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Properties

Label 84.3.h.e
Level $84$
Weight $3$
Character orbit 84.h
Analytic conductor $2.289$
Analytic rank $0$
Dimension $24$
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] = [84,3,Mod(83,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.83");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 84.h (of order \(2\), degree \(1\), 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: \(2.28883422063\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} - 40 q^{9} - 60 q^{16} - 44 q^{18} + 56 q^{21} + 112 q^{22} + 104 q^{25} + 68 q^{28} + 28 q^{30} - 208 q^{36} - 48 q^{37} + 92 q^{42} - 40 q^{46} - 264 q^{49} - 144 q^{57} + 280 q^{58} + 28 q^{60} - 68 q^{64} - 472 q^{70} + 484 q^{72} + 372 q^{78} - 328 q^{81} + 52 q^{84} + 128 q^{85} + 528 q^{88} + 720 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −1.79242 − 0.887252i −0.162333 + 2.99560i 2.42557 + 3.18066i −4.56888 2.94883 − 5.22536i −4.97544 − 4.92392i −1.52559 − 7.85319i −8.94730 − 0.972571i 8.18938 + 4.05375i
83.2 −1.79242 − 0.887252i 0.162333 − 2.99560i 2.42557 + 3.18066i 4.56888 −2.94883 + 5.22536i 4.97544 − 4.92392i −1.52559 − 7.85319i −8.94730 − 0.972571i −8.18938 − 4.05375i
83.3 −1.79242 + 0.887252i −0.162333 − 2.99560i 2.42557 − 3.18066i −4.56888 2.94883 + 5.22536i −4.97544 + 4.92392i −1.52559 + 7.85319i −8.94730 + 0.972571i 8.18938 − 4.05375i
83.4 −1.79242 + 0.887252i 0.162333 + 2.99560i 2.42557 − 3.18066i 4.56888 −2.94883 − 5.22536i 4.97544 + 4.92392i −1.52559 + 7.85319i −8.94730 + 0.972571i −8.18938 + 4.05375i
83.5 −1.13899 − 1.64399i −2.60176 − 1.49360i −1.40542 + 3.74497i −1.12104 0.507909 + 5.97846i −4.78713 + 5.10719i 7.75746 − 1.95495i 4.53834 + 7.77197i 1.27685 + 1.84299i
83.6 −1.13899 − 1.64399i 2.60176 + 1.49360i −1.40542 + 3.74497i 1.12104 −0.507909 − 5.97846i 4.78713 + 5.10719i 7.75746 − 1.95495i 4.53834 + 7.77197i −1.27685 − 1.84299i
83.7 −1.13899 + 1.64399i −2.60176 + 1.49360i −1.40542 − 3.74497i −1.12104 0.507909 − 5.97846i −4.78713 − 5.10719i 7.75746 + 1.95495i 4.53834 − 7.77197i 1.27685 − 1.84299i
83.8 −1.13899 + 1.64399i 2.60176 − 1.49360i −1.40542 − 3.74497i 1.12104 −0.507909 + 5.97846i 4.78713 − 5.10719i 7.75746 + 1.95495i 4.53834 − 7.77197i −1.27685 + 1.84299i
83.9 −0.489825 − 1.93909i −2.05048 + 2.18987i −3.52014 + 1.89963i 8.11595 5.25073 + 2.90342i 3.05424 − 6.29854i 5.40781 + 5.89539i −0.591047 − 8.98057i −3.97539 − 15.7376i
83.10 −0.489825 − 1.93909i 2.05048 − 2.18987i −3.52014 + 1.89963i −8.11595 −5.25073 − 2.90342i −3.05424 − 6.29854i 5.40781 + 5.89539i −0.591047 − 8.98057i 3.97539 + 15.7376i
83.11 −0.489825 + 1.93909i −2.05048 − 2.18987i −3.52014 − 1.89963i 8.11595 5.25073 − 2.90342i 3.05424 + 6.29854i 5.40781 − 5.89539i −0.591047 + 8.98057i −3.97539 + 15.7376i
83.12 −0.489825 + 1.93909i 2.05048 + 2.18987i −3.52014 − 1.89963i −8.11595 −5.25073 + 2.90342i −3.05424 + 6.29854i 5.40781 − 5.89539i −0.591047 + 8.98057i 3.97539 − 15.7376i
83.13 0.489825 − 1.93909i −2.05048 + 2.18987i −3.52014 − 1.89963i −8.11595 3.24197 + 5.04872i 3.05424 + 6.29854i −5.40781 + 5.89539i −0.591047 − 8.98057i −3.97539 + 15.7376i
83.14 0.489825 − 1.93909i 2.05048 − 2.18987i −3.52014 − 1.89963i 8.11595 −3.24197 − 5.04872i −3.05424 + 6.29854i −5.40781 + 5.89539i −0.591047 − 8.98057i 3.97539 − 15.7376i
83.15 0.489825 + 1.93909i −2.05048 − 2.18987i −3.52014 + 1.89963i −8.11595 3.24197 − 5.04872i 3.05424 − 6.29854i −5.40781 − 5.89539i −0.591047 + 8.98057i −3.97539 − 15.7376i
83.16 0.489825 + 1.93909i 2.05048 + 2.18987i −3.52014 + 1.89963i 8.11595 −3.24197 + 5.04872i −3.05424 − 6.29854i −5.40781 − 5.89539i −0.591047 + 8.98057i 3.97539 + 15.7376i
83.17 1.13899 − 1.64399i −2.60176 − 1.49360i −1.40542 − 3.74497i 1.12104 −5.41883 + 2.57610i −4.78713 − 5.10719i −7.75746 − 1.95495i 4.53834 + 7.77197i 1.27685 − 1.84299i
83.18 1.13899 − 1.64399i 2.60176 + 1.49360i −1.40542 − 3.74497i −1.12104 5.41883 − 2.57610i 4.78713 − 5.10719i −7.75746 − 1.95495i 4.53834 + 7.77197i −1.27685 + 1.84299i
83.19 1.13899 + 1.64399i −2.60176 + 1.49360i −1.40542 + 3.74497i 1.12104 −5.41883 − 2.57610i −4.78713 + 5.10719i −7.75746 + 1.95495i 4.53834 − 7.77197i 1.27685 + 1.84299i
83.20 1.13899 + 1.64399i 2.60176 − 1.49360i −1.40542 + 3.74497i −1.12104 5.41883 + 2.57610i 4.78713 + 5.10719i −7.75746 + 1.95495i 4.53834 − 7.77197i −1.27685 − 1.84299i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.3.h.e ✓ 24
3.b odd 2 1 inner 84.3.h.e ✓ 24
4.b odd 2 1 inner 84.3.h.e ✓ 24
7.b odd 2 1 inner 84.3.h.e ✓ 24
12.b even 2 1 inner 84.3.h.e ✓ 24
21.c even 2 1 inner 84.3.h.e ✓ 24
28.d even 2 1 inner 84.3.h.e ✓ 24
84.h odd 2 1 inner 84.3.h.e ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.h.e ✓ 24 1.a even 1 1 trivial
84.3.h.e ✓ 24 3.b odd 2 1 inner
84.3.h.e ✓ 24 4.b odd 2 1 inner
84.3.h.e ✓ 24 7.b odd 2 1 inner
84.3.h.e ✓ 24 12.b even 2 1 inner
84.3.h.e ✓ 24 21.c even 2 1 inner
84.3.h.e ✓ 24 28.d even 2 1 inner
84.3.h.e ✓ 24 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(84, [\chi])\):

\( T_{5}^{6} - 88T_{5}^{4} + 1484T_{5}^{2} - 1728 \) Copy content Toggle raw display
\( T_{11}^{6} - 256T_{11}^{4} + 16128T_{11}^{2} - 65536 \) Copy content Toggle raw display