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Properties

Label 600.2.w.j
Level $600$
Weight $2$
Character orbit 600.w
Analytic conductor $4.791$
Analytic rank $0$
Dimension $32$
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(293,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.w (of order \(4\), degree \(2\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 4 q^{6} + 8 q^{12} + 28 q^{16} + 20 q^{18} + 52 q^{22} - 12 q^{28} - 32 q^{31} - 8 q^{33} - 20 q^{36} - 16 q^{42} + 24 q^{46} - 44 q^{48} - 8 q^{52} + 16 q^{57} - 28 q^{58} - 48 q^{63} + 16 q^{66} - 32 q^{72} + 64 q^{73} - 88 q^{76} - 64 q^{78} + 48 q^{81} - 64 q^{82} + 8 q^{87} + 52 q^{88} - 52 q^{96} - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1 −1.41107 − 0.0941764i 1.68122 + 0.416519i 1.98226 + 0.265780i 0 −2.33310 − 0.746071i 0.361989 − 0.361989i −2.77209 − 0.561717i 2.65302 + 1.40052i 0
293.2 −1.39193 − 0.250043i −1.01856 + 1.40091i 1.87496 + 0.696087i 0 1.76805 − 1.69529i 2.29041 − 2.29041i −2.43576 − 1.43773i −0.925085 − 2.85381i 0
293.3 −1.30986 + 0.533177i 0.667305 − 1.59834i 1.43144 − 1.39677i 0 −0.0218716 + 2.44939i −0.582772 + 0.582772i −1.13026 + 2.59278i −2.10941 − 2.13317i 0
293.4 −1.11940 − 0.864261i −0.170116 − 1.72368i 0.506107 + 1.93490i 0 −1.29928 + 2.07651i −2.06963 + 2.06963i 1.10573 − 2.60334i −2.94212 + 0.586449i 0
293.5 −0.864261 − 1.11940i 0.170116 + 1.72368i −0.506107 + 1.93490i 0 1.78246 − 1.68013i −2.06963 + 2.06963i 2.60334 − 1.10573i −2.94212 + 0.586449i 0
293.6 −0.533177 + 1.30986i −1.59834 + 0.667305i −1.43144 − 1.39677i 0 −0.0218716 − 2.44939i −0.582772 + 0.582772i 2.59278 − 1.13026i 2.10941 − 2.13317i 0
293.7 −0.250043 − 1.39193i 1.01856 − 1.40091i −1.87496 + 0.696087i 0 −2.20465 − 1.06748i 2.29041 − 2.29041i 1.43773 + 2.43576i −0.925085 − 2.85381i 0
293.8 −0.0941764 − 1.41107i −1.68122 − 0.416519i −1.98226 + 0.265780i 0 −0.429408 + 2.41156i 0.361989 − 0.361989i 0.561717 + 2.77209i 2.65302 + 1.40052i 0
293.9 0.0941764 + 1.41107i 0.416519 + 1.68122i −1.98226 + 0.265780i 0 −2.33310 + 0.746071i 0.361989 − 0.361989i −0.561717 − 2.77209i −2.65302 + 1.40052i 0
293.10 0.250043 + 1.39193i 1.40091 − 1.01856i −1.87496 + 0.696087i 0 1.76805 + 1.69529i 2.29041 − 2.29041i −1.43773 − 2.43576i 0.925085 − 2.85381i 0
293.11 0.533177 − 1.30986i −0.667305 + 1.59834i −1.43144 − 1.39677i 0 1.73781 + 1.72627i −0.582772 + 0.582772i −2.59278 + 1.13026i −2.10941 − 2.13317i 0
293.12 0.864261 + 1.11940i −1.72368 − 0.170116i −0.506107 + 1.93490i 0 −1.29928 − 2.07651i −2.06963 + 2.06963i −2.60334 + 1.10573i 2.94212 + 0.586449i 0
293.13 1.11940 + 0.864261i 1.72368 + 0.170116i 0.506107 + 1.93490i 0 1.78246 + 1.68013i −2.06963 + 2.06963i −1.10573 + 2.60334i 2.94212 + 0.586449i 0
293.14 1.30986 − 0.533177i 1.59834 − 0.667305i 1.43144 − 1.39677i 0 1.73781 − 1.72627i −0.582772 + 0.582772i 1.13026 − 2.59278i 2.10941 − 2.13317i 0
293.15 1.39193 + 0.250043i −1.40091 + 1.01856i 1.87496 + 0.696087i 0 −2.20465 + 1.06748i 2.29041 − 2.29041i 2.43576 + 1.43773i 0.925085 − 2.85381i 0
293.16 1.41107 + 0.0941764i −0.416519 − 1.68122i 1.98226 + 0.265780i 0 −0.429408 − 2.41156i 0.361989 − 0.361989i 2.77209 + 0.561717i −2.65302 + 1.40052i 0
557.1 −1.41107 + 0.0941764i 1.68122 − 0.416519i 1.98226 − 0.265780i 0 −2.33310 + 0.746071i 0.361989 + 0.361989i −2.77209 + 0.561717i 2.65302 − 1.40052i 0
557.2 −1.39193 + 0.250043i −1.01856 − 1.40091i 1.87496 − 0.696087i 0 1.76805 + 1.69529i 2.29041 + 2.29041i −2.43576 + 1.43773i −0.925085 + 2.85381i 0
557.3 −1.30986 − 0.533177i 0.667305 + 1.59834i 1.43144 + 1.39677i 0 −0.0218716 − 2.44939i −0.582772 − 0.582772i −1.13026 − 2.59278i −2.10941 + 2.13317i 0
557.4 −1.11940 + 0.864261i −0.170116 + 1.72368i 0.506107 − 1.93490i 0 −1.29928 − 2.07651i −2.06963 − 2.06963i 1.10573 + 2.60334i −2.94212 − 0.586449i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
24.h odd 2 1 inner
40.i odd 4 1 inner
120.w 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.w.j 32
3.b odd 2 1 inner 600.2.w.j 32
5.b even 2 1 120.2.w.c ✓ 32
5.c odd 4 1 120.2.w.c ✓ 32
5.c odd 4 1 inner 600.2.w.j 32
8.b even 2 1 inner 600.2.w.j 32
15.d odd 2 1 120.2.w.c ✓ 32
15.e even 4 1 120.2.w.c ✓ 32
15.e even 4 1 inner 600.2.w.j 32
20.d odd 2 1 480.2.bi.c 32
20.e even 4 1 480.2.bi.c 32
24.h odd 2 1 inner 600.2.w.j 32
40.e odd 2 1 480.2.bi.c 32
40.f even 2 1 120.2.w.c ✓ 32
40.i odd 4 1 120.2.w.c ✓ 32
40.i odd 4 1 inner 600.2.w.j 32
40.k even 4 1 480.2.bi.c 32
60.h even 2 1 480.2.bi.c 32
60.l odd 4 1 480.2.bi.c 32
120.i odd 2 1 120.2.w.c ✓ 32
120.m even 2 1 480.2.bi.c 32
120.q odd 4 1 480.2.bi.c 32
120.w even 4 1 120.2.w.c ✓ 32
120.w even 4 1 inner 600.2.w.j 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.w.c ✓ 32 5.b even 2 1
120.2.w.c ✓ 32 5.c odd 4 1
120.2.w.c ✓ 32 15.d odd 2 1
120.2.w.c ✓ 32 15.e even 4 1
120.2.w.c ✓ 32 40.f even 2 1
120.2.w.c ✓ 32 40.i odd 4 1
120.2.w.c ✓ 32 120.i odd 2 1
120.2.w.c ✓ 32 120.w even 4 1
480.2.bi.c 32 20.d odd 2 1
480.2.bi.c 32 20.e even 4 1
480.2.bi.c 32 40.e odd 2 1
480.2.bi.c 32 40.k even 4 1
480.2.bi.c 32 60.h even 2 1
480.2.bi.c 32 60.l odd 4 1
480.2.bi.c 32 120.m even 2 1
480.2.bi.c 32 120.q odd 4 1
600.2.w.j 32 1.a even 1 1 trivial
600.2.w.j 32 3.b odd 2 1 inner
600.2.w.j 32 5.c odd 4 1 inner
600.2.w.j 32 8.b even 2 1 inner
600.2.w.j 32 15.e even 4 1 inner
600.2.w.j 32 24.h odd 2 1 inner
600.2.w.j 32 40.i odd 4 1 inner
600.2.w.j 32 120.w even 4 1 inner

Hecke kernels

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

\( T_{7}^{8} + 4T_{7}^{5} + 92T_{7}^{4} + 40T_{7}^{3} + 8T_{7}^{2} - 16T_{7} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 26T_{11}^{6} + 208T_{11}^{4} - 544T_{11}^{2} + 128 \) Copy content Toggle raw display
\( T_{17}^{16} + 1420T_{17}^{12} + 639792T_{17}^{8} + 93389888T_{17}^{4} + 220463104 \) Copy content Toggle raw display