# Properties

 Label 175.2.n.a Level 175 Weight 2 Character orbit 175.n Analytic conductor 1.397 Analytic rank 0 Dimension 56 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.n (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56q + 12q^{4} - 6q^{5} + 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q + 12q^{4} - 6q^{5} + 6q^{9} - 4q^{10} + 8q^{11} - 40q^{12} - 4q^{14} - 18q^{15} - 32q^{16} + 12q^{19} + 12q^{20} + 4q^{21} - 30q^{22} + 10q^{23} - 28q^{24} + 4q^{25} + 12q^{26} + 30q^{27} - 2q^{29} + 28q^{30} + 12q^{31} + 20q^{33} + 2q^{35} - 14q^{36} - 70q^{37} - 70q^{38} - 4q^{39} - 30q^{40} + 4q^{41} + 50q^{42} + 22q^{44} - 52q^{45} - 4q^{46} - 10q^{47} + 30q^{48} - 56q^{49} - 54q^{50} - 44q^{51} - 20q^{53} + 54q^{54} - 2q^{55} + 12q^{56} + 10q^{58} - 6q^{59} + 16q^{60} - 4q^{61} + 50q^{62} + 20q^{63} + 24q^{64} - 18q^{65} - 74q^{66} + 10q^{67} - 78q^{69} + 8q^{70} - 8q^{71} + 140q^{72} + 40q^{73} + 60q^{74} - 8q^{75} + 52q^{76} - 20q^{77} - 90q^{78} + 124q^{80} - 72q^{81} - 30q^{83} - 12q^{84} + 96q^{85} - 20q^{86} + 30q^{87} + 140q^{88} + 38q^{89} - 8q^{90} + 8q^{91} + 80q^{92} + 88q^{94} - 70q^{95} - 28q^{96} - 30q^{97} + 60q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −2.54065 0.825507i −0.247551 + 0.340724i 4.15540 + 3.01908i −1.54620 1.61532i 0.910210 0.661306i 1.00000i −4.92473 6.77832i 0.872239 + 2.68448i 2.59489 + 5.38036i
29.2 −2.05165 0.666622i 0.505071 0.695171i 2.14685 + 1.55978i 2.21022 0.338990i −1.49965 + 1.08956i 1.00000i −0.828832 1.14079i 0.698885 + 2.15095i −4.76059 0.777894i
29.3 −1.82157 0.591863i −1.68900 + 2.32470i 1.34977 + 0.980668i 2.12640 0.691667i 4.45253 3.23495i 1.00000i 0.373298 + 0.513801i −1.62449 4.99966i −4.28276 + 0.00137701i
29.4 −1.51762 0.493105i −1.52541 + 2.09954i 0.441990 + 0.321125i −2.23549 + 0.0510350i 3.35029 2.43413i 1.00000i 1.36346 + 1.87664i −1.15416 3.55215i 3.41779 + 1.02488i
29.5 −0.977435 0.317588i 1.68681 2.32169i −0.763516 0.554727i 0.852197 2.06731i −2.38608 + 1.73359i 1.00000i 1.77829 + 2.44761i −1.61787 4.97930i −1.48952 + 1.75001i
29.6 −0.798956 0.259597i −0.142668 + 0.196366i −1.04709 0.760758i 1.61677 + 1.54469i 0.164962 0.119852i 1.00000i 1.62666 + 2.23890i 0.908846 + 2.79714i −0.890733 1.65384i
29.7 −0.387846 0.126019i −0.531527 + 0.731584i −1.48349 1.07782i 0.0211801 + 2.23597i 0.298344 0.216760i 1.00000i 0.918945 + 1.26482i 0.674357 + 2.07546i 0.273560 0.869881i
29.8 0.225611 + 0.0733055i 1.21485 1.67209i −1.57251 1.14249i −2.18117 0.492433i 0.396656 0.288188i 1.00000i −0.549895 0.756865i −0.392990 1.20950i −0.455999 0.270990i
29.9 1.03051 + 0.334833i −1.58321 + 2.17910i −0.668196 0.485473i −2.00240 + 0.995194i −2.36115 + 1.71547i 1.00000i −1.79981 2.47723i −1.31487 4.04676i −2.39671 + 0.355089i
29.10 1.09195 + 0.354796i 1.51811 2.08949i −0.551558 0.400730i 0.904522 + 2.04495i 2.39904 1.74301i 1.00000i −1.80982 2.49101i −1.13429 3.49098i 0.262151 + 2.55391i
29.11 1.19597 + 0.388594i 0.374566 0.515546i −0.338697 0.246078i 1.94508 1.10303i 0.648308 0.471023i 1.00000i −1.78775 2.46062i 0.801563 + 2.46696i 2.75488 0.563341i
29.12 1.99559 + 0.648406i −0.448050 + 0.616687i 1.94391 + 1.41233i −0.794894 + 2.09001i −1.29399 + 0.940136i 1.00000i 0.496791 + 0.683774i 0.747496 + 2.30056i −2.94146 + 3.65539i
29.13 2.11404 + 0.686892i 0.515049 0.708904i 2.37929 + 1.72866i −1.55449 1.60734i 1.57577 1.14486i 1.00000i 1.22941 + 1.69214i 0.689782 + 2.12293i −2.18219 4.46575i
29.14 2.44207 + 0.793475i −1.88310 + 2.59187i 3.71604 + 2.69986i 0.256301 2.22133i −6.65525 + 4.83532i 1.00000i 3.91399 + 5.38714i −2.24466 6.90836i 2.38847 5.22127i
64.1 −1.61734 2.22607i −2.13699 0.694351i −1.72159 + 5.29851i −2.23415 + 0.0925149i 1.91056 + 5.88011i 1.00000i 9.34546 3.03652i 1.65757 + 1.20429i 3.81932 + 4.82376i
64.2 −1.40576 1.93486i 2.61714 + 0.850362i −1.14949 + 3.53776i −0.290501 + 2.21712i −2.03374 6.25920i 1.00000i 3.91185 1.27104i 3.69928 + 2.68768i 4.69818 2.55465i
64.3 −1.11095 1.52910i 1.72878 + 0.561715i −0.485887 + 1.49541i −1.28703 1.82854i −1.06168 3.26751i 1.00000i −0.768705 + 0.249767i 0.246106 + 0.178807i −1.36619 + 3.99942i
64.4 −0.635734 0.875013i −3.03925 0.987513i 0.256544 0.789562i 0.574707 2.16095i 1.06807 + 3.28718i 1.00000i −2.91125 + 0.945922i 5.83482 + 4.23925i −2.25622 + 0.870914i
64.5 −0.560222 0.771080i −1.03930 0.337689i 0.337319 1.03816i −2.23402 0.0956332i 0.321853 + 0.990563i 1.00000i −2.80240 + 0.910553i −1.46094 1.06144i 1.17781 + 1.77618i
64.6 −0.455964 0.627580i 1.65172 + 0.536677i 0.432080 1.32981i 0.928918 2.03399i −0.416318 1.28129i 1.00000i −2.50710 + 0.814607i 0.0131148 + 0.00952845i −1.70004 + 0.344454i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 169.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.n.a 56
5.b even 2 1 875.2.n.c 56
5.c odd 4 1 875.2.h.d 56
5.c odd 4 1 875.2.h.e 56
25.d even 5 1 875.2.n.c 56
25.e even 10 1 inner 175.2.n.a 56
25.f odd 20 1 875.2.h.d 56
25.f odd 20 1 875.2.h.e 56
25.f odd 20 1 4375.2.a.o 28
25.f odd 20 1 4375.2.a.p 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.n.a 56 1.a even 1 1 trivial
175.2.n.a 56 25.e even 10 1 inner
875.2.h.d 56 5.c odd 4 1
875.2.h.d 56 25.f odd 20 1
875.2.h.e 56 5.c odd 4 1
875.2.h.e 56 25.f odd 20 1
875.2.n.c 56 5.b even 2 1
875.2.n.c 56 25.d even 5 1
4375.2.a.o 28 25.f odd 20 1
4375.2.a.p 28 25.f odd 20 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database