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Properties

Label 1690.2.e.c
Level $1690$
Weight $2$
Character orbit 1690.e
Analytic conductor $13.495$
Analytic rank $0$
Dimension $2$
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 = [2,-1,0,-1,2,0,0] 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content 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 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + q^{5} + q^{8} + 3 \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + (\zeta_{6} - 1) q^{16} - 2 \zeta_{6} q^{17} - 3 q^{18} + 8 \zeta_{6} q^{19} - \zeta_{6} q^{20} + \cdots + 7 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{5} + 2 q^{8} + 3 q^{9} - q^{10} - q^{16} - 2 q^{17} - 6 q^{18} + 8 q^{19} - q^{20} + 4 q^{23} + 2 q^{25} + 2 q^{29} - 8 q^{31} - q^{32} + 4 q^{34} + 3 q^{36} - 6 q^{37}+ \cdots + 7 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.00000 0 0 1.00000 1.50000 + 2.59808i −0.500000 + 0.866025i
991.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 0 1.00000 1.50000 − 2.59808i −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.c 2
13.b even 2 1 1690.2.e.i 2
13.c even 3 1 130.2.a.b ✓ 1
13.c even 3 1 inner 1690.2.e.c 2
13.d odd 4 2 1690.2.l.f 4
13.e even 6 1 1690.2.a.b 1
13.e even 6 1 1690.2.e.i 2
13.f odd 12 2 1690.2.d.d 2
13.f odd 12 2 1690.2.l.f 4
39.i odd 6 1 1170.2.a.b 1
52.j odd 6 1 1040.2.a.e 1
65.l even 6 1 8450.2.a.r 1
65.n even 6 1 650.2.a.d 1
65.q odd 12 2 650.2.b.e 2
91.n odd 6 1 6370.2.a.r 1
104.n odd 6 1 4160.2.a.h 1
104.r even 6 1 4160.2.a.i 1
156.p even 6 1 9360.2.a.l 1
195.x odd 6 1 5850.2.a.bq 1
195.bl even 12 2 5850.2.e.q 2
260.v odd 6 1 5200.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.a.b ✓ 1 13.c even 3 1
650.2.a.d 1 65.n even 6 1
650.2.b.e 2 65.q odd 12 2
1040.2.a.e 1 52.j odd 6 1
1170.2.a.b 1 39.i odd 6 1
1690.2.a.b 1 13.e even 6 1
1690.2.d.d 2 13.f odd 12 2
1690.2.e.c 2 1.a even 1 1 trivial
1690.2.e.c 2 13.c even 3 1 inner
1690.2.e.i 2 13.b even 2 1
1690.2.e.i 2 13.e even 6 1
1690.2.l.f 4 13.d odd 4 2
1690.2.l.f 4 13.f odd 12 2
4160.2.a.h 1 104.n odd 6 1
4160.2.a.i 1 104.r even 6 1
5200.2.a.r 1 260.v odd 6 1
5850.2.a.bq 1 195.x odd 6 1
5850.2.e.q 2 195.bl even 12 2
6370.2.a.r 1 91.n odd 6 1
8450.2.a.r 1 65.l even 6 1
9360.2.a.l 1 156.p even 6 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19}^{2} - 8T_{19} + 64 \) Copy content Toggle raw display
\( T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) 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
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 64 \) 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
$71$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
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