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Properties

Label 546.2.bq.c
Level $546$
Weight $2$
Character orbit 546.bq
Analytic conductor $4.360$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [546,2,Mod(419,546)]
 
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, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("546.419");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.bq (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: \(64\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} + 16 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 32 q^{4} + 16 q^{7} + 16 q^{9} - 12 q^{15} - 32 q^{16} - 24 q^{18} + 16 q^{21} - 16 q^{28} + 8 q^{30} - 16 q^{36} - 48 q^{39} + 20 q^{42} - 8 q^{43} + 8 q^{46} + 8 q^{49} - 8 q^{51} + 8 q^{57} + 16 q^{58} - 24 q^{60} + 32 q^{63} - 64 q^{64} + 24 q^{67} + 48 q^{70} - 12 q^{72} - 12 q^{78} + 96 q^{79} + 68 q^{81} + 8 q^{84} - 112 q^{85} - 16 q^{91} - 56 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1 −0.866025 + 0.500000i −1.73204 − 0.00466741i 0.500000 − 0.866025i 0.915374 1.50233 − 0.861980i −2.64454 + 0.0800636i 1.00000i 2.99996 + 0.0161683i −0.792737 + 0.457687i
419.2 −0.866025 + 0.500000i −1.73041 + 0.0753636i 0.500000 − 0.866025i −2.24165 1.46090 − 0.930472i −0.301604 − 2.62850i 1.00000i 2.98864 − 0.260820i 1.94133 − 1.12083i
419.3 −0.866025 + 0.500000i −1.69217 + 0.369559i 0.500000 − 0.866025i −3.03607 1.28068 − 1.16613i 2.25111 + 1.39014i 1.00000i 2.72685 − 1.25071i 2.62932 − 1.51804i
419.4 −0.866025 + 0.500000i −1.66375 − 0.481611i 0.500000 − 0.866025i 4.01418 1.68165 − 0.414785i 1.77617 − 1.96093i 1.00000i 2.53610 + 1.60256i −3.47639 + 2.00709i
419.5 −0.866025 + 0.500000i −1.22965 + 1.21982i 0.500000 − 0.866025i 1.88020 0.454992 − 1.67122i −1.36777 + 2.26478i 1.00000i 0.0240567 − 2.99990i −1.62830 + 0.940099i
419.6 −0.866025 + 0.500000i −0.998793 − 1.41507i 0.500000 − 0.866025i 1.30549 1.57251 + 0.726086i 2.26007 + 1.37552i 1.00000i −1.00482 + 2.82672i −1.13058 + 0.652743i
419.7 −0.866025 + 0.500000i −0.344709 + 1.69740i 0.500000 − 0.866025i 1.64213 −0.550175 − 1.64235i −1.55066 − 2.14370i 1.00000i −2.76235 − 1.17022i −1.42213 + 0.821065i
419.8 −0.866025 + 0.500000i −0.211712 + 1.71906i 0.500000 − 0.866025i −0.932541 −0.676184 − 1.59461i 2.18650 − 1.48970i 1.00000i −2.91036 − 0.727891i 0.807604 − 0.466271i
419.9 −0.866025 + 0.500000i 0.211712 − 1.71906i 0.500000 − 0.866025i 0.932541 0.676184 + 1.59461i 0.196871 − 2.63842i 1.00000i −2.91036 − 0.727891i −0.807604 + 0.466271i
419.10 −0.866025 + 0.500000i 0.344709 − 1.69740i 0.500000 − 0.866025i −1.64213 0.550175 + 1.64235i 2.63183 + 0.271064i 1.00000i −2.76235 − 1.17022i 1.42213 − 0.821065i
419.11 −0.866025 + 0.500000i 0.998793 + 1.41507i 0.500000 − 0.866025i −1.30549 −1.57251 − 0.726086i −2.32128 − 1.26952i 1.00000i −1.00482 + 2.82672i 1.13058 − 0.652743i
419.12 −0.866025 + 0.500000i 1.22965 − 1.21982i 0.500000 − 0.866025i −1.88020 −0.454992 + 1.67122i −1.27747 + 2.31691i 1.00000i 0.0240567 − 2.99990i 1.62830 − 0.940099i
419.13 −0.866025 + 0.500000i 1.66375 + 0.481611i 0.500000 − 0.866025i −4.01418 −1.68165 + 0.414785i 0.810131 − 2.51867i 1.00000i 2.53610 + 1.60256i 3.47639 − 2.00709i
419.14 −0.866025 + 0.500000i 1.69217 − 0.369559i 0.500000 − 0.866025i 3.03607 −1.28068 + 1.16613i −2.32945 − 1.25445i 1.00000i 2.72685 − 1.25071i −2.62932 + 1.51804i
419.15 −0.866025 + 0.500000i 1.73041 − 0.0753636i 0.500000 − 0.866025i 2.24165 −1.46090 + 0.930472i 2.42715 − 1.05306i 1.00000i 2.98864 − 0.260820i −1.94133 + 1.12083i
419.16 −0.866025 + 0.500000i 1.73204 + 0.00466741i 0.500000 − 0.866025i −0.915374 −1.50233 + 0.861980i 1.25293 + 2.33027i 1.00000i 2.99996 + 0.0161683i 0.792737 − 0.457687i
419.17 0.866025 − 0.500000i −1.67122 + 0.454992i 0.500000 − 0.866025i −1.88020 −1.21982 + 1.22965i −1.36777 + 2.26478i − 1.00000i 2.58596 − 1.52079i −1.62830 + 0.940099i
419.18 0.866025 − 0.500000i −1.64235 − 0.550175i 0.500000 − 0.866025i −1.64213 −1.69740 + 0.344709i −1.55066 − 2.14370i − 1.00000i 2.39462 + 1.80716i −1.42213 + 0.821065i
419.19 0.866025 − 0.500000i −1.59461 − 0.676184i 0.500000 − 0.866025i 0.932541 −1.71906 + 0.211712i 2.18650 − 1.48970i − 1.00000i 2.08555 + 2.15650i 0.807604 − 0.466271i
419.20 0.866025 − 0.500000i −1.16613 + 1.28068i 0.500000 − 0.866025i 3.03607 −0.369559 + 1.69217i 2.25111 + 1.39014i − 1.00000i −0.280280 − 2.98688i 2.62932 − 1.51804i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
13.c even 3 1 inner
21.c even 2 1 inner
39.i odd 6 1 inner
91.n odd 6 1 inner
273.bn 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.bq.c ✓ 64
3.b odd 2 1 inner 546.2.bq.c ✓ 64
7.b odd 2 1 inner 546.2.bq.c ✓ 64
13.c even 3 1 inner 546.2.bq.c ✓ 64
21.c even 2 1 inner 546.2.bq.c ✓ 64
39.i odd 6 1 inner 546.2.bq.c ✓ 64
91.n odd 6 1 inner 546.2.bq.c ✓ 64
273.bn even 6 1 inner 546.2.bq.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.bq.c ✓ 64 1.a even 1 1 trivial
546.2.bq.c ✓ 64 3.b odd 2 1 inner
546.2.bq.c ✓ 64 7.b odd 2 1 inner
546.2.bq.c ✓ 64 13.c even 3 1 inner
546.2.bq.c ✓ 64 21.c even 2 1 inner
546.2.bq.c ✓ 64 39.i odd 6 1 inner
546.2.bq.c ✓ 64 91.n odd 6 1 inner
546.2.bq.c ✓ 64 273.bn even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\):

\( T_{5}^{16} - 40T_{5}^{14} + 603T_{5}^{12} - 4508T_{5}^{10} + 18451T_{5}^{8} - 42568T_{5}^{6} + 54189T_{5}^{4} - 34932T_{5}^{2} + 8836 \) Copy content Toggle raw display
\( T_{61}^{32} - 242 T_{61}^{30} + 46172 T_{61}^{28} - 2648516 T_{61}^{26} + 110141918 T_{61}^{24} + \cdots + 35153041 \) Copy content Toggle raw display