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Properties

Label 88.6.i.b
Level $88$
Weight $6$
Character orbit 88.i
Analytic conductor $14.114$
Analytic rank $0$
Dimension $32$
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] = [88,6,Mod(9,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.9");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 88.i (of order \(5\), degree \(4\), 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: \(14.1137761435\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 21 q^{3} + 37 q^{5} - 181 q^{7} - 1123 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 21 q^{3} + 37 q^{5} - 181 q^{7} - 1123 q^{9} + 563 q^{11} - 753 q^{13} - 1731 q^{15} + 3153 q^{17} - 2022 q^{19} - 2282 q^{21} + 4516 q^{23} - 2551 q^{25} + 11538 q^{27} + 7829 q^{29} - 7643 q^{31} - 20846 q^{33} - 47237 q^{35} - 30139 q^{37} + 15155 q^{39} + 17377 q^{41} + 37862 q^{43} + 46556 q^{45} - 7053 q^{47} - 43539 q^{49} - 55812 q^{51} - 17637 q^{53} + 81511 q^{55} + 137684 q^{57} + 58662 q^{59} + 32619 q^{61} - 29128 q^{63} - 103034 q^{65} + 20982 q^{67} - 183982 q^{69} + 43641 q^{71} + 94725 q^{73} - 14162 q^{75} + 143529 q^{77} + 111121 q^{79} - 97990 q^{81} - 117894 q^{83} + 123843 q^{85} - 154438 q^{87} + 13394 q^{89} - 378075 q^{91} - 83153 q^{93} + 362641 q^{95} - 332002 q^{97} + 689815 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −23.1107 + 16.7909i 0 −2.41351 − 7.42801i 0 −183.543 − 133.352i 0 177.080 − 544.995i 0
9.2 0 −17.1412 + 12.4538i 0 26.4202 + 81.3130i 0 174.345 + 126.669i 0 63.6321 − 195.839i 0
9.3 0 −9.46729 + 6.87839i 0 −7.68277 − 23.6451i 0 61.8852 + 44.9623i 0 −32.7737 + 100.867i 0
9.4 0 −3.91381 + 2.84355i 0 −25.3488 − 78.0157i 0 −9.17147 − 6.66346i 0 −67.8590 + 208.849i 0
9.5 0 3.05815 − 2.22187i 0 27.7854 + 85.5146i 0 −208.040 − 151.150i 0 −70.6756 + 217.517i 0
9.6 0 4.28396 − 3.11248i 0 1.95669 + 6.02206i 0 19.9658 + 14.5060i 0 −66.4263 + 204.439i 0
9.7 0 16.3689 − 11.8927i 0 18.3395 + 56.4432i 0 89.0437 + 64.6941i 0 51.4139 − 158.236i 0
9.8 0 19.6408 − 14.2699i 0 −15.8312 − 48.7235i 0 −20.4815 − 14.8806i 0 107.041 − 329.439i 0
25.1 0 −8.38198 − 25.7971i 0 −71.7771 − 52.1491i 0 56.8840 − 175.071i 0 −398.641 + 289.630i 0
25.2 0 −7.55065 − 23.2385i 0 46.4402 + 33.7408i 0 −48.4576 + 149.137i 0 −286.424 + 208.100i 0
25.3 0 −3.40722 − 10.4863i 0 24.8294 + 18.0396i 0 19.3096 − 59.4288i 0 98.2368 − 71.3732i 0
25.4 0 −1.49749 − 4.60879i 0 −9.51767 − 6.91500i 0 −17.0105 + 52.3529i 0 177.593 − 129.029i 0
25.5 0 1.86043 + 5.72580i 0 −61.9819 − 45.0325i 0 −34.8453 + 107.243i 0 167.267 − 121.527i 0
25.6 0 3.92898 + 12.0922i 0 45.2219 + 32.8556i 0 60.1876 − 185.239i 0 65.8076 − 47.8120i 0
25.7 0 6.70898 + 20.6481i 0 67.3844 + 48.9576i 0 −74.4380 + 229.097i 0 −184.743 + 134.224i 0
25.8 0 8.12010 + 24.9911i 0 −45.3247 − 32.9303i 0 23.8662 − 73.4525i 0 −362.028 + 263.029i 0
49.1 0 −23.1107 − 16.7909i 0 −2.41351 + 7.42801i 0 −183.543 + 133.352i 0 177.080 + 544.995i 0
49.2 0 −17.1412 − 12.4538i 0 26.4202 − 81.3130i 0 174.345 − 126.669i 0 63.6321 + 195.839i 0
49.3 0 −9.46729 − 6.87839i 0 −7.68277 + 23.6451i 0 61.8852 − 44.9623i 0 −32.7737 − 100.867i 0
49.4 0 −3.91381 − 2.84355i 0 −25.3488 + 78.0157i 0 −9.17147 + 6.66346i 0 −67.8590 − 208.849i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.6.i.b ✓ 32
11.c even 5 1 inner 88.6.i.b ✓ 32
11.c even 5 1 968.6.a.q 16
11.d odd 10 1 968.6.a.p 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.6.i.b ✓ 32 1.a even 1 1 trivial
88.6.i.b ✓ 32 11.c even 5 1 inner
968.6.a.p 16 11.d odd 10 1
968.6.a.q 16 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 21 T_{3}^{31} + 1754 T_{3}^{30} + 28443 T_{3}^{29} + 1724760 T_{3}^{28} + \cdots + 27\!\cdots\!25 \) acting on \(S_{6}^{\mathrm{new}}(88, [\chi])\). Copy content Toggle raw display