LMFDB - Newform orbit 735.2.y.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(128,735)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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 $

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

$ T_{13}^{8} + 146T_{13}^{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 $

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
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\( q + 2 \zeta_{24}^{7} q^{2} + (\zeta_{24}^{7} + \cdots + \zeta_{24}^{2}) q^{3} - 2 \zeta_{24}^{2} q^{4} + ( - \zeta_{24}^{6} - 2 \zeta_{24}^{5} + \cdots + \zeta_{24}) q^{5} + (2 \zeta_{24}^{7} + \cdots - 2 \zeta_{24}^{2}) q^{6} + \cdots + (4 \zeta_{24}^{7} + \cdots - 7 \zeta_{24}^{2}) q^{99} +O(q^{100}) \) 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

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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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	$4$

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:	$\mathrm{SU}(2)[C_{12}]$

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 128.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - 16T^{4} + 256 \) 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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