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Properties

Label 735.2.i.j
Level $735$
Weight $2$
Character orbit 735.i
Analytic conductor $5.869$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [735,2,Mod(226,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_{3} q^{6} - 2 \beta_{3} q^{8} + ( - \beta_{2} - 1) q^{9} + ( - \beta_{3} - \beta_1) q^{10} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{11}+ \cdots + (\beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} - 2 q^{9} - 4 q^{11} + 12 q^{13} + 4 q^{15} + 8 q^{16} - 4 q^{17} - 2 q^{19} + 8 q^{22} + 4 q^{23} - 2 q^{25} + 4 q^{26} + 4 q^{27} + 16 q^{29} - 6 q^{31} - 4 q^{33} + 24 q^{34}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.41421 0 −2.82843 −0.500000 + 0.866025i −0.707107 − 1.22474i
226.2 0.707107 − 1.22474i −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.41421 0 2.82843 −0.500000 + 0.866025i 0.707107 + 1.22474i
361.1 −0.707107 − 1.22474i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.41421 0 −2.82843 −0.500000 − 0.866025i −0.707107 + 1.22474i
361.2 0.707107 + 1.22474i −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.41421 0 2.82843 −0.500000 − 0.866025i 0.707107 − 1.22474i
\(n\): 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.j 4
7.b odd 2 1 105.2.i.c ✓ 4
7.c even 3 1 735.2.a.j 2
7.c even 3 1 inner 735.2.i.j 4
7.d odd 6 1 105.2.i.c ✓ 4
7.d odd 6 1 735.2.a.i 2
21.c even 2 1 315.2.j.d 4
21.g even 6 1 315.2.j.d 4
21.g even 6 1 2205.2.a.u 2
21.h odd 6 1 2205.2.a.s 2
28.d even 2 1 1680.2.bg.p 4
28.f even 6 1 1680.2.bg.p 4
35.c odd 2 1 525.2.i.g 4
35.f even 4 2 525.2.r.g 8
35.i odd 6 1 525.2.i.g 4
35.i odd 6 1 3675.2.a.z 2
35.j even 6 1 3675.2.a.x 2
35.k even 12 2 525.2.r.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.c ✓ 4 7.b odd 2 1
105.2.i.c ✓ 4 7.d odd 6 1
315.2.j.d 4 21.c even 2 1
315.2.j.d 4 21.g even 6 1
525.2.i.g 4 35.c odd 2 1
525.2.i.g 4 35.i odd 6 1
525.2.r.g 8 35.f even 4 2
525.2.r.g 8 35.k even 12 2
735.2.a.i 2 7.d odd 6 1
735.2.a.j 2 7.c even 3 1
735.2.i.j 4 1.a even 1 1 trivial
735.2.i.j 4 7.c even 3 1 inner
1680.2.bg.p 4 28.d even 2 1
1680.2.bg.p 4 28.f even 6 1
2205.2.a.s 2 21.h odd 6 1
2205.2.a.u 2 21.g even 6 1
3675.2.a.x 2 35.j even 6 1
3675.2.a.z 2 35.i odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 2T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 6T_{13} + 7 \) Copy content Toggle raw display
\( T_{17}^{4} + 4T_{17}^{3} + 30T_{17}^{2} - 56T_{17} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 6 T + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{4} - 14 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 14)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18 T + 79)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + \cdots + 15376 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + \cdots + 25921 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + \cdots + 625 \) Copy content Toggle raw display
$79$ \( T^{4} + 2 T^{3} + \cdots + 961 \) Copy content Toggle raw display
$83$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 16 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
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