# Properties

 Label 1089.4.a.v Level $1089$ Weight $4$ Character orbit 1089.a Self dual yes Analytic conductor $64.253$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1089.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.2530799963$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( -4 + 2 \beta ) q^{4} + ( -1 - 8 \beta ) q^{5} + ( -10 + 4 \beta ) q^{7} + ( -6 - 10 \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( -4 + 2 \beta ) q^{4} + ( -1 - 8 \beta ) q^{5} + ( -10 + 4 \beta ) q^{7} + ( -6 - 10 \beta ) q^{8} + ( -25 - 9 \beta ) q^{10} + ( -40 + 20 \beta ) q^{13} + ( 2 - 6 \beta ) q^{14} + ( -4 - 32 \beta ) q^{16} + ( -62 + 12 \beta ) q^{17} + ( -36 - 60 \beta ) q^{19} + ( -44 + 30 \beta ) q^{20} + ( 49 + 36 \beta ) q^{23} + ( 68 + 16 \beta ) q^{25} + ( 20 - 20 \beta ) q^{26} + ( 64 - 36 \beta ) q^{28} + ( 72 - 56 \beta ) q^{29} + ( -17 + 28 \beta ) q^{31} + ( -52 + 44 \beta ) q^{32} + ( -26 - 50 \beta ) q^{34} + ( -86 + 76 \beta ) q^{35} + ( 27 - 8 \beta ) q^{37} + ( -216 - 96 \beta ) q^{38} + ( 246 + 58 \beta ) q^{40} + ( 268 - 4 \beta ) q^{41} + ( 30 + 16 \beta ) q^{43} + ( 157 + 85 \beta ) q^{46} + ( 136 + 120 \beta ) q^{47} + ( -195 - 80 \beta ) q^{49} + ( 116 + 84 \beta ) q^{50} + ( 280 - 160 \beta ) q^{52} + ( 246 + 56 \beta ) q^{53} + ( -60 + 76 \beta ) q^{56} + ( -96 + 16 \beta ) q^{58} + ( -317 + 132 \beta ) q^{59} + ( -420 - 184 \beta ) q^{61} + ( 67 + 11 \beta ) q^{62} + ( 112 + 248 \beta ) q^{64} + ( -440 + 300 \beta ) q^{65} + ( 377 - 20 \beta ) q^{67} + ( 320 - 172 \beta ) q^{68} + ( 142 - 10 \beta ) q^{70} + ( 339 - 76 \beta ) q^{71} + ( 200 + 468 \beta ) q^{73} + ( 3 + 19 \beta ) q^{74} + ( -216 + 168 \beta ) q^{76} + ( -158 - 656 \beta ) q^{79} + ( 772 + 64 \beta ) q^{80} + ( 256 + 264 \beta ) q^{82} + ( 234 + 120 \beta ) q^{83} + ( -226 + 484 \beta ) q^{85} + ( 78 + 46 \beta ) q^{86} + ( 921 + 328 \beta ) q^{89} + ( 640 - 360 \beta ) q^{91} + ( 20 - 46 \beta ) q^{92} + ( 496 + 256 \beta ) q^{94} + ( 1476 + 348 \beta ) q^{95} + ( 1097 + 144 \beta ) q^{97} + ( -435 - 275 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 8q^{4} - 2q^{5} - 20q^{7} - 12q^{8} + O(q^{10})$$ $$2q + 2q^{2} - 8q^{4} - 2q^{5} - 20q^{7} - 12q^{8} - 50q^{10} - 80q^{13} + 4q^{14} - 8q^{16} - 124q^{17} - 72q^{19} - 88q^{20} + 98q^{23} + 136q^{25} + 40q^{26} + 128q^{28} + 144q^{29} - 34q^{31} - 104q^{32} - 52q^{34} - 172q^{35} + 54q^{37} - 432q^{38} + 492q^{40} + 536q^{41} + 60q^{43} + 314q^{46} + 272q^{47} - 390q^{49} + 232q^{50} + 560q^{52} + 492q^{53} - 120q^{56} - 192q^{58} - 634q^{59} - 840q^{61} + 134q^{62} + 224q^{64} - 880q^{65} + 754q^{67} + 640q^{68} + 284q^{70} + 678q^{71} + 400q^{73} + 6q^{74} - 432q^{76} - 316q^{79} + 1544q^{80} + 512q^{82} + 468q^{83} - 452q^{85} + 156q^{86} + 1842q^{89} + 1280q^{91} + 40q^{92} + 992q^{94} + 2952q^{95} + 2194q^{97} - 870q^{98} + O(q^{100})$$

## 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
−0.732051 0 −7.46410 12.8564 0 −16.9282 11.3205 0 −9.41154
1.2 2.73205 0 −0.535898 −14.8564 0 −3.07180 −23.3205 0 −40.5885
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$

## 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 1089.4.a.v 2
3.b odd 2 1 121.4.a.c 2
11.b odd 2 1 99.4.a.c 2
12.b even 2 1 1936.4.a.w 2
33.d even 2 1 11.4.a.a 2
33.f even 10 4 121.4.c.c 8
33.h odd 10 4 121.4.c.f 8
44.c even 2 1 1584.4.a.bc 2
55.d odd 2 1 2475.4.a.q 2
132.d odd 2 1 176.4.a.i 2
165.d even 2 1 275.4.a.b 2
165.l odd 4 2 275.4.b.c 4
231.h odd 2 1 539.4.a.e 2
264.m even 2 1 704.4.a.p 2
264.p odd 2 1 704.4.a.n 2
429.e even 2 1 1859.4.a.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 33.d even 2 1
99.4.a.c 2 11.b odd 2 1
121.4.a.c 2 3.b odd 2 1
121.4.c.c 8 33.f even 10 4
121.4.c.f 8 33.h odd 10 4
176.4.a.i 2 132.d odd 2 1
275.4.a.b 2 165.d even 2 1
275.4.b.c 4 165.l odd 4 2
539.4.a.e 2 231.h odd 2 1
704.4.a.n 2 264.p odd 2 1
704.4.a.p 2 264.m even 2 1
1089.4.a.v 2 1.a even 1 1 trivial
1584.4.a.bc 2 44.c even 2 1
1859.4.a.a 2 429.e even 2 1
1936.4.a.w 2 12.b even 2 1
2475.4.a.q 2 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1089))$$:

 $$T_{2}^{2} - 2 T_{2} - 2$$ $$T_{5}^{2} + 2 T_{5} - 191$$ $$T_{7}^{2} + 20 T_{7} + 52$$

## Hecke characteristic polynomials

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