Properties

Label 140.3.t.a
Level $140$
Weight $3$
Character orbit 140.t
Analytic conductor $3.815$
Analytic rank $0$
Dimension $64$
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] = [140,3,Mod(11,140)]
 
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([3, 0, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 140.t (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: \(3.81472370104\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{2} + 6 q^{4} + 20 q^{8} + 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{2} + 6 q^{4} + 20 q^{8} + 96 q^{9} + 10 q^{12} + 32 q^{13} - 38 q^{14} - 22 q^{16} - 80 q^{18} - 40 q^{20} + 104 q^{21} - 112 q^{22} + 104 q^{24} - 160 q^{25} - 66 q^{26} - 30 q^{28} - 112 q^{29} + 162 q^{32} + 408 q^{34} + 140 q^{36} - 176 q^{37} - 80 q^{38} - 16 q^{41} + 54 q^{42} - 138 q^{44} - 40 q^{45} - 206 q^{46} - 780 q^{48} - 96 q^{49} - 20 q^{50} - 132 q^{52} + 144 q^{53} - 452 q^{54} + 104 q^{56} + 288 q^{57} + 142 q^{58} + 70 q^{60} - 176 q^{61} + 536 q^{62} - 300 q^{64} + 40 q^{65} + 60 q^{66} + 176 q^{68} + 288 q^{69} + 180 q^{70} - 120 q^{72} + 240 q^{73} - 198 q^{74} - 588 q^{76} + 272 q^{77} - 120 q^{78} - 248 q^{81} + 126 q^{82} + 556 q^{84} + 196 q^{86} + 40 q^{88} - 8 q^{89} + 180 q^{90} + 1292 q^{92} - 304 q^{93} - 354 q^{94} + 468 q^{96} - 1344 q^{97} + 454 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.98057 0.278096i −3.59758 2.07707i 3.84533 + 1.10158i 1.11803 + 1.93649i 6.54764 + 5.11425i −5.74208 + 4.00357i −7.30960 3.25112i 4.12840 + 7.15060i −1.67582 4.14628i
11.2 −1.97671 + 0.304335i 0.0515150 + 0.0297422i 3.81476 1.20316i −1.11803 1.93649i −0.110882 0.0431139i 6.36655 + 2.90982i −7.17451 + 3.53927i −4.49823 7.79116i 2.79937 + 3.48762i
11.3 −1.84602 0.769551i 4.25107 + 2.45436i 2.81558 + 2.84121i 1.11803 + 1.93649i −5.95882 7.80221i 4.48383 + 5.37543i −3.01117 7.41167i 7.54776 + 13.0731i −0.573685 4.43519i
11.4 −1.83325 + 0.799499i −3.67944 2.12433i 2.72160 2.93136i −1.11803 1.93649i 8.44374 + 0.952710i −2.95680 6.34487i −2.64575 + 7.54983i 4.52554 + 7.83847i 3.59786 + 2.65620i
11.5 −1.78604 0.900030i −0.304256 0.175662i 2.37989 + 3.21498i −1.11803 1.93649i 0.385312 + 0.587580i −3.95608 + 5.77489i −1.35700 7.88407i −4.43829 7.68734i 0.253954 + 4.46492i
11.6 −1.64012 + 1.14455i 2.36556 + 1.36575i 1.38001 3.75441i 1.11803 + 1.93649i −5.44298 + 0.467488i −2.85107 + 6.39308i 2.03371 + 7.73719i −0.769427 1.33269i −4.05012 1.89644i
11.7 −1.60242 + 1.19677i −3.02406 1.74594i 1.13550 3.83545i 1.11803 + 1.93649i 6.93529 0.821360i 6.46101 + 2.69358i 2.77058 + 7.50492i 1.59661 + 2.76541i −4.10909 1.76505i
11.8 −1.59436 1.20748i 3.26501 + 1.88505i 1.08398 + 3.85032i −1.11803 1.93649i −2.92944 6.94789i 0.821623 6.95161i 2.92093 7.44769i 2.60684 + 4.51519i −0.555726 + 4.43747i
11.9 −1.39243 + 1.43566i 4.67673 + 2.70011i −0.122262 3.99813i −1.11803 1.93649i −10.3885 + 2.95449i 6.64462 2.20205i 5.91021 + 5.39160i 10.0812 + 17.4612i 4.33694 + 1.09131i
11.10 −1.21722 1.58694i −0.0501702 0.0289658i −1.03674 + 3.86331i 1.11803 + 1.93649i 0.0151014 + 0.114875i −5.51812 4.30701i 7.39278 3.05726i −4.49832 7.79132i 1.71220 4.13139i
11.11 −1.05997 + 1.69602i 0.809998 + 0.467653i −1.75294 3.59544i −1.11803 1.93649i −1.65172 + 0.878074i −6.99108 + 0.353216i 7.95599 + 0.838020i −4.06260 7.03663i 4.46940 + 0.156411i
11.12 −0.803353 + 1.83156i −1.11560 0.644090i −2.70925 2.94278i 1.11803 + 1.93649i 2.07591 1.52585i 1.11922 6.90995i 7.56638 2.59807i −3.67030 6.35714i −4.44498 + 0.492064i
11.13 −0.765717 1.84761i 0.0501702 + 0.0289658i −2.82735 + 2.82950i 1.11803 + 1.93649i 0.0151014 0.114875i 5.51812 + 4.30701i 7.39278 + 3.05726i −4.49832 7.79132i 2.72179 3.54850i
11.14 −0.422124 + 1.95495i −3.79087 2.18866i −3.64362 1.65046i −1.11803 1.93649i 5.87893 6.48706i 2.80797 + 6.41212i 4.76462 6.42638i 5.08046 + 8.79962i 4.25768 1.36826i
11.15 −0.248528 1.98450i −3.26501 1.88505i −3.87647 + 0.986407i −1.11803 1.93649i −2.92944 + 6.94789i −0.821623 + 6.95161i 2.92093 + 7.44769i 2.60684 + 4.51519i −3.56510 + 2.70001i
11.16 0.0819421 + 1.99832i 4.93310 + 2.84813i −3.98657 + 0.327493i 1.11803 + 1.93649i −5.28724 + 10.0913i −6.48796 2.62800i −0.981105 7.93961i 11.7237 + 20.3060i −3.77812 + 2.39287i
11.17 0.113572 1.99677i 0.304256 + 0.175662i −3.97420 0.453554i −1.11803 1.93649i 0.385312 0.587580i 3.95608 5.77489i −1.35700 + 7.88407i −4.43829 7.68734i −3.99371 + 2.01253i
11.18 0.256560 1.98348i −4.25107 2.45436i −3.86835 1.01776i 1.11803 + 1.93649i −5.95882 + 7.80221i −4.48383 5.37543i −3.01117 + 7.41167i 7.54776 + 13.0731i 4.12783 1.72077i
11.19 0.553243 + 1.92196i −2.00413 1.15708i −3.38784 + 2.12662i 1.11803 + 1.93649i 1.11510 4.49200i −5.87344 + 3.80825i −5.96157 5.33476i −1.82232 3.15635i −3.10331 + 3.22016i
11.20 0.749447 1.85427i 3.59758 + 2.07707i −2.87666 2.77936i 1.11803 + 1.93649i 6.54764 5.11425i 5.74208 4.00357i −7.30960 + 3.25112i 4.12840 + 7.15060i 4.42869 0.621842i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.3.t.a 64
4.b odd 2 1 inner 140.3.t.a 64
7.c even 3 1 inner 140.3.t.a 64
28.g odd 6 1 inner 140.3.t.a 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.3.t.a 64 1.a even 1 1 trivial
140.3.t.a 64 4.b odd 2 1 inner
140.3.t.a 64 7.c even 3 1 inner
140.3.t.a 64 28.g odd 6 1 inner

Hecke kernels

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