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Properties

Label 735.2.i.k
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 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")
 
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);
 
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,-6] 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: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_1 q^{3} + 3 \beta_1 q^{4} + (\beta_1 + 1) q^{5} + \beta_{3} q^{6} - \beta_{3} q^{8} + ( - \beta_1 - 1) q^{9} + ( - \beta_{3} - \beta_{2}) q^{10} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{11}+ \cdots + (2 \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 6 q^{4} + 2 q^{5} - 2 q^{9} - 4 q^{11} + 6 q^{12} + 4 q^{15} + 2 q^{16} + 4 q^{17} - 4 q^{19} - 12 q^{20} - 40 q^{22} - 8 q^{23} - 2 q^{25} + 20 q^{26} - 4 q^{27} - 8 q^{29} - 12 q^{31}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display

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

Character values

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

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

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
0.809017 − 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 − 0.535233i
−1.11803 + 1.93649i 0.500000 + 0.866025i −1.50000 − 2.59808i 0.500000 − 0.866025i −2.23607 0 2.23607 −0.500000 + 0.866025i 1.11803 + 1.93649i
226.2 1.11803 − 1.93649i 0.500000 + 0.866025i −1.50000 − 2.59808i 0.500000 − 0.866025i 2.23607 0 −2.23607 −0.500000 + 0.866025i −1.11803 − 1.93649i
361.1 −1.11803 − 1.93649i 0.500000 − 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i −2.23607 0 2.23607 −0.500000 − 0.866025i 1.11803 − 1.93649i
361.2 1.11803 + 1.93649i 0.500000 − 0.866025i −1.50000 + 2.59808i 0.500000 + 0.866025i 2.23607 0 −2.23607 −0.500000 − 0.866025i −1.11803 + 1.93649i
\(n\): 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.k 4
7.b odd 2 1 735.2.i.i 4
7.c even 3 1 105.2.a.b ✓ 2
7.c even 3 1 inner 735.2.i.k 4
7.d odd 6 1 735.2.a.k 2
7.d odd 6 1 735.2.i.i 4
21.g even 6 1 2205.2.a.w 2
21.h odd 6 1 315.2.a.d 2
28.g odd 6 1 1680.2.a.v 2
35.i odd 6 1 3675.2.a.y 2
35.j even 6 1 525.2.a.g 2
35.l odd 12 2 525.2.d.c 4
56.k odd 6 1 6720.2.a.cs 2
56.p even 6 1 6720.2.a.cx 2
84.n even 6 1 5040.2.a.bw 2
105.o odd 6 1 1575.2.a.r 2
105.x even 12 2 1575.2.d.d 4
140.p odd 6 1 8400.2.a.cx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b ✓ 2 7.c even 3 1
315.2.a.d 2 21.h odd 6 1
525.2.a.g 2 35.j even 6 1
525.2.d.c 4 35.l odd 12 2
735.2.a.k 2 7.d odd 6 1
735.2.i.i 4 7.b odd 2 1
735.2.i.i 4 7.d odd 6 1
735.2.i.k 4 1.a even 1 1 trivial
735.2.i.k 4 7.c even 3 1 inner
1575.2.a.r 2 105.o odd 6 1
1575.2.d.d 4 105.x even 12 2
1680.2.a.v 2 28.g odd 6 1
2205.2.a.w 2 21.g even 6 1
3675.2.a.y 2 35.i odd 6 1
5040.2.a.bw 2 84.n even 6 1
6720.2.a.cs 2 56.k odd 6 1
6720.2.a.cx 2 56.p even 6 1
8400.2.a.cx 2 140.p odd 6 1

Hecke kernels

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

\( T_{2}^{4} + 5T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5T^{2} + 25 \) 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
$11$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 4 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 5776 \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$59$ \( T^{4} + 80T^{2} + 6400 \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 20 T + 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( (T^{2} + 16 T - 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
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