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Properties

Label 1388.4.a.a
Level $1388$
Weight $4$
Character orbit 1388.a
Self dual yes
Analytic conductor $81.895$
Analytic rank $1$
Dimension $41$
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: \(1\)
Dimension: \(41\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 41 q - 26 q^{3} - q^{5} - 106 q^{7} + 333 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 41 q - 26 q^{3} - q^{5} - 106 q^{7} + 333 q^{9} - 44 q^{11} + 24 q^{13} - 126 q^{15} - 200 q^{17} - 399 q^{19} - 10 q^{21} - 330 q^{23} + 500 q^{25} - 1010 q^{27} + 22 q^{29} - 567 q^{31} - 408 q^{33} - 438 q^{35} + 179 q^{37} - 316 q^{39} - 166 q^{41} - 990 q^{43} + 223 q^{45} - 658 q^{47} + 1325 q^{49} - 362 q^{51} + 642 q^{53} - 1882 q^{55} - 106 q^{57} - 990 q^{59} - 418 q^{61} - 3178 q^{63} - 1318 q^{65} - 3234 q^{67} - 800 q^{69} - 1243 q^{71} - 2331 q^{73} - 2160 q^{75} + 250 q^{77} - 2866 q^{79} + 1885 q^{81} - 800 q^{83} - 362 q^{85} - 4521 q^{87} - 2518 q^{89} - 6956 q^{91} - 4023 q^{93} - 6524 q^{95} - 7506 q^{97} - 7819 q^{99}+O(q^{100}) \) 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 −10.3300 0 5.04279 0 −28.2678 0 79.7090 0
1.2 0 −9.97252 0 2.01774 0 18.7731 0 72.4511 0
1.3 0 −9.83454 0 10.8629 0 −22.1815 0 69.7182 0
1.4 0 −8.64386 0 15.7702 0 14.2161 0 47.7164 0
1.5 0 −8.36579 0 3.43294 0 33.0571 0 42.9864 0
1.6 0 −7.46987 0 −16.3108 0 −26.8428 0 28.7990 0
1.7 0 −7.16763 0 −18.6823 0 3.15889 0 24.3750 0
1.8 0 −7.13255 0 −5.57171 0 14.8839 0 23.8733 0
1.9 0 −7.03842 0 16.3091 0 −15.9257 0 22.5394 0
1.10 0 −6.70020 0 −12.8270 0 −31.0060 0 17.8926 0
1.11 0 −5.73068 0 −12.8107 0 11.3371 0 5.84071 0
1.12 0 −4.85103 0 1.26887 0 30.0394 0 −3.46750 0
1.13 0 −4.59975 0 14.1075 0 −29.7930 0 −5.84228 0
1.14 0 −3.78594 0 1.26819 0 −15.7894 0 −12.6667 0
1.15 0 −3.77917 0 −7.89863 0 5.98676 0 −12.7179 0
1.16 0 −2.92587 0 −1.14101 0 −28.5986 0 −18.4393 0
1.17 0 −2.92353 0 13.4856 0 6.07671 0 −18.4530 0
1.18 0 −2.37164 0 −20.8926 0 −27.1286 0 −21.3753 0
1.19 0 −2.20800 0 10.2962 0 4.93566 0 −22.1247 0
1.20 0 −1.63012 0 −16.5345 0 3.87069 0 −24.3427 0
See all 41 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.41
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.a ✓ 41
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1388.4.a.a ✓ 41 1.a even 1 1 trivial