# Properties

 Label 2496.4.a.v 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: $$\Q(\sqrt{113})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 28$$ x^2 - x - 28 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 312) 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 $$\beta = \sqrt{113}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + ( - \beta - 3) q^{5} + (\beta + 5) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (-b - 3) * q^5 + (b + 5) * q^7 + 9 * q^9 $$q - 3 q^{3} + ( - \beta - 3) q^{5} + (\beta + 5) q^{7} + 9 q^{9} + ( - 4 \beta - 8) q^{11} - 13 q^{13} + (3 \beta + 9) q^{15} + 2 q^{17} + (5 \beta - 35) q^{19} + ( - 3 \beta - 15) q^{21} + 64 q^{23} + (6 \beta - 3) q^{25} - 27 q^{27} + (6 \beta + 40) q^{29} + ( - 15 \beta + 125) q^{31} + (12 \beta + 24) q^{33} + ( - 8 \beta - 128) q^{35} + (18 \beta + 76) q^{37} + 39 q^{39} + ( - 7 \beta - 73) q^{41} + (8 \beta - 252) q^{43} + ( - 9 \beta - 27) q^{45} + (6 \beta + 262) q^{47} + (10 \beta - 205) q^{49} - 6 q^{51} + (8 \beta + 26) q^{53} + (20 \beta + 476) q^{55} + ( - 15 \beta + 105) q^{57} + (10 \beta - 82) q^{59} + (46 \beta + 152) q^{61} + (9 \beta + 45) q^{63} + (13 \beta + 39) q^{65} + (7 \beta - 457) q^{67} - 192 q^{69} - 48 \beta q^{71} + ( - 6 \beta - 228) q^{73} + ( - 18 \beta + 9) q^{75} + ( - 28 \beta - 492) q^{77} + (36 \beta + 412) q^{79} + 81 q^{81} + (50 \beta + 414) q^{83} + ( - 2 \beta - 6) q^{85} + ( - 18 \beta - 120) q^{87} + ( - 9 \beta + 413) q^{89} + ( - 13 \beta - 65) q^{91} + (45 \beta - 375) q^{93} + (20 \beta - 460) q^{95} + (122 \beta + 276) q^{97} + ( - 36 \beta - 72) q^{99} +O(q^{100})$$ q - 3 * q^3 + (-b - 3) * q^5 + (b + 5) * q^7 + 9 * q^9 + (-4*b - 8) * q^11 - 13 * q^13 + (3*b + 9) * q^15 + 2 * q^17 + (5*b - 35) * q^19 + (-3*b - 15) * q^21 + 64 * q^23 + (6*b - 3) * q^25 - 27 * q^27 + (6*b + 40) * q^29 + (-15*b + 125) * q^31 + (12*b + 24) * q^33 + (-8*b - 128) * q^35 + (18*b + 76) * q^37 + 39 * q^39 + (-7*b - 73) * q^41 + (8*b - 252) * q^43 + (-9*b - 27) * q^45 + (6*b + 262) * q^47 + (10*b - 205) * q^49 - 6 * q^51 + (8*b + 26) * q^53 + (20*b + 476) * q^55 + (-15*b + 105) * q^57 + (10*b - 82) * q^59 + (46*b + 152) * q^61 + (9*b + 45) * q^63 + (13*b + 39) * q^65 + (7*b - 457) * q^67 - 192 * q^69 - 48*b * q^71 + (-6*b - 228) * q^73 + (-18*b + 9) * q^75 + (-28*b - 492) * q^77 + (36*b + 412) * q^79 + 81 * q^81 + (50*b + 414) * q^83 + (-2*b - 6) * q^85 + (-18*b - 120) * q^87 + (-9*b + 413) * q^89 + (-13*b - 65) * q^91 + (45*b - 375) * q^93 + (20*b - 460) * q^95 + (122*b + 276) * q^97 + (-36*b - 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 6 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 6 * q^5 + 10 * q^7 + 18 * q^9 $$2 q - 6 q^{3} - 6 q^{5} + 10 q^{7} + 18 q^{9} - 16 q^{11} - 26 q^{13} + 18 q^{15} + 4 q^{17} - 70 q^{19} - 30 q^{21} + 128 q^{23} - 6 q^{25} - 54 q^{27} + 80 q^{29} + 250 q^{31} + 48 q^{33} - 256 q^{35} + 152 q^{37} + 78 q^{39} - 146 q^{41} - 504 q^{43} - 54 q^{45} + 524 q^{47} - 410 q^{49} - 12 q^{51} + 52 q^{53} + 952 q^{55} + 210 q^{57} - 164 q^{59} + 304 q^{61} + 90 q^{63} + 78 q^{65} - 914 q^{67} - 384 q^{69} - 456 q^{73} + 18 q^{75} - 984 q^{77} + 824 q^{79} + 162 q^{81} + 828 q^{83} - 12 q^{85} - 240 q^{87} + 826 q^{89} - 130 q^{91} - 750 q^{93} - 920 q^{95} + 552 q^{97} - 144 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 6 * q^5 + 10 * q^7 + 18 * q^9 - 16 * q^11 - 26 * q^13 + 18 * q^15 + 4 * q^17 - 70 * q^19 - 30 * q^21 + 128 * q^23 - 6 * q^25 - 54 * q^27 + 80 * q^29 + 250 * q^31 + 48 * q^33 - 256 * q^35 + 152 * q^37 + 78 * q^39 - 146 * q^41 - 504 * q^43 - 54 * q^45 + 524 * q^47 - 410 * q^49 - 12 * q^51 + 52 * q^53 + 952 * q^55 + 210 * q^57 - 164 * q^59 + 304 * q^61 + 90 * q^63 + 78 * q^65 - 914 * q^67 - 384 * q^69 - 456 * q^73 + 18 * q^75 - 984 * q^77 + 824 * q^79 + 162 * q^81 + 828 * q^83 - 12 * q^85 - 240 * q^87 + 826 * q^89 - 130 * q^91 - 750 * q^93 - 920 * q^95 + 552 * q^97 - 144 * 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
 5.81507 −4.81507
0 −3.00000 0 −13.6301 0 15.6301 0 9.00000 0
1.2 0 −3.00000 0 7.63015 0 −5.63015 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.v 2
4.b odd 2 1 2496.4.a.be 2
8.b even 2 1 624.4.a.q 2
8.d odd 2 1 312.4.a.c 2
24.f even 2 1 936.4.a.d 2
24.h odd 2 1 1872.4.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.c 2 8.d odd 2 1
624.4.a.q 2 8.b even 2 1
936.4.a.d 2 24.f even 2 1
1872.4.a.x 2 24.h odd 2 1
2496.4.a.v 2 1.a even 1 1 trivial
2496.4.a.be 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2496))$$:

 $$T_{5}^{2} + 6T_{5} - 104$$ T5^2 + 6*T5 - 104 $$T_{7}^{2} - 10T_{7} - 88$$ T7^2 - 10*T7 - 88 $$T_{11}^{2} + 16T_{11} - 1744$$ T11^2 + 16*T11 - 1744

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 6T - 104$$
$7$ $$T^{2} - 10T - 88$$
$11$ $$T^{2} + 16T - 1744$$
$13$ $$(T + 13)^{2}$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 70T - 1600$$
$23$ $$(T - 64)^{2}$$
$29$ $$T^{2} - 80T - 2468$$
$31$ $$T^{2} - 250T - 9800$$
$37$ $$T^{2} - 152T - 30836$$
$41$ $$T^{2} + 146T - 208$$
$43$ $$T^{2} + 504T + 56272$$
$47$ $$T^{2} - 524T + 64576$$
$53$ $$T^{2} - 52T - 6556$$
$59$ $$T^{2} + 164T - 4576$$
$61$ $$T^{2} - 304T - 216004$$
$67$ $$T^{2} + 914T + 203312$$
$71$ $$T^{2} - 260352$$
$73$ $$T^{2} + 456T + 47916$$
$79$ $$T^{2} - 824T + 23296$$
$83$ $$T^{2} - 828T - 111104$$
$89$ $$T^{2} - 826T + 161416$$
$97$ $$T^{2} - 552 T - 1605716$$