Properties

Label 570.2.bb.b
Level $570$
Weight $2$
Character orbit 570.bb
Analytic conductor $4.551$
Analytic rank $0$
Dimension $84$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [570,2,Mod(41,570)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("570.41"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(570, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [84,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{6} + 42 q^{8} + 24 q^{13} - 24 q^{14} + 12 q^{17} - 12 q^{19} + 36 q^{22} + 24 q^{27} - 12 q^{28} - 12 q^{29} + 36 q^{33} + 12 q^{34} + 6 q^{36} + 18 q^{38} + 12 q^{39} + 6 q^{41} + 24 q^{43}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.766044 − 0.642788i −1.71584 − 0.236392i 0.173648 + 0.984808i −0.984808 − 0.173648i 1.16246 + 1.28401i −1.76528 − 3.05755i 0.500000 − 0.866025i 2.88824 + 0.811224i 0.642788 + 0.766044i
41.2 −0.766044 − 0.642788i −1.50714 − 0.853538i 0.173648 + 0.984808i 0.984808 + 0.173648i 0.605893 + 1.62262i 1.94656 + 3.37154i 0.500000 − 0.866025i 1.54295 + 2.57280i −0.642788 − 0.766044i
41.3 −0.766044 − 0.642788i −1.15586 − 1.28996i 0.173648 + 0.984808i 0.984808 + 0.173648i 0.0562681 + 1.73114i −0.561471 − 0.972497i 0.500000 − 0.866025i −0.327989 + 2.98202i −0.642788 − 0.766044i
41.4 −0.766044 − 0.642788i −1.07847 + 1.35533i 0.173648 + 0.984808i −0.984808 − 0.173648i 1.69734 − 0.345017i −0.267952 − 0.464106i 0.500000 − 0.866025i −0.673825 − 2.92335i 0.642788 + 0.766044i
41.5 −0.766044 − 0.642788i −1.01882 + 1.40072i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.68082 − 0.418131i 0.0247885 + 0.0429350i 0.500000 − 0.866025i −0.924029 − 2.85415i −0.642788 − 0.766044i
41.6 −0.766044 − 0.642788i −0.169275 − 1.72376i 0.173648 + 0.984808i −0.984808 − 0.173648i −0.978339 + 1.42928i 1.36690 + 2.36755i 0.500000 − 0.866025i −2.94269 + 0.583578i 0.642788 + 0.766044i
41.7 −0.766044 − 0.642788i 0.0507095 + 1.73131i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.07402 − 1.35885i 0.847833 + 1.46849i 0.500000 − 0.866025i −2.99486 + 0.175587i −0.642788 − 0.766044i
41.8 −0.766044 − 0.642788i 0.446223 + 1.67358i 0.173648 + 0.984808i −0.984808 − 0.173648i 0.733933 − 1.56887i 2.13041 + 3.68998i 0.500000 − 0.866025i −2.60177 + 1.49358i 0.642788 + 0.766044i
41.9 −0.766044 − 0.642788i 0.902121 − 1.47857i 0.173648 + 0.984808i −0.984808 − 0.173648i −1.64147 + 0.552780i −1.68197 − 2.91326i 0.500000 − 0.866025i −1.37235 − 2.66770i 0.642788 + 0.766044i
41.10 −0.766044 − 0.642788i 0.964794 + 1.43846i 0.173648 + 0.984808i −0.984808 − 0.173648i 0.185551 − 1.72208i −1.20034 − 2.07905i 0.500000 − 0.866025i −1.13835 + 2.77564i 0.642788 + 0.766044i
41.11 −0.766044 − 0.642788i 1.46696 − 0.920884i 0.173648 + 0.984808i 0.984808 + 0.173648i −1.71569 − 0.237506i −2.15391 − 3.73068i 0.500000 − 0.866025i 1.30395 − 2.70180i −0.642788 − 0.766044i
41.12 −0.766044 − 0.642788i 1.48528 + 0.891036i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.565043 − 1.63729i −2.04901 − 3.54899i 0.500000 − 0.866025i 1.41211 + 2.64687i −0.642788 − 0.766044i
41.13 −0.766044 − 0.642788i 1.59014 − 0.686630i 0.173648 + 0.984808i −0.984808 − 0.173648i −1.65947 − 0.496132i 1.07093 + 1.85491i 0.500000 − 0.866025i 2.05708 − 2.18367i 0.642788 + 0.766044i
41.14 −0.766044 − 0.642788i 1.61856 − 0.616663i 0.173648 + 0.984808i 0.984808 + 0.173648i −1.63627 − 0.567997i 1.59792 + 2.76767i 0.500000 − 0.866025i 2.23945 − 1.99621i −0.642788 − 0.766044i
71.1 0.939693 − 0.342020i −1.66038 − 0.493094i 0.766044 − 0.642788i 0.642788 − 0.766044i −1.72889 + 0.104526i 0.222334 + 0.385093i 0.500000 − 0.866025i 2.51372 + 1.63745i 0.342020 − 0.939693i
71.2 0.939693 − 0.342020i −1.63144 + 0.581714i 0.766044 − 0.642788i −0.642788 + 0.766044i −1.33410 + 1.10462i −0.223800 − 0.387633i 0.500000 − 0.866025i 2.32322 − 1.89807i −0.342020 + 0.939693i
71.3 0.939693 − 0.342020i −1.49197 − 0.879786i 0.766044 − 0.642788i −0.642788 + 0.766044i −1.70290 − 0.316445i −1.40312 − 2.43028i 0.500000 − 0.866025i 1.45195 + 2.62523i −0.342020 + 0.939693i
71.4 0.939693 − 0.342020i −1.12629 − 1.31585i 0.766044 − 0.642788i −0.642788 + 0.766044i −1.50842 − 0.851282i 2.21271 + 3.83252i 0.500000 − 0.866025i −0.462935 + 2.96407i −0.342020 + 0.939693i
71.5 0.939693 − 0.342020i −0.950784 + 1.44776i 0.766044 − 0.642788i 0.642788 − 0.766044i −0.398282 + 1.68564i −2.26841 − 3.92899i 0.500000 − 0.866025i −1.19202 − 2.75301i 0.342020 − 0.939693i
71.6 0.939693 − 0.342020i −0.936498 + 1.45704i 0.766044 − 0.642788i 0.642788 − 0.766044i −0.381683 + 1.68947i 0.199417 + 0.345401i 0.500000 − 0.866025i −1.24594 − 2.72903i 0.342020 − 0.939693i
See all 84 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 41.14
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.bb.b yes 84
3.b odd 2 1 570.2.bb.a ✓ 84
19.f odd 18 1 570.2.bb.a ✓ 84
57.j even 18 1 inner 570.2.bb.b yes 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bb.a ✓ 84 3.b odd 2 1
570.2.bb.a ✓ 84 19.f odd 18 1
570.2.bb.b yes 84 1.a even 1 1 trivial
570.2.bb.b yes 84 57.j even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{84} - 264 T_{11}^{82} + 38286 T_{11}^{80} + 3150 T_{11}^{79} - 3827326 T_{11}^{78} + \cdots + 12\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display