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Properties

Label 1035.2.j.b
Level $1035$
Weight $2$
Character orbit 1035.j
Analytic conductor $8.265$
Analytic rank $0$
Dimension $44$
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] = [1035,2,Mod(323,1035)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1035, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1035.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1035.j (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: \(8.26451660920\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 12 q^{7} - 20 q^{10} + 4 q^{13} - 44 q^{16} + 16 q^{22} - 8 q^{25} + 40 q^{28} - 32 q^{31} + 56 q^{37} - 16 q^{40} + 72 q^{43} - 4 q^{46} + 76 q^{52} + 56 q^{55} - 12 q^{58} - 96 q^{61} + 12 q^{67} - 48 q^{70} + 68 q^{73} - 112 q^{76} + 52 q^{82} + 32 q^{85} + 56 q^{88} - 176 q^{91} + 76 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1 −1.79212 − 1.79212i 0 4.42339i −0.646671 − 2.14052i 0 −0.345144 + 0.345144i 4.34301 − 4.34301i 0 −2.67715 + 4.99498i
323.2 −1.74829 − 1.74829i 0 4.11305i 0.818718 − 2.08079i 0 −1.90749 + 1.90749i 3.69423 − 3.69423i 0 −5.06919 + 2.20648i
323.3 −1.65714 − 1.65714i 0 3.49225i 1.35878 + 1.77587i 0 2.43785 − 2.43785i 2.47288 − 2.47288i 0 0.691179 − 5.19457i
323.4 −1.56692 − 1.56692i 0 2.91045i 2.12072 + 0.708917i 0 1.66463 − 1.66463i 1.42659 − 1.42659i 0 −2.21217 − 4.43380i
323.5 −1.47325 − 1.47325i 0 2.34094i −0.834294 + 2.07460i 0 0.188870 − 0.188870i 0.502291 − 0.502291i 0 4.28553 − 1.82728i
323.6 −1.27425 − 1.27425i 0 1.24744i 2.19469 − 0.428183i 0 0.936657 − 0.936657i −0.958947 + 0.958947i 0 −3.34220 − 2.25098i
323.7 −0.842836 − 0.842836i 0 − 0.579255i −0.954547 + 2.02209i 0 2.82060 − 2.82060i −2.17389 + 2.17389i 0 2.50881 − 0.899762i
323.8 −0.637433 − 0.637433i 0 − 1.18736i −2.21909 − 0.275025i 0 1.16972 − 1.16972i −2.03173 + 2.03173i 0 1.23921 + 1.58983i
323.9 −0.246102 − 0.246102i 0 − 1.87887i 1.93545 + 1.11984i 0 −1.89566 + 1.89566i −0.954598 + 0.954598i 0 −0.200722 − 0.751914i
323.10 −0.221928 − 0.221928i 0 − 1.90150i 0.110029 − 2.23336i 0 −2.97861 + 2.97861i −0.865851 + 0.865851i 0 −0.520063 + 0.471226i
323.11 −0.0986143 − 0.0986143i 0 − 1.98055i −1.99036 + 1.01906i 0 0.908571 − 0.908571i −0.392539 + 0.392539i 0 0.296772 + 0.0957835i
323.12 0.0986143 + 0.0986143i 0 − 1.98055i 1.99036 − 1.01906i 0 0.908571 − 0.908571i 0.392539 − 0.392539i 0 0.296772 + 0.0957835i
323.13 0.221928 + 0.221928i 0 − 1.90150i −0.110029 + 2.23336i 0 −2.97861 + 2.97861i 0.865851 − 0.865851i 0 −0.520063 + 0.471226i
323.14 0.246102 + 0.246102i 0 − 1.87887i −1.93545 − 1.11984i 0 −1.89566 + 1.89566i 0.954598 − 0.954598i 0 −0.200722 − 0.751914i
323.15 0.637433 + 0.637433i 0 − 1.18736i 2.21909 + 0.275025i 0 1.16972 − 1.16972i 2.03173 − 2.03173i 0 1.23921 + 1.58983i
323.16 0.842836 + 0.842836i 0 − 0.579255i 0.954547 − 2.02209i 0 2.82060 − 2.82060i 2.17389 − 2.17389i 0 2.50881 − 0.899762i
323.17 1.27425 + 1.27425i 0 1.24744i −2.19469 + 0.428183i 0 0.936657 − 0.936657i 0.958947 − 0.958947i 0 −3.34220 − 2.25098i
323.18 1.47325 + 1.47325i 0 2.34094i 0.834294 − 2.07460i 0 0.188870 − 0.188870i −0.502291 + 0.502291i 0 4.28553 − 1.82728i
323.19 1.56692 + 1.56692i 0 2.91045i −2.12072 − 0.708917i 0 1.66463 − 1.66463i −1.42659 + 1.42659i 0 −2.21217 − 4.43380i
323.20 1.65714 + 1.65714i 0 3.49225i −1.35878 − 1.77587i 0 2.43785 − 2.43785i −2.47288 + 2.47288i 0 0.691179 − 5.19457i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1035.2.j.b ✓ 44
3.b odd 2 1 inner 1035.2.j.b ✓ 44
5.c odd 4 1 inner 1035.2.j.b ✓ 44
15.e even 4 1 inner 1035.2.j.b ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1035.2.j.b ✓ 44 1.a even 1 1 trivial
1035.2.j.b ✓ 44 3.b odd 2 1 inner
1035.2.j.b ✓ 44 5.c odd 4 1 inner
1035.2.j.b ✓ 44 15.e even 4 1 inner