# Properties

 Label 784.6.a.bf Level $784$ Weight $6$ Character orbit 784.a Self dual yes Analytic conductor $125.741$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$125.740914733$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{113})$$ Defining polynomial: $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ x^4 - 2*x^3 - 59*x^2 + 60*x + 674 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}\cdot 7$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_{2}) q^{3} + (4 \beta_{3} - 2 \beta_{2}) q^{5} + (12 \beta_1 + 55) q^{9}+O(q^{10})$$ q + (b3 - b2) * q^3 + (4*b3 - 2*b2) * q^5 + (12*b1 + 55) * q^9 $$q + (\beta_{3} - \beta_{2}) q^{3} + (4 \beta_{3} - 2 \beta_{2}) q^{5} + (12 \beta_1 + 55) q^{9} + ( - 3 \beta_1 + 488) q^{11} + ( - 28 \beta_{3} - 64 \beta_{2}) q^{13} + (34 \beta_1 + 1024) q^{15} + (76 \beta_{3} - 7 \beta_{2}) q^{17} + ( - 39 \beta_{3} - 53 \beta_{2}) q^{19} + (86 \beta_1 + 1784) q^{23} + (80 \beta_1 + 691) q^{25} + ( - 44 \beta_{3} - 220 \beta_{2}) q^{27} + ( - 196 \beta_1 - 838) q^{29} + (386 \beta_{3} + 246 \beta_{2}) q^{31} + (452 \beta_{3} - 386 \beta_{2}) q^{33} + (12 \beta_1 - 2302) q^{37} + (308 \beta_1 - 616) q^{39} + (112 \beta_{3} + 1103 \beta_{2}) q^{41} + ( - 245 \beta_1 - 5112) q^{43} + (460 \beta_{3} - 1694 \beta_{2}) q^{45} + (822 \beta_{3} - 590 \beta_{2}) q^{47} + (429 \beta_1 + 16852) q^{51} + ( - 648 \beta_1 - 25730) q^{53} + (1892 \beta_{3} - 580 \beta_{2}) q^{55} + (176 \beta_1 - 3894) q^{57} + (1457 \beta_{3} - 1425 \beta_{2}) q^{59} + (340 \beta_{3} + 3326 \beta_{2}) q^{61} + (1624 \beta_1 - 15792) q^{65} + ( - 2908 \beta_1 + 5724) q^{67} + (2816 \beta_{3} - 4708 \beta_{2}) q^{69} + ( - 1078 \beta_1 + 38456) q^{71} + (2540 \beta_{3} - 4407 \beta_{2}) q^{73} + (1651 \beta_{3} - 3411 \beta_{2}) q^{75} + (1066 \beta_1 + 22672) q^{79} + ( - 1596 \beta_1 - 4301) q^{81} + ( - 3115 \beta_{3} - 4425 \beta_{2}) q^{83} + (652 \beta_1 + 68164) q^{85} + ( - 3190 \beta_{3} + 7502 \beta_{2}) q^{87} + (508 \beta_{3} - 6259 \beta_{2}) q^{89} + (208 \beta_1 + 61940) q^{93} + (1250 \beta_1 - 27056) q^{95} + ( - 3724 \beta_{3} + 5873 \beta_{2}) q^{97} + (5691 \beta_1 + 10568) q^{99}+O(q^{100})$$ q + (b3 - b2) * q^3 + (4*b3 - 2*b2) * q^5 + (12*b1 + 55) * q^9 + (-3*b1 + 488) * q^11 + (-28*b3 - 64*b2) * q^13 + (34*b1 + 1024) * q^15 + (76*b3 - 7*b2) * q^17 + (-39*b3 - 53*b2) * q^19 + (86*b1 + 1784) * q^23 + (80*b1 + 691) * q^25 + (-44*b3 - 220*b2) * q^27 + (-196*b1 - 838) * q^29 + (386*b3 + 246*b2) * q^31 + (452*b3 - 386*b2) * q^33 + (12*b1 - 2302) * q^37 + (308*b1 - 616) * q^39 + (112*b3 + 1103*b2) * q^41 + (-245*b1 - 5112) * q^43 + (460*b3 - 1694*b2) * q^45 + (822*b3 - 590*b2) * q^47 + (429*b1 + 16852) * q^51 + (-648*b1 - 25730) * q^53 + (1892*b3 - 580*b2) * q^55 + (176*b1 - 3894) * q^57 + (1457*b3 - 1425*b2) * q^59 + (340*b3 + 3326*b2) * q^61 + (1624*b1 - 15792) * q^65 + (-2908*b1 + 5724) * q^67 + (2816*b3 - 4708*b2) * q^69 + (-1078*b1 + 38456) * q^71 + (2540*b3 - 4407*b2) * q^73 + (1651*b3 - 3411*b2) * q^75 + (1066*b1 + 22672) * q^79 + (-1596*b1 - 4301) * q^81 + (-3115*b3 - 4425*b2) * q^83 + (652*b1 + 68164) * q^85 + (-3190*b3 + 7502*b2) * q^87 + (508*b3 - 6259*b2) * q^89 + (208*b1 + 61940) * q^93 + (1250*b1 - 27056) * q^95 + (-3724*b3 + 5873*b2) * q^97 + (5691*b1 + 10568) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 220 q^{9}+O(q^{10})$$ 4 * q + 220 * q^9 $$4 q + 220 q^{9} + 1952 q^{11} + 4096 q^{15} + 7136 q^{23} + 2764 q^{25} - 3352 q^{29} - 9208 q^{37} - 2464 q^{39} - 20448 q^{43} + 67408 q^{51} - 102920 q^{53} - 15576 q^{57} - 63168 q^{65} + 22896 q^{67} + 153824 q^{71} + 90688 q^{79} - 17204 q^{81} + 272656 q^{85} + 247760 q^{93} - 108224 q^{95} + 42272 q^{99}+O(q^{100})$$ 4 * q + 220 * q^9 + 1952 * q^11 + 4096 * q^15 + 7136 * q^23 + 2764 * q^25 - 3352 * q^29 - 9208 * q^37 - 2464 * q^39 - 20448 * q^43 + 67408 * q^51 - 102920 * q^53 - 15576 * q^57 - 63168 * q^65 + 22896 * q^67 + 153824 * q^71 + 90688 * q^79 - 17204 * q^81 + 272656 * q^85 + 247760 * q^93 - 108224 * q^95 + 42272 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 59x^{2} + 60x + 674$$ :

 $$\beta_{1}$$ $$=$$ $$( 8\nu^{3} - 12\nu^{2} - 688\nu + 346 ) / 105$$ (8*v^3 - 12*v^2 - 688*v + 346) / 105 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} - 3\nu^{2} - 67\nu + 34 ) / 15$$ (2*v^3 - 3*v^2 - 67*v + 34) / 15 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + 102\nu^{2} - 172\nu - 3116 ) / 105$$ (2*v^3 + 102*v^2 - 172*v - 3116) / 105
 $$\nu$$ $$=$$ $$( 4\beta_{2} - 7\beta _1 + 14 ) / 28$$ (4*b2 - 7*b1 + 14) / 28 $$\nu^{2}$$ $$=$$ $$( 4\beta_{3} - \beta _1 + 122 ) / 4$$ (4*b3 - b1 + 122) / 4 $$\nu^{3}$$ $$=$$ $$( 42\beta_{3} + 344\beta_{2} - 245\beta _1 + 1274 ) / 28$$ (42*b3 + 344*b2 - 245*b1 + 1274) / 28

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.40086 4.40086 7.22929 −6.22929
0 −23.5186 0 −74.2753 0 0 0 310.123 0
1.2 0 −6.54802 0 −45.9910 0 0 0 −200.123 0
1.3 0 6.54802 0 45.9910 0 0 0 −200.123 0
1.4 0 23.5186 0 74.2753 0 0 0 310.123 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.6.a.bf 4
4.b odd 2 1 49.6.a.g 4
7.b odd 2 1 inner 784.6.a.bf 4
12.b even 2 1 441.6.a.z 4
28.d even 2 1 49.6.a.g 4
28.f even 6 2 49.6.c.h 8
28.g odd 6 2 49.6.c.h 8
84.h odd 2 1 441.6.a.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.a.g 4 4.b odd 2 1
49.6.a.g 4 28.d even 2 1
49.6.c.h 8 28.f even 6 2
49.6.c.h 8 28.g odd 6 2
441.6.a.z 4 12.b even 2 1
441.6.a.z 4 84.h odd 2 1
784.6.a.bf 4 1.a even 1 1 trivial
784.6.a.bf 4 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 596T_{3}^{2} + 23716$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(784))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 596 T^{2} + 23716$$
$5$ $$T^{4} - 7632 T^{2} + \cdots + 11669056$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 976 T + 234076)^{2}$$
$13$ $$T^{4} - 1260672 T^{2} + \cdots + 76158337024$$
$17$ $$T^{4} - 2613668 T^{2} + \cdots + 1700202150724$$
$19$ $$T^{4} - 1359892 T^{2} + \cdots + 56942116$$
$23$ $$(T^{2} - 3568 T - 160336)^{2}$$
$29$ $$(T^{2} + 1676 T - 16661788)^{2}$$
$31$ $$T^{4} + \cdots + 614334295349824$$
$37$ $$(T^{2} + 4604 T + 5234116)^{2}$$
$41$ $$T^{4} - 251093444 T^{2} + \cdots + 14\!\cdots\!56$$
$43$ $$(T^{2} + 10224 T - 998756)^{2}$$
$47$ $$T^{4} - 349180624 T^{2} + \cdots + 17\!\cdots\!36$$
$53$ $$(T^{2} + 51460 T + 472236292)^{2}$$
$59$ $$T^{4} - 1249753044 T^{2} + \cdots + 11\!\cdots\!76$$
$61$ $$T^{4} - 2284246736 T^{2} + \cdots + 11\!\cdots\!24$$
$67$ $$(T^{2} - 11448 T - 3789557552)^{2}$$
$71$ $$(T^{2} - 76912 T + 953601968)^{2}$$
$73$ $$T^{4} - 6121721124 T^{2} + \cdots + 20\!\cdots\!44$$
$79$ $$(T^{2} - 45344 T + 386672)^{2}$$
$83$ $$T^{4} - 9034370100 T^{2} + \cdots + 17\!\cdots\!00$$
$89$ $$T^{4} - 7617937028 T^{2} + \cdots + 13\!\cdots\!96$$
$97$ $$T^{4} - 11859566628 T^{2} + \cdots + 11\!\cdots\!44$$