Properties

Label 4400.2.b.y
Level $4400$
Weight $2$
Character orbit 4400.b
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4400,2,Mod(4049,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4400.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), 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: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (3 \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (3 \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 1) q^{9} + q^{11} + 5 \beta_{2} q^{13} + 3 \beta_1 q^{17} - q^{19} + ( - 2 \beta_{3} - 1) q^{21} + (5 \beta_{2} - \beta_1) q^{23} + (3 \beta_{2} + 2 \beta_1) q^{27} + (3 \beta_{3} + 3) q^{29} + (4 \beta_{3} - 5) q^{31} + \beta_1 q^{33} + ( - 7 \beta_{2} - 2 \beta_1) q^{37} + ( - 5 \beta_{3} + 5) q^{39} + (2 \beta_{3} - 3) q^{41} + (2 \beta_{2} + 4 \beta_1) q^{43} - 3 \beta_{2} q^{47} - 5 \beta_{3} q^{49} + (3 \beta_{3} - 12) q^{51} + (\beta_{2} + \beta_1) q^{53} - \beta_1 q^{57} + (4 \beta_{3} - 9) q^{59} + (3 \beta_{3} - 4) q^{61} + (3 \beta_{2} + 2 \beta_1) q^{63} - 4 \beta_{2} q^{67} + ( - 6 \beta_{3} + 9) q^{69} - 2 \beta_{3} q^{71} + ( - \beta_{2} + 3 \beta_1) q^{73} + (3 \beta_{2} + \beta_1) q^{77} + (3 \beta_{3} - 7) q^{79} + (2 \beta_{3} - 8) q^{81} + ( - 8 \beta_{2} - 5 \beta_1) q^{83} + (9 \beta_{2} + 3 \beta_1) q^{87} + ( - \beta_{3} - 3) q^{89} + ( - 5 \beta_{3} - 10) q^{91} + (12 \beta_{2} - 5 \beta_1) q^{93} + ( - 13 \beta_{2} + \beta_1) q^{97} + (\beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} - 4\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
2.30278i
1.30278i
1.30278i
2.30278i
0 2.30278i 0 0 0 0.697224i 0 −2.30278 0
4049.2 0 1.30278i 0 0 0 4.30278i 0 1.30278 0
4049.3 0 1.30278i 0 0 0 4.30278i 0 1.30278 0
4049.4 0 2.30278i 0 0 0 0.697224i 0 −2.30278 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4400.2.b.y 4
4.b odd 2 1 275.2.b.c 4
5.b even 2 1 inner 4400.2.b.y 4
5.c odd 4 1 4400.2.a.bh 2
5.c odd 4 1 4400.2.a.bs 2
12.b even 2 1 2475.2.c.k 4
20.d odd 2 1 275.2.b.c 4
20.e even 4 1 275.2.a.e 2
20.e even 4 1 275.2.a.f yes 2
60.h even 2 1 2475.2.c.k 4
60.l odd 4 1 2475.2.a.o 2
60.l odd 4 1 2475.2.a.t 2
220.i odd 4 1 3025.2.a.h 2
220.i odd 4 1 3025.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 20.e even 4 1
275.2.a.f yes 2 20.e even 4 1
275.2.b.c 4 4.b odd 2 1
275.2.b.c 4 20.d odd 2 1
2475.2.a.o 2 60.l odd 4 1
2475.2.a.t 2 60.l odd 4 1
2475.2.c.k 4 12.b even 2 1
2475.2.c.k 4 60.h even 2 1
3025.2.a.h 2 220.i odd 4 1
3025.2.a.n 2 220.i odd 4 1
4400.2.a.bh 2 5.c odd 4 1
4400.2.a.bs 2 5.c odd 4 1
4400.2.b.y 4 1.a even 1 1 trivial
4400.2.b.y 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4400, [\chi])\):

\( T_{3}^{4} + 7T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 19T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{2} + 25 \) Copy content Toggle raw display
\( T_{17}^{4} + 63T_{17}^{2} + 729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 19T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 63T^{2} + 729 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 67T^{2} + 729 \) Copy content Toggle raw display
$29$ \( (T^{2} - 9 T - 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 43)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 98T^{2} + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 9)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 7T^{2} + 9 \) Copy content Toggle raw display
$59$ \( (T^{2} + 14 T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T - 23)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 71T^{2} + 529 \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 223T^{2} + 2601 \) Copy content Toggle raw display
$89$ \( (T^{2} + 7 T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 371 T^{2} + 32041 \) Copy content Toggle raw display
show more
show less