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Properties

Label 109.2.h.a
Level $109$
Weight $2$
Character orbit 109.h
Analytic conductor $0.870$
Analytic rank $0$
Dimension $48$
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] = [109,2,Mod(4,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 109.h (of order \(18\), 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: \(0.870369382032\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(8\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 9 q^{2} - 6 q^{3} + 15 q^{4} - 6 q^{5} + 3 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 9 q^{2} - 6 q^{3} + 15 q^{4} - 6 q^{5} + 3 q^{7} + 12 q^{9} - 9 q^{10} + 9 q^{11} - 3 q^{12} - 15 q^{14} - 12 q^{15} - 3 q^{16} - 9 q^{17} - 69 q^{18} - 9 q^{19} - 42 q^{20} + 15 q^{21} + 12 q^{22} - 18 q^{23} + 60 q^{24} - 18 q^{25} - 12 q^{26} - 45 q^{27} - 54 q^{28} + 33 q^{29} - 33 q^{30} - 18 q^{31} + 9 q^{32} + 3 q^{34} + 18 q^{35} + 105 q^{36} + 15 q^{37} - 33 q^{38} - 6 q^{39} - 12 q^{40} - 36 q^{42} + 9 q^{43} + 66 q^{44} - 6 q^{45} - 60 q^{46} - 6 q^{47} + 57 q^{48} + 39 q^{49} + 87 q^{50} - 48 q^{51} - 36 q^{52} + 15 q^{53} + 90 q^{54} - 9 q^{55} + 3 q^{56} + 15 q^{57} - 27 q^{58} - 6 q^{59} - 33 q^{60} + 78 q^{61} + 36 q^{62} + 72 q^{63} + 126 q^{64} - 54 q^{65} + 9 q^{66} + 54 q^{67} - 3 q^{69} + 39 q^{70} + 15 q^{71} - 99 q^{72} - 21 q^{73} + 24 q^{74} - 30 q^{75} + 81 q^{77} + 27 q^{78} - 126 q^{79} + 45 q^{80} - 75 q^{81} - 3 q^{82} - 21 q^{83} - 63 q^{84} - 6 q^{85} - 189 q^{86} + 114 q^{87} + 12 q^{88} + 3 q^{89} + 225 q^{90} - 72 q^{91} - 27 q^{92} + 33 q^{93} - 42 q^{94} - 39 q^{95} - 99 q^{96} + 51 q^{97} - 150 q^{98} - 93 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.20553 − 1.27336i −3.21410 + 1.16984i 2.24290 + 3.88482i −1.45755 − 1.22303i 8.57841 + 1.51261i 0.588695 + 0.493974i − 6.33067i 6.66378 − 5.59158i 1.65731 + 4.55342i
4.2 −2.06568 − 1.19262i 1.06255 − 0.386737i 1.84468 + 3.19508i 1.65716 + 1.39053i −2.65612 − 0.468345i 2.16880 + 1.81984i − 4.02953i −1.31868 + 1.10651i −1.76480 − 4.84874i
4.3 −1.30942 − 0.755993i 2.57756 − 0.938155i 0.143051 + 0.247772i −2.97738 − 2.49832i −4.08434 − 0.720180i 0.695750 + 0.583804i 2.59139i 3.46555 − 2.90794i 2.00993 + 5.52223i
4.4 −0.364013 − 0.210163i −2.12370 + 0.772965i −0.911663 − 1.57905i 3.24085 + 2.71940i 0.935505 + 0.164955i 2.24631 + 1.88488i 1.60704i 1.61451 − 1.35473i −0.608196 − 1.67100i
4.5 −0.178921 − 0.103300i −1.38561 + 0.504320i −0.978658 − 1.69509i −1.84592 − 1.54891i 0.300010 + 0.0528999i −1.07611 − 0.902960i 0.817581i −0.632562 + 0.530783i 0.170271 + 0.467815i
4.6 0.570691 + 0.329489i 1.29137 − 0.470019i −0.782875 − 1.35598i −0.326574 − 0.274028i 0.891837 + 0.157255i 1.90667 + 1.59988i − 2.34975i −0.851424 + 0.714429i −0.0960836 − 0.263987i
4.7 1.51067 + 0.872184i 0.346822 − 0.126233i 0.521409 + 0.903107i 0.764799 + 0.641742i 0.634031 + 0.111797i −2.29364 − 1.92459i − 1.66968i −2.19378 + 1.84080i 0.595639 + 1.63650i
4.8 1.94980 + 1.12572i −2.20032 + 0.800851i 1.53449 + 2.65782i 0.537004 + 0.450600i −5.19173 − 0.915442i 0.441032 + 0.370070i 2.40675i 1.90191 − 1.59589i 0.539804 + 1.48310i
34.1 −2.06261 − 1.19085i 0.0732338 + 0.0614504i 1.83623 + 3.18044i 0.437618 − 2.48185i −0.0778744 − 0.213958i 0.0446013 − 0.252946i − 3.98329i −0.519358 − 2.94542i −3.85814 + 4.59795i
34.2 −1.59549 − 0.921155i 0.737722 + 0.619022i 0.697052 + 1.20733i −0.672574 + 3.81436i −0.606810 − 1.66720i −0.221838 + 1.25810i 1.11625i −0.359899 − 2.04109i 4.58670 − 5.46621i
34.3 −1.07880 − 0.622846i −2.15996 − 1.81242i −0.224125 − 0.388196i −0.0885784 + 0.502353i 1.20131 + 3.30056i −0.427582 + 2.42494i 3.04977i 0.859605 + 4.87506i 0.408447 − 0.486768i
34.4 −0.879527 − 0.507795i 2.62285 + 2.20083i −0.484288 − 0.838812i 0.337855 − 1.91607i −1.18929 − 3.26756i 0.344106 − 1.95152i 3.01486i 1.51473 + 8.59046i −1.27012 + 1.51368i
34.5 0.142717 + 0.0823978i −1.17786 − 0.988341i −0.986421 − 1.70853i −0.154521 + 0.876330i −0.0866636 − 0.238106i 0.721147 − 4.08983i − 0.654707i −0.110411 − 0.626169i −0.0942605 + 0.112335i
34.6 0.147002 + 0.0848719i −0.223847 − 0.187830i −0.985594 − 1.70710i 0.686476 − 3.89320i −0.0169646 − 0.0466098i −0.693883 + 3.93520i − 0.674085i −0.506117 − 2.87033i 0.431337 − 0.514048i
34.7 0.752135 + 0.434245i 1.32951 + 1.11559i −0.622862 − 1.07883i −0.358713 + 2.03436i 0.515531 + 1.41641i −0.0963838 + 0.546620i − 2.81888i 0.00210829 + 0.0119567i −1.15321 + 1.37435i
34.8 1.96123 + 1.13231i −0.843209 − 0.707537i 1.56427 + 2.70940i −0.0742219 + 0.420934i −0.852570 − 2.34242i −0.181313 + 1.02827i 2.55573i −0.310551 − 1.76122i −0.622195 + 0.741503i
43.1 −2.06402 + 1.19166i −0.0173879 − 0.0986119i 1.84012 − 3.18718i −1.68628 − 0.613754i 0.153401 + 0.182816i −4.42168 − 1.60936i 4.00454i 2.80966 − 1.02263i 4.21189 − 0.742671i
43.2 −1.57726 + 0.910629i −0.201050 − 1.14021i 0.658491 − 1.14054i −0.689021 − 0.250783i 1.35542 + 1.61532i 4.49333 + 1.63544i − 1.24395i 1.55942 − 0.567582i 1.31513 − 0.231894i
43.3 −1.15759 + 0.668336i 0.448253 + 2.54217i −0.106654 + 0.184731i 0.845392 + 0.307697i −2.21792 − 2.64321i −1.24373 − 0.452681i − 2.95847i −3.44263 + 1.25301i −1.18426 + 0.208818i
43.4 −0.257893 + 0.148894i −0.192157 − 1.08977i −0.955661 + 1.65525i 3.58003 + 1.30303i 0.211817 + 0.252434i −1.04767 − 0.381319i − 1.16475i 1.66839 − 0.607246i −1.11728 + 0.197006i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.h even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.2.h.a ✓ 48
3.b odd 2 1 981.2.bn.b 48
109.h even 18 1 inner 109.2.h.a ✓ 48
327.n odd 18 1 981.2.bn.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.2.h.a ✓ 48 1.a even 1 1 trivial
109.2.h.a ✓ 48 109.h even 18 1 inner
981.2.bn.b 48 3.b odd 2 1
981.2.bn.b 48 327.n odd 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(109, [\chi])\).