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Properties

Label 169.3.f.e
Level $169$
Weight $3$
Character orbit 169.f
Analytic conductor $4.605$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,3,Mod(19,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 169.f (of order \(12\), degree \(4\), 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: \(4.60491646769\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.77720518656.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 121x^{4} + 14641 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 \beta_{4} q^{3} + 7 \beta_{2} q^{4} + ( - \beta_{5} + \beta_1) q^{5} + 3 \beta_{5} q^{6} + (3 \beta_{7} - 3 \beta_{3}) q^{7} + 3 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 \beta_{4} q^{3} + 7 \beta_{2} q^{4} + ( - \beta_{5} + \beta_1) q^{5} + 3 \beta_{5} q^{6} + (3 \beta_{7} - 3 \beta_{3}) q^{7} + 3 \beta_{3} q^{8} + ( - 11 \beta_{6} + 11 \beta_{2}) q^{10} - 2 \beta_{7} q^{11} + 21 \beta_{6} q^{12} - 33 q^{14} + 3 \beta_1 q^{15} + 5 \beta_{4} q^{16} + 3 \beta_{2} q^{17} - 6 \beta_{5} q^{19} + ( - 7 \beta_{7} + 7 \beta_{3}) q^{20} - 9 \beta_{3} q^{21} + ( - 22 \beta_{4} + 22) q^{22} + (12 \beta_{6} - 12 \beta_{2}) q^{23} + 9 \beta_{7} q^{24} + 14 \beta_{6} q^{25} + 27 q^{27} - 21 \beta_1 q^{28} + 42 \beta_{4} q^{29} + 33 \beta_{2} q^{30} + (12 \beta_{5} - 12 \beta_1) q^{31} - 7 \beta_{5} q^{32} + ( - 6 \beta_{7} + 6 \beta_{3}) q^{33} + 3 \beta_{3} q^{34} + (33 \beta_{4} - 33) q^{35} - 15 \beta_{7} q^{37} - 66 \beta_{6} q^{38} + 33 q^{40} + 14 \beta_1 q^{41} - 99 \beta_{4} q^{42} - 49 \beta_{2} q^{43} + ( - 14 \beta_{5} + 14 \beta_1) q^{44} + (12 \beta_{7} - 12 \beta_{3}) q^{46} - \beta_{3} q^{47} + (15 \beta_{4} - 15) q^{48} + ( - 50 \beta_{6} + 50 \beta_{2}) q^{49} + 14 \beta_{7} q^{50} + 9 \beta_{6} q^{51} - 24 q^{53} + 27 \beta_1 q^{54} - 22 \beta_{4} q^{55} - 99 \beta_{2} q^{56} + ( - 18 \beta_{5} + 18 \beta_1) q^{57} + 42 \beta_{5} q^{58} + (16 \beta_{7} - 16 \beta_{3}) q^{59} + 21 \beta_{3} q^{60} + (30 \beta_{4} - 30) q^{61} + (132 \beta_{6} - 132 \beta_{2}) q^{62} - 97 \beta_{6} q^{64} + 66 q^{66} - 12 \beta_1 q^{67} + 21 \beta_{4} q^{68} - 36 \beta_{2} q^{69} + (33 \beta_{5} - 33 \beta_1) q^{70} + 7 \beta_{5} q^{71} + 12 \beta_{3} q^{73} + ( - 165 \beta_{4} + 165) q^{74} + (42 \beta_{6} - 42 \beta_{2}) q^{75} - 42 \beta_{7} q^{76} + 66 \beta_{6} q^{77} + 54 q^{79} + 5 \beta_1 q^{80} + 81 \beta_{4} q^{81} + 154 \beta_{2} q^{82} + ( - 14 \beta_{5} + 14 \beta_1) q^{83} - 63 \beta_{5} q^{84} + ( - 3 \beta_{7} + 3 \beta_{3}) q^{85} - 49 \beta_{3} q^{86} + (126 \beta_{4} - 126) q^{87} + ( - 66 \beta_{6} + 66 \beta_{2}) q^{88} + 50 \beta_{7} q^{89} - 84 q^{92} - 36 \beta_1 q^{93} - 11 \beta_{4} q^{94} - 66 \beta_{2} q^{95} + ( - 21 \beta_{5} + 21 \beta_1) q^{96} - 6 \beta_{5} q^{97} + ( - 50 \beta_{7} + 50 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{3} - 264 q^{14} + 20 q^{16} + 88 q^{22} + 216 q^{27} + 168 q^{29} - 132 q^{35} + 264 q^{40} - 396 q^{42} - 60 q^{48} - 192 q^{53} - 88 q^{55} - 120 q^{61} + 528 q^{66} + 84 q^{68} + 660 q^{74} + 432 q^{79} + 324 q^{81} - 504 q^{87} - 672 q^{92} - 44 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 121x^{4} + 14641 \) : 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
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 121 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 121 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 1331 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 1331 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) 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
\(\nu^{4}\)\(=\) \( 121\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 121\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1331\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1331\beta_{7} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.858406 − 3.20361i
0.858406 + 3.20361i
−3.20361 − 0.858406i
3.20361 + 0.858406i
−0.858406 + 3.20361i
0.858406 − 3.20361i
−3.20361 + 0.858406i
3.20361 − 0.858406i
−0.858406 − 3.20361i 1.50000 − 2.59808i −6.06218 + 3.50000i 2.34521 − 2.34521i −9.61084 − 2.57522i 2.57522 − 9.61084i 7.03562 + 7.03562i 0 −9.52628 − 5.50000i
19.2 0.858406 + 3.20361i 1.50000 − 2.59808i −6.06218 + 3.50000i −2.34521 + 2.34521i 9.61084 + 2.57522i −2.57522 + 9.61084i −7.03562 − 7.03562i 0 −9.52628 − 5.50000i
80.1 −3.20361 − 0.858406i 1.50000 + 2.59808i 6.06218 + 3.50000i −2.34521 + 2.34521i −2.57522 − 9.61084i 9.61084 − 2.57522i −7.03562 − 7.03562i 0 9.52628 − 5.50000i
80.2 3.20361 + 0.858406i 1.50000 + 2.59808i 6.06218 + 3.50000i 2.34521 − 2.34521i 2.57522 + 9.61084i −9.61084 + 2.57522i 7.03562 + 7.03562i 0 9.52628 − 5.50000i
89.1 −0.858406 + 3.20361i 1.50000 + 2.59808i −6.06218 − 3.50000i 2.34521 + 2.34521i −9.61084 + 2.57522i 2.57522 + 9.61084i 7.03562 − 7.03562i 0 −9.52628 + 5.50000i
89.2 0.858406 − 3.20361i 1.50000 + 2.59808i −6.06218 − 3.50000i −2.34521 − 2.34521i 9.61084 − 2.57522i −2.57522 − 9.61084i −7.03562 + 7.03562i 0 −9.52628 + 5.50000i
150.1 −3.20361 + 0.858406i 1.50000 − 2.59808i 6.06218 − 3.50000i −2.34521 − 2.34521i −2.57522 + 9.61084i 9.61084 + 2.57522i −7.03562 + 7.03562i 0 9.52628 + 5.50000i
150.2 3.20361 − 0.858406i 1.50000 − 2.59808i 6.06218 − 3.50000i 2.34521 + 2.34521i 2.57522 − 9.61084i −9.61084 − 2.57522i 7.03562 − 7.03562i 0 9.52628 + 5.50000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.2
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.d odd 4 2 inner
13.e even 6 1 inner
13.f odd 12 2 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 121T_{2}^{4} + 14641 \) acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 121 T^{4} + 14641 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 121)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 9801 T^{4} + \cdots + 96059601 \) Copy content Toggle raw display
$11$ \( T^{8} - 1936 T^{4} + \cdots + 3748096 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 156816 T^{4} + \cdots + 24591257856 \) Copy content Toggle raw display
$23$ \( (T^{4} - 144 T^{2} + 20736)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 42 T + 1764)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2509056)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} - 6125625 T^{4} + \cdots + 37523281640625 \) Copy content Toggle raw display
$41$ \( T^{8} - 4648336 T^{4} + \cdots + 21607027568896 \) Copy content Toggle raw display
$43$ \( (T^{4} - 2401 T^{2} + 5764801)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 121)^{2} \) Copy content Toggle raw display
$53$ \( (T + 24)^{8} \) Copy content Toggle raw display
$59$ \( T^{8} - 7929856 T^{4} + \cdots + 62882616180736 \) Copy content Toggle raw display
$61$ \( (T^{2} + 30 T + 900)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 2509056 T^{4} + \cdots + 6295362011136 \) Copy content Toggle raw display
$71$ \( T^{8} - 290521 T^{4} + \cdots + 84402451441 \) Copy content Toggle raw display
$73$ \( (T^{4} + 2509056)^{2} \) Copy content Toggle raw display
$79$ \( (T - 54)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4648336)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 756250000 T^{4} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} - 156816 T^{4} + \cdots + 24591257856 \) Copy content Toggle raw display
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