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Properties

Label 624.2.bm.c
Level $624$
Weight $2$
Character orbit 624.bm
Analytic conductor $4.983$
Analytic rank $0$
Dimension $208$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(317,624)]
 
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, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.317");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.bm (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: \(208\)
Relative dimension: \(104\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{3} + 8 q^{6} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 208 q - 2 q^{3} + 8 q^{6} + 12 q^{7} - 16 q^{10} + 4 q^{12} - 20 q^{13} - 6 q^{15} + 24 q^{16} + 20 q^{18} - 16 q^{19} - 20 q^{21} - 8 q^{22} - 32 q^{24} + 216 q^{25} - 20 q^{27} - 56 q^{28} - 24 q^{30} - 24 q^{31} - 28 q^{33} - 8 q^{34} + 52 q^{36} + 32 q^{37} + 10 q^{39} + 16 q^{40} - 4 q^{43} - 44 q^{45} + 72 q^{46} - 32 q^{48} + 18 q^{51} + 64 q^{52} - 80 q^{54} - 32 q^{57} - 24 q^{58} + 20 q^{60} + 24 q^{61} + 46 q^{63} - 48 q^{64} + 16 q^{66} - 8 q^{69} - 88 q^{70} - 4 q^{72} + 48 q^{73} - 28 q^{75} - 24 q^{76} - 72 q^{78} + 48 q^{79} - 36 q^{81} + 8 q^{82} - 64 q^{85} + 80 q^{88} - 12 q^{90} - 12 q^{91} - 100 q^{93} - 32 q^{94} - 80 q^{96} + 56 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
317.1 −1.41418 + 0.00952869i −0.521759 + 1.65160i 1.99982 − 0.0269506i 0.816879 0.722124 − 2.34063i 0.669039 − 0.669039i −2.82785 + 0.0571687i −2.45554 − 1.72347i −1.15522 + 0.00778379i
317.2 −1.41351 − 0.0446258i −1.73179 − 0.0299631i 1.99602 + 0.126158i 0.317962 2.44657 + 0.119636i −0.0919918 + 0.0919918i −2.81576 − 0.267399i 2.99820 + 0.103780i −0.449442 − 0.0141893i
317.3 −1.40713 − 0.141342i 1.11325 − 1.32690i 1.96004 + 0.397774i −2.31361 −1.75404 + 1.70978i 1.14179 − 1.14179i −2.70182 − 0.836758i −0.521349 − 2.95435i 3.25556 + 0.327011i
317.4 −1.39734 − 0.217797i −0.608775 − 1.62154i 1.90513 + 0.608675i −3.73796 0.497500 + 2.39844i −2.84347 + 2.84347i −2.52955 − 1.26546i −2.25879 + 1.97431i 5.22321 + 0.814118i
317.5 −1.38769 − 0.272618i 0.970060 + 1.43492i 1.85136 + 0.756617i 1.97066 −0.954958 − 2.25567i −0.0346010 + 0.0346010i −2.36284 − 1.55466i −1.11797 + 2.78391i −2.73467 − 0.537238i
317.6 −1.38403 + 0.290615i 1.54175 + 0.789304i 1.83109 − 0.804439i 1.60722 −2.36322 − 0.644366i 3.63444 − 3.63444i −2.30050 + 1.64551i 1.75400 + 2.43382i −2.22445 + 0.467083i
317.7 −1.38232 + 0.298633i −1.19782 − 1.25109i 1.82164 − 0.825616i −0.826970 2.02939 + 1.37171i 2.81241 − 2.81241i −2.27153 + 1.68527i −0.130463 + 2.99716i 1.14314 − 0.246961i
317.8 −1.38167 − 0.301648i 1.21898 − 1.23048i 1.81802 + 0.833556i 2.20675 −2.05540 + 1.33242i −2.93336 + 2.93336i −2.26046 − 1.70010i −0.0281855 − 2.99987i −3.04900 − 0.665663i
317.9 −1.37650 + 0.324413i 1.72443 + 0.162349i 1.78951 − 0.893111i −2.24586 −2.42634 + 0.335953i −1.87064 + 1.87064i −2.17353 + 1.80991i 2.94729 + 0.559917i 3.09143 − 0.728586i
317.10 −1.36033 − 0.386643i −1.58661 + 0.694735i 1.70101 + 1.05193i −2.36257 2.42694 − 0.331618i 1.19537 − 1.19537i −1.90723 − 2.08866i 2.03469 − 2.20455i 3.21388 + 0.913471i
317.11 −1.34884 + 0.425019i −1.19852 + 1.25042i 1.63872 − 1.14656i 4.08345 1.08515 − 2.19601i −0.134690 + 0.134690i −1.72305 + 2.24301i −0.127102 − 2.99731i −5.50790 + 1.73554i
317.12 −1.33291 − 0.472587i −0.122569 − 1.72771i 1.55332 + 1.25984i 3.73128 −0.653119 + 2.36081i 2.93667 − 2.93667i −1.47506 − 2.41334i −2.96995 + 0.423528i −4.97348 − 1.76336i
317.13 −1.32887 + 0.483843i 0.181931 + 1.72247i 1.53179 − 1.28593i −1.13548 −1.07517 − 2.20091i −2.62458 + 2.62458i −1.41337 + 2.44998i −2.93380 + 0.626742i 1.50891 − 0.549394i
317.14 −1.32264 + 0.500620i 0.429730 − 1.67790i 1.49876 − 1.32428i 2.06962 0.271609 + 2.43438i −1.51263 + 1.51263i −1.31936 + 2.50186i −2.63066 − 1.44208i −2.73737 + 1.03609i
317.15 −1.32187 − 0.502656i −1.38794 − 1.03616i 1.49467 + 1.32889i 1.91108 1.31384 + 2.06733i −1.53759 + 1.53759i −1.30779 − 2.50793i 0.852734 + 2.87626i −2.52619 − 0.960615i
317.16 −1.27779 + 0.606013i −0.379797 − 1.68990i 1.26550 − 1.54872i 1.18761 1.50940 + 1.92917i −0.812035 + 0.812035i −0.678496 + 2.74584i −2.71151 + 1.28364i −1.51751 + 0.719705i
317.17 −1.23941 − 0.681072i −0.837156 + 1.51630i 1.07228 + 1.68826i −2.33402 2.07029 − 1.30916i −3.61996 + 3.61996i −0.179177 − 2.82275i −1.59834 − 2.53876i 2.89281 + 1.58963i
317.18 −1.21804 − 0.718597i 1.61331 − 0.630262i 0.967237 + 1.75056i −0.114943 −2.41798 − 0.391636i 1.84806 − 1.84806i 0.0798126 − 2.82730i 2.20554 − 2.03362i 0.140005 + 0.0825975i
317.19 −1.21446 − 0.724624i 1.47346 + 0.910452i 0.949840 + 1.76006i −2.82604 −1.12972 − 2.17341i 0.429428 − 0.429428i 0.121835 − 2.82580i 1.34216 + 2.68302i 3.43213 + 2.04782i
317.20 −1.20098 + 0.746753i −1.67529 − 0.439762i 0.884719 − 1.79368i −3.91902 2.34039 − 0.722884i −0.857197 + 0.857197i 0.276902 + 2.81484i 2.61322 + 1.47346i 4.70668 − 2.92654i
See next 80 embeddings (of 208 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 317.104
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
208.m odd 4 1 inner
624.bm 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.bm.c yes 208
3.b odd 2 1 inner 624.2.bm.c yes 208
13.d odd 4 1 624.2.u.c ✓ 208
16.e even 4 1 624.2.u.c ✓ 208
39.f even 4 1 624.2.u.c ✓ 208
48.i odd 4 1 624.2.u.c ✓ 208
208.m odd 4 1 inner 624.2.bm.c yes 208
624.bm even 4 1 inner 624.2.bm.c yes 208
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.u.c ✓ 208 13.d odd 4 1
624.2.u.c ✓ 208 16.e even 4 1
624.2.u.c ✓ 208 39.f even 4 1
624.2.u.c ✓ 208 48.i odd 4 1
624.2.bm.c yes 208 1.a even 1 1 trivial
624.2.bm.c yes 208 3.b odd 2 1 inner
624.2.bm.c yes 208 208.m odd 4 1 inner
624.2.bm.c yes 208 624.bm even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{104} - 314 T_{5}^{102} + 47587 T_{5}^{100} - 4637600 T_{5}^{98} + 326709961 T_{5}^{96} + \cdots + 71\!\cdots\!56 \) acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display