Properties

Label 624.2.x.a
Level $624$
Weight $2$
Character orbit 624.x
Analytic conductor $4.983$
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] = [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: \(40\)
Relative dimension: \(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 4 q^{4} + 12 q^{8} + 8 q^{10} + 8 q^{11} - 8 q^{12} + 20 q^{14} + 8 q^{15} + 32 q^{17} - 4 q^{18} + 8 q^{19} + 36 q^{20} - 44 q^{22} - 4 q^{24} - 32 q^{28} + 16 q^{29} + 8 q^{30} - 24 q^{31} + 20 q^{34} - 24 q^{35} + 4 q^{36} + 16 q^{37} - 12 q^{38} - 16 q^{40} + 16 q^{43} - 12 q^{44} + 60 q^{46} + 16 q^{48} + 24 q^{49} + 16 q^{50} - 8 q^{51} + 8 q^{52} - 16 q^{53} - 4 q^{54} + 16 q^{58} - 44 q^{60} - 16 q^{61} + 76 q^{62} - 8 q^{63} - 8 q^{64} - 8 q^{65} - 24 q^{66} + 16 q^{67} - 8 q^{68} - 16 q^{69} - 84 q^{70} + 4 q^{72} - 72 q^{74} - 24 q^{75} - 80 q^{76} - 16 q^{77} + 4 q^{78} - 16 q^{79} - 8 q^{80} - 40 q^{81} - 40 q^{82} + 24 q^{84} + 16 q^{85} - 16 q^{86} - 32 q^{88} + 8 q^{90} - 48 q^{92} + 36 q^{94} + 48 q^{95} - 96 q^{97} - 32 q^{98} + 8 q^{99}+O(q^{100}) \) 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 −1.41187 + 0.0813196i −0.707107 0.707107i 1.98677 0.229626i 0.430937 0.430937i 1.05585 + 0.940844i 0.332860i −2.78640 + 0.485767i 1.00000i −0.573385 + 0.643473i
157.2 −1.37720 0.321439i 0.707107 + 0.707107i 1.79335 + 0.885370i −0.616150 + 0.616150i −0.746535 1.20112i 2.54951i −2.18521 1.79578i 1.00000i 1.04661 0.650506i
157.3 −1.28294 + 0.595032i 0.707107 + 0.707107i 1.29187 1.52678i −2.23094 + 2.23094i −1.32793 0.486425i 0.469035i −0.748916 + 2.72748i 1.00000i 1.53469 4.18965i
157.4 −1.14904 0.824448i −0.707107 0.707107i 0.640572 + 1.89464i 1.44135 1.44135i 0.229519 + 1.39546i 2.43406i 0.825993 2.70513i 1.00000i −2.84448 + 0.467847i
157.5 −1.09850 + 0.890678i −0.707107 0.707107i 0.413387 1.95681i 1.72155 1.72155i 1.40656 + 0.146950i 3.99681i 1.28878 + 2.51774i 1.00000i −0.357769 + 3.42446i
157.6 −0.862181 1.12100i 0.707107 + 0.707107i −0.513289 + 1.93301i 2.38716 2.38716i 0.183014 1.40232i 2.28372i 2.60946 1.09121i 1.00000i −4.73418 0.617847i
157.7 −0.856472 + 1.12537i −0.707107 0.707107i −0.532911 1.92769i −2.33677 + 2.33677i 1.40137 0.190139i 1.80314i 2.62579 + 1.05130i 1.00000i −0.628351 4.63111i
157.8 −0.620977 1.27059i −0.707107 0.707107i −1.22878 + 1.57801i −2.40184 + 2.40184i −0.459343 + 1.33754i 0.488115i 2.76804 + 0.581357i 1.00000i 4.54323 + 1.56026i
157.9 −0.434856 + 1.34570i 0.707107 + 0.707107i −1.62180 1.17037i −0.929776 + 0.929776i −1.25904 + 0.644062i 3.22395i 2.28021 1.67351i 1.00000i −0.846879 1.65552i
157.10 −0.159779 + 1.40516i −0.707107 0.707107i −1.94894 0.449029i −0.624160 + 0.624160i 1.10658 0.880617i 0.519233i 0.942356 2.66683i 1.00000i −0.777317 0.976772i
157.11 −0.0467634 + 1.41344i 0.707107 + 0.707107i −1.99563 0.132195i 2.42463 2.42463i −1.03252 + 0.966386i 3.56880i 0.280172 2.81452i 1.00000i 3.31369 + 3.54046i
157.12 0.366592 1.36587i 0.707107 + 0.707107i −1.73122 1.00144i −0.366436 + 0.366436i 1.22504 0.706599i 3.68431i −2.00249 + 1.99751i 1.00000i 0.366172 + 0.634837i
157.13 0.528198 + 1.31187i 0.707107 + 0.707107i −1.44201 + 1.38586i −1.06983 + 1.06983i −0.554141 + 1.30113i 0.473561i −2.57973 1.15973i 1.00000i −1.96857 0.838400i
157.14 0.707327 1.22462i −0.707107 0.707107i −0.999377 1.73241i −1.67831 + 1.67831i −1.36609 + 0.365780i 4.31119i −2.82843 0.00152607i 1.00000i 0.868176 + 3.24241i
157.15 1.09789 0.891425i 0.707107 + 0.707107i 0.410724 1.95737i 0.927746 0.927746i 1.40666 + 0.145993i 3.33268i −1.29392 2.51511i 1.00000i 0.191547 1.84558i
157.16 1.21541 0.723027i −0.707107 0.707107i 0.954464 1.75755i 0.790012 0.790012i −1.37069 0.348170i 1.56424i −0.110691 2.82626i 1.00000i 0.388992 1.53139i
157.17 1.26768 + 0.626887i −0.707107 0.707107i 1.21402 + 1.58938i 2.76434 2.76434i −0.453109 1.33966i 1.16682i 0.542630 + 2.77589i 1.00000i 5.23722 1.77137i
157.18 1.32653 + 0.490219i 0.707107 + 0.707107i 1.51937 + 1.30058i 1.28803 1.28803i 0.591362 + 1.28464i 2.72177i 1.37792 + 2.47009i 1.00000i 2.34003 1.07720i
157.19 1.39184 + 0.250586i 0.707107 + 0.707107i 1.87441 + 0.697549i −0.400225 + 0.400225i 0.806986 + 1.16137i 2.81691i 2.43408 + 1.44057i 1.00000i −0.657339 + 0.456757i
157.20 1.39911 + 0.206159i −0.707107 0.707107i 1.91500 + 0.576877i −1.52131 + 1.52131i −0.843541 1.13509i 2.12662i 2.56036 + 1.20191i 1.00000i −2.44210 + 1.81484i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.20
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.a 40
4.b odd 2 1 2496.2.x.a 40
16.e even 4 1 inner 624.2.x.a 40
16.f odd 4 1 2496.2.x.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.x.a 40 1.a even 1 1 trivial
624.2.x.a 40 16.e even 4 1 inner
2496.2.x.a 40 4.b odd 2 1
2496.2.x.a 40 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} + 16 T_{5}^{37} + 492 T_{5}^{36} + 192 T_{5}^{35} + 128 T_{5}^{34} + 5456 T_{5}^{33} + 90236 T_{5}^{32} + 64032 T_{5}^{31} + 42752 T_{5}^{30} + 637568 T_{5}^{29} + 7521440 T_{5}^{28} + 7145216 T_{5}^{27} + \cdots + 879952896 \) acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display