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Properties

Label 110.5.h.a
Level $110$
Weight $5$
Character orbit 110.h
Analytic conductor $11.371$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [110,5,Mod(41,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("110.41");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 110.h (of order \(10\), 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: \(11.3706959392\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 18 q^{3} + 64 q^{4} - 200 q^{5} + 80 q^{6} - 60 q^{7} - 262 q^{9} - 258 q^{11} - 96 q^{12} - 570 q^{13} - 48 q^{14} + 150 q^{15} - 512 q^{16} + 150 q^{17} - 480 q^{18} - 1020 q^{19} + 1600 q^{20}+ \cdots + 65268 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −1.66251 − 2.28825i −4.27558 − 13.1589i −2.47214 + 7.60845i −9.04508 − 6.57164i −23.0026 + 31.6603i −19.6159 − 6.37359i 21.5200 − 6.99226i −89.3451 + 64.9130i 31.6228i
41.2 −1.66251 − 2.28825i −2.06757 − 6.36334i −2.47214 + 7.60845i −9.04508 − 6.57164i −11.1235 + 15.3102i 87.9902 + 28.5898i 21.5200 − 6.99226i 29.3131 − 21.2972i 31.6228i
41.3 −1.66251 − 2.28825i 1.90371 + 5.85901i −2.47214 + 7.60845i −9.04508 − 6.57164i 10.2419 − 14.0968i 3.91028 + 1.27053i 21.5200 − 6.99226i 34.8265 − 25.3030i 31.6228i
41.4 −1.66251 − 2.28825i 4.89398 + 15.0621i −2.47214 + 7.60845i −9.04508 − 6.57164i 26.3296 − 36.2396i −75.4276 − 24.5079i 21.5200 − 6.99226i −137.386 + 99.8171i 31.6228i
41.5 1.66251 + 2.28825i −4.16806 − 12.8280i −2.47214 + 7.60845i −9.04508 − 6.57164i 22.4241 − 30.8641i 17.3927 + 5.65125i −21.5200 + 6.99226i −81.6534 + 59.3247i − 31.6228i
41.6 1.66251 + 2.28825i −1.15021 − 3.54000i −2.47214 + 7.60845i −9.04508 − 6.57164i 6.18814 − 8.51724i 15.3085 + 4.97404i −21.5200 + 6.99226i 54.3218 − 39.4671i − 31.6228i
41.7 1.66251 + 2.28825i −0.417925 − 1.28624i −2.47214 + 7.60845i −9.04508 − 6.57164i 2.24843 − 3.09470i −34.7724 − 11.2982i −21.5200 + 6.99226i 64.0506 − 46.5355i − 31.6228i
41.8 1.66251 + 2.28825i 4.13576 + 12.7286i −2.47214 + 7.60845i −9.04508 − 6.57164i −22.2503 + 30.6250i −43.3269 − 14.0778i −21.5200 + 6.99226i −79.3813 + 57.6739i − 31.6228i
51.1 −1.66251 + 2.28825i −4.27558 + 13.1589i −2.47214 − 7.60845i −9.04508 + 6.57164i −23.0026 − 31.6603i −19.6159 + 6.37359i 21.5200 + 6.99226i −89.3451 − 64.9130i − 31.6228i
51.2 −1.66251 + 2.28825i −2.06757 + 6.36334i −2.47214 − 7.60845i −9.04508 + 6.57164i −11.1235 − 15.3102i 87.9902 − 28.5898i 21.5200 + 6.99226i 29.3131 + 21.2972i − 31.6228i
51.3 −1.66251 + 2.28825i 1.90371 − 5.85901i −2.47214 − 7.60845i −9.04508 + 6.57164i 10.2419 + 14.0968i 3.91028 − 1.27053i 21.5200 + 6.99226i 34.8265 + 25.3030i − 31.6228i
51.4 −1.66251 + 2.28825i 4.89398 − 15.0621i −2.47214 − 7.60845i −9.04508 + 6.57164i 26.3296 + 36.2396i −75.4276 + 24.5079i 21.5200 + 6.99226i −137.386 − 99.8171i − 31.6228i
51.5 1.66251 − 2.28825i −4.16806 + 12.8280i −2.47214 − 7.60845i −9.04508 + 6.57164i 22.4241 + 30.8641i 17.3927 − 5.65125i −21.5200 − 6.99226i −81.6534 − 59.3247i 31.6228i
51.6 1.66251 − 2.28825i −1.15021 + 3.54000i −2.47214 − 7.60845i −9.04508 + 6.57164i 6.18814 + 8.51724i 15.3085 − 4.97404i −21.5200 − 6.99226i 54.3218 + 39.4671i 31.6228i
51.7 1.66251 − 2.28825i −0.417925 + 1.28624i −2.47214 − 7.60845i −9.04508 + 6.57164i 2.24843 + 3.09470i −34.7724 + 11.2982i −21.5200 − 6.99226i 64.0506 + 46.5355i 31.6228i
51.8 1.66251 − 2.28825i 4.13576 − 12.7286i −2.47214 − 7.60845i −9.04508 + 6.57164i −22.2503 − 30.6250i −43.3269 + 14.0778i −21.5200 − 6.99226i −79.3813 − 57.6739i 31.6228i
61.1 −2.68999 + 0.874032i −14.0428 − 10.2027i 6.47214 − 4.70228i −3.45492 + 10.6331i 46.6926 + 15.1713i 17.2371 + 23.7249i −13.3001 + 18.3060i 68.0752 + 209.514i − 31.6228i
61.2 −2.68999 + 0.874032i −5.99233 − 4.35368i 6.47214 − 4.70228i −3.45492 + 10.6331i 19.9246 + 6.47389i −10.1130 − 13.9193i −13.3001 + 18.3060i −8.07693 − 24.8582i − 31.6228i
61.3 −2.68999 + 0.874032i −1.67962 − 1.22032i 6.47214 − 4.70228i −3.45492 + 10.6331i 5.58477 + 1.81460i 6.11394 + 8.41512i −13.3001 + 18.3060i −23.6984 − 72.9362i − 31.6228i
61.4 −2.68999 + 0.874032i 13.4352 + 9.76125i 6.47214 − 4.70228i −3.45492 + 10.6331i −44.6723 − 14.5149i −5.87268 − 8.08305i −13.3001 + 18.3060i 60.1925 + 185.253i − 31.6228i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.5.h.a ✓ 32
11.d odd 10 1 inner 110.5.h.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.5.h.a ✓ 32 1.a even 1 1 trivial
110.5.h.a ✓ 32 11.d odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 18 T_{3}^{31} + 617 T_{3}^{30} + 9912 T_{3}^{29} + 251358 T_{3}^{28} + \cdots + 34\!\cdots\!16 \) acting on \(S_{5}^{\mathrm{new}}(110, [\chi])\). Copy content Toggle raw display