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Properties

Label 855.2.bu.b
Level $855$
Weight $2$
Character orbit 855.bu
Analytic conductor $6.827$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(16,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([12, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.bu (of order \(9\), degree \(6\), 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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 3 q^{3} - 3 q^{6} - 9 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 3 q^{3} - 3 q^{6} - 9 q^{8} - 15 q^{9} - 3 q^{13} - 3 q^{15} + 3 q^{19} - 120 q^{20} - 12 q^{21} + 9 q^{22} + 18 q^{24} + 12 q^{26} - 15 q^{27} + 15 q^{28} + 9 q^{29} - 12 q^{30} - 48 q^{31} - 33 q^{33} - 9 q^{34} - 3 q^{35} + 15 q^{36} + 96 q^{37} - 6 q^{38} - 18 q^{39} + 21 q^{41} + 12 q^{42} - 6 q^{43} + 15 q^{44} - 60 q^{46} + 18 q^{47} - 54 q^{48} + 240 q^{49} - 6 q^{50} + 24 q^{51} - 12 q^{52} - 24 q^{53} + 3 q^{55} - 42 q^{57} + 42 q^{59} + 18 q^{60} + 42 q^{61} - 36 q^{62} + 18 q^{63} - 141 q^{64} - 24 q^{65} + 168 q^{66} - 33 q^{67} + 42 q^{68} + 15 q^{69} - 6 q^{71} - 141 q^{72} - 75 q^{73} - 84 q^{74} + 54 q^{76} - 132 q^{78} - 3 q^{79} - 63 q^{81} + 33 q^{82} - 24 q^{83} + 66 q^{84} - 54 q^{86} + 30 q^{87} + 216 q^{88} + 18 q^{89} + 39 q^{90} - 36 q^{91} + 105 q^{92} - 66 q^{93} - 60 q^{94} + 9 q^{95} - 81 q^{96} + 63 q^{97} - 66 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.62042 + 0.953754i −0.485526 − 1.66261i 4.42486 − 3.71290i −0.939693 + 0.342020i 2.85800 + 3.89366i −0.365609 −5.26520 + 9.11959i −2.52853 + 1.61448i 2.13618 − 1.79247i
16.2 −2.53580 + 0.922955i 1.10797 + 1.33131i 4.04634 − 3.39528i −0.939693 + 0.342020i −4.03834 − 2.35333i 3.39766 −4.42846 + 7.67032i −0.544791 + 2.95012i 2.06720 − 1.73459i
16.3 −2.41350 + 0.878442i −1.73072 + 0.0677643i 3.52124 − 2.95467i −0.939693 + 0.342020i 4.11758 − 1.68389i −1.03468 −3.33461 + 5.77571i 2.99082 − 0.234563i 1.96750 − 1.65093i
16.4 −2.23789 + 0.814524i 1.60143 − 0.659877i 2.81259 − 2.36005i −0.939693 + 0.342020i −3.04632 + 2.78113i 0.454592 −1.99044 + 3.44755i 2.12913 − 2.11349i 1.82434 − 1.53080i
16.5 −2.08481 + 0.758807i −0.640617 + 1.60923i 2.23854 − 1.87836i −0.939693 + 0.342020i 0.114469 − 3.84103i 2.81905 −1.02300 + 1.77189i −2.17922 − 2.06180i 1.69955 − 1.42609i
16.6 −2.01359 + 0.732888i −1.66727 − 0.469274i 1.98534 − 1.66590i −0.939693 + 0.342020i 3.70112 − 0.276995i 4.68152 −0.633928 + 1.09800i 2.55956 + 1.56481i 1.64150 − 1.37738i
16.7 −1.97243 + 0.717905i 1.02633 + 1.39522i 1.84299 − 1.54645i −0.939693 + 0.342020i −3.02600 − 2.01516i −3.64433 −0.425949 + 0.737766i −0.893287 + 2.86392i 1.60794 − 1.34922i
16.8 −1.92959 + 0.702314i 0.835486 − 1.51722i 1.69799 − 1.42478i −0.939693 + 0.342020i −0.546579 + 3.51439i −5.02512 −0.222354 + 0.385128i −1.60393 − 2.53523i 1.57302 − 1.31992i
16.9 −1.74712 + 0.635900i −0.246142 + 1.71447i 1.11597 − 0.936410i −0.939693 + 0.342020i −0.660193 − 3.15191i −1.14772 0.504976 − 0.874643i −2.87883 − 0.844006i 1.42426 − 1.19510i
16.10 −1.69588 + 0.617249i −1.24529 − 1.20385i 0.962920 − 0.807985i −0.939693 + 0.342020i 2.85493 + 1.27293i −2.96255 0.670450 − 1.16125i 0.101488 + 2.99828i 1.38249 − 1.16005i
16.11 −1.56794 + 0.570683i 0.287424 − 1.70804i 0.600663 − 0.504016i −0.939693 + 0.342020i 0.524084 + 2.84212i 4.58482 1.01440 − 1.75699i −2.83478 − 0.981860i 1.27819 − 1.07253i
16.12 −1.38510 + 0.504133i 1.67420 + 0.443926i 0.132249 − 0.110970i −0.939693 + 0.342020i −2.54272 + 0.229138i −1.09429 1.34675 − 2.33265i 2.60586 + 1.48644i 1.12914 − 0.947461i
16.13 −1.19004 + 0.433138i −1.57767 − 0.714806i −0.303511 + 0.254676i −0.939693 + 0.342020i 2.18710 + 0.167296i −0.296812 1.51729 − 2.62802i 1.97810 + 2.25546i 0.970126 − 0.814033i
16.14 −1.06483 + 0.387565i 1.55437 − 0.764149i −0.548439 + 0.460195i −0.939693 + 0.342020i −1.35898 + 1.41611i 2.45454 1.53880 − 2.66528i 1.83215 − 2.37554i 0.868055 − 0.728385i
16.15 −0.881901 + 0.320986i −0.824027 + 1.52348i −0.857371 + 0.719420i −0.939693 + 0.342020i 0.237696 − 1.60806i 2.15978 1.46369 − 2.53519i −1.64196 − 2.51077i 0.718933 − 0.603256i
16.16 −0.582514 + 0.212018i 0.517113 + 1.65306i −1.23772 + 1.03857i −0.939693 + 0.342020i −0.651703 − 0.853291i −1.22323 1.12069 − 1.94109i −2.46519 + 1.70963i 0.474870 − 0.398463i
16.17 −0.578418 + 0.210527i −1.47190 + 0.912965i −1.24184 + 1.04203i −0.939693 + 0.342020i 0.659171 − 0.837951i −5.18026 1.11447 − 1.93032i 1.33299 − 2.68759i 0.471531 − 0.395662i
16.18 −0.465670 + 0.169490i −0.0116985 − 1.73201i −1.34397 + 1.12772i −0.939693 + 0.342020i 0.299006 + 0.804563i −3.60298 0.930263 − 1.61126i −2.99973 + 0.0405240i 0.379618 − 0.318537i
16.19 −0.320293 + 0.116577i −0.964027 − 1.43898i −1.44309 + 1.21090i −0.939693 + 0.342020i 0.476523 + 0.348511i 1.05788 0.661899 − 1.14644i −1.14131 + 2.77442i 0.261106 − 0.219094i
16.20 −0.262125 + 0.0954056i 1.72917 + 0.0997779i −1.47248 + 1.23556i −0.939693 + 0.342020i −0.462779 + 0.138819i −2.58248 0.547042 − 0.947505i 2.98009 + 0.345067i 0.213686 − 0.179304i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.w even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.bu.b yes 240
9.c even 3 1 855.2.bt.b ✓ 240
19.e even 9 1 855.2.bt.b ✓ 240
171.w even 9 1 inner 855.2.bu.b yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.bt.b ✓ 240 9.c even 3 1
855.2.bt.b ✓ 240 19.e even 9 1
855.2.bu.b yes 240 1.a even 1 1 trivial
855.2.bu.b yes 240 171.w even 9 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{240} + 11 T_{2}^{237} + 1784 T_{2}^{234} - 6 T_{2}^{233} + 24 T_{2}^{232} + \cdots + 156363445584369 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display