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Properties

Label 141.2.g.a
Level $141$
Weight $2$
Character orbit 141.g
Analytic conductor $1.126$
Analytic rank $0$
Dimension $308$
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] = [141,2,Mod(5,141)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(141, base_ring=CyclotomicField(46))
 
chi = DirichletCharacter(H, H._module([23, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("141.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 141 = 3 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 141.g (of order \(46\), degree \(22\), 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: \(1.12589066850\)
Analytic rank: \(0\)
Dimension: \(308\)
Relative dimension: \(14\) over \(\Q(\zeta_{46})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{46}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 308 q - 19 q^{3} - 26 q^{4} - 15 q^{6} - 42 q^{7} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 308 q - 19 q^{3} - 26 q^{4} - 15 q^{6} - 42 q^{7} - 19 q^{9} - 46 q^{10} - 28 q^{12} - 46 q^{13} - 23 q^{15} - 70 q^{16} - 14 q^{18} - 46 q^{19} - 27 q^{21} - 46 q^{22} - 42 q^{24} - 36 q^{25} - 43 q^{27} - 46 q^{28} - 23 q^{30} - 46 q^{31} - 23 q^{33} - 38 q^{34} + 69 q^{36} - 66 q^{37} + 46 q^{39} - 46 q^{40} + 162 q^{42} - 46 q^{43} + 92 q^{45} + 178 q^{48} - 92 q^{49} + 80 q^{51} - 46 q^{52} + 58 q^{54} + 14 q^{55} + 46 q^{57} - 46 q^{58} + 69 q^{60} - 78 q^{61} + 13 q^{63} + 66 q^{64} - 23 q^{66} - 46 q^{67} - 23 q^{69} - 46 q^{70} - 105 q^{72} - 46 q^{73} + 17 q^{75} - 23 q^{78} + 50 q^{79} + 5 q^{81} + 368 q^{82} + 62 q^{84} + 138 q^{85} - 23 q^{87} + 230 q^{88} - 23 q^{90} + 322 q^{91} + 198 q^{94} + 93 q^{96} + 26 q^{97} - 23 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.12444 + 0.755025i 1.31641 − 1.12564i 2.39174 − 1.94583i −0.0804473 + 1.17610i −1.94673 + 3.38528i 0.588354 − 0.833507i −1.26903 + 2.08684i 0.465848 − 2.96361i −0.717078 − 2.55928i
5.2 −2.06929 + 0.735425i −0.943659 − 1.45241i 2.18968 − 1.78144i 0.133165 − 1.94680i 3.02085 + 2.31147i −2.43469 + 3.44917i −0.938864 + 1.54390i −1.21901 + 2.74117i 1.15617 + 4.12643i
5.3 −1.65893 + 0.589585i 0.151464 + 1.72542i 0.853028 − 0.693989i −0.158247 + 2.31349i −1.26855 − 2.77305i −0.0790737 + 0.112022i 0.823597 − 1.35435i −2.95412 + 0.522677i −1.10148 − 3.93122i
5.4 −1.35677 + 0.482197i 1.59914 + 0.665404i 0.0568971 − 0.0462892i 0.288862 − 4.22302i −2.49052 − 0.131702i 1.67731 − 2.37620i 1.44144 − 2.37034i 2.11448 + 2.12814i 1.64441 + 5.86897i
5.5 −0.920765 + 0.327240i −1.72800 − 0.118326i −0.810701 + 0.659554i 0.0433257 − 0.633400i 1.62981 − 0.456521i 1.63998 − 2.32333i 1.54609 − 2.54244i 2.97200 + 0.408936i 0.167381 + 0.597390i
5.6 −0.581201 + 0.206559i −0.470364 − 1.66696i −1.25629 + 1.02207i −0.265119 + 3.87591i 0.617701 + 0.871681i −0.0201057 + 0.0284832i 1.16002 − 1.90757i −2.55751 + 1.56816i −0.646516 − 2.30744i
5.7 −0.216384 + 0.0769029i 1.63641 + 0.567592i −1.51051 + 1.22889i −0.0834835 + 1.22049i −0.397743 + 0.00302699i −2.03690 + 2.88562i 0.470984 − 0.774500i 2.35568 + 1.85763i −0.0757944 − 0.270514i
5.8 0.216384 − 0.0769029i −1.27481 + 1.17255i −1.51051 + 1.22889i 0.0834835 − 1.22049i −0.185675 + 0.351757i −2.03690 + 2.88562i −0.470984 + 0.774500i 0.250257 − 2.98954i −0.0757944 − 0.270514i
5.9 0.581201 − 0.206559i −0.232695 − 1.71635i −1.25629 + 1.02207i 0.265119 − 3.87591i −0.489770 − 0.949478i −0.0201057 + 0.0284832i −1.16002 + 1.90757i −2.89171 + 0.798773i −0.646516 − 2.30744i
5.10 0.920765 − 0.327240i 1.53780 − 0.796969i −0.810701 + 0.659554i −0.0433257 + 0.633400i 1.15516 − 1.23705i 1.63998 − 2.32333i −1.54609 + 2.54244i 1.72968 − 2.45116i 0.167381 + 0.597390i
5.11 1.35677 − 0.482197i −1.20165 + 1.24741i 0.0568971 − 0.0462892i −0.288862 + 4.22302i −1.02887 + 2.27189i 1.67731 − 2.37620i −1.44144 + 2.37034i −0.112081 − 2.99791i 1.64441 + 5.86897i
5.12 1.65893 − 0.589585i 0.548483 + 1.64291i 0.853028 − 0.693989i 0.158247 − 2.31349i 1.87853 + 2.40211i −0.0790737 + 0.112022i −0.823597 + 1.35435i −2.39833 + 1.80222i −1.10148 − 3.93122i
5.13 2.06929 − 0.735425i 0.286892 − 1.70813i 2.18968 − 1.78144i −0.133165 + 1.94680i −0.662537 − 3.74559i −2.43469 + 3.44917i 0.938864 − 1.54390i −2.83539 − 0.980094i 1.15617 + 4.12643i
5.14 2.12444 − 0.755025i −1.65588 − 0.507997i 2.39174 − 1.94583i 0.0804473 − 1.17610i −3.90136 + 0.171024i 0.588354 − 0.833507i 1.26903 − 2.08684i 2.48388 + 1.68236i −0.717078 − 2.55928i
11.1 −2.01954 − 1.88612i −1.72013 + 0.202853i 0.384613 + 5.62284i −0.100193 + 0.193363i 3.85648 + 2.83470i 0.480921 − 0.208894i 6.34080 − 7.79389i 2.91770 − 0.697866i 0.567048 − 0.201529i
11.2 −1.52475 − 1.42401i 0.558658 + 1.63948i 0.160552 + 2.34719i −0.519134 + 1.00188i 1.48283 − 3.29533i −4.03098 + 1.75090i 0.464345 − 0.570757i −2.37580 + 1.83182i 2.21824 − 0.788364i
11.3 −1.24966 − 1.16710i −0.802147 − 1.53511i 0.0630391 + 0.921599i 1.43987 − 2.77881i −0.789214 + 2.85454i 1.21328 − 0.527003i −1.16137 + 1.42752i −1.71312 + 2.46277i −5.04248 + 1.79210i
11.4 −0.966726 − 0.902859i −1.00778 + 1.40868i −0.0170801 − 0.249702i 0.242160 − 0.467347i 2.24609 − 0.451931i 3.06467 − 1.33117i −1.87850 + 2.30899i −0.968776 − 2.83927i −0.656051 + 0.233160i
11.5 −0.840871 − 0.785319i 1.70711 − 0.292854i −0.0461463 − 0.674635i 1.24240 − 2.39773i −1.66545 − 1.09438i −2.61153 + 1.13435i −1.94321 + 2.38853i 2.82847 − 0.999870i −2.92768 + 1.04050i
11.6 −0.596212 − 0.556824i −1.66862 − 0.464434i −0.0910680 − 1.33137i −1.53452 + 2.96148i 0.736246 + 1.20603i −1.61001 + 0.699326i −1.71672 + 2.11013i 2.56860 + 1.54993i 2.56392 − 0.911218i
See next 80 embeddings (of 308 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
47.d odd 46 1 inner
141.g even 46 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 141.2.g.a ✓ 308
3.b odd 2 1 inner 141.2.g.a ✓ 308
47.d odd 46 1 inner 141.2.g.a ✓ 308
141.g even 46 1 inner 141.2.g.a ✓ 308
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
141.2.g.a ✓ 308 1.a even 1 1 trivial
141.2.g.a ✓ 308 3.b odd 2 1 inner
141.2.g.a ✓ 308 47.d odd 46 1 inner
141.2.g.a ✓ 308 141.g even 46 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(141, [\chi])\).