Properties

Label 4235.2.a.l
Level $4235$
Weight $2$
Character orbit 4235.a
Self dual yes
Analytic conductor $33.817$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4235,2,Mod(1,4235)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4235, 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("4235.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4235 = 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4235.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: \(33.8166452560\)
Analytic rank: \(0\)
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: no (minimal twist has level 385)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - \beta + 1) q^{3} + q^{4} - q^{5} + (\beta - 3) q^{6} - q^{7} - \beta q^{8} + ( - 2 \beta + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - \beta + 1) q^{3} + q^{4} - q^{5} + (\beta - 3) q^{6} - q^{7} - \beta q^{8} + ( - 2 \beta + 1) q^{9} - \beta q^{10} + ( - \beta + 1) q^{12} + (\beta + 1) q^{13} - \beta q^{14} + (\beta - 1) q^{15} - 5 q^{16} + ( - 3 \beta - 3) q^{17} + (\beta - 6) q^{18} + 4 q^{19} - q^{20} + (\beta - 1) q^{21} + 6 q^{23} + ( - \beta + 3) q^{24} + q^{25} + (\beta + 3) q^{26} + 4 q^{27} - q^{28} - 2 \beta q^{29} + ( - \beta + 3) q^{30} + (\beta + 5) q^{31} - 3 \beta q^{32} + ( - 3 \beta - 9) q^{34} + q^{35} + ( - 2 \beta + 1) q^{36} + (4 \beta - 4) q^{37} + 4 \beta q^{38} - 2 q^{39} + \beta q^{40} + (\beta - 9) q^{41} + ( - \beta + 3) q^{42} + ( - 2 \beta + 4) q^{43} + (2 \beta - 1) q^{45} + 6 \beta q^{46} + (\beta + 3) q^{47} + (5 \beta - 5) q^{48} + q^{49} + \beta q^{50} + 6 q^{51} + (\beta + 1) q^{52} - 4 \beta q^{53} + 4 \beta q^{54} + \beta q^{56} + ( - 4 \beta + 4) q^{57} - 6 q^{58} + (3 \beta + 3) q^{59} + (\beta - 1) q^{60} + ( - \beta + 1) q^{61} + (5 \beta + 3) q^{62} + (2 \beta - 1) q^{63} + q^{64} + ( - \beta - 1) q^{65} + (4 \beta + 2) q^{67} + ( - 3 \beta - 3) q^{68} + ( - 6 \beta + 6) q^{69} + \beta q^{70} + ( - 2 \beta + 6) q^{71} + ( - \beta + 6) q^{72} + (3 \beta - 5) q^{73} + ( - 4 \beta + 12) q^{74} + ( - \beta + 1) q^{75} + 4 q^{76} - 2 \beta q^{78} + (4 \beta + 10) q^{79} + 5 q^{80} + (2 \beta + 1) q^{81} + ( - 9 \beta + 3) q^{82} + (\beta - 1) q^{84} + (3 \beta + 3) q^{85} + (4 \beta - 6) q^{86} + ( - 2 \beta + 6) q^{87} - 6 \beta q^{89} + ( - \beta + 6) q^{90} + ( - \beta - 1) q^{91} + 6 q^{92} + ( - 4 \beta + 2) q^{93} + (3 \beta + 3) q^{94} - 4 q^{95} + ( - 3 \beta + 9) q^{96} + 2 q^{97} + \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} - 6 q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{4} - 2 q^{5} - 6 q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{12} + 2 q^{13} - 2 q^{15} - 10 q^{16} - 6 q^{17} - 12 q^{18} + 8 q^{19} - 2 q^{20} - 2 q^{21} + 12 q^{23} + 6 q^{24} + 2 q^{25} + 6 q^{26} + 8 q^{27} - 2 q^{28} + 6 q^{30} + 10 q^{31} - 18 q^{34} + 2 q^{35} + 2 q^{36} - 8 q^{37} - 4 q^{39} - 18 q^{41} + 6 q^{42} + 8 q^{43} - 2 q^{45} + 6 q^{47} - 10 q^{48} + 2 q^{49} + 12 q^{51} + 2 q^{52} + 8 q^{57} - 12 q^{58} + 6 q^{59} - 2 q^{60} + 2 q^{61} + 6 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{65} + 4 q^{67} - 6 q^{68} + 12 q^{69} + 12 q^{71} + 12 q^{72} - 10 q^{73} + 24 q^{74} + 2 q^{75} + 8 q^{76} + 20 q^{79} + 10 q^{80} + 2 q^{81} + 6 q^{82} - 2 q^{84} + 6 q^{85} - 12 q^{86} + 12 q^{87} + 12 q^{90} - 2 q^{91} + 12 q^{92} + 4 q^{93} + 6 q^{94} - 8 q^{95} + 18 q^{96} + 4 q^{97}+O(q^{100}) \) 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 2.73205 1.00000 −1.00000 −4.73205 −1.00000 1.73205 4.46410 1.73205
1.2 1.73205 −0.732051 1.00000 −1.00000 −1.26795 −1.00000 −1.73205 −2.46410 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

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 4235.2.a.l 2
11.b odd 2 1 385.2.a.c 2
33.d even 2 1 3465.2.a.x 2
44.c even 2 1 6160.2.a.u 2
55.d odd 2 1 1925.2.a.q 2
55.e even 4 2 1925.2.b.k 4
77.b even 2 1 2695.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.c 2 11.b odd 2 1
1925.2.a.q 2 55.d odd 2 1
1925.2.b.k 4 55.e even 4 2
2695.2.a.d 2 77.b even 2 1
3465.2.a.x 2 33.d even 2 1
4235.2.a.l 2 1.a even 1 1 trivial
6160.2.a.u 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4235))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 18 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 12 \) Copy content Toggle raw display
$31$ \( T^{2} - 10T + 22 \) Copy content Toggle raw display
$37$ \( T^{2} + 8T - 32 \) Copy content Toggle raw display
$41$ \( T^{2} + 18T + 78 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$53$ \( T^{2} - 48 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T - 18 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$71$ \( T^{2} - 12T + 24 \) Copy content Toggle raw display
$73$ \( T^{2} + 10T - 2 \) Copy content Toggle raw display
$79$ \( T^{2} - 20T + 52 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 108 \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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