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Properties

Label 124.3.b.a
Level $124$
Weight $3$
Character orbit 124.b
Analytic conductor $3.379$
Analytic rank $0$
Dimension $30$
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] = [124,3,Mod(63,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, 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("124.63");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 124.b (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: \(3.37875527807\)
Analytic rank: \(0\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{2} - 6 q^{4} - 4 q^{5} + 13 q^{8} - 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 2 q^{2} - 6 q^{4} - 4 q^{5} + 13 q^{8} - 82 q^{9} + q^{10} - 14 q^{12} + 12 q^{13} + 29 q^{14} + 50 q^{16} - 4 q^{17} - 34 q^{18} - 63 q^{20} - 16 q^{21} - 24 q^{22} - 20 q^{24} + 90 q^{25} + 38 q^{26} + 3 q^{28} - 4 q^{29} - 6 q^{30} + 118 q^{32} + 80 q^{33} + 4 q^{34} - 2 q^{36} + 76 q^{37} + 37 q^{38} - 180 q^{40} - 4 q^{41} - 38 q^{42} + 184 q^{44} - 20 q^{45} - 54 q^{46} - 172 q^{48} - 258 q^{49} - 31 q^{50} - 88 q^{52} - 132 q^{53} - 84 q^{54} - 28 q^{56} + 176 q^{57} + 164 q^{58} + 108 q^{60} - 100 q^{61} + 381 q^{64} - 104 q^{65} + 60 q^{66} + 214 q^{68} + 112 q^{69} + 45 q^{70} - 167 q^{72} - 132 q^{73} + 398 q^{74} - 317 q^{76} + 176 q^{77} - 188 q^{78} - 203 q^{80} + 158 q^{81} - 81 q^{82} + 176 q^{84} + 248 q^{85} - 78 q^{86} + 98 q^{88} - 20 q^{89} - 567 q^{90} - 260 q^{92} - 244 q^{94} - 90 q^{96} + 300 q^{97} - 371 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1 −1.96532 − 0.370831i − 2.27228i 3.72497 + 1.45760i −6.91201 −0.842630 + 4.46576i 10.2860i −6.78024 − 4.24599i 3.83675 13.5843 + 2.56318i
63.2 −1.96532 + 0.370831i 2.27228i 3.72497 − 1.45760i −6.91201 −0.842630 − 4.46576i − 10.2860i −6.78024 + 4.24599i 3.83675 13.5843 − 2.56318i
63.3 −1.96086 − 0.393756i 3.47301i 3.68991 + 1.54420i 6.23763 1.36752 − 6.81006i − 1.04310i −6.62735 − 4.48088i −3.06177 −12.2311 − 2.45611i
63.4 −1.96086 + 0.393756i − 3.47301i 3.68991 − 1.54420i 6.23763 1.36752 + 6.81006i 1.04310i −6.62735 + 4.48088i −3.06177 −12.2311 + 2.45611i
63.5 −1.66997 − 1.10055i − 3.18220i 1.57758 + 3.67576i 1.05927 −3.50217 + 5.31417i − 9.87284i 1.41085 − 7.87461i −1.12641 −1.76894 − 1.16578i
63.6 −1.66997 + 1.10055i 3.18220i 1.57758 − 3.67576i 1.05927 −3.50217 − 5.31417i 9.87284i 1.41085 + 7.87461i −1.12641 −1.76894 + 1.16578i
63.7 −1.37549 − 1.45191i 5.29882i −0.216076 + 3.99416i −4.75394 7.69340 − 7.28845i 1.34774i 6.09636 − 5.18019i −19.0775 6.53898 + 6.90228i
63.8 −1.37549 + 1.45191i − 5.29882i −0.216076 − 3.99416i −4.75394 7.69340 + 7.28845i − 1.34774i 6.09636 + 5.18019i −19.0775 6.53898 − 6.90228i
63.9 −1.23553 − 1.57273i 0.706826i −0.946952 + 3.88629i −0.787341 1.11164 − 0.873301i − 0.940771i 7.28207 − 3.31232i 8.50040 0.972780 + 1.23827i
63.10 −1.23553 + 1.57273i − 0.706826i −0.946952 − 3.88629i −0.787341 1.11164 + 0.873301i 0.940771i 7.28207 + 3.31232i 8.50040 0.972780 − 1.23827i
63.11 −0.785439 − 1.83932i − 0.629544i −2.76617 + 2.88934i 8.36044 −1.15793 + 0.494468i 10.6307i 7.48707 + 2.81846i 8.60367 −6.56662 − 15.3775i
63.12 −0.785439 + 1.83932i 0.629544i −2.76617 − 2.88934i 8.36044 −1.15793 − 0.494468i − 10.6307i 7.48707 − 2.81846i 8.60367 −6.56662 + 15.3775i
63.13 −0.616722 − 1.90254i − 5.19866i −3.23931 + 2.34667i −4.08317 −9.89065 + 3.20613i 2.88507i 6.46239 + 4.71567i −18.0260 2.51818 + 7.76838i
63.14 −0.616722 + 1.90254i 5.19866i −3.23931 − 2.34667i −4.08317 −9.89065 − 3.20613i − 2.88507i 6.46239 − 4.71567i −18.0260 2.51818 − 7.76838i
63.15 −0.0170245 − 1.99993i 1.51810i −3.99942 + 0.0680956i −6.79879 3.03609 − 0.0258450i 4.18800i 0.204275 + 7.99739i 6.69537 0.115746 + 13.5971i
63.16 −0.0170245 + 1.99993i − 1.51810i −3.99942 − 0.0680956i −6.79879 3.03609 + 0.0258450i − 4.18800i 0.204275 − 7.99739i 6.69537 0.115746 − 13.5971i
63.17 0.153814 − 1.99408i 2.07170i −3.95268 − 0.613434i 1.93675 4.13114 + 0.318657i − 13.3450i −1.83121 + 7.78760i 4.70804 0.297900 − 3.86204i
63.18 0.153814 + 1.99408i − 2.07170i −3.95268 + 0.613434i 1.93675 4.13114 − 0.318657i 13.3450i −1.83121 − 7.78760i 4.70804 0.297900 + 3.86204i
63.19 0.778186 − 1.84240i 5.21571i −2.78885 − 2.86745i 3.92621 9.60941 + 4.05879i 8.65127i −7.45324 + 2.90676i −18.2037 3.05532 − 7.23364i
63.20 0.778186 + 1.84240i − 5.21571i −2.78885 + 2.86745i 3.92621 9.60941 − 4.05879i − 8.65127i −7.45324 − 2.90676i −18.2037 3.05532 + 7.23364i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.30
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 124.3.b.a ✓ 30
4.b odd 2 1 inner 124.3.b.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.3.b.a ✓ 30 1.a even 1 1 trivial
124.3.b.a ✓ 30 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(124, [\chi])\).