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Properties

Label 129.2.m.a
Level $129$
Weight $2$
Character orbit 129.m
Analytic conductor $1.030$
Analytic rank $0$
Dimension $36$
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] = [129,2,Mod(10,129)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(129, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 129.m (of order \(21\), 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: \(1.03007018607\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(3\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 3 q^{3} - 6 q^{4} + 3 q^{5} + q^{6} - 11 q^{7} - 14 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 3 q^{3} - 6 q^{4} + 3 q^{5} + q^{6} - 11 q^{7} - 14 q^{8} + 3 q^{9} - 2 q^{10} - q^{11} + 4 q^{12} - 8 q^{13} - 70 q^{14} - 10 q^{15} + 10 q^{16} - 3 q^{17} - q^{18} - 3 q^{19} + 13 q^{20} + 13 q^{21} - 8 q^{22} - 32 q^{23} + 7 q^{24} + 12 q^{25} + 11 q^{26} + 6 q^{27} + 35 q^{28} + 8 q^{29} + 2 q^{30} - 24 q^{31} + 40 q^{32} + 17 q^{33} + 54 q^{34} - 14 q^{35} - 11 q^{36} + 2 q^{37} + 44 q^{38} + 12 q^{39} - 66 q^{40} + 55 q^{41} + 28 q^{42} - 23 q^{43} - 10 q^{44} + 8 q^{45} + 30 q^{46} + 26 q^{47} + 5 q^{48} - q^{49} + 19 q^{50} - 6 q^{51} - 64 q^{52} - 59 q^{53} - 2 q^{54} + 19 q^{55} - 11 q^{56} - 39 q^{57} - 69 q^{58} + 16 q^{59} - 48 q^{60} - 87 q^{61} - 67 q^{62} + 3 q^{63} + 18 q^{64} - 19 q^{65} + 10 q^{66} - 29 q^{67} + 3 q^{68} + 4 q^{69} - 83 q^{70} + 34 q^{71} + 8 q^{73} - 70 q^{74} - 32 q^{75} + 99 q^{76} + 101 q^{77} - 20 q^{78} - 10 q^{79} + 42 q^{80} + 3 q^{81} + 134 q^{82} - 33 q^{83} + 21 q^{84} + 92 q^{85} + 15 q^{86} + 16 q^{87} + 16 q^{88} + 80 q^{89} + 18 q^{90} - 22 q^{91} + 75 q^{92} - 4 q^{93} + 54 q^{94} - 69 q^{95} - 50 q^{96} + 91 q^{97} + 165 q^{98} - 17 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −0.518727 − 2.27269i 0.733052 + 0.680173i −3.09411 + 1.49005i 2.38077 − 0.358843i 1.16557 − 2.01883i −2.58250 − 4.47301i 2.08454 + 2.61393i 0.0747301 + 0.997204i −2.05051 − 5.22460i
10.2 −0.127617 − 0.559125i 0.733052 + 0.680173i 1.50560 − 0.725060i −0.474050 + 0.0714515i 0.286752 − 0.496669i 0.489376 + 0.847624i −1.31269 − 1.64606i 0.0747301 + 0.997204i 0.100447 + 0.255935i
10.3 0.483752 + 2.11945i 0.733052 + 0.680173i −2.45614 + 1.18281i −1.31885 + 0.198785i −1.08698 + 1.88271i −0.485942 − 0.841677i −0.984194 − 1.23414i 0.0747301 + 0.997204i −1.05931 − 2.69909i
13.1 −0.518727 + 2.27269i 0.733052 − 0.680173i −3.09411 − 1.49005i 2.38077 + 0.358843i 1.16557 + 2.01883i −2.58250 + 4.47301i 2.08454 − 2.61393i 0.0747301 − 0.997204i −2.05051 + 5.22460i
13.2 −0.127617 + 0.559125i 0.733052 − 0.680173i 1.50560 + 0.725060i −0.474050 − 0.0714515i 0.286752 + 0.496669i 0.489376 − 0.847624i −1.31269 + 1.64606i 0.0747301 − 0.997204i 0.100447 − 0.255935i
13.3 0.483752 − 2.11945i 0.733052 − 0.680173i −2.45614 − 1.18281i −1.31885 − 0.198785i −1.08698 − 1.88271i −0.485942 + 0.841677i −0.984194 + 1.23414i 0.0747301 − 0.997204i −1.05931 + 2.69909i
25.1 −2.39067 − 1.15128i −0.826239 − 0.563320i 3.14285 + 3.94102i 1.30558 − 1.21140i 1.32672 + 2.29795i 1.22240 − 2.11726i −1.79540 − 7.86615i 0.365341 + 0.930874i −4.51588 + 1.39297i
25.2 −1.12790 − 0.543170i −0.826239 − 0.563320i −0.269844 − 0.338374i −1.91668 + 1.77841i 0.625940 + 1.08416i −0.378126 + 0.654934i 0.677703 + 2.96921i 0.365341 + 0.930874i 3.12781 − 0.964801i
25.3 1.79669 + 0.865240i −0.826239 − 0.563320i 1.23247 + 1.54547i 2.46763 − 2.28963i −0.997088 − 1.72701i −1.69648 + 2.93840i −0.0103242 − 0.0452331i 0.365341 + 0.930874i 6.41465 − 1.97866i
31.1 −2.39067 + 1.15128i −0.826239 + 0.563320i 3.14285 − 3.94102i 1.30558 + 1.21140i 1.32672 − 2.29795i 1.22240 + 2.11726i −1.79540 + 7.86615i 0.365341 − 0.930874i −4.51588 − 1.39297i
31.2 −1.12790 + 0.543170i −0.826239 + 0.563320i −0.269844 + 0.338374i −1.91668 − 1.77841i 0.625940 − 1.08416i −0.378126 − 0.654934i 0.677703 − 2.96921i 0.365341 − 0.930874i 3.12781 + 0.964801i
31.3 1.79669 − 0.865240i −0.826239 + 0.563320i 1.23247 − 1.54547i 2.46763 + 2.28963i −0.997088 + 1.72701i −1.69648 − 2.93840i −0.0103242 + 0.0452331i 0.365341 − 0.930874i 6.41465 + 1.97866i
40.1 −0.560252 − 0.269803i −0.0747301 + 0.997204i −1.00589 − 1.26135i −3.07271 − 0.947806i 0.310916 − 0.538523i −1.75779 − 3.04459i 0.499979 + 2.19055i −0.988831 − 0.149042i 1.46577 + 1.36004i
40.2 0.0952184 + 0.0458548i −0.0747301 + 0.997204i −1.24002 − 1.55493i 2.83097 + 0.873239i −0.0528422 + 0.0915254i 1.47849 + 2.56082i −0.0938053 − 0.410988i −0.988831 − 0.149042i 0.229518 + 0.212962i
40.3 1.78595 + 0.860067i −0.0747301 + 0.997204i 1.20291 + 1.50841i 0.409656 + 0.126362i −0.991126 + 1.71668i −1.32442 − 2.29396i −0.0311748 − 0.136586i −0.988831 − 0.149042i 0.622945 + 0.578008i
52.1 −0.691120 − 0.866638i 0.988831 + 0.149042i 0.171629 − 0.751954i 2.06843 + 1.41023i −0.554236 − 0.959964i 0.151485 − 0.262380i −2.76768 + 1.33285i 0.955573 + 0.294755i −0.207375 − 2.76723i
52.2 0.0239568 + 0.0300409i 0.988831 + 0.149042i 0.444713 − 1.94842i −2.52876 − 1.72408i 0.0192119 + 0.0332759i 1.38332 − 2.39599i 0.138423 − 0.0666611i 0.955573 + 0.294755i −0.00878813 − 0.117269i
52.3 0.760351 + 0.953449i 0.988831 + 0.149042i 0.114109 − 0.499944i −0.0884386 − 0.0602964i 0.609754 + 1.05612i −1.99918 + 3.46268i 2.76091 − 1.32958i 0.955573 + 0.294755i −0.00975475 − 0.130168i
58.1 −0.951870 + 1.19361i −0.365341 + 0.930874i −0.0736000 − 0.322463i 0.174476 + 2.32822i −0.763341 − 1.32214i 0.760524 − 1.31727i −2.29603 − 1.10571i −0.733052 − 0.680173i −2.94506 − 2.00791i
58.2 0.539867 − 0.676972i −0.365341 + 0.930874i 0.278207 + 1.21891i 0.0863108 + 1.15174i 0.432940 + 0.749874i −0.657282 + 1.13845i 2.53562 + 1.22109i −0.733052 − 0.680173i 0.826291 + 0.563355i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.2.m.a ✓ 36
3.b odd 2 1 387.2.y.b 36
43.g even 21 1 inner 129.2.m.a ✓ 36
43.g even 21 1 5547.2.a.z 18
43.h odd 42 1 5547.2.a.y 18
129.o odd 42 1 387.2.y.b 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.2.m.a ✓ 36 1.a even 1 1 trivial
129.2.m.a ✓ 36 43.g even 21 1 inner
387.2.y.b 36 3.b odd 2 1
387.2.y.b 36 129.o odd 42 1
5547.2.a.y 18 43.h odd 42 1
5547.2.a.z 18 43.g even 21 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 2 T_{2}^{35} + 11 T_{2}^{34} - 8 T_{2}^{33} + 34 T_{2}^{32} - 2 T_{2}^{31} + 248 T_{2}^{30} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display