# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.m (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$i$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
197.1
 1.00000i − 1.00000i
−1.00000 + 1.00000i 0 2.00000i −2.00000 2.00000i 0 0 2.00000 + 2.00000i 3.00000i 4.00000
589.1 −1.00000 1.00000i 0 2.00000i −2.00000 + 2.00000i 0 0 2.00000 2.00000i 3.00000i 4.00000
 $$n$$: 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
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.m.a 2
7.b odd 2 1 112.2.m.b 2
7.c even 3 2 784.2.x.e 4
7.d odd 6 2 784.2.x.d 4
16.e even 4 1 inner 784.2.m.a 2
28.d even 2 1 448.2.m.a 2
56.e even 2 1 896.2.m.c 2
56.h odd 2 1 896.2.m.b 2
112.j even 4 1 448.2.m.a 2
112.j even 4 1 896.2.m.c 2
112.l odd 4 1 112.2.m.b 2
112.l odd 4 1 896.2.m.b 2
112.w even 12 2 784.2.x.e 4
112.x odd 12 2 784.2.x.d 4
224.v odd 8 2 7168.2.a.b 2
224.x even 8 2 7168.2.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.b 2 7.b odd 2 1
112.2.m.b 2 112.l odd 4 1
448.2.m.a 2 28.d even 2 1
448.2.m.a 2 112.j even 4 1
784.2.m.a 2 1.a even 1 1 trivial
784.2.m.a 2 16.e even 4 1 inner
784.2.x.d 4 7.d odd 6 2
784.2.x.d 4 112.x odd 12 2
784.2.x.e 4 7.c even 3 2
784.2.x.e 4 112.w even 12 2
896.2.m.b 2 56.h odd 2 1
896.2.m.b 2 112.l odd 4 1
896.2.m.c 2 56.e even 2 1
896.2.m.c 2 112.j even 4 1
7168.2.a.b 2 224.v odd 8 2
7168.2.a.k 2 224.x even 8 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 $$T_{5}^{2} + 4T_{5} + 8$$ T5^2 + 4*T5 + 8

## Hecke characteristic polynomials

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