Properties

Label 570.2.bb.b
Level $570$
Weight $2$
Character orbit 570.bb
Analytic conductor $4.551$
Analytic rank $0$
Dimension $84$
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] = [570,2,Mod(41,570)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

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: \(4.55147291521\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{6} + 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{6} + 42 q^{8} + 24 q^{13} - 24 q^{14} + 12 q^{17} - 12 q^{19} + 36 q^{22} + 24 q^{27} - 12 q^{28} - 12 q^{29} + 36 q^{33} + 12 q^{34} + 6 q^{36} + 18 q^{38} + 12 q^{39} + 6 q^{41} + 24 q^{43} + 36 q^{44} + 12 q^{47} - 12 q^{48} - 54 q^{49} - 42 q^{50} + 96 q^{51} + 12 q^{52} - 60 q^{53} - 18 q^{54} - 96 q^{57} - 24 q^{58} - 18 q^{59} - 48 q^{61} - 12 q^{62} - 114 q^{63} - 42 q^{64} - 24 q^{66} + 6 q^{67} - 54 q^{68} - 48 q^{69} + 48 q^{71} + 84 q^{73} + 24 q^{74} - 12 q^{79} - 36 q^{81} - 6 q^{82} + 36 q^{83} + 18 q^{84} + 12 q^{86} + 6 q^{87} - 12 q^{89} - 24 q^{90} + 24 q^{91} - 6 q^{93} - 12 q^{95} - 42 q^{97} + 36 q^{98} - 6 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 990-1000
Embeddings: e.g. 1-3 or 41.14
Significant digits:
Format:

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 \( T_{11}^{84} - 264 T_{11}^{82} + 38286 T_{11}^{80} + 3150 T_{11}^{79} - 3827326 T_{11}^{78} - 742986 T_{11}^{77} + 291243165 T_{11}^{76} + 97206804 T_{11}^{75} - 17721561573 T_{11}^{74} + \cdots + 12\!\cdots\!49 \) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\). Copy content Toggle raw display