Properties

Label 169.2.a.c
Level $169$
Weight $2$
Character orbit 169.a
Self dual yes
Analytic conductor $1.349$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + (\beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{2} - 1) q^{3} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{2} + 2 \beta_1) q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + (\beta_{2} + \beta_1 - 1) q^{9} + ( - \beta_1 + 2) q^{10} + (\beta_{2} + 3) q^{11} + (\beta_{2} + \beta_1) q^{12} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{14} + ( - \beta_{2} + \beta_1) q^{15} + ( - \beta_{2} + \beta_1) q^{16} + ( - \beta_{2} + 3 \beta_1 - 2) q^{17} + ( - \beta_{2} + 2 \beta_1 - 4) q^{18} + (2 \beta_{2} - 3 \beta_1 + 3) q^{19} + (3 \beta_{2} - 3 \beta_1 + 2) q^{20} + ( - 2 \beta_{2} - \beta_1 - 1) q^{21} + ( - 3 \beta_1 + 2) q^{22} + (2 \beta_{2} - 1) q^{23} + ( - \beta_{2} - \beta_1 - 3) q^{24} + ( - 3 \beta_{2} + \beta_1 - 3) q^{25} + (3 \beta_{2} - 2 \beta_1 + 2) q^{27} + (\beta_1 - 5) q^{28} + (5 \beta_{2} - 2 \beta_1 + 2) q^{29} + ( - \beta_{2} + \beta_1 - 1) q^{30} + (5 \beta_{2} - 2 \beta_1 + 4) q^{31} + ( - 5 \beta_{2} + 3 \beta_1 - 5) q^{32} + ( - 3 \beta_{2} - \beta_1 - 4) q^{33} + ( - 3 \beta_{2} + 5 \beta_1 - 7) q^{34} + ( - 4 \beta_{2} + 3 \beta_1 - 1) q^{35} + ( - 4 \beta_{2} + 4 \beta_1 - 5) q^{36} + ( - 2 \beta_{2} + \beta_1 - 5) q^{37} + (3 \beta_{2} - 6 \beta_1 + 7) q^{38} + (3 \beta_{2} - 3 \beta_1 + 1) q^{40} + (4 \beta_{2} + 2 \beta_1 + 3) q^{41} + (\beta_{2} + 3) q^{42} + ( - 2 \beta_{2} - \beta_1 + 4) q^{43} + (\beta_{2} - 5 \beta_1 + 2) q^{44} + (2 \beta_{2} - 3) q^{45} + (\beta_1 - 3) q^{46} + ( - \beta_{2} + 2 \beta_1 + 5) q^{47} - \beta_{2} q^{48} + ( - \beta_{2} + \beta_1 - 2) q^{49} + ( - \beta_{2} + 4 \beta_1 - 2) q^{50} + ( - \beta_{2} - 2 \beta_1) q^{51} + ( - 4 \beta_{2} - 3 \beta_1) q^{53} + (2 \beta_{2} - 4 \beta_1 + 3) q^{54} + ( - \beta_{2} - \beta_1 + 2) q^{55} + (3 \beta_{2} + 2 \beta_1 - 1) q^{56} + (\beta_1 - 2) q^{57} + (2 \beta_{2} - 4 \beta_1 + 1) q^{58} + (4 \beta_{2} - 4 \beta_1 + 9) q^{59} + (\beta_{2} - 2) q^{60} + (6 \beta_{2} - 5 \beta_1 + 5) q^{61} + (2 \beta_{2} - 6 \beta_1 + 3) q^{62} + (5 \beta_{2} - 3 \beta_1 + 4) q^{63} + ( - \beta_{2} + 6 \beta_1 - 6) q^{64} + (\beta_{2} + 3 \beta_1 + 1) q^{66} + ( - 6 \beta_{2} + \beta_1 - 2) q^{67} + ( - 3 \beta_{2} + 6 \beta_1 - 10) q^{68} + (\beta_{2} - 2 \beta_1 - 1) q^{69} + ( - 3 \beta_{2} + 4 \beta_1 - 3) q^{70} + ( - 3 \beta_{2} + 3 \beta_1 + 7) q^{71} + ( - 2 \beta_{2} + 5 \beta_1 - 1) q^{72} + (6 \beta_{2} - 9 \beta_1 + 2) q^{73} + ( - \beta_{2} + 6 \beta_1 - 5) q^{74} + (2 \beta_{2} + 2 \beta_1 + 5) q^{75} + (2 \beta_{2} - 7 \beta_1 + 10) q^{76} + (5 \beta_1 + 1) q^{77} + ( - \beta_{2} + 9 \beta_1 - 5) q^{79} + ( - 3 \beta_{2} + 2 \beta_1) q^{80} + ( - 3 \beta_{2} - 4 \beta_1) q^{81} + ( - 2 \beta_{2} - \beta_1 - 5) q^{82} + ( - 9 \beta_{2} + 7 \beta_1 - 3) q^{83} + (4 \beta_{2} - \beta_1 + 4) q^{84} + ( - 3 \beta_{2} + 4 \beta_1 - 4) q^{85} + (\beta_{2} - 5 \beta_1 + 8) q^{86} + ( - 3 \beta_1 - 5) q^{87} + (5 \beta_{2} - \beta_1 + 7) q^{88} + ( - 7 \beta_{2} + 7 \beta_1 - 1) q^{89} + (3 \beta_1 - 5) q^{90} + ( - 5 \beta_{2} + 4 \beta_1 - 3) q^{92} + ( - 2 \beta_{2} - 3 \beta_1 - 7) q^{93} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{94} + (4 \beta_{2} - 5 \beta_1 + 4) q^{95} + (2 \beta_{2} + 2 \beta_1 + 7) q^{96} + (\beta_{2} + 6 \beta_1 - 4) q^{97} + ( - \beta_{2} + 3 \beta_1 - 3) q^{98} + (2 \beta_{2} + 4 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} + 5 q^{10} + 8 q^{11} - 5 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 9 q^{18} + 4 q^{19} - 2 q^{21} + 3 q^{22} - 5 q^{23} - 9 q^{24} - 5 q^{25} + q^{27} - 14 q^{28} - q^{29} - q^{30} + 5 q^{31} - 7 q^{32} - 10 q^{33} - 13 q^{34} + 4 q^{35} - 7 q^{36} - 12 q^{37} + 12 q^{38} - 3 q^{40} + 7 q^{41} + 8 q^{42} + 13 q^{43} - 11 q^{45} - 8 q^{46} + 18 q^{47} + q^{48} - 4 q^{49} - q^{50} - q^{51} + q^{53} + 3 q^{54} + 6 q^{55} - 4 q^{56} - 5 q^{57} - 3 q^{58} + 19 q^{59} - 7 q^{60} + 4 q^{61} + q^{62} + 4 q^{63} - 11 q^{64} + 5 q^{66} + q^{67} - 21 q^{68} - 6 q^{69} - 2 q^{70} + 27 q^{71} + 4 q^{72} - 9 q^{73} - 8 q^{74} + 15 q^{75} + 21 q^{76} + 8 q^{77} - 5 q^{79} + 5 q^{80} - q^{81} - 14 q^{82} + 7 q^{83} + 7 q^{84} - 5 q^{85} + 18 q^{86} - 18 q^{87} + 15 q^{88} + 11 q^{89} - 12 q^{90} - 22 q^{93} + 5 q^{94} + 3 q^{95} + 21 q^{96} - 7 q^{97} - 5 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
1.80194
0.445042
−1.24698
−0.801938 −2.24698 −1.35690 −0.246980 1.80194 2.35690 2.69202 2.04892 0.198062
1.2 0.554958 0.801938 −1.69202 2.80194 0.445042 2.69202 −2.04892 −2.35690 1.55496
1.3 2.24698 −0.554958 3.04892 1.44504 −1.24698 −2.04892 2.35690 −2.69202 3.24698
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.2.a.c yes 3
3.b odd 2 1 1521.2.a.o 3
4.b odd 2 1 2704.2.a.ba 3
5.b even 2 1 4225.2.a.bb 3
7.b odd 2 1 8281.2.a.bj 3
13.b even 2 1 169.2.a.b 3
13.c even 3 2 169.2.c.b 6
13.d odd 4 2 169.2.b.b 6
13.e even 6 2 169.2.c.c 6
13.f odd 12 4 169.2.e.b 12
39.d odd 2 1 1521.2.a.r 3
39.f even 4 2 1521.2.b.l 6
52.b odd 2 1 2704.2.a.z 3
52.f even 4 2 2704.2.f.o 6
65.d even 2 1 4225.2.a.bg 3
91.b odd 2 1 8281.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.2.a.b 3 13.b even 2 1
169.2.a.c yes 3 1.a even 1 1 trivial
169.2.b.b 6 13.d odd 4 2
169.2.c.b 6 13.c even 3 2
169.2.c.c 6 13.e even 6 2
169.2.e.b 12 13.f odd 12 4
1521.2.a.o 3 3.b odd 2 1
1521.2.a.r 3 39.d odd 2 1
1521.2.b.l 6 39.f even 4 2
2704.2.a.z 3 52.b odd 2 1
2704.2.a.ba 3 4.b odd 2 1
2704.2.f.o 6 52.f even 4 2
4225.2.a.bb 3 5.b even 2 1
4225.2.a.bg 3 65.d even 2 1
8281.2.a.bf 3 91.b odd 2 1
8281.2.a.bj 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 2T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} + 2T^{2} - T - 1 \) Copy content Toggle raw display
$5$ \( T^{3} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{3} - 3 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$11$ \( T^{3} - 8 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{3} + 5T^{2} - T - 13 \) Copy content Toggle raw display
$29$ \( T^{3} + T^{2} + \cdots + 83 \) Copy content Toggle raw display
$31$ \( T^{3} - 5 T^{2} + \cdots + 167 \) Copy content Toggle raw display
$37$ \( T^{3} + 12 T^{2} + \cdots + 29 \) Copy content Toggle raw display
$41$ \( T^{3} - 7 T^{2} + \cdots - 49 \) Copy content Toggle raw display
$43$ \( T^{3} - 13 T^{2} + \cdots + 13 \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + \cdots - 167 \) Copy content Toggle raw display
$53$ \( T^{3} - T^{2} + \cdots + 337 \) Copy content Toggle raw display
$59$ \( T^{3} - 19 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$61$ \( T^{3} - 4 T^{2} + \cdots + 239 \) Copy content Toggle raw display
$67$ \( T^{3} - T^{2} + \cdots - 41 \) Copy content Toggle raw display
$71$ \( T^{3} - 27 T^{2} + \cdots - 547 \) Copy content Toggle raw display
$73$ \( T^{3} + 9 T^{2} + \cdots - 911 \) Copy content Toggle raw display
$79$ \( T^{3} + 5 T^{2} + \cdots + 127 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} + \cdots - 203 \) Copy content Toggle raw display
$89$ \( T^{3} - 11 T^{2} + \cdots + 281 \) Copy content Toggle raw display
$97$ \( T^{3} + 7 T^{2} + \cdots - 301 \) Copy content Toggle raw display
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