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Properties

Label 8450.2.a.cj
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $1$
Dimension $4$
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] = [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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + \beta_{2} q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + \beta_{2} q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + (\beta_{3} - \beta_1) q^{11} + \beta_1 q^{12} - \beta_{2} q^{14} + q^{16} + ( - \beta_{3} - 2 \beta_1) q^{17} + ( - \beta_{2} - 1) q^{18} + ( - \beta_{3} - \beta_1) q^{19} + (\beta_{3} + 2 \beta_1) q^{21} + ( - \beta_{3} + \beta_1) q^{22} + ( - \beta_{3} + 3 \beta_1) q^{23} - \beta_1 q^{24} + \beta_{3} q^{27} + \beta_{2} q^{28} + ( - 2 \beta_{2} + 2) q^{29} + ( - \beta_{3} - \beta_1) q^{31} - q^{32} - 4 q^{33} + (\beta_{3} + 2 \beta_1) q^{34} + (\beta_{2} + 1) q^{36} + ( - 3 \beta_{2} - 2) q^{37} + (\beta_{3} + \beta_1) q^{38} - 4 \beta_1 q^{41} + ( - \beta_{3} - 2 \beta_1) q^{42} + (2 \beta_{3} - \beta_1) q^{43} + (\beta_{3} - \beta_1) q^{44} + (\beta_{3} - 3 \beta_1) q^{46} + ( - \beta_{2} - 8) q^{47} + \beta_1 q^{48} + ( - \beta_{2} + 1) q^{49} + ( - 3 \beta_{2} - 8) q^{51} + 2 \beta_1 q^{53} - \beta_{3} q^{54} - \beta_{2} q^{56} + ( - 2 \beta_{2} - 4) q^{57} + (2 \beta_{2} - 2) q^{58} + (\beta_{3} - 3 \beta_1) q^{59} + ( - 2 \beta_{2} - 2) q^{61} + (\beta_{3} + \beta_1) q^{62} + 8 q^{63} + q^{64} + 4 q^{66} - 4 q^{67} + ( - \beta_{3} - 2 \beta_1) q^{68} + (2 \beta_{2} + 12) q^{69} + 5 \beta_1 q^{71} + ( - \beta_{2} - 1) q^{72} - 10 q^{73} + (3 \beta_{2} + 2) q^{74} + ( - \beta_{3} - \beta_1) q^{76} - 4 \beta_{3} q^{77} - 2 \beta_{2} q^{79} + ( - 2 \beta_{2} - 3) q^{81} + 4 \beta_1 q^{82} + (2 \beta_{2} + 4) q^{83} + (\beta_{3} + 2 \beta_1) q^{84} + ( - 2 \beta_{3} + \beta_1) q^{86} + ( - 2 \beta_{3} - 2 \beta_1) q^{87} + ( - \beta_{3} + \beta_1) q^{88} - 2 \beta_{3} q^{89} + ( - \beta_{3} + 3 \beta_1) q^{92} + ( - 2 \beta_{2} - 4) q^{93} + (\beta_{2} + 8) q^{94} - \beta_1 q^{96} + ( - 2 \beta_{2} - 2) q^{97} + (\beta_{2} - 1) q^{98} + ( - 3 \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 2 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{4} - 2 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{14} + 4 q^{16} - 2 q^{18} - 2 q^{28} + 12 q^{29} - 4 q^{32} - 16 q^{33} + 2 q^{36} - 2 q^{37} - 30 q^{47} + 6 q^{49} - 26 q^{51} + 2 q^{56} - 12 q^{57} - 12 q^{58} - 4 q^{61} + 32 q^{63} + 4 q^{64} + 16 q^{66} - 16 q^{67} + 44 q^{69} - 2 q^{72} - 40 q^{73} + 2 q^{74} + 4 q^{79} - 8 q^{81} + 12 q^{83} - 12 q^{93} + 30 q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) 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
−2.52434
−0.792287
0.792287
2.52434
−1.00000 −2.52434 1.00000 0 2.52434 2.37228 −1.00000 3.37228 0
1.2 −1.00000 −0.792287 1.00000 0 0.792287 −3.37228 −1.00000 −2.37228 0
1.3 −1.00000 0.792287 1.00000 0 −0.792287 −3.37228 −1.00000 −2.37228 0
1.4 −1.00000 2.52434 1.00000 0 −2.52434 2.37228 −1.00000 3.37228 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8450.2.a.cj 4
5.b even 2 1 8450.2.a.cn 4
5.c odd 4 2 1690.2.b.d 8
13.b even 2 1 8450.2.a.cn 4
13.d odd 4 2 650.2.d.e 8
65.d even 2 1 inner 8450.2.a.cj 4
65.f even 4 2 130.2.c.b yes 4
65.g odd 4 2 650.2.d.e 8
65.h odd 4 2 1690.2.b.d 8
65.k even 4 2 130.2.c.a ✓ 4
195.j odd 4 2 1170.2.f.b 4
195.u odd 4 2 1170.2.f.a 4
260.l odd 4 2 1040.2.f.c 4
260.s odd 4 2 1040.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.c.a ✓ 4 65.k even 4 2
130.2.c.b yes 4 65.f even 4 2
650.2.d.e 8 13.d odd 4 2
650.2.d.e 8 65.g odd 4 2
1040.2.f.c 4 260.l odd 4 2
1040.2.f.d 4 260.s odd 4 2
1170.2.f.a 4 195.u odd 4 2
1170.2.f.b 4 195.j odd 4 2
1690.2.b.d 8 5.c odd 4 2
1690.2.b.d 8 65.h odd 4 2
8450.2.a.cj 4 1.a even 1 1 trivial
8450.2.a.cj 4 65.d even 2 1 inner
8450.2.a.cn 4 5.b even 2 1
8450.2.a.cn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{4} - 7T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{4} - 28T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{4} - 43T_{17}^{2} + 256 \) Copy content Toggle raw display
\( T_{31}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 7T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 28T^{2} + 64 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 43T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + T - 74)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 112T^{2} + 1024 \) Copy content Toggle raw display
$43$ \( T^{4} - 87T^{2} + 36 \) Copy content Toggle raw display
$47$ \( (T^{2} + 15 T + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 28T^{2} + 64 \) Copy content Toggle raw display
$59$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 175T^{2} + 2500 \) Copy content Toggle raw display
$73$ \( (T + 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T - 24)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 76T^{2} + 256 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
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