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Properties

Label 770.2.l.b
Level $770$
Weight $2$
Character orbit 770.l
Analytic conductor $6.148$
Analytic rank $0$
Dimension $24$
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] = [770,2,Mod(573,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.573");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.l (of order \(4\), 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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{11} - 24 q^{15} - 24 q^{16} - 24 q^{18} - 28 q^{21} + 16 q^{23} + 16 q^{25} - 24 q^{30} + 16 q^{35} - 64 q^{36} - 8 q^{37} + 16 q^{42} + 40 q^{43} - 16 q^{46} + 16 q^{50} - 24 q^{51} - 88 q^{53} + 4 q^{56} - 112 q^{57} + 8 q^{58} - 8 q^{60} + 48 q^{63} - 8 q^{65} - 88 q^{67} - 28 q^{70} + 40 q^{71} - 24 q^{72} + 64 q^{78} - 136 q^{81} + 64 q^{85} + 96 q^{86} + 16 q^{92} - 104 q^{93} + 64 q^{95} - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
573.1 −0.707107 + 0.707107i −2.35787 + 2.35787i − 1.00000i 2.00068 − 0.998647i − 3.33453i 0.252130 + 2.63371i 0.707107 + 0.707107i − 8.11912i −0.708541 + 2.12084i
573.2 −0.707107 + 0.707107i −0.878808 + 0.878808i − 1.00000i 0.142676 + 2.23151i − 1.24282i 0.993466 − 2.45215i 0.707107 + 0.707107i 1.45539i −1.67880 − 1.47703i
573.3 −0.707107 + 0.707107i −0.216369 + 0.216369i − 1.00000i 1.84832 + 1.25846i − 0.305992i 1.68383 + 2.04076i 0.707107 + 0.707107i 2.90637i −2.19682 + 0.417093i
573.4 −0.707107 + 0.707107i 0.216369 − 0.216369i − 1.00000i −1.84832 − 1.25846i 0.305992i −2.04076 − 1.68383i 0.707107 + 0.707107i 2.90637i 2.19682 − 0.417093i
573.5 −0.707107 + 0.707107i 0.878808 − 0.878808i − 1.00000i −0.142676 − 2.23151i 1.24282i 2.45215 − 0.993466i 0.707107 + 0.707107i 1.45539i 1.67880 + 1.47703i
573.6 −0.707107 + 0.707107i 2.35787 − 2.35787i − 1.00000i −2.00068 + 0.998647i 3.33453i −2.63371 − 0.252130i 0.707107 + 0.707107i − 8.11912i 0.708541 − 2.12084i
573.7 0.707107 − 0.707107i −2.10618 + 2.10618i − 1.00000i −2.14674 − 0.625704i 2.97859i −2.64445 − 0.0831086i −0.707107 − 0.707107i − 5.87198i −1.96041 + 1.07553i
573.8 0.707107 − 0.707107i −1.96534 + 1.96534i − 1.00000i 1.73312 + 1.41290i 2.77941i 2.19900 − 1.47118i −0.707107 − 0.707107i − 4.72512i 2.22457 − 0.226432i
573.9 0.707107 − 0.707107i −1.52406 + 1.52406i − 1.00000i −1.39587 + 1.74687i 2.15535i 0.727740 + 2.54370i −0.707107 − 0.707107i − 1.64554i 0.248193 + 2.22225i
573.10 0.707107 − 0.707107i 1.52406 − 1.52406i − 1.00000i 1.39587 − 1.74687i − 2.15535i −2.54370 − 0.727740i −0.707107 − 0.707107i − 1.64554i −0.248193 − 2.22225i
573.11 0.707107 − 0.707107i 1.96534 − 1.96534i − 1.00000i −1.73312 − 1.41290i − 2.77941i 1.47118 − 2.19900i −0.707107 − 0.707107i − 4.72512i −2.22457 + 0.226432i
573.12 0.707107 − 0.707107i 2.10618 − 2.10618i − 1.00000i 2.14674 + 0.625704i − 2.97859i 0.0831086 + 2.64445i −0.707107 − 0.707107i − 5.87198i 1.96041 − 1.07553i
727.1 −0.707107 − 0.707107i −2.35787 − 2.35787i 1.00000i 2.00068 + 0.998647i 3.33453i 0.252130 − 2.63371i 0.707107 − 0.707107i 8.11912i −0.708541 − 2.12084i
727.2 −0.707107 − 0.707107i −0.878808 − 0.878808i 1.00000i 0.142676 − 2.23151i 1.24282i 0.993466 + 2.45215i 0.707107 − 0.707107i − 1.45539i −1.67880 + 1.47703i
727.3 −0.707107 − 0.707107i −0.216369 − 0.216369i 1.00000i 1.84832 − 1.25846i 0.305992i 1.68383 − 2.04076i 0.707107 − 0.707107i − 2.90637i −2.19682 − 0.417093i
727.4 −0.707107 − 0.707107i 0.216369 + 0.216369i 1.00000i −1.84832 + 1.25846i − 0.305992i −2.04076 + 1.68383i 0.707107 − 0.707107i − 2.90637i 2.19682 + 0.417093i
727.5 −0.707107 − 0.707107i 0.878808 + 0.878808i 1.00000i −0.142676 + 2.23151i − 1.24282i 2.45215 + 0.993466i 0.707107 − 0.707107i − 1.45539i 1.67880 − 1.47703i
727.6 −0.707107 − 0.707107i 2.35787 + 2.35787i 1.00000i −2.00068 − 0.998647i − 3.33453i −2.63371 + 0.252130i 0.707107 − 0.707107i 8.11912i 0.708541 + 2.12084i
727.7 0.707107 + 0.707107i −2.10618 − 2.10618i 1.00000i −2.14674 + 0.625704i − 2.97859i −2.64445 + 0.0831086i −0.707107 + 0.707107i 5.87198i −1.96041 − 1.07553i
727.8 0.707107 + 0.707107i −1.96534 − 1.96534i 1.00000i 1.73312 − 1.41290i − 2.77941i 2.19900 + 1.47118i −0.707107 + 0.707107i 4.72512i 2.22457 + 0.226432i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 573.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.l.b ✓ 24
5.c odd 4 1 inner 770.2.l.b ✓ 24
7.b odd 2 1 inner 770.2.l.b ✓ 24
35.f even 4 1 inner 770.2.l.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.l.b ✓ 24 1.a even 1 1 trivial
770.2.l.b ✓ 24 5.c odd 4 1 inner
770.2.l.b ✓ 24 7.b odd 2 1 inner
770.2.l.b ✓ 24 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 286T_{3}^{20} + 28141T_{3}^{16} + 1117140T_{3}^{12} + 15051460T_{3}^{8} + 30033920T_{3}^{4} + 262144 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display