# Properties

 Label 2496.4.a.c Level $2496$ Weight $4$ Character orbit 2496.a Self dual yes Analytic conductor $147.269$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 4 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 4 * q^5 + 4 * q^7 + 9 * q^9 $$q - 3 q^{3} - 4 q^{5} + 4 q^{7} + 9 q^{9} - 2 q^{11} + 13 q^{13} + 12 q^{15} - 6 q^{17} + 36 q^{19} - 12 q^{21} - 20 q^{23} - 109 q^{25} - 27 q^{27} + 14 q^{29} - 152 q^{31} + 6 q^{33} - 16 q^{35} + 258 q^{37} - 39 q^{39} + 84 q^{41} + 188 q^{43} - 36 q^{45} + 254 q^{47} - 327 q^{49} + 18 q^{51} - 366 q^{53} + 8 q^{55} - 108 q^{57} - 550 q^{59} + 14 q^{61} + 36 q^{63} - 52 q^{65} - 448 q^{67} + 60 q^{69} + 926 q^{71} + 254 q^{73} + 327 q^{75} - 8 q^{77} + 1328 q^{79} + 81 q^{81} - 186 q^{83} + 24 q^{85} - 42 q^{87} - 336 q^{89} + 52 q^{91} + 456 q^{93} - 144 q^{95} + 614 q^{97} - 18 q^{99}+O(q^{100})$$ q - 3 * q^3 - 4 * q^5 + 4 * q^7 + 9 * q^9 - 2 * q^11 + 13 * q^13 + 12 * q^15 - 6 * q^17 + 36 * q^19 - 12 * q^21 - 20 * q^23 - 109 * q^25 - 27 * q^27 + 14 * q^29 - 152 * q^31 + 6 * q^33 - 16 * q^35 + 258 * q^37 - 39 * q^39 + 84 * q^41 + 188 * q^43 - 36 * q^45 + 254 * q^47 - 327 * q^49 + 18 * q^51 - 366 * q^53 + 8 * q^55 - 108 * q^57 - 550 * q^59 + 14 * q^61 + 36 * q^63 - 52 * q^65 - 448 * q^67 + 60 * q^69 + 926 * q^71 + 254 * q^73 + 327 * q^75 - 8 * q^77 + 1328 * q^79 + 81 * q^81 - 186 * q^83 + 24 * q^85 - 42 * q^87 - 336 * q^89 + 52 * q^91 + 456 * q^93 - 144 * q^95 + 614 * q^97 - 18 * 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
 0
0 −3.00000 0 −4.00000 0 4.00000 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.c 1
4.b odd 2 1 2496.4.a.l 1
8.b even 2 1 78.4.a.f 1
8.d odd 2 1 624.4.a.c 1
24.f even 2 1 1872.4.a.f 1
24.h odd 2 1 234.4.a.c 1
40.f even 2 1 1950.4.a.a 1
104.e even 2 1 1014.4.a.e 1
104.j odd 4 2 1014.4.b.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.f 1 8.b even 2 1
234.4.a.c 1 24.h odd 2 1
624.4.a.c 1 8.d odd 2 1
1014.4.a.e 1 104.e even 2 1
1014.4.b.g 2 104.j odd 4 2
1872.4.a.f 1 24.f even 2 1
1950.4.a.a 1 40.f even 2 1
2496.4.a.c 1 1.a even 1 1 trivial
2496.4.a.l 1 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2496))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{7} - 4$$ T7 - 4 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 4$$
$7$ $$T - 4$$
$11$ $$T + 2$$
$13$ $$T - 13$$
$17$ $$T + 6$$
$19$ $$T - 36$$
$23$ $$T + 20$$
$29$ $$T - 14$$
$31$ $$T + 152$$
$37$ $$T - 258$$
$41$ $$T - 84$$
$43$ $$T - 188$$
$47$ $$T - 254$$
$53$ $$T + 366$$
$59$ $$T + 550$$
$61$ $$T - 14$$
$67$ $$T + 448$$
$71$ $$T - 926$$
$73$ $$T - 254$$
$79$ $$T - 1328$$
$83$ $$T + 186$$
$89$ $$T + 336$$
$97$ $$T - 614$$