Properties

Label 169.3.d.b
Level $169$
Weight $3$
Character orbit 169.d
Analytic conductor $4.605$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,3,Mod(70,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.70");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 169.d (of order \(4\), degree \(2\), 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: \(4.60491646769\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + 7 \beta_{2} q^{4} - \beta_1 q^{5} - 3 \beta_1 q^{6} + 3 \beta_{3} q^{7} + 3 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + 7 \beta_{2} q^{4} - \beta_1 q^{5} - 3 \beta_1 q^{6} + 3 \beta_{3} q^{7} + 3 \beta_{3} q^{8} - 11 \beta_{2} q^{10} + 2 \beta_{3} q^{11} - 21 \beta_{2} q^{12} - 33 q^{14} + 3 \beta_1 q^{15} - 5 q^{16} + 3 \beta_{2} q^{17} + 6 \beta_1 q^{19} - 7 \beta_{3} q^{20} - 9 \beta_{3} q^{21} - 22 q^{22} + 12 \beta_{2} q^{23} - 9 \beta_{3} q^{24} - 14 \beta_{2} q^{25} + 27 q^{27} - 21 \beta_1 q^{28} - 42 q^{29} + 33 \beta_{2} q^{30} + 12 \beta_1 q^{31} + 7 \beta_1 q^{32} - 6 \beta_{3} q^{33} + 3 \beta_{3} q^{34} + 33 q^{35} + 15 \beta_{3} q^{37} + 66 \beta_{2} q^{38} + 33 q^{40} + 14 \beta_1 q^{41} + 99 q^{42} - 49 \beta_{2} q^{43} - 14 \beta_1 q^{44} + 12 \beta_{3} q^{46} - \beta_{3} q^{47} + 15 q^{48} - 50 \beta_{2} q^{49} - 14 \beta_{3} q^{50} - 9 \beta_{2} q^{51} - 24 q^{53} + 27 \beta_1 q^{54} + 22 q^{55} - 99 \beta_{2} q^{56} - 18 \beta_1 q^{57} - 42 \beta_1 q^{58} + 16 \beta_{3} q^{59} + 21 \beta_{3} q^{60} + 30 q^{61} + 132 \beta_{2} q^{62} + 97 \beta_{2} q^{64} + 66 q^{66} - 12 \beta_1 q^{67} - 21 q^{68} - 36 \beta_{2} q^{69} + 33 \beta_1 q^{70} - 7 \beta_1 q^{71} + 12 \beta_{3} q^{73} - 165 q^{74} + 42 \beta_{2} q^{75} + 42 \beta_{3} q^{76} - 66 \beta_{2} q^{77} + 54 q^{79} + 5 \beta_1 q^{80} - 81 q^{81} + 154 \beta_{2} q^{82} - 14 \beta_1 q^{83} + 63 \beta_1 q^{84} - 3 \beta_{3} q^{85} - 49 \beta_{3} q^{86} + 126 q^{87} - 66 \beta_{2} q^{88} - 50 \beta_{3} q^{89} - 84 q^{92} - 36 \beta_1 q^{93} + 11 q^{94} - 66 \beta_{2} q^{95} - 21 \beta_1 q^{96} + 6 \beta_1 q^{97} - 50 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 132 q^{14} - 20 q^{16} - 88 q^{22} + 108 q^{27} - 168 q^{29} + 132 q^{35} + 132 q^{40} + 396 q^{42} + 60 q^{48} - 96 q^{53} + 88 q^{55} + 120 q^{61} + 264 q^{66} - 84 q^{68} - 660 q^{74} + 216 q^{79} - 324 q^{81} + 504 q^{87} - 336 q^{92} + 44 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 11\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1
−2.34521 + 2.34521i
2.34521 2.34521i
−2.34521 2.34521i
2.34521 + 2.34521i
−2.34521 + 2.34521i −3.00000 7.00000i 2.34521 2.34521i 7.03562 7.03562i 7.03562 + 7.03562i 7.03562 + 7.03562i 0 11.0000i
70.2 2.34521 2.34521i −3.00000 7.00000i −2.34521 + 2.34521i −7.03562 + 7.03562i −7.03562 7.03562i −7.03562 7.03562i 0 11.0000i
99.1 −2.34521 2.34521i −3.00000 7.00000i 2.34521 + 2.34521i 7.03562 + 7.03562i 7.03562 7.03562i 7.03562 7.03562i 0 11.0000i
99.2 2.34521 + 2.34521i −3.00000 7.00000i −2.34521 2.34521i −7.03562 7.03562i −7.03562 + 7.03562i −7.03562 + 7.03562i 0 11.0000i
\(n\): 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.d odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.d.b 4
13.b even 2 1 inner 169.3.d.b 4
13.c even 3 2 169.3.f.e 8
13.d odd 4 2 inner 169.3.d.b 4
13.e even 6 2 169.3.f.e 8
13.f odd 12 4 169.3.f.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.3.d.b 4 1.a even 1 1 trivial
169.3.d.b 4 13.b even 2 1 inner
169.3.d.b 4 13.d odd 4 2 inner
169.3.f.e 8 13.c even 3 2
169.3.f.e 8 13.e even 6 2
169.3.f.e 8 13.f odd 12 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 121 \) acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 121 \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 121 \) Copy content Toggle raw display
$7$ \( T^{4} + 9801 \) Copy content Toggle raw display
$11$ \( T^{4} + 1936 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 156816 \) Copy content Toggle raw display
$23$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( (T + 42)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 2509056 \) Copy content Toggle raw display
$37$ \( T^{4} + 6125625 \) Copy content Toggle raw display
$41$ \( T^{4} + 4648336 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2401)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 121 \) Copy content Toggle raw display
$53$ \( (T + 24)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 7929856 \) Copy content Toggle raw display
$61$ \( (T - 30)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 2509056 \) Copy content Toggle raw display
$71$ \( T^{4} + 290521 \) Copy content Toggle raw display
$73$ \( T^{4} + 2509056 \) Copy content Toggle raw display
$79$ \( (T - 54)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 4648336 \) Copy content Toggle raw display
$89$ \( T^{4} + 756250000 \) Copy content Toggle raw display
$97$ \( T^{4} + 156816 \) Copy content Toggle raw display
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