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Properties

Label 164.3.c.a
Level $164$
Weight $3$
Character orbit 164.c
Analytic conductor $4.469$
Analytic rank $0$
Dimension $40$
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] = [164,3,Mod(83,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(2))
 
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("164.83");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.c (of order \(2\), degree \(1\), 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: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 2 q^{4} + 12 q^{6} + 10 q^{8} - 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} - 2 q^{4} + 12 q^{6} + 10 q^{8} - 120 q^{9} - 20 q^{10} + 10 q^{12} - 16 q^{13} + 36 q^{14} - 42 q^{16} - 34 q^{18} + 8 q^{20} + 48 q^{21} - 38 q^{22} + 52 q^{24} + 216 q^{25} + 54 q^{26} - 54 q^{28} - 32 q^{29} + 66 q^{30} - 2 q^{32} - 48 q^{33} - 124 q^{34} - 70 q^{36} + 16 q^{37} - 140 q^{38} + 16 q^{40} + 88 q^{42} + 100 q^{44} - 48 q^{45} + 196 q^{46} + 78 q^{48} - 152 q^{49} + 198 q^{50} + 26 q^{52} - 32 q^{53} - 74 q^{54} + 96 q^{56} - 112 q^{57} + 38 q^{58} - 102 q^{60} + 96 q^{61} - 112 q^{62} + 70 q^{64} - 96 q^{65} - 528 q^{66} - 148 q^{68} - 80 q^{69} - 82 q^{70} - 458 q^{72} + 128 q^{73} + 372 q^{74} - 50 q^{76} + 192 q^{77} + 144 q^{78} - 100 q^{80} + 520 q^{81} + 344 q^{84} - 176 q^{85} - 460 q^{86} + 66 q^{88} + 16 q^{89} + 80 q^{90} + 572 q^{92} + 32 q^{93} + 262 q^{94} - 304 q^{96} - 304 q^{97} + 174 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −1.99946 − 0.0466052i 2.58485i 3.99566 + 0.186370i 6.26907 0.120467 − 5.16829i − 11.4160i −7.98046 − 0.558857i 2.31858 −12.5347 − 0.292171i
83.2 −1.99946 + 0.0466052i − 2.58485i 3.99566 − 0.186370i 6.26907 0.120467 + 5.16829i 11.4160i −7.98046 + 0.558857i 2.31858 −12.5347 + 0.292171i
83.3 −1.99123 − 0.187081i − 3.90867i 3.93000 + 0.745044i −3.21306 −0.731240 + 7.78307i − 7.36065i −7.68616 − 2.21879i −6.27772 6.39794 + 0.601103i
83.4 −1.99123 + 0.187081i 3.90867i 3.93000 − 0.745044i −3.21306 −0.731240 − 7.78307i 7.36065i −7.68616 + 2.21879i −6.27772 6.39794 − 0.601103i
83.5 −1.74626 − 0.974974i 0.487501i 2.09885 + 3.40512i −6.86061 0.475301 − 0.851304i 1.80145i −0.345239 − 7.99255i 8.76234 11.9804 + 6.68892i
83.6 −1.74626 + 0.974974i − 0.487501i 2.09885 − 3.40512i −6.86061 0.475301 + 0.851304i − 1.80145i −0.345239 + 7.99255i 8.76234 11.9804 − 6.68892i
83.7 −1.54213 − 1.27351i 2.47677i 0.756327 + 3.92785i −1.28625 3.15420 − 3.81950i − 3.58567i 3.83581 − 7.02044i 2.86563 1.98356 + 1.63805i
83.8 −1.54213 + 1.27351i − 2.47677i 0.756327 − 3.92785i −1.28625 3.15420 + 3.81950i 3.58567i 3.83581 + 7.02044i 2.86563 1.98356 − 1.63805i
83.9 −1.50188 − 1.32074i − 2.23825i 0.511316 + 3.96718i 4.89966 −2.95614 + 3.36160i − 0.843158i 4.47166 − 6.63357i 3.99022 −7.35873 − 6.47116i
83.10 −1.50188 + 1.32074i 2.23825i 0.511316 − 3.96718i 4.89966 −2.95614 − 3.36160i 0.843158i 4.47166 + 6.63357i 3.99022 −7.35873 + 6.47116i
83.11 −1.40870 − 1.41971i 5.22032i −0.0311413 + 3.99988i 8.34602 7.41133 − 7.35385i 10.3463i 5.72253 − 5.59041i −18.2517 −11.7570 − 11.8489i
83.12 −1.40870 + 1.41971i − 5.22032i −0.0311413 − 3.99988i 8.34602 7.41133 + 7.35385i − 10.3463i 5.72253 + 5.59041i −18.2517 −11.7570 + 11.8489i
83.13 −1.29148 − 1.52711i − 5.51459i −0.664159 + 3.94448i −3.66998 −8.42141 + 7.12198i 8.18794i 6.88142 − 4.07997i −21.4107 4.73971 + 5.60448i
83.14 −1.29148 + 1.52711i 5.51459i −0.664159 − 3.94448i −3.66998 −8.42141 − 7.12198i − 8.18794i 6.88142 + 4.07997i −21.4107 4.73971 − 5.60448i
83.15 −0.688190 − 1.87787i − 1.75883i −3.05279 + 2.58466i 5.36632 −3.30284 + 1.21041i − 7.30850i 6.95456 + 3.95400i 5.90653 −3.69305 − 10.0773i
83.16 −0.688190 + 1.87787i 1.75883i −3.05279 − 2.58466i 5.36632 −3.30284 − 1.21041i 7.30850i 6.95456 − 3.95400i 5.90653 −3.69305 + 10.0773i
83.17 −0.687445 − 1.87814i 4.43708i −3.05484 + 2.58224i −5.62408 8.33347 − 3.05025i − 6.78235i 6.94985 + 3.96227i −10.6877 3.86624 + 10.5628i
83.18 −0.687445 + 1.87814i − 4.43708i −3.05484 − 2.58224i −5.62408 8.33347 + 3.05025i 6.78235i 6.94985 − 3.96227i −10.6877 3.86624 − 10.5628i
83.19 −0.387677 − 1.96207i − 0.338353i −3.69941 + 1.52129i −3.21167 −0.663871 + 0.131171i 12.6392i 4.41906 + 6.66873i 8.88552 1.24509 + 6.30151i
83.20 −0.387677 + 1.96207i 0.338353i −3.69941 − 1.52129i −3.21167 −0.663871 − 0.131171i − 12.6392i 4.41906 − 6.66873i 8.88552 1.24509 − 6.30151i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.3.c.a ✓ 40
4.b odd 2 1 inner 164.3.c.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.3.c.a ✓ 40 1.a even 1 1 trivial
164.3.c.a ✓ 40 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(164, [\chi])\).