Properties

Label 775.2.j.c
Level $775$
Weight $2$
Character orbit 775.j
Analytic conductor $6.188$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(156,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.156");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.j (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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + q^{2} + 6 q^{3} - 43 q^{4} + 2 q^{5} + 6 q^{6} - 10 q^{7} + 6 q^{8} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + q^{2} + 6 q^{3} - 43 q^{4} + 2 q^{5} + 6 q^{6} - 10 q^{7} + 6 q^{8} - 62 q^{9} - 16 q^{10} - 4 q^{11} - 16 q^{12} + 6 q^{13} - 20 q^{14} + 39 q^{15} - 45 q^{16} - 4 q^{17} - 66 q^{18} + 4 q^{19} + 7 q^{20} - 20 q^{21} - 24 q^{22} + 5 q^{23} + 58 q^{24} + 2 q^{25} + 22 q^{26} + 9 q^{27} - 84 q^{28} - 31 q^{29} - 30 q^{30} - 40 q^{31} + 48 q^{32} + 33 q^{33} - 15 q^{34} - 10 q^{35} - 89 q^{36} + 29 q^{37} + 64 q^{38} - 50 q^{39} + 79 q^{40} - 43 q^{41} + 10 q^{42} - 54 q^{43} - 66 q^{44} - 46 q^{45} - 34 q^{46} - 10 q^{47} - 83 q^{48} + 234 q^{49} - 12 q^{50} + 84 q^{51} + 44 q^{52} - 32 q^{53} + 55 q^{54} - 83 q^{55} + 6 q^{56} - 42 q^{57} + 112 q^{58} - 53 q^{59} - 129 q^{60} - 23 q^{61} + q^{62} + 63 q^{63} + 40 q^{64} + 14 q^{65} - 44 q^{66} + 101 q^{67} - 30 q^{68} - 2 q^{69} - 123 q^{70} - 35 q^{71} - 32 q^{72} - 38 q^{73} + 88 q^{74} + 112 q^{75} + 90 q^{76} + 39 q^{77} - 147 q^{78} - 5 q^{79} + 47 q^{80} - 36 q^{81} - 74 q^{82} + 42 q^{83} - 34 q^{84} - 6 q^{85} - 22 q^{86} - 86 q^{87} + 142 q^{88} - 34 q^{89} + 301 q^{90} + 16 q^{92} - 4 q^{93} + 56 q^{94} - 96 q^{95} - 56 q^{96} - 19 q^{97} + 3 q^{98} + 224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
156.1 −0.848625 2.61180i 2.44944 1.77962i −4.48329 + 3.25730i 2.19781 + 0.411887i −6.72666 4.88720i 1.14447 7.86859 + 5.71687i 1.90564 5.86496i −0.789347 6.08976i
156.2 −0.823075 2.53316i −1.25286 + 0.910254i −4.12144 + 2.99440i −2.23433 + 0.0880920i 3.33702 + 2.42449i −0.296470 6.66788 + 4.84450i −0.185963 + 0.572335i 2.06217 + 5.58742i
156.3 −0.801339 2.46627i 0.547645 0.397887i −3.82229 + 2.77706i 0.533421 2.17151i −1.42015 1.03180i −3.54137 5.71605 + 4.15295i −0.785450 + 2.41737i −5.78298 + 0.424558i
156.4 −0.749608 2.30706i −0.693590 + 0.503923i −3.14256 + 2.28320i −1.28236 1.83182i 1.68250 + 1.22241i 2.66727 3.69817 + 2.68688i −0.699922 + 2.15414i −3.26485 + 4.33161i
156.5 −0.745963 2.29584i 1.42406 1.03464i −3.09638 + 2.24965i −1.72323 + 1.42495i −3.43767 2.49762i 3.97778 3.56871 + 2.59282i 0.0304220 0.0936293i 4.55692 + 2.89330i
156.6 −0.743746 2.28902i −2.09240 + 1.52022i −3.06840 + 2.22932i 0.487900 + 2.18219i 5.03601 + 3.65888i 4.93338 3.49076 + 2.53618i 1.14002 3.50863i 4.63219 2.73981i
156.7 −0.683199 2.10267i 2.53065 1.83863i −2.33643 + 1.69751i −1.64075 1.51919i −5.59496 4.06498i −0.194679 1.58828 + 1.15395i 2.09660 6.45267i −2.07339 + 4.48787i
156.8 −0.609561 1.87604i −1.75820 + 1.27741i −1.52991 + 1.11155i 1.46351 + 1.69060i 3.46820 + 2.51980i −0.863422 −0.173831 0.126295i 0.532455 1.63873i 2.27953 3.77612i
156.9 −0.555168 1.70863i −1.92643 + 1.39963i −0.993180 + 0.721587i −1.73604 + 1.40931i 3.46096 + 2.51453i −4.33155 −1.12259 0.815609i 0.825112 2.53943i 3.37179 + 2.18385i
156.10 −0.524831 1.61527i 1.13868 0.827302i −0.715599 + 0.519913i 2.09379 + 0.784886i −1.93393 1.40508i −4.31341 −1.53269 1.11356i −0.314879 + 0.969099i 0.168912 3.79396i
156.11 −0.459923 1.41550i 0.828343 0.601826i −0.174068 + 0.126468i 2.03639 0.923634i −1.23286 0.895723i 3.39926 −2.14911 1.56142i −0.603094 + 1.85613i −2.24398 2.45771i
156.12 −0.381577 1.17437i 0.756089 0.549331i 0.384480 0.279341i 0.0519802 + 2.23546i −0.933627 0.678320i −0.771186 −2.47272 1.79654i −0.657145 + 2.02248i 2.60544 0.914047i
156.13 −0.379931 1.16931i −2.41947 + 1.75785i 0.395100 0.287057i 1.77514 1.35973i 2.97470 + 2.16124i 4.68592 −2.47511 1.79828i 1.83675 5.65293i −2.26438 1.55908i
156.14 −0.369570 1.13742i 0.759672 0.551934i 0.460895 0.334860i −1.65224 1.50669i −0.908532 0.660087i −1.94021 −2.48630 1.80640i −0.654580 + 2.01459i −1.10312 + 2.43611i
156.15 −0.322877 0.993713i −2.20206 + 1.59989i 0.734818 0.533877i −0.985462 2.00720i 2.30083 + 1.67165i −3.45698 −2.45838 1.78612i 1.36237 4.19296i −1.67640 + 1.62735i
156.16 −0.218896 0.673692i 2.00537 1.45699i 1.21209 0.880634i 2.00345 0.993066i −1.42053 1.03207i −0.368794 −2.00475 1.45654i 0.971653 2.99044i −1.10757 1.13233i
156.17 −0.200194 0.616134i 1.52398 1.10724i 1.27849 0.928878i −1.99743 + 1.00513i −0.987297 0.717313i 3.29259 −1.87649 1.36335i 0.169491 0.521639i 1.01917 + 1.02946i
156.18 −0.145907 0.449056i −1.13168 + 0.822213i 1.43767 1.04453i −2.14491 + 0.631944i 0.534339 + 0.388220i 0.465253 −1.44280 1.04825i −0.322388 + 0.992209i 0.596736 + 0.870980i
156.19 −0.0759340 0.233701i 2.63647 1.91551i 1.56918 1.14008i −2.22553 + 0.216824i −0.647853 0.470693i −4.97983 −0.783187 0.569019i 2.35475 7.24717i 0.219665 + 0.503644i
156.20 −0.0232911 0.0716827i 0.686947 0.499096i 1.61344 1.17223i 1.74588 1.39711i −0.0517764 0.0376177i −0.320307 −0.243561 0.176958i −0.704252 + 2.16746i −0.140812 0.0926090i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 156.40
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 775.2.j.c 160
25.d even 5 1 inner 775.2.j.c 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.j.c 160 1.a even 1 1 trivial
775.2.j.c 160 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - T_{2}^{159} + 62 T_{2}^{158} - 65 T_{2}^{157} + 2081 T_{2}^{156} - 2267 T_{2}^{155} + 50180 T_{2}^{154} - 56474 T_{2}^{153} + 974476 T_{2}^{152} - 1127114 T_{2}^{151} + 16126729 T_{2}^{150} + \cdots + 7307701132176 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display