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Properties

Label 546.2.bi.f
Level $546$
Weight $2$
Character orbit 546.bi
Analytic conductor $4.360$
Analytic rank $0$
Dimension $34$
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] = [546,2,Mod(17,546)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(546, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bi (of order \(6\), degree \(2\), 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: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 34 q^{2} + 6 q^{3} + 34 q^{4} + 9 q^{5} + 6 q^{6} + 4 q^{7} + 34 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 34 q^{2} + 6 q^{3} + 34 q^{4} + 9 q^{5} + 6 q^{6} + 4 q^{7} + 34 q^{8} + 4 q^{9} + 9 q^{10} + 9 q^{11} + 6 q^{12} + 8 q^{13} + 4 q^{14} - 17 q^{15} + 34 q^{16} + 12 q^{17} + 4 q^{18} - 5 q^{19} + 9 q^{20} - 7 q^{21} + 9 q^{22} + 6 q^{24} + 16 q^{25} + 8 q^{26} - 18 q^{27} + 4 q^{28} + 27 q^{29} - 17 q^{30} - q^{31} + 34 q^{32} + 12 q^{34} - 3 q^{35} + 4 q^{36} - 5 q^{38} - 10 q^{39} + 9 q^{40} - 3 q^{41} - 7 q^{42} - 3 q^{43} + 9 q^{44} + 9 q^{45} - 27 q^{47} + 6 q^{48} - 2 q^{49} + 16 q^{50} - 36 q^{51} + 8 q^{52} - 21 q^{53} - 18 q^{54} - 57 q^{55} + 4 q^{56} - 17 q^{57} + 27 q^{58} - 17 q^{60} - 51 q^{61} - q^{62} - 24 q^{63} + 34 q^{64} - 21 q^{65} - 21 q^{67} + 12 q^{68} + 30 q^{69} - 3 q^{70} - 15 q^{71} + 4 q^{72} - 19 q^{73} - 54 q^{75} - 5 q^{76} + 9 q^{77} - 10 q^{78} - 9 q^{79} + 9 q^{80} + 28 q^{81} - 3 q^{82} - 7 q^{84} - 42 q^{85} - 3 q^{86} - 81 q^{87} + 9 q^{88} + 9 q^{90} - 72 q^{91} - 17 q^{93} - 27 q^{94} + 6 q^{96} + 19 q^{97} - 2 q^{98} - 27 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 1.00000 −1.72563 − 0.148954i 1.00000 3.72094 − 2.14828i −1.72563 − 0.148954i 1.54641 + 2.14677i 1.00000 2.95563 + 0.514080i 3.72094 − 2.14828i
17.2 1.00000 −1.66986 − 0.459961i 1.00000 0.567570 − 0.327687i −1.66986 − 0.459961i −2.37289 + 1.17020i 1.00000 2.57687 + 1.53614i 0.567570 − 0.327687i
17.3 1.00000 −1.63421 + 0.573890i 1.00000 −1.62172 + 0.936303i −1.63421 + 0.573890i 2.04281 − 1.68135i 1.00000 2.34130 − 1.87572i −1.62172 + 0.936303i
17.4 1.00000 −1.23948 + 1.20983i 1.00000 1.58996 − 0.917964i −1.23948 + 1.20983i −0.364289 − 2.62055i 1.00000 0.0726040 − 2.99912i 1.58996 − 0.917964i
17.5 1.00000 −0.942473 − 1.45318i 1.00000 3.27919 − 1.89324i −0.942473 − 1.45318i 0.475130 − 2.60274i 1.00000 −1.22349 + 2.73917i 3.27919 − 1.89324i
17.6 1.00000 −0.889878 − 1.48597i 1.00000 −2.84717 + 1.64381i −0.889878 − 1.48597i 1.87202 + 1.86964i 1.00000 −1.41623 + 2.64467i −2.84717 + 1.64381i
17.7 1.00000 0.166804 + 1.72400i 1.00000 1.41302 − 0.815806i 0.166804 + 1.72400i 2.62951 + 0.292738i 1.00000 −2.94435 + 0.575141i 1.41302 − 0.815806i
17.8 1.00000 0.243110 + 1.71490i 1.00000 −2.60318 + 1.50295i 0.243110 + 1.71490i −2.60776 − 0.446759i 1.00000 −2.88179 + 0.833822i −2.60318 + 1.50295i
17.9 1.00000 0.403394 − 1.68442i 1.00000 −1.80315 + 1.04105i 0.403394 − 1.68442i −0.800654 − 2.52170i 1.00000 −2.67455 − 1.35897i −1.80315 + 1.04105i
17.10 1.00000 0.517534 − 1.65292i 1.00000 0.870413 − 0.502533i 0.517534 − 1.65292i 2.64571 − 0.0151415i 1.00000 −2.46432 − 1.71089i 0.870413 − 0.502533i
17.11 1.00000 0.942850 − 1.45294i 1.00000 1.98183 − 1.14421i 0.942850 − 1.45294i −0.877809 + 2.49589i 1.00000 −1.22207 − 2.73981i 1.98183 − 1.14421i
17.12 1.00000 1.06823 + 1.36341i 1.00000 2.88000 − 1.66277i 1.06823 + 1.36341i −2.37914 − 1.15746i 1.00000 −0.717779 + 2.91287i 2.88000 − 1.66277i
17.13 1.00000 1.13542 + 1.30799i 1.00000 −1.26448 + 0.730045i 1.13542 + 1.30799i −1.08820 + 2.41160i 1.00000 −0.421657 + 2.97022i −1.26448 + 0.730045i
17.14 1.00000 1.48743 + 0.887438i 1.00000 0.511132 − 0.295102i 1.48743 + 0.887438i 2.62812 − 0.304939i 1.00000 1.42491 + 2.64001i 0.511132 − 0.295102i
17.15 1.00000 1.67564 − 0.438430i 1.00000 1.57344 − 0.908426i 1.67564 − 0.438430i −2.47008 − 0.947995i 1.00000 2.61556 − 1.46930i 1.57344 − 0.908426i
17.16 1.00000 1.73046 − 0.0741172i 1.00000 −1.09866 + 0.634311i 1.73046 − 0.0741172i 0.151485 + 2.64141i 1.00000 2.98901 − 0.256514i −1.09866 + 0.634311i
17.17 1.00000 1.73066 + 0.0694407i 1.00000 −2.64914 + 1.52948i 1.73066 + 0.0694407i 0.969634 − 2.46167i 1.00000 2.99036 + 0.240356i −2.64914 + 1.52948i
257.1 1.00000 −1.72563 + 0.148954i 1.00000 3.72094 + 2.14828i −1.72563 + 0.148954i 1.54641 − 2.14677i 1.00000 2.95563 − 0.514080i 3.72094 + 2.14828i
257.2 1.00000 −1.66986 + 0.459961i 1.00000 0.567570 + 0.327687i −1.66986 + 0.459961i −2.37289 − 1.17020i 1.00000 2.57687 − 1.53614i 0.567570 + 0.327687i
257.3 1.00000 −1.63421 − 0.573890i 1.00000 −1.62172 − 0.936303i −1.63421 − 0.573890i 2.04281 + 1.68135i 1.00000 2.34130 + 1.87572i −1.62172 − 0.936303i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
273.br 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.bi.f yes 34
3.b odd 2 1 546.2.bi.e ✓ 34
7.d odd 6 1 546.2.bn.e yes 34
13.e even 6 1 546.2.bn.f yes 34
21.g even 6 1 546.2.bn.f yes 34
39.h odd 6 1 546.2.bn.e yes 34
91.l odd 6 1 546.2.bi.e ✓ 34
273.br even 6 1 inner 546.2.bi.f yes 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bi.e ✓ 34 3.b odd 2 1
546.2.bi.e ✓ 34 91.l odd 6 1
546.2.bi.f yes 34 1.a even 1 1 trivial
546.2.bi.f yes 34 273.br even 6 1 inner
546.2.bn.e yes 34 7.d odd 6 1
546.2.bn.e yes 34 39.h odd 6 1
546.2.bn.f yes 34 13.e even 6 1
546.2.bn.f yes 34 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{34} - 9 T_{5}^{33} - 10 T_{5}^{32} + 333 T_{5}^{31} - 234 T_{5}^{30} - 7575 T_{5}^{29} + \cdots + 3270763083 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\). Copy content Toggle raw display