# Properties

 Label 704.4.a.p Level $704$ Weight $4$ Character orbit 704.a Self dual yes Analytic conductor $41.537$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 704.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$41.5373446440$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( -1 - 2 \beta ) q^{5} + ( 10 - \beta ) q^{7} + ( 22 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( -1 - 2 \beta ) q^{5} + ( 10 - \beta ) q^{7} + ( 22 + 2 \beta ) q^{9} + 11 q^{11} + ( -40 + 5 \beta ) q^{13} + ( -97 - 3 \beta ) q^{15} + ( -62 + 3 \beta ) q^{17} + ( -36 - 15 \beta ) q^{19} + ( -38 + 9 \beta ) q^{21} + ( -49 - 9 \beta ) q^{23} + ( 68 + 4 \beta ) q^{25} + ( 91 - 3 \beta ) q^{27} + ( -72 + 14 \beta ) q^{29} + ( -17 + 7 \beta ) q^{31} + ( 11 + 11 \beta ) q^{33} + ( 86 - 19 \beta ) q^{35} + ( -27 + 2 \beta ) q^{37} + ( 200 - 35 \beta ) q^{39} + ( 268 - \beta ) q^{41} + ( 30 + 4 \beta ) q^{43} + ( -214 - 46 \beta ) q^{45} + ( -136 - 30 \beta ) q^{47} + ( -195 - 20 \beta ) q^{49} + ( 82 - 59 \beta ) q^{51} + ( 246 + 14 \beta ) q^{53} + ( -11 - 22 \beta ) q^{55} + ( -756 - 51 \beta ) q^{57} + ( -317 + 33 \beta ) q^{59} + ( -420 - 46 \beta ) q^{61} + ( 124 - 2 \beta ) q^{63} + ( -440 + 75 \beta ) q^{65} + ( -377 + 5 \beta ) q^{67} + ( -481 - 58 \beta ) q^{69} + ( -339 + 19 \beta ) q^{71} + ( -200 - 117 \beta ) q^{73} + ( 260 + 72 \beta ) q^{75} + ( 110 - 11 \beta ) q^{77} + ( 158 + 164 \beta ) q^{79} + ( -647 + 34 \beta ) q^{81} + ( -234 - 30 \beta ) q^{83} + ( -226 + 121 \beta ) q^{85} + ( 600 - 58 \beta ) q^{87} + ( -921 - 82 \beta ) q^{89} + ( -640 + 90 \beta ) q^{91} + ( 319 - 10 \beta ) q^{93} + ( 1476 + 87 \beta ) q^{95} + ( 1097 + 36 \beta ) q^{97} + ( 242 + 22 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{5} + 20q^{7} + 44q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{5} + 20q^{7} + 44q^{9} + 22q^{11} - 80q^{13} - 194q^{15} - 124q^{17} - 72q^{19} - 76q^{21} - 98q^{23} + 136q^{25} + 182q^{27} - 144q^{29} - 34q^{31} + 22q^{33} + 172q^{35} - 54q^{37} + 400q^{39} + 536q^{41} + 60q^{43} - 428q^{45} - 272q^{47} - 390q^{49} + 164q^{51} + 492q^{53} - 22q^{55} - 1512q^{57} - 634q^{59} - 840q^{61} + 248q^{63} - 880q^{65} - 754q^{67} - 962q^{69} - 678q^{71} - 400q^{73} + 520q^{75} + 220q^{77} + 316q^{79} - 1294q^{81} - 468q^{83} - 452q^{85} + 1200q^{87} - 1842q^{89} - 1280q^{91} + 638q^{93} + 2952q^{95} + 2194q^{97} + 484q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −5.92820 0 12.8564 0 16.9282 0 8.14359 0
1.2 0 7.92820 0 −14.8564 0 3.07180 0 35.8564 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
$$2$$ $$1$$
$$11$$ $$-1$$

## 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 704.4.a.p 2
4.b odd 2 1 704.4.a.n 2
8.b even 2 1 11.4.a.a 2
8.d odd 2 1 176.4.a.i 2
24.f even 2 1 1584.4.a.bc 2
24.h odd 2 1 99.4.a.c 2
40.f even 2 1 275.4.a.b 2
40.i odd 4 2 275.4.b.c 4
56.h odd 2 1 539.4.a.e 2
88.b odd 2 1 121.4.a.c 2
88.g even 2 1 1936.4.a.w 2
88.o even 10 4 121.4.c.c 8
88.p odd 10 4 121.4.c.f 8
104.e even 2 1 1859.4.a.a 2
120.i odd 2 1 2475.4.a.q 2
264.m even 2 1 1089.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 8.b even 2 1
99.4.a.c 2 24.h odd 2 1
121.4.a.c 2 88.b odd 2 1
121.4.c.c 8 88.o even 10 4
121.4.c.f 8 88.p odd 10 4
176.4.a.i 2 8.d odd 2 1
275.4.a.b 2 40.f even 2 1
275.4.b.c 4 40.i odd 4 2
539.4.a.e 2 56.h odd 2 1
704.4.a.n 2 4.b odd 2 1
704.4.a.p 2 1.a even 1 1 trivial
1089.4.a.v 2 264.m even 2 1
1584.4.a.bc 2 24.f even 2 1
1859.4.a.a 2 104.e even 2 1
1936.4.a.w 2 88.g even 2 1
2475.4.a.q 2 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(704))$$:

 $$T_{3}^{2} - 2 T_{3} - 47$$ $$T_{5}^{2} + 2 T_{5} - 191$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-47 - 2 T + T^{2}$$
$5$ $$-191 + 2 T + T^{2}$$
$7$ $$52 - 20 T + T^{2}$$
$11$ $$( -11 + T )^{2}$$
$13$ $$400 + 80 T + T^{2}$$
$17$ $$3412 + 124 T + T^{2}$$
$19$ $$-9504 + 72 T + T^{2}$$
$23$ $$-1487 + 98 T + T^{2}$$
$29$ $$-4224 + 144 T + T^{2}$$
$31$ $$-2063 + 34 T + T^{2}$$
$37$ $$537 + 54 T + T^{2}$$
$41$ $$71776 - 536 T + T^{2}$$
$43$ $$132 - 60 T + T^{2}$$
$47$ $$-24704 + 272 T + T^{2}$$
$53$ $$51108 - 492 T + T^{2}$$
$59$ $$48217 + 634 T + T^{2}$$
$61$ $$74832 + 840 T + T^{2}$$
$67$ $$140929 + 754 T + T^{2}$$
$71$ $$97593 + 678 T + T^{2}$$
$73$ $$-617072 + 400 T + T^{2}$$
$79$ $$-1266044 - 316 T + T^{2}$$
$83$ $$11556 + 468 T + T^{2}$$
$89$ $$525489 + 1842 T + T^{2}$$
$97$ $$1141201 - 2194 T + T^{2}$$