# Properties

 Label 784.2.x.b 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + 4 \zeta_{12}^{3} q^{6} + (2 \zeta_{12}^{3} + 2) q^{8} + 5 \zeta_{12} q^{9}+O(q^{10})$$ q + (-z^2 - z + 1) * q^2 + (2*z^3 - 2*z^2 - 2*z) * q^3 + (2*z^3 - 2*z) * q^4 + (2*z^2 - 2*z - 2) * q^5 + 4*z^3 * q^6 + (2*z^3 + 2) * q^8 + 5*z * q^9 $$q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{5} + 4 \zeta_{12}^{3} q^{6} + (2 \zeta_{12}^{3} + 2) q^{8} + 5 \zeta_{12} q^{9} + 4 \zeta_{12}^{2} q^{10} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{11} + ( - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 4) q^{12} + 8 q^{15} + ( - 4 \zeta_{12}^{2} + 4) q^{16} - 6 \zeta_{12}^{2} q^{17} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12}) q^{18} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{19} + ( - 4 \zeta_{12}^{3} + 4) q^{20} + 6 q^{22} - 2 \zeta_{12} q^{23} - 8 \zeta_{12}^{2} q^{24} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{25} + ( - 4 \zeta_{12}^{3} - 4) q^{27} + (\zeta_{12}^{3} - 1) q^{29} + ( - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 8) q^{30} + 4 \zeta_{12}^{2} q^{31} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{32} + ( - 12 \zeta_{12}^{2} + 12) q^{33} + (6 \zeta_{12}^{3} - 6) q^{34} - 10 q^{36} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{37} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{38} - 8 \zeta_{12} q^{40} - 2 \zeta_{12}^{3} q^{41} + ( - 5 \zeta_{12}^{3} - 5) q^{43} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{44} + (10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 10 \zeta_{12}) q^{45} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{46} + (8 \zeta_{12}^{2} - 8) q^{47} + (8 \zeta_{12}^{3} - 8) q^{48} + ( - 3 \zeta_{12}^{3} - 3) q^{50} + (12 \zeta_{12}^{2} + 12 \zeta_{12} - 12) q^{51} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}^{2} + 7 \zeta_{12}) q^{53} + (8 \zeta_{12}^{2} - 8) q^{54} - 12 \zeta_{12}^{3} q^{55} - 16 \zeta_{12}^{3} q^{57} + 2 \zeta_{12} q^{58} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{59} + (16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{60} + ( - 6 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{61} + ( - 4 \zeta_{12}^{3} + 4) q^{62} + 8 \zeta_{12}^{3} q^{64} + (12 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 12 \zeta_{12}) q^{66} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12}) q^{67} + 12 \zeta_{12} q^{68} + (4 \zeta_{12}^{3} + 4) q^{69} - 8 \zeta_{12}^{3} q^{71} + (10 \zeta_{12}^{2} + 10 \zeta_{12} - 10) q^{72} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}) q^{73} + 6 \zeta_{12}^{2} q^{74} + (6 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{75} + ( - 8 \zeta_{12}^{3} - 8) q^{76} + ( - 14 \zeta_{12}^{2} + 14) q^{79} + (8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{80} + \zeta_{12}^{2} q^{81} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{82} + (12 \zeta_{12}^{3} + 12) q^{85} + (10 \zeta_{12}^{2} - 10) q^{86} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{87} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{88} + 6 \zeta_{12} q^{89} + 20 \zeta_{12}^{3} q^{90} + 4 q^{92} + ( - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 8) q^{93} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} + 8 \zeta_{12}) q^{94} - 16 \zeta_{12}^{2} q^{95} + 16 \zeta_{12} q^{96} + 6 q^{97} + (15 \zeta_{12}^{3} - 15) q^{99} +O(q^{100})$$ q + (-z^2 - z + 1) * q^2 + (2*z^3 - 2*z^2 - 2*z) * q^3 + (2*z^3 - 2*z) * q^4 + (2*z^2 - 2*z - 2) * q^5 + 4*z^3 * q^6 + (2*z^3 + 2) * q^8 + 5*z * q^9 + 4*z^2 * q^10 + (3*z^3 + 3*z^2 - 3*z) * q^11 + (-4*z^2 + 4*z + 4) * q^12 + 8 * q^15 + (-4*z^2 + 4) * q^16 - 6*z^2 * q^17 + (-5*z^3 - 5*z^2 + 5*z) * q^18 + (4*z^2 + 4*z - 4) * q^19 + (-4*z^3 + 4) * q^20 + 6 * q^22 - 2*z * q^23 - 8*z^2 * q^24 + (-3*z^3 + 3*z) * q^25 + (-4*z^3 - 4) * q^27 + (z^3 - 1) * q^29 + (-8*z^2 - 8*z + 8) * q^30 + 4*z^2 * q^31 + (4*z^3 - 4*z^2 - 4*z) * q^32 + (-12*z^2 + 12) * q^33 + (6*z^3 - 6) * q^34 - 10 * q^36 + (3*z^2 - 3*z - 3) * q^37 + (-8*z^3 + 8*z) * q^38 - 8*z * q^40 - 2*z^3 * q^41 + (-5*z^3 - 5) * q^43 + (-6*z^2 - 6*z + 6) * q^44 + (10*z^3 - 10*z^2 - 10*z) * q^45 + (2*z^3 + 2*z^2 - 2*z) * q^46 + (8*z^2 - 8) * q^47 + (8*z^3 - 8) * q^48 + (-3*z^3 - 3) * q^50 + (12*z^2 + 12*z - 12) * q^51 + (-7*z^3 - 7*z^2 + 7*z) * q^53 + (8*z^2 - 8) * q^54 - 12*z^3 * q^55 - 16*z^3 * q^57 + 2*z * q^58 + (2*z^3 + 2*z^2 - 2*z) * q^59 + (16*z^3 - 16*z) * q^60 + (-6*z^2 - 6*z + 6) * q^61 + (-4*z^3 + 4) * q^62 + 8*z^3 * q^64 + (12*z^3 - 12*z^2 - 12*z) * q^66 + (-5*z^3 + 5*z^2 + 5*z) * q^67 + 12*z * q^68 + (4*z^3 + 4) * q^69 - 8*z^3 * q^71 + (10*z^2 + 10*z - 10) * q^72 + (-10*z^3 + 10*z) * q^73 + 6*z^2 * q^74 + (6*z^2 - 6*z - 6) * q^75 + (-8*z^3 - 8) * q^76 + (-14*z^2 + 14) * q^79 + (8*z^3 + 8*z^2 - 8*z) * q^80 + z^2 * q^81 + (2*z^2 - 2*z - 2) * q^82 + (12*z^3 + 12) * q^85 + (10*z^2 - 10) * q^86 + (-4*z^3 + 4*z) * q^87 + (12*z^3 - 12*z) * q^88 + 6*z * q^89 + 20*z^3 * q^90 + 4 * q^92 + (-8*z^2 - 8*z + 8) * q^93 + (-8*z^3 + 8*z^2 + 8*z) * q^94 - 16*z^2 * q^95 + 16*z * q^96 + 6 * q^97 + (15*z^3 - 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{3} - 4 q^{5} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 4 * q^3 - 4 * q^5 + 8 * q^8 $$4 q + 2 q^{2} - 4 q^{3} - 4 q^{5} + 8 q^{8} + 8 q^{10} + 6 q^{11} + 8 q^{12} + 32 q^{15} + 8 q^{16} - 12 q^{17} - 10 q^{18} - 8 q^{19} + 16 q^{20} + 24 q^{22} - 16 q^{24} - 16 q^{27} - 4 q^{29} + 16 q^{30} + 8 q^{31} - 8 q^{32} + 24 q^{33} - 24 q^{34} - 40 q^{36} - 6 q^{37} - 20 q^{43} + 12 q^{44} - 20 q^{45} + 4 q^{46} - 16 q^{47} - 32 q^{48} - 12 q^{50} - 24 q^{51} - 14 q^{53} - 16 q^{54} + 4 q^{59} + 12 q^{61} + 16 q^{62} - 24 q^{66} + 10 q^{67} + 16 q^{69} - 20 q^{72} + 12 q^{74} - 12 q^{75} - 32 q^{76} + 28 q^{79} + 16 q^{80} + 2 q^{81} - 4 q^{82} + 48 q^{85} - 20 q^{86} + 16 q^{92} + 16 q^{93} + 16 q^{94} - 32 q^{95} + 24 q^{97} - 60 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 4 * q^3 - 4 * q^5 + 8 * q^8 + 8 * q^10 + 6 * q^11 + 8 * q^12 + 32 * q^15 + 8 * q^16 - 12 * q^17 - 10 * q^18 - 8 * q^19 + 16 * q^20 + 24 * q^22 - 16 * q^24 - 16 * q^27 - 4 * q^29 + 16 * q^30 + 8 * q^31 - 8 * q^32 + 24 * q^33 - 24 * q^34 - 40 * q^36 - 6 * q^37 - 20 * q^43 + 12 * q^44 - 20 * q^45 + 4 * q^46 - 16 * q^47 - 32 * q^48 - 12 * q^50 - 24 * q^51 - 14 * q^53 - 16 * q^54 + 4 * q^59 + 12 * q^61 + 16 * q^62 - 24 * q^66 + 10 * q^67 + 16 * q^69 - 20 * q^72 + 12 * q^74 - 12 * q^75 - 32 * q^76 + 28 * q^79 + 16 * q^80 + 2 * q^81 - 4 * q^82 + 48 * q^85 - 20 * q^86 + 16 * q^92 + 16 * q^93 + 16 * q^94 - 32 * q^95 + 24 * q^97 - 60 * 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 0.732051 + 2.73205i 1.73205 + 1.00000i 0.732051 2.73205i 4.00000i 0 2.00000 + 2.00000i −4.33013 + 2.50000i 2.00000 3.46410i
373.1 −0.366025 1.36603i −2.73205 0.732051i −1.73205 + 1.00000i −2.73205 + 0.732051i 4.00000i 0 2.00000 + 2.00000i 4.33013 + 2.50000i 2.00000 + 3.46410i
557.1 −0.366025 + 1.36603i −2.73205 + 0.732051i −1.73205 1.00000i −2.73205 0.732051i 4.00000i 0 2.00000 2.00000i 4.33013 2.50000i 2.00000 3.46410i
765.1 1.36603 0.366025i 0.732051 2.73205i 1.73205 1.00000i 0.732051 + 2.73205i 4.00000i 0 2.00000 2.00000i −4.33013 2.50000i 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.b 4
7.b odd 2 1 784.2.x.g 4
7.c even 3 1 784.2.m.c 2
7.c even 3 1 inner 784.2.x.b 4
7.d odd 6 1 112.2.m.a 2
7.d odd 6 1 784.2.x.g 4
16.e even 4 1 inner 784.2.x.b 4
28.f even 6 1 448.2.m.b 2
56.j odd 6 1 896.2.m.d 2
56.m even 6 1 896.2.m.a 2
112.l odd 4 1 784.2.x.g 4
112.v even 12 1 448.2.m.b 2
112.v even 12 1 896.2.m.a 2
112.w even 12 1 784.2.m.c 2
112.w even 12 1 inner 784.2.x.b 4
112.x odd 12 1 112.2.m.a 2
112.x odd 12 1 784.2.x.g 4
112.x odd 12 1 896.2.m.d 2
224.bc odd 24 2 7168.2.a.q 2
224.be even 24 2 7168.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.a 2 7.d odd 6 1
112.2.m.a 2 112.x odd 12 1
448.2.m.b 2 28.f even 6 1
448.2.m.b 2 112.v even 12 1
784.2.m.c 2 7.c even 3 1
784.2.m.c 2 112.w even 12 1
784.2.x.b 4 1.a even 1 1 trivial
784.2.x.b 4 7.c even 3 1 inner
784.2.x.b 4 16.e even 4 1 inner
784.2.x.b 4 112.w even 12 1 inner
784.2.x.g 4 7.b odd 2 1
784.2.x.g 4 7.d odd 6 1
784.2.x.g 4 112.l odd 4 1
784.2.x.g 4 112.x odd 12 1
896.2.m.a 2 56.m even 6 1
896.2.m.a 2 112.v even 12 1
896.2.m.d 2 56.j odd 6 1
896.2.m.d 2 112.x odd 12 1
7168.2.a.i 2 224.be even 24 2
7168.2.a.q 2 224.bc odd 24 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64$$
$5$ $$T^{4} + 4 T^{3} + 8 T^{2} + 32 T + 64$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 6 T + 36)^{2}$$
$19$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 1024$$
$23$ $$T^{4} - 4T^{2} + 16$$
$29$ $$(T^{2} + 2 T + 2)^{2}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324$$
$41$ $$(T^{2} + 4)^{2}$$
$43$ $$(T^{2} + 10 T + 50)^{2}$$
$47$ $$(T^{2} + 8 T + 64)^{2}$$
$53$ $$T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 9604$$
$59$ $$T^{4} - 4 T^{3} + 8 T^{2} - 32 T + 64$$
$61$ $$T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 5184$$
$67$ $$T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500$$
$71$ $$(T^{2} + 64)^{2}$$
$73$ $$T^{4} - 100 T^{2} + 10000$$
$79$ $$(T^{2} - 14 T + 196)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 36T^{2} + 1296$$
$97$ $$(T - 6)^{4}$$