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Properties

Label 585.2.j.g
Level $585$
Weight $2$
Character orbit 585.j
Analytic conductor $4.671$
Analytic rank $0$
Dimension $6$
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] = [585,2,Mod(406,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.j (of order \(3\), 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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.591408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{4} + q^{5} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{7}+ \cdots - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{4} + q^{5} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - 8 \beta_{5} + 8 \beta_{3} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} + 6 q^{5} + 5 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} + 6 q^{5} + 5 q^{7} - 12 q^{8} + 4 q^{11} + 3 q^{13} + 24 q^{14} + 4 q^{16} - 4 q^{17} - 2 q^{20} - 12 q^{22} + 8 q^{23} + 6 q^{25} + 18 q^{26} + 6 q^{29} - 18 q^{31} - 8 q^{32} - 8 q^{34} + 5 q^{35} + 14 q^{37} - 28 q^{38} - 12 q^{40} - 20 q^{41} + 15 q^{43} - 48 q^{44} + 28 q^{46} - 30 q^{49} + 16 q^{52} + 32 q^{53} + 4 q^{55} - 10 q^{56} + 6 q^{58} + 10 q^{59} - 9 q^{61} - 2 q^{62} - 24 q^{64} + 3 q^{65} + 11 q^{67} - 18 q^{68} + 24 q^{70} + 4 q^{71} - 30 q^{73} + 8 q^{74} + 8 q^{76} + 34 q^{79} + 4 q^{80} + 14 q^{82} - 4 q^{83} - 4 q^{85} + 20 q^{86} - 8 q^{88} - 22 q^{89} - 31 q^{91} + 56 q^{92} - 8 q^{94} + q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 4x^{4} + x^{3} + 10x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 16\nu^{3} + 10\nu^{2} - 3\nu + 12 ) / 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{5} + 16\nu^{4} - 27\nu^{3} + 40\nu^{2} - 12\nu + 85 ) / 37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{5} - 11\nu^{4} + 44\nu^{3} + 28\nu^{2} + 110\nu + 4 ) / 37 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{5} + 26\nu^{4} - 104\nu^{3} - 9\nu^{2} - 260\nu + 78 ) / 37 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{5} - 7\beta_{4} + 4\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + 17\beta_{2} - 17\beta _1 + 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
−0.740597 + 1.28275i
1.08504 − 1.87935i
0.155554 − 0.269427i
−0.740597 − 1.28275i
1.08504 + 1.87935i
0.155554 + 0.269427i
−0.837565 − 1.45071i 0 −0.403032 + 0.698071i 1.00000 0 −1.81876 + 3.15018i −2.00000 0 −0.837565 − 1.45071i
406.2 −0.269594 − 0.466951i 0 0.854638 − 1.48028i 1.00000 0 2.40049 − 4.15777i −2.00000 0 −0.269594 − 0.466951i
406.3 1.10716 + 1.91766i 0 −1.45161 + 2.51426i 1.00000 0 1.91827 − 3.32254i −2.00000 0 1.10716 + 1.91766i
451.1 −0.837565 + 1.45071i 0 −0.403032 − 0.698071i 1.00000 0 −1.81876 − 3.15018i −2.00000 0 −0.837565 + 1.45071i
451.2 −0.269594 + 0.466951i 0 0.854638 + 1.48028i 1.00000 0 2.40049 + 4.15777i −2.00000 0 −0.269594 + 0.466951i
451.3 1.10716 − 1.91766i 0 −1.45161 − 2.51426i 1.00000 0 1.91827 + 3.32254i −2.00000 0 1.10716 − 1.91766i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.g 6
3.b odd 2 1 195.2.i.e ✓ 6
13.c even 3 1 inner 585.2.j.g 6
13.c even 3 1 7605.2.a.bu 3
13.e even 6 1 7605.2.a.bt 3
15.d odd 2 1 975.2.i.m 6
15.e even 4 2 975.2.bb.j 12
39.h odd 6 1 2535.2.a.z 3
39.i odd 6 1 195.2.i.e ✓ 6
39.i odd 6 1 2535.2.a.y 3
195.x odd 6 1 975.2.i.m 6
195.bl even 12 2 975.2.bb.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.e ✓ 6 3.b odd 2 1
195.2.i.e ✓ 6 39.i odd 6 1
585.2.j.g 6 1.a even 1 1 trivial
585.2.j.g 6 13.c even 3 1 inner
975.2.i.m 6 15.d odd 2 1
975.2.i.m 6 195.x odd 6 1
975.2.bb.j 12 15.e even 4 2
975.2.bb.j 12 195.bl even 12 2
2535.2.a.y 3 39.i odd 6 1
2535.2.a.z 3 39.h odd 6 1
7605.2.a.bt 3 13.e even 6 1
7605.2.a.bu 3 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 4T_{2}^{4} + 4T_{2}^{3} + 16T_{2}^{2} + 8T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 4 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 5 T^{5} + \cdots + 4489 \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{6} - 3 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 4 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$19$ \( T^{6} + 40 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$23$ \( T^{6} - 8 T^{5} + \cdots + 92416 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + \cdots + 2916 \) Copy content Toggle raw display
$31$ \( (T^{3} + 9 T^{2} + \cdots - 109)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 14 T^{5} + \cdots + 23104 \) Copy content Toggle raw display
$41$ \( T^{6} + 20 T^{5} + \cdots + 45796 \) Copy content Toggle raw display
$43$ \( T^{6} - 15 T^{5} + \cdots + 11449 \) Copy content Toggle raw display
$47$ \( (T^{3} - 4 T + 2)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 16 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 10 T^{5} + \cdots + 940900 \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + \cdots + 11881 \) Copy content Toggle raw display
$67$ \( T^{6} - 11 T^{5} + \cdots + 61009 \) Copy content Toggle raw display
$71$ \( T^{6} - 4 T^{5} + \cdots + 2116 \) Copy content Toggle raw display
$73$ \( (T^{3} + 15 T^{2} + \cdots + 103)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 17 T^{2} + \cdots - 67)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 2 T^{2} - 12 T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 22 T^{5} + \cdots + 17956 \) Copy content Toggle raw display
$97$ \( T^{6} - T^{5} + \cdots + 299209 \) Copy content Toggle raw display
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