LMFDB - Newform orbit 57.3.l.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,3,Mod(5,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 16]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("57.5");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	57.l (of order $18$, degree $6$, 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:	$1.55313750685$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

$q$-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 $\zeta_{18}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - 3 \zeta_{18}^{4} q^{3} + 4 \zeta_{18}^{2} q^{4} + ( - 8 \zeta_{18}^{5} + \cdots + 3 \zeta_{18}) q^{7} + \cdots + (9 \zeta_{18}^{5} - 9 \zeta_{18}^{2}) q^{9} +O(q^{10}) $ q - 3z^4 q^3 + 4z^2 q^4 + (-8z^5 - 8z^4 + 5z^2 + 3z) * q^7 + (9z^5 - 9z^2) * q^9	$ q - 3 \zeta_{18}^{4} q^{3} + 4 \zeta_{18}^{2} q^{4} + ( - 8 \zeta_{18}^{5} + \cdots + 3 \zeta_{18}) q^{7} + \cdots - 167 \zeta_{18} q^{97} +O(q^{100}) $ q - 3z^4 q^3 + 4z^2 q^4 + (-8z^5 - 8z^4 + 5z^2 + 3z) * q^7 + (9z^5 - 9z^2) * q^9 + (-12z^3 + 12) q^12 + (15z^4 + 15z^3 - 7z - 8) q^13 + 16z^4 q^16 + (5z^3 - 21) q^19 + (15z^5 - 15z^3 - 24z^2 - 9) q^21 - 25z^5 q^25 + 27z^3 q^27 + (-12z^4 - 20z^3 + 32z + 32) q^28 + (11z^5 + 11z^4 + 24z^2 - 35z) * q^31 - 36z q^36 + (33z^5 - 40z^4 + 7z^2 + 7z) * q^37 + (-24z^5 - 21z^4 + 45z^2 + 45z) * q^39 + (-35z^5 + 13z^3 - 13z^2 + 35) q^43 + (-48z^5 + 48z^2) * q^48 + (16z^5 - 55z^4 - 49z^3 - 55z^2 + 16z) q^49 + (60z^5 + 32z^3 - 32z^2 - 60) q^52 + (48z^4 + 15z) * q^57 + (65z^4 - 56z^3 - 56z + 65) q^61 + (72z^4 + 72z^3 - 45z - 27) q^63 + (64z^3 - 64) q^64 + (45z^4 - 77z^3 + 32z + 32) q^67 + (80z^5 + 17z^3 - 63z^2 + 63) q^73 - 75 * q^75 + (20z^5 - 84z^2) * q^76 + (-91z^5 - 40z^3 + 51z^2 - 51) q^79 + (-81z^4 + 81z) * q^81 + (-60z^5 - 36z^4 - 36z^2 - 60z) * q^84 + (-11z^4 + 96z^3 + 179z^2 + 96z - 11) * q^91 + (72z^5 - 72z^3 + 33z^2 + 105) q^93 - 167z q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 36 q^{12} - 3 q^{13} - 111 q^{19} - 99 q^{21} + 81 q^{27} + 132 q^{28} + 249 q^{43} - 147 q^{49} - 264 q^{52} + 222 q^{61} + 54 q^{63} - 192 q^{64} - 39 q^{67} + 429 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) $ 6 * q + 36 * q^12 - 3 * q^13 - 111 * q^19 - 99 * q^21 + 81 * q^27 + 132 * q^28 + 249 * q^43 - 147 * q^49 - 264 * q^52 + 222 * q^61 + 54 * q^63 - 192 * q^64 - 39 * q^67 + 429 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$.

$n$	$20$	$40$
$\chi(n)$	$-1$	$\zeta_{18}^{2}$

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} $

5.1

0.939693	−	0.342020i
−0.173648	+	0.984808i
0.939693	+	0.342020i
−0.766044	−	0.642788i
−0.766044	+	0.642788i
−0.173648	−	0.984808i

−0.520945

2.95442i

3.06418

−

2.57115i

6.64930

11.5169i

−8.45723

−

3.07818i

17.1

−2.29813

−

1.92836i

−3.75877

−

1.36808i

−5.21941

−

9.04028i

1.56283

8.86327i

23.1

−0.520945

−

2.95442i

3.06418

2.57115i

6.64930

−

11.5169i

−8.45723

3.07818i

35.1

2.81908

−

1.02606i

0.694593

3.93923i

−1.42989

−

2.47665i

6.89440

−

5.78509i

44.1

2.81908

1.02606i

0.694593

−

3.93923i

−1.42989

2.47665i

6.89440

5.78509i

47.1

−2.29813

1.92836i

−3.75877

1.36808i

−5.21941

9.04028i

1.56283

−

8.86327i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
19.e	even	9	1	inner
57.l	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	57.3.l.a	✓	6
3.b	odd	2	1	CM	57.3.l.a	✓	6
19.e	even	9	1	inner	57.3.l.a	✓	6
57.l	odd	18	1	inner	57.3.l.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
57.3.l.a	✓	6	1.a	even	1	1	trivial
57.3.l.a	✓	6	3.b	odd	2	1	CM
57.3.l.a	✓	6	19.e	even	9	1	inner
57.3.l.a	✓	6	57.l	odd	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} $ T2 acting on $S_{3}^{\mathrm{new}}(57, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 27T^{3} + 729 $ T^6 - 27*T^3 + 729
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 147 T^{4} + \cdots + 157609 $ T^6 + 147T^4 + 794T^3 + 21609T^2 + 58359T + 157609
$11$	$ T^{6} $ T^6
$13$	$ T^{6} + 3 T^{5} + \cdots + 16265089 $ T^6 + 3T^5 + 510T^4 - 2512T^3 - 21228T^2 + 1766454*T + 16265089
$17$	$ T^{6} $ T^6
$19$	$ (T^{2} + 37 T + 361)^{3} $ (T^2 + 37*T + 361)^3
$23$	$ T^{6} $ T^6
$29$	$ T^{6} $ T^6
$31$	$ T^{6} + \cdots + 3507126841 $ T^6 + 2883T^4 + 118442T^3 + 8311689T^2 + 170734143T + 3507126841
$37$	$ (T^{3} - 4107 T - 3023)^{2} $ (T^3 - 4107*T - 3023)^2
$41$	$ T^{6} $ T^6
$43$	$ T^{6} + \cdots + 23707684729 $ T^6 - 249T^5 + 26214T^4 - 1535176T^3 + 58946268T^2 - 1484607666*T + 23707684729
$47$	$ T^{6} $ T^6
$53$	$ T^{6} $ T^6
$59$	$ T^{6} $ T^6
$61$	$ T^{6} + \cdots + 3968874001 $ T^6 - 222T^5 + 27591T^4 - 2415187T^3 + 101743932T^2 + 989021301*T + 3968874001
$67$	$ T^{6} + \cdots + 170175225529 $ T^6 + 39T^5 + 13974T^4 + 937736T^3 + 46465068T^2 - 7309082514*T + 170175225529
$71$	$ T^{6} $ T^6
$73$	$ T^{6} + \cdots + 496451295649 $ T^6 - 429T^5 + 77334T^4 - 7563016T^3 + 460620732T^2 - 18055900218*T + 496451295649
$79$	$ T^{6} + \cdots + 879264411481 $ T^6 + 426T^5 + 79215T^4 + 8913689T^3 + 719063292T^2 + 34772395353*T + 879264411481
$83$	$ T^{6} $ T^6
$89$	$ T^{6} $ T^6
$97$	$ T^{6} + \cdots + 21691961596369 $ T^6 + 4657463*T^3 + 21691961596369
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	57.l (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q - 3 \zeta_{18}^{4} q^{3} + 4 \zeta_{18}^{2} q^{4} + ( - 8 \zeta_{18}^{5} + \cdots + 3 \zeta_{18}) q^{7} + \cdots + (9 \zeta_{18}^{5} - 9 \zeta_{18}^{2}) q^{9} +O(q^{10}) \) q - 3z^4 q^3 + 4z^2 q^4 + (-8z^5 - 8z^4 + 5z^2 + 3z) * q^7 + (9z^5 - 9z^2) * q^9	\( q - 3 \zeta_{18}^{4} q^{3} + 4 \zeta_{18}^{2} q^{4} + ( - 8 \zeta_{18}^{5} + \cdots + 3 \zeta_{18}) q^{7} + \cdots - 167 \zeta_{18} q^{97} +O(q^{100}) \) q - 3z^4 q^3 + 4z^2 q^4 + (-8z^5 - 8z^4 + 5z^2 + 3z) * q^7 + (9z^5 - 9z^2) * q^9 + (-12z^3 + 12) q^12 + (15z^4 + 15z^3 - 7z - 8) q^13 + 16z^4 q^16 + (5z^3 - 21) q^19 + (15z^5 - 15z^3 - 24z^2 - 9) q^21 - 25z^5 q^25 + 27z^3 q^27 + (-12z^4 - 20z^3 + 32z + 32) q^28 + (11z^5 + 11z^4 + 24z^2 - 35z) * q^31 - 36z q^36 + (33z^5 - 40z^4 + 7z^2 + 7z) * q^37 + (-24z^5 - 21z^4 + 45z^2 + 45z) * q^39 + (-35z^5 + 13z^3 - 13z^2 + 35) q^43 + (-48z^5 + 48z^2) * q^48 + (16z^5 - 55z^4 - 49z^3 - 55z^2 + 16z) q^49 + (60z^5 + 32z^3 - 32z^2 - 60) q^52 + (48z^4 + 15z) * q^57 + (65z^4 - 56z^3 - 56z + 65) q^61 + (72z^4 + 72z^3 - 45z - 27) q^63 + (64z^3 - 64) q^64 + (45z^4 - 77z^3 + 32z + 32) q^67 + (80z^5 + 17z^3 - 63z^2 + 63) q^73 - 75 * q^75 + (20z^5 - 84z^2) * q^76 + (-91z^5 - 40z^3 + 51z^2 - 51) q^79 + (-81z^4 + 81z) * q^81 + (-60z^5 - 36z^4 - 36z^2 - 60z) * q^84 + (-11z^4 + 96z^3 + 179z^2 + 96z - 11) * q^91 + (72z^5 - 72z^3 + 33z^2 + 105) q^93 - 167z q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 36 q^{12} - 3 q^{13} - 111 q^{19} - 99 q^{21} + 81 q^{27} + 132 q^{28} + 249 q^{43} - 147 q^{49} - 264 q^{52} + 222 q^{61} + 54 q^{63} - 192 q^{64} - 39 q^{67} + 429 q^{73} - 450 q^{75} - 426 q^{79} + 222 q^{91} + 414 q^{93}+O(q^{100}) \) 6 * q + 36 * q^12 - 3 * q^13 - 111 * q^19 - 99 * q^21 + 81 * q^27 + 132 * q^28 + 249 * q^43 - 147 * q^49 - 264 * q^52 + 222 * q^61 + 54 * q^63 - 192 * q^64 - 39 * q^67 + 429 * q^73 - 450 * q^75 - 426 * q^79 + 222 * q^91 + 414 * q^93

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