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Properties

Label 1690.2.e.s
Level $1690$
Weight $2$
Character orbit 1690.e
Analytic conductor $13.495$
Analytic rank $0$
Dimension $8$
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 = [8,-4,-2,-4,8,-2,0] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.22581504.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{4} - 1) q^{4} + q^{5} + \beta_{2} q^{6} - \beta_{7} q^{7} + q^{8} + (\beta_{7} + 2 \beta_{2}) q^{9} - \beta_{4} q^{10} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{11}+ \cdots + (3 \beta_{5} + 2 \beta_{3} - 5 \beta_1 - 13) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 2 q^{3} - 4 q^{4} + 8 q^{5} - 2 q^{6} + 8 q^{8} - 4 q^{9} - 4 q^{10} + 6 q^{11} + 4 q^{12} - 2 q^{15} - 4 q^{16} - 6 q^{17} + 8 q^{18} - 6 q^{19} - 4 q^{20} + 12 q^{21} + 6 q^{22} - 12 q^{23}+ \cdots - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} + 3\nu^{5} + 10\nu^{4} - 13\nu^{3} - 8\nu^{2} + 32\nu - 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 22\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - \nu^{5} + 4\nu^{4} - \nu^{3} - 6\nu^{2} + 10\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 9\nu^{6} + 5\nu^{5} + 13\nu^{4} - 21\nu^{3} - 13\nu^{2} + 54\nu - 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 24\nu^{6} + 13\nu^{5} + 38\nu^{4} - 51\nu^{3} - 32\nu^{2} + 132\nu - 104 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6\nu^{7} - 13\nu^{6} + 2\nu^{5} + 23\nu^{4} - 20\nu^{3} - 33\nu^{2} + 68\nu - 36 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{4} + 2\beta_{3} + 2\beta_{2} - 2\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + 2\beta_{6} - 2\beta_{5} - 3\beta_{4} + 3\beta_{3} + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{6} - 4\beta_{5} + 6\beta_{4} + 3\beta_{3} + 6\beta_{2} - 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 10\beta_{6} - 4\beta_{5} + 5\beta_{4} + 2\beta_{3} - 4\beta_{2} - 8\beta _1 - 9 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 8\beta_{6} - 8\beta_{5} + 11\beta_{4} - 7\beta_{3} + 8\beta_{2} - 2\beta _1 + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{6} - 10\beta_{5} - 18\beta_{4} - 3\beta_{3} - 24\beta _1 - 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{7} - 8\beta_{6} - 28\beta_{5} - 47\beta_{4} - 8\beta_{3} + 10\beta_{2} + 2\beta _1 + 51 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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
−1.27597 − 0.609843i
1.20036 − 0.747754i
1.40994 + 0.109843i
0.665665 + 1.24775i
−1.27597 + 0.609843i
1.20036 + 0.747754i
1.40994 − 0.109843i
0.665665 − 1.24775i
−0.500000 + 0.866025i −1.66612 + 2.88581i −0.500000 − 0.866025i 1.00000 −1.66612 − 2.88581i 0.719687 + 1.24653i 1.00000 −4.05193 − 7.01815i −0.500000 + 0.866025i
191.2 −0.500000 + 0.866025i −0.547394 + 0.948114i −0.500000 − 0.866025i 1.00000 −0.547394 − 0.948114i −1.99551 − 3.45632i 1.00000 0.900720 + 1.56009i −0.500000 + 0.866025i
191.3 −0.500000 + 0.866025i 0.300098 − 0.519785i −0.500000 − 0.866025i 1.00000 0.300098 + 0.519785i −0.719687 − 1.24653i 1.00000 1.31988 + 2.28610i −0.500000 + 0.866025i
191.4 −0.500000 + 0.866025i 0.913419 − 1.58209i −0.500000 − 0.866025i 1.00000 0.913419 + 1.58209i 1.99551 + 3.45632i 1.00000 −0.168669 − 0.292144i −0.500000 + 0.866025i
991.1 −0.500000 − 0.866025i −1.66612 − 2.88581i −0.500000 + 0.866025i 1.00000 −1.66612 + 2.88581i 0.719687 − 1.24653i 1.00000 −4.05193 + 7.01815i −0.500000 − 0.866025i
991.2 −0.500000 − 0.866025i −0.547394 − 0.948114i −0.500000 + 0.866025i 1.00000 −0.547394 + 0.948114i −1.99551 + 3.45632i 1.00000 0.900720 − 1.56009i −0.500000 − 0.866025i
991.3 −0.500000 − 0.866025i 0.300098 + 0.519785i −0.500000 + 0.866025i 1.00000 0.300098 − 0.519785i −0.719687 + 1.24653i 1.00000 1.31988 − 2.28610i −0.500000 − 0.866025i
991.4 −0.500000 − 0.866025i 0.913419 + 1.58209i −0.500000 + 0.866025i 1.00000 0.913419 − 1.58209i 1.99551 − 3.45632i 1.00000 −0.168669 + 0.292144i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.4
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.s 8
13.b even 2 1 1690.2.e.t 8
13.c even 3 1 1690.2.a.u 4
13.c even 3 1 inner 1690.2.e.s 8
13.d odd 4 1 130.2.l.b ✓ 8
13.d odd 4 1 1690.2.l.j 8
13.e even 6 1 1690.2.a.t 4
13.e even 6 1 1690.2.e.t 8
13.f odd 12 1 130.2.l.b ✓ 8
13.f odd 12 2 1690.2.d.k 8
13.f odd 12 1 1690.2.l.j 8
39.f even 4 1 1170.2.bs.g 8
39.k even 12 1 1170.2.bs.g 8
52.f even 4 1 1040.2.da.d 8
52.l even 12 1 1040.2.da.d 8
65.f even 4 1 650.2.n.e 8
65.g odd 4 1 650.2.m.c 8
65.k even 4 1 650.2.n.d 8
65.l even 6 1 8450.2.a.cm 4
65.n even 6 1 8450.2.a.ci 4
65.o even 12 1 650.2.n.d 8
65.s odd 12 1 650.2.m.c 8
65.t even 12 1 650.2.n.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.b ✓ 8 13.d odd 4 1
130.2.l.b ✓ 8 13.f odd 12 1
650.2.m.c 8 65.g odd 4 1
650.2.m.c 8 65.s odd 12 1
650.2.n.d 8 65.k even 4 1
650.2.n.d 8 65.o even 12 1
650.2.n.e 8 65.f even 4 1
650.2.n.e 8 65.t even 12 1
1040.2.da.d 8 52.f even 4 1
1040.2.da.d 8 52.l even 12 1
1170.2.bs.g 8 39.f even 4 1
1170.2.bs.g 8 39.k even 12 1
1690.2.a.t 4 13.e even 6 1
1690.2.a.u 4 13.c even 3 1
1690.2.d.k 8 13.f odd 12 2
1690.2.e.s 8 1.a even 1 1 trivial
1690.2.e.s 8 13.c even 3 1 inner
1690.2.e.t 8 13.b even 2 1
1690.2.e.t 8 13.e even 6 1
1690.2.l.j 8 13.d odd 4 1
1690.2.l.j 8 13.f odd 12 1
8450.2.a.ci 4 65.n even 6 1
8450.2.a.cm 4 65.l even 6 1

Hecke kernels

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

\( T_{3}^{8} + 2T_{3}^{7} + 10T_{3}^{6} - 4T_{3}^{5} + 40T_{3}^{4} + 8T_{3}^{3} + 40T_{3}^{2} - 16T_{3} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} + 18T_{7}^{6} + 291T_{7}^{4} + 594T_{7}^{2} + 1089 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{7} + 42T_{11}^{6} - 72T_{11}^{5} + 351T_{11}^{4} - 216T_{11}^{3} + 2970T_{11}^{2} + 486T_{11} + 81 \) Copy content Toggle raw display
\( T_{19}^{8} + 6T_{19}^{7} + 30T_{19}^{6} + 48T_{19}^{5} + 75T_{19}^{4} + 54T_{19}^{2} + 18T_{19} + 9 \) Copy content Toggle raw display
\( T_{31}^{4} - 18T_{31}^{3} + 90T_{31}^{2} - 132T_{31} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 18 T^{6} + \cdots + 1089 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{8} + 96 T^{6} + \cdots + 4460544 \) Copy content Toggle raw display
$31$ \( (T^{4} - 18 T^{3} + \cdots - 12)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 9801 \) Copy content Toggle raw display
$41$ \( T^{8} + 60 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$43$ \( T^{8} + 4 T^{7} + \cdots + 135424 \) Copy content Toggle raw display
$47$ \( (T^{4} - 114 T^{2} + \cdots - 639)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 30 T^{3} + \cdots - 579)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 26 T^{7} + \cdots + 394384 \) Copy content Toggle raw display
$67$ \( T^{8} - 24 T^{7} + \cdots + 9437184 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 6 T^{3} - 114 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} + \cdots - 188)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 18 T^{3} + \cdots - 2124)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 78 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$97$ \( T^{8} + 18 T^{7} + \cdots + 18558864 \) Copy content Toggle raw display
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