# Properties

 Label 784.2.x.d Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.x (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$1$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + 2) q^{8} - 3 \zeta_{12} q^{9} +O(q^{10})$$ q + (-z^2 - z + 1) * q^2 + (2*z^3 - 2*z) * q^4 + (2*z^2 - 2*z - 2) * q^5 + (2*z^3 + 2) * q^8 - 3*z * q^9 $$q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + (2 \zeta_{12}^{3} + 2) q^{8} - 3 \zeta_{12} q^{9} + 4 \zeta_{12}^{2} q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + 2 \zeta_{12}^{2} q^{17} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{18} + (2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{19} + ( - 4 \zeta_{12}^{3} + 4) q^{20} - 2 q^{22} + 6 \zeta_{12} q^{23} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{25} + ( - 7 \zeta_{12}^{3} + 7) q^{29} + 8 \zeta_{12}^{2} q^{31} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{32} + ( - 2 \zeta_{12}^{3} + 2) q^{34} + 6 q^{36} + ( - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{37} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{38} - 8 \zeta_{12} q^{40} + 10 \zeta_{12}^{3} q^{41} + ( - \zeta_{12}^{3} - 1) q^{43} + (2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{44} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12}) q^{45} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 6 \zeta_{12}) q^{46} + ( - 12 \zeta_{12}^{2} + 12) q^{47} + ( - 3 \zeta_{12}^{3} - 3) q^{50} + (\zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{53} + 4 \zeta_{12}^{3} q^{55} - 14 \zeta_{12} q^{58} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12}) q^{59} + (6 \zeta_{12}^{2} + 6 \zeta_{12} - 6) q^{61} + ( - 8 \zeta_{12}^{3} + 8) q^{62} + 8 \zeta_{12}^{3} q^{64} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{67} - 4 \zeta_{12} q^{68} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{72} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{73} - 10 \zeta_{12}^{2} q^{74} + ( - 4 \zeta_{12}^{3} - 4) q^{76} + (10 \zeta_{12}^{2} - 10) q^{79} + (8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{80} + 9 \zeta_{12}^{2} q^{81} + ( - 10 \zeta_{12}^{2} + 10 \zeta_{12} + 10) q^{82} + (10 \zeta_{12}^{3} - 10) q^{83} + ( - 4 \zeta_{12}^{3} - 4) q^{85} + (2 \zeta_{12}^{2} - 2) q^{86} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{88} + 14 \zeta_{12} q^{89} - 12 \zeta_{12}^{3} q^{90} - 12 q^{92} + (12 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 12 \zeta_{12}) q^{94} - 8 \zeta_{12}^{2} q^{95} - 2 q^{97} + (3 \zeta_{12}^{3} - 3) q^{99} +O(q^{100})$$ q + (-z^2 - z + 1) * q^2 + (2*z^3 - 2*z) * q^4 + (2*z^2 - 2*z - 2) * q^5 + (2*z^3 + 2) * q^8 - 3*z * q^9 + 4*z^2 * q^10 + (-z^3 - z^2 + z) * q^11 + (-4*z^2 + 4) * q^16 + 2*z^2 * q^17 + (3*z^3 + 3*z^2 - 3*z) * q^18 + (2*z^2 + 2*z - 2) * q^19 + (-4*z^3 + 4) * q^20 - 2 * q^22 + 6*z * q^23 + (-3*z^3 + 3*z) * q^25 + (-7*z^3 + 7) * q^29 + 8*z^2 * q^31 + (4*z^3 - 4*z^2 - 4*z) * q^32 + (-2*z^3 + 2) * q^34 + 6 * q^36 + (-5*z^2 + 5*z + 5) * q^37 + (-4*z^3 + 4*z) * q^38 - 8*z * q^40 + 10*z^3 * q^41 + (-z^3 - 1) * q^43 + (2*z^2 + 2*z - 2) * q^44 + (-6*z^3 + 6*z^2 + 6*z) * q^45 + (-6*z^3 - 6*z^2 + 6*z) * q^46 + (-12*z^2 + 12) * q^47 + (-3*z^3 - 3) * q^50 + (z^3 + z^2 - z) * q^53 + 4*z^3 * q^55 - 14*z * q^58 + (-8*z^3 - 8*z^2 + 8*z) * q^59 + (6*z^2 + 6*z - 6) * q^61 + (-8*z^3 + 8) * q^62 + 8*z^3 * q^64 + (3*z^3 - 3*z^2 - 3*z) * q^67 - 4*z * q^68 + (-6*z^2 - 6*z + 6) * q^72 + (6*z^3 - 6*z) * q^73 - 10*z^2 * q^74 + (-4*z^3 - 4) * q^76 + (10*z^2 - 10) * q^79 + (8*z^3 + 8*z^2 - 8*z) * q^80 + 9*z^2 * q^81 + (-10*z^2 + 10*z + 10) * q^82 + (10*z^3 - 10) * q^83 + (-4*z^3 - 4) * q^85 + (2*z^2 - 2) * q^86 + (-4*z^3 + 4*z) * q^88 + 14*z * q^89 - 12*z^3 * q^90 - 12 * q^92 + (12*z^3 - 12*z^2 - 12*z) * q^94 - 8*z^2 * q^95 - 2 * q^97 + (3*z^3 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{5} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 $$4 q + 2 q^{2} - 4 q^{5} + 8 q^{8} + 8 q^{10} - 2 q^{11} + 8 q^{16} + 4 q^{17} + 6 q^{18} - 4 q^{19} + 16 q^{20} - 8 q^{22} + 28 q^{29} + 16 q^{31} - 8 q^{32} + 8 q^{34} + 24 q^{36} + 10 q^{37} - 4 q^{43} - 4 q^{44} + 12 q^{45} - 12 q^{46} + 24 q^{47} - 12 q^{50} + 2 q^{53} - 16 q^{59} - 12 q^{61} + 32 q^{62} - 6 q^{67} + 12 q^{72} - 20 q^{74} - 16 q^{76} - 20 q^{79} + 16 q^{80} + 18 q^{81} + 20 q^{82} - 40 q^{83} - 16 q^{85} - 4 q^{86} - 48 q^{92} - 24 q^{94} - 16 q^{95} - 8 q^{97} - 12 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 + 8 * q^10 - 2 * q^11 + 8 * q^16 + 4 * q^17 + 6 * q^18 - 4 * q^19 + 16 * q^20 - 8 * q^22 + 28 * q^29 + 16 * q^31 - 8 * q^32 + 8 * q^34 + 24 * q^36 + 10 * q^37 - 4 * q^43 - 4 * q^44 + 12 * q^45 - 12 * q^46 + 24 * q^47 - 12 * q^50 + 2 * q^53 - 16 * q^59 - 12 * q^61 + 32 * q^62 - 6 * q^67 + 12 * q^72 - 20 * q^74 - 16 * q^76 - 20 * q^79 + 16 * q^80 + 18 * q^81 + 20 * q^82 - 40 * q^83 - 16 * q^85 - 4 * q^86 - 48 * q^92 - 24 * q^94 - 16 * q^95 - 8 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.36603 + 0.366025i 0 1.73205 + 1.00000i 0.732051 2.73205i 0 0 2.00000 + 2.00000i 2.59808 1.50000i 2.00000 3.46410i
373.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −2.73205 + 0.732051i 0 0 2.00000 + 2.00000i −2.59808 1.50000i 2.00000 + 3.46410i
557.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i −2.73205 0.732051i 0 0 2.00000 2.00000i −2.59808 + 1.50000i 2.00000 3.46410i
765.1 1.36603 0.366025i 0 1.73205 1.00000i 0.732051 + 2.73205i 0 0 2.00000 2.00000i 2.59808 + 1.50000i 2.00000 + 3.46410i
 $$n$$: 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
16.e even 4 1 inner
112.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.d 4
7.b odd 2 1 784.2.x.e 4
7.c even 3 1 112.2.m.b 2
7.c even 3 1 inner 784.2.x.d 4
7.d odd 6 1 784.2.m.a 2
7.d odd 6 1 784.2.x.e 4
16.e even 4 1 inner 784.2.x.d 4
28.g odd 6 1 448.2.m.a 2
56.k odd 6 1 896.2.m.c 2
56.p even 6 1 896.2.m.b 2
112.l odd 4 1 784.2.x.e 4
112.u odd 12 1 448.2.m.a 2
112.u odd 12 1 896.2.m.c 2
112.w even 12 1 112.2.m.b 2
112.w even 12 1 inner 784.2.x.d 4
112.w even 12 1 896.2.m.b 2
112.x odd 12 1 784.2.m.a 2
112.x odd 12 1 784.2.x.e 4
224.bd even 24 2 7168.2.a.b 2
224.bf odd 24 2 7168.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 7.c even 3 1
112.2.m.b 2 112.w even 12 1
448.2.m.a 2 28.g odd 6 1
448.2.m.a 2 112.u odd 12 1
784.2.m.a 2 7.d odd 6 1
784.2.m.a 2 112.x odd 12 1
784.2.x.d 4 1.a even 1 1 trivial
784.2.x.d 4 7.c even 3 1 inner
784.2.x.d 4 16.e even 4 1 inner
784.2.x.d 4 112.w even 12 1 inner
784.2.x.e 4 7.b odd 2 1
784.2.x.e 4 7.d odd 6 1
784.2.x.e 4 112.l odd 4 1
784.2.x.e 4 112.x odd 12 1
896.2.m.b 2 56.p even 6 1
896.2.m.b 2 112.w even 12 1
896.2.m.c 2 56.k odd 6 1
896.2.m.c 2 112.u odd 12 1
7168.2.a.b 2 224.bd even 24 2
7168.2.a.k 2 224.bf odd 24 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{4} + 4T_{5}^{3} + 8T_{5}^{2} + 32T_{5} + 64$$ T5^4 + 4*T5^3 + 8*T5^2 + 32*T5 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 2 T + 4)^{2}$$
$19$ $$T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} - 14 T + 98)^{2}$$
$31$ $$(T^{2} - 8 T + 64)^{2}$$
$37$ $$T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500$$
$41$ $$(T^{2} + 100)^{2}$$
$43$ $$(T^{2} + 2 T + 2)^{2}$$
$47$ $$(T^{2} - 12 T + 144)^{2}$$
$53$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$59$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 16384$$
$61$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 5184$$
$67$ $$T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 36T^{2} + 1296$$
$79$ $$(T^{2} + 10 T + 100)^{2}$$
$83$ $$(T^{2} + 20 T + 200)^{2}$$
$89$ $$T^{4} - 196 T^{2} + 38416$$
$97$ $$(T + 2)^{4}$$