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Properties

Label 1690.2.e.m
Level $1690$
Weight $2$
Character orbit 1690.e
Analytic conductor $13.495$
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] = [1690,2,Mod(191,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) 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("1690.191"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4947179416\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) 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 130)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_1 q^{3} + \beta_{2} q^{4} + q^{5} + (\beta_{3} + \beta_1) q^{6} + \beta_{2} q^{7} - q^{8} + 7 \beta_{2} q^{9} + (\beta_{2} + 1) q^{10} + ( - \beta_{2} - \beta_1 - 1) q^{11}+ \cdots + ( - 7 \beta_{3} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 2 q^{7} - 4 q^{8} - 14 q^{9} + 2 q^{10} - 2 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{17} - 28 q^{18} - 10 q^{19} - 2 q^{20} + 2 q^{22} - 12 q^{23} + 4 q^{25} - 2 q^{28} - 8 q^{29}+ \cdots + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−1.58114 + 2.73861i
1.58114 − 2.73861i
−1.58114 − 2.73861i
1.58114 + 2.73861i
0.500000 − 0.866025i −1.58114 + 2.73861i −0.500000 − 0.866025i 1.00000 1.58114 + 2.73861i −0.500000 − 0.866025i −1.00000 −3.50000 − 6.06218i 0.500000 − 0.866025i
191.2 0.500000 − 0.866025i 1.58114 − 2.73861i −0.500000 − 0.866025i 1.00000 −1.58114 − 2.73861i −0.500000 − 0.866025i −1.00000 −3.50000 − 6.06218i 0.500000 − 0.866025i
991.1 0.500000 + 0.866025i −1.58114 − 2.73861i −0.500000 + 0.866025i 1.00000 1.58114 − 2.73861i −0.500000 + 0.866025i −1.00000 −3.50000 + 6.06218i 0.500000 + 0.866025i
991.2 0.500000 + 0.866025i 1.58114 + 2.73861i −0.500000 + 0.866025i 1.00000 −1.58114 + 2.73861i −0.500000 + 0.866025i −1.00000 −3.50000 + 6.06218i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1690.2.e.m 4
13.b even 2 1 130.2.e.c ✓ 4
13.c even 3 1 1690.2.a.k 2
13.c even 3 1 inner 1690.2.e.m 4
13.d odd 4 2 1690.2.l.k 8
13.e even 6 1 130.2.e.c ✓ 4
13.e even 6 1 1690.2.a.n 2
13.f odd 12 2 1690.2.d.g 4
13.f odd 12 2 1690.2.l.k 8
39.d odd 2 1 1170.2.i.q 4
39.h odd 6 1 1170.2.i.q 4
52.b odd 2 1 1040.2.q.m 4
52.i odd 6 1 1040.2.q.m 4
65.d even 2 1 650.2.e.h 4
65.h odd 4 2 650.2.o.g 8
65.l even 6 1 650.2.e.h 4
65.l even 6 1 8450.2.a.bc 2
65.n even 6 1 8450.2.a.bj 2
65.r odd 12 2 650.2.o.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.c ✓ 4 13.b even 2 1
130.2.e.c ✓ 4 13.e even 6 1
650.2.e.h 4 65.d even 2 1
650.2.e.h 4 65.l even 6 1
650.2.o.g 8 65.h odd 4 2
650.2.o.g 8 65.r odd 12 2
1040.2.q.m 4 52.b odd 2 1
1040.2.q.m 4 52.i odd 6 1
1170.2.i.q 4 39.d odd 2 1
1170.2.i.q 4 39.h odd 6 1
1690.2.a.k 2 13.c even 3 1
1690.2.a.n 2 13.e even 6 1
1690.2.d.g 4 13.f odd 12 2
1690.2.e.m 4 1.a even 1 1 trivial
1690.2.e.m 4 13.c even 3 1 inner
1690.2.l.k 8 13.d odd 4 2
1690.2.l.k 8 13.f odd 12 2
8450.2.a.bc 2 65.l even 6 1
8450.2.a.bj 2 65.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1690, [\chi])\):

\( T_{3}^{4} + 10T_{3}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} + 13T_{11}^{2} - 18T_{11} + 81 \) Copy content Toggle raw display
\( T_{19}^{4} + 10T_{19}^{3} + 85T_{19}^{2} + 150T_{19} + 225 \) Copy content Toggle raw display
\( T_{31}^{2} + 8T_{31} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 10T^{2} + 100 \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + \cdots + 225 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 6 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T - 3)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 7396 \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$71$ \( T^{4} - 4 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( (T^{2} + 12 T + 26)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T - 74)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 2 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + \cdots + 7396 \) Copy content Toggle raw display
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