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Properties

Label 722.2.i.a
Level $722$
Weight $2$
Character orbit 722.i
Analytic conductor $5.765$
Analytic rank $0$
Dimension $576$
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] = [722,2,Mod(7,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(114))
 
chi = DirichletCharacter(H, H._module([50]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 722.i (of order \(57\), degree \(36\), 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: \(5.76519902594\)
Analytic rank: \(0\)
Dimension: \(576\)
Relative dimension: \(16\) over \(\Q(\zeta_{57})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{57}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 576 q - 16 q^{2} + q^{3} + 16 q^{4} - q^{5} - q^{6} + 10 q^{7} + 32 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q - 16 q^{2} + q^{3} + 16 q^{4} - q^{5} - q^{6} + 10 q^{7} + 32 q^{8} + 13 q^{9} + q^{10} + 34 q^{11} - 2 q^{12} + 2 q^{13} - 14 q^{14} + 38 q^{15} + 16 q^{16} + 12 q^{17} + 26 q^{18} + 7 q^{19} + 2 q^{20} + 75 q^{21} + 36 q^{22} - 45 q^{23} - q^{24} + 9 q^{25} + 4 q^{26} - 110 q^{27} + 33 q^{28} - 36 q^{29} - 57 q^{30} - 4 q^{31} - 16 q^{32} + 81 q^{33} + 7 q^{34} - 2 q^{35} + 13 q^{36} + 14 q^{37} - 30 q^{38} + 40 q^{39} + q^{40} + 11 q^{41} + q^{42} - 9 q^{43} + 2 q^{44} - 85 q^{45} - 33 q^{46} - 213 q^{47} + q^{48} - 76 q^{49} - 20 q^{50} - 3 q^{51} + 2 q^{52} - 144 q^{53} + 78 q^{54} - 6 q^{55} - 29 q^{56} - 8 q^{57} + 4 q^{58} - 5 q^{59} + 69 q^{61} - 2 q^{62} - 95 q^{63} - 32 q^{64} - 84 q^{65} - 5 q^{66} - 5 q^{67} - 62 q^{68} + 2 q^{69} + 211 q^{70} + 129 q^{71} - 13 q^{72} - 8 q^{73} - 31 q^{74} - 36 q^{75} - 18 q^{76} - 10 q^{77} + q^{78} - 2 q^{79} - q^{80} - 64 q^{81} - 49 q^{82} - 72 q^{83} + 2 q^{84} - 6 q^{85} - 10 q^{86} - 81 q^{87} + 4 q^{88} - 7 q^{89} + 81 q^{90} + 35 q^{91} - 7 q^{92} - 80 q^{93} - 27 q^{94} + 285 q^{95} + 2 q^{96} + 38 q^{97} - 171 q^{98} + 84 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 0.191711 + 0.981451i −3.02839 + 0.335182i −0.926494 + 0.376309i 0.284542 − 0.276806i −0.909540 − 2.90796i −0.809355 + 0.629946i −0.546948 − 0.837166i 6.13143 − 1.37409i 0.326222 + 0.226197i
7.2 0.191711 + 0.981451i −2.99885 + 0.331913i −0.926494 + 0.376309i 0.116249 − 0.113089i −0.900668 − 2.87959i 1.28638 − 1.00123i −0.546948 − 0.837166i 5.95555 − 1.33467i 0.133278 + 0.0924128i
7.3 0.191711 + 0.981451i −2.14695 + 0.237624i −0.926494 + 0.376309i −2.05631 + 2.00041i −0.644809 − 2.06157i −0.971052 + 0.755800i −0.546948 − 0.837166i 1.62552 − 0.364288i −2.35752 − 1.63467i
7.4 0.191711 + 0.981451i −1.75789 + 0.194563i −0.926494 + 0.376309i 3.09134 − 3.00731i −0.527961 − 1.68798i −2.69413 + 2.09693i −0.546948 − 0.837166i 0.124936 − 0.0279988i 3.54417 + 2.45747i
7.5 0.191711 + 0.981451i −1.38274 + 0.153041i −0.926494 + 0.376309i −0.998374 + 0.971233i −0.415288 − 1.32775i −1.81335 + 1.41139i −0.546948 − 0.837166i −1.03884 + 0.232811i −1.14462 − 0.793660i
7.6 0.191711 + 0.981451i −1.37418 + 0.152095i −0.926494 + 0.376309i 1.50204 − 1.46121i −0.412719 − 1.31954i 3.06618 − 2.38650i −0.546948 − 0.837166i −1.06214 + 0.238031i 1.72206 + 1.19405i
7.7 0.191711 + 0.981451i −0.850977 + 0.0941861i −0.926494 + 0.376309i 0.830084 − 0.807518i −0.255580 − 0.817136i 1.16599 − 0.907530i −0.546948 − 0.837166i −2.21210 + 0.495743i 0.951676 + 0.659878i
7.8 0.191711 + 0.981451i −0.0242700 + 0.00268620i −0.926494 + 0.376309i −0.488894 + 0.475604i −0.00728919 − 0.0233048i 2.64759 − 2.06070i −0.546948 − 0.837166i −2.92681 + 0.655913i −0.560508 − 0.388648i
7.9 0.191711 + 0.981451i 0.399618 − 0.0442297i −0.926494 + 0.376309i −2.76873 + 2.69346i 0.120020 + 0.383726i 1.62426 − 1.26421i −0.546948 − 0.837166i −2.76965 + 0.620693i −3.17429 − 2.20101i
7.10 0.191711 + 0.981451i 1.40541 − 0.155550i −0.926494 + 0.376309i 2.84434 − 2.76702i 0.422096 + 1.34952i 0.725554 − 0.564722i −0.546948 − 0.837166i −0.976419 + 0.218821i 3.26099 + 2.26112i
7.11 0.191711 + 0.981451i 1.46818 − 0.162498i −0.926494 + 0.376309i 0.343366 − 0.334031i 0.440950 + 1.40980i −2.66000 + 2.07036i −0.546948 − 0.837166i −0.798239 + 0.178890i 0.393662 + 0.272959i
7.12 0.191711 + 0.981451i 1.52731 − 0.169043i −0.926494 + 0.376309i −0.766833 + 0.745986i 0.458709 + 1.46657i −1.82182 + 1.41798i −0.546948 − 0.837166i −0.623291 + 0.139683i −0.879159 − 0.609596i
7.13 0.191711 + 0.981451i 1.70900 − 0.189152i −0.926494 + 0.376309i 0.366238 − 0.356282i 0.513276 + 1.64103i 1.51668 − 1.18048i −0.546948 − 0.837166i −0.0424980 + 0.00952402i 0.419886 + 0.291142i
7.14 0.191711 + 0.981451i 2.10054 − 0.232488i −0.926494 + 0.376309i −2.78638 + 2.71063i 0.630871 + 2.01701i −3.43214 + 2.67134i −0.546948 − 0.837166i 1.43082 − 0.320655i −3.19453 − 2.21504i
7.15 0.191711 + 0.981451i 2.88188 − 0.318967i −0.926494 + 0.376309i −1.48049 + 1.44024i 0.865538 + 2.76728i 2.02514 − 1.57623i −0.546948 − 0.837166i 5.27611 − 1.18240i −1.69735 − 1.17692i
7.16 0.191711 + 0.981451i 3.06625 − 0.339372i −0.926494 + 0.376309i 1.25101 − 1.21700i 0.920910 + 2.94431i 0.0242932 − 0.0189081i −0.546948 − 0.837166i 6.35932 − 1.42516i 1.43426 + 0.994492i
11.1 0.592235 + 0.805765i −0.415715 − 2.99792i −0.298515 + 0.954405i 0.199286 + 0.138182i 2.16942 − 2.11044i −1.45470 + 1.58023i −0.945817 + 0.324699i −5.92791 + 1.67625i 0.00668209 + 0.242414i
11.2 0.592235 + 0.805765i −0.350802 − 2.52980i −0.298515 + 0.954405i 1.32680 + 0.919980i 1.83067 − 1.78090i 0.609129 − 0.661690i −0.945817 + 0.324699i −3.39004 + 0.958612i 0.0444877 + 1.61393i
11.3 0.592235 + 0.805765i −0.317278 − 2.28804i −0.298515 + 0.954405i −3.43417 − 2.38120i 1.65572 − 1.61071i −1.04262 + 1.13258i −0.945817 + 0.324699i −2.24766 + 0.635578i −0.115148 − 4.17737i
11.4 0.592235 + 0.805765i −0.271767 − 1.95984i −0.298515 + 0.954405i 3.51033 + 2.43401i 1.41822 − 1.37967i 1.70655 − 1.85381i −0.945817 + 0.324699i −0.880305 + 0.248926i 0.117702 + 4.27001i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
361.i even 57 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.2.i.a ✓ 576
361.i even 57 1 inner 722.2.i.a ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.i.a ✓ 576 1.a even 1 1 trivial
722.2.i.a ✓ 576 361.i even 57 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{576} - T_{3}^{575} - 30 T_{3}^{574} + 71 T_{3}^{573} + 324 T_{3}^{572} - 1359 T_{3}^{571} + \cdots + 29\!\cdots\!09 \) acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\). Copy content Toggle raw display