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Properties

Label 169.4.e.f
Level $169$
Weight $4$
Character orbit 169.e
Analytic conductor $9.971$
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] = [169,4,Mod(23,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.e (of order \(6\), degree \(2\), 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: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.1731891456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 3 \beta_{7} + \beta_{2}) q^{3} + (\beta_{7} - \beta_{4} - 3 \beta_{2} - 4) q^{4} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 6 \beta_{6} - \beta_{5}) q^{6} + (5 \beta_{6} + 11 \beta_{5}) q^{7} + (2 \beta_{6} + 11 \beta_{5} + \cdots - 11 \beta_1) q^{8}+ \cdots + (15 \beta_{7} - 15 \beta_{4} + \cdots - 25) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 3 \beta_{7} + \beta_{2}) q^{3} + (\beta_{7} - \beta_{4} - 3 \beta_{2} - 4) q^{4} + (\beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{5} + ( - 6 \beta_{6} - \beta_{5}) q^{6} + (5 \beta_{6} + 11 \beta_{5}) q^{7} + (2 \beta_{6} + 11 \beta_{5} + \cdots - 11 \beta_1) q^{8}+ \cdots + (65 \beta_{6} + 390 \beta_{5} + \cdots - 390 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} - 14 q^{4} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{3} - 14 q^{4} - 70 q^{9} + 14 q^{10} + 172 q^{12} + 356 q^{14} + 78 q^{16} + 38 q^{17} - 284 q^{22} + 392 q^{23} + 948 q^{25} + 1340 q^{27} + 88 q^{29} + 86 q^{30} - 214 q^{35} - 500 q^{36} + 1256 q^{38} - 356 q^{40} - 394 q^{42} + 574 q^{43} - 570 q^{48} + 766 q^{49} - 1924 q^{51} - 472 q^{53} + 36 q^{55} - 2030 q^{56} + 2116 q^{61} - 664 q^{62} - 3076 q^{64} - 3272 q^{66} + 422 q^{68} + 1592 q^{69} - 294 q^{74} - 1032 q^{75} - 3048 q^{77} - 4032 q^{79} - 244 q^{81} + 144 q^{82} + 5116 q^{87} - 2484 q^{88} - 1000 q^{90} - 3152 q^{92} + 1622 q^{94} - 292 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 181\nu ) / 130 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 116 ) / 65 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 65\nu^{5} - 585\nu^{3} + 1296\nu ) / 1040 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -29\nu^{7} + 325\nu^{5} - 1885\nu^{3} + 4176\nu ) / 2080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} + 5\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} - 5\beta_{5} - 2\beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{7} + 29\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{6} - 29\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 65\beta_{4} - 116 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 130\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−2.21837 + 1.28078i
−1.35234 + 0.780776i
1.35234 − 0.780776i
2.21837 − 1.28078i
−2.21837 − 1.28078i
−1.35234 − 0.780776i
1.35234 + 0.780776i
2.21837 + 1.28078i
−2.21837 + 1.28078i 1.84233 + 3.19101i −0.719224 + 1.24573i − 0.561553i −8.17394 − 4.71922i −15.7418 − 9.08854i − 24.1771i 6.71165 − 11.6249i 0.719224 + 1.24573i
23.2 −1.35234 + 0.780776i −4.34233 − 7.52113i −2.78078 + 4.81645i − 3.56155i 11.7446 + 6.78078i −23.5360 − 13.5885i − 21.1771i −24.2116 + 41.9358i 2.78078 + 4.81645i
23.3 1.35234 − 0.780776i −4.34233 − 7.52113i −2.78078 + 4.81645i 3.56155i −11.7446 − 6.78078i 23.5360 + 13.5885i 21.1771i −24.2116 + 41.9358i 2.78078 + 4.81645i
23.4 2.21837 − 1.28078i 1.84233 + 3.19101i −0.719224 + 1.24573i 0.561553i 8.17394 + 4.71922i 15.7418 + 9.08854i 24.1771i 6.71165 − 11.6249i 0.719224 + 1.24573i
147.1 −2.21837 − 1.28078i 1.84233 − 3.19101i −0.719224 − 1.24573i 0.561553i −8.17394 + 4.71922i −15.7418 + 9.08854i 24.1771i 6.71165 + 11.6249i 0.719224 − 1.24573i
147.2 −1.35234 − 0.780776i −4.34233 + 7.52113i −2.78078 − 4.81645i 3.56155i 11.7446 − 6.78078i −23.5360 + 13.5885i 21.1771i −24.2116 − 41.9358i 2.78078 − 4.81645i
147.3 1.35234 + 0.780776i −4.34233 + 7.52113i −2.78078 − 4.81645i − 3.56155i −11.7446 + 6.78078i 23.5360 − 13.5885i − 21.1771i −24.2116 − 41.9358i 2.78078 − 4.81645i
147.4 2.21837 + 1.28078i 1.84233 − 3.19101i −0.719224 − 1.24573i − 0.561553i 8.17394 − 4.71922i 15.7418 − 9.08854i − 24.1771i 6.71165 + 11.6249i 0.719224 − 1.24573i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.4
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.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.e.f 8
13.b even 2 1 inner 169.4.e.f 8
13.c even 3 1 169.4.b.f 4
13.c even 3 1 inner 169.4.e.f 8
13.d odd 4 1 169.4.c.g 4
13.d odd 4 1 169.4.c.j 4
13.e even 6 1 169.4.b.f 4
13.e even 6 1 inner 169.4.e.f 8
13.f odd 12 1 13.4.a.b ✓ 2
13.f odd 12 1 169.4.a.g 2
13.f odd 12 1 169.4.c.g 4
13.f odd 12 1 169.4.c.j 4
39.k even 12 1 117.4.a.d 2
39.k even 12 1 1521.4.a.r 2
52.l even 12 1 208.4.a.h 2
65.o even 12 1 325.4.b.e 4
65.s odd 12 1 325.4.a.f 2
65.t even 12 1 325.4.b.e 4
91.bc even 12 1 637.4.a.b 2
104.u even 12 1 832.4.a.z 2
104.x odd 12 1 832.4.a.s 2
143.o even 12 1 1573.4.a.b 2
156.v odd 12 1 1872.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.b ✓ 2 13.f odd 12 1
117.4.a.d 2 39.k even 12 1
169.4.a.g 2 13.f odd 12 1
169.4.b.f 4 13.c even 3 1
169.4.b.f 4 13.e even 6 1
169.4.c.g 4 13.d odd 4 1
169.4.c.g 4 13.f odd 12 1
169.4.c.j 4 13.d odd 4 1
169.4.c.j 4 13.f odd 12 1
169.4.e.f 8 1.a even 1 1 trivial
169.4.e.f 8 13.b even 2 1 inner
169.4.e.f 8 13.c even 3 1 inner
169.4.e.f 8 13.e even 6 1 inner
208.4.a.h 2 52.l even 12 1
325.4.a.f 2 65.s odd 12 1
325.4.b.e 4 65.o even 12 1
325.4.b.e 4 65.t even 12 1
637.4.a.b 2 91.bc even 12 1
832.4.a.s 2 104.x odd 12 1
832.4.a.z 2 104.u even 12 1
1521.4.a.r 2 39.k even 12 1
1573.4.a.b 2 143.o even 12 1
1872.4.a.bb 2 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 9T_{2}^{6} + 65T_{2}^{4} - 144T_{2}^{2} + 256 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 9 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} + 5 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 13 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 59553569296 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 952857108736 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 19 T^{3} + \cdots + 1295044)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 44859774689536 \) Copy content Toggle raw display
$23$ \( (T^{4} - 196 T^{3} + \cdots + 80856064)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 44 T^{3} + \cdots + 1496451856)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 13524 T^{2} + 9388096)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} - 287 T^{3} + \cdots + 4397811856)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 219061 T^{2} + 222546724)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 118 T - 344)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 98\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{4} - 1058 T^{3} + \cdots + 15981005056)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + 678568 T^{2} + 55373619856)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1008 T + 247216)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2198436 T^{2} + 668574416896)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 67\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
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