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Properties

Label 93.4.f.b
Level $93$
Weight $4$
Character orbit 93.f
Analytic conductor $5.487$
Analytic rank $0$
Dimension $32$
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] = [93,4,Mod(4,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 93.f (of order \(5\), degree \(4\), 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: \(5.48717763053\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{3} - 22 q^{4} + 54 q^{5} - 30 q^{6} - 25 q^{7} + 11 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 24 q^{3} - 22 q^{4} + 54 q^{5} - 30 q^{6} - 25 q^{7} + 11 q^{8} - 72 q^{9} - 71 q^{10} - 55 q^{11} + 111 q^{12} + 90 q^{13} - 119 q^{14} - 12 q^{15} - 162 q^{16} + 52 q^{17} - 272 q^{19} - 634 q^{20} + 15 q^{21} - 205 q^{22} + 374 q^{23} - 33 q^{24} + 938 q^{25} - 476 q^{26} + 216 q^{27} - 435 q^{28} - 823 q^{29} - 102 q^{30} - 162 q^{31} + 588 q^{32} - 150 q^{33} + 616 q^{34} + 498 q^{35} + 1062 q^{36} + 14 q^{37} - 1030 q^{38} + 225 q^{39} + 1190 q^{40} + 936 q^{41} + 357 q^{42} - 791 q^{43} + 1234 q^{44} - 279 q^{45} + 1051 q^{46} - 838 q^{47} + 51 q^{48} + 1711 q^{49} + 14 q^{50} - 156 q^{51} + 1008 q^{52} + 1036 q^{53} + 135 q^{54} + 1458 q^{55} + 250 q^{56} - 714 q^{57} - 2895 q^{58} - 67 q^{59} - 258 q^{60} + 158 q^{61} - 5144 q^{62} + 540 q^{63} - 5265 q^{64} + 634 q^{65} + 615 q^{66} + 1122 q^{67} - 3702 q^{68} - 1122 q^{69} - 4583 q^{70} + 952 q^{71} - 396 q^{72} + 1414 q^{73} + 374 q^{74} + 471 q^{75} + 5569 q^{76} + 235 q^{77} + 183 q^{78} - 2611 q^{79} + 1936 q^{80} - 648 q^{81} - 207 q^{82} + 3238 q^{83} + 195 q^{84} + 7056 q^{85} - 408 q^{86} - 6 q^{87} - 13130 q^{88} - 3366 q^{89} - 639 q^{90} + 4185 q^{91} + 5398 q^{92} - 3264 q^{93} + 8118 q^{94} - 8408 q^{95} + 1551 q^{96} + 1311 q^{97} + 5992 q^{98} + 90 q^{99}+O(q^{100}) \) 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 −1.51707 − 4.66906i −0.927051 + 2.85317i −13.0265 + 9.46428i 16.5894 14.7280 13.8981 − 10.0975i 32.1774 + 23.3783i −7.28115 − 5.29007i −25.1673 − 77.4570i
4.2 −1.02904 − 3.16706i −0.927051 + 2.85317i −2.49921 + 1.81579i 0.841583 9.99013 −22.1053 + 16.0605i −13.2300 − 9.61216i −7.28115 − 5.29007i −0.866022 − 2.66534i
4.3 −0.533450 − 1.64179i −0.927051 + 2.85317i 4.06123 − 2.95066i −8.37608 5.17884 13.4494 − 9.77155i −18.1835 − 13.2111i −7.28115 − 5.29007i 4.46821 + 13.7517i
4.4 0.118470 + 0.364614i −0.927051 + 2.85317i 6.35323 − 4.61589i 7.86848 −1.15013 −2.24442 + 1.63067i 4.91695 + 3.57238i −7.28115 − 5.29007i 0.932180 + 2.86896i
4.5 0.404669 + 1.24544i −0.927051 + 2.85317i 5.08476 − 3.69430i −15.8836 −3.92861 −17.4941 + 12.7102i 15.1342 + 10.9956i −7.28115 − 5.29007i −6.42760 − 19.7821i
4.6 0.789306 + 2.42923i −0.927051 + 2.85317i 1.19396 − 0.867466i 11.9874 −7.66274 22.6549 − 16.4597i 19.5811 + 14.2265i −7.28115 − 5.29007i 9.46175 + 29.1203i
4.7 1.42950 + 4.39955i −0.927051 + 2.85317i −10.8404 + 7.87602i 18.8251 −13.8779 −27.6633 + 20.0985i −20.2075 − 14.6816i −7.28115 − 5.29007i 26.9105 + 82.8219i
4.8 1.45565 + 4.48002i −0.927051 + 2.85317i −11.4796 + 8.34039i −10.5261 −14.1317 3.75148 − 2.72561i −23.5879 − 17.1376i −7.28115 − 5.29007i −15.3223 − 47.1573i
16.1 −4.39980 + 3.19664i 2.42705 + 1.76336i 6.66758 − 20.5207i 1.38207 −16.3153 −0.898947 + 2.76667i 22.8167 + 70.2226i 2.78115 + 8.55951i −6.08085 + 4.41799i
16.2 −2.81445 + 2.04482i 2.42705 + 1.76336i 1.26771 − 3.90160i 19.4880 −10.4365 7.55577 − 23.2543i −4.19003 − 12.8956i 2.78115 + 8.55951i −54.8479 + 39.8493i
16.3 −2.46379 + 1.79004i 2.42705 + 1.76336i 0.393841 − 1.21212i −14.3902 −9.13622 5.24255 − 16.1349i −6.32925 − 19.4794i 2.78115 + 8.55951i 35.4544 − 25.7591i
16.4 −1.14478 + 0.831728i 2.42705 + 1.76336i −1.85340 + 5.70417i 2.61969 −4.24506 −4.79928 + 14.7707i −6.12072 − 18.8376i 2.78115 + 8.55951i −2.99896 + 2.17887i
16.5 1.16752 − 0.848253i 2.42705 + 1.76336i −1.82857 + 5.62775i 7.01679 4.32940 3.44250 − 10.5949i 6.20649 + 19.1016i 2.78115 + 8.55951i 8.19224 − 5.95201i
16.6 1.60545 − 1.16642i 2.42705 + 1.76336i −1.25523 + 3.86319i −19.7218 5.95332 −7.47610 + 23.0091i 7.39673 + 22.7648i 2.78115 + 8.55951i −31.6623 + 23.0040i
16.7 2.84694 − 2.06842i 2.42705 + 1.76336i 1.35456 − 4.16892i 12.4809 10.5570 −3.24078 + 9.97410i 3.93275 + 12.1038i 2.78115 + 8.55951i 35.5324 − 25.8158i
16.8 4.08486 − 2.96783i 2.42705 + 1.76336i 5.40598 − 16.6379i −3.20169 15.1475 3.42757 − 10.5490i −14.8135 − 45.5912i 2.78115 + 8.55951i −13.0785 + 9.50207i
64.1 −4.39980 − 3.19664i 2.42705 − 1.76336i 6.66758 + 20.5207i 1.38207 −16.3153 −0.898947 − 2.76667i 22.8167 − 70.2226i 2.78115 − 8.55951i −6.08085 − 4.41799i
64.2 −2.81445 − 2.04482i 2.42705 − 1.76336i 1.26771 + 3.90160i 19.4880 −10.4365 7.55577 + 23.2543i −4.19003 + 12.8956i 2.78115 − 8.55951i −54.8479 − 39.8493i
64.3 −2.46379 − 1.79004i 2.42705 − 1.76336i 0.393841 + 1.21212i −14.3902 −9.13622 5.24255 + 16.1349i −6.32925 + 19.4794i 2.78115 − 8.55951i 35.4544 + 25.7591i
64.4 −1.14478 − 0.831728i 2.42705 − 1.76336i −1.85340 − 5.70417i 2.61969 −4.24506 −4.79928 − 14.7707i −6.12072 + 18.8376i 2.78115 − 8.55951i −2.99896 − 2.17887i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.4.f.b ✓ 32
31.d even 5 1 inner 93.4.f.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.4.f.b ✓ 32 1.a even 1 1 trivial
93.4.f.b ✓ 32 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 43 T_{2}^{30} + 23 T_{2}^{29} + 1443 T_{2}^{28} - 191 T_{2}^{27} + 44734 T_{2}^{26} + \cdots + 10703890022400 \) acting on \(S_{4}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display