# Properties

 Label 784.4.a.ba Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$46.2574974445$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{37})$$ Defining polynomial: $$x^{2} - x - 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 28) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{37}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + ( 7 - 2 \beta ) q^{5} + 10 q^{9} +O(q^{10})$$ $$q -\beta q^{3} + ( 7 - 2 \beta ) q^{5} + 10 q^{9} + ( -16 - 7 \beta ) q^{11} + ( 14 + 4 \beta ) q^{13} + ( 74 - 7 \beta ) q^{15} + ( 77 - 4 \beta ) q^{17} + ( -112 - 3 \beta ) q^{19} + ( -34 - 7 \beta ) q^{23} + ( 72 - 28 \beta ) q^{25} + 17 \beta q^{27} + ( -118 - 28 \beta ) q^{29} + ( -98 - 7 \beta ) q^{31} + ( 259 + 16 \beta ) q^{33} + ( 173 - 42 \beta ) q^{37} + ( -148 - 14 \beta ) q^{39} + ( 210 + 12 \beta ) q^{41} + 172 q^{43} + ( 70 - 20 \beta ) q^{45} + ( 42 + 15 \beta ) q^{47} + ( 148 - 77 \beta ) q^{51} + ( -219 + 42 \beta ) q^{53} + ( 406 - 17 \beta ) q^{55} + ( 111 + 112 \beta ) q^{57} + ( -28 - 37 \beta ) q^{59} + ( -49 - 74 \beta ) q^{61} -198 q^{65} + ( -168 - 7 \beta ) q^{67} + ( 259 + 34 \beta ) q^{69} + ( -448 + 56 \beta ) q^{71} + ( -483 + 20 \beta ) q^{73} + ( 1036 - 72 \beta ) q^{75} + ( 26 + 133 \beta ) q^{79} -899 q^{81} + ( 196 + 88 \beta ) q^{83} + ( 835 - 182 \beta ) q^{85} + ( 1036 + 118 \beta ) q^{87} + ( -147 + 60 \beta ) q^{89} + ( 259 + 98 \beta ) q^{93} + ( -562 + 203 \beta ) q^{95} + ( 210 - 76 \beta ) q^{97} + ( -160 - 70 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{5} + 20q^{9} + O(q^{10})$$ $$2q + 14q^{5} + 20q^{9} - 32q^{11} + 28q^{13} + 148q^{15} + 154q^{17} - 224q^{19} - 68q^{23} + 144q^{25} - 236q^{29} - 196q^{31} + 518q^{33} + 346q^{37} - 296q^{39} + 420q^{41} + 344q^{43} + 140q^{45} + 84q^{47} + 296q^{51} - 438q^{53} + 812q^{55} + 222q^{57} - 56q^{59} - 98q^{61} - 396q^{65} - 336q^{67} + 518q^{69} - 896q^{71} - 966q^{73} + 2072q^{75} + 52q^{79} - 1798q^{81} + 392q^{83} + 1670q^{85} + 2072q^{87} - 294q^{89} + 518q^{93} - 1124q^{95} + 420q^{97} - 320q^{99} + O(q^{100})$$

## 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
 3.54138 −2.54138
0 −6.08276 0 −5.16553 0 0 0 10.0000 0
1.2 0 6.08276 0 19.1655 0 0 0 10.0000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.ba 2
4.b odd 2 1 196.4.a.g 2
7.b odd 2 1 784.4.a.u 2
7.d odd 6 2 112.4.i.d 4
12.b even 2 1 1764.4.a.n 2
28.d even 2 1 196.4.a.e 2
28.f even 6 2 28.4.e.a 4
28.g odd 6 2 196.4.e.g 4
56.j odd 6 2 448.4.i.g 4
56.m even 6 2 448.4.i.h 4
84.h odd 2 1 1764.4.a.z 2
84.j odd 6 2 252.4.k.c 4
84.n even 6 2 1764.4.k.ba 4
140.s even 6 2 700.4.i.g 4
140.x odd 12 4 700.4.r.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.e.a 4 28.f even 6 2
112.4.i.d 4 7.d odd 6 2
196.4.a.e 2 28.d even 2 1
196.4.a.g 2 4.b odd 2 1
196.4.e.g 4 28.g odd 6 2
252.4.k.c 4 84.j odd 6 2
448.4.i.g 4 56.j odd 6 2
448.4.i.h 4 56.m even 6 2
700.4.i.g 4 140.s even 6 2
700.4.r.d 8 140.x odd 12 4
784.4.a.u 2 7.b odd 2 1
784.4.a.ba 2 1.a even 1 1 trivial
1764.4.a.n 2 12.b even 2 1
1764.4.a.z 2 84.h odd 2 1
1764.4.k.ba 4 84.n even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} - 37$$ $$T_{5}^{2} - 14 T_{5} - 99$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-37 + T^{2}$$
$5$ $$-99 - 14 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1557 + 32 T + T^{2}$$
$13$ $$-396 - 28 T + T^{2}$$
$17$ $$5337 - 154 T + T^{2}$$
$19$ $$12211 + 224 T + T^{2}$$
$23$ $$-657 + 68 T + T^{2}$$
$29$ $$-15084 + 236 T + T^{2}$$
$31$ $$7791 + 196 T + T^{2}$$
$37$ $$-35339 - 346 T + T^{2}$$
$41$ $$38772 - 420 T + T^{2}$$
$43$ $$( -172 + T )^{2}$$
$47$ $$-6561 - 84 T + T^{2}$$
$53$ $$-17307 + 438 T + T^{2}$$
$59$ $$-49869 + 56 T + T^{2}$$
$61$ $$-200211 + 98 T + T^{2}$$
$67$ $$26411 + 336 T + T^{2}$$
$71$ $$84672 + 896 T + T^{2}$$
$73$ $$218489 + 966 T + T^{2}$$
$79$ $$-653817 - 52 T + T^{2}$$
$83$ $$-248112 - 392 T + T^{2}$$
$89$ $$-111591 + 294 T + T^{2}$$
$97$ $$-169612 - 420 T + T^{2}$$