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

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.k (of order \(20\), degree \(8\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 80q + 4q^{2} + 4q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q + 4q^{2} + 4q^{5} - 4q^{7} - 12q^{8} - 4q^{10} - 24q^{12} + 32q^{13} - 24q^{15} + 80q^{16} - 100q^{17} - 48q^{18} - 100q^{19} - 244q^{20} - 100q^{22} - 96q^{23} - 16q^{25} - 40q^{26} + 196q^{28} + 200q^{29} + 264q^{30} + 636q^{32} + 216q^{33} + 100q^{34} + 260q^{35} - 120q^{36} - 184q^{37} - 564q^{38} - 948q^{40} + 160q^{41} - 12q^{42} - 472q^{43} - 700q^{44} - 36q^{45} - 288q^{47} - 48q^{48} + 16q^{50} + 620q^{52} + 304q^{53} + 604q^{55} + 72q^{57} + 1272q^{58} + 800q^{59} + 84q^{60} - 240q^{61} + 1212q^{62} - 12q^{63} + 100q^{64} + 272q^{65} - 80q^{67} + 104q^{68} - 260q^{70} + 36q^{72} - 116q^{73} - 24q^{75} - 88q^{77} - 120q^{78} + 200q^{79} - 164q^{80} + 180q^{81} - 168q^{82} - 1264q^{83} - 1200q^{84} - 212q^{85} - 876q^{87} - 212q^{88} - 1500q^{89} - 444q^{90} - 1504q^{92} - 648q^{93} - 200q^{94} - 784q^{95} + 60q^{96} - 260q^{97} - 92q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 73.10
Significant digits:
Format:

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 \(S_{3}^{\mathrm{new}}(75, [\chi])\).