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Properties

Label 625.2.h.b
Level $625$
Weight $2$
Character orbit 625.h
Analytic conductor $4.991$
Analytic rank $0$
Dimension $440$
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] = [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: \(440\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 440 q + 40 q^{4} - 40 q^{6} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 440 q + 40 q^{4} - 40 q^{6} + 40 q^{9} - 30 q^{11} + 20 q^{14} + 20 q^{19} - 10 q^{21} - 30 q^{24} + 30 q^{26} + 20 q^{29} - 20 q^{31} - 10 q^{34} - 420 q^{36} - 10 q^{41} - 30 q^{44} + 40 q^{49} + 50 q^{51} - 120 q^{54} - 210 q^{56} - 10 q^{59} - 90 q^{64} + 110 q^{66} - 80 q^{69} - 170 q^{71} - 70 q^{74} + 70 q^{76} + 110 q^{81} - 230 q^{84} + 80 q^{86} - 110 q^{89} - 210 q^{91} - 120 q^{94} + 270 q^{96} - 90 q^{99}+O(q^{100}) \) 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 −2.11763 − 1.75186i 0.240189 − 1.90129i 1.04060 + 5.45502i 0 −3.83943 + 3.60546i −1.00583 + 1.38440i 4.70477 − 8.55795i −0.651466 − 0.167268i 0
24.2 −1.88195 − 1.55688i −0.0216097 + 0.171059i 0.743080 + 3.89536i 0 0.306986 − 0.288279i 0.116216 − 0.159958i 2.31285 − 4.20707i 2.87696 + 0.738677i 0
24.3 −1.78418 − 1.47601i 0.342413 − 2.71048i 0.629958 + 3.30236i 0 −4.61161 + 4.33059i 1.04390 − 1.43680i 1.51926 − 2.76352i −4.32370 − 1.11014i 0
24.4 −1.60169 − 1.32504i −0.307726 + 2.43590i 0.434938 + 2.28002i 0 3.72054 − 3.49382i −0.457412 + 0.629574i 0.321599 − 0.584987i −2.93319 − 0.753115i 0
24.5 −1.36565 − 1.12976i −0.154256 + 1.22106i 0.213871 + 1.12115i 0 1.59017 − 1.49327i 0.550283 − 0.757399i −0.733147 + 1.33359i 1.43855 + 0.369356i 0
24.6 −1.05725 − 0.874630i 0.0217764 − 0.172378i −0.0219713 − 0.115178i 0 −0.173790 + 0.163200i −0.241629 + 0.332574i −1.39957 + 2.54580i 2.87651 + 0.738562i 0
24.7 −0.957548 − 0.792152i −0.299035 + 2.36710i −0.0853705 − 0.447528i 0 2.16144 − 2.02973i −1.90398 + 2.62061i −1.47015 + 2.67420i −2.60800 − 0.669621i 0
24.8 −0.739941 − 0.612133i 0.237679 − 1.88142i −0.201956 − 1.05869i 0 −1.32755 + 1.24665i −1.57358 + 2.16585i −1.42390 + 2.59007i −0.577510 − 0.148279i 0
24.9 −0.500422 − 0.413985i 0.204813 − 1.62126i −0.295724 − 1.55024i 0 −0.773671 + 0.726525i 2.46338 − 3.39055i −1.11955 + 2.03646i 0.319211 + 0.0819594i 0
24.10 −0.390792 − 0.323291i 0.382186 − 3.02532i −0.326561 − 1.71189i 0 −1.12741 + 1.05871i 0.745451 − 1.02603i −0.914499 + 1.66347i −6.10072 − 1.56640i 0
24.11 −0.0418508 − 0.0346220i 0.0279170 − 0.220986i −0.374210 − 1.96168i 0 −0.00881931 + 0.00828189i 2.46479 − 3.39249i −0.104590 + 0.190248i 2.85769 + 0.733731i 0
24.12 0.0418508 + 0.0346220i −0.0279170 + 0.220986i −0.374210 − 1.96168i 0 −0.00881931 + 0.00828189i −2.46479 + 3.39249i 0.104590 − 0.190248i 2.85769 + 0.733731i 0
24.13 0.390792 + 0.323291i −0.382186 + 3.02532i −0.326561 − 1.71189i 0 −1.12741 + 1.05871i −0.745451 + 1.02603i 0.914499 − 1.66347i −6.10072 − 1.56640i 0
24.14 0.500422 + 0.413985i −0.204813 + 1.62126i −0.295724 − 1.55024i 0 −0.773671 + 0.726525i −2.46338 + 3.39055i 1.11955 − 2.03646i 0.319211 + 0.0819594i 0
24.15 0.739941 + 0.612133i −0.237679 + 1.88142i −0.201956 − 1.05869i 0 −1.32755 + 1.24665i 1.57358 − 2.16585i 1.42390 − 2.59007i −0.577510 − 0.148279i 0
24.16 0.957548 + 0.792152i 0.299035 − 2.36710i −0.0853705 − 0.447528i 0 2.16144 − 2.02973i 1.90398 − 2.62061i 1.47015 − 2.67420i −2.60800 − 0.669621i 0
24.17 1.05725 + 0.874630i −0.0217764 + 0.172378i −0.0219713 − 0.115178i 0 −0.173790 + 0.163200i 0.241629 − 0.332574i 1.39957 − 2.54580i 2.87651 + 0.738562i 0
24.18 1.36565 + 1.12976i 0.154256 − 1.22106i 0.213871 + 1.12115i 0 1.59017 − 1.49327i −0.550283 + 0.757399i 0.733147 − 1.33359i 1.43855 + 0.369356i 0
24.19 1.60169 + 1.32504i 0.307726 − 2.43590i 0.434938 + 2.28002i 0 3.72054 − 3.49382i 0.457412 − 0.629574i −0.321599 + 0.584987i −2.93319 − 0.753115i 0
24.20 1.78418 + 1.47601i −0.342413 + 2.71048i 0.629958 + 3.30236i 0 −4.61161 + 4.33059i −1.04390 + 1.43680i −1.51926 + 2.76352i −4.32370 − 1.11014i 0
See next 80 embeddings (of 440 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
125.g even 25 1 inner
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.b 440
5.b even 2 1 inner 625.2.h.b 440
5.c odd 4 1 125.2.g.a ✓ 220
5.c odd 4 1 625.2.g.a 220
125.g even 25 1 inner 625.2.h.b 440
125.h even 50 1 inner 625.2.h.b 440
125.i odd 100 1 125.2.g.a ✓ 220
125.i odd 100 1 625.2.g.a 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.g.a ✓ 220 5.c odd 4 1
125.2.g.a ✓ 220 125.i odd 100 1
625.2.g.a 220 5.c odd 4 1
625.2.g.a 220 125.i odd 100 1
625.2.h.b 440 1.a even 1 1 trivial
625.2.h.b 440 5.b even 2 1 inner
625.2.h.b 440 125.g even 25 1 inner
625.2.h.b 440 125.h even 50 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{440} - 20 T_{2}^{438} + 200 T_{2}^{436} - 1235 T_{2}^{434} + 3860 T_{2}^{432} + \cdots + 75\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\). Copy content Toggle raw display