↑

Properties

Label 1232.2.bn.d
Level $1232$
Weight $2$
Character orbit 1232.bn
Analytic conductor $9.838$
Analytic rank $0$
Dimension $48$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1232,2,Mod(241,1232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1232.241"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 616)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{9} + 2 q^{11} + 8 q^{15} - 12 q^{23} + 20 q^{25} - 12 q^{31} - 30 q^{33} + 4 q^{37} + 48 q^{45} - 48 q^{49} + 20 q^{53} - 8 q^{67} + 16 q^{71} + 24 q^{75} + 34 q^{77} - 24 q^{81} - 24 q^{89}+ \cdots + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1 0 −2.52465 − 1.45761i 0 −2.19238 + 1.26577i 0 −2.62002 + 0.368113i 0 2.74924 + 4.76183i 0
241.2 0 −2.52465 − 1.45761i 0 −2.19238 + 1.26577i 0 2.62002 − 0.368113i 0 2.74924 + 4.76183i 0
241.3 0 −2.42681 − 1.40112i 0 0.927262 − 0.535355i 0 1.12707 + 2.39368i 0 2.42627 + 4.20243i 0
241.4 0 −2.42681 − 1.40112i 0 0.927262 − 0.535355i 0 −1.12707 − 2.39368i 0 2.42627 + 4.20243i 0
241.5 0 −1.98542 − 1.14628i 0 2.90486 − 1.67712i 0 0.435818 + 2.60961i 0 1.12794 + 1.95364i 0
241.6 0 −1.98542 − 1.14628i 0 2.90486 − 1.67712i 0 −0.435818 − 2.60961i 0 1.12794 + 1.95364i 0
241.7 0 −1.19884 − 0.692153i 0 −2.04863 + 1.18278i 0 2.46903 − 0.950736i 0 −0.541847 − 0.938507i 0
241.8 0 −1.19884 − 0.692153i 0 −2.04863 + 1.18278i 0 −2.46903 + 0.950736i 0 −0.541847 − 0.938507i 0
241.9 0 −0.318149 − 0.183684i 0 2.31054 − 1.33399i 0 −2.22342 + 1.43402i 0 −1.43252 − 2.48120i 0
241.10 0 −0.318149 − 0.183684i 0 2.31054 − 1.33399i 0 2.22342 − 1.43402i 0 −1.43252 − 2.48120i 0
241.11 0 −0.309930 − 0.178938i 0 −1.70809 + 0.986168i 0 −0.955814 − 2.46707i 0 −1.43596 − 2.48716i 0
241.12 0 −0.309930 − 0.178938i 0 −1.70809 + 0.986168i 0 0.955814 + 2.46707i 0 −1.43596 − 2.48716i 0
241.13 0 −0.260234 − 0.150246i 0 −0.391190 + 0.225854i 0 −1.21502 + 2.35026i 0 −1.45485 − 2.51988i 0
241.14 0 −0.260234 − 0.150246i 0 −0.391190 + 0.225854i 0 1.21502 − 2.35026i 0 −1.45485 − 2.51988i 0
241.15 0 1.07273 + 0.619341i 0 2.68786 − 1.55184i 0 −2.52052 − 0.804357i 0 −0.732832 − 1.26930i 0
241.16 0 1.07273 + 0.619341i 0 2.68786 − 1.55184i 0 2.52052 + 0.804357i 0 −0.732832 − 1.26930i 0
241.17 0 1.58110 + 0.912851i 0 −3.66553 + 2.11630i 0 −0.791894 + 2.52446i 0 0.166593 + 0.288547i 0
241.18 0 1.58110 + 0.912851i 0 −3.66553 + 2.11630i 0 0.791894 − 2.52446i 0 0.166593 + 0.288547i 0
241.19 0 1.58602 + 0.915689i 0 −0.631951 + 0.364857i 0 −2.46576 − 0.959177i 0 0.176973 + 0.306526i 0
241.20 0 1.58602 + 0.915689i 0 −0.631951 + 0.364857i 0 2.46576 + 0.959177i 0 0.176973 + 0.306526i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 241.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.b odd 2 1 inner
77.i 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.bn.d 48
4.b odd 2 1 616.2.bf.a ✓ 48
7.d odd 6 1 inner 1232.2.bn.d 48
11.b odd 2 1 inner 1232.2.bn.d 48
28.f even 6 1 616.2.bf.a ✓ 48
44.c even 2 1 616.2.bf.a ✓ 48
77.i even 6 1 inner 1232.2.bn.d 48
308.m odd 6 1 616.2.bf.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.bf.a ✓ 48 4.b odd 2 1
616.2.bf.a ✓ 48 28.f even 6 1
616.2.bf.a ✓ 48 44.c even 2 1
616.2.bf.a ✓ 48 308.m odd 6 1
1232.2.bn.d 48 1.a even 1 1 trivial
1232.2.bn.d 48 7.d odd 6 1 inner
1232.2.bn.d 48 11.b odd 2 1 inner
1232.2.bn.d 48 77.i even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 24 T_{3}^{22} + 372 T_{3}^{20} + 18 T_{3}^{19} - 3386 T_{3}^{18} - 78 T_{3}^{17} + \cdots + 961 \) acting on \(S_{2}^{\mathrm{new}}(1232, [\chi])\). Copy content Toggle raw display