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Properties

Label 625.2.g.a
Level $625$
Weight $2$
Character orbit 625.g
Analytic conductor $4.991$
Analytic rank $0$
Dimension $220$
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(26,625)]
 
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([14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.g (of order \(25\), 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: \(220\)
Relative dimension: \(11\) over \(\Q(\zeta_{25})\)
Twist minimal: no (minimal twist has level 125)
Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220 q + 20 q^{2} + 20 q^{3} - 20 q^{4} - 20 q^{6} + 15 q^{7} + 5 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 220 q + 20 q^{2} + 20 q^{3} - 20 q^{4} - 20 q^{6} + 15 q^{7} + 5 q^{8} - 20 q^{9} - 15 q^{11} + 100 q^{12} + 20 q^{13} - 10 q^{14} + 15 q^{17} - 10 q^{18} - 10 q^{19} - 5 q^{21} - 25 q^{22} + 90 q^{23} + 15 q^{24} + 15 q^{26} - 10 q^{27} - 30 q^{28} - 10 q^{29} - 10 q^{31} - 20 q^{32} - 15 q^{33} + 5 q^{34} - 210 q^{36} - 20 q^{37} - 30 q^{38} - 5 q^{41} - 70 q^{42} - 25 q^{43} + 15 q^{44} + 100 q^{47} - 105 q^{48} - 20 q^{49} + 25 q^{51} - 85 q^{52} - 20 q^{53} + 60 q^{54} - 105 q^{56} - 40 q^{57} + 225 q^{58} + 5 q^{59} - 65 q^{62} - 85 q^{63} + 45 q^{64} + 55 q^{66} + 105 q^{67} - 80 q^{68} + 40 q^{69} - 85 q^{71} - 160 q^{72} - 40 q^{73} + 35 q^{74} + 35 q^{76} - 75 q^{77} + 10 q^{78} + 55 q^{81} + 95 q^{82} - 50 q^{83} + 115 q^{84} + 40 q^{86} - 95 q^{87} - 165 q^{88} + 55 q^{89} - 105 q^{91} - 155 q^{92} + 185 q^{93} + 60 q^{94} + 135 q^{96} - 65 q^{97} - 135 q^{98} + 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 −1.55688 − 1.88195i −0.171059 + 0.0216097i −0.743080 + 3.89536i 0 0.306986 + 0.288279i −0.159958 + 0.116216i 4.20707 − 2.31285i −2.87696 + 0.738677i 0
26.2 −1.47601 − 1.78418i 2.71048 − 0.342413i −0.629958 + 3.30236i 0 −4.61161 − 4.33059i −1.43680 + 1.04390i 2.76352 − 1.51926i 4.32370 − 1.11014i 0
26.3 −0.874630 − 1.05725i 0.172378 − 0.0217764i 0.0219713 − 0.115178i 0 −0.173790 − 0.163200i 0.332574 − 0.241629i −2.54580 + 1.39957i −2.87651 + 0.738562i 0
26.4 −0.792152 − 0.957548i −2.36710 + 0.299035i 0.0853705 − 0.447528i 0 2.16144 + 2.02973i 2.62061 − 1.90398i −2.67420 + 1.47015i 2.60800 − 0.669621i 0
26.5 −0.323291 − 0.390792i 3.02532 − 0.382186i 0.326561 − 1.71189i 0 −1.12741 − 1.05871i −1.02603 + 0.745451i −1.66347 + 0.914499i 6.10072 − 1.56640i 0
26.6 −0.0346220 − 0.0418508i 0.220986 − 0.0279170i 0.374210 − 1.96168i 0 −0.00881931 − 0.00828189i −3.39249 + 2.46479i −0.190248 + 0.104590i −2.85769 + 0.733731i 0
26.7 0.413985 + 0.500422i −1.62126 + 0.204813i 0.295724 − 1.55024i 0 −0.773671 − 0.726525i 3.39055 − 2.46338i 2.03646 − 1.11955i −0.319211 + 0.0819594i 0
26.8 0.612133 + 0.739941i −1.88142 + 0.237679i 0.201956 − 1.05869i 0 −1.32755 − 1.24665i −2.16585 + 1.57358i 2.59007 − 1.42390i 0.577510 − 0.148279i 0
26.9 1.12976 + 1.36565i 1.22106 − 0.154256i −0.213871 + 1.12115i 0 1.59017 + 1.49327i 0.757399 − 0.550283i 1.33359 − 0.733147i −1.43855 + 0.369356i 0
26.10 1.32504 + 1.60169i 2.43590 − 0.307726i −0.434938 + 2.28002i 0 3.72054 + 3.49382i −0.629574 + 0.457412i −0.584987 + 0.321599i 2.93319 − 0.753115i 0
26.11 1.75186 + 2.11763i −1.90129 + 0.240189i −1.04060 + 5.45502i 0 −3.83943 − 3.60546i −1.38440 + 1.00583i −8.55795 + 4.70477i 0.651466 − 0.167268i 0
51.1 −0.521661 + 2.73464i −2.18811 + 0.561810i −5.34659 − 2.11686i 0 −0.394900 − 6.27676i 0.406765 − 1.25189i 5.59453 − 8.81558i 1.84326 − 1.01334i 0
51.2 −0.434494 + 2.27770i 1.35096 − 0.346869i −3.13957 − 1.24304i 0 0.203076 + 3.22780i −0.594400 + 1.82938i 1.71048 − 2.69529i −0.924132 + 0.508046i 0
51.3 −0.307043 + 1.60957i −0.830194 + 0.213158i −0.636901 − 0.252167i 0 −0.0881880 − 1.40171i 1.52191 − 4.68395i −1.15457 + 1.81931i −1.98513 + 1.09134i 0
51.4 −0.256356 + 1.34387i 0.420177 − 0.107883i 0.119296 + 0.0472327i 0 0.0372655 + 0.592318i −1.05184 + 3.23722i −1.56018 + 2.45846i −2.46401 + 1.35460i 0
51.5 −0.122403 + 0.641658i 2.48816 − 0.638852i 1.46281 + 0.579167i 0 0.105366 + 1.67475i 1.12083 − 3.44955i −1.25071 + 1.97081i 3.15390 − 1.73387i 0
51.6 0.0692385 − 0.362961i −1.39738 + 0.358785i 1.73261 + 0.685987i 0 0.0334727 + 0.532034i −0.193835 + 0.596563i 0.764932 − 1.20534i −0.804989 + 0.442547i 0
51.7 0.0733210 − 0.384362i −0.539828 + 0.138604i 1.71719 + 0.679885i 0 0.0136935 + 0.217652i −0.613589 + 1.88843i 0.806560 − 1.27093i −2.35672 + 1.29562i 0
51.8 0.205875 − 1.07923i 1.61945 − 0.415805i 0.737195 + 0.291876i 0 −0.115346 − 1.83337i 0.665850 − 2.04927i 1.64419 − 2.59083i −0.179188 + 0.0985092i 0
51.9 0.346649 − 1.81720i 2.50111 − 0.642175i −1.32249 − 0.523612i 0 −0.299953 − 4.76762i −0.819982 + 2.52365i 0.572575 − 0.902233i 3.21422 − 1.76703i 0
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.g even 25 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{220} - 20 T_{2}^{219} + 210 T_{2}^{218} - 1535 T_{2}^{217} + 8750 T_{2}^{216} - 41302 T_{2}^{215} + \cdots + 870309001 \) acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\). Copy content Toggle raw display