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Properties

Label 546.4.g.a
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
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 192 q^{4} + 6 q^{7} + 28 q^{9} - 48 q^{14} - 168 q^{15} + 768 q^{16} - 36 q^{17} + 188 q^{21} + 240 q^{22} + 1080 q^{25} - 1248 q^{26} + 462 q^{27} - 24 q^{28} - 364 q^{30} - 146 q^{33} + 1350 q^{35} - 112 q^{36} + 60 q^{37} + 192 q^{38} - 1008 q^{41} - 584 q^{42} + 1344 q^{43} - 380 q^{45} - 1224 q^{46} - 660 q^{47} - 1134 q^{49} - 734 q^{51} + 372 q^{54} + 192 q^{56} - 2320 q^{57} + 1512 q^{58} + 264 q^{59} + 672 q^{60} - 1512 q^{62} - 312 q^{63} - 3072 q^{64} - 480 q^{66} + 144 q^{68} + 1260 q^{69} + 96 q^{70} + 2114 q^{75} + 1152 q^{77} + 924 q^{79} - 268 q^{81} - 504 q^{83} - 752 q^{84} + 3696 q^{85} - 1220 q^{87} - 960 q^{88} - 1476 q^{89} + 84 q^{90} - 312 q^{91} + 2584 q^{93} + 1200 q^{98} - 4452 q^{99}+O(q^{100}) \) 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 5.19530 + 0.0942712i −4.00000 −12.3865 0.188542 − 10.3906i 7.97412 − 16.7157i 8.00000i 26.9822 + 0.979534i 24.7730i
209.2 2.00000i 5.19530 − 0.0942712i −4.00000 −12.3865 0.188542 + 10.3906i 7.97412 + 16.7157i − 8.00000i 26.9822 − 0.979534i − 24.7730i
209.3 − 2.00000i −0.632799 − 5.15748i −4.00000 1.96309 −10.3150 + 1.26560i −0.179916 − 18.5194i 8.00000i −26.1991 + 6.52730i − 3.92618i
209.4 2.00000i −0.632799 + 5.15748i −4.00000 1.96309 −10.3150 − 1.26560i −0.179916 + 18.5194i − 8.00000i −26.1991 − 6.52730i 3.92618i
209.5 − 2.00000i 4.62938 − 2.35984i −4.00000 −11.4268 −4.71969 − 9.25876i 11.9519 + 14.1475i 8.00000i 15.8623 − 21.8492i 22.8536i
209.6 2.00000i 4.62938 + 2.35984i −4.00000 −11.4268 −4.71969 + 9.25876i 11.9519 − 14.1475i − 8.00000i 15.8623 + 21.8492i − 22.8536i
209.7 − 2.00000i −2.91987 + 4.29818i −4.00000 −5.11604 8.59636 + 5.83975i −4.51282 + 17.9620i 8.00000i −9.94868 − 25.1003i 10.2321i
209.8 2.00000i −2.91987 − 4.29818i −4.00000 −5.11604 8.59636 − 5.83975i −4.51282 − 17.9620i − 8.00000i −9.94868 + 25.1003i − 10.2321i
209.9 − 2.00000i −4.21728 − 3.03555i −4.00000 8.54585 −6.07110 + 8.43456i −10.1832 − 15.4694i 8.00000i 8.57090 + 25.6035i − 17.0917i
209.10 2.00000i −4.21728 + 3.03555i −4.00000 8.54585 −6.07110 − 8.43456i −10.1832 + 15.4694i − 8.00000i 8.57090 − 25.6035i 17.0917i
209.11 − 2.00000i 4.62054 + 2.37710i −4.00000 14.5512 4.75419 − 9.24109i −4.39777 + 17.9905i 8.00000i 15.6988 + 21.9669i − 29.1023i
209.12 2.00000i 4.62054 − 2.37710i −4.00000 14.5512 4.75419 + 9.24109i −4.39777 − 17.9905i − 8.00000i 15.6988 − 21.9669i 29.1023i
209.13 − 2.00000i −5.16769 + 0.543148i −4.00000 −3.34819 1.08630 + 10.3354i −17.8991 + 4.75635i 8.00000i 26.4100 − 5.61364i 6.69639i
209.14 2.00000i −5.16769 − 0.543148i −4.00000 −3.34819 1.08630 − 10.3354i −17.8991 − 4.75635i − 8.00000i 26.4100 + 5.61364i − 6.69639i
209.15 − 2.00000i 1.87533 + 4.84594i −4.00000 9.48384 9.69188 − 3.75066i −17.8721 − 4.85661i 8.00000i −19.9663 + 18.1755i − 18.9677i
209.16 2.00000i 1.87533 − 4.84594i −4.00000 9.48384 9.69188 + 3.75066i −17.8721 + 4.85661i − 8.00000i −19.9663 − 18.1755i 18.9677i
209.17 − 2.00000i −0.411683 + 5.17982i −4.00000 −4.86798 10.3596 + 0.823366i 10.2779 − 15.4066i 8.00000i −26.6610 − 4.26489i 9.73596i
209.18 2.00000i −0.411683 − 5.17982i −4.00000 −4.86798 10.3596 − 0.823366i 10.2779 + 15.4066i − 8.00000i −26.6610 + 4.26489i − 9.73596i
209.19 − 2.00000i 3.25797 + 4.04792i −4.00000 −6.18835 8.09583 − 6.51594i 16.3538 + 8.69215i 8.00000i −5.77126 + 26.3760i 12.3767i
209.20 2.00000i 3.25797 − 4.04792i −4.00000 −6.18835 8.09583 + 6.51594i 16.3538 − 8.69215i − 8.00000i −5.77126 − 26.3760i − 12.3767i
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.a ✓ 48
3.b odd 2 1 546.4.g.b yes 48
7.b odd 2 1 546.4.g.b yes 48
21.c even 2 1 inner 546.4.g.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.4.g.a ✓ 48 1.a even 1 1 trivial
546.4.g.a ✓ 48 21.c even 2 1 inner
546.4.g.b yes 48 3.b odd 2 1
546.4.g.b yes 48 7.b odd 2 1