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Properties

Label 88.6.c.a
Level $88$
Weight $6$
Character orbit 88.c
Analytic conductor $14.114$
Analytic rank $0$
Dimension $50$
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(45,88)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("88.45");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 88 = 2^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 88.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: \(14.1137761435\)
Analytic rank: \(0\)
Dimension: \(50\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 20 q^{4} + 294 q^{6} + 672 q^{8} - 4050 q^{9} - 634 q^{10} - 1876 q^{12} + 3004 q^{14} + 1800 q^{15} + 1600 q^{16} - 404 q^{17} - 8278 q^{18} - 1892 q^{20} - 6348 q^{23} + 8096 q^{24} - 34366 q^{25}+ \cdots + 541672 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −5.64233 − 0.405172i 23.7812i 31.6717 + 4.57222i 29.2444i 9.63548 − 134.181i −155.388 −176.849 − 38.6304i −322.547 11.8490 − 165.007i
45.2 −5.64233 + 0.405172i − 23.7812i 31.6717 − 4.57222i − 29.2444i 9.63548 + 134.181i −155.388 −176.849 + 38.6304i −322.547 11.8490 + 165.007i
45.3 −5.46626 − 1.45602i 10.1508i 27.7600 + 15.9180i 11.8120i 14.7798 − 55.4868i 105.581 −128.566 − 127.431i 139.962 17.1986 − 64.5677i
45.4 −5.46626 + 1.45602i − 10.1508i 27.7600 − 15.9180i − 11.8120i 14.7798 + 55.4868i 105.581 −128.566 + 127.431i 139.962 17.1986 + 64.5677i
45.5 −5.37187 − 1.77287i − 1.79889i 25.7139 + 19.0472i − 96.2311i −3.18921 + 9.66342i 15.2477 −104.363 − 147.906i 239.764 −170.605 + 516.941i
45.6 −5.37187 + 1.77287i 1.79889i 25.7139 − 19.0472i 96.2311i −3.18921 − 9.66342i 15.2477 −104.363 + 147.906i 239.764 −170.605 − 516.941i
45.7 −5.25739 − 2.08803i − 15.3551i 23.2803 + 21.9551i 106.563i −32.0619 + 80.7278i −90.6215 −76.5507 − 164.037i 7.22042 222.506 − 560.242i
45.8 −5.25739 + 2.08803i 15.3551i 23.2803 − 21.9551i − 106.563i −32.0619 − 80.7278i −90.6215 −76.5507 + 164.037i 7.22042 222.506 + 560.242i
45.9 −4.22471 − 3.76190i − 2.50880i 3.69627 + 31.7858i − 0.695436i −9.43785 + 10.5989i −230.038 103.959 − 148.191i 236.706 −2.61616 + 2.93801i
45.10 −4.22471 + 3.76190i 2.50880i 3.69627 − 31.7858i 0.695436i −9.43785 − 10.5989i −230.038 103.959 + 148.191i 236.706 −2.61616 − 2.93801i
45.11 −4.04438 − 3.95512i 30.7355i 0.713979 + 31.9920i − 80.1293i 121.563 − 124.306i 31.2790 123.645 − 132.212i −701.671 −316.921 + 324.073i
45.12 −4.04438 + 3.95512i − 30.7355i 0.713979 − 31.9920i 80.1293i 121.563 + 124.306i 31.2790 123.645 + 132.212i −701.671 −316.921 − 324.073i
45.13 −3.59816 − 4.36501i 19.0047i −6.10654 + 31.4119i 96.1103i 82.9554 − 68.3817i 182.403 159.086 − 86.3700i −118.177 419.522 − 345.820i
45.14 −3.59816 + 4.36501i − 19.0047i −6.10654 − 31.4119i − 96.1103i 82.9554 + 68.3817i 182.403 159.086 + 86.3700i −118.177 419.522 + 345.820i
45.15 −3.58284 − 4.37758i 9.64835i −6.32647 + 31.3684i − 18.4321i 42.2365 − 34.5685i −36.8402 159.984 − 84.6934i 149.909 −80.6880 + 66.0393i
45.16 −3.58284 + 4.37758i − 9.64835i −6.32647 − 31.3684i 18.4321i 42.2365 + 34.5685i −36.8402 159.984 + 84.6934i 149.909 −80.6880 − 66.0393i
45.17 −2.36733 − 5.13768i − 24.4516i −20.7915 + 24.3252i − 74.8784i −125.625 + 57.8852i −108.926 174.195 + 49.2339i −354.882 −384.701 + 177.262i
45.18 −2.36733 + 5.13768i 24.4516i −20.7915 − 24.3252i 74.8784i −125.625 − 57.8852i −108.926 174.195 − 49.2339i −354.882 −384.701 − 177.262i
45.19 −2.35408 − 5.14376i − 12.4417i −20.9166 + 24.2177i 50.7268i −63.9971 + 29.2887i 63.3730 173.809 + 50.5798i 88.2044 260.927 − 119.415i
45.20 −2.35408 + 5.14376i 12.4417i −20.9166 − 24.2177i − 50.7268i −63.9971 − 29.2887i 63.3730 173.809 − 50.5798i 88.2044 260.927 + 119.415i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.50
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.c.a ✓ 50
4.b odd 2 1 352.6.c.a 50
8.b even 2 1 inner 88.6.c.a ✓ 50
8.d odd 2 1 352.6.c.a 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.6.c.a ✓ 50 1.a even 1 1 trivial
88.6.c.a ✓ 50 8.b even 2 1 inner
352.6.c.a 50 4.b odd 2 1
352.6.c.a 50 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(88, [\chi])\).