Properties

Label 735.2.i.b
Level $735$
Weight $2$
Character orbit 735.i
Analytic conductor $5.869$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,2,Mod(226,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.i (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(5.86900454856\)
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: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + ( - \zeta_{6} + 1) q^{4} + \zeta_{6} q^{5} - q^{6} - 3 q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} - \zeta_{6} q^{12} + 6 q^{13} + q^{15} + \zeta_{6} q^{16} + \cdots + 18 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} + q^{4} + q^{5} - 2 q^{6} - 6 q^{8} - q^{9} + q^{10} - q^{12} + 12 q^{13} + 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + 2 q^{20} - 8 q^{23} - 3 q^{24} - q^{25} - 6 q^{26}+ \cdots + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −1.00000 0 −3.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
361.1 −0.500000 0.866025i 0.500000 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −1.00000 0 −3.00000 −0.500000 0.866025i 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.i.b 2
7.b odd 2 1 735.2.i.a 2
7.c even 3 1 735.2.a.f 1
7.c even 3 1 inner 735.2.i.b 2
7.d odd 6 1 105.2.a.a 1
7.d odd 6 1 735.2.i.a 2
21.g even 6 1 315.2.a.a 1
21.h odd 6 1 2205.2.a.b 1
28.f even 6 1 1680.2.a.f 1
35.i odd 6 1 525.2.a.a 1
35.j even 6 1 3675.2.a.f 1
35.k even 12 2 525.2.d.b 2
56.j odd 6 1 6720.2.a.p 1
56.m even 6 1 6720.2.a.bk 1
84.j odd 6 1 5040.2.a.d 1
105.p even 6 1 1575.2.a.h 1
105.w odd 12 2 1575.2.d.b 2
140.s even 6 1 8400.2.a.co 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.a 1 7.d odd 6 1
315.2.a.a 1 21.g even 6 1
525.2.a.a 1 35.i odd 6 1
525.2.d.b 2 35.k even 12 2
735.2.a.f 1 7.c even 3 1
735.2.i.a 2 7.b odd 2 1
735.2.i.a 2 7.d odd 6 1
735.2.i.b 2 1.a even 1 1 trivial
735.2.i.b 2 7.c even 3 1 inner
1575.2.a.h 1 105.p even 6 1
1575.2.d.b 2 105.w odd 12 2
1680.2.a.f 1 28.f even 6 1
2205.2.a.b 1 21.h odd 6 1
3675.2.a.f 1 35.j even 6 1
5040.2.a.d 1 84.j odd 6 1
6720.2.a.p 1 56.j odd 6 1
6720.2.a.bk 1 56.m even 6 1
8400.2.a.co 1 140.s even 6 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$71$ \( (T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( (T - 18)^{2} \) Copy content Toggle raw display
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