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Properties

Label 324.2.m.a
Level $324$
Weight $2$
Character orbit 324.m
Analytic conductor $2.587$
Analytic rank $0$
Dimension $162$
Inner twists $2$

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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] = [324,2,Mod(13,324)] 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("324.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.m (of order \(27\), degree \(18\), 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: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(9\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 162 q + 27 q^{21} + 27 q^{23} + 27 q^{27} + 27 q^{29} + 27 q^{33} + 27 q^{35} - 18 q^{41} - 54 q^{45} - 54 q^{47} - 63 q^{51} - 54 q^{53} - 54 q^{57} - 63 q^{59} - 54 q^{63} - 90 q^{65} + 27 q^{67} - 90 q^{69}+ \cdots - 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −1.73190 − 0.0227943i 0 −1.16619 − 3.89534i 0 1.26893 + 2.94171i 0 2.99896 + 0.0789550i 0
13.2 0 −1.48240 + 0.895824i 0 0.0229196 + 0.0765569i 0 −1.39529 − 3.23464i 0 1.39500 − 2.65593i 0
13.3 0 −1.37167 − 1.05760i 0 0.465459 + 1.55474i 0 0.273810 + 0.634762i 0 0.762961 + 2.90136i 0
13.4 0 −0.429742 + 1.67789i 0 0.575939 + 1.92377i 0 0.833877 + 1.93314i 0 −2.63064 − 1.44212i 0
13.5 0 −0.355946 − 1.69508i 0 −0.474145 − 1.58376i 0 −0.187557 − 0.434806i 0 −2.74661 + 1.20671i 0
13.6 0 1.19354 − 1.25517i 0 0.462190 + 1.54382i 0 1.59904 + 3.70700i 0 −0.150919 − 2.99620i 0
13.7 0 1.26242 + 1.18587i 0 −0.909501 − 3.03794i 0 0.410517 + 0.951685i 0 0.187417 + 2.99414i 0
13.8 0 1.50798 + 0.852058i 0 1.16819 + 3.90202i 0 −1.69197 − 3.92242i 0 1.54799 + 2.56977i 0
13.9 0 1.55696 − 0.758860i 0 −0.491276 − 1.64098i 0 −1.11136 − 2.57643i 0 1.84826 − 2.36303i 0
25.1 0 −1.73190 + 0.0227943i 0 −1.16619 + 3.89534i 0 1.26893 − 2.94171i 0 2.99896 − 0.0789550i 0
25.2 0 −1.48240 − 0.895824i 0 0.0229196 − 0.0765569i 0 −1.39529 + 3.23464i 0 1.39500 + 2.65593i 0
25.3 0 −1.37167 + 1.05760i 0 0.465459 − 1.55474i 0 0.273810 − 0.634762i 0 0.762961 − 2.90136i 0
25.4 0 −0.429742 − 1.67789i 0 0.575939 − 1.92377i 0 0.833877 − 1.93314i 0 −2.63064 + 1.44212i 0
25.5 0 −0.355946 + 1.69508i 0 −0.474145 + 1.58376i 0 −0.187557 + 0.434806i 0 −2.74661 − 1.20671i 0
25.6 0 1.19354 + 1.25517i 0 0.462190 − 1.54382i 0 1.59904 − 3.70700i 0 −0.150919 + 2.99620i 0
25.7 0 1.26242 − 1.18587i 0 −0.909501 + 3.03794i 0 0.410517 − 0.951685i 0 0.187417 − 2.99414i 0
25.8 0 1.50798 − 0.852058i 0 1.16819 − 3.90202i 0 −1.69197 + 3.92242i 0 1.54799 − 2.56977i 0
25.9 0 1.55696 + 0.758860i 0 −0.491276 + 1.64098i 0 −1.11136 + 2.57643i 0 1.84826 + 2.36303i 0
49.1 0 −1.71035 + 0.273337i 0 0.101566 − 0.235457i 0 0.0645556 + 1.10838i 0 2.85057 − 0.935001i 0
49.2 0 −1.50806 − 0.851908i 0 −1.59595 + 3.69982i 0 −0.240156 − 4.12332i 0 1.54850 + 2.56946i 0
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
81.g even 27 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.m.a ✓ 162
3.b odd 2 1 972.2.m.a 162
81.g even 27 1 inner 324.2.m.a ✓ 162
81.h odd 54 1 972.2.m.a 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.2.m.a ✓ 162 1.a even 1 1 trivial
324.2.m.a ✓ 162 81.g even 27 1 inner
972.2.m.a 162 3.b odd 2 1
972.2.m.a 162 81.h odd 54 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(324, [\chi])\).