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Properties

Label 210.2.t.b
Level $210$
Weight $2$
Character orbit 210.t
Analytic conductor $1.677$
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] = [210,2,Mod(59,210)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(210, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) 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("210.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.67685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} - \beta_1 q^{3} - \beta_{2} q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + \beta_{3} q^{6} + (\beta_{2} + 2) q^{7} + q^{8} + (\beta_{3} + 3 \beta_{2}) q^{9} + (\beta_{3} - \beta_1 + 2) q^{10}+ \cdots + (3 \beta_{3} + 3 \beta_{2} + \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 3 q^{5} - q^{6} + 10 q^{7} + 4 q^{8} + 5 q^{9} + 6 q^{10} + 9 q^{11} + 2 q^{12} + 8 q^{13} - 8 q^{14} + 14 q^{15} - 2 q^{16} - 10 q^{18} - 12 q^{19} - 3 q^{20} - 4 q^{21}+ \cdots + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
1.68614 − 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 − 1.26217i
−0.500000 − 0.866025i −1.68614 + 0.396143i −0.500000 + 0.866025i −2.18614 + 0.469882i 1.18614 + 1.26217i 2.50000 − 0.866025i 1.00000 2.68614 − 1.33591i 1.50000 + 1.65831i
59.2 −0.500000 − 0.866025i 1.18614 − 1.26217i −0.500000 + 0.866025i 0.686141 + 2.12819i −1.68614 − 0.396143i 2.50000 − 0.866025i 1.00000 −0.186141 − 2.99422i 1.50000 − 1.65831i
89.1 −0.500000 + 0.866025i −1.68614 − 0.396143i −0.500000 − 0.866025i −2.18614 − 0.469882i 1.18614 − 1.26217i 2.50000 + 0.866025i 1.00000 2.68614 + 1.33591i 1.50000 − 1.65831i
89.2 −0.500000 + 0.866025i 1.18614 + 1.26217i −0.500000 − 0.866025i 0.686141 − 2.12819i −1.68614 + 0.396143i 2.50000 + 0.866025i 1.00000 −0.186141 + 2.99422i 1.50000 + 1.65831i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.2.t.b yes 4
3.b odd 2 1 210.2.t.d yes 4
5.b even 2 1 210.2.t.c yes 4
5.c odd 4 2 1050.2.s.d 8
7.c even 3 1 1470.2.d.c 4
7.d odd 6 1 210.2.t.a ✓ 4
7.d odd 6 1 1470.2.d.d 4
15.d odd 2 1 210.2.t.a ✓ 4
15.e even 4 2 1050.2.s.e 8
21.g even 6 1 210.2.t.c yes 4
21.g even 6 1 1470.2.d.b 4
21.h odd 6 1 1470.2.d.a 4
35.i odd 6 1 210.2.t.d yes 4
35.i odd 6 1 1470.2.d.a 4
35.j even 6 1 1470.2.d.b 4
35.k even 12 2 1050.2.s.e 8
105.o odd 6 1 1470.2.d.d 4
105.p even 6 1 inner 210.2.t.b yes 4
105.p even 6 1 1470.2.d.c 4
105.w odd 12 2 1050.2.s.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a ✓ 4 7.d odd 6 1
210.2.t.a ✓ 4 15.d odd 2 1
210.2.t.b yes 4 1.a even 1 1 trivial
210.2.t.b yes 4 105.p even 6 1 inner
210.2.t.c yes 4 5.b even 2 1
210.2.t.c yes 4 21.g even 6 1
210.2.t.d yes 4 3.b odd 2 1
210.2.t.d yes 4 35.i odd 6 1
1050.2.s.d 8 5.c odd 4 2
1050.2.s.d 8 105.w odd 12 2
1050.2.s.e 8 15.e even 4 2
1050.2.s.e 8 35.k even 12 2
1470.2.d.a 4 21.h odd 6 1
1470.2.d.a 4 35.i odd 6 1
1470.2.d.b 4 21.g even 6 1
1470.2.d.b 4 35.j even 6 1
1470.2.d.c 4 7.c even 3 1
1470.2.d.c 4 105.p even 6 1
1470.2.d.d 4 7.d odd 6 1
1470.2.d.d 4 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(210, [\chi])\):

\( T_{11}^{4} - 9T_{11}^{3} + 31T_{11}^{2} - 36T_{11} + 16 \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{23}^{4} + 3T_{23}^{3} + 15T_{23}^{2} - 18T_{23} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 2 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 9 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 44T^{2} + 1936 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 9 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 123T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{4} + 18 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$59$ \( T^{4} + 9 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$61$ \( T^{4} + 27 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} + 9 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$71$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + T^{3} + \cdots + 5476 \) Copy content Toggle raw display
$83$ \( T^{4} + 142T^{2} + 289 \) Copy content Toggle raw display
$89$ \( T^{4} - 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$97$ \( (T^{2} - 13 T - 32)^{2} \) Copy content Toggle raw display
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