Properties

Label 735.4.a.l
Level $735$
Weight $4$
Character orbit 735.a
Self dual yes
Analytic conductor $43.366$
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] = [735,4,Mod(1,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.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: \(43.3664038542\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 2) q^{2} - 3 q^{3} + ( - 4 \beta - 2) q^{4} - 5 q^{5} + ( - 3 \beta + 6) q^{6} + ( - 2 \beta + 12) q^{8} + 9 q^{9} + ( - 5 \beta + 10) q^{10} + ( - 38 \beta - 16) q^{11} + (12 \beta + 6) q^{12}+ \cdots + ( - 342 \beta - 144) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 6 q^{3} - 4 q^{4} - 10 q^{5} + 12 q^{6} + 24 q^{8} + 18 q^{9} + 20 q^{10} - 32 q^{11} + 12 q^{12} + 14 q^{13} + 30 q^{15} - 24 q^{16} - 20 q^{17} - 36 q^{18} + 18 q^{19} + 20 q^{20} - 88 q^{22}+ \cdots - 288 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−3.41421 −3.00000 3.65685 −5.00000 10.2426 0 14.8284 9.00000 17.0711
1.2 −0.585786 −3.00000 −7.65685 −5.00000 1.75736 0 9.17157 9.00000 2.92893
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

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 735.4.a.l 2
3.b odd 2 1 2205.4.a.bd 2
7.b odd 2 1 735.4.a.n 2
7.c even 3 2 105.4.i.b 4
21.c even 2 1 2205.4.a.bc 2
21.h odd 6 2 315.4.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.b 4 7.c even 3 2
315.4.j.d 4 21.h odd 6 2
735.4.a.l 2 1.a even 1 1 trivial
735.4.a.n 2 7.b odd 2 1
2205.4.a.bc 2 21.c even 2 1
2205.4.a.bd 2 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):

\( T_{2}^{2} + 4T_{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 32T_{11} - 2632 \) Copy content Toggle raw display
\( T_{13}^{2} - 14T_{13} - 1303 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32T - 2632 \) Copy content Toggle raw display
$13$ \( T^{2} - 14T - 1303 \) Copy content Toggle raw display
$17$ \( T^{2} + 20T - 2212 \) Copy content Toggle raw display
$19$ \( T^{2} - 18T - 8111 \) Copy content Toggle raw display
$23$ \( T^{2} + 68T - 196 \) Copy content Toggle raw display
$29$ \( T^{2} + 332T + 27484 \) Copy content Toggle raw display
$31$ \( T^{2} - 66T - 1503 \) Copy content Toggle raw display
$37$ \( T^{2} - 18T - 98487 \) Copy content Toggle raw display
$41$ \( T^{2} - 152T - 23992 \) Copy content Toggle raw display
$43$ \( T^{2} + 842T + 174353 \) Copy content Toggle raw display
$47$ \( T^{2} + 212T - 17564 \) Copy content Toggle raw display
$53$ \( T^{2} + 368T + 27584 \) Copy content Toggle raw display
$59$ \( T^{2} + 140T - 64292 \) Copy content Toggle raw display
$61$ \( T^{2} + 732T + 126756 \) Copy content Toggle raw display
$67$ \( T^{2} - 1066 T + 159089 \) Copy content Toggle raw display
$71$ \( T^{2} + 1208 T + 243784 \) Copy content Toggle raw display
$73$ \( T^{2} - 1654 T + 683729 \) Copy content Toggle raw display
$79$ \( T^{2} - 1134 T + 319441 \) Copy content Toggle raw display
$83$ \( T^{2} - 968T + 213448 \) Copy content Toggle raw display
$89$ \( T^{2} - 204 T - 2177828 \) Copy content Toggle raw display
$97$ \( T^{2} - 1692 T - 367676 \) Copy content Toggle raw display
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