LMFDB - Newform orbit 8624.2.a.dc

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,1,0,4,0,0,0,0,0,5,0,1,0,-7,0,9,0,-1,0,0,0,8,0,5,0,4,0,9, 0,-3,0,1,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.559701.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 10x - 3 $ x^5 - x^4 - 7x^3 + 4x^2 + 10*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 616)
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 $1,\beta_1,\beta_2,\beta_3,\beta_4$ 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_{2} - \beta_1 + 1) q^{5} + \beta_{2} q^{9} + q^{11} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{13} + (\beta_{3} + 2 \beta_1 - 2) q^{15} + (\beta_{4} + \beta_{2} + 2) q^{17} + ( - \beta_{3} + \beta_{2}) q^{19}+ \cdots + \beta_{2} q^{99}+O(q^{100}) $ q + b1 * q^3 + (b2 - b1 + 1) * q^5 + b2 * q^9 + q^11 + (-b4 - b3 + b1) * q^13 + (b3 + 2b1 - 2) q^15 + (b4 + b2 + 2) * q^17 + (-b3 + b2) * q^19 + (b4 - b3 + b2 + 2) * q^23 + (b4 - b3 + 2b2 - 3b1 + 2) * q^25 + (b3 + b2 - 2b1 + 1) q^27 + (-b4 - b3 + b2 - b1 + 2) * q^29 + (b3 - b2 + b1 - 1) * q^31 + b1 * q^33 + (-b4 + b3 + 2b2) q^37 + (-b4 - b3 - b2 + 2b1 + 1) q^39 + (b4 + 2b3 - b1 + 3) q^41 + (-b4 - 3b3 + 3b1 - 3) * q^43 + (b4 + b2 + 4) * q^45 + (b4 + b2 - 2) * q^47 + (2b3 + b2 + 2b1 + 2) * q^51 + (-b4 - 2b3 - 3b1 - 1) * q^53 + (b2 - b1 + 1) * q^55 + (-b4 + b3 - b2 + 2b1) q^57 + (b4 - 2b3 - 2b2 - b1) * q^59 + (-b4 + 2b3 - 2b2 - 1) * q^61 + (2b4 + b3 - 2b1 + 2) * q^65 + (-b3 + 4b1 + 1) q^67 + (-b4 + 2b3 - b2 + 3b1 + 1) * q^69 + (3b3 - b2 + 2b1 - 4) * q^71 + (-2b3 + 3b1 + 5) * q^73 + (-b4 + 3b3 - 3b2 + 4b1 - 7) q^75 + (-2b4 - 3b2 + b1) * q^79 + (b4 + b3 - 2b2 + b1 - 4) q^81 + (-3b4 - b2 - 2b1) * q^83 + (-b3 + 4b2 - b1 + 3) q^85 + (-b4 - 2b2 + 5b1 - 4) * q^87 + (-b3 + b2 - 3b1 + 9) q^89 + (b4 - b3 + 2b2 - 3b1 + 3) * q^93 + (2b4 - b3 + 2b2 - 3b1 + 5) q^95 + (-b4 - b3 + 2b2 - 3b1 + 2) * q^97 + b2 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + 4 q^{5} + 5 q^{11} + q^{13} - 7 q^{15} + 9 q^{17} - q^{19} + 8 q^{23} + 5 q^{25} + 4 q^{27} + 9 q^{29} - 3 q^{31} + q^{33} + 2 q^{37} + 7 q^{39} + 15 q^{41} - 14 q^{43} + 19 q^{45} - 11 q^{47}+ \cdots + 7 q^{97}+O(q^{100}) $ 5 * q + q^3 + 4 * q^5 + 5 * q^11 + q^13 - 7 * q^15 + 9 * q^17 - q^19 + 8 * q^23 + 5 * q^25 + 4 * q^27 + 9 * q^29 - 3 * q^31 + q^33 + 2 * q^37 + 7 * q^39 + 15 * q^41 - 14 * q^43 + 19 * q^45 - 11 * q^47 + 14 * q^51 - 9 * q^53 + 4 * q^55 + 4 * q^57 - 4 * q^59 - 2 * q^61 + 7 * q^65 + 8 * q^67 + 11 * q^69 - 15 * q^71 + 26 * q^73 - 27 * q^75 + 3 * q^79 - 19 * q^81 + q^83 + 13 * q^85 - 14 * q^87 + 41 * q^89 + 10 * q^93 + 19 * q^95 + 7 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 10x - 3 $ x^5 - x^4 - 7*x^3 + 4*x^2 + 10*x - 3 :

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

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} + 3 $ b2 + 3
$\nu^{3}$	$=$	$ \beta_{3} + \beta_{2} + 4\beta _1 + 1 $ b3 + b2 + 4*b1 + 1
$\nu^{4}$	$=$	$ \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 $ b4 + b3 + 7*b2 + b1 + 14

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

−2.04091
−1.33366
0.284226
1.49300
2.59735

−2.04091

4.20624

1.16532

1.2

−1.33366

1.11230

−1.22136

1.3

0.284226

−2.20344

−2.91922

1.4

1.49300

−1.26396

−0.770961

1.5

2.59735

2.14886

3.74621

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$7$	$ +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	8624.2.a.dc		5
4.b	odd	2	1		4312.2.a.bf		5
7.b	odd	2	1		8624.2.a.db		5
7.c	even	3	2		1232.2.q.o		10
28.d	even	2	1		4312.2.a.bg		5
28.g	odd	6	2		616.2.q.f	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
616.2.q.f	✓	10	28.g	odd	6	2
1232.2.q.o		10	7.c	even	3	2
4312.2.a.bf		5	4.b	odd	2	1
4312.2.a.bg		5	28.d	even	2	1
8624.2.a.db		5	7.b	odd	2	1
8624.2.a.dc		5	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 $S_{2}^{\mathrm{new}}(\Gamma_0(8624))$:

$ T_{3}^{5} - T_{3}^{4} - 7T_{3}^{3} + 4T_{3}^{2} + 10T_{3} - 3 $

T3^5 - T3^4 - 7*T3^3 + 4*T3^2 + 10*T3 - 3

$ T_{5}^{5} - 4T_{5}^{4} - 7T_{5}^{3} + 25T_{5}^{2} + 10T_{5} - 28 $

T5^5 - 4*T5^4 - 7*T5^3 + 25*T5^2 + 10*T5 - 28

$ T_{13}^{5} - T_{13}^{4} - 39T_{13}^{3} + 10T_{13}^{2} + 12T_{13} + 1 $

T13^5 - T13^4 - 39*T13^3 + 10*T13^2 + 12*T13 + 1

$ T_{17}^{5} - 9T_{17}^{4} + 2T_{17}^{3} + 127T_{17}^{2} - 144T_{17} - 268 $

T17^5 - 9*T17^4 + 2*T17^3 + 127*T17^2 - 144*T17 - 268

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{5} $ T^5
$3$	$ T^{5} - T^{4} - 7 T^{3} + \cdots - 3 $ T^5 - T^4 - 7T^3 + 4T^2 + 10*T - 3
$5$	$ T^{5} - 4 T^{4} + \cdots - 28 $ T^5 - 4T^4 - 7T^3 + 25T^2 + 10T - 28
$7$	$ T^{5} $ T^5
$11$	$ (T - 1)^{5} $ (T - 1)^5
$13$	$ T^{5} - T^{4} - 39 T^{3} + \cdots + 1 $ T^5 - T^4 - 39T^3 + 10T^2 + 12*T + 1
$17$	$ T^{5} - 9 T^{4} + \cdots - 268 $ T^5 - 9T^4 + 2T^3 + 127T^2 - 144T - 268
$19$	$ T^{5} + T^{4} + \cdots - 172 $ T^5 + T^4 - 26T^3 - T^2 + 170T - 172
$23$	$ T^{5} - 8 T^{4} + \cdots + 868 $ T^5 - 8T^4 - 19T^3 + 329T^2 - 974T + 868
$29$	$ T^{5} - 9 T^{4} + \cdots + 401 $ T^5 - 9T^4 - 35T^3 + 474T^2 - 1084T + 401
$31$	$ T^{5} + 3 T^{4} + \cdots - 16 $ T^5 + 3T^4 - 26T^3 + 3T^2 + 68T - 16
$37$	$ T^{5} - 2 T^{4} + \cdots - 96 $ T^5 - 2T^4 - 112T^3 + 272T^2 + 1024T - 96
$41$	$ T^{5} - 15 T^{4} + \cdots - 516 $ T^5 - 15T^4 + 8T^3 + 645T^2 - 2114T - 516
$43$	$ T^{5} + 14 T^{4} + \cdots - 5488 $ T^5 + 14T^4 - 109T^3 - 2099T^2 - 6668T - 5488
$47$	$ T^{5} + 11 T^{4} + \cdots + 36 $ T^5 + 11T^4 + 18T^3 - 73T^2 - 56T + 36
$53$	$ T^{5} + 9 T^{4} + \cdots - 2468 $ T^5 + 9T^4 - 152T^3 - 873T^2 + 3962T - 2468
$59$	$ T^{5} + 4 T^{4} + \cdots - 14353 $ T^5 + 4T^4 - 180T^3 + 77T^2 + 5703T - 14353
$61$	$ T^{5} + 2 T^{4} + \cdots + 1256 $ T^5 + 2T^4 - 115T^3 - 596T^2 - 356T + 1256
$67$	$ T^{5} - 8 T^{4} + \cdots - 4136 $ T^5 - 8T^4 - 97T^3 + 560T^2 + 2548T - 4136
$71$	$ T^{5} + 15 T^{4} + \cdots - 5348 $ T^5 + 15T^4 - 88T^3 - 703T^2 + 4350T - 5348
$73$	$ T^{5} - 26 T^{4} + \cdots + 10796 $ T^5 - 26T^4 + 159T^3 + 383T^2 - 5370T + 10796
$79$	$ T^{5} - 3 T^{4} + \cdots + 6787 $ T^5 - 3T^4 - 143T^3 + 156T^2 + 4718T + 6787
$83$	$ T^{5} - T^{4} + \cdots - 37116 $ T^5 - T^4 - 292T^3 - 23T^2 + 13972*T - 37116
$89$	$ T^{5} - 41 T^{4} + \cdots - 12148 $ T^5 - 41T^4 + 592T^3 - 3853T^2 + 11372T - 12148
$97$	$ T^{5} - 7 T^{4} + \cdots - 709 $ T^5 - 7T^4 - 133T^3 + 670T^2 + 70T - 709
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Level:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

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

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