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Properties

Label 129.4.i.a
Level $129$
Weight $4$
Character orbit 129.i
Analytic conductor $7.611$
Analytic rank $0$
Dimension $66$
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(4,129)]
 
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, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("129.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 129 = 3 \cdot 43 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 129.i (of order \(7\), 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: \(7.61124639074\)
Analytic rank: \(0\)
Dimension: \(66\)
Relative dimension: \(11\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 2 q^{2} - 33 q^{3} - 54 q^{4} + 12 q^{5} + 6 q^{6} - 118 q^{7} - 74 q^{8} - 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 2 q^{2} - 33 q^{3} - 54 q^{4} + 12 q^{5} + 6 q^{6} - 118 q^{7} - 74 q^{8} - 99 q^{9} - 12 q^{10} - 88 q^{11} - 57 q^{12} - 42 q^{13} - 205 q^{14} - 6 q^{15} - 530 q^{16} + 66 q^{17} + 18 q^{18} + 50 q^{19} - 2 q^{20} - 18 q^{21} + 238 q^{22} + 16 q^{23} + 303 q^{24} - 47 q^{25} + 278 q^{26} - 297 q^{27} - 1032 q^{28} + 560 q^{29} - 36 q^{30} + 989 q^{31} + 250 q^{32} + 534 q^{33} + 326 q^{34} + 424 q^{35} + 2538 q^{36} - 894 q^{37} - 286 q^{38} + 567 q^{39} - 1472 q^{40} + 1384 q^{41} - 48 q^{42} - 663 q^{43} - 12358 q^{44} - 144 q^{45} + 1327 q^{46} + 506 q^{47} - 1590 q^{48} + 3376 q^{49} + 2524 q^{50} + 198 q^{51} + 2700 q^{52} + 2314 q^{53} + 54 q^{54} + 1140 q^{55} + 2857 q^{56} + 297 q^{57} - 204 q^{58} - 1208 q^{59} + 981 q^{60} - 2170 q^{61} + 1472 q^{62} - 180 q^{63} + 1044 q^{64} - 1660 q^{65} - 315 q^{66} + 108 q^{67} + 2988 q^{68} + 888 q^{69} + 2897 q^{70} - 632 q^{71} - 162 q^{72} - 884 q^{73} - 2740 q^{74} - 1149 q^{75} + 3535 q^{76} + 2288 q^{77} + 834 q^{78} - 1822 q^{79} - 2682 q^{80} - 891 q^{81} - 906 q^{82} - 1188 q^{83} + 1167 q^{84} + 4488 q^{85} - 4008 q^{86} - 2100 q^{87} - 3690 q^{88} + 2588 q^{89} + 2034 q^{90} - 4204 q^{91} - 2598 q^{92} - 330 q^{93} + 424 q^{94} - 2094 q^{95} - 3450 q^{96} + 1538 q^{97} - 1791 q^{98} + 1602 q^{99}+O(q^{100}) \) 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 −3.29617 + 4.13326i 1.87047 + 2.34549i −4.43898 − 19.4484i −0.781544 − 0.376372i −15.8599 −26.0438 56.9124 + 27.4075i −2.00269 + 8.77435i 4.13174 − 1.98974i
4.2 −2.99586 + 3.75669i 1.87047 + 2.34549i −3.35737 − 14.7096i 2.51238 + 1.20990i −14.4150 30.4877 30.6845 + 14.7769i −2.00269 + 8.77435i −12.0720 + 5.81356i
4.3 −2.03118 + 2.54702i 1.87047 + 2.34549i −0.581453 − 2.54751i −12.3762 − 5.96006i −9.77329 9.18859 −15.8115 − 7.61444i −2.00269 + 8.77435i 40.3187 − 19.4165i
4.4 −1.69100 + 2.12045i 1.87047 + 2.34549i 0.143346 + 0.628040i 3.56835 + 1.71843i −8.13648 −17.9647 −21.1227 − 10.1722i −2.00269 + 8.77435i −9.67794 + 4.66065i
4.5 −0.463188 + 0.580820i 1.87047 + 2.34549i 1.65736 + 7.26137i 12.5463 + 6.04196i −2.22869 17.3536 −10.3398 − 4.97940i −2.00269 + 8.77435i −9.32057 + 4.48855i
4.6 0.292620 − 0.366934i 1.87047 + 2.34549i 1.73115 + 7.58468i −11.8714 − 5.71696i 1.40798 −19.1402 6.67243 + 3.21327i −2.00269 + 8.77435i −5.57156 + 2.68312i
4.7 0.934501 − 1.17183i 1.87047 + 2.34549i 1.28028 + 5.60928i −5.12960 − 2.47028i 4.49647 3.27932 18.5727 + 8.94412i −2.00269 + 8.77435i −7.68836 + 3.70252i
4.8 1.72957 − 2.16882i 1.87047 + 2.34549i 0.0678282 + 0.297175i 17.6434 + 8.49663i 8.32206 −30.2128 20.7563 + 9.99569i −2.00269 + 8.77435i 48.9432 − 23.5698i
4.9 2.25057 − 2.82213i 1.87047 + 2.34549i −1.11916 − 4.90338i 0.980146 + 0.472013i 10.8289 19.1992 9.66064 + 4.65232i −2.00269 + 8.77435i 3.53797 − 1.70380i
4.10 2.88137 − 3.61313i 1.87047 + 2.34549i −2.97222 − 13.0221i −18.2100 − 8.76946i 13.8641 −25.4959 −22.3050 − 10.7415i −2.00269 + 8.77435i −84.1549 + 40.5269i
4.11 3.51225 − 4.40423i 1.87047 + 2.34549i −5.28112 − 23.1381i 14.8102 + 7.13219i 16.8997 0.813942 −79.8512 − 38.4543i −2.00269 + 8.77435i 83.4288 − 40.1772i
16.1 −1.13786 − 4.98530i −0.667563 + 2.92478i −16.3507 + 7.87409i 0.995300 + 1.24807i 15.3405 26.9381 32.3538 + 40.5704i −8.10872 − 3.90495i 5.08947 − 6.38199i
16.2 −0.920066 − 4.03107i −0.667563 + 2.92478i −8.19528 + 3.94664i −7.92848 − 9.94200i 12.4042 −0.529241 2.82564 + 3.54324i −8.10872 − 3.90495i −32.7822 + 41.1076i
16.3 −0.786182 − 3.44449i −0.667563 + 2.92478i −4.03867 + 1.94492i 3.23201 + 4.05281i 10.5992 −10.1171 −7.74828 − 9.71603i −8.10872 − 3.90495i 11.4189 − 14.3189i
16.4 −0.463135 − 2.02913i −0.667563 + 2.92478i 3.30490 − 1.59155i 1.61945 + 2.03073i 6.24392 8.09684 −15.1415 − 18.9868i −8.10872 − 3.90495i 3.37058 − 4.22657i
16.5 −0.0837684 − 0.367013i −0.667563 + 2.92478i 7.08007 − 3.40958i −5.04248 − 6.32307i 1.12935 −33.7731 −3.72216 − 4.66744i −8.10872 − 3.90495i −1.89825 + 2.38033i
16.6 −0.000350715 − 0.00153658i −0.667563 + 2.92478i 7.20775 − 3.47107i −9.39555 − 11.7816i 0.00472830 18.7520 −0.0157229 − 0.0197159i −8.10872 − 3.90495i −0.0148083 + 0.0185690i
16.7 0.000408730 0.00179076i −0.667563 + 2.92478i 7.20775 − 3.47107i 12.6775 + 15.8971i −0.00551045 −5.03329 0.0183238 + 0.0229773i −8.10872 − 3.90495i −0.0232862 + 0.0292000i
16.8 0.636876 + 2.79034i −0.667563 + 2.92478i −0.172614 + 0.0831267i 3.63012 + 4.55202i −8.58628 30.9413 13.9340 + 17.4727i −8.10872 − 3.90495i −10.3897 + 13.0283i
16.9 0.839634 + 3.67868i −0.667563 + 2.92478i −5.61993 + 2.70642i 2.69661 + 3.38144i −11.3198 −25.3130 4.14610 + 5.19905i −8.10872 − 3.90495i −10.1751 + 12.7591i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 129.4.i.a ✓ 66
43.e even 7 1 inner 129.4.i.a ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
129.4.i.a ✓ 66 1.a even 1 1 trivial
129.4.i.a ✓ 66 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} - 2 T_{2}^{65} + 73 T_{2}^{64} - 92 T_{2}^{63} + 3453 T_{2}^{62} - 3998 T_{2}^{61} + \cdots + 658299301134336 \) acting on \(S_{4}^{\mathrm{new}}(129, [\chi])\). Copy content Toggle raw display