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Properties

Label 1690.2.b.d
Level $1690$
Weight $2$
Character orbit 1690.b
Analytic conductor $13.495$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1690,2,Mod(339,1690)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1690.339");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.b (of order \(2\), degree \(1\), 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: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{2} q^{3} - q^{4} + ( - \beta_{7} + \beta_{4}) q^{5} - \beta_{6} q^{6} + ( - \beta_{7} - \beta_{3}) q^{7} + \beta_{4} q^{8} + (\beta_{5} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{2} q^{3} - q^{4} + ( - \beta_{7} + \beta_{4}) q^{5} - \beta_{6} q^{6} + ( - \beta_{7} - \beta_{3}) q^{7} + \beta_{4} q^{8} + (\beta_{5} - \beta_1 - 1) q^{9} + ( - \beta_{5} + 1) q^{10} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{11} + \beta_{2} q^{12} + ( - \beta_{5} + \beta_1) q^{14} + (2 \beta_{6} + \beta_{3}) q^{15} + q^{16} + ( - \beta_{5} - 2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{18} + ( - \beta_{7} + \beta_{6} + \beta_{3}) q^{19} + (\beta_{7} - \beta_{4}) q^{20} + ( - \beta_{7} + 2 \beta_{6} + \beta_{3}) q^{21} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{22} + (\beta_{5} - 3 \beta_{2} + \beta_1) q^{23} + \beta_{6} q^{24} + (\beta_{5} + \beta_{2} - 2 \beta_1 - 1) q^{25} + (\beta_{5} + \beta_1) q^{27} + (\beta_{7} + \beta_{3}) q^{28} + ( - 2 \beta_{5} + 2 \beta_1 - 2) q^{29} + ( - 2 \beta_{2} - \beta_1) q^{30} + (\beta_{7} - \beta_{6} - \beta_{3}) q^{31} - \beta_{4} q^{32} + 4 \beta_{4} q^{33} + (\beta_{7} - 2 \beta_{6} - \beta_{3}) q^{34} + ( - \beta_{5} + \beta_{2} - 2 \beta_1 - 4) q^{35} + ( - \beta_{5} + \beta_1 + 1) q^{36} + (3 \beta_{7} - 2 \beta_{4} + 3 \beta_{3}) q^{37} + ( - \beta_{5} - \beta_{2} - \beta_1) q^{38} + (\beta_{5} - 1) q^{40} - 4 \beta_{6} q^{41} + ( - \beta_{5} - 2 \beta_{2} - \beta_1) q^{42} + ( - 2 \beta_{5} + \beta_{2} - 2 \beta_1) q^{43} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{44} + ( - \beta_{6} - 5 \beta_{4} + 2 \beta_{3}) q^{45} + ( - \beta_{7} - 3 \beta_{6} + \beta_{3}) q^{46} + (\beta_{7} - 8 \beta_{4} + \beta_{3}) q^{47} - \beta_{2} q^{48} + ( - \beta_{5} + \beta_1 - 1) q^{49} + ( - \beta_{7} + \beta_{6} + \cdots - 2 \beta_{3}) q^{50}+ \cdots + ( - 3 \beta_{7} + \beta_{6} + 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 4 q^{9} + 6 q^{10} - 4 q^{14} + 8 q^{16} - 2 q^{25} - 24 q^{29} + 2 q^{30} - 30 q^{35} + 4 q^{36} - 6 q^{40} - 12 q^{49} - 52 q^{51} + 40 q^{55} + 4 q^{56} - 8 q^{61} - 8 q^{64} + 32 q^{66} - 88 q^{69} - 4 q^{74} + 30 q^{75} - 8 q^{79} - 16 q^{81} - 36 q^{90} - 60 q^{94} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 4\nu^{4} + 16\nu^{2} + 9 ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 20\nu^{2} + 27 ) / 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} + 45\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{6} + 16\nu^{4} + 44\nu^{2} + 117 ) / 36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 16\nu^{3} + 18\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} + 33\nu ) / 18 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 8\beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{2} - 4\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -9\beta_{7} + 7\beta_{6} + 16\beta_{4} + 7\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{5} - 12\beta_{2} + 4\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11\beta_{7} - 11\beta_{6} + 48\beta_{4} - 37\beta_{3} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
339.1
−1.26217 − 1.18614i
−0.396143 + 1.68614i
0.396143 + 1.68614i
1.26217 − 1.18614i
1.26217 + 1.18614i
0.396143 − 1.68614i
−0.396143 − 1.68614i
−1.26217 + 1.18614i
− 1.00000i − 2.52434i −1.00000 0.469882 + 2.18614i −2.52434 2.37228i 1.00000i −3.37228 2.18614 − 0.469882i
339.2 − 1.00000i − 0.792287i −1.00000 −2.12819 − 0.686141i −0.792287 − 3.37228i 1.00000i 2.37228 −0.686141 + 2.12819i
339.3 − 1.00000i 0.792287i −1.00000 2.12819 − 0.686141i 0.792287 − 3.37228i 1.00000i 2.37228 −0.686141 − 2.12819i
339.4 − 1.00000i 2.52434i −1.00000 −0.469882 + 2.18614i 2.52434 2.37228i 1.00000i −3.37228 2.18614 + 0.469882i
339.5 1.00000i − 2.52434i −1.00000 −0.469882 − 2.18614i 2.52434 − 2.37228i − 1.00000i −3.37228 2.18614 − 0.469882i
339.6 1.00000i − 0.792287i −1.00000 2.12819 + 0.686141i 0.792287 3.37228i − 1.00000i 2.37228 −0.686141 + 2.12819i
339.7 1.00000i 0.792287i −1.00000 −2.12819 + 0.686141i −0.792287 3.37228i − 1.00000i 2.37228 −0.686141 − 2.12819i
339.8 1.00000i 2.52434i −1.00000 0.469882 − 2.18614i −2.52434 − 2.37228i − 1.00000i −3.37228 2.18614 + 0.469882i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 339.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.b even 2 1 inner
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.b.d 8
5.b even 2 1 inner 1690.2.b.d 8
5.c odd 4 1 8450.2.a.cj 4
5.c odd 4 1 8450.2.a.cn 4
13.b even 2 1 inner 1690.2.b.d 8
13.d odd 4 1 130.2.c.a ✓ 4
13.d odd 4 1 130.2.c.b yes 4
39.f even 4 1 1170.2.f.a 4
39.f even 4 1 1170.2.f.b 4
52.f even 4 1 1040.2.f.c 4
52.f even 4 1 1040.2.f.d 4
65.d even 2 1 inner 1690.2.b.d 8
65.f even 4 2 650.2.d.e 8
65.g odd 4 1 130.2.c.a ✓ 4
65.g odd 4 1 130.2.c.b yes 4
65.h odd 4 1 8450.2.a.cj 4
65.h odd 4 1 8450.2.a.cn 4
65.k even 4 2 650.2.d.e 8
195.n even 4 1 1170.2.f.a 4
195.n even 4 1 1170.2.f.b 4
260.u even 4 1 1040.2.f.c 4
260.u even 4 1 1040.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.c.a ✓ 4 13.d odd 4 1
130.2.c.a ✓ 4 65.g odd 4 1
130.2.c.b yes 4 13.d odd 4 1
130.2.c.b yes 4 65.g odd 4 1
650.2.d.e 8 65.f even 4 2
650.2.d.e 8 65.k even 4 2
1040.2.f.c 4 52.f even 4 1
1040.2.f.c 4 260.u even 4 1
1040.2.f.d 4 52.f even 4 1
1040.2.f.d 4 260.u even 4 1
1170.2.f.a 4 39.f even 4 1
1170.2.f.a 4 195.n even 4 1
1170.2.f.b 4 39.f even 4 1
1170.2.f.b 4 195.n even 4 1
1690.2.b.d 8 1.a even 1 1 trivial
1690.2.b.d 8 5.b even 2 1 inner
1690.2.b.d 8 13.b even 2 1 inner
1690.2.b.d 8 65.d even 2 1 inner
8450.2.a.cj 4 5.c odd 4 1
8450.2.a.cj 4 65.h odd 4 1
8450.2.a.cn 4 5.c odd 4 1
8450.2.a.cn 4 65.h odd 4 1

Hecke kernels

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

\( T_{3}^{4} + 7T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 28T_{11}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 17 T^{2} + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 28 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 43 T^{2} + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 44)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 24)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 149 T^{2} + 5476)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 112 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 87 T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 129 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 28 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 44)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 32)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 175 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T - 32)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 84 T^{2} + 576)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 68 T^{2} + 1024)^{2} \) Copy content Toggle raw display
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