Properties

Label 1100.2.k.c
Level $1100$
Weight $2$
Character orbit 1100.k
Analytic conductor $8.784$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,2,Mod(593,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.593");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.k (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: \(8.78354422234\)
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: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{7} q^{7} + \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{7} q^{7} + \beta_1 q^{9} + ( - \beta_{5} - 1) q^{11} - \beta_{4} q^{13} - \beta_{7} q^{17} + 2 \beta_{6} q^{19} - 2 \beta_{5} q^{21} - \beta_{2} q^{23} + 2 \beta_{3} q^{27} + 6 q^{31} + ( - 2 \beta_{4} - \beta_{2}) q^{33} - 2 \beta_{3} q^{37} + 2 \beta_{6} q^{39} - 2 \beta_{5} q^{41} + 3 \beta_{4} q^{43} + 5 \beta_{3} q^{47} + 3 \beta_1 q^{49} - 2 \beta_{5} q^{51} + 4 \beta_{7} q^{57} + 8 \beta_1 q^{59} - 2 \beta_{5} q^{61} - \beta_{4} q^{63} - 7 \beta_{3} q^{67} - 4 \beta_1 q^{69} + 4 q^{71} + \beta_{4} q^{73} + (\beta_{7} - 5 \beta_{3}) q^{77} + 2 \beta_{6} q^{79} + 11 q^{81} + 5 \beta_{4} q^{83} - 8 \beta_1 q^{89} - 10 q^{91} + 6 \beta_{2} q^{93} + 2 \beta_{3} q^{97} + (\beta_{6} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{11} + 48 q^{31} + 32 q^{71} + 88 q^{81} - 80 q^{91}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} - 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} + 26\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 2\nu^{4} - 18\nu^{2} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{7} + \nu^{5} + 29\nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{7} + \nu^{5} - 29\nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{6} - 2\nu^{4} - 18\nu^{2} - 7 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{4} + 6\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} + \beta_{5} - 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + 3\beta_{4} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{6} - 5\beta_{5} - 11\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} - 2\beta_{4} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{6} - 13\beta_{5} + 29\beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
−1.14412 1.14412i
0.437016 + 0.437016i
1.14412 + 1.14412i
−0.437016 0.437016i
−1.14412 + 1.14412i
0.437016 0.437016i
1.14412 1.14412i
−0.437016 + 0.437016i
0 −1.41421 1.41421i 0 0 0 −2.23607 2.23607i 0 1.00000i 0
593.2 0 −1.41421 1.41421i 0 0 0 2.23607 + 2.23607i 0 1.00000i 0
593.3 0 1.41421 + 1.41421i 0 0 0 −2.23607 2.23607i 0 1.00000i 0
593.4 0 1.41421 + 1.41421i 0 0 0 2.23607 + 2.23607i 0 1.00000i 0
857.1 0 −1.41421 + 1.41421i 0 0 0 −2.23607 + 2.23607i 0 1.00000i 0
857.2 0 −1.41421 + 1.41421i 0 0 0 2.23607 2.23607i 0 1.00000i 0
857.3 0 1.41421 1.41421i 0 0 0 −2.23607 + 2.23607i 0 1.00000i 0
857.4 0 1.41421 1.41421i 0 0 0 2.23607 2.23607i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.4
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
11.b odd 2 1 inner
55.d odd 2 1 inner
55.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.2.k.c 8
5.b even 2 1 inner 1100.2.k.c 8
5.c odd 4 2 inner 1100.2.k.c 8
11.b odd 2 1 inner 1100.2.k.c 8
55.d odd 2 1 inner 1100.2.k.c 8
55.e even 4 2 inner 1100.2.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.k.c 8 1.a even 1 1 trivial
1100.2.k.c 8 5.b even 2 1 inner
1100.2.k.c 8 5.c odd 4 2 inner
1100.2.k.c 8 11.b odd 2 1 inner
1100.2.k.c 8 55.d odd 2 1 inner
1100.2.k.c 8 55.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 \(S_{2}^{\mathrm{new}}(1100, [\chi])\):

\( T_{3}^{4} + 16 \) Copy content Toggle raw display
\( T_{19}^{2} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 11)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T - 6)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 10000)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 38416)^{2} \) Copy content Toggle raw display
$71$ \( (T - 4)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 62500)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
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