Properties

Label 128.13.f.b
Level $128$
Weight $13$
Character orbit 128.f
Analytic conductor $116.991$
Analytic rank $0$
Dimension $46$
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] = [128,13,Mod(31,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.31");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 128.f (of order \(4\), degree \(2\), not 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: \(116.991208611\)
Analytic rank: \(0\)
Dimension: \(46\)
Relative dimension: \(23\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 46 q + 2 q^{3} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q + 2 q^{3} + 2 q^{5} - 4 q^{7} - 2668318 q^{11} + 2 q^{13} - 4 q^{17} - 51868606 q^{19} + 1062884 q^{21} + 298270076 q^{23} - 970053760 q^{27} - 704570398 q^{29} - 4 q^{33} + 3815032900 q^{35} - 364298398 q^{37} + 15553507196 q^{39} - 363863518 q^{43} - 489344130 q^{45} + 67229109258 q^{49} - 33806024892 q^{51} + 11168756642 q^{53} - 74491808260 q^{55} - 104334793054 q^{59} + 106371743810 q^{61} - 75186419620 q^{65} + 43778233922 q^{67} + 214340079908 q^{69} - 188251854340 q^{71} - 308961520610 q^{75} + 341607754084 q^{77} - 941431788274 q^{81} + 1025936323202 q^{83} - 436332718748 q^{85} + 2368412421756 q^{87} + 2028231531652 q^{91} - 1534541270080 q^{93} - 4 q^{97} - 4950023059646 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −949.435 949.435i 0 −10755.1 10755.1i 0 35819.9 0 1.27141e6i 0
31.2 0 −837.232 837.232i 0 −5671.68 5671.68i 0 −4585.41 0 870475.i 0
31.3 0 −769.939 769.939i 0 17644.5 + 17644.5i 0 15384.5 0 654171.i 0
31.4 0 −739.301 739.301i 0 805.082 + 805.082i 0 −97582.7 0 561690.i 0
31.5 0 −652.763 652.763i 0 7919.92 + 7919.92i 0 182451. 0 320757.i 0
31.6 0 −543.347 543.347i 0 12801.7 + 12801.7i 0 −188146. 0 59011.2i 0
31.7 0 −398.082 398.082i 0 −14857.0 14857.0i 0 113515. 0 214502.i 0
31.8 0 −386.327 386.327i 0 −20822.4 20822.4i 0 −171983. 0 232944.i 0
31.9 0 −252.451 252.451i 0 9569.73 + 9569.73i 0 168743. 0 403978.i 0
31.10 0 −193.300 193.300i 0 10355.8 + 10355.8i 0 −108466. 0 456712.i 0
31.11 0 −87.0633 87.0633i 0 −8119.76 8119.76i 0 149756. 0 516281.i 0
31.12 0 −78.2777 78.2777i 0 −5406.83 5406.83i 0 −30061.3 0 519186.i 0
31.13 0 −37.8347 37.8347i 0 −5270.38 5270.38i 0 −115235. 0 528578.i 0
31.14 0 234.930 + 234.930i 0 18167.9 + 18167.9i 0 112043. 0 421057.i 0
31.15 0 322.288 + 322.288i 0 16463.9 + 16463.9i 0 21051.9 0 323701.i 0
31.16 0 429.462 + 429.462i 0 2473.08 + 2473.08i 0 −198740. 0 162567.i 0
31.17 0 435.597 + 435.597i 0 −17337.6 17337.6i 0 102408. 0 151952.i 0
31.18 0 513.579 + 513.579i 0 −208.882 208.882i 0 42912.4 0 3914.51i 0
31.19 0 623.967 + 623.967i 0 4768.39 + 4768.39i 0 −21612.8 0 247229.i 0
31.20 0 640.742 + 640.742i 0 −10737.8 10737.8i 0 25233.9 0 289658.i 0
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.13.f.b 46
4.b odd 2 1 128.13.f.a 46
8.b even 2 1 16.13.f.a 46
8.d odd 2 1 64.13.f.a 46
16.e even 4 1 64.13.f.a 46
16.e even 4 1 128.13.f.a 46
16.f odd 4 1 16.13.f.a 46
16.f odd 4 1 inner 128.13.f.b 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.13.f.a 46 8.b even 2 1
16.13.f.a 46 16.f odd 4 1
64.13.f.a 46 8.d odd 2 1
64.13.f.a 46 16.e even 4 1
128.13.f.a 46 4.b odd 2 1
128.13.f.a 46 16.e even 4 1
128.13.f.b 46 1.a even 1 1 trivial
128.13.f.b 46 16.f odd 4 1 inner