LMFDB - Newform orbit 4576.2.a.p

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4576,2,Mod(1,4576)] 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("4576.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4576 = 2^{5} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4576.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,-2,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$36.5395439649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.106740016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 9x^{4} + 7x^{3} + 15x^{2} + x - 2 $ x^6 - x^5 - 9x^4 + 7x^3 + 15*x^2 + x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
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 q^{3} - \beta_{5} q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 1) q^{7} + (\beta_{4} - \beta_{3} + \beta_1) q^{9} - q^{11} - q^{13} + (\beta_{3} + \beta_{2} + \beta_1) q^{15}+ \cdots + ( - \beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) $ q - b1 * q^3 - b5 * q^5 + (-b5 + b4 - b3 + b1 + 1) * q^7 + (b4 - b3 + b1) * q^9 - q^11 - q^13 + (b3 + b2 + b1) * q^15 + (-b5 + b4 + 1) * q^17 + (-b5 + b4 - b2) * q^19 + (b4 - 2b1) q^21 + (-b5 + b4 + b2 + 3) * q^23 + (-b5 + b3 + 2b2) q^25 + (b4 - b3 - b2 + b1) * q^27 + (b5 - b3 - b2 - b1) * q^29 + (-2b3 - b2 - b1 + 3) q^31 + b1 * q^33 + (-b5 - 2b4 + b3 + b2 - b1 + 2) q^35 + (b5 + b4 + b3 - b2 - b1 - 1) * q^37 + b1 * q^39 + (-2b5 - 2b3 + b1 + 1) * q^41 + (-b5 - 2b4 + b2 + b1 - 1) q^43 + (b5 - 2b4 - b2 - b1 - 3) q^45 + (b5 - b4 - 2b2 + 2b1 - 1) * q^47 + (-b4 - b3 - b1 + 1) * q^49 + (-b5 + b4 - b3 + 2) * q^51 + (-b3 + b2 - 2b1) q^53 + b5 * q^55 + (b4 + b2 + 2b1 + 1) q^57 + (-b5 - 2b4 + b3 + 2b2 + 1) * q^59 + (-3b4 + b3 + b2 - 3b1 - 1) * q^61 + (2b5 - b3 - b2 - b1 + 5) q^63 + b5 * q^65 + (-2b5 + b4 + b3 + 2b1 + 2) * q^67 + (-2b5 + b4 - 2b3 - b2 - 3b1 + 3) q^69 + (2b4 - b2 - b1 + 3) q^71 + (b4 - 2b3 - 2b2 - b1 + 1) * q^73 + (-3b5 - b4 - b3 - b2 + 1) q^75 + (b5 - b4 + b3 - b1 - 1) * q^77 + (b5 - b4 + b3 - 2b1 + 6) q^79 + (b5 - 2b4 + 3b3 - 4b1 - 1) q^81 + (b5 + 3b3 - b2 - 2b1 - 1) * q^83 + (-2b5 - 2b4 + 2b3 + 2b2 - 2b1 + 4) q^85 + (2b5 + 2b4 - b3 + 3) * q^87 + (4b4 - 3b3 - 5b2 + 3b1) * q^89 + (b5 - b4 + b3 - b1 - 1) * q^91 + (3b5 + 3b4 + b2 - 3b1 + 4) q^93 + (b5 - b4 + 2b3 + 2b2 + 5) * q^95 + (-2b3 - b2 - b1 + 1) q^97 + (-b4 + b3 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - 2 q^{5} + 5 q^{7} + q^{9} - 6 q^{11} - 6 q^{13} + 5 q^{15} + 6 q^{17} - 2 q^{19} + 20 q^{23} + 4 q^{25} - q^{27} - 3 q^{29} + 11 q^{31} + q^{33} + 9 q^{35} - 3 q^{37} + q^{39} - q^{41}+ \cdots - q^{99}+O(q^{100}) $ 6 * q - q^3 - 2 * q^5 + 5 * q^7 + q^9 - 6 * q^11 - 6 * q^13 + 5 * q^15 + 6 * q^17 - 2 * q^19 + 20 * q^23 + 4 * q^25 - q^27 - 3 * q^29 + 11 * q^31 + q^33 + 9 * q^35 - 3 * q^37 + q^39 - q^41 - 9 * q^43 - 23 * q^45 - 8 * q^47 + q^49 + 10 * q^51 - 2 * q^53 + 2 * q^55 + 12 * q^57 + 6 * q^59 - 11 * q^61 + 29 * q^63 + 2 * q^65 + 14 * q^67 + 7 * q^69 + 19 * q^71 - q^73 - 6 * q^75 - 5 * q^77 + 36 * q^79 - 6 * q^81 - 2 * q^83 + 22 * q^85 + 24 * q^87 - 5 * q^89 - 5 * q^91 + 35 * q^93 + 38 * q^95 - q^97 - q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 9x^{4} + 7x^{3} + 15x^{2} + x - 2 $ x^6 - x^5 - 9*x^4 + 7*x^3 + 15*x^2 + x - 2 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{3} + \nu^{2} - 6\nu - 3 $ v^3 + v^2 - 6*v - 3
$\beta_{3}$	$=$	$ \nu^{5} - \nu^{4} - 8\nu^{3} + 7\nu^{2} + 9\nu + 1 $ v^5 - v^4 - 8v^3 + 7v^2 + 9*v + 1
$\beta_{4}$	$=$	$ \nu^{5} - \nu^{4} - 8\nu^{3} + 8\nu^{2} + 8\nu - 2 $ v^5 - v^4 - 8v^3 + 8v^2 + 8*v - 2
$\beta_{5}$	$=$	$ -\nu^{5} + 2\nu^{4} + 8\nu^{3} - 14\nu^{2} - 7\nu + 3 $ -v^5 + 2v^4 + 8v^3 - 14v^2 - 7v + 3

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{4} - \beta_{3} + \beta _1 + 3 $ b4 - b3 + b1 + 3
$\nu^{3}$	$=$	$ -\beta_{4} + \beta_{3} + \beta_{2} + 5\beta_1 $ -b4 + b3 + b2 + 5*b1
$\nu^{4}$	$=$	$ \beta_{5} + 7\beta_{4} - 6\beta_{3} + 5\beta _1 + 17 $ b5 + 7b4 - 6b3 + 5*b1 + 17
$\nu^{5}$	$=$	$ \beta_{5} - 8\beta_{4} + 10\beta_{3} + 8\beta_{2} + 29\beta _1 - 5 $ b5 - 8b4 + 10b3 + 8b2 + 29b1 - 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

2.53526
2.13437
0.320976
−0.531122
−0.822033
−2.63744

−2.53526

−3.51816

0.909368

3.42753

1.2

−2.13437

0.721145

3.27666

1.55552

1.3

−0.320976

0.406820

−1.49015

−2.89697

1.4

0.531122

−1.77141

−3.48932

−2.71791

1.5

0.822033

3.86135

2.53709

−2.32426

1.6

2.63744

−1.69975

3.25636

3.95610

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$11$	$ +1 $
$13$	$ +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	4576.2.a.p	✓	6
4.b	odd	2	1		4576.2.a.q	yes	6
8.b	even	2	1		9152.2.a.cr		6
8.d	odd	2	1		9152.2.a.cp		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4576.2.a.p	✓	6	1.a	even	1	1	trivial
4576.2.a.q	yes	6	4.b	odd	2	1
9152.2.a.cp		6	8.d	odd	2	1
9152.2.a.cr		6	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4576))$:

$ T_{3}^{6} + T_{3}^{5} - 9T_{3}^{4} - 7T_{3}^{3} + 15T_{3}^{2} - T_{3} - 2 $

T3^6 + T3^5 - 9*T3^4 - 7*T3^3 + 15*T3^2 - T3 - 2

$ T_{5}^{6} + 2T_{5}^{5} - 15T_{5}^{4} - 34T_{5}^{3} + 10T_{5}^{2} + 32T_{5} - 12 $

T5^6 + 2*T5^5 - 15*T5^4 - 34*T5^3 + 10*T5^2 + 32*T5 - 12

$ T_{7}^{6} - 5T_{7}^{5} - 9T_{7}^{4} + 73T_{7}^{3} - 49T_{7}^{2} - 147T_{7} + 128 $

T7^6 - 5*T7^5 - 9*T7^4 + 73*T7^3 - 49*T7^2 - 147*T7 + 128

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 - 7T^3 + 15*T^2 - T - 2
$5$	$ T^{6} + 2 T^{5} + \cdots - 12 $ T^6 + 2T^5 - 15T^4 - 34T^3 + 10T^2 + 32*T - 12
$7$	$ T^{6} - 5 T^{5} + \cdots + 128 $ T^6 - 5T^5 - 9T^4 + 73T^3 - 49T^2 - 147*T + 128
$11$	$ (T + 1)^{6} $ (T + 1)^6
$13$	$ (T + 1)^{6} $ (T + 1)^6
$17$	$ T^{6} - 6 T^{5} + \cdots + 32 $ T^6 - 6T^5 - 10T^4 + 88T^3 - 60T^2 - 120*T + 32
$19$	$ T^{6} + 2 T^{5} + \cdots - 32 $ T^6 + 2T^5 - 47T^4 - 94T^3 + 355T^2 + 118*T - 32
$23$	$ T^{6} - 20 T^{5} + \cdots - 73 $ T^6 - 20T^5 + 114T^4 - 62T^3 - 634T^2 - 438*T - 73
$29$	$ T^{6} + 3 T^{5} + \cdots - 144 $ T^6 + 3T^5 - 58T^4 - 100T^3 + 720T^2 - 316*T - 144
$31$	$ T^{6} - 11 T^{5} + \cdots - 6696 $ T^6 - 11T^5 - 64T^4 + 760T^3 + 192T^2 - 6508*T - 6696
$37$	$ T^{6} + 3 T^{5} + \cdots - 128 $ T^6 + 3T^5 - 70T^4 - 280T^3 + 208T^2 + 1152*T - 128
$41$	$ T^{6} + T^{5} + \cdots - 4154 $ T^6 + T^5 - 97T^4 - 131T^3 + 1909T^2 + 453T - 4154
$43$	$ T^{6} + 9 T^{5} + \cdots - 12184 $ T^6 + 9T^5 - 72T^4 - 446T^3 + 1924T^2 + 3300*T - 12184
$47$	$ T^{6} + 8 T^{5} + \cdots - 63552 $ T^6 + 8T^5 - 162T^4 - 924T^3 + 7884T^2 + 21088*T - 63552
$53$	$ T^{6} + 2 T^{5} + \cdots - 3204 $ T^6 + 2T^5 - 109T^4 - 226T^3 + 1101T^2 + 1144*T - 3204
$59$	$ T^{6} - 6 T^{5} + \cdots + 13136 $ T^6 - 6T^5 - 99T^4 + 508T^3 + 1804T^2 - 11376*T + 13136
$61$	$ T^{6} + 11 T^{5} + \cdots + 31744 $ T^6 + 11T^5 - 76T^4 - 1042T^3 - 876T^2 + 14740*T + 31744
$67$	$ T^{6} - 14 T^{5} + \cdots + 51364 $ T^6 - 14T^5 - 53T^4 + 1176T^3 - 646T^2 - 23796*T + 51364
$71$	$ T^{6} - 19 T^{5} + \cdots + 3312 $ T^6 - 19T^5 + 62T^4 + 388T^3 - 960T^2 - 1780*T + 3312
$73$	$ T^{6} + T^{5} + \cdots - 3746 $ T^6 + T^5 - 149T^4 + 3T^3 + 3295T^2 - 4797T - 3746
$79$	$ T^{6} - 36 T^{5} + \cdots + 17152 $ T^6 - 36T^5 + 512T^4 - 3660T^3 + 13732T^2 - 25208*T + 17152
$83$	$ T^{6} + 2 T^{5} + \cdots + 496 $ T^6 + 2T^5 - 197T^4 - 1218T^3 - 1373T^2 + 306*T + 496
$89$	$ T^{6} + 5 T^{5} + \cdots - 2541536 $ T^6 + 5T^5 - 466T^4 - 1758T^3 + 62112T^2 + 118508*T - 2541536
$97$	$ T^{6} + T^{5} + \cdots - 14176 $ T^6 + T^5 - 114T^4 - 32T^3 + 2576T^2 + 644T - 14176
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Level:	\( N \)	\(=\)	\( 4576 = 2^{5} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4576.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{3} - \beta_{5} q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 1) q^{7} + (\beta_{4} - \beta_{3} + \beta_1) q^{9} - q^{11} - q^{13} + (\beta_{3} + \beta_{2} + \beta_1) q^{15}+ \cdots + ( - \beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) q - b1 * q^3 - b5 * q^5 + (-b5 + b4 - b3 + b1 + 1) * q^7 + (b4 - b3 + b1) * q^9 - q^11 - q^13 + (b3 + b2 + b1) * q^15 + (-b5 + b4 + 1) * q^17 + (-b5 + b4 - b2) * q^19 + (b4 - 2b1) q^21 + (-b5 + b4 + b2 + 3) * q^23 + (-b5 + b3 + 2b2) q^25 + (b4 - b3 - b2 + b1) * q^27 + (b5 - b3 - b2 - b1) * q^29 + (-2b3 - b2 - b1 + 3) q^31 + b1 * q^33 + (-b5 - 2b4 + b3 + b2 - b1 + 2) q^35 + (b5 + b4 + b3 - b2 - b1 - 1) * q^37 + b1 * q^39 + (-2b5 - 2b3 + b1 + 1) * q^41 + (-b5 - 2b4 + b2 + b1 - 1) q^43 + (b5 - 2b4 - b2 - b1 - 3) q^45 + (b5 - b4 - 2b2 + 2b1 - 1) * q^47 + (-b4 - b3 - b1 + 1) * q^49 + (-b5 + b4 - b3 + 2) * q^51 + (-b3 + b2 - 2b1) q^53 + b5 * q^55 + (b4 + b2 + 2b1 + 1) q^57 + (-b5 - 2b4 + b3 + 2b2 + 1) * q^59 + (-3b4 + b3 + b2 - 3b1 - 1) * q^61 + (2b5 - b3 - b2 - b1 + 5) q^63 + b5 * q^65 + (-2b5 + b4 + b3 + 2b1 + 2) * q^67 + (-2b5 + b4 - 2b3 - b2 - 3b1 + 3) q^69 + (2b4 - b2 - b1 + 3) q^71 + (b4 - 2b3 - 2b2 - b1 + 1) * q^73 + (-3b5 - b4 - b3 - b2 + 1) q^75 + (b5 - b4 + b3 - b1 - 1) * q^77 + (b5 - b4 + b3 - 2b1 + 6) q^79 + (b5 - 2b4 + 3b3 - 4b1 - 1) q^81 + (b5 + 3b3 - b2 - 2b1 - 1) * q^83 + (-2b5 - 2b4 + 2b3 + 2b2 - 2b1 + 4) q^85 + (2b5 + 2b4 - b3 + 3) * q^87 + (4b4 - 3b3 - 5b2 + 3b1) * q^89 + (b5 - b4 + b3 - b1 - 1) * q^91 + (3b5 + 3b4 + b2 - 3b1 + 4) q^93 + (b5 - b4 + 2b3 + 2b2 + 5) * q^95 + (-2b3 - b2 - b1 + 1) q^97 + (-b4 + b3 - b1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - 2 q^{5} + 5 q^{7} + q^{9} - 6 q^{11} - 6 q^{13} + 5 q^{15} + 6 q^{17} - 2 q^{19} + 20 q^{23} + 4 q^{25} - q^{27} - 3 q^{29} + 11 q^{31} + q^{33} + 9 q^{35} - 3 q^{37} + q^{39} - q^{41}+ \cdots - q^{99}+O(q^{100}) \) 6 * q - q^3 - 2 * q^5 + 5 * q^7 + q^9 - 6 * q^11 - 6 * q^13 + 5 * q^15 + 6 * q^17 - 2 * q^19 + 20 * q^23 + 4 * q^25 - q^27 - 3 * q^29 + 11 * q^31 + q^33 + 9 * q^35 - 3 * q^37 + q^39 - q^41 - 9 * q^43 - 23 * q^45 - 8 * q^47 + q^49 + 10 * q^51 - 2 * q^53 + 2 * q^55 + 12 * q^57 + 6 * q^59 - 11 * q^61 + 29 * q^63 + 2 * q^65 + 14 * q^67 + 7 * q^69 + 19 * q^71 - q^73 - 6 * q^75 - 5 * q^77 + 36 * q^79 - 6 * q^81 - 2 * q^83 + 22 * q^85 + 24 * q^87 - 5 * q^89 - 5 * q^91 + 35 * q^93 + 38 * q^95 - q^97 - q^99

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