Properties

Label 435.2.a.g
Level $435$
Weight $2$
Character orbit 435.a
Self dual yes
Analytic conductor $3.473$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + 3 q^{4} - q^{5} - \beta q^{6} + 2 q^{7} - \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + 3 q^{4} - q^{5} - \beta q^{6} + 2 q^{7} - \beta q^{8} + q^{9} + \beta q^{10} - 2 q^{11} + 3 q^{12} + 2 q^{13} - 2 \beta q^{14} - q^{15} - q^{16} + 2 \beta q^{17} - \beta q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + 2 \beta q^{22} - 2 q^{23} - \beta q^{24} + q^{25} - 2 \beta q^{26} + q^{27} + 6 q^{28} - q^{29} + \beta q^{30} - 2 q^{31} + 3 \beta q^{32} - 2 q^{33} - 10 q^{34} - 2 q^{35} + 3 q^{36} + (2 \beta + 4) q^{37} - 2 \beta q^{38} + 2 q^{39} + \beta q^{40} + 2 q^{41} - 2 \beta q^{42} + 4 q^{43} - 6 q^{44} - q^{45} + 2 \beta q^{46} + (4 \beta + 4) q^{47} - q^{48} - 3 q^{49} - \beta q^{50} + 2 \beta q^{51} + 6 q^{52} + 2 q^{53} - \beta q^{54} + 2 q^{55} - 2 \beta q^{56} + 2 q^{57} + \beta q^{58} + 8 q^{59} - 3 q^{60} + (4 \beta - 2) q^{61} + 2 \beta q^{62} + 2 q^{63} - 13 q^{64} - 2 q^{65} + 2 \beta q^{66} + (4 \beta - 6) q^{67} + 6 \beta q^{68} - 2 q^{69} + 2 \beta q^{70} + 4 q^{71} - \beta q^{72} + ( - 2 \beta + 8) q^{73} + ( - 4 \beta - 10) q^{74} + q^{75} + 6 q^{76} - 4 q^{77} - 2 \beta q^{78} + ( - 4 \beta + 6) q^{79} + q^{80} + q^{81} - 2 \beta q^{82} + ( - 4 \beta - 6) q^{83} + 6 q^{84} - 2 \beta q^{85} - 4 \beta q^{86} - q^{87} + 2 \beta q^{88} - 6 q^{89} + \beta q^{90} + 4 q^{91} - 6 q^{92} - 2 q^{93} + ( - 4 \beta - 20) q^{94} - 2 q^{95} + 3 \beta q^{96} + ( - 6 \beta - 4) q^{97} + 3 \beta q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{4} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 6 q^{12} + 4 q^{13} - 2 q^{15} - 2 q^{16} + 4 q^{19} - 6 q^{20} + 4 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 12 q^{28} - 2 q^{29} - 4 q^{31} - 4 q^{33} - 20 q^{34} - 4 q^{35} + 6 q^{36} + 8 q^{37} + 4 q^{39} + 4 q^{41} + 8 q^{43} - 12 q^{44} - 2 q^{45} + 8 q^{47} - 2 q^{48} - 6 q^{49} + 12 q^{52} + 4 q^{53} + 4 q^{55} + 4 q^{57} + 16 q^{59} - 6 q^{60} - 4 q^{61} + 4 q^{63} - 26 q^{64} - 4 q^{65} - 12 q^{67} - 4 q^{69} + 8 q^{71} + 16 q^{73} - 20 q^{74} + 2 q^{75} + 12 q^{76} - 8 q^{77} + 12 q^{79} + 2 q^{80} + 2 q^{81} - 12 q^{83} + 12 q^{84} - 2 q^{87} - 12 q^{89} + 8 q^{91} - 12 q^{92} - 4 q^{93} - 40 q^{94} - 4 q^{95} - 8 q^{97} - 4 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−2.23607 1.00000 3.00000 −1.00000 −2.23607 2.00000 −2.23607 1.00000 2.23607
1.2 2.23607 1.00000 3.00000 −1.00000 2.23607 2.00000 2.23607 1.00000 −2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(29\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.2.a.g 2
3.b odd 2 1 1305.2.a.j 2
4.b odd 2 1 6960.2.a.bp 2
5.b even 2 1 2175.2.a.o 2
5.c odd 4 2 2175.2.c.g 4
15.d odd 2 1 6525.2.a.y 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.g 2 1.a even 1 1 trivial
1305.2.a.j 2 3.b odd 2 1
2175.2.a.o 2 5.b even 2 1
2175.2.c.g 4 5.c odd 4 2
6525.2.a.y 2 15.d odd 2 1
6960.2.a.bp 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(435))\):

\( T_{2}^{2} - 5 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( (T + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T - 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$71$ \( (T - 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 16T + 44 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$89$ \( (T + 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 8T - 164 \) Copy content Toggle raw display
show more
show less