# Properties

 Label 784.2.x.i Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $8$ CM discriminant -7 Inner twists $8$

# 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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} + (\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_1) q^{8} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b6 + b4 - b2 - 1) * q^4 + (b7 + 2*b5 - b3 - b1) * q^8 + (-3*b5 - 3*b3) * q^9 $$q + \beta_1 q^{2} + (\beta_{6} + \beta_{4} - \beta_{2} - 1) q^{4} + (\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_1) q^{8} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{9} + ( - 2 \beta_{7} + 2 \beta_{6} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3) q^{11} + (\beta_{4} + 3 \beta_{2}) q^{16} + (3 \beta_{6} - 3 \beta_{4} - 3 \beta_{2} + 3) q^{18} + ( - 3 \beta_{7} + \beta_{6} + 4 \beta_{5} + 3 \beta_{3} + 3 \beta_1 + 5) q^{22} + (2 \beta_{4} + 4 \beta_{2}) q^{23} - 5 \beta_{3} q^{25} + (4 \beta_{7} + 4 \beta_{6} - \beta_{5} - 4 \beta_{3} - 4 \beta_1 + 1) q^{29} + (\beta_{7} + 5 \beta_{3}) q^{32} + ( - 3 \beta_{7} + 6 \beta_{5} + 3 \beta_{3} + 3 \beta_1) q^{36} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{37} + ( - 2 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} + 2 \beta_{3} + 2 \beta_1 - 5) q^{43} + (2 \beta_{5} + 7 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1) q^{44} + (2 \beta_{7} + 6 \beta_{3}) q^{46} + (5 \beta_{6} + 5) q^{50} + ( - 4 \beta_{7} + 4 \beta_{6} - 7 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 7) q^{53} + (8 \beta_{5} - 5 \beta_{4} + 8 \beta_{3} + 3 \beta_{2} + \beta_1) q^{58} + ( - 5 \beta_{6} - 7) q^{64} + ( - 6 \beta_{7} - 6 \beta_{6} + 5 \beta_{4} + \beta_{3} + 6 \beta_{2} - 5) q^{67} - 16 \beta_{5} q^{71} + (9 \beta_{4} + 3 \beta_{2}) q^{72} + ( - \beta_{7} + 5 \beta_{6} + 3 \beta_{4} + 9 \beta_{3} - 5 \beta_{2} - 3) q^{74} + ( - 6 \beta_{5} - 6 \beta_{3} - 12 \beta_1) q^{79} + ( - 9 \beta_{4} + 9) q^{81} + (4 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 7 \beta_{2} - 5 \beta_1) q^{86} + (7 \beta_{7} + 3 \beta_{6} + 7 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 7) q^{88} + ( - 6 \beta_{6} - 10) q^{92} + ( - 6 \beta_{7} - 6 \beta_{6} - 3 \beta_{5} + 6 \beta_{3} + 6 \beta_1 + 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b6 + b4 - b2 - 1) * q^4 + (b7 + 2*b5 - b3 - b1) * q^8 + (-3*b5 - 3*b3) * q^9 + (-2*b7 + 2*b6 - 3*b4 - b3 - 2*b2 + 3) * q^11 + (b4 + 3*b2) * q^16 + (3*b6 - 3*b4 - 3*b2 + 3) * q^18 + (-3*b7 + b6 + 4*b5 + 3*b3 + 3*b1 + 5) * q^22 + (2*b4 + 4*b2) * q^23 - 5*b3 * q^25 + (4*b7 + 4*b6 - b5 - 4*b3 - 4*b1 + 1) * q^29 + (b7 + 5*b3) * q^32 + (-3*b7 + 6*b5 + 3*b3 + 3*b1) * q^36 + (-b5 - b4 - b3 + 4*b2 + 4*b1) * q^37 + (-2*b7 + 2*b6 - 5*b5 + 2*b3 + 2*b1 - 5) * q^43 + (2*b5 + 7*b4 + 2*b3 + b2 + 5*b1) * q^44 + (2*b7 + 6*b3) * q^46 + (5*b6 + 5) * q^50 + (-4*b7 + 4*b6 - 7*b4 - 3*b3 - 4*b2 + 7) * q^53 + (8*b5 - 5*b4 + 8*b3 + 3*b2 + b1) * q^58 + (-5*b6 - 7) * q^64 + (-6*b7 - 6*b6 + 5*b4 + b3 + 6*b2 - 5) * q^67 - 16*b5 * q^71 + (9*b4 + 3*b2) * q^72 + (-b7 + 5*b6 + 3*b4 + 9*b3 - 5*b2 - 3) * q^74 + (-6*b5 - 6*b3 - 12*b1) * q^79 + (-9*b4 + 9) * q^81 + (4*b5 - 3*b4 + 4*b3 - 7*b2 - 5*b1) * q^86 + (7*b7 + 3*b6 + 7*b4 - 5*b3 - 3*b2 - 7) * q^88 + (-6*b6 - 10) * q^92 + (-6*b7 - 6*b6 - 3*b5 + 6*b3 + 6*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{4}+O(q^{10})$$ 8 * q - 6 * q^4 $$8 q - 6 q^{4} + 8 q^{11} - 2 q^{16} + 6 q^{18} + 36 q^{22} - 8 q^{29} - 12 q^{37} - 48 q^{43} + 26 q^{44} + 20 q^{50} + 20 q^{53} - 26 q^{58} - 36 q^{64} - 8 q^{67} + 30 q^{72} - 22 q^{74} + 36 q^{81} + 2 q^{86} - 34 q^{88} - 56 q^{92} + 48 q^{99}+O(q^{100})$$ 8 * q - 6 * q^4 + 8 * q^11 - 2 * q^16 + 6 * q^18 + 36 * q^22 - 8 * q^29 - 12 * q^37 - 48 * q^43 + 26 * q^44 + 20 * q^50 + 20 * q^53 - 26 * q^58 - 36 * q^64 - 8 * q^67 + 30 * q^72 - 22 * q^74 + 36 * q^81 + 2 * q^86 - 34 * q^88 - 56 * q^92 + 48 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{6} + 5\nu^{4} - 5\nu^{2} - 12 ) / 20$$ (-v^6 + 5*v^4 - 5*v^2 - 12) / 20 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40$$ (-v^7 + 5*v^5 - 5*v^3 - 12*v) / 40 $$\beta_{4}$$ $$=$$ $$( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20$$ (3*v^6 + 5*v^4 + 15*v^2 + 36) / 20 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40$$ (-3*v^7 - 5*v^5 + 5*v^3 - 16*v) / 40 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 7 ) / 5$$ (-v^6 - 7) / 5 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 3\nu^{5} + 5\nu^{3} + 12\nu ) / 8$$ (v^7 + 3*v^5 + 5*v^3 + 12*v) / 8
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} - \beta_{2} - 1$$ b6 + b4 - b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{5} - \beta_{3} - \beta_1$$ b7 + 2*b5 - b3 - b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3\beta_{2}$$ b4 + 3*b2 $$\nu^{5}$$ $$=$$ $$\beta_{7} + 5\beta_{3}$$ b7 + 5*b3 $$\nu^{6}$$ $$=$$ $$-5\beta_{6} - 7$$ -5*b6 - 7 $$\nu^{7}$$ $$=$$ $$-10\beta_{5} - 10\beta_{3} - 7\beta_1$$ -10*b5 - 10*b3 - 7*b1

## Character values

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

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$\beta_{5}$$ $$1$$ $$-\beta_{4}$$

## 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
 −0.228425 − 1.39564i 1.09445 + 0.895644i −1.09445 + 0.895644i 0.228425 − 1.39564i −1.09445 − 0.895644i 0.228425 + 1.39564i −0.228425 + 1.39564i 1.09445 − 0.895644i
−0.228425 1.39564i 0 −1.89564 + 0.637600i 0 0 0 1.32288 + 2.50000i 2.59808 1.50000i 0
165.2 1.09445 + 0.895644i 0 0.395644 + 1.96048i 0 0 0 −1.32288 + 2.50000i 2.59808 1.50000i 0
373.1 −1.09445 + 0.895644i 0 0.395644 1.96048i 0 0 0 1.32288 + 2.50000i −2.59808 1.50000i 0
373.2 0.228425 1.39564i 0 −1.89564 0.637600i 0 0 0 −1.32288 + 2.50000i −2.59808 1.50000i 0
557.1 −1.09445 0.895644i 0 0.395644 + 1.96048i 0 0 0 1.32288 2.50000i −2.59808 + 1.50000i 0
557.2 0.228425 + 1.39564i 0 −1.89564 + 0.637600i 0 0 0 −1.32288 2.50000i −2.59808 + 1.50000i 0
765.1 −0.228425 + 1.39564i 0 −1.89564 0.637600i 0 0 0 1.32288 2.50000i 2.59808 + 1.50000i 0
765.2 1.09445 0.895644i 0 0.395644 1.96048i 0 0 0 −1.32288 2.50000i 2.59808 + 1.50000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 765.2 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d odd 6 1 inner
16.e even 4 1 inner
112.l odd 4 1 inner
112.w even 12 1 inner
112.x odd 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.i 8
7.b odd 2 1 CM 784.2.x.i 8
7.c even 3 1 784.2.m.f 4
7.c even 3 1 inner 784.2.x.i 8
7.d odd 6 1 784.2.m.f 4
7.d odd 6 1 inner 784.2.x.i 8
16.e even 4 1 inner 784.2.x.i 8
112.l odd 4 1 inner 784.2.x.i 8
112.w even 12 1 784.2.m.f 4
112.w even 12 1 inner 784.2.x.i 8
112.x odd 12 1 784.2.m.f 4
112.x odd 12 1 inner 784.2.x.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.m.f 4 7.c even 3 1
784.2.m.f 4 7.d odd 6 1
784.2.m.f 4 112.w even 12 1
784.2.m.f 4 112.x odd 12 1
784.2.x.i 8 1.a even 1 1 trivial
784.2.x.i 8 7.b odd 2 1 CM
784.2.x.i 8 7.c even 3 1 inner
784.2.x.i 8 7.d odd 6 1 inner
784.2.x.i 8 16.e even 4 1 inner
784.2.x.i 8 112.l odd 4 1 inner
784.2.x.i 8 112.w even 12 1 inner
784.2.x.i 8 112.x odd 12 1 inner

## 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 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 3 T^{6} + 5 T^{4} + 12 T^{2} + \cdots + 16$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 1296$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$(T^{4} - 28 T^{2} + 784)^{2}$$
$29$ $$(T^{4} + 4 T^{3} + 8 T^{2} - 216 T + 2916)^{2}$$
$31$ $$T^{8}$$
$37$ $$T^{8} + 12 T^{7} + 72 T^{6} + \cdots + 2085136$$
$41$ $$T^{8}$$
$43$ $$(T^{4} + 24 T^{3} + 288 T^{2} + 1392 T + 3364)^{2}$$
$47$ $$T^{8}$$
$53$ $$T^{8} - 20 T^{7} + 200 T^{6} + \cdots + 1296$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 193877776$$
$71$ $$(T^{2} + 256)^{4}$$
$73$ $$T^{8}$$
$79$ $$(T^{4} + 252 T^{2} + 63504)^{2}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$