Properties

Label 624.2.bu.j
Level $624$
Weight $2$
Character orbit 624.bu
Analytic conductor $4.983$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,2,Mod(191,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.bu (of order \(6\), degree \(2\), 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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (4 \beta_1 - 2) q^{5} + ( - 2 \beta_{3} + \beta_{2}) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (4 \beta_1 - 2) q^{5} + ( - 2 \beta_{3} + \beta_{2}) q^{7} + 3 q^{9} + ( - \beta_{3} + \beta_{2}) q^{11} + ( - 4 \beta_1 + 1) q^{13} + (2 \beta_{3} - 4 \beta_{2}) q^{15} + (\beta_1 - 2) q^{17} + ( - 2 \beta_{3} + \beta_{2}) q^{19} + ( - 3 \beta_1 + 6) q^{21} + (5 \beta_{3} - 5 \beta_{2}) q^{23} - 7 q^{25} - 3 \beta_{3} q^{27} + (\beta_1 + 1) q^{29} + (2 \beta_{3} - 4 \beta_{2}) q^{31} + ( - 3 \beta_1 + 3) q^{33} - 6 \beta_{2} q^{35} + (5 \beta_1 - 5) q^{37} + ( - \beta_{3} + 4 \beta_{2}) q^{39} + ( - 5 \beta_1 - 5) q^{41} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{43} + (12 \beta_1 - 6) q^{45} - 2 \beta_{3} q^{47} + ( - 2 \beta_1 + 2) q^{49} + (2 \beta_{3} - \beta_{2}) q^{51} + ( - 4 \beta_1 + 2) q^{53} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{55} + ( - 3 \beta_1 + 6) q^{57} + 7 \beta_{2} q^{59} - 11 \beta_1 q^{61} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{63} + ( - 4 \beta_1 + 14) q^{65} + (\beta_{3} + \beta_{2}) q^{67} + (15 \beta_1 - 15) q^{69} + 9 \beta_{2} q^{71} - 8 q^{73} + 7 \beta_{3} q^{75} + ( - 6 \beta_1 + 3) q^{77} + (4 \beta_{3} - 8 \beta_{2}) q^{79} + 9 q^{81} - 4 \beta_{3} q^{83} - 6 \beta_1 q^{85} + ( - \beta_{3} - \beta_{2}) q^{87} + ( - 3 \beta_1 - 3) q^{89} + (2 \beta_{3} + 5 \beta_{2}) q^{91} + (12 \beta_1 - 6) q^{93} - 6 \beta_{2} q^{95} - 7 \beta_1 q^{97} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} - 4 q^{13} - 6 q^{17} + 18 q^{21} - 28 q^{25} + 6 q^{29} + 6 q^{33} - 10 q^{37} - 30 q^{41} + 4 q^{49} + 18 q^{57} - 22 q^{61} + 48 q^{65} - 30 q^{69} - 32 q^{73} + 36 q^{81} - 12 q^{85} - 18 q^{89} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 3.46410i 0 −2.59808 + 1.50000i 0 3.00000 0
191.2 0 1.73205 0 3.46410i 0 2.59808 1.50000i 0 3.00000 0
575.1 0 −1.73205 0 3.46410i 0 −2.59808 1.50000i 0 3.00000 0
575.2 0 1.73205 0 3.46410i 0 2.59808 + 1.50000i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
39.i odd 6 1 inner
156.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bu.j yes 4
3.b odd 2 1 624.2.bu.i 4
4.b odd 2 1 inner 624.2.bu.j yes 4
12.b even 2 1 624.2.bu.i 4
13.c even 3 1 624.2.bu.i 4
39.i odd 6 1 inner 624.2.bu.j yes 4
52.j odd 6 1 624.2.bu.i 4
156.p even 6 1 inner 624.2.bu.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.bu.i 4 3.b odd 2 1
624.2.bu.i 4 12.b even 2 1
624.2.bu.i 4 13.c even 3 1
624.2.bu.i 4 52.j odd 6 1
624.2.bu.j yes 4 1.a even 1 1 trivial
624.2.bu.j yes 4 4.b odd 2 1 inner
624.2.bu.j yes 4 39.i odd 6 1 inner
624.2.bu.j yes 4 156.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7}^{4} - 9T_{7}^{2} + 81 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{2} + 9 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$11$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$23$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 15 T + 75)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( T^{4} + 243 T^{2} + 59049 \) Copy content Toggle raw display
$73$ \( (T + 8)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
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