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Properties

Label 1690.2.e.q
Level $1690$
Weight $2$
Character orbit 1690.e
Analytic conductor $13.495$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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 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);
 
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")
 
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 = [6,3,-1,-3,6,1,5] 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 1) q^{2} - \beta_1 q^{3} - \beta_{5} q^{4} + q^{5} + (\beta_{2} - \beta_1) q^{6} + (2 \beta_{5} + \beta_{4} + \beta_{3}) q^{7} - q^{8} + (2 \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{9}+ \cdots + ( - 2 \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - q^{3} - 3 q^{4} + 6 q^{5} + q^{6} + 5 q^{7} - 6 q^{8} + 4 q^{9} + 3 q^{10} + 4 q^{11} + 2 q^{12} + 10 q^{14} - q^{15} - 3 q^{16} - 10 q^{17} + 8 q^{18} + 8 q^{19} - 3 q^{20} - 8 q^{21}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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
0.900969 − 1.56052i
0.222521 − 0.385418i
−0.623490 + 1.07992i
0.900969 + 1.56052i
0.222521 + 0.385418i
−0.623490 − 1.07992i
0.500000 − 0.866025i −0.900969 + 1.56052i −0.500000 − 0.866025i 1.00000 0.900969 + 1.56052i 0.777479 + 1.34663i −1.00000 −0.123490 − 0.213891i 0.500000 − 0.866025i
191.2 0.500000 − 0.866025i −0.222521 + 0.385418i −0.500000 − 0.866025i 1.00000 0.222521 + 0.385418i 1.62349 + 2.81197i −1.00000 1.40097 + 2.42655i 0.500000 − 0.866025i
191.3 0.500000 − 0.866025i 0.623490 − 1.07992i −0.500000 − 0.866025i 1.00000 −0.623490 − 1.07992i 0.0990311 + 0.171527i −1.00000 0.722521 + 1.25144i 0.500000 − 0.866025i
991.1 0.500000 + 0.866025i −0.900969 − 1.56052i −0.500000 + 0.866025i 1.00000 0.900969 − 1.56052i 0.777479 − 1.34663i −1.00000 −0.123490 + 0.213891i 0.500000 + 0.866025i
991.2 0.500000 + 0.866025i −0.222521 − 0.385418i −0.500000 + 0.866025i 1.00000 0.222521 − 0.385418i 1.62349 − 2.81197i −1.00000 1.40097 − 2.42655i 0.500000 + 0.866025i
991.3 0.500000 + 0.866025i 0.623490 + 1.07992i −0.500000 + 0.866025i 1.00000 −0.623490 + 1.07992i 0.0990311 − 0.171527i −1.00000 0.722521 − 1.25144i 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.3
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.q 6
13.b even 2 1 1690.2.e.o 6
13.c even 3 1 1690.2.a.q ✓ 3
13.c even 3 1 inner 1690.2.e.q 6
13.d odd 4 2 1690.2.l.l 12
13.e even 6 1 1690.2.a.s yes 3
13.e even 6 1 1690.2.e.o 6
13.f odd 12 2 1690.2.d.j 6
13.f odd 12 2 1690.2.l.l 12
65.l even 6 1 8450.2.a.bo 3
65.n even 6 1 8450.2.a.bz 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1690.2.a.q ✓ 3 13.c even 3 1
1690.2.a.s yes 3 13.e even 6 1
1690.2.d.j 6 13.f odd 12 2
1690.2.e.o 6 13.b even 2 1
1690.2.e.o 6 13.e even 6 1
1690.2.e.q 6 1.a even 1 1 trivial
1690.2.e.q 6 13.c even 3 1 inner
1690.2.l.l 12 13.d odd 4 2
1690.2.l.l 12 13.f odd 12 2
8450.2.a.bo 3 65.l even 6 1
8450.2.a.bz 3 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 \(S_{2}^{\mathrm{new}}(1690, [\chi])\):

\( T_{3}^{6} + T_{3}^{5} + 3T_{3}^{4} + 5T_{3}^{2} + 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} - 5T_{7}^{5} + 19T_{7}^{4} - 28T_{7}^{3} + 31T_{7}^{2} - 6T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} - 4T_{11}^{5} + 20T_{11}^{4} + 48T_{11}^{2} - 32T_{11} + 64 \) Copy content Toggle raw display
\( T_{19}^{6} - 8T_{19}^{5} + 52T_{19}^{4} - 112T_{19}^{3} + 208T_{19}^{2} + 96T_{19} + 64 \) Copy content Toggle raw display
\( T_{31}^{3} + 8T_{31}^{2} - 44T_{31} - 344 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} - 4 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 10 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{6} - 8 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + \cdots + 19321 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{3} + 8 T^{2} + \cdots - 344)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 8 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{6} - 11 T^{5} + \cdots + 683929 \) Copy content Toggle raw display
$43$ \( T^{6} - T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$47$ \( (T^{3} + 15 T^{2} + \cdots - 71)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 4 T^{2} + \cdots + 568)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 24 T^{5} + \cdots + 118336 \) Copy content Toggle raw display
$61$ \( T^{6} - 15 T^{5} + \cdots + 284089 \) Copy content Toggle raw display
$67$ \( T^{6} - 5 T^{5} + \cdots + 28561 \) Copy content Toggle raw display
$71$ \( T^{6} - 32 T^{5} + \cdots + 1236544 \) Copy content Toggle raw display
$73$ \( (T^{3} - 26 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 2 T^{2} + \cdots + 344)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 7 T^{2} + \cdots - 287)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 17 T^{5} + \cdots + 4826809 \) Copy content Toggle raw display
$97$ \( T^{6} + 6 T^{5} + \cdots + 1236544 \) Copy content Toggle raw display
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