# Properties

 Label 2496.4.a.s 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: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 39) 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 $$\beta = 2\sqrt{14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + (\beta - 12) q^{5} + \beta q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (b - 12) * q^5 + b * q^7 + 9 * q^9 $$q - 3 q^{3} + (\beta - 12) q^{5} + \beta q^{7} + 9 q^{9} + ( - 6 \beta - 22) q^{11} + 13 q^{13} + ( - 3 \beta + 36) q^{15} + (2 \beta + 82) q^{17} + (\beta + 24) q^{19} - 3 \beta q^{21} + ( - 24 \beta - 4) q^{23} + ( - 24 \beta + 75) q^{25} - 27 q^{27} + (12 \beta - 202) q^{29} + (13 \beta - 20) q^{31} + (18 \beta + 66) q^{33} + ( - 12 \beta + 56) q^{35} + ( - 14 \beta + 50) q^{37} - 39 q^{39} + (47 \beta + 100) q^{41} + (26 \beta - 308) q^{43} + (9 \beta - 108) q^{45} + ( - 16 \beta + 162) q^{47} - 287 q^{49} + ( - 6 \beta - 246) q^{51} + (60 \beta + 82) q^{53} + (50 \beta - 72) q^{55} + ( - 3 \beta - 72) q^{57} + (20 \beta + 70) q^{59} + ( - 68 \beta - 314) q^{61} + 9 \beta q^{63} + (13 \beta - 156) q^{65} + ( - 85 \beta - 236) q^{67} + (72 \beta + 12) q^{69} + (42 \beta - 214) q^{71} + (38 \beta - 450) q^{73} + (72 \beta - 225) q^{75} + ( - 22 \beta - 336) q^{77} + (44 \beta + 216) q^{79} + 81 q^{81} + (32 \beta - 694) q^{83} + (58 \beta - 872) q^{85} + ( - 36 \beta + 606) q^{87} + ( - 95 \beta + 480) q^{89} + 13 \beta q^{91} + ( - 39 \beta + 60) q^{93} + (12 \beta - 232) q^{95} + ( - 110 \beta - 266) q^{97} + ( - 54 \beta - 198) q^{99} +O(q^{100})$$ q - 3 * q^3 + (b - 12) * q^5 + b * q^7 + 9 * q^9 + (-6*b - 22) * q^11 + 13 * q^13 + (-3*b + 36) * q^15 + (2*b + 82) * q^17 + (b + 24) * q^19 - 3*b * q^21 + (-24*b - 4) * q^23 + (-24*b + 75) * q^25 - 27 * q^27 + (12*b - 202) * q^29 + (13*b - 20) * q^31 + (18*b + 66) * q^33 + (-12*b + 56) * q^35 + (-14*b + 50) * q^37 - 39 * q^39 + (47*b + 100) * q^41 + (26*b - 308) * q^43 + (9*b - 108) * q^45 + (-16*b + 162) * q^47 - 287 * q^49 + (-6*b - 246) * q^51 + (60*b + 82) * q^53 + (50*b - 72) * q^55 + (-3*b - 72) * q^57 + (20*b + 70) * q^59 + (-68*b - 314) * q^61 + 9*b * q^63 + (13*b - 156) * q^65 + (-85*b - 236) * q^67 + (72*b + 12) * q^69 + (42*b - 214) * q^71 + (38*b - 450) * q^73 + (72*b - 225) * q^75 + (-22*b - 336) * q^77 + (44*b + 216) * q^79 + 81 * q^81 + (32*b - 694) * q^83 + (58*b - 872) * q^85 + (-36*b + 606) * q^87 + (-95*b + 480) * q^89 + 13*b * q^91 + (-39*b + 60) * q^93 + (12*b - 232) * q^95 + (-110*b - 266) * q^97 + (-54*b - 198) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 24 q^{5} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 24 * q^5 + 18 * q^9 $$2 q - 6 q^{3} - 24 q^{5} + 18 q^{9} - 44 q^{11} + 26 q^{13} + 72 q^{15} + 164 q^{17} + 48 q^{19} - 8 q^{23} + 150 q^{25} - 54 q^{27} - 404 q^{29} - 40 q^{31} + 132 q^{33} + 112 q^{35} + 100 q^{37} - 78 q^{39} + 200 q^{41} - 616 q^{43} - 216 q^{45} + 324 q^{47} - 574 q^{49} - 492 q^{51} + 164 q^{53} - 144 q^{55} - 144 q^{57} + 140 q^{59} - 628 q^{61} - 312 q^{65} - 472 q^{67} + 24 q^{69} - 428 q^{71} - 900 q^{73} - 450 q^{75} - 672 q^{77} + 432 q^{79} + 162 q^{81} - 1388 q^{83} - 1744 q^{85} + 1212 q^{87} + 960 q^{89} + 120 q^{93} - 464 q^{95} - 532 q^{97} - 396 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 24 * q^5 + 18 * q^9 - 44 * q^11 + 26 * q^13 + 72 * q^15 + 164 * q^17 + 48 * q^19 - 8 * q^23 + 150 * q^25 - 54 * q^27 - 404 * q^29 - 40 * q^31 + 132 * q^33 + 112 * q^35 + 100 * q^37 - 78 * q^39 + 200 * q^41 - 616 * q^43 - 216 * q^45 + 324 * q^47 - 574 * q^49 - 492 * q^51 + 164 * q^53 - 144 * q^55 - 144 * q^57 + 140 * q^59 - 628 * q^61 - 312 * q^65 - 472 * q^67 + 24 * q^69 - 428 * q^71 - 900 * q^73 - 450 * q^75 - 672 * q^77 + 432 * q^79 + 162 * q^81 - 1388 * q^83 - 1744 * q^85 + 1212 * q^87 + 960 * q^89 + 120 * q^93 - 464 * q^95 - 532 * q^97 - 396 * 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
 −3.74166 3.74166
0 −3.00000 0 −19.4833 0 −7.48331 0 9.00000 0
1.2 0 −3.00000 0 −4.51669 0 7.48331 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.s 2
4.b odd 2 1 2496.4.a.bc 2
8.b even 2 1 624.4.a.r 2
8.d odd 2 1 39.4.a.b 2
24.f even 2 1 117.4.a.c 2
24.h odd 2 1 1872.4.a.t 2
40.e odd 2 1 975.4.a.j 2
56.e even 2 1 1911.4.a.h 2
104.h odd 2 1 507.4.a.f 2
104.m even 4 2 507.4.b.f 4
312.h even 2 1 1521.4.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 8.d odd 2 1
117.4.a.c 2 24.f even 2 1
507.4.a.f 2 104.h odd 2 1
507.4.b.f 4 104.m even 4 2
624.4.a.r 2 8.b even 2 1
975.4.a.j 2 40.e odd 2 1
1521.4.a.s 2 312.h even 2 1
1872.4.a.t 2 24.h odd 2 1
1911.4.a.h 2 56.e even 2 1
2496.4.a.s 2 1.a even 1 1 trivial
2496.4.a.bc 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2496))$$:

 $$T_{5}^{2} + 24T_{5} + 88$$ T5^2 + 24*T5 + 88 $$T_{7}^{2} - 56$$ T7^2 - 56 $$T_{11}^{2} + 44T_{11} - 1532$$ T11^2 + 44*T11 - 1532

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 24T + 88$$
$7$ $$T^{2} - 56$$
$11$ $$T^{2} + 44T - 1532$$
$13$ $$(T - 13)^{2}$$
$17$ $$T^{2} - 164T + 6500$$
$19$ $$T^{2} - 48T + 520$$
$23$ $$T^{2} + 8T - 32240$$
$29$ $$T^{2} + 404T + 32740$$
$31$ $$T^{2} + 40T - 9064$$
$37$ $$T^{2} - 100T - 8476$$
$41$ $$T^{2} - 200T - 113704$$
$43$ $$T^{2} + 616T + 57008$$
$47$ $$T^{2} - 324T + 11908$$
$53$ $$T^{2} - 164T - 194876$$
$59$ $$T^{2} - 140T - 17500$$
$61$ $$T^{2} + 628T - 160348$$
$67$ $$T^{2} + 472T - 348904$$
$71$ $$T^{2} + 428T - 52988$$
$73$ $$T^{2} + 900T + 121636$$
$79$ $$T^{2} - 432T - 61760$$
$83$ $$T^{2} + 1388 T + 424292$$
$89$ $$T^{2} - 960T - 275000$$
$97$ $$T^{2} + 532T - 606844$$