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Properties

Label 2300.2.i.a
Level $2300$
Weight $2$
Character orbit 2300.i
Analytic conductor $18.366$
Analytic rank $0$
Dimension $8$
CM discriminant -115
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: 8.0.928445276160000.122
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 353x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{7} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{7} - 3 \beta_{2} q^{9} + ( - 2 \beta_{5} + \beta_1) q^{17} + (\beta_{6} + \beta_{3}) q^{23} - \beta_{7} q^{29} + ( - \beta_{4} + 2) q^{31} + (2 \beta_{5} - 3 \beta_1) q^{37} + (\beta_{4} - 4) q^{41} + (2 \beta_{6} + 2 \beta_{3}) q^{43} + (\beta_{7} + 3 \beta_{2}) q^{49} + (2 \beta_{6} + \beta_{3}) q^{53} + (\beta_{7} + 6 \beta_{2}) q^{59} + 3 \beta_{3} q^{63} + ( - 4 \beta_{5} + 3 \beta_1) q^{67} + (\beta_{4} + 6) q^{71} - 9 q^{81} + (4 \beta_{6} - \beta_{3}) q^{83} + ( - 4 \beta_{5} - 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{31} - 28 q^{41} + 52 q^{71} - 72 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 353x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 357\nu^{2} ) / 76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 357\nu^{3} ) / 76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 186 ) / 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 357\nu ) / 38 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 353\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{6} - 1747\nu^{2} ) / 38 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} + 19\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 19\beta_{4} - 186 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 38\beta_{5} - 357\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -357\beta_{7} - 3494\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 714\beta_{6} - 6707\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1057.1
3.06489 − 3.06489i
0.326276 − 0.326276i
−0.326276 + 0.326276i
−3.06489 + 3.06489i
3.06489 + 3.06489i
0.326276 + 0.326276i
−0.326276 − 0.326276i
−3.06489 − 3.06489i
0 0 0 0 0 −3.06489 + 3.06489i 0 3.00000i 0
1057.2 0 0 0 0 0 −0.326276 + 0.326276i 0 3.00000i 0
1057.3 0 0 0 0 0 0.326276 − 0.326276i 0 3.00000i 0
1057.4 0 0 0 0 0 3.06489 − 3.06489i 0 3.00000i 0
1793.1 0 0 0 0 0 −3.06489 − 3.06489i 0 − 3.00000i 0
1793.2 0 0 0 0 0 −0.326276 − 0.326276i 0 − 3.00000i 0
1793.3 0 0 0 0 0 0.326276 + 0.326276i 0 − 3.00000i 0
1793.4 0 0 0 0 0 3.06489 + 3.06489i 0 − 3.00000i 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
115.c odd 2 1 CM by \(\Q(\sqrt{-115}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
23.b 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.a ✓ 8
5.b even 2 1 inner 2300.2.i.a ✓ 8
5.c odd 4 2 inner 2300.2.i.a ✓ 8
23.b odd 2 1 inner 2300.2.i.a ✓ 8
115.c odd 2 1 CM 2300.2.i.a ✓ 8
115.e even 4 2 inner 2300.2.i.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2300.2.i.a ✓ 8 1.a even 1 1 trivial
2300.2.i.a ✓ 8 5.b even 2 1 inner
2300.2.i.a ✓ 8 5.c odd 4 2 inner
2300.2.i.a ✓ 8 23.b odd 2 1 inner
2300.2.i.a ✓ 8 115.c odd 2 1 CM
2300.2.i.a ✓ 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 \(S_{2}^{\mathrm{new}}(2300, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 353T^{4} + 16 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 4673 T^{4} + 614656 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 529)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 173 T^{2} + 7396)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3 T - 84)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 24113 T^{4} + 59969536 \) Copy content Toggle raw display
$41$ \( (T^{2} + 7 T - 74)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8464)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 7713 T^{4} + 5308416 \) Copy content Toggle raw display
$59$ \( (T^{4} + 233 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 1003875856 \) Copy content Toggle raw display
$71$ \( (T^{2} - 13 T - 44)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 81153 T^{4} + 3111696 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 135424)^{2} \) Copy content Toggle raw display
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