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Properties

Label 117.3.s.a
Level $117$
Weight $3$
Character orbit 117.s
Analytic conductor $3.188$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,3,Mod(14,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.14");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 117.s (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: \(3.18801909302\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} + 48 q^{4} - 18 q^{5} - 24 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2 q^{3} + 48 q^{4} - 18 q^{5} - 24 q^{6} + 6 q^{9} - 36 q^{11} + 54 q^{12} - 18 q^{14} - 10 q^{15} - 96 q^{16} + 46 q^{18} - 24 q^{19} - 18 q^{20} - 10 q^{21} - 24 q^{22} - 54 q^{23} + 210 q^{24} + 114 q^{25} - 88 q^{27} - 216 q^{29} - 110 q^{30} + 30 q^{31} - 180 q^{32} - 76 q^{33} + 60 q^{34} + 32 q^{36} + 84 q^{37} + 162 q^{38} - 60 q^{40} + 108 q^{41} + 72 q^{42} - 60 q^{43} + 256 q^{45} - 216 q^{46} + 288 q^{47} + 176 q^{48} - 216 q^{49} + 288 q^{50} - 308 q^{51} - 228 q^{54} - 84 q^{55} + 324 q^{56} - 114 q^{57} + 48 q^{58} + 252 q^{59} - 336 q^{60} + 96 q^{61} - 500 q^{63} - 120 q^{64} - 450 q^{66} + 6 q^{67} + 378 q^{68} + 194 q^{69} - 216 q^{70} + 132 q^{72} - 72 q^{73} - 774 q^{74} - 94 q^{75} + 84 q^{76} - 378 q^{77} - 130 q^{78} + 120 q^{79} + 198 q^{81} + 192 q^{82} + 792 q^{83} + 890 q^{84} + 396 q^{86} + 82 q^{87} + 156 q^{88} + 186 q^{90} - 540 q^{92} - 168 q^{93} + 108 q^{94} - 648 q^{95} + 464 q^{96} - 78 q^{97} - 60 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −3.22484 + 1.86186i 0.496158 − 2.95869i 4.93306 − 8.54431i 0.733765 + 0.423639i 3.90864 + 10.4651i −2.32019 − 4.01868i 21.8438i −8.50766 − 2.93595i −3.15503
14.2 −3.11753 + 1.79991i 2.11830 + 2.12434i 4.47935 − 7.75845i −2.22107 − 1.28233i −10.4275 − 2.80995i 5.05104 + 8.74866i 17.8504i −0.0256148 + 8.99996i 9.23234
14.3 −3.00319 + 1.73389i −1.65318 + 2.50339i 4.01277 − 6.95032i 2.01759 + 1.16486i 0.624209 − 10.3846i −5.32532 − 9.22372i 13.9597i −3.53397 − 8.27714i −8.07896
14.4 −2.60452 + 1.50372i −2.96002 − 0.488115i 2.52236 − 4.36886i −5.98512 − 3.45551i 8.44344 − 3.17975i 1.16593 + 2.01946i 3.14196i 8.52349 + 2.88967i 20.7845
14.5 −2.28108 + 1.31698i 2.99804 + 0.108415i 1.46888 − 2.54417i 7.21734 + 4.16693i −6.98154 + 3.70106i −2.13432 − 3.69674i − 2.79791i 8.97649 + 0.650063i −21.9511
14.6 −2.06366 + 1.19146i 1.77538 − 2.41827i 0.839131 − 1.45342i −1.90645 − 1.10069i −0.782523 + 7.10577i 5.12019 + 8.86842i − 5.53249i −2.69605 − 8.58669i 5.24570
14.7 −1.87167 + 1.08061i −1.94021 + 2.28814i 0.335440 − 0.580998i 6.24677 + 3.60658i 1.15884 − 6.37926i 6.55090 + 11.3465i − 7.19497i −1.47121 − 8.87894i −15.5892
14.8 −1.77151 + 1.02278i 2.99100 − 0.232264i 0.0921684 − 0.159640i −6.10945 − 3.52729i −5.06103 + 3.47060i −4.97929 − 8.62438i − 7.80519i 8.89211 − 1.38940i 14.4306
14.9 −1.30007 + 0.750594i −0.0792791 + 2.99895i −0.873217 + 1.51246i −6.06839 − 3.50359i −2.14793 − 3.95834i 0.489533 + 0.847896i − 8.62648i −8.98743 − 0.475509i 10.5191
14.10 −0.752157 + 0.434258i −2.94987 + 0.546119i −1.62284 + 2.81084i 0.439114 + 0.253523i 1.98161 − 1.69177i −3.60069 − 6.23658i − 6.29299i 8.40351 − 3.22197i −0.440377
14.11 −0.446054 + 0.257529i 1.28237 + 2.71211i −1.86736 + 3.23436i 3.26144 + 1.88299i −1.27045 − 0.879499i −1.26265 − 2.18697i − 3.98383i −5.71106 + 6.95585i −1.93971
14.12 0.289796 − 0.167314i 2.95040 + 0.543258i −1.94401 + 3.36713i −1.25705 − 0.725758i 0.945909 − 0.336209i 3.58721 + 6.21323i 2.63955i 8.40974 + 3.20566i −0.485717
14.13 0.386021 − 0.222869i −1.42956 − 2.63749i −1.90066 + 3.29204i −6.36049 − 3.67223i −1.13966 − 0.699521i 5.87473 + 10.1753i 3.47735i −4.91270 + 7.54092i −3.27371
14.14 0.449733 − 0.259653i 0.689185 − 2.91976i −1.86516 + 3.23055i −4.45907 − 2.57445i −0.448177 − 1.49206i −6.04441 − 10.4692i 4.01441i −8.05005 − 4.02452i −2.67386
14.15 0.658120 − 0.379966i 1.82350 − 2.38219i −1.71125 + 2.96398i 7.70917 + 4.45089i 0.294929 − 2.26064i −0.0636420 − 0.110231i 5.64060i −2.34970 − 8.68786i 6.76475
14.16 1.10079 − 0.635542i −2.90552 − 0.746967i −1.19217 + 2.06490i 4.29328 + 2.47873i −3.67310 + 1.02433i 2.07751 + 3.59836i 8.11504i 7.88408 + 4.34065i 6.30135
14.17 1.12806 − 0.651287i −2.09532 + 2.14701i −1.15165 + 1.99472i −4.65732 − 2.68891i −0.965333 + 3.78661i −2.66094 − 4.60889i 8.21052i −0.219277 − 8.99733i −7.00500
14.18 1.86662 − 1.07769i −0.693574 + 2.91872i 0.322836 − 0.559168i 1.08497 + 0.626410i 1.85085 + 6.19560i 3.26882 + 5.66177i 7.22986i −8.03791 − 4.04870i 2.70031
14.19 2.23217 − 1.28874i 2.24130 + 1.99413i 1.32172 − 2.28929i 2.98624 + 1.72410i 7.57289 + 1.56279i −6.28185 − 10.8805i 3.49651i 1.04685 + 8.93891i 8.88772
14.20 2.31527 − 1.33672i 2.75356 − 1.19077i 1.57366 − 2.72565i −2.65644 − 1.53369i 4.78351 − 6.43769i 0.203159 + 0.351882i 2.27962i 6.16415 − 6.55769i −8.20049
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.3.s.a ✓ 48
3.b odd 2 1 351.3.s.a 48
9.c even 3 1 351.3.s.a 48
9.c even 3 1 1053.3.c.b 48
9.d odd 6 1 inner 117.3.s.a ✓ 48
9.d odd 6 1 1053.3.c.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.3.s.a ✓ 48 1.a even 1 1 trivial
117.3.s.a ✓ 48 9.d odd 6 1 inner
351.3.s.a 48 3.b odd 2 1
351.3.s.a 48 9.c even 3 1
1053.3.c.b 48 9.c even 3 1
1053.3.c.b 48 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).