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Properties

Label 693.2.cg.b
Level $693$
Weight $2$
Character orbit 693.cg
Analytic conductor $5.534$
Analytic rank $0$
Dimension $128$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(19,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 25, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.cg (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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 20 q^{4} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 20 q^{4} + 10 q^{7} + 76 q^{22} - 12 q^{25} + 30 q^{28} - 18 q^{31} + 16 q^{37} - 90 q^{40} - 70 q^{46} + 58 q^{49} - 20 q^{58} - 30 q^{61} - 96 q^{64} - 40 q^{67} - 118 q^{70} - 90 q^{73} - 10 q^{79} + 24 q^{82} - 180 q^{85} - 56 q^{88} - 56 q^{91} - 120 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.44091 + 0.256551i 0 3.93595 − 0.836613i −0.822522 − 1.84741i 0 −0.673347 − 2.55863i −4.72422 + 1.53499i 0 2.48166 + 4.29836i
19.2 −2.34777 + 0.246761i 0 3.49484 − 0.742851i 0.712786 + 1.60094i 0 2.64216 + 0.137763i −3.53145 + 1.14744i 0 −2.06851 − 3.58276i
19.3 −2.24029 + 0.235464i 0 3.00716 − 0.639191i 1.49939 + 3.36768i 0 −2.46592 − 0.958768i −2.30164 + 0.747850i 0 −4.15203 − 7.19152i
19.4 −1.59354 + 0.167488i 0 0.555038 − 0.117977i −0.461594 − 1.03676i 0 −0.982705 + 2.45648i 2.18308 − 0.709326i 0 0.909216 + 1.57481i
19.5 −1.30213 + 0.136859i 0 −0.279493 + 0.0594081i −1.77962 − 3.99709i 0 2.52240 − 0.798438i 2.84624 − 0.924799i 0 2.86433 + 4.96116i
19.6 −0.921700 + 0.0968746i 0 −1.11615 + 0.237245i 0.746663 + 1.67703i 0 1.46149 − 2.20546i 2.76861 − 0.899575i 0 −0.850662 − 1.47339i
19.7 −0.743253 + 0.0781190i 0 −1.40997 + 0.299699i 0.772273 + 1.73455i 0 1.16296 + 2.37645i 2.44609 − 0.794784i 0 −0.709496 − 1.22888i
19.8 −0.107490 + 0.0112976i 0 −1.94487 + 0.413395i 0.916676 + 2.05889i 0 −2.49791 − 0.872043i 0.409967 − 0.133206i 0 −0.121794 − 0.210953i
19.9 0.107490 − 0.0112976i 0 −1.94487 + 0.413395i −0.916676 − 2.05889i 0 −2.49791 − 0.872043i −0.409967 + 0.133206i 0 −0.121794 − 0.210953i
19.10 0.743253 − 0.0781190i 0 −1.40997 + 0.299699i −0.772273 − 1.73455i 0 1.16296 + 2.37645i −2.44609 + 0.794784i 0 −0.709496 − 1.22888i
19.11 0.921700 − 0.0968746i 0 −1.11615 + 0.237245i −0.746663 − 1.67703i 0 1.46149 − 2.20546i −2.76861 + 0.899575i 0 −0.850662 − 1.47339i
19.12 1.30213 − 0.136859i 0 −0.279493 + 0.0594081i 1.77962 + 3.99709i 0 2.52240 − 0.798438i −2.84624 + 0.924799i 0 2.86433 + 4.96116i
19.13 1.59354 − 0.167488i 0 0.555038 − 0.117977i 0.461594 + 1.03676i 0 −0.982705 + 2.45648i −2.18308 + 0.709326i 0 0.909216 + 1.57481i
19.14 2.24029 − 0.235464i 0 3.00716 − 0.639191i −1.49939 − 3.36768i 0 −2.46592 − 0.958768i 2.30164 − 0.747850i 0 −4.15203 − 7.19152i
19.15 2.34777 − 0.246761i 0 3.49484 − 0.742851i −0.712786 − 1.60094i 0 2.64216 + 0.137763i 3.53145 − 1.14744i 0 −2.06851 − 3.58276i
19.16 2.44091 − 0.256551i 0 3.93595 − 0.836613i 0.822522 + 1.84741i 0 −0.673347 − 2.55863i 4.72422 − 1.53499i 0 2.48166 + 4.29836i
73.1 −2.44091 − 0.256551i 0 3.93595 + 0.836613i −0.822522 + 1.84741i 0 −0.673347 + 2.55863i −4.72422 − 1.53499i 0 2.48166 − 4.29836i
73.2 −2.34777 − 0.246761i 0 3.49484 + 0.742851i 0.712786 − 1.60094i 0 2.64216 − 0.137763i −3.53145 − 1.14744i 0 −2.06851 + 3.58276i
73.3 −2.24029 − 0.235464i 0 3.00716 + 0.639191i 1.49939 − 3.36768i 0 −2.46592 + 0.958768i −2.30164 − 0.747850i 0 −4.15203 + 7.19152i
73.4 −1.59354 − 0.167488i 0 0.555038 + 0.117977i −0.461594 + 1.03676i 0 −0.982705 − 2.45648i 2.18308 + 0.709326i 0 0.909216 − 1.57481i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
11.d odd 10 1 inner
21.g even 6 1 inner
33.f even 10 1 inner
77.n even 30 1 inner
231.bf odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.cg.b ✓ 128
3.b odd 2 1 inner 693.2.cg.b ✓ 128
7.d odd 6 1 inner 693.2.cg.b ✓ 128
11.d odd 10 1 inner 693.2.cg.b ✓ 128
21.g even 6 1 inner 693.2.cg.b ✓ 128
33.f even 10 1 inner 693.2.cg.b ✓ 128
77.n even 30 1 inner 693.2.cg.b ✓ 128
231.bf odd 30 1 inner 693.2.cg.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.cg.b ✓ 128 1.a even 1 1 trivial
693.2.cg.b ✓ 128 3.b odd 2 1 inner
693.2.cg.b ✓ 128 7.d odd 6 1 inner
693.2.cg.b ✓ 128 11.d odd 10 1 inner
693.2.cg.b ✓ 128 21.g even 6 1 inner
693.2.cg.b ✓ 128 33.f even 10 1 inner
693.2.cg.b ✓ 128 77.n even 30 1 inner
693.2.cg.b ✓ 128 231.bf odd 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{128} + 26 T_{2}^{126} + 291 T_{2}^{124} + 1390 T_{2}^{122} - 5089 T_{2}^{120} - 147014 T_{2}^{118} + \cdots + 214358881 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display