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Properties

Label 93.4.k.b
Level $93$
Weight $4$
Character orbit 93.k
Analytic conductor $5.487$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [93,4,Mod(23,93)]
 
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([5, 9]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 93.k (of order \(10\), 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: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 5 q^{3} + 134 q^{4} + 24 q^{7} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 5 q^{3} + 134 q^{4} + 24 q^{7} - 25 q^{9} + 78 q^{10} - 240 q^{12} - 10 q^{13} - 40 q^{15} - 18 q^{16} - 93 q^{18} - 258 q^{19} + 805 q^{21} + 410 q^{22} + 315 q^{24} - 3180 q^{25} + 655 q^{27} - 430 q^{28} - 458 q^{31} + 925 q^{33} - 370 q^{34} + 1610 q^{36} + 463 q^{39} + 1676 q^{40} + 3330 q^{42} - 1630 q^{43} + 138 q^{45} - 160 q^{46} - 870 q^{48} - 1456 q^{49} + 845 q^{51} - 90 q^{52} - 280 q^{54} + 340 q^{55} + 1880 q^{58} - 5485 q^{60} - 6510 q^{63} - 648 q^{64} - 3612 q^{66} + 728 q^{67} + 4033 q^{69} + 3308 q^{70} + 233 q^{72} - 2410 q^{73} + 1895 q^{75} - 5440 q^{76} + 917 q^{78} - 2650 q^{79} - 69 q^{81} + 166 q^{82} + 2705 q^{84} + 5380 q^{85} - 3738 q^{87} + 7992 q^{90} - 7330 q^{91} + 3343 q^{93} + 13872 q^{94} + 11120 q^{96} + 8238 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −5.13061 − 1.66704i −0.109573 + 5.19500i 17.0720 + 12.4035i 18.7291i 9.22242 − 26.4708i 15.0325 + 10.9218i −41.5454 − 57.1824i −26.9760 − 1.13846i 31.2221 − 96.0917i
23.2 −5.08957 − 1.65370i −5.10131 + 0.988247i 16.6968 + 12.1309i − 12.5277i 27.5977 + 3.40629i −23.5075 − 17.0792i −39.7544 − 54.7173i 25.0467 − 10.0827i −20.7171 + 63.7608i
23.3 −4.56023 − 1.48171i 2.81805 − 4.36562i 12.1281 + 8.81156i 7.75477i −19.3195 + 15.7327i −17.3242 − 12.5868i −19.7036 − 27.1197i −11.1172 − 24.6050i 11.4903 − 35.3635i
23.4 −4.43481 − 1.44096i 4.92796 + 1.64779i 11.1190 + 8.07846i − 10.4171i −19.4802 − 14.4086i 1.45150 + 1.05458i −15.7432 − 21.6687i 21.5696 + 16.2405i −15.0106 + 46.1977i
23.5 −4.13956 − 1.34503i −2.99354 − 4.24720i 8.85476 + 6.43336i 1.96046i 6.67934 + 21.6080i 13.5314 + 9.83117i −7.53467 − 10.3706i −9.07748 + 25.4283i 2.63688 − 8.11547i
23.6 −3.42859 − 1.11402i −2.70143 + 4.43873i 4.04207 + 2.93674i − 14.3741i 14.2069 − 12.2091i 27.1845 + 19.7507i 6.36485 + 8.76047i −12.4046 − 23.9818i −16.0130 + 49.2829i
23.7 −2.92470 − 0.950292i −5.04753 + 1.23386i 1.17866 + 0.856350i 18.0734i 15.9350 + 1.18796i −7.64897 − 5.55730i 11.8270 + 16.2785i 23.9552 − 12.4559i 17.1750 − 52.8593i
23.8 −2.81532 − 0.914752i 3.55345 + 3.79117i 0.617109 + 0.448356i 7.85472i −6.53611 − 13.9239i −8.96045 − 6.51015i 12.5925 + 17.3321i −1.74597 + 26.9435i 7.18512 − 22.1135i
23.9 −2.65470 − 0.862566i −1.43051 + 4.99536i −0.168701 − 0.122569i − 3.02396i 8.10640 − 12.0273i −16.9658 − 12.3264i 13.4677 + 18.5367i −22.9073 − 14.2918i −2.60837 + 8.02773i
23.10 −2.60443 − 0.846231i 2.17478 − 4.71915i −0.405188 − 0.294386i − 19.5661i −9.65754 + 10.4503i −0.925677 − 0.672544i 13.6832 + 18.8333i −17.5407 − 20.5262i −16.5575 + 50.9586i
23.11 −2.26947 − 0.737396i 4.74190 − 2.12471i −1.86538 − 1.35528i 9.94906i −12.3284 + 1.32532i 17.9675 + 13.0541i 14.4549 + 19.8955i 17.9712 − 20.1504i 7.33640 − 22.5791i
23.12 −1.36821 − 0.444559i −4.53720 − 2.53255i −4.79776 − 3.48578i − 9.57502i 5.08199 + 5.48212i −8.88150 − 6.45279i 11.7795 + 16.2132i 14.1724 + 22.9814i −4.25666 + 13.1007i
23.13 −0.557181 − 0.181039i −0.967430 − 5.10530i −6.19446 − 4.50054i 20.4220i −0.385225 + 3.01972i −4.46473 − 3.24381i 5.39152 + 7.42079i −25.1282 + 9.87804i 3.69718 − 11.3787i
23.14 −0.427120 − 0.138780i 3.44026 + 3.89418i −6.30896 − 4.58373i 1.85232i −0.928973 − 2.14072i 14.2755 + 10.3717i 4.17036 + 5.74000i −3.32920 + 26.7940i 0.257065 − 0.791164i
23.15 0.427120 + 0.138780i 0.494291 − 5.17259i −6.30896 − 4.58373i − 1.85232i 0.928973 − 2.14072i 14.2755 + 10.3717i −4.17036 − 5.74000i −26.5114 − 5.11353i 0.257065 − 0.791164i
23.16 0.557181 + 0.181039i 2.21815 + 4.69891i −6.19446 − 4.50054i − 20.4220i 0.385225 + 3.01972i −4.46473 − 3.24381i −5.39152 − 7.42079i −17.1596 + 20.8458i 3.69718 − 11.3787i
23.17 1.36821 + 0.444559i −2.18208 + 4.71578i −4.79776 − 3.48578i 9.57502i −5.08199 + 5.48212i −8.88150 − 6.45279i −11.7795 − 16.2132i −17.4771 − 20.5804i −4.25666 + 13.1007i
23.18 2.26947 + 0.737396i 5.08515 − 1.06829i −1.86538 − 1.35528i − 9.94906i 12.3284 + 1.32532i 17.9675 + 13.0541i −14.4549 − 19.8955i 24.7175 − 10.8648i 7.33640 − 22.5791i
23.19 2.60443 + 0.846231i 4.53328 + 2.53957i −0.405188 − 0.294386i 19.5661i 9.65754 + 10.4503i −0.925677 − 0.672544i −13.6832 − 18.8333i 14.1012 + 23.0251i −16.5575 + 50.9586i
23.20 2.65470 + 0.862566i −4.09351 − 3.20050i −0.168701 − 0.122569i 3.02396i −8.10640 − 12.0273i −16.9658 − 12.3264i −13.4677 − 18.5367i 6.51358 + 26.2025i −2.60837 + 8.02773i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.f odd 10 1 inner
93.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.4.k.b ✓ 112
3.b odd 2 1 inner 93.4.k.b ✓ 112
31.f odd 10 1 inner 93.4.k.b ✓ 112
93.k even 10 1 inner 93.4.k.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.4.k.b ✓ 112 1.a even 1 1 trivial
93.4.k.b ✓ 112 3.b odd 2 1 inner
93.4.k.b ✓ 112 31.f odd 10 1 inner
93.4.k.b ✓ 112 93.k even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} - 179 T_{2}^{110} + 17605 T_{2}^{108} - 1262418 T_{2}^{106} + 74079046 T_{2}^{104} + \cdots + 41\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display