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Properties

Label 585.2.c.b
Level $585$
Weight $2$
Character orbit 585.c
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(469,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.c (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{4} - \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{4} - 4 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{2} - \beta_1 - 2) q^{4} + (\beta_{4} - \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{4} - 4 \beta_{3}) q^{8} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 1) q^{10} + (\beta_1 + 2) q^{11} - \beta_{3} q^{13} + (\beta_{2} + \beta_1 - 1) q^{14} + (4 \beta_{2} + 2 \beta_1 + 3) q^{16} + (2 \beta_{5} + 2 \beta_{3}) q^{17} + \beta_1 q^{19} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \cdots + 4) q^{20}+ \cdots + ( - 3 \beta_{5} - 2 \beta_{4} + 9 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 2 q^{10} + 12 q^{11} - 8 q^{14} + 10 q^{16} + 20 q^{20} + 2 q^{25} + 6 q^{26} + 12 q^{29} - 20 q^{31} + 20 q^{34} - 8 q^{35} - 34 q^{40} + 8 q^{41} - 40 q^{44} + 32 q^{46} + 18 q^{49} - 16 q^{50} - 16 q^{55} - 40 q^{56} - 16 q^{59} + 12 q^{61} - 66 q^{64} + 2 q^{65} - 20 q^{70} + 24 q^{71} - 4 q^{74} - 20 q^{76} + 32 q^{79} - 48 q^{80} - 12 q^{85} + 32 q^{86} + 20 q^{89} - 4 q^{91} - 32 q^{94} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
0.403032 − 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.45161 − 1.45161i
−0.854638 − 0.854638i
0.403032 + 0.403032i
− 2.67513i 0 −5.15633 −1.67513 + 1.48119i 0 0.806063i 8.44358i 0 3.96239 + 4.48119i
469.2 − 1.53919i 0 −0.369102 −0.539189 − 2.17009i 0 − 1.70928i − 2.51026i 0 −3.34017 + 0.829914i
469.3 − 1.21432i 0 0.525428 2.21432 + 0.311108i 0 − 2.90321i − 3.06668i 0 0.377784 − 2.68889i
469.4 1.21432i 0 0.525428 2.21432 − 0.311108i 0 2.90321i 3.06668i 0 0.377784 + 2.68889i
469.5 1.53919i 0 −0.369102 −0.539189 + 2.17009i 0 1.70928i 2.51026i 0 −3.34017 − 0.829914i
469.6 2.67513i 0 −5.15633 −1.67513 − 1.48119i 0 − 0.806063i − 8.44358i 0 3.96239 − 4.48119i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.c.b 6
3.b odd 2 1 65.2.b.a ✓ 6
5.b even 2 1 inner 585.2.c.b 6
5.c odd 4 1 2925.2.a.bf 3
5.c odd 4 1 2925.2.a.bj 3
12.b even 2 1 1040.2.d.c 6
15.d odd 2 1 65.2.b.a ✓ 6
15.e even 4 1 325.2.a.j 3
15.e even 4 1 325.2.a.k 3
39.d odd 2 1 845.2.b.c 6
39.f even 4 1 845.2.d.a 6
39.f even 4 1 845.2.d.b 6
39.h odd 6 2 845.2.n.g 12
39.i odd 6 2 845.2.n.f 12
39.k even 12 2 845.2.l.d 12
39.k even 12 2 845.2.l.e 12
60.h even 2 1 1040.2.d.c 6
60.l odd 4 1 5200.2.a.cb 3
60.l odd 4 1 5200.2.a.cj 3
195.e odd 2 1 845.2.b.c 6
195.n even 4 1 845.2.d.a 6
195.n even 4 1 845.2.d.b 6
195.s even 4 1 4225.2.a.ba 3
195.s even 4 1 4225.2.a.bh 3
195.x odd 6 2 845.2.n.f 12
195.y odd 6 2 845.2.n.g 12
195.bh even 12 2 845.2.l.d 12
195.bh even 12 2 845.2.l.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a ✓ 6 3.b odd 2 1
65.2.b.a ✓ 6 15.d odd 2 1
325.2.a.j 3 15.e even 4 1
325.2.a.k 3 15.e even 4 1
585.2.c.b 6 1.a even 1 1 trivial
585.2.c.b 6 5.b even 2 1 inner
845.2.b.c 6 39.d odd 2 1
845.2.b.c 6 195.e odd 2 1
845.2.d.a 6 39.f even 4 1
845.2.d.a 6 195.n even 4 1
845.2.d.b 6 39.f even 4 1
845.2.d.b 6 195.n even 4 1
845.2.l.d 12 39.k even 12 2
845.2.l.d 12 195.bh even 12 2
845.2.l.e 12 39.k even 12 2
845.2.l.e 12 195.bh even 12 2
845.2.n.f 12 39.i odd 6 2
845.2.n.f 12 195.x odd 6 2
845.2.n.g 12 39.h odd 6 2
845.2.n.g 12 195.y odd 6 2
1040.2.d.c 6 12.b even 2 1
1040.2.d.c 6 60.h even 2 1
2925.2.a.bf 3 5.c odd 4 1
2925.2.a.bj 3 5.c odd 4 1
4225.2.a.ba 3 195.s even 4 1
4225.2.a.bh 3 195.s even 4 1
5200.2.a.cb 3 60.l odd 4 1
5200.2.a.cj 3 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 11T_{2}^{4} + 31T_{2}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 11 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{3} - 6 T^{2} + 8 T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 44 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 72 T^{4} + \cdots + 7396 \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 10 T^{2} + \cdots - 26)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 56 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$41$ \( (T^{3} - 4 T^{2} - 32 T - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 128 T^{4} + \cdots + 77284 \) Copy content Toggle raw display
$47$ \( T^{6} + 44 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$53$ \( T^{6} + 144 T^{4} + \cdots + 92416 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} + \cdots - 262)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 6 T^{2} - 16 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 220 T^{4} + \cdots + 364816 \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T^{2} + \cdots + 754)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 248 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 180 T^{4} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( (T^{3} - 10 T^{2} + \cdots + 200)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 364 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
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