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Properties

Label 600.2.v.b
Level $600$
Weight $2$
Character orbit 600.v
Analytic conductor $4.791$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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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] = [600,2,Mod(43,600)] 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("600.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.v (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 120)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{6} + 12 q^{8} + 8 q^{12} - 20 q^{16} - 8 q^{17} + 28 q^{22} - 16 q^{26} - 4 q^{28} - 20 q^{32} + 4 q^{36} - 40 q^{38} - 20 q^{42} + 32 q^{43} + 48 q^{46} - 16 q^{48} - 16 q^{51} + 48 q^{52}+ \cdots + 88 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.31581 + 0.518298i −0.707107 − 0.707107i 1.46273 − 1.36397i 0 1.29691 + 0.563929i −0.645414 − 0.645414i −1.21775 + 2.55286i 1.00000i 0
43.2 −1.25738 + 0.647304i 0.707107 + 0.707107i 1.16200 − 1.62781i 0 −1.34681 − 0.431387i 1.45533 + 1.45533i −0.407381 + 2.79894i 1.00000i 0
43.3 −1.08304 − 0.909406i −0.707107 − 0.707107i 0.345961 + 1.96985i 0 0.122779 + 1.40887i 2.10796 + 2.10796i 1.41670 − 2.44805i 1.00000i 0
43.4 −0.647304 + 1.25738i 0.707107 + 0.707107i −1.16200 − 1.62781i 0 −1.34681 + 0.431387i −1.45533 − 1.45533i 2.79894 − 0.407381i 1.00000i 0
43.5 −0.624608 − 1.26880i 0.707107 + 0.707107i −1.21973 + 1.58501i 0 0.455516 − 1.33884i −1.93078 − 1.93078i 2.77292 + 0.557590i 1.00000i 0
43.6 −0.518298 + 1.31581i −0.707107 − 0.707107i −1.46273 − 1.36397i 0 1.29691 − 0.563929i 0.645414 + 0.645414i 2.55286 − 1.21775i 1.00000i 0
43.7 −0.109339 − 1.40998i −0.707107 − 0.707107i −1.97609 + 0.308331i 0 −0.919693 + 1.07432i 1.21782 + 1.21782i 0.650804 + 2.75254i 1.00000i 0
43.8 0.804501 − 1.16309i 0.707107 + 0.707107i −0.705556 − 1.87141i 0 1.39130 − 0.253561i 3.43671 + 3.43671i −2.74424 − 0.684930i 1.00000i 0
43.9 0.909406 + 1.08304i −0.707107 − 0.707107i −0.345961 + 1.96985i 0 0.122779 − 1.40887i −2.10796 − 2.10796i −2.44805 + 1.41670i 1.00000i 0
43.10 1.16309 − 0.804501i 0.707107 + 0.707107i 0.705556 − 1.87141i 0 1.39130 + 0.253561i −3.43671 − 3.43671i −0.684930 − 2.74424i 1.00000i 0
43.11 1.26880 + 0.624608i 0.707107 + 0.707107i 1.21973 + 1.58501i 0 0.455516 + 1.33884i 1.93078 + 1.93078i 0.557590 + 2.77292i 1.00000i 0
43.12 1.40998 + 0.109339i −0.707107 − 0.707107i 1.97609 + 0.308331i 0 −0.919693 − 1.07432i −1.21782 − 1.21782i 2.75254 + 0.650804i 1.00000i 0
307.1 −1.31581 − 0.518298i −0.707107 + 0.707107i 1.46273 + 1.36397i 0 1.29691 − 0.563929i −0.645414 + 0.645414i −1.21775 − 2.55286i − 1.00000i 0
307.2 −1.25738 − 0.647304i 0.707107 − 0.707107i 1.16200 + 1.62781i 0 −1.34681 + 0.431387i 1.45533 − 1.45533i −0.407381 − 2.79894i − 1.00000i 0
307.3 −1.08304 + 0.909406i −0.707107 + 0.707107i 0.345961 − 1.96985i 0 0.122779 − 1.40887i 2.10796 − 2.10796i 1.41670 + 2.44805i − 1.00000i 0
307.4 −0.647304 − 1.25738i 0.707107 − 0.707107i −1.16200 + 1.62781i 0 −1.34681 − 0.431387i −1.45533 + 1.45533i 2.79894 + 0.407381i − 1.00000i 0
307.5 −0.624608 + 1.26880i 0.707107 − 0.707107i −1.21973 − 1.58501i 0 0.455516 + 1.33884i −1.93078 + 1.93078i 2.77292 − 0.557590i − 1.00000i 0
307.6 −0.518298 − 1.31581i −0.707107 + 0.707107i −1.46273 + 1.36397i 0 1.29691 + 0.563929i 0.645414 − 0.645414i 2.55286 + 1.21775i − 1.00000i 0
307.7 −0.109339 + 1.40998i −0.707107 + 0.707107i −1.97609 − 0.308331i 0 −0.919693 − 1.07432i 1.21782 − 1.21782i 0.650804 − 2.75254i − 1.00000i 0
307.8 0.804501 + 1.16309i 0.707107 − 0.707107i −0.705556 + 1.87141i 0 1.39130 + 0.253561i 3.43671 − 3.43671i −2.74424 + 0.684930i − 1.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.d odd 2 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.v.b 24
4.b odd 2 1 2400.2.bh.b 24
5.b even 2 1 120.2.v.a ✓ 24
5.c odd 4 1 120.2.v.a ✓ 24
5.c odd 4 1 inner 600.2.v.b 24
8.b even 2 1 2400.2.bh.b 24
8.d odd 2 1 inner 600.2.v.b 24
15.d odd 2 1 360.2.w.e 24
15.e even 4 1 360.2.w.e 24
20.d odd 2 1 480.2.bh.a 24
20.e even 4 1 480.2.bh.a 24
20.e even 4 1 2400.2.bh.b 24
40.e odd 2 1 120.2.v.a ✓ 24
40.f even 2 1 480.2.bh.a 24
40.i odd 4 1 480.2.bh.a 24
40.i odd 4 1 2400.2.bh.b 24
40.k even 4 1 120.2.v.a ✓ 24
40.k even 4 1 inner 600.2.v.b 24
60.h even 2 1 1440.2.bi.e 24
60.l odd 4 1 1440.2.bi.e 24
120.i odd 2 1 1440.2.bi.e 24
120.m even 2 1 360.2.w.e 24
120.q odd 4 1 360.2.w.e 24
120.w even 4 1 1440.2.bi.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.v.a ✓ 24 5.b even 2 1
120.2.v.a ✓ 24 5.c odd 4 1
120.2.v.a ✓ 24 40.e odd 2 1
120.2.v.a ✓ 24 40.k even 4 1
360.2.w.e 24 15.d odd 2 1
360.2.w.e 24 15.e even 4 1
360.2.w.e 24 120.m even 2 1
360.2.w.e 24 120.q odd 4 1
480.2.bh.a 24 20.d odd 2 1
480.2.bh.a 24 20.e even 4 1
480.2.bh.a 24 40.f even 2 1
480.2.bh.a 24 40.i odd 4 1
600.2.v.b 24 1.a even 1 1 trivial
600.2.v.b 24 5.c odd 4 1 inner
600.2.v.b 24 8.d odd 2 1 inner
600.2.v.b 24 40.k even 4 1 inner
1440.2.bi.e 24 60.h even 2 1
1440.2.bi.e 24 60.l odd 4 1
1440.2.bi.e 24 120.i odd 2 1
1440.2.bi.e 24 120.w even 4 1
2400.2.bh.b 24 4.b odd 2 1
2400.2.bh.b 24 8.b even 2 1
2400.2.bh.b 24 20.e even 4 1
2400.2.bh.b 24 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 720T_{7}^{20} + 98656T_{7}^{16} + 4752640T_{7}^{12} + 81309952T_{7}^{8} + 440926208T_{7}^{4} + 268435456 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display