Properties

Label 735.2.a.b
Level $735$
Weight $2$
Character orbit 735.a
Self dual yes
Analytic conductor $5.869$
Analytic rank $1$
Dimension $1$
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] = [735,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(5.86900454856\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{9} - 2 q^{10} - 6 q^{11} + 2 q^{12} - 3 q^{13} + q^{15} - 4 q^{16} - 4 q^{17} - 2 q^{18} + q^{19} + 2 q^{20} + 12 q^{22} - 4 q^{23} + q^{25} + 6 q^{26} + q^{27} - 8 q^{29} - 2 q^{30} + q^{31} + 8 q^{32} - 6 q^{33} + 8 q^{34} + 2 q^{36} + 7 q^{37} - 2 q^{38} - 3 q^{39} - 6 q^{41} + q^{43} - 12 q^{44} + q^{45} + 8 q^{46} + 2 q^{47} - 4 q^{48} - 2 q^{50} - 4 q^{51} - 6 q^{52} + 4 q^{53} - 2 q^{54} - 6 q^{55} + q^{57} + 16 q^{58} - 8 q^{59} + 2 q^{60} - 14 q^{61} - 2 q^{62} - 8 q^{64} - 3 q^{65} + 12 q^{66} + 7 q^{67} - 8 q^{68} - 4 q^{69} + 6 q^{71} + q^{73} - 14 q^{74} + q^{75} + 2 q^{76} + 6 q^{78} - q^{79} - 4 q^{80} + q^{81} + 12 q^{82} + 2 q^{83} - 4 q^{85} - 2 q^{86} - 8 q^{87} - 12 q^{89} - 2 q^{90} - 8 q^{92} + q^{93} - 4 q^{94} + q^{95} + 8 q^{96} - 6 q^{97} - 6 q^{99}+O(q^{100}) \) 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
0
−2.00000 1.00000 2.00000 1.00000 −2.00000 0 0 1.00000 −2.00000
\(n\): 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 735.2.a.b 1
3.b odd 2 1 2205.2.a.k 1
5.b even 2 1 3675.2.a.o 1
7.b odd 2 1 735.2.a.a 1
7.c even 3 2 105.2.i.b 2
7.d odd 6 2 735.2.i.f 2
21.c even 2 1 2205.2.a.m 1
21.h odd 6 2 315.2.j.a 2
28.g odd 6 2 1680.2.bg.l 2
35.c odd 2 1 3675.2.a.p 1
35.j even 6 2 525.2.i.a 2
35.l odd 12 4 525.2.r.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.b 2 7.c even 3 2
315.2.j.a 2 21.h odd 6 2
525.2.i.a 2 35.j even 6 2
525.2.r.d 4 35.l odd 12 4
735.2.a.a 1 7.b odd 2 1
735.2.a.b 1 1.a even 1 1 trivial
735.2.i.f 2 7.d odd 6 2
1680.2.bg.l 2 28.g odd 6 2
2205.2.a.k 1 3.b odd 2 1
2205.2.a.m 1 21.c even 2 1
3675.2.a.o 1 5.b even 2 1
3675.2.a.p 1 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T + 3 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T + 8 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T - 7 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 1 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T - 4 \) Copy content Toggle raw display
$59$ \( T + 8 \) Copy content Toggle raw display
$61$ \( T + 14 \) Copy content Toggle raw display
$67$ \( T - 7 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 1 \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T - 2 \) Copy content Toggle raw display
$89$ \( T + 12 \) Copy content Toggle raw display
$97$ \( T + 6 \) Copy content Toggle raw display
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