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Properties

Label 140.15.d.a
Level $140$
Weight $15$
Character orbit 140.d
Analytic conductor $174.061$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [140,15,Mod(41,140)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.41");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 140.d (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: \(174.060555413\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 1364266 q^{7} - 54790830 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 1364266 q^{7} - 54790830 q^{9} - 26192606 q^{11} + 44843750 q^{15} + 1512952694 q^{21} - 8670648636 q^{23} - 43945312500 q^{25} - 43956395706 q^{29} + 44839531250 q^{35} - 169523027308 q^{37} + 805671747486 q^{39} + 554691319560 q^{43} + 1095688125176 q^{49} + 1032170625826 q^{51} - 4262050556480 q^{53} - 3162001614828 q^{57} - 15828953775898 q^{63} - 3014492656250 q^{65} - 23495876471600 q^{67} + 22887953193352 q^{71} + 56411959501488 q^{77} + 8995204220854 q^{79} + 132868621377344 q^{81} - 2034215156250 q^{85} - 53912825209186 q^{91} + 101093199187348 q^{93} + 3862990000000 q^{95} - 416078903388420 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 − 4265.32i 0 34938.6i 0 584126. − 580535.i 0 −1.34099e7 0
41.2 0 − 3896.14i 0 − 34938.6i 0 578123. + 586512.i 0 −1.03970e7 0
41.3 0 − 3891.45i 0 − 34938.6i 0 −806096. + 168618.i 0 −1.03604e7 0
41.4 0 − 3489.14i 0 34938.6i 0 189719. + 801392.i 0 −7.39111e6 0
41.5 0 − 3089.94i 0 − 34938.6i 0 775860. − 276159.i 0 −4.76477e6 0
41.6 0 − 3008.95i 0 34938.6i 0 −209395. − 796478.i 0 −4.27082e6 0
41.7 0 − 2691.62i 0 34938.6i 0 −810472. + 146144.i 0 −2.46186e6 0
41.8 0 − 2472.82i 0 − 34938.6i 0 463971. + 680407.i 0 −1.33187e6 0
41.9 0 − 2419.29i 0 − 34938.6i 0 57888.7 − 821506.i 0 −1.06999e6 0
41.10 0 − 2045.44i 0 34938.6i 0 −672278. − 475673.i 0 599139. 0
41.11 0 − 2006.39i 0 34938.6i 0 770567. + 290602.i 0 757366. 0
41.12 0 − 1565.13i 0 − 34938.6i 0 −816438. + 107948.i 0 2.33333e6 0
41.13 0 − 1557.79i 0 − 34938.6i 0 −188159. + 801760.i 0 2.35627e6 0
41.14 0 − 970.232i 0 34938.6i 0 823180. − 24435.3i 0 3.84162e6 0
41.15 0 − 770.711i 0 34938.6i 0 503431. − 651752.i 0 4.18897e6 0
41.16 0 − 376.610i 0 − 34938.6i 0 489531. − 662255.i 0 4.64113e6 0
41.17 0 − 359.529i 0 34938.6i 0 −635855. + 523366.i 0 4.65371e6 0
41.18 0 − 303.599i 0 34938.6i 0 −415571. + 711002.i 0 4.69080e6 0
41.19 0 303.599i 0 − 34938.6i 0 −415571. − 711002.i 0 4.69080e6 0
41.20 0 359.529i 0 − 34938.6i 0 −635855. − 523366.i 0 4.65371e6 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.36
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.15.d.a ✓ 36
7.b odd 2 1 inner 140.15.d.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.15.d.a ✓ 36 1.a even 1 1 trivial
140.15.d.a ✓ 36 7.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(140, [\chi])\).