↑

Properties

Label 145.2.k.a
Level $145$
Weight $2$
Character orbit 145.k
Analytic conductor $1.158$
Analytic rank $0$
Dimension $24$
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] = [145,2,Mod(16,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(14))
 
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("145.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.k (of order \(7\), degree \(6\), 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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 2 q^{3} - 4 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 2 q^{3} - 4 q^{4} + 4 q^{5} - 4 q^{6} + 4 q^{7} + 9 q^{8} - 5 q^{10} + 12 q^{11} + 2 q^{12} + 4 q^{13} - 32 q^{14} - 2 q^{15} + 10 q^{16} - 40 q^{17} + 10 q^{18} + 2 q^{19} - 3 q^{20} - 35 q^{21} + 13 q^{22} + 12 q^{23} - 22 q^{24} - 4 q^{25} - 8 q^{26} - 13 q^{27} + 20 q^{28} + 5 q^{29} + 18 q^{30} - 7 q^{31} + 22 q^{32} - 12 q^{33} + 12 q^{34} - 4 q^{35} - 16 q^{36} - 14 q^{37} + 14 q^{38} - 9 q^{39} - 9 q^{40} + 8 q^{41} + 19 q^{42} - 47 q^{43} + 17 q^{44} - 48 q^{46} + 22 q^{47} + 31 q^{48} - 10 q^{49} - 2 q^{50} + 34 q^{51} + 13 q^{52} - 53 q^{53} + 23 q^{54} + 16 q^{55} + 10 q^{56} + 20 q^{57} + 10 q^{58} - 38 q^{59} - 23 q^{60} - 19 q^{62} + 11 q^{63} + 37 q^{64} + 3 q^{65} + 2 q^{66} + 43 q^{67} - 50 q^{68} + 48 q^{69} - 10 q^{70} - 61 q^{71} - 37 q^{72} + 24 q^{73} + 109 q^{74} + 2 q^{75} + 24 q^{76} + 41 q^{77} - 42 q^{78} + 2 q^{79} + 4 q^{80} - 15 q^{81} - 40 q^{82} + 37 q^{83} - 6 q^{84} - 2 q^{85} - 18 q^{86} + 9 q^{87} - 20 q^{88} + 16 q^{89} + 18 q^{90} - q^{91} + 29 q^{92} - 26 q^{93} - 54 q^{94} - 2 q^{95} + 15 q^{96} - 57 q^{97} - 26 q^{98} + 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.94373 − 0.936051i 0.429035 − 0.537992i 1.65491 + 2.07520i 0.900969 + 0.433884i −1.33752 + 0.644113i 1.97327 − 2.47441i −0.314092 − 1.37613i 0.562198 + 2.46315i −1.34510 − 1.68671i
16.2 −0.994366 − 0.478861i −0.939621 + 1.17825i −0.487524 − 0.611336i 0.900969 + 0.433884i 1.49854 − 0.721661i −0.880793 + 1.10448i 0.683208 + 2.99333i 0.162183 + 0.710572i −0.688123 − 0.862878i
16.3 0.0878682 + 0.0423151i 1.48831 − 1.86628i −1.24105 − 1.55623i 0.900969 + 0.433884i 0.209747 − 0.101009i −1.00514 + 1.26040i −0.0866002 − 0.379420i −0.600378 − 2.63043i 0.0608067 + 0.0762492i
16.4 1.72674 + 0.831553i −0.700245 + 0.878079i 1.04316 + 1.30808i 0.900969 + 0.433884i −1.93931 + 0.933921i −1.05684 + 1.32524i −0.139412 − 0.610805i 0.386882 + 1.69504i 1.19494 + 1.49841i
36.1 −0.446022 − 1.95415i 2.33473 + 1.12435i −1.81783 + 0.875423i 0.222521 + 0.974928i 1.15580 − 5.06391i 0.000629566 0 0.000303183i 0.0220508 + 0.0276508i 2.31635 + 2.90461i 1.80591 − 0.869679i
36.2 0.0486603 + 0.213195i 1.07523 + 0.517805i 1.75885 − 0.847019i 0.222521 + 0.974928i −0.0580721 + 0.254431i −1.52525 − 0.734524i 0.538852 + 0.675700i −0.982464 − 1.23197i −0.197021 + 0.0948805i
36.3 0.263278 + 1.15350i −2.67743 − 1.28938i 0.540697 − 0.260386i 0.222521 + 0.974928i 0.782391 − 3.42788i 3.34803 + 1.61232i 1.91809 + 2.40520i 3.63566 + 4.55898i −1.06599 + 0.513355i
36.4 0.535053 + 2.34422i 0.390955 + 0.188274i −3.40714 + 1.64079i 0.222521 + 0.974928i −0.232174 + 1.01722i 1.10203 + 0.530707i −2.67101 − 3.34934i −1.75307 − 2.19828i −2.16638 + 1.04328i
81.1 −1.36814 + 1.71559i −0.530661 + 2.32498i −0.626413 − 2.74450i −0.623490 + 0.781831i −3.26270 − 4.09130i −0.423324 + 1.85470i 1.61142 + 0.776019i −2.42102 − 1.16590i −0.488284 − 2.13931i
81.2 −0.388637 + 0.487336i 0.153777 − 0.673740i 0.358585 + 1.57106i −0.623490 + 0.781831i 0.268574 + 0.336781i −0.617778 + 2.70666i −2.02819 − 0.976724i 2.27263 + 1.09444i −0.138703 − 0.607698i
81.3 0.552702 − 0.693066i 0.484696 − 2.12359i 0.270180 + 1.18374i −0.623490 + 0.781831i −1.20390 − 1.50964i 0.932055 − 4.08360i 2.56709 + 1.23625i −1.57180 − 0.756939i 0.197257 + 0.864239i
81.4 0.926597 − 1.16192i −0.508780 + 2.22911i −0.0464250 − 0.203401i −0.623490 + 0.781831i 2.11861 + 2.65665i 0.153120 − 0.670862i 2.39859 + 1.15510i −2.00718 − 0.966605i 0.330699 + 1.44889i
111.1 −1.36814 − 1.71559i −0.530661 − 2.32498i −0.626413 + 2.74450i −0.623490 − 0.781831i −3.26270 + 4.09130i −0.423324 − 1.85470i 1.61142 − 0.776019i −2.42102 + 1.16590i −0.488284 + 2.13931i
111.2 −0.388637 − 0.487336i 0.153777 + 0.673740i 0.358585 − 1.57106i −0.623490 − 0.781831i 0.268574 − 0.336781i −0.617778 − 2.70666i −2.02819 + 0.976724i 2.27263 − 1.09444i −0.138703 + 0.607698i
111.3 0.552702 + 0.693066i 0.484696 + 2.12359i 0.270180 − 1.18374i −0.623490 − 0.781831i −1.20390 + 1.50964i 0.932055 + 4.08360i 2.56709 − 1.23625i −1.57180 + 0.756939i 0.197257 − 0.864239i
111.4 0.926597 + 1.16192i −0.508780 − 2.22911i −0.0464250 + 0.203401i −0.623490 − 0.781831i 2.11861 − 2.65665i 0.153120 + 0.670862i 2.39859 − 1.15510i −2.00718 + 0.966605i 0.330699 − 1.44889i
136.1 −1.94373 + 0.936051i 0.429035 + 0.537992i 1.65491 − 2.07520i 0.900969 − 0.433884i −1.33752 − 0.644113i 1.97327 + 2.47441i −0.314092 + 1.37613i 0.562198 − 2.46315i −1.34510 + 1.68671i
136.2 −0.994366 + 0.478861i −0.939621 − 1.17825i −0.487524 + 0.611336i 0.900969 − 0.433884i 1.49854 + 0.721661i −0.880793 − 1.10448i 0.683208 − 2.99333i 0.162183 − 0.710572i −0.688123 + 0.862878i
136.3 0.0878682 − 0.0423151i 1.48831 + 1.86628i −1.24105 + 1.55623i 0.900969 − 0.433884i 0.209747 + 0.101009i −1.00514 − 1.26040i −0.0866002 + 0.379420i −0.600378 + 2.63043i 0.0608067 − 0.0762492i
136.4 1.72674 − 0.831553i −0.700245 − 0.878079i 1.04316 − 1.30808i 0.900969 − 0.433884i −1.93931 − 0.933921i −1.05684 − 1.32524i −0.139412 + 0.610805i 0.386882 − 1.69504i 1.19494 − 1.49841i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.2.k.a ✓ 24
5.b even 2 1 725.2.l.d 24
5.c odd 4 2 725.2.r.c 48
29.d even 7 1 inner 145.2.k.a ✓ 24
29.d even 7 1 4205.2.a.q 12
29.e even 14 1 4205.2.a.r 12
145.n even 14 1 725.2.l.d 24
145.p odd 28 2 725.2.r.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.k.a ✓ 24 1.a even 1 1 trivial
145.2.k.a ✓ 24 29.d even 7 1 inner
725.2.l.d 24 5.b even 2 1
725.2.l.d 24 145.n even 14 1
725.2.r.c 48 5.c odd 4 2
725.2.r.c 48 145.p odd 28 2
4205.2.a.q 12 29.d even 7 1
4205.2.a.r 12 29.e even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 2 T_{2}^{23} + 8 T_{2}^{22} + 13 T_{2}^{21} + 31 T_{2}^{20} + 16 T_{2}^{19} + 64 T_{2}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(145, [\chi])\). Copy content Toggle raw display