Properties

Label 605.2.s.a
Level $605$
Weight $2$
Character orbit 605.s
Analytic conductor $4.831$
Analytic rank $0$
Dimension $880$
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] = [605,2,Mod(16,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.s (of order \(55\), degree \(40\), 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: \(880\)
Relative dimension: \(22\) over \(\Q(\zeta_{55})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{55}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 880 q + 2 q^{2} + 5 q^{3} + 24 q^{4} - 22 q^{5} + 7 q^{6} + q^{7} - 4 q^{8} - 217 q^{9} - 2 q^{10} - 3 q^{11} - 49 q^{12} + 2 q^{13} + 60 q^{14} - 16 q^{15} + 18 q^{16} + 13 q^{17} - 15 q^{19} - 19 q^{20} + 42 q^{21} - 301 q^{22} + 31 q^{23} - 409 q^{24} + 22 q^{25} - 4 q^{26} - 4 q^{27} + 6 q^{28} + 9 q^{29} + 19 q^{30} - 10 q^{31} + 94 q^{32} - 5 q^{33} + 12 q^{34} + 4 q^{35} + 35 q^{36} + 73 q^{37} - 8 q^{38} - 21 q^{39} + 4 q^{40} - 8 q^{41} - 17 q^{42} + 93 q^{43} + q^{44} + 80 q^{45} + 129 q^{46} - 178 q^{47} + 153 q^{48} - 177 q^{49} + 2 q^{50} - 12 q^{51} - 126 q^{52} - 99 q^{53} - 93 q^{54} - 7 q^{55} - 25 q^{56} + 100 q^{57} - 26 q^{58} + 25 q^{59} - 4 q^{60} - 6 q^{61} + 63 q^{62} - 25 q^{63} + 8 q^{64} - 2 q^{65} + 64 q^{66} + 47 q^{67} - 11 q^{68} + 67 q^{69} - 60 q^{70} + 18 q^{71} - 641 q^{72} - 35 q^{73} - 108 q^{74} - 5 q^{75} - 132 q^{76} - 166 q^{77} + 115 q^{78} - 103 q^{79} - 23 q^{80} - 230 q^{81} - 72 q^{82} + 149 q^{83} - 28 q^{84} - 21 q^{85} + 49 q^{86} + 6 q^{87} - 85 q^{88} + 44 q^{89} - 34 q^{90} - 135 q^{92} - 172 q^{93} + 49 q^{94} - 106 q^{95} - 115 q^{96} - 421 q^{97} - 192 q^{98} - 156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.64787 0.303815i 1.98682 + 1.44351i 4.97089 + 1.15593i 0.564443 0.825472i −4.82227 4.42584i 0.0313353 + 0.0322433i −7.79061 2.77968i 0.936675 + 2.88279i −1.74536 + 2.01426i
16.2 −2.40732 0.276215i −0.917619 0.666690i 3.77088 + 0.876881i 0.564443 0.825472i 2.02486 + 1.85840i 0.271254 + 0.279114i −4.27113 1.52393i −0.529500 1.62963i −1.58680 + 1.83127i
16.3 −2.25925 0.259225i 0.692604 + 0.503206i 3.08898 + 0.718312i 0.564443 0.825472i −1.43432 1.31641i −1.93384 1.98987i −2.50894 0.895186i −0.700567 2.15612i −1.48920 + 1.71863i
16.4 −2.03514 0.233510i 2.04797 + 1.48794i 2.13922 + 0.497455i 0.564443 0.825472i −3.82045 3.50637i 2.83652 + 2.91871i −0.378738 0.135133i 1.05317 + 3.24133i −1.34147 + 1.54814i
16.5 −1.96587 0.225563i −1.14488 0.831801i 1.86575 + 0.433862i 0.564443 0.825472i 2.06306 + 1.89346i 2.64315 + 2.71973i 0.157420 + 0.0561672i −0.308203 0.948551i −1.29582 + 1.49546i
16.6 −1.41473 0.162326i −2.46787 1.79301i 0.0270973 + 0.00630121i 0.564443 0.825472i 3.20032 + 2.93723i −2.76991 2.85017i 2.64509 + 0.943766i 1.94843 + 5.99666i −0.932532 + 1.07620i
16.7 −1.18251 0.135680i 0.544716 + 0.395759i −0.568105 0.132107i 0.564443 0.825472i −0.590435 0.541896i 2.13190 + 2.19368i 2.89596 + 1.03328i −0.786961 2.42202i −0.779459 + 0.899544i
16.8 −1.07976 0.123891i 0.718084 + 0.521719i −0.797485 0.185447i 0.564443 0.825472i −0.710724 0.652297i −0.657495 0.676547i 2.88540 + 1.02951i −0.683596 2.10389i −0.711734 + 0.821384i
16.9 −0.949888 0.108989i −1.30238 0.946237i −1.05762 0.245938i 0.564443 0.825472i 1.13399 + 1.04077i −0.316454 0.325624i 2.77884 + 0.991490i −0.126212 0.388441i −0.626126 + 0.722587i
16.10 0.144844 + 0.0166193i −2.68544 1.95109i −1.92732 0.448179i 0.564443 0.825472i −0.356545 0.327234i 2.53859 + 2.61215i −0.546344 0.194935i 2.47780 + 7.62590i 0.0954751 0.110184i
16.11 0.217809 + 0.0249913i 1.53233 + 1.11330i −1.90121 0.442107i 0.564443 0.825472i 0.305933 + 0.280782i −1.17803 1.21217i −0.816028 0.291158i 0.181542 + 0.558728i 0.143570 0.165689i
16.12 0.229050 + 0.0262811i −1.48776 1.08092i −1.89625 0.440954i 0.564443 0.825472i −0.312365 0.286686i −1.06136 1.09211i −0.857039 0.305791i 0.117995 + 0.363151i 0.150980 0.174240i
16.13 0.329324 + 0.0377864i 1.61702 + 1.17484i −1.84100 0.428106i 0.564443 0.825472i 0.488132 + 0.448003i −2.98313 3.06957i −1.21452 0.433340i 0.307474 + 0.946309i 0.217076 0.250519i
16.14 0.710733 + 0.0815490i 2.35793 + 1.71314i −1.44953 0.337074i 0.564443 0.825472i 1.53616 + 1.40987i 2.60666 + 2.68219i −2.35033 0.838595i 1.69796 + 5.22577i 0.468485 0.540661i
16.15 1.06134 + 0.121777i −0.605637 0.440021i −0.836420 0.194501i 0.564443 0.825472i −0.589199 0.540762i −1.40253 1.44317i −2.87638 1.02629i −0.753874 2.32018i 0.699587 0.807367i
16.16 1.50155 + 0.172287i −0.190847 0.138659i 0.276953 + 0.0644028i 0.564443 0.825472i −0.262678 0.241084i −0.620747 0.638734i −2.44225 0.871393i −0.909855 2.80024i 0.989759 1.14224i
16.17 1.66802 + 0.191388i −1.24319 0.903228i 0.797652 + 0.185486i 0.564443 0.825472i −1.90080 1.74454i 3.32658 + 3.42297i −1.86765 0.666376i −0.197358 0.607406i 1.09949 1.26888i
16.18 1.90115 + 0.218137i 1.85412 + 1.34710i 1.61876 + 0.376427i 0.564443 0.825472i 3.23112 + 2.96549i 0.529131 + 0.544463i −0.609275 0.217389i 0.696047 + 2.14221i 1.25316 1.44622i
16.19 1.92192 + 0.220520i −2.15448 1.56532i 1.69713 + 0.394651i 0.564443 0.825472i −3.79556 3.48353i −0.812019 0.835548i −0.469333 0.167458i 1.26451 + 3.89175i 1.26685 1.46202i
16.20 2.46804 + 0.283181i 2.06854 + 1.50288i 4.06299 + 0.944807i 0.564443 0.825472i 4.67964 + 4.29494i −3.42528 3.52452i 5.08053 + 1.81273i 1.09315 + 3.36437i 1.62682 1.87745i
See next 80 embeddings (of 880 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.g even 55 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.s.a 880
121.g even 55 1 inner 605.2.s.a 880
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.s.a 880 1.a even 1 1 trivial
605.2.s.a 880 121.g even 55 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{880} - 2 T_{2}^{879} - 32 T_{2}^{878} + 72 T_{2}^{877} + 423 T_{2}^{876} - 1164 T_{2}^{875} - 2711 T_{2}^{874} + 9338 T_{2}^{873} + 7446 T_{2}^{872} - 7598 T_{2}^{871} - 20117 T_{2}^{870} - 692064 T_{2}^{869} + \cdots + 61\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\). Copy content Toggle raw display