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Properties

Label 693.2.bu.f
Level $693$
Weight $2$
Character orbit 693.bu
Analytic conductor $5.534$
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] = [693,2,Mod(118,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.bu (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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 231)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 12 q^{4} + 10 q^{7} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 12 q^{4} + 10 q^{7} + 20 q^{8} + 16 q^{11} - 12 q^{14} - 16 q^{16} - 40 q^{22} + 24 q^{23} + 44 q^{25} - 30 q^{28} + 40 q^{29} + 40 q^{35} + 32 q^{37} - 22 q^{44} - 70 q^{46} - 50 q^{49} + 64 q^{53} - 80 q^{56} + 2 q^{58} + 72 q^{64} - 8 q^{67} - 26 q^{70} - 68 q^{71} - 80 q^{74} - 90 q^{77} + 40 q^{79} - 40 q^{85} + 62 q^{86} + 140 q^{88} + 54 q^{91} - 18 q^{92} - 20 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1 −1.55625 − 2.14199i 0 −1.54818 + 4.76480i −0.689984 + 0.949681i 0 −0.00284148 + 2.64575i 7.57938 − 2.46269i 0 3.10799
118.2 −1.55625 − 2.14199i 0 −1.54818 + 4.76480i 0.689984 − 0.949681i 0 1.55283 − 2.14213i 7.57938 − 2.46269i 0 −3.10799
118.3 −1.29930 − 1.78834i 0 −0.891932 + 2.74508i −2.07229 + 2.85226i 0 −2.48888 + 0.897495i 1.86340 − 0.605454i 0 7.79336
118.4 −1.29930 − 1.78834i 0 −0.891932 + 2.74508i 2.07229 − 2.85226i 0 −1.48601 − 2.18901i 1.86340 − 0.605454i 0 −7.79336
118.5 −0.895378 − 1.23238i 0 −0.0990298 + 0.304782i −1.73657 + 2.39018i 0 1.93204 − 1.80755i −2.43323 + 0.790603i 0 4.50049
118.6 −0.895378 − 1.23238i 0 −0.0990298 + 0.304782i 1.73657 − 2.39018i 0 0.500603 + 2.59796i −2.43323 + 0.790603i 0 −4.50049
118.7 −0.588635 − 0.810187i 0 0.308123 − 0.948304i −0.977076 + 1.34483i 0 −0.674930 + 2.55822i −2.85454 + 0.927496i 0 1.66470
118.8 −0.588635 − 0.810187i 0 0.308123 − 0.948304i 0.977076 − 1.34483i 0 0.957652 − 2.46635i −2.85454 + 0.927496i 0 −1.66470
118.9 0.0467407 + 0.0643331i 0 0.616080 − 1.89610i −1.04717 + 1.44131i 0 −0.522131 + 2.59372i 0.302034 − 0.0981368i 0 −0.141669
118.10 0.0467407 + 0.0643331i 0 0.616080 − 1.89610i 1.04717 − 1.44131i 0 1.10214 − 2.40526i 0.302034 − 0.0981368i 0 0.141669
118.11 0.811699 + 1.11721i 0 0.0287353 − 0.0884381i −1.25522 + 1.72766i 0 2.62494 − 0.331189i 2.74884 − 0.893153i 0 −2.94902
118.12 0.811699 + 1.11721i 0 0.0287353 − 0.0884381i 1.25522 − 1.72766i 0 1.92895 + 1.81084i 2.74884 − 0.893153i 0 2.94902
118.13 0.917545 + 1.26289i 0 −0.134974 + 0.415407i −0.118521 + 0.163130i 0 −2.40543 + 1.10177i 2.32078 − 0.754067i 0 −0.314765
118.14 0.917545 + 1.26289i 0 −0.134974 + 0.415407i 0.118521 − 0.163130i 0 −1.29844 − 2.30523i 2.32078 − 0.754067i 0 0.314765
118.15 1.44554 + 1.98962i 0 −1.25096 + 3.85006i −1.57908 + 2.17341i 0 2.53314 − 0.763690i −4.79060 + 1.55656i 0 −6.60689
118.16 1.44554 + 1.98962i 0 −1.25096 + 3.85006i 1.57908 − 2.17341i 0 1.60046 + 2.10678i −4.79060 + 1.55656i 0 6.60689
244.1 −2.20622 − 0.716844i 0 2.73550 + 1.98746i −0.0209622 + 0.00681102i 0 1.69347 + 2.03277i −1.88338 − 2.59224i 0 0.0511296
244.2 −2.20622 − 0.716844i 0 2.73550 + 1.98746i 0.0209622 − 0.00681102i 0 −2.45659 − 0.982422i −1.88338 − 2.59224i 0 −0.0511296
244.3 −1.94435 − 0.631758i 0 1.76335 + 1.28115i −2.31266 + 0.751430i 0 −1.00115 − 2.44902i −0.215840 − 0.297079i 0 4.97135
244.4 −1.94435 − 0.631758i 0 1.76335 + 1.28115i 2.31266 − 0.751430i 0 2.63853 + 0.195359i −0.215840 − 0.297079i 0 −4.97135
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.bu.f 64
3.b odd 2 1 231.2.w.a ✓ 64
7.b odd 2 1 inner 693.2.bu.f 64
11.d odd 10 1 inner 693.2.bu.f 64
21.c even 2 1 231.2.w.a ✓ 64
33.f even 10 1 231.2.w.a ✓ 64
77.l even 10 1 inner 693.2.bu.f 64
231.r odd 10 1 231.2.w.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.w.a ✓ 64 3.b odd 2 1
231.2.w.a ✓ 64 21.c even 2 1
231.2.w.a ✓ 64 33.f even 10 1
231.2.w.a ✓ 64 231.r odd 10 1
693.2.bu.f 64 1.a even 1 1 trivial
693.2.bu.f 64 7.b odd 2 1 inner
693.2.bu.f 64 11.d odd 10 1 inner
693.2.bu.f 64 77.l even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 11 T_{2}^{30} - 10 T_{2}^{29} + 95 T_{2}^{28} + 110 T_{2}^{27} - 690 T_{2}^{26} + \cdots + 961 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display