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Properties

Label 240.8.h.b
Level $240$
Weight $8$
Character orbit 240.h
Analytic conductor $74.972$
Analytic rank $0$
Dimension $36$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,8,Mod(191,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.191");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 240.h (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: \(74.9724061162\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3324 q^{9} - 432 q^{13} - 22032 q^{21} - 562500 q^{25} - 386232 q^{33} + 1200048 q^{37} - 1710000 q^{45} - 3237972 q^{49} - 9331200 q^{57} + 11262360 q^{61} - 11352552 q^{69} + 37210296 q^{73} - 12564252 q^{81}+ \cdots + 94537320 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1 0 −45.6703 − 10.0611i 0 − 125.000i 0 − 760.932i 0 1984.55 + 918.983i 0
191.2 0 −45.6703 + 10.0611i 0 125.000i 0 760.932i 0 1984.55 − 918.983i 0
191.3 0 −44.4287 − 14.5975i 0 125.000i 0 48.6804i 0 1760.83 + 1297.10i 0
191.4 0 −44.4287 + 14.5975i 0 − 125.000i 0 − 48.6804i 0 1760.83 − 1297.10i 0
191.5 0 −42.2282 − 20.0943i 0 125.000i 0 − 1615.14i 0 1379.44 + 1697.09i 0
191.6 0 −42.2282 + 20.0943i 0 − 125.000i 0 1615.14i 0 1379.44 − 1697.09i 0
191.7 0 −41.2400 − 22.0513i 0 125.000i 0 1218.04i 0 1214.48 + 1818.79i 0
191.8 0 −41.2400 + 22.0513i 0 − 125.000i 0 − 1218.04i 0 1214.48 − 1818.79i 0
191.9 0 −35.0344 − 30.9773i 0 − 125.000i 0 201.250i 0 267.817 + 2170.54i 0
191.10 0 −35.0344 + 30.9773i 0 125.000i 0 − 201.250i 0 267.817 − 2170.54i 0
191.11 0 −27.9167 − 37.5188i 0 − 125.000i 0 1233.84i 0 −628.314 + 2094.80i 0
191.12 0 −27.9167 + 37.5188i 0 125.000i 0 − 1233.84i 0 −628.314 − 2094.80i 0
191.13 0 −20.1503 − 42.2015i 0 125.000i 0 400.709i 0 −1374.93 + 1700.74i 0
191.14 0 −20.1503 + 42.2015i 0 − 125.000i 0 − 400.709i 0 −1374.93 − 1700.74i 0
191.15 0 −15.4346 − 44.1449i 0 125.000i 0 − 3.33310i 0 −1710.54 + 1362.72i 0
191.16 0 −15.4346 + 44.1449i 0 − 125.000i 0 3.33310i 0 −1710.54 − 1362.72i 0
191.17 0 −7.89558 − 46.0940i 0 125.000i 0 − 1350.65i 0 −2062.32 + 727.879i 0
191.18 0 −7.89558 + 46.0940i 0 − 125.000i 0 1350.65i 0 −2062.32 − 727.879i 0
191.19 0 7.89558 − 46.0940i 0 − 125.000i 0 − 1350.65i 0 −2062.32 − 727.879i 0
191.20 0 7.89558 + 46.0940i 0 125.000i 0 1350.65i 0 −2062.32 + 727.879i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.8.h.b ✓ 36
3.b odd 2 1 inner 240.8.h.b ✓ 36
4.b odd 2 1 inner 240.8.h.b ✓ 36
12.b even 2 1 inner 240.8.h.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.8.h.b ✓ 36 1.a even 1 1 trivial
240.8.h.b ✓ 36 3.b odd 2 1 inner
240.8.h.b ✓ 36 4.b odd 2 1 inner
240.8.h.b ✓ 36 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{18} + 8221380 T_{7}^{16} + 26288308795320 T_{7}^{14} + \cdots + 10\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display