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Properties

Label 93.2.p.b
Level $93$
Weight $2$
Character orbit 93.p
Analytic conductor $0.743$
Analytic rank $0$
Dimension $64$
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] = [93,2,Mod(11,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 23]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 93.p (of order \(30\), degree \(8\), 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: \(0.742608738798\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 10 q^{3} + 12 q^{4} - 9 q^{6} - 26 q^{7} - 8 q^{9} - 36 q^{10} + 15 q^{12} - 32 q^{13} - 20 q^{15} - 24 q^{16} - 6 q^{18} + 5 q^{21} - 24 q^{22} - 48 q^{24} + 38 q^{25} + 5 q^{27} + 76 q^{28} + 30 q^{31} - 7 q^{33} - 4 q^{34} - 5 q^{36} + 48 q^{37} - 7 q^{39} + 8 q^{40} + 15 q^{42} - 92 q^{43} - 63 q^{45} - 70 q^{46} + 12 q^{48} - 2 q^{49} + 58 q^{51} + 72 q^{52} + 100 q^{54} + 10 q^{55} + 93 q^{57} + 50 q^{58} + 85 q^{60} - 18 q^{63} + 46 q^{64} + 6 q^{66} - 46 q^{67} + 110 q^{69} - 158 q^{70} + 163 q^{72} - 30 q^{73} + 55 q^{75} + 34 q^{76} - 11 q^{78} + 24 q^{79} - 108 q^{81} - 116 q^{82} - 80 q^{84} - 130 q^{85} - 9 q^{87} - 222 q^{88} - 93 q^{90} - 20 q^{91} - 121 q^{93} + 128 q^{94} - 122 q^{96} + 18 q^{97} - 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.49541 + 0.810809i 0.448212 + 1.67305i 3.95165 − 2.87104i 2.63849 − 1.52333i −2.47500 − 3.81155i 0.249973 + 2.37834i −4.44863 + 6.12302i −2.59821 + 1.49976i −5.34899 + 5.94065i
11.2 −1.72336 + 0.559954i −1.73134 + 0.0496529i 1.03839 − 0.754435i 0.322909 − 0.186432i 2.95592 − 1.05504i −0.413561 − 3.93477i 0.763119 − 1.05034i 2.99507 − 0.171932i −0.452096 + 0.502103i
11.3 −1.25999 + 0.409396i 1.20011 − 1.24889i −0.198061 + 0.143900i 1.61219 − 0.930798i −1.00084 + 2.06491i 0.00741560 + 0.0705547i 1.74808 − 2.40602i −0.119450 − 2.99762i −1.65028 + 1.83282i
11.4 −1.13134 + 0.367594i 1.09350 + 1.34323i −0.473233 + 0.343824i −3.25867 + 1.88139i −1.73088 − 1.11768i −0.121266 − 1.15377i 1.80741 − 2.48769i −0.608530 + 2.93763i 2.99507 − 3.32636i
11.5 1.13134 − 0.367594i −1.22157 − 1.22791i −0.473233 + 0.343824i 3.25867 − 1.88139i −1.83338 − 0.940142i −0.121266 − 1.15377i −1.80741 + 2.48769i −0.0155363 + 2.99996i 2.99507 − 3.32636i
11.6 1.25999 − 0.409396i 1.36749 − 1.06300i −0.198061 + 0.143900i −1.61219 + 0.930798i 1.28784 − 1.89921i 0.00741560 + 0.0705547i −1.74808 + 2.40602i 0.740080 − 2.90728i −1.65028 + 1.83282i
11.7 1.72336 − 0.559954i −0.230355 + 1.71666i 1.03839 − 0.754435i −0.322909 + 0.186432i 0.564268 + 3.08742i −0.413561 − 3.93477i −0.763119 + 1.05034i −2.89387 − 0.790885i −0.452096 + 0.502103i
11.8 2.49541 − 0.810809i −1.61704 − 0.620638i 3.95165 − 2.87104i −2.63849 + 1.52333i −4.53840 − 0.237641i 0.249973 + 2.37834i 4.44863 − 6.12302i 2.22962 + 2.00719i −5.34899 + 5.94065i
17.1 −2.49541 − 0.810809i 0.448212 − 1.67305i 3.95165 + 2.87104i 2.63849 + 1.52333i −2.47500 + 3.81155i 0.249973 − 2.37834i −4.44863 − 6.12302i −2.59821 − 1.49976i −5.34899 − 5.94065i
17.2 −1.72336 − 0.559954i −1.73134 − 0.0496529i 1.03839 + 0.754435i 0.322909 + 0.186432i 2.95592 + 1.05504i −0.413561 + 3.93477i 0.763119 + 1.05034i 2.99507 + 0.171932i −0.452096 − 0.502103i
17.3 −1.25999 − 0.409396i 1.20011 + 1.24889i −0.198061 − 0.143900i 1.61219 + 0.930798i −1.00084 − 2.06491i 0.00741560 − 0.0705547i 1.74808 + 2.40602i −0.119450 + 2.99762i −1.65028 − 1.83282i
17.4 −1.13134 − 0.367594i 1.09350 − 1.34323i −0.473233 − 0.343824i −3.25867 − 1.88139i −1.73088 + 1.11768i −0.121266 + 1.15377i 1.80741 + 2.48769i −0.608530 − 2.93763i 2.99507 + 3.32636i
17.5 1.13134 + 0.367594i −1.22157 + 1.22791i −0.473233 − 0.343824i 3.25867 + 1.88139i −1.83338 + 0.940142i −0.121266 + 1.15377i −1.80741 − 2.48769i −0.0155363 − 2.99996i 2.99507 + 3.32636i
17.6 1.25999 + 0.409396i 1.36749 + 1.06300i −0.198061 − 0.143900i −1.61219 − 0.930798i 1.28784 + 1.89921i 0.00741560 − 0.0705547i −1.74808 − 2.40602i 0.740080 + 2.90728i −1.65028 − 1.83282i
17.7 1.72336 + 0.559954i −0.230355 − 1.71666i 1.03839 + 0.754435i −0.322909 − 0.186432i 0.564268 − 3.08742i −0.413561 + 3.93477i −0.763119 − 1.05034i −2.89387 + 0.790885i −0.452096 − 0.502103i
17.8 2.49541 + 0.810809i −1.61704 + 0.620638i 3.95165 + 2.87104i −2.63849 − 1.52333i −4.53840 + 0.237641i 0.249973 − 2.37834i 4.44863 + 6.12302i 2.22962 − 2.00719i −5.34899 − 5.94065i
44.1 −1.59081 − 2.18957i −1.15089 − 1.29440i −1.64548 + 5.06426i −1.13653 + 0.656178i −1.00332 + 4.57908i −1.89531 − 2.10496i 8.55821 − 2.78073i −0.350919 + 2.97941i 3.24476 + 1.44466i
44.2 −1.13925 − 1.56805i 1.68300 − 0.409301i −0.542842 + 1.67070i 1.08227 − 0.624852i −2.55916 − 2.17272i 0.706282 + 0.784405i −0.448535 + 0.145738i 2.66495 − 1.37770i −2.21278 − 0.985195i
44.3 −0.888042 − 1.22228i −1.27658 + 1.17061i −0.0873279 + 0.268768i 3.07140 − 1.77328i 2.56448 + 0.520794i −2.50044 − 2.77703i −2.46770 + 0.801805i 0.259328 − 2.98877i −4.89498 − 2.17939i
44.4 −0.399926 − 0.550451i 0.304138 − 1.70514i 0.474979 − 1.46183i −1.60778 + 0.928255i −1.06023 + 0.514516i 0.402313 + 0.446814i −2.28881 + 0.743680i −2.81500 − 1.03720i 1.15395 + 0.513773i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.h odd 30 1 inner
93.p even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.2.p.b ✓ 64
3.b odd 2 1 inner 93.2.p.b ✓ 64
31.h odd 30 1 inner 93.2.p.b ✓ 64
93.p even 30 1 inner 93.2.p.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.p.b ✓ 64 1.a even 1 1 trivial
93.2.p.b ✓ 64 3.b odd 2 1 inner
93.2.p.b ✓ 64 31.h odd 30 1 inner
93.2.p.b ✓ 64 93.p even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 22 T_{2}^{62} + 313 T_{2}^{60} - 3623 T_{2}^{58} + 36308 T_{2}^{56} - 305891 T_{2}^{54} + \cdots + 492884401 \) acting on \(S_{2}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display