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Properties

Label 1386.4.c.b
Level $1386$
Weight $4$
Character orbit 1386.c
Analytic conductor $81.777$
Analytic rank $0$
Dimension $36$
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] = [1386,4,Mod(197,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1386.197");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1386.c (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: \(81.7766472680\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 72 q^{2} + 144 q^{4} + 288 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 72 q^{2} + 144 q^{4} + 288 q^{8} + 36 q^{11} + 576 q^{16} - 144 q^{17} + 72 q^{22} - 444 q^{25} - 432 q^{29} - 48 q^{31} + 1152 q^{32} - 288 q^{34} - 504 q^{35} + 24 q^{37} + 144 q^{41} + 144 q^{44} - 1764 q^{49} - 888 q^{50} + 2448 q^{55} - 864 q^{58} - 96 q^{62} + 2304 q^{64} + 2400 q^{65} + 624 q^{67} - 576 q^{68} - 1008 q^{70} + 48 q^{74} + 168 q^{77} + 288 q^{82} - 1296 q^{83} + 288 q^{88} + 6096 q^{95} + 768 q^{97} - 3528 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 2.00000 0 4.00000 − 21.0789i 0 − 7.00000i 8.00000 0 − 42.1579i
197.2 2.00000 0 4.00000 − 16.1024i 0 − 7.00000i 8.00000 0 − 32.2048i
197.3 2.00000 0 4.00000 − 15.7104i 0 7.00000i 8.00000 0 − 31.4209i
197.4 2.00000 0 4.00000 − 15.2228i 0 − 7.00000i 8.00000 0 − 30.4455i
197.5 2.00000 0 4.00000 − 15.1163i 0 − 7.00000i 8.00000 0 − 30.2327i
197.6 2.00000 0 4.00000 − 14.3512i 0 7.00000i 8.00000 0 − 28.7024i
197.7 2.00000 0 4.00000 − 13.1336i 0 − 7.00000i 8.00000 0 − 26.2672i
197.8 2.00000 0 4.00000 − 12.7025i 0 7.00000i 8.00000 0 − 25.4049i
197.9 2.00000 0 4.00000 − 12.6122i 0 − 7.00000i 8.00000 0 − 25.2243i
197.10 2.00000 0 4.00000 − 12.2189i 0 7.00000i 8.00000 0 − 24.4379i
197.11 2.00000 0 4.00000 − 10.2351i 0 7.00000i 8.00000 0 − 20.4702i
197.12 2.00000 0 4.00000 − 6.55239i 0 7.00000i 8.00000 0 − 13.1048i
197.13 2.00000 0 4.00000 − 6.08249i 0 − 7.00000i 8.00000 0 − 12.1650i
197.14 2.00000 0 4.00000 − 4.17356i 0 − 7.00000i 8.00000 0 − 8.34711i
197.15 2.00000 0 4.00000 − 2.82166i 0 − 7.00000i 8.00000 0 − 5.64331i
197.16 2.00000 0 4.00000 − 1.14450i 0 − 7.00000i 8.00000 0 − 2.28900i
197.17 2.00000 0 4.00000 − 1.07648i 0 − 7.00000i 8.00000 0 − 2.15296i
197.18 2.00000 0 4.00000 − 0.794361i 0 7.00000i 8.00000 0 − 1.58872i
197.19 2.00000 0 4.00000 0.794361i 0 − 7.00000i 8.00000 0 1.58872i
197.20 2.00000 0 4.00000 1.07648i 0 7.00000i 8.00000 0 2.15296i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.4.c.b yes 36
3.b odd 2 1 1386.4.c.a ✓ 36
11.b odd 2 1 1386.4.c.a ✓ 36
33.d even 2 1 inner 1386.4.c.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.4.c.a ✓ 36 3.b odd 2 1
1386.4.c.a ✓ 36 11.b odd 2 1
1386.4.c.b yes 36 1.a even 1 1 trivial
1386.4.c.b yes 36 33.d even 2 1 inner