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Properties

Label 300.3.l.g
Level $300$
Weight $3$
Character orbit 300.l
Analytic conductor $8.174$
Analytic rank $0$
Dimension $40$
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] = [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: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 60)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{6} + 20 q^{12} + 8 q^{13} - 36 q^{16} + 24 q^{18} - 24 q^{21} + 76 q^{22} + 84 q^{28} + 40 q^{33} + 172 q^{36} + 40 q^{37} - 236 q^{42} + 240 q^{46} - 196 q^{48} - 304 q^{52} + 72 q^{57} - 180 q^{58} + 48 q^{61} - 552 q^{66} + 600 q^{72} - 104 q^{73} - 736 q^{76} + 408 q^{78} + 72 q^{81} + 720 q^{82} + 580 q^{88} - 368 q^{93} + 884 q^{96} - 72 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.99497 + 0.141758i 2.17477 + 2.06649i 3.95981 − 0.565605i 0 −4.63154 − 3.81429i 5.18766 − 5.18766i −7.81952 + 1.68970i 0.459255 + 8.98827i 0
107.2 −1.84549 − 0.770813i −1.12501 + 2.78107i 2.81170 + 2.84506i 0 4.21989 − 4.26527i −4.75159 + 4.75159i −2.99596 − 7.41783i −6.46869 − 6.25748i 0
107.3 −1.81610 + 0.837725i 1.32197 − 2.69303i 2.59643 − 3.04278i 0 −0.144815 + 5.99825i −3.54241 + 3.54241i −2.16636 + 7.70110i −5.50478 − 7.12021i 0
107.4 −1.75394 + 0.961083i −2.86057 − 0.903948i 2.15264 − 3.37137i 0 5.88605 − 1.16377i 7.30016 − 7.30016i −0.535443 + 7.98206i 7.36576 + 5.17162i 0
107.5 −1.68375 − 1.07935i 0.130491 − 2.99716i 1.67002 + 3.63470i 0 −3.45469 + 4.90562i −1.91561 + 1.91561i 1.11122 − 7.92245i −8.96594 − 0.782204i 0
107.6 −1.07935 − 1.68375i 2.99716 − 0.130491i −1.67002 + 3.63470i 0 −3.45469 − 4.90562i 1.91561 − 1.91561i 7.92245 − 1.11122i 8.96594 − 0.782204i 0
107.7 −0.961083 + 1.75394i 2.86057 + 0.903948i −2.15264 − 3.37137i 0 −4.33472 + 4.14852i −7.30016 + 7.30016i 7.98206 − 0.535443i 7.36576 + 5.17162i 0
107.8 −0.837725 + 1.81610i −1.32197 + 2.69303i −2.59643 − 3.04278i 0 −3.78336 − 4.65685i 3.54241 − 3.54241i 7.70110 − 2.16636i −5.50478 − 7.12021i 0
107.9 −0.770813 − 1.84549i −2.78107 + 1.12501i −2.81170 + 2.84506i 0 4.21989 + 4.26527i 4.75159 − 4.75159i 7.41783 + 2.99596i 6.46869 − 6.25748i 0
107.10 −0.141758 + 1.99497i −2.17477 − 2.06649i −3.95981 − 0.565605i 0 4.43087 − 4.04566i −5.18766 + 5.18766i 1.68970 − 7.81952i 0.459255 + 8.98827i 0
107.11 0.141758 − 1.99497i −2.06649 − 2.17477i −3.95981 − 0.565605i 0 −4.63154 + 3.81429i −5.18766 + 5.18766i −1.68970 + 7.81952i −0.459255 + 8.98827i 0
107.12 0.770813 + 1.84549i 1.12501 − 2.78107i −2.81170 + 2.84506i 0 5.99962 − 0.0674770i 4.75159 − 4.75159i −7.41783 − 2.99596i −6.46869 − 6.25748i 0
107.13 0.837725 − 1.81610i 2.69303 − 1.32197i −2.59643 − 3.04278i 0 −0.144815 − 5.99825i 3.54241 − 3.54241i −7.70110 + 2.16636i 5.50478 − 7.12021i 0
107.14 0.961083 − 1.75394i 0.903948 + 2.86057i −2.15264 − 3.37137i 0 5.88605 + 1.16377i −7.30016 + 7.30016i −7.98206 + 0.535443i −7.36576 + 5.17162i 0
107.15 1.07935 + 1.68375i −0.130491 + 2.99716i −1.67002 + 3.63470i 0 −5.18731 + 3.01526i 1.91561 − 1.91561i −7.92245 + 1.11122i −8.96594 − 0.782204i 0
107.16 1.68375 + 1.07935i −2.99716 + 0.130491i 1.67002 + 3.63470i 0 −5.18731 − 3.01526i −1.91561 + 1.91561i −1.11122 + 7.92245i 8.96594 − 0.782204i 0
107.17 1.75394 − 0.961083i −0.903948 − 2.86057i 2.15264 − 3.37137i 0 −4.33472 − 4.14852i 7.30016 − 7.30016i 0.535443 − 7.98206i −7.36576 + 5.17162i 0
107.18 1.81610 − 0.837725i −2.69303 + 1.32197i 2.59643 − 3.04278i 0 −3.78336 + 4.65685i −3.54241 + 3.54241i 2.16636 − 7.70110i 5.50478 − 7.12021i 0
107.19 1.84549 + 0.770813i 2.78107 − 1.12501i 2.81170 + 2.84506i 0 5.99962 + 0.0674770i −4.75159 + 4.75159i 2.99596 + 7.41783i 6.46869 − 6.25748i 0
107.20 1.99497 − 0.141758i 2.06649 + 2.17477i 3.95981 − 0.565605i 0 4.43087 + 4.04566i 5.18766 − 5.18766i 7.81952 − 1.68970i −0.459255 + 8.98827i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.20
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.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.l.g 40
3.b odd 2 1 inner 300.3.l.g 40
4.b odd 2 1 inner 300.3.l.g 40
5.b even 2 1 60.3.l.a ✓ 40
5.c odd 4 1 60.3.l.a ✓ 40
5.c odd 4 1 inner 300.3.l.g 40
12.b even 2 1 inner 300.3.l.g 40
15.d odd 2 1 60.3.l.a ✓ 40
15.e even 4 1 60.3.l.a ✓ 40
15.e even 4 1 inner 300.3.l.g 40
20.d odd 2 1 60.3.l.a ✓ 40
20.e even 4 1 60.3.l.a ✓ 40
20.e even 4 1 inner 300.3.l.g 40
60.h even 2 1 60.3.l.a ✓ 40
60.l odd 4 1 60.3.l.a ✓ 40
60.l odd 4 1 inner 300.3.l.g 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.l.a ✓ 40 5.b even 2 1
60.3.l.a ✓ 40 5.c odd 4 1
60.3.l.a ✓ 40 15.d odd 2 1
60.3.l.a ✓ 40 15.e even 4 1
60.3.l.a ✓ 40 20.d odd 2 1
60.3.l.a ✓ 40 20.e even 4 1
60.3.l.a ✓ 40 60.h even 2 1
60.3.l.a ✓ 40 60.l odd 4 1
300.3.l.g 40 1.a even 1 1 trivial
300.3.l.g 40 3.b odd 2 1 inner
300.3.l.g 40 4.b odd 2 1 inner
300.3.l.g 40 5.c odd 4 1 inner
300.3.l.g 40 12.b even 2 1 inner
300.3.l.g 40 15.e even 4 1 inner
300.3.l.g 40 20.e even 4 1 inner
300.3.l.g 40 60.l odd 4 1 inner

Hecke kernels

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

\( T_{7}^{20} + 16980T_{7}^{16} + 73157408T_{7}^{12} + 110036323456T_{7}^{8} + 47984849132800T_{7}^{4} + 2276663967360000 \) Copy content Toggle raw display
\( T_{17}^{20} + 401292 T_{17}^{16} + 33078944688 T_{17}^{12} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
\( T_{19}^{10} - 1364T_{19}^{8} + 304112T_{19}^{6} - 20814528T_{19}^{4} + 364098560T_{19}^{2} - 707788800 \) Copy content Toggle raw display