Properties

Label 169.2.c.a
Level $169$
Weight $2$
Character orbit 169.c
Analytic conductor $1.349$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,Mod(22,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), 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: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 1.50000i −1.00000 + 1.73205i −0.500000 0.866025i −1.73205 −1.73205 3.00000i 0 −1.73205 −0.500000 0.866025i 1.50000 2.59808i
22.2 0.866025 1.50000i −1.00000 + 1.73205i −0.500000 0.866025i 1.73205 1.73205 + 3.00000i 0 1.73205 −0.500000 0.866025i 1.50000 2.59808i
146.1 −0.866025 1.50000i −1.00000 1.73205i −0.500000 + 0.866025i −1.73205 −1.73205 + 3.00000i 0 −1.73205 −0.500000 + 0.866025i 1.50000 + 2.59808i
146.2 0.866025 + 1.50000i −1.00000 1.73205i −0.500000 + 0.866025i 1.73205 1.73205 3.00000i 0 1.73205 −0.500000 + 0.866025i 1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$41$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T + 3)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$73$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$97$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
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