# Properties

 Label 676.2.e.b Level $676$ Weight $2$ Character orbit 676.e Analytic conductor $5.398$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 676.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.39788717664$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{5} - 2 \zeta_{6} q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q - 2 * q^5 - 2*z * q^7 + 3*z * q^9 $$q - 2 q^{5} - 2 \zeta_{6} q^{7} + 3 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} - 6 \zeta_{6} q^{17} - 6 \zeta_{6} q^{19} + (8 \zeta_{6} - 8) q^{23} - q^{25} + (2 \zeta_{6} - 2) q^{29} - 10 q^{31} + 4 \zeta_{6} q^{35} + (6 \zeta_{6} - 6) q^{37} + (6 \zeta_{6} - 6) q^{41} - 4 \zeta_{6} q^{43} - 6 \zeta_{6} q^{45} + 2 q^{47} + ( - 3 \zeta_{6} + 3) q^{49} + 6 q^{53} + ( - 4 \zeta_{6} + 4) q^{55} - 10 \zeta_{6} q^{59} + 2 \zeta_{6} q^{61} + ( - 6 \zeta_{6} + 6) q^{63} + ( - 10 \zeta_{6} + 10) q^{67} + 10 \zeta_{6} q^{71} - 2 q^{73} + 4 q^{77} - 4 q^{79} + (9 \zeta_{6} - 9) q^{81} + 6 q^{83} + 12 \zeta_{6} q^{85} + (6 \zeta_{6} - 6) q^{89} + 12 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} - 6 q^{99} +O(q^{100})$$ q - 2 * q^5 - 2*z * q^7 + 3*z * q^9 + (2*z - 2) * q^11 - 6*z * q^17 - 6*z * q^19 + (8*z - 8) * q^23 - q^25 + (2*z - 2) * q^29 - 10 * q^31 + 4*z * q^35 + (6*z - 6) * q^37 + (6*z - 6) * q^41 - 4*z * q^43 - 6*z * q^45 + 2 * q^47 + (-3*z + 3) * q^49 + 6 * q^53 + (-4*z + 4) * q^55 - 10*z * q^59 + 2*z * q^61 + (-6*z + 6) * q^63 + (-10*z + 10) * q^67 + 10*z * q^71 - 2 * q^73 + 4 * q^77 - 4 * q^79 + (9*z - 9) * q^81 + 6 * q^83 + 12*z * q^85 + (6*z - 6) * q^89 + 12*z * q^95 + 2*z * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - 4 * q^5 - 2 * q^7 + 3 * q^9 $$2 q - 4 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{17} - 6 q^{19} - 8 q^{23} - 2 q^{25} - 2 q^{29} - 20 q^{31} + 4 q^{35} - 6 q^{37} - 6 q^{41} - 4 q^{43} - 6 q^{45} + 4 q^{47} + 3 q^{49} + 12 q^{53} + 4 q^{55} - 10 q^{59} + 2 q^{61} + 6 q^{63} + 10 q^{67} + 10 q^{71} - 4 q^{73} + 8 q^{77} - 8 q^{79} - 9 q^{81} + 12 q^{83} + 12 q^{85} - 6 q^{89} + 12 q^{95} + 2 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - 4 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^17 - 6 * q^19 - 8 * q^23 - 2 * q^25 - 2 * q^29 - 20 * q^31 + 4 * q^35 - 6 * q^37 - 6 * q^41 - 4 * q^43 - 6 * q^45 + 4 * q^47 + 3 * q^49 + 12 * q^53 + 4 * q^55 - 10 * q^59 + 2 * q^61 + 6 * q^63 + 10 * q^67 + 10 * q^71 - 4 * q^73 + 8 * q^77 - 8 * q^79 - 9 * q^81 + 12 * q^83 + 12 * q^85 - 6 * q^89 + 12 * q^95 + 2 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$509$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
529.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −2.00000 0 −1.00000 1.73205i 0 1.50000 + 2.59808i 0
653.1 0 0 0 −2.00000 0 −1.00000 + 1.73205i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.2.e.b 2
13.b even 2 1 676.2.e.c 2
13.c even 3 1 676.2.a.c 1
13.c even 3 1 inner 676.2.e.b 2
13.d odd 4 2 676.2.h.c 4
13.e even 6 1 52.2.a.a 1
13.e even 6 1 676.2.e.c 2
13.f odd 12 2 676.2.d.c 2
13.f odd 12 2 676.2.h.c 4
39.h odd 6 1 468.2.a.b 1
39.i odd 6 1 6084.2.a.m 1
39.k even 12 2 6084.2.b.m 2
52.i odd 6 1 208.2.a.c 1
52.j odd 6 1 2704.2.a.g 1
52.l even 12 2 2704.2.f.f 2
65.l even 6 1 1300.2.a.d 1
65.r odd 12 2 1300.2.c.c 2
91.k even 6 1 2548.2.j.e 2
91.l odd 6 1 2548.2.j.f 2
91.p odd 6 1 2548.2.j.f 2
91.t odd 6 1 2548.2.a.e 1
91.u even 6 1 2548.2.j.e 2
104.p odd 6 1 832.2.a.f 1
104.s even 6 1 832.2.a.e 1
117.l even 6 1 4212.2.i.d 2
117.m odd 6 1 4212.2.i.i 2
117.r even 6 1 4212.2.i.d 2
117.v odd 6 1 4212.2.i.i 2
143.i odd 6 1 6292.2.a.g 1
156.r even 6 1 1872.2.a.f 1
208.bh even 12 2 3328.2.b.q 2
208.bi odd 12 2 3328.2.b.e 2
260.w odd 6 1 5200.2.a.q 1
312.ba even 6 1 7488.2.a.bw 1
312.bg odd 6 1 7488.2.a.bn 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.a.a 1 13.e even 6 1
208.2.a.c 1 52.i odd 6 1
468.2.a.b 1 39.h odd 6 1
676.2.a.c 1 13.c even 3 1
676.2.d.c 2 13.f odd 12 2
676.2.e.b 2 1.a even 1 1 trivial
676.2.e.b 2 13.c even 3 1 inner
676.2.e.c 2 13.b even 2 1
676.2.e.c 2 13.e even 6 1
676.2.h.c 4 13.d odd 4 2
676.2.h.c 4 13.f odd 12 2
832.2.a.e 1 104.s even 6 1
832.2.a.f 1 104.p odd 6 1
1300.2.a.d 1 65.l even 6 1
1300.2.c.c 2 65.r odd 12 2
1872.2.a.f 1 156.r even 6 1
2548.2.a.e 1 91.t odd 6 1
2548.2.j.e 2 91.k even 6 1
2548.2.j.e 2 91.u even 6 1
2548.2.j.f 2 91.l odd 6 1
2548.2.j.f 2 91.p odd 6 1
2704.2.a.g 1 52.j odd 6 1
2704.2.f.f 2 52.l even 12 2
3328.2.b.e 2 208.bi odd 12 2
3328.2.b.q 2 208.bh even 12 2
4212.2.i.d 2 117.l even 6 1
4212.2.i.d 2 117.r even 6 1
4212.2.i.i 2 117.m odd 6 1
4212.2.i.i 2 117.v odd 6 1
5200.2.a.q 1 260.w odd 6 1
6084.2.a.m 1 39.i odd 6 1
6084.2.b.m 2 39.k even 12 2
6292.2.a.g 1 143.i odd 6 1
7488.2.a.bn 1 312.bg odd 6 1
7488.2.a.bw 1 312.ba even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(676, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5} + 2$$ T5 + 2

## Hecke characteristic polynomials

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