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Properties

Label 161.3.h.a
Level $161$
Weight $3$
Character orbit 161.h
Analytic conductor $4.387$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [161,3,Mod(24,161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(161, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("161.24");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 161 = 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 161.h (of order \(6\), degree \(2\), 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: \(4.38693225620\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} - 64 q^{4} - 10 q^{7} + 12 q^{8} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} - 64 q^{4} - 10 q^{7} + 12 q^{8} + 92 q^{9} - 6 q^{10} + 12 q^{11} + 18 q^{12} + 68 q^{14} - 76 q^{15} - 120 q^{16} - 6 q^{17} + 50 q^{18} - 78 q^{19} - 42 q^{21} + 40 q^{22} - 108 q^{24} + 146 q^{25} + 70 q^{28} + 140 q^{29} - 68 q^{30} + 12 q^{31} - 54 q^{32} + 186 q^{33} - 194 q^{35} - 616 q^{36} + 46 q^{37} + 78 q^{38} - 40 q^{39} - 72 q^{40} + 34 q^{42} + 60 q^{43} + 74 q^{44} + 390 q^{45} - 300 q^{47} + 12 q^{49} + 76 q^{50} - 36 q^{51} + 252 q^{52} - 158 q^{53} + 306 q^{54} - 296 q^{56} + 112 q^{57} + 50 q^{58} + 258 q^{59} + 248 q^{60} + 372 q^{61} + 124 q^{63} + 188 q^{64} - 262 q^{65} - 504 q^{66} + 30 q^{67} - 372 q^{68} + 662 q^{70} - 240 q^{71} + 308 q^{72} - 378 q^{73} - 30 q^{74} - 420 q^{75} + 52 q^{77} + 1092 q^{78} + 306 q^{79} - 1128 q^{80} - 262 q^{81} + 144 q^{82} - 520 q^{84} - 172 q^{85} - 210 q^{86} - 570 q^{87} - 558 q^{88} - 486 q^{89} + 322 q^{91} - 378 q^{93} + 156 q^{94} + 32 q^{95} + 1524 q^{96} + 190 q^{98} + 576 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −1.91511 − 3.31706i 1.34733 + 0.777881i −5.33526 + 9.24094i 0.519197 − 0.299759i − 5.95890i 4.10948 + 5.66676i 25.5495 −3.28980 − 5.69811i −1.98864 − 1.14814i
24.2 −1.90960 − 3.30753i −4.84318 − 2.79621i −5.29315 + 9.16800i 4.75546 − 2.74556i 21.3586i −6.79924 + 1.66443i 25.1544 11.1376 + 19.2909i −18.1620 − 10.4859i
24.3 −1.74621 − 3.02452i 1.71882 + 0.992363i −4.09847 + 7.09875i −2.00132 + 1.15546i − 6.93148i −4.18806 − 5.60893i 14.6574 −2.53043 − 4.38284i 6.98944 + 4.03535i
24.4 −1.56061 − 2.70305i −3.28927 − 1.89906i −2.87098 + 4.97269i −7.46694 + 4.31104i 11.8547i 5.84708 − 3.84859i 5.43704 2.71285 + 4.69880i 23.3059 + 13.4557i
24.5 −1.48793 − 2.57717i 5.07259 + 2.92866i −2.42788 + 4.20521i 6.12326 − 3.53527i − 17.4306i −6.87350 − 1.32474i 2.54661 12.6541 + 21.9176i −18.2220 − 10.5205i
24.6 −1.47596 − 2.55643i −1.29048 − 0.745061i −2.35690 + 4.08227i 5.27881 − 3.04772i 4.39871i 4.89459 − 5.00429i 2.10706 −3.38977 − 5.87125i −15.5826 − 8.99662i
24.7 −1.29378 − 2.24090i 3.85035 + 2.22300i −1.34776 + 2.33438i −7.51711 + 4.34000i − 11.5043i 2.02538 + 6.70058i −3.37546 5.38344 + 9.32439i 19.4510 + 11.2301i
24.8 −1.16030 − 2.00970i −2.61145 − 1.50772i −0.692602 + 1.19962i −1.20982 + 0.698488i 6.99765i 0.354679 + 6.99101i −6.06791 0.0464459 + 0.0804466i 2.80750 + 1.62091i
24.9 −1.14783 − 1.98810i 0.233812 + 0.134991i −0.635021 + 1.09989i 1.08673 − 0.627422i − 0.619788i −6.08041 + 3.46823i −6.26705 −4.46355 − 7.73110i −2.49475 − 1.44035i
24.10 −0.770447 − 1.33445i 3.07729 + 1.77667i 0.812822 − 1.40785i 3.51691 − 2.03049i − 5.47533i 6.99874 − 0.132869i −8.66852 1.81312 + 3.14042i −5.41919 − 3.12877i
24.11 −0.710330 − 1.23033i 1.77682 + 1.02584i 0.990861 − 1.71622i −4.88925 + 2.82281i − 2.91476i −1.48060 − 6.84162i −8.49800 −2.39529 − 4.14876i 6.94597 + 4.01026i
24.12 −0.630311 − 1.09173i −4.22159 − 2.43734i 1.20542 − 2.08784i −2.12168 + 1.22495i 6.14512i −5.58373 − 4.22161i −8.08164 7.38121 + 12.7846i 2.67464 + 1.54420i
24.13 −0.247825 − 0.429245i −3.27660 − 1.89175i 1.87717 − 3.25135i 7.93213 − 4.57962i 1.87529i 3.05089 + 6.30017i −3.84343 2.65740 + 4.60275i −3.93156 − 2.26989i
24.14 −0.124040 − 0.214844i 0.816232 + 0.471252i 1.96923 − 3.41080i 6.34206 − 3.66159i − 0.233817i −6.65100 − 2.18270i −1.96938 −4.05584 − 7.02493i −1.57334 − 0.908370i
24.15 −0.107504 − 0.186203i −3.51694 − 2.03051i 1.97689 − 3.42407i 0.963454 − 0.556251i 0.873153i 2.76467 − 6.43091i −1.71013 3.74591 + 6.48811i −0.207151 − 0.119599i
24.16 0.119685 + 0.207301i −1.00155 − 0.578248i 1.97135 − 3.41448i −3.87246 + 2.23577i − 0.276831i 6.60024 + 2.33171i 1.90125 −3.83126 − 6.63594i −0.926952 − 0.535176i
24.17 0.189627 + 0.328443i −0.496435 − 0.286617i 1.92808 − 3.33954i −7.10646 + 4.10292i − 0.217401i −5.73748 + 4.01016i 2.97948 −4.33570 − 7.50965i −2.69515 − 1.55605i
24.18 0.310784 + 0.538294i 3.35407 + 1.93647i 1.80683 − 3.12952i 0.905092 − 0.522555i 2.40730i −0.808672 + 6.95313i 4.73240 2.99984 + 5.19587i 0.562576 + 0.324803i
24.19 0.370409 + 0.641567i 4.60177 + 2.65683i 1.72559 − 2.98882i −2.06838 + 1.19418i 3.93646i 0.229372 − 6.99624i 5.51998 9.61753 + 16.6581i −1.53229 − 0.884670i
24.20 0.882755 + 1.52898i −0.244166 − 0.140969i 0.441488 − 0.764680i 1.98069 − 1.14355i − 0.497765i 3.63819 − 5.98027i 8.62094 −4.46026 − 7.72539i 3.49693 + 2.01896i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 161.3.h.a ✓ 60
7.d odd 6 1 inner 161.3.h.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.3.h.a ✓ 60 1.a even 1 1 trivial
161.3.h.a ✓ 60 7.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(161, [\chi])\).