# Properties

 Label 2496.4.a.w Level $2496$ Weight $4$ Character orbit 2496.a Self dual yes Analytic conductor $147.269$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2496, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2496.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2496.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: $$147.268767374$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 22$$ x^2 - 22 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 156) 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 $$\beta = 2\sqrt{22}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + \beta q^{5} + ( - 3 \beta - 4) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + b * q^5 + (-3*b - 4) * q^7 + 9 * q^9 $$q - 3 q^{3} + \beta q^{5} + ( - 3 \beta - 4) q^{7} + 9 q^{9} + ( - 2 \beta - 30) q^{11} + 13 q^{13} - 3 \beta q^{15} + ( - 6 \beta - 54) q^{17} + ( - 3 \beta - 108) q^{19} + (9 \beta + 12) q^{21} + ( - 8 \beta + 108) q^{23} - 37 q^{25} - 27 q^{27} + ( - 20 \beta + 54) q^{29} + ( - 15 \beta - 40) q^{31} + (6 \beta + 90) q^{33} + ( - 4 \beta - 264) q^{35} + (18 \beta - 54) q^{37} - 39 q^{39} + (15 \beta - 24) q^{41} + ( - 54 \beta + 4) q^{43} + 9 \beta q^{45} + (44 \beta + 114) q^{47} + (24 \beta + 465) q^{49} + (18 \beta + 162) q^{51} + ( - 12 \beta - 270) q^{53} + ( - 30 \beta - 176) q^{55} + (9 \beta + 324) q^{57} + ( - 40 \beta - 426) q^{59} + ( - 12 \beta - 154) q^{61} + ( - 27 \beta - 36) q^{63} + 13 \beta q^{65} + (39 \beta - 152) q^{67} + (24 \beta - 324) q^{69} + ( - 82 \beta + 114) q^{71} + (6 \beta + 710) q^{73} + 111 q^{75} + (98 \beta + 648) q^{77} + (60 \beta - 248) q^{79} + 81 q^{81} + (12 \beta + 618) q^{83} + ( - 54 \beta - 528) q^{85} + (60 \beta - 162) q^{87} + ( - 63 \beta - 708) q^{89} + ( - 39 \beta - 52) q^{91} + (45 \beta + 120) q^{93} + ( - 108 \beta - 264) q^{95} + ( - 30 \beta + 302) q^{97} + ( - 18 \beta - 270) q^{99} +O(q^{100})$$ q - 3 * q^3 + b * q^5 + (-3*b - 4) * q^7 + 9 * q^9 + (-2*b - 30) * q^11 + 13 * q^13 - 3*b * q^15 + (-6*b - 54) * q^17 + (-3*b - 108) * q^19 + (9*b + 12) * q^21 + (-8*b + 108) * q^23 - 37 * q^25 - 27 * q^27 + (-20*b + 54) * q^29 + (-15*b - 40) * q^31 + (6*b + 90) * q^33 + (-4*b - 264) * q^35 + (18*b - 54) * q^37 - 39 * q^39 + (15*b - 24) * q^41 + (-54*b + 4) * q^43 + 9*b * q^45 + (44*b + 114) * q^47 + (24*b + 465) * q^49 + (18*b + 162) * q^51 + (-12*b - 270) * q^53 + (-30*b - 176) * q^55 + (9*b + 324) * q^57 + (-40*b - 426) * q^59 + (-12*b - 154) * q^61 + (-27*b - 36) * q^63 + 13*b * q^65 + (39*b - 152) * q^67 + (24*b - 324) * q^69 + (-82*b + 114) * q^71 + (6*b + 710) * q^73 + 111 * q^75 + (98*b + 648) * q^77 + (60*b - 248) * q^79 + 81 * q^81 + (12*b + 618) * q^83 + (-54*b - 528) * q^85 + (60*b - 162) * q^87 + (-63*b - 708) * q^89 + (-39*b - 52) * q^91 + (45*b + 120) * q^93 + (-108*b - 264) * q^95 + (-30*b + 302) * q^97 + (-18*b - 270) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 8 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 8 * q^7 + 18 * q^9 $$2 q - 6 q^{3} - 8 q^{7} + 18 q^{9} - 60 q^{11} + 26 q^{13} - 108 q^{17} - 216 q^{19} + 24 q^{21} + 216 q^{23} - 74 q^{25} - 54 q^{27} + 108 q^{29} - 80 q^{31} + 180 q^{33} - 528 q^{35} - 108 q^{37} - 78 q^{39} - 48 q^{41} + 8 q^{43} + 228 q^{47} + 930 q^{49} + 324 q^{51} - 540 q^{53} - 352 q^{55} + 648 q^{57} - 852 q^{59} - 308 q^{61} - 72 q^{63} - 304 q^{67} - 648 q^{69} + 228 q^{71} + 1420 q^{73} + 222 q^{75} + 1296 q^{77} - 496 q^{79} + 162 q^{81} + 1236 q^{83} - 1056 q^{85} - 324 q^{87} - 1416 q^{89} - 104 q^{91} + 240 q^{93} - 528 q^{95} + 604 q^{97} - 540 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 8 * q^7 + 18 * q^9 - 60 * q^11 + 26 * q^13 - 108 * q^17 - 216 * q^19 + 24 * q^21 + 216 * q^23 - 74 * q^25 - 54 * q^27 + 108 * q^29 - 80 * q^31 + 180 * q^33 - 528 * q^35 - 108 * q^37 - 78 * q^39 - 48 * q^41 + 8 * q^43 + 228 * q^47 + 930 * q^49 + 324 * q^51 - 540 * q^53 - 352 * q^55 + 648 * q^57 - 852 * q^59 - 308 * q^61 - 72 * q^63 - 304 * q^67 - 648 * q^69 + 228 * q^71 + 1420 * q^73 + 222 * q^75 + 1296 * q^77 - 496 * q^79 + 162 * q^81 + 1236 * q^83 - 1056 * q^85 - 324 * q^87 - 1416 * q^89 - 104 * q^91 + 240 * q^93 - 528 * q^95 + 604 * q^97 - 540 * q^99

## 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
 −4.69042 4.69042
0 −3.00000 0 −9.38083 0 24.1425 0 9.00000 0
1.2 0 −3.00000 0 9.38083 0 −32.1425 0 9.00000 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
$$2$$ $$-1$$
$$3$$ $$+1$$
$$13$$ $$-1$$

## 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 2496.4.a.w 2
4.b odd 2 1 2496.4.a.bf 2
8.b even 2 1 624.4.a.p 2
8.d odd 2 1 156.4.a.c 2
24.f even 2 1 468.4.a.g 2
24.h odd 2 1 1872.4.a.y 2
104.h odd 2 1 2028.4.a.d 2
104.m even 4 2 2028.4.b.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.a.c 2 8.d odd 2 1
468.4.a.g 2 24.f even 2 1
624.4.a.p 2 8.b even 2 1
1872.4.a.y 2 24.h odd 2 1
2028.4.a.d 2 104.h odd 2 1
2028.4.b.e 4 104.m even 4 2
2496.4.a.w 2 1.a even 1 1 trivial
2496.4.a.bf 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2496))$$:

 $$T_{5}^{2} - 88$$ T5^2 - 88 $$T_{7}^{2} + 8T_{7} - 776$$ T7^2 + 8*T7 - 776 $$T_{11}^{2} + 60T_{11} + 548$$ T11^2 + 60*T11 + 548

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} - 88$$
$7$ $$T^{2} + 8T - 776$$
$11$ $$T^{2} + 60T + 548$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} + 108T - 252$$
$19$ $$T^{2} + 216T + 10872$$
$23$ $$T^{2} - 216T + 6032$$
$29$ $$T^{2} - 108T - 32284$$
$31$ $$T^{2} + 80T - 18200$$
$37$ $$T^{2} + 108T - 25596$$
$41$ $$T^{2} + 48T - 19224$$
$43$ $$T^{2} - 8T - 256592$$
$47$ $$T^{2} - 228T - 157372$$
$53$ $$T^{2} + 540T + 60228$$
$59$ $$T^{2} + 852T + 40676$$
$61$ $$T^{2} + 308T + 11044$$
$67$ $$T^{2} + 304T - 110744$$
$71$ $$T^{2} - 228T - 578716$$
$73$ $$T^{2} - 1420 T + 500932$$
$79$ $$T^{2} + 496T - 255296$$
$83$ $$T^{2} - 1236 T + 369252$$
$89$ $$T^{2} + 1416 T + 151992$$
$97$ $$T^{2} - 604T + 12004$$