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Properties

Label 182.2.bc.a
Level $182$
Weight $2$
Character orbit 182.bc
Analytic conductor $1.453$
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] = [182,2,Mod(45,182)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(182, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("182.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 182.bc (of order \(12\), degree \(4\), 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: \(1.45327731679\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{7} + 24 q^{9} + 4 q^{12} - 4 q^{14} - 16 q^{15} - 40 q^{16} - 8 q^{18} - 16 q^{19} - 4 q^{21} + 4 q^{22} - 8 q^{28} + 4 q^{29} + 24 q^{30} - 4 q^{31} + 36 q^{33} + 28 q^{35} - 24 q^{36} - 8 q^{37} - 24 q^{39} - 24 q^{41} - 12 q^{42} - 72 q^{43} + 12 q^{44} + 4 q^{49} - 24 q^{51} + 4 q^{52} - 4 q^{53} + 36 q^{54} - 12 q^{55} - 12 q^{56} - 12 q^{57} - 8 q^{58} - 8 q^{60} - 12 q^{61} - 36 q^{62} - 108 q^{63} - 16 q^{65} + 44 q^{67} + 84 q^{69} + 8 q^{70} + 28 q^{71} + 8 q^{72} - 44 q^{73} + 40 q^{74} + 56 q^{75} + 32 q^{76} + 8 q^{78} - 16 q^{79} + 4 q^{81} + 48 q^{82} - 108 q^{83} + 24 q^{84} + 56 q^{85} + 52 q^{86} - 12 q^{89} + 52 q^{91} + 32 q^{92} - 24 q^{93} + 48 q^{94} + 88 q^{97} + 88 q^{98} + 28 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −0.707107 + 0.707107i −2.27550 + 1.31376i − 1.00000i −1.07072 + 3.99600i 0.680053 − 2.53799i −0.465421 − 2.60449i 0.707107 + 0.707107i 1.95193 − 3.38085i −2.06848 − 3.58271i
45.2 −0.707107 + 0.707107i −2.19775 + 1.26887i − 1.00000i 0.791799 − 2.95503i 0.656817 − 2.45127i 2.57725 − 0.598143i 0.707107 + 0.707107i 1.72008 − 2.97926i 1.52964 + 2.64941i
45.3 −0.707107 + 0.707107i −0.624408 + 0.360502i − 1.00000i −0.0497600 + 0.185707i 0.186610 − 0.696436i −0.622077 + 2.57158i 0.707107 + 0.707107i −1.24008 + 2.14788i −0.0961290 − 0.166500i
45.4 −0.707107 + 0.707107i 1.36811 − 0.789879i − 1.00000i −0.00830053 + 0.0309780i −0.408872 + 1.52593i 1.62925 − 2.08460i 0.707107 + 0.707107i −0.252182 + 0.436792i −0.0160354 − 0.0277741i
45.5 −0.707107 + 0.707107i 2.86352 − 1.65326i − 1.00000i 0.336986 − 1.25765i −0.855788 + 3.19385i −1.36932 + 2.26384i 0.707107 + 0.707107i 3.96651 − 6.87020i 0.651008 + 1.12758i
45.6 0.707107 − 0.707107i −2.57736 + 1.48804i − 1.00000i 0.611706 − 2.28292i −0.770265 + 2.87467i −1.69102 − 2.03481i −0.707107 − 0.707107i 2.92851 − 5.07233i −1.18173 − 2.04681i
45.7 0.707107 − 0.707107i −1.73766 + 1.00324i − 1.00000i −0.777903 + 2.90317i −0.519313 + 1.93810i 2.46461 + 0.962121i −0.707107 − 0.707107i 0.512966 − 0.888483i 1.50279 + 2.60291i
45.8 0.707107 − 0.707107i −0.0813346 + 0.0469586i − 1.00000i 0.391050 − 1.45942i −0.0243075 + 0.0907170i 2.47334 − 0.939450i −0.707107 − 0.707107i −1.49559 + 2.59044i −0.755451 − 1.30848i
45.9 0.707107 − 0.707107i 1.36122 − 0.785899i − 1.00000i 0.584789 − 2.18246i 0.406811 − 1.51824i −2.26688 + 1.36428i −0.707107 − 0.707107i −0.264726 + 0.458518i −1.12973 − 1.95674i
45.10 0.707107 − 0.707107i 2.16910 − 1.25233i − 1.00000i −0.809642 + 3.02163i 0.648255 − 2.41932i −0.265655 − 2.63238i −0.707107 − 0.707107i 1.63668 − 2.83481i 1.56411 + 2.70912i
59.1 −0.707107 − 0.707107i −2.36484 + 1.36534i 1.00000i 1.43364 + 0.384143i 2.63763 + 0.706751i −2.15363 − 1.53684i 0.707107 − 0.707107i 2.22830 − 3.85953i −0.742107 − 1.28537i
59.2 −0.707107 − 0.707107i −0.817271 + 0.471852i 1.00000i 0.361635 + 0.0968998i 0.911547 + 0.244248i 1.45367 − 2.21062i 0.707107 − 0.707107i −1.05471 + 1.82681i −0.187196 − 0.324233i
59.3 −0.707107 − 0.707107i 0.0314033 − 0.0181307i 1.00000i −3.24551 − 0.869631i −0.0350259 − 0.00938515i −2.06794 + 1.65034i 0.707107 − 0.707107i −1.49934 + 2.59694i 1.68000 + 2.90984i
59.4 −0.707107 − 0.707107i 1.32104 − 0.762704i 1.00000i 2.72206 + 0.729372i −1.47343 − 0.394805i 0.574274 + 2.58267i 0.707107 − 0.707107i −0.336565 + 0.582947i −1.40904 − 2.44053i
59.5 −0.707107 − 0.707107i 2.69569 − 1.55636i 1.00000i −1.27182 − 0.340784i −3.00665 − 0.805629i −1.97027 − 1.76580i 0.707107 − 0.707107i 3.34448 − 5.79281i 0.658344 + 1.14028i
59.6 0.707107 + 0.707107i −1.21580 + 0.701941i 1.00000i −3.67993 − 0.986033i −1.35605 − 0.363351i −0.836168 + 2.51014i −0.707107 + 0.707107i −0.514557 + 0.891239i −1.90487 − 3.29933i
59.7 0.707107 + 0.707107i −1.14719 + 0.662331i 1.00000i 3.76310 + 1.00832i −1.27953 − 0.342848i −2.15089 − 1.54068i −0.707107 + 0.707107i −0.622635 + 1.07844i 1.94792 + 3.37390i
59.8 0.707107 + 0.707107i −0.702817 + 0.405771i 1.00000i 1.15249 + 0.308809i −0.783890 − 0.210043i 2.61276 + 0.416516i −0.707107 + 0.707107i −1.17070 + 2.02771i 0.596574 + 1.03330i
59.9 0.707107 + 0.707107i 1.88594 − 1.08885i 1.00000i −2.10776 − 0.564774i 2.10349 + 0.563629i 2.44224 − 1.01757i −0.707107 + 0.707107i 0.871174 − 1.50892i −1.09106 − 1.88977i
59.10 0.707107 + 0.707107i 2.04589 − 1.18120i 1.00000i 0.872102 + 0.233679i 2.28190 + 0.611433i −2.36815 + 1.17977i −0.707107 + 0.707107i 1.29045 − 2.23513i 0.451433 + 0.781905i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.ba even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.2.bc.a yes 40
7.d odd 6 1 182.2.w.a ✓ 40
13.f odd 12 1 182.2.w.a ✓ 40
91.ba even 12 1 inner 182.2.bc.a yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.w.a ✓ 40 7.d odd 6 1
182.2.w.a ✓ 40 13.f odd 12 1
182.2.bc.a yes 40 1.a even 1 1 trivial
182.2.bc.a yes 40 91.ba even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(182, [\chi])\).