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Properties

Label 29.3.f.a
Level $29$
Weight $3$
Character orbit 29.f
Analytic conductor $0.790$
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] = [29,3,Mod(2,29)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(29, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("29.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 29.f (of order \(28\), degree \(12\), 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: \(0.790192766645\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(4\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9} - 20 q^{10} - 8 q^{11} - 68 q^{12} - 14 q^{13} + 26 q^{14} - 4 q^{15} + 18 q^{16} - 26 q^{17} - 34 q^{18} + 2 q^{19} + 46 q^{20} + 218 q^{21} + 154 q^{22} + 56 q^{23} + 154 q^{24} - 34 q^{25} + 110 q^{26} + 126 q^{27} - 170 q^{29} + 24 q^{30} - 88 q^{31} - 132 q^{32} - 224 q^{33} - 224 q^{34} - 210 q^{35} - 434 q^{36} - 56 q^{37} - 294 q^{38} - 232 q^{39} - 492 q^{40} - 34 q^{41} - 14 q^{42} + 176 q^{43} + 126 q^{44} + 114 q^{45} + 744 q^{46} + 208 q^{47} + 640 q^{48} + 506 q^{49} + 732 q^{50} + 322 q^{51} + 690 q^{52} - 14 q^{53} - 36 q^{54} + 284 q^{55} + 332 q^{56} - 508 q^{58} - 44 q^{59} - 316 q^{60} - 30 q^{61} - 504 q^{62} - 686 q^{63} - 896 q^{64} - 554 q^{65} - 608 q^{66} - 574 q^{67} - 796 q^{68} - 806 q^{69} - 1066 q^{70} + 224 q^{71} + 748 q^{72} - 22 q^{73} + 820 q^{74} + 768 q^{75} + 514 q^{76} + 436 q^{77} + 282 q^{78} + 564 q^{79} + 1162 q^{80} + 670 q^{81} - 18 q^{82} - 126 q^{83} + 572 q^{84} + 38 q^{85} - 118 q^{87} - 384 q^{88} - 160 q^{89} - 828 q^{90} - 434 q^{91} - 1022 q^{92} - 406 q^{93} - 2 q^{94} - 642 q^{95} - 1176 q^{96} + 604 q^{97} - 102 q^{98} + 316 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.56136 − 0.401269i 1.78374 + 1.12080i 8.62256 + 1.96804i 5.24106 − 4.17960i −5.90279 − 4.70732i 2.11977 + 9.28734i −16.3872 − 5.73413i −1.97942 − 4.11031i −20.3424 + 12.7820i
2.2 −1.68783 − 0.190173i −3.78594 − 2.37886i −1.08711 − 0.248126i 0.141728 − 0.113024i 5.93762 + 4.73509i −1.55116 − 6.79606i 8.20045 + 2.86946i 4.76938 + 9.90371i −0.260707 + 0.163813i
2.3 0.415096 + 0.0467701i 2.68233 + 1.68542i −3.72959 − 0.851255i 0.738700 − 0.589093i 1.03460 + 0.825065i −0.577468 − 2.53005i −3.08546 − 1.07965i 0.449302 + 0.932984i 0.334184 − 0.209981i
2.4 2.58169 + 0.290886i −2.11401 − 1.32832i 2.68078 + 0.611870i −2.49104 + 1.98654i −5.07133 − 4.04425i 1.30161 + 5.70272i −3.06598 − 1.07283i −1.20034 − 2.49253i −7.00893 + 4.40400i
3.1 −3.24379 − 2.03821i −2.23035 + 0.780434i 4.63233 + 9.61914i −5.12808 + 1.17045i 8.82547 + 2.01436i −6.56728 − 3.16264i 2.86375 − 25.4165i −2.67109 + 2.13013i 19.0200 + 6.65539i
3.2 −1.29187 − 0.811733i 2.15095 − 0.752648i −0.725528 − 1.50658i 3.36294 − 0.767569i −3.38968 − 0.773673i −0.255710 − 0.123143i −0.968958 + 8.59974i −2.97640 + 2.37360i −4.96753 − 1.73821i
3.3 1.44470 + 0.907767i 1.22940 − 0.430186i −0.472410 − 0.980970i −8.15497 + 1.86132i 2.16663 + 0.494518i 7.53715 + 3.62970i 0.972146 − 8.62804i −5.71012 + 4.55367i −13.4711 − 4.71376i
3.4 2.05804 + 1.29315i −2.93183 + 1.02589i 0.827740 + 1.71882i 5.87720 − 1.34143i −7.36043 − 1.67997i −9.36468 − 4.50979i 0.569384 − 5.05343i 0.506667 − 0.404053i 13.8301 + 4.83938i
8.1 −2.32673 − 0.814157i 0.184193 − 1.63476i 1.62348 + 1.29468i −3.83207 − 7.95737i −1.75952 + 3.65367i 2.23777 + 2.80607i 2.52264 + 4.01476i 6.13584 + 1.40047i 2.43763 + 21.6345i
8.2 −2.20133 − 0.770280i −0.552425 + 4.90291i 1.12521 + 0.897324i 2.64264 + 5.48750i 4.99268 − 10.3674i −3.22936 − 4.04949i 3.17747 + 5.05691i −14.9590 − 3.41428i −1.59042 − 14.1154i
8.3 −0.0310749 − 0.0108736i 0.529545 − 4.69985i −3.12648 − 2.49328i 3.79007 + 7.87017i −0.0675598 + 0.140289i 3.62612 + 4.54701i 0.140107 + 0.222980i −13.0338 − 2.97487i −0.0321993 − 0.285777i
8.4 1.65381 + 0.578694i −0.186386 + 1.65422i −0.727117 − 0.579856i −0.825315 − 1.71379i −1.26553 + 2.62791i −1.24782 − 1.56472i −4.59573 − 7.31406i 6.07265 + 1.38604i −0.373160 − 3.31188i
10.1 −3.24379 + 2.03821i −2.23035 − 0.780434i 4.63233 − 9.61914i −5.12808 − 1.17045i 8.82547 − 2.01436i −6.56728 + 3.16264i 2.86375 + 25.4165i −2.67109 − 2.13013i 19.0200 − 6.65539i
10.2 −1.29187 + 0.811733i 2.15095 + 0.752648i −0.725528 + 1.50658i 3.36294 + 0.767569i −3.38968 + 0.773673i −0.255710 + 0.123143i −0.968958 − 8.59974i −2.97640 − 2.37360i −4.96753 + 1.73821i
10.3 1.44470 − 0.907767i 1.22940 + 0.430186i −0.472410 + 0.980970i −8.15497 − 1.86132i 2.16663 − 0.494518i 7.53715 − 3.62970i 0.972146 + 8.62804i −5.71012 − 4.55367i −13.4711 + 4.71376i
10.4 2.05804 − 1.29315i −2.93183 − 1.02589i 0.827740 − 1.71882i 5.87720 + 1.34143i −7.36043 + 1.67997i −9.36468 + 4.50979i 0.569384 + 5.05343i 0.506667 + 0.404053i 13.8301 − 4.83938i
11.1 −2.32673 + 0.814157i 0.184193 + 1.63476i 1.62348 − 1.29468i −3.83207 + 7.95737i −1.75952 − 3.65367i 2.23777 − 2.80607i 2.52264 − 4.01476i 6.13584 − 1.40047i 2.43763 − 21.6345i
11.2 −2.20133 + 0.770280i −0.552425 − 4.90291i 1.12521 − 0.897324i 2.64264 − 5.48750i 4.99268 + 10.3674i −3.22936 + 4.04949i 3.17747 − 5.05691i −14.9590 + 3.41428i −1.59042 + 14.1154i
11.3 −0.0310749 + 0.0108736i 0.529545 + 4.69985i −3.12648 + 2.49328i 3.79007 − 7.87017i −0.0675598 − 0.140289i 3.62612 − 4.54701i 0.140107 − 0.222980i −13.0338 + 2.97487i −0.0321993 + 0.285777i
11.4 1.65381 − 0.578694i −0.186386 − 1.65422i −0.727117 + 0.579856i −0.825315 + 1.71379i −1.26553 − 2.62791i −1.24782 + 1.56472i −4.59573 + 7.31406i 6.07265 − 1.38604i −0.373160 + 3.31188i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.3.f.a ✓ 48
3.b odd 2 1 261.3.s.a 48
29.f odd 28 1 inner 29.3.f.a ✓ 48
87.k even 28 1 261.3.s.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.f.a ✓ 48 1.a even 1 1 trivial
29.3.f.a ✓ 48 29.f odd 28 1 inner
261.3.s.a 48 3.b odd 2 1
261.3.s.a 48 87.k even 28 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(29, [\chi])\).