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Properties

Label 1500.2.m.c
Level $1500$
Weight $2$
Character orbit 1500.m
Analytic conductor $11.978$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1500,2,Mod(301,1500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1500, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1500.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.m (of order \(5\), degree \(4\), 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: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} + 16 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} + 16 q^{7} - 6 q^{9} - 6 q^{11} - 4 q^{17} - 10 q^{19} - 4 q^{21} - 14 q^{23} - 6 q^{27} - 4 q^{29} + 6 q^{31} + 4 q^{33} + 8 q^{37} - 10 q^{41} + 56 q^{43} - 26 q^{47} + 56 q^{49} + 16 q^{51} + 32 q^{53} + 20 q^{57} + 36 q^{59} - 12 q^{61} - 4 q^{63} - 36 q^{67} - 4 q^{69} + 40 q^{71} - 32 q^{73} + 46 q^{77} - 8 q^{79} - 6 q^{81} + 6 q^{83} - 4 q^{87} - 30 q^{91} - 4 q^{93} - 48 q^{97} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1 0 0.309017 − 0.951057i 0 0 0 −3.54704 0 −0.809017 − 0.587785i 0
301.2 0 0.309017 − 0.951057i 0 0 0 −1.31873 0 −0.809017 − 0.587785i 0
301.3 0 0.309017 − 0.951057i 0 0 0 −0.595901 0 −0.809017 − 0.587785i 0
301.4 0 0.309017 − 0.951057i 0 0 0 1.04684 0 −0.809017 − 0.587785i 0
301.5 0 0.309017 − 0.951057i 0 0 0 3.78808 0 −0.809017 − 0.587785i 0
301.6 0 0.309017 − 0.951057i 0 0 0 4.62675 0 −0.809017 − 0.587785i 0
601.1 0 −0.809017 − 0.587785i 0 0 0 −4.13266 0 0.309017 + 0.951057i 0
601.2 0 −0.809017 − 0.587785i 0 0 0 −1.57893 0 0.309017 + 0.951057i 0
601.3 0 −0.809017 − 0.587785i 0 0 0 −0.957526 0 0.309017 + 0.951057i 0
601.4 0 −0.809017 − 0.587785i 0 0 0 2.44380 0 0.309017 + 0.951057i 0
601.5 0 −0.809017 − 0.587785i 0 0 0 3.80992 0 0.309017 + 0.951057i 0
601.6 0 −0.809017 − 0.587785i 0 0 0 4.41540 0 0.309017 + 0.951057i 0
901.1 0 −0.809017 + 0.587785i 0 0 0 −4.13266 0 0.309017 − 0.951057i 0
901.2 0 −0.809017 + 0.587785i 0 0 0 −1.57893 0 0.309017 − 0.951057i 0
901.3 0 −0.809017 + 0.587785i 0 0 0 −0.957526 0 0.309017 − 0.951057i 0
901.4 0 −0.809017 + 0.587785i 0 0 0 2.44380 0 0.309017 − 0.951057i 0
901.5 0 −0.809017 + 0.587785i 0 0 0 3.80992 0 0.309017 − 0.951057i 0
901.6 0 −0.809017 + 0.587785i 0 0 0 4.41540 0 0.309017 − 0.951057i 0
1201.1 0 0.309017 + 0.951057i 0 0 0 −3.54704 0 −0.809017 + 0.587785i 0
1201.2 0 0.309017 + 0.951057i 0 0 0 −1.31873 0 −0.809017 + 0.587785i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1500.2.m.c 24
5.b even 2 1 1500.2.m.d 24
5.c odd 4 1 300.2.o.a ✓ 24
5.c odd 4 1 1500.2.o.c 24
15.e even 4 1 900.2.w.c 24
25.d even 5 1 inner 1500.2.m.c 24
25.d even 5 1 7500.2.a.n 12
25.e even 10 1 1500.2.m.d 24
25.e even 10 1 7500.2.a.m 12
25.f odd 20 1 300.2.o.a ✓ 24
25.f odd 20 1 1500.2.o.c 24
25.f odd 20 2 7500.2.d.g 24
75.l even 20 1 900.2.w.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.o.a ✓ 24 5.c odd 4 1
300.2.o.a ✓ 24 25.f odd 20 1
900.2.w.c 24 15.e even 4 1
900.2.w.c 24 75.l even 20 1
1500.2.m.c 24 1.a even 1 1 trivial
1500.2.m.c 24 25.d even 5 1 inner
1500.2.m.d 24 5.b even 2 1
1500.2.m.d 24 25.e even 10 1
1500.2.o.c 24 5.c odd 4 1
1500.2.o.c 24 25.f odd 20 1
7500.2.a.m 12 25.e even 10 1
7500.2.a.n 12 25.d even 5 1
7500.2.d.g 24 25.f odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 8 T_{7}^{11} - 24 T_{7}^{10} + 300 T_{7}^{9} + 10 T_{7}^{8} - 3768 T_{7}^{7} + 2289 T_{7}^{6} + \cdots + 13136 \) acting on \(S_{2}^{\mathrm{new}}(1500, [\chi])\). Copy content Toggle raw display