Properties

Label 4400.2.b.bb
Level $4400$
Weight $2$
Character orbit 4400.b
Analytic conductor $35.134$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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: \(6\)
Coefficient field: 6.0.96668224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 15x^{4} + 61x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2200)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{3} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2}) q^{7} + (\beta_{3} - 2) q^{9} - q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{5} + 3) q^{19} + (3 \beta_{5} - \beta_{3} - 1) q^{21} + (3 \beta_{2} + \beta_1) q^{23} + ( - \beta_{4} + \beta_{2} - \beta_1) q^{27} + (2 \beta_{5} - \beta_{3} - 4) q^{29} + ( - 2 \beta_{3} + 1) q^{31} - \beta_1 q^{33} + ( - \beta_{4} - 3 \beta_{2} + \beta_1) q^{37} + \beta_{5} q^{39} + (\beta_{5} + \beta_{3} + 1) q^{41} + ( - \beta_{4} + 2 \beta_{2} + 3 \beta_1) q^{43} + (\beta_{4} - \beta_{2} + 3 \beta_1) q^{47} + ( - \beta_{5} - 3) q^{49} + (2 \beta_{5} + \beta_{3} - 5) q^{51} + ( - 3 \beta_{4} - \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{4} - 10 \beta_{2} + 3 \beta_1) q^{57} + ( - 3 \beta_{5} + \beta_{3} + 1) q^{59} + (2 \beta_{5} - \beta_{3} - 1) q^{61} + (\beta_{4} + 11 \beta_{2} + \beta_1) q^{63} + (2 \beta_{4} + 4 \beta_{2} + 2 \beta_1) q^{67} + ( - 3 \beta_{5} + \beta_{3} - 5) q^{69} + (\beta_{5} + \beta_{3} - 6) q^{71} + (3 \beta_{4} + 7 \beta_{2}) q^{73} + (\beta_{4} + \beta_{2}) q^{77} + ( - \beta_{5} + 2 \beta_{3} + 4) q^{79} + (\beta_{5} + \beta_{3} - 2) q^{81} + ( - \beta_{4} + 8 \beta_{2} - 4 \beta_1) q^{83} + (3 \beta_{4} + 9 \beta_{2} - 2 \beta_1) q^{87} + ( - \beta_{3} - 10) q^{89} + (\beta_{3} - 1) q^{91} + (2 \beta_{4} - 2 \beta_{2} + 5 \beta_1) q^{93} + (2 \beta_{4} - 3 \beta_{2} - \beta_1) q^{97} + ( - \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 15x^{4} + 61x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 9\nu^{3} + 13\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 29\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 8\nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 8\beta_{3} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{4} - 3\beta_{2} + 50\beta_1 \) 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
2.75153i
2.59261i
0.841083i
0.841083i
2.59261i
2.75153i
0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
4049.2 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.3 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.4 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.5 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.6 0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.6
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.bb 6
4.b odd 2 1 2200.2.b.m 6
5.b even 2 1 inner 4400.2.b.bb 6
5.c odd 4 1 4400.2.a.by 3
5.c odd 4 1 4400.2.a.bz 3
20.d odd 2 1 2200.2.b.m 6
20.e even 4 1 2200.2.a.u 3
20.e even 4 1 2200.2.a.v yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.u 3 20.e even 4 1
2200.2.a.v yes 3 20.e even 4 1
2200.2.b.m 6 4.b odd 2 1
2200.2.b.m 6 20.d odd 2 1
4400.2.a.by 3 5.c odd 4 1
4400.2.a.bz 3 5.c odd 4 1
4400.2.b.bb 6 1.a even 1 1 trivial
4400.2.b.bb 6 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 \(S_{2}^{\mathrm{new}}(4400, [\chi])\):

\( T_{3}^{6} + 15T_{3}^{4} + 61T_{3}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{6} + 31T_{7}^{4} + 313T_{7}^{2} + 1024 \) Copy content Toggle raw display
\( T_{13}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{6} + 23T_{17}^{4} + 41T_{17}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 15 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 31 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 23 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{3} - 7 T^{2} - 13 T + 27)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 48 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( (T^{3} + 10 T^{2} + \cdots - 201)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} + \cdots + 207)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 66 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( (T^{3} - 4 T^{2} - 17 T + 24)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 193 T^{4} + \cdots + 258064 \) Copy content Toggle raw display
$47$ \( T^{6} + 154 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( T^{6} + 307 T^{4} + \cdots + 20164 \) Copy content Toggle raw display
$59$ \( (T^{3} - 77 T + 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + T^{2} - 41 T - 114)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 228 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$71$ \( (T^{3} + 17 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 399 T^{4} + \cdots + 1106704 \) Copy content Toggle raw display
$79$ \( (T^{3} - 11 T^{2} + \cdots + 144)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 388 T^{4} + \cdots + 687241 \) Copy content Toggle raw display
$89$ \( (T^{3} + 30 T^{2} + \cdots + 879)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 164 T^{4} + \cdots + 529 \) Copy content Toggle raw display
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