Properties

Label 624.2.x.b
Level $624$
Weight $2$
Character orbit 624.x
Analytic conductor $4.983$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(157,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.x (of order \(4\), 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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{4} + 12 q^{8} + 8 q^{10} + 8 q^{11} - 12 q^{14} + 8 q^{15} - 32 q^{17} - 4 q^{18} + 8 q^{19} - 36 q^{20} + 28 q^{22} - 4 q^{24} + 32 q^{28} + 16 q^{29} - 24 q^{31} + 20 q^{34} - 24 q^{35} + 4 q^{36} + 16 q^{37} + 68 q^{38} - 16 q^{40} - 16 q^{43} + 60 q^{44} - 52 q^{46} + 16 q^{48} - 120 q^{49} + 16 q^{50} - 8 q^{51} - 16 q^{53} - 4 q^{54} + 16 q^{58} + 44 q^{60} - 16 q^{61} - 100 q^{62} - 8 q^{63} - 8 q^{64} + 8 q^{65} - 24 q^{66} + 16 q^{67} - 72 q^{68} - 16 q^{69} + 108 q^{70} + 4 q^{72} - 72 q^{74} + 8 q^{75} + 96 q^{76} - 16 q^{77} - 4 q^{78} - 16 q^{79} - 8 q^{80} - 56 q^{81} - 40 q^{82} + 24 q^{84} + 16 q^{85} + 80 q^{86} - 40 q^{88} + 128 q^{92} - 92 q^{94} + 48 q^{95} + 96 q^{97} + 40 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 −1.40239 0.182491i 0.707107 + 0.707107i 1.93339 + 0.511848i −1.29029 + 1.29029i −0.862598 1.12068i 4.74894i −2.61796 1.07064i 1.00000i 2.04496 1.57402i
157.2 −1.36726 + 0.361394i 0.707107 + 0.707107i 1.73879 0.988237i 2.24248 2.24248i −1.22234 0.711253i 0.711007i −2.02023 + 1.97956i 1.00000i −2.25563 + 3.87647i
157.3 −1.33481 0.467208i −0.707107 0.707107i 1.56343 + 1.24727i 1.08546 1.08546i 0.613487 + 1.27422i 3.56874i −1.50415 2.39531i 1.00000i −1.95602 + 0.941749i
157.4 −1.30024 0.556219i −0.707107 0.707107i 1.38124 + 1.44644i −2.42862 + 2.42862i 0.526101 + 1.31271i 5.20571i −0.991405 2.64898i 1.00000i 4.50863 1.80694i
157.5 −1.29421 + 0.570097i −0.707107 0.707107i 1.34998 1.47566i −2.02025 + 2.02025i 1.31827 + 0.512028i 0.634897i −0.905894 + 2.67943i 1.00000i 1.46289 3.76637i
157.6 −1.14499 + 0.830062i −0.707107 0.707107i 0.621993 1.90082i 1.63652 1.63652i 1.39657 + 0.222686i 2.06925i 0.865628 + 2.69271i 1.00000i −0.515380 + 3.23220i
157.7 −1.01862 + 0.981027i 0.707107 + 0.707107i 0.0751712 1.99859i 0.918279 0.918279i −1.41396 0.0265817i 3.67302i 1.88410 + 2.10954i 1.00000i −0.0345202 + 1.83623i
157.8 −0.913674 1.07944i 0.707107 + 0.707107i −0.330401 + 1.97252i −0.470608 + 0.470608i 0.117218 1.40935i 2.14920i 2.43111 1.44559i 1.00000i 0.937977 + 0.0780132i
157.9 −0.881982 1.10549i −0.707107 0.707107i −0.444217 + 1.95004i −0.687536 + 0.687536i −0.158044 + 1.40535i 2.87123i 2.54755 1.22883i 1.00000i 1.36646 + 0.153670i
157.10 −0.779416 + 1.18005i 0.707107 + 0.707107i −0.785021 1.83950i −0.439534 + 0.439534i −1.38555 + 0.283289i 3.49065i 2.78255 + 0.507371i 1.00000i −0.176091 0.861251i
157.11 −0.546564 + 1.30433i −0.707107 0.707107i −1.40253 1.42580i −0.741100 + 0.741100i 1.30878 0.535819i 3.54681i 2.62628 1.05007i 1.00000i −0.561578 1.37170i
157.12 −0.327918 + 1.37567i 0.707107 + 0.707107i −1.78494 0.902215i −3.01052 + 3.01052i −1.20462 + 0.740873i 4.33033i 1.82646 2.15964i 1.00000i −3.15428 5.12869i
157.13 −0.326306 1.37605i 0.707107 + 0.707107i −1.78705 + 0.898028i −0.534609 + 0.534609i 0.742284 1.20375i 1.29164i 1.81886 + 2.16604i 1.00000i 0.910097 + 0.561205i
157.14 −0.173487 1.40353i −0.707107 0.707107i −1.93980 + 0.486990i −0.503626 + 0.503626i −0.869773 + 1.11512i 0.824309i 1.02004 + 2.63809i 1.00000i 0.794228 + 0.619483i
157.15 −0.125504 + 1.40863i −0.707107 0.707107i −1.96850 0.353578i 2.13999 2.13999i 1.08480 0.907310i 2.18307i 0.745117 2.72852i 1.00000i 2.74588 + 3.28304i
157.16 0.179347 1.40280i 0.707107 + 0.707107i −1.93567 0.503174i 2.10708 2.10708i 1.11874 0.865109i 0.641112i −1.05301 + 2.62511i 1.00000i −2.57791 3.33371i
157.17 0.511838 + 1.31834i −0.707107 0.707107i −1.47604 + 1.34955i 1.19874 1.19874i 0.570284 1.29413i 0.592345i −2.53467 1.25518i 1.00000i 2.19392 + 0.966791i
157.18 0.512178 + 1.31821i 0.707107 + 0.707107i −1.47535 + 1.35031i 2.44843 2.44843i −0.569950 + 1.29428i 5.13203i −2.53564 1.25322i 1.00000i 4.48157 + 1.97351i
157.19 0.617594 1.27223i −0.707107 0.707107i −1.23715 1.57145i 3.01500 3.01500i −1.33631 + 0.462899i 2.10775i −2.76331 + 0.603431i 1.00000i −1.97374 5.69783i
157.20 0.989564 + 1.01033i 0.707107 + 0.707107i −0.0415264 + 1.99957i 0.996563 0.996563i −0.0146826 + 1.41414i 3.61917i −2.06131 + 1.93675i 1.00000i 1.99302 + 0.0206930i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.x.b 56
4.b odd 2 1 2496.2.x.b 56
16.e even 4 1 inner 624.2.x.b 56
16.f odd 4 1 2496.2.x.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.x.b 56 1.a even 1 1 trivial
624.2.x.b 56 16.e even 4 1 inner
2496.2.x.b 56 4.b odd 2 1
2496.2.x.b 56 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{56} + 16 T_{5}^{53} + 1092 T_{5}^{52} + 224 T_{5}^{51} + 128 T_{5}^{50} + 13552 T_{5}^{49} + 478260 T_{5}^{48} + 176096 T_{5}^{47} + 102144 T_{5}^{46} + 4553088 T_{5}^{45} + 109377472 T_{5}^{44} + \cdots + 23\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display