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Properties

Label 546.2.bb.a
Level $546$
Weight $2$
Character orbit 546.bb
Analytic conductor $4.360$
Analytic rank $0$
Dimension $76$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [546,2,Mod(269,546)] 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("546.269"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 76 q^{4} + 10 q^{7} + 4 q^{9} - 2 q^{13} + 12 q^{15} + 76 q^{16} + 18 q^{19} - 8 q^{21} - 42 q^{25} - 10 q^{28} - 10 q^{30} + 12 q^{31} + 72 q^{33} - 4 q^{36} + 12 q^{37} - 42 q^{39} + 2 q^{42} - 2 q^{43}+ \cdots + 66 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 − 1.00000i −1.68782 − 0.388948i −1.00000 −0.386492 − 0.669424i −0.388948 + 1.68782i 1.88739 + 1.85412i 1.00000i 2.69744 + 1.31294i −0.669424 + 0.386492i
269.2 − 1.00000i −1.59766 + 0.668950i −1.00000 −0.356048 − 0.616693i 0.668950 + 1.59766i 2.34738 − 1.22057i 1.00000i 2.10501 − 2.13750i −0.616693 + 0.356048i
269.3 − 1.00000i −1.54137 + 0.790045i −1.00000 0.886718 + 1.53584i 0.790045 + 1.54137i −1.67305 + 2.04961i 1.00000i 1.75166 − 2.43551i 1.53584 − 0.886718i
269.4 − 1.00000i −1.48241 − 0.895807i −1.00000 0.699535 + 1.21163i −0.895807 + 1.48241i −2.51159 + 0.831830i 1.00000i 1.39506 + 2.65590i 1.21163 − 0.699535i
269.5 − 1.00000i −1.42260 − 0.988029i −1.00000 1.41378 + 2.44875i −0.988029 + 1.42260i 0.383892 − 2.61775i 1.00000i 1.04760 + 2.81115i 2.44875 − 1.41378i
269.6 − 1.00000i −0.926039 + 1.46371i −1.00000 −2.14723 − 3.71911i 1.46371 + 0.926039i −0.786623 + 2.52611i 1.00000i −1.28490 − 2.71091i −3.71911 + 2.14723i
269.7 − 1.00000i −0.834971 + 1.51751i −1.00000 0.277881 + 0.481304i 1.51751 + 0.834971i −1.71510 − 2.01455i 1.00000i −1.60565 − 2.53415i 0.481304 − 0.277881i
269.8 − 1.00000i −0.592023 + 1.62773i −1.00000 2.12998 + 3.68924i 1.62773 + 0.592023i 2.63074 + 0.281454i 1.00000i −2.29902 − 1.92731i 3.68924 − 2.12998i
269.9 − 1.00000i −0.459307 − 1.67004i −1.00000 −1.40279 − 2.42970i −1.67004 + 0.459307i 1.21733 + 2.34907i 1.00000i −2.57807 + 1.53412i −2.42970 + 1.40279i
269.10 − 1.00000i −0.215340 − 1.71861i −1.00000 −1.07511 − 1.86215i −1.71861 + 0.215340i −2.48199 − 0.916357i 1.00000i −2.90726 + 0.740172i −1.86215 + 1.07511i
269.11 − 1.00000i 0.201730 − 1.72026i −1.00000 1.75377 + 3.03762i −1.72026 − 0.201730i 1.27169 + 2.32009i 1.00000i −2.91861 − 0.694059i 3.03762 − 1.75377i
269.12 − 1.00000i 0.691481 + 1.58803i −1.00000 −0.349437 − 0.605242i 1.58803 − 0.691481i 2.58515 − 0.563029i 1.00000i −2.04371 + 2.19619i −0.605242 + 0.349437i
269.13 − 1.00000i 1.03468 − 1.38904i −1.00000 0.587127 + 1.01693i −1.38904 − 1.03468i −0.859975 − 2.50209i 1.00000i −0.858887 − 2.87442i 1.01693 − 0.587127i
269.14 − 1.00000i 1.15993 + 1.28630i −1.00000 −0.0272832 − 0.0472559i 1.28630 − 1.15993i 0.114956 + 2.64325i 1.00000i −0.309116 + 2.98403i −0.0472559 + 0.0272832i
269.15 − 1.00000i 1.32763 + 1.11238i −1.00000 −1.76347 − 3.05441i 1.11238 − 1.32763i −2.64100 + 0.158422i 1.00000i 0.525225 + 2.95367i −3.05441 + 1.76347i
269.16 − 1.00000i 1.34890 − 1.08650i −1.00000 −0.279399 − 0.483934i −1.08650 − 1.34890i 2.15416 + 1.53609i 1.00000i 0.639049 − 2.93115i −0.483934 + 0.279399i
269.17 − 1.00000i 1.63536 − 0.570612i −1.00000 −1.99252 − 3.45115i −0.570612 − 1.63536i 2.09752 − 1.61258i 1.00000i 2.34880 − 1.86631i −3.45115 + 1.99252i
269.18 − 1.00000i 1.63815 + 0.562548i −1.00000 0.604752 + 1.04746i 0.562548 − 1.63815i 0.374319 − 2.61914i 1.00000i 2.36708 + 1.84308i 1.04746 − 0.604752i
269.19 − 1.00000i 1.72167 − 0.189348i −1.00000 1.42622 + 2.47029i −0.189348 − 1.72167i −1.89518 + 1.84615i 1.00000i 2.92829 − 0.651988i 2.47029 − 1.42622i
269.20 1.00000i −1.73168 − 0.0356471i −1.00000 −0.277881 − 0.481304i 0.0356471 − 1.73168i −1.71510 − 2.01455i − 1.00000i 2.99746 + 0.123459i 0.481304 − 0.277881i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.38
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.v odd 6 1 inner
273.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.bb.a yes 76
3.b odd 2 1 inner 546.2.bb.a yes 76
7.d odd 6 1 546.2.u.a ✓ 76
13.c even 3 1 546.2.u.a ✓ 76
21.g even 6 1 546.2.u.a ✓ 76
39.i odd 6 1 546.2.u.a ✓ 76
91.v odd 6 1 inner 546.2.bb.a yes 76
273.r even 6 1 inner 546.2.bb.a yes 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.u.a ✓ 76 7.d odd 6 1
546.2.u.a ✓ 76 13.c even 3 1
546.2.u.a ✓ 76 21.g even 6 1
546.2.u.a ✓ 76 39.i odd 6 1
546.2.bb.a yes 76 1.a even 1 1 trivial
546.2.bb.a yes 76 3.b odd 2 1 inner
546.2.bb.a yes 76 91.v odd 6 1 inner
546.2.bb.a yes 76 273.r even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(546, [\chi])\).