Properties

Label 300.3.l.h
Level $300$
Weight $3$
Character orbit 300.l
Analytic conductor $8.174$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,3,Mod(107,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.107");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.l (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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 12 q^{6} + 24 q^{16} + 32 q^{21} - 52 q^{36} - 112 q^{46} - 352 q^{61} + 316 q^{66} + 808 q^{76} - 720 q^{81} - 908 q^{96}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.99335 + 0.162980i −2.88075 + 0.837414i 3.94688 0.649751i 0 5.60586 2.13876i −1.76134 + 1.76134i −7.76160 + 1.93844i 7.59748 4.82477i 0
107.2 −1.99335 + 0.162980i −0.837414 + 2.88075i 3.94688 0.649751i 0 1.19975 5.87883i 1.76134 1.76134i −7.76160 + 1.93844i −7.59748 4.82477i 0
107.3 −1.95438 0.424734i 1.76787 2.42377i 3.63920 + 1.66018i 0 −4.48454 + 3.98609i 8.59996 8.59996i −6.40725 4.79032i −2.74930 8.56979i 0
107.4 −1.95438 0.424734i 2.42377 1.76787i 3.63920 + 1.66018i 0 −5.48784 + 2.42562i −8.59996 + 8.59996i −6.40725 4.79032i 2.74930 8.56979i 0
107.5 −1.72739 1.00803i −2.43345 1.75452i 1.96775 + 3.48252i 0 2.43491 + 5.48372i 3.08744 3.08744i 0.111402 7.99922i 2.84335 + 8.53905i 0
107.6 −1.72739 1.00803i 1.75452 + 2.43345i 1.96775 + 3.48252i 0 −0.577744 5.97212i −3.08744 + 3.08744i 0.111402 7.99922i −2.84335 + 8.53905i 0
107.7 −1.45557 + 1.37162i −0.658656 2.92680i 0.237343 3.99295i 0 4.97316 + 3.35673i −4.73351 + 4.73351i 5.13133 + 6.13755i −8.13235 + 3.85551i 0
107.8 −1.45557 + 1.37162i 2.92680 + 0.658656i 0.237343 3.99295i 0 −5.16358 + 3.05573i 4.73351 4.73351i 5.13133 + 6.13755i 8.13235 + 3.85551i 0
107.9 −1.37162 + 1.45557i −2.92680 0.658656i −0.237343 3.99295i 0 4.97316 3.35673i −4.73351 + 4.73351i 6.13755 + 5.13133i 8.13235 + 3.85551i 0
107.10 −1.37162 + 1.45557i 0.658656 + 2.92680i −0.237343 3.99295i 0 −5.16358 3.05573i 4.73351 4.73351i 6.13755 + 5.13133i −8.13235 + 3.85551i 0
107.11 −1.00803 1.72739i −2.43345 1.75452i −1.96775 + 3.48252i 0 −0.577744 + 5.97212i 3.08744 3.08744i 7.99922 0.111402i 2.84335 + 8.53905i 0
107.12 −1.00803 1.72739i 1.75452 + 2.43345i −1.96775 + 3.48252i 0 2.43491 5.48372i −3.08744 + 3.08744i 7.99922 0.111402i −2.84335 + 8.53905i 0
107.13 −0.424734 1.95438i 1.76787 2.42377i −3.63920 + 1.66018i 0 −5.48784 2.42562i 8.59996 8.59996i 4.79032 + 6.40725i −2.74930 8.56979i 0
107.14 −0.424734 1.95438i 2.42377 1.76787i −3.63920 + 1.66018i 0 −4.48454 3.98609i −8.59996 + 8.59996i 4.79032 + 6.40725i 2.74930 8.56979i 0
107.15 −0.162980 + 1.99335i 0.837414 2.88075i −3.94688 0.649751i 0 5.60586 + 2.13876i −1.76134 + 1.76134i 1.93844 7.76160i −7.59748 4.82477i 0
107.16 −0.162980 + 1.99335i 2.88075 0.837414i −3.94688 0.649751i 0 1.19975 + 5.87883i 1.76134 1.76134i 1.93844 7.76160i 7.59748 4.82477i 0
107.17 0.162980 1.99335i −2.88075 + 0.837414i −3.94688 0.649751i 0 1.19975 + 5.87883i −1.76134 + 1.76134i −1.93844 + 7.76160i 7.59748 4.82477i 0
107.18 0.162980 1.99335i −0.837414 + 2.88075i −3.94688 0.649751i 0 5.60586 + 2.13876i 1.76134 1.76134i −1.93844 + 7.76160i −7.59748 4.82477i 0
107.19 0.424734 + 1.95438i −2.42377 + 1.76787i −3.63920 + 1.66018i 0 −4.48454 3.98609i 8.59996 8.59996i −4.79032 6.40725i 2.74930 8.56979i 0
107.20 0.424734 + 1.95438i −1.76787 + 2.42377i −3.63920 + 1.66018i 0 −5.48784 2.42562i −8.59996 + 8.59996i −4.79032 6.40725i −2.74930 8.56979i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
12.b even 2 1 inner
15.d odd 2 1 inner
15.e even 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner
60.h even 2 1 inner
60.l odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.l.h 64
3.b odd 2 1 inner 300.3.l.h 64
4.b odd 2 1 inner 300.3.l.h 64
5.b even 2 1 inner 300.3.l.h 64
5.c odd 4 2 inner 300.3.l.h 64
12.b even 2 1 inner 300.3.l.h 64
15.d odd 2 1 inner 300.3.l.h 64
15.e even 4 2 inner 300.3.l.h 64
20.d odd 2 1 inner 300.3.l.h 64
20.e even 4 2 inner 300.3.l.h 64
60.h even 2 1 inner 300.3.l.h 64
60.l odd 4 2 inner 300.3.l.h 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.l.h 64 1.a even 1 1 trivial
300.3.l.h 64 3.b odd 2 1 inner
300.3.l.h 64 4.b odd 2 1 inner
300.3.l.h 64 5.b even 2 1 inner
300.3.l.h 64 5.c odd 4 2 inner
300.3.l.h 64 12.b even 2 1 inner
300.3.l.h 64 15.d odd 2 1 inner
300.3.l.h 64 15.e even 4 2 inner
300.3.l.h 64 20.d odd 2 1 inner
300.3.l.h 64 20.e even 4 2 inner
300.3.l.h 64 60.h even 2 1 inner
300.3.l.h 64 60.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{16} + 24290T_{7}^{12} + 53553793T_{7}^{8} + 17995252896T_{7}^{4} + 614781446400 \) Copy content Toggle raw display
\( T_{17}^{16} + 597042T_{17}^{12} + 109730300913T_{17}^{8} + 5774819750046144T_{17}^{4} + 313717924683878400 \) Copy content Toggle raw display
\( T_{19}^{8} - 2024T_{19}^{6} + 1426642T_{19}^{4} - 416756448T_{19}^{2} + 43164127845 \) Copy content Toggle raw display