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Properties

Label 129.4.m.a
Level $129$
Weight $4$
Character orbit 129.m
Analytic conductor $7.611$
Analytic rank $0$
Dimension $120$
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] = [129,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.10");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(7.61124639074\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 2 q^{2} + 30 q^{3} - 58 q^{4} - 6 q^{5} + 3 q^{6} + 74 q^{7} - 28 q^{8} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 2 q^{2} + 30 q^{3} - 58 q^{4} - 6 q^{5} + 3 q^{6} + 74 q^{7} - 28 q^{8} + 90 q^{9} - 36 q^{10} + 124 q^{11} - 18 q^{12} - 32 q^{13} - 722 q^{14} + 24 q^{15} + 178 q^{16} - 162 q^{17} + 9 q^{18} - 76 q^{19} - 67 q^{20} + 102 q^{21} + 1172 q^{22} + 740 q^{23} - 231 q^{24} - 68 q^{25} - 65 q^{26} - 540 q^{27} + 553 q^{28} - 296 q^{29} - 108 q^{30} - 1944 q^{31} + 1940 q^{32} + 276 q^{33} + 400 q^{34} + 1940 q^{35} - 1755 q^{36} - 146 q^{37} - 938 q^{38} - 102 q^{39} - 2572 q^{40} - 1036 q^{41} - 1464 q^{42} - 622 q^{43} + 4246 q^{44} - 396 q^{45} - 1426 q^{46} - 1232 q^{47} - 1653 q^{48} - 1188 q^{49} + 773 q^{50} + 972 q^{51} + 3302 q^{52} + 344 q^{53} - 54 q^{54} + 1764 q^{55} + 5333 q^{56} - 1110 q^{57} - 567 q^{58} - 226 q^{59} - 1188 q^{60} - 6100 q^{61} + 6739 q^{62} - 216 q^{63} - 154 q^{64} + 1732 q^{65} - 540 q^{66} + 178 q^{67} - 2217 q^{68} + 3270 q^{69} + 2563 q^{70} + 3764 q^{71} + 378 q^{72} + 2634 q^{73} - 5696 q^{74} + 1164 q^{75} - 1695 q^{76} + 1738 q^{77} - 2382 q^{78} + 1232 q^{79} - 6906 q^{80} + 810 q^{81} + 3192 q^{82} - 13242 q^{83} + 1869 q^{84} - 11004 q^{85} + 2205 q^{86} + 1272 q^{87} + 3564 q^{88} - 7910 q^{89} - 1494 q^{90} - 9360 q^{91} + 4851 q^{92} + 342 q^{93} + 2948 q^{94} + 5568 q^{95} - 2742 q^{96} - 310 q^{97} + 13047 q^{98} + 828 q^{99}+O(q^{100}) \) 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 −1.06210 − 4.65336i −2.19916 − 2.04052i −13.3179 + 6.41358i −6.12704 + 0.923503i −7.15954 + 12.4007i 12.8775 + 22.3044i 20.1822 + 25.3077i 0.672571 + 8.97483i 10.8049 + 27.5305i
10.2 −0.945388 − 4.14201i −2.19916 − 2.04052i −9.05478 + 4.36055i −6.65003 + 1.00233i −6.37280 + 11.0380i −10.1145 − 17.5188i 5.43040 + 6.80951i 0.672571 + 8.97483i 10.4385 + 26.5969i
10.3 −0.582570 − 2.55240i −2.19916 − 2.04052i 1.03237 − 0.497163i 4.67908 − 0.705258i −3.92707 + 6.80188i −6.53094 − 11.3119i −14.9290 − 18.7203i 0.672571 + 8.97483i −4.52600 − 11.5321i
10.4 −0.360225 − 1.57825i −2.19916 − 2.04052i 4.84664 − 2.33402i 13.7139 − 2.06704i −2.42826 + 4.20586i 17.5233 + 30.3513i −13.5042 − 16.9337i 0.672571 + 8.97483i −8.20239 − 20.8993i
10.5 −0.152368 − 0.667567i −2.19916 − 2.04052i 6.78532 − 3.26764i −13.2515 + 1.99734i −1.02710 + 1.77899i −1.30940 − 2.26794i −6.63064 − 8.31455i 0.672571 + 8.97483i 3.35245 + 8.54191i
10.6 0.174488 + 0.764482i −2.19916 − 2.04052i 6.65376 − 3.20428i 10.9649 − 1.65269i 1.17621 − 2.03726i −6.96952 − 12.0716i 7.52186 + 9.43211i 0.672571 + 8.97483i 3.17669 + 8.09407i
10.7 0.265381 + 1.16271i −2.19916 − 2.04052i 5.92629 − 2.85395i −14.0668 + 2.12023i 1.78891 − 3.09849i 6.21578 + 10.7660i 10.8397 + 13.5925i 0.672571 + 8.97483i −6.19826 − 15.7929i
10.8 0.721046 + 3.15911i −2.19916 − 2.04052i −2.25230 + 1.08465i −3.38421 + 0.510087i 4.86053 − 8.41868i 8.83861 + 15.3089i 11.1121 + 13.9341i 0.672571 + 8.97483i −4.05159 − 10.3233i
10.9 0.930580 + 4.07714i −2.19916 − 2.04052i −8.54932 + 4.11714i 16.3472 − 2.46394i 6.27298 − 10.8651i −2.07434 − 3.59286i −3.88254 − 4.86856i 0.672571 + 8.97483i 25.2582 + 64.3568i
10.10 1.17375 + 5.14252i −2.19916 − 2.04052i −17.8601 + 8.60097i −3.69018 + 0.556205i 7.91216 − 13.7043i −0.0402941 − 0.0697914i −38.8838 − 48.7587i 0.672571 + 8.97483i −7.19164 − 18.3240i
13.1 −1.06210 + 4.65336i −2.19916 + 2.04052i −13.3179 − 6.41358i −6.12704 − 0.923503i −7.15954 − 12.4007i 12.8775 − 22.3044i 20.1822 − 25.3077i 0.672571 − 8.97483i 10.8049 − 27.5305i
13.2 −0.945388 + 4.14201i −2.19916 + 2.04052i −9.05478 − 4.36055i −6.65003 − 1.00233i −6.37280 − 11.0380i −10.1145 + 17.5188i 5.43040 − 6.80951i 0.672571 − 8.97483i 10.4385 − 26.5969i
13.3 −0.582570 + 2.55240i −2.19916 + 2.04052i 1.03237 + 0.497163i 4.67908 + 0.705258i −3.92707 − 6.80188i −6.53094 + 11.3119i −14.9290 + 18.7203i 0.672571 − 8.97483i −4.52600 + 11.5321i
13.4 −0.360225 + 1.57825i −2.19916 + 2.04052i 4.84664 + 2.33402i 13.7139 + 2.06704i −2.42826 − 4.20586i 17.5233 − 30.3513i −13.5042 + 16.9337i 0.672571 − 8.97483i −8.20239 + 20.8993i
13.5 −0.152368 + 0.667567i −2.19916 + 2.04052i 6.78532 + 3.26764i −13.2515 − 1.99734i −1.02710 − 1.77899i −1.30940 + 2.26794i −6.63064 + 8.31455i 0.672571 − 8.97483i 3.35245 − 8.54191i
13.6 0.174488 − 0.764482i −2.19916 + 2.04052i 6.65376 + 3.20428i 10.9649 + 1.65269i 1.17621 + 2.03726i −6.96952 + 12.0716i 7.52186 − 9.43211i 0.672571 − 8.97483i 3.17669 − 8.09407i
13.7 0.265381 − 1.16271i −2.19916 + 2.04052i 5.92629 + 2.85395i −14.0668 − 2.12023i 1.78891 + 3.09849i 6.21578 − 10.7660i 10.8397 − 13.5925i 0.672571 − 8.97483i −6.19826 + 15.7929i
13.8 0.721046 − 3.15911i −2.19916 + 2.04052i −2.25230 − 1.08465i −3.38421 − 0.510087i 4.86053 + 8.41868i 8.83861 − 15.3089i 11.1121 − 13.9341i 0.672571 − 8.97483i −4.05159 + 10.3233i
13.9 0.930580 − 4.07714i −2.19916 + 2.04052i −8.54932 − 4.11714i 16.3472 + 2.46394i 6.27298 + 10.8651i −2.07434 + 3.59286i −3.88254 + 4.86856i 0.672571 − 8.97483i 25.2582 − 64.3568i
13.10 1.17375 − 5.14252i −2.19916 + 2.04052i −17.8601 − 8.60097i −3.69018 − 0.556205i 7.91216 + 13.7043i −0.0402941 + 0.0697914i −38.8838 + 48.7587i 0.672571 − 8.97483i −7.19164 + 18.3240i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.10
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 129.4.m.a ✓ 120
43.g even 21 1 inner 129.4.m.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.4.m.a ✓ 120 1.a even 1 1 trivial
129.4.m.a ✓ 120 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{120} + 2 T_{2}^{119} + 111 T_{2}^{118} + 202 T_{2}^{117} + 6984 T_{2}^{116} + \cdots + 51\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display