Properties

Label 169.2.a.a
Level $169$
Weight $2$
Character orbit 169.a
Self dual yes
Analytic conductor $1.349$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,Mod(1,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

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: yes
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 2 q^{3} + q^{4} - \beta q^{5} + 2 \beta q^{6} - \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 2 q^{3} + q^{4} - \beta q^{5} + 2 \beta q^{6} - \beta q^{8} + q^{9} - 3 q^{10} + 2 q^{12} - 2 \beta q^{15} - 5 q^{16} + 3 q^{17} + \beta q^{18} - 2 \beta q^{19} - \beta q^{20} + 6 q^{23} - 2 \beta q^{24} - 2 q^{25} - 4 q^{27} + 3 q^{29} - 6 q^{30} + 2 \beta q^{31} - 3 \beta q^{32} + 3 \beta q^{34} + q^{36} + 5 \beta q^{37} - 6 q^{38} + 3 q^{40} + 3 \beta q^{41} - 8 q^{43} - \beta q^{45} + 6 \beta q^{46} + 2 \beta q^{47} - 10 q^{48} - 7 q^{49} - 2 \beta q^{50} + 6 q^{51} - 3 q^{53} - 4 \beta q^{54} - 4 \beta q^{57} + 3 \beta q^{58} - 4 \beta q^{59} - 2 \beta q^{60} + q^{61} + 6 q^{62} + q^{64} - 2 \beta q^{67} + 3 q^{68} + 12 q^{69} + 2 \beta q^{71} - \beta q^{72} - \beta q^{73} + 15 q^{74} - 4 q^{75} - 2 \beta q^{76} + 4 q^{79} + 5 \beta q^{80} - 11 q^{81} + 9 q^{82} + 8 \beta q^{83} - 3 \beta q^{85} - 8 \beta q^{86} + 6 q^{87} - 4 \beta q^{89} - 3 q^{90} + 6 q^{92} + 4 \beta q^{93} + 6 q^{94} + 6 q^{95} - 6 \beta q^{96} + 4 \beta q^{97} - 7 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 2 q^{4} + 2 q^{9} - 6 q^{10} + 4 q^{12} - 10 q^{16} + 6 q^{17} + 12 q^{23} - 4 q^{25} - 8 q^{27} + 6 q^{29} - 12 q^{30} + 2 q^{36} - 12 q^{38} + 6 q^{40} - 16 q^{43} - 20 q^{48} - 14 q^{49} + 12 q^{51} - 6 q^{53} + 2 q^{61} + 12 q^{62} + 2 q^{64} + 6 q^{68} + 24 q^{69} + 30 q^{74} - 8 q^{75} + 8 q^{79} - 22 q^{81} + 18 q^{82} + 12 q^{87} - 6 q^{90} + 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−1.73205 2.00000 1.00000 1.73205 −3.46410 0 1.73205 1.00000 −3.00000
1.2 1.73205 2.00000 1.00000 −1.73205 3.46410 0 −1.73205 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.a.a 2
3.b odd 2 1 1521.2.a.k 2
4.b odd 2 1 2704.2.a.o 2
5.b even 2 1 4225.2.a.v 2
7.b odd 2 1 8281.2.a.q 2
13.b even 2 1 inner 169.2.a.a 2
13.c even 3 2 169.2.c.a 4
13.d odd 4 2 169.2.b.a 2
13.e even 6 2 169.2.c.a 4
13.f odd 12 2 13.2.e.a 2
13.f odd 12 2 169.2.e.a 2
39.d odd 2 1 1521.2.a.k 2
39.f even 4 2 1521.2.b.a 2
39.k even 12 2 117.2.q.c 2
52.b odd 2 1 2704.2.a.o 2
52.f even 4 2 2704.2.f.b 2
52.l even 12 2 208.2.w.b 2
65.d even 2 1 4225.2.a.v 2
65.o even 12 2 325.2.m.a 4
65.s odd 12 2 325.2.n.a 2
65.t even 12 2 325.2.m.a 4
91.b odd 2 1 8281.2.a.q 2
91.w even 12 2 637.2.u.b 2
91.x odd 12 2 637.2.k.a 2
91.ba even 12 2 637.2.k.c 2
91.bc even 12 2 637.2.q.a 2
91.bd odd 12 2 637.2.u.c 2
104.u even 12 2 832.2.w.a 2
104.x odd 12 2 832.2.w.d 2
156.v odd 12 2 1872.2.by.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 13.f odd 12 2
117.2.q.c 2 39.k even 12 2
169.2.a.a 2 1.a even 1 1 trivial
169.2.a.a 2 13.b even 2 1 inner
169.2.b.a 2 13.d odd 4 2
169.2.c.a 4 13.c even 3 2
169.2.c.a 4 13.e even 6 2
169.2.e.a 2 13.f odd 12 2
208.2.w.b 2 52.l even 12 2
325.2.m.a 4 65.o even 12 2
325.2.m.a 4 65.t even 12 2
325.2.n.a 2 65.s odd 12 2
637.2.k.a 2 91.x odd 12 2
637.2.k.c 2 91.ba even 12 2
637.2.q.a 2 91.bc even 12 2
637.2.u.b 2 91.w even 12 2
637.2.u.c 2 91.bd odd 12 2
832.2.w.a 2 104.u even 12 2
832.2.w.d 2 104.x odd 12 2
1521.2.a.k 2 3.b odd 2 1
1521.2.a.k 2 39.d odd 2 1
1521.2.b.a 2 39.f even 4 2
1872.2.by.d 2 156.v odd 12 2
2704.2.a.o 2 4.b odd 2 1
2704.2.a.o 2 52.b odd 2 1
2704.2.f.b 2 52.f even 4 2
4225.2.a.v 2 5.b even 2 1
4225.2.a.v 2 65.d even 2 1
8281.2.a.q 2 7.b odd 2 1
8281.2.a.q 2 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 12 \) Copy content Toggle raw display
$37$ \( T^{2} - 75 \) Copy content Toggle raw display
$41$ \( T^{2} - 27 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 12 \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 48 \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 3 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 48 \) Copy content Toggle raw display
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