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Properties

Label 570.2.bb.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{3} - 42 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{3} - 42 q^{8} + 6 q^{9} + 24 q^{13} + 24 q^{14} - 12 q^{17} - 12 q^{19} + 36 q^{22} - 6 q^{24} - 6 q^{27} - 12 q^{28} + 12 q^{29} - 6 q^{33} + 12 q^{34} - 18 q^{38} + 12 q^{39} - 6 q^{41} + 24 q^{43} - 36 q^{44} - 12 q^{47} - 54 q^{49} + 42 q^{50} - 54 q^{51} + 12 q^{52} + 60 q^{53} - 54 q^{54} - 60 q^{57} - 24 q^{58} + 18 q^{59} - 48 q^{61} + 12 q^{62} + 18 q^{63} - 42 q^{64} + 54 q^{66} + 6 q^{67} + 54 q^{68} - 60 q^{69} - 48 q^{71} - 12 q^{72} + 84 q^{73} - 24 q^{74} + 36 q^{78} - 12 q^{79} + 114 q^{81} - 6 q^{82} - 36 q^{83} - 18 q^{84} - 12 q^{86} + 6 q^{87} + 12 q^{89} + 24 q^{91} + 6 q^{93} + 12 q^{95} - 42 q^{97} - 36 q^{98} + 102 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0.766044 + 0.642788i −1.71569 + 0.237506i 0.173648 + 0.984808i −0.984808 − 0.173648i −1.46696 − 0.920884i −2.15391 − 3.73068i −0.500000 + 0.866025i 2.88718 − 0.814973i −0.642788 − 0.766044i
41.2 0.766044 + 0.642788i −1.65947 + 0.496132i 0.173648 + 0.984808i 0.984808 + 0.173648i −1.59014 − 0.686630i 1.07093 + 1.85491i −0.500000 + 0.866025i 2.50771 − 1.64664i 0.642788 + 0.766044i
41.3 0.766044 + 0.642788i −1.64147 − 0.552780i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.902121 − 1.47857i −1.68197 − 2.91326i −0.500000 + 0.866025i 2.38887 + 1.81475i 0.642788 + 0.766044i
41.4 0.766044 + 0.642788i −1.63627 + 0.567997i 0.173648 + 0.984808i −0.984808 − 0.173648i −1.61856 − 0.616663i 1.59792 + 2.76767i −0.500000 + 0.866025i 2.35476 − 1.85879i −0.642788 − 0.766044i
41.5 0.766044 + 0.642788i −0.978339 − 1.42928i 0.173648 + 0.984808i 0.984808 + 0.173648i 0.169275 − 1.72376i 1.36690 + 2.36755i −0.500000 + 0.866025i −1.08571 + 2.79665i 0.642788 + 0.766044i
41.6 0.766044 + 0.642788i −0.565043 + 1.63729i 0.173648 + 0.984808i −0.984808 − 0.173648i −1.48528 + 0.891036i −2.04901 − 3.54899i −0.500000 + 0.866025i −2.36145 − 1.85028i −0.642788 − 0.766044i
41.7 0.766044 + 0.642788i 0.0562681 − 1.73114i 0.173648 + 0.984808i −0.984808 − 0.173648i 1.15586 − 1.28996i −0.561471 − 0.972497i −0.500000 + 0.866025i −2.99367 − 0.194816i −0.642788 − 0.766044i
41.8 0.766044 + 0.642788i 0.185551 + 1.72208i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.964794 + 1.43846i −1.20034 − 2.07905i −0.500000 + 0.866025i −2.93114 + 0.639068i 0.642788 + 0.766044i
41.9 0.766044 + 0.642788i 0.605893 − 1.62262i 0.173648 + 0.984808i −0.984808 − 0.173648i 1.50714 − 0.853538i 1.94656 + 3.37154i −0.500000 + 0.866025i −2.26579 − 1.96627i −0.642788 − 0.766044i
41.10 0.766044 + 0.642788i 0.733933 + 1.56887i 0.173648 + 0.984808i 0.984808 + 0.173648i −0.446223 + 1.67358i 2.13041 + 3.68998i −0.500000 + 0.866025i −1.92269 + 2.30289i 0.642788 + 0.766044i
41.11 0.766044 + 0.642788i 1.07402 + 1.35885i 0.173648 + 0.984808i −0.984808 − 0.173648i −0.0507095 + 1.73131i 0.847833 + 1.46849i −0.500000 + 0.866025i −0.692971 + 2.91887i −0.642788 − 0.766044i
41.12 0.766044 + 0.642788i 1.16246 − 1.28401i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.71584 − 0.236392i −1.76528 − 3.05755i −0.500000 + 0.866025i −0.297363 − 2.98523i 0.642788 + 0.766044i
41.13 0.766044 + 0.642788i 1.68082 + 0.418131i 0.173648 + 0.984808i −0.984808 − 0.173648i 1.01882 + 1.40072i 0.0247885 + 0.0429350i −0.500000 + 0.866025i 2.65033 + 1.40561i −0.642788 − 0.766044i
41.14 0.766044 + 0.642788i 1.69734 + 0.345017i 0.173648 + 0.984808i 0.984808 + 0.173648i 1.07847 + 1.35533i −0.267952 − 0.464106i −0.500000 + 0.866025i 2.76193 + 1.17122i 0.642788 + 0.766044i
71.1 −0.939693 + 0.342020i −1.72889 − 0.104526i 0.766044 − 0.642788i −0.642788 + 0.766044i 1.66038 − 0.493094i 0.222334 + 0.385093i −0.500000 + 0.866025i 2.97815 + 0.361430i 0.342020 − 0.939693i
71.2 −0.939693 + 0.342020i −1.70290 + 0.316445i 0.766044 − 0.642788i 0.642788 − 0.766044i 1.49197 − 0.879786i −1.40312 − 2.43028i −0.500000 + 0.866025i 2.79973 − 1.07775i −0.342020 + 0.939693i
71.3 −0.939693 + 0.342020i −1.50842 + 0.851282i 0.766044 − 0.642788i 0.642788 − 0.766044i 1.12629 − 1.31585i 2.21271 + 3.83252i −0.500000 + 0.866025i 1.55064 − 2.56818i −0.342020 + 0.939693i
71.4 −0.939693 + 0.342020i −1.33410 − 1.10462i 0.766044 − 0.642788i 0.642788 − 0.766044i 1.63144 + 0.581714i −0.223800 − 0.387633i −0.500000 + 0.866025i 0.559632 + 2.94734i −0.342020 + 0.939693i
71.5 −0.939693 + 0.342020i −0.939713 + 1.45497i 0.766044 − 0.642788i −0.642788 + 0.766044i 0.385412 − 1.68863i −1.27478 − 2.20798i −0.500000 + 0.866025i −1.23388 − 2.73451i 0.342020 − 0.939693i
71.6 −0.939693 + 0.342020i −0.398282 − 1.68564i 0.766044 − 0.642788i −0.642788 + 0.766044i 0.950784 + 1.44776i −2.26841 − 3.92899i −0.500000 + 0.866025i −2.68274 + 1.34272i 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.a ✓ 84
3.b odd 2 1 570.2.bb.b yes 84
19.f odd 18 1 570.2.bb.b yes 84
57.j even 18 1 inner 570.2.bb.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.bb.a ✓ 84 1.a even 1 1 trivial
570.2.bb.a ✓ 84 57.j even 18 1 inner
570.2.bb.b yes 84 3.b odd 2 1
570.2.bb.b yes 84 19.f odd 18 1

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