LMFDB - Newform orbit 6664.2.a.u

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 = [6,0,4,0,2] 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:	$0$
Dimension:	$6$
Coefficient field:	6.6.13431004.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} - 3x^{3} + 15x^{2} + 8x - 2 $ x^6 - 8x^4 - 3x^3 + 15x^2 + 8x - 2
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_1 + 1) q^{3} + \beta_{3} q^{5} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{9} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{4}) q^{13} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots + 1) q^{15}+ \cdots + (\beta_{5} + 6 \beta_{4} + 3 \beta_{3} + \cdots + 6) q^{99}+O(q^{100}) $ q + (b1 + 1) * q^3 + b3 * q^5 + (b4 + b3 + b1 + 1) * q^9 + (b4 + b3 - b2) * q^11 + (-b5 - b4) * q^13 + (-b5 - b4 - b2 + 2b1 + 1) q^15 + q^17 + (2b4 + b3 + b2 - b1) q^19 + (b3 + b2 - b1) * q^23 + (-b5 + b3 + b2 - b1) * q^25 + (2b4 + b3 - 2b2 + b1 + 3) * q^27 + (b4 - b3 + b2) * q^29 + (-b5 + b3 + b1 + 2) * q^31 + (2b4 + b3 - 3b2 + 3b1) q^33 + (b5 + b3 - b2 + 2b1 + 2) q^37 + (-b5 - 3b4 + b3 + b2 - b1 - 2) q^39 + (b5 - 2b2 + 1) q^41 + (b5 - 2b4 - b3 + b2 + b1 + 1) q^43 + (-b5 + b3 + 3) * q^45 + (b5 - b4 - b3 + b2 - b1) * q^47 + (b1 + 1) * q^51 + (-b4 - b3 + 2b2 - b1 - 2) q^53 + (b5 + b4 + 2b2 - 2b1 + 2) * q^55 + (b5 + b4 - 2b3 - 2b2 + 4b1 + 2) q^57 + (2b1 + 4) q^59 + (2b4 + b1 - 3) q^61 + (-2b4 + b3 + b2 + 3b1 + 2) * q^65 + (2b5 - b4 + b3 - 3b1 - 2) * q^67 + (-b5 - 3b4 - 2b3 + 2b1) q^69 + (-2b5 + b3 + 3b2 + b1 + 2) * q^71 + (3b5 - 2b3 - 2b1 + 3) q^73 + (-b5 - 4b4 - b3 + 2b1 - 1) * q^75 + (-b5 - b4 + b3 - b2 + b1) * q^79 + (b5 + 3b4 - 5b2 + 4b1 + 2) q^81 + (b5 - 3b4 + 2b2 - 2b1 + 6) q^83 + b3 * q^85 + (2b5 + 2b4 - b3 + b2 - b1 + 2) * q^87 + (-b5 + b4 + b3 + b2 - b1 - 2) * q^89 + (-b5 - b4 + 2b3 - b2 + 4b1 + 5) * q^93 + (b5 + 3b4 + 3b3 + b2 - 5b1 - 2) q^95 + (b5 + 2b4 - b3 + 4b2 - 5b1 - 2) q^97 + (b5 + 6b4 + 3b3 - 3b2 + 4b1 + 6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 2 q^{5} + 8 q^{9} + 4 q^{11} - 2 q^{13} + 6 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{25} + 22 q^{27} + 12 q^{31} + 10 q^{37} - 14 q^{39} + 6 q^{41} - 2 q^{43} + 20 q^{45} - 2 q^{47} + 4 q^{51}+ \cdots + 46 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 2 * q^5 + 8 * q^9 + 4 * q^11 - 2 * q^13 + 6 * q^17 + 8 * q^19 + 4 * q^23 + 4 * q^25 + 22 * q^27 + 12 * q^31 + 10 * q^37 - 14 * q^39 + 6 * q^41 - 2 * q^43 + 20 * q^45 - 2 * q^47 + 4 * q^51 - 14 * q^53 + 18 * q^55 + 2 * q^57 + 20 * q^59 - 16 * q^61 + 4 * q^65 - 6 * q^67 - 14 * q^69 + 12 * q^71 + 18 * q^73 - 20 * q^75 - 2 * q^79 + 10 * q^81 + 34 * q^83 + 2 * q^85 + 16 * q^87 - 6 * q^89 + 24 * q^93 + 10 * q^95 + 46 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 8x^{4} - 3x^{3} + 15x^{2} + 8x - 2 $ x^6 - 8*x^4 - 3*x^3 + 15*x^2 + 8*x - 2 :

$\beta_{1}$	$=$	$ \nu^{2} - \nu - 3 $ v^2 - v - 3
$\beta_{2}$	$=$	$ \nu^{5} - \nu^{4} - 6\nu^{3} + 3\nu^{2} + 8\nu - 1 $ v^5 - v^4 - 6v^3 + 3v^2 + 8*v - 1
$\beta_{3}$	$=$	$ \nu^{5} - \nu^{4} - 6\nu^{3} + 2\nu^{2} + 7\nu + 2 $ v^5 - v^4 - 6v^3 + 2v^2 + 7*v + 2
$\beta_{4}$	$=$	$ -\nu^{5} + 2\nu^{4} + 4\nu^{3} - 6\nu^{2} - 2\nu + 1 $ -v^5 + 2v^4 + 4v^3 - 6v^2 - 2v + 1
$\beta_{5}$	$=$	$ -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 8\nu^{2} - 10\nu + 3 $ -v^5 + 2v^4 + 6v^3 - 8v^2 - 10v + 3

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

$\nu$	$=$	$ ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 $ (-b3 + b2 - b1) / 2
$\nu^{2}$	$=$	$ ( -\beta_{3} + \beta_{2} + \beta _1 + 6 ) / 2 $ (-b3 + b2 + b1 + 6) / 2
$\nu^{3}$	$=$	$ ( \beta_{5} - \beta_{4} - 5\beta_{3} + 5\beta_{2} - 3\beta _1 + 4 ) / 2 $ (b5 - b4 - 5b3 + 5b2 - 3*b1 + 4) / 2
$\nu^{4}$	$=$	$ ( 2\beta_{5} - 7\beta_{3} + 9\beta_{2} + 3\beta _1 + 26 ) / 2 $ (2b5 - 7b3 + 9b2 + 3b1 + 26) / 2
$\nu^{5}$	$=$	$ 4\beta_{5} - 3\beta_{4} - 13\beta_{3} + 15\beta_{2} - 5\beta _1 + 17 $ 4b5 - 3b4 - 13b3 + 15b2 - 5*b1 + 17

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

0.187674
1.63501
−0.838529
2.52882
−1.64932
−1.86366

−2.15245

3.34349

1.63305

1.2

−0.961752

−2.89526

−2.07503

1.3

−0.458339

0.165178

−2.78993

1.4

1.86611

−2.01690

0.482357

1.5

2.36956

3.21031

2.61483

1.6

3.33687

0.193178

8.13472

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.u	yes	6
7.b	odd	2	1		6664.2.a.p	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6664.2.a.p	✓	6	7.b	odd	2	1
6664.2.a.u	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} - 4T_{3}^{5} - 5T_{3}^{4} + 26T_{3}^{3} + 5T_{3}^{2} - 34T_{3} - 14 $

T3^6 - 4*T3^5 - 5*T3^4 + 26*T3^3 + 5*T3^2 - 34*T3 - 14

$ T_{5}^{6} - 2T_{5}^{5} - 15T_{5}^{4} + 20T_{5}^{3} + 57T_{5}^{2} - 22T_{5} + 2 $

T5^6 - 2*T5^5 - 15*T5^4 + 20*T5^3 + 57*T5^2 - 22*T5 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 4 T^{5} + \cdots - 14 $ T^6 - 4T^5 - 5T^4 + 26T^3 + 5T^2 - 34*T - 14
$5$	$ T^{6} - 2 T^{5} + \cdots + 2 $ T^6 - 2T^5 - 15T^4 + 20T^3 + 57T^2 - 22*T + 2
$7$	$ T^{6} $ T^6
$11$	$ T^{6} - 4 T^{5} + \cdots + 32 $ T^6 - 4T^5 - 30T^4 + 108T^3 + 16T^2 - 96*T + 32
$13$	$ T^{6} + 2 T^{5} + \cdots + 512 $ T^6 + 2T^5 - 44T^4 - 72T^3 + 368T^2 + 896*T + 512
$17$	$ (T - 1)^{6} $ (T - 1)^6
$19$	$ T^{6} - 8 T^{5} + \cdots + 1984 $ T^6 - 8T^5 - 44T^4 + 376T^3 + 80T^2 - 2336*T + 1984
$23$	$ T^{6} - 4 T^{5} + \cdots - 256 $ T^6 - 4T^5 - 32T^4 + 40T^3 + 240T^2 - 256
$29$	$ T^{6} - 62 T^{4} + \cdots - 4192 $ T^6 - 62T^4 - 12T^3 + 1008T^2 - 96T - 4192
$31$	$ T^{6} - 12 T^{5} + \cdots + 328 $ T^6 - 12T^5 + 19T^4 + 114T^3 - 273T^2 - 100*T + 328
$37$	$ T^{6} - 10 T^{5} + \cdots - 352 $ T^6 - 10T^5 - 66T^4 + 848T^3 - 1536T^2 - 1904*T - 352
$41$	$ T^{6} - 6 T^{5} + \cdots - 1252 $ T^6 - 6T^5 - 71T^4 + 120T^3 + 1215T^2 + 720*T - 1252
$43$	$ T^{6} + 2 T^{5} + \cdots - 45256 $ T^6 + 2T^5 - 125T^4 - 170T^3 + 4317T^2 + 2988*T - 45256
$47$	$ T^{6} + 2 T^{5} + \cdots - 1728 $ T^6 + 2T^5 - 64T^4 - 112T^3 + 960T^2 + 1440*T - 1728
$53$	$ T^{6} + 14 T^{5} + \cdots - 1004 $ T^6 + 14T^5 - T^4 - 474T^3 - 1735T^2 - 2272T - 1004
$59$	$ T^{6} - 20 T^{5} + \cdots - 64 $ T^6 - 20T^5 + 120T^4 - 112T^3 - 768T^2 + 896*T - 64
$61$	$ T^{6} + 16 T^{5} + \cdots - 3902 $ T^6 + 16T^5 + 3T^4 - 1050T^3 - 5519T^2 - 8502*T - 3902
$67$	$ T^{6} + 6 T^{5} + \cdots + 274232 $ T^6 + 6T^5 - 277T^4 - 1890T^3 + 17117T^2 + 149724*T + 274232
$71$	$ T^{6} - 12 T^{5} + \cdots + 40576 $ T^6 - 12T^5 - 208T^4 + 2984T^3 - 6768T^2 - 16000*T + 40576
$73$	$ T^{6} - 18 T^{5} + \cdots + 172556 $ T^6 - 18T^5 - 147T^4 + 3308T^3 - 4021T^2 - 72080*T + 172556
$79$	$ T^{6} + 2 T^{5} + \cdots - 128 $ T^6 + 2T^5 - 68T^4 - 184T^3 + 80T^2 + 256*T - 128
$83$	$ T^{6} - 34 T^{5} + \cdots + 1662400 $ T^6 - 34T^5 + 144T^4 + 6096T^3 - 65696T^2 - 20640*T + 1662400
$89$	$ T^{6} + 6 T^{5} + \cdots - 256 $ T^6 + 6T^5 - 48T^4 - 112T^3 + 368T^2 + 640*T - 256
$97$	$ T^{6} - 389 T^{4} + \cdots - 405748 $ T^6 - 389T^4 - 558T^3 + 38743T^2 + 128448T - 405748
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Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta_1 + 1) q^{3} + \beta_{3} q^{5} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{9} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{11} + ( - \beta_{5} - \beta_{4}) q^{13} + ( - \beta_{5} - \beta_{4} - \beta_{2} + \cdots + 1) q^{15}+ \cdots + (\beta_{5} + 6 \beta_{4} + 3 \beta_{3} + \cdots + 6) q^{99}+O(q^{100}) \) q + (b1 + 1) * q^3 + b3 * q^5 + (b4 + b3 + b1 + 1) * q^9 + (b4 + b3 - b2) * q^11 + (-b5 - b4) * q^13 + (-b5 - b4 - b2 + 2b1 + 1) q^15 + q^17 + (2b4 + b3 + b2 - b1) q^19 + (b3 + b2 - b1) * q^23 + (-b5 + b3 + b2 - b1) * q^25 + (2b4 + b3 - 2b2 + b1 + 3) * q^27 + (b4 - b3 + b2) * q^29 + (-b5 + b3 + b1 + 2) * q^31 + (2b4 + b3 - 3b2 + 3b1) q^33 + (b5 + b3 - b2 + 2b1 + 2) q^37 + (-b5 - 3b4 + b3 + b2 - b1 - 2) q^39 + (b5 - 2b2 + 1) q^41 + (b5 - 2b4 - b3 + b2 + b1 + 1) q^43 + (-b5 + b3 + 3) * q^45 + (b5 - b4 - b3 + b2 - b1) * q^47 + (b1 + 1) * q^51 + (-b4 - b3 + 2b2 - b1 - 2) q^53 + (b5 + b4 + 2b2 - 2b1 + 2) * q^55 + (b5 + b4 - 2b3 - 2b2 + 4b1 + 2) q^57 + (2b1 + 4) q^59 + (2b4 + b1 - 3) q^61 + (-2b4 + b3 + b2 + 3b1 + 2) * q^65 + (2b5 - b4 + b3 - 3b1 - 2) * q^67 + (-b5 - 3b4 - 2b3 + 2b1) q^69 + (-2b5 + b3 + 3b2 + b1 + 2) * q^71 + (3b5 - 2b3 - 2b1 + 3) q^73 + (-b5 - 4b4 - b3 + 2b1 - 1) * q^75 + (-b5 - b4 + b3 - b2 + b1) * q^79 + (b5 + 3b4 - 5b2 + 4b1 + 2) q^81 + (b5 - 3b4 + 2b2 - 2b1 + 6) q^83 + b3 * q^85 + (2b5 + 2b4 - b3 + b2 - b1 + 2) * q^87 + (-b5 + b4 + b3 + b2 - b1 - 2) * q^89 + (-b5 - b4 + 2b3 - b2 + 4b1 + 5) * q^93 + (b5 + 3b4 + 3b3 + b2 - 5b1 - 2) q^95 + (b5 + 2b4 - b3 + 4b2 - 5b1 - 2) q^97 + (b5 + 6b4 + 3b3 - 3b2 + 4b1 + 6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 2 q^{5} + 8 q^{9} + 4 q^{11} - 2 q^{13} + 6 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{25} + 22 q^{27} + 12 q^{31} + 10 q^{37} - 14 q^{39} + 6 q^{41} - 2 q^{43} + 20 q^{45} - 2 q^{47} + 4 q^{51}+ \cdots + 46 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 2 * q^5 + 8 * q^9 + 4 * q^11 - 2 * q^13 + 6 * q^17 + 8 * q^19 + 4 * q^23 + 4 * q^25 + 22 * q^27 + 12 * q^31 + 10 * q^37 - 14 * q^39 + 6 * q^41 - 2 * q^43 + 20 * q^45 - 2 * q^47 + 4 * q^51 - 14 * q^53 + 18 * q^55 + 2 * q^57 + 20 * q^59 - 16 * q^61 + 4 * q^65 - 6 * q^67 - 14 * q^69 + 12 * q^71 + 18 * q^73 - 20 * q^75 - 2 * q^79 + 10 * q^81 + 34 * q^83 + 2 * q^85 + 16 * q^87 - 6 * q^89 + 24 * q^93 + 10 * q^95 + 46 * q^99

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