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Properties

Label 300.3.l.f
Level $300$
Weight $3$
Character orbit 300.l
Analytic conductor $8.174$
Analytic rank $0$
Dimension $8$
CM discriminant -20
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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{3}) q^{3} - 4 \beta_{2} q^{4} + ( - \beta_{5} + 4) q^{6} - 6 \beta_1 q^{7} + 4 \beta_{3} q^{8} + ( - 2 \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{7} - \beta_{3}) q^{3} - 4 \beta_{2} q^{4} + ( - \beta_{5} + 4) q^{6} - 6 \beta_1 q^{7} + 4 \beta_{3} q^{8} + ( - 2 \beta_{4} - \beta_{2}) q^{9} + ( - 4 \beta_{6} + 4 \beta_1) q^{12} - 6 \beta_{4} q^{14} - 16 q^{16} + (8 \beta_{7} + \beta_{3}) q^{18} + (6 \beta_{5} + 30) q^{21} + 22 \beta_{3} q^{23} + (4 \beta_{4} - 16 \beta_{2}) q^{24} + ( - 11 \beta_{6} - 7 \beta_1) q^{27} + 24 \beta_{7} q^{28} + 12 \beta_{4} q^{29} + 16 \beta_{6} q^{32} + ( - 8 \beta_{5} - 4) q^{36} + 12 \beta_{5} q^{41} + ( - 30 \beta_{6} - 24 \beta_1) q^{42} + 18 \beta_{7} q^{43} - 88 q^{46} + 2 \beta_{6} q^{47} + ( - 16 \beta_{7} + 16 \beta_{3}) q^{48} + 131 \beta_{2} q^{49} + ( - 7 \beta_{4} - 44 \beta_{2}) q^{54} - 24 \beta_{5} q^{56} - 48 \beta_{7} q^{58} + 58 q^{61} + (6 \beta_{7} - 60 \beta_{3}) q^{63} + 64 \beta_{2} q^{64} + 30 \beta_1 q^{67} + (22 \beta_{4} - 88 \beta_{2}) q^{69} + (4 \beta_{6} + 32 \beta_1) q^{72} + ( - 4 \beta_{5} + 79) q^{81} - 48 \beta_1 q^{82} + 38 \beta_{3} q^{83} + ( - 24 \beta_{4} - 120 \beta_{2}) q^{84} - 18 \beta_{5} q^{86} + (60 \beta_{6} + 48 \beta_1) q^{87} - 24 \beta_{4} q^{89} + 88 \beta_{6} q^{92} + 8 \beta_{2} q^{94} + (16 \beta_{5} - 64) q^{96} - 131 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{6} - 128 q^{16} + 240 q^{21} - 32 q^{36} - 704 q^{46} + 464 q^{61} + 632 q^{81} - 512 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{4} + 14 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\nu^{6} - 12\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{7} + 26\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} + 29\nu^{3} ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 6\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{4} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{3} - 10\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{5} - 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -26\beta_{7} + 29\beta_{6} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
1.14412 − 1.14412i
−0.437016 + 0.437016i
0.437016 − 0.437016i
−1.14412 + 1.14412i
1.14412 + 1.14412i
−0.437016 − 0.437016i
0.437016 + 0.437016i
−1.14412 − 1.14412i
−1.41421 − 1.41421i −2.99535 − 0.166925i 4.00000i 0 4.00000 + 4.47214i −9.48683 + 9.48683i 5.65685 − 5.65685i 8.94427 + 1.00000i 0
107.2 −1.41421 − 1.41421i 0.166925 + 2.99535i 4.00000i 0 4.00000 − 4.47214i 9.48683 − 9.48683i 5.65685 − 5.65685i −8.94427 + 1.00000i 0
107.3 1.41421 + 1.41421i −0.166925 − 2.99535i 4.00000i 0 4.00000 − 4.47214i −9.48683 + 9.48683i −5.65685 + 5.65685i −8.94427 + 1.00000i 0
107.4 1.41421 + 1.41421i 2.99535 + 0.166925i 4.00000i 0 4.00000 + 4.47214i 9.48683 − 9.48683i −5.65685 + 5.65685i 8.94427 + 1.00000i 0
143.1 −1.41421 + 1.41421i −2.99535 + 0.166925i − 4.00000i 0 4.00000 − 4.47214i −9.48683 − 9.48683i 5.65685 + 5.65685i 8.94427 − 1.00000i 0
143.2 −1.41421 + 1.41421i 0.166925 − 2.99535i − 4.00000i 0 4.00000 + 4.47214i 9.48683 + 9.48683i 5.65685 + 5.65685i −8.94427 − 1.00000i 0
143.3 1.41421 − 1.41421i −0.166925 + 2.99535i − 4.00000i 0 4.00000 + 4.47214i −9.48683 − 9.48683i −5.65685 − 5.65685i −8.94427 − 1.00000i 0
143.4 1.41421 − 1.41421i 2.99535 − 0.166925i − 4.00000i 0 4.00000 − 4.47214i 9.48683 + 9.48683i −5.65685 − 5.65685i 8.94427 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
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.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.f ✓ 8
3.b odd 2 1 inner 300.3.l.f ✓ 8
4.b odd 2 1 inner 300.3.l.f ✓ 8
5.b even 2 1 inner 300.3.l.f ✓ 8
5.c odd 4 2 inner 300.3.l.f ✓ 8
12.b even 2 1 inner 300.3.l.f ✓ 8
15.d odd 2 1 inner 300.3.l.f ✓ 8
15.e even 4 2 inner 300.3.l.f ✓ 8
20.d odd 2 1 CM 300.3.l.f ✓ 8
20.e even 4 2 inner 300.3.l.f ✓ 8
60.h even 2 1 inner 300.3.l.f ✓ 8
60.l odd 4 2 inner 300.3.l.f ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.l.f ✓ 8 1.a even 1 1 trivial
300.3.l.f ✓ 8 3.b odd 2 1 inner
300.3.l.f ✓ 8 4.b odd 2 1 inner
300.3.l.f ✓ 8 5.b even 2 1 inner
300.3.l.f ✓ 8 5.c odd 4 2 inner
300.3.l.f ✓ 8 12.b even 2 1 inner
300.3.l.f ✓ 8 15.d odd 2 1 inner
300.3.l.f ✓ 8 15.e even 4 2 inner
300.3.l.f ✓ 8 20.d odd 2 1 CM
300.3.l.f ✓ 8 20.e even 4 2 inner
300.3.l.f ✓ 8 60.h even 2 1 inner
300.3.l.f ✓ 8 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 \(S_{3}^{\mathrm{new}}(300, [\chi])\):

\( T_{7}^{4} + 32400 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 158T^{4} + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 32400)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 3748096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 2880)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2880)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2624400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T - 58)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 20250000)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 33362176)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 11520)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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