LMFDB - Newform orbit 6664.2.a.t

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Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.15751800.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 7x - 1 $ x^6 - x^5 - 8x^4 + 6x^3 + 16x^2 - 7x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{2} q^{3} + ( - \beta_{4} + \beta_{2}) q^{5} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{4} - \beta_{3} - 1) q^{13} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 2) q^{15}+ \cdots + (\beta_{5} - 4 \beta_{4} - 2 \beta_{3} + \cdots + 1) q^{99}+O(q^{100}) $ q - b2 * q^3 + (-b4 + b2) * q^5 + (-b5 + b4 - b2 - b1) * q^9 + (-b5 + b4 + b3) * q^11 + (-b4 - b3 - 1) * q^13 + (-b4 + 2b2 + b1 - 2) q^15 + q^17 + (2b5 - b4 - b3 + b1 + 1) q^19 + (b4 - b3 + 2b2 + b1 - 1) q^23 + (b5 + 2b4 - 3b2) * q^25 + (b4 - 2b2 - 2b1 + 2) * q^27 + (b5 + b4 + b3 + 2b2) q^29 + (b5 + b3 - b2 - b1 - 2) * q^31 + (-b4 + b3 + b1 - 1) * q^33 + (b5 + 2b3 - 1) q^37 + (-b1 + 2) * q^39 + (b5 + b4 - b3 + b2 + b1 - 4) * q^41 + (-b5 + b2 - 2b1 + 1) q^43 + (b5 + b3 + 4b2 + 3b1 - 4) * q^45 + (2b5 - 2b4 + 2b2 + b1 + 2) q^47 - b2 * q^51 + (-b5 + b4 - 3b2 + b1 - 1) q^53 + (-2b5 + b4 + b3 - 2b2 - 2b1 - 1) q^55 + (b4 - b3 + 2b2 + 1) q^57 + (-2b5 - 2b3 + 2b2 - 4) q^59 + (2b4 - 2b3 + b2) * q^61 + (2b5 + b4 - b3 + 3b1 + 1) * q^65 + (b5 + b4 - b2 + b1 - 1) * q^67 + (4b5 - 3b4 + b3 + 2b2 + 2b1 - 5) * q^69 + (-3b4 + b3 + 2b2 - b1 + 1) * q^71 + (b5 - b4 + b3 - b2 - b1 - 2) * q^73 + (-b5 + 4b4 - b3 - 4b2 - 3b1 + 6) q^75 + (2b4 - 3b1 + 2) * q^79 + (2b5 + b4 - 2b3 - 6b2 - b1 + 3) q^81 + (b4 - b3 - 2b2 + 2b1 - 3) * q^83 + (-b4 + b2) * q^85 + (2b5 - b4 - b3 + 4b2 + 3b1 - 9) q^87 + (-2b5 + 2b4 + 2b3 + b1 - 2) q^89 + (-2b5 + 3b4 - 2b3 + 2b2 - b1) * q^93 + (-4b4 - 2b3 + 2b2 - b1) q^95 + (-3b5 + 3b3 - b2 - b1 - 6) * q^97 + (b5 - 4b4 - 2b3 + 6b2 + 2b1 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 4 q^{5} + 4 q^{9} + 2 q^{11} - 6 q^{13} - 16 q^{15} + 6 q^{17} + 4 q^{19} - 4 q^{23} + 8 q^{25} + 14 q^{27} - 6 q^{29} - 16 q^{31} - 8 q^{33} - 12 q^{37} + 10 q^{39} - 22 q^{41} + 2 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 4 * q^5 + 4 * q^9 + 2 * q^11 - 6 * q^13 - 16 * q^15 + 6 * q^17 + 4 * q^19 - 4 * q^23 + 8 * q^25 + 14 * q^27 - 6 * q^29 - 16 * q^31 - 8 * q^33 - 12 * q^37 + 10 * q^39 - 22 * q^41 + 2 * q^43 - 30 * q^45 + 2 * q^47 + 2 * q^51 + 6 * q^53 - 2 * q^55 + 6 * q^57 - 20 * q^59 + 6 * q^61 + 12 * q^65 - 2 * q^67 - 46 * q^69 - 8 * q^71 - 18 * q^73 + 50 * q^75 + 10 * q^79 + 30 * q^81 - 6 * q^83 - 4 * q^85 - 60 * q^87 - 6 * q^89 + 8 * q^93 - 10 * q^95 - 36 * q^97 - 8 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 7x - 1 $ x^6 - x^5 - 8*x^4 + 6*x^3 + 16*x^2 - 7*x - 1 :

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

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

$\nu$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{4} + \beta_{3} - 2\beta_{2} + 5 ) / 2 $ (b4 + b3 - 2*b2 + 5) / 2
$\nu^{3}$	$=$	$ ( \beta_{4} + \beta_{3} + 4\beta _1 + 1 ) / 2 $ (b4 + b3 + 4*b1 + 1) / 2
$\nu^{4}$	$=$	$ ( 6\beta_{4} + 4\beta_{3} - 10\beta_{2} + \beta _1 + 20 ) / 2 $ (6b4 + 4b3 - 10*b2 + b1 + 20) / 2
$\nu^{5}$	$=$	$ \beta_{5} + 3\beta_{4} + 3\beta_{3} + 9\beta _1 + 5 $ b5 + 3b4 + 3b3 + 9*b1 + 5

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

−0.114413
−1.66208
0.527344
2.27955
2.07639
−2.10679

−2.44306

−0.0622230

2.96856

1.2

−1.29428

1.75649

−1.32483

1.3

0.240816

1.56237

−2.94201

1.4

0.469186

−3.93720

−2.77986

1.5

1.66484

0.733917

−0.228318

1.6

3.36251

−4.05336

8.30646

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$7$	$ -1 $
$17$	$ -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	6664.2.a.t	yes	6
7.b	odd	2	1		6664.2.a.q	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6664.2.a.q	✓	6	7.b	odd	2	1
6664.2.a.t	yes	6	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6664))$:

$ T_{3}^{6} - 2T_{3}^{5} - 9T_{3}^{4} + 12T_{3}^{3} + 13T_{3}^{2} - 12T_{3} + 2 $

T3^6 - 2*T3^5 - 9*T3^4 + 12*T3^3 + 13*T3^2 - 12*T3 + 2

$ T_{5}^{6} + 4T_{5}^{5} - 11T_{5}^{4} - 26T_{5}^{3} + 65T_{5}^{2} - 28T_{5} - 2 $

T5^6 + 4*T5^5 - 11*T5^4 - 26*T5^3 + 65*T5^2 - 28*T5 - 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 2 T^{5} + \cdots + 2 $ T^6 - 2T^5 - 9T^4 + 12T^3 + 13T^2 - 12*T + 2
$5$	$ T^{6} + 4 T^{5} + \cdots - 2 $ T^6 + 4T^5 - 11T^4 - 26T^3 + 65T^2 - 28*T - 2
$7$	$ T^{6} $ T^6
$11$	$ T^{6} - 2 T^{5} + \cdots + 96 $ T^6 - 2T^5 - 34T^4 + 76T^3 + 192T^2 - 480*T + 96
$13$	$ T^{6} + 6 T^{5} + \cdots + 192 $ T^6 + 6T^5 - 16T^4 - 112T^3 + 288T + 192
$17$	$ (T - 1)^{6} $ (T - 1)^6
$19$	$ T^{6} - 4 T^{5} + \cdots - 128 $ T^6 - 4T^5 - 56T^4 + 104T^3 + 848T^2 + 64*T - 128
$23$	$ T^{6} + 4 T^{5} + \cdots - 9344 $ T^6 + 4T^5 - 104T^4 - 440T^3 + 2384T^2 + 8768*T - 9344
$29$	$ T^{6} + 6 T^{5} + \cdots - 38880 $ T^6 + 6T^5 - 126T^4 - 612T^3 + 4320T^2 + 12960*T - 38880
$31$	$ T^{6} + 16 T^{5} + \cdots + 13136 $ T^6 + 16T^5 + 19T^4 - 714T^3 - 2897T^2 + 1624*T + 13136
$37$	$ T^{6} + 12 T^{5} + \cdots + 3872 $ T^6 + 12T^5 - 70T^4 - 1152T^3 - 1008T^2 + 13904*T + 3872
$41$	$ T^{6} + 22 T^{5} + \cdots + 2764 $ T^6 + 22T^5 + 113T^4 - 364T^3 - 3305T^2 - 2140*T + 2764
$43$	$ T^{6} - 2 T^{5} + \cdots - 18856 $ T^6 - 2T^5 - 125T^4 + 450T^3 + 2845T^2 - 8312*T - 18856
$47$	$ T^{6} - 2 T^{5} + \cdots - 10688 $ T^6 - 2T^5 - 120T^4 + 352T^3 + 2432T^2 - 3744*T - 10688
$53$	$ T^{6} - 6 T^{5} + \cdots - 19300 $ T^6 - 6T^5 - 145T^4 + 954T^3 + 2265T^2 - 13300*T - 19300
$59$	$ T^{6} + 20 T^{5} + \cdots + 24832 $ T^6 + 20T^5 - 44T^4 - 2560T^3 - 11536T^2 - 6272*T + 24832
$61$	$ T^{6} - 6 T^{5} + \cdots + 55758 $ T^6 - 6T^5 - 181T^4 + 1064T^3 + 4725T^2 - 38088*T + 55758
$67$	$ T^{6} + 2 T^{5} + \cdots - 376 $ T^6 + 2T^5 - 61T^4 - 390T^3 - 931T^2 - 976*T - 376
$71$	$ T^{6} + 8 T^{5} + \cdots - 28032 $ T^6 + 8T^5 - 176T^4 - 1112T^3 + 3792T^2 + 13632*T - 28032
$73$	$ T^{6} + 18 T^{5} + \cdots + 10804 $ T^6 + 18T^5 + 29T^4 - 832T^3 - 4005T^2 - 1732*T + 10804
$79$	$ T^{6} - 10 T^{5} + \cdots - 470144 $ T^6 - 10T^5 - 308T^4 + 2696T^3 + 18896T^2 - 95552*T - 470144
$83$	$ T^{6} + 6 T^{5} + \cdots - 768 $ T^6 + 6T^5 - 172T^4 + 40T^3 + 1872T^2 - 768*T - 768
$89$	$ T^{6} + 6 T^{5} + \cdots - 168192 $ T^6 + 6T^5 - 232T^4 - 1088T^3 + 13776T^2 + 31104*T - 168192
$97$	$ T^{6} + 36 T^{5} + \cdots - 431700 $ T^6 + 36T^5 + 243T^4 - 4706T^3 - 76185T^2 - 352500*T - 431700
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Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{3} + ( - \beta_{4} + \beta_{2}) q^{5} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{9} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{4} - \beta_{3} - 1) q^{13} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 2) q^{15}+ \cdots + (\beta_{5} - 4 \beta_{4} - 2 \beta_{3} + \cdots + 1) q^{99}+O(q^{100}) \) q - b2 * q^3 + (-b4 + b2) * q^5 + (-b5 + b4 - b2 - b1) * q^9 + (-b5 + b4 + b3) * q^11 + (-b4 - b3 - 1) * q^13 + (-b4 + 2b2 + b1 - 2) q^15 + q^17 + (2b5 - b4 - b3 + b1 + 1) q^19 + (b4 - b3 + 2b2 + b1 - 1) q^23 + (b5 + 2b4 - 3b2) * q^25 + (b4 - 2b2 - 2b1 + 2) * q^27 + (b5 + b4 + b3 + 2b2) q^29 + (b5 + b3 - b2 - b1 - 2) * q^31 + (-b4 + b3 + b1 - 1) * q^33 + (b5 + 2b3 - 1) q^37 + (-b1 + 2) * q^39 + (b5 + b4 - b3 + b2 + b1 - 4) * q^41 + (-b5 + b2 - 2b1 + 1) q^43 + (b5 + b3 + 4b2 + 3b1 - 4) * q^45 + (2b5 - 2b4 + 2b2 + b1 + 2) q^47 - b2 * q^51 + (-b5 + b4 - 3b2 + b1 - 1) q^53 + (-2b5 + b4 + b3 - 2b2 - 2b1 - 1) q^55 + (b4 - b3 + 2b2 + 1) q^57 + (-2b5 - 2b3 + 2b2 - 4) q^59 + (2b4 - 2b3 + b2) * q^61 + (2b5 + b4 - b3 + 3b1 + 1) * q^65 + (b5 + b4 - b2 + b1 - 1) * q^67 + (4b5 - 3b4 + b3 + 2b2 + 2b1 - 5) * q^69 + (-3b4 + b3 + 2b2 - b1 + 1) * q^71 + (b5 - b4 + b3 - b2 - b1 - 2) * q^73 + (-b5 + 4b4 - b3 - 4b2 - 3b1 + 6) q^75 + (2b4 - 3b1 + 2) * q^79 + (2b5 + b4 - 2b3 - 6b2 - b1 + 3) q^81 + (b4 - b3 - 2b2 + 2b1 - 3) * q^83 + (-b4 + b2) * q^85 + (2b5 - b4 - b3 + 4b2 + 3b1 - 9) q^87 + (-2b5 + 2b4 + 2b3 + b1 - 2) q^89 + (-2b5 + 3b4 - 2b3 + 2b2 - b1) * q^93 + (-4b4 - 2b3 + 2b2 - b1) q^95 + (-3b5 + 3b3 - b2 - b1 - 6) * q^97 + (b5 - 4b4 - 2b3 + 6b2 + 2b1 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 4 q^{5} + 4 q^{9} + 2 q^{11} - 6 q^{13} - 16 q^{15} + 6 q^{17} + 4 q^{19} - 4 q^{23} + 8 q^{25} + 14 q^{27} - 6 q^{29} - 16 q^{31} - 8 q^{33} - 12 q^{37} + 10 q^{39} - 22 q^{41} + 2 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 4 * q^5 + 4 * q^9 + 2 * q^11 - 6 * q^13 - 16 * q^15 + 6 * q^17 + 4 * q^19 - 4 * q^23 + 8 * q^25 + 14 * q^27 - 6 * q^29 - 16 * q^31 - 8 * q^33 - 12 * q^37 + 10 * q^39 - 22 * q^41 + 2 * q^43 - 30 * q^45 + 2 * q^47 + 2 * q^51 + 6 * q^53 - 2 * q^55 + 6 * q^57 - 20 * q^59 + 6 * q^61 + 12 * q^65 - 2 * q^67 - 46 * q^69 - 8 * q^71 - 18 * q^73 + 50 * q^75 + 10 * q^79 + 30 * q^81 - 6 * q^83 - 4 * q^85 - 60 * q^87 - 6 * q^89 + 8 * q^93 - 10 * q^95 - 36 * q^97 - 8 * q^99

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