LMFDB - Newform orbit 8379.2.a.cn

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] 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("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,4,0,0,0,0,0,-4,0,0,16,0,0,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.8446345216.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 $ x^8 - 8x^6 + 19x^4 - 14*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{4} q^{2} - \beta_{5} q^{4} - \beta_{7} q^{5} - \beta_{2} q^{8} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{10} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{3} + 2) q^{13}+ \cdots + ( - \beta_{6} + 3 \beta_{3} + 1) q^{97}+O(q^{100}) $ q - b4 * q^2 - b5 * q^4 - b7 * q^5 - b2 * q^8 + (-b6 + b5 + b3) * q^10 + (-b7 - b4 - b1) * q^11 + (-b5 - b3 + 2) * q^13 + (b5 + b3 - 1) * q^16 + (-b4 - 2b2 + b1) q^17 - q^19 + (-b7 - b2 - b1) * q^20 + (-b6 + b3 + 1) * q^22 + (-b7 + b4 + b1) * q^23 + (b6 - b5 - 2b3 + 2) q^25 + (b7 - 3b4 + b2) q^26 + (-b7 - 2b2) q^29 + (b6 + b5 + 3) * q^31 + (-b7 + 2b4 + b2) q^32 + (-3b5 + 2b3 + 1) * q^34 + (-2b6 + 2b5) * q^37 + b4 * q^38 + (b6 - 2b5 - 2) q^40 + (-2b7 - 2b2 - 2b1) q^41 + (-b6 + b5 - 1) * q^43 + (-b7 - b4 - 2b2 + b1) q^44 + (-b6 + 2b5 + b3 - 1) q^46 + (-b7 - 2b4 + 2b2 - 2b1) q^47 + (4b7 - b4 + 3b2 + b1) * q^50 + (b6 - b5 + 3) * q^52 + (-b4 - 2b2 + 3b1) * q^53 + (-2b5 - 2b3 + 4) * q^55 + (-b6 - b5 + 3b3 - 2) q^58 + (-6b4 - 2b1) * q^59 - 4b5 q^61 + (2b7 + b2 + b1) q^62 + (-b6 + 2b5 - 2b3 - 1) * q^64 + (-5b7 + b4 - 3b1) * q^65 + (b6 - 3b3 + 3) q^67 + (-2b7 - 7b4 - 3b2 - 2b1) * q^68 + (2b7 - b4 + 2b2 + b1) * q^71 + (2b5 - 2b3 + 4) * q^73 + (-4b7 + 2b4 + 2b2 - 2b1) * q^74 + b5 * q^76 + (b6 - 2b5 - b3 + 3) q^79 + (4b7 - b4 + 3b1) * q^80 + (-2b6 + 4b3 - 4) * q^82 + (2b7 + 3b4 + 2b2 + 3b1) * q^83 + (b6 - b5 + 2b3 - 1) q^85 + (-2b7 + 2b4 + b2 - b1) * q^86 + (b6 - 2b5 + b3 - 1) q^88 + (2b7 - 2b4 - 2b2 + 4b1) * q^89 + (-b7 + b4 - 3b1) q^92 + (-b6 + b5 - b3 + 4) * q^94 + b7 * q^95 + (-b6 + 3b3 + 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 4 q^{10} + 16 q^{13} - 8 q^{16} - 8 q^{19} + 8 q^{22} + 16 q^{25} + 24 q^{31} + 28 q^{34} - 16 q^{37} - 4 q^{40} - 16 q^{43} - 16 q^{46} + 32 q^{52} + 32 q^{55} - 4 q^{58} + 16 q^{61} - 28 q^{64}+ \cdots + 16 q^{97}+O(q^{100}) $ 8 * q + 4 * q^4 - 4 * q^10 + 16 * q^13 - 8 * q^16 - 8 * q^19 + 8 * q^22 + 16 * q^25 + 24 * q^31 + 28 * q^34 - 16 * q^37 - 4 * q^40 - 16 * q^43 - 16 * q^46 + 32 * q^52 + 32 * q^55 - 4 * q^58 + 16 * q^61 - 28 * q^64 + 16 * q^67 + 16 * q^73 - 4 * q^76 + 32 * q^79 - 24 * q^82 + 8 * q^85 + 8 * q^88 + 20 * q^94 + 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 $ x^8 - 8*x^6 + 19*x^4 - 14*x^2 + 1 :

$\beta_{1}$	$=$	$ \nu^{3} - 2\nu $ v^3 - 2*v
$\beta_{2}$	$=$	$ \nu^{3} - 4\nu $ v^3 - 4*v
$\beta_{3}$	$=$	$ -\nu^{6} + 6\nu^{4} - 6\nu^{2} - 2 $ -v^6 + 6v^4 - 6v^2 - 2
$\beta_{4}$	$=$	$ \nu^{5} - 6\nu^{3} + 7\nu $ v^5 - 6v^3 + 7v
$\beta_{5}$	$=$	$ \nu^{6} - 6\nu^{4} + 8\nu^{2} - 2 $ v^6 - 6v^4 + 8v^2 - 2
$\beta_{6}$	$=$	$ -\nu^{6} + 8\nu^{4} - 16\nu^{2} + 5 $ -v^6 + 8v^4 - 16v^2 + 5
$\beta_{7}$	$=$	$ 2\nu^{7} - 15\nu^{5} + 31\nu^{3} - 17\nu $ 2v^7 - 15v^5 + 31v^3 - 17v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( -\beta_{2} + \beta_1 ) / 2 $ (-b2 + b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{5} + \beta_{3} + 4 ) / 2 $ (b5 + b3 + 4) / 2
$\nu^{3}$	$=$	$ -\beta_{2} + 2\beta_1 $ -b2 + 2*b1
$\nu^{4}$	$=$	$ ( \beta_{6} + 5\beta_{5} + 4\beta_{3} + 13 ) / 2 $ (b6 + 5b5 + 4b3 + 13) / 2
$\nu^{5}$	$=$	$ ( 2\beta_{4} - 5\beta_{2} + 17\beta_1 ) / 2 $ (2b4 - 5b2 + 17*b1) / 2
$\nu^{6}$	$=$	$ 3\beta_{6} + 12\beta_{5} + 8\beta_{3} + 25 $ 3b6 + 12b5 + 8*b3 + 25
$\nu^{7}$	$=$	$ ( \beta_{7} + 15\beta_{4} - 15\beta_{2} + 74\beta_1 ) / 2 $ (b7 + 15b4 - 15b2 + 74*b1) / 2

Display $\beta_i$ in terms of $\nu^j$

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} $

1.1

−1.51373
0.282452
1.11758
−2.09282
2.09282
−1.11758
−0.282452
1.51373

−2.26732

3.14073

−1.00216

−2.58640

2.27221

1.2

−1.84376

1.39945

4.12982

1.10727

−7.61439

1.3

−1.19143

−0.580491

−2.47632

3.07448

2.95037

1.4

−0.200777

−1.95969

−1.95144

0.795015

0.391805

1.5

0.200777

−1.95969

1.95144

−0.795015

0.391805

1.6

1.19143

−0.580491

2.47632

−3.07448

2.95037

1.7

1.84376

1.39945

−4.12982

−1.10727

−7.61439

1.8

2.26732

3.14073

1.00216

2.58640

2.27221

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$7$	$ -1 $
$19$	$ +1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8379.2.a.cn	✓	8
3.b	odd	2	1	inner	8379.2.a.cn	✓	8
7.b	odd	2	1		8379.2.a.co	yes	8
21.c	even	2	1		8379.2.a.co	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8379.2.a.cn	✓	8	1.a	even	1	1	trivial
8379.2.a.cn	✓	8	3.b	odd	2	1	inner
8379.2.a.co	yes	8	7.b	odd	2	1
8379.2.a.co	yes	8	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8379))$:

$ T_{2}^{8} - 10T_{2}^{6} + 30T_{2}^{4} - 26T_{2}^{2} + 1 $

T2^8 - 10*T2^6 + 30*T2^4 - 26*T2^2 + 1

$ T_{5}^{8} - 28T_{5}^{6} + 220T_{5}^{4} - 592T_{5}^{2} + 400 $

T5^8 - 28*T5^6 + 220*T5^4 - 592*T5^2 + 400

$ T_{11}^{2} - 8 $

T11^2 - 8

$ T_{13}^{4} - 8T_{13}^{3} + 4T_{13}^{2} + 64T_{13} - 80 $

T13^4 - 8*T13^3 + 4*T13^2 + 64*T13 - 80

$ T_{17}^{8} - 92T_{17}^{6} + 2076T_{17}^{4} - 13648T_{17}^{2} + 400 $

T17^8 - 92*T17^6 + 2076*T17^4 - 13648*T17^2 + 400

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - 10 T^{6} + \cdots + 1 $ T^8 - 10T^6 + 30T^4 - 26*T^2 + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - 28 T^{6} + \cdots + 400 $ T^8 - 28T^6 + 220T^4 - 592*T^2 + 400
$7$	$ T^{8} $ T^8
$11$	$ (T^{2} - 8)^{4} $ (T^2 - 8)^4
$13$	$ (T^{4} - 8 T^{3} + 4 T^{2} + \cdots - 80)^{2} $ (T^4 - 8T^3 + 4T^2 + 64*T - 80)^2
$17$	$ T^{8} - 92 T^{6} + \cdots + 400 $ T^8 - 92T^6 + 2076T^4 - 13648*T^2 + 400
$19$	$ (T + 1)^{8} $ (T + 1)^8
$23$	$ T^{8} - 80 T^{6} + \cdots + 4096 $ T^8 - 80T^6 + 1728T^4 - 7168*T^2 + 4096
$29$	$ T^{8} - 92 T^{6} + \cdots + 2704 $ T^8 - 92T^6 + 2604T^4 - 21040*T^2 + 2704
$31$	$ (T^{4} - 12 T^{3} + \cdots + 80)^{2} $ (T^4 - 12T^3 + 12T^2 + 112*T + 80)^2
$37$	$ (T^{4} + 8 T^{3} + \cdots - 1456)^{2} $ (T^4 + 8T^3 - 88T^2 - 800*T - 1456)^2
$41$	$ T^{8} - 240 T^{6} + \cdots + 2560000 $ T^8 - 240T^6 + 17344T^4 - 437248*T^2 + 2560000
$43$	$ (T^{4} + 8 T^{3} + \cdots - 208)^{2} $ (T^4 + 8T^3 - 4T^2 - 128*T - 208)^2
$47$	$ T^{8} - 124 T^{6} + \cdots + 400 $ T^8 - 124T^6 + 3244T^4 - 2352*T^2 + 400
$53$	$ T^{8} - 268 T^{6} + \cdots + 132496 $ T^8 - 268T^6 + 16076T^4 - 108656*T^2 + 132496
$59$	$ T^{8} - 368 T^{6} + \cdots + 36966400 $ T^8 - 368T^6 + 46272T^4 - 2317312*T^2 + 36966400
$61$	$ (T^{4} - 8 T^{3} + \cdots + 1280)^{2} $ (T^4 - 8T^3 - 96T^2 + 384*T + 1280)^2
$67$	$ (T^{4} - 8 T^{3} + \cdots + 688)^{2} $ (T^4 - 8T^3 - 128T^2 + 544*T + 688)^2
$71$	$ T^{8} - 204 T^{6} + \cdots + 283024 $ T^8 - 204T^6 + 6844T^4 - 77456*T^2 + 283024
$73$	$ (T^{4} - 8 T^{3} + \cdots - 1280)^{2} $ (T^4 - 8T^3 - 80T^2 + 768*T - 1280)^2
$79$	$ (T^{4} - 16 T^{3} + \cdots - 16)^{2} $ (T^4 - 16T^3 + 48T^2 - 16)^2
$83$	$ T^{8} - 348 T^{6} + \cdots + 19600 $ T^8 - 348T^6 + 36332T^4 - 1000752*T^2 + 19600
$89$	$ T^{8} - 480 T^{6} + \cdots + 80281600 $ T^8 - 480T^6 + 76544T^4 - 4620288*T^2 + 80281600
$97$	$ (T^{4} - 8 T^{3} + \cdots + 560)^{2} $ (T^4 - 8T^3 - 128T^2 + 608*T + 560)^2
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Elliptic curves over $\Q$
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Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_{4} q^{2} - \beta_{5} q^{4} - \beta_{7} q^{5} - \beta_{2} q^{8} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{10} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{3} + 2) q^{13}+ \cdots + ( - \beta_{6} + 3 \beta_{3} + 1) q^{97}+O(q^{100}) \) q - b4 * q^2 - b5 * q^4 - b7 * q^5 - b2 * q^8 + (-b6 + b5 + b3) * q^10 + (-b7 - b4 - b1) * q^11 + (-b5 - b3 + 2) * q^13 + (b5 + b3 - 1) * q^16 + (-b4 - 2b2 + b1) q^17 - q^19 + (-b7 - b2 - b1) * q^20 + (-b6 + b3 + 1) * q^22 + (-b7 + b4 + b1) * q^23 + (b6 - b5 - 2b3 + 2) q^25 + (b7 - 3b4 + b2) q^26 + (-b7 - 2b2) q^29 + (b6 + b5 + 3) * q^31 + (-b7 + 2b4 + b2) q^32 + (-3b5 + 2b3 + 1) * q^34 + (-2b6 + 2b5) * q^37 + b4 * q^38 + (b6 - 2b5 - 2) q^40 + (-2b7 - 2b2 - 2b1) q^41 + (-b6 + b5 - 1) * q^43 + (-b7 - b4 - 2b2 + b1) q^44 + (-b6 + 2b5 + b3 - 1) q^46 + (-b7 - 2b4 + 2b2 - 2b1) q^47 + (4b7 - b4 + 3b2 + b1) * q^50 + (b6 - b5 + 3) * q^52 + (-b4 - 2b2 + 3b1) * q^53 + (-2b5 - 2b3 + 4) * q^55 + (-b6 - b5 + 3b3 - 2) q^58 + (-6b4 - 2b1) * q^59 - 4b5 q^61 + (2b7 + b2 + b1) q^62 + (-b6 + 2b5 - 2b3 - 1) * q^64 + (-5b7 + b4 - 3b1) * q^65 + (b6 - 3b3 + 3) q^67 + (-2b7 - 7b4 - 3b2 - 2b1) * q^68 + (2b7 - b4 + 2b2 + b1) * q^71 + (2b5 - 2b3 + 4) * q^73 + (-4b7 + 2b4 + 2b2 - 2b1) * q^74 + b5 * q^76 + (b6 - 2b5 - b3 + 3) q^79 + (4b7 - b4 + 3b1) * q^80 + (-2b6 + 4b3 - 4) * q^82 + (2b7 + 3b4 + 2b2 + 3b1) * q^83 + (b6 - b5 + 2b3 - 1) q^85 + (-2b7 + 2b4 + b2 - b1) * q^86 + (b6 - 2b5 + b3 - 1) q^88 + (2b7 - 2b4 - 2b2 + 4b1) * q^89 + (-b7 + b4 - 3b1) q^92 + (-b6 + b5 - b3 + 4) * q^94 + b7 * q^95 + (-b6 + 3b3 + 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 4 q^{10} + 16 q^{13} - 8 q^{16} - 8 q^{19} + 8 q^{22} + 16 q^{25} + 24 q^{31} + 28 q^{34} - 16 q^{37} - 4 q^{40} - 16 q^{43} - 16 q^{46} + 32 q^{52} + 32 q^{55} - 4 q^{58} + 16 q^{61} - 28 q^{64}+ \cdots + 16 q^{97}+O(q^{100}) \) 8 * q + 4 * q^4 - 4 * q^10 + 16 * q^13 - 8 * q^16 - 8 * q^19 + 8 * q^22 + 16 * q^25 + 24 * q^31 + 28 * q^34 - 16 * q^37 - 4 * q^40 - 16 * q^43 - 16 * q^46 + 32 * q^52 + 32 * q^55 - 4 * q^58 + 16 * q^61 - 28 * q^64 + 16 * q^67 + 16 * q^73 - 4 * q^76 + 32 * q^79 - 24 * q^82 + 8 * q^85 + 8 * q^88 + 20 * q^94 + 16 * q^97

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