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Properties

Label 1690.2.c.f
Level $1690$
Weight $2$
Character orbit 1690.c
Analytic conductor $13.495$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1690,2,Mod(1689,1690)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1690, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1690.1689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.4947179416\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.50027374224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + (\beta_{6} - \beta_{5} - 2) q^{7} + q^{8} + (\beta_{6} - \beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - \beta_{5} q^{5} + \beta_1 q^{6} + (\beta_{6} - \beta_{5} - 2) q^{7} + q^{8} + (\beta_{6} - \beta_{4} - 2) q^{9} - \beta_{5} q^{10} + (\beta_{7} - \beta_1) q^{11} + \beta_1 q^{12} + (\beta_{6} - \beta_{5} - 2) q^{14} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1) q^{15}+ \cdots + ( - 2 \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 3 q^{5} - 10 q^{7} + 8 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 3 q^{5} - 10 q^{7} + 8 q^{8} - 16 q^{9} + 3 q^{10} - 10 q^{14} - 8 q^{15} + 8 q^{16} - 16 q^{18} + 3 q^{20} + 5 q^{25} - 10 q^{28} + 6 q^{29} - 8 q^{30} + 8 q^{32} + 20 q^{33} + 18 q^{35} - 16 q^{36} - 40 q^{37} + 3 q^{40} - 27 q^{45} + 6 q^{47} + 30 q^{49} + 5 q^{50} - 20 q^{51} - 8 q^{55} - 10 q^{56} + 24 q^{57} + 6 q^{58} - 8 q^{60} + 10 q^{61} + 50 q^{63} + 8 q^{64} + 20 q^{66} + 32 q^{67} + 4 q^{69} + 18 q^{70} - 16 q^{72} + 26 q^{73} - 40 q^{74} - 48 q^{75} + 4 q^{79} + 3 q^{80} - 16 q^{81} - 48 q^{83} + 5 q^{85} - 27 q^{90} - 36 q^{93} + 6 q^{94} - 24 q^{95} + 8 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} + 132x^{4} + 332x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 10\nu^{2} + 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 20\nu^{5} - 116\nu^{3} - 172\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{6} + 20\nu^{5} + 128\nu^{4} + 132\nu^{3} + 544\nu^{2} + 268\nu + 544 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 8\nu^{6} + 20\nu^{5} - 128\nu^{4} + 132\nu^{3} - 544\nu^{2} + 268\nu - 544 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 8\nu^{6} + 20\nu^{5} + 128\nu^{4} + 132\nu^{3} + 608\nu^{2} + 268\nu + 864 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 16\nu^{5} - 72\nu^{3} - 84\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{6} + 10\beta_{4} + 2\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 22\beta_{5} - 22\beta_{4} - 30\beta_{3} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 92\beta_{6} - 4\beta_{5} - 88\beta_{4} - 32\beta_{2} - 272 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -40\beta_{7} + 208\beta_{5} + 208\beta_{4} + 336\beta_{3} - 356\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1689.1
− 3.07108i
− 2.40987i
− 1.83766i
− 1.17644i
1.17644i
1.83766i
2.40987i
3.07108i
1.00000 − 3.07108i 1.00000 1.36822 − 1.76861i − 3.07108i −3.69510 1.00000 −6.43154 1.36822 − 1.76861i
1689.2 1.00000 − 2.40987i 1.00000 2.10653 − 0.750022i − 2.40987i 1.40561 1.00000 −2.80745 2.10653 − 0.750022i
1689.3 1.00000 − 1.83766i 1.00000 −2.21022 + 0.339024i − 1.83766i −4.79742 1.00000 −0.376989 −2.21022 + 0.339024i
1689.4 1.00000 − 1.17644i 1.00000 0.235468 + 2.22364i − 1.17644i 2.08692 1.00000 1.61598 0.235468 + 2.22364i
1689.5 1.00000 1.17644i 1.00000 0.235468 − 2.22364i 1.17644i 2.08692 1.00000 1.61598 0.235468 − 2.22364i
1689.6 1.00000 1.83766i 1.00000 −2.21022 − 0.339024i 1.83766i −4.79742 1.00000 −0.376989 −2.21022 − 0.339024i
1689.7 1.00000 2.40987i 1.00000 2.10653 + 0.750022i 2.40987i 1.40561 1.00000 −2.80745 2.10653 + 0.750022i
1689.8 1.00000 3.07108i 1.00000 1.36822 + 1.76861i 3.07108i −3.69510 1.00000 −6.43154 1.36822 + 1.76861i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1689.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1690.2.c.f 8
5.b even 2 1 1690.2.c.e 8
13.b even 2 1 1690.2.c.e 8
13.c even 3 1 130.2.m.a ✓ 8
13.d odd 4 2 1690.2.b.e 16
13.e even 6 1 130.2.m.b yes 8
39.h odd 6 1 1170.2.bj.a 8
39.i odd 6 1 1170.2.bj.b 8
52.i odd 6 1 1040.2.df.a 8
52.j odd 6 1 1040.2.df.c 8
65.d even 2 1 inner 1690.2.c.f 8
65.f even 4 2 8450.2.a.cr 8
65.g odd 4 2 1690.2.b.e 16
65.k even 4 2 8450.2.a.cs 8
65.l even 6 1 130.2.m.a ✓ 8
65.n even 6 1 130.2.m.b yes 8
65.q odd 12 2 650.2.m.e 16
65.r odd 12 2 650.2.m.e 16
195.x odd 6 1 1170.2.bj.a 8
195.y odd 6 1 1170.2.bj.b 8
260.v odd 6 1 1040.2.df.a 8
260.w odd 6 1 1040.2.df.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.m.a ✓ 8 13.c even 3 1
130.2.m.a ✓ 8 65.l even 6 1
130.2.m.b yes 8 13.e even 6 1
130.2.m.b yes 8 65.n even 6 1
650.2.m.e 16 65.q odd 12 2
650.2.m.e 16 65.r odd 12 2
1040.2.df.a 8 52.i odd 6 1
1040.2.df.a 8 260.v odd 6 1
1040.2.df.c 8 52.j odd 6 1
1040.2.df.c 8 260.w odd 6 1
1170.2.bj.a 8 39.h odd 6 1
1170.2.bj.a 8 195.x odd 6 1
1170.2.bj.b 8 39.i odd 6 1
1170.2.bj.b 8 195.y odd 6 1
1690.2.b.e 16 13.d odd 4 2
1690.2.b.e 16 65.g odd 4 2
1690.2.c.e 8 5.b even 2 1
1690.2.c.e 8 13.b even 2 1
1690.2.c.f 8 1.a even 1 1 trivial
1690.2.c.f 8 65.d even 2 1 inner
8450.2.a.cr 8 65.f even 4 2
8450.2.a.cs 8 65.k even 4 2

Hecke kernels

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

\( T_{3}^{8} + 20T_{3}^{6} + 132T_{3}^{4} + 332T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{3} - 9T_{7}^{2} - 37T_{7} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 20 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 5 T^{3} - 9 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 41 T^{6} + \cdots + 784 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 59 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} + 45 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$23$ \( T^{8} + 44 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{4} - 3 T^{3} - 18 T^{2} + \cdots - 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 60 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$37$ \( (T^{4} + 20 T^{3} + \cdots + 109)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 203 T^{6} + \cdots + 1106704 \) Copy content Toggle raw display
$43$ \( T^{8} + 180 T^{6} + \cdots + 2359296 \) Copy content Toggle raw display
$47$ \( (T^{4} - 3 T^{3} - 117 T^{2} + \cdots - 96)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 356 T^{6} + \cdots + 17783089 \) Copy content Toggle raw display
$59$ \( T^{8} + 212 T^{6} + \cdots + 369664 \) Copy content Toggle raw display
$61$ \( (T^{4} - 5 T^{3} + \cdots + 5362)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 320 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$73$ \( (T^{4} - 13 T^{3} + \cdots - 1406)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots + 1384)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 24 T^{3} + \cdots - 9744)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 425 T^{6} + \cdots + 59474944 \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots - 572)^{2} \) Copy content Toggle raw display
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