Properties

Label 169.4.a.c
Level $169$
Weight $4$
Character orbit 169.a
Self dual yes
Analytic conductor $9.971$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(9.97132279097\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} - q^{3} + q^{4} - 9 q^{5} - 3 q^{6} + 15 q^{7} - 21 q^{8} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} - q^{3} + q^{4} - 9 q^{5} - 3 q^{6} + 15 q^{7} - 21 q^{8} - 26 q^{9} - 27 q^{10} - 48 q^{11} - q^{12} + 45 q^{14} + 9 q^{15} - 71 q^{16} + 45 q^{17} - 78 q^{18} + 6 q^{19} - 9 q^{20} - 15 q^{21} - 144 q^{22} - 162 q^{23} + 21 q^{24} - 44 q^{25} + 53 q^{27} + 15 q^{28} - 144 q^{29} + 27 q^{30} + 264 q^{31} - 45 q^{32} + 48 q^{33} + 135 q^{34} - 135 q^{35} - 26 q^{36} + 303 q^{37} + 18 q^{38} + 189 q^{40} - 192 q^{41} - 45 q^{42} + 97 q^{43} - 48 q^{44} + 234 q^{45} - 486 q^{46} + 111 q^{47} + 71 q^{48} - 118 q^{49} - 132 q^{50} - 45 q^{51} - 414 q^{53} + 159 q^{54} + 432 q^{55} - 315 q^{56} - 6 q^{57} - 432 q^{58} + 522 q^{59} + 9 q^{60} + 376 q^{61} + 792 q^{62} - 390 q^{63} + 433 q^{64} + 144 q^{66} - 36 q^{67} + 45 q^{68} + 162 q^{69} - 405 q^{70} + 357 q^{71} + 546 q^{72} - 1098 q^{73} + 909 q^{74} + 44 q^{75} + 6 q^{76} - 720 q^{77} - 830 q^{79} + 639 q^{80} + 649 q^{81} - 576 q^{82} - 438 q^{83} - 15 q^{84} - 405 q^{85} + 291 q^{86} + 144 q^{87} + 1008 q^{88} - 438 q^{89} + 702 q^{90} - 162 q^{92} - 264 q^{93} + 333 q^{94} - 54 q^{95} + 45 q^{96} - 852 q^{97} - 354 q^{98} + 1248 q^{99}+O(q^{100}) \) 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
0
3.00000 −1.00000 1.00000 −9.00000 −3.00000 15.0000 −21.0000 −26.0000 −27.0000
\(n\): 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 169.4.a.c 1
3.b odd 2 1 1521.4.a.d 1
13.b even 2 1 169.4.a.b 1
13.c even 3 2 169.4.c.b 2
13.d odd 4 2 13.4.b.a 2
13.e even 6 2 169.4.c.c 2
13.f odd 12 4 169.4.e.d 4
39.d odd 2 1 1521.4.a.i 1
39.f even 4 2 117.4.b.a 2
52.f even 4 2 208.4.f.b 2
65.f even 4 2 325.4.d.a 2
65.g odd 4 2 325.4.c.b 2
65.k even 4 2 325.4.d.b 2
104.j odd 4 2 832.4.f.e 2
104.m even 4 2 832.4.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 13.d odd 4 2
117.4.b.a 2 39.f even 4 2
169.4.a.b 1 13.b even 2 1
169.4.a.c 1 1.a even 1 1 trivial
169.4.c.b 2 13.c even 3 2
169.4.c.c 2 13.e even 6 2
169.4.e.d 4 13.f odd 12 4
208.4.f.b 2 52.f even 4 2
325.4.c.b 2 65.g odd 4 2
325.4.d.a 2 65.f even 4 2
325.4.d.b 2 65.k even 4 2
832.4.f.c 2 104.m even 4 2
832.4.f.e 2 104.j odd 4 2
1521.4.a.d 1 3.b odd 2 1
1521.4.a.i 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T + 9 \) Copy content Toggle raw display
$7$ \( T - 15 \) Copy content Toggle raw display
$11$ \( T + 48 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 45 \) Copy content Toggle raw display
$19$ \( T - 6 \) Copy content Toggle raw display
$23$ \( T + 162 \) Copy content Toggle raw display
$29$ \( T + 144 \) Copy content Toggle raw display
$31$ \( T - 264 \) Copy content Toggle raw display
$37$ \( T - 303 \) Copy content Toggle raw display
$41$ \( T + 192 \) Copy content Toggle raw display
$43$ \( T - 97 \) Copy content Toggle raw display
$47$ \( T - 111 \) Copy content Toggle raw display
$53$ \( T + 414 \) Copy content Toggle raw display
$59$ \( T - 522 \) Copy content Toggle raw display
$61$ \( T - 376 \) Copy content Toggle raw display
$67$ \( T + 36 \) Copy content Toggle raw display
$71$ \( T - 357 \) Copy content Toggle raw display
$73$ \( T + 1098 \) Copy content Toggle raw display
$79$ \( T + 830 \) Copy content Toggle raw display
$83$ \( T + 438 \) Copy content Toggle raw display
$89$ \( T + 438 \) Copy content Toggle raw display
$97$ \( T + 852 \) Copy content Toggle raw display
show more
show less