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Properties

Label 93.3.h.b
Level $93$
Weight $3$
Character orbit 93.h
Analytic conductor $2.534$
Analytic rank $0$
Dimension $36$
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] = [93,3,Mod(5,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 93.h (of order \(6\), degree \(2\), 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: \(2.53406645855\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} - 84 q^{4} - 14 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 2 q^{3} - 84 q^{4} - 14 q^{7} - 2 q^{9} - 20 q^{10} - 64 q^{12} + 16 q^{13} + 24 q^{15} + 60 q^{16} - 34 q^{18} - 16 q^{19} - 8 q^{21} - 18 q^{22} + 8 q^{24} + 114 q^{25} + 172 q^{27} + 54 q^{28} - 120 q^{30} + 38 q^{31} + 48 q^{33} - 28 q^{34} + 186 q^{36} + 228 q^{37} - 96 q^{39} + 300 q^{40} - 16 q^{42} - 80 q^{43} - 6 q^{45} - 272 q^{46} + 182 q^{48} + 12 q^{49} - 84 q^{51} - 188 q^{52} - 100 q^{54} - 184 q^{55} + 50 q^{57} - 20 q^{58} - 740 q^{60} - 280 q^{61} + 4 q^{63} - 1020 q^{64} - 316 q^{66} + 8 q^{67} + 30 q^{69} + 440 q^{70} + 572 q^{72} + 372 q^{73} - 20 q^{75} + 600 q^{76} + 724 q^{78} - 520 q^{79} + 402 q^{81} + 200 q^{82} - 324 q^{84} - 392 q^{85} - 146 q^{87} + 46 q^{88} - 72 q^{90} + 520 q^{91} - 234 q^{93} + 776 q^{94} + 460 q^{96} + 804 q^{97} - 568 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 − 3.93005i 2.42349 − 1.76825i −11.4453 5.08783 − 2.93746i −6.94930 − 9.52443i −2.37089 + 4.10650i 29.2603i 2.74659 − 8.57066i −11.5444 − 19.9954i
5.2 − 3.36524i −2.90793 + 0.737527i −7.32483 −2.02074 + 1.16668i 2.48195 + 9.78587i −1.83962 + 3.18631i 11.1888i 7.91211 − 4.28935i 3.92614 + 6.80028i
5.3 − 2.86392i −0.510403 − 2.95626i −4.20205 −6.09010 + 3.51612i −8.46651 + 1.46175i 0.307822 − 0.533164i 0.578665i −8.47898 + 3.01777i 10.0699 + 17.4416i
5.4 − 2.82153i −0.314058 + 2.98352i −3.96101 4.29989 − 2.48254i 8.41807 + 0.886123i 3.13699 − 5.43342i − 0.110006i −8.80274 − 1.87399i −7.00455 − 12.1322i
5.5 − 2.32878i 2.99359 − 0.196035i −1.42321 −3.71912 + 2.14724i −0.456522 − 6.97140i 4.74004 − 8.20998i − 6.00078i 8.92314 − 1.17370i 5.00043 + 8.66100i
5.6 − 2.09121i −1.45488 − 2.62361i −0.373152 6.15174 − 3.55171i −5.48652 + 3.04245i −0.812682 + 1.40761i − 7.58450i −4.76668 + 7.63405i −7.42737 − 12.8646i
5.7 − 1.66353i 2.64779 + 1.41039i 1.23266 2.95746 − 1.70749i 2.34623 − 4.40469i −6.01712 + 10.4220i − 8.70470i 5.02161 + 7.46883i −2.84046 − 4.91982i
5.8 − 1.03966i −0.838897 + 2.88032i 2.91910 −8.09441 + 4.67331i 2.99457 + 0.872172i −4.21247 + 7.29622i − 7.19354i −7.59250 − 4.83259i 4.85867 + 8.41547i
5.9 − 0.649797i −2.99701 + 0.133827i 3.57776 0.102380 − 0.0591093i 0.0869607 + 1.94745i 3.56794 − 6.17985i − 4.92401i 8.96418 − 0.802165i −0.0384091 − 0.0665265i
5.10 0.649797i 1.38261 + 2.66240i 3.57776 −0.102380 + 0.0591093i −1.73002 + 0.898416i 3.56794 − 6.17985i 4.92401i −5.17679 + 7.36213i −0.0384091 − 0.0665265i
5.11 1.03966i −2.07498 + 2.16667i 2.91910 8.09441 − 4.67331i −2.25261 − 2.15729i −4.21247 + 7.29622i 7.19354i −0.388894 − 8.99159i 4.85867 + 8.41547i
5.12 1.66353i −2.54533 − 1.58786i 1.23266 −2.95746 + 1.70749i 2.64146 − 4.23423i −6.01712 + 10.4220i 8.70470i 3.95739 + 8.08326i −2.84046 − 4.91982i
5.13 2.09121i 2.99955 − 0.0518471i −0.373152 −6.15174 + 3.55171i 0.108423 + 6.27269i −0.812682 + 1.40761i 7.58450i 8.99462 − 0.311036i −7.42737 − 12.8646i
5.14 2.32878i −1.32702 − 2.69054i −1.42321 3.71912 − 2.14724i 6.26567 − 3.09034i 4.74004 − 8.20998i 6.00078i −5.47802 + 7.14082i 5.00043 + 8.66100i
5.15 2.82153i −2.42677 + 1.76374i −3.96101 −4.29989 + 2.48254i −4.97644 − 6.84720i 3.13699 − 5.43342i 0.110006i 2.77844 − 8.56039i −7.00455 − 12.1322i
5.16 2.86392i 2.81540 − 1.03611i −4.20205 6.09010 − 3.51612i 2.96734 + 8.06309i 0.307822 − 0.533164i − 0.578665i 6.85295 − 5.83413i 10.0699 + 17.4416i
5.17 3.36524i 0.815248 + 2.88710i −7.32483 2.02074 − 1.16668i −9.71579 + 2.74350i −1.83962 + 3.18631i − 11.1888i −7.67074 + 4.70741i 3.92614 + 6.80028i
5.18 3.93005i 0.319604 − 2.98293i −11.4453 −5.08783 + 2.93746i 11.7230 + 1.25606i −2.37089 + 4.10650i − 29.2603i −8.79571 − 1.90671i −11.5444 − 19.9954i
56.1 − 3.93005i 0.319604 + 2.98293i −11.4453 −5.08783 − 2.93746i 11.7230 − 1.25606i −2.37089 − 4.10650i 29.2603i −8.79571 + 1.90671i −11.5444 + 19.9954i
56.2 − 3.36524i 0.815248 − 2.88710i −7.32483 2.02074 + 1.16668i −9.71579 − 2.74350i −1.83962 − 3.18631i 11.1888i −7.67074 − 4.70741i 3.92614 − 6.80028i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.c even 3 1 inner
93.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.3.h.b ✓ 36
3.b odd 2 1 inner 93.3.h.b ✓ 36
31.c even 3 1 inner 93.3.h.b ✓ 36
93.h odd 6 1 inner 93.3.h.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.3.h.b ✓ 36 1.a even 1 1 trivial
93.3.h.b ✓ 36 3.b odd 2 1 inner
93.3.h.b ✓ 36 31.c even 3 1 inner
93.3.h.b ✓ 36 93.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 57 T_{2}^{16} + 1347 T_{2}^{14} + 17201 T_{2}^{12} + 129461 T_{2}^{10} + 586383 T_{2}^{8} + \cdots + 342116 \) acting on \(S_{3}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display