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Properties

Label 93.4.f.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{3} - 30 q^{4} - 54 q^{5} - 30 q^{6} - 25 q^{7} + 31 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{3} - 30 q^{4} - 54 q^{5} - 30 q^{6} - 25 q^{7} + 31 q^{8} - 72 q^{9} - 7 q^{10} - 5 q^{11} - 135 q^{12} - 34 q^{13} + 141 q^{14} - 12 q^{15} - 258 q^{16} - 96 q^{17} + 180 q^{19} - 190 q^{20} + 195 q^{21} + 187 q^{22} + 110 q^{23} + 93 q^{24} + 1114 q^{25} - 192 q^{26} - 216 q^{27} + 813 q^{28} + 239 q^{29} - 426 q^{30} - 224 q^{31} - 4752 q^{32} - 30 q^{33} + 852 q^{34} + 402 q^{35} + 1350 q^{36} - 746 q^{37} + 1588 q^{38} + 303 q^{39} + 118 q^{40} + 32 q^{41} + 423 q^{42} + 1315 q^{43} + 1686 q^{44} + 279 q^{45} - 1053 q^{46} - 350 q^{47} - 339 q^{48} + 191 q^{49} + 2154 q^{50} - 288 q^{51} - 2204 q^{52} + 624 q^{53} + 135 q^{54} - 1166 q^{55} - 246 q^{56} - 810 q^{57} - 791 q^{58} + 1391 q^{59} + 720 q^{60} - 1618 q^{61} - 1106 q^{62} - 720 q^{63} - 245 q^{64} - 798 q^{65} + 561 q^{66} + 886 q^{67} + 3654 q^{68} + 330 q^{69} - 51 q^{70} + 320 q^{71} + 774 q^{72} - 2194 q^{73} + 842 q^{74} - 1263 q^{75} + 1929 q^{76} + 93 q^{77} + 1989 q^{78} + 1395 q^{79} - 3186 q^{80} - 648 q^{81} - 1447 q^{82} - 2342 q^{83} - 2511 q^{84} + 452 q^{85} - 844 q^{86} - 558 q^{87} + 574 q^{88} + 722 q^{89} - 63 q^{90} + 1219 q^{91} - 3074 q^{92} - 342 q^{93} - 21690 q^{94} - 488 q^{95} + 3399 q^{96} + 7239 q^{97} - 2612 q^{98} + 270 q^{99}+O(q^{100}) \) 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 −1.74543 − 5.37189i 0.927051 − 2.85317i −19.3385 + 14.0503i 2.70847 −16.9450 −14.6494 + 10.6434i 72.6736 + 52.8005i −7.28115 − 5.29007i −4.72745 − 14.5496i
4.2 −1.10548 − 3.40232i 0.927051 − 2.85317i −3.88157 + 2.82013i 11.0457 −10.7322 23.6480 − 17.1812i −9.26752 − 6.73325i −7.28115 − 5.29007i −12.2108 − 37.5811i
4.3 −1.04055 − 3.20248i 0.927051 − 2.85317i −2.70097 + 1.96237i −8.78017 −10.1018 −0.411143 + 0.298713i −12.6986 − 9.22605i −7.28115 − 5.29007i 9.13618 + 28.1183i
4.4 −0.428291 − 1.31814i 0.927051 − 2.85317i 4.91807 − 3.57318i −17.7845 −4.15794 −21.8369 + 15.8654i −15.7866 − 11.4696i −7.28115 − 5.29007i 7.61695 + 23.4426i
4.5 −0.0431063 − 0.132668i 0.927051 − 2.85317i 6.45639 − 4.69084i 9.64187 −0.418485 −3.21428 + 2.33531i −1.80347 − 1.31029i −7.28115 − 5.29007i −0.415626 − 1.27916i
4.6 0.633444 + 1.94954i 0.927051 − 2.85317i 3.07268 − 2.23243i −15.4983 6.14960 23.3733 − 16.9817i 19.5656 + 14.2153i −7.28115 − 5.29007i −9.81730 − 30.2145i
4.7 1.19491 + 3.67757i 0.927051 − 2.85317i −5.62456 + 4.08648i 11.9998 11.6005 8.00882 − 5.81875i 3.27742 + 2.38119i −7.28115 − 5.29007i 14.3387 + 44.1301i
4.8 1.41647 + 4.35943i 0.927051 − 2.85317i −10.5261 + 7.64768i −14.6591 13.7513 −15.0191 + 10.9120i −18.5826 − 13.5011i −7.28115 − 5.29007i −20.7641 − 63.9054i
16.1 −3.78710 + 2.75149i −2.42705 − 1.76336i 4.29929 − 13.2319i −19.0221 14.0433 2.16251 − 6.65551i 8.55312 + 26.3238i 2.78115 + 8.55951i 72.0387 − 52.3392i
16.2 −3.29167 + 2.39154i −2.42705 − 1.76336i 2.64350 − 8.13587i 9.99239 12.2062 −8.53930 + 26.2813i 0.697250 + 2.14592i 2.78115 + 8.55951i −32.8917 + 23.8972i
16.3 −2.09804 + 1.52431i −2.42705 − 1.76336i −0.393905 + 1.21231i 9.32353 7.77995 5.04717 − 15.5336i −7.43255 − 22.8751i 2.78115 + 8.55951i −19.5611 + 14.2120i
16.4 −0.820690 + 0.596267i −2.42705 − 1.76336i −2.15414 + 6.62975i −1.98616 3.04329 3.29129 − 10.1295i −4.69303 − 14.4436i 2.78115 + 8.55951i 1.63002 − 1.18428i
16.5 1.30749 − 0.949949i −2.42705 − 1.76336i −1.66500 + 5.12435i 9.77321 −4.84845 −9.52849 + 29.3257i 6.68623 + 20.5781i 2.78115 + 8.55951i 12.7784 − 9.28405i
16.6 1.55833 − 1.13219i −2.42705 − 1.76336i −1.32560 + 4.07978i −9.73219 −5.77861 −1.83205 + 5.63846i 7.31522 + 22.5139i 2.78115 + 8.55951i −15.1660 + 11.0187i
16.7 3.80189 − 2.76223i −2.42705 − 1.76336i 4.35228 − 13.3949i −19.2063 −14.0982 3.74349 − 11.5213i −8.83553 − 27.1930i 2.78115 + 8.55951i −73.0202 + 53.0523i
16.8 4.44782 − 3.23153i −2.42705 − 1.76336i 6.86818 − 21.1381i 15.1839 −16.4934 −6.74380 + 20.7553i −24.1686 − 74.3834i 2.78115 + 8.55951i 67.5354 − 49.0673i
64.1 −3.78710 − 2.75149i −2.42705 + 1.76336i 4.29929 + 13.2319i −19.0221 14.0433 2.16251 + 6.65551i 8.55312 − 26.3238i 2.78115 − 8.55951i 72.0387 + 52.3392i
64.2 −3.29167 − 2.39154i −2.42705 + 1.76336i 2.64350 + 8.13587i 9.99239 12.2062 −8.53930 − 26.2813i 0.697250 − 2.14592i 2.78115 − 8.55951i −32.8917 − 23.8972i
64.3 −2.09804 − 1.52431i −2.42705 + 1.76336i −0.393905 − 1.21231i 9.32353 7.77995 5.04717 + 15.5336i −7.43255 + 22.8751i 2.78115 − 8.55951i −19.5611 − 14.2120i
64.4 −0.820690 − 0.596267i −2.42705 + 1.76336i −2.15414 − 6.62975i −1.98616 3.04329 3.29129 + 10.1295i −4.69303 + 14.4436i 2.78115 − 8.55951i 1.63002 + 1.18428i
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.a ✓ 32
31.d even 5 1 inner 93.4.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.4.f.a ✓ 32 1.a even 1 1 trivial
93.4.f.a ✓ 32 31.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 47 T_{2}^{30} - 37 T_{2}^{29} + 1695 T_{2}^{28} + 1349 T_{2}^{27} + 55236 T_{2}^{26} + \cdots + 3710030708736 \) acting on \(S_{4}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display