# Properties

 Label 128.4.a.h Level 128 Weight 4 Character orbit 128.a Self dual yes Analytic conductor 7.552 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 128.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.55224448073$$ Analytic rank: $$0$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + 4 \beta ) q^{3} + ( 2 - 8 \beta ) q^{5} + ( -4 + 8 \beta ) q^{7} + ( 25 + 16 \beta ) q^{9} +O(q^{10})$$ $$q + ( 2 + 4 \beta ) q^{3} + ( 2 - 8 \beta ) q^{5} + ( -4 + 8 \beta ) q^{7} + ( 25 + 16 \beta ) q^{9} + ( 46 - 4 \beta ) q^{11} + ( 50 + 24 \beta ) q^{13} + ( -92 - 8 \beta ) q^{15} + ( 46 - 48 \beta ) q^{17} + ( 2 - 28 \beta ) q^{19} + 88 q^{21} + ( 4 - 72 \beta ) q^{23} + ( 71 - 32 \beta ) q^{25} + ( 188 + 24 \beta ) q^{27} + ( 42 - 40 \beta ) q^{29} + ( -192 + 64 \beta ) q^{31} + ( 44 + 176 \beta ) q^{33} + ( -200 + 48 \beta ) q^{35} + ( -86 + 56 \beta ) q^{37} + ( 388 + 248 \beta ) q^{39} + ( -150 - 32 \beta ) q^{41} + ( 150 - 20 \beta ) q^{43} + ( -334 - 168 \beta ) q^{45} + ( -8 - 176 \beta ) q^{47} + ( -135 - 64 \beta ) q^{49} + ( -484 + 88 \beta ) q^{51} + ( -6 + 56 \beta ) q^{53} + ( 188 - 376 \beta ) q^{55} + ( -332 - 48 \beta ) q^{57} + ( -322 - 132 \beta ) q^{59} + ( 146 + 280 \beta ) q^{61} + ( 284 + 136 \beta ) q^{63} + ( -476 - 352 \beta ) q^{65} + ( -86 - 332 \beta ) q^{67} + ( -856 - 128 \beta ) q^{69} + ( 204 + 168 \beta ) q^{71} + ( 206 + 208 \beta ) q^{73} + ( -242 + 220 \beta ) q^{75} + ( -280 + 384 \beta ) q^{77} + ( -200 + 144 \beta ) q^{79} + ( -11 + 368 \beta ) q^{81} + ( 474 + 52 \beta ) q^{83} + ( 1244 - 464 \beta ) q^{85} + ( -396 + 88 \beta ) q^{87} + ( 286 + 464 \beta ) q^{89} + ( 376 + 304 \beta ) q^{91} + ( 384 - 640 \beta ) q^{93} + ( 676 - 72 \beta ) q^{95} + ( 1102 - 368 \beta ) q^{97} + ( 958 + 636 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} + 4q^{5} - 8q^{7} + 50q^{9} + O(q^{10})$$ $$2q + 4q^{3} + 4q^{5} - 8q^{7} + 50q^{9} + 92q^{11} + 100q^{13} - 184q^{15} + 92q^{17} + 4q^{19} + 176q^{21} + 8q^{23} + 142q^{25} + 376q^{27} + 84q^{29} - 384q^{31} + 88q^{33} - 400q^{35} - 172q^{37} + 776q^{39} - 300q^{41} + 300q^{43} - 668q^{45} - 16q^{47} - 270q^{49} - 968q^{51} - 12q^{53} + 376q^{55} - 664q^{57} - 644q^{59} + 292q^{61} + 568q^{63} - 952q^{65} - 172q^{67} - 1712q^{69} + 408q^{71} + 412q^{73} - 484q^{75} - 560q^{77} - 400q^{79} - 22q^{81} + 948q^{83} + 2488q^{85} - 792q^{87} + 572q^{89} + 752q^{91} + 768q^{93} + 1352q^{95} + 2204q^{97} + 1916q^{99} + O(q^{100})$$

## 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 −4.92820 0 15.8564 0 −17.8564 0 −2.71281 0
1.2 0 8.92820 0 −11.8564 0 9.85641 0 52.7128 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 128.4.a.h yes 2
3.b odd 2 1 1152.4.a.q 2
4.b odd 2 1 128.4.a.f yes 2
8.b even 2 1 128.4.a.e 2
8.d odd 2 1 128.4.a.g yes 2
12.b even 2 1 1152.4.a.r 2
16.e even 4 2 256.4.b.i 4
16.f odd 4 2 256.4.b.h 4
24.f even 2 1 1152.4.a.t 2
24.h odd 2 1 1152.4.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.e 2 8.b even 2 1
128.4.a.f yes 2 4.b odd 2 1
128.4.a.g yes 2 8.d odd 2 1
128.4.a.h yes 2 1.a even 1 1 trivial
256.4.b.h 4 16.f odd 4 2
256.4.b.i 4 16.e even 4 2
1152.4.a.q 2 3.b odd 2 1
1152.4.a.r 2 12.b even 2 1
1152.4.a.s 2 24.h odd 2 1
1152.4.a.t 2 24.f even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(128))$$:

 $$T_{3}^{2} - 4 T_{3} - 44$$ $$T_{5}^{2} - 4 T_{5} - 188$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T + 10 T^{2} - 108 T^{3} + 729 T^{4}$$
$5$ $$1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4}$$
$7$ $$1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4}$$
$11$ $$1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 100 T + 5166 T^{2} - 219700 T^{3} + 4826809 T^{4}$$
$17$ $$1 - 92 T + 5030 T^{2} - 451996 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 4 T + 11370 T^{2} - 27436 T^{3} + 47045881 T^{4}$$
$23$ $$1 - 8 T + 8798 T^{2} - 97336 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4}$$
$37$ $$1 + 172 T + 99294 T^{2} + 8712316 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 300 T + 157270 T^{2} + 20676300 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 300 T + 180314 T^{2} - 23852100 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 16 T + 114782 T^{2} + 1661168 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4}$$
$59$ $$1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 292 T + 240078 T^{2} - 66278452 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 172 T + 278250 T^{2} + 51731236 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 408 T + 672766 T^{2} - 146027688 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4}$$
$89$ $$1 - 572 T + 845846 T^{2} - 403242268 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4}$$