↑

Properties

Label 140.3.b.a
Level $140$
Weight $3$
Character orbit 140.b
Analytic conductor $3.815$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [140,3,Mod(71,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.71");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 140.b (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: \(3.81472370104\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 2 q^{4} + 26 q^{8} - 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 2 q^{4} + 26 q^{8} - 88 q^{9} + 20 q^{10} - 60 q^{12} - 14 q^{14} + 18 q^{16} + 48 q^{17} - 114 q^{18} + 68 q^{22} - 52 q^{24} + 120 q^{25} + 60 q^{26} + 70 q^{28} + 32 q^{29} + 80 q^{30} + 2 q^{32} - 160 q^{33} - 88 q^{34} + 202 q^{36} - 96 q^{37} + 100 q^{38} - 80 q^{40} + 240 q^{41} - 32 q^{44} - 144 q^{46} - 60 q^{48} - 168 q^{49} + 10 q^{50} - 376 q^{52} + 160 q^{53} - 140 q^{54} + 14 q^{56} - 288 q^{57} + 56 q^{58} + 60 q^{60} + 64 q^{61} - 376 q^{62} + 482 q^{64} + 548 q^{66} + 36 q^{68} - 192 q^{69} - 462 q^{72} - 48 q^{73} + 36 q^{74} + 84 q^{76} - 224 q^{77} - 140 q^{78} + 160 q^{80} + 920 q^{81} + 20 q^{82} + 196 q^{84} + 320 q^{85} + 384 q^{86} - 176 q^{89} - 240 q^{90} + 40 q^{92} + 192 q^{93} + 540 q^{94} + 180 q^{96} - 144 q^{97} - 14 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1 −1.93152 − 0.518886i − 1.17068i 3.46151 + 2.00447i −2.23607 −0.607450 + 2.26119i − 2.64575i −5.64588 − 5.66781i 7.62951 4.31900 + 1.16026i
71.2 −1.93152 + 0.518886i 1.17068i 3.46151 − 2.00447i −2.23607 −0.607450 − 2.26119i 2.64575i −5.64588 + 5.66781i 7.62951 4.31900 − 1.16026i
71.3 −1.89507 − 0.639293i 2.04825i 3.18261 + 2.42302i 2.23607 1.30943 − 3.88159i − 2.64575i −4.48226 − 6.62641i 4.80467 −4.23751 − 1.42950i
71.4 −1.89507 + 0.639293i − 2.04825i 3.18261 − 2.42302i 2.23607 1.30943 + 3.88159i 2.64575i −4.48226 + 6.62641i 4.80467 −4.23751 + 1.42950i
71.5 −1.29284 − 1.52596i 0.0358808i −0.657137 + 3.94565i −2.23607 0.0547529 − 0.0463881i 2.64575i 6.87050 − 4.09832i 8.99871 2.89087 + 3.41216i
71.6 −1.29284 + 1.52596i − 0.0358808i −0.657137 − 3.94565i −2.23607 0.0547529 + 0.0463881i − 2.64575i 6.87050 + 4.09832i 8.99871 2.89087 − 3.41216i
71.7 −1.04389 − 1.70596i 5.03307i −1.82058 + 3.56167i −2.23607 8.58620 − 5.25398i − 2.64575i 7.97654 − 0.612157i −16.3318 2.33421 + 3.81464i
71.8 −1.04389 + 1.70596i − 5.03307i −1.82058 − 3.56167i −2.23607 8.58620 + 5.25398i 2.64575i 7.97654 + 0.612157i −16.3318 2.33421 − 3.81464i
71.9 −0.644828 − 1.89320i − 5.17316i −3.16839 + 2.44157i −2.23607 −9.79381 + 3.33580i − 2.64575i 6.66545 + 4.42400i −17.7616 1.44188 + 4.23332i
71.10 −0.644828 + 1.89320i 5.17316i −3.16839 − 2.44157i −2.23607 −9.79381 − 3.33580i 2.64575i 6.66545 − 4.42400i −17.7616 1.44188 − 4.23332i
71.11 −0.583151 − 1.91310i 1.28336i −3.31987 + 2.23125i 2.23607 2.45519 − 0.748392i − 2.64575i 6.20458 + 5.05007i 7.35299 −1.30397 − 4.27781i
71.12 −0.583151 + 1.91310i − 1.28336i −3.31987 − 2.23125i 2.23607 2.45519 + 0.748392i 2.64575i 6.20458 − 5.05007i 7.35299 −1.30397 + 4.27781i
71.13 0.452596 − 1.94812i 5.20930i −3.59031 − 1.76342i 2.23607 10.1483 + 2.35771i 2.64575i −5.06031 + 6.19623i −18.1369 1.01203 − 4.35612i
71.14 0.452596 + 1.94812i − 5.20930i −3.59031 + 1.76342i 2.23607 10.1483 − 2.35771i − 2.64575i −5.06031 − 6.19623i −18.1369 1.01203 + 4.35612i
71.15 0.992782 − 1.73620i − 3.02629i −2.02877 − 3.44733i 2.23607 −5.25425 − 3.00445i − 2.64575i −7.99938 + 0.0998882i −0.158458 2.21993 − 3.88226i
71.16 0.992782 + 1.73620i 3.02629i −2.02877 + 3.44733i 2.23607 −5.25425 + 3.00445i 2.64575i −7.99938 − 0.0998882i −0.158458 2.21993 + 3.88226i
71.17 1.20405 − 1.59696i − 3.96180i −1.10055 − 3.84562i −2.23607 −6.32683 − 4.77019i 2.64575i −7.46641 − 2.87277i −6.69586 −2.69233 + 3.57091i
71.18 1.20405 + 1.59696i 3.96180i −1.10055 + 3.84562i −2.23607 −6.32683 + 4.77019i − 2.64575i −7.46641 + 2.87277i −6.69586 −2.69233 − 3.57091i
71.19 1.77085 − 0.929558i 0.852380i 2.27184 − 3.29222i 2.23607 0.792337 + 1.50944i 2.64575i 0.962787 − 7.94185i 8.27345 3.95975 − 2.07856i
71.20 1.77085 + 0.929558i − 0.852380i 2.27184 + 3.29222i 2.23607 0.792337 − 1.50944i − 2.64575i 0.962787 + 7.94185i 8.27345 3.95975 + 2.07856i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.3.b.a ✓ 24
4.b odd 2 1 inner 140.3.b.a ✓ 24
8.b even 2 1 2240.3.d.c 24
8.d odd 2 1 2240.3.d.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.3.b.a ✓ 24 1.a even 1 1 trivial
140.3.b.a ✓ 24 4.b odd 2 1 inner
2240.3.d.c 24 8.b even 2 1
2240.3.d.c 24 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(140, [\chi])\).