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Properties

Label 672.4.h.a
Level $672$
Weight $4$
Character orbit 672.h
Analytic conductor $39.649$
Analytic rank $0$
Dimension $36$
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] = [672,4,Mod(575,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.575");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.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: \(39.6492835239\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 132 q^{9} - 120 q^{13} + 28 q^{21} - 756 q^{25} + 40 q^{33} + 672 q^{37} + 304 q^{45} - 1764 q^{49} + 1624 q^{57} + 2472 q^{61} - 1224 q^{69} - 2376 q^{73} + 468 q^{81} + 5160 q^{85} - 648 q^{93} - 4488 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1 0 −5.18300 − 0.369441i 0 − 10.4214i 0 − 7.00000i 0 26.7270 + 3.82963i 0
575.2 0 −5.18300 + 0.369441i 0 10.4214i 0 7.00000i 0 26.7270 − 3.82963i 0
575.3 0 −4.99974 − 1.41514i 0 9.76395i 0 − 7.00000i 0 22.9948 + 14.1506i 0
575.4 0 −4.99974 + 1.41514i 0 − 9.76395i 0 7.00000i 0 22.9948 − 14.1506i 0
575.5 0 −4.41319 − 2.74295i 0 14.6482i 0 7.00000i 0 11.9525 + 24.2103i 0
575.6 0 −4.41319 + 2.74295i 0 − 14.6482i 0 − 7.00000i 0 11.9525 − 24.2103i 0
575.7 0 −3.68812 − 3.66030i 0 − 1.63405i 0 7.00000i 0 0.204455 + 26.9992i 0
575.8 0 −3.68812 + 3.66030i 0 1.63405i 0 − 7.00000i 0 0.204455 − 26.9992i 0
575.9 0 −2.77808 − 4.39116i 0 − 6.94562i 0 − 7.00000i 0 −11.5645 + 24.3980i 0
575.10 0 −2.77808 + 4.39116i 0 6.94562i 0 7.00000i 0 −11.5645 − 24.3980i 0
575.11 0 −2.55007 − 4.52737i 0 − 10.7846i 0 − 7.00000i 0 −13.9942 + 23.0903i 0
575.12 0 −2.55007 + 4.52737i 0 10.7846i 0 7.00000i 0 −13.9942 − 23.0903i 0
575.13 0 −1.85075 − 4.85538i 0 15.3410i 0 − 7.00000i 0 −20.1494 + 17.9722i 0
575.14 0 −1.85075 + 4.85538i 0 − 15.3410i 0 7.00000i 0 −20.1494 − 17.9722i 0
575.15 0 −1.49716 − 4.97579i 0 − 20.4606i 0 7.00000i 0 −22.5170 + 14.8991i 0
575.16 0 −1.49716 + 4.97579i 0 20.4606i 0 − 7.00000i 0 −22.5170 − 14.8991i 0
575.17 0 −0.416222 − 5.17946i 0 − 8.61908i 0 7.00000i 0 −26.6535 + 4.31160i 0
575.18 0 −0.416222 + 5.17946i 0 8.61908i 0 − 7.00000i 0 −26.6535 − 4.31160i 0
575.19 0 0.416222 − 5.17946i 0 8.61908i 0 7.00000i 0 −26.6535 − 4.31160i 0
575.20 0 0.416222 + 5.17946i 0 − 8.61908i 0 − 7.00000i 0 −26.6535 + 4.31160i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.36
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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.h.a ✓ 36
3.b odd 2 1 inner 672.4.h.a ✓ 36
4.b odd 2 1 inner 672.4.h.a ✓ 36
12.b even 2 1 inner 672.4.h.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.h.a ✓ 36 1.a even 1 1 trivial
672.4.h.a ✓ 36 3.b odd 2 1 inner
672.4.h.a ✓ 36 4.b odd 2 1 inner
672.4.h.a ✓ 36 12.b even 2 1 inner