Properties

 Label 504.6.a.m Level $504$ Weight $6$ Character orbit 504.a Self dual yes Analytic conductor $80.833$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("504.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 504.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: $$80.8334451857$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{193})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 48$$ x^2 - x - 48 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 56) 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 $$\beta = \sqrt{193}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 5 \beta - 21) q^{5} - 49 q^{7}+O(q^{10})$$ q + (-5*b - 21) * q^5 - 49 * q^7 $$q + ( - 5 \beta - 21) q^{5} - 49 q^{7} + (14 \beta + 358) q^{11} + ( - 17 \beta - 357) q^{13} + (54 \beta + 672) q^{17} + (93 \beta - 973) q^{19} + ( - 112 \beta + 896) q^{23} + (210 \beta + 2141) q^{25} + (546 \beta + 600) q^{29} + ( - 446 \beta - 3402) q^{31} + (245 \beta + 1029) q^{35} + ( - 154 \beta + 7320) q^{37} + (442 \beta - 3948) q^{41} + (518 \beta + 262) q^{43} + (746 \beta + 9198) q^{47} + 2401 q^{49} + ( - 1008 \beta - 22566) q^{53} + ( - 2084 \beta - 21028) q^{55} + ( - 365 \beta - 11291) q^{59} + (1117 \beta - 26411) q^{61} + (2142 \beta + 23902) q^{65} + (224 \beta + 4924) q^{67} + ( - 4060 \beta + 420) q^{71} + ( - 1204 \beta - 61026) q^{73} + ( - 686 \beta - 17542) q^{77} + ( - 1372 \beta + 15852) q^{79} + ( - 4185 \beta - 18487) q^{83} + ( - 4494 \beta - 66222) q^{85} + ( - 1512 \beta + 105294) q^{89} + (833 \beta + 17493) q^{91} + (2912 \beta - 69312) q^{95} + (8882 \beta - 22120) q^{97}+O(q^{100})$$ q + (-5*b - 21) * q^5 - 49 * q^7 + (14*b + 358) * q^11 + (-17*b - 357) * q^13 + (54*b + 672) * q^17 + (93*b - 973) * q^19 + (-112*b + 896) * q^23 + (210*b + 2141) * q^25 + (546*b + 600) * q^29 + (-446*b - 3402) * q^31 + (245*b + 1029) * q^35 + (-154*b + 7320) * q^37 + (442*b - 3948) * q^41 + (518*b + 262) * q^43 + (746*b + 9198) * q^47 + 2401 * q^49 + (-1008*b - 22566) * q^53 + (-2084*b - 21028) * q^55 + (-365*b - 11291) * q^59 + (1117*b - 26411) * q^61 + (2142*b + 23902) * q^65 + (224*b + 4924) * q^67 + (-4060*b + 420) * q^71 + (-1204*b - 61026) * q^73 + (-686*b - 17542) * q^77 + (-1372*b + 15852) * q^79 + (-4185*b - 18487) * q^83 + (-4494*b - 66222) * q^85 + (-1512*b + 105294) * q^89 + (833*b + 17493) * q^91 + (2912*b - 69312) * q^95 + (8882*b - 22120) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 42 q^{5} - 98 q^{7}+O(q^{10})$$ 2 * q - 42 * q^5 - 98 * q^7 $$2 q - 42 q^{5} - 98 q^{7} + 716 q^{11} - 714 q^{13} + 1344 q^{17} - 1946 q^{19} + 1792 q^{23} + 4282 q^{25} + 1200 q^{29} - 6804 q^{31} + 2058 q^{35} + 14640 q^{37} - 7896 q^{41} + 524 q^{43} + 18396 q^{47} + 4802 q^{49} - 45132 q^{53} - 42056 q^{55} - 22582 q^{59} - 52822 q^{61} + 47804 q^{65} + 9848 q^{67} + 840 q^{71} - 122052 q^{73} - 35084 q^{77} + 31704 q^{79} - 36974 q^{83} - 132444 q^{85} + 210588 q^{89} + 34986 q^{91} - 138624 q^{95} - 44240 q^{97}+O(q^{100})$$ 2 * q - 42 * q^5 - 98 * q^7 + 716 * q^11 - 714 * q^13 + 1344 * q^17 - 1946 * q^19 + 1792 * q^23 + 4282 * q^25 + 1200 * q^29 - 6804 * q^31 + 2058 * q^35 + 14640 * q^37 - 7896 * q^41 + 524 * q^43 + 18396 * q^47 + 4802 * q^49 - 45132 * q^53 - 42056 * q^55 - 22582 * q^59 - 52822 * q^61 + 47804 * q^65 + 9848 * q^67 + 840 * q^71 - 122052 * q^73 - 35084 * q^77 + 31704 * q^79 - 36974 * q^83 - 132444 * q^85 + 210588 * q^89 + 34986 * q^91 - 138624 * q^95 - 44240 * q^97

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
 7.44622 −6.44622
0 0 0 −90.4622 0 −49.0000 0 0 0
1.2 0 0 0 48.4622 0 −49.0000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$+1$$

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 504.6.a.m 2
3.b odd 2 1 56.6.a.d 2
4.b odd 2 1 1008.6.a.bi 2
12.b even 2 1 112.6.a.j 2
21.c even 2 1 392.6.a.e 2
21.g even 6 2 392.6.i.h 4
21.h odd 6 2 392.6.i.k 4
24.f even 2 1 448.6.a.r 2
24.h odd 2 1 448.6.a.x 2
84.h odd 2 1 784.6.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.d 2 3.b odd 2 1
112.6.a.j 2 12.b even 2 1
392.6.a.e 2 21.c even 2 1
392.6.i.h 4 21.g even 6 2
392.6.i.k 4 21.h odd 6 2
448.6.a.r 2 24.f even 2 1
448.6.a.x 2 24.h odd 2 1
504.6.a.m 2 1.a even 1 1 trivial
784.6.a.q 2 84.h odd 2 1
1008.6.a.bi 2 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(504))$$:

 $$T_{5}^{2} + 42T_{5} - 4384$$ T5^2 + 42*T5 - 4384 $$T_{11}^{2} - 716T_{11} + 90336$$ T11^2 - 716*T11 + 90336

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 42T - 4384$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} - 716T + 90336$$
$13$ $$T^{2} + 714T + 71672$$
$17$ $$T^{2} - 1344 T - 111204$$
$19$ $$T^{2} + 1946 T - 722528$$
$23$ $$T^{2} - 1792 T - 1618176$$
$29$ $$T^{2} - 1200 T - 57176388$$
$31$ $$T^{2} + 6804 T - 26817184$$
$37$ $$T^{2} - 14640 T + 49005212$$
$41$ $$T^{2} + 7896 T - 22118548$$
$43$ $$T^{2} - 524 T - 51717888$$
$47$ $$T^{2} - 18396 T - 22804384$$
$53$ $$T^{2} + 45132 T + 313124004$$
$59$ $$T^{2} + 22582 T + 101774256$$
$61$ $$T^{2} + 52822 T + 456736944$$
$67$ $$T^{2} - 9848 T + 14561808$$
$71$ $$T^{2} + \cdots - 3181158400$$
$73$ $$T^{2} + \cdots + 3444396788$$
$79$ $$T^{2} - 31704 T - 112014208$$
$83$ $$T^{2} + \cdots - 3038476256$$
$89$ $$T^{2} + \cdots + 10645600644$$
$97$ $$T^{2} + \cdots - 14736460932$$
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