LMFDB - Newform orbit 3024.2.q.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(2305,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3024.2305");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3024 = 2^{4} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3024.q (of order $3$ , degree $2$ , not minimal)

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:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$3$
Twist minimal:	no (minimal twist has level 126)
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 basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{7} + (\beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{5} + 3 \beta_{4} + \cdots + 2 \beta_1) q^{13} + (2 \beta_{5} - 2 \beta_{4} + 2) q^{17} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{19}+ \cdots + ( - 2 \beta_{5} - 8 \beta_{4} + \cdots + 8) q^{97}+O(q^{100})$ q - b2 * q^5 + (b5 - b4 - b3) * q^7 + (b3 + b2) * q^11 + (-2b5 + 3b4 - b3 - b2 + 2b1) q^13 + (2b5 - 2b4 + 2) * q^17 + (-3b5 + 2b4 + 3b1) q^19 + (b5 + 2b4 - 2) q^23 + (b5 + b4 + 2b3 + 2b2 - b1) * q^25 + (-3b5 + 2b2) * q^29 + (-b3 - b1 + 6) * q^31 + (3b4 - b3 - 2b2 - b1 - 4) * q^35 + b4 * q^37 + (b5 - b4 - 2b3 - 2b2 - b1) * q^41 + (-3b5 - 2b4 - 3b2 + 2) q^43 + (3b3 - 3b1 + 3) * q^47 + (-3b5 - b3 + b2 + 1) q^49 + (b5 + 5b4 + b2 - 5) q^53 + (-2b3 + b1 + 4) q^55 + (2b3 - b1 + 5) q^59 + (-2b3 + b1 - 3) q^61 + (5b3 + b1 - 2) q^65 + (4b3 + b1 + 2) q^67 + (-2b3 + 7b1 + 4) * q^71 + (b5 - 8b4 - 4b2 + 8) * q^73 + (b5 - 4b4 + 2b3 + b2 + 1) * q^77 + (-b3 - 4b1) q^79 + (-2b5 - b4 - 3b2 + 1) * q^83 + (2b5 - 2b4 - 2b3 - 2b2 - 2b1) q^85 + (4b5 + 2b4 + b3 + b2 - 4b1) q^89 + (3b5 - b4 - 3b3 - 4b2 - b1 + 8) q^91 + (2b3 + 3b1 + 3) * q^95 + (-2b5 - 8b4 + 2b2 + 8) q^97
$\operatorname{Tr}(f)(q)$	$=$	$6 q - q^{5} - 2 q^{7} - q^{11} + 8 q^{13} + 4 q^{17} + 3 q^{19} - 7 q^{23} + 2 q^{25} + 5 q^{29} + 40 q^{31} - 13 q^{35} + 3 q^{37} + 6 q^{43} + 18 q^{47} + 12 q^{49} - 15 q^{53} + 26 q^{55} + 28 q^{59}+ \cdots + 28 q^{97}+O(q^{100})$ 6 * q - q^5 - 2 * q^7 - q^11 + 8 * q^13 + 4 * q^17 + 3 * q^19 - 7 * q^23 + 2 * q^25 + 5 * q^29 + 40 * q^31 - 13 * q^35 + 3 * q^37 + 6 * q^43 + 18 * q^47 + 12 * q^49 - 15 * q^53 + 26 * q^55 + 28 * q^59 - 16 * q^61 - 24 * q^65 + 2 * q^67 + 14 * q^71 + 19 * q^73 - 10 * q^77 + 10 * q^79 + 2 * q^83 - 2 * q^85 + 9 * q^89 + 46 * q^91 + 8 * q^95 + 28 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 :

$\beta_{1}$	$=$	$\nu^{2} - \nu + 2$ v^2 - v + 2
$\beta_{2}$	$=$	$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3$ (-v^5 + v^4 - 8v^3 + 5v^2 - 18*v + 6) / 3
$\beta_{3}$	$=$	$\nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3$ v^4 - 2v^3 + 6v^2 - 5*v + 3
$\beta_{4}$	$=$	$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3$ (-2v^5 + 5v^4 - 16v^3 + 19v^2 - 21*v + 9) / 3
$\beta_{5}$	$=$	$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3$ (2v^5 - 5v^4 + 19v^3 - 22v^2 + 30*v - 9) / 3

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$ (-2b5 - b4 - b3 - 2b2 + b1 + 2) / 3
$\nu^{2}$	$=$	$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3$ (-2b5 - b4 - b3 - 2b2 + 4*b1 - 4) / 3
$\nu^{3}$	$=$	$( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3$ (7b5 + 5b4 + 2b3 + 4b2 + b1 - 10) / 3
$\nu^{4}$	$=$	$( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3$ (16b5 + 11b4 + 8b3 + 10b2 - 17*b1 + 5) / 3
$\nu^{5}$	$=$	$( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3$ (-14b5 - 16b4 + 5b3 - 5b2 - 23*b1 + 47) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$ .

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-\beta_{4}$	$1$	$-\beta_{4}$

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}

2305.1

0.500000	−	1.41036i
0.500000	+	2.05195i
0.500000	+	0.224437i
0.500000	+	1.41036i
0.500000	−	2.05195i
0.500000	−	0.224437i

−1.59097

−

2.75564i

2.56238

0.658939i

2305.2

0.296790

0.514055i

−2.32383

−

1.26483i

2305.3

0.794182

1.37556i

−1.23855

2.33795i

2881.1

−1.59097

2.75564i

2.56238

−

0.658939i

2881.2

0.296790

−

0.514055i

−2.32383

1.26483i

2881.3

0.794182

−

1.37556i

−1.23855

−

2.33795i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3024.2.q.g		6
3.b	odd	2	1		1008.2.q.g		6
4.b	odd	2	1		378.2.e.d		6
7.c	even	3	1		3024.2.t.h		6
9.c	even	3	1		3024.2.t.h		6
9.d	odd	6	1		1008.2.t.h		6
12.b	even	2	1		126.2.e.c	✓	6
21.h	odd	6	1		1008.2.t.h		6
28.d	even	2	1		2646.2.e.p		6
28.f	even	6	1		2646.2.f.m		6
28.f	even	6	1		2646.2.h.o		6
28.g	odd	6	1		378.2.h.c		6
28.g	odd	6	1		2646.2.f.l		6
36.f	odd	6	1		378.2.h.c		6
36.f	odd	6	1		1134.2.g.l		6
36.h	even	6	1		126.2.h.d	yes	6
36.h	even	6	1		1134.2.g.m		6
63.h	even	3	1	inner	3024.2.q.g		6
63.j	odd	6	1		1008.2.q.g		6
84.h	odd	2	1		882.2.e.o		6
84.j	odd	6	1		882.2.f.o		6
84.j	odd	6	1		882.2.h.p		6
84.n	even	6	1		126.2.h.d	yes	6
84.n	even	6	1		882.2.f.n		6
252.n	even	6	1		2646.2.f.m		6
252.o	even	6	1		882.2.f.n		6
252.o	even	6	1		1134.2.g.m		6
252.r	odd	6	1		882.2.e.o		6
252.r	odd	6	1		7938.2.a.bw		3
252.s	odd	6	1		882.2.h.p		6
252.u	odd	6	1		378.2.e.d		6
252.u	odd	6	1		7938.2.a.ca		3
252.bb	even	6	1		126.2.e.c	✓	6
252.bb	even	6	1		7938.2.a.bv		3
252.bi	even	6	1		2646.2.h.o		6
252.bj	even	6	1		2646.2.e.p		6
252.bj	even	6	1		7938.2.a.bz		3
252.bl	odd	6	1		1134.2.g.l		6
252.bl	odd	6	1		2646.2.f.l		6
252.bn	odd	6	1		882.2.f.o		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.2.e.c	✓	6	12.b	even	2	1
126.2.e.c	✓	6	252.bb	even	6	1
126.2.h.d	yes	6	36.h	even	6	1
126.2.h.d	yes	6	84.n	even	6	1
378.2.e.d		6	4.b	odd	2	1
378.2.e.d		6	252.u	odd	6	1
378.2.h.c		6	28.g	odd	6	1
378.2.h.c		6	36.f	odd	6	1
882.2.e.o		6	84.h	odd	2	1
882.2.e.o		6	252.r	odd	6	1
882.2.f.n		6	84.n	even	6	1
882.2.f.n		6	252.o	even	6	1
882.2.f.o		6	84.j	odd	6	1
882.2.f.o		6	252.bn	odd	6	1
882.2.h.p		6	84.j	odd	6	1
882.2.h.p		6	252.s	odd	6	1
1008.2.q.g		6	3.b	odd	2	1
1008.2.q.g		6	63.j	odd	6	1
1008.2.t.h		6	9.d	odd	6	1
1008.2.t.h		6	21.h	odd	6	1
1134.2.g.l		6	36.f	odd	6	1
1134.2.g.l		6	252.bl	odd	6	1
1134.2.g.m		6	36.h	even	6	1
1134.2.g.m		6	252.o	even	6	1
2646.2.e.p		6	28.d	even	2	1
2646.2.e.p		6	252.bj	even	6	1
2646.2.f.l		6	28.g	odd	6	1
2646.2.f.l		6	252.bl	odd	6	1
2646.2.f.m		6	28.f	even	6	1
2646.2.f.m		6	252.n	even	6	1
2646.2.h.o		6	28.f	even	6	1
2646.2.h.o		6	252.bi	even	6	1
3024.2.q.g		6	1.a	even	1	1	trivial
3024.2.q.g		6	63.h	even	3	1	inner
3024.2.t.h		6	7.c	even	3	1
3024.2.t.h		6	9.c	even	3	1
7938.2.a.bv		3	252.bb	even	6	1
7938.2.a.bw		3	252.r	odd	6	1
7938.2.a.bz		3	252.bj	even	6	1
7938.2.a.ca		3	252.u	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3024, [\chi])$ :

T_{5}^{6} + T_{5}^{5} + 7T_{5}^{4} - 12T_{5}^{3} + 33T_{5}^{2} - 18T_{5} + 9

T_{11}^{6} + T_{11}^{5} + 7T_{11}^{4} - 12T_{11}^{3} + 33T_{11}^{2} - 18T_{11} + 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6}$ T^6
$5$	$T^{6} + T^{5} + 7 T^{4} + \cdots + 9$ T^6 + T^5 + 7T^4 - 12T^3 + 33T^2 - 18T + 9
$7$	$T^{6} + 2 T^{5} + \cdots + 343$ T^6 + 2T^5 - 4T^4 - 31T^3 - 28T^2 + 98*T + 343
$11$	$T^{6} + T^{5} + 7 T^{4} + \cdots + 9$ T^6 + T^5 + 7T^4 - 12T^3 + 33T^2 - 18T + 9
$13$	$T^{6} - 8 T^{5} + \cdots + 4761$ T^6 - 8T^5 + 63T^4 - 146T^3 + 553T^2 + 69*T + 4761
$17$	$T^{6} - 4 T^{5} + \cdots + 576$ T^6 - 4T^5 + 28T^4 + 240T^2 - 288T + 576
$19$	$T^{6} - 3 T^{5} + \cdots + 2401$ T^6 - 3T^5 + 45T^4 + 10T^3 + 1443T^2 - 1764*T + 2401
$23$	$T^{6} + 7 T^{5} + \cdots + 9$ T^6 + 7T^5 + 37T^4 + 78T^3 + 123T^2 + 36*T + 9
$29$	$T^{6} - 5 T^{5} + \cdots + 131769$ T^6 - 5T^5 + 91T^4 - 396T^3 + 6171T^2 - 23958*T + 131769
$31$	$(T^{3} - 20 T^{2} + \cdots - 201)^{2}$ (T^3 - 20T^2 + 121T - 201)^2
$37$	$(T^{2} - T + 1)^{3}$ (T^2 - T + 1)^3
$41$	$T^{6} + 33 T^{4} + \cdots + 81$ T^6 + 33T^4 - 18T^3 + 1089T^2 - 297T + 81
$43$	$T^{6} - 6 T^{5} + \cdots + 16129$ T^6 - 6T^5 + 105T^4 + 160T^3 + 5523T^2 - 8763*T + 16129
$47$	$(T^{3} - 9 T^{2} + \cdots + 189)^{2}$ (T^3 - 9T^2 - 54T + 189)^2
$53$	$T^{6} + 15 T^{5} + \cdots + 6561$ T^6 + 15T^5 + 159T^4 + 828T^3 + 3141T^2 + 5346*T + 6561
$59$	$(T^{3} - 14 T^{2} + \cdots + 63)^{2}$ (T^3 - 14T^2 + 39T + 63)^2
$61$	$(T^{3} + 8 T^{2} - 5 T - 93)^{2}$ (T^3 + 8T^2 - 5T - 93)^2
$67$	$(T^{3} - T^{2} - 112 T + 211)^{2}$ (T^3 - T^2 - 112*T + 211)^2
$71$	$(T^{3} - 7 T^{2} + \cdots + 1593)^{2}$ (T^3 - 7T^2 - 198T + 1593)^2
$73$	$T^{6} - 19 T^{5} + \cdots + 398161$ T^6 - 19T^5 + 353T^4 - 1414T^3 + 12053T^2 + 5048*T + 398161
$79$	$(T^{3} - 5 T^{2} + \cdots + 321)^{2}$ (T^3 - 5T^2 - 74T + 321)^2
$83$	$T^{6} - 2 T^{5} + \cdots + 21609$ T^6 - 2T^5 + 67T^4 - 168T^3 + 4263T^2 - 9261*T + 21609
$89$	$T^{6} - 9 T^{5} + \cdots + 81$ T^6 - 9T^5 + 123T^4 + 396T^3 + 1683T^2 + 378*T + 81
$97$	$T^{6} - 28 T^{5} + \cdots + 61504$ T^6 - 28T^5 + 572T^4 - 5440T^3 + 38000T^2 - 52576*T + 61504
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Elliptic curves over $\Q$
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$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2305.3
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