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Properties

Label 88.6.g.b
Level $88$
Weight $6$
Character orbit 88.g
Analytic conductor $14.114$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [88,6,Mod(43,88)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(88, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.43");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 88.g (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: \(14.1137761435\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{3} + 84 q^{4} + 4848 q^{9} - 1080 q^{11} - 1152 q^{12} + 392 q^{14} - 1968 q^{16} - 3264 q^{20} + 1744 q^{22} - 37504 q^{25} - 8656 q^{26} + 952 q^{27} + 1400 q^{33} - 38664 q^{34} - 46332 q^{36}+ \cdots - 419680 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −5.62982 − 0.552410i 15.8790 31.3897 + 6.21994i − 74.0890i −89.3961 − 8.77175i −107.629 −173.282 − 52.3571i 9.14385 −40.9275 + 417.108i
43.2 −5.62982 + 0.552410i 15.8790 31.3897 − 6.21994i 74.0890i −89.3961 + 8.77175i −107.629 −173.282 + 52.3571i 9.14385 −40.9275 − 417.108i
43.3 −5.60568 − 0.759211i −4.76318 30.8472 + 8.51179i − 77.8059i 26.7008 + 3.61626i 220.263 −166.457 − 71.1339i −220.312 −59.0712 + 436.155i
43.4 −5.60568 + 0.759211i −4.76318 30.8472 − 8.51179i 77.8059i 26.7008 − 3.61626i 220.263 −166.457 + 71.1339i −220.312 −59.0712 − 436.155i
43.5 −5.50295 − 1.31056i −16.3781 28.5649 + 14.4239i − 21.0020i 90.1278 + 21.4644i −70.9880 −138.288 − 116.810i 25.2417 −27.5243 + 115.573i
43.6 −5.50295 + 1.31056i −16.3781 28.5649 − 14.4239i 21.0020i 90.1278 − 21.4644i −70.9880 −138.288 + 116.810i 25.2417 −27.5243 − 115.573i
43.7 −5.15446 − 2.33057i −28.6305 21.1369 + 24.0257i 76.6651i 147.575 + 66.7254i −1.31858 −52.9557 − 173.100i 576.707 178.673 − 395.167i
43.8 −5.15446 + 2.33057i −28.6305 21.1369 − 24.0257i − 76.6651i 147.575 − 66.7254i −1.31858 −52.9557 + 173.100i 576.707 178.673 + 395.167i
43.9 −5.14270 − 2.35640i 24.2747 20.8948 + 24.2365i 5.84531i −124.838 − 57.2009i 82.7069 −50.3445 − 173.878i 346.261 13.7739 − 30.0607i
43.10 −5.14270 + 2.35640i 24.2747 20.8948 − 24.2365i − 5.84531i −124.838 + 57.2009i 82.7069 −50.3445 + 173.878i 346.261 13.7739 + 30.0607i
43.11 −4.88483 − 2.85279i 1.60981 15.7232 + 27.8708i 41.3820i −7.86363 − 4.59244i −175.453 2.70444 − 180.999i −240.409 118.054 − 202.144i
43.12 −4.88483 + 2.85279i 1.60981 15.7232 − 27.8708i − 41.3820i −7.86363 + 4.59244i −175.453 2.70444 + 180.999i −240.409 118.054 + 202.144i
43.13 −4.07708 − 3.92140i 8.04581 1.24517 + 31.9758i 59.2406i −32.8034 − 31.5509i 154.480 120.313 − 135.251i −178.265 232.306 − 241.529i
43.14 −4.07708 + 3.92140i 8.04581 1.24517 − 31.9758i − 59.2406i −32.8034 + 31.5509i 154.480 120.313 + 135.251i −178.265 232.306 + 241.529i
43.15 −4.02296 − 3.97691i −6.67736 0.368403 + 31.9979i − 46.7458i 26.8627 + 26.5552i −19.9373 125.771 − 130.191i −198.413 −185.904 + 188.056i
43.16 −4.02296 + 3.97691i −6.67736 0.368403 − 31.9979i 46.7458i 26.8627 − 26.5552i −19.9373 125.771 + 130.191i −198.413 −185.904 − 188.056i
43.17 −3.20033 − 4.66453i −24.1739 −11.5157 + 29.8561i − 41.4331i 77.3645 + 112.760i 127.804 176.119 − 41.8341i 341.376 −193.266 + 132.600i
43.18 −3.20033 + 4.66453i −24.1739 −11.5157 − 29.8561i 41.4331i 77.3645 − 112.760i 127.804 176.119 + 41.8341i 341.376 −193.266 − 132.600i
43.19 −2.96677 − 4.81646i 17.9061 −14.3965 + 28.5787i − 101.069i −53.1231 − 86.2437i −119.327 180.359 − 15.4460i 77.6267 −486.792 + 299.847i
43.20 −2.96677 + 4.81646i 17.9061 −14.3965 − 28.5787i 101.069i −53.1231 + 86.2437i −119.327 180.359 + 15.4460i 77.6267 −486.792 − 299.847i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.56
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.b odd 2 1 inner
88.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 88.6.g.b ✓ 56
4.b odd 2 1 352.6.g.b 56
8.b even 2 1 352.6.g.b 56
8.d odd 2 1 inner 88.6.g.b ✓ 56
11.b odd 2 1 inner 88.6.g.b ✓ 56
44.c even 2 1 352.6.g.b 56
88.b odd 2 1 352.6.g.b 56
88.g even 2 1 inner 88.6.g.b ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.6.g.b ✓ 56 1.a even 1 1 trivial
88.6.g.b ✓ 56 8.d odd 2 1 inner
88.6.g.b ✓ 56 11.b odd 2 1 inner
88.6.g.b ✓ 56 88.g even 2 1 inner
352.6.g.b 56 4.b odd 2 1
352.6.g.b 56 8.b even 2 1
352.6.g.b 56 44.c even 2 1
352.6.g.b 56 88.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 2 T_{3}^{13} - 2305 T_{3}^{12} - 4368 T_{3}^{11} + 2011302 T_{3}^{10} + \cdots - 34\!\cdots\!72 \) acting on \(S_{6}^{\mathrm{new}}(88, [\chi])\). Copy content Toggle raw display