Properties

 Label 75.3.k.a Level $75$ Weight $3$ Character orbit 75.k Analytic conductor $2.044$ Analytic rank $0$ Dimension $80$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(13,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 19]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("75.13");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 75.k (of order $$20$$, degree $$8$$, 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: $$2.04360198270$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$10$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 4 q^{2} + 4 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10})$$ 80 * q + 4 * q^2 + 4 * q^5 - 4 * q^7 - 12 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$80 q + 4 q^{2} + 4 q^{5} - 4 q^{7} - 12 q^{8} - 4 q^{10} - 24 q^{12} + 32 q^{13} - 24 q^{15} + 80 q^{16} - 100 q^{17} - 48 q^{18} - 100 q^{19} - 244 q^{20} - 100 q^{22} - 96 q^{23} - 16 q^{25} - 40 q^{26} + 196 q^{28} + 200 q^{29} + 264 q^{30} + 636 q^{32} + 216 q^{33} + 100 q^{34} + 260 q^{35} - 120 q^{36} - 184 q^{37} - 564 q^{38} - 948 q^{40} + 160 q^{41} - 12 q^{42} - 472 q^{43} - 700 q^{44} - 36 q^{45} - 288 q^{47} - 48 q^{48} + 16 q^{50} + 620 q^{52} + 304 q^{53} + 604 q^{55} + 72 q^{57} + 1272 q^{58} + 800 q^{59} + 84 q^{60} - 240 q^{61} + 1212 q^{62} - 12 q^{63} + 100 q^{64} + 272 q^{65} - 80 q^{67} + 104 q^{68} - 260 q^{70} + 36 q^{72} - 116 q^{73} - 24 q^{75} - 88 q^{77} - 120 q^{78} + 200 q^{79} - 164 q^{80} + 180 q^{81} - 168 q^{82} - 1264 q^{83} - 1200 q^{84} - 212 q^{85} - 876 q^{87} - 212 q^{88} - 1500 q^{89} - 444 q^{90} - 1504 q^{92} - 648 q^{93} - 200 q^{94} - 784 q^{95} + 60 q^{96} - 260 q^{97} - 92 q^{98}+O(q^{100})$$ 80 * q + 4 * q^2 + 4 * q^5 - 4 * q^7 - 12 * q^8 - 4 * q^10 - 24 * q^12 + 32 * q^13 - 24 * q^15 + 80 * q^16 - 100 * q^17 - 48 * q^18 - 100 * q^19 - 244 * q^20 - 100 * q^22 - 96 * q^23 - 16 * q^25 - 40 * q^26 + 196 * q^28 + 200 * q^29 + 264 * q^30 + 636 * q^32 + 216 * q^33 + 100 * q^34 + 260 * q^35 - 120 * q^36 - 184 * q^37 - 564 * q^38 - 948 * q^40 + 160 * q^41 - 12 * q^42 - 472 * q^43 - 700 * q^44 - 36 * q^45 - 288 * q^47 - 48 * q^48 + 16 * q^50 + 620 * q^52 + 304 * q^53 + 604 * q^55 + 72 * q^57 + 1272 * q^58 + 800 * q^59 + 84 * q^60 - 240 * q^61 + 1212 * q^62 - 12 * q^63 + 100 * q^64 + 272 * q^65 - 80 * q^67 + 104 * q^68 - 260 * q^70 + 36 * q^72 - 116 * q^73 - 24 * q^75 - 88 * q^77 - 120 * q^78 + 200 * q^79 - 164 * q^80 + 180 * q^81 - 168 * q^82 - 1264 * q^83 - 1200 * q^84 - 212 * q^85 - 876 * q^87 - 212 * q^88 - 1500 * q^89 - 444 * q^90 - 1504 * q^92 - 648 * q^93 - 200 * q^94 - 784 * q^95 + 60 * q^96 - 260 * q^97 - 92 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −3.44703 + 0.545956i −0.786335 + 1.54327i 7.77971 2.52778i −0.831800 + 4.93033i 1.86796 5.74899i −0.992761 + 0.992761i −12.9984 + 6.62301i −1.76336 2.42705i 0.175498 17.4491i
13.2 −2.60745 + 0.412980i 0.786335 1.54327i 2.82402 0.917581i −1.42490 + 4.79267i −1.41299 + 4.34874i 5.98338 5.98338i 2.42430 1.23524i −1.76336 2.42705i 1.73607 13.0851i
13.3 −1.75107 + 0.277342i −0.786335 + 1.54327i −0.814895 + 0.264776i −2.37217 4.40146i 0.948914 2.92046i 5.49856 5.49856i 7.67216 3.90916i −1.76336 2.42705i 5.37454 + 7.04936i
13.4 −1.20062 + 0.190159i 0.786335 1.54327i −2.39891 + 0.779452i −4.66031 1.81149i −0.650620 + 2.00240i −8.35473 + 8.35473i 7.06431 3.59945i −1.76336 2.42705i 5.93972 + 1.28870i
13.5 −0.247004 + 0.0391215i 0.786335 1.54327i −3.74475 + 1.21674i 3.63583 3.43231i −0.133852 + 0.411956i 8.31728 8.31728i 1.76867 0.901180i −1.76336 2.42705i −0.763787 + 0.990032i
13.6 0.367346 0.0581819i −0.786335 + 1.54327i −3.67267 + 1.19332i −3.52059 + 3.55042i −0.199067 + 0.612664i −1.57845 + 1.57845i −2.60526 + 1.32745i −1.76336 2.42705i −1.08670 + 1.50906i
13.7 2.47604 0.392165i −0.786335 + 1.54327i 2.17273 0.705963i 4.85676 + 1.18822i −1.34178 + 4.12956i 3.46290 3.46290i −3.83175 + 1.95238i −1.76336 2.42705i 12.4915 + 1.03743i
13.8 2.52458 0.399854i 0.786335 1.54327i 2.40940 0.782861i 2.98039 4.01464i 1.36808 4.21053i −7.99303 + 7.99303i −3.34014 + 1.70189i −1.76336 2.42705i 5.91896 11.3270i
13.9 2.92729 0.463638i 0.786335 1.54327i 4.54985 1.47834i −2.61452 + 4.26196i 1.58631 4.88217i 2.66748 2.66748i 2.07035 1.05489i −1.76336 2.42705i −5.67746 + 13.6882i
13.10 3.75152 0.594183i −0.786335 + 1.54327i 9.91663 3.22211i −4.60913 1.93801i −2.03297 + 6.25683i −5.76987 + 5.76987i 21.7507 11.0826i −1.76336 2.42705i −18.4428 4.53182i
22.1 −3.41817 + 1.74164i −1.71073 + 0.270952i 6.29941 8.67040i 4.97139 + 0.534117i 5.37565 3.90564i −0.908106 0.908106i −4.03119 + 25.4519i 2.85317 0.927051i −17.9233 + 6.83269i
22.2 −3.18220 + 1.62141i 1.71073 0.270952i 5.14628 7.08325i −4.97907 + 0.457044i −5.00455 + 3.63602i −6.93420 6.93420i −2.65683 + 16.7746i 2.85317 0.927051i 15.1033 9.52752i
22.3 −2.16137 + 1.10127i 1.71073 0.270952i 1.10757 1.52445i 2.52133 4.31774i −3.39912 + 2.46960i 6.20191 + 6.20191i 0.802843 5.06895i 2.85317 0.927051i −0.694505 + 12.1089i
22.4 −1.53414 + 0.781681i −1.71073 + 0.270952i −0.608595 + 0.837659i 0.173747 4.99698i 2.41269 1.75292i −1.91431 1.91431i 1.35628 8.56322i 2.85317 0.927051i 3.63949 + 7.80186i
22.5 −0.186256 + 0.0949024i 1.71073 0.270952i −2.32546 + 3.20072i −1.31868 + 4.82297i −0.292920 + 0.212819i 5.11319 + 5.11319i 0.260180 1.64271i 2.85317 0.927051i −0.212098 1.02346i
22.6 0.250128 0.127447i −1.71073 + 0.270952i −2.30482 + 3.17231i −4.47793 + 2.22444i −0.393368 + 0.285799i −6.72381 6.72381i −0.347860 + 2.19630i 2.85317 0.927051i −0.836558 + 1.12709i
22.7 0.510204 0.259962i −1.71073 + 0.270952i −2.15841 + 2.97080i 4.69871 + 1.70942i −0.802382 + 0.582965i 6.06980 + 6.06980i −0.687243 + 4.33908i 2.85317 0.927051i 2.84168 0.349330i
22.8 1.53105 0.780110i 1.71073 0.270952i −0.615594 + 0.847292i 4.34778 2.46917i 2.40784 1.74940i −3.26334 3.26334i −1.35675 + 8.56621i 2.85317 0.927051i 4.73046 7.17217i
22.9 2.73870 1.39544i 1.71073 0.270952i 3.20210 4.40731i −4.71895 + 1.65274i 4.30707 3.12927i −0.250082 0.250082i 0.696119 4.39513i 2.85317 0.927051i −10.6175 + 11.1114i
22.10 2.93190 1.49388i −1.71073 + 0.270952i 4.01322 5.52372i −0.890939 4.91998i −4.61091 + 3.35002i 4.34391 + 4.34391i 1.45557 9.19012i 2.85317 0.927051i −9.96199 13.0939i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.10 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.f odd 20 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.k.a 80
3.b odd 2 1 225.3.r.b 80
5.b even 2 1 375.3.k.a 80
5.c odd 4 1 375.3.k.b 80
5.c odd 4 1 375.3.k.c 80
25.d even 5 1 375.3.k.c 80
25.e even 10 1 375.3.k.b 80
25.f odd 20 1 inner 75.3.k.a 80
25.f odd 20 1 375.3.k.a 80
75.l even 20 1 225.3.r.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.k.a 80 1.a even 1 1 trivial
75.3.k.a 80 25.f odd 20 1 inner
225.3.r.b 80 3.b odd 2 1
225.3.r.b 80 75.l even 20 1
375.3.k.a 80 5.b even 2 1
375.3.k.a 80 25.f odd 20 1
375.3.k.b 80 5.c odd 4 1
375.3.k.b 80 25.e even 10 1
375.3.k.c 80 5.c odd 4 1
375.3.k.c 80 25.d even 5 1

Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(75, [\chi])$$.