Properties

Label 1200.2.v.l
Level $1200$
Weight $2$
Character orbit 1200.v
Analytic conductor $9.582$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,2,Mod(257,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.v (of order \(4\), degree \(2\), 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: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - 2 \beta_1 q^{7} + ( - \beta_{6} + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - 2 \beta_1 q^{7} + ( - \beta_{6} + 2 \beta_{2}) q^{9} + (2 \beta_{4} - 1) q^{11} + ( - 2 \beta_{5} + \beta_1) q^{17} + \beta_{2} q^{19} + (2 \beta_{4} - 4) q^{21} + (2 \beta_{7} + 2 \beta_{3}) q^{23} + 3 \beta_1 q^{27} + 2 q^{31} + (6 \beta_{7} - 3 \beta_{3}) q^{33} - 2 \beta_1 q^{37} + (2 \beta_{4} - 1) q^{41} + (2 \beta_{7} - 2 \beta_{3}) q^{43} + 5 \beta_{2} q^{49} + (\beta_{4} + 4) q^{51} + (2 \beta_{7} + 2 \beta_{3}) q^{53} + \beta_{5} q^{57} + ( - 4 \beta_{6} + 2 \beta_{2}) q^{59} + 14 q^{61} + 6 \beta_{7} q^{63} - 3 \beta_1 q^{67} + (2 \beta_{6} - 10 \beta_{2}) q^{69} + ( - 5 \beta_{7} + 5 \beta_{3}) q^{73} + (12 \beta_{5} - 6 \beta_1) q^{77} + 14 \beta_{2} q^{79} + ( - 3 \beta_{4} + 6) q^{81} + ( - \beta_{7} - \beta_{3}) q^{83} + ( - 6 \beta_{6} + 3 \beta_{2}) q^{89} - 2 \beta_{3} q^{93} + 4 \beta_1 q^{97} + ( - 3 \beta_{6} - 12 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{21} + 16 q^{31} + 36 q^{51} + 112 q^{61} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 3\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} + 3\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{5} + 2\beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
0 −1.67303 0.448288i 0 0 0 2.44949 + 2.44949i 0 2.59808 + 1.50000i 0
257.2 0 −0.448288 1.67303i 0 0 0 −2.44949 2.44949i 0 −2.59808 + 1.50000i 0
257.3 0 0.448288 + 1.67303i 0 0 0 2.44949 + 2.44949i 0 −2.59808 + 1.50000i 0
257.4 0 1.67303 + 0.448288i 0 0 0 −2.44949 2.44949i 0 2.59808 + 1.50000i 0
593.1 0 −1.67303 + 0.448288i 0 0 0 2.44949 2.44949i 0 2.59808 1.50000i 0
593.2 0 −0.448288 + 1.67303i 0 0 0 −2.44949 + 2.44949i 0 −2.59808 1.50000i 0
593.3 0 0.448288 1.67303i 0 0 0 2.44949 2.44949i 0 −2.59808 1.50000i 0
593.4 0 1.67303 0.448288i 0 0 0 −2.44949 + 2.44949i 0 2.59808 1.50000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.v.l 8
3.b odd 2 1 inner 1200.2.v.l 8
4.b odd 2 1 150.2.e.b 8
5.b even 2 1 inner 1200.2.v.l 8
5.c odd 4 2 inner 1200.2.v.l 8
12.b even 2 1 150.2.e.b 8
15.d odd 2 1 inner 1200.2.v.l 8
15.e even 4 2 inner 1200.2.v.l 8
20.d odd 2 1 150.2.e.b 8
20.e even 4 2 150.2.e.b 8
60.h even 2 1 150.2.e.b 8
60.l odd 4 2 150.2.e.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.e.b 8 4.b odd 2 1
150.2.e.b 8 12.b even 2 1
150.2.e.b 8 20.d odd 2 1
150.2.e.b 8 20.e even 4 2
150.2.e.b 8 60.h even 2 1
150.2.e.b 8 60.l odd 4 2
1200.2.v.l 8 1.a even 1 1 trivial
1200.2.v.l 8 3.b odd 2 1 inner
1200.2.v.l 8 5.b even 2 1 inner
1200.2.v.l 8 5.c odd 4 2 inner
1200.2.v.l 8 15.d odd 2 1 inner
1200.2.v.l 8 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} + 144 \) Copy content Toggle raw display
\( T_{11}^{2} + 27 \) Copy content Toggle raw display
\( T_{17}^{4} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T - 2)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$61$ \( (T - 14)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 5625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 196)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 243)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
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