Properties

Label 475.2.h.b
Level $475$
Weight $2$
Character orbit 475.h
Analytic conductor $3.793$
Analytic rank $0$
Dimension $100$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(96,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.96");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.h (of order \(5\), degree \(4\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(25\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q - q^{2} - 2 q^{3} - 29 q^{4} + q^{5} - 6 q^{6} - 10 q^{7} - 3 q^{8} - 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q - q^{2} - 2 q^{3} - 29 q^{4} + q^{5} - 6 q^{6} - 10 q^{7} - 3 q^{8} - 37 q^{9} + 23 q^{10} - 5 q^{11} + 43 q^{12} - 8 q^{13} - 10 q^{14} - 12 q^{15} - 25 q^{16} + 4 q^{17} + 20 q^{18} - 25 q^{19} + 4 q^{20} - 16 q^{21} - 40 q^{22} + 19 q^{23} + 72 q^{24} + 7 q^{25} + 74 q^{26} - 29 q^{27} - 22 q^{28} - 34 q^{29} - 20 q^{30} - 8 q^{31} - 22 q^{32} + 31 q^{33} - 9 q^{34} - 3 q^{35} - 53 q^{36} + 24 q^{37} - q^{38} - 30 q^{39} - 47 q^{40} - 23 q^{41} + 106 q^{42} - 46 q^{43} - 64 q^{44} - 60 q^{45} - 30 q^{46} - 30 q^{47} + 99 q^{48} + 142 q^{49} + 12 q^{50} + 36 q^{51} - 37 q^{52} - 13 q^{53} + 15 q^{54} - 65 q^{55} - 48 q^{56} - 2 q^{57} - 26 q^{58} - 19 q^{59} + 244 q^{60} - 68 q^{61} + 54 q^{62} - 42 q^{63} - 29 q^{64} - 16 q^{65} + 17 q^{66} + 23 q^{67} - 142 q^{68} - 12 q^{69} - 48 q^{70} - 38 q^{71} - 143 q^{72} + 12 q^{73} + 172 q^{74} + 81 q^{75} + 116 q^{76} - 50 q^{77} + 65 q^{78} - 16 q^{79} - 68 q^{80} - 66 q^{81} - 22 q^{82} + 52 q^{83} - 46 q^{84} + 73 q^{85} - 64 q^{86} + 50 q^{87} + 32 q^{88} - q^{89} - 149 q^{90} - 16 q^{91} + 35 q^{92} - 94 q^{93} + 38 q^{94} - 4 q^{95} - 50 q^{96} + 6 q^{97} - 79 q^{98} + 108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
96.1 −0.863118 + 2.65640i 1.40079 + 1.01774i −4.69347 3.41001i −0.827159 2.07745i −3.91257 + 2.84265i −1.84253 8.59003 6.24102i −0.000614660 0.00189173i 6.23249 0.404182i
96.2 −0.772517 + 2.37756i −1.20343 0.874341i −3.43799 2.49784i 2.02265 0.953348i 3.00847 2.18578i −0.767667 4.54973 3.30557i −0.243286 0.748756i 0.704111 + 5.54546i
96.3 −0.752126 + 2.31480i −0.941470 0.684018i −3.17459 2.30648i 1.37508 + 1.76328i 2.29147 1.66485i 1.65826 3.78856 2.75255i −0.508566 1.56520i −5.11588 + 1.85682i
96.4 −0.646124 + 1.98857i −2.59805 1.88759i −1.91889 1.39415i −1.25953 + 1.84759i 5.43227 3.94678i 3.14260 0.629051 0.457032i 2.25980 + 6.95496i −2.86024 3.69843i
96.5 −0.634072 + 1.95147i 1.87612 + 1.36308i −1.78817 1.29918i −1.84159 + 1.26828i −3.84962 + 2.79691i −1.52192 0.349094 0.253632i 0.734790 + 2.26145i −1.30732 4.39800i
96.6 −0.587814 + 1.80910i 2.71967 + 1.97596i −1.30930 0.951263i 2.23564 + 0.0439146i −5.17338 + 3.75868i −1.25099 −0.587274 + 0.426680i 2.56516 + 7.89476i −1.39358 + 4.01869i
96.7 −0.493239 + 1.51803i 0.186082 + 0.135196i −0.443103 0.321933i 0.773173 2.09814i −0.297015 + 0.215794i −4.71192 −1.87537 + 1.36254i −0.910703 2.80285i 2.80369 + 2.20859i
96.8 −0.369958 + 1.13861i 0.747313 + 0.542954i 0.458460 + 0.333091i 2.01735 + 0.964530i −0.894690 + 0.650031i 1.71390 −2.48600 + 1.80618i −0.663374 2.04166i −1.84456 + 1.94014i
96.9 −0.341598 + 1.05133i −2.01270 1.46231i 0.629426 + 0.457304i −2.06527 0.857119i 2.22491 1.61649i 0.994577 −2.48442 + 1.80504i 0.985556 + 3.03323i 1.60661 1.87849i
96.10 −0.261155 + 0.803753i 2.09467 + 1.52187i 1.04022 + 0.755762i −1.94820 1.09750i −1.77024 + 1.28616i 3.94187 −2.24653 + 1.63220i 1.14452 + 3.52246i 1.39090 1.27926i
96.11 −0.178800 + 0.550289i −1.40972 1.02422i 1.34719 + 0.978787i −1.32915 + 1.79815i 0.815673 0.592621i −2.88374 −1.71570 + 1.24653i 0.0112241 + 0.0345441i −0.751852 1.05293i
96.12 −0.0113872 + 0.0350462i 0.448582 + 0.325914i 1.61694 + 1.17477i −2.20559 0.367944i −0.0165301 + 0.0120098i −4.38426 −0.119208 + 0.0866095i −0.832045 2.56077i 0.0380105 0.0731075i
96.13 0.0528673 0.162709i −1.93493 1.40581i 1.59435 + 1.15837i 1.36930 + 1.76777i −0.331033 + 0.240509i −0.881017 0.549582 0.399295i 0.840612 + 2.58714i 0.360023 0.129341i
96.14 0.137823 0.424175i 1.13004 + 0.821019i 1.45711 + 1.05865i 0.932667 + 2.03227i 0.504000 0.366177i 2.05496 1.37152 0.996470i −0.324143 0.997609i 0.990581 0.115520i
96.15 0.193750 0.596301i −2.19174 1.59239i 1.30000 + 0.944504i 1.06522 1.96604i −1.37419 + 0.998411i 3.99247 1.82957 1.32926i 1.34096 + 4.12706i −0.965962 1.01611i
96.16 0.282940 0.870798i 2.53514 + 1.84188i 0.939799 + 0.682804i −0.358646 2.20712i 2.32120 1.68645i −2.52925 2.34198 1.70155i 2.10733 + 6.48568i −2.02343 0.312173i
96.17 0.287240 0.884034i 0.765559 + 0.556211i 0.919025 + 0.667710i 0.547190 2.16808i 0.711609 0.517014i 1.70926 2.35827 1.71338i −0.650342 2.00155i −1.75948 1.10649i
96.18 0.448526 1.38042i −1.35592 0.985131i −0.0863536 0.0627396i −1.86087 + 1.23983i −1.96806 + 1.42988i 3.51266 2.22317 1.61523i −0.0590237 0.181656i 0.876840 + 3.12488i
96.19 0.468232 1.44107i −0.519720 0.377598i −0.239411 0.173942i 1.99648 1.00701i −0.787496 + 0.572149i −4.33676 2.08893 1.51770i −0.799523 2.46068i −0.516351 3.34859i
96.20 0.509658 1.56857i 0.899154 + 0.653274i −0.582611 0.423292i −0.0865774 + 2.23439i 1.48296 1.07744i 0.691992 1.70771 1.24072i −0.545340 1.67838i 3.46066 + 1.27458i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 96.25
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.h.b 100
25.d even 5 1 inner 475.2.h.b 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.h.b 100 1.a even 1 1 trivial
475.2.h.b 100 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{100} + T_{2}^{99} + 40 T_{2}^{98} + 42 T_{2}^{97} + 902 T_{2}^{96} + 949 T_{2}^{95} + 15105 T_{2}^{94} + 15750 T_{2}^{93} + 209951 T_{2}^{92} + 216105 T_{2}^{91} + 2517136 T_{2}^{90} + 2548686 T_{2}^{89} + \cdots + 4000000 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display