Properties

Label 605.2.a.e
Level $605$
Weight $2$
Character orbit 605.a
Self dual yes
Analytic conductor $4.831$
Analytic rank $1$
Dimension $2$
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] = [605,2,Mod(1,605)]
 
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([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

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: yes
Analytic conductor: \(4.83094932229\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + q^{4} - q^{5} - \beta q^{6} - \beta q^{7} - \beta q^{8} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + q^{4} - q^{5} - \beta q^{6} - \beta q^{7} - \beta q^{8} - 2 q^{9} - \beta q^{10} - q^{12} + 2 \beta q^{13} - 3 q^{14} + q^{15} - 5 q^{16} - 4 \beta q^{17} - 2 \beta q^{18} - 2 \beta q^{19} - q^{20} + \beta q^{21} + \beta q^{24} + q^{25} + 6 q^{26} + 5 q^{27} - \beta q^{28} + \beta q^{30} - 8 q^{31} - 3 \beta q^{32} - 12 q^{34} + \beta q^{35} - 2 q^{36} - 8 q^{37} - 6 q^{38} - 2 \beta q^{39} + \beta q^{40} + 7 \beta q^{41} + 3 q^{42} + 5 \beta q^{43} + 2 q^{45} + 9 q^{47} + 5 q^{48} - 4 q^{49} + \beta q^{50} + 4 \beta q^{51} + 2 \beta q^{52} + 6 q^{53} + 5 \beta q^{54} + 3 q^{56} + 2 \beta q^{57} - 12 q^{59} + q^{60} + 5 \beta q^{61} - 8 \beta q^{62} + 2 \beta q^{63} + q^{64} - 2 \beta q^{65} - 5 q^{67} - 4 \beta q^{68} + 3 q^{70} - 12 q^{71} + 2 \beta q^{72} - 8 \beta q^{74} - q^{75} - 2 \beta q^{76} - 6 q^{78} - 6 \beta q^{79} + 5 q^{80} + q^{81} + 21 q^{82} + 2 \beta q^{83} + \beta q^{84} + 4 \beta q^{85} + 15 q^{86} + 3 q^{89} + 2 \beta q^{90} - 6 q^{91} + 8 q^{93} + 9 \beta q^{94} + 2 \beta q^{95} + 3 \beta q^{96} - 10 q^{97} - 4 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{9} - 2 q^{12} - 6 q^{14} + 2 q^{15} - 10 q^{16} - 2 q^{20} + 2 q^{25} + 12 q^{26} + 10 q^{27} - 16 q^{31} - 24 q^{34} - 4 q^{36} - 16 q^{37} - 12 q^{38} + 6 q^{42} + 4 q^{45} + 18 q^{47} + 10 q^{48} - 8 q^{49} + 12 q^{53} + 6 q^{56} - 24 q^{59} + 2 q^{60} + 2 q^{64} - 10 q^{67} + 6 q^{70} - 24 q^{71} - 2 q^{75} - 12 q^{78} + 10 q^{80} + 2 q^{81} + 42 q^{82} + 30 q^{86} + 6 q^{89} - 12 q^{91} + 16 q^{93} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−1.73205 −1.00000 1.00000 −1.00000 1.73205 1.73205 1.73205 −2.00000 1.73205
1.2 1.73205 −1.00000 1.00000 −1.00000 −1.73205 −1.73205 −1.73205 −2.00000 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b 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.a.e 2
3.b odd 2 1 5445.2.a.u 2
4.b odd 2 1 9680.2.a.bu 2
5.b even 2 1 3025.2.a.l 2
11.b odd 2 1 inner 605.2.a.e 2
11.c even 5 4 605.2.g.i 8
11.d odd 10 4 605.2.g.i 8
33.d even 2 1 5445.2.a.u 2
44.c even 2 1 9680.2.a.bu 2
55.d odd 2 1 3025.2.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.e 2 1.a even 1 1 trivial
605.2.a.e 2 11.b odd 2 1 inner
605.2.g.i 8 11.c even 5 4
605.2.g.i 8 11.d odd 10 4
3025.2.a.l 2 5.b even 2 1
3025.2.a.l 2 55.d odd 2 1
5445.2.a.u 2 3.b odd 2 1
5445.2.a.u 2 33.d even 2 1
9680.2.a.bu 2 4.b odd 2 1
9680.2.a.bu 2 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(605))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 12 \) Copy content Toggle raw display
$17$ \( T^{2} - 48 \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 147 \) Copy content Toggle raw display
$43$ \( T^{2} - 75 \) Copy content Toggle raw display
$47$ \( (T - 9)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 75 \) Copy content Toggle raw display
$67$ \( (T + 5)^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 108 \) Copy content Toggle raw display
$83$ \( T^{2} - 12 \) Copy content Toggle raw display
$89$ \( (T - 3)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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