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Properties

Label 8450.2.a.ci
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8450,2,Mod(1,8450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)

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: yes
Analytic conductor: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{3} - \beta_{2} - 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_{2} + 1) q^{6} - \beta_1 q^{7} - q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{3} - \beta_{2} - 1) q^{3} + q^{4} + ( - \beta_{3} + \beta_{2} + 1) q^{6} - \beta_1 q^{7} - q^{8} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{9}+ \cdots + (8 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 2 q^{6} - 4 q^{8} + 4 q^{9} - 6 q^{11} - 2 q^{12} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 6 q^{19} + 6 q^{21} + 6 q^{22} - 12 q^{23} + 2 q^{24} - 20 q^{27} + 18 q^{31} - 4 q^{32} + 6 q^{33} + 6 q^{34} + 4 q^{36} + 6 q^{37} - 6 q^{38} - 6 q^{42} - 4 q^{43} - 6 q^{44} + 12 q^{46} - 2 q^{48} + 8 q^{49} - 30 q^{53} + 20 q^{54} + 6 q^{57} - 26 q^{61} - 18 q^{62} + 24 q^{63} + 4 q^{64} - 6 q^{66} + 24 q^{67} - 6 q^{68} + 12 q^{69} - 24 q^{71} - 4 q^{72} + 6 q^{73} - 6 q^{74} + 6 q^{76} - 30 q^{77} + 10 q^{79} + 28 q^{81} + 18 q^{83} + 6 q^{84} + 4 q^{86} + 48 q^{87} + 6 q^{88} - 12 q^{92} + 12 q^{93} + 2 q^{96} - 18 q^{97} - 8 q^{98} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.219687
2.49551
1.21969
−1.49551
−1.00000 −3.33225 1.00000 0 3.33225 1.43937 −1.00000 8.10387 0
1.2 −1.00000 −1.09479 1.00000 0 1.09479 −3.99102 −1.00000 −1.80144 0
1.3 −1.00000 0.600196 1.00000 0 −0.600196 −1.43937 −1.00000 −2.63977 0
1.4 −1.00000 1.82684 1.00000 0 −1.82684 3.99102 −1.00000 0.337339 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8450.2.a.ci 4
5.b even 2 1 1690.2.a.u 4
13.b even 2 1 8450.2.a.cm 4
13.f odd 12 2 650.2.m.c 8
65.d even 2 1 1690.2.a.t 4
65.g odd 4 2 1690.2.d.k 8
65.l even 6 2 1690.2.e.t 8
65.n even 6 2 1690.2.e.s 8
65.o even 12 2 650.2.n.e 8
65.s odd 12 2 130.2.l.b ✓ 8
65.s odd 12 2 1690.2.l.j 8
65.t even 12 2 650.2.n.d 8
195.bh even 12 2 1170.2.bs.g 8
260.bc even 12 2 1040.2.da.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.b ✓ 8 65.s odd 12 2
650.2.m.c 8 13.f odd 12 2
650.2.n.d 8 65.t even 12 2
650.2.n.e 8 65.o even 12 2
1040.2.da.d 8 260.bc even 12 2
1170.2.bs.g 8 195.bh even 12 2
1690.2.a.t 4 65.d even 2 1
1690.2.a.u 4 5.b even 2 1
1690.2.d.k 8 65.g odd 4 2
1690.2.e.s 8 65.n even 6 2
1690.2.e.t 8 65.l even 6 2
1690.2.l.j 8 65.s odd 12 2
8450.2.a.ci 4 1.a even 1 1 trivial
8450.2.a.cm 4 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{4} + 2T_{3}^{3} - 6T_{3}^{2} - 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 18T_{7}^{2} + 33 \) Copy content Toggle raw display
\( T_{11}^{4} + 6T_{11}^{3} - 6T_{11}^{2} - 54T_{11} + 9 \) Copy content Toggle raw display
\( T_{17}^{4} + 6T_{17}^{3} - 18T_{17}^{2} - 108T_{17} - 108 \) Copy content Toggle raw display
\( T_{31}^{4} - 18T_{31}^{3} + 90T_{31}^{2} - 132T_{31} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 18T^{2} + 33 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 6 T^{3} + \cdots - 108 \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + \cdots - 3 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + \cdots - 48 \) Copy content Toggle raw display
$29$ \( T^{4} - 96T^{2} + 2112 \) Copy content Toggle raw display
$31$ \( T^{4} - 18 T^{3} + \cdots - 12 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots - 99 \) Copy content Toggle raw display
$41$ \( T^{4} - 60 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots - 368 \) Copy content Toggle raw display
$47$ \( T^{4} - 114 T^{2} + \cdots - 639 \) Copy content Toggle raw display
$53$ \( T^{4} + 30 T^{3} + \cdots - 579 \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 26 T^{3} + \cdots + 628 \) Copy content Toggle raw display
$67$ \( T^{4} - 24 T^{3} + \cdots - 3072 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$79$ \( T^{4} - 10 T^{3} + \cdots - 188 \) Copy content Toggle raw display
$83$ \( T^{4} - 18 T^{3} + \cdots - 2124 \) Copy content Toggle raw display
$89$ \( T^{4} - 78 T^{2} + \cdots - 207 \) Copy content Toggle raw display
$97$ \( T^{4} + 18 T^{3} + \cdots + 4308 \) Copy content Toggle raw display
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