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Properties

Label 605.2.k.b
Level $605$
Weight $2$
Character orbit 605.k
Analytic conductor $4.831$
Analytic rank $0$
Dimension $220$
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] = [605,2,Mod(56,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.56");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.k (of order \(11\), degree \(10\), 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(220\)
Relative dimension: \(22\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 220 q - 2 q^{2} - 24 q^{4} + 22 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 212 q^{9} + 2 q^{10} - 2 q^{11} + 49 q^{12} + 8 q^{13} - 40 q^{14} + 11 q^{15} - 28 q^{16} - 8 q^{17} - 10 q^{18} + 24 q^{20} - 22 q^{21} - 79 q^{22} - 31 q^{23} - 36 q^{24} - 22 q^{25} - 6 q^{26} - 6 q^{27} + 4 q^{28} - 4 q^{29} - 19 q^{30} + 20 q^{31} - 104 q^{32} - 12 q^{34} - 4 q^{35} - 30 q^{36} - 93 q^{37} + 8 q^{38} + 16 q^{39} + 6 q^{40} - 12 q^{41} - 8 q^{42} - 43 q^{43} + 9 q^{44} + 30 q^{45} - 124 q^{46} - 42 q^{47} - 158 q^{48} - 38 q^{49} - 2 q^{50} + 27 q^{51} + 146 q^{52} + 74 q^{53} + 93 q^{54} + 2 q^{55} + 25 q^{56} - 55 q^{57} + 26 q^{58} + 10 q^{59} - 16 q^{60} - 4 q^{61} - 33 q^{62} + 20 q^{63} + 32 q^{64} - 8 q^{65} - 69 q^{66} - 47 q^{67} - 24 q^{68} - 82 q^{69} - 15 q^{70} + 2 q^{71} - 294 q^{72} + 30 q^{73} - 112 q^{74} + 132 q^{76} + 136 q^{77} - 115 q^{78} + 58 q^{79} + 28 q^{80} + 220 q^{81} + 32 q^{82} - 164 q^{83} - 32 q^{84} + 41 q^{85} - 34 q^{86} - 76 q^{87} + 115 q^{88} - 44 q^{89} + 54 q^{90} - 60 q^{91} + 140 q^{92} - 68 q^{93} - 74 q^{94} - 44 q^{95} + 140 q^{96} - 39 q^{97} + 182 q^{98} - 274 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −2.41398 − 0.708808i 3.07614 3.64238 + 2.34081i 0.654861 − 0.755750i −7.42572 − 2.18039i 0.721882 − 1.58070i −3.83832 − 4.42965i 6.46261 −2.11650 + 1.36019i
56.2 −2.31434 − 0.679550i −0.0870913 3.21185 + 2.06413i 0.654861 − 0.755750i 0.201559 + 0.0591829i −1.78239 + 3.90290i −2.87152 − 3.31392i −2.99242 −2.02914 + 1.30405i
56.3 −2.28582 − 0.671178i 0.714225 3.09200 + 1.98711i 0.654861 − 0.755750i −1.63259 − 0.479373i −0.0293477 + 0.0642624i −2.61389 − 3.01659i −2.48988 −2.00414 + 1.28798i
56.4 −2.21684 − 0.650922i −2.65600 2.80815 + 1.80469i 0.654861 − 0.755750i 5.88791 + 1.72885i 1.70360 − 3.73037i −2.02449 − 2.33638i 4.05433 −1.94365 + 1.24911i
56.5 −1.87250 − 0.549815i −1.56075 1.52144 + 0.977773i 0.654861 − 0.755750i 2.92251 + 0.858125i −0.922489 + 2.01997i 0.244679 + 0.282374i −0.564050 −1.64175 + 1.05509i
56.6 −1.47955 − 0.434434i 1.25158 0.317819 + 0.204250i 0.654861 − 0.755750i −1.85177 − 0.543727i 1.94824 − 4.26605i 1.63811 + 1.89048i −1.43356 −1.29722 + 0.833673i
56.7 −1.20161 − 0.352823i 3.05910 −0.363136 − 0.233373i 0.654861 − 0.755750i −3.67583 − 1.07932i −1.77893 + 3.89532i 1.99421 + 2.30145i 6.35811 −1.05353 + 0.677063i
56.8 −0.960803 − 0.282117i −1.66611 −0.838954 − 0.539163i 0.654861 − 0.755750i 1.60080 + 0.470037i −0.0635449 + 0.139144i 1.96547 + 2.26828i −0.224089 −0.842402 + 0.541379i
56.9 −0.926188 − 0.271953i 0.536603 −0.898642 − 0.577522i 0.654861 − 0.755750i −0.496995 − 0.145931i 0.519083 − 1.13663i 1.93951 + 2.23832i −2.71206 −0.812053 + 0.521875i
56.10 −0.309263 − 0.0908078i 1.15033 −1.59511 − 1.02511i 0.654861 − 0.755750i −0.355754 − 0.104459i −1.76485 + 3.86448i 0.822368 + 0.949063i −1.67674 −0.271152 + 0.174259i
56.11 −0.127482 − 0.0374322i −3.30973 −1.66766 − 1.07174i 0.654861 − 0.755750i 0.421933 + 0.123891i 1.07269 − 2.34886i 0.346495 + 0.399876i 7.95431 −0.111773 + 0.0718319i
56.12 −0.0357503 − 0.0104972i 2.77455 −1.68134 − 1.08053i 0.654861 − 0.755750i −0.0991912 − 0.0291252i 0.491662 − 1.07659i 0.0975655 + 0.112597i 4.69815 −0.0313448 + 0.0201441i
56.13 0.105289 + 0.0309156i −0.0677895 −1.67238 − 1.07477i 0.654861 − 0.755750i −0.00713748 − 0.00209575i −0.157614 + 0.345127i −0.286576 − 0.330727i −2.99540 0.0923140 − 0.0593266i
56.14 0.685415 + 0.201256i −1.63608 −1.25322 − 0.805394i 0.654861 − 0.755750i −1.12140 − 0.329271i −1.36643 + 2.99207i −1.63249 − 1.88399i −0.323235 0.600951 − 0.386208i
56.15 0.701517 + 0.205984i 2.32552 −1.23281 − 0.792279i 0.654861 − 0.755750i 1.63139 + 0.479020i 1.16427 − 2.54940i −1.65922 − 1.91484i 2.40803 0.615068 − 0.395280i
56.16 1.59296 + 0.467736i −2.26440 0.636241 + 0.408888i 0.654861 − 0.755750i −3.60711 − 1.05914i −0.226450 + 0.495857i −1.35216 − 1.56047i 2.12752 1.39666 − 0.897578i
56.17 1.60588 + 0.471530i −1.30135 0.674012 + 0.433161i 0.654861 − 0.755750i −2.08982 − 0.613626i 1.11835 − 2.44885i −1.31392 − 1.51634i −1.30648 1.40799 − 0.904859i
56.18 1.76280 + 0.517603i 1.27989 1.15703 + 0.743575i 0.654861 − 0.755750i 2.25619 + 0.662476i 0.990961 − 2.16990i −0.751518 − 0.867297i −1.36188 1.54556 − 0.993273i
56.19 1.92400 + 0.564938i 2.93193 1.70012 + 1.09260i 0.654861 − 0.755750i 5.64104 + 1.65636i −0.651570 + 1.42674i 0.0274874 + 0.0317221i 5.59624 1.68690 − 1.08411i
56.20 2.11159 + 0.620020i −3.15222 2.39189 + 1.53718i 0.654861 − 0.755750i −6.65621 − 1.95444i −0.373094 + 0.816963i 1.21527 + 1.40250i 6.93651 1.85138 − 1.18981i
See next 80 embeddings (of 220 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.k.b ✓ 220
121.e even 11 1 inner 605.2.k.b ✓ 220
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.k.b ✓ 220 1.a even 1 1 trivial
605.2.k.b ✓ 220 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{220} + 2 T_{2}^{219} + 36 T_{2}^{218} + 74 T_{2}^{217} + 743 T_{2}^{216} + 1582 T_{2}^{215} + \cdots + 33723931887361 \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\). Copy content Toggle raw display