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Properties

Label 600.2.m.e
Level $600$
Weight $2$
Character orbit 600.m
Analytic conductor $4.791$
Analytic rank $0$
Dimension $24$
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] = [600,2,Mod(299,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.299");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.m (of order \(2\), degree \(1\), not 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} + 14 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} + 14 q^{6} + 4 q^{9} - 12 q^{16} + 8 q^{19} - 10 q^{24} + 4 q^{34} + 38 q^{36} - 32 q^{46} + 72 q^{49} - 60 q^{51} + 60 q^{54} - 20 q^{64} + 14 q^{66} - 76 q^{76} - 20 q^{81} + 68 q^{84} - 48 q^{91} - 56 q^{94} - 62 q^{96} - 116 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1 −1.13191 − 0.847808i −1.71500 − 0.242431i 0.562443 + 1.91929i 0 1.73569 + 1.72840i 3.08957 0.990551 − 2.64930i 2.88245 + 0.831539i 0
299.2 −1.13191 − 0.847808i −1.71500 + 0.242431i 0.562443 + 1.91929i 0 2.14676 + 1.17958i −3.08957 0.990551 − 2.64930i 2.88245 − 0.831539i 0
299.3 −1.13191 + 0.847808i −1.71500 − 0.242431i 0.562443 − 1.91929i 0 2.14676 − 1.17958i −3.08957 0.990551 + 2.64930i 2.88245 + 0.831539i 0
299.4 −1.13191 + 0.847808i −1.71500 + 0.242431i 0.562443 − 1.91929i 0 1.73569 − 1.72840i 3.08957 0.990551 + 2.64930i 2.88245 − 0.831539i 0
299.5 −0.639662 − 1.26128i 0.730070 − 1.57067i −1.18166 + 1.61359i 0 −2.44805 + 0.0838735i 1.25539 2.79106 + 0.458259i −1.93400 − 2.29339i 0
299.6 −0.639662 − 1.26128i 0.730070 + 1.57067i −1.18166 + 1.61359i 0 1.51406 − 1.92552i −1.25539 2.79106 + 0.458259i −1.93400 + 2.29339i 0
299.7 −0.639662 + 1.26128i 0.730070 − 1.57067i −1.18166 − 1.61359i 0 1.51406 + 1.92552i −1.25539 2.79106 − 0.458259i −1.93400 − 2.29339i 0
299.8 −0.639662 + 1.26128i 0.730070 + 1.57067i −1.18166 − 1.61359i 0 −2.44805 − 0.0838735i 1.25539 2.79106 − 0.458259i −1.93400 + 2.29339i 0
299.9 −0.244153 − 1.39298i −1.12950 − 1.31310i −1.88078 + 0.680200i 0 −1.55335 + 1.89397i 4.34495 1.40670 + 2.45381i −0.448458 + 2.96629i 0
299.10 −0.244153 − 1.39298i −1.12950 + 1.31310i −1.88078 + 0.680200i 0 2.10489 + 1.25277i −4.34495 1.40670 + 2.45381i −0.448458 − 2.96629i 0
299.11 −0.244153 + 1.39298i −1.12950 − 1.31310i −1.88078 − 0.680200i 0 2.10489 − 1.25277i −4.34495 1.40670 − 2.45381i −0.448458 + 2.96629i 0
299.12 −0.244153 + 1.39298i −1.12950 + 1.31310i −1.88078 − 0.680200i 0 −1.55335 − 1.89397i 4.34495 1.40670 − 2.45381i −0.448458 − 2.96629i 0
299.13 0.244153 − 1.39298i 1.12950 − 1.31310i −1.88078 − 0.680200i 0 −1.55335 − 1.89397i −4.34495 −1.40670 + 2.45381i −0.448458 − 2.96629i 0
299.14 0.244153 − 1.39298i 1.12950 + 1.31310i −1.88078 − 0.680200i 0 2.10489 − 1.25277i 4.34495 −1.40670 + 2.45381i −0.448458 + 2.96629i 0
299.15 0.244153 + 1.39298i 1.12950 − 1.31310i −1.88078 + 0.680200i 0 2.10489 + 1.25277i 4.34495 −1.40670 − 2.45381i −0.448458 − 2.96629i 0
299.16 0.244153 + 1.39298i 1.12950 + 1.31310i −1.88078 + 0.680200i 0 −1.55335 + 1.89397i −4.34495 −1.40670 − 2.45381i −0.448458 + 2.96629i 0
299.17 0.639662 − 1.26128i −0.730070 − 1.57067i −1.18166 − 1.61359i 0 −2.44805 − 0.0838735i −1.25539 −2.79106 + 0.458259i −1.93400 + 2.29339i 0
299.18 0.639662 − 1.26128i −0.730070 + 1.57067i −1.18166 − 1.61359i 0 1.51406 + 1.92552i 1.25539 −2.79106 + 0.458259i −1.93400 − 2.29339i 0
299.19 0.639662 + 1.26128i −0.730070 − 1.57067i −1.18166 + 1.61359i 0 1.51406 − 1.92552i 1.25539 −2.79106 − 0.458259i −1.93400 + 2.29339i 0
299.20 0.639662 + 1.26128i −0.730070 + 1.57067i −1.18166 + 1.61359i 0 −2.44805 + 0.0838735i −1.25539 −2.79106 − 0.458259i −1.93400 − 2.29339i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.24
Significant digits:
Format:

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
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.m.e 24
3.b odd 2 1 inner 600.2.m.e 24
4.b odd 2 1 2400.2.m.e 24
5.b even 2 1 inner 600.2.m.e 24
5.c odd 4 1 600.2.b.g ✓ 12
5.c odd 4 1 600.2.b.h yes 12
8.b even 2 1 2400.2.m.e 24
8.d odd 2 1 inner 600.2.m.e 24
12.b even 2 1 2400.2.m.e 24
15.d odd 2 1 inner 600.2.m.e 24
15.e even 4 1 600.2.b.g ✓ 12
15.e even 4 1 600.2.b.h yes 12
20.d odd 2 1 2400.2.m.e 24
20.e even 4 1 2400.2.b.g 12
20.e even 4 1 2400.2.b.h 12
24.f even 2 1 inner 600.2.m.e 24
24.h odd 2 1 2400.2.m.e 24
40.e odd 2 1 inner 600.2.m.e 24
40.f even 2 1 2400.2.m.e 24
40.i odd 4 1 2400.2.b.g 12
40.i odd 4 1 2400.2.b.h 12
40.k even 4 1 600.2.b.g ✓ 12
40.k even 4 1 600.2.b.h yes 12
60.h even 2 1 2400.2.m.e 24
60.l odd 4 1 2400.2.b.g 12
60.l odd 4 1 2400.2.b.h 12
120.i odd 2 1 2400.2.m.e 24
120.m even 2 1 inner 600.2.m.e 24
120.q odd 4 1 600.2.b.g ✓ 12
120.q odd 4 1 600.2.b.h yes 12
120.w even 4 1 2400.2.b.g 12
120.w even 4 1 2400.2.b.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.b.g ✓ 12 5.c odd 4 1
600.2.b.g ✓ 12 15.e even 4 1
600.2.b.g ✓ 12 40.k even 4 1
600.2.b.g ✓ 12 120.q odd 4 1
600.2.b.h yes 12 5.c odd 4 1
600.2.b.h yes 12 15.e even 4 1
600.2.b.h yes 12 40.k even 4 1
600.2.b.h yes 12 120.q odd 4 1
600.2.m.e 24 1.a even 1 1 trivial
600.2.m.e 24 3.b odd 2 1 inner
600.2.m.e 24 5.b even 2 1 inner
600.2.m.e 24 8.d odd 2 1 inner
600.2.m.e 24 15.d odd 2 1 inner
600.2.m.e 24 24.f even 2 1 inner
600.2.m.e 24 40.e odd 2 1 inner
600.2.m.e 24 120.m even 2 1 inner
2400.2.b.g 12 20.e even 4 1
2400.2.b.g 12 40.i odd 4 1
2400.2.b.g 12 60.l odd 4 1
2400.2.b.g 12 120.w even 4 1
2400.2.b.h 12 20.e even 4 1
2400.2.b.h 12 40.i odd 4 1
2400.2.b.h 12 60.l odd 4 1
2400.2.b.h 12 120.w even 4 1
2400.2.m.e 24 4.b odd 2 1
2400.2.m.e 24 8.b even 2 1
2400.2.m.e 24 12.b even 2 1
2400.2.m.e 24 20.d odd 2 1
2400.2.m.e 24 24.h odd 2 1
2400.2.m.e 24 40.f even 2 1
2400.2.m.e 24 60.h even 2 1
2400.2.m.e 24 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{6} - 30T_{7}^{4} + 225T_{7}^{2} - 284 \) Copy content Toggle raw display
\( T_{11}^{6} + 19T_{11}^{4} + 112T_{11}^{2} + 200 \) Copy content Toggle raw display
\( T_{29}^{6} - 140T_{29}^{4} + 5752T_{29}^{2} - 56800 \) Copy content Toggle raw display