# Properties

 Label 2496.4.a.bb Level $2496$ Weight $4$ Character orbit 2496.a Self dual yes Analytic conductor $147.269$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 10$$ x^2 - 10 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 $$\beta = 2\sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + (\beta - 12) q^{5} + ( - 3 \beta - 4) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (b - 12) * q^5 + (-3*b - 4) * q^7 + 9 * q^9 $$q + 3 q^{3} + (\beta - 12) q^{5} + ( - 3 \beta - 4) q^{7} + 9 q^{9} + ( - 8 \beta + 18) q^{11} + 13 q^{13} + (3 \beta - 36) q^{15} + (18 \beta - 6) q^{17} + (9 \beta + 60) q^{19} + ( - 9 \beta - 12) q^{21} + (16 \beta - 36) q^{23} + ( - 24 \beta + 59) q^{25} + 27 q^{27} + ( - 8 \beta - 42) q^{29} + (21 \beta - 184) q^{31} + ( - 24 \beta + 54) q^{33} + (32 \beta - 72) q^{35} + ( - 30 \beta - 78) q^{37} + 39 q^{39} + ( - 45 \beta - 108) q^{41} + (42 \beta + 52) q^{43} + (9 \beta - 108) q^{45} + ( - 34 \beta + 330) q^{47} + (24 \beta + 33) q^{49} + (54 \beta - 18) q^{51} + 618 q^{53} + (114 \beta - 536) q^{55} + (27 \beta + 180) q^{57} + ( - 58 \beta - 234) q^{59} + (84 \beta + 326) q^{61} + ( - 27 \beta - 36) q^{63} + (13 \beta - 156) q^{65} + (87 \beta - 128) q^{67} + (48 \beta - 108) q^{69} + (8 \beta + 378) q^{71} + (54 \beta - 562) q^{73} + ( - 72 \beta + 177) q^{75} + ( - 22 \beta + 888) q^{77} + (12 \beta + 160) q^{79} + 81 q^{81} + ( - 42 \beta + 330) q^{83} + ( - 222 \beta + 792) q^{85} + ( - 24 \beta - 126) q^{87} + ( - 3 \beta - 1056) q^{89} + ( - 39 \beta - 52) q^{91} + (63 \beta - 552) q^{93} + ( - 48 \beta - 360) q^{95} + ( - 102 \beta - 58) q^{97} + ( - 72 \beta + 162) q^{99}+O(q^{100})$$ q + 3 * q^3 + (b - 12) * q^5 + (-3*b - 4) * q^7 + 9 * q^9 + (-8*b + 18) * q^11 + 13 * q^13 + (3*b - 36) * q^15 + (18*b - 6) * q^17 + (9*b + 60) * q^19 + (-9*b - 12) * q^21 + (16*b - 36) * q^23 + (-24*b + 59) * q^25 + 27 * q^27 + (-8*b - 42) * q^29 + (21*b - 184) * q^31 + (-24*b + 54) * q^33 + (32*b - 72) * q^35 + (-30*b - 78) * q^37 + 39 * q^39 + (-45*b - 108) * q^41 + (42*b + 52) * q^43 + (9*b - 108) * q^45 + (-34*b + 330) * q^47 + (24*b + 33) * q^49 + (54*b - 18) * q^51 + 618 * q^53 + (114*b - 536) * q^55 + (27*b + 180) * q^57 + (-58*b - 234) * q^59 + (84*b + 326) * q^61 + (-27*b - 36) * q^63 + (13*b - 156) * q^65 + (87*b - 128) * q^67 + (48*b - 108) * q^69 + (8*b + 378) * q^71 + (54*b - 562) * q^73 + (-72*b + 177) * q^75 + (-22*b + 888) * q^77 + (12*b + 160) * q^79 + 81 * q^81 + (-42*b + 330) * q^83 + (-222*b + 792) * q^85 + (-24*b - 126) * q^87 + (-3*b - 1056) * q^89 + (-39*b - 52) * q^91 + (63*b - 552) * q^93 + (-48*b - 360) * q^95 + (-102*b - 58) * q^97 + (-72*b + 162) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 24 q^{5} - 8 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 24 * q^5 - 8 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 24 q^{5} - 8 q^{7} + 18 q^{9} + 36 q^{11} + 26 q^{13} - 72 q^{15} - 12 q^{17} + 120 q^{19} - 24 q^{21} - 72 q^{23} + 118 q^{25} + 54 q^{27} - 84 q^{29} - 368 q^{31} + 108 q^{33} - 144 q^{35} - 156 q^{37} + 78 q^{39} - 216 q^{41} + 104 q^{43} - 216 q^{45} + 660 q^{47} + 66 q^{49} - 36 q^{51} + 1236 q^{53} - 1072 q^{55} + 360 q^{57} - 468 q^{59} + 652 q^{61} - 72 q^{63} - 312 q^{65} - 256 q^{67} - 216 q^{69} + 756 q^{71} - 1124 q^{73} + 354 q^{75} + 1776 q^{77} + 320 q^{79} + 162 q^{81} + 660 q^{83} + 1584 q^{85} - 252 q^{87} - 2112 q^{89} - 104 q^{91} - 1104 q^{93} - 720 q^{95} - 116 q^{97} + 324 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 24 * q^5 - 8 * q^7 + 18 * q^9 + 36 * q^11 + 26 * q^13 - 72 * q^15 - 12 * q^17 + 120 * q^19 - 24 * q^21 - 72 * q^23 + 118 * q^25 + 54 * q^27 - 84 * q^29 - 368 * q^31 + 108 * q^33 - 144 * q^35 - 156 * q^37 + 78 * q^39 - 216 * q^41 + 104 * q^43 - 216 * q^45 + 660 * q^47 + 66 * q^49 - 36 * q^51 + 1236 * q^53 - 1072 * q^55 + 360 * q^57 - 468 * q^59 + 652 * q^61 - 72 * q^63 - 312 * q^65 - 256 * q^67 - 216 * q^69 + 756 * q^71 - 1124 * q^73 + 354 * q^75 + 1776 * q^77 + 320 * q^79 + 162 * q^81 + 660 * q^83 + 1584 * q^85 - 252 * q^87 - 2112 * q^89 - 104 * q^91 - 1104 * q^93 - 720 * q^95 - 116 * q^97 + 324 * 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
 −3.16228 3.16228
0 3.00000 0 −18.3246 0 14.9737 0 9.00000 0
1.2 0 3.00000 0 −5.67544 0 −22.9737 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.bb 2
4.b odd 2 1 2496.4.a.t 2
8.b even 2 1 624.4.a.m 2
8.d odd 2 1 156.4.a.d 2
24.f even 2 1 468.4.a.d 2
24.h odd 2 1 1872.4.a.s 2
104.h odd 2 1 2028.4.a.e 2
104.m even 4 2 2028.4.b.f 4

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

 $$T_{5}^{2} + 24T_{5} + 104$$ T5^2 + 24*T5 + 104 $$T_{7}^{2} + 8T_{7} - 344$$ T7^2 + 8*T7 - 344 $$T_{11}^{2} - 36T_{11} - 2236$$ T11^2 - 36*T11 - 2236

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 24T + 104$$
$7$ $$T^{2} + 8T - 344$$
$11$ $$T^{2} - 36T - 2236$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} + 12T - 12924$$
$19$ $$T^{2} - 120T + 360$$
$23$ $$T^{2} + 72T - 8944$$
$29$ $$T^{2} + 84T - 796$$
$31$ $$T^{2} + 368T + 16216$$
$37$ $$T^{2} + 156T - 29916$$
$41$ $$T^{2} + 216T - 69336$$
$43$ $$T^{2} - 104T - 67856$$
$47$ $$T^{2} - 660T + 62660$$
$53$ $$(T - 618)^{2}$$
$59$ $$T^{2} + 468T - 79804$$
$61$ $$T^{2} - 652T - 175964$$
$67$ $$T^{2} + 256T - 286376$$
$71$ $$T^{2} - 756T + 140324$$
$73$ $$T^{2} + 1124 T + 199204$$
$79$ $$T^{2} - 320T + 19840$$
$83$ $$T^{2} - 660T + 38340$$
$89$ $$T^{2} + 2112 T + 1114776$$
$97$ $$T^{2} + 116T - 412796$$