LMFDB - Newform orbit 6664.2.a.r

Show commands: Magma / Pari/GP / SageMath

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,-1,0,-1] 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:	6.6.30091192.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 10x^{3} + 18x^{2} - 23x + 2 $ x^6 - x^5 - 9x^4 + 10x^3 + 18x^2 - 23x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
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 q^{3} + \beta_{4} q^{5} + \beta_{2} q^{9} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{4} - \beta_{2}) q^{15} + q^{17} + (\beta_{5} + 2 \beta_{3} + \cdots + \beta_1) q^{19}+ \cdots + (\beta_{4} + 2 \beta_{3} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) $ q - b1 * q^3 + b4 * q^5 + b2 * q^9 + (b4 + b3 + b1 - 1) * q^11 + (-b4 - b3 + b2) * q^13 + (-b4 - b2) * q^15 + q^17 + (b5 + 2b3 - b2 + b1) q^19 + (-b5 - b3 - b1 - 1) * q^23 + (2b5 - b2 + 1) q^25 + (-b4 - b3 + b1 + 1) * q^27 + (-2b5 - b3 - b2 - 3) q^29 + (-3b5 - b3 + b2 + 3b1 - 3) * q^31 + (-b5 - b4 + b3 - 3b2 + b1 - 2) q^33 + (b5 - b4 - 3b1 - 2) q^37 + (b5 - 2b3 + 2b2 - 2b1) q^39 + (2b5 - b4 + 2b3 - b2) * q^41 + (b5 - 2b4 - 2b3 + b2 - b1 + 4) * q^43 + (-b4 + b3 + b2 + 2b1 - 1) q^45 + (2b5 - 3b3 + b2 - 2b1 + 1) q^47 - b1 * q^51 + (-2b5 - b3 - 2b2 + 2b1 - 6) q^53 + (b5 - b4 + b2 + b1 + 4) * q^55 + (-3b5 + b4 + 2b3 - 3b2 + 3b1 - 4) * q^57 + (b4 - b3 - b2 + 2b1 + 1) q^59 + (-3b5 - 3b4 + b3 - 2b2 + b1 - 3) q^61 + (-b5 + b3 + b2 + b1 - 5) * q^65 + (4b5 + b4 + 2b2 - 2b1 + 6) q^67 + (2b5 + 2b2 + 4) * q^69 + (-b4 - 2b2 + b1 - 2) q^71 + (-3b5 - b4 + 2b3 - b2 + 3b1 - 4) q^73 + (-2b5 + b4 - b3 + 3b1 - 5) * q^75 + (b5 + 3b3 - b2 - 3) q^79 + (b5 + b4 - b3 - 2b2 - b1 - 4) q^81 + (-2b5 - b4 + 2b3 - b2 + 4b1 + 2) q^83 + b4 * q^85 + (3b5 + b4 + 2b3 + b2 + 3b1 + 2) q^87 + (2b5 - 2b4 - b3 + b2 - 2b1 + 6) q^89 + (4b5 - b4 + b3 - 2b2 - 2b1 - 3) q^93 + (-4b5 + b4 - 2b3 + 2b2 - 4) q^95 + (4b5 + b4 + 3b3 + 3) * q^97 + (b4 + 2b3 - b2 + 4b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} + q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} - 5 q^{23} + q^{25} + 8 q^{27} - 15 q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{37} - 2 q^{39} - 4 q^{41} + 24 q^{43} - 2 q^{45} + q^{47}+ \cdots + 2 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 + q^9 - 6 * q^11 + 2 * q^13 + 6 * q^17 - 2 * q^19 - 5 * q^23 + q^25 + 8 * q^27 - 15 * q^29 - 8 * q^31 - 11 * q^33 - 16 * q^37 - 2 * q^39 - 4 * q^41 + 24 * q^43 - 2 * q^45 + q^47 - q^51 - 32 * q^53 + 25 * q^55 - 19 * q^57 + 6 * q^59 - 10 * q^61 - 26 * q^65 + 27 * q^67 + 22 * q^69 - 12 * q^71 - 15 * q^73 - 24 * q^75 - 21 * q^79 - 30 * q^81 + 20 * q^83 - q^85 + 9 * q^87 + 33 * q^89 - 29 * q^93 - 15 * q^95 + 9 * q^97 + 2 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 9x^{4} + 10x^{3} + 18x^{2} - 23x + 2 $ x^6 - x^5 - 9*x^4 + 10*x^3 + 18*x^2 - 23*x + 2 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} - 3 $ v^2 - 3
$\beta_{3}$	$=$	$ \nu^{5} - 8\nu^{3} + \nu^{2} + 13\nu - 4 $ v^5 - 8v^3 + v^2 + 13v - 4
$\beta_{4}$	$=$	$ -\nu^{5} + 9\nu^{3} - \nu^{2} - 18\nu + 5 $ -v^5 + 9v^3 - v^2 - 18v + 5
$\beta_{5}$	$=$	$ 2\nu^{5} + \nu^{4} - 17\nu^{3} - 5\nu^{2} + 32\nu - 2 $ 2v^5 + v^4 - 17v^3 - 5v^2 + 32v - 2

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{4} + \beta_{3} + 5\beta _1 - 1 $ b4 + b3 + 5*b1 - 1
$\nu^{4}$	$=$	$ \beta_{5} + \beta_{4} - \beta_{3} + 7\beta_{2} - \beta _1 + 14 $ b5 + b4 - b3 + 7*b2 - b1 + 14
$\nu^{5}$	$=$	$ 8\beta_{4} + 9\beta_{3} - \beta_{2} + 27\beta _1 - 7 $ 8b4 + 9b3 - b2 + 27*b1 - 7

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

2.36506
1.49411
1.33445
0.0942397
−1.83437
−2.45349

−2.36506

1.89992

2.59349

1.2

−1.49411

−1.55355

−0.767628

1.3

−1.33445

−3.64559

−1.21925

1.4

−0.0942397

3.30233

−2.99112

1.5

1.83437

−0.128742

0.364902

1.6

2.45349

−0.874364

3.01960

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.r		6
7.b	odd	2	1		6664.2.a.s		6
7.d	odd	6	2		952.2.q.c	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
952.2.q.c	✓	12	7.d	odd	6	2
6664.2.a.r		6	1.a	even	1	1	trivial
6664.2.a.s		6	7.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 $S_{2}^{\mathrm{new}}(\Gamma_0(6664))$:

$ T_{3}^{6} + T_{3}^{5} - 9T_{3}^{4} - 10T_{3}^{3} + 18T_{3}^{2} + 23T_{3} + 2 $

T3^6 + T3^5 - 9*T3^4 - 10*T3^3 + 18*T3^2 + 23*T3 + 2

$ T_{5}^{6} + T_{5}^{5} - 15T_{5}^{4} - 12T_{5}^{3} + 37T_{5}^{2} + 36T_{5} + 4 $

T5^6 + T5^5 - 15*T5^4 - 12*T5^3 + 37*T5^2 + 36*T5 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + T^{5} - 9 T^{4} + \cdots + 2 $ T^6 + T^5 - 9T^4 - 10T^3 + 18T^2 + 23T + 2
$5$	$ T^{6} + T^{5} - 15 T^{4} + \cdots + 4 $ T^6 + T^5 - 15T^4 - 12T^3 + 37T^2 + 36T + 4
$7$	$ T^{6} $ T^6
$11$	$ T^{6} + 6 T^{5} + \cdots + 64 $ T^6 + 6T^5 - 10T^4 - 95T^3 - 69T^2 + 157*T + 64
$13$	$ T^{6} - 2 T^{5} + \cdots + 54 $ T^6 - 2T^5 - 26T^4 + 47T^3 + 153T^2 - 315*T + 54
$17$	$ (T - 1)^{6} $ (T - 1)^6
$19$	$ T^{6} + 2 T^{5} + \cdots - 652 $ T^6 + 2T^5 - 60T^4 - 46T^3 + 567T^2 + 316*T - 652
$23$	$ T^{6} + 5 T^{5} + \cdots - 32 $ T^6 + 5T^5 - 18T^4 - 60T^3 + 112T^2 - 32
$29$	$ T^{6} + 15 T^{5} + \cdots + 968 $ T^6 + 15T^5 + 45T^4 - 187T^3 - 862T^2 - 308*T + 968
$31$	$ T^{6} + 8 T^{5} + \cdots + 37936 $ T^6 + 8T^5 - 101T^4 - 877T^3 + 1196T^2 + 20780*T + 37936
$37$	$ T^{6} + 16 T^{5} + \cdots - 1024 $ T^6 + 16T^5 + 19T^4 - 523T^3 - 1113T^2 + 2496*T - 1024
$41$	$ T^{6} + 4 T^{5} + \cdots + 10552 $ T^6 + 4T^5 - 105T^4 - 833T^3 - 1186T^2 + 4500*T + 10552
$43$	$ T^{6} - 24 T^{5} + \cdots + 1312 $ T^6 - 24T^5 + 134T^4 + 416T^3 - 3583T^2 + 960*T + 1312
$47$	$ T^{6} - T^{5} + \cdots - 11072 $ T^6 - T^5 - 161T^4 + 509T^3 + 3104T^2 - 2212T - 11072
$53$	$ T^{6} + 32 T^{5} + \cdots + 30206 $ T^6 + 32T^5 + 310T^4 + 391T^3 - 7671T^2 - 22245*T + 30206
$59$	$ T^{6} - 6 T^{5} + \cdots + 10384 $ T^6 - 6T^5 - 109T^4 + 551T^3 + 2513T^2 - 14232*T + 10384
$61$	$ T^{6} + 10 T^{5} + \cdots - 183104 $ T^6 + 10T^5 - 189T^4 - 1455T^3 + 11266T^2 + 45312*T - 183104
$67$	$ T^{6} - 27 T^{5} + \cdots + 56644 $ T^6 - 27T^5 + 151T^4 + 1572T^3 - 18345T^2 + 39508*T + 56644
$71$	$ T^{6} + 12 T^{5} + \cdots - 3107 $ T^6 + 12T^5 - 25T^4 - 508T^3 + 27T^2 + 4304*T - 3107
$73$	$ T^{6} + 15 T^{5} + \cdots + 15016 $ T^6 + 15T^5 - 80T^4 - 1359T^3 + 2214T^2 + 23756*T + 15016
$79$	$ T^{6} + 21 T^{5} + \cdots + 34616 $ T^6 + 21T^5 + 81T^4 - 816T^3 - 5538T^2 + 417*T + 34616
$83$	$ T^{6} - 20 T^{5} + \cdots + 245696 $ T^6 - 20T^5 - 9T^4 + 2183T^3 - 8868T^2 - 45712*T + 245696
$89$	$ T^{6} - 33 T^{5} + \cdots - 64 $ T^6 - 33T^5 + 327T^4 - 903T^3 + 704T^2 - 48*T - 64
$97$	$ T^{6} - 9 T^{5} + \cdots + 31328 $ T^6 - 9T^5 - 152T^4 + 1425T^3 + 606T^2 - 22080*T + 31328
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Elliptic curves over $\Q$ over number fields
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Belyi maps

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

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{2} q^{9} + (\beta_{4} + \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{13} + ( - \beta_{4} - \beta_{2}) q^{15} + q^{17} + (\beta_{5} + 2 \beta_{3} + \cdots + \beta_1) q^{19}+ \cdots + (\beta_{4} + 2 \beta_{3} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) q - b1 * q^3 + b4 * q^5 + b2 * q^9 + (b4 + b3 + b1 - 1) * q^11 + (-b4 - b3 + b2) * q^13 + (-b4 - b2) * q^15 + q^17 + (b5 + 2b3 - b2 + b1) q^19 + (-b5 - b3 - b1 - 1) * q^23 + (2b5 - b2 + 1) q^25 + (-b4 - b3 + b1 + 1) * q^27 + (-2b5 - b3 - b2 - 3) q^29 + (-3b5 - b3 + b2 + 3b1 - 3) * q^31 + (-b5 - b4 + b3 - 3b2 + b1 - 2) q^33 + (b5 - b4 - 3b1 - 2) q^37 + (b5 - 2b3 + 2b2 - 2b1) q^39 + (2b5 - b4 + 2b3 - b2) * q^41 + (b5 - 2b4 - 2b3 + b2 - b1 + 4) * q^43 + (-b4 + b3 + b2 + 2b1 - 1) q^45 + (2b5 - 3b3 + b2 - 2b1 + 1) q^47 - b1 * q^51 + (-2b5 - b3 - 2b2 + 2b1 - 6) q^53 + (b5 - b4 + b2 + b1 + 4) * q^55 + (-3b5 + b4 + 2b3 - 3b2 + 3b1 - 4) * q^57 + (b4 - b3 - b2 + 2b1 + 1) q^59 + (-3b5 - 3b4 + b3 - 2b2 + b1 - 3) q^61 + (-b5 + b3 + b2 + b1 - 5) * q^65 + (4b5 + b4 + 2b2 - 2b1 + 6) q^67 + (2b5 + 2b2 + 4) * q^69 + (-b4 - 2b2 + b1 - 2) q^71 + (-3b5 - b4 + 2b3 - b2 + 3b1 - 4) q^73 + (-2b5 + b4 - b3 + 3b1 - 5) * q^75 + (b5 + 3b3 - b2 - 3) q^79 + (b5 + b4 - b3 - 2b2 - b1 - 4) q^81 + (-2b5 - b4 + 2b3 - b2 + 4b1 + 2) q^83 + b4 * q^85 + (3b5 + b4 + 2b3 + b2 + 3b1 + 2) q^87 + (2b5 - 2b4 - b3 + b2 - 2b1 + 6) q^89 + (4b5 - b4 + b3 - 2b2 - 2b1 - 3) q^93 + (-4b5 + b4 - 2b3 + 2b2 - 4) q^95 + (4b5 + b4 + 3b3 + 3) * q^97 + (b4 + 2b3 - b2 + 4b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} + q^{9} - 6 q^{11} + 2 q^{13} + 6 q^{17} - 2 q^{19} - 5 q^{23} + q^{25} + 8 q^{27} - 15 q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{37} - 2 q^{39} - 4 q^{41} + 24 q^{43} - 2 q^{45} + q^{47}+ \cdots + 2 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 + q^9 - 6 * q^11 + 2 * q^13 + 6 * q^17 - 2 * q^19 - 5 * q^23 + q^25 + 8 * q^27 - 15 * q^29 - 8 * q^31 - 11 * q^33 - 16 * q^37 - 2 * q^39 - 4 * q^41 + 24 * q^43 - 2 * q^45 + q^47 - q^51 - 32 * q^53 + 25 * q^55 - 19 * q^57 + 6 * q^59 - 10 * q^61 - 26 * q^65 + 27 * q^67 + 22 * q^69 - 12 * q^71 - 15 * q^73 - 24 * q^75 - 21 * q^79 - 30 * q^81 + 20 * q^83 - q^85 + 9 * q^87 + 33 * q^89 - 29 * q^93 - 15 * q^95 + 9 * q^97 + 2 * q^99

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