Properties

Label 2300.2.i.b
Level $2300$
Weight $2$
Character orbit 2300.i
Analytic conductor $18.366$
Analytic rank $0$
Dimension $8$
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] = [2300,2,Mod(1057,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2300.1057");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.i (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: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{5} q^{7} + 4 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{5} q^{7} + 4 \beta_{3} q^{9} - \beta_1 q^{11} + \beta_{2} q^{13} + \beta_{6} q^{19} - \beta_1 q^{21} + (2 \beta_{4} - \beta_{2}) q^{23} + \beta_{7} q^{27} - \beta_{3} q^{29} + 3 q^{31} + 7 \beta_{4} q^{33} + 5 \beta_{5} q^{37} + 7 \beta_{3} q^{39} + 9 q^{41} - 3 \beta_{4} q^{43} + \beta_{7} q^{47} - 3 \beta_{3} q^{49} - 2 \beta_{4} q^{53} + 7 \beta_{5} q^{57} - \beta_1 q^{61} + 4 \beta_{4} q^{63} + \beta_{5} q^{67} + (2 \beta_{6} - 7 \beta_{3}) q^{69} - 9 q^{71} + 5 \beta_{2} q^{73} + 4 \beta_{7} q^{77} + \beta_{6} q^{79} + 5 q^{81} + 6 \beta_{4} q^{83} - \beta_{7} q^{87} - 3 \beta_{6} q^{89} - \beta_1 q^{91} + 3 \beta_{2} q^{93} + \beta_{5} q^{97} + 4 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{31} + 72 q^{41} - 72 q^{71} + 40 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{4} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 11\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 7\nu^{3} ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 3\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 13\nu^{3} ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 6\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 5\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{4} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{6} + 18\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14\beta_{7} - 13\beta_{5} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1
−0.581861 + 1.28897i
−1.28897 + 0.581861i
1.28897 0.581861i
0.581861 1.28897i
−0.581861 1.28897i
−1.28897 0.581861i
1.28897 + 0.581861i
0.581861 + 1.28897i
0 −1.87083 + 1.87083i 0 0 0 −1.41421 + 1.41421i 0 4.00000i 0
1057.2 0 −1.87083 + 1.87083i 0 0 0 1.41421 1.41421i 0 4.00000i 0
1057.3 0 1.87083 1.87083i 0 0 0 −1.41421 + 1.41421i 0 4.00000i 0
1057.4 0 1.87083 1.87083i 0 0 0 1.41421 1.41421i 0 4.00000i 0
1793.1 0 −1.87083 1.87083i 0 0 0 −1.41421 1.41421i 0 4.00000i 0
1793.2 0 −1.87083 1.87083i 0 0 0 1.41421 + 1.41421i 0 4.00000i 0
1793.3 0 1.87083 + 1.87083i 0 0 0 −1.41421 1.41421i 0 4.00000i 0
1793.4 0 1.87083 + 1.87083i 0 0 0 1.41421 + 1.41421i 0 4.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1057.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
23.b odd 2 1 inner
115.c odd 2 1 inner
115.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2300.2.i.b 8
5.b even 2 1 inner 2300.2.i.b 8
5.c odd 4 2 inner 2300.2.i.b 8
23.b odd 2 1 inner 2300.2.i.b 8
115.c odd 2 1 inner 2300.2.i.b 8
115.e even 4 2 inner 2300.2.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.i.b 8 1.a even 1 1 trivial
2300.2.i.b 8 5.b even 2 1 inner
2300.2.i.b 8 5.c odd 4 2 inner
2300.2.i.b 8 23.b odd 2 1 inner
2300.2.i.b 8 115.c odd 2 1 inner
2300.2.i.b 8 115.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 \(S_{2}^{\mathrm{new}}(2300, [\chi])\):

\( T_{3}^{4} + 49 \) Copy content Toggle raw display
\( T_{19}^{2} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 734 T^{4} + 279841 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$31$ \( (T - 3)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 10000)^{2} \) Copy content Toggle raw display
$41$ \( (T - 9)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T + 9)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 30625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 252)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
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