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Properties

Label 546.4.g.b
Level $546$
Weight $4$
Character orbit 546.g
Analytic conductor $32.215$
Analytic rank $0$
Dimension $48$
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] = [546,4,Mod(209,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.209");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 546.g (of order \(2\), degree \(1\), 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: \(32.2150428631\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 192 q^{4} + 6 q^{7} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 192 q^{4} + 6 q^{7} + 28 q^{9} + 48 q^{14} - 168 q^{15} + 768 q^{16} + 36 q^{17} + 108 q^{21} + 240 q^{22} + 1080 q^{25} + 1248 q^{26} - 462 q^{27} - 24 q^{28} - 364 q^{30} + 146 q^{33} - 1350 q^{35} - 112 q^{36} + 60 q^{37} - 192 q^{38} + 1008 q^{41} + 108 q^{42} + 1344 q^{43} + 380 q^{45} - 1224 q^{46} + 660 q^{47} - 1134 q^{49} - 734 q^{51} - 372 q^{54} - 192 q^{56} - 2320 q^{57} + 1512 q^{58} - 264 q^{59} + 672 q^{60} + 1512 q^{62} + 2908 q^{63} - 3072 q^{64} + 480 q^{66} - 144 q^{68} - 1260 q^{69} + 96 q^{70} - 2114 q^{75} - 1152 q^{77} + 924 q^{79} - 268 q^{81} + 504 q^{83} - 432 q^{84} + 3696 q^{85} + 1220 q^{87} - 960 q^{88} + 1476 q^{89} - 84 q^{90} - 312 q^{91} + 2584 q^{93} - 1200 q^{98} - 4452 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 − 2.00000i −5.11345 − 0.923397i −4.00000 −4.81067 −1.84679 + 10.2269i 6.24945 + 17.4340i 8.00000i 25.2947 + 9.44348i 9.62133i
209.2 2.00000i −5.11345 + 0.923397i −4.00000 −4.81067 −1.84679 − 10.2269i 6.24945 − 17.4340i − 8.00000i 25.2947 − 9.44348i − 9.62133i
209.3 − 2.00000i −0.904159 + 5.11688i −4.00000 −11.7291 10.2338 + 1.80832i −14.6699 − 11.3046i 8.00000i −25.3650 − 9.25295i 23.4581i
209.4 2.00000i −0.904159 − 5.11688i −4.00000 −11.7291 10.2338 − 1.80832i −14.6699 + 11.3046i − 8.00000i −25.3650 + 9.25295i − 23.4581i
209.5 − 2.00000i −1.82108 − 4.86659i −4.00000 20.4454 −9.73317 + 3.64216i −18.3264 + 2.67255i 8.00000i −20.3673 + 17.7249i − 40.8908i
209.6 2.00000i −1.82108 + 4.86659i −4.00000 20.4454 −9.73317 − 3.64216i −18.3264 − 2.67255i − 8.00000i −20.3673 − 17.7249i 40.8908i
209.7 − 2.00000i 5.01424 − 1.36287i −4.00000 −2.67543 −2.72575 − 10.0285i 18.0402 + 4.18930i 8.00000i 23.2852 − 13.6675i 5.35086i
209.8 2.00000i 5.01424 + 1.36287i −4.00000 −2.67543 −2.72575 + 10.0285i 18.0402 − 4.18930i − 8.00000i 23.2852 + 13.6675i − 5.35086i
209.9 − 2.00000i 2.95899 + 4.27135i −4.00000 19.1878 8.54269 − 5.91797i 7.18366 − 17.0703i 8.00000i −9.48881 + 25.2777i − 38.3755i
209.10 2.00000i 2.95899 − 4.27135i −4.00000 19.1878 8.54269 + 5.91797i 7.18366 + 17.0703i − 8.00000i −9.48881 − 25.2777i 38.3755i
209.11 − 2.00000i 4.90131 − 1.72544i −4.00000 15.6889 −3.45087 − 9.80263i 0.725043 + 18.5061i 8.00000i 21.0457 − 16.9138i − 31.3779i
209.12 2.00000i 4.90131 + 1.72544i −4.00000 15.6889 −3.45087 + 9.80263i 0.725043 − 18.5061i − 8.00000i 21.0457 + 16.9138i 31.3779i
209.13 − 2.00000i 4.00795 + 3.30701i −4.00000 −21.5424 6.61402 − 8.01591i 17.8710 − 4.86079i 8.00000i 5.12738 + 26.5087i 43.0848i
209.14 2.00000i 4.00795 − 3.30701i −4.00000 −21.5424 6.61402 + 8.01591i 17.8710 + 4.86079i − 8.00000i 5.12738 − 26.5087i − 43.0848i
209.15 − 2.00000i 3.03994 − 4.21411i −4.00000 −15.2954 −8.42823 − 6.07989i −7.27280 + 17.0325i 8.00000i −8.51749 − 25.6213i 30.5908i
209.16 2.00000i 3.03994 + 4.21411i −4.00000 −15.2954 −8.42823 + 6.07989i −7.27280 − 17.0325i − 8.00000i −8.51749 + 25.6213i − 30.5908i
209.17 − 2.00000i −4.44296 + 2.69445i −4.00000 −19.4917 5.38891 + 8.88592i 14.2146 + 11.8721i 8.00000i 12.4798 − 23.9427i 38.9833i
209.18 2.00000i −4.44296 − 2.69445i −4.00000 −19.4917 5.38891 − 8.88592i 14.2146 − 11.8721i − 8.00000i 12.4798 + 23.9427i − 38.9833i
209.19 − 2.00000i −1.28501 + 5.03475i −4.00000 −8.08948 10.0695 + 2.57002i 15.2888 − 10.4524i 8.00000i −23.6975 − 12.9394i 16.1790i
209.20 2.00000i −1.28501 − 5.03475i −4.00000 −8.08948 10.0695 − 2.57002i 15.2888 + 10.4524i − 8.00000i −23.6975 + 12.9394i − 16.1790i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.48
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.4.g.b yes 48
3.b odd 2 1 546.4.g.a ✓ 48
7.b odd 2 1 546.4.g.a ✓ 48
21.c even 2 1 inner 546.4.g.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.g.a ✓ 48 3.b odd 2 1
546.4.g.a ✓ 48 7.b odd 2 1
546.4.g.b yes 48 1.a even 1 1 trivial
546.4.g.b yes 48 21.c even 2 1 inner