LMFDB - Newform orbit 3364.2.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3364,2,Mod(1,3364)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3364, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3364.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3364 = 2^{2} \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3364.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.8616752400$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6456289.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 12x^{4} + 3x^{3} + 40x^{2} + 6x - 29$ x^6 - x^5 - 12x^4 + 3x^3 + 40x^2 + 6x - 29
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 116)
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.

$f(q)$	$=$	$q + ( - \beta_1 + 1) q^{3} + ( - \beta_{5} - \beta_{2} + 1) q^{5} + (\beta_1 + 1) q^{7} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 4) q^{13}+ \cdots + (\beta_{5} - \beta_{4} + \cdots - 4 \beta_1) q^{99}+O(q^{100})$ q + (-b1 + 1) * q^3 + (-b5 - b2 + 1) * q^5 + (b1 + 1) * q^7 + (b3 - b2 - b1 + 2) * q^9 + (-b3 - b2 - b1 + 1) * q^11 + (-2b4 + b3 + 2b2 + 4) * q^13 + (-b5 + 3b4 - 2b2 - 2b1) q^15 + (-b5 + 2b4 - b3 - b2 + b1 - 1) q^17 + (2b5 + 2b4 - 2b2 - b1 + 1) q^19 + (-b3 + b2 - b1 - 3) * q^21 + (b5 + b4 + b3 + b2 + b1 + 3) * q^23 + (-3b5 - b3 - b2 + 1) q^25 + (-b5 - b4 + b3 + b2 + 4) * q^27 + (-b5 - 3b4 - b3 + b2 - b1 - 1) q^31 + (b5 + b4 - b3 - 5b2 - 2b1 + 2) * q^33 + (-b5 - 3b4 + 2b1 + 2) * q^35 + (-2b4 + b3 + 2b1 - 2) * q^37 + (b5 - 3b4 + 3b3 + 5b2 - 2b1 + 8) * q^39 + (2b5 - 4b4 + 2b3 + b2 + b1 + 3) q^41 + (b5 - 3b4 + b3 + 3b2 + b1 + 5) * q^43 + (-b5 + 6b4 + b3 - 2b2 - 2b1 + 3) q^45 + (2b5 - 4b4 + b3 + 3b2 - b1 - 1) q^47 + (b3 - b2 + 3b1 - 2) q^49 + (-2b5 + 6b4 - 2b3 - 4b2 - 8) * q^51 + (-3b4 + 4) q^53 + (-2b5 + 2b4 - 3b3 - 7b2 - 2b1 - 2) q^55 + (-4b4 - 3b3 - b2 - 3b1 + 3) q^57 + (-2b3 + 2b1 + 2) * q^59 + (-b5 + 2b4 + b3 + b2 - 4) q^61 + (b5 + b4 + b3 - 3b2 + b1 - 3) q^63 + (-3b5 - 2b4 + 2b3 + 5b2 + 2b1 + 6) q^65 + (4b5 - b3 + b2 - b1 + 3) q^67 + (-b5 - 3b4 + 6b2 - 2b1 + 2) q^69 + (-b3 + 7b2 - b1 + 1) q^71 + (b4 + 4b2 + 2b1 + 6) * q^73 + (-2b5 + 10b4 + b3 - 7b2 - 2b1 - 2) * q^75 + (-b5 - b4 - b3 + 3b2) q^77 + (6b4 - 3b3 + 3b2 + b1 - 7) q^79 + (-b5 + b4 + 6b2 + 1) q^81 + (-2b4 - 2b3 - 3b1 + 1) q^83 + (-2b5 - 3b4 - b3 - 3b2 + 3) q^85 + (3b5 + 2b3 - 3b2 - 2b1 - 1) * q^89 + (-b5 - b4 - b3 - b2 + 2b1) q^91 + (3b5 + b4 + 2b3 - 4b2 + 2b1 + 2) * q^93 + (3b5 + 9b4 - 12b2 - 4b1 - 10) * q^95 + (-4b3 - b2 + 8) q^97 + (b5 - b4 - 2b3 - 6b2 - 4b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 5 q^{3} + 10 q^{5} + 7 q^{7} + 11 q^{9} + 9 q^{11} + 14 q^{13} + 10 q^{15} + 5 q^{17} + 9 q^{19} - 19 q^{21} + 15 q^{23} + 16 q^{25} + 20 q^{27} - 11 q^{31} + 22 q^{33} + 10 q^{35} - 16 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100})$ 6 * q + 5 * q^3 + 10 * q^5 + 7 * q^7 + 11 * q^9 + 9 * q^11 + 14 * q^13 + 10 * q^15 + 5 * q^17 + 9 * q^19 - 19 * q^21 + 15 * q^23 + 16 * q^25 + 20 * q^27 - 11 * q^31 + 22 * q^33 + 10 * q^35 - 16 * q^37 + 22 * q^39 + q^41 + 15 * q^43 + 32 * q^45 - 27 * q^47 - 9 * q^49 - 20 * q^51 + 18 * q^53 + 14 * q^55 + 15 * q^57 + 18 * q^59 - 22 * q^61 - 13 * q^63 + 26 * q^65 + 9 * q^67 - 6 * q^69 - 7 * q^71 + 32 * q^73 + 22 * q^75 - 4 * q^77 - 29 * q^79 - 2 * q^81 + 3 * q^83 + 24 * q^85 - 12 * q^89 + 6 * q^91 + 14 * q^93 - 28 * q^95 + 58 * q^97 + 8 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - x^{5} - 12x^{4} + 3x^{3} + 40x^{2} + 6x - 29$ x^6 - x^5 - 12*x^4 + 3*x^3 + 40*x^2 + 6*x - 29 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 13\nu^{2} + 23\nu - 17 ) / 4$ (v^5 - 2v^4 - 10v^3 + 13v^2 + 23v - 17) / 4
$\beta_{3}$	$=$	$( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 17\nu^{2} + 19\nu - 33 ) / 4$ (v^5 - 2v^4 - 10v^3 + 17v^2 + 19v - 33) / 4
$\beta_{4}$	$=$	$( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 18\nu^{2} - 13\nu + 22 ) / 2$ (-v^5 + 3v^4 + 7v^3 - 18v^2 - 13v + 22) / 2
$\beta_{5}$	$=$	$( 2\nu^{5} - 5\nu^{4} - 15\nu^{3} + 27\nu^{2} + 28\nu - 29 ) / 2$ (2v^5 - 5v^4 - 15v^3 + 27v^2 + 28*v - 29) / 2

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} - \beta_{2} + \beta _1 + 4$ b3 - b2 + b1 + 4
$\nu^{3}$	$=$	$\beta_{5} + \beta_{4} + 2\beta_{3} - 4\beta_{2} + 6\beta _1 + 3$ b5 + b4 + 2b3 - 4b2 + 6*b1 + 3
$\nu^{4}$	$=$	$3\beta_{5} + 5\beta_{4} + 11\beta_{3} - 13\beta_{2} + 13\beta _1 + 24$ 3b5 + 5b4 + 11b3 - 13b2 + 13*b1 + 24
$\nu^{5}$	$=$	$16\beta_{5} + 20\beta_{4} + 29\beta_{3} - 49\beta_{2} + 50\beta _1 + 43$ 16b5 + 20b4 + 29b3 - 49b2 + 50*b1 + 43

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

3.27029
2.09189
0.839985
−1.46835
−1.64685
−2.08697

−2.27029

1.34652

4.27029

2.15422

1.2

−1.09189

4.05359

3.09189

−1.80777

1.3

0.160015

−1.76058

1.83999

−2.97440

1.4

2.46835

3.45542

−0.468352

3.09276

1.5

2.64685

−0.608550

−0.646853

4.00583

1.6

3.08697

3.51360

−1.08697

6.52935

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$29$	$+1$

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	3364.2.a.p		6
29.b	even	2	1		3364.2.a.m		6
29.c	odd	4	2		3364.2.c.j		12
29.e	even	14	2		116.2.g.b	✓	12
87.h	odd	14	2		1044.2.u.c		12
116.h	odd	14	2		464.2.u.g		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
116.2.g.b	✓	12	29.e	even	14	2
464.2.u.g		12	116.h	odd	14	2
1044.2.u.c		12	87.h	odd	14	2
3364.2.a.m		6	29.b	even	2	1
3364.2.a.p		6	1.a	even	1	1	trivial
3364.2.c.j		12	29.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3364))$ :

T_{3}^{6} - 5T_{3}^{5} - 2T_{3}^{4} + 35T_{3}^{3} - 18T_{3}^{2} - 48T_{3} + 8

T_{5}^{6} - 10T_{5}^{5} + 27T_{5}^{4} + 14T_{5}^{3} - 120T_{5}^{2} + 46T_{5} + 71

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} - 5 T^{5} + \cdots + 8$ T^6 - 5T^5 - 2T^4 + 35T^3 - 18T^2 - 48*T + 8
$5$	$T^{6} - 10 T^{5} + \cdots + 71$ T^6 - 10T^5 + 27T^4 + 14T^3 - 120T^2 + 46*T + 71
$7$	$T^{6} - 7 T^{5} + \cdots - 8$ T^6 - 7T^5 + 8T^4 + 21T^3 - 16T^2 - 28*T - 8
$11$	$T^{6} - 9 T^{5} + \cdots + 216$ T^6 - 9T^5 + 8T^4 + 73T^3 - 102T^2 - 144*T + 216
$13$	$T^{6} - 14 T^{5} + \cdots - 1$ T^6 - 14T^5 + 51T^4 + 42T^3 - 428T^2 + 378*T - 1
$17$	$T^{6} - 5 T^{5} + \cdots + 344$ T^6 - 5T^5 - 30T^4 + 245T^3 - 480T^2 + 92*T + 344
$19$	$T^{6} - 9 T^{5} + \cdots + 16904$ T^6 - 9T^5 - 48T^4 + 675T^3 - 1096T^2 - 6276*T + 16904
$23$	$T^{6} - 15 T^{5} + \cdots + 1016$ T^6 - 15T^5 + 52T^4 + 105T^3 - 590T^2 - 72*T + 1016
$29$	$T^{6}$ T^6
$31$	$T^{6} + 11 T^{5} + \cdots - 232$ T^6 + 11T^5 - 24T^4 - 325T^3 - 36T^2 + 1444*T - 232
$37$	$T^{6} + 16 T^{5} + \cdots + 12713$ T^6 + 16T^5 + 33T^4 - 532T^3 - 2160T^2 + 2486*T + 12713
$41$	$T^{6} - T^{5} + \cdots - 1624$ T^6 - T^5 - 142T^4 - 167T^3 + 5320T^2 + 16436T - 1624
$43$	$T^{6} - 15 T^{5} + \cdots - 3352$ T^6 - 15T^5 + 38T^4 + 399T^3 - 2634T^2 + 5360*T - 3352
$47$	$T^{6} + 27 T^{5} + \cdots - 128584$ T^6 + 27T^5 + 198T^4 - 711T^3 - 16648T^2 - 80172*T - 128584
$53$	$(T^{3} - 9 T^{2} + 6 T + 29)^{2}$ (T^3 - 9T^2 + 6T + 29)^2
$59$	$T^{6} - 18 T^{5} + \cdots + 19264$ T^6 - 18T^5 - 4T^4 + 1576T^3 - 8176T^2 + 6048*T + 19264
$61$	$T^{6} + 22 T^{5} + \cdots - 53033$ T^6 + 22T^5 + 107T^4 - 906T^3 - 11356T^2 - 41978*T - 53033
$67$	$T^{6} - 9 T^{5} + \cdots + 70904$ T^6 - 9T^5 - 226T^4 + 1785T^3 + 10650T^2 - 61748*T + 70904
$71$	$T^{6} + 7 T^{5} + \cdots + 9848$ T^6 + 7T^5 - 248T^4 - 903T^3 + 11714T^2 + 30352*T + 9848
$73$	$T^{6} - 32 T^{5} + \cdots + 6119$ T^6 - 32T^5 + 321T^4 - 742T^3 - 4614T^2 + 17678*T + 6119
$79$	$T^{6} + 29 T^{5} + \cdots - 630328$ T^6 + 29T^5 + 10T^4 - 7091T^3 - 85416T^2 - 389772*T - 630328
$83$	$T^{6} - 3 T^{5} + \cdots + 3688$ T^6 - 3T^5 - 164T^4 + 795T^3 + 2568T^2 - 7628*T + 3688
$89$	$T^{6} + 12 T^{5} + \cdots - 29609$ T^6 + 12T^5 - 157T^4 - 1268T^3 + 5210T^2 + 24972*T - 29609
$97$	$T^{6} - 58 T^{5} + \cdots - 1348999$ T^6 - 58T^5 + 1121T^4 - 5870T^3 - 54422T^2 + 617816*T - 1348999
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