LMFDB - Newform orbit 7744.2.a.de

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7744,2,Mod(1,7744)] 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("7744.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$7744 = 2^{6} \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7744.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,3,0,-4,0,7,0,0,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$61.8361513253$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 5x - 1$ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 3872)
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 $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{2} q^{3} + (\beta_{2} + 1) q^{5} + (\beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_1 + 2) q^{13} + (2 \beta_{2} - \beta_1 + 5) q^{15} + ( - \beta_{2} - \beta_1) q^{17} + (\beta_{2} + 4) q^{19}+ \cdots + (3 \beta_{2} + 3 \beta_1 + 4) q^{97}+O(q^{100})$ q + b2 * q^3 + (b2 + 1) * q^5 + (b1 - 1) * q^7 + (b2 - b1 + 2) * q^9 + (-b1 + 2) * q^13 + (2b2 - b1 + 5) q^15 + (-b2 - b1) * q^17 + (b2 + 4) * q^19 + (-3b2 - b1 + 1) q^21 + (b1 - 1) * q^23 + (3b2 - b1 + 1) q^25 + (2b2 + 4) q^27 + (-2b2 + b1) q^29 + (2b2 - b1 - 3) q^31 - 3b2 q^35 + b1 * q^37 + (4b2 + b1 - 1) q^39 + (2b2 - 1) q^41 + (-2b2 + 4) q^43 + (6b2 - b1 + 6) q^45 + (b1 + 3) * q^47 + (-3b2 - b1 + 6) q^49 + (b2 + 2b1 - 6) q^51 + (-b2 + 5) * q^53 + (5b2 - b1 + 5) q^57 + (-2b1 - 6) q^59 - 6 * q^61 + (b1 - 13) * q^63 + (4b2 + 1) q^65 + (-3b2 + 2b1 - 2) * q^67 + (-3b2 - b1 + 1) q^69 - 8 * q^71 + (3b2 + b1 + 5) q^73 + (6b2 - 2b1 + 14) * q^75 + (-4b2 - b1 - 3) q^79 + (3b2 + b1 + 4) q^81 + (3b2 + 2b1 + 2) * q^83 + (b1 - 6) * q^85 + (-4b2 + b1 - 9) q^87 + (b2 - b1 + 8) * q^89 + (3b2 + 2b1 - 14) * q^91 + (b2 - b1 + 9) * q^93 + (6b2 - b1 + 9) q^95 + (3b2 + 3b1 + 4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{5} - 4 q^{7} + 7 q^{9} + 7 q^{13} + 16 q^{15} + q^{17} + 12 q^{19} + 4 q^{21} - 4 q^{23} + 4 q^{25} + 12 q^{27} - q^{29} - 8 q^{31} - q^{37} - 4 q^{39} - 3 q^{41} + 12 q^{43} + 19 q^{45} + 8 q^{47}+ \cdots + 9 q^{97}+O(q^{100})$ 3 * q + 3 * q^5 - 4 * q^7 + 7 * q^9 + 7 * q^13 + 16 * q^15 + q^17 + 12 * q^19 + 4 * q^21 - 4 * q^23 + 4 * q^25 + 12 * q^27 - q^29 - 8 * q^31 - q^37 - 4 * q^39 - 3 * q^41 + 12 * q^43 + 19 * q^45 + 8 * q^47 + 19 * q^49 - 20 * q^51 + 15 * q^53 + 16 * q^57 - 16 * q^59 - 18 * q^61 - 40 * q^63 + 3 * q^65 - 8 * q^67 + 4 * q^69 - 24 * q^71 + 14 * q^73 + 44 * q^75 - 8 * q^79 + 11 * q^81 + 4 * q^83 - 19 * q^85 - 28 * q^87 + 25 * q^89 - 44 * q^91 + 28 * q^93 + 28 * q^95 + 9 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - x^{2} - 5x - 1$ x^3 - x^2 - 5*x - 1 :

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

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

$\nu$	$=$	$( -\beta_{2} + \beta _1 + 1 ) / 2$ (-b2 + b1 + 1) / 2
$\nu^{2}$	$=$	$\beta _1 + 4$ b1 + 4

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

−0.210756
2.86620
−1.65544

−2.53407

−1.53407

−4.95558

3.42151

1.2

−0.517304

0.482696

3.21509

−2.73240

1.3

3.05137

4.05137

−2.25951

6.31088

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$11$	$-1$

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	7744.2.a.de		3
4.b	odd	2	1		7744.2.a.dg		3
8.b	even	2	1		3872.2.a.bb	✓	3
8.d	odd	2	1		3872.2.a.bd	yes	3
11.b	odd	2	1		7744.2.a.df		3
44.c	even	2	1		7744.2.a.dd		3
88.b	odd	2	1		3872.2.a.be	yes	3
88.g	even	2	1		3872.2.a.bc	yes	3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3872.2.a.bb	✓	3	8.b	even	2	1
3872.2.a.bc	yes	3	88.g	even	2	1
3872.2.a.bd	yes	3	8.d	odd	2	1
3872.2.a.be	yes	3	88.b	odd	2	1
7744.2.a.dd		3	44.c	even	2	1
7744.2.a.de		3	1.a	even	1	1	trivial
7744.2.a.df		3	11.b	odd	2	1
7744.2.a.dg		3	4.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 $S_{2}^{\mathrm{new}}(\Gamma_0(7744))$ :

T_{3}^{3} - 8T_{3} - 4

T3^3 - 8*T3 - 4

T_{5}^{3} - 3T_{5}^{2} - 5T_{5} + 3

T5^3 - 3*T5^2 - 5*T5 + 3

T_{7}^{3} + 4T_{7}^{2} - 12T_{7} - 36

T7^3 + 4*T7^2 - 12*T7 - 36

T_{13}^{3} - 7T_{13}^{2} - T_{13} + 43

T13^3 - 7*T13^2 - T13 + 43

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} - 8T - 4$ T^3 - 8*T - 4
$5$	$T^{3} - 3 T^{2} + \cdots + 3$ T^3 - 3T^2 - 5T + 3
$7$	$T^{3} + 4 T^{2} + \cdots - 36$ T^3 + 4T^2 - 12T - 36
$11$	$T^{3}$ T^3
$13$	$T^{3} - 7T^{2} - T + 43$ T^3 - 7*T^2 - T + 43
$17$	$T^{3} - T^{2} + \cdots - 43$ T^3 - T^2 - 29*T - 43
$19$	$T^{3} - 12 T^{2} + \cdots - 36$ T^3 - 12T^2 + 40T - 36
$23$	$T^{3} + 4 T^{2} + \cdots - 36$ T^3 + 4T^2 - 12T - 36
$29$	$T^{3} + T^{2} + \cdots + 43$ T^3 + T^2 - 41*T + 43
$31$	$T^{3} + 8 T^{2} + \cdots - 148$ T^3 + 8T^2 - 20T - 148
$37$	$T^{3} + T^{2} + \cdots - 21$ T^3 + T^2 - 17*T - 21
$41$	$T^{3} + 3 T^{2} + \cdots - 63$ T^3 + 3T^2 - 29T - 63
$43$	$T^{3} - 12 T^{2} + \cdots + 96$ T^3 - 12T^2 + 16T + 96
$47$	$T^{3} - 8 T^{2} + \cdots + 12$ T^3 - 8T^2 + 4T + 12
$53$	$T^{3} - 15 T^{2} + \cdots - 81$ T^3 - 15T^2 + 67T - 81
$59$	$T^{3} + 16 T^{2} + \cdots - 96$ T^3 + 16T^2 + 16T - 96
$61$	$(T + 6)^{3}$ (T + 6)^3
$67$	$T^{3} + 8 T^{2} + \cdots - 252$ T^3 + 8T^2 - 96T - 252
$71$	$(T + 8)^{3}$ (T + 8)^3
$73$	$T^{3} - 14 T^{2} + \cdots + 648$ T^3 - 14T^2 - 36T + 648
$79$	$T^{3} + 8 T^{2} + \cdots - 796$ T^3 + 8T^2 - 140T - 796
$83$	$T^{3} - 4 T^{2} + \cdots + 1036$ T^3 - 4T^2 - 160T + 1036
$89$	$T^{3} - 25 T^{2} + \cdots - 379$ T^3 - 25T^2 + 187T - 379
$97$	$T^{3} - 9 T^{2} + \cdots + 2189$ T^3 - 9T^2 - 237T + 2189
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