LMFDB - Newform orbit 784.2.u.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(113,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 0, 10]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("784.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.u (of order $7$, degree $6$, 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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$1$	$-\zeta_{14}$

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} $

113.1

0.222521	+	0.974928i
0.900969	−	0.433884i
−0.623490	−	0.781831i
−0.623490	+	0.781831i
0.900969	+	0.433884i
0.222521	−	0.974928i

0.500000

−

0.626980i

0.153989

−

0.193096i

−2.06853

1.64960i

0.524459

2.29780i

225.1

0.500000

2.19064i

0.321552

1.40881i

−2.57942

−

0.588735i

−1.84601

0.888992i

337.1

0.500000

0.240787i

2.52446

1.21572i

1.14795

2.38374i

−1.67845

−

2.10471i

449.1

0.500000

−

0.240787i

2.52446

−

1.21572i

1.14795

−

2.38374i

−1.67845

2.10471i

561.1

0.500000

−

2.19064i

0.321552

−

1.40881i

−2.57942

0.588735i

−1.84601

−

0.888992i

673.1

0.500000

0.626980i

0.153989

0.193096i

−2.06853

−

1.64960i

0.524459

−

2.29780i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.e	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.2.u.a		6
4.b	odd	2	1		49.2.e.a	✓	6
12.b	even	2	1		441.2.u.a		6
28.d	even	2	1		343.2.e.a		6
28.f	even	6	2		343.2.g.e		12
28.g	odd	6	2		343.2.g.f		12
49.e	even	7	1	inner	784.2.u.a		6
196.j	even	14	1		343.2.e.a		6
196.j	even	14	1		2401.2.a.a		3
196.k	odd	14	1		49.2.e.a	✓	6
196.k	odd	14	1		2401.2.a.b		3
196.o	odd	42	2		343.2.g.f		12
196.p	even	42	2		343.2.g.e		12
588.u	even	14	1		441.2.u.a		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
49.2.e.a	✓	6	4.b	odd	2	1
49.2.e.a	✓	6	196.k	odd	14	1
343.2.e.a		6	28.d	even	2	1
343.2.e.a		6	196.j	even	14	1
343.2.g.e		12	28.f	even	6	2
343.2.g.e		12	196.p	even	42	2
343.2.g.f		12	28.g	odd	6	2
343.2.g.f		12	196.o	odd	42	2
441.2.u.a		6	12.b	even	2	1
441.2.u.a		6	588.u	even	14	1
784.2.u.a		6	1.a	even	1	1	trivial
784.2.u.a		6	49.e	even	7	1	inner
2401.2.a.a		3	196.j	even	14	1
2401.2.a.b		3	196.k	odd	14	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 13T_{3}^{3} + 11T_{3}^{2} - 5T_{3} + 1 $ T3^6 - 3*T3^5 + 9*T3^4 - 13*T3^3 + 11*T3^2 - 5*T3 + 1 acting on $S_{2}^{\mathrm{new}}(784, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 3 T^{5} + \cdots + 1 $ T^6 - 3T^5 + 9T^4 - 13T^3 + 11T^2 - 5*T + 1
$5$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 15T^4 - 20T^3 + 22T^2 - 6*T + 1
$7$	$ T^{6} + 7 T^{5} + \cdots + 343 $ T^6 + 7T^5 + 21T^4 + 49T^3 + 147T^2 + 343*T + 343
$11$	$ T^{6} + 2 T^{5} + \cdots + 169 $ T^6 + 2T^5 - 3T^4 + 22T^3 + 142T^2 - 52*T + 169
$13$	$ T^{6} + 14 T^{4} + \cdots + 41209 $ T^6 + 14T^4 + 259T^3 + 784T^2 + 1421T + 41209
$17$	$ T^{6} - 4 T^{5} + \cdots + 1681 $ T^6 - 4T^5 + 2T^4 - 29T^3 + 228T^2 + 369*T + 1681
$19$	$ (T^{3} - 4 T^{2} - 11 T + 1)^{2} $ (T^3 - 4T^2 - 11T + 1)^2
$23$	$ T^{6} - 10 T^{5} + \cdots + 169 $ T^6 - 10T^5 + 30T^4 + T^3 + 88T^2 + 65T + 169
$29$	$ T^{6} + 16 T^{5} + \cdots + 6889 $ T^6 + 16T^5 + 158T^4 + 827T^3 + 3264T^2 + 7221*T + 6889
$31$	$ (T^{3} - 5 T^{2} + 6 T - 1)^{2} $ (T^3 - 5T^2 + 6T - 1)^2
$37$	$ T^{6} - 4 T^{5} + \cdots + 10816 $ T^6 - 4T^5 + 72T^4 - 680T^3 + 1600T^2 + 4576*T + 10816
$41$	$ T^{6} + 56 T^{4} + \cdots + 3136 $ T^6 + 56T^4 + 112T^3 + 784T^2 - 3136T + 3136
$43$	$ T^{6} + 12 T^{5} + \cdots + 841 $ T^6 + 12T^5 + 74T^4 + 265T^3 + 604T^2 + 899*T + 841
$47$	$ T^{6} - 15 T^{5} + \cdots + 9409 $ T^6 - 15T^5 + 162T^4 - 1079T^3 + 4180T^2 - 8730*T + 9409
$53$	$ T^{6} + 26 T^{5} + \cdots + 53824 $ T^6 + 26T^5 + 396T^4 + 2680T^3 + 12784T^2 + 38048*T + 53824
$59$	$ T^{6} + 11 T^{5} + \cdots + 57121 $ T^6 + 11T^5 + 65T^4 + 295T^3 + 1257T^2 + 2151*T + 57121
$61$	$ T^{6} + 8 T^{5} + \cdots + 1849 $ T^6 + 8T^5 + 43T^4 + 190T^3 + 680T^2 + 1548*T + 1849
$67$	$ (T^{3} - 6 T^{2} + 5 T + 13)^{2} $ (T^3 - 6T^2 + 5T + 13)^2
$71$	$ T^{6} + 5 T^{5} + \cdots + 85849 $ T^6 + 5T^5 + 137T^4 + 97T^3 + 1101T^2 + 23733*T + 85849
$73$	$ T^{6} - 4 T^{5} + \cdots + 5041 $ T^6 - 4T^5 + 184T^4 + 370T^3 + 1019T^2 - 5112*T + 5041
$79$	$ (T^{3} + 30 T^{2} + \cdots + 937)^{2} $ (T^3 + 30T^2 + 293T + 937)^2
$83$	$ T^{6} - 14 T^{5} + \cdots + 625681 $ T^6 - 14T^5 + 252T^4 - 280T^3 + 7105T^2 + 155036*T + 625681
$89$	$ T^{6} - 13 T^{5} + \cdots + 187489 $ T^6 - 13T^5 + 281T^4 - 4955T^3 + 40419T^2 - 115611*T + 187489
$97$	$ (T^{3} - 7 T - 7)^{2} $ (T^3 - 7*T - 7)^2
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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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.u (of order \(7\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{14}^{5} - \zeta_{14}^{3} + \cdots + 1) q^{3}+ \cdots + (\zeta_{14}^{5} - \zeta_{14}^{4} + \cdots - 2) q^{9}+O(q^{10}) \) q + (-z^5 - z^3 - z + 1) * q^3 + (-z^4 - z + 1) * q^5 + (2z^5 - 2z^4 + z^3 - 2) * q^7 + (z^5 - z^4 - 2z^2 + 2z - 2) * q^9	\( q + ( - \zeta_{14}^{5} - \zeta_{14}^{3} + \cdots + 1) q^{3}+ \cdots + ( - 5 \zeta_{14}^{5} + 6 \zeta_{14}^{4} + \cdots + 4) q^{99}+O(q^{100}) \) q + (-z^5 - z^3 - z + 1) * q^3 + (-z^4 - z + 1) * q^5 + (2z^5 - 2z^4 + z^3 - 2) * q^7 + (z^5 - z^4 - 2z^2 + 2z - 2) * q^9 + (2z^5 - z^4 + 3z^3 - 3z^2 + z - 2) q^11 + (2z^5 + 5z^3 - 5z^2 - 2) q^13 + (z^5 - z^4 - z^2 - z - 1) * q^15 + (2z^4 - 3z^3 - 3z + 2) q^17 + (2z^5 + z^4 - z^3 - 2z^2 + 1) * q^19 + (3z^5 + 2z^3 + z^2 + 2z - 1) q^21 + (3z^5 - z^3 - z^2 + z + 1) q^23 + (2z^5 - 2z^4 + z^2 + 2z + 1) q^25 + (-z^3 - 2z^2 + 2z + 1) * q^27 + (-2z^4 - 3z^3 - 3z - 2) q^29 + (-z^5 + z^2 + 2) * q^31 + (2z^5 + z^4 + 4z^3 + z^2 + 2z) q^33 + (2z^5 + z^4 - z^3 + 4z^2 - 2z + 1) q^35 + (4z^4 - 6z^3 + 4z^2 - 6z + 4) * q^37 + (2z^5 + 2z^4 + 7z^3 + 2z^2 + 2z) q^39 + (-2z^5 - 4z^4 - 2z^3) q^41 + (-3z^5 + 2z^4 - 2z^3 + 2z^2 - 3z) q^43 + (z^5 + 3z^3 - 4z^2 + 4z - 3) q^45 + (-5z^5 + 4z^4 - z^3 + z^2 - 4z + 5) q^47 + (-7z^5 + 7z^4 - 7z^3 + 7z^2 - 7z + 7) q^49 + (2z^5 + 2z^4 + z^3 - z^2 - 2z - 2) q^51 + (6z^5 + 8z^3 - 4z^2 + 4z - 8) * q^53 + (3z^5 - 3z^4 + 7z^3 - 3z^2 + 3z) q^55 + (z^5 + 3z^4 + z^3 - z + 1) q^57 + (-z^5 + z^4 - 7z^3 + z^2 - z) q^59 + (-3z^4 + 2z^3 - 3z^2 + 2z - 3) * q^61 + (-4z^5 + 4z^3 + z + 1) * q^63 + (5z^5 - 6z^4 + 13z^3 - 6z^2 + 5z) q^65 + (2z^5 - z^4 + z^3 - 2z^2 + 1) * q^67 + (3z^4 - z^3 + z^2 - z + 3) q^69 + (-6z^5 + z^3 - 3z^2 + 3z - 1) q^71 + (5z^5 - 5z^4 - 3z^2 + 9z - 3) * q^73 + (-2z^5 - 4z^3 + z^2 - z + 4) * q^75 + (-2z^5 - 4z^4 - z^3 + z^2 - z + 6) * q^77 + (2z^5 - z^4 + z^3 - 2z^2 - 11) * q^79 + (-3z^4 - 4z^3 - 4z - 3) q^81 + (9z^5 - 9z^4 - z^2 + z - 1) * q^83 + (z^5 + 3z^4 - 3z^3 + 3z^2 - 3z - 1) * q^85 + (10z^5 - 2z^4 + 5z^3 - 5z^2 + 2z - 10) q^87 + (-9z^5 + 9z^4 + 8z^2 - 9z + 8) * q^89 + (2z^5 - 11z^4 - z^3 - z^2 + 3z + 9) q^91 + (-3z^5 - z^4 - 3z^3 - 2z + 2) q^93 + (z^5 + z^4 + z^3) * q^95 + (z^5 - 2z^4 + 2z^3 - z^2 - 1) * q^97 + (-5z^5 + 6z^4 - 6z^3 + 5z^2 + 4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 6 q^{5} - 7 q^{7} - 6 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 + 6 * q^5 - 7 * q^7 - 6 * q^9	\( 6 q + 3 q^{3} + 6 q^{5} - 7 q^{7} - 6 q^{9} - 2 q^{11} - 4 q^{15} + 4 q^{17} + 8 q^{19} + 10 q^{23} + 11 q^{25} + 9 q^{27} - 16 q^{29} + 10 q^{31} + 6 q^{33} + 4 q^{37} + 7 q^{39} - 12 q^{43} - 6 q^{45} + 15 q^{47} + 7 q^{49} - 12 q^{51} - 26 q^{53} + 19 q^{55} + 4 q^{57} - 11 q^{59} - 8 q^{61} + 7 q^{63} + 35 q^{65} + 12 q^{67} + 12 q^{69} - 5 q^{71} + 4 q^{73} + 16 q^{75} + 35 q^{77} - 60 q^{79} - 23 q^{81} + 14 q^{83} - 17 q^{85} - 36 q^{87} + 13 q^{89} + 70 q^{91} + 5 q^{93} + q^{95} + 2 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 6 * q^5 - 7 * q^7 - 6 * q^9 - 2 * q^11 - 4 * q^15 + 4 * q^17 + 8 * q^19 + 10 * q^23 + 11 * q^25 + 9 * q^27 - 16 * q^29 + 10 * q^31 + 6 * q^33 + 4 * q^37 + 7 * q^39 - 12 * q^43 - 6 * q^45 + 15 * q^47 + 7 * q^49 - 12 * q^51 - 26 * q^53 + 19 * q^55 + 4 * q^57 - 11 * q^59 - 8 * q^61 + 7 * q^63 + 35 * q^65 + 12 * q^67 + 12 * q^69 - 5 * q^71 + 4 * q^73 + 16 * q^75 + 35 * q^77 - 60 * q^79 - 23 * q^81 + 14 * q^83 - 17 * q^85 - 36 * q^87 + 13 * q^89 + 70 * q^91 + 5 * q^93 + q^95 + 2 * q^99

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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	784.2.u.a

Level	$784$
Weight	$2$
Character orbit	784.u
Analytic conductor	$6.260$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{14})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 49)
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

\(n\)	\(197\)	\(687\)	\(689\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{14}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 113.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} \) T^6
$3$	\( T^{6} - 3 T^{5} + \cdots + 1 \) T^6 - 3T^5 + 9T^4 - 13T^3 + 11T^2 - 5*T + 1
$5$	\( T^{6} - 6 T^{5} + \cdots + 1 \) T^6 - 6T^5 + 15T^4 - 20T^3 + 22T^2 - 6*T + 1
$7$	\( T^{6} + 7 T^{5} + \cdots + 343 \) T^6 + 7T^5 + 21T^4 + 49T^3 + 147T^2 + 343*T + 343
$11$	\( T^{6} + 2 T^{5} + \cdots + 169 \) T^6 + 2T^5 - 3T^4 + 22T^3 + 142T^2 - 52*T + 169
$13$	\( T^{6} + 14 T^{4} + \cdots + 41209 \) T^6 + 14T^4 + 259T^3 + 784T^2 + 1421T + 41209
$17$	\( T^{6} - 4 T^{5} + \cdots + 1681 \) T^6 - 4T^5 + 2T^4 - 29T^3 + 228T^2 + 369*T + 1681
$19$	\( (T^{3} - 4 T^{2} - 11 T + 1)^{2} \) (T^3 - 4T^2 - 11T + 1)^2
$23$	\( T^{6} - 10 T^{5} + \cdots + 169 \) T^6 - 10T^5 + 30T^4 + T^3 + 88T^2 + 65T + 169
$29$	\( T^{6} + 16 T^{5} + \cdots + 6889 \) T^6 + 16T^5 + 158T^4 + 827T^3 + 3264T^2 + 7221*T + 6889
$31$	\( (T^{3} - 5 T^{2} + 6 T - 1)^{2} \) (T^3 - 5T^2 + 6T - 1)^2
$37$	\( T^{6} - 4 T^{5} + \cdots + 10816 \) T^6 - 4T^5 + 72T^4 - 680T^3 + 1600T^2 + 4576*T + 10816
$41$	\( T^{6} + 56 T^{4} + \cdots + 3136 \) T^6 + 56T^4 + 112T^3 + 784T^2 - 3136T + 3136
$43$	\( T^{6} + 12 T^{5} + \cdots + 841 \) T^6 + 12T^5 + 74T^4 + 265T^3 + 604T^2 + 899*T + 841
$47$	\( T^{6} - 15 T^{5} + \cdots + 9409 \) T^6 - 15T^5 + 162T^4 - 1079T^3 + 4180T^2 - 8730*T + 9409
$53$	\( T^{6} + 26 T^{5} + \cdots + 53824 \) T^6 + 26T^5 + 396T^4 + 2680T^3 + 12784T^2 + 38048*T + 53824
$59$	\( T^{6} + 11 T^{5} + \cdots + 57121 \) T^6 + 11T^5 + 65T^4 + 295T^3 + 1257T^2 + 2151*T + 57121
$61$	\( T^{6} + 8 T^{5} + \cdots + 1849 \) T^6 + 8T^5 + 43T^4 + 190T^3 + 680T^2 + 1548*T + 1849
$67$	\( (T^{3} - 6 T^{2} + 5 T + 13)^{2} \) (T^3 - 6T^2 + 5T + 13)^2
$71$	\( T^{6} + 5 T^{5} + \cdots + 85849 \) T^6 + 5T^5 + 137T^4 + 97T^3 + 1101T^2 + 23733*T + 85849
$73$	\( T^{6} - 4 T^{5} + \cdots + 5041 \) T^6 - 4T^5 + 184T^4 + 370T^3 + 1019T^2 - 5112*T + 5041
$79$	\( (T^{3} + 30 T^{2} + \cdots + 937)^{2} \) (T^3 + 30T^2 + 293T + 937)^2
$83$	\( T^{6} - 14 T^{5} + \cdots + 625681 \) T^6 - 14T^5 + 252T^4 - 280T^3 + 7105T^2 + 155036*T + 625681
$89$	\( T^{6} - 13 T^{5} + \cdots + 187489 \) T^6 - 13T^5 + 281T^4 - 4955T^3 + 40419T^2 - 115611*T + 187489
$97$	\( (T^{3} - 7 T - 7)^{2} \) (T^3 - 7*T - 7)^2
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