# Properties

 Label 175.2.x.a Level $175$ Weight $2$ Character orbit 175.x Analytic conductor $1.397$ Analytic rank $0$ Dimension $288$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,2,Mod(3,175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(60))

chi = DirichletCharacter(H, H._module([21, 10]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("175.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.x (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$288$$ Relative dimension: $$18$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$288 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 30 q^{5} - 10 q^{7} - 36 q^{8} - 10 q^{9}+O(q^{10})$$ 288 * q - 8 * q^2 - 24 * q^3 - 10 * q^4 - 30 * q^5 - 10 * q^7 - 36 * q^8 - 10 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$288 q - 8 q^{2} - 24 q^{3} - 10 q^{4} - 30 q^{5} - 10 q^{7} - 36 q^{8} - 10 q^{9} - 36 q^{10} - 6 q^{11} - 36 q^{12} - 20 q^{14} - 28 q^{15} - 30 q^{16} - 42 q^{17} - 14 q^{18} - 30 q^{19} - 12 q^{21} + 32 q^{22} - 40 q^{23} + 2 q^{25} - 48 q^{26} + 22 q^{28} - 58 q^{30} - 18 q^{31} + 8 q^{32} - 30 q^{33} - 2 q^{35} + 40 q^{36} - 10 q^{37} + 72 q^{38} + 30 q^{39} - 48 q^{40} + 6 q^{42} - 108 q^{43} - 10 q^{44} + 186 q^{45} - 6 q^{46} - 54 q^{47} - 248 q^{50} - 16 q^{51} + 216 q^{52} + 50 q^{53} - 30 q^{54} + 4 q^{56} - 216 q^{57} - 4 q^{58} + 90 q^{59} + 96 q^{60} - 18 q^{61} - 66 q^{63} - 100 q^{64} + 14 q^{65} - 90 q^{66} + 4 q^{67} + 342 q^{68} - 60 q^{70} - 24 q^{71} + 58 q^{72} - 6 q^{73} + 216 q^{75} - 80 q^{77} - 132 q^{78} - 10 q^{79} - 6 q^{80} - 10 q^{81} + 216 q^{82} + 20 q^{84} - 48 q^{85} - 6 q^{86} - 48 q^{87} - 122 q^{88} + 120 q^{89} - 12 q^{91} - 4 q^{92} + 106 q^{93} - 30 q^{94} - 98 q^{95} - 90 q^{96} + 222 q^{98}+O(q^{100})$$ 288 * q - 8 * q^2 - 24 * q^3 - 10 * q^4 - 30 * q^5 - 10 * q^7 - 36 * q^8 - 10 * q^9 - 36 * q^10 - 6 * q^11 - 36 * q^12 - 20 * q^14 - 28 * q^15 - 30 * q^16 - 42 * q^17 - 14 * q^18 - 30 * q^19 - 12 * q^21 + 32 * q^22 - 40 * q^23 + 2 * q^25 - 48 * q^26 + 22 * q^28 - 58 * q^30 - 18 * q^31 + 8 * q^32 - 30 * q^33 - 2 * q^35 + 40 * q^36 - 10 * q^37 + 72 * q^38 + 30 * q^39 - 48 * q^40 + 6 * q^42 - 108 * q^43 - 10 * q^44 + 186 * q^45 - 6 * q^46 - 54 * q^47 - 248 * q^50 - 16 * q^51 + 216 * q^52 + 50 * q^53 - 30 * q^54 + 4 * q^56 - 216 * q^57 - 4 * q^58 + 90 * q^59 + 96 * q^60 - 18 * q^61 - 66 * q^63 - 100 * q^64 + 14 * q^65 - 90 * q^66 + 4 * q^67 + 342 * q^68 - 60 * q^70 - 24 * q^71 + 58 * q^72 - 6 * q^73 + 216 * q^75 - 80 * q^77 - 132 * q^78 - 10 * q^79 - 6 * q^80 - 10 * q^81 + 216 * q^82 + 20 * q^84 - 48 * q^85 - 6 * q^86 - 48 * q^87 - 122 * q^88 + 120 * q^89 - 12 * q^91 - 4 * q^92 + 106 * q^93 - 30 * q^94 - 98 * q^95 - 90 * q^96 + 222 * q^98

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.36631 + 2.10393i 0.553395 1.44164i −1.74625 3.92214i 0.644473 + 2.14118i 2.27701 + 3.13403i 0.377880 2.61863i 5.68228 + 0.899985i 0.457347 + 0.411797i −5.38544 1.56958i
3.2 −1.20294 + 1.85237i 0.944233 2.45981i −1.17073 2.62951i −1.77209 1.36371i 3.42063 + 4.70809i −2.43764 + 1.02854i 1.91615 + 0.303488i −2.92966 2.63788i 4.65782 1.64211i
3.3 −1.15251 + 1.77471i −0.686737 + 1.78901i −1.00784 2.26365i −0.960426 + 2.01930i −2.38350 3.28061i 1.78481 + 1.95307i 0.998771 + 0.158190i −0.499520 0.449770i −2.47677 4.03174i
3.4 −1.09978 + 1.69351i 0.0195422 0.0509090i −0.844998 1.89790i 1.92424 1.13899i 0.0647231 + 0.0890837i −0.0453616 + 2.64536i 0.154579 + 0.0244829i 2.22722 + 2.00540i −0.187337 + 4.51137i
3.5 −0.863911 + 1.33031i −0.702377 + 1.82975i −0.209899 0.471441i −2.03789 0.920336i −1.82734 2.51512i −2.08368 1.63042i −2.32486 0.368222i −0.625235 0.562964i 2.98488 1.91592i
3.6 −0.467632 + 0.720090i 1.11331 2.90028i 0.513623 + 1.15362i 1.62243 + 1.53874i 1.56784 + 2.15795i 1.61121 + 2.09857i −2.76697 0.438245i −4.94272 4.45045i −1.86673 + 0.448731i
3.7 −0.423319 + 0.651855i 0.402507 1.04857i 0.567758 + 1.27521i 0.567586 2.16283i 0.513123 + 0.706254i 1.24404 2.33503i −2.60695 0.412900i 1.29196 + 1.16328i 1.16958 + 1.28555i
3.8 −0.365924 + 0.563474i −0.784887 + 2.04470i 0.629871 + 1.41471i 1.68811 + 1.46639i −0.864927 1.19047i 0.965333 2.46336i −2.35483 0.372968i −1.33532 1.20233i −1.44399 + 0.414616i
3.9 −0.000514176 0 0.000791762i −0.955405 + 2.48892i 0.813473 + 1.82709i 1.57098 1.59123i −0.00147938 0.00203619i −1.45449 + 2.21008i −0.00372978 0.000590739i −3.05247 2.74845i 0.000452119 0.00206202i
3.10 0.0204066 0.0314234i −0.0391619 + 0.102020i 0.812902 + 1.82581i −2.17394 0.523442i 0.00240666 + 0.00331248i 1.46457 + 2.20341i 0.147975 + 0.0234370i 2.22056 + 1.99940i −0.0608110 + 0.0576309i
3.11 0.437609 0.673859i 1.12462 2.92974i 0.550889 + 1.23732i −2.22451 + 0.227106i −1.48209 2.03992i −0.782993 2.52724i 2.66204 + 0.421625i −5.08915 4.58229i −0.820427 + 1.59839i
3.12 0.466637 0.718557i 0.476012 1.24005i 0.514898 + 1.15648i 2.23345 + 0.108225i −0.668925 0.920697i −2.64513 + 0.0574557i 2.76373 + 0.437732i 0.918288 + 0.826830i 1.11997 1.55436i
3.13 0.668598 1.02955i −0.0902073 + 0.234998i 0.200523 + 0.450382i −0.532453 + 2.17175i 0.181630 + 0.249992i 1.56284 2.13484i 3.02273 + 0.478753i 2.18235 + 1.96499i 1.87993 + 2.00021i
3.14 0.882496 1.35892i −0.978941 + 2.55023i −0.254403 0.571398i −1.48415 + 1.67252i 2.60166 + 3.58087i −2.64114 + 0.156130i 2.19977 + 0.348409i −3.31591 2.98566i 0.963067 + 3.49283i
3.15 0.986015 1.51833i −0.756771 + 1.97146i −0.519626 1.16710i 0.208311 2.22634i 2.24713 + 3.09291i 2.61446 0.405696i 1.29182 + 0.204604i −1.08450 0.976490i −3.17493 2.51149i
3.16 1.13877 1.75355i 0.297680 0.775483i −0.964662 2.16667i −1.08500 1.95519i −1.02086 1.40509i −2.15744 + 1.53148i −0.767632 0.121581i 1.71667 + 1.54570i −4.66408 0.323915i
3.17 1.32350 2.03802i 0.732885 1.90923i −1.58838 3.56756i −0.600501 + 2.15393i −2.92107 4.02051i 1.28591 + 2.31224i −4.57269 0.724243i −0.878604 0.791099i 3.59497 + 4.07456i
3.18 1.51333 2.33033i −0.475498 + 1.23871i −2.32679 5.22605i 2.19854 + 0.407943i 2.16703 + 2.98265i −1.54641 2.14677i −10.2108 1.61724i 0.921121 + 0.829381i 4.27777 4.50597i
12.1 −2.43914 0.936298i 2.98133 0.156245i 3.58646 + 3.22926i −1.61949 1.54184i −7.41816 2.41031i −1.17017 2.37291i −3.35206 6.57879i 5.88033 0.618047i 2.50654 + 5.27708i
12.2 −2.43845 0.936034i −2.16962 + 0.113705i 3.58360 + 3.22669i −2.10335 + 0.758905i 5.39694 + 1.75357i 2.32567 + 1.26145i −3.34656 6.56798i 1.71075 0.179807i 5.83927 + 0.118249i
See next 80 embeddings (of 288 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
25.f odd 20 1 inner
175.x even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.x.a 288
5.b even 2 1 875.2.bb.c 288
5.c odd 4 1 875.2.bb.a 288
5.c odd 4 1 875.2.bb.b 288
7.d odd 6 1 inner 175.2.x.a 288
25.d even 5 1 875.2.bb.b 288
25.e even 10 1 875.2.bb.a 288
25.f odd 20 1 inner 175.2.x.a 288
25.f odd 20 1 875.2.bb.c 288
35.i odd 6 1 875.2.bb.c 288
35.k even 12 1 875.2.bb.a 288
35.k even 12 1 875.2.bb.b 288
175.u odd 30 1 875.2.bb.a 288
175.v odd 30 1 875.2.bb.b 288
175.x even 60 1 inner 175.2.x.a 288
175.x even 60 1 875.2.bb.c 288

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.x.a 288 1.a even 1 1 trivial
175.2.x.a 288 7.d odd 6 1 inner
175.2.x.a 288 25.f odd 20 1 inner
175.2.x.a 288 175.x even 60 1 inner
875.2.bb.a 288 5.c odd 4 1
875.2.bb.a 288 25.e even 10 1
875.2.bb.a 288 35.k even 12 1
875.2.bb.a 288 175.u odd 30 1
875.2.bb.b 288 5.c odd 4 1
875.2.bb.b 288 25.d even 5 1
875.2.bb.b 288 35.k even 12 1
875.2.bb.b 288 175.v odd 30 1
875.2.bb.c 288 5.b even 2 1
875.2.bb.c 288 25.f odd 20 1
875.2.bb.c 288 35.i odd 6 1
875.2.bb.c 288 175.x even 60 1

## Hecke kernels

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