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Properties

Label 625.2.h.a
Level $625$
Weight $2$
Character orbit 625.h
Analytic conductor $4.991$
Analytic rank $0$
Dimension $240$
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] = [625,2,Mod(24,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.h (of order \(50\), degree \(20\), not 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: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(12\) over \(\Q(\zeta_{50})\)
Twist minimal: no (minimal twist has level 125)
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 20 q^{2} + 20 q^{3} - 20 q^{4} - 20 q^{6} + 25 q^{7} + 35 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 20 q^{2} + 20 q^{3} - 20 q^{4} - 20 q^{6} + 25 q^{7} + 35 q^{8} - 20 q^{9} - 25 q^{11} - 60 q^{12} + 20 q^{13} - 30 q^{14} - 40 q^{16} + 15 q^{17} + 25 q^{18} - 10 q^{19} - 35 q^{21} + 25 q^{22} - 70 q^{23} + 15 q^{24} - 45 q^{26} + 20 q^{27} + 10 q^{28} - 10 q^{29} - 30 q^{31} + 25 q^{32} + 35 q^{33} - 20 q^{34} + 170 q^{36} + 55 q^{37} + 40 q^{38} - 35 q^{41} + 10 q^{42} + 25 q^{43} + 15 q^{44} - 40 q^{46} - 100 q^{47} - 5 q^{48} + 35 q^{49} - 55 q^{51} + 15 q^{52} + 15 q^{53} + 30 q^{54} + 65 q^{56} - 255 q^{58} + 5 q^{59} - 40 q^{61} - 5 q^{62} + 35 q^{63} + 25 q^{64} - 95 q^{66} - 105 q^{67} - 10 q^{69} + 45 q^{71} + 30 q^{72} + 40 q^{73} + 35 q^{74} - 65 q^{76} + 35 q^{77} - 100 q^{78} - 95 q^{81} - 175 q^{82} - 20 q^{83} + 45 q^{84} - 80 q^{86} + 5 q^{87} + 5 q^{88} + 30 q^{89} + 65 q^{91} + 55 q^{92} - 275 q^{93} + 60 q^{94} - 135 q^{96} - 35 q^{97} + 15 q^{98} + 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −1.98459 − 1.64179i −0.361034 + 2.85788i 0.868341 + 4.55200i 0 5.40855 − 5.07897i −1.02828 + 1.41531i 3.26848 − 5.94534i −5.13136 − 1.31751i 0
24.2 −1.51704 − 1.25500i 0.365042 − 2.88960i 0.351611 + 1.84321i 0 −4.18025 + 3.92552i −2.26589 + 3.11873i −0.117191 + 0.213170i −5.31081 − 1.36358i 0
24.3 −1.34158 − 1.10985i 0.125748 − 0.995400i 0.193306 + 1.01335i 0 −1.27345 + 1.19585i 0.990155 − 1.36283i −0.812285 + 1.47754i 1.93074 + 0.495730i 0
24.4 −1.19217 − 0.986251i −0.195971 + 1.55127i 0.0738221 + 0.386989i 0 1.76358 − 1.65611i 2.66804 − 3.67224i −1.19712 + 2.17756i 0.537707 + 0.138060i 0
24.5 −0.369274 − 0.305490i 0.0227881 − 0.180386i −0.331724 − 1.73896i 0 −0.0635211 + 0.0596503i −0.964889 + 1.32806i −0.870504 + 1.58344i 2.87373 + 0.737848i 0
24.6 0.0285177 + 0.0235919i −0.374846 + 2.96721i −0.374506 − 1.96323i 0 −0.0806918 + 0.0757746i 0.938062 − 1.29113i 0.0712968 − 0.129688i −5.75808 − 1.47842i 0
24.7 0.434858 + 0.359746i 0.166855 − 1.32079i −0.315078 − 1.65170i 0 0.547708 − 0.514332i −0.551442 + 0.758994i 1.00096 − 1.82073i 1.18910 + 0.305308i 0
24.8 0.752221 + 0.622291i 0.335353 − 2.65459i −0.196173 − 1.02837i 0 1.90419 − 1.78815i 0.477458 − 0.657165i 1.43302 − 2.60665i −4.02866 − 1.03438i 0
24.9 0.972495 + 0.804518i −0.195914 + 1.55082i −0.0762652 − 0.399796i 0 −1.43819 + 1.35055i 1.13416 − 1.56104i 1.46356 − 2.66220i 0.539082 + 0.138413i 0
24.10 1.57464 + 1.30265i −0.295889 + 2.34220i 0.407815 + 2.13784i 0 −3.51699 + 3.30268i −2.45750 + 3.38245i −0.173661 + 0.315888i −2.49260 − 0.639991i 0
24.11 1.77307 + 1.46681i 0.0227892 − 0.180395i 0.617477 + 3.23693i 0 0.305011 − 0.286425i 2.80184 − 3.85640i −1.43595 + 2.61199i 2.87373 + 0.737848i 0
24.12 2.05624 + 1.70107i 0.166324 − 1.31659i 0.959725 + 5.03106i 0 2.58161 − 2.42429i −2.47576 + 3.40759i −4.01348 + 7.30050i 1.20001 + 0.308111i 0
49.1 −2.31593 + 0.441787i −0.0749067 + 0.291742i 3.30880 − 1.31005i 0 0.0445906 − 0.708747i 3.14708 − 1.02255i −3.10286 + 1.96914i 2.54942 + 1.40155i 0
49.2 −2.11316 + 0.403106i 0.740930 − 2.88573i 2.44338 − 0.967402i 0 −0.402445 + 6.39667i 0.0354597 − 0.0115215i −1.14053 + 0.723805i −5.14955 − 2.83099i 0
49.3 −1.46917 + 0.280260i −0.666016 + 2.59396i 0.220369 − 0.0872504i 0 0.251510 − 3.99763i 2.93500 − 0.953639i 2.22636 − 1.41289i −3.65614 − 2.00998i 0
49.4 −1.24154 + 0.236836i −0.223760 + 0.871489i −0.374224 + 0.148166i 0 0.0714071 − 1.13498i −1.20297 + 0.390868i 2.56386 − 1.62708i 1.91949 + 1.05525i 0
49.5 −0.670014 + 0.127812i 0.413431 − 1.61021i −1.42697 + 0.564977i 0 −0.0712008 + 1.13170i −2.46772 + 0.801811i 2.03570 − 1.29190i 0.207073 + 0.113840i 0
49.6 0.179456 − 0.0342330i 0.513270 − 1.99905i −1.82852 + 0.723963i 0 0.0236755 − 0.376312i 4.39822 − 1.42907i −0.611858 + 0.388297i −1.10385 − 0.606847i 0
49.7 0.563283 − 0.107452i −0.789691 + 3.07564i −1.55381 + 0.615197i 0 −0.114336 + 1.81731i −1.04902 + 0.340847i −1.77748 + 1.12802i −6.20705 − 3.41236i 0
49.8 0.625049 − 0.119235i −0.260060 + 1.01287i −1.48308 + 0.587194i 0 −0.0417816 + 0.664099i −2.56059 + 0.831988i −1.93151 + 1.22577i 1.67065 + 0.918450i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.h even 50 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.h.a 240
5.b even 2 1 125.2.h.a ✓ 240
5.c odd 4 2 625.2.g.b 480
125.g even 25 1 125.2.h.a ✓ 240
125.h even 50 1 inner 625.2.h.a 240
125.i odd 100 2 625.2.g.b 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.h.a ✓ 240 5.b even 2 1
125.2.h.a ✓ 240 125.g even 25 1
625.2.g.b 480 5.c odd 4 2
625.2.g.b 480 125.i odd 100 2
625.2.h.a 240 1.a even 1 1 trivial
625.2.h.a 240 125.h even 50 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{240} - 20 T_{2}^{239} + 210 T_{2}^{238} - 1545 T_{2}^{237} + 8960 T_{2}^{236} + \cdots + 12248257062001 \) acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\). Copy content Toggle raw display