↑

Properties

Label 539.2.bf.a
Level $539$
Weight $2$
Character orbit 539.bf
Analytic conductor $4.304$
Analytic rank $0$
Dimension $2592$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(17,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(210))
 
chi = DirichletCharacter(H, H._module([125, 189]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.bf (of order \(210\), degree \(48\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(2592\)
Relative dimension: \(54\) over \(\Q(\zeta_{210})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{210}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2592 q - 65 q^{2} - 33 q^{3} + 13 q^{4} - 27 q^{5} - 70 q^{6} - 65 q^{7} - 50 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 2592 q - 65 q^{2} - 33 q^{3} + 13 q^{4} - 27 q^{5} - 70 q^{6} - 65 q^{7} - 50 q^{8} + 11 q^{9} - 62 q^{11} - 100 q^{12} - 70 q^{13} - 34 q^{14} - 72 q^{15} - 79 q^{16} - 120 q^{17} - 30 q^{18} - 90 q^{19} - 98 q^{20} - 10 q^{22} - 160 q^{23} + 65 q^{24} + 17 q^{25} - 97 q^{26} + 21 q^{27} - 30 q^{28} - 30 q^{29} - 60 q^{30} - 72 q^{31} - 98 q^{33} - 224 q^{34} - 75 q^{35} + 132 q^{36} - 37 q^{37} - 47 q^{38} - 65 q^{39} + 275 q^{40} - 70 q^{41} - 50 q^{42} - 147 q^{44} - 350 q^{45} - 85 q^{46} - 80 q^{47} + 11 q^{49} - 170 q^{50} + 195 q^{51} - 55 q^{52} - 9 q^{53} - 49 q^{55} - 552 q^{56} - 130 q^{57} - 175 q^{58} - 11 q^{59} - 153 q^{60} - 40 q^{61} + 70 q^{62} - 30 q^{63} - 218 q^{64} - 61 q^{66} - 72 q^{67} - 30 q^{68} + 126 q^{69} + 27 q^{70} - 50 q^{71} - 35 q^{72} - 115 q^{73} - 115 q^{74} + 15 q^{75} - 35 q^{77} + 108 q^{78} - 15 q^{79} + 12 q^{80} + 109 q^{81} + 87 q^{82} - 70 q^{83} + 55 q^{84} + 116 q^{86} - 28 q^{88} + 106 q^{89} - 70 q^{90} + 82 q^{91} - 122 q^{92} + 32 q^{93} + 595 q^{94} - 60 q^{95} - 145 q^{96} - 484 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.05816 − 2.58189i −0.660545 + 0.495179i −4.12172 + 4.06052i −3.25694 + 1.50886i 1.97746 + 1.18148i 2.64545 + 0.0397822i 9.71373 + 4.15185i −0.650165 + 2.22545i 7.34209 + 6.81247i
17.2 −1.01564 − 2.47814i 1.84701 − 1.38462i −3.68492 + 3.63020i 1.49741 − 0.693713i −5.30718 − 3.17089i 1.35556 − 2.27211i 7.81336 + 3.33959i 0.653009 − 2.23519i −3.23994 − 3.00623i
17.3 −0.995649 − 2.42937i −0.644229 + 0.482948i −3.48576 + 3.43400i 2.98658 − 1.38361i 1.81469 + 1.08422i −2.32727 + 1.25850i 6.98465 + 2.98538i −0.659489 + 2.25737i −6.33489 − 5.87792i
17.4 −0.973949 − 2.37642i 1.85242 − 1.38867i −3.27405 + 3.22543i −2.09906 + 0.972442i −5.10424 − 3.04964i −2.35894 + 1.19809i 6.13056 + 2.62033i 0.661778 − 2.26520i 4.35531 + 4.04113i
17.5 −0.929676 − 2.26840i −1.26542 + 0.948622i −2.85657 + 2.81415i 3.08470 − 1.42907i 3.32828 + 1.98855i 2.45589 − 0.984167i 4.53083 + 1.93657i −0.139888 + 0.478821i −6.10946 − 5.66875i
17.6 −0.883535 − 2.15581i −2.57975 + 1.93392i −2.44214 + 2.40587i 0.350075 − 0.162181i 6.44847 + 3.85278i −1.03677 − 2.43416i 3.05962 + 1.30774i 2.07381 − 7.09844i −0.658936 − 0.611403i
17.7 −0.858780 − 2.09541i −1.01387 + 0.760053i −2.22849 + 2.19540i −1.92813 + 0.893254i 2.46332 + 1.47176i −1.61904 − 2.09254i 2.34937 + 1.00417i −0.391022 + 1.33843i 3.52757 + 3.27311i
17.8 −0.840099 − 2.04983i 0.654421 − 0.490588i −2.07128 + 2.04052i −0.616172 + 0.285458i −1.55540 − 0.929308i 1.57306 + 2.12732i 1.84873 + 0.790187i −0.653692 + 2.23752i 1.10278 + 1.02323i
17.9 −0.826190 − 2.01589i −0.0854536 + 0.0640605i −1.95648 + 1.92742i 0.976267 − 0.452281i 0.199740 + 0.119339i −1.18120 + 2.36744i 1.49529 + 0.639116i −0.838084 + 2.86868i −1.71833 − 1.59438i
17.10 −0.758157 − 1.84989i −1.98863 + 1.49078i −1.42254 + 1.40142i −0.622779 + 0.288518i 4.26547 + 2.54850i 1.36556 + 2.26611i −0.00570736 − 0.00243944i 0.890935 − 3.04958i 1.00589 + 0.933330i
17.11 −0.744980 − 1.81774i 1.86502 − 1.39812i −1.32443 + 1.30477i 3.40595 − 1.57789i −3.93082 − 2.34856i −2.08112 − 1.63369i −0.254383 − 0.108728i 0.682294 − 2.33543i −5.40557 − 5.01563i
17.12 −0.729061 − 1.77890i 2.32591 − 1.74362i −1.20820 + 1.19026i −1.17818 + 0.545822i −4.79746 − 2.86635i 2.58713 + 0.553841i −0.537391 − 0.229692i 1.52835 − 5.23139i 1.82993 + 1.69792i
17.13 −0.571242 − 1.39382i 1.31422 − 0.985204i −0.191670 + 0.188824i −2.27363 + 1.05332i −2.12393 − 1.26899i −2.64424 − 0.0894104i −2.39756 − 1.02477i −0.0847484 + 0.290085i 2.76693 + 2.56733i
17.14 −0.542992 − 1.32489i −0.0275723 + 0.0206697i −0.0357464 + 0.0352156i 1.71069 − 0.792522i 0.0423566 + 0.0253069i 2.03845 − 1.68663i −2.56718 − 1.09726i −0.840949 + 2.87849i −1.97890 − 1.83615i
17.15 −0.535354 − 1.30626i 1.48620 − 1.11414i 0.00505123 − 0.00497622i 3.14952 − 1.45910i −2.25099 − 1.34491i 1.20841 + 2.35367i −2.60541 − 1.11360i 0.126221 − 0.432042i −3.59206 − 3.33295i
17.16 −0.514005 − 1.25416i −1.89896 + 1.42356i 0.116026 − 0.114303i −2.20896 + 1.02336i 2.76145 + 1.64989i 2.54357 − 0.728191i −2.69566 − 1.15218i 0.738246 − 2.52694i 2.41888 + 2.24439i
17.17 −0.496384 − 1.21117i 0.425283 − 0.318814i 0.204219 − 0.201186i −3.57393 + 1.65572i −0.597241 − 0.356835i 0.770271 − 2.53114i −2.75226 − 1.17637i −0.762059 + 2.60845i 3.77939 + 3.50677i
17.18 −0.468092 − 1.14214i −1.20193 + 0.901030i 0.339385 − 0.334345i 0.888509 − 0.411625i 1.59171 + 0.951005i −2.50881 + 0.840167i −2.81075 − 1.20137i −0.208500 + 0.713674i −0.886036 − 0.822121i
17.19 −0.447743 − 1.09249i −2.44486 + 1.83279i 0.431699 − 0.425289i 3.06821 − 1.42143i 3.09697 + 1.85035i 0.321365 + 2.62616i −2.82925 − 1.20928i 1.77692 − 6.08221i −2.92666 − 2.71555i
17.20 −0.418144 − 1.02026i 2.49666 − 1.87163i 0.558657 − 0.550362i −0.743718 + 0.344547i −2.95352 − 1.76464i −0.0499942 − 2.64528i −2.82290 − 1.20657i 1.88905 − 6.46601i 0.662510 + 0.614719i
See next 80 embeddings (of 2592 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
49.h odd 42 1 inner
539.bf even 210 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.bf.a ✓ 2592
11.d odd 10 1 inner 539.2.bf.a ✓ 2592
49.h odd 42 1 inner 539.2.bf.a ✓ 2592
539.bf even 210 1 inner 539.2.bf.a ✓ 2592
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.bf.a ✓ 2592 1.a even 1 1 trivial
539.2.bf.a ✓ 2592 11.d odd 10 1 inner
539.2.bf.a ✓ 2592 49.h odd 42 1 inner
539.2.bf.a ✓ 2592 539.bf even 210 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(539, [\chi])\).