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Properties

Label 1232.2.be.b
Level $1232$
Weight $2$
Character orbit 1232.be
Analytic conductor $9.838$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1232,2,Mod(815,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.815");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.be (of order \(6\), 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: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{3} + 2 q^{7} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 2 q^{3} + 2 q^{7} - 16 q^{9} + 18 q^{17} - 14 q^{19} - 2 q^{21} + 12 q^{23} + 14 q^{25} + 16 q^{27} + 16 q^{29} - 10 q^{31} - 6 q^{35} - 4 q^{37} - 42 q^{39} - 42 q^{45} - 6 q^{47} - 4 q^{49} + 24 q^{51} - 2 q^{53} - 48 q^{57} + 12 q^{59} - 12 q^{61} + 26 q^{63} + 6 q^{65} + 48 q^{67} + 6 q^{73} + 14 q^{75} + 2 q^{77} - 48 q^{79} - 22 q^{81} + 84 q^{83} + 16 q^{85} - 18 q^{87} + 6 q^{89} + 18 q^{91} + 28 q^{93} + 36 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
815.1 0 −1.64223 + 2.84442i 0 2.86914 − 1.65650i 0 −2.38609 − 1.14306i 0 −3.89383 − 6.74431i 0
815.2 0 −1.51694 + 2.62742i 0 −0.766106 + 0.442311i 0 0.296809 + 2.62905i 0 −3.10224 − 5.37323i 0
815.3 0 −1.14753 + 1.98758i 0 −2.00611 + 1.15823i 0 2.45312 + 0.991064i 0 −1.13364 − 1.96352i 0
815.4 0 −0.949508 + 1.64460i 0 −0.525125 + 0.303181i 0 −0.0911016 − 2.64418i 0 −0.303130 − 0.525037i 0
815.5 0 −0.714866 + 1.23818i 0 3.79415 − 2.19056i 0 2.58584 + 0.559873i 0 0.477933 + 0.827805i 0
815.6 0 −0.359161 + 0.622084i 0 1.00312 − 0.579151i 0 −2.15481 + 1.53519i 0 1.24201 + 2.15122i 0
815.7 0 −0.229364 + 0.397271i 0 1.48979 − 0.860132i 0 −0.601552 + 2.57646i 0 1.39478 + 2.41584i 0
815.8 0 −0.147074 + 0.254740i 0 −2.57282 + 1.48542i 0 −2.48401 + 0.910888i 0 1.45674 + 2.52315i 0
815.9 0 0.239666 − 0.415114i 0 −1.69398 + 0.978020i 0 2.64082 − 0.161422i 0 1.38512 + 2.39910i 0
815.10 0 0.534042 − 0.924987i 0 −2.90816 + 1.67903i 0 0.00296709 − 2.64575i 0 0.929599 + 1.61011i 0
815.11 0 1.06364 − 1.84229i 0 0.859921 − 0.496476i 0 0.981549 + 2.45694i 0 −0.762680 − 1.32100i 0
815.12 0 1.06788 − 1.84962i 0 2.29169 − 1.32311i 0 1.91105 − 1.82973i 0 −0.780725 − 1.35225i 0
815.13 0 1.27636 − 2.21072i 0 −2.73947 + 1.58164i 0 0.409288 + 2.61390i 0 −1.75820 − 3.04529i 0
815.14 0 1.52508 − 2.64152i 0 0.903957 − 0.521900i 0 −2.56388 − 0.653071i 0 −3.15174 − 5.45898i 0
1167.1 0 −1.64223 − 2.84442i 0 2.86914 + 1.65650i 0 −2.38609 + 1.14306i 0 −3.89383 + 6.74431i 0
1167.2 0 −1.51694 − 2.62742i 0 −0.766106 − 0.442311i 0 0.296809 − 2.62905i 0 −3.10224 + 5.37323i 0
1167.3 0 −1.14753 − 1.98758i 0 −2.00611 − 1.15823i 0 2.45312 − 0.991064i 0 −1.13364 + 1.96352i 0
1167.4 0 −0.949508 − 1.64460i 0 −0.525125 − 0.303181i 0 −0.0911016 + 2.64418i 0 −0.303130 + 0.525037i 0
1167.5 0 −0.714866 − 1.23818i 0 3.79415 + 2.19056i 0 2.58584 − 0.559873i 0 0.477933 − 0.827805i 0
1167.6 0 −0.359161 − 0.622084i 0 1.00312 + 0.579151i 0 −2.15481 − 1.53519i 0 1.24201 − 2.15122i 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 815.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1232.2.be.b ✓ 28
4.b odd 2 1 1232.2.be.c yes 28
7.d odd 6 1 1232.2.be.c yes 28
28.f even 6 1 inner 1232.2.be.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1232.2.be.b ✓ 28 1.a even 1 1 trivial
1232.2.be.b ✓ 28 28.f even 6 1 inner
1232.2.be.c yes 28 4.b odd 2 1
1232.2.be.c yes 28 7.d odd 6 1