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Properties

Label 1218.2.m.b
Level $1218$
Weight $2$
Character orbit 1218.m
Analytic conductor $9.726$
Analytic rank $0$
Dimension $40$
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] = [1218,2,Mod(307,1218)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1218, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1218.307"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1218 = 2 \cdot 3 \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1218.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.72577896619\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 40 q^{6} - 4 q^{10} + 4 q^{14} - 4 q^{15} - 40 q^{16} - 8 q^{19} + 4 q^{21} + 24 q^{25} + 12 q^{28} + 8 q^{29} + 24 q^{31} + 12 q^{35} + 40 q^{36} - 16 q^{37} + 4 q^{40} - 16 q^{41} - 20 q^{43} + 4 q^{46}+ \cdots - 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i 3.66176 1.00000 −1.87946 + 1.86216i 0.707107 + 0.707107i 1.00000i −2.58925 + 2.58925i
307.2 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i 2.31151 1.00000 2.48366 + 0.911828i 0.707107 + 0.707107i 1.00000i −1.63448 + 1.63448i
307.3 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i 0.964064 1.00000 −1.12003 − 2.39698i 0.707107 + 0.707107i 1.00000i −0.681696 + 0.681696i
307.4 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i −0.447637 1.00000 2.17349 − 1.50862i 0.707107 + 0.707107i 1.00000i 0.316527 − 0.316527i
307.5 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i 1.12959 1.00000 −2.63276 − 0.261871i 0.707107 + 0.707107i 1.00000i −0.798741 + 0.798741i
307.6 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i −1.44011 1.00000 2.28150 − 1.33969i 0.707107 + 0.707107i 1.00000i 1.01831 − 1.01831i
307.7 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i 2.17708 1.00000 −0.0332084 + 2.64554i 0.707107 + 0.707107i 1.00000i −1.53943 + 1.53943i
307.8 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i −2.29304 1.00000 −2.56094 − 0.664510i 0.707107 + 0.707107i 1.00000i 1.62142 − 1.62142i
307.9 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i −1.75408 1.00000 −1.92667 + 1.81327i 0.707107 + 0.707107i 1.00000i 1.24032 − 1.24032i
307.10 −0.707107 + 0.707107i −0.707107 − 0.707107i − 1.00000i −2.89493 1.00000 1.80021 + 1.93888i 0.707107 + 0.707107i 1.00000i 2.04702 − 2.04702i
307.11 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i 3.88041 1.00000 −0.687128 + 2.55497i −0.707107 − 0.707107i 1.00000i 2.74387 − 2.74387i
307.12 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i 3.35007 1.00000 2.47992 + 0.921964i −0.707107 − 0.707107i 1.00000i 2.36886 − 2.36886i
307.13 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i −1.77140 1.00000 −1.44420 + 2.21682i −0.707107 − 0.707107i 1.00000i −1.25257 + 1.25257i
307.14 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i 1.19809 1.00000 2.55829 − 0.674664i −0.707107 − 0.707107i 1.00000i 0.847181 − 0.847181i
307.15 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i −0.948587 1.00000 −2.30612 − 1.29684i −0.707107 − 0.707107i 1.00000i −0.670752 + 0.670752i
307.16 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i 0.0557326 1.00000 −0.842198 + 2.50813i −0.707107 − 0.707107i 1.00000i 0.0394089 − 0.0394089i
307.17 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i −2.82382 1.00000 1.44991 − 2.21309i −0.707107 − 0.707107i 1.00000i −1.99674 + 1.99674i
307.18 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i 2.30923 1.00000 0.637885 − 2.56770i −0.707107 − 0.707107i 1.00000i 1.63287 − 1.63287i
307.19 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i −3.17981 1.00000 −2.64392 + 0.0985113i −0.707107 − 0.707107i 1.00000i −2.24847 + 2.24847i
307.20 0.707107 − 0.707107i 0.707107 + 0.707107i − 1.00000i −3.48413 1.00000 2.21178 + 1.45191i −0.707107 − 0.707107i 1.00000i −2.46365 + 2.46365i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
203.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1218.2.m.b yes 40
7.b odd 2 1 1218.2.m.a ✓ 40
29.c odd 4 1 1218.2.m.a ✓ 40
203.g even 4 1 inner 1218.2.m.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1218.2.m.a ✓ 40 7.b odd 2 1
1218.2.m.a ✓ 40 29.c odd 4 1
1218.2.m.b yes 40 1.a even 1 1 trivial
1218.2.m.b yes 40 203.g even 4 1 inner