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Properties

Label 4410.2.b.e
Level $4410$
Weight $2$
Character orbit 4410.b
Analytic conductor $35.214$
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] = [4410,2,Mod(881,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4410.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.b (of order \(2\), degree \(1\), 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: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 630)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} + q^{5} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} + q^{5} + \beta_1 q^{8} - \beta_1 q^{10} + (\beta_{4} + 2 \beta_{2}) q^{11} + (\beta_{7} + 2 \beta_1) q^{13} + q^{16} + ( - 2 \beta_{6} + 2 \beta_{3}) q^{17} + ( - 2 \beta_{4} - \beta_1) q^{19} - q^{20} + (\beta_{5} + 2 \beta_{3}) q^{22} + ( - \beta_{2} - 2 \beta_1) q^{23} + q^{25} + (\beta_{6} + 2) q^{26} + (2 \beta_{7} - 2 \beta_{4} + 4 \beta_1) q^{29} + ( - 2 \beta_{7} - 2 \beta_{4} + 2 \beta_1) q^{31} - \beta_1 q^{32} + (2 \beta_{7} - 2 \beta_{2}) q^{34} + ( - \beta_{6} - 6) q^{37} + ( - 2 \beta_{5} - 1) q^{38} + \beta_1 q^{40} + ( - 2 \beta_{6} - 3 \beta_{5} - 4) q^{41} + (2 \beta_{5} - 2) q^{43} + ( - \beta_{4} - 2 \beta_{2}) q^{44} + ( - \beta_{3} - 2) q^{46} + (4 \beta_{5} - \beta_{3} - 2) q^{47} - \beta_1 q^{50} + ( - \beta_{7} - 2 \beta_1) q^{52} + ( - 4 \beta_{7} + 2 \beta_{4} + \cdots + \beta_1) q^{53}+ \cdots + ( - 2 \beta_{7} - 6 \beta_{4} + \cdots + 6 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{5} + 8 q^{16} - 8 q^{20} + 8 q^{25} + 16 q^{26} - 48 q^{37} - 8 q^{38} - 32 q^{41} - 16 q^{43} - 16 q^{46} - 16 q^{47} + 32 q^{58} - 48 q^{59} + 16 q^{62} - 8 q^{64} + 48 q^{67} + 48 q^{79} + 8 q^{80} + 16 q^{83} - 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{5} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2647\) \(3431\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.965926 + 0.258819i
0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
− 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 − 1.00000i
881.2 − 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 − 1.00000i
881.3 − 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 − 1.00000i
881.4 − 1.00000i 0 −1.00000 1.00000 0 0 1.00000i 0 − 1.00000i
881.5 1.00000i 0 −1.00000 1.00000 0 0 − 1.00000i 0 1.00000i
881.6 1.00000i 0 −1.00000 1.00000 0 0 − 1.00000i 0 1.00000i
881.7 1.00000i 0 −1.00000 1.00000 0 0 − 1.00000i 0 1.00000i
881.8 1.00000i 0 −1.00000 1.00000 0 0 − 1.00000i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4410.2.b.e 8
3.b odd 2 1 4410.2.b.b 8
7.b odd 2 1 4410.2.b.b 8
7.c even 3 1 630.2.be.a ✓ 8
7.d odd 6 1 630.2.be.b yes 8
21.c even 2 1 inner 4410.2.b.e 8
21.g even 6 1 630.2.be.a ✓ 8
21.h odd 6 1 630.2.be.b yes 8
35.i odd 6 1 3150.2.bf.c 8
35.j even 6 1 3150.2.bf.b 8
35.k even 12 1 3150.2.bp.c 8
35.k even 12 1 3150.2.bp.f 8
35.l odd 12 1 3150.2.bp.a 8
35.l odd 12 1 3150.2.bp.d 8
105.o odd 6 1 3150.2.bf.c 8
105.p even 6 1 3150.2.bf.b 8
105.w odd 12 1 3150.2.bp.a 8
105.w odd 12 1 3150.2.bp.d 8
105.x even 12 1 3150.2.bp.c 8
105.x even 12 1 3150.2.bp.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.be.a ✓ 8 7.c even 3 1
630.2.be.a ✓ 8 21.g even 6 1
630.2.be.b yes 8 7.d odd 6 1
630.2.be.b yes 8 21.h odd 6 1
3150.2.bf.b 8 35.j even 6 1
3150.2.bf.b 8 105.p even 6 1
3150.2.bf.c 8 35.i odd 6 1
3150.2.bf.c 8 105.o odd 6 1
3150.2.bp.a 8 35.l odd 12 1
3150.2.bp.a 8 105.w odd 12 1
3150.2.bp.c 8 35.k even 12 1
3150.2.bp.c 8 105.x even 12 1
3150.2.bp.d 8 35.l odd 12 1
3150.2.bp.d 8 105.w odd 12 1
3150.2.bp.f 8 35.k even 12 1
3150.2.bp.f 8 105.x even 12 1
4410.2.b.b 8 3.b odd 2 1
4410.2.b.b 8 7.b odd 2 1
4410.2.b.e 8 1.a even 1 1 trivial
4410.2.b.e 8 21.c 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 \(S_{2}^{\mathrm{new}}(4410, [\chi])\):

\( T_{11}^{8} + 56T_{11}^{6} + 978T_{11}^{4} + 6008T_{11}^{2} + 9409 \) Copy content Toggle raw display
\( T_{17}^{4} - 40T_{17}^{2} - 96T_{17} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 56 T^{6} + \cdots + 9409 \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 36 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{4} + 14 T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 80 T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 24 T^{3} + \cdots + 1153)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 16 T^{3} + \cdots - 71)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + 8 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots - 575)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 276 T^{6} + \cdots + 2745649 \) Copy content Toggle raw display
$59$ \( (T^{4} + 24 T^{3} + \cdots - 4700)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 240 T^{6} + \cdots + 2262016 \) Copy content Toggle raw display
$67$ \( (T^{4} - 24 T^{3} + \cdots - 21024)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 288 T^{6} + \cdots + 1364224 \) Copy content Toggle raw display
$73$ \( (T^{4} + 136 T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + \cdots - 13088)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 8 T^{3} + \cdots + 7648)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + \cdots - 1436)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 800 T^{6} + \cdots + 610287616 \) Copy content Toggle raw display
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