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Properties

Label 800.2.by.a
Level $800$
Weight $2$
Character orbit 800.by
Analytic conductor $6.388$
Analytic rank $0$
Dimension $1888$
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] = [800,2,Mod(29,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([0, 15, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.by (of order \(40\), degree \(16\), 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(1888\)
Relative dimension: \(118\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1888 q - 20 q^{2} - 20 q^{3} - 12 q^{4} - 16 q^{5} - 12 q^{6} - 20 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1888 q - 20 q^{2} - 20 q^{3} - 12 q^{4} - 16 q^{5} - 12 q^{6} - 20 q^{8} - 12 q^{9} - 16 q^{10} - 12 q^{11} - 20 q^{12} - 20 q^{13} - 12 q^{14} - 12 q^{16} - 12 q^{19} - 16 q^{20} - 12 q^{21} - 20 q^{22} - 20 q^{23} - 32 q^{24} - 16 q^{25} - 72 q^{26} - 20 q^{27} - 20 q^{28} - 12 q^{29} + 32 q^{30} + 24 q^{31} - 40 q^{33} + 4 q^{34} - 40 q^{35} + 108 q^{36} - 20 q^{37} - 160 q^{38} - 12 q^{39} - 64 q^{40} - 12 q^{41} - 20 q^{42} - 12 q^{44} - 4 q^{45} - 12 q^{46} + 240 q^{48} + 24 q^{50} - 8 q^{51} - 20 q^{52} - 20 q^{53} - 12 q^{54} - 48 q^{55} - 12 q^{56} - 20 q^{58} - 12 q^{59} - 192 q^{60} - 12 q^{61} - 20 q^{62} - 40 q^{63} + 24 q^{64} - 32 q^{65} - 28 q^{66} - 20 q^{67} - 12 q^{69} - 100 q^{70} - 12 q^{71} - 20 q^{72} - 20 q^{73} - 120 q^{74} - 48 q^{75} - 32 q^{76} - 20 q^{77} - 20 q^{78} - 68 q^{80} - 20 q^{83} - 164 q^{84} + 4 q^{85} - 12 q^{86} - 20 q^{87} - 240 q^{88} - 12 q^{89} - 100 q^{90} - 12 q^{91} - 200 q^{92} - 92 q^{94} - 32 q^{95} - 132 q^{96} - 40 q^{97} + 20 q^{98} - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.41308 − 0.0565690i −1.52971 + 0.937406i 1.99360 + 0.159873i −0.0652890 − 2.23511i 2.21463 − 1.23810i −0.696707 − 0.696707i −2.80808 − 0.338690i 0.0993042 − 0.194895i −0.0341795 + 3.16209i
29.2 −1.40928 − 0.117997i −1.39511 + 0.854923i 1.97215 + 0.332581i 2.22929 − 0.173978i 2.06698 − 1.04021i 1.04264 + 1.04264i −2.74008 − 0.701408i −0.146539 + 0.287599i −3.16223 − 0.0178648i
29.3 −1.40690 − 0.143619i −0.0217502 + 0.0133285i 1.95875 + 0.404116i −2.19258 − 0.438868i 0.0325146 − 0.0156282i −2.35072 − 2.35072i −2.69773 − 0.849864i −1.36168 + 2.67244i 3.02171 + 0.932340i
29.4 −1.40510 − 0.160254i 1.58148 − 0.969131i 1.94864 + 0.450346i −2.03763 − 0.920911i −2.37745 + 1.10829i 0.0976139 + 0.0976139i −2.66587 − 0.945060i 0.199886 − 0.392299i 2.71550 + 1.62051i
29.5 −1.40044 − 0.196882i 2.92159 − 1.79036i 1.92248 + 0.551443i −0.511737 + 2.17672i −4.44401 + 1.93208i −3.24808 − 3.24808i −2.58375 − 1.15076i 3.96837 − 7.78837i 1.14522 − 2.94762i
29.6 −1.39719 + 0.218764i −2.40155 + 1.47167i 1.90428 − 0.611311i 1.65411 + 1.50463i 3.03347 − 2.58158i −2.79072 − 2.79072i −2.52692 + 1.27071i 2.23964 − 4.39554i −2.64027 − 1.74040i
29.7 −1.38876 + 0.267115i −2.12626 + 1.30297i 1.85730 − 0.741915i −0.836275 + 2.07380i 2.60481 − 2.37747i 1.42371 + 1.42371i −2.38116 + 1.52645i 1.46126 − 2.86788i 0.607442 − 3.10339i
29.8 −1.38382 + 0.291635i 1.59995 − 0.980452i 1.82990 − 0.807138i 0.933684 + 2.03181i −1.92811 + 1.82337i 1.37370 + 1.37370i −2.29685 + 1.65059i 0.236590 − 0.464333i −1.88459 − 2.53935i
29.9 −1.38153 − 0.302280i 0.705667 − 0.432433i 1.81725 + 0.835218i 1.79719 + 1.33046i −1.10562 + 0.384111i 1.81664 + 1.81664i −2.25812 − 1.70320i −1.05100 + 2.06271i −2.08070 − 2.38132i
29.10 −1.37427 − 0.333727i 1.87060 − 1.14630i 1.77725 + 0.917264i 0.517798 − 2.17529i −2.95327 + 0.951065i −0.777091 − 0.777091i −2.13631 − 1.85369i 0.823153 − 1.61553i −1.43755 + 2.81664i
29.11 −1.35772 + 0.395719i 2.48380 − 1.52207i 1.68681 − 1.07455i 1.47718 − 1.67867i −2.76999 + 3.04944i 1.86250 + 1.86250i −1.86500 + 2.12645i 2.49057 − 4.88802i −1.34131 + 2.86372i
29.12 −1.34885 + 0.424964i 0.572069 − 0.350564i 1.63881 − 1.14643i 2.23522 − 0.0614721i −0.622660 + 0.715968i −3.63393 − 3.63393i −1.72333 + 2.24280i −1.15760 + 2.27193i −2.98886 + 1.03281i
29.13 −1.34547 + 0.435550i −0.136161 + 0.0834397i 1.62059 − 1.17204i −1.24728 − 1.85588i 0.146859 − 0.171571i 3.30595 + 3.30595i −1.66998 + 2.28280i −1.35039 + 2.65030i 2.48651 + 1.95379i
29.14 −1.34476 − 0.437749i −0.978201 + 0.599442i 1.61675 + 1.17733i −1.50829 + 1.65078i 1.57785 − 0.377898i 3.29298 + 3.29298i −1.65876 − 2.29096i −0.764426 + 1.50027i 2.75091 − 1.55965i
29.15 −1.34111 + 0.448798i −0.571249 + 0.350062i 1.59716 − 1.20378i −1.89503 + 1.18696i 0.609002 − 0.725848i −1.65263 − 1.65263i −1.60172 + 2.33120i −1.15819 + 2.27307i 2.00874 − 2.44233i
29.16 −1.29636 + 0.565207i 1.62773 − 0.997476i 1.36108 − 1.46542i −1.94488 + 1.10337i −1.54634 + 2.21309i 0.804565 + 0.804565i −0.936182 + 2.66900i 0.292584 − 0.574229i 1.89763 − 2.52963i
29.17 −1.29613 − 0.565734i −2.48730 + 1.52422i 1.35989 + 1.46653i −0.493865 − 2.18085i 4.08615 − 0.568432i 1.37170 + 1.37170i −0.932929 − 2.67014i 2.50144 − 4.90935i −0.593667 + 3.10605i
29.18 −1.29303 − 0.572788i −2.38710 + 1.46282i 1.34383 + 1.48126i −2.16207 + 0.570478i 3.92446 − 0.524158i −2.13142 − 2.13142i −0.889158 − 2.68503i 2.19644 − 4.31076i 3.12238 + 0.500765i
29.19 −1.28971 − 0.580204i 0.137225 − 0.0840917i 1.32673 + 1.49659i 0.123822 + 2.23264i −0.225772 + 0.0288357i −1.72777 − 1.72777i −0.842768 − 2.69995i −1.35021 + 2.64994i 1.13569 − 2.95131i
29.20 −1.26008 − 0.642024i 1.02229 − 0.626459i 1.17561 + 1.61800i 2.21847 − 0.279952i −1.69037 + 0.133056i −1.67907 − 1.67907i −0.442568 − 2.79359i −0.709349 + 1.39218i −2.97519 − 1.07155i
See next 80 embeddings (of 1888 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.118
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner
32.g even 8 1 inner
800.by even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.by.a ✓ 1888
25.e even 10 1 inner 800.2.by.a ✓ 1888
32.g even 8 1 inner 800.2.by.a ✓ 1888
800.by even 40 1 inner 800.2.by.a ✓ 1888
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.by.a ✓ 1888 1.a even 1 1 trivial
800.2.by.a ✓ 1888 25.e even 10 1 inner
800.2.by.a ✓ 1888 32.g even 8 1 inner
800.2.by.a ✓ 1888 800.by even 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(800, [\chi])\).