Properties

Label 128.5.h.a
Level $128$
Weight $5$
Character orbit 128.h
Analytic conductor $13.231$
Analytic rank $0$
Dimension $60$
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] = [128,5,Mod(15,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.15");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.h (of order \(8\), degree \(4\), 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: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(15\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 8 q^{15} + 4 q^{19} - 4 q^{21} + 1156 q^{23} - 4 q^{25} - 3644 q^{27} - 4 q^{29} - 8 q^{33} + 5188 q^{35} - 4 q^{37} + 2692 q^{39} - 4 q^{41} - 5564 q^{43} - 328 q^{45} + 8 q^{47} - 8384 q^{51} + 956 q^{53} + 11780 q^{55} - 4 q^{57} + 13060 q^{59} + 7548 q^{61} - 8 q^{65} - 18876 q^{67} - 19588 q^{69} - 19964 q^{71} - 4 q^{73} + 200 q^{75} + 9404 q^{77} + 50184 q^{79} - 10556 q^{83} + 2496 q^{85} - 49276 q^{87} - 4 q^{89} - 31868 q^{91} + 320 q^{93} - 8 q^{97} + 46920 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 0 −15.8039 + 6.54620i 0 21.3091 + 8.82651i 0 18.2180 18.2180i 0 149.636 149.636i 0
15.2 0 −12.3594 + 5.11942i 0 −32.9803 13.6609i 0 −59.2707 + 59.2707i 0 69.2702 69.2702i 0
15.3 0 −10.5427 + 4.36694i 0 −27.2781 11.2990i 0 51.6577 51.6577i 0 34.8033 34.8033i 0
15.4 0 −8.76479 + 3.63049i 0 8.51003 + 3.52497i 0 −21.5633 + 21.5633i 0 6.36534 6.36534i 0
15.5 0 −7.43335 + 3.07899i 0 8.82167 + 3.65406i 0 −10.5220 + 10.5220i 0 −11.5012 + 11.5012i 0
15.6 0 −4.81945 + 1.99628i 0 42.1637 + 17.4648i 0 27.8831 27.8831i 0 −38.0337 + 38.0337i 0
15.7 0 −0.421454 + 0.174572i 0 −33.4127 13.8400i 0 −1.19092 + 1.19092i 0 −57.1285 + 57.1285i 0
15.8 0 0.239783 0.0993213i 0 −10.4583 4.33197i 0 51.7271 51.7271i 0 −57.2280 + 57.2280i 0
15.9 0 3.21611 1.33215i 0 23.1494 + 9.58878i 0 −30.5337 + 30.5337i 0 −48.7069 + 48.7069i 0
15.10 0 4.25478 1.76239i 0 −19.2280 7.96449i 0 −0.0963230 + 0.0963230i 0 −42.2785 + 42.2785i 0
15.11 0 5.05753 2.09490i 0 14.0253 + 5.80946i 0 −61.2653 + 61.2653i 0 −36.0856 + 36.0856i 0
15.12 0 10.9321 4.52821i 0 −0.302610 0.125345i 0 32.5244 32.5244i 0 41.7299 41.7299i 0
15.13 0 10.9680 4.54309i 0 35.3175 + 14.6290i 0 47.3057 47.3057i 0 42.3816 42.3816i 0
15.14 0 12.3801 5.12803i 0 −36.8127 15.2483i 0 −16.3058 + 16.3058i 0 69.6958 69.6958i 0
15.15 0 14.8038 6.13192i 0 5.46900 + 2.26533i 0 −27.5680 + 27.5680i 0 124.275 124.275i 0
47.1 0 −6.02892 + 14.5551i 0 1.24804 + 3.01304i 0 −48.9953 48.9953i 0 −118.227 118.227i 0
47.2 0 −5.09563 + 12.3019i 0 −6.07195 14.6590i 0 −3.01401 3.01401i 0 −68.0967 68.0967i 0
47.3 0 −4.50708 + 10.8811i 0 −0.960017 2.31769i 0 43.4088 + 43.4088i 0 −40.8081 40.8081i 0
47.4 0 −3.05748 + 7.38140i 0 −13.2656 32.0260i 0 14.4988 + 14.4988i 0 12.1387 + 12.1387i 0
47.5 0 −3.03024 + 7.31565i 0 12.4171 + 29.9775i 0 51.5981 + 51.5981i 0 12.9392 + 12.9392i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.h odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 128.5.h.a 60
4.b odd 2 1 32.5.h.a 60
32.g even 8 1 32.5.h.a 60
32.h odd 8 1 inner 128.5.h.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.5.h.a 60 4.b odd 2 1
32.5.h.a 60 32.g even 8 1
128.5.h.a 60 1.a even 1 1 trivial
128.5.h.a 60 32.h odd 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(128, [\chi])\).