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Properties

Label 387.3.bf.a
Level $387$
Weight $3$
Character orbit 387.bf
Analytic conductor $10.545$
Analytic rank $0$
Dimension $1032$
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] = [387,3,Mod(22,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([14, 15]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.22");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.bf (of order \(42\), degree \(12\), 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: \(10.5449862307\)
Analytic rank: \(0\)
Dimension: \(1032\)
Relative dimension: \(86\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1032 q - 7 q^{2} - 14 q^{3} - 173 q^{4} - 7 q^{5} + 8 q^{6} - 28 q^{8} + 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1032 q - 7 q^{2} - 14 q^{3} - 173 q^{4} - 7 q^{5} + 8 q^{6} - 28 q^{8} + 42 q^{9} - 4 q^{10} - 17 q^{11} + 14 q^{12} + 5 q^{13} + 15 q^{14} + 8 q^{15} + 307 q^{16} - 92 q^{17} - 14 q^{18} + 56 q^{19} - 7 q^{20} - 63 q^{21} + 77 q^{22} - 5 q^{23} + 280 q^{24} - 395 q^{25} - 28 q^{26} - 14 q^{27} - 252 q^{28} - 7 q^{29} + 231 q^{30} + 75 q^{31} + 203 q^{32} - 224 q^{33} + 161 q^{34} - 186 q^{35} - 396 q^{36} + 35 q^{38} - 14 q^{39} - 7 q^{40} + 151 q^{41} + 56 q^{43} - 240 q^{44} + 798 q^{45} - 28 q^{46} - 35 q^{47} - 518 q^{48} + 3074 q^{49} + 574 q^{51} + 123 q^{52} - 380 q^{53} + 80 q^{54} - 28 q^{55} + 354 q^{56} - 14 q^{57} + 11 q^{58} - 140 q^{59} - 138 q^{60} - 7 q^{61} - 28 q^{62} - 532 q^{63} + 1084 q^{64} - 7 q^{65} + 188 q^{66} - 19 q^{67} + 148 q^{68} + 210 q^{69} + 336 q^{70} - 28 q^{71} - 1351 q^{72} - 28 q^{73} + 489 q^{74} - 1414 q^{75} - 7 q^{76} - 693 q^{77} + 644 q^{78} + 22 q^{79} - 490 q^{81} - 28 q^{82} + 151 q^{83} + 513 q^{84} - 436 q^{86} - 240 q^{87} + 469 q^{88} - 1078 q^{89} - 202 q^{90} - 714 q^{91} - 588 q^{92} - 7 q^{94} + 53 q^{95} - 1850 q^{96} + 213 q^{97} - 658 q^{98} + 154 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −3.91905 + 0.293692i 0.856322 − 2.87519i 11.3174 − 1.70582i 0.0741423 − 0.240364i −2.51155 + 11.5195i 1.98839 − 1.14800i −28.5264 + 6.51097i −7.53343 − 4.92418i −0.219975 + 0.963772i
22.2 −3.84567 + 0.288193i 2.97379 + 0.395727i 10.7508 − 1.62042i 0.466062 − 1.51094i −11.5502 − 0.664812i −10.4492 + 6.03283i −25.8379 + 5.89733i 8.68680 + 2.35362i −1.35688 + 5.94487i
22.3 −3.77984 + 0.283260i −1.52962 + 2.58075i 10.2516 − 1.54518i 0.913217 − 2.96058i 5.05071 − 10.1881i 5.34199 − 3.08420i −23.5302 + 5.37062i −4.32050 − 7.89514i −2.61320 + 11.4492i
22.4 −3.69192 + 0.276671i −2.99889 − 0.0815464i 9.59842 − 1.44673i −2.53128 + 8.20621i 11.0942 − 0.528644i 5.99850 − 3.46324i −20.5986 + 4.70149i 8.98670 + 0.489098i 7.07487 − 30.9970i
22.5 −3.59897 + 0.269706i −2.37917 − 1.82744i 8.92454 − 1.34516i 1.77669 − 5.75988i 9.05544 + 5.93525i 0.940192 − 0.542820i −17.6821 + 4.03582i 2.32089 + 8.69560i −4.84078 + 21.2088i
22.6 −3.54155 + 0.265403i 1.98190 + 2.25213i 8.51683 − 1.28371i −1.34320 + 4.35454i −7.61672 − 7.45003i 4.99275 − 2.88257i −15.9724 + 3.64559i −1.14416 + 8.92698i 3.60130 − 15.7783i
22.7 −3.48258 + 0.260983i −2.87441 + 0.858935i 8.10492 − 1.22162i 0.258238 − 0.837188i 9.78619 − 3.74148i −11.4540 + 6.61296i −14.2881 + 3.26116i 7.52446 − 4.93786i −0.680843 + 2.98297i
22.8 −3.40674 + 0.255300i −1.01935 − 2.82151i 7.58538 − 1.14331i −2.51903 + 8.16648i 4.19299 + 9.35192i −8.41875 + 4.86057i −12.2270 + 2.79072i −6.92185 + 5.75222i 6.49677 − 28.4642i
22.9 −3.28680 + 0.246312i 0.948866 + 2.84599i 6.78708 − 1.02299i 1.86544 − 6.04760i −3.81973 − 9.12049i −1.61790 + 0.934095i −9.20228 + 2.10036i −7.19931 + 5.40092i −4.64173 + 20.3367i
22.10 −3.28367 + 0.246077i 2.32269 − 1.89871i 6.76662 − 1.01990i −1.43776 + 4.66112i −7.15972 + 6.80631i 9.43407 − 5.44676i −9.12710 + 2.08320i 1.78979 − 8.82024i 3.57415 − 15.6594i
22.11 −3.23446 + 0.242389i −1.37679 + 2.66542i 6.44765 − 0.971827i −0.511604 + 1.65858i 3.80711 − 8.95490i −2.14051 + 1.23582i −7.97029 + 1.81917i −5.20888 − 7.33945i 1.25274 − 5.48861i
22.12 −3.13066 + 0.234610i 2.99093 + 0.233112i 5.79066 − 0.872801i 2.16474 − 7.01791i −9.41827 − 0.0280907i 8.11090 − 4.68283i −5.68090 + 1.29663i 8.89132 + 1.39444i −5.13059 + 22.4786i
22.13 −3.06571 + 0.229743i 1.22454 − 2.73871i 5.39046 − 0.812481i 2.69039 − 8.72203i −3.12487 + 8.67740i −2.57912 + 1.48906i −4.35000 + 0.992859i −6.00102 − 6.70729i −6.24413 + 27.3573i
22.14 −3.03906 + 0.227746i 2.98332 − 0.315935i 5.22869 − 0.788098i −2.30841 + 7.48370i −8.99453 + 1.63958i −4.31854 + 2.49331i −3.82612 + 0.873287i 8.80037 − 1.88507i 5.31103 − 23.2691i
22.15 −2.85595 + 0.214024i −0.632321 − 2.93260i 4.15533 − 0.626315i −0.254368 + 0.824642i 2.43353 + 8.24004i −0.310220 + 0.179106i −0.564753 + 0.128901i −8.20034 + 3.70869i 0.549970 − 2.40958i
22.16 −2.79367 + 0.209357i −2.81813 − 1.02865i 3.80546 − 0.573580i 0.381143 − 1.23564i 8.08830 + 2.28372i 10.9798 − 6.33921i 0.413949 − 0.0944811i 6.88375 + 5.79775i −0.806100 + 3.53176i
22.17 −2.74144 + 0.205442i 2.13591 − 2.10663i 3.51795 − 0.530245i −0.124744 + 0.404411i −5.42266 + 6.21401i −6.70926 + 3.87359i 1.18550 − 0.270583i 0.124194 − 8.99914i 0.258895 − 1.13430i
22.18 −2.58143 + 0.193451i 0.716945 + 2.91307i 2.67102 − 0.402591i −1.02539 + 3.32422i −2.41428 − 7.38119i −6.76670 + 3.90676i 3.27790 − 0.748158i −7.97198 + 4.17703i 2.00388 − 8.77959i
22.19 −2.52861 + 0.189493i −2.99906 − 0.0752685i 2.40264 − 0.362140i −0.568510 + 1.84306i 7.59771 − 0.377976i −3.48985 + 2.01487i 3.88178 − 0.885991i 8.98867 + 0.451469i 1.08829 − 4.76812i
22.20 −2.44466 + 0.183202i −0.810922 − 2.88832i 1.98745 − 0.299560i −0.521409 + 1.69037i 2.51157 + 6.91239i 8.08727 − 4.66919i 4.75643 − 1.08562i −7.68481 + 4.68441i 0.964987 − 4.22788i
See next 80 embeddings (of 1032 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
43.f odd 14 1 inner
387.bf odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.bf.a ✓ 1032
9.c even 3 1 inner 387.3.bf.a ✓ 1032
43.f odd 14 1 inner 387.3.bf.a ✓ 1032
387.bf odd 42 1 inner 387.3.bf.a ✓ 1032
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.3.bf.a ✓ 1032 1.a even 1 1 trivial
387.3.bf.a ✓ 1032 9.c even 3 1 inner
387.3.bf.a ✓ 1032 43.f odd 14 1 inner
387.3.bf.a ✓ 1032 387.bf odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(387, [\chi])\).