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Properties

Label 1690.2.b.c
Level $1690$
Weight $2$
Character orbit 1690.b
Analytic conductor $13.495$
Analytic rank $0$
Dimension $6$
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(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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{5} q^{3} - q^{4} + ( - \beta_{4} - \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{7} - \beta_{3} q^{8} + ( - \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{5} q^{3} - q^{4} + ( - \beta_{4} - \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3}) q^{7} - \beta_{3} q^{8} + ( - \beta_{2} + \beta_1) q^{9} + (\beta_{5} - \beta_{2}) q^{10} + (3 \beta_{2} + 2) q^{11} + \beta_{5} q^{12} + (\beta_{2} - \beta_1 + 2) q^{14} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots - 1) q^{15}+ \cdots + (4 \beta_{2} - \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} + 2 q^{9} + 2 q^{10} + 6 q^{11} + 10 q^{14} - 4 q^{15} + 6 q^{16} + 26 q^{19} - 12 q^{21} + 2 q^{25} + 28 q^{29} + 16 q^{30} + 12 q^{31} - 8 q^{34} + 6 q^{35} - 2 q^{36} - 2 q^{40} + 4 q^{41} - 6 q^{44} - 12 q^{45} + 4 q^{49} + 8 q^{50} + 4 q^{51} - 12 q^{54} - 12 q^{55} - 10 q^{56} - 16 q^{59} + 4 q^{60} + 8 q^{61} - 6 q^{64} - 12 q^{66} - 28 q^{69} + 12 q^{70} - 40 q^{71} + 10 q^{74} + 8 q^{75} - 26 q^{76} + 28 q^{79} - 26 q^{81} + 12 q^{84} + 16 q^{85} + 12 q^{86} - 14 q^{89} + 10 q^{90} + 14 q^{94} + 8 q^{95} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) 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
0.403032 − 0.403032i
−0.854638 + 0.854638i
1.45161 − 1.45161i
1.45161 + 1.45161i
−0.854638 − 0.854638i
0.403032 + 0.403032i
− 1.00000i − 1.67513i −1.00000 −1.67513 − 1.48119i −1.67513 1.80606i 1.00000i 0.193937 −1.48119 + 1.67513i
339.2 − 1.00000i − 0.539189i −1.00000 −0.539189 + 2.17009i −0.539189 − 0.709275i 1.00000i 2.70928 2.17009 + 0.539189i
339.3 − 1.00000i 2.21432i −1.00000 2.21432 + 0.311108i 2.21432 3.90321i 1.00000i −1.90321 0.311108 − 2.21432i
339.4 1.00000i − 2.21432i −1.00000 2.21432 − 0.311108i 2.21432 − 3.90321i − 1.00000i −1.90321 0.311108 + 2.21432i
339.5 1.00000i 0.539189i −1.00000 −0.539189 − 2.17009i −0.539189 0.709275i − 1.00000i 2.70928 2.17009 − 0.539189i
339.6 1.00000i 1.67513i −1.00000 −1.67513 + 1.48119i −1.67513 − 1.80606i − 1.00000i 0.193937 −1.48119 − 1.67513i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 339.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.c 6
5.b even 2 1 inner 1690.2.b.c 6
5.c odd 4 1 8450.2.a.bu 3
5.c odd 4 1 8450.2.a.cb 3
13.b even 2 1 1690.2.b.b 6
13.c even 3 2 130.2.n.a ✓ 12
13.d odd 4 1 1690.2.c.b 6
13.d odd 4 1 1690.2.c.c 6
39.i odd 6 2 1170.2.bp.h 12
52.j odd 6 2 1040.2.dh.b 12
65.d even 2 1 1690.2.b.b 6
65.g odd 4 1 1690.2.c.b 6
65.g odd 4 1 1690.2.c.c 6
65.h odd 4 1 8450.2.a.bt 3
65.h odd 4 1 8450.2.a.ca 3
65.n even 6 2 130.2.n.a ✓ 12
65.q odd 12 2 650.2.e.j 6
65.q odd 12 2 650.2.e.k 6
195.x odd 6 2 1170.2.bp.h 12
260.v odd 6 2 1040.2.dh.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.n.a ✓ 12 13.c even 3 2
130.2.n.a ✓ 12 65.n even 6 2
650.2.e.j 6 65.q odd 12 2
650.2.e.k 6 65.q odd 12 2
1040.2.dh.b 12 52.j odd 6 2
1040.2.dh.b 12 260.v odd 6 2
1170.2.bp.h 12 39.i odd 6 2
1170.2.bp.h 12 195.x odd 6 2
1690.2.b.b 6 13.b even 2 1
1690.2.b.b 6 65.d even 2 1
1690.2.b.c 6 1.a even 1 1 trivial
1690.2.b.c 6 5.b even 2 1 inner
1690.2.c.b 6 13.d odd 4 1
1690.2.c.b 6 65.g odd 4 1
1690.2.c.c 6 13.d odd 4 1
1690.2.c.c 6 65.g odd 4 1
8450.2.a.bt 3 65.h odd 4 1
8450.2.a.bu 3 5.c odd 4 1
8450.2.a.ca 3 65.h odd 4 1
8450.2.a.cb 3 5.c odd 4 1

Hecke kernels

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

\( T_{3}^{6} + 8T_{3}^{4} + 16T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 27T_{11} + 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 19 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( (T^{3} - 3 T^{2} - 27 T + 31)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( (T^{3} - 13 T^{2} + 43 T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 80 T^{4} + \cdots + 5776 \) Copy content Toggle raw display
$29$ \( (T^{3} - 14 T^{2} + \cdots + 152)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} + \cdots + 218)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 83 T^{4} + \cdots + 11449 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} - 44 T + 20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$47$ \( T^{6} + 27 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{6} + 95 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} + \cdots - 272)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 4 T^{2} + \cdots + 610)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 80 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( (T^{3} + 20 T^{2} + \cdots - 464)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 288 T^{4} + \cdots + 72900 \) Copy content Toggle raw display
$79$ \( (T^{3} - 14 T^{2} + \cdots + 158)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 272 T^{4} + \cdots + 2116 \) Copy content Toggle raw display
$89$ \( (T^{3} + 7 T^{2} + 7 T - 19)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 272 T^{4} + \cdots + 217156 \) Copy content Toggle raw display
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