Properties

Label 169.2.b.b
Level $169$
Weight $2$
Character orbit 169.b
Analytic conductor $1.349$
Analytic rank $0$
Dimension $6$
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(168,169)]
 
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([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{2} + (\beta_{4} - 1) q^{3} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_1) q^{5} - \beta_{3} q^{6} + (2 \beta_{3} - \beta_1) q^{7} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{8} + ( - 2 \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3}) q^{2} + (\beta_{4} - 1) q^{3} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{4} + (\beta_{5} + \beta_1) q^{5} - \beta_{3} q^{6} + (2 \beta_{3} - \beta_1) q^{7} + ( - 2 \beta_{5} - \beta_{3} + 2 \beta_1) q^{8} + ( - 2 \beta_{4} - \beta_{2}) q^{9} + ( - \beta_{4} - \beta_{2} - 1) q^{10} + ( - 3 \beta_{5} + \beta_1) q^{11} + (2 \beta_{4} + \beta_{2} - 1) q^{12} + ( - 2 \beta_{2} - 1) q^{14} + ( - \beta_{3} + \beta_1) q^{15} + ( - \beta_{2} + 1) q^{16} + (2 \beta_{4} + 3 \beta_{2} - 1) q^{17} + ( - 4 \beta_{5} - 2 \beta_{3} + \beta_1) q^{18} + (3 \beta_{5} + 3 \beta_{3} - 2 \beta_1) q^{19} + ( - 2 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{20} + (\beta_{5} - \beta_{3} - 2 \beta_1) q^{21} + (3 \beta_{4} + 3 \beta_{2} - 1) q^{22} + (2 \beta_{4} + 1) q^{23} + (3 \beta_{5} - \beta_{3} - \beta_1) q^{24} + ( - 2 \beta_{4} + \beta_{2} + 2) q^{25} + ( - \beta_{4} + 2 \beta_{2}) q^{27} + ( - 5 \beta_{5} - \beta_{3}) q^{28} + ( - 3 \beta_{4} + 2 \beta_{2}) q^{29} + \beta_{2} q^{30} + (4 \beta_{5} + 2 \beta_{3} - 5 \beta_1) q^{31} + ( - 5 \beta_{5} - 3 \beta_{3} + 5 \beta_1) q^{32} + (4 \beta_{5} - \beta_{3} - 3 \beta_1) q^{33} + (7 \beta_{5} + 5 \beta_{3} - 3 \beta_1) q^{34} + (\beta_{4} - 3 \beta_{2} + 2) q^{35} + (4 \beta_{2} + 1) q^{36} + (5 \beta_{5} + \beta_{3} - 2 \beta_1) q^{37} + ( - 3 \beta_{4} - 6 \beta_{2} - 1) q^{38} + (3 \beta_{2} - 2) q^{40} + (3 \beta_{5} - 2 \beta_{3} - 4 \beta_1) q^{41} + ( - \beta_{4} + 3) q^{42} + ( - 3 \beta_{4} - \beta_{2} - 3) q^{43} + (2 \beta_{5} + 5 \beta_{3} - \beta_1) q^{44} + ( - 3 \beta_{5} - 2 \beta_1) q^{45} + (3 \beta_{5} + \beta_{3}) q^{46} + ( - 5 \beta_{5} + 2 \beta_{3} - \beta_1) q^{47} + \beta_{4} q^{48} + (\beta_{2} + 1) q^{49} + (2 \beta_{5} + 4 \beta_{3} - \beta_1) q^{50} + ( - 3 \beta_{4} - 2 \beta_{2} + 2) q^{51} + (7 \beta_{4} + 3 \beta_{2} - 3) q^{53} + (3 \beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{54} + (2 \beta_{4} + \beta_{2} + 1) q^{55} + (5 \beta_{4} + 2 \beta_{2} - 1) q^{56} + ( - 2 \beta_{5} - \beta_{3}) q^{57} + (\beta_{5} + 4 \beta_{3} - 2 \beta_1) q^{58} + ( - 9 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{59} + (2 \beta_{5} + \beta_1) q^{60} + ( - \beta_{4} + 5 \beta_{2}) q^{61} + ( - 4 \beta_{4} - 6 \beta_{2} + 3) q^{62} + ( - 4 \beta_{5} - 3 \beta_{3} + 5 \beta_1) q^{63} + (5 \beta_{4} + 6 \beta_{2}) q^{64} + ( - 4 \beta_{4} - 3 \beta_{2} + 4) q^{66} + ( - 2 \beta_{5} - \beta_{3} + 6 \beta_1) q^{67} + ( - 3 \beta_{4} - 6 \beta_{2} - 4) q^{68} + ( - \beta_{4} - 2 \beta_{2} + 3) q^{69} + ( - 3 \beta_{5} - 4 \beta_{3} + 3 \beta_1) q^{70} + (7 \beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{71} + (\beta_{5} + 5 \beta_{3} - 2 \beta_1) q^{72} + ( - 2 \beta_{5} - 9 \beta_{3} + 6 \beta_1) q^{73} + ( - 5 \beta_{4} - 6 \beta_{2} + 1) q^{74} + (4 \beta_{4} + 2 \beta_{2} - 7) q^{75} + ( - 10 \beta_{5} - 7 \beta_{3} + 2 \beta_1) q^{76} + (5 \beta_{4} + 5 \beta_{2} - 6) q^{77} + ( - 8 \beta_{4} - 9 \beta_{2} + 4) q^{79} + ( - 2 \beta_{3} + 3 \beta_1) q^{80} + (7 \beta_{4} + 4 \beta_{2} - 4) q^{81} + ( - 3 \beta_{4} - \beta_{2} + 6) q^{82} + ( - 3 \beta_{5} - 7 \beta_{3} + 9 \beta_1) q^{83} + (4 \beta_{5} + \beta_{3} - 4 \beta_1) q^{84} + (4 \beta_{5} + 4 \beta_{3} - 3 \beta_1) q^{85} + ( - 8 \beta_{5} - 5 \beta_{3} + \beta_1) q^{86} + (3 \beta_{4} + 3 \beta_{2} - 8) q^{87} + (4 \beta_{4} - \beta_{2} - 6) q^{88} + (\beta_{5} + 7 \beta_{3} - 7 \beta_1) q^{89} + (3 \beta_{4} + 3 \beta_{2} + 2) q^{90} + (\beta_{4} - 4 \beta_{2} + 1) q^{92} + ( - 7 \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{93} + (5 \beta_{4} + 3 \beta_{2} - 1) q^{94} + ( - \beta_{4} - 5 \beta_{2} + 1) q^{95} + (7 \beta_{5} - 2 \beta_{3} - 2 \beta_1) q^{96} + ( - 4 \beta_{5} - 6 \beta_{3} - \beta_1) q^{97} + (3 \beta_{5} + 3 \beta_{3} - \beta_1) q^{98} + (\beta_{5} + 4 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
168.1
0.445042i
1.24698i
1.80194i
1.80194i
1.24698i
0.445042i
2.24698i −0.554958 −3.04892 1.44504i 1.24698i 2.04892i 2.35690i −2.69202 −3.24698
168.2 0.801938i −2.24698 1.35690 0.246980i 1.80194i 2.35690i 2.69202i 2.04892 −0.198062
168.3 0.554958i 0.801938 1.69202 2.80194i 0.445042i 2.69202i 2.04892i −2.35690 −1.55496
168.4 0.554958i 0.801938 1.69202 2.80194i 0.445042i 2.69202i 2.04892i −2.35690 −1.55496
168.5 0.801938i −2.24698 1.35690 0.246980i 1.80194i 2.35690i 2.69202i 2.04892 −0.198062
168.6 2.24698i −0.554958 −3.04892 1.44504i 1.24698i 2.04892i 2.35690i −2.69202 −3.24698
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 168.6
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 6T_{2}^{4} + 5T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 6 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{3} + 2 T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 10 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} + 17 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$11$ \( T^{6} + 26 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} - 15 T - 13)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 38 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{3} - 5 T^{2} - T + 13)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + T^{2} - 44 T + 83)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 97 T^{4} + \cdots + 27889 \) Copy content Toggle raw display
$37$ \( T^{6} + 62 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$41$ \( T^{6} + 147 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$43$ \( (T^{3} + 13 T^{2} + \cdots - 13)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 122 T^{4} + \cdots + 27889 \) Copy content Toggle raw display
$53$ \( (T^{3} - T^{2} - 86 T + 337)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 195 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( (T^{3} - 4 T^{2} + \cdots + 239)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 145 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$71$ \( T^{6} + 285 T^{4} + \cdots + 299209 \) Copy content Toggle raw display
$73$ \( T^{6} + 321 T^{4} + \cdots + 829921 \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} + \cdots + 127)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 329 T^{4} + \cdots + 41209 \) Copy content Toggle raw display
$89$ \( T^{6} + 269 T^{4} + \cdots + 78961 \) Copy content Toggle raw display
$97$ \( T^{6} + 217 T^{4} + \cdots + 90601 \) Copy content Toggle raw display
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