Properties

Label 252.9.bg.a
Level $252$
Weight $9$
Character orbit 252.bg
Analytic conductor $102.659$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.bg (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 0 −80.9976 0.623037i 0 25.8029 14.8973i 0 453.746 785.912i 0 6560.22 + 100.929i 0
29.2 0 −80.2339 11.1142i 0 −207.070 + 119.552i 0 −453.746 + 785.912i 0 6313.95 + 1783.47i 0
29.3 0 −79.9711 12.8694i 0 −908.444 + 524.491i 0 453.746 785.912i 0 6229.76 + 2058.36i 0
29.4 0 −79.7801 14.0049i 0 225.541 130.216i 0 −453.746 + 785.912i 0 6168.73 + 2234.62i 0
29.5 0 −75.7614 + 28.6568i 0 425.975 245.937i 0 453.746 785.912i 0 4918.57 4342.16i 0
29.6 0 −75.7417 + 28.7087i 0 820.829 473.906i 0 453.746 785.912i 0 4912.62 4348.90i 0
29.7 0 −73.8531 + 33.2674i 0 −290.393 + 167.658i 0 −453.746 + 785.912i 0 4347.56 4913.80i 0
29.8 0 −71.6703 + 37.7408i 0 900.834 520.096i 0 −453.746 + 785.912i 0 3712.26 5409.79i 0
29.9 0 −71.3437 38.3547i 0 330.062 190.561i 0 453.746 785.912i 0 3618.84 + 5472.73i 0
29.10 0 −68.4056 43.3783i 0 645.917 372.921i 0 −453.746 + 785.912i 0 2797.64 + 5934.64i 0
29.11 0 −63.2760 + 50.5682i 0 −405.178 + 233.930i 0 −453.746 + 785.912i 0 1446.71 6399.51i 0
29.12 0 −62.9498 50.9738i 0 −401.865 + 232.017i 0 453.746 785.912i 0 1364.35 + 6417.57i 0
29.13 0 −55.9608 58.5610i 0 −883.740 + 510.228i 0 −453.746 + 785.912i 0 −297.778 + 6554.24i 0
29.14 0 −55.8369 + 58.6791i 0 −400.624 + 231.301i 0 453.746 785.912i 0 −325.483 6552.92i 0
29.15 0 −45.2321 67.1942i 0 −4.45748 + 2.57353i 0 −453.746 + 785.912i 0 −2469.12 + 6078.67i 0
29.16 0 −41.3930 + 69.6248i 0 −865.452 + 499.669i 0 453.746 785.912i 0 −3134.24 5763.96i 0
29.17 0 −31.9190 74.4458i 0 867.572 500.893i 0 453.746 785.912i 0 −4523.36 + 4752.47i 0
29.18 0 −26.4497 + 76.5599i 0 151.995 87.7545i 0 −453.746 + 785.912i 0 −5161.83 4049.97i 0
29.19 0 −24.5606 77.1866i 0 −248.615 + 143.538i 0 453.746 785.912i 0 −5354.55 + 3791.50i 0
29.20 0 −21.6670 + 78.0483i 0 514.619 297.115i 0 453.746 785.912i 0 −5622.08 3382.15i 0
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.bg.a 96
9.d odd 6 1 inner 252.9.bg.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.9.bg.a 96 1.a even 1 1 trivial
252.9.bg.a 96 9.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(252, [\chi])\).