Properties

Label 8450.2.a.bq
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2808.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 9x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 650)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_1 - 1) q^{7} - q^{8} + (\beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_1 - 1) q^{7} - q^{8} + (\beta_{2} + 3) q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{11} + \beta_1 q^{12} + (\beta_1 + 1) q^{14} + q^{16} + 3 q^{17} + ( - \beta_{2} - 3) q^{18} + (\beta_1 - 2) q^{19} + ( - \beta_{2} - \beta_1 - 6) q^{21} + (\beta_{2} - \beta_1 + 1) q^{22} + ( - \beta_{2} - \beta_1) q^{23} - \beta_1 q^{24} + (3 \beta_1 + 2) q^{27} + ( - \beta_1 - 1) q^{28} + (\beta_{2} - 2 \beta_1 - 3) q^{29} + ( - \beta_{2} - 5) q^{31} - q^{32} + (\beta_{2} - 4 \beta_1 + 4) q^{33} - 3 q^{34} + (\beta_{2} + 3) q^{36} + (\beta_{2} - \beta_1 + 4) q^{37} + ( - \beta_1 + 2) q^{38} + (\beta_{2} + 2 \beta_1 - 2) q^{41} + (\beta_{2} + \beta_1 + 6) q^{42} + 4 q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{44} + (\beta_{2} + \beta_1) q^{46} + (\beta_{2} + 2 \beta_1 + 1) q^{47} + \beta_1 q^{48} + (\beta_{2} + 2 \beta_1) q^{49} + 3 \beta_1 q^{51} + (\beta_{2} - 2 \beta_1 + 3) q^{53} + ( - 3 \beta_1 - 2) q^{54} + (\beta_1 + 1) q^{56} + (\beta_{2} - 2 \beta_1 + 6) q^{57} + ( - \beta_{2} + 2 \beta_1 + 3) q^{58} + (\beta_{2} - 4 \beta_1 - 5) q^{59} + ( - 3 \beta_1 - 5) q^{61} + (\beta_{2} + 5) q^{62} + ( - \beta_{2} - 6 \beta_1 - 5) q^{63} + q^{64} + ( - \beta_{2} + 4 \beta_1 - 4) q^{66} + ( - \beta_{2} - \beta_1 + 3) q^{67} + 3 q^{68} + ( - \beta_{2} - 3 \beta_1 - 8) q^{69} + ( - \beta_{2} - 3) q^{72} + ( - \beta_{2} + 4 \beta_1 + 2) q^{73} + ( - \beta_{2} + \beta_1 - 4) q^{74} + (\beta_1 - 2) q^{76} + (3 \beta_1 - 3) q^{77} + ( - \beta_{2} + \beta_1 - 6) q^{79} + (2 \beta_1 + 9) q^{81} + ( - \beta_{2} - 2 \beta_1 + 2) q^{82} + (\beta_{2} - \beta_1 - 5) q^{83} + ( - \beta_{2} - \beta_1 - 6) q^{84} - 4 q^{86} + ( - 2 \beta_{2} - 10) q^{87} + (\beta_{2} - \beta_1 + 1) q^{88} + ( - \beta_{2} - 2 \beta_1 + 8) q^{89} + ( - \beta_{2} - \beta_1) q^{92} + ( - 8 \beta_1 - 2) q^{93} + ( - \beta_{2} - 2 \beta_1 - 1) q^{94} - \beta_1 q^{96} + (2 \beta_1 - 10) q^{97} + ( - \beta_{2} - 2 \beta_1) q^{98} + ( - \beta_{2} + 4 \beta_1 - 19) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{7} - 3 q^{8} + 9 q^{9} - 3 q^{11} + 3 q^{14} + 3 q^{16} + 9 q^{17} - 9 q^{18} - 6 q^{19} - 18 q^{21} + 3 q^{22} + 6 q^{27} - 3 q^{28} - 9 q^{29} - 15 q^{31} - 3 q^{32} + 12 q^{33} - 9 q^{34} + 9 q^{36} + 12 q^{37} + 6 q^{38} - 6 q^{41} + 18 q^{42} + 12 q^{43} - 3 q^{44} + 3 q^{47} + 9 q^{53} - 6 q^{54} + 3 q^{56} + 18 q^{57} + 9 q^{58} - 15 q^{59} - 15 q^{61} + 15 q^{62} - 15 q^{63} + 3 q^{64} - 12 q^{66} + 9 q^{67} + 9 q^{68} - 24 q^{69} - 9 q^{72} + 6 q^{73} - 12 q^{74} - 6 q^{76} - 9 q^{77} - 18 q^{79} + 27 q^{81} + 6 q^{82} - 15 q^{83} - 18 q^{84} - 12 q^{86} - 30 q^{87} + 3 q^{88} + 24 q^{89} - 6 q^{93} - 3 q^{94} - 30 q^{97} - 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 9x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 6 \) Copy content Toggle raw display

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
−2.88202
−0.223462
3.10548
−1.00000 −2.88202 1.00000 0 2.88202 1.88202 −1.00000 5.30604 0
1.2 −1.00000 −0.223462 1.00000 0 0.223462 −0.776538 −1.00000 −2.95006 0
1.3 −1.00000 3.10548 1.00000 0 −3.10548 −4.10548 −1.00000 6.64402 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.bq 3
5.b even 2 1 8450.2.a.cd 3
13.b even 2 1 8450.2.a.ce 3
13.d odd 4 2 650.2.d.d yes 6
65.d even 2 1 8450.2.a.br 3
65.f even 4 2 650.2.c.f 6
65.g odd 4 2 650.2.d.c 6
65.k even 4 2 650.2.c.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.c.e 6 65.k even 4 2
650.2.c.f 6 65.f even 4 2
650.2.d.c 6 65.g odd 4 2
650.2.d.d yes 6 13.d odd 4 2
8450.2.a.bq 3 1.a even 1 1 trivial
8450.2.a.br 3 65.d even 2 1
8450.2.a.cd 3 5.b even 2 1
8450.2.a.ce 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{3} - 9T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 6T_{7} - 6 \) Copy content Toggle raw display
\( T_{11}^{3} + 3T_{11}^{2} - 27T_{11} - 45 \) Copy content Toggle raw display
\( T_{17} - 3 \) Copy content Toggle raw display
\( T_{31}^{3} + 15T_{31}^{2} + 48T_{31} - 60 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 9T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} - 6 T - 6 \) Copy content Toggle raw display
$11$ \( T^{3} + 3 T^{2} - 27 T - 45 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( (T - 3)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 6 T^{2} + 3 T - 12 \) Copy content Toggle raw display
$23$ \( T^{3} - 42T + 24 \) Copy content Toggle raw display
$29$ \( T^{3} + 9 T^{2} - 24 T - 240 \) Copy content Toggle raw display
$31$ \( T^{3} + 15 T^{2} + 48 T - 60 \) Copy content Toggle raw display
$37$ \( T^{3} - 12 T^{2} + 18 T + 72 \) Copy content Toggle raw display
$41$ \( T^{3} + 6 T^{2} - 63 T - 360 \) Copy content Toggle raw display
$43$ \( (T - 4)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 3 T^{2} - 72 T - 144 \) Copy content Toggle raw display
$53$ \( T^{3} - 9 T^{2} - 24 T + 12 \) Copy content Toggle raw display
$59$ \( T^{3} + 15 T^{2} - 72 T - 1224 \) Copy content Toggle raw display
$61$ \( T^{3} + 15 T^{2} - 6 T - 226 \) Copy content Toggle raw display
$67$ \( T^{3} - 9 T^{2} - 15 T + 123 \) Copy content Toggle raw display
$71$ \( T^{3} \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 135 T + 900 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + 78 T + 20 \) Copy content Toggle raw display
$83$ \( T^{3} + 15 T^{2} + 45 T - 9 \) Copy content Toggle raw display
$89$ \( T^{3} - 24 T^{2} + 117 T + 306 \) Copy content Toggle raw display
$97$ \( T^{3} + 30 T^{2} + 264 T + 624 \) Copy content Toggle raw display
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