Properties

Label 440.2.y.a
Level $440$
Weight $2$
Character orbit 440.y
Analytic conductor $3.513$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(81,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.y (of order \(5\), degree \(4\), 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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{3} + \beta_{7} q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{3} + \beta_{7} q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 2 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 2 q^{5} + q^{7} + 7 q^{9} + 3 q^{11} - 4 q^{13} + q^{15} - 3 q^{17} + 9 q^{19} - 4 q^{21} - 22 q^{23} - 2 q^{25} - 8 q^{27} - 17 q^{29} - 4 q^{31} + 21 q^{33} + 6 q^{35} + 24 q^{37} - 13 q^{39} - 4 q^{41} - 14 q^{43} - 8 q^{45} - 12 q^{47} - 15 q^{49} - 17 q^{51} + 35 q^{53} + 3 q^{55} - q^{57} + 21 q^{59} - 22 q^{61} + 5 q^{63} + 6 q^{65} + 14 q^{67} + 3 q^{69} + 40 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 41 q^{79} + 24 q^{81} - 7 q^{83} - 8 q^{85} - 46 q^{87} - 24 q^{89} - 18 q^{91} + 3 q^{93} + 9 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
1.69513 + 1.23158i
−0.386111 0.280526i
1.69513 1.23158i
−0.386111 + 0.280526i
−0.227943 + 0.701538i
0.418926 1.28932i
−0.227943 0.701538i
0.418926 + 1.28932i
0 −0.400166 + 1.23158i 0 −0.809017 + 0.587785i 0 −0.0703870 0.216629i 0 1.07039 + 0.777682i 0
81.2 0 0.0911485 0.280526i 0 −0.809017 + 0.587785i 0 −1.35666 4.17538i 0 2.35666 + 1.71222i 0
201.1 0 −0.400166 1.23158i 0 −0.809017 0.587785i 0 −0.0703870 + 0.216629i 0 1.07039 0.777682i 0
201.2 0 0.0911485 + 0.280526i 0 −0.809017 0.587785i 0 −1.35666 + 4.17538i 0 2.35666 1.71222i 0
361.1 0 −0.965584 + 0.701538i 0 0.309017 + 0.951057i 0 1.48685 + 1.08026i 0 −0.486854 + 1.49838i 0
361.2 0 1.77460 1.28932i 0 0.309017 + 0.951057i 0 0.440197 + 0.319822i 0 0.559803 1.72290i 0
401.1 0 −0.965584 0.701538i 0 0.309017 0.951057i 0 1.48685 1.08026i 0 −0.486854 1.49838i 0
401.2 0 1.77460 + 1.28932i 0 0.309017 0.951057i 0 0.440197 0.319822i 0 0.559803 + 1.72290i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.y.a 8
4.b odd 2 1 880.2.bo.d 8
11.c even 5 1 inner 440.2.y.a 8
11.c even 5 1 4840.2.a.y 4
11.d odd 10 1 4840.2.a.z 4
44.g even 10 1 9680.2.a.ct 4
44.h odd 10 1 880.2.bo.d 8
44.h odd 10 1 9680.2.a.cu 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.y.a 8 1.a even 1 1 trivial
440.2.y.a 8 11.c even 5 1 inner
880.2.bo.d 8 4.b odd 2 1
880.2.bo.d 8 44.h odd 10 1
4840.2.a.y 4 11.c even 5 1
4840.2.a.z 4 11.d odd 10 1
9680.2.a.ct 4 44.g even 10 1
9680.2.a.cu 4 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{7} + T_{3}^{5} + 9T_{3}^{4} + 11T_{3}^{3} + 10T_{3}^{2} - T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + 15 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots + 1681 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$23$ \( (T^{4} + 11 T^{3} + \cdots + 31)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 17 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} + \cdots + 143641 \) Copy content Toggle raw display
$37$ \( T^{8} - 24 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots + 292681 \) Copy content Toggle raw display
$43$ \( (T^{4} + 7 T^{3} - 17 T^{2} + \cdots + 61)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 22201 \) Copy content Toggle raw display
$53$ \( T^{8} - 35 T^{7} + \cdots + 1852321 \) Copy content Toggle raw display
$59$ \( T^{8} - 21 T^{7} + \cdots + 2825761 \) Copy content Toggle raw display
$61$ \( T^{8} + 22 T^{7} + \cdots + 657721 \) Copy content Toggle raw display
$67$ \( (T^{4} - 7 T^{3} + \cdots + 431)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 40 T^{7} + \cdots + 97436641 \) Copy content Toggle raw display
$73$ \( T^{8} - 9 T^{7} + \cdots + 14070001 \) Copy content Toggle raw display
$79$ \( T^{8} - 41 T^{7} + \cdots + 5669161 \) Copy content Toggle raw display
$83$ \( T^{8} + 7 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + \cdots - 479)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} + \cdots + 4748041 \) Copy content Toggle raw display
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