# Properties

 Label 2548.2.j.e Level $2548$ Weight $2$ Character orbit 2548.j Analytic conductor $20.346$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2548 = 2^{2} \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2548.j (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.3458824350$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q - 2*z * q^5 + 3*z * q^9 $$q - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - q^{13} + (6 \zeta_{6} - 6) q^{17} + 6 \zeta_{6} q^{19} - 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + 2 q^{29} + (10 \zeta_{6} - 10) q^{31} + 6 \zeta_{6} q^{37} - 6 q^{41} + 4 q^{43} + ( - 6 \zeta_{6} + 6) q^{45} + 2 \zeta_{6} q^{47} + (6 \zeta_{6} - 6) q^{53} - 4 q^{55} + ( - 10 \zeta_{6} + 10) q^{59} + 2 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + (10 \zeta_{6} - 10) q^{67} + 10 q^{71} + (2 \zeta_{6} - 2) q^{73} + 4 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} - 6 q^{83} + 12 q^{85} + 6 \zeta_{6} q^{89} + ( - 12 \zeta_{6} + 12) q^{95} + 2 q^{97} + 6 q^{99} +O(q^{100})$$ q - 2*z * q^5 + 3*z * q^9 + (-2*z + 2) * q^11 - q^13 + (6*z - 6) * q^17 + 6*z * q^19 - 8*z * q^23 + (-z + 1) * q^25 + 2 * q^29 + (10*z - 10) * q^31 + 6*z * q^37 - 6 * q^41 + 4 * q^43 + (-6*z + 6) * q^45 + 2*z * q^47 + (6*z - 6) * q^53 - 4 * q^55 + (-10*z + 10) * q^59 + 2*z * q^61 + 2*z * q^65 + (10*z - 10) * q^67 + 10 * q^71 + (2*z - 2) * q^73 + 4*z * q^79 + (9*z - 9) * q^81 - 6 * q^83 + 12 * q^85 + 6*z * q^89 + (-12*z + 12) * q^95 + 2 * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 + 3 * q^9 $$2 q - 2 q^{5} + 3 q^{9} + 2 q^{11} - 2 q^{13} - 6 q^{17} + 6 q^{19} - 8 q^{23} + q^{25} + 4 q^{29} - 10 q^{31} + 6 q^{37} - 12 q^{41} + 8 q^{43} + 6 q^{45} + 2 q^{47} - 6 q^{53} - 8 q^{55} + 10 q^{59} + 2 q^{61} + 2 q^{65} - 10 q^{67} + 20 q^{71} - 2 q^{73} + 4 q^{79} - 9 q^{81} - 12 q^{83} + 24 q^{85} + 6 q^{89} + 12 q^{95} + 4 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 + 3 * q^9 + 2 * q^11 - 2 * q^13 - 6 * q^17 + 6 * q^19 - 8 * q^23 + q^25 + 4 * q^29 - 10 * q^31 + 6 * q^37 - 12 * q^41 + 8 * q^43 + 6 * q^45 + 2 * q^47 - 6 * q^53 - 8 * q^55 + 10 * q^59 + 2 * q^61 + 2 * q^65 - 10 * q^67 + 20 * q^71 - 2 * q^73 + 4 * q^79 - 9 * q^81 - 12 * q^83 + 24 * q^85 + 6 * q^89 + 12 * q^95 + 4 * q^97 + 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$885$$ $$1275$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1145.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.00000 1.73205i 0 0 0 1.50000 + 2.59808i 0
1353.1 0 0 0 −1.00000 + 1.73205i 0 0 0 1.50000 2.59808i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2548.2.j.e 2
7.b odd 2 1 2548.2.j.f 2
7.c even 3 1 52.2.a.a 1
7.c even 3 1 inner 2548.2.j.e 2
7.d odd 6 1 2548.2.a.e 1
7.d odd 6 1 2548.2.j.f 2
21.h odd 6 1 468.2.a.b 1
28.g odd 6 1 208.2.a.c 1
35.j even 6 1 1300.2.a.d 1
35.l odd 12 2 1300.2.c.c 2
56.k odd 6 1 832.2.a.f 1
56.p even 6 1 832.2.a.e 1
63.g even 3 1 4212.2.i.d 2
63.h even 3 1 4212.2.i.d 2
63.j odd 6 1 4212.2.i.i 2
63.n odd 6 1 4212.2.i.i 2
77.h odd 6 1 6292.2.a.g 1
84.n even 6 1 1872.2.a.f 1
91.g even 3 1 676.2.e.c 2
91.h even 3 1 676.2.e.c 2
91.k even 6 1 676.2.e.b 2
91.r even 6 1 676.2.a.c 1
91.u even 6 1 676.2.e.b 2
91.x odd 12 2 676.2.h.c 4
91.z odd 12 2 676.2.d.c 2
91.bd odd 12 2 676.2.h.c 4
112.u odd 12 2 3328.2.b.e 2
112.w even 12 2 3328.2.b.q 2
140.p odd 6 1 5200.2.a.q 1
168.s odd 6 1 7488.2.a.bn 1
168.v even 6 1 7488.2.a.bw 1
273.w odd 6 1 6084.2.a.m 1
273.cd even 12 2 6084.2.b.m 2
364.bl odd 6 1 2704.2.a.g 1
364.ce even 12 2 2704.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 7.c even 3 1
208.2.a.c 1 28.g odd 6 1
468.2.a.b 1 21.h odd 6 1
676.2.a.c 1 91.r even 6 1
676.2.d.c 2 91.z odd 12 2
676.2.e.b 2 91.k even 6 1
676.2.e.b 2 91.u even 6 1
676.2.e.c 2 91.g even 3 1
676.2.e.c 2 91.h even 3 1
676.2.h.c 4 91.x odd 12 2
676.2.h.c 4 91.bd odd 12 2
832.2.a.e 1 56.p even 6 1
832.2.a.f 1 56.k odd 6 1
1300.2.a.d 1 35.j even 6 1
1300.2.c.c 2 35.l odd 12 2
1872.2.a.f 1 84.n even 6 1
2548.2.a.e 1 7.d odd 6 1
2548.2.j.e 2 1.a even 1 1 trivial
2548.2.j.e 2 7.c even 3 1 inner
2548.2.j.f 2 7.b odd 2 1
2548.2.j.f 2 7.d odd 6 1
2704.2.a.g 1 364.bl odd 6 1
2704.2.f.f 2 364.ce even 12 2
3328.2.b.e 2 112.u odd 12 2
3328.2.b.q 2 112.w even 12 2
4212.2.i.d 2 63.g even 3 1
4212.2.i.d 2 63.h even 3 1
4212.2.i.i 2 63.j odd 6 1
4212.2.i.i 2 63.n odd 6 1
5200.2.a.q 1 140.p odd 6 1
6084.2.a.m 1 273.w odd 6 1
6084.2.b.m 2 273.cd even 12 2
6292.2.a.g 1 77.h odd 6 1
7488.2.a.bn 1 168.s odd 6 1
7488.2.a.bw 1 168.v even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2548, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 10T + 100$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 10T + 100$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 10T + 100$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$(T - 2)^{2}$$