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Properties

Label 585.2.b.e
Level $585$
Weight $2$
Character orbit 585.b
Analytic conductor $4.671$
Analytic rank $0$
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] = [585,2,Mod(181,585)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.b (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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} - \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} - \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + (\beta_{3} - 1) q^{10} + (3 \beta_{2} - \beta_1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + ( - 2 \beta_{3} + 6) q^{14} + ( - 3 \beta_{3} + 3) q^{16} + (\beta_{3} + 2) q^{17} + (4 \beta_{2} + 2 \beta_1) q^{19} + (2 \beta_{2} - \beta_1) q^{20} + ( - 4 \beta_{3} + 8) q^{22} + (\beta_{3} - 4) q^{23} - q^{25} + ( - 2 \beta_{3} + 4 \beta_{2} + \cdots + 6) q^{26}+ \cdots + (12 \beta_{2} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} - 2 q^{10} + 6 q^{13} + 20 q^{14} + 6 q^{16} + 10 q^{17} + 24 q^{22} - 14 q^{23} - 4 q^{25} + 20 q^{26} + 6 q^{35} - 28 q^{38} + 18 q^{40} + 12 q^{43} + 2 q^{49} + 2 q^{52} - 2 q^{53} + 14 q^{55} - 44 q^{56} - 42 q^{61} + 12 q^{62} - 14 q^{64} + 6 q^{65} - 8 q^{68} - 24 q^{74} - 38 q^{77} - 14 q^{79} + 12 q^{82} - 80 q^{88} - 26 q^{91} + 52 q^{92} - 8 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
− 2.56155i
− 1.56155i
1.56155i
2.56155i
− 2.56155i 0 −4.56155 − 1.00000i 0 3.56155i 6.56155i 0 −2.56155
181.2 − 1.56155i 0 −0.438447 1.00000i 0 0.561553i − 2.43845i 0 1.56155
181.3 1.56155i 0 −0.438447 − 1.00000i 0 − 0.561553i 2.43845i 0 1.56155
181.4 2.56155i 0 −4.56155 1.00000i 0 − 3.56155i − 6.56155i 0 −2.56155
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.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.b.e 4
3.b odd 2 1 195.2.b.c ✓ 4
12.b even 2 1 3120.2.g.n 4
13.b even 2 1 inner 585.2.b.e 4
13.d odd 4 1 7605.2.a.bc 2
13.d odd 4 1 7605.2.a.bh 2
15.d odd 2 1 975.2.b.f 4
15.e even 4 1 975.2.h.e 4
15.e even 4 1 975.2.h.g 4
39.d odd 2 1 195.2.b.c ✓ 4
39.f even 4 1 2535.2.a.p 2
39.f even 4 1 2535.2.a.q 2
156.h even 2 1 3120.2.g.n 4
195.e odd 2 1 975.2.b.f 4
195.s even 4 1 975.2.h.e 4
195.s even 4 1 975.2.h.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.c ✓ 4 3.b odd 2 1
195.2.b.c ✓ 4 39.d odd 2 1
585.2.b.e 4 1.a even 1 1 trivial
585.2.b.e 4 13.b even 2 1 inner
975.2.b.f 4 15.d odd 2 1
975.2.b.f 4 195.e odd 2 1
975.2.h.e 4 15.e even 4 1
975.2.h.e 4 195.s even 4 1
975.2.h.g 4 15.e even 4 1
975.2.h.g 4 195.s even 4 1
2535.2.a.p 2 39.f even 4 1
2535.2.a.q 2 39.f even 4 1
3120.2.g.n 4 12.b even 2 1
3120.2.g.n 4 156.h even 2 1
7605.2.a.bc 2 13.d odd 4 1
7605.2.a.bh 2 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} + 13T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 5T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 33T^{2} + 64 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 5 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$23$ \( (T^{2} + 7 T + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 33T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T - 8)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + T - 106)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 21 T + 106)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 132T^{2} + 1024 \) Copy content Toggle raw display
$71$ \( T^{4} + 89T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} + 132T^{2} + 1024 \) Copy content Toggle raw display
$79$ \( (T^{2} + 7 T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 308 T^{2} + 23104 \) Copy content Toggle raw display
$89$ \( T^{4} + 189T^{2} + 324 \) Copy content Toggle raw display
$97$ \( T^{4} + 417 T^{2} + 43264 \) Copy content Toggle raw display
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