LMFDB - Newform orbit 7232.2.a.be

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7232,2,Mod(1,7232)] 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("7232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7232 = 2^{6} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,6,0,0] 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:	$57.7478107418$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.54033678848.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 22x^{4} - 13x^{2} + 2 $ x^8 - 9x^6 + 22x^4 - 13*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3616)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{3} q^{3} + ( - \beta_{5} + \beta_1) q^{5} - \beta_{7} q^{7} - \beta_{2} q^{9} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{11} + (\beta_{2} - \beta_1 - 1) q^{13} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{15}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{3}) q^{99}+O(q^{100}) $ q - b3 * q^3 + (-b5 + b1) * q^5 - b7 * q^7 - b2 * q^9 + (b6 + b4 + b3) * q^11 + (b2 - b1 - 1) * q^13 + (b7 + b6 - b3) * q^15 + (-b5 + b2 - b1 - 3) * q^17 + (-b6 - 2b4) q^19 + (2b5 - 2b2 - b1 + 2) * q^21 + (2b7 + b4 - b3) q^23 + (-b5 + 2b2 + 3b1 - 1) * q^25 + (-b7 + 2b3) q^27 + (3b5 - b2 - 2b1 + 3) * q^29 + (-2b6 - b4 + 2b3) * q^31 + (b5 + 2b2 + 2b1 - 4) * q^33 + (-b7 - 2b6 + 2b3) * q^35 + (b5 - 2b2 - 2b1 + 2) * q^37 + (b7 - b6 - b4 + 3b3) q^39 + (2b5 - b1 - 2) q^41 + (-2b6 + b4 + b3) q^43 + (b5 + b2 - b1 + 1) * q^45 + (-2b6 - 3b3) * q^47 + (b5 - 2b2 + 1) q^49 + (2b7 - b6 - 2b4 + 5b3) q^51 + (4b5 + 2b2 + 3b1 + 4) q^53 + (b6 + 3b4 - b3) q^55 + (-2b5 - 2b2 - 3b1 + 2) q^57 + (-b7 - 2b4 + 5b3) * q^59 + (b5 + 3b2 - 5) q^61 + (-b7 - b6 + b4 - 3b3) q^63 + (-3b2 - 3b1 - 3) * q^65 + (b7 + 2b4 - b3) q^67 + (-3b5 + 4b2 + 3b1 - 2) q^69 + (b7 + 3b6 - 4b4) * q^71 + (2b5 - b2 - 7) q^73 + (3b7 + 3b6 + 2b4) q^75 + (-3b5 + b2 - 2b1 - 1) * q^77 + (3b7 + 3b6 - 3b4 + 2b3) * q^79 + (2b5 + 3b2 - b1 - 4) * q^81 + (b7 + 3b6 - b4 - 3b3) * q^83 + (b5 - 3b2 - 5b1 - 1) * q^85 + (-4b7 - 2b6 + b4 - 2b3) q^87 + (-b5 + 2b2 + 2b1 - 14) * q^89 + (3b7 + 3b6 + 3b3) q^91 + (-b5 + b2 - 3b1 - 5) q^93 + (-b7 - 4b6 - 4b4 + 2b3) q^95 + (3b5 - b2 + b1 - 5) q^97 + (b7 - b6 + b3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{5} - 2 q^{9} - 8 q^{13} - 20 q^{17} + 2 q^{21} + 6 q^{25} + 6 q^{29} - 28 q^{33} + 4 q^{37} - 26 q^{41} + 4 q^{45} + 26 q^{53} + 14 q^{57} - 38 q^{61} - 36 q^{65} + 10 q^{69} - 66 q^{73} + 2 q^{77}+ \cdots - 52 q^{97}+O(q^{100}) $ 8 * q + 6 * q^5 - 2 * q^9 - 8 * q^13 - 20 * q^17 + 2 * q^21 + 6 * q^25 + 6 * q^29 - 28 * q^33 + 4 * q^37 - 26 * q^41 + 4 * q^45 + 26 * q^53 + 14 * q^57 - 38 * q^61 - 36 * q^65 + 10 * q^69 - 66 * q^73 + 2 * q^77 - 36 * q^81 - 28 * q^85 - 100 * q^89 - 40 * q^93 - 52 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 9x^{6} + 22x^{4} - 13x^{2} + 2 $ x^8 - 9*x^6 + 22*x^4 - 13*x^2 + 2 :

$\beta_{1}$	$=$	$ \nu^{2} - 2 $ v^2 - 2
$\beta_{2}$	$=$	$ \nu^{6} - 8\nu^{4} + 15\nu^{2} - 3 $ v^6 - 8v^4 + 15v^2 - 3
$\beta_{3}$	$=$	$ \nu^{7} - 9\nu^{5} + 21\nu^{3} - 8\nu $ v^7 - 9v^5 + 21v^3 - 8*v
$\beta_{4}$	$=$	$ \nu^{7} - 9\nu^{5} + 22\nu^{3} - 12\nu $ v^7 - 9v^5 + 22v^3 - 12*v
$\beta_{5}$	$=$	$ \nu^{6} - 9\nu^{4} + 21\nu^{2} - 8 $ v^6 - 9v^4 + 21v^2 - 8
$\beta_{6}$	$=$	$ 2\nu^{7} - 17\nu^{5} + 36\nu^{3} - 10\nu $ 2v^7 - 17v^5 + 36v^3 - 10v
$\beta_{7}$	$=$	$ -2\nu^{7} + 17\nu^{5} - 36\nu^{3} + 12\nu $ -2v^7 + 17v^5 - 36v^3 + 12v

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

$\nu$	$=$	$ ( \beta_{7} + \beta_{6} ) / 2 $ (b7 + b6) / 2
$\nu^{2}$	$=$	$ \beta _1 + 2 $ b1 + 2
$\nu^{3}$	$=$	$ 2\beta_{7} + 2\beta_{6} + \beta_{4} - \beta_{3} $ 2b7 + 2b6 + b4 - b3
$\nu^{4}$	$=$	$ -\beta_{5} + \beta_{2} + 6\beta _1 + 7 $ -b5 + b2 + 6*b1 + 7
$\nu^{5}$	$=$	$ 9\beta_{7} + 10\beta_{6} + 6\beta_{4} - 8\beta_{3} $ 9b7 + 10b6 + 6b4 - 8b3
$\nu^{6}$	$=$	$ -8\beta_{5} + 9\beta_{2} + 33\beta _1 + 29 $ -8b5 + 9b2 + 33*b1 + 29
$\nu^{7}$	$=$	$ 43\beta_{7} + 52\beta_{6} + 33\beta_{4} - 50\beta_{3} $ 43b7 + 52b6 + 33b4 - 50b3

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

1.71353
−0.496801
−2.30128
0.721892
−0.721892
2.30128
0.496801
−1.71353

−2.36923

−0.446484

−3.82220

2.61326

1.2

−1.66437

1.59698

2.04697

−0.229876

1.3

−1.54999

3.96941

2.47616

−0.597538

1.4

−0.462764

−2.11992

1.75195

−2.78585

1.5

0.462764

−2.11992

−1.75195

−2.78585

1.6

1.54999

3.96941

−2.47616

−0.597538

1.7

1.66437

1.59698

−2.04697

−0.229876

1.8

2.36923

−0.446484

3.82220

2.61326

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$113$	$ -1 $

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7232.2.a.be		8
4.b	odd	2	1	inner	7232.2.a.be		8
8.b	even	2	1		3616.2.a.c	✓	8
8.d	odd	2	1		3616.2.a.c	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3616.2.a.c	✓	8	8.b	even	2	1
3616.2.a.c	✓	8	8.d	odd	2	1
7232.2.a.be		8	1.a	even	1	1	trivial
7232.2.a.be		8	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7232))$:

$ T_{3}^{8} - 11T_{3}^{6} + 38T_{3}^{4} - 45T_{3}^{2} + 8 $

T3^8 - 11*T3^6 + 38*T3^4 - 45*T3^2 + 8

$ T_{5}^{4} - 3T_{5}^{3} - 7T_{5}^{2} + 11T_{5} + 6 $

T5^4 - 3*T5^3 - 7*T5^2 + 11*T5 + 6

$ T_{7}^{8} - 28T_{7}^{6} + 253T_{7}^{4} - 917T_{7}^{2} + 1152 $

T7^8 - 28*T7^6 + 253*T7^4 - 917*T7^2 + 1152

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - 11 T^{6} + \cdots + 8 $ T^8 - 11T^6 + 38T^4 - 45*T^2 + 8
$5$	$ (T^{4} - 3 T^{3} - 7 T^{2} + \cdots + 6)^{2} $ (T^4 - 3T^3 - 7T^2 + 11*T + 6)^2
$7$	$ T^{8} - 28 T^{6} + \cdots + 1152 $ T^8 - 28T^6 + 253T^4 - 917*T^2 + 1152
$11$	$ T^{8} - 52 T^{6} + \cdots + 2592 $ T^8 - 52T^6 + 559T^4 - 2117*T^2 + 2592
$13$	$ (T^{4} + 4 T^{3} - 15 T^{2} + \cdots + 54)^{2} $ (T^4 + 4T^3 - 15T^2 - 45*T + 54)^2
$17$	$ (T^{4} + 10 T^{3} + \cdots + 58)^{2} $ (T^4 + 10T^3 + 3T^2 - 99*T + 58)^2
$19$	$ T^{8} - 92 T^{6} + \cdots + 648 $ T^8 - 92T^6 + 1757T^4 - 2165*T^2 + 648
$23$	$ T^{8} - 138 T^{6} + \cdots + 383688 $ T^8 - 138T^6 + 6299T^4 - 105221*T^2 + 383688
$29$	$ (T^{4} - 3 T^{3} + \cdots + 942)^{2} $ (T^4 - 3T^3 - 66T^2 + 97*T + 942)^2
$31$	$ T^{8} - 119 T^{6} + \cdots + 119072 $ T^8 - 119T^6 + 3712T^4 - 38909*T^2 + 119072
$37$	$ (T^{4} - 2 T^{3} - 43 T^{2} + \cdots - 2)^{2} $ (T^4 - 2T^3 - 43T^2 + 75*T - 2)^2
$41$	$ (T^{4} + 13 T^{3} + 38 T^{2} + \cdots - 6)^{2} $ (T^4 + 13T^3 + 38T^2 - 29*T - 6)^2
$43$	$ T^{8} - 106 T^{6} + \cdots + 159048 $ T^8 - 106T^6 + 3271T^4 - 39101*T^2 + 159048
$47$	$ T^{8} - 151 T^{6} + \cdots + 19208 $ T^8 - 151T^6 + 7502T^4 - 122941*T^2 + 19208
$53$	$ (T^{4} - 13 T^{3} + \cdots - 9634)^{2} $ (T^4 - 13T^3 - 148T^2 + 2705*T - 9634)^2
$59$	$ T^{8} - 297 T^{6} + \cdots + 6013512 $ T^8 - 297T^6 + 25988T^4 - 799085*T^2 + 6013512
$61$	$ (T^{4} + 19 T^{3} + \cdots - 1362)^{2} $ (T^4 + 19T^3 + 70T^2 - 307*T - 1362)^2
$67$	$ T^{8} - 105 T^{6} + \cdots + 137288 $ T^8 - 105T^6 + 3508T^4 - 39781*T^2 + 137288
$71$	$ T^{8} - 436 T^{6} + \cdots + 86528 $ T^8 - 436T^6 + 47696T^4 - 434368*T^2 + 86528
$73$	$ (T^{4} + 33 T^{3} + \cdots + 1718)^{2} $ (T^4 + 33T^3 + 370T^2 + 1577*T + 1718)^2
$79$	$ T^{8} - 455 T^{6} + \cdots + 100309448 $ T^8 - 455T^6 + 72692T^4 - 4729053*T^2 + 100309448
$83$	$ T^{8} - 298 T^{6} + \cdots + 14536832 $ T^8 - 298T^6 + 29351T^4 - 1143445*T^2 + 14536832
$89$	$ (T^{4} + 50 T^{3} + \cdots + 17098)^{2} $ (T^4 + 50T^3 + 893T^2 + 6669*T + 17098)^2
$97$	$ (T^{4} + 26 T^{3} + \cdots - 1458)^{2} $ (T^4 + 26T^3 + 153T^2 - 81*T - 1458)^2
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Level:	\( N \)	\(=\)	\( 7232 = 2^{6} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7232.a (trivial)

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{3} + ( - \beta_{5} + \beta_1) q^{5} - \beta_{7} q^{7} - \beta_{2} q^{9} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{11} + (\beta_{2} - \beta_1 - 1) q^{13} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{15}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{3}) q^{99}+O(q^{100}) \) q - b3 * q^3 + (-b5 + b1) * q^5 - b7 * q^7 - b2 * q^9 + (b6 + b4 + b3) * q^11 + (b2 - b1 - 1) * q^13 + (b7 + b6 - b3) * q^15 + (-b5 + b2 - b1 - 3) * q^17 + (-b6 - 2b4) q^19 + (2b5 - 2b2 - b1 + 2) * q^21 + (2b7 + b4 - b3) q^23 + (-b5 + 2b2 + 3b1 - 1) * q^25 + (-b7 + 2b3) q^27 + (3b5 - b2 - 2b1 + 3) * q^29 + (-2b6 - b4 + 2b3) * q^31 + (b5 + 2b2 + 2b1 - 4) * q^33 + (-b7 - 2b6 + 2b3) * q^35 + (b5 - 2b2 - 2b1 + 2) * q^37 + (b7 - b6 - b4 + 3b3) q^39 + (2b5 - b1 - 2) q^41 + (-2b6 + b4 + b3) q^43 + (b5 + b2 - b1 + 1) * q^45 + (-2b6 - 3b3) * q^47 + (b5 - 2b2 + 1) q^49 + (2b7 - b6 - 2b4 + 5b3) q^51 + (4b5 + 2b2 + 3b1 + 4) q^53 + (b6 + 3b4 - b3) q^55 + (-2b5 - 2b2 - 3b1 + 2) q^57 + (-b7 - 2b4 + 5b3) * q^59 + (b5 + 3b2 - 5) q^61 + (-b7 - b6 + b4 - 3b3) q^63 + (-3b2 - 3b1 - 3) * q^65 + (b7 + 2b4 - b3) q^67 + (-3b5 + 4b2 + 3b1 - 2) q^69 + (b7 + 3b6 - 4b4) * q^71 + (2b5 - b2 - 7) q^73 + (3b7 + 3b6 + 2b4) q^75 + (-3b5 + b2 - 2b1 - 1) * q^77 + (3b7 + 3b6 - 3b4 + 2b3) * q^79 + (2b5 + 3b2 - b1 - 4) * q^81 + (b7 + 3b6 - b4 - 3b3) * q^83 + (b5 - 3b2 - 5b1 - 1) * q^85 + (-4b7 - 2b6 + b4 - 2b3) q^87 + (-b5 + 2b2 + 2b1 - 14) * q^89 + (3b7 + 3b6 + 3b3) q^91 + (-b5 + b2 - 3b1 - 5) q^93 + (-b7 - 4b6 - 4b4 + 2b3) q^95 + (3b5 - b2 + b1 - 5) q^97 + (b7 - b6 + b3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{5} - 2 q^{9} - 8 q^{13} - 20 q^{17} + 2 q^{21} + 6 q^{25} + 6 q^{29} - 28 q^{33} + 4 q^{37} - 26 q^{41} + 4 q^{45} + 26 q^{53} + 14 q^{57} - 38 q^{61} - 36 q^{65} + 10 q^{69} - 66 q^{73} + 2 q^{77}+ \cdots - 52 q^{97}+O(q^{100}) \) 8 * q + 6 * q^5 - 2 * q^9 - 8 * q^13 - 20 * q^17 + 2 * q^21 + 6 * q^25 + 6 * q^29 - 28 * q^33 + 4 * q^37 - 26 * q^41 + 4 * q^45 + 26 * q^53 + 14 * q^57 - 38 * q^61 - 36 * q^65 + 10 * q^69 - 66 * q^73 + 2 * q^77 - 36 * q^81 - 28 * q^85 - 100 * q^89 - 40 * q^93 - 52 * q^97

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