# Properties

 Label 784.4.f.i Level $784$ Weight $4$ Character orbit 784.f Analytic conductor $46.257$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.2574974445$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.16928550682624.32 Defining polynomial: $$x^{8} - 4 x^{7} + 62 x^{6} - 152 x^{5} + 1187 x^{4} - 1424 x^{3} + 7038 x^{2} + 1452 x + 5287$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{5}\cdot 7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( 2 \beta_{2} + \beta_{4} ) q^{5} + ( 43 + 3 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + ( 2 \beta_{2} + \beta_{4} ) q^{5} + ( 43 + 3 \beta_{5} ) q^{9} -\beta_{7} q^{11} + ( -16 \beta_{2} - \beta_{4} ) q^{13} + ( \beta_{6} + 2 \beta_{7} ) q^{15} + ( -13 \beta_{2} + 12 \beta_{4} ) q^{17} + ( \beta_{1} + \beta_{3} ) q^{19} + ( 5 \beta_{6} - 4 \beta_{7} ) q^{23} + ( 57 - 2 \beta_{5} ) q^{25} + ( 3 \beta_{1} + 28 \beta_{3} ) q^{27} + ( -112 - 2 \beta_{5} ) q^{29} + ( -2 \beta_{1} + 12 \beta_{3} ) q^{31} + ( -100 \beta_{2} + 3 \beta_{4} ) q^{33} -28 \beta_{5} q^{37} + ( -\beta_{6} - 16 \beta_{7} ) q^{39} + ( 57 \beta_{2} + 41 \beta_{4} ) q^{41} + ( 10 \beta_{6} + 9 \beta_{7} ) q^{43} + ( 152 \beta_{2} + 7 \beta_{4} ) q^{45} + ( 4 \beta_{1} - 10 \beta_{3} ) q^{47} + ( 12 \beta_{6} - 13 \beta_{7} ) q^{51} + ( 26 + 34 \beta_{5} ) q^{53} + ( 2 \beta_{1} + 30 \beta_{3} ) q^{55} + ( 84 + 61 \beta_{5} ) q^{57} + ( \beta_{1} + 43 \beta_{3} ) q^{59} + ( 28 \beta_{2} - 25 \beta_{4} ) q^{61} + ( 376 + 30 \beta_{5} ) q^{65} + ( 4 \beta_{6} + 10 \beta_{7} ) q^{67} + ( -370 \beta_{2} + 212 \beta_{4} ) q^{69} + ( -12 \beta_{6} + 12 \beta_{7} ) q^{71} + ( -47 \beta_{2} + 141 \beta_{4} ) q^{73} + ( -2 \beta_{1} + 49 \beta_{3} ) q^{75} + ( -12 \beta_{6} + 26 \beta_{7} ) q^{79} + ( 841 + 177 \beta_{5} ) q^{81} + ( -\beta_{1} - 67 \beta_{3} ) q^{83} + ( -2 + 50 \beta_{5} ) q^{85} + ( -2 \beta_{1} - 120 \beta_{3} ) q^{87} + ( -11 \beta_{2} - 5 \beta_{4} ) q^{89} + ( 812 - 80 \beta_{5} ) q^{93} + ( -15 \beta_{6} + 16 \beta_{7} ) q^{95} + ( -151 \beta_{2} + 262 \beta_{4} ) q^{97} + ( 3 \beta_{6} - 73 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 344q^{9} + O(q^{10})$$ $$8q + 344q^{9} + 456q^{25} - 896q^{29} + 208q^{53} + 672q^{57} + 3008q^{65} + 6728q^{81} - 16q^{85} + 6496q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 62 x^{6} - 152 x^{5} + 1187 x^{4} - 1424 x^{3} + 7038 x^{2} + 1452 x + 5287$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4164 \nu^{7} - 358345 \nu^{6} - 650409 \nu^{5} - 16098218 \nu^{4} - 15747163 \nu^{3} - 245807314 \nu^{2} - 102163903 \nu - 788088157$$$$)/10361911$$ $$\beta_{2}$$ $$=$$ $$($$$$-6093932 \nu^{7} + 25171336 \nu^{6} - 423467291 \nu^{5} + 1234423939 \nu^{4} - 9271817955 \nu^{3} + 15018125350 \nu^{2} - 55844883984 \nu + 2237396676$$$$)/ 10703854063$$ $$\beta_{3}$$ $$=$$ $$($$$$-6106 \nu^{7} + 66782 \nu^{6} - 431171 \nu^{5} + 2452981 \nu^{4} - 6194717 \nu^{3} + 14622770 \nu^{2} + 1073804 \nu - 91499108$$$$)/10361911$$ $$\beta_{4}$$ $$=$$ $$($$$$-10107994 \nu^{7} + 19374127 \nu^{6} - 457850077 \nu^{5} + 577333526 \nu^{4} - 5688721391 \nu^{3} + 4029751416 \nu^{2} - 7580517255 \nu + 3307731061$$$$)/ 10703854063$$ $$\beta_{5}$$ $$=$$ $$($$$$1630 \nu^{7} + 4476 \nu^{6} + 47706 \nu^{5} + 326529 \nu^{4} + 461416 \nu^{3} + 5257845 \nu^{2} + 2260794 \nu + 18437249$$$$)/1480273$$ $$\beta_{6}$$ $$=$$ $$($$$$53734552 \nu^{7} - 275479040 \nu^{6} + 2941079496 \nu^{5} - 8802476688 \nu^{4} + 41209290032 \nu^{3} - 72973086916 \nu^{2} + 54821054056 \nu - 52627155866$$$$)/ 10703854063$$ $$\beta_{7}$$ $$=$$ $$($$$$108735962 \nu^{7} - 493132366 \nu^{6} + 6381684922 \nu^{5} - 17094460415 \nu^{4} + 109403899092 \nu^{3} - 153489402447 \nu^{2} + 568341608774 \nu - 27356882655$$$$)/ 10703854063$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 12 \beta_{2} + 14$$$$)/28$$ $$\nu^{2}$$ $$=$$ $$($$$$-7 \beta_{6} + 8 \beta_{5} - 8 \beta_{4} - 28 \beta_{3} - 4 \beta_{2} - 378$$$$)/28$$ $$\nu^{3}$$ $$=$$ $$($$$$-17 \beta_{7} + 22 \beta_{6} + 49 \beta_{5} + 89 \beta_{4} - 9 \beta_{3} - 155 \beta_{2} + 3 \beta_{1} - 392$$$$)/14$$ $$\nu^{4}$$ $$=$$ $$($$$$245 \beta_{6} + 12 \beta_{5} + 824 \beta_{4} + 640 \beta_{3} + 160 \beta_{2} - 8 \beta_{1} + 5810$$$$)/28$$ $$\nu^{5}$$ $$=$$ $$($$$$642 \beta_{7} - 417 \beta_{6} - 2918 \beta_{5} - 3138 \beta_{4} + 1210 \beta_{3} + 6306 \beta_{2} - 310 \beta_{1} + 22694$$$$)/28$$ $$\nu^{6}$$ $$=$$ $$($$$$-53 \beta_{7} - 3235 \beta_{6} - 2653 \beta_{5} - 16240 \beta_{4} - 5246 \beta_{3} - 2142 \beta_{2} - 132 \beta_{1} - 33894$$$$)/14$$ $$\nu^{7}$$ $$=$$ $$($$$$-11648 \beta_{7} - 4823 \beta_{6} + 72224 \beta_{5} - 3070 \beta_{4} - 39396 \beta_{3} - 120836 \beta_{2} + 9702 \beta_{1} - 536830$$$$)/28$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
783.1
 −0.207107 + 0.854139i −0.207107 − 0.854139i 1.20711 − 3.38189i 1.20711 + 3.38189i 1.20711 + 4.91262i 1.20711 − 4.91262i −0.207107 − 4.54966i −0.207107 + 4.54966i
0 −9.98491 0 9.37011i 0 0 0 72.6985 0
783.2 0 −9.98491 0 9.37011i 0 0 0 72.6985 0
783.3 0 −6.34835 0 6.94269i 0 0 0 13.3015 0
783.4 0 −6.34835 0 6.94269i 0 0 0 13.3015 0
783.5 0 6.34835 0 6.94269i 0 0 0 13.3015 0
783.6 0 6.34835 0 6.94269i 0 0 0 13.3015 0
783.7 0 9.98491 0 9.37011i 0 0 0 72.6985 0
783.8 0 9.98491 0 9.37011i 0 0 0 72.6985 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 783.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.f.i 8
4.b odd 2 1 inner 784.4.f.i 8
7.b odd 2 1 inner 784.4.f.i 8
28.d even 2 1 inner 784.4.f.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.4.f.i 8 1.a even 1 1 trivial
784.4.f.i 8 4.b odd 2 1 inner
784.4.f.i 8 7.b odd 2 1 inner
784.4.f.i 8 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 140 T_{3}^{2} + 4018$$ acting on $$S_{4}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 4018 - 140 T^{2} + T^{4} )^{2}$$
$5$ $$( 4232 + 136 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 8036 + 1988 T^{2} + T^{4} )^{2}$$
$13$ $$( 668168 + 5288 T^{2} + T^{4} )^{2}$$
$17$ $$( 14183138 + 7892 T^{2} + T^{4} )^{2}$$
$19$ $$( 31826578 - 11452 T^{2} + T^{4} )^{2}$$
$23$ $$( 896464016 + 61208 T^{2} + T^{4} )^{2}$$
$29$ $$( 12152 + 224 T + T^{2} )^{4}$$
$31$ $$( 257152 - 63840 T^{2} + T^{4} )^{2}$$
$37$ $$( -76832 + T^{2} )^{4}$$
$41$ $$( 5692658402 + 150916 T^{2} + T^{4} )^{2}$$
$43$ $$( 34800388196 + 373828 T^{2} + T^{4} )^{2}$$
$47$ $$( 3196977952 - 191856 T^{2} + T^{4} )^{2}$$
$53$ $$( -112612 - 52 T + T^{2} )^{4}$$
$59$ $$( 8136164722 - 272524 T^{2} + T^{4} )^{2}$$
$61$ $$( 286370312 + 35080 T^{2} + T^{4} )^{2}$$
$67$ $$( 6980262464 + 247184 T^{2} + T^{4} )^{2}$$
$71$ $$( 48157369344 + 439488 T^{2} + T^{4} )^{2}$$
$73$ $$( 16405487522 + 786404 T^{2} + T^{4} )^{2}$$
$79$ $$( 223691510336 + 1403024 T^{2} + T^{4} )^{2}$$
$83$ $$( 60208287538 - 643468 T^{2} + T^{4} )^{2}$$
$89$ $$( 3230882 + 3860 T^{2} + T^{4} )^{2}$$
$97$ $$( 796703169602 + 2885284 T^{2} + T^{4} )^{2}$$