# Properties

 Label 676.2.h.c Level $676$ Weight $2$ Character orbit 676.h Analytic conductor $5.398$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 676.h (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.39788717664$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 52) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} - \beta_1 q^{7} + ( - 3 \beta_{2} + 3) q^{9}+O(q^{10})$$ q + b3 * q^5 - b1 * q^7 + (-3*b2 + 3) * q^9 $$q + \beta_{3} q^{5} - \beta_1 q^{7} + ( - 3 \beta_{2} + 3) q^{9} + ( - \beta_{3} + \beta_1) q^{11} + ( - 6 \beta_{2} + 6) q^{17} + 3 \beta_1 q^{19} + 8 \beta_{2} q^{23} + q^{25} - 2 \beta_{2} q^{29} + 5 \beta_{3} q^{31} + ( - 4 \beta_{2} + 4) q^{35} + ( - 3 \beta_{3} + 3 \beta_1) q^{37} + (3 \beta_{3} - 3 \beta_1) q^{41} + ( - 4 \beta_{2} + 4) q^{43} + 3 \beta_1 q^{45} + \beta_{3} q^{47} - 3 \beta_{2} q^{49} + 6 q^{53} + 4 \beta_{2} q^{55} - 5 \beta_1 q^{59} + ( - 2 \beta_{2} + 2) q^{61} + (3 \beta_{3} - 3 \beta_1) q^{63} + ( - 5 \beta_{3} + 5 \beta_1) q^{67} - 5 \beta_1 q^{71} - \beta_{3} q^{73} - 4 q^{77} - 4 q^{79} - 9 \beta_{2} q^{81} - 3 \beta_{3} q^{83} + 6 \beta_1 q^{85} + ( - 3 \beta_{3} + 3 \beta_1) q^{89} + (12 \beta_{2} - 12) q^{95} - \beta_1 q^{97} - 3 \beta_{3} q^{99}+O(q^{100})$$ q + b3 * q^5 - b1 * q^7 + (-3*b2 + 3) * q^9 + (-b3 + b1) * q^11 + (-6*b2 + 6) * q^17 + 3*b1 * q^19 + 8*b2 * q^23 + q^25 - 2*b2 * q^29 + 5*b3 * q^31 + (-4*b2 + 4) * q^35 + (-3*b3 + 3*b1) * q^37 + (3*b3 - 3*b1) * q^41 + (-4*b2 + 4) * q^43 + 3*b1 * q^45 + b3 * q^47 - 3*b2 * q^49 + 6 * q^53 + 4*b2 * q^55 - 5*b1 * q^59 + (-2*b2 + 2) * q^61 + (3*b3 - 3*b1) * q^63 + (-5*b3 + 5*b1) * q^67 - 5*b1 * q^71 - b3 * q^73 - 4 * q^77 - 4 * q^79 - 9*b2 * q^81 - 3*b3 * q^83 + 6*b1 * q^85 + (-3*b3 + 3*b1) * q^89 + (12*b2 - 12) * q^95 - b1 * q^97 - 3*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{9}+O(q^{10})$$ 4 * q + 6 * q^9 $$4 q + 6 q^{9} + 12 q^{17} + 16 q^{23} + 4 q^{25} - 4 q^{29} + 8 q^{35} + 8 q^{43} - 6 q^{49} + 24 q^{53} + 8 q^{55} + 4 q^{61} - 16 q^{77} - 16 q^{79} - 18 q^{81} - 24 q^{95}+O(q^{100})$$ 4 * q + 6 * q^9 + 12 * q^17 + 16 * q^23 + 4 * q^25 - 4 * q^29 + 8 * q^35 + 8 * q^43 - 6 * q^49 + 24 * q^53 + 8 * q^55 + 4 * q^61 - 16 * q^77 - 16 * q^79 - 18 * q^81 - 24 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$339$$ $$509$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 2.00000i 0 1.73205 + 1.00000i 0 1.50000 2.59808i 0
361.2 0 0 0 2.00000i 0 −1.73205 1.00000i 0 1.50000 2.59808i 0
485.1 0 0 0 2.00000i 0 −1.73205 + 1.00000i 0 1.50000 + 2.59808i 0
485.2 0 0 0 2.00000i 0 1.73205 1.00000i 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(676, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 4)^{2}$$
$7$ $$T^{4} - 4T^{2} + 16$$
$11$ $$T^{4} - 4T^{2} + 16$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 6 T + 36)^{2}$$
$19$ $$T^{4} - 36T^{2} + 1296$$
$23$ $$(T^{2} - 8 T + 64)^{2}$$
$29$ $$(T^{2} + 2 T + 4)^{2}$$
$31$ $$(T^{2} + 100)^{2}$$
$37$ $$T^{4} - 36T^{2} + 1296$$
$41$ $$T^{4} - 36T^{2} + 1296$$
$43$ $$(T^{2} - 4 T + 16)^{2}$$
$47$ $$(T^{2} + 4)^{2}$$
$53$ $$(T - 6)^{4}$$
$59$ $$T^{4} - 100 T^{2} + 10000$$
$61$ $$(T^{2} - 2 T + 4)^{2}$$
$67$ $$T^{4} - 100 T^{2} + 10000$$
$71$ $$T^{4} - 100 T^{2} + 10000$$
$73$ $$(T^{2} + 4)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$T^{4} - 36T^{2} + 1296$$
$97$ $$T^{4} - 4T^{2} + 16$$