↑

Properties

Label 71.2.d.a
Level $71$
Weight $2$
Character orbit 71.d
Analytic conductor $0.567$
Analytic rank $0$
Dimension $30$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [71,2,Mod(20,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.20");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 71.d (of order \(7\), degree \(6\), 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: \(0.566937854351\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} - 10 q^{3} - 7 q^{4} - 10 q^{5} + 5 q^{6} - 3 q^{7} + 17 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} - 10 q^{3} - 7 q^{4} - 10 q^{5} + 5 q^{6} - 3 q^{7} + 17 q^{8} - 9 q^{9} - 22 q^{10} + 7 q^{11} + 2 q^{12} + q^{13} - 2 q^{14} - 2 q^{15} - 15 q^{16} - 24 q^{17} - 37 q^{18} + q^{19} + 8 q^{20} + 2 q^{21} + 19 q^{22} + 21 q^{23} + 42 q^{24} + 12 q^{25} - 32 q^{26} + 17 q^{27} + 13 q^{28} - 11 q^{29} - 5 q^{30} - 30 q^{31} - 5 q^{32} + 9 q^{33} + 15 q^{34} - 44 q^{35} - 3 q^{36} + 37 q^{37} + 24 q^{38} + 29 q^{39} + 10 q^{40} + 16 q^{41} + 74 q^{42} - 43 q^{43} + 33 q^{44} - 36 q^{45} - 54 q^{46} - 16 q^{47} + 59 q^{48} - 8 q^{49} + 20 q^{50} + 6 q^{51} - 53 q^{52} + 65 q^{53} + 50 q^{54} + 13 q^{55} - 21 q^{56} - 20 q^{57} - 12 q^{58} + 30 q^{59} - 150 q^{60} - 18 q^{61} + 8 q^{62} - q^{63} + 17 q^{64} + 14 q^{65} - 16 q^{66} + 29 q^{67} - 13 q^{68} + 10 q^{69} + 64 q^{70} - 5 q^{71} - 156 q^{72} - 61 q^{73} - 35 q^{74} - 82 q^{75} - 58 q^{76} - 82 q^{77} + 3 q^{78} + 55 q^{79} - 22 q^{80} + 3 q^{81} + 18 q^{82} + 45 q^{83} + 52 q^{84} - 22 q^{85} - 89 q^{86} + 40 q^{87} + 112 q^{88} - 8 q^{89} + 77 q^{90} + 35 q^{91} + 17 q^{92} + 42 q^{93} + q^{94} + 50 q^{95} + 90 q^{96} - 26 q^{97} + 7 q^{98} + 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −0.512695 − 2.24626i −2.31785 + 1.11622i −2.98090 + 1.43552i −3.13999 3.69567 + 4.63422i 0.916758 − 4.01658i 1.87978 + 2.35717i 2.25602 − 2.82895i 1.60985 + 7.05323i
20.2 −0.469495 − 2.05699i 1.38273 − 0.665888i −2.20886 + 1.06373i 0.419938 −2.01891 − 2.53164i −0.760462 + 3.33180i 0.594137 + 0.745024i −0.401933 + 0.504008i −0.197159 − 0.863809i
20.3 −0.0812940 − 0.356172i −1.94214 + 0.935283i 1.68169 − 0.809858i 3.28338 0.491006 + 0.615702i −0.0815359 + 0.357232i −0.880722 − 1.10439i 1.02667 − 1.28740i −0.266919 − 1.16945i
20.4 0.0919395 + 0.402813i 0.860332 − 0.414314i 1.64813 − 0.793699i −2.73724 0.245990 + 0.308461i 0.0252603 − 0.110672i 0.986458 + 1.23698i −1.30195 + 1.63260i −0.251661 − 1.10260i
20.5 0.471544 + 2.06597i −1.23006 + 0.592364i −2.24394 + 1.08063i 0.618957 −1.80383 − 2.26194i 0.245991 − 1.07776i −0.648183 − 0.812795i −0.708324 + 0.888211i 0.291866 + 1.27875i
30.1 −2.12218 − 1.02199i 0.593598 + 0.744349i 2.21220 + 2.77401i 1.05635 −0.499006 − 2.18629i 2.02367 − 0.974546i −0.811409 − 3.55501i 0.465867 − 2.04110i −2.24176 − 1.07957i
30.2 −0.774825 − 0.373136i −0.865413 − 1.08519i −0.785856 − 0.985433i −3.62688 0.265619 + 1.16375i 2.52089 − 1.21399i 0.623933 + 2.73363i 0.238858 − 1.04650i 2.81020 + 1.35332i
30.3 −0.154599 − 0.0744510i 1.72136 + 2.15851i −1.22862 − 1.54064i 0.566787 −0.105417 − 0.461861i −1.38741 + 0.668143i 0.151607 + 0.664234i −1.02855 + 4.50637i −0.0876248 − 0.0421979i
30.4 0.196395 + 0.0945787i −1.21380 − 1.52206i −1.21735 − 1.52651i 3.36610 −0.0944299 − 0.413725i −0.907124 + 0.436848i −0.191717 − 0.839967i −0.175789 + 0.770183i 0.661083 + 0.318361i
30.5 2.35521 + 1.13421i −1.79070 − 2.24546i 3.01359 + 3.77892i −1.56041 −1.67064 − 7.31956i −2.07157 + 0.997613i 1.64817 + 7.22110i −1.16795 + 5.11712i −3.67509 − 1.76983i
32.1 −0.512695 + 2.24626i −2.31785 − 1.11622i −2.98090 − 1.43552i −3.13999 3.69567 − 4.63422i 0.916758 + 4.01658i 1.87978 − 2.35717i 2.25602 + 2.82895i 1.60985 − 7.05323i
32.2 −0.469495 + 2.05699i 1.38273 + 0.665888i −2.20886 − 1.06373i 0.419938 −2.01891 + 2.53164i −0.760462 − 3.33180i 0.594137 − 0.745024i −0.401933 − 0.504008i −0.197159 + 0.863809i
32.3 −0.0812940 + 0.356172i −1.94214 − 0.935283i 1.68169 + 0.809858i 3.28338 0.491006 − 0.615702i −0.0815359 − 0.357232i −0.880722 + 1.10439i 1.02667 + 1.28740i −0.266919 + 1.16945i
32.4 0.0919395 − 0.402813i 0.860332 + 0.414314i 1.64813 + 0.793699i −2.73724 0.245990 − 0.308461i 0.0252603 + 0.110672i 0.986458 − 1.23698i −1.30195 − 1.63260i −0.251661 + 1.10260i
32.5 0.471544 − 2.06597i −1.23006 − 0.592364i −2.24394 − 1.08063i 0.618957 −1.80383 + 2.26194i 0.245991 + 1.07776i −0.648183 + 0.812795i −0.708324 − 0.888211i 0.291866 − 1.27875i
37.1 −1.55593 + 1.95108i −0.618192 − 2.70847i −0.940734 − 4.12162i 2.61195 6.24630 + 3.00806i −1.63739 − 2.05323i 5.00855 + 2.41199i −4.25077 + 2.04706i −4.06401 + 5.09611i
37.2 −1.18346 + 1.48402i 0.379212 + 1.66144i −0.356677 − 1.56270i −0.529055 −2.91438 − 1.40349i 0.212498 + 0.266464i −0.679118 − 0.327046i 0.0863348 − 0.0415766i 0.626117 − 0.785126i
37.3 0.191306 − 0.239890i 0.312972 + 1.37122i 0.424093 + 1.85807i −0.883903 0.388815 + 0.187244i −2.40928 − 3.02114i 1.07975 + 0.519982i 0.920613 − 0.443344i −0.169096 + 0.212040i
37.4 0.599543 − 0.751803i −0.513729 − 2.25079i 0.239286 + 1.04838i −0.464535 −2.00016 − 0.963225i −0.437866 − 0.549067i 2.66437 + 1.28309i −2.09925 + 1.01095i −0.278509 + 0.349239i
37.5 1.44855 − 1.81642i 0.241674 + 1.05884i −0.756051 − 3.31248i −3.98144 2.27338 + 1.09480i 2.24758 + 2.81838i −2.92560 − 1.40890i 1.64016 − 0.789861i −5.76729 + 7.23196i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.2.d.a ✓ 30
3.b odd 2 1 639.2.j.c 30
71.d even 7 1 inner 71.2.d.a ✓ 30
71.d even 7 1 5041.2.a.l 15
71.f odd 14 1 5041.2.a.m 15
213.k odd 14 1 639.2.j.c 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.2.d.a ✓ 30 1.a even 1 1 trivial
71.2.d.a ✓ 30 71.d even 7 1 inner
639.2.j.c 30 3.b odd 2 1
639.2.j.c 30 213.k odd 14 1
5041.2.a.l 15 71.d even 7 1
5041.2.a.m 15 71.f odd 14 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(71, [\chi])\).