Properties

Label 138.5.c.a
Level $138$
Weight $5$
Character orbit 138.c
Analytic conductor $14.265$
Analytic rank $0$
Dimension $28$
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] = [138,5,Mod(47,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.47");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 138.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: \(14.2650549056\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 8 q^{3} - 224 q^{4} - 32 q^{6} - 104 q^{7} + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 8 q^{3} - 224 q^{4} - 32 q^{6} - 104 q^{7} + 80 q^{9} + 192 q^{10} + 64 q^{12} + 148 q^{15} + 1792 q^{16} + 448 q^{18} + 912 q^{19} - 1412 q^{21} - 1984 q^{22} + 256 q^{24} - 3028 q^{25} - 1700 q^{27} + 832 q^{28} + 768 q^{30} - 2400 q^{31} + 2772 q^{33} + 2944 q^{34} - 640 q^{36} - 2080 q^{37} + 468 q^{39} - 1536 q^{40} - 4480 q^{42} + 5536 q^{43} + 13852 q^{45} - 512 q^{48} + 13444 q^{49} - 16972 q^{51} + 6112 q^{54} - 624 q^{55} - 3304 q^{57} - 20160 q^{58} - 1184 q^{60} + 6376 q^{61} + 276 q^{63} - 14336 q^{64} + 960 q^{66} + 2168 q^{67} + 26624 q^{70} - 3584 q^{72} - 13568 q^{73} - 2464 q^{75} - 7296 q^{76} - 9920 q^{78} - 46064 q^{79} + 19184 q^{81} + 23168 q^{82} + 11296 q^{84} + 27584 q^{85} + 42140 q^{87} + 15872 q^{88} - 11008 q^{90} - 18040 q^{91} - 23892 q^{93} + 3840 q^{94} - 2048 q^{96} + 56848 q^{97} - 40092 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 2.82843i −8.97110 + 0.720642i −8.00000 29.0082i 2.03828 + 25.3741i 75.1852 22.6274i 79.9613 12.9299i −82.0477
47.2 2.82843i −8.21093 + 3.68520i −8.00000 42.4258i 10.4233 + 23.2240i −2.59538 22.6274i 53.8386 60.5178i 119.998
47.3 2.82843i −7.96199 + 4.19604i −8.00000 18.2777i 11.8682 + 22.5199i −45.3951 22.6274i 45.7866 66.8176i −51.6971
47.4 2.82843i −7.90779 4.29732i −8.00000 10.5993i −12.1546 + 22.3666i −56.5485 22.6274i 44.0662 + 67.9645i 29.9794
47.5 2.82843i −6.26270 6.46363i −8.00000 3.82580i −18.2819 + 17.7136i 80.3647 22.6274i −2.55712 + 80.9596i 10.8210
47.6 2.82843i −2.30705 8.69928i −8.00000 39.0683i −24.6053 + 6.52531i −28.2665 22.6274i −70.3551 + 40.1393i −110.502
47.7 2.82843i −0.694207 + 8.97319i −8.00000 10.5332i 25.3800 + 1.96351i −61.3754 22.6274i −80.0362 12.4585i 29.7925
47.8 2.82843i 1.63185 8.85082i −8.00000 5.43082i −25.0339 4.61556i −23.1901 22.6274i −75.6741 28.8864i 15.3607
47.9 2.82843i 2.12978 8.74437i −8.00000 37.2957i −24.7328 6.02393i 61.4020 22.6274i −71.9281 37.2472i 105.488
47.10 2.82843i 3.06562 + 8.46180i −8.00000 31.7511i 23.9336 8.67088i 50.3621 22.6274i −62.2040 + 51.8813i 89.8057
47.11 2.82843i 6.27270 + 6.45393i −8.00000 43.2073i 18.2545 17.7419i −83.1420 22.6274i −2.30643 + 80.9672i −122.209
47.12 2.82843i 7.99491 + 4.13297i −8.00000 7.26641i 11.6898 22.6130i 48.6143 22.6274i 46.8371 + 66.0854i −20.5525
47.13 2.82843i 8.55653 2.79032i −8.00000 33.9236i −7.89220 24.2015i −34.3722 22.6274i 65.4283 47.7508i 95.9504
47.14 2.82843i 8.66438 2.43487i −8.00000 5.01640i −6.88684 24.5066i −33.0433 22.6274i 69.1428 42.1932i −14.1885
47.15 2.82843i −8.97110 0.720642i −8.00000 29.0082i 2.03828 25.3741i 75.1852 22.6274i 79.9613 + 12.9299i −82.0477
47.16 2.82843i −8.21093 3.68520i −8.00000 42.4258i 10.4233 23.2240i −2.59538 22.6274i 53.8386 + 60.5178i 119.998
47.17 2.82843i −7.96199 4.19604i −8.00000 18.2777i 11.8682 22.5199i −45.3951 22.6274i 45.7866 + 66.8176i −51.6971
47.18 2.82843i −7.90779 + 4.29732i −8.00000 10.5993i −12.1546 22.3666i −56.5485 22.6274i 44.0662 67.9645i 29.9794
47.19 2.82843i −6.26270 + 6.46363i −8.00000 3.82580i −18.2819 17.7136i 80.3647 22.6274i −2.55712 80.9596i 10.8210
47.20 2.82843i −2.30705 + 8.69928i −8.00000 39.0683i −24.6053 6.52531i −28.2665 22.6274i −70.3551 40.1393i −110.502
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.28
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.5.c.a 28
3.b odd 2 1 inner 138.5.c.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.5.c.a 28 1.a even 1 1 trivial
138.5.c.a 28 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(138, [\chi])\).