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Properties

Label 140.6.q.a
Level $140$
Weight $6$
Character orbit 140.q
Analytic conductor $22.454$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [140,6,Mod(9,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.9");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 140.q (of order \(6\), degree \(2\), 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: \(22.4537347738\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 11 q^{5} + 1700 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 11 q^{5} + 1700 q^{9} - 338 q^{11} + 994 q^{15} + 566 q^{19} + 4484 q^{21} - 579 q^{25} + 92 q^{29} - 16614 q^{31} - 13755 q^{35} - 748 q^{39} - 52644 q^{41} + 26320 q^{45} + 37826 q^{49} + 50158 q^{51} + 35054 q^{55} - 53294 q^{59} + 19444 q^{61} + 29396 q^{65} + 250200 q^{69} - 102536 q^{71} - 1067 q^{75} - 198010 q^{79} - 143876 q^{81} + 284570 q^{85} - 167384 q^{89} - 253084 q^{91} - 20209 q^{95} - 323152 q^{99}+O(q^{100}) \) 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 −26.3427 + 15.2090i 0 −55.1017 + 9.42361i 0 −93.8337 − 89.4552i 0 341.126 − 590.848i 0
9.2 0 −20.4750 + 11.8212i 0 −17.9534 − 52.9403i 0 124.137 + 37.3767i 0 157.983 − 273.634i 0
9.3 0 −20.3682 + 11.7596i 0 28.3146 + 48.2005i 0 118.363 − 52.8896i 0 155.074 − 268.597i 0
9.4 0 −18.6125 + 10.7459i 0 44.8931 − 33.3108i 0 −61.5653 + 114.091i 0 109.449 − 189.571i 0
9.5 0 −16.7852 + 9.69094i 0 26.7024 + 49.1119i 0 −125.654 + 31.9078i 0 66.3287 − 114.885i 0
9.6 0 −10.2510 + 5.91839i 0 22.2839 − 51.2682i 0 54.4300 − 117.662i 0 −51.4452 + 89.1057i 0
9.7 0 −10.0374 + 5.79508i 0 −43.8978 + 34.6119i 0 24.0289 + 127.395i 0 −54.3340 + 94.1092i 0
9.8 0 −6.20815 + 3.58428i 0 55.8994 − 0.511204i 0 −57.0587 − 116.410i 0 −95.8059 + 165.941i 0
9.9 0 −5.98937 + 3.45796i 0 −50.3517 + 24.2839i 0 99.6943 − 82.8737i 0 −97.5850 + 169.022i 0
9.10 0 −4.85422 + 2.80258i 0 −40.5786 − 38.4496i 0 −119.268 − 50.8138i 0 −105.791 + 183.235i 0
9.11 0 4.85422 − 2.80258i 0 53.5877 + 15.9173i 0 119.268 + 50.8138i 0 −105.791 + 183.235i 0
9.12 0 5.98937 − 3.45796i 0 4.14535 + 55.7478i 0 −99.6943 + 82.8737i 0 −97.5850 + 169.022i 0
9.13 0 6.20815 − 3.58428i 0 −27.5070 − 48.6659i 0 57.0587 + 116.410i 0 −95.8059 + 165.941i 0
9.14 0 10.0374 − 5.79508i 0 −8.02591 + 55.3226i 0 −24.0289 − 127.395i 0 −54.3340 + 94.1092i 0
9.15 0 10.2510 − 5.91839i 0 33.2577 − 44.9325i 0 −54.4300 + 117.662i 0 −51.4452 + 89.1057i 0
9.16 0 16.7852 − 9.69094i 0 −55.8834 + 1.43102i 0 125.654 − 31.9078i 0 66.3287 − 114.885i 0
9.17 0 18.6125 − 10.7459i 0 6.40151 − 55.5340i 0 61.5653 − 114.091i 0 109.449 − 189.571i 0
9.18 0 20.3682 − 11.7596i 0 −55.9001 − 0.420909i 0 −118.363 + 52.8896i 0 155.074 − 268.597i 0
9.19 0 20.4750 − 11.8212i 0 54.8243 − 10.9220i 0 −124.137 − 37.3767i 0 157.983 − 273.634i 0
9.20 0 26.3427 − 15.2090i 0 19.3898 + 52.4313i 0 93.8337 + 89.4552i 0 341.126 − 590.848i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.6.q.a ✓ 40
5.b even 2 1 inner 140.6.q.a ✓ 40
7.c even 3 1 inner 140.6.q.a ✓ 40
35.j even 6 1 inner 140.6.q.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.q.a ✓ 40 1.a even 1 1 trivial
140.6.q.a ✓ 40 5.b even 2 1 inner
140.6.q.a ✓ 40 7.c even 3 1 inner
140.6.q.a ✓ 40 35.j even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(140, [\chi])\).