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Properties

Label 1388.4.a.b
Level $1388$
Weight $4$
Character orbit 1388.a
Self dual yes
Analytic conductor $81.895$
Analytic rank $0$
Dimension $46$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1388,4,Mod(1,1388)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1388, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1388.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1388 = 2^{2} \cdot 347 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1388.a (trivial)

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: yes
Analytic conductor: \(81.8946510880\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 46 q + 28 q^{3} - q^{5} + 90 q^{7} + 468 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q + 28 q^{3} - q^{5} + 90 q^{7} + 468 q^{9} + 66 q^{11} + 24 q^{13} + 234 q^{15} + 106 q^{17} + 247 q^{19} - 10 q^{21} + 314 q^{23} + 1375 q^{25} + 934 q^{27} + 22 q^{29} + 983 q^{31} + 648 q^{33} + 542 q^{35} + 179 q^{37} + 620 q^{39} + 572 q^{41} + 1332 q^{43} + 223 q^{45} + 1222 q^{47} + 3530 q^{49} + 1270 q^{51} + 642 q^{53} + 1638 q^{55} + 1946 q^{57} + 1016 q^{59} - 418 q^{61} + 2114 q^{63} - 18 q^{65} + 2796 q^{67} - 800 q^{69} + 887 q^{71} + 1757 q^{73} + 4590 q^{75} + 250 q^{77} + 2190 q^{79} + 5530 q^{81} + 2686 q^{83} - 362 q^{85} + 3391 q^{87} + 3844 q^{89} + 6024 q^{91} + 4867 q^{93} + 4960 q^{95} + 6888 q^{97} + 8291 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −9.48823 0 −17.2095 0 15.7274 0 63.0266 0
1.2 0 −9.35050 0 −12.9343 0 −9.27149 0 60.4318 0
1.3 0 −8.91666 0 2.38769 0 16.7628 0 52.5068 0
1.4 0 −8.15915 0 −2.53318 0 −16.7680 0 39.5717 0
1.5 0 −7.95524 0 20.2521 0 7.20124 0 36.2858 0
1.6 0 −7.53038 0 −6.52853 0 −13.4582 0 29.7067 0
1.7 0 −7.00085 0 5.17933 0 1.88382 0 22.0119 0
1.8 0 −6.85818 0 2.20615 0 −12.2556 0 20.0347 0
1.9 0 −6.32527 0 5.71989 0 −5.06275 0 13.0090 0
1.10 0 −6.13767 0 −14.8970 0 31.1530 0 10.6710 0
1.11 0 −4.92968 0 6.42037 0 27.2424 0 −2.69821 0
1.12 0 −4.56621 0 14.4197 0 −31.7610 0 −6.14976 0
1.13 0 −4.40261 0 7.06538 0 −5.46750 0 −7.61702 0
1.14 0 −3.90920 0 −18.7929 0 26.1678 0 −11.7182 0
1.15 0 −3.62276 0 19.8859 0 35.4582 0 −13.8756 0
1.16 0 −3.28437 0 −9.76615 0 −7.90600 0 −16.2129 0
1.17 0 −1.74683 0 −17.2246 0 13.9751 0 −23.9486 0
1.18 0 −1.16715 0 14.6228 0 17.9077 0 −25.6378 0
1.19 0 −1.14298 0 −4.19972 0 −5.17757 0 −25.6936 0
1.20 0 −0.991244 0 −1.84711 0 −14.6592 0 −26.0174 0
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(347\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1388.4.a.b ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1388.4.a.b ✓ 46 1.a even 1 1 trivial