Properties

Label 605.2.b.a
Level $605$
Weight $2$
Character orbit 605.b
Analytic conductor $4.831$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(364,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("605.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta - 1) q^{3} + 2 q^{4} + (\beta + 1) q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta - 1) q^{3} + 2 q^{4} + (\beta + 1) q^{5} - 8 q^{9} + (4 \beta - 2) q^{12} + (3 \beta - 7) q^{15} + 4 q^{16} + (2 \beta + 2) q^{20} + ( - 2 \beta + 1) q^{23} + (3 \beta - 2) q^{25} + ( - 10 \beta + 5) q^{27} + 5 q^{31} - 16 q^{36} + ( - 6 \beta + 3) q^{37} + ( - 8 \beta - 8) q^{45} + ( - 4 \beta + 2) q^{47} + (8 \beta - 4) q^{48} + 7 q^{49} + (8 \beta - 4) q^{53} - 15 q^{59} + (6 \beta - 14) q^{60} + 8 q^{64} + (6 \beta - 3) q^{67} + 11 q^{69} - 3 q^{71} + ( - \beta - 16) q^{75} + (4 \beta + 4) q^{80} + 31 q^{81} + 9 q^{89} + ( - 4 \beta + 2) q^{92} + (10 \beta - 5) q^{93} + ( - 6 \beta + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 3 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 3 q^{5} - 16 q^{9} - 11 q^{15} + 8 q^{16} + 6 q^{20} - q^{25} + 10 q^{31} - 32 q^{36} - 24 q^{45} + 14 q^{49} - 30 q^{59} - 22 q^{60} + 16 q^{64} + 22 q^{69} - 6 q^{71} - 33 q^{75} + 12 q^{80} + 62 q^{81} + 18 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
0.500000 1.65831i
0.500000 + 1.65831i
0 3.31662i 2.00000 1.50000 1.65831i 0 0 0 −8.00000 0
364.2 0 3.31662i 2.00000 1.50000 + 1.65831i 0 0 0 −8.00000 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.b even 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.b.a 2
5.b even 2 1 inner 605.2.b.a 2
5.c odd 4 2 3025.2.a.k 2
11.b odd 2 1 CM 605.2.b.a 2
11.c even 5 4 605.2.j.b 8
11.d odd 10 4 605.2.j.b 8
55.d odd 2 1 inner 605.2.b.a 2
55.e even 4 2 3025.2.a.k 2
55.h odd 10 4 605.2.j.b 8
55.j even 10 4 605.2.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.a 2 1.a even 1 1 trivial
605.2.b.a 2 5.b even 2 1 inner
605.2.b.a 2 11.b odd 2 1 CM
605.2.b.a 2 55.d odd 2 1 inner
605.2.j.b 8 11.c even 5 4
605.2.j.b 8 11.d odd 10 4
605.2.j.b 8 55.h odd 10 4
605.2.j.b 8 55.j even 10 4
3025.2.a.k 2 5.c odd 4 2
3025.2.a.k 2 55.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 11 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 99 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 44 \) Copy content Toggle raw display
$53$ \( T^{2} + 176 \) Copy content Toggle raw display
$59$ \( (T + 15)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 99 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 99 \) Copy content Toggle raw display
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