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Properties

Label 104.3.g.a
Level $104$
Weight $3$
Character orbit 104.g
Analytic conductor $2.834$
Analytic rank $0$
Dimension $24$
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] = [104,3,Mod(27,104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(104, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("104.27");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 104.g (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: \(2.83379474935\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 10 q^{4} - 12 q^{6} - 4 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 10 q^{4} - 12 q^{6} - 4 q^{8} + 72 q^{9} + 2 q^{10} - 32 q^{11} - 22 q^{12} + 14 q^{14} - 38 q^{16} - 6 q^{18} + 64 q^{19} + 60 q^{22} - 104 q^{24} - 120 q^{25} - 6 q^{30} + 172 q^{32} + 16 q^{33} - 20 q^{34} + 96 q^{35} - 48 q^{36} - 236 q^{38} - 106 q^{40} + 48 q^{41} - 58 q^{42} + 32 q^{43} + 16 q^{44} + 12 q^{46} + 118 q^{48} - 168 q^{49} - 66 q^{50} - 32 q^{51} + 52 q^{52} + 112 q^{54} + 90 q^{56} - 80 q^{57} + 184 q^{58} + 132 q^{60} - 224 q^{62} + 254 q^{64} - 224 q^{66} - 128 q^{67} + 102 q^{68} + 232 q^{70} - 288 q^{72} + 78 q^{74} - 32 q^{75} + 332 q^{76} + 130 q^{78} + 56 q^{80} + 72 q^{81} - 4 q^{82} - 480 q^{83} - 240 q^{84} + 412 q^{86} + 312 q^{88} - 96 q^{89} + 684 q^{90} + 152 q^{92} + 94 q^{94} - 448 q^{96} + 32 q^{97} - 166 q^{98} - 480 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.93503 − 0.505640i 3.66227 3.48866 + 1.95686i 5.19686i −7.08660 − 1.85179i 9.37638i −5.76118 − 5.55057i 4.41224 2.62774 − 10.0561i
27.2 −1.93503 + 0.505640i 3.66227 3.48866 − 1.95686i − 5.19686i −7.08660 + 1.85179i − 9.37638i −5.76118 + 5.55057i 4.41224 2.62774 + 10.0561i
27.3 −1.60726 − 1.19026i −1.48928 1.16656 + 3.82611i − 7.88659i 2.39365 + 1.77263i 5.80546i 2.67911 − 7.53806i −6.78205 −9.38710 + 12.6758i
27.4 −1.60726 + 1.19026i −1.48928 1.16656 − 3.82611i 7.88659i 2.39365 − 1.77263i − 5.80546i 2.67911 + 7.53806i −6.78205 −9.38710 − 12.6758i
27.5 −1.45554 − 1.37164i −5.08271 0.237219 + 3.99296i 3.73505i 7.39811 + 6.97163i − 5.95606i 5.13161 − 6.13731i 16.8339 5.12314 − 5.43653i
27.6 −1.45554 + 1.37164i −5.08271 0.237219 − 3.99296i − 3.73505i 7.39811 − 6.97163i 5.95606i 5.13161 + 6.13731i 16.8339 5.12314 + 5.43653i
27.7 −1.14903 − 1.63699i 4.05934 −1.35945 + 3.76190i − 3.50498i −4.66431 − 6.64508i − 3.71323i 7.72023 − 2.09716i 7.47822 −5.73761 + 4.02734i
27.8 −1.14903 + 1.63699i 4.05934 −1.35945 − 3.76190i 3.50498i −4.66431 + 6.64508i 3.71323i 7.72023 + 2.09716i 7.47822 −5.73761 − 4.02734i
27.9 −0.555918 − 1.92119i −0.169737 −3.38191 + 2.13604i 6.61888i 0.0943599 + 0.326097i 6.46265i 5.98380 + 5.30981i −8.97119 12.7161 − 3.67955i
27.10 −0.555918 + 1.92119i −0.169737 −3.38191 − 2.13604i − 6.61888i 0.0943599 − 0.326097i − 6.46265i 5.98380 − 5.30981i −8.97119 12.7161 + 3.67955i
27.11 −0.0299033 − 1.99978i −0.741155 −3.99821 + 0.119600i − 3.13434i 0.0221629 + 1.48214i − 6.36831i 0.358732 + 7.99195i −8.45069 −6.26798 + 0.0937270i
27.12 −0.0299033 + 1.99978i −0.741155 −3.99821 − 0.119600i 3.13434i 0.0221629 − 1.48214i 6.36831i 0.358732 − 7.99195i −8.45069 −6.26798 − 0.0937270i
27.13 0.652784 − 1.89047i 5.53553 −3.14775 − 2.46814i 5.99565i 3.61351 − 10.4648i − 6.12782i −6.72073 + 4.33955i 21.6421 11.3346 + 3.91387i
27.14 0.652784 + 1.89047i 5.53553 −3.14775 + 2.46814i − 5.99565i 3.61351 + 10.4648i 6.12782i −6.72073 − 4.33955i 21.6421 11.3346 − 3.91387i
27.15 0.884987 − 1.79354i −4.89707 −2.43360 − 3.17453i 1.60970i −4.33384 + 8.78310i 10.1578i −7.84735 + 1.55534i 14.9813 2.88706 + 1.42456i
27.16 0.884987 + 1.79354i −4.89707 −2.43360 + 3.17453i − 1.60970i −4.33384 − 8.78310i − 10.1578i −7.84735 − 1.55534i 14.9813 2.88706 − 1.42456i
27.17 1.02197 − 1.71918i 2.77694 −1.91116 − 3.51389i − 7.79847i 2.83794 − 4.77406i 11.3494i −7.99417 − 0.305451i −1.28861 −13.4070 − 7.96978i
27.18 1.02197 + 1.71918i 2.77694 −1.91116 + 3.51389i 7.79847i 2.83794 + 4.77406i − 11.3494i −7.99417 + 0.305451i −1.28861 −13.4070 + 7.96978i
27.19 1.32373 − 1.49925i −1.94132 −0.495497 − 3.96919i 0.451692i −2.56977 + 2.91052i − 11.9733i −6.60671 − 4.51125i −5.23128 0.677198 + 0.597916i
27.20 1.32373 + 1.49925i −1.94132 −0.495497 + 3.96919i − 0.451692i −2.56977 − 2.91052i 11.9733i −6.60671 + 4.51125i −5.23128 0.677198 − 0.597916i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.3.g.a ✓ 24
4.b odd 2 1 416.3.g.a 24
8.b even 2 1 416.3.g.a 24
8.d odd 2 1 inner 104.3.g.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.3.g.a ✓ 24 1.a even 1 1 trivial
104.3.g.a ✓ 24 8.d odd 2 1 inner
416.3.g.a 24 4.b odd 2 1
416.3.g.a 24 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(104, [\chi])\).