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Properties

Label 240.9.c.f
Level $240$
Weight $9$
Character orbit 240.c
Analytic conductor $97.771$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [240,9,Mod(209,240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 9, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("240.209"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,0,0,0,0,0,2528] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(97.7708664147\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 2528 q^{9} - 132352 q^{15} + 116176 q^{21} + 56976 q^{25} - 1395648 q^{31} - 6888832 q^{39} - 4287056 q^{45} - 30813552 q^{49} + 22815168 q^{51} + 6062784 q^{55} + 14031936 q^{61} + 2522608 q^{69}+ \cdots + 21719360 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −80.8401 − 5.08629i 0 615.961 + 105.912i 0 − 2674.51i 0 6509.26 + 822.353i 0
209.2 0 −80.8401 + 5.08629i 0 615.961 − 105.912i 0 2674.51i 0 6509.26 − 822.353i 0
209.3 0 −77.3427 − 24.0647i 0 −566.843 + 263.275i 0 3447.79i 0 5402.78 + 3722.46i 0
209.4 0 −77.3427 + 24.0647i 0 −566.843 − 263.275i 0 − 3447.79i 0 5402.78 − 3722.46i 0
209.5 0 −76.6243 − 26.2625i 0 −461.945 − 420.989i 0 − 982.121i 0 5181.56 + 4024.69i 0
209.6 0 −76.6243 + 26.2625i 0 −461.945 + 420.989i 0 982.121i 0 5181.56 − 4024.69i 0
209.7 0 −76.4075 − 26.8867i 0 266.192 + 565.479i 0 − 424.367i 0 5115.21 + 4108.69i 0
209.8 0 −76.4075 + 26.8867i 0 266.192 − 565.479i 0 424.367i 0 5115.21 − 4108.69i 0
209.9 0 −63.1416 − 50.7359i 0 94.4932 − 617.816i 0 2120.56i 0 1412.73 + 6407.10i 0
209.10 0 −63.1416 + 50.7359i 0 94.4932 + 617.816i 0 − 2120.56i 0 1412.73 − 6407.10i 0
209.11 0 −58.3695 − 56.1605i 0 617.515 − 96.4387i 0 − 3509.51i 0 252.988 + 6556.12i 0
209.12 0 −58.3695 + 56.1605i 0 617.515 + 96.4387i 0 3509.51i 0 252.988 − 6556.12i 0
209.13 0 −51.8771 − 62.2074i 0 254.792 + 570.707i 0 4202.69i 0 −1178.52 + 6454.29i 0
209.14 0 −51.8771 + 62.2074i 0 254.792 − 570.707i 0 − 4202.69i 0 −1178.52 − 6454.29i 0
209.15 0 −48.4629 − 64.9026i 0 −236.231 + 578.636i 0 − 3308.40i 0 −1863.70 + 6290.74i 0
209.16 0 −48.4629 + 64.9026i 0 −236.231 − 578.636i 0 3308.40i 0 −1863.70 − 6290.74i 0
209.17 0 −47.7055 − 65.4613i 0 326.077 − 533.197i 0 535.371i 0 −2009.38 + 6245.73i 0
209.18 0 −47.7055 + 65.4613i 0 326.077 + 533.197i 0 − 535.371i 0 −2009.38 − 6245.73i 0
209.19 0 −24.3113 − 77.2655i 0 −547.246 − 301.906i 0 − 1536.33i 0 −5378.92 + 3756.85i 0
209.20 0 −24.3113 + 77.2655i 0 −547.246 + 301.906i 0 1536.33i 0 −5378.92 − 3756.85i 0
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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.9.c.f 48
3.b odd 2 1 inner 240.9.c.f 48
4.b odd 2 1 120.9.c.a ✓ 48
5.b even 2 1 inner 240.9.c.f 48
12.b even 2 1 120.9.c.a ✓ 48
15.d odd 2 1 inner 240.9.c.f 48
20.d odd 2 1 120.9.c.a ✓ 48
60.h even 2 1 120.9.c.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.9.c.a ✓ 48 4.b odd 2 1
120.9.c.a ✓ 48 12.b even 2 1
120.9.c.a ✓ 48 20.d odd 2 1
120.9.c.a ✓ 48 60.h even 2 1
240.9.c.f 48 1.a even 1 1 trivial
240.9.c.f 48 3.b odd 2 1 inner
240.9.c.f 48 5.b even 2 1 inner
240.9.c.f 48 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{24} + 76881000 T_{7}^{22} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
\( T_{17}^{24} - 84688503264 T_{17}^{22} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display