LMFDB - Newform orbit 7128.2.a.w

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7128,2,Mod(1,7128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7128.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7128 = 2^{3} \cdot 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7128.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-4,0,5,0,0,0,-6] 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:	yes
Analytic conductor:	$56.9173665608$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.30796308.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - x^{3} + 10x^{2} + 2x - 2 $ x^6 - 7x^4 - x^3 + 10x^2 + 2*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 792)
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_1 - 1) q^{5} + ( - \beta_{4} + 1) q^{7} - q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{5} - 1) q^{17} + (\beta_{4} - \beta_1 + 1) q^{19} + (\beta_{5} + \beta_{3} - \beta_{2} + \cdots - 1) q^{23}+ \cdots + (2 \beta_{5} + 4 \beta_{3} - 2 \beta_1 - 1) q^{97}+O(q^{100}) $ q + (b1 - 1) * q^5 + (-b4 + 1) * q^7 - q^11 + (-b3 + b2 - b1) * q^13 + (-b5 - 1) * q^17 + (b4 - b1 + 1) * q^19 + (b5 + b3 - b2 + b1 - 1) * q^23 + b3 * q^25 + (2b5 + b3 - 2b2 - b1) * q^29 + (b4 - b3 + b2 - b1) * q^31 + (-b5 + b4 - b3 + b1 - 2) * q^35 + (-b5 + b3 - 2b2 + 2) q^37 + (b5 + 2b4 + b2 - 1) q^41 + (b4 + b3 + 1) * q^43 + (-3b5 + b4 + b2 - 3) q^47 + (2b5 - b3 - b2 - 2b1 + 1) * q^49 + (-3b5 + b4 - b3 + 2b2 - b1 - 3) * q^53 + (-b1 + 1) * q^55 + (3b5 + b4 + 3b3 - 2b2 - b1 + 1) q^59 + (2b5 + 2b3 + b2) * q^61 + (b5 + 2b3 - 2b2 - 1) * q^65 + (-3b4 - b3 - b2) q^67 + (b5 + b4 - 3b3 + b2 - 3) q^71 + (-4b5 - 2b4 - 2b3 + b1 - 2) q^73 + (b4 - 1) * q^77 + (2b5 + 2b4 + 2b3 - 3b2 - b1 + 2) * q^79 + (-3b5 + 3b2 + b1 - 5) * q^83 + (b5 - b4 - b3 - b2 - b1) * q^85 + (b5 - b4 - 3b3 - 2b1 - 4) * q^89 + (b5 - b4 - 3b3 + 3b2 - 2b1) q^91 + (b5 - b4 - 4) * q^95 + (2b5 + 4b3 - 2b1 - 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 5 q^{7} - 6 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - 8 q^{29} + 4 q^{31} - 8 q^{35} + 3 q^{37} + 5 q^{43} - 17 q^{47} + 3 q^{49} - 14 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) $ 6 * q - 4 * q^5 + 5 * q^7 - 6 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - 8 * q^29 + 4 * q^31 - 8 * q^35 + 3 * q^37 + 5 * q^43 - 17 * q^47 + 3 * q^49 - 14 * q^53 + 4 * q^55 - 4 * q^59 + q^61 - 15 * q^65 - 4 * q^67 - 7 * q^71 - 12 * q^73 - 5 * q^77 + q^79 - 22 * q^83 - 3 * q^85 - 22 * q^89 + 11 * q^91 - 24 * q^95 - 16 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 7x^{4} - x^{3} + 10x^{2} + 2x - 2 $ x^6 - 7*x^4 - x^3 + 10*x^2 + 2*x - 2 :

$\beta_{1}$	$=$	$ \nu^{2} - 2 $ v^2 - 2
$\beta_{2}$	$=$	$ \nu^{3} - 4\nu $ v^3 - 4*v
$\beta_{3}$	$=$	$ \nu^{4} - 6\nu^{2} + 4 $ v^4 - 6*v^2 + 4
$\beta_{4}$	$=$	$ -\nu^{5} + 6\nu^{3} + \nu^{2} - 4\nu - 1 $ -v^5 + 6v^3 + v^2 - 4v - 1
$\beta_{5}$	$=$	$ \nu^{5} - \nu^{4} - 6\nu^{3} + 5\nu^{2} + 6\nu - 3 $ v^5 - v^4 - 6v^3 + 5v^2 + 6*v - 3

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

$\nu$	$=$	$ ( \beta_{5} + \beta_{4} + \beta_{3} ) / 2 $ (b5 + b4 + b3) / 2
$\nu^{2}$	$=$	$ \beta _1 + 2 $ b1 + 2
$\nu^{3}$	$=$	$ 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} $ 2b5 + 2b4 + 2*b3 + b2
$\nu^{4}$	$=$	$ \beta_{3} + 6\beta _1 + 8 $ b3 + 6*b1 + 8
$\nu^{5}$	$=$	$ 10\beta_{5} + 9\beta_{4} + 10\beta_{3} + 6\beta_{2} + \beta _1 + 1 $ 10b5 + 9b4 + 10b3 + 6b2 + b1 + 1

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

0.379195
−0.645179
−1.22634
1.30831
−2.16714
2.35114

−2.85621

3.05369

1.2

−2.58374

0.502596

1.3

−1.49610

3.88286

1.4

−1.28831

−4.08178

1.5

1.69649

1.90201

1.6

2.52788

−0.259373

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$11$	$ +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	7128.2.a.w		6
3.b	odd	2	1		7128.2.a.ba		6
9.c	even	3	2		2376.2.q.f		12
9.d	odd	6	2		792.2.q.f	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
792.2.q.f	✓	12	9.d	odd	6	2
2376.2.q.f		12	9.c	even	3	2
7128.2.a.w		6	1.a	even	1	1	trivial
7128.2.a.ba		6	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7128))$:

$ T_{5}^{6} + 4T_{5}^{5} - 6T_{5}^{4} - 37T_{5}^{3} - 12T_{5}^{2} + 73T_{5} + 61 $

T5^6 + 4*T5^5 - 6*T5^4 - 37*T5^3 - 12*T5^2 + 73*T5 + 61

$ T_{7}^{6} - 5T_{7}^{5} - 10T_{7}^{4} + 83T_{7}^{3} - 110T_{7}^{2} + 12T_{7} + 12 $

T7^6 - 5*T7^5 - 10*T7^4 + 83*T7^3 - 110*T7^2 + 12*T7 + 12

$ T_{13}^{6} - 3T_{13}^{5} - 25T_{13}^{4} + 7T_{13}^{3} + 112T_{13}^{2} + 40T_{13} - 68 $

T13^6 - 3*T13^5 - 25*T13^4 + 7*T13^3 + 112*T13^2 + 40*T13 - 68

$ T_{17}^{6} + 7T_{17}^{5} + 6T_{17}^{4} - 41T_{17}^{3} - 74T_{17}^{2} - 12T_{17} + 12 $

T17^6 + 7*T17^5 + 6*T17^4 - 41*T17^3 - 74*T17^2 - 12*T17 + 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 4 T^{5} + \cdots + 61 $ T^6 + 4T^5 - 6T^4 - 37T^3 - 12T^2 + 73*T + 61
$7$	$ T^{6} - 5 T^{5} + \cdots + 12 $ T^6 - 5T^5 - 10T^4 + 83T^3 - 110T^2 + 12*T + 12
$11$	$ (T + 1)^{6} $ (T + 1)^6
$13$	$ T^{6} - 3 T^{5} + \cdots - 68 $ T^6 - 3T^5 - 25T^4 + 7T^3 + 112T^2 + 40*T - 68
$17$	$ T^{6} + 7 T^{5} + \cdots + 12 $ T^6 + 7T^5 + 6T^4 - 41T^3 - 74T^2 - 12*T + 12
$19$	$ T^{6} - 5 T^{5} + \cdots - 72 $ T^6 - 5T^5 - 18T^4 + 43T^3 + 106T^2 - 12*T - 72
$23$	$ T^{6} + 8 T^{5} + \cdots + 16 $ T^6 + 8T^5 - 3T^4 - 77T^3 + 24T^2 + 80*T + 16
$29$	$ T^{6} + 8 T^{5} + \cdots + 13012 $ T^6 + 8T^5 - 65T^4 - 583T^3 + 356T^2 + 8928*T + 13012
$31$	$ T^{6} - 4 T^{5} + \cdots - 51 $ T^6 - 4T^5 - 34T^4 + 121T^3 + 232T^2 - 753*T - 51
$37$	$ T^{6} - 3 T^{5} + \cdots + 19819 $ T^6 - 3T^5 - 118T^4 + 523T^3 + 2206T^2 - 14840*T + 19819
$41$	$ T^{6} - 91 T^{4} + \cdots + 1164 $ T^6 - 91T^4 - 69T^3 + 2044T^2 + 3324T + 1164
$43$	$ T^{6} - 5 T^{5} + \cdots + 628 $ T^6 - 5T^5 - 18T^4 + 131T^3 - 96T^2 - 452*T + 628
$47$	$ T^{6} + 17 T^{5} + \cdots - 15623 $ T^6 + 17T^5 - 16T^4 - 1531T^3 - 9358T^2 - 21178*T - 15623
$53$	$ T^{6} + 14 T^{5} + \cdots - 36261 $ T^6 + 14T^5 - 58T^4 - 2021T^3 - 13394T^2 - 36549*T - 36261
$59$	$ T^{6} + 4 T^{5} + \cdots - 48021 $ T^6 + 4T^5 - 172T^4 - 841T^3 + 5296T^2 + 19689*T - 48021
$61$	$ T^{6} - T^{5} + \cdots - 14336 $ T^6 - T^5 - 138T^4 + 73T^3 + 4224T^2 - 6400T - 14336
$67$	$ T^{6} + 4 T^{5} + \cdots - 101081 $ T^6 + 4T^5 - 150T^4 - 415T^3 + 7062T^2 + 9367*T - 101081
$71$	$ T^{6} + 7 T^{5} + \cdots + 1161 $ T^6 + 7T^5 - 234T^4 - 629T^3 + 11932T^2 + 7578*T + 1161
$73$	$ T^{6} + 12 T^{5} + \cdots + 146764 $ T^6 + 12T^5 - 151T^4 - 2137T^3 + 34T^2 + 59336*T + 146764
$79$	$ T^{6} - T^{5} + \cdots + 155404 $ T^6 - T^5 - 284T^4 - 889T^3 + 15710T^2 + 99972T + 155404
$83$	$ T^{6} + 22 T^{5} + \cdots + 99552 $ T^6 + 22T^5 - 37T^4 - 2749T^3 - 10076T^2 + 25968*T + 99552
$89$	$ T^{6} + 22 T^{5} + \cdots + 106432 $ T^6 + 22T^5 - 16T^4 - 2888T^3 - 15520T^2 + 8512*T + 106432
$97$	$ T^{6} + 16 T^{5} + \cdots - 45113 $ T^6 + 16T^5 - 219T^4 - 3520T^3 - 237T^2 + 38320*T - 45113
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Level:	\( N \)	\(=\)	\( 7128 = 2^{3} \cdot 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7128.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta_1 - 1) q^{5} + ( - \beta_{4} + 1) q^{7} - q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{5} - 1) q^{17} + (\beta_{4} - \beta_1 + 1) q^{19} + (\beta_{5} + \beta_{3} - \beta_{2} + \cdots - 1) q^{23}+ \cdots + (2 \beta_{5} + 4 \beta_{3} - 2 \beta_1 - 1) q^{97}+O(q^{100}) \) q + (b1 - 1) * q^5 + (-b4 + 1) * q^7 - q^11 + (-b3 + b2 - b1) * q^13 + (-b5 - 1) * q^17 + (b4 - b1 + 1) * q^19 + (b5 + b3 - b2 + b1 - 1) * q^23 + b3 * q^25 + (2b5 + b3 - 2b2 - b1) * q^29 + (b4 - b3 + b2 - b1) * q^31 + (-b5 + b4 - b3 + b1 - 2) * q^35 + (-b5 + b3 - 2b2 + 2) q^37 + (b5 + 2b4 + b2 - 1) q^41 + (b4 + b3 + 1) * q^43 + (-3b5 + b4 + b2 - 3) q^47 + (2b5 - b3 - b2 - 2b1 + 1) * q^49 + (-3b5 + b4 - b3 + 2b2 - b1 - 3) * q^53 + (-b1 + 1) * q^55 + (3b5 + b4 + 3b3 - 2b2 - b1 + 1) q^59 + (2b5 + 2b3 + b2) * q^61 + (b5 + 2b3 - 2b2 - 1) * q^65 + (-3b4 - b3 - b2) q^67 + (b5 + b4 - 3b3 + b2 - 3) q^71 + (-4b5 - 2b4 - 2b3 + b1 - 2) q^73 + (b4 - 1) * q^77 + (2b5 + 2b4 + 2b3 - 3b2 - b1 + 2) * q^79 + (-3b5 + 3b2 + b1 - 5) * q^83 + (b5 - b4 - b3 - b2 - b1) * q^85 + (b5 - b4 - 3b3 - 2b1 - 4) * q^89 + (b5 - b4 - 3b3 + 3b2 - 2b1) q^91 + (b5 - b4 - 4) * q^95 + (2b5 + 4b3 - 2b1 - 1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 5 q^{7} - 6 q^{11} + 3 q^{13} - 7 q^{17} + 5 q^{19} - 8 q^{23} - 2 q^{25} - 8 q^{29} + 4 q^{31} - 8 q^{35} + 3 q^{37} + 5 q^{43} - 17 q^{47} + 3 q^{49} - 14 q^{53} + 4 q^{55} - 4 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) 6 * q - 4 * q^5 + 5 * q^7 - 6 * q^11 + 3 * q^13 - 7 * q^17 + 5 * q^19 - 8 * q^23 - 2 * q^25 - 8 * q^29 + 4 * q^31 - 8 * q^35 + 3 * q^37 + 5 * q^43 - 17 * q^47 + 3 * q^49 - 14 * q^53 + 4 * q^55 - 4 * q^59 + q^61 - 15 * q^65 - 4 * q^67 - 7 * q^71 - 12 * q^73 - 5 * q^77 + q^79 - 22 * q^83 - 3 * q^85 - 22 * q^89 + 11 * q^91 - 24 * q^95 - 16 * q^97

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