LMFDB - Newform orbit 8784.2.a.ca

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8784,2,Mod(1,8784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8784, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8784.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8784 = 2^{4} \cdot 3^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8784.a (trivial)

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:	yes
Analytic conductor:	$70.1405931355$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.91407488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 11x^{4} - 2x^{3} + 31x^{2} + 10x - 17 $ x^6 - 11x^4 - 2x^3 + 31x^2 + 10x - 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 183)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{3} q^{5} - \beta_{5} q^{7}+O(q^{10}) $ q + b3 * q^5 - b5 * q^7	$ q + \beta_{3} q^{5} - \beta_{5} q^{7} + (\beta_{5} - \beta_{4} - 2) q^{11} + (\beta_{4} - \beta_{2} + 1) q^{13} + (\beta_{5} - \beta_{2} - \beta_1 - 2) q^{17} + (\beta_{4} - \beta_{3} + \beta_{2} - 1) q^{19} + (\beta_{2} + \beta_1) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 3) q^{25}+ \cdots + ( - \beta_{5} + 3 \beta_{3} - 3 \beta_1 + 5) q^{97}+O(q^{100}) $ q + b3 * q^5 - b5 * q^7 + (b5 - b4 - 2) * q^11 + (b4 - b2 + 1) * q^13 + (b5 - b2 - b1 - 2) * q^17 + (b4 - b3 + b2 - 1) * q^19 + (b2 + b1) * q^23 + (-b5 - b4 - b3 + b2 + b1 + 3) * q^25 + (-b4 + 2b2 + 2) q^29 + (-b4 - b3 - b2 - 1) * q^31 + (-b5 + 3b4 + b3 - b2 - 1) q^35 - 2b1 q^37 + (b5 - b4 - b3 - b2 + b1) * q^41 + (b4 - b3 - b2 - 1) * q^43 + (b5 - b4 - 2b3 + b2 + b1 - 2) q^47 + (-b5 - b4 + b3 + b2 - b1 + 3) * q^49 + (-b5 + 2b4 + 2b3 - b2 - b1 + 2) * q^53 + (3b5 - b4 - 3b3 + b2 + 2b1 - 1) q^55 + (b2 - b1 - 2) * q^59 - q^61 + (-3b5 - b4 + b3 + b2 - b1 + 2) q^65 + (b5 + 2b2 + 2b1) * q^67 + (2b4 - b3 - b2 - 3) q^71 + (b5 + b4 - b3 - b2 + b1 + 4) * q^73 + (2b5 - 3b3 - 6) * q^77 + (b5 - 2b2 - 2) q^79 + (b5 - 3b4 - b2 + b1 - 4) q^83 + (-4b3 + 2b2 - 2) * q^85 + (-2b5 + b4 - 2b3 + 2b2 + 2b1 - 4) * q^89 + (-b5 + b4 + 3b3 + b2 + 1) q^91 + (-2b4 - 2b2 - 4b1 - 6) q^95 + (-b5 + 3b3 - 3b1 + 5) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 6 * q - 2 * q^5 - 2 * q^7	$ 6 q - 2 q^{5} - 2 q^{7} - 8 q^{11} + 6 q^{13} - 10 q^{17} - 8 q^{19} + 20 q^{25} + 10 q^{29} - 14 q^{35} - 4 q^{37} + 10 q^{41} - 4 q^{43} - 4 q^{47} + 12 q^{49} + 2 q^{53} + 10 q^{55} - 16 q^{59} - 6 q^{61} + 2 q^{65} + 2 q^{67} - 18 q^{71} + 30 q^{73} - 26 q^{77} - 6 q^{79} - 12 q^{83} - 8 q^{85} - 26 q^{89} - 6 q^{91} - 36 q^{95} + 16 q^{97}+O(q^{100}) $ 6 * q - 2 * q^5 - 2 * q^7 - 8 * q^11 + 6 * q^13 - 10 * q^17 - 8 * q^19 + 20 * q^25 + 10 * q^29 - 14 * q^35 - 4 * q^37 + 10 * q^41 - 4 * q^43 - 4 * q^47 + 12 * q^49 + 2 * q^53 + 10 * q^55 - 16 * q^59 - 6 * q^61 + 2 * q^65 + 2 * q^67 - 18 * q^71 + 30 * q^73 - 26 * q^77 - 6 * q^79 - 12 * q^83 - 8 * q^85 - 26 * q^89 - 6 * q^91 - 36 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 11x^{4} - 2x^{3} + 31x^{2} + 10x - 17 $ x^6 - 11*x^4 - 2*x^3 + 31*x^2 + 10*x - 17 :

$\beta_{1}$	$=$	$ ( -\nu^{5} + \nu^{4} + 8\nu^{3} - 6\nu^{2} - 9\nu + 5 ) / 2 $ (-v^5 + v^4 + 8v^3 - 6v^2 - 9*v + 5) / 2
$\beta_{2}$	$=$	$ ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 16\nu^{2} + 11\nu - 22 ) / 2 $ (v^5 - 2v^4 - 8v^3 + 16v^2 + 11v - 22) / 2
$\beta_{3}$	$=$	$ ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 16\nu^{2} + 21\nu - 20 ) / 2 $ (v^5 - 2v^4 - 10v^3 + 16v^2 + 21v - 20) / 2
$\beta_{4}$	$=$	$ \nu^{5} - \nu^{4} - 9\nu^{3} + 8\nu^{2} + 16\nu - 11 $ v^5 - v^4 - 9v^3 + 8v^2 + 16*v - 11
$\beta_{5}$	$=$	$ ( 2\nu^{5} - 3\nu^{4} - 18\nu^{3} + 22\nu^{2} + 34\nu - 23 ) / 2 $ (2v^5 - 3v^4 - 18v^3 + 22v^2 + 34*v - 23) / 2

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

$\nu$	$=$	$ ( \beta_{5} - \beta_{3} + \beta _1 - 1 ) / 2 $ (b5 - b3 + b1 - 1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 8 ) / 2 $ (-b5 + b4 + b2 + b1 + 8) / 2
$\nu^{3}$	$=$	$ ( 5\beta_{5} - 7\beta_{3} + 2\beta_{2} + 5\beta _1 - 3 ) / 2 $ (5b5 - 7b3 + 2b2 + 5b1 - 3) / 2
$\nu^{4}$	$=$	$ -4\beta_{5} + 5\beta_{4} - \beta_{3} + 3\beta_{2} + 4\beta _1 + 22 $ -4b5 + 5b4 - b3 + 3b2 + 4b1 + 22
$\nu^{5}$	$=$	$ ( 29\beta_{5} + 4\beta_{4} - 49\beta_{3} + 16\beta_{2} + 29\beta _1 - 9 ) / 2 $ (29b5 + 4b4 - 49b3 + 16b2 + 29*b1 - 9) / 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

−1.27291
2.70416
0.645598
−1.52769
−2.52911
1.97995

−4.38695

4.03359

1.2

−3.15051

−1.33027

1.3

−1.34991

−1.48975

1.4

0.849522

−3.79865

1.5

2.84983

3.38611

1.6

3.18802

−2.80102

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$61$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8784.2.a.ca		6
3.b	odd	2	1		2928.2.a.bd		6
4.b	odd	2	1		549.2.a.h		6
12.b	even	2	1		183.2.a.c	✓	6
60.h	even	2	1		4575.2.a.s		6
84.h	odd	2	1		8967.2.a.y		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
183.2.a.c	✓	6	12.b	even	2	1
549.2.a.h		6	4.b	odd	2	1
2928.2.a.bd		6	3.b	odd	2	1
4575.2.a.s		6	60.h	even	2	1
8784.2.a.ca		6	1.a	even	1	1	trivial
8967.2.a.y		6	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8784))$:

$ T_{5}^{6} + 2T_{5}^{5} - 23T_{5}^{4} - 28T_{5}^{3} + 144T_{5}^{2} + 80T_{5} - 144 $

$ T_{7}^{6} + 2T_{7}^{5} - 25T_{7}^{4} - 60T_{7}^{3} + 128T_{7}^{2} + 432T_{7} + 288 $

$ T_{11}^{6} + 8T_{11}^{5} - 5T_{11}^{4} - 110T_{11}^{3} - 68T_{11}^{2} + 8T_{11} + 4 $

$ T_{13}^{6} - 6T_{13}^{5} - 23T_{13}^{4} + 116T_{13}^{3} + 168T_{13}^{2} - 464T_{13} - 608 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 2 T^{5} + \cdots - 144 $ T^6 + 2T^5 - 23T^4 - 28T^3 + 144T^2 + 80*T - 144
$7$	$ T^{6} + 2 T^{5} + \cdots + 288 $ T^6 + 2T^5 - 25T^4 - 60T^3 + 128T^2 + 432*T + 288
$11$	$ T^{6} + 8 T^{5} + \cdots + 4 $ T^6 + 8T^5 - 5T^4 - 110T^3 - 68T^2 + 8*T + 4
$13$	$ T^{6} - 6 T^{5} + \cdots - 608 $ T^6 - 6T^5 - 23T^4 + 116T^3 + 168T^2 - 464*T - 608
$17$	$ T^{6} + 10 T^{5} + \cdots + 5968 $ T^6 + 10T^5 - 12T^4 - 368T^3 - 684T^2 + 2352*T + 5968
$19$	$ T^{6} + 8 T^{5} + \cdots + 15808 $ T^6 + 8T^5 - 60T^4 - 656T^3 - 592T^2 + 7232*T + 15808
$23$	$ T^{6} - 45 T^{4} + \cdots - 204 $ T^6 - 45T^4 + 2T^3 + 420T^2 + 208T - 204
$29$	$ T^{6} - 10 T^{5} + \cdots + 10368 $ T^6 - 10T^5 - 56T^4 + 832T^3 - 1740T^2 - 4032*T + 10368
$31$	$ T^{6} - 108 T^{4} + \cdots + 7296 $ T^6 - 108T^4 - 256T^3 + 2016T^2 + 8128T + 7296
$37$	$ T^{6} + 4 T^{5} + \cdots - 7488 $ T^6 + 4T^5 - 124T^4 - 416T^3 + 2096T^2 + 5184*T - 7488
$41$	$ T^{6} - 10 T^{5} + \cdots + 2864 $ T^6 - 10T^5 - 55T^4 + 700T^3 - 144T^2 - 7568*T + 2864
$43$	$ T^{6} + 4 T^{5} + \cdots + 1152 $ T^6 + 4T^5 - 52T^4 - 176T^3 + 512T^2 + 1728*T + 1152
$47$	$ T^{6} + 4 T^{5} + \cdots - 9216 $ T^6 + 4T^5 - 104T^4 - 32T^3 + 2048T^2 - 9216
$53$	$ T^{6} - 2 T^{5} + \cdots - 22416 $ T^6 - 2T^5 - 172T^4 + 208T^3 + 4612T^2 - 5680*T - 22416
$59$	$ T^{6} + 16 T^{5} + \cdots - 4332 $ T^6 + 16T^5 + 35T^4 - 614T^3 - 3892T^2 - 7376*T - 4332
$61$	$ (T + 1)^{6} $ (T + 1)^6
$67$	$ T^{6} - 2 T^{5} + \cdots - 51104 $ T^6 - 2T^5 - 241T^4 + 64T^3 + 14152T^2 + 9008*T - 51104
$71$	$ T^{6} + 18 T^{5} + \cdots + 27952 $ T^6 + 18T^5 + 12T^4 - 984T^3 - 2292T^2 + 11632*T + 27952
$73$	$ T^{6} - 30 T^{5} + \cdots + 2192 $ T^6 - 30T^5 + 281T^4 - 664T^3 - 1784T^2 + 4480*T + 2192
$79$	$ T^{6} + 6 T^{5} + \cdots + 1632 $ T^6 + 6T^5 - 73T^4 - 720T^3 - 1848T^2 - 656*T + 1632
$83$	$ T^{6} + 12 T^{5} + \cdots - 228352 $ T^6 + 12T^5 - 264T^4 - 2592T^3 + 18944T^2 + 110080*T - 228352
$89$	$ T^{6} + 26 T^{5} + \cdots + 22144 $ T^6 + 26T^5 + 16T^4 - 3584T^3 - 20188T^2 + 4672*T + 22144
$97$	$ T^{6} - 16 T^{5} + \cdots + 16768 $ T^6 - 16T^5 - 180T^4 + 2208T^3 + 4864T^2 - 22976*T + 16768
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8784 = 2^{4} \cdot 3^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8784.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{5} - \beta_{5} q^{7}+O(q^{10}) \) q + b3 * q^5 - b5 * q^7	\( q + \beta_{3} q^{5} - \beta_{5} q^{7} + (\beta_{5} - \beta_{4} - 2) q^{11} + (\beta_{4} - \beta_{2} + 1) q^{13} + (\beta_{5} - \beta_{2} - \beta_1 - 2) q^{17} + (\beta_{4} - \beta_{3} + \beta_{2} - 1) q^{19} + (\beta_{2} + \beta_1) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 3) q^{25}+ \cdots + ( - \beta_{5} + 3 \beta_{3} - 3 \beta_1 + 5) q^{97}+O(q^{100}) \) q + b3 * q^5 - b5 * q^7 + (b5 - b4 - 2) * q^11 + (b4 - b2 + 1) * q^13 + (b5 - b2 - b1 - 2) * q^17 + (b4 - b3 + b2 - 1) * q^19 + (b2 + b1) * q^23 + (-b5 - b4 - b3 + b2 + b1 + 3) * q^25 + (-b4 + 2b2 + 2) q^29 + (-b4 - b3 - b2 - 1) * q^31 + (-b5 + 3b4 + b3 - b2 - 1) q^35 - 2b1 q^37 + (b5 - b4 - b3 - b2 + b1) * q^41 + (b4 - b3 - b2 - 1) * q^43 + (b5 - b4 - 2b3 + b2 + b1 - 2) q^47 + (-b5 - b4 + b3 + b2 - b1 + 3) * q^49 + (-b5 + 2b4 + 2b3 - b2 - b1 + 2) * q^53 + (3b5 - b4 - 3b3 + b2 + 2b1 - 1) q^55 + (b2 - b1 - 2) * q^59 - q^61 + (-3b5 - b4 + b3 + b2 - b1 + 2) q^65 + (b5 + 2b2 + 2b1) * q^67 + (2b4 - b3 - b2 - 3) q^71 + (b5 + b4 - b3 - b2 + b1 + 4) * q^73 + (2b5 - 3b3 - 6) * q^77 + (b5 - 2b2 - 2) q^79 + (b5 - 3b4 - b2 + b1 - 4) q^83 + (-4b3 + 2b2 - 2) * q^85 + (-2b5 + b4 - 2b3 + 2b2 + 2b1 - 4) * q^89 + (-b5 + b4 + 3b3 + b2 + 1) q^91 + (-2b4 - 2b2 - 4b1 - 6) q^95 + (-b5 + 3b3 - 3b1 + 5) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 6 * q - 2 * q^5 - 2 * q^7	\( 6 q - 2 q^{5} - 2 q^{7} - 8 q^{11} + 6 q^{13} - 10 q^{17} - 8 q^{19} + 20 q^{25} + 10 q^{29} - 14 q^{35} - 4 q^{37} + 10 q^{41} - 4 q^{43} - 4 q^{47} + 12 q^{49} + 2 q^{53} + 10 q^{55} - 16 q^{59} - 6 q^{61} + 2 q^{65} + 2 q^{67} - 18 q^{71} + 30 q^{73} - 26 q^{77} - 6 q^{79} - 12 q^{83} - 8 q^{85} - 26 q^{89} - 6 q^{91} - 36 q^{95} + 16 q^{97}+O(q^{100}) \) 6 * q - 2 * q^5 - 2 * q^7 - 8 * q^11 + 6 * q^13 - 10 * q^17 - 8 * q^19 + 20 * q^25 + 10 * q^29 - 14 * q^35 - 4 * q^37 + 10 * q^41 - 4 * q^43 - 4 * q^47 + 12 * q^49 + 2 * q^53 + 10 * q^55 - 16 * q^59 - 6 * q^61 + 2 * q^65 + 2 * q^67 - 18 * q^71 + 30 * q^73 - 26 * q^77 - 6 * q^79 - 12 * q^83 - 8 * q^85 - 26 * q^89 - 6 * q^91 - 36 * q^95 + 16 * q^97

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