1	7.d	odd	6	1
294.4.e.b		2	7.b	odd	2	1
294.4.e.b		2	7.d	odd	6	1
294.4.e.c		2	1.a	even	1	1	trivial
294.4.e.c		2	7.c	even	3	1	inner
336.4.a.l		1	28.g	odd	6	1
882.4.a.g		1	21.g	even	6	1
882.4.g.o		2	21.c	even	2	1
882.4.g.o		2	21.g	even	6	1
882.4.g.w		2	3.b	odd	2	1
882.4.g.w		2	21.h	odd	6	1
1008.4.a.b		1	84.n	even	6	1
1050.4.a.g		1	35.j	even	6	1
1050.4.g.a		2	35.l	odd	12	2
1344.4.a.a		1	56.k	odd	6	1
1344.4.a.o		1	56.p	even	6	1
2352.4.a.a		1	28.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(294, [\chi])$ :

T_{5}^{2} + 18T_{5} + 324

T5^2 + 18*T5 + 324

T_{11}^{2} - 72T_{11} + 5184

T11^2 - 72*T11 + 5184

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$3$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$5$	$T^{2} + 18T + 324$ T^2 + 18*T + 324
$7$	$T^{2}$ T^2
$11$	$T^{2} - 72T + 5184$ T^2 - 72*T + 5184
$13$	$(T + 34)^{2}$ (T + 34)^2
$17$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$19$	$T^{2} + 92T + 8464$ T^2 + 92*T + 8464
$23$	$T^{2} - 180T + 32400$ T^2 - 180*T + 32400
$29$	$(T + 114)^{2}$ (T + 114)^2
$31$	$T^{2} + 56T + 3136$ T^2 + 56*T + 3136
$37$	$T^{2} - 34T + 1156$ T^2 - 34*T + 1156
$41$	$(T - 6)^{2}$ (T - 6)^2
$43$	$(T - 164)^{2}$ (T - 164)^2
$47$	$T^{2} + 168T + 28224$ T^2 + 168*T + 28224
$53$	$T^{2} + 654T + 427716$ T^2 + 654*T + 427716
$59$	$T^{2} - 492T + 242064$ T^2 - 492*T + 242064
$61$	$T^{2} - 250T + 62500$ T^2 - 250*T + 62500
$67$	$T^{2} - 124T + 15376$ T^2 - 124*T + 15376
$71$	$(T - 36)^{2}$ (T - 36)^2
$73$	$T^{2} + 1010 T + 1020100$ T^2 + 1010*T + 1020100
$79$	$T^{2} + 56T + 3136$ T^2 + 56*T + 3136
$83$	$(T - 228)^{2}$ (T - 228)^2
$89$	$T^{2} + 390T + 152100$ T^2 + 390*T + 152100
$97$	$(T + 70)^{2}$ (T + 70)^2
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