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Properties

Label 799.2.k.a
Level $799$
Weight $2$
Character orbit 799.k
Analytic conductor $6.380$
Analytic rank $0$
Dimension $704$
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] = [799,2,Mod(18,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.18");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 799.k (of order \(23\), degree \(22\), 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: \(6.38004712150\)
Analytic rank: \(0\)
Dimension: \(704\)
Relative dimension: \(32\) over \(\Q(\zeta_{23})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{23}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 704 q - 2 q^{3} - 32 q^{4} - 6 q^{5} - 10 q^{6} + 21 q^{7} - 24 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 704 q - 2 q^{3} - 32 q^{4} - 6 q^{5} - 10 q^{6} + 21 q^{7} - 24 q^{8} - 42 q^{9} - 36 q^{10} - 12 q^{11} - 40 q^{12} - 12 q^{13} - 30 q^{14} - 34 q^{15} - 48 q^{16} - 32 q^{17} - 6 q^{18} - 12 q^{19} - 26 q^{20} - 32 q^{21} - 28 q^{22} - 30 q^{23} - 24 q^{24} - 42 q^{25} - 46 q^{26} - 56 q^{27} - 14 q^{28} - 44 q^{29} + 136 q^{30} - 44 q^{31} - 48 q^{32} - 28 q^{33} - 46 q^{35} - 42 q^{36} - 24 q^{37} + 520 q^{38} - 11 q^{39} + 50 q^{40} + 31 q^{41} - 94 q^{42} - 35 q^{43} - 68 q^{44} - 110 q^{45} + 280 q^{46} - 17 q^{47} - 128 q^{48} - 59 q^{49} + 20 q^{50} - 2 q^{51} - 84 q^{52} - 11 q^{53} + 415 q^{54} - 17 q^{55} - 18 q^{56} - 37 q^{57} - 40 q^{58} - 54 q^{59} - 128 q^{60} - 66 q^{61} + 96 q^{62} - 74 q^{63} - 124 q^{64} - 52 q^{65} - 68 q^{66} - 56 q^{67} - 32 q^{68} + 4 q^{69} + 514 q^{70} - 42 q^{71} - 130 q^{72} - 62 q^{73} - 110 q^{74} - 80 q^{75} - 186 q^{76} + 450 q^{77} - 82 q^{78} - 64 q^{79} - 164 q^{80} + 18 q^{81} + 57 q^{82} - 56 q^{83} - 38 q^{84} - 6 q^{85} + 72 q^{86} - 42 q^{87} - 178 q^{88} - 56 q^{89} + 94 q^{90} + 110 q^{91} + 31 q^{92} + 290 q^{93} - 136 q^{94} - 96 q^{95} - 188 q^{96} - 74 q^{97} - 11 q^{98} + 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1 −1.56089 + 2.21128i −1.30677 − 1.06313i −1.78361 − 5.01859i 0.721752 + 0.772808i 4.39060 − 1.23019i 1.05655 + 2.03905i 8.66890 + 2.42891i −0.0329823 − 0.158720i −2.83547 + 0.389726i
18.2 −1.50481 + 2.13184i −1.01419 − 0.825101i −1.61050 − 4.53151i −0.406288 − 0.435028i 3.28514 − 0.920453i −1.81598 − 3.50468i 7.05859 + 1.97772i −0.262588 − 1.26364i 1.53880 − 0.211503i
18.3 −1.42103 + 2.01314i 1.87111 + 1.52226i −1.36366 − 3.83696i 0.934242 + 1.00033i −5.72345 + 1.60363i 1.95354 + 3.77015i 4.91659 + 1.37756i 0.573410 + 2.75940i −3.34139 + 0.459264i
18.4 −1.36007 + 1.92677i 0.621116 + 0.505315i −1.19292 − 3.35656i 2.71807 + 2.91034i −1.81839 + 0.509488i −1.83138 − 3.53441i 3.54781 + 0.994051i −0.479927 − 2.30953i −9.30432 + 1.27885i
18.5 −1.30803 + 1.85305i 1.51201 + 1.23011i −1.05311 − 2.96317i −2.49517 − 2.67167i −4.25720 + 1.19281i 0.641145 + 1.23735i 2.50022 + 0.700529i 0.162627 + 0.782605i 8.21450 − 1.12906i
18.6 −1.24786 + 1.76781i −0.239771 − 0.195069i −0.898244 − 2.52742i −2.22897 − 2.38665i 0.644044 − 0.180453i −1.58272 − 3.05451i 1.42163 + 0.398322i −0.590929 − 2.84371i 7.00057 − 0.962206i
18.7 −0.979101 + 1.38707i −1.80843 − 1.47127i −0.295565 − 0.831639i 0.978916 + 1.04816i 3.81138 − 1.06790i −0.105379 − 0.203372i −1.82680 − 0.511844i 0.495424 + 2.38411i −2.41233 + 0.331567i
18.8 −0.864465 + 1.22467i −0.241942 − 0.196834i −0.0827523 − 0.232843i −0.657374 − 0.703876i 0.450206 − 0.126142i 0.639471 + 1.23412i −2.53021 − 0.708930i −0.590576 − 2.84201i 1.43029 − 0.196589i
18.9 −0.809496 + 1.14679i −1.79403 − 1.45955i 0.00990521 + 0.0278706i −2.52562 − 2.70428i 3.12606 − 0.875881i 1.89488 + 3.65696i −2.74331 − 0.768639i 0.477881 + 2.29969i 5.14573 − 0.707264i
18.10 −0.680961 + 0.964702i −2.63033 − 2.13993i 0.202817 + 0.570673i −1.35606 − 1.45199i 3.85555 − 1.08027i −1.75570 − 3.38835i −2.96272 − 0.830116i 1.72896 + 8.32021i 2.32416 − 0.319448i
18.11 −0.668618 + 0.947216i −0.370094 − 0.301094i 0.219591 + 0.617871i 2.07133 + 2.21786i 0.532653 − 0.149242i 1.80830 + 3.48986i −2.96494 − 0.830738i −0.564056 − 2.71439i −3.48572 + 0.479101i
18.12 −0.661550 + 0.937202i 1.55234 + 1.26292i 0.229059 + 0.644510i 2.03438 + 2.17829i −2.21056 + 0.619371i 0.0996636 + 0.192342i −2.96483 − 0.830706i 0.204418 + 0.983715i −3.38734 + 0.465579i
18.13 −0.461477 + 0.653763i 2.60394 + 2.11846i 0.455313 + 1.28113i −0.797214 − 0.853607i −2.58663 + 0.724739i 1.39339 + 2.68913i −2.58878 − 0.725342i 1.68225 + 8.09542i 0.925953 − 0.127269i
18.14 −0.431020 + 0.610616i 0.675842 + 0.549838i 0.482686 + 1.35815i −0.342996 − 0.367259i −0.627041 + 0.175689i −1.72639 − 3.33179i −2.47675 − 0.693953i −0.455928 − 2.19405i 0.372092 − 0.0511429i
18.15 0.0129426 − 0.0183356i 2.11633 + 1.72176i 0.669591 + 1.88405i 0.327686 + 0.350866i 0.0589602 − 0.0165199i −0.746463 − 1.44061i 0.0864335 + 0.0242175i 0.904013 + 4.35035i 0.0106744 − 0.00146717i
18.16 0.0144505 − 0.0204717i −0.934755 − 0.760479i 0.669549 + 1.88393i −0.412144 − 0.441298i −0.0290759 + 0.00814669i −0.344803 − 0.665439i 0.0965001 + 0.0270381i −0.314931 − 1.51553i −0.0149898 + 0.00206030i
18.17 0.0188788 − 0.0267452i −0.614940 − 0.500291i 0.669400 + 1.88351i 2.30884 + 2.47216i −0.0249897 + 0.00700179i 1.03681 + 2.00096i 0.126058 + 0.0353199i −0.482508 − 2.32195i 0.109706 − 0.0150788i
18.18 0.156505 − 0.221717i −2.44274 − 1.98731i 0.645095 + 1.81512i 2.75658 + 2.95158i −0.822923 + 0.230572i −0.339943 − 0.656060i 1.02606 + 0.287487i 1.40718 + 6.77172i 1.08583 − 0.149245i
18.19 0.190036 − 0.269219i 1.34437 + 1.09373i 0.633394 + 1.78220i −2.08607 − 2.23364i 0.549930 − 0.154083i −0.166034 − 0.320431i 1.23480 + 0.345974i 0.000725599 0.00349177i −0.997767 + 0.137140i
18.20 0.464222 − 0.657653i −0.644232 − 0.524121i 0.452754 + 1.27393i 1.45649 + 1.55952i −0.643756 + 0.180372i −1.76207 − 3.40064i 2.59826 + 0.727998i −0.470037 − 2.26194i 1.70175 − 0.233901i
See next 80 embeddings (of 704 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.c even 23 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.2.k.a ✓ 704
47.c even 23 1 inner 799.2.k.a ✓ 704
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.k.a ✓ 704 1.a even 1 1 trivial
799.2.k.a ✓ 704 47.c even 23 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{704} + 48 T_{2}^{702} + 8 T_{2}^{701} + 1276 T_{2}^{700} + 432 T_{2}^{699} + \cdots + 58\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display