Properties

Label 169.4.b.d
Level $169$
Weight $4$
Character orbit 169.b
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
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 \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 2 q^{3} + 5 q^{4} - \beta q^{5} - 2 \beta q^{6} - 8 \beta q^{7} - 13 \beta q^{8} - 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 2 q^{3} + 5 q^{4} - \beta q^{5} - 2 \beta q^{6} - 8 \beta q^{7} - 13 \beta q^{8} - 23 q^{9} - 3 q^{10} - 8 \beta q^{11} + 10 q^{12} - 24 q^{14} - 2 \beta q^{15} + q^{16} + 117 q^{17} + 23 \beta q^{18} - 66 \beta q^{19} - 5 \beta q^{20} - 16 \beta q^{21} - 24 q^{22} - 78 q^{23} - 26 \beta q^{24} + 122 q^{25} - 100 q^{27} - 40 \beta q^{28} - 141 q^{29} - 6 q^{30} + 90 \beta q^{31} - 105 \beta q^{32} - 16 \beta q^{33} - 117 \beta q^{34} - 24 q^{35} - 115 q^{36} + 83 \beta q^{37} - 198 q^{38} - 39 q^{40} - 157 \beta q^{41} - 48 q^{42} + 104 q^{43} - 40 \beta q^{44} + 23 \beta q^{45} + 78 \beta q^{46} + 174 \beta q^{47} + 2 q^{48} + 151 q^{49} - 122 \beta q^{50} + 234 q^{51} + 93 q^{53} + 100 \beta q^{54} - 24 q^{55} - 312 q^{56} - 132 \beta q^{57} + 141 \beta q^{58} - 164 \beta q^{59} - 10 \beta q^{60} + 145 q^{61} + 270 q^{62} + 184 \beta q^{63} - 307 q^{64} - 48 q^{66} + 454 \beta q^{67} + 585 q^{68} - 156 q^{69} + 24 \beta q^{70} + 610 \beta q^{71} + 299 \beta q^{72} + 265 \beta q^{73} + 249 q^{74} + 244 q^{75} - 330 \beta q^{76} - 192 q^{77} + 1276 q^{79} - \beta q^{80} + 421 q^{81} - 471 q^{82} - 456 \beta q^{83} - 80 \beta q^{84} - 117 \beta q^{85} - 104 \beta q^{86} - 282 q^{87} - 312 q^{88} + 564 \beta q^{89} + 69 q^{90} - 390 q^{92} + 180 \beta q^{93} + 522 q^{94} - 198 q^{95} - 210 \beta q^{96} + 116 \beta q^{97} - 151 \beta q^{98} + 184 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 10 q^{4} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 10 q^{4} - 46 q^{9} - 6 q^{10} + 20 q^{12} - 48 q^{14} + 2 q^{16} + 234 q^{17} - 48 q^{22} - 156 q^{23} + 244 q^{25} - 200 q^{27} - 282 q^{29} - 12 q^{30} - 48 q^{35} - 230 q^{36} - 396 q^{38} - 78 q^{40} - 96 q^{42} + 208 q^{43} + 4 q^{48} + 302 q^{49} + 468 q^{51} + 186 q^{53} - 48 q^{55} - 624 q^{56} + 290 q^{61} + 540 q^{62} - 614 q^{64} - 96 q^{66} + 1170 q^{68} - 312 q^{69} + 498 q^{74} + 488 q^{75} - 384 q^{77} + 2552 q^{79} + 842 q^{81} - 942 q^{82} - 564 q^{87} - 624 q^{88} + 138 q^{90} - 780 q^{92} + 1044 q^{94} - 396 q^{95}+O(q^{100}) \) 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
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i 2.00000 5.00000 1.73205i 3.46410i 13.8564i 22.5167i −23.0000 −3.00000
168.2 1.73205i 2.00000 5.00000 1.73205i 3.46410i 13.8564i 22.5167i −23.0000 −3.00000
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.b.d 2
13.b even 2 1 inner 169.4.b.d 2
13.c even 3 1 13.4.e.b 2
13.c even 3 1 169.4.e.a 2
13.d odd 4 2 169.4.a.i 2
13.e even 6 1 13.4.e.b 2
13.e even 6 1 169.4.e.a 2
13.f odd 12 4 169.4.c.h 4
39.f even 4 2 1521.4.a.o 2
39.h odd 6 1 117.4.q.a 2
39.i odd 6 1 117.4.q.a 2
52.i odd 6 1 208.4.w.b 2
52.j odd 6 1 208.4.w.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 13.c even 3 1
13.4.e.b 2 13.e even 6 1
117.4.q.a 2 39.h odd 6 1
117.4.q.a 2 39.i odd 6 1
169.4.a.i 2 13.d odd 4 2
169.4.b.d 2 1.a even 1 1 trivial
169.4.b.d 2 13.b even 2 1 inner
169.4.c.h 4 13.f odd 12 4
169.4.e.a 2 13.c even 3 1
169.4.e.a 2 13.e even 6 1
208.4.w.b 2 52.i odd 6 1
208.4.w.b 2 52.j odd 6 1
1521.4.a.o 2 39.f even 4 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} + 192 \) Copy content Toggle raw display
$11$ \( T^{2} + 192 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 117)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 13068 \) Copy content Toggle raw display
$23$ \( (T + 78)^{2} \) Copy content Toggle raw display
$29$ \( (T + 141)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 24300 \) Copy content Toggle raw display
$37$ \( T^{2} + 20667 \) Copy content Toggle raw display
$41$ \( T^{2} + 73947 \) Copy content Toggle raw display
$43$ \( (T - 104)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 90828 \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 80688 \) Copy content Toggle raw display
$61$ \( (T - 145)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 618348 \) Copy content Toggle raw display
$71$ \( T^{2} + 1116300 \) Copy content Toggle raw display
$73$ \( T^{2} + 210675 \) Copy content Toggle raw display
$79$ \( (T - 1276)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 623808 \) Copy content Toggle raw display
$89$ \( T^{2} + 954288 \) Copy content Toggle raw display
$97$ \( T^{2} + 40368 \) Copy content Toggle raw display
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