LMFDB - Newform orbit 544.2.bx.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [544,2,Mod(31,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("544.31"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 0, 9])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 544 = 2^{5} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	544.bx (of order $16$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-8,0,8,0,-8,0,8,0,-8] 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:	$4.34386186996$
Analytic rank:	$0$
Dimension:	$8$
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:	$\mathrm{SU}(2)[C_{16}]$

$q$-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}^{7} - \zeta_{16}^{6} + \cdots - 1) q^{3} + (\zeta_{16}^{5} + 1) q^{5} + (\zeta_{16}^{6} - \zeta_{16}^{5} + \cdots - 1) q^{7} + (2 \zeta_{16}^{6} - 2 \zeta_{16}^{4} + \cdots + 1) q^{9}+ \cdots + ( - 3 \zeta_{16}^{7} - 2 \zeta_{16}^{6} + \cdots + 3) q^{99}+O(q^{100}) $ q + (-z^7 - z^6 - z^3 - z^2 + z - 1) * q^3 + (z^5 + 1) * q^5 + (z^6 - z^5 - z^3 - z^2 - z - 1) * q^7 + (2z^6 - 2z^4 + z^2 - 3z + 1) q^9 + (z^7 + z^6 + z^5 - z^2 - z - 1) * q^11 + (-2z^7 + 2z^5 - z^4 - 1) * q^13 + (-2z^7 - z^5 + z^4 - z^2 + z) q^15 + (2z^6 - 2z^5 - 2z^3 - 2z^2 + 1) * q^17 + (-2z^5 - z^4 - 2z^3 - z^2 - 2z) q^19 + (4z^7 + 4z^5 + z^4 + z^3 - z - 1) * q^21 + (z^7 - z^6 + z^5 + 2z^4 + 2z^3 + z^2 - z + 1) * q^23 + (-3z^5 - z^2 + 1) q^25 + (3z^7 - z^6 - z^5 + 3z^4 - 2z^3 - 4z^2 + 4z + 2) q^27 + (2z^7 + 2z^6 - z^5 + 2z^4 - 2z + 1) * q^29 + (-5z^7 - z^6 - 2z^5 + 2z^4 + z^3 + 5z^2 - z - 1) * q^31 + (z^7 + 2z^6 + 2z^5 + 4z^4 + 2z^3 + 2z^2 + z) q^33 + (-z^7 - 2z^5 - 2z^3 - z) * q^35 + (2z^7 + 2z^6 - 4z^5 + 4z^4 - 2z^3 - 2z^2 + z + 1) * q^37 + (2z^7 + 2z^6 - 5z^5 + 3z^4 + 2z^3 - 2z^2 - 3z + 5) q^39 + (-2z^6 - 2z^5 - z^3 + 2z^2 - 2z + 1) * q^41 + (4z^7 - z^6 + 2z^5 - z^4 + 4z^3 - 4z^2 + 4) * q^43 + (z^7 - z^6 + z^5 - 2z^4 - 2z^3 + z^2 - z + 1) * q^45 + (-4z^7 - 4z^5 + z^4 + 2z^3 - 2z - 1) * q^47 + (z^6 + 4z^5 + 2z^4 - z^3 + 2z^2 + 4z + 1) * q^49 + (3z^7 - 3z^6 + 8z^5 - 3z^3 - z^2 - z - 3) * q^51 + (-4z^7 + z^6 - 2z^4 + 2z^2 - 1) q^53 + (-z^4 - z^3 - 2z^2 - z - 1) q^55 + (5z^7 + z^6 + 4z^5 - 4z^2 - z - 5) q^57 + (5z^6 - 5z^4 - 2z) q^59 + (2z^7 + 4z^6 - 4z^5 - 2z^4 - z^3 - 4z^2 - 4z - 1) * q^61 + (-4z^7 + 4z^6 + 2z^4 + 3z^3 + 3z^2 + 2z) * q^63 + (-2z^7 + z^5 + z^4 - 2z^2 + z - 1) * q^65 + (-6z^7 - 2z^6 + 2z^5 - 2z^3 + 2z^2 + 6z) * q^67 + (-5z^7 - 2z^6 - 2z^5 + 2z^3 + 2z^2 + 5z - 2) * q^69 + (3z^7 + 3z^6 + 3z^3 + 3z^2 - 3z + 3) q^71 + (2z^7 - 2z^6 - z^5 + 2z^4 + 2z^3 + 2z^2 + 2z - 1) * q^73 + (2z^7 - 4z^6 + 4z^5 - 2z^4 - 5z^3 - 5) q^75 + (-3z^7 - 2z^6 + 2z^4 + 3z^3 + 5z^2 + 4z + 5) * q^77 + (3z^7 - z^6 - z^5 + 2z^4 - 2z^3 + z^2 + z - 3) q^79 + (-6z^7 + 6z^5 - 7z^4 - 3z^3 + z^2 - 3z - 7) q^81 + (2z^7 + 4z^6 - 4z^5 - z^4 + z^2 + 4z - 4) * q^83 + (-2z^7 + 2z^6 - z^5 - 4z^3 + 3) q^85 + (-4z^6 + 7z^5 + z^4 + 2z^3 + z^2 + 7z - 4) * q^87 + (-12z^6 - 3z^4 + 2z^3 - 2z + 3) * q^89 + (z^7 - 2z^6 + 2z^5 - z^4 - z^3 + 2z^2 - 2z + 1) * q^91 + (6z^7 - 8z^6 - 8z^5 - 8z^4 + 6z^3 - 7z^2 + 7) * q^93 + (-z^7 - 2z^6 - 2z^5 - z^4 - 2z^3 + z^2 - z + 2) q^95 + (-4z^7 - 4z^6 - z^5 - 2z^4 + 6z^3 - 6z^2 + 2z + 1) * q^97 + (-3z^7 - 2z^6 + z^5 - z^4 + 2z^3 + 3z^2 + 3z + 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 8 q^{5} - 8 q^{7} + 8 q^{9} - 8 q^{11} - 8 q^{13} + 8 q^{17} - 8 q^{21} + 8 q^{23} + 8 q^{25} + 16 q^{27} + 8 q^{29} - 8 q^{31} + 8 q^{37} + 40 q^{39} + 8 q^{41} + 32 q^{43} + 8 q^{45} - 8 q^{47}+ \cdots + 24 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 8 * q^5 - 8 * q^7 + 8 * q^9 - 8 * q^11 - 8 * q^13 + 8 * q^17 - 8 * q^21 + 8 * q^23 + 8 * q^25 + 16 * q^27 + 8 * q^29 - 8 * q^31 + 8 * q^37 + 40 * q^39 + 8 * q^41 + 32 * q^43 + 8 * q^45 - 8 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^53 - 8 * q^55 - 40 * q^57 - 8 * q^61 - 8 * q^65 - 16 * q^69 + 24 * q^71 - 8 * q^73 - 40 * q^75 + 40 * q^77 - 24 * q^79 - 56 * q^81 - 32 * q^83 + 24 * q^85 - 32 * q^87 + 24 * q^89 + 8 * q^91 + 56 * q^93 + 16 * q^95 + 8 * q^97 + 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/544\mathbb{Z}\right)^\times$.

$n$	$69$	$511$	$513$
$\chi(n)$	$1$	$-1$	$\zeta_{16}$

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} $

31.1

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

−2.46508

−

0.490334i

1.38268

−

0.923880i

−1.49033

2.23044i

3.06453

1.26937i

63.1

0.689246

1.03153i

1.92388

0.382683i

0.0315301

0.158513i

0.559056

−

1.34968i

95.1

0.689246

−

1.03153i

1.92388

−

0.382683i

0.0315301

−

0.158513i

0.559056

1.34968i

159.1

0.465076

2.33809i

0.617317

−

0.923880i

−3.33809

2.23044i

−2.47875

1.02673i

351.1

−2.46508

0.490334i

1.38268

0.923880i

−1.49033

−

2.23044i

3.06453

−

1.26937i

415.1

−2.68925

−

1.79690i

0.0761205

0.382683i

0.796897

0.158513i

2.85516

6.89296i

447.1

−2.68925

1.79690i

0.0761205

−

0.382683i

0.796897

−

0.158513i

2.85516

−

6.89296i

479.1

0.465076

−

2.33809i

0.617317

0.923880i

−3.33809

−

2.23044i

−2.47875

−

1.02673i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
68.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	544.2.bx.b	✓	8
4.b	odd	2	1		544.2.bx.h	yes	8
17.e	odd	16	1		544.2.bx.h	yes	8
68.i	even	16	1	inner	544.2.bx.b	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
544.2.bx.b	✓	8	1.a	even	1	1	trivial
544.2.bx.b	✓	8	68.i	even	16	1	inner
544.2.bx.h	yes	8	4.b	odd	2	1
544.2.bx.h	yes	8	17.e	odd	16	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(544, [\chi])$:

$ T_{3}^{8} + 8T_{3}^{7} + 28T_{3}^{6} + 64T_{3}^{5} + 150T_{3}^{4} + 264T_{3}^{3} + 148T_{3}^{2} + 136T_{3} + 578 $

T3^8 + 8*T3^7 + 28*T3^6 + 64*T3^5 + 150*T3^4 + 264*T3^3 + 148*T3^2 + 136*T3 + 578

$ T_{5}^{8} - 8T_{5}^{7} + 28T_{5}^{6} - 56T_{5}^{5} + 70T_{5}^{4} - 56T_{5}^{3} + 28T_{5}^{2} - 8T_{5} + 2 $

T5^8 - 8*T5^7 + 28*T5^6 - 56*T5^5 + 70*T5^4 - 56*T5^3 + 28*T5^2 - 8*T5 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 8 T^{7} + \cdots + 578 $ T^8 + 8T^7 + 28T^6 + 64T^5 + 150T^4 + 264T^3 + 148T^2 + 136*T + 578
$5$	$ T^{8} - 8 T^{7} + \cdots + 2 $ T^8 - 8T^7 + 28T^6 - 56T^5 + 70T^4 - 56T^3 + 28T^2 - 8*T + 2
$7$	$ T^{8} + 8 T^{7} + \cdots + 2 $ T^8 + 8T^7 + 28T^6 + 32T^5 - 10T^4 - 120T^3 + 84T^2 - 8*T + 2
$11$	$ T^{8} + 8 T^{7} + \cdots + 2 $ T^8 + 8T^7 + 12T^6 - 16T^5 + 22T^4 - 24T^3 + 20T^2 - 8*T + 2
$13$	$ T^{8} + 8 T^{7} + \cdots + 16 $ T^8 + 8T^7 + 32T^6 + 16T^5 + 8T^4 + 160T^3 + 1152T^2 - 192*T + 16
$17$	$ (T^{4} - 4 T^{3} + \cdots + 289)^{2} $ (T^4 - 4T^3 + 6T^2 - 68*T + 289)^2
$19$	$ T^{8} + 36 T^{6} + \cdots + 4 $ T^8 + 36T^6 - 168T^5 + 648T^4 - 1232T^3 + 856T^2 + 112T + 4
$23$	$ T^{8} - 8 T^{7} + \cdots + 12482 $ T^8 - 8T^7 + 44T^6 - 208T^5 + 630T^4 - 2424T^3 + 9716T^2 - 17064*T + 12482
$29$	$ T^{8} - 8 T^{7} + \cdots + 578 $ T^8 - 8T^7 + 12T^6 + 88T^5 - 186T^4 - 3208T^3 + 13212T^2 + 952*T + 578
$31$	$ T^{8} + 8 T^{7} + \cdots + 14439938 $ T^8 + 8T^7 + 28T^6 - 768T^5 - 7498T^4 - 36712T^3 + 244052T^2 + 2558024*T + 14439938
$37$	$ T^{8} - 8 T^{7} + \cdots + 132098 $ T^8 - 8T^7 + 140T^6 - 328T^5 + 7590T^4 + 9288T^3 - 41188T^2 + 6168*T + 132098
$41$	$ T^{8} - 8 T^{7} + \cdots + 4802 $ T^8 - 8T^7 + 12T^6 + 184T^5 - 506T^4 + 168T^3 + 9212T^2 + 2744*T + 4802
$43$	$ T^{8} - 32 T^{7} + \cdots + 1110916 $ T^8 - 32T^7 + 548T^6 - 5688T^5 + 33160T^4 - 87984T^3 + 202200T^2 - 649264*T + 1110916
$47$	$ T^{8} + 8 T^{7} + \cdots + 1993744 $ T^8 + 8T^7 + 32T^6 - 240T^5 + 1800T^4 + 9376T^3 + 46208T^2 - 429248*T + 1993744
$53$	$ T^{8} + 8 T^{7} + \cdots + 3844 $ T^8 + 8T^7 + 60T^6 - 56T^5 - 1464T^4 - 224T^3 + 17896T^2 + 6448*T + 3844
$59$	$ T^{8} + 100 T^{6} + \cdots + 749956 $ T^8 + 100T^6 + 1160T^5 + 5000T^4 + 41360T^3 + 339800T^2 + 866000T + 749956
$61$	$ T^{8} + 8 T^{7} + \cdots + 1276802 $ T^8 + 8T^7 - 116T^6 - 1992T^5 + 6950T^4 + 56584T^3 + 375516T^2 + 492184*T + 1276802
$67$	$ (T^{4} - 176 T^{2} + \cdots + 5056)^{2} $ (T^4 - 176T^2 - 128T + 5056)^2
$71$	$ T^{8} - 24 T^{7} + \cdots + 3792258 $ T^8 - 24T^7 + 252T^6 - 1728T^5 + 12150T^4 - 64152T^3 + 107892T^2 - 297432*T + 3792258
$73$	$ T^{8} + 8 T^{7} + \cdots + 25538 $ T^8 + 8T^7 + 28T^6 + 264T^5 + 998T^4 + 4264T^3 + 30972T^2 - 47912*T + 25538
$79$	$ T^{8} + 24 T^{7} + \cdots + 293378 $ T^8 + 24T^7 + 300T^6 + 2256T^5 + 11030T^4 + 38376T^3 + 109268T^2 + 248184*T + 293378
$83$	$ T^{8} + 32 T^{7} + \cdots + 1110916 $ T^8 + 32T^7 + 548T^6 + 5688T^5 + 33160T^4 + 87984T^3 + 202200T^2 + 649264*T + 1110916
$89$	$ T^{8} - 24 T^{7} + \cdots + 147962896 $ T^8 - 24T^7 + 288T^6 + 1104T^5 + 14088T^4 - 413664T^3 + 6480000T^2 + 43790400*T + 147962896
$97$	$ T^{8} - 8 T^{7} + \cdots + 77900162 $ T^8 - 8T^7 + 124T^6 - 3608T^5 + 49990T^4 - 434184T^3 + 3969276T^2 - 27610184*T + 77900162
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 544 = 2^{5} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	544.bx (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{16}^{7} - \zeta_{16}^{6} + \cdots - 1) q^{3} + (\zeta_{16}^{5} + 1) q^{5} + (\zeta_{16}^{6} - \zeta_{16}^{5} + \cdots - 1) q^{7} + (2 \zeta_{16}^{6} - 2 \zeta_{16}^{4} + \cdots + 1) q^{9}+ \cdots + ( - 3 \zeta_{16}^{7} - 2 \zeta_{16}^{6} + \cdots + 3) q^{99}+O(q^{100}) \) q + (-z^7 - z^6 - z^3 - z^2 + z - 1) * q^3 + (z^5 + 1) * q^5 + (z^6 - z^5 - z^3 - z^2 - z - 1) * q^7 + (2z^6 - 2z^4 + z^2 - 3z + 1) q^9 + (z^7 + z^6 + z^5 - z^2 - z - 1) * q^11 + (-2z^7 + 2z^5 - z^4 - 1) * q^13 + (-2z^7 - z^5 + z^4 - z^2 + z) q^15 + (2z^6 - 2z^5 - 2z^3 - 2z^2 + 1) * q^17 + (-2z^5 - z^4 - 2z^3 - z^2 - 2z) q^19 + (4z^7 + 4z^5 + z^4 + z^3 - z - 1) * q^21 + (z^7 - z^6 + z^5 + 2z^4 + 2z^3 + z^2 - z + 1) * q^23 + (-3z^5 - z^2 + 1) q^25 + (3z^7 - z^6 - z^5 + 3z^4 - 2z^3 - 4z^2 + 4z + 2) q^27 + (2z^7 + 2z^6 - z^5 + 2z^4 - 2z + 1) * q^29 + (-5z^7 - z^6 - 2z^5 + 2z^4 + z^3 + 5z^2 - z - 1) * q^31 + (z^7 + 2z^6 + 2z^5 + 4z^4 + 2z^3 + 2z^2 + z) q^33 + (-z^7 - 2z^5 - 2z^3 - z) * q^35 + (2z^7 + 2z^6 - 4z^5 + 4z^4 - 2z^3 - 2z^2 + z + 1) * q^37 + (2z^7 + 2z^6 - 5z^5 + 3z^4 + 2z^3 - 2z^2 - 3z + 5) q^39 + (-2z^6 - 2z^5 - z^3 + 2z^2 - 2z + 1) * q^41 + (4z^7 - z^6 + 2z^5 - z^4 + 4z^3 - 4z^2 + 4) * q^43 + (z^7 - z^6 + z^5 - 2z^4 - 2z^3 + z^2 - z + 1) * q^45 + (-4z^7 - 4z^5 + z^4 + 2z^3 - 2z - 1) * q^47 + (z^6 + 4z^5 + 2z^4 - z^3 + 2z^2 + 4z + 1) * q^49 + (3z^7 - 3z^6 + 8z^5 - 3z^3 - z^2 - z - 3) * q^51 + (-4z^7 + z^6 - 2z^4 + 2z^2 - 1) q^53 + (-z^4 - z^3 - 2z^2 - z - 1) q^55 + (5z^7 + z^6 + 4z^5 - 4z^2 - z - 5) q^57 + (5z^6 - 5z^4 - 2z) q^59 + (2z^7 + 4z^6 - 4z^5 - 2z^4 - z^3 - 4z^2 - 4z - 1) * q^61 + (-4z^7 + 4z^6 + 2z^4 + 3z^3 + 3z^2 + 2z) * q^63 + (-2z^7 + z^5 + z^4 - 2z^2 + z - 1) * q^65 + (-6z^7 - 2z^6 + 2z^5 - 2z^3 + 2z^2 + 6z) * q^67 + (-5z^7 - 2z^6 - 2z^5 + 2z^3 + 2z^2 + 5z - 2) * q^69 + (3z^7 + 3z^6 + 3z^3 + 3z^2 - 3z + 3) q^71 + (2z^7 - 2z^6 - z^5 + 2z^4 + 2z^3 + 2z^2 + 2z - 1) * q^73 + (2z^7 - 4z^6 + 4z^5 - 2z^4 - 5z^3 - 5) q^75 + (-3z^7 - 2z^6 + 2z^4 + 3z^3 + 5z^2 + 4z + 5) * q^77 + (3z^7 - z^6 - z^5 + 2z^4 - 2z^3 + z^2 + z - 3) q^79 + (-6z^7 + 6z^5 - 7z^4 - 3z^3 + z^2 - 3z - 7) q^81 + (2z^7 + 4z^6 - 4z^5 - z^4 + z^2 + 4z - 4) * q^83 + (-2z^7 + 2z^6 - z^5 - 4z^3 + 3) q^85 + (-4z^6 + 7z^5 + z^4 + 2z^3 + z^2 + 7z - 4) * q^87 + (-12z^6 - 3z^4 + 2z^3 - 2z + 3) * q^89 + (z^7 - 2z^6 + 2z^5 - z^4 - z^3 + 2z^2 - 2z + 1) * q^91 + (6z^7 - 8z^6 - 8z^5 - 8z^4 + 6z^3 - 7z^2 + 7) * q^93 + (-z^7 - 2z^6 - 2z^5 - z^4 - 2z^3 + z^2 - z + 2) q^95 + (-4z^7 - 4z^6 - z^5 - 2z^4 + 6z^3 - 6z^2 + 2z + 1) * q^97 + (-3z^7 - 2z^6 + z^5 - z^4 + 2z^3 + 3z^2 + 3z + 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 8 q^{5} - 8 q^{7} + 8 q^{9} - 8 q^{11} - 8 q^{13} + 8 q^{17} - 8 q^{21} + 8 q^{23} + 8 q^{25} + 16 q^{27} + 8 q^{29} - 8 q^{31} + 8 q^{37} + 40 q^{39} + 8 q^{41} + 32 q^{43} + 8 q^{45} - 8 q^{47}+ \cdots + 24 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 8 * q^5 - 8 * q^7 + 8 * q^9 - 8 * q^11 - 8 * q^13 + 8 * q^17 - 8 * q^21 + 8 * q^23 + 8 * q^25 + 16 * q^27 + 8 * q^29 - 8 * q^31 + 8 * q^37 + 40 * q^39 + 8 * q^41 + 32 * q^43 + 8 * q^45 - 8 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^53 - 8 * q^55 - 40 * q^57 - 8 * q^61 - 8 * q^65 - 16 * q^69 + 24 * q^71 - 8 * q^73 - 40 * q^75 + 40 * q^77 - 24 * q^79 - 56 * q^81 - 32 * q^83 + 24 * q^85 - 32 * q^87 + 24 * q^89 + 8 * q^91 + 56 * q^93 + 16 * q^95 + 8 * q^97 + 24 * q^99

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	544.2.bx.b

Level	$544$
Weight	$2$
Character orbit	544.bx
Analytic conductor	$4.344$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(4.34386186996\)
Analytic rank:	\(0\)
Dimension:	\(8\)
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:	$\mathrm{SU}(2)[C_{16}]$

\(n\)	\(69\)	\(511\)	\(513\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\zeta_{16}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + 8 T^{7} + \cdots + 578 \) T^8 + 8T^7 + 28T^6 + 64T^5 + 150T^4 + 264T^3 + 148T^2 + 136*T + 578
$5$	\( T^{8} - 8 T^{7} + \cdots + 2 \) T^8 - 8T^7 + 28T^6 - 56T^5 + 70T^4 - 56T^3 + 28T^2 - 8*T + 2
$7$	\( T^{8} + 8 T^{7} + \cdots + 2 \) T^8 + 8T^7 + 28T^6 + 32T^5 - 10T^4 - 120T^3 + 84T^2 - 8*T + 2
$11$	\( T^{8} + 8 T^{7} + \cdots + 2 \) T^8 + 8T^7 + 12T^6 - 16T^5 + 22T^4 - 24T^3 + 20T^2 - 8*T + 2
$13$	\( T^{8} + 8 T^{7} + \cdots + 16 \) T^8 + 8T^7 + 32T^6 + 16T^5 + 8T^4 + 160T^3 + 1152T^2 - 192*T + 16
$17$	\( (T^{4} - 4 T^{3} + \cdots + 289)^{2} \) (T^4 - 4T^3 + 6T^2 - 68*T + 289)^2
$19$	\( T^{8} + 36 T^{6} + \cdots + 4 \) T^8 + 36T^6 - 168T^5 + 648T^4 - 1232T^3 + 856T^2 + 112T + 4
$23$	\( T^{8} - 8 T^{7} + \cdots + 12482 \) T^8 - 8T^7 + 44T^6 - 208T^5 + 630T^4 - 2424T^3 + 9716T^2 - 17064*T + 12482
$29$	\( T^{8} - 8 T^{7} + \cdots + 578 \) T^8 - 8T^7 + 12T^6 + 88T^5 - 186T^4 - 3208T^3 + 13212T^2 + 952*T + 578
$31$	\( T^{8} + 8 T^{7} + \cdots + 14439938 \) T^8 + 8T^7 + 28T^6 - 768T^5 - 7498T^4 - 36712T^3 + 244052T^2 + 2558024*T + 14439938
$37$	\( T^{8} - 8 T^{7} + \cdots + 132098 \) T^8 - 8T^7 + 140T^6 - 328T^5 + 7590T^4 + 9288T^3 - 41188T^2 + 6168*T + 132098
$41$	\( T^{8} - 8 T^{7} + \cdots + 4802 \) T^8 - 8T^7 + 12T^6 + 184T^5 - 506T^4 + 168T^3 + 9212T^2 + 2744*T + 4802
$43$	\( T^{8} - 32 T^{7} + \cdots + 1110916 \) T^8 - 32T^7 + 548T^6 - 5688T^5 + 33160T^4 - 87984T^3 + 202200T^2 - 649264*T + 1110916
$47$	\( T^{8} + 8 T^{7} + \cdots + 1993744 \) T^8 + 8T^7 + 32T^6 - 240T^5 + 1800T^4 + 9376T^3 + 46208T^2 - 429248*T + 1993744
$53$	\( T^{8} + 8 T^{7} + \cdots + 3844 \) T^8 + 8T^7 + 60T^6 - 56T^5 - 1464T^4 - 224T^3 + 17896T^2 + 6448*T + 3844
$59$	\( T^{8} + 100 T^{6} + \cdots + 749956 \) T^8 + 100T^6 + 1160T^5 + 5000T^4 + 41360T^3 + 339800T^2 + 866000T + 749956
$61$	\( T^{8} + 8 T^{7} + \cdots + 1276802 \) T^8 + 8T^7 - 116T^6 - 1992T^5 + 6950T^4 + 56584T^3 + 375516T^2 + 492184*T + 1276802
$67$	\( (T^{4} - 176 T^{2} + \cdots + 5056)^{2} \) (T^4 - 176T^2 - 128T + 5056)^2
$71$	\( T^{8} - 24 T^{7} + \cdots + 3792258 \) T^8 - 24T^7 + 252T^6 - 1728T^5 + 12150T^4 - 64152T^3 + 107892T^2 - 297432*T + 3792258
$73$	\( T^{8} + 8 T^{7} + \cdots + 25538 \) T^8 + 8T^7 + 28T^6 + 264T^5 + 998T^4 + 4264T^3 + 30972T^2 - 47912*T + 25538
$79$	\( T^{8} + 24 T^{7} + \cdots + 293378 \) T^8 + 24T^7 + 300T^6 + 2256T^5 + 11030T^4 + 38376T^3 + 109268T^2 + 248184*T + 293378
$83$	\( T^{8} + 32 T^{7} + \cdots + 1110916 \) T^8 + 32T^7 + 548T^6 + 5688T^5 + 33160T^4 + 87984T^3 + 202200T^2 + 649264*T + 1110916
$89$	\( T^{8} - 24 T^{7} + \cdots + 147962896 \) T^8 - 24T^7 + 288T^6 + 1104T^5 + 14088T^4 - 413664T^3 + 6480000T^2 + 43790400*T + 147962896
$97$	\( T^{8} - 8 T^{7} + \cdots + 77900162 \) T^8 - 8T^7 + 124T^6 - 3608T^5 + 49990T^4 - 434184T^3 + 3969276T^2 - 27610184*T + 77900162
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