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Properties

Label 387.2.ba.a
Level $387$
Weight $2$
Character orbit 387.ba
Analytic conductor $3.090$
Analytic rank $0$
Dimension $504$
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] = [387,2,Mod(4,387)]
 
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, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.ba (of order \(21\), 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: \(3.09021055822\)
Analytic rank: \(0\)
Dimension: \(504\)
Relative dimension: \(42\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 504 q - 7 q^{2} - 12 q^{3} + 35 q^{4} - 9 q^{5} - 22 q^{6} - 14 q^{7} - 16 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 504 q - 7 q^{2} - 12 q^{3} + 35 q^{4} - 9 q^{5} - 22 q^{6} - 14 q^{7} - 16 q^{8} - 6 q^{9} - 28 q^{10} - 3 q^{11} + 6 q^{12} - 7 q^{13} + 5 q^{14} - 16 q^{15} + 35 q^{16} - 32 q^{17} + 18 q^{18} + 14 q^{19} - 15 q^{20} - 11 q^{21} - 28 q^{22} - 5 q^{23} + 7 q^{24} + 29 q^{25} - 4 q^{26} - 18 q^{27} + 28 q^{28} - 5 q^{29} + 67 q^{30} + 15 q^{31} - 69 q^{32} + 5 q^{33} - 16 q^{34} - 14 q^{35} - 46 q^{36} - 44 q^{37} - 57 q^{38} - 4 q^{39} + 5 q^{40} + 21 q^{41} + 66 q^{42} - 7 q^{43} - 84 q^{44} - 26 q^{45} - 52 q^{46} + 5 q^{47} - 142 q^{48} - 182 q^{49} + 60 q^{50} - 132 q^{51} + 3 q^{52} + 20 q^{53} - 6 q^{54} - 40 q^{55} - 26 q^{56} - 20 q^{57} - 13 q^{58} - 32 q^{59} - 34 q^{60} - 7 q^{61} - 20 q^{62} + 24 q^{63} - 100 q^{64} - 49 q^{65} + 26 q^{66} - 13 q^{67} - 108 q^{68} + 28 q^{69} - 44 q^{70} + 4 q^{71} + 161 q^{72} - 28 q^{73} + 15 q^{74} + 29 q^{75} + 5 q^{76} + 29 q^{77} - 74 q^{78} - 14 q^{79} - 420 q^{80} + 126 q^{81} - 40 q^{82} + 13 q^{83} + 161 q^{84} - 2 q^{85} - 53 q^{86} - 132 q^{87} - 84 q^{88} + 26 q^{89} - 8 q^{90} + 94 q^{91} - 80 q^{92} + 98 q^{93} - 25 q^{94} + 39 q^{95} - 2 q^{96} + 11 q^{97} - 42 q^{98} - 127 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.66219 − 0.401261i −0.0427016 + 1.73152i 5.01511 + 1.54696i 0.0460785 − 0.0314158i 0.808473 − 4.59252i −0.227279 − 0.393659i −7.87917 − 3.79441i −2.99635 − 0.147878i −0.135276 + 0.0651454i
4.2 −2.66008 − 0.400943i 0.152221 − 1.72535i 5.00414 + 1.54357i −3.42683 + 2.33637i −1.09669 + 4.52854i −0.863678 − 1.49593i −7.84509 − 3.77800i −2.95366 − 0.525267i 10.0524 − 4.84098i
4.3 −2.52858 − 0.381122i 1.69433 − 0.359519i 4.33732 + 1.33788i 0.798458 − 0.544380i −4.42126 + 0.263327i 0.713735 + 1.23623i −5.84954 − 2.81699i 2.74149 − 1.21829i −2.22644 + 1.07220i
4.4 −2.31102 − 0.348331i −1.65913 + 0.497278i 3.30835 + 1.02049i −1.72318 + 1.17484i 4.00751 − 0.571294i 1.92290 + 3.33056i −3.07886 − 1.48270i 2.50543 − 1.65010i 4.39154 − 2.11485i
4.5 −2.29174 − 0.345425i −1.62909 + 0.588263i 3.22162 + 0.993739i 3.27823 − 2.23506i 3.93666 − 0.785419i −1.19776 − 2.07458i −2.86364 − 1.37906i 2.30789 − 1.91667i −8.28489 + 3.98979i
4.6 −2.21378 − 0.333674i −1.41499 − 0.998898i 2.87835 + 0.887853i −0.139359 + 0.0950137i 2.79918 + 2.68349i −1.17716 − 2.03890i −2.04163 − 0.983196i 1.00441 + 2.82687i 0.340215 − 0.163839i
4.7 −1.93902 − 0.292261i 1.69111 + 0.374356i 1.76326 + 0.543892i −1.59558 + 1.08785i −3.16970 − 1.22013i −2.50844 − 4.34475i 0.273431 + 0.131677i 2.71972 + 1.26616i 3.41181 − 1.64304i
4.8 −1.82627 − 0.275267i 1.29923 + 1.14543i 1.34836 + 0.415915i 2.14269 − 1.46086i −2.05745 − 2.44950i 1.12430 + 1.94735i 0.980011 + 0.471948i 0.375985 + 2.97635i −4.31526 + 2.07812i
4.9 −1.79517 − 0.270578i −0.0155002 − 1.73198i 1.23828 + 0.381959i 1.70242 − 1.16069i −0.440811 + 3.11340i −0.919364 − 1.59238i 1.15175 + 0.554653i −2.99952 + 0.0536922i −3.37020 + 1.62300i
4.10 −1.77662 − 0.267782i 0.280923 − 1.70912i 1.17352 + 0.361983i 0.375866 − 0.256261i −0.956763 + 2.96122i 2.46766 + 4.27412i 1.24955 + 0.601753i −2.84217 − 0.960259i −0.736393 + 0.354628i
4.11 −1.68527 − 0.254013i 0.674735 + 1.59522i 0.864460 + 0.266651i −3.30023 + 2.25006i −0.731902 − 2.85977i 1.30659 + 2.26309i 1.68194 + 0.809977i −2.08947 + 2.15270i 6.13331 − 2.95365i
4.12 −1.53314 − 0.231084i 1.24000 − 1.20930i 0.385978 + 0.119058i −2.29067 + 1.56175i −2.18054 + 1.56748i 0.606173 + 1.04992i 2.22958 + 1.07371i 0.0751862 − 2.99906i 3.87281 − 1.86505i
4.13 −1.45425 − 0.219193i −0.645948 + 1.60709i 0.155660 + 0.0480147i −0.338955 + 0.231096i 1.29164 − 2.19553i −1.26760 − 2.19555i 2.43423 + 1.17226i −2.16550 − 2.07620i 0.543581 − 0.261775i
4.14 −0.849256 − 0.128005i −1.51118 + 0.846367i −1.20630 − 0.372093i 0.794765 − 0.541862i 1.39172 − 0.525345i 0.397920 + 0.689218i 2.52442 + 1.21569i 1.56732 − 2.55803i −0.744320 + 0.358446i
4.15 −0.806513 − 0.121562i −1.53138 − 0.809252i −1.27546 − 0.393427i 2.89313 − 1.97250i 1.13670 + 0.838830i 1.41110 + 2.44409i 2.45055 + 1.18012i 1.69022 + 2.47854i −2.57313 + 1.23915i
4.16 −0.728914 − 0.109866i 1.57998 − 0.709701i −1.39190 − 0.429345i 3.07578 − 2.09703i −1.22964 + 0.343725i −0.523725 − 0.907118i 2.29570 + 1.10555i 1.99265 − 2.24262i −2.47237 + 1.19063i
4.17 −0.664825 − 0.100206i −0.377568 − 1.69040i −1.47919 − 0.456271i 0.266348 − 0.181593i 0.0816287 + 1.16165i −1.69054 − 2.92810i 2.14919 + 1.03500i −2.71488 + 1.27648i −0.195272 + 0.0940380i
4.18 −0.456952 − 0.0688744i 1.71835 − 0.217414i −1.70708 − 0.526566i −2.06422 + 1.40736i −0.800178 − 0.0190028i 0.823415 + 1.42620i 1.57649 + 0.759197i 2.90546 − 0.747187i 1.04018 − 0.500925i
4.19 −0.453818 − 0.0684020i −1.71461 + 0.245187i −1.70987 − 0.527426i −3.34851 + 2.28298i 0.794891 + 0.00601266i −1.43200 − 2.48030i 1.56688 + 0.754571i 2.87977 − 0.840798i 1.67578 − 0.807011i
4.20 −0.282381 − 0.0425621i 1.33471 + 1.10388i −1.83322 − 0.565473i 0.615771 − 0.419826i −0.329914 − 0.368521i −1.88349 − 3.26231i 1.00818 + 0.485513i 0.562919 + 2.94671i −0.191751 + 0.0923422i
See next 80 embeddings (of 504 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
43.e even 7 1 inner
387.ba even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.2.ba.a ✓ 504
9.c even 3 1 inner 387.2.ba.a ✓ 504
43.e even 7 1 inner 387.2.ba.a ✓ 504
387.ba even 21 1 inner 387.2.ba.a ✓ 504
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.2.ba.a ✓ 504 1.a even 1 1 trivial
387.2.ba.a ✓ 504 9.c even 3 1 inner
387.2.ba.a ✓ 504 43.e even 7 1 inner
387.2.ba.a ✓ 504 387.ba even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(387, [\chi])\).