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Properties

Label 800.2.ca.a
Level $800$
Weight $2$
Character orbit 800.ca
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(21,800)]
 
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, 25, 24]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.21");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.ca (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.

\(\operatorname{Tr}(f)(q) = \) \( 1888 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 16 q^{5} - 12 q^{6} - 32 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1888 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 16 q^{5} - 12 q^{6} - 32 q^{7} - 12 q^{8} - 12 q^{9} - 16 q^{10} - 12 q^{11} - 44 q^{12} - 12 q^{13} - 12 q^{14} - 12 q^{16} - 32 q^{18} - 12 q^{19} - 16 q^{20} - 12 q^{21} - 44 q^{22} - 12 q^{23} - 32 q^{24} - 16 q^{25} + 8 q^{26} - 12 q^{27} - 12 q^{28} - 12 q^{29} + 16 q^{30} - 72 q^{31} - 32 q^{32} - 24 q^{33} - 28 q^{34} - 40 q^{35} - 132 q^{36} - 12 q^{37} + 72 q^{38} - 12 q^{39} - 128 q^{40} - 12 q^{41} - 12 q^{42} - 32 q^{43} - 12 q^{44} - 28 q^{45} - 12 q^{46} + 144 q^{48} - 56 q^{50} - 56 q^{51} - 12 q^{52} - 12 q^{53} - 12 q^{54} + 16 q^{55} - 12 q^{56} - 32 q^{57} - 76 q^{58} - 12 q^{59} + 128 q^{60} - 12 q^{61} + 4 q^{62} - 24 q^{63} + 24 q^{64} - 32 q^{65} + 4 q^{66} - 12 q^{67} - 120 q^{68} - 12 q^{69} - 4 q^{70} - 12 q^{71} - 132 q^{72} - 12 q^{73} - 120 q^{74} + 16 q^{75} - 32 q^{76} - 12 q^{77} - 164 q^{78} - 44 q^{80} - 32 q^{82} - 12 q^{83} - 164 q^{84} - 36 q^{85} - 12 q^{86} - 124 q^{87} + 120 q^{88} - 12 q^{89} - 100 q^{90} - 12 q^{91} + 248 q^{92} - 8 q^{93} - 44 q^{94} - 32 q^{95} + 108 q^{96} - 24 q^{97} - 12 q^{98} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −1.41246 + 0.0703721i −0.663470 − 0.159285i 1.99010 − 0.198796i 1.68790 + 1.46663i 0.948335 + 0.178294i 2.81823 + 2.81823i −2.79694 + 0.420839i −2.25820 − 1.15061i −2.48730 − 1.95278i
21.2 −1.41240 − 0.0716844i −2.02927 − 0.487184i 1.98972 + 0.202493i −2.11048 − 0.738835i 2.83121 + 0.833564i −2.16002 − 2.16002i −2.79576 − 0.428633i 1.20757 + 0.615285i 2.92787 + 1.19482i
21.3 −1.41114 − 0.0932563i 0.00445115 + 0.00106863i 1.98261 + 0.263195i −2.10724 − 0.748023i −0.00618152 − 0.00192308i 1.71432 + 1.71432i −2.77318 − 0.556294i −2.67300 − 1.36196i 2.90384 + 1.25208i
21.4 −1.41002 + 0.108889i 1.55817 + 0.374083i 1.97629 − 0.307070i −0.991743 + 2.00411i −2.23777 − 0.357795i −1.92598 − 1.92598i −2.75316 + 0.648170i −0.385075 − 0.196205i 1.18015 − 2.93381i
21.5 −1.40599 + 0.152286i −1.36880 − 0.328619i 1.95362 − 0.428226i 1.73884 − 1.40586i 1.97456 + 0.253586i 0.432853 + 0.432853i −2.68155 + 0.899591i −0.907404 − 0.462345i −2.23070 + 2.24142i
21.6 −1.40084 + 0.194008i 1.01115 + 0.242755i 1.92472 − 0.543549i 0.684637 − 2.12868i −1.46355 − 0.143891i −0.952020 − 0.952020i −2.59078 + 1.13484i −1.70953 − 0.871051i −0.546088 + 3.11477i
21.7 −1.39798 + 0.213663i −3.06997 − 0.737035i 1.90870 − 0.597393i 0.215509 − 2.22566i 4.44924 + 0.374421i −1.21268 − 1.21268i −2.54068 + 1.24296i 6.20849 + 3.16338i 0.174264 + 3.15747i
21.8 −1.38912 − 0.265206i 1.87437 + 0.449995i 1.85933 + 0.736809i 2.20743 − 0.356742i −2.48439 − 1.12219i 1.62986 + 1.62986i −2.38744 − 1.51663i 0.637733 + 0.324941i −3.16100 − 0.0898645i
21.9 −1.38186 − 0.300748i 2.92815 + 0.702987i 1.81910 + 0.831186i −2.23551 − 0.0498622i −3.83489 − 1.85207i −1.02313 − 1.02313i −2.26377 − 1.69568i 5.40687 + 2.75494i 3.07418 + 0.741229i
21.10 −1.36415 + 0.372946i −0.424266 − 0.101857i 1.72182 − 1.01751i 1.75032 + 1.39154i 0.616751 − 0.0192796i −3.11082 − 3.11082i −1.96935 + 2.03019i −2.50339 − 1.27554i −2.90667 − 1.24549i
21.11 −1.33864 − 0.456130i −3.16571 − 0.760021i 1.58389 + 1.22118i −1.45493 + 1.69799i 3.89107 + 2.46137i 2.94421 + 2.94421i −1.56323 − 2.35718i 6.77109 + 3.45004i 2.72213 − 1.60935i
21.12 −1.32965 + 0.481693i 2.67877 + 0.643117i 1.53594 − 1.28097i −0.396184 − 2.20069i −3.87162 + 0.435225i 2.82138 + 2.82138i −1.42524 + 2.44309i 4.08922 + 2.08356i 1.58684 + 2.73531i
21.13 −1.32713 + 0.488591i 3.16962 + 0.760959i 1.52256 − 1.29685i 1.46341 + 1.69069i −4.57830 + 0.538755i 0.0527159 + 0.0527159i −1.38701 + 2.46500i 6.79442 + 3.46193i −2.76819 − 1.52876i
21.14 −1.32225 − 0.501649i −2.36642 − 0.568126i 1.49670 + 1.32661i 1.84169 + 1.26814i 2.84400 + 1.93832i −1.22414 − 1.22414i −1.31351 − 2.50493i 2.60414 + 1.32687i −1.79901 − 2.60068i
21.15 −1.30743 − 0.539091i 0.433170 + 0.103995i 1.41876 + 1.40965i −0.931156 − 2.03297i −0.510278 − 0.369485i −0.0748549 − 0.0748549i −1.09500 − 2.60787i −2.49620 − 1.27188i 0.121469 + 3.15994i
21.16 −1.29354 − 0.571620i 1.88839 + 0.453362i 1.34650 + 1.47883i 2.16256 + 0.568624i −2.18356 − 1.66588i −3.04877 − 3.04877i −0.896427 − 2.68261i 0.687459 + 0.350278i −2.47233 − 1.97170i
21.17 −1.29199 − 0.575112i 0.357446 + 0.0858152i 1.33849 + 1.48608i 0.142570 + 2.23152i −0.412464 − 0.316444i 0.327205 + 0.327205i −0.874660 − 2.68979i −2.55262 − 1.30062i 1.09917 − 2.96510i
21.18 −1.27944 + 0.602522i 0.613353 + 0.147253i 1.27394 − 1.54178i −1.49087 + 1.66653i −0.873471 + 0.181157i 0.193755 + 0.193755i −0.700967 + 2.74019i −2.31850 − 1.18134i 0.903361 − 3.03050i
21.19 −1.27123 − 0.619656i −2.07113 − 0.497235i 1.23205 + 1.57545i 0.381308 − 2.20332i 2.32477 + 1.91549i 3.44384 + 3.44384i −0.589988 − 2.76621i 1.36934 + 0.697712i −1.85003 + 2.56464i
21.20 −1.24725 + 0.666605i −1.85623 − 0.445642i 1.11127 − 1.66285i −2.21861 + 0.278883i 2.61225 − 0.681546i 1.32006 + 1.32006i −0.277574 + 2.81477i 0.573978 + 0.292456i 2.58126 − 1.82677i
See next 80 embeddings (of 1888 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.118
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner
32.g even 8 1 inner
800.ca 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.ca.a ✓ 1888
25.d even 5 1 inner 800.2.ca.a ✓ 1888
32.g even 8 1 inner 800.2.ca.a ✓ 1888
800.ca even 40 1 inner 800.2.ca.a ✓ 1888
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.ca.a ✓ 1888 1.a even 1 1 trivial
800.2.ca.a ✓ 1888 25.d even 5 1 inner
800.2.ca.a ✓ 1888 32.g even 8 1 inner
800.2.ca.a ✓ 1888 800.ca even 40 1 inner

Hecke kernels

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