# Properties

 Label 128.3.h.a Level 128 Weight 3 Character orbit 128.h Analytic conductor 3.488 Analytic rank 0 Dimension 28 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$128 = 2^{7}$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 128.h (of order $$8$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$3.48774738381$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{8})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$28q$$ $$\mathstrut +\mathstrut 4q^{3}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut +\mathstrut 4q^{7}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q$$ $$\mathstrut +\mathstrut 4q^{3}$$ $$\mathstrut -\mathstrut 4q^{5}$$ $$\mathstrut +\mathstrut 4q^{7}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut 4q^{11}$$ $$\mathstrut -\mathstrut 4q^{13}$$ $$\mathstrut +\mathstrut 8q^{15}$$ $$\mathstrut +\mathstrut 4q^{19}$$ $$\mathstrut -\mathstrut 4q^{21}$$ $$\mathstrut +\mathstrut 68q^{23}$$ $$\mathstrut -\mathstrut 4q^{25}$$ $$\mathstrut +\mathstrut 100q^{27}$$ $$\mathstrut -\mathstrut 4q^{29}$$ $$\mathstrut -\mathstrut 8q^{33}$$ $$\mathstrut -\mathstrut 92q^{35}$$ $$\mathstrut -\mathstrut 4q^{37}$$ $$\mathstrut -\mathstrut 188q^{39}$$ $$\mathstrut -\mathstrut 4q^{41}$$ $$\mathstrut -\mathstrut 92q^{43}$$ $$\mathstrut -\mathstrut 40q^{45}$$ $$\mathstrut +\mathstrut 8q^{47}$$ $$\mathstrut -\mathstrut 224q^{51}$$ $$\mathstrut -\mathstrut 164q^{53}$$ $$\mathstrut -\mathstrut 252q^{55}$$ $$\mathstrut -\mathstrut 4q^{57}$$ $$\mathstrut -\mathstrut 124q^{59}$$ $$\mathstrut -\mathstrut 68q^{61}$$ $$\mathstrut -\mathstrut 8q^{65}$$ $$\mathstrut +\mathstrut 164q^{67}$$ $$\mathstrut +\mathstrut 188q^{69}$$ $$\mathstrut +\mathstrut 260q^{71}$$ $$\mathstrut -\mathstrut 4q^{73}$$ $$\mathstrut +\mathstrut 488q^{75}$$ $$\mathstrut +\mathstrut 220q^{77}$$ $$\mathstrut +\mathstrut 520q^{79}$$ $$\mathstrut +\mathstrut 484q^{83}$$ $$\mathstrut +\mathstrut 96q^{85}$$ $$\mathstrut +\mathstrut 452q^{87}$$ $$\mathstrut -\mathstrut 4q^{89}$$ $$\mathstrut +\mathstrut 196q^{91}$$ $$\mathstrut +\mathstrut 32q^{93}$$ $$\mathstrut -\mathstrut 8q^{97}$$ $$\mathstrut -\mathstrut 216q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1 0 −4.35131 + 1.80237i 0 −2.81639 1.16659i 0 6.23443 6.23443i 0 9.32143 9.32143i 0
15.2 0 −2.49683 + 1.03422i 0 −0.452310 0.187353i 0 −0.429965 + 0.429965i 0 −1.19943 + 1.19943i 0
15.3 0 −1.58190 + 0.655246i 0 4.18866 + 1.73500i 0 −3.93197 + 3.93197i 0 −4.29089 + 4.29089i 0
15.4 0 0.374985 0.155324i 0 −7.60625 3.15061i 0 −6.84161 + 6.84161i 0 −6.24747 + 6.24747i 0
15.5 0 1.37292 0.568682i 0 2.28872 + 0.948019i 0 6.37744 6.37744i 0 −4.80245 + 4.80245i 0
15.6 0 3.70255 1.53365i 0 7.20074 + 2.98264i 0 −4.26150 + 4.26150i 0 4.99283 4.99283i 0
15.7 0 4.68670 1.94129i 0 −4.51028 1.86822i 0 3.85317 3.85317i 0 11.8326 11.8326i 0
47.1 0 −2.10187 + 5.07436i 0 1.74699 + 4.21761i 0 0.392379 + 0.392379i 0 −14.9674 14.9674i 0
47.2 0 −0.936461 + 2.26082i 0 −3.18221 7.68254i 0 −3.67370 3.67370i 0 2.12963 + 2.12963i 0
47.3 0 −0.527719 + 1.27403i 0 −0.642823 1.55191i 0 4.95044 + 4.95044i 0 5.01930 + 5.01930i 0
47.4 0 −0.299792 + 0.723762i 0 1.34740 + 3.25291i 0 −0.583225 0.583225i 0 5.93000 + 5.93000i 0
47.5 0 1.10785 2.67458i 0 2.95565 + 7.13556i 0 4.18452 + 4.18452i 0 0.437918 + 0.437918i 0
47.6 0 1.31872 3.18367i 0 −0.659338 1.59178i 0 −9.54718 9.54718i 0 −2.03276 2.03276i 0
47.7 0 1.73217 4.18183i 0 −1.85856 4.48696i 0 5.27676 + 5.27676i 0 −8.12333 8.12333i 0
79.1 0 −2.10187 5.07436i 0 1.74699 4.21761i 0 0.392379 0.392379i 0 −14.9674 + 14.9674i 0
79.2 0 −0.936461 2.26082i 0 −3.18221 + 7.68254i 0 −3.67370 + 3.67370i 0 2.12963 2.12963i 0
79.3 0 −0.527719 1.27403i 0 −0.642823 + 1.55191i 0 4.95044 4.95044i 0 5.01930 5.01930i 0
79.4 0 −0.299792 0.723762i 0 1.34740 3.25291i 0 −0.583225 + 0.583225i 0 5.93000 5.93000i 0
79.5 0 1.10785 + 2.67458i 0 2.95565 7.13556i 0 4.18452 4.18452i 0 0.437918 0.437918i 0
79.6 0 1.31872 + 3.18367i 0 −0.659338 + 1.59178i 0 −9.54718 + 9.54718i 0 −2.03276 + 2.03276i 0
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 111.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

There are no other newforms in $$S_{3}^{\mathrm{new}}(128, [\chi])$$.