LMFDB - Newform orbit 735.2.y.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [735,2,Mod(128,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 4])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("735.128"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$735 = 3 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	735.y (of order $12$ , degree $4$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{4} + 1$ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$ -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 $\zeta_{24}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 2 \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \cdots + \zeta_{24}^{2}) q^{3} - 2 \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{6} - 2 \zeta_{24}^{5} + \cdots + \zeta_{24}) q^{5} + (2 \zeta_{24}^{7} + \cdots - 2 \zeta_{24}^{2}) q^{6} + \cdots + (4 \zeta_{24}^{7} + \cdots - 7 \zeta_{24}^{2}) q^{99} +O(q^{100})$ q + 2z^7 q^2 + (z^7 - z^6 + z^4 + z^2) * q^3 - 2z^2 q^4 + (-z^6 - 2z^5 + z^4 + z^2 + z) q^5 + (2z^7 + 2z^5 - 2z^3 - 2z^2) * q^6 + (2z^7 + 2z^5 - 2z^3 + z^2) q^9 + (2z^7 + 2z^5 + 2z^4 - 2z^3 + 2) * q^10 + (-2z^7 + 2z^5 + z^4 + 2z^3 + 1) q^11 + (-2z^6 - 2z^5 + 2z - 2) q^12 + (-z^6 + 2z^4 - z^3 + 2z^2 - 1) * q^13 + (z^4 - 2z^3 + 2z^2 + 2z + 1) q^15 - 4z^4 q^16 + (-3z^5 - z^4 + z^2 + 6z + 1) * q^17 + (-4z^6 + 2z^5 - 2z - 4) q^18 + (3z^7 - 2z^4 - 3z + 4) q^19 + (4z^7 - 2z^6 - 2z^3 - 2) q^20 + (4z^7 + 4z^6 - 2z^3 - 4) q^22 + (z^6 - z^4 + z^2 - 2z + 2) q^23 + (-2z^7 - 2z^5 - 2z^3 - z^2 + 4z) * q^25 + (2z^7 - 2z^6 + 4z^5 - 4z^3 + 2z^2 - 2z) * q^26 + (-z^6 + 5z^5 - 5z - 1) * q^27 + (5z^6 - 10z^2) * q^29 + (4z^7 - 4z^6 + 4z^5 + 4z^4 - 2z^3 + 4z^2 - 4z - 4) q^30 + (4z^6 - 2z^2) * q^31 + (-8z^7 + 8z^3) * q^32 + (2z^7 + z^6 + 2z^4 + 3z^3 + 2z^2 - 3) * q^33 + (2z^5 + 12z^4 + 2z^3 - 2z - 6) * q^34 + (-4z^7 - 2z^4 + 4z) q^36 + (-2z^7 + z^6 - z^4 + 4z^3 - z^2) * q^37 + (4z^7 - 6z^4 + 4z^3 - 6z^2 + 6) * q^38 + (z^6 + 2z^5 - 2z^3 + 3z^2 - 2z) * q^39 + (4z^7 - 2z^6 + 2z^5 - 2z^3 + 2z) q^41 + (-3z^6 - 4z^5 - 4z - 3) q^43 + (-4z^7 - 2z^6 - 2z^2 - 4z) * q^44 + (-2z^7 - z^6 + 4z^5 + 4z^4 + z^3 + 4z^2 - 4z - 1) q^45 + (2z^7 + 2z^5 - 4z^4 + 2z^3 - 4z + 4) q^46 + (z^6 + 6z^5 + z^4 - z^2 - 3z) * q^47 + (-4z^7 - 4z^4 + 4z^3 - 4z^2 + 4) * q^48 + (-4z^6 - 2z^5 + 8z^4 + 8z^2 + 2z - 4) q^50 + (-6z^7 + 4z^5 + 6z^4 + 4z^3 + 2z - 1) q^51 + (-4z^6 + 2z^5 - 2z^4 + 2z^2 - 2) * q^52 + (-4z^6 - 2z^5 - 2z^4 + 2z^2 - 2) * q^53 + (-2z^7 - 10z^4 + 2z) q^54 + (-3z^6 - 3z^5 - 2z^4 + 4z^3 + 6z^2 + 3z + 1) * q^55 + (8z^7 - 4z^6 - z^4 - 4z^3 - z^2 + 5) q^57 + (-20z^5 + 10z) * q^58 + (10z^4 - 10) q^59 + (-2z^6 + 4z^5 - 4z^4 - 4z^3 - 2z^2) q^60 + (-6z^7 - 6z) * q^61 + (-4z^5 - 4z) * q^62 + 8z^6 q^64 + (-4z^7 + 2z^6 + z^4 + 2z^2 + 2z - 2) * q^65 + (-2z^7 + 6z^6 + 4z^5 - 4z^3 - 10z^2 - 6z) * q^66 + (-2z^7 - 4z^4 - 2z^3 - 4z^2 + 4) * q^67 + (6z^7 + 2z^6 - 2z^4 - 12z^3 - 2z^2) q^68 + (3z^7 - z^5 - z^3 - 2z + 4) * q^69 + (-2z^5 + 8z^4 - 2z^3 + 2z - 4) * q^71 + (-10z^7 + 10z^3) * q^73 + (-2z^7 + 8z^6 - 2z^5 + 2z^3 - 4z^2) q^74 + (-8z^7 - 3z^6 - z^5 + 4z^4 + 4z^3 + 4z^2 + z - 3) q^75 + (4z^6 - 6z^5 + 6z^3 - 8z^2 + 6z) q^76 + (-4z^6 + 6z^5 - 4z^4 + 4z^2 - 8z) q^78 + (-2z^7 + 7z^6 - 4z^5 + 4z^3 - 7z^2 + 2z) * q^79 + (4z^5 - 4z^4 - 4z^2 - 8z + 4) * q^80 + (4z^7 - 7z^4 - 4z) q^81 + (-4z^6 + 4z^4 - 4z^2 + 4z - 8) * q^82 + (-12z^7 - 2z^6 + 6z^3 + 2) q^83 + (-8z^7 - 9z^6 + 2z^5 + 4z^3 + 2z + 2) q^85 + (-6z^7 - 8z^4 + 6z + 16) q^86 + (-5z^6 - 10z^5 + 5z^4 - 5z^2 + 5z - 10) q^87 + (2z^7 - 4z^5 + 12z^4 - 4z^3 + 2z) q^89 + (6z^7 + 2z^6 + 8z^5 - 8z^4 - 8z^3 + 2z^2 - 6z) q^90 + (2z^6 - 4z^4 + 4z^3 - 4z^2 + 2) * q^92 + (2z^6 - 2z^5 + 4z^4 - 4z^2 - 2z - 2) q^93 + (2z^7 - 2z^5 - 6z^4 - 2z^3 - 6) * q^94 + (6z^7 + 2z^6 - 6z^5 + 5z^4 - 6z^3 - z^2 + 5) q^95 + (8z^6 - 8z^5 + 8z^3 + 8z) * q^96 + (-z^6 + z^5 - 2z^4 + 2z^2 - z + 1) * q^97 + (4z^7 + 9z^6 + 4z^5 - 4z^3 - 7z^2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{3} + 4 q^{5} + 24 q^{10} + 12 q^{11} - 16 q^{12} + 12 q^{15} - 16 q^{16} + 4 q^{17} - 32 q^{18} + 24 q^{19} - 16 q^{20} - 32 q^{22} + 12 q^{23} - 8 q^{27} - 16 q^{30} - 16 q^{33} - 8 q^{36} - 4 q^{37}+ \cdots + 60 q^{95}+O(q^{100})$ 8 * q + 4 * q^3 + 4 * q^5 + 24 * q^10 + 12 * q^11 - 16 * q^12 + 12 * q^15 - 16 * q^16 + 4 * q^17 - 32 * q^18 + 24 * q^19 - 16 * q^20 - 32 * q^22 + 12 * q^23 - 8 * q^27 - 16 * q^30 - 16 * q^33 - 8 * q^36 - 4 * q^37 + 24 * q^38 - 24 * q^43 + 8 * q^45 + 16 * q^46 + 4 * q^47 + 16 * q^48 + 16 * q^51 - 24 * q^52 - 24 * q^53 - 40 * q^54 + 36 * q^57 - 40 * q^59 - 16 * q^60 - 12 * q^65 + 16 * q^67 - 8 * q^68 + 32 * q^69 - 8 * q^75 - 16 * q^78 + 16 * q^80 - 28 * q^81 - 48 * q^82 + 16 * q^83 + 16 * q^85 + 96 * q^86 - 60 * q^87 + 48 * q^89 - 32 * q^90 - 72 * q^94 + 60 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$ .

$n$	$346$	$442$	$491$
$\chi(n)$	$-\zeta_{24}^{4}$	$-\zeta_{24}^{6}$	$-1$

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}

128.1

0.258819	+	0.965926i
−0.258819	−	0.965926i
0.965926	+	0.258819i
−0.965926	−	0.258819i
0.965926	−	0.258819i
−0.965926	+	0.258819i
0.258819	−	0.965926i
−0.258819	+	0.965926i

−1.93185

0.517638i

−1.33195

−

1.10721i

1.73205

−

1.00000i

−2.03906

−

0.917738i

3.14626

1.44949i

0.548188

2.94949i

4.41421

0.717439i

128.2

1.93185

−

0.517638i

0.599900

−

1.62484i

1.73205

−

1.00000i

1.30701

−

1.81431i

0.317837

−

3.44949i

−2.28024

−

1.94949i

1.58579

−

4.18154i

263.1

−0.517638

1.93185i

1.10721

1.33195i

−1.73205

−

1.00000i

1.81431

−

1.30701i

−3.14626

1.44949i

−0.548188

2.94949i

1.58579

4.18154i

263.2

0.517638

−

1.93185i

1.62484

−

0.599900i

−1.73205

−

1.00000i

0.917738

2.03906i

−0.317837

−

3.44949i

2.28024

−

1.94949i

4.41421

−

0.717439i

422.1

−0.517638

−

1.93185i

1.10721

−

1.33195i

−1.73205

1.00000i

1.81431

1.30701i

−3.14626

−

1.44949i

−0.548188

−

2.94949i

1.58579

−

4.18154i

422.2

0.517638

1.93185i

1.62484

0.599900i

−1.73205

1.00000i

0.917738

−

2.03906i

−0.317837

3.44949i

2.28024

1.94949i

4.41421

0.717439i

557.1

−1.93185

−

0.517638i

−1.33195

1.10721i

1.73205

1.00000i

−2.03906

0.917738i

3.14626

−

1.44949i

0.548188

−

2.94949i

4.41421

−

0.717439i

557.2

1.93185

0.517638i

0.599900

1.62484i

1.73205

1.00000i

1.30701

1.81431i

0.317837

3.44949i

−2.28024

1.94949i

1.58579

4.18154i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
21.g	even	6	1	inner
35.f	even	4	1	inner
105.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.y.c		8
3.b	odd	2	1		735.2.y.d		8
5.c	odd	4	1		735.2.y.b		8
7.b	odd	2	1		735.2.y.b		8
7.c	even	3	1		735.2.j.b	yes	8
7.c	even	3	1		735.2.y.a		8
7.d	odd	6	1		735.2.j.a	✓	8
7.d	odd	6	1		735.2.y.d		8
15.e	even	4	1		735.2.y.a		8
21.c	even	2	1		735.2.y.a		8
21.g	even	6	1		735.2.j.b	yes	8
21.g	even	6	1	inner	735.2.y.c		8
21.h	odd	6	1		735.2.j.a	✓	8
21.h	odd	6	1		735.2.y.b		8
35.f	even	4	1	inner	735.2.y.c		8
35.k	even	12	1		735.2.j.b	yes	8
35.k	even	12	1		735.2.y.a		8
35.l	odd	12	1		735.2.j.a	✓	8
35.l	odd	12	1		735.2.y.d		8
105.k	odd	4	1		735.2.y.d		8
105.w	odd	12	1		735.2.j.a	✓	8
105.w	odd	12	1		735.2.y.b		8
105.x	even	12	1		735.2.j.b	yes	8
105.x	even	12	1	inner	735.2.y.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.j.a	✓	8	7.d	odd	6	1
735.2.j.a	✓	8	21.h	odd	6	1
735.2.j.a	✓	8	35.l	odd	12	1
735.2.j.a	✓	8	105.w	odd	12	1
735.2.j.b	yes	8	7.c	even	3	1
735.2.j.b	yes	8	21.g	even	6	1
735.2.j.b	yes	8	35.k	even	12	1
735.2.j.b	yes	8	105.x	even	12	1
735.2.y.a		8	7.c	even	3	1
735.2.y.a		8	15.e	even	4	1
735.2.y.a		8	21.c	even	2	1
735.2.y.a		8	35.k	even	12	1
735.2.y.b		8	5.c	odd	4	1
735.2.y.b		8	7.b	odd	2	1
735.2.y.b		8	21.h	odd	6	1
735.2.y.b		8	105.w	odd	12	1
735.2.y.c		8	1.a	even	1	1	trivial
735.2.y.c		8	21.g	even	6	1	inner
735.2.y.c		8	35.f	even	4	1	inner
735.2.y.c		8	105.x	even	12	1	inner
735.2.y.d		8	3.b	odd	2	1
735.2.y.d		8	7.d	odd	6	1
735.2.y.d		8	35.l	odd	12	1
735.2.y.d		8	105.k	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(735, [\chi])$ :

T_{2}^{8} - 16T_{2}^{4} + 256

T2^8 - 16*T2^4 + 256

T_{11}^{4} - 6T_{11}^{3} + 7T_{11}^{2} + 30T_{11} + 25

T11^4 - 6*T11^3 + 7*T11^2 + 30*T11 + 25

T_{13}^{8} + 146T_{13}^{4} + 625

T13^8 + 146*T13^4 + 625

T_{17}^{8} - 4T_{17}^{7} + 8T_{17}^{6} - 232T_{17}^{5} - 161T_{17}^{4} + 5800T_{17}^{3} + 5000T_{17}^{2} + 62500T_{17} + 390625

T17^8 - 4*T17^7 + 8*T17^6 - 232*T17^5 - 161*T17^4 + 5800*T17^3 + 5000*T17^2 + 62500*T17 + 390625

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8} - 16T^{4} + 256$ T^8 - 16*T^4 + 256
$3$	$T^{8} - 4 T^{7} + \cdots + 81$ T^8 - 4T^7 + 8T^6 - 8T^5 + 7T^4 - 24T^3 + 72T^2 - 108*T + 81
$5$	$T^{8} - 4 T^{7} + \cdots + 625$ T^8 - 4T^7 + 8T^6 + 8T^5 - 41T^4 + 40T^3 + 200T^2 - 500*T + 625
$7$	$T^{8}$ T^8
$11$	$(T^{4} - 6 T^{3} + 7 T^{2} + \cdots + 25)^{2}$ (T^4 - 6T^3 + 7T^2 + 30*T + 25)^2
$13$	$T^{8} + 146T^{4} + 625$ T^8 + 146*T^4 + 625
$17$	$T^{8} - 4 T^{7} + \cdots + 390625$ T^8 - 4T^7 + 8T^6 - 232T^5 - 161T^4 + 5800T^3 + 5000T^2 + 62500*T + 390625
$19$	$(T^{4} - 12 T^{3} + \cdots + 36)^{2}$ (T^4 - 12T^3 + 42T^2 + 72*T + 36)^2
$23$	$T^{8} - 12 T^{7} + \cdots + 16$ T^8 - 12T^7 + 72T^6 - 288T^5 + 668T^4 - 576T^3 + 288T^2 - 96*T + 16
$29$	$(T^{2} - 75)^{4}$ (T^2 - 75)^4
$31$	$(T^{4} + 12 T^{2} + 144)^{2}$ (T^4 + 12*T^2 + 144)^2
$37$	$T^{8} + 4 T^{7} + \cdots + 10000$ T^8 + 4T^7 + 8T^6 + 112T^5 + 124T^4 - 1120T^3 + 800T^2 - 4000*T + 10000
$41$	$(T^{4} + 56 T^{2} + 400)^{2}$ (T^4 + 56*T^2 + 400)^2
$43$	$(T^{4} + 12 T^{3} + \cdots + 900)^{2}$ (T^4 + 12T^3 + 72T^2 - 360*T + 900)^2
$47$	$T^{8} - 4 T^{7} + \cdots + 390625$ T^8 - 4T^7 + 8T^6 - 232T^5 - 161T^4 + 5800T^3 + 5000T^2 + 62500*T + 390625
$53$	$T^{8} + 24 T^{7} + \cdots + 160000$ T^8 + 24T^7 + 288T^6 + 2304T^5 + 12656T^4 + 46080T^3 + 115200T^2 + 192000*T + 160000
$59$	$(T^{2} + 10 T + 100)^{4}$ (T^2 + 10*T + 100)^4
$61$	$(T^{4} + 72 T^{2} + 5184)^{2}$ (T^4 + 72*T^2 + 5184)^2
$67$	$T^{8} - 16 T^{7} + \cdots + 160000$ T^8 - 16T^7 + 128T^6 - 1408T^5 + 10864T^4 - 28160T^3 + 51200T^2 - 128000*T + 160000
$71$	$(T^{4} + 112 T^{2} + 1600)^{2}$ (T^4 + 112*T^2 + 1600)^2
$73$	$T^{8} - 10000 T^{4} + 100000000$ T^8 - 10000*T^4 + 100000000
$79$	$T^{8} - 146 T^{6} + \cdots + 390625$ T^8 - 146T^6 + 20691T^4 - 91250*T^2 + 390625
$83$	$(T^{4} - 8 T^{3} + \cdots + 10000)^{2}$ (T^4 - 8T^3 + 32T^2 + 800*T + 10000)^2
$89$	$(T^{4} - 24 T^{3} + \cdots + 14400)^{2}$ (T^4 - 24T^3 + 456T^2 - 2880*T + 14400)^2
$97$	$T^{8} + 146T^{4} + 625$ T^8 + 146*T^4 + 625
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 128.2
Significant digits:
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Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	735.2.y.c

Level	$735$
Weight	$2$
Character orbit	735.y
Analytic conductor	$5.869$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$