Properties

Label 129.3.k.a
Level $129$
Weight $3$
Character orbit 129.k
Analytic conductor $3.515$
Analytic rank $0$
Dimension $84$
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] = [129,3,Mod(22,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.22");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 129.k (of order \(14\), 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: \(3.51499541025\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 36 q^{4} - 56 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 36 q^{4} - 56 q^{8} + 42 q^{9} - 8 q^{10} + 14 q^{11} + 16 q^{13} + 12 q^{14} - 30 q^{15} - 4 q^{16} + 18 q^{17} - 280 q^{20} + 48 q^{21} - 94 q^{23} + 118 q^{25} - 112 q^{26} - 140 q^{29} + 64 q^{31} - 280 q^{32} - 126 q^{33} + 40 q^{35} + 396 q^{36} - 108 q^{38} - 84 q^{39} + 112 q^{40} + 36 q^{41} + 206 q^{43} - 128 q^{44} + 42 q^{45} + 266 q^{46} - 82 q^{47} + 336 q^{48} - 480 q^{49} + 126 q^{51} + 56 q^{52} + 598 q^{53} + 630 q^{55} + 190 q^{56} - 300 q^{57} + 20 q^{58} + 244 q^{59} + 354 q^{60} + 336 q^{61} + 168 q^{62} - 852 q^{64} - 336 q^{65} + 216 q^{66} + 594 q^{67} + 16 q^{68} - 1610 q^{70} - 546 q^{71} - 154 q^{73} - 186 q^{74} - 84 q^{75} - 1442 q^{76} - 910 q^{77} - 204 q^{78} + 60 q^{79} - 126 q^{81} - 504 q^{82} - 714 q^{83} - 312 q^{84} - 1140 q^{86} - 252 q^{87} + 756 q^{88} + 532 q^{89} + 24 q^{90} + 728 q^{91} + 2056 q^{92} + 364 q^{94} + 84 q^{95} + 480 q^{96} + 1132 q^{97} + 1722 q^{98} - 252 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −1.41223 + 2.93252i −0.751509 1.56052i −4.11131 5.15543i 5.65471 1.29065i 5.63756 9.63817i 8.23151 1.87879i −1.87047 + 2.34549i −4.20087 + 18.4052i
22.2 −1.16061 + 2.41002i −0.751509 1.56052i −1.96725 2.46685i −1.61798 + 0.369294i 4.63311 9.66910i −2.20305 + 0.502833i −1.87047 + 2.34549i 0.987833 4.32798i
22.3 −1.14117 + 2.36966i 0.751509 + 1.56052i −1.81907 2.28104i 0.108463 0.0247561i −4.55551 5.42580i −2.77557 + 0.633505i −1.87047 + 2.34549i −0.0651116 + 0.285272i
22.4 −0.826539 + 1.71632i 0.751509 + 1.56052i 0.231354 + 0.290109i −8.96045 + 2.04516i −3.29952 8.80602i −8.11800 + 1.85288i −1.87047 + 2.34549i 3.89599 17.0694i
22.5 −0.352919 + 0.732845i 0.751509 + 1.56052i 2.08145 + 2.61006i 5.98914 1.36698i −1.40884 0.499685i −5.81936 + 1.32823i −1.87047 + 2.34549i −1.11190 + 4.87155i
22.6 −0.310075 + 0.643878i −0.751509 1.56052i 2.17553 + 2.72802i −5.26232 + 1.20109i 1.23781 3.62578i −5.21803 + 1.19098i −1.87047 + 2.34549i 0.858360 3.76072i
22.7 0.212773 0.441828i 0.751509 + 1.56052i 2.34402 + 2.93931i 1.18678 0.270874i 0.849383 1.46169i 3.70980 0.846737i −1.87047 + 2.34549i 0.132835 0.581986i
22.8 0.422999 0.878366i −0.751509 1.56052i 1.90136 + 2.38423i 5.11914 1.16841i −1.68860 9.90706i 6.70038 1.52932i −1.87047 + 2.34549i 1.13910 4.99071i
22.9 0.516112 1.07172i −0.751509 1.56052i 1.61175 + 2.02108i 0.417208 0.0952250i −2.06030 10.4524i 7.63663 1.74301i −1.87047 + 2.34549i 0.113272 0.496275i
22.10 0.641871 1.33286i 0.751509 + 1.56052i 1.12944 + 1.41628i −8.38778 + 1.91446i 2.56233 12.6855i 8.38174 1.91308i −1.87047 + 2.34549i −2.83217 + 12.4086i
22.11 1.13308 2.35286i 0.751509 + 1.56052i −1.75812 2.20461i 0.515964 0.117765i 4.52321 8.20311i 3.00479 0.685825i −1.87047 + 2.34549i 0.307541 1.34743i
22.12 1.17892 2.44805i 0.751509 + 1.56052i −2.10913 2.64476i 6.59618 1.50554i 4.70620 4.23961i 1.63502 0.373182i −1.87047 + 2.34549i 4.09073 17.9227i
22.13 1.32508 2.75156i −0.751509 1.56052i −3.32127 4.16474i −8.70094 + 1.98593i −5.28968 2.32386i −3.95074 + 0.901730i −1.87047 + 2.34549i −6.06503 + 26.5726i
22.14 1.46473 3.04154i −0.751509 1.56052i −4.61158 5.78274i 5.64984 1.28954i −5.84715 2.51254i −11.1783 + 2.55137i −1.87047 + 2.34549i 4.35330 19.0730i
70.1 −3.76661 + 0.859705i 1.68862 + 0.385418i 9.84441 4.74082i 3.59221 2.86470i −6.69174 7.95256i −20.9220 + 16.6848i 2.70291 + 1.30165i −11.0677 + 13.8784i
70.2 −3.21984 + 0.734907i −1.68862 0.385418i 6.22340 2.99703i −2.14187 + 1.70808i 5.72035 7.45032i −7.50738 + 5.98693i 2.70291 + 1.30165i 5.64119 7.07383i
70.3 −2.69435 + 0.614968i −1.68862 0.385418i 3.27747 1.57835i 1.94272 1.54927i 4.78677 9.96218i 0.782782 0.624248i 2.70291 + 1.30165i −4.28162 + 5.36899i
70.4 −2.24438 + 0.512265i 1.68862 + 0.385418i 1.17095 0.563898i 1.48452 1.18387i −3.98735 5.24089i 4.86023 3.87590i 2.70291 + 1.30165i −2.72538 + 3.41751i
70.5 −2.16778 + 0.494783i 1.68862 + 0.385418i 0.850604 0.409629i −7.32813 + 5.84399i −3.85127 10.2781i 5.31246 4.23655i 2.70291 + 1.30165i 12.9943 16.2943i
70.6 −1.21121 + 0.276451i −1.68862 0.385418i −2.21327 + 1.06585i −0.564929 + 0.450516i 2.15183 4.88887i 6.27134 5.00123i 2.70291 + 1.30165i 0.559703 0.701845i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.3.k.a 84
3.b odd 2 1 387.3.w.d 84
43.f odd 14 1 inner 129.3.k.a 84
129.j even 14 1 387.3.w.d 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.3.k.a 84 1.a even 1 1 trivial
129.3.k.a 84 43.f odd 14 1 inner
387.3.w.d 84 3.b odd 2 1
387.3.w.d 84 129.j even 14 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(129, [\chi])\).