# Properties

 Label 784.4.f.h Level $784$ Weight $4$ Character orbit 784.f Analytic conductor $46.257$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# 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: $$6$$ Coefficient field: 6.0.12258833328.1 Defining polynomial: $$x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: no (minimal twist has level 112) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + ( 26 - \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \beta_{2} ) q^{3} + \beta_{3} q^{5} + ( 26 - \beta_{1} - \beta_{2} ) q^{9} + ( -\beta_{3} - 11 \beta_{4} + \beta_{5} ) q^{11} + ( -\beta_{3} - 23 \beta_{4} ) q^{13} + ( 7 \beta_{3} - 19 \beta_{4} + \beta_{5} ) q^{15} + ( -5 \beta_{3} + 8 \beta_{5} ) q^{17} + ( -48 - \beta_{1} - 2 \beta_{2} ) q^{19} + ( 6 \beta_{3} - \beta_{5} ) q^{23} + ( -94 + 24 \beta_{2} ) q^{25} + ( 60 - 7 \beta_{1} - 8 \beta_{2} ) q^{27} + ( -50 - \beta_{1} + 23 \beta_{2} ) q^{29} + ( -76 - \beta_{1} - 24 \beta_{2} ) q^{31} + ( -4 \beta_{3} - 51 \beta_{4} + 8 \beta_{5} ) q^{33} + ( 75 - 2 \beta_{1} - 26 \beta_{2} ) q^{37} + ( -7 \beta_{3} - 27 \beta_{4} + 22 \beta_{5} ) q^{39} + ( -9 \beta_{3} + 141 \beta_{4} + 8 \beta_{5} ) q^{41} + ( -16 \beta_{3} + 20 \beta_{4} + 22 \beta_{5} ) q^{43} + ( 25 \beta_{3} - 219 \beta_{4} + 24 \beta_{5} ) q^{45} + ( -32 - \beta_{1} - 22 \beta_{2} ) q^{47} + ( -11 \beta_{3} - 289 \beta_{4} - 21 \beta_{5} ) q^{51} + ( 231 + 6 \beta_{1} + 54 \beta_{2} ) q^{53} + ( 146 + 16 \beta_{1} - 5 \beta_{2} ) q^{55} + ( 15 - 8 \beta_{1} + 40 \beta_{2} ) q^{57} + ( -106 - 2 \beta_{1} + \beta_{2} ) q^{59} + ( -6 \beta_{3} - 189 \beta_{4} + 8 \beta_{5} ) q^{61} + ( 196 + 23 \beta_{1} - \beta_{2} ) q^{65} + ( 42 \beta_{3} - 44 \beta_{4} - 25 \beta_{5} ) q^{67} + ( 39 \beta_{3} - 66 \beta_{4} + 8 \beta_{5} ) q^{69} + ( -14 \beta_{3} - 322 \beta_{4} - 26 \beta_{5} ) q^{71} + ( -42 \beta_{3} - 95 \beta_{4} - 8 \beta_{5} ) q^{73} + ( -1364 + 24 \beta_{1} + 70 \beta_{2} ) q^{75} + ( -14 \beta_{3} - 372 \beta_{4} + 47 \beta_{5} ) q^{79} + ( -99 - 23 \beta_{1} - 95 \beta_{2} ) q^{81} + ( 20 + 22 \beta_{1} + 70 \beta_{2} ) q^{83} + ( 599 + 40 \beta_{1} - 56 \beta_{2} ) q^{85} + ( -1214 + 17 \beta_{1} + 17 \beta_{2} ) q^{87} + ( 28 \beta_{3} - 291 \beta_{4} - 32 \beta_{5} ) q^{89} + ( 1037 - 30 \beta_{1} + 90 \beta_{2} ) q^{93} + ( -44 \beta_{3} - 238 \beta_{4} + 25 \beta_{5} ) q^{95} + ( -\beta_{3} + 45 \beta_{4} - 48 \beta_{5} ) q^{97} + ( 23 \beta_{3} - 113 \beta_{4} + 4 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 14q^{3} + 156q^{9} + O(q^{10})$$ $$6q + 14q^{3} + 156q^{9} - 286q^{19} - 612q^{25} + 362q^{27} - 348q^{29} - 410q^{31} + 498q^{37} - 150q^{47} + 1290q^{53} + 918q^{55} - 6q^{57} - 642q^{59} + 1224q^{65} - 8276q^{75} - 450q^{81} + 24q^{83} + 3786q^{85} - 7284q^{87} + 5982q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$17 \nu^{5} - 493 \nu^{4} - 8463 \nu^{3} - 13736 \nu^{2} + 11424 \nu - 285776$$$$)/45520$$ $$\beta_{2}$$ $$=$$ $$($$$$-29 \nu^{5} + 841 \nu^{4} - 1629 \nu^{3} + 23432 \nu^{2} - 19488 \nu + 428592$$$$)/45520$$ $$\beta_{3}$$ $$=$$ $$($$$$-197 \nu^{5} + 23 \nu^{4} - 667 \nu^{3} - 5834 \nu^{2} + 117976 \nu - 52824$$$$)/68280$$ $$\beta_{4}$$ $$=$$ $$($$$$203 \nu^{5} - 197 \nu^{4} + 5713 \nu^{3} + 986 \nu^{2} + 159176 \nu - 64104$$$$)/68280$$ $$\beta_{5}$$ $$=$$ $$($$$$1191 \nu^{5} - 399 \nu^{4} + 34331 \nu^{3} - 17788 \nu^{2} + 982432 \nu - 404688$$$$)/45520$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_{1} + 2$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$-12 \beta_{5} + 111 \beta_{4} + 3 \beta_{3} + 11 \beta_{2} - \beta_{1} - 110$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-17 \beta_{2} - 29 \beta_{1} - 22$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$116 \beta_{5} - 1029 \beta_{4} - 33 \beta_{3} + 105 \beta_{2} - 11 \beta_{1} - 1018$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$396 \beta_{5} - 1851 \beta_{4} - 2463 \beta_{3} + 425 \beta_{2} + 821 \beta_{1} + 1030$$$$)/12$$

## 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
 0.426664 − 0.739004i 0.426664 + 0.739004i 2.68858 − 4.65676i 2.68858 + 4.65676i −2.61524 − 4.52973i −2.61524 + 4.52973i
0 −7.06258 0 1.22397i 0 0 0 22.8800 0
783.2 0 −7.06258 0 1.22397i 0 0 0 22.8800 0
783.3 0 4.76834 0 16.8950i 0 0 0 −4.26295 0
783.4 0 4.76834 0 16.8950i 0 0 0 −4.26295 0
783.5 0 9.29424 0 19.8510i 0 0 0 59.3829 0
783.6 0 9.29424 0 19.8510i 0 0 0 59.3829 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 783.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.h 6
4.b odd 2 1 784.4.f.g 6
7.b odd 2 1 784.4.f.g 6
7.c even 3 1 112.4.p.f 6
7.d odd 6 1 112.4.p.g yes 6
28.d even 2 1 inner 784.4.f.h 6
28.f even 6 1 112.4.p.f 6
28.g odd 6 1 112.4.p.g yes 6
56.j odd 6 1 448.4.p.f 6
56.k odd 6 1 448.4.p.f 6
56.m even 6 1 448.4.p.g 6
56.p even 6 1 448.4.p.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.f 6 7.c even 3 1
112.4.p.f 6 28.f even 6 1
112.4.p.g yes 6 7.d odd 6 1
112.4.p.g yes 6 28.g odd 6 1
448.4.p.f 6 56.j odd 6 1
448.4.p.f 6 56.k odd 6 1
448.4.p.g 6 56.m even 6 1
448.4.p.g 6 56.p even 6 1
784.4.f.g 6 4.b odd 2 1
784.4.f.g 6 7.b odd 2 1
784.4.f.h 6 1.a even 1 1 trivial
784.4.f.h 6 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - 7 T_{3}^{2} - 55 T_{3} + 313$$ acting on $$S_{4}^{\mathrm{new}}(784, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 313 - 55 T - 7 T^{2} + T^{3} )^{2}$$
$5$ $$168507 + 113499 T^{2} + 681 T^{4} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$4876875 + 270279 T^{2} + 1821 T^{4} + T^{6}$$
$13$ $$2167603200 + 7385616 T^{2} + 5304 T^{4} + T^{6}$$
$17$ $$710833347 + 215397963 T^{2} + 29361 T^{4} + T^{6}$$
$19$ $$( 33271 + 5537 T + 143 T^{2} + T^{3} )^{2}$$
$23$ $$8915001507 + 129347991 T^{2} + 22677 T^{4} + T^{6}$$
$29$ $$( -4622400 - 27504 T + 174 T^{2} + T^{3} )^{2}$$
$31$ $$( -1532779 - 29071 T + 205 T^{2} + T^{3} )^{2}$$
$37$ $$( 3813125 - 33525 T - 249 T^{2} + T^{3} )^{2}$$
$41$ $$38591270836992 + 14002779024 T^{2} + 235176 T^{4} + T^{6}$$
$43$ $$83474849412288 + 16477161840 T^{2} + 251796 T^{4} + T^{6}$$
$47$ $$( -143829 - 34551 T + 75 T^{2} + T^{3} )^{2}$$
$53$ $$( 61174089 - 116541 T - 645 T^{2} + T^{3} )^{2}$$
$59$ $$( 908793 + 30753 T + 321 T^{2} + T^{3} )^{2}$$
$61$ $$912934776095523 + 34600686795 T^{2} + 357585 T^{4} + T^{6}$$
$67$ $$34861164967167075 + 360473787351 T^{2} + 1094421 T^{4} + T^{6}$$
$71$ $$1026472074810048 + 436555595376 T^{2} + 1416948 T^{4} + T^{6}$$
$73$ $$48340453761263403 + 565120567611 T^{2} + 1408473 T^{4} + T^{6}$$
$79$ $$6567218609777523 + 1140744027447 T^{2} + 2151525 T^{4} + T^{6}$$
$83$ $$( 308212992 - 856512 T - 12 T^{2} + T^{3} )^{2}$$
$89$ $$6090737949537003 + 426701516283 T^{2} + 1389657 T^{4} + T^{6}$$
$97$ $$19259496900926208 + 308008959120 T^{2} + 1038696 T^{4} + T^{6}$$