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Properties

Label 58.6.d.a
Level $58$
Weight $6$
Character orbit 58.d
Analytic conductor $9.302$
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] = [58,6,Mod(7,58)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(58, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([6]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 58.d (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: \(9.30226154915\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 24 q^{2} + q^{3} - 96 q^{4} - 228 q^{5} - 136 q^{6} - 9 q^{7} - 384 q^{8} - 435 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 24 q^{2} + q^{3} - 96 q^{4} - 228 q^{5} - 136 q^{6} - 9 q^{7} - 384 q^{8} - 435 q^{9} - 44 q^{10} + 1002 q^{11} + 1024 q^{12} + 1169 q^{13} - 428 q^{14} + 1423 q^{15} - 1536 q^{16} + 3076 q^{17} - 1740 q^{18} + 2384 q^{19} + 2512 q^{20} - 1145 q^{21} - 2656 q^{22} - 6191 q^{23} - 2176 q^{24} + 1482 q^{25} - 840 q^{26} - 26153 q^{27} + 416 q^{28} + 12252 q^{29} + 49848 q^{30} + 478 q^{31} - 6144 q^{32} - 22635 q^{33} + 2364 q^{34} + 5562 q^{35} - 17600 q^{36} - 26269 q^{37} - 13004 q^{38} + 10986 q^{39} + 10048 q^{40} + 30706 q^{41} - 4580 q^{42} + 7583 q^{43} - 15216 q^{44} + 73454 q^{45} - 4352 q^{46} + 69793 q^{47} + 256 q^{48} - 55295 q^{49} - 83756 q^{50} - 84857 q^{51} - 19488 q^{52} - 31117 q^{53} + 63808 q^{54} - 181270 q^{55} - 6848 q^{56} + 22514 q^{57} + 2248 q^{58} + 330930 q^{59} + 22768 q^{60} + 95927 q^{61} - 32080 q^{62} - 212748 q^{63} - 24576 q^{64} + 262824 q^{65} + 17960 q^{66} - 183085 q^{67} - 45872 q^{68} - 230020 q^{69} + 363400 q^{70} + 92215 q^{71} + 21440 q^{72} + 74390 q^{73} - 105076 q^{74} - 252426 q^{75} - 52016 q^{76} - 12722 q^{77} + 53548 q^{78} - 161746 q^{79} - 2816 q^{80} + 356304 q^{81} - 180920 q^{82} + 293637 q^{83} - 146000 q^{84} + 204690 q^{85} + 190520 q^{86} - 38179 q^{87} + 78464 q^{88} - 529801 q^{89} + 117080 q^{90} - 261193 q^{91} - 99056 q^{92} - 20830 q^{93} - 43864 q^{94} + 2973 q^{95} + 1024 q^{96} + 27745 q^{97} - 221180 q^{98} + 6916 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.890084 − 3.89971i −21.5191 − 10.3630i −14.4155 + 6.94214i −21.1087 − 92.4834i −21.2591 + 93.1422i −138.771 − 66.8284i 39.9033 + 50.0372i 204.170 + 256.021i −341.870 + 164.636i
7.2 −0.890084 − 3.89971i −19.8767 − 9.57210i −14.4155 + 6.94214i 12.0244 + 52.6823i −19.6365 + 86.0333i −35.1227 − 16.9142i 39.9033 + 50.0372i 151.949 + 190.538i 194.743 − 93.7833i
7.3 −0.890084 − 3.89971i −4.43755 − 2.13701i −14.4155 + 6.94214i 4.41541 + 19.3452i −4.38393 + 19.2073i 196.591 + 94.6733i 39.9033 + 50.0372i −136.383 − 171.019i 71.5105 − 34.4377i
7.4 −0.890084 − 3.89971i −1.36726 − 0.658438i −14.4155 + 6.94214i −11.3678 − 49.8056i −1.35074 + 5.91799i 39.4149 + 18.9812i 39.9033 + 50.0372i −150.072 − 188.185i −184.109 + 88.6623i
7.5 −0.890084 − 3.89971i 13.8497 + 6.66968i −14.4155 + 6.94214i −6.76939 − 29.6586i 13.6824 − 59.9465i −132.869 − 63.9865i 39.9033 + 50.0372i −4.17750 − 5.23842i −109.635 + 52.7973i
7.6 −0.890084 − 3.89971i 23.2373 + 11.1905i −14.4155 + 6.94214i 7.51365 + 32.9195i 22.9565 − 100.579i 42.6625 + 20.5452i 39.9033 + 50.0372i 263.236 + 330.087i 121.689 − 58.6022i
23.1 2.49396 − 3.12733i −4.81913 + 21.1140i −3.56033 − 15.5988i −9.98480 + 12.5205i 54.0116 + 67.7284i 4.60639 − 20.1819i −57.6620 − 27.7686i −203.641 − 98.0685i 14.2541 + 62.4515i
23.2 2.49396 − 3.12733i −4.00359 + 17.5409i −3.56033 − 15.5988i 60.6777 − 76.0875i 44.8712 + 56.2667i −31.4888 + 137.961i −57.6620 − 27.7686i −72.7179 − 35.0191i −86.6225 − 379.518i
23.3 2.49396 − 3.12733i −2.36919 + 10.3801i −3.56033 − 15.5988i −16.5689 + 20.7768i 26.5533 + 33.2967i 50.4185 − 220.898i −57.6620 − 27.7686i 116.802 + 56.2490i 23.6535 + 103.633i
23.4 2.49396 − 3.12733i 0.390591 − 1.71129i −3.56033 − 15.5988i −36.6772 + 45.9917i −4.37765 − 5.48940i −43.8484 + 192.112i −57.6620 − 27.7686i 216.159 + 104.097i 52.3597 + 229.403i
23.5 2.49396 − 3.12733i 3.63750 − 15.9369i −3.56033 − 15.5988i 34.4349 − 43.1800i −40.7682 − 51.1217i 4.62013 − 20.2421i −57.6620 − 27.7686i −21.8190 − 10.5075i −49.1587 − 215.378i
23.6 2.49396 − 3.12733i 6.54852 − 28.6909i −3.56033 − 15.5988i −67.3601 + 84.4669i −73.3942 − 92.0334i 24.8798 − 109.005i −57.6620 − 27.7686i −561.352 − 270.333i 96.1622 + 421.314i
25.1 −0.890084 + 3.89971i −21.5191 + 10.3630i −14.4155 − 6.94214i −21.1087 + 92.4834i −21.2591 − 93.1422i −138.771 + 66.8284i 39.9033 − 50.0372i 204.170 − 256.021i −341.870 − 164.636i
25.2 −0.890084 + 3.89971i −19.8767 + 9.57210i −14.4155 − 6.94214i 12.0244 − 52.6823i −19.6365 − 86.0333i −35.1227 + 16.9142i 39.9033 − 50.0372i 151.949 − 190.538i 194.743 + 93.7833i
25.3 −0.890084 + 3.89971i −4.43755 + 2.13701i −14.4155 − 6.94214i 4.41541 − 19.3452i −4.38393 − 19.2073i 196.591 − 94.6733i 39.9033 − 50.0372i −136.383 + 171.019i 71.5105 + 34.4377i
25.4 −0.890084 + 3.89971i −1.36726 + 0.658438i −14.4155 − 6.94214i −11.3678 + 49.8056i −1.35074 − 5.91799i 39.4149 − 18.9812i 39.9033 − 50.0372i −150.072 + 188.185i −184.109 − 88.6623i
25.5 −0.890084 + 3.89971i 13.8497 − 6.66968i −14.4155 − 6.94214i −6.76939 + 29.6586i 13.6824 + 59.9465i −132.869 + 63.9865i 39.9033 − 50.0372i −4.17750 + 5.23842i −109.635 − 52.7973i
25.6 −0.890084 + 3.89971i 23.2373 − 11.1905i −14.4155 − 6.94214i 7.51365 − 32.9195i 22.9565 + 100.579i 42.6625 − 20.5452i 39.9033 − 50.0372i 263.236 − 330.087i 121.689 + 58.6022i
45.1 −3.60388 − 1.73553i −11.6096 + 14.5580i 9.97584 + 12.5093i 56.8231 + 27.3646i 67.1053 − 32.3162i 40.1947 − 50.4026i −14.2413 − 62.3954i −23.0791 − 101.116i −157.291 − 197.237i
45.2 −3.60388 − 1.73553i −6.00585 + 7.53110i 9.97584 + 12.5093i −87.8894 − 42.3253i 34.7148 − 16.7178i 75.3239 − 94.4532i −14.2413 − 62.3954i 33.4254 + 146.446i 243.285 + 305.070i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.6.d.a ✓ 36
29.d even 7 1 inner 58.6.d.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.6.d.a ✓ 36 1.a even 1 1 trivial
58.6.d.a ✓ 36 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - T_{3}^{35} + 947 T_{3}^{34} + 13279 T_{3}^{33} + 461196 T_{3}^{32} + 7592068 T_{3}^{31} + \cdots + 13\!\cdots\!16 \) acting on \(S_{6}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display