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Properties

Label 169.3.f.f
Level $169$
Weight $3$
Character orbit 169.f
Analytic conductor $4.605$
Analytic rank $0$
Dimension $8$
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(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: 8.0.3317760000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 1) q^{2}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 1) q^{2}+ \cdots + ( - 38 \beta_{6} - 4 \beta_{5} + \cdots + 38) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 4 q^{3} + 16 q^{5} + 16 q^{6} + 12 q^{7} + 72 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 4 q^{3} + 16 q^{5} + 16 q^{6} + 12 q^{7} + 72 q^{8} - 8 q^{9} + 4 q^{11} + 8 q^{14} + 28 q^{15} + 84 q^{16} + 64 q^{18} - 16 q^{20} + 64 q^{21} - 88 q^{22} - 24 q^{24} - 104 q^{27} - 4 q^{28} - 40 q^{29} + 80 q^{31} - 20 q^{32} + 76 q^{33} - 216 q^{34} + 68 q^{35} - 40 q^{37} + 168 q^{40} - 32 q^{41} - 76 q^{42} - 344 q^{44} - 56 q^{45} - 132 q^{46} - 8 q^{47} + 76 q^{48} + 128 q^{50} - 160 q^{53} - 152 q^{54} - 64 q^{55} - 80 q^{57} + 140 q^{58} - 56 q^{59} - 232 q^{60} + 296 q^{61} + 64 q^{63} + 224 q^{66} + 84 q^{67} - 444 q^{68} - 64 q^{70} - 284 q^{71} + 48 q^{72} - 100 q^{74} - 80 q^{76} + 128 q^{79} + 88 q^{80} + 220 q^{81} - 104 q^{83} + 184 q^{84} + 144 q^{85} - 48 q^{86} - 160 q^{87} - 200 q^{89} - 912 q^{92} - 80 q^{93} + 452 q^{94} + 640 q^{96} + 68 q^{97} - 224 q^{98} + 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2} - \beta_{6}\)

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
0.578737 − 2.15988i
−0.578737 + 2.15988i
2.15988 − 0.578737i
−2.15988 + 0.578737i
0.578737 + 2.15988i
−0.578737 − 2.15988i
2.15988 + 0.578737i
−2.15988 − 0.578737i
−0.944762 − 3.52590i −1.08114 + 1.87259i −8.07530 + 4.66228i 0.418861 − 0.418861i 7.62398 + 2.04284i −0.519339 + 1.93820i 13.7434 + 13.7434i 2.16228 + 3.74517i −1.87259 − 1.08114i
19.2 0.212712 + 0.793850i 2.08114 − 3.60464i 2.87915 − 1.66228i 3.58114 − 3.58114i 3.30423 + 0.885365i −1.67681 + 6.25795i 4.25658 + 4.25658i −4.16228 − 7.20928i 3.60464 + 2.08114i
80.1 −0.793850 − 0.212712i 2.08114 + 3.60464i −2.87915 − 1.66228i 3.58114 − 3.58114i −0.885365 − 3.30423i 6.25795 − 1.67681i 4.25658 + 4.25658i −4.16228 + 7.20928i −3.60464 + 2.08114i
80.2 3.52590 + 0.944762i −1.08114 − 1.87259i 8.07530 + 4.66228i 0.418861 − 0.418861i −2.04284 − 7.62398i 1.93820 − 0.519339i 13.7434 + 13.7434i 2.16228 − 3.74517i 1.87259 − 1.08114i
89.1 −0.944762 + 3.52590i −1.08114 − 1.87259i −8.07530 − 4.66228i 0.418861 + 0.418861i 7.62398 − 2.04284i −0.519339 − 1.93820i 13.7434 − 13.7434i 2.16228 − 3.74517i −1.87259 + 1.08114i
89.2 0.212712 − 0.793850i 2.08114 + 3.60464i 2.87915 + 1.66228i 3.58114 + 3.58114i 3.30423 − 0.885365i −1.67681 − 6.25795i 4.25658 − 4.25658i −4.16228 + 7.20928i 3.60464 − 2.08114i
150.1 −0.793850 + 0.212712i 2.08114 − 3.60464i −2.87915 + 1.66228i 3.58114 + 3.58114i −0.885365 + 3.30423i 6.25795 + 1.67681i 4.25658 − 4.25658i −4.16228 − 7.20928i −3.60464 − 2.08114i
150.2 3.52590 − 0.944762i −1.08114 + 1.87259i 8.07530 − 4.66228i 0.418861 + 0.418861i −2.04284 + 7.62398i 1.93820 + 0.519339i 13.7434 − 13.7434i 2.16228 + 3.74517i 1.87259 + 1.08114i
\(n\): 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.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.3.f.f 8
13.b even 2 1 169.3.f.d 8
13.c even 3 1 13.3.d.a ✓ 4
13.c even 3 1 inner 169.3.f.f 8
13.d odd 4 1 169.3.f.d 8
13.d odd 4 1 inner 169.3.f.f 8
13.e even 6 1 169.3.d.d 4
13.e even 6 1 169.3.f.d 8
13.f odd 12 1 13.3.d.a ✓ 4
13.f odd 12 1 169.3.d.d 4
13.f odd 12 1 169.3.f.d 8
13.f odd 12 1 inner 169.3.f.f 8
39.i odd 6 1 117.3.j.a 4
39.k even 12 1 117.3.j.a 4
52.j odd 6 1 208.3.t.c 4
52.l even 12 1 208.3.t.c 4
65.n even 6 1 325.3.j.a 4
65.o even 12 1 325.3.g.a 4
65.q odd 12 1 325.3.g.a 4
65.q odd 12 1 325.3.g.b 4
65.s odd 12 1 325.3.j.a 4
65.t even 12 1 325.3.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.3.d.a ✓ 4 13.c even 3 1
13.3.d.a ✓ 4 13.f odd 12 1
117.3.j.a 4 39.i odd 6 1
117.3.j.a 4 39.k even 12 1
169.3.d.d 4 13.e even 6 1
169.3.d.d 4 13.f odd 12 1
169.3.f.d 8 13.b even 2 1
169.3.f.d 8 13.d odd 4 1
169.3.f.d 8 13.e even 6 1
169.3.f.d 8 13.f odd 12 1
169.3.f.f 8 1.a even 1 1 trivial
169.3.f.f 8 13.c even 3 1 inner
169.3.f.f 8 13.d odd 4 1 inner
169.3.f.f 8 13.f odd 12 1 inner
208.3.t.c 4 52.j odd 6 1
208.3.t.c 4 52.l even 12 1
325.3.g.a 4 65.o even 12 1
325.3.g.a 4 65.q odd 12 1
325.3.g.b 4 65.q odd 12 1
325.3.g.b 4 65.t even 12 1
325.3.j.a 4 65.n even 6 1
325.3.j.a 4 65.s odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 4T_{2}^{7} + 8T_{2}^{6} - 56T_{2}^{5} + 103T_{2}^{4} + 168T_{2}^{3} + 72T_{2}^{2} + 108T_{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} + 13 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 12 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + \cdots + 37015056 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 15178486401 \) Copy content Toggle raw display
$19$ \( T^{8} - 400 T^{4} + 160000 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 2998219536 \) Copy content Toggle raw display
$29$ \( (T^{4} + 20 T^{3} + \cdots + 22500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 40 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 1766100625 \) Copy content Toggle raw display
$41$ \( T^{8} + 32 T^{7} + \cdots + 136048896 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 15178486401 \) Copy content Toggle raw display
$47$ \( (T^{4} + 4 T^{3} + \cdots + 6985449)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 40 T + 150)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 154922431942656 \) Copy content Toggle raw display
$61$ \( (T^{4} - 148 T^{3} + \cdots + 29550096)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 72\!\cdots\!61 \) Copy content Toggle raw display
$73$ \( (T^{4} + 4000000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 32 T - 954)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 52 T^{3} + \cdots + 2762244)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 6887475360000 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
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