LMFDB - Newform orbit 714.2.w.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [714,2,Mod(83,714)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(714, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 4, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("714.83"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 714 = 2 \cdot 3 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	714.w (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,8,0,-16,0,8,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$5.70131870432$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-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 $\zeta_{16}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \zeta_{16}^{2} q^{2} + ( - \zeta_{16}^{7} + \cdots + \zeta_{16}^{3}) q^{3} + \zeta_{16}^{4} q^{4} + (\zeta_{16}^{7} + \cdots + \zeta_{16}^{3}) q^{5} + (\zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}) q^{6}+ \cdots + (4 \zeta_{16}^{6} - 5 \zeta_{16}^{2} - 7) q^{99}+O(q^{100}) $ q + z^2 * q^2 + (-z^7 + z^5 + z^3) * q^3 + z^4 * q^4 + (z^7 - 2z^5 + z^3) q^5 + (z^7 + z^5 + z) * q^6 + (z^7 + z^6 + z^5 - z^4 - z^2 - z + 1) * q^7 + z^6 * q^8 + (2z^4 + z^2 - 2) q^9 + (-2z^7 + z^5 - z) q^10 + (z^6 + 2z^4 + 2z^2 + 1) * q^11 + (z^7 + z^3 - z) * q^12 + (z^7 - z^5 + z^3 - z) * q^13 + (z^7 - z^6 - z^4 - z^3 + z^2 - z - 1) * q^14 + (2z^6 - 3z^4 + 2z^2 + 1) q^15 - q^16 + (2z^7 + 3z^5 - 2z^3) q^17 + (2z^6 + z^4 - 2z^2) * q^18 + (3z^7 - 3z^5 + 2z^3 + 2z) * q^19 + (z^7 - z^3 + 2z) q^20 + (-3z^7 + z^5 - z^4 - z^3 - 2z^2 - z - 2) * q^21 + (2z^6 + 2z^4 + z^2 - 1) * q^22 + (z^6 - 4z^4 - 4z^2 + 1) * q^23 + (z^5 - z^3 - z) * q^24 + (4z^4 - z^2 + 4) q^25 + (-z^7 + z^5 - z^3 - z) * q^26 + (5z^7 - z^5 - z) q^27 + (-z^6 - z^5 + z^4 - z^3 - z^2 - z + 1) * q^28 + (-z^6 + z^4 + 5z^2 + 5) q^29 + (-3z^6 + 2z^4 + z^2 - 2) * q^30 + (3z^7 + 2z^5 + 3z^3) q^31 - z^2 * q^32 + (3z^7 + 4z^5 + 2z^3 - z) q^33 + (3z^7 - 2z^5 - 2z) q^34 + (2z^7 + z^6 - 4z^5 + 4z^3 + z^2 - 2z) * q^35 + (z^6 - 2z^4 - 2) q^36 + (z^6 - z^4 + 4z^2 + 4) q^37 + (-3z^7 + 2z^5 + 2z^3 - 3z) * q^38 + (z^6 - 3z^4 + z^2 - 1) q^39 + (-z^5 + 2z^3 - z) q^40 + (2z^5 - 6z^3 + 2z) q^41 + (z^7 - z^6 - z^5 - 2z^4 - z^3 - 2z^2 + 3z) q^42 + 2z^2 q^43 + (2z^6 + z^4 - z^2 - 2) q^44 + (-2z^7 + 5z^5 - 4z^3 + 3z) * q^45 + (-4z^6 - 4z^4 + z^2 - 1) * q^46 + (2z^7 - 6z^5 - 6z^3 + 2z) * q^47 + (z^7 - z^5 - z^3) * q^48 + (-2z^7 + z^6 + 2z^5 - 4z^4 + 2z^3 + 2z + 4) q^49 + (4z^6 - z^4 + 4z^2) * q^50 + (z^4 - 7z^2 - 1) q^51 + (z^7 - z^5 - z^3 + z) * q^52 + (-8z^4 - 8) q^53 + (-z^7 - z^3 - 5z) q^54 + (-z^7 - z^5 + z^3 + z) * q^55 + (-z^7 + z^6 - z^5 - z^4 - z^3 + z^2 + 1) * q^56 + (7z^6 - 4z^4 + 2z^2 + 3) q^57 + (z^6 + 5z^4 + 5z^2 + 1) * q^58 + (-5z^7 + 5z^5 + z^3 + z) * q^59 + (2z^6 + z^4 - 2z^2 + 3) * q^60 + (-12z^7 - 2z^5 + 2z) q^61 + (2z^7 + 3z^5 - 3z) q^62 + (-z^7 - 5z^6 - 4z^5 + 3z^4 - 3z^3 + z^2 - z - 1) * q^63 - z^4 * q^64 + (2z^6 + 2z^4 - 4z^2 + 4) q^65 + (4z^7 + 2z^5 - z^3 - 3z) q^66 + (2z^6 - 2z^2 + 12) * q^67 + (-2z^7 - 2z^3 - 3z) q^68 + (-9z^7 - 2z^5 - 4z^3 - z) q^69 + (-4z^7 + 4z^5 + z^4 - 2z^3 - 2z - 1) * q^70 + (-4z^6 - 4z^4 + 7z^2 - 7) q^71 + (-2z^6 - 2z^2 - 1) * q^72 + (6z^7 - 6z^3) * q^73 + (-z^6 + 4z^4 + 4z^2 - 1) * q^74 + (-z^7 + 3z^5 + 8z^3 - 5z) q^75 + (2z^7 + 2z^5 - 3z^3 + 3z) * q^76 + (2z^7 - 2z^6 - 2z^5 - 2z^4 - 5z^3 - 5z + 2) * q^77 + (-3z^6 + z^4 - z^2 - 1) q^78 + (6z^6 - 2z^4 + 2z^2 - 6) q^79 + (-z^7 + 2z^5 - z^3) q^80 + (4z^6 - 7z^4 - 4z^2) q^81 + (2z^7 - 6z^5 + 2z^3) q^82 + (3z^7 + 3z^5 + 3z^3 - 3z) * q^83 + (-z^7 - 2z^6 - z^5 - 2z^4 + 3z^3 - z + 1) q^84 + (-4z^6 + z^4 + 6z^2 - 7) * q^85 + 2z^4 q^86 + (z^7 + 9z^5 + 7z^3 + 5z) q^87 + (z^6 - z^4 - 2z^2 - 2) q^88 + (z^7 - 4z^5 - 4z^3 + z) * q^89 + (5z^7 - 4z^5 + 3z^3 + 2z) * q^90 + (-z^6 - 2z^5 - z^4 + 4z^3 + z^2 - 2z + 1) q^91 + (-4z^6 + z^4 - z^2 + 4) q^92 + (6z^6 - z^4 - 2z^2 - 5) * q^93 + (-6z^7 - 6z^5 + 2z^3 - 2z) * q^94 + (-5z^6 + 11z^4 - 11z^2 + 5) q^95 + (-z^7 - z^5 - z) * q^96 + (-8z^7 + 8z^3 + 2z) q^97 + (2z^7 - 4z^6 + 2z^5 + 2z^3 + 4z^2 + 2z - 1) * q^98 + (4z^6 - 5z^2 - 7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7} - 16 q^{9} + 8 q^{11} - 8 q^{14} + 8 q^{15} - 8 q^{16} - 16 q^{21} - 8 q^{22} + 8 q^{23} + 32 q^{25} + 8 q^{28} + 40 q^{29} - 16 q^{30} - 16 q^{36} + 32 q^{37} - 8 q^{39} - 16 q^{44} - 8 q^{46}+ \cdots - 56 q^{99}+O(q^{100}) $ 8 * q + 8 * q^7 - 16 * q^9 + 8 * q^11 - 8 * q^14 + 8 * q^15 - 8 * q^16 - 16 * q^21 - 8 * q^22 + 8 * q^23 + 32 * q^25 + 8 * q^28 + 40 * q^29 - 16 * q^30 - 16 * q^36 + 32 * q^37 - 8 * q^39 - 16 * q^44 - 8 * q^46 + 32 * q^49 - 8 * q^51 - 64 * q^53 + 8 * q^56 + 24 * q^57 + 8 * q^58 + 24 * q^60 - 8 * q^63 + 32 * q^65 + 96 * q^67 - 8 * q^70 - 56 * q^71 - 8 * q^72 - 8 * q^74 + 16 * q^77 - 8 * q^78 - 48 * q^79 + 8 * q^84 - 56 * q^85 - 16 * q^88 + 8 * q^91 + 32 * q^92 - 40 * q^93 + 40 * q^95 - 8 * q^98 - 56 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/714\mathbb{Z}\right)^\times$.

$n$	$239$	$409$	$547$
$\chi(n)$	$-1$	$-1$	$\zeta_{16}^{2}$

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} $

83.1

−0.382683	−	0.923880i
0.382683	+	0.923880i
−0.923880	+	0.382683i
0.923880	−	0.382683i
−0.923880	−	0.382683i
0.923880	+	0.382683i
−0.382683	+	0.923880i
0.382683	−	0.923880i

−0.707107

0.707107i

−0.382683

1.68925i

−

1.00000i

3.15432

−

1.30656i

−0.923880

−

1.46508i

2.25570

1.38268i

0.707107

0.707107i

−2.70711

−

1.29289i

−1.30656

3.15432i

83.2

−0.707107

0.707107i

0.382683

−

1.68925i

−

1.00000i

−3.15432

1.30656i

0.923880

1.46508i

2.57273

0.617317i

0.707107

0.707107i

−2.70711

−

1.29289i

1.30656

−

3.15432i

461.1

0.707107

−

0.707107i

−0.923880

1.46508i

−

1.00000i

−0.224171

−

0.541196i

0.382683

1.68925i

1.81623

1.92388i

−0.707107

−

0.707107i

−1.29289

−

2.70711i

−0.541196

−

0.224171i

461.2

0.707107

−

0.707107i

0.923880

−

1.46508i

−

1.00000i

0.224171

0.541196i

−0.382683

−

1.68925i

−2.64466

0.0761205i

−0.707107

−

0.707107i

−1.29289

−

2.70711i

0.541196

0.224171i

587.1

0.707107

0.707107i

−0.923880

−

1.46508i

1.00000i

−0.224171

0.541196i

0.382683

−

1.68925i

1.81623

−

1.92388i

−0.707107

0.707107i

−1.29289

2.70711i

−0.541196

0.224171i

587.2

0.707107

0.707107i

0.923880

1.46508i

1.00000i

0.224171

−

0.541196i

−0.382683

1.68925i

−2.64466

−

0.0761205i

−0.707107

0.707107i

−1.29289

2.70711i

0.541196

−

0.224171i

671.1

−0.707107

−

0.707107i

−0.382683

−

1.68925i

1.00000i

3.15432

1.30656i

−0.923880

1.46508i

2.25570

−

1.38268i

0.707107

−

0.707107i

−2.70711

1.29289i

−1.30656

−

3.15432i

671.2

−0.707107

−

0.707107i

0.382683

1.68925i

1.00000i

−3.15432

−

1.30656i

0.923880

−

1.46508i

2.57273

−

0.617317i

0.707107

−

0.707107i

−2.70711

1.29289i

1.30656

3.15432i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
51.g	odd	8	1	inner
357.w	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	714.2.w.a	✓	8
3.b	odd	2	1		714.2.w.d	yes	8
7.b	odd	2	1	inner	714.2.w.a	✓	8
17.d	even	8	1		714.2.w.d	yes	8
21.c	even	2	1		714.2.w.d	yes	8
51.g	odd	8	1	inner	714.2.w.a	✓	8
119.l	odd	8	1		714.2.w.d	yes	8
357.w	even	8	1	inner	714.2.w.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
714.2.w.a	✓	8	1.a	even	1	1	trivial
714.2.w.a	✓	8	7.b	odd	2	1	inner
714.2.w.a	✓	8	51.g	odd	8	1	inner
714.2.w.a	✓	8	357.w	even	8	1	inner
714.2.w.d	yes	8	3.b	odd	2	1
714.2.w.d	yes	8	17.d	even	8	1
714.2.w.d	yes	8	21.c	even	2	1
714.2.w.d	yes	8	119.l	odd	8	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(714, [\chi])$:

$ T_{5}^{8} - 16T_{5}^{6} + 128T_{5}^{4} + 64T_{5}^{2} + 16 $

T5^8 - 16*T5^6 + 128*T5^4 + 64*T5^2 + 16

$ T_{11}^{4} - 4T_{11}^{3} + 22T_{11}^{2} - 12T_{11} + 2 $

T11^4 - 4*T11^3 + 22*T11^2 - 12*T11 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$3$	$ T^{8} + 8 T^{6} + \cdots + 81 $ T^8 + 8T^6 + 32T^4 + 72*T^2 + 81
$5$	$ T^{8} - 16 T^{6} + \cdots + 16 $ T^8 - 16T^6 + 128T^4 + 64*T^2 + 16
$7$	$ T^{8} - 8 T^{7} + \cdots + 2401 $ T^8 - 8T^7 + 16T^6 + 56T^5 - 320T^4 + 392T^3 + 784T^2 - 2744*T + 2401
$11$	$ (T^{4} - 4 T^{3} + 22 T^{2} + \cdots + 2)^{2} $ (T^4 - 4T^3 + 22T^2 - 12*T + 2)^2
$13$	$ (T^{4} - 8 T^{2} + 8)^{2} $ (T^4 - 8*T^2 + 8)^2
$17$	$ T^{8} - 48 T^{6} + \cdots + 83521 $ T^8 - 48T^6 + 1152T^4 - 13872*T^2 + 83521
$19$	$ T^{8} + 2508 T^{4} + 9604 $ T^8 + 2508*T^4 + 9604
$23$	$ (T^{4} - 4 T^{3} + \cdots + 1058)^{2} $ (T^4 - 4T^3 + 22T^2 - 276*T + 1058)^2
$29$	$ (T^{4} - 20 T^{3} + \cdots + 392)^{2} $ (T^4 - 20T^3 + 132T^2 - 224*T + 392)^2
$31$	$ T^{8} + 48 T^{6} + \cdots + 38416 $ T^8 + 48T^6 + 1152T^4 - 9408*T^2 + 38416
$37$	$ (T^{4} - 16 T^{3} + \cdots + 1058)^{2} $ (T^4 - 16T^3 + 114T^2 - 460*T + 1058)^2
$41$	$ T^{8} - 96 T^{6} + \cdots + 614656 $ T^8 - 96T^6 + 4608T^4 + 75264*T^2 + 614656
$43$	$ (T^{4} + 16)^{2} $ (T^4 + 16)^2
$47$	$ (T^{4} + 160 T^{2} + 6272)^{2} $ (T^4 + 160*T^2 + 6272)^2
$53$	$ (T^{2} + 16 T + 128)^{4} $ (T^2 + 16*T + 128)^4
$59$	$ T^{8} + 6192 T^{4} + 5345344 $ T^8 + 6192*T^4 + 5345344
$61$	$ T^{8} - 192 T^{6} + \cdots + 342102016 $ T^8 - 192T^6 + 18432T^4 + 3551232*T^2 + 342102016
$67$	$ (T^{2} - 24 T + 136)^{4} $ (T^2 - 24*T + 136)^4
$71$	$ (T^{4} + 28 T^{3} + \cdots + 1058)^{2} $ (T^4 + 28T^3 + 214T^2 - 276*T + 1058)^2
$73$	$ T^{8} + 26873856 $ T^8 + 26873856
$79$	$ (T^{4} + 24 T^{3} + \cdots + 6272)^{2} $ (T^4 + 24T^3 + 272T^2 + 1792*T + 6272)^2
$83$	$ T^{8} + 3888 T^{4} + 419904 $ T^8 + 3888*T^4 + 419904
$89$	$ (T^{4} + 68 T^{2} + 1058)^{2} $ (T^4 + 68*T^2 + 1058)^2
$97$	$ T^{8} - 128 T^{6} + \cdots + 236421376 $ T^8 - 128T^6 + 8192T^4 + 1968128*T^2 + 236421376
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Level:	\( N \)	\(=\)	\( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	714.w (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{16}^{2} q^{2} + ( - \zeta_{16}^{7} + \cdots + \zeta_{16}^{3}) q^{3} + \zeta_{16}^{4} q^{4} + (\zeta_{16}^{7} + \cdots + \zeta_{16}^{3}) q^{5} + (\zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}) q^{6}+ \cdots + (4 \zeta_{16}^{6} - 5 \zeta_{16}^{2} - 7) q^{99}+O(q^{100}) \) q + z^2 * q^2 + (-z^7 + z^5 + z^3) * q^3 + z^4 * q^4 + (z^7 - 2z^5 + z^3) q^5 + (z^7 + z^5 + z) * q^6 + (z^7 + z^6 + z^5 - z^4 - z^2 - z + 1) * q^7 + z^6 * q^8 + (2z^4 + z^2 - 2) q^9 + (-2z^7 + z^5 - z) q^10 + (z^6 + 2z^4 + 2z^2 + 1) * q^11 + (z^7 + z^3 - z) * q^12 + (z^7 - z^5 + z^3 - z) * q^13 + (z^7 - z^6 - z^4 - z^3 + z^2 - z - 1) * q^14 + (2z^6 - 3z^4 + 2z^2 + 1) q^15 - q^16 + (2z^7 + 3z^5 - 2z^3) q^17 + (2z^6 + z^4 - 2z^2) * q^18 + (3z^7 - 3z^5 + 2z^3 + 2z) * q^19 + (z^7 - z^3 + 2z) q^20 + (-3z^7 + z^5 - z^4 - z^3 - 2z^2 - z - 2) * q^21 + (2z^6 + 2z^4 + z^2 - 1) * q^22 + (z^6 - 4z^4 - 4z^2 + 1) * q^23 + (z^5 - z^3 - z) * q^24 + (4z^4 - z^2 + 4) q^25 + (-z^7 + z^5 - z^3 - z) * q^26 + (5z^7 - z^5 - z) q^27 + (-z^6 - z^5 + z^4 - z^3 - z^2 - z + 1) * q^28 + (-z^6 + z^4 + 5z^2 + 5) q^29 + (-3z^6 + 2z^4 + z^2 - 2) * q^30 + (3z^7 + 2z^5 + 3z^3) q^31 - z^2 * q^32 + (3z^7 + 4z^5 + 2z^3 - z) q^33 + (3z^7 - 2z^5 - 2z) q^34 + (2z^7 + z^6 - 4z^5 + 4z^3 + z^2 - 2z) * q^35 + (z^6 - 2z^4 - 2) q^36 + (z^6 - z^4 + 4z^2 + 4) q^37 + (-3z^7 + 2z^5 + 2z^3 - 3z) * q^38 + (z^6 - 3z^4 + z^2 - 1) q^39 + (-z^5 + 2z^3 - z) q^40 + (2z^5 - 6z^3 + 2z) q^41 + (z^7 - z^6 - z^5 - 2z^4 - z^3 - 2z^2 + 3z) q^42 + 2z^2 q^43 + (2z^6 + z^4 - z^2 - 2) q^44 + (-2z^7 + 5z^5 - 4z^3 + 3z) * q^45 + (-4z^6 - 4z^4 + z^2 - 1) * q^46 + (2z^7 - 6z^5 - 6z^3 + 2z) * q^47 + (z^7 - z^5 - z^3) * q^48 + (-2z^7 + z^6 + 2z^5 - 4z^4 + 2z^3 + 2z + 4) q^49 + (4z^6 - z^4 + 4z^2) * q^50 + (z^4 - 7z^2 - 1) q^51 + (z^7 - z^5 - z^3 + z) * q^52 + (-8z^4 - 8) q^53 + (-z^7 - z^3 - 5z) q^54 + (-z^7 - z^5 + z^3 + z) * q^55 + (-z^7 + z^6 - z^5 - z^4 - z^3 + z^2 + 1) * q^56 + (7z^6 - 4z^4 + 2z^2 + 3) q^57 + (z^6 + 5z^4 + 5z^2 + 1) * q^58 + (-5z^7 + 5z^5 + z^3 + z) * q^59 + (2z^6 + z^4 - 2z^2 + 3) * q^60 + (-12z^7 - 2z^5 + 2z) q^61 + (2z^7 + 3z^5 - 3z) q^62 + (-z^7 - 5z^6 - 4z^5 + 3z^4 - 3z^3 + z^2 - z - 1) * q^63 - z^4 * q^64 + (2z^6 + 2z^4 - 4z^2 + 4) q^65 + (4z^7 + 2z^5 - z^3 - 3z) q^66 + (2z^6 - 2z^2 + 12) * q^67 + (-2z^7 - 2z^3 - 3z) q^68 + (-9z^7 - 2z^5 - 4z^3 - z) q^69 + (-4z^7 + 4z^5 + z^4 - 2z^3 - 2z - 1) * q^70 + (-4z^6 - 4z^4 + 7z^2 - 7) q^71 + (-2z^6 - 2z^2 - 1) * q^72 + (6z^7 - 6z^3) * q^73 + (-z^6 + 4z^4 + 4z^2 - 1) * q^74 + (-z^7 + 3z^5 + 8z^3 - 5z) q^75 + (2z^7 + 2z^5 - 3z^3 + 3z) * q^76 + (2z^7 - 2z^6 - 2z^5 - 2z^4 - 5z^3 - 5z + 2) * q^77 + (-3z^6 + z^4 - z^2 - 1) q^78 + (6z^6 - 2z^4 + 2z^2 - 6) q^79 + (-z^7 + 2z^5 - z^3) q^80 + (4z^6 - 7z^4 - 4z^2) q^81 + (2z^7 - 6z^5 + 2z^3) q^82 + (3z^7 + 3z^5 + 3z^3 - 3z) * q^83 + (-z^7 - 2z^6 - z^5 - 2z^4 + 3z^3 - z + 1) q^84 + (-4z^6 + z^4 + 6z^2 - 7) * q^85 + 2z^4 q^86 + (z^7 + 9z^5 + 7z^3 + 5z) q^87 + (z^6 - z^4 - 2z^2 - 2) q^88 + (z^7 - 4z^5 - 4z^3 + z) * q^89 + (5z^7 - 4z^5 + 3z^3 + 2z) * q^90 + (-z^6 - 2z^5 - z^4 + 4z^3 + z^2 - 2z + 1) q^91 + (-4z^6 + z^4 - z^2 + 4) q^92 + (6z^6 - z^4 - 2z^2 - 5) * q^93 + (-6z^7 - 6z^5 + 2z^3 - 2z) * q^94 + (-5z^6 + 11z^4 - 11z^2 + 5) q^95 + (-z^7 - z^5 - z) * q^96 + (-8z^7 + 8z^3 + 2z) q^97 + (2z^7 - 4z^6 + 2z^5 + 2z^3 + 4z^2 + 2z - 1) * q^98 + (4z^6 - 5z^2 - 7) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7} - 16 q^{9} + 8 q^{11} - 8 q^{14} + 8 q^{15} - 8 q^{16} - 16 q^{21} - 8 q^{22} + 8 q^{23} + 32 q^{25} + 8 q^{28} + 40 q^{29} - 16 q^{30} - 16 q^{36} + 32 q^{37} - 8 q^{39} - 16 q^{44} - 8 q^{46}+ \cdots - 56 q^{99}+O(q^{100}) \) 8 * q + 8 * q^7 - 16 * q^9 + 8 * q^11 - 8 * q^14 + 8 * q^15 - 8 * q^16 - 16 * q^21 - 8 * q^22 + 8 * q^23 + 32 * q^25 + 8 * q^28 + 40 * q^29 - 16 * q^30 - 16 * q^36 + 32 * q^37 - 8 * q^39 - 16 * q^44 - 8 * q^46 + 32 * q^49 - 8 * q^51 - 64 * q^53 + 8 * q^56 + 24 * q^57 + 8 * q^58 + 24 * q^60 - 8 * q^63 + 32 * q^65 + 96 * q^67 - 8 * q^70 - 56 * q^71 - 8 * q^72 - 8 * q^74 + 16 * q^77 - 8 * q^78 - 48 * q^79 + 8 * q^84 - 56 * q^85 - 16 * q^88 + 8 * q^91 + 32 * q^92 - 40 * q^93 + 40 * q^95 - 8 * q^98 - 56 * q^99

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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	714.2.w.a

Level	$714$
Weight	$2$
Character orbit	714.w
Analytic conductor	$5.701$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(5.70131870432\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{8})\)
Coefficient field:	\(\Q(\zeta_{16})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

\(n\)	\(239\)	\(409\)	\(547\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{16}^{2}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 83.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{4} + 1)^{2} \) (T^4 + 1)^2
$3$	\( T^{8} + 8 T^{6} + \cdots + 81 \) T^8 + 8T^6 + 32T^4 + 72*T^2 + 81
$5$	\( T^{8} - 16 T^{6} + \cdots + 16 \) T^8 - 16T^6 + 128T^4 + 64*T^2 + 16
$7$	\( T^{8} - 8 T^{7} + \cdots + 2401 \) T^8 - 8T^7 + 16T^6 + 56T^5 - 320T^4 + 392T^3 + 784T^2 - 2744*T + 2401
$11$	\( (T^{4} - 4 T^{3} + 22 T^{2} + \cdots + 2)^{2} \) (T^4 - 4T^3 + 22T^2 - 12*T + 2)^2
$13$	\( (T^{4} - 8 T^{2} + 8)^{2} \) (T^4 - 8*T^2 + 8)^2
$17$	\( T^{8} - 48 T^{6} + \cdots + 83521 \) T^8 - 48T^6 + 1152T^4 - 13872*T^2 + 83521
$19$	\( T^{8} + 2508 T^{4} + 9604 \) T^8 + 2508*T^4 + 9604
$23$	\( (T^{4} - 4 T^{3} + \cdots + 1058)^{2} \) (T^4 - 4T^3 + 22T^2 - 276*T + 1058)^2
$29$	\( (T^{4} - 20 T^{3} + \cdots + 392)^{2} \) (T^4 - 20T^3 + 132T^2 - 224*T + 392)^2
$31$	\( T^{8} + 48 T^{6} + \cdots + 38416 \) T^8 + 48T^6 + 1152T^4 - 9408*T^2 + 38416
$37$	\( (T^{4} - 16 T^{3} + \cdots + 1058)^{2} \) (T^4 - 16T^3 + 114T^2 - 460*T + 1058)^2
$41$	\( T^{8} - 96 T^{6} + \cdots + 614656 \) T^8 - 96T^6 + 4608T^4 + 75264*T^2 + 614656
$43$	\( (T^{4} + 16)^{2} \) (T^4 + 16)^2
$47$	\( (T^{4} + 160 T^{2} + 6272)^{2} \) (T^4 + 160*T^2 + 6272)^2
$53$	\( (T^{2} + 16 T + 128)^{4} \) (T^2 + 16*T + 128)^4
$59$	\( T^{8} + 6192 T^{4} + 5345344 \) T^8 + 6192*T^4 + 5345344
$61$	\( T^{8} - 192 T^{6} + \cdots + 342102016 \) T^8 - 192T^6 + 18432T^4 + 3551232*T^2 + 342102016
$67$	\( (T^{2} - 24 T + 136)^{4} \) (T^2 - 24*T + 136)^4
$71$	\( (T^{4} + 28 T^{3} + \cdots + 1058)^{2} \) (T^4 + 28T^3 + 214T^2 - 276*T + 1058)^2
$73$	\( T^{8} + 26873856 \) T^8 + 26873856
$79$	\( (T^{4} + 24 T^{3} + \cdots + 6272)^{2} \) (T^4 + 24T^3 + 272T^2 + 1792*T + 6272)^2
$83$	\( T^{8} + 3888 T^{4} + 419904 \) T^8 + 3888*T^4 + 419904
$89$	\( (T^{4} + 68 T^{2} + 1058)^{2} \) (T^4 + 68*T^2 + 1058)^2
$97$	\( T^{8} - 128 T^{6} + \cdots + 236421376 \) T^8 - 128T^6 + 8192T^4 + 1968128*T^2 + 236421376
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