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Properties

Label 712.4.e.a
Level $712$
Weight $4$
Character orbit 712.e
Analytic conductor $42.009$
Analytic rank $0$
Dimension $34$
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] = [712,4,Mod(177,712)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(712, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("712.177");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 712 = 2^{3} \cdot 89 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 712.e (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: \(42.0093599241\)
Analytic rank: \(0\)
Dimension: \(34\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 16 q^{5} - 330 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 16 q^{5} - 330 q^{9} - 28 q^{11} + 48 q^{17} - 236 q^{21} + 822 q^{25} + 180 q^{39} + 396 q^{45} - 8 q^{47} - 2082 q^{49} + 104 q^{53} - 804 q^{55} + 944 q^{57} + 540 q^{67} + 780 q^{69} - 1396 q^{71} + 1548 q^{73} + 332 q^{79} + 3226 q^{81} + 520 q^{85} - 8 q^{87} + 734 q^{89} - 1868 q^{91} - 2640 q^{93} - 456 q^{97} + 5372 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1 0 − 10.3736i 0 8.07548 0 16.6641i 0 −80.6108 0
177.2 0 − 9.37392i 0 −16.8576 0 − 28.8055i 0 −60.8703 0
177.3 0 − 8.53900i 0 14.7702 0 − 19.0585i 0 −45.9145 0
177.4 0 − 7.83836i 0 −9.81868 0 0.367453i 0 −34.4399 0
177.5 0 − 7.46545i 0 −14.0934 0 32.2538i 0 −28.7330 0
177.6 0 − 7.01212i 0 −5.84394 0 − 6.40198i 0 −22.1698 0
177.7 0 − 6.43825i 0 10.0341 0 3.99009i 0 −14.4510 0
177.8 0 − 5.89848i 0 −11.1314 0 − 11.8931i 0 −7.79206 0
177.9 0 − 5.69603i 0 12.0034 0 − 36.0183i 0 −5.44476 0
177.10 0 − 5.62981i 0 −4.51593 0 20.9527i 0 −4.69473 0
177.11 0 − 5.15814i 0 13.8730 0 11.6623i 0 0.393596 0
177.12 0 − 2.99780i 0 20.4090 0 16.0126i 0 18.0132 0
177.13 0 − 2.77088i 0 0.0358063 0 − 18.5588i 0 19.3222 0
177.14 0 − 1.78296i 0 −13.1427 0 − 1.16524i 0 23.8210 0
177.15 0 − 1.38119i 0 5.03847 0 26.2959i 0 25.0923 0
177.16 0 − 0.715973i 0 3.85762 0 − 19.3335i 0 26.4874 0
177.17 0 − 0.0943157i 0 −20.6934 0 − 24.9547i 0 26.9911 0
177.18 0 0.0943157i 0 −20.6934 0 24.9547i 0 26.9911 0
177.19 0 0.715973i 0 3.85762 0 19.3335i 0 26.4874 0
177.20 0 1.38119i 0 5.03847 0 − 26.2959i 0 25.0923 0
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 177.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
89.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 712.4.e.a ✓ 34
89.b even 2 1 inner 712.4.e.a ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
712.4.e.a ✓ 34 1.a even 1 1 trivial
712.4.e.a ✓ 34 89.b even 2 1 inner