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Properties

Label 575.2.g.c
Level $575$
Weight $2$
Character orbit 575.g
Analytic conductor $4.591$
Analytic rank $0$
Dimension $112$
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] = [575,2,Mod(116,575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(575, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("575.116");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.g (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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + q^{2} + 4 q^{3} - 29 q^{4} - 2 q^{5} - 2 q^{6} - 36 q^{7} + 12 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + q^{2} + 4 q^{3} - 29 q^{4} - 2 q^{5} - 2 q^{6} - 36 q^{7} + 12 q^{8} - 22 q^{9} - 5 q^{10} + 4 q^{12} + 8 q^{13} + 18 q^{14} - 25 q^{15} - 19 q^{16} - 34 q^{18} + 6 q^{19} + 24 q^{20} - 6 q^{21} + 2 q^{22} + 28 q^{23} - 198 q^{24} - 6 q^{25} + 26 q^{26} + 13 q^{27} + 36 q^{28} - 6 q^{29} - 36 q^{30} - 18 q^{31} - 60 q^{32} + 15 q^{33} + 45 q^{34} + q^{35} + 27 q^{36} + 43 q^{37} + 14 q^{38} + 6 q^{39} - 30 q^{40} - 41 q^{41} + 20 q^{42} - 88 q^{43} + 91 q^{44} + 60 q^{45} - q^{46} + 11 q^{47} - 2 q^{48} + 36 q^{49} + 36 q^{50} + 26 q^{51} + 10 q^{52} + 14 q^{53} + 56 q^{54} - 51 q^{55} - 75 q^{56} - 72 q^{57} + 43 q^{58} - 2 q^{59} + 131 q^{60} - 3 q^{61} - q^{63} + 22 q^{64} - 13 q^{65} - 5 q^{66} + 40 q^{67} - 70 q^{68} - 4 q^{69} + 50 q^{70} + 57 q^{71} + 52 q^{72} - 4 q^{73} - 274 q^{74} + 4 q^{75} + 52 q^{76} + 18 q^{77} - 36 q^{78} + 2 q^{79} + 30 q^{80} - 128 q^{81} - 148 q^{82} + 22 q^{83} + 112 q^{84} + 77 q^{85} - 5 q^{86} + 33 q^{87} + 121 q^{88} + 69 q^{89} - 167 q^{90} - 64 q^{91} + 29 q^{92} - 92 q^{93} + 64 q^{94} + 6 q^{95} + 172 q^{96} + 68 q^{97} + 28 q^{98} - 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1 −2.10507 − 1.52942i −0.525564 − 1.61752i 1.47414 + 4.53694i 2.03096 − 0.935525i −1.36752 + 4.20879i −1.87561 2.22760 − 6.85584i 0.0869004 − 0.0631368i −5.70611 − 1.13685i
116.2 −2.09068 − 1.51897i −0.839892 − 2.58492i 1.44564 + 4.44922i −1.06680 − 1.96518i −2.17047 + 6.68000i 1.92555 2.13871 − 6.58228i −3.54935 + 2.57876i −0.754703 + 5.72899i
116.3 −1.95531 − 1.42062i 0.481235 + 1.48109i 1.18705 + 3.65338i −1.37069 + 1.76669i 1.16310 − 3.57964i −4.40282 1.37526 − 4.23262i 0.465011 − 0.337850i 5.18992 − 1.50721i
116.4 −1.84243 − 1.33861i 0.705243 + 2.17051i 0.984661 + 3.03048i 1.54657 + 1.61497i 1.60610 − 4.94307i −0.934388 0.834948 − 2.56970i −1.78671 + 1.29812i −0.687650 − 5.04572i
116.5 −1.77137 − 1.28697i −0.311045 − 0.957298i 0.863409 + 2.65730i 1.28105 + 1.83273i −0.681043 + 2.09603i −0.151634 0.537257 − 1.65351i 1.60738 − 1.16783i 0.0894567 − 4.89513i
116.6 −1.46816 − 1.06668i 0.480902 + 1.48007i 0.399652 + 1.23000i 0.146224 − 2.23128i 0.872716 − 2.68594i 3.67120 −0.396308 + 1.21971i 0.467724 − 0.339821i −2.59474 + 3.11990i
116.7 −1.42372 − 1.03439i −0.376304 − 1.15814i 0.338976 + 1.04326i −1.80368 + 1.32165i −0.662225 + 2.03812i 2.98715 −0.491090 + 1.51142i 1.22736 − 0.891729i 3.93503 − 0.0159427i
116.8 −1.12481 − 0.817223i −0.455123 − 1.40073i −0.0206870 − 0.0636682i 1.86324 − 1.23625i −0.632778 + 1.94749i 1.17187 −0.888042 + 2.73311i 0.672157 − 0.488350i −3.10609 − 0.132133i
116.9 −0.931389 − 0.676694i 0.983670 + 3.02742i −0.208463 − 0.641584i 2.23371 + 0.102624i 1.13246 − 3.48535i −1.84155 −0.951513 + 2.92846i −5.77064 + 4.19262i −2.01101 − 1.60712i
116.10 −0.914427 − 0.664370i −0.746599 − 2.29780i −0.223245 − 0.687077i 0.955397 + 2.02169i −0.843877 + 2.59718i −4.78732 −0.950892 + 2.92655i −2.29541 + 1.66771i 0.469507 − 2.48342i
116.11 −0.818592 − 0.594742i 0.161365 + 0.496629i −0.301659 − 0.928410i −1.99201 − 1.01582i 0.163274 − 0.502507i −1.44980 −0.930578 + 2.86402i 2.20645 − 1.60308i 1.02650 + 2.01627i
116.12 −0.560041 − 0.406893i 0.714176 + 2.19801i −0.469951 − 1.44636i 0.435906 − 2.19317i 0.494388 − 1.52157i −3.11552 −0.753156 + 2.31797i −1.89415 + 1.37618i −1.13651 + 1.05090i
116.13 0.0849340 + 0.0617082i −0.432193 − 1.33015i −0.614628 − 1.89163i −0.379623 + 2.20361i 0.0453735 − 0.139645i 1.91621 0.129410 − 0.398283i 0.844533 − 0.613589i −0.168224 + 0.163735i
116.14 0.148493 + 0.107886i 0.974599 + 2.99951i −0.607623 − 1.87007i −2.02372 − 0.951075i −0.178885 + 0.550552i 3.45702 0.224966 − 0.692375i −5.62016 + 4.08328i −0.197900 − 0.359560i
116.15 0.189361 + 0.137579i −1.05915 − 3.25972i −0.601104 − 1.85001i −2.19525 − 0.425283i 0.247908 − 0.762982i 0.351742 0.285356 − 0.878235i −7.07694 + 5.14170i −0.357186 − 0.382553i
116.16 0.271290 + 0.197104i 0.425255 + 1.30880i −0.583286 − 1.79517i 1.35705 + 1.77719i −0.142602 + 0.438884i 0.536907 0.402842 − 1.23982i 0.894933 − 0.650207i 0.0178620 + 0.749614i
116.17 0.306636 + 0.222784i −0.00716704 − 0.0220579i −0.573641 − 1.76549i 1.28487 − 1.83006i 0.00271647 − 0.00836044i −0.738278 0.451672 − 1.39010i 2.42662 − 1.76304i 0.801694 − 0.274913i
116.18 0.755402 + 0.548831i −0.515730 − 1.58725i −0.348618 − 1.07294i −2.20629 + 0.363739i 0.481552 − 1.48206i −2.77396 0.902590 − 2.77789i 0.173653 − 0.126166i −1.86626 − 0.936110i
116.19 0.957137 + 0.695401i −0.815610 − 2.51019i −0.185505 − 0.570924i 0.932165 − 2.03250i 0.964938 − 2.96977i 4.61296 0.950656 − 2.92582i −3.20878 + 2.33132i 2.30561 − 1.29716i
116.20 1.12082 + 0.814327i −0.282256 − 0.868694i −0.0249145 − 0.0766788i −0.832185 + 2.07544i 0.391042 − 1.20350i −3.98554 0.890751 − 2.74145i 1.75209 − 1.27297i −2.62282 + 1.64854i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.28
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 575.2.g.c ✓ 112
25.d even 5 1 inner 575.2.g.c ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
575.2.g.c ✓ 112 1.a even 1 1 trivial
575.2.g.c ✓ 112 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} - T_{2}^{111} + 43 T_{2}^{110} - 48 T_{2}^{109} + 1034 T_{2}^{108} - 1187 T_{2}^{107} + \cdots + 29241 \) acting on \(S_{2}^{\mathrm{new}}(575, [\chi])\). Copy content Toggle raw display