# Properties

 Label 2496.2.c.d Level $2496$ Weight $2$ Character orbit 2496.c Analytic conductor $19.931$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2496 = 2^{6} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2496.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.9306603445$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 + b * q^7 + q^9 $$q - q^{3} + \beta q^{7} + q^{9} - \beta q^{11} + ( - \beta + 1) q^{13} - 6 q^{17} + \beta q^{19} - \beta q^{21} + 5 q^{25} - q^{27} - 6 q^{29} + \beta q^{31} + \beta q^{33} - 2 \beta q^{37} + (\beta - 1) q^{39} - 2 \beta q^{41} - 4 q^{43} - \beta q^{47} - 5 q^{49} + 6 q^{51} - 6 q^{53} - \beta q^{57} + 3 \beta q^{59} + 2 q^{61} + \beta q^{63} - 3 \beta q^{67} - \beta q^{71} - 5 q^{75} + 12 q^{77} + 8 q^{79} + q^{81} - \beta q^{83} + 6 q^{87} - 2 \beta q^{89} + (\beta + 12) q^{91} - \beta q^{93} - 4 \beta q^{97} - \beta q^{99} +O(q^{100})$$ q - q^3 + b * q^7 + q^9 - b * q^11 + (-b + 1) * q^13 - 6 * q^17 + b * q^19 - b * q^21 + 5 * q^25 - q^27 - 6 * q^29 + b * q^31 + b * q^33 - 2*b * q^37 + (b - 1) * q^39 - 2*b * q^41 - 4 * q^43 - b * q^47 - 5 * q^49 + 6 * q^51 - 6 * q^53 - b * q^57 + 3*b * q^59 + 2 * q^61 + b * q^63 - 3*b * q^67 - b * q^71 - 5 * q^75 + 12 * q^77 + 8 * q^79 + q^81 - b * q^83 + 6 * q^87 - 2*b * q^89 + (b + 12) * q^91 - b * q^93 - 4*b * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} + 2 q^{13} - 12 q^{17} + 10 q^{25} - 2 q^{27} - 12 q^{29} - 2 q^{39} - 8 q^{43} - 10 q^{49} + 12 q^{51} - 12 q^{53} + 4 q^{61} - 10 q^{75} + 24 q^{77} + 16 q^{79} + 2 q^{81} + 12 q^{87} + 24 q^{91}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 2 * q^13 - 12 * q^17 + 10 * q^25 - 2 * q^27 - 12 * q^29 - 2 * q^39 - 8 * q^43 - 10 * q^49 + 12 * q^51 - 12 * q^53 + 4 * q^61 - 10 * q^75 + 24 * q^77 + 16 * q^79 + 2 * q^81 + 12 * q^87 + 24 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$$.

 $$n$$ $$703$$ $$769$$ $$833$$ $$1093$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 0 0 0 3.46410i 0 1.00000 0
961.2 0 −1.00000 0 0 0 3.46410i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2496.2.c.d 2
4.b odd 2 1 2496.2.c.k 2
8.b even 2 1 624.2.c.e 2
8.d odd 2 1 39.2.b.a 2
13.b even 2 1 inner 2496.2.c.d 2
24.f even 2 1 117.2.b.a 2
24.h odd 2 1 1872.2.c.e 2
40.e odd 2 1 975.2.b.d 2
40.k even 4 2 975.2.h.f 4
52.b odd 2 1 2496.2.c.k 2
56.e even 2 1 1911.2.c.d 2
104.e even 2 1 624.2.c.e 2
104.h odd 2 1 39.2.b.a 2
104.j odd 4 2 8112.2.a.bv 2
104.m even 4 2 507.2.a.f 2
104.n odd 6 1 507.2.j.a 2
104.n odd 6 1 507.2.j.c 2
104.p odd 6 1 507.2.j.a 2
104.p odd 6 1 507.2.j.c 2
104.u even 12 4 507.2.e.e 4
312.b odd 2 1 1872.2.c.e 2
312.h even 2 1 117.2.b.a 2
312.w odd 4 2 1521.2.a.l 2
520.b odd 2 1 975.2.b.d 2
520.bc even 4 2 975.2.h.f 4
728.b even 2 1 1911.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 8.d odd 2 1
39.2.b.a 2 104.h odd 2 1
117.2.b.a 2 24.f even 2 1
117.2.b.a 2 312.h even 2 1
507.2.a.f 2 104.m even 4 2
507.2.e.e 4 104.u even 12 4
507.2.j.a 2 104.n odd 6 1
507.2.j.a 2 104.p odd 6 1
507.2.j.c 2 104.n odd 6 1
507.2.j.c 2 104.p odd 6 1
624.2.c.e 2 8.b even 2 1
624.2.c.e 2 104.e even 2 1
975.2.b.d 2 40.e odd 2 1
975.2.b.d 2 520.b odd 2 1
975.2.h.f 4 40.k even 4 2
975.2.h.f 4 520.bc even 4 2
1521.2.a.l 2 312.w odd 4 2
1872.2.c.e 2 24.h odd 2 1
1872.2.c.e 2 312.b odd 2 1
1911.2.c.d 2 56.e even 2 1
1911.2.c.d 2 728.b even 2 1
2496.2.c.d 2 1.a even 1 1 trivial
2496.2.c.d 2 13.b even 2 1 inner
2496.2.c.k 2 4.b odd 2 1
2496.2.c.k 2 52.b odd 2 1
8112.2.a.bv 2 104.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2496, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 12$$ T7^2 + 12 $$T_{11}^{2} + 12$$ T11^2 + 12 $$T_{23}$$ T23 $$T_{43} + 4$$ T43 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 12$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} - 2T + 13$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 12$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 12$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2} + 48$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 108$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 108$$
$71$ $$T^{2} + 12$$
$73$ $$T^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 192$$