Properties

Label 169.4.c.d
Level $169$
Weight $4$
Character orbit 169.c
Analytic conductor $9.971$
Analytic rank $0$
Dimension $2$
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(22,169)]
 
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([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.22");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), 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: \(2\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 4 \zeta_{6} + 4) q^{2} + (2 \zeta_{6} - 2) q^{3} - 8 \zeta_{6} q^{4} - 17 q^{5} + 8 \zeta_{6} q^{6} + 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \zeta_{6} + 4) q^{2} + (2 \zeta_{6} - 2) q^{3} - 8 \zeta_{6} q^{4} - 17 q^{5} + 8 \zeta_{6} q^{6} + 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} + (68 \zeta_{6} - 68) q^{10} + (32 \zeta_{6} - 32) q^{11} + 16 q^{12} + 80 q^{14} + ( - 34 \zeta_{6} + 34) q^{15} + ( - 64 \zeta_{6} + 64) q^{16} + 13 \zeta_{6} q^{17} + 92 q^{18} + 30 \zeta_{6} q^{19} + 136 \zeta_{6} q^{20} - 40 q^{21} + 128 \zeta_{6} q^{22} + (78 \zeta_{6} - 78) q^{23} + 164 q^{25} - 100 q^{27} + ( - 160 \zeta_{6} + 160) q^{28} + (197 \zeta_{6} - 197) q^{29} - 136 \zeta_{6} q^{30} + 74 q^{31} - 256 \zeta_{6} q^{32} - 64 \zeta_{6} q^{33} + 52 q^{34} - 340 \zeta_{6} q^{35} + ( - 184 \zeta_{6} + 184) q^{36} + (227 \zeta_{6} - 227) q^{37} + 120 q^{38} + (165 \zeta_{6} - 165) q^{41} + (160 \zeta_{6} - 160) q^{42} + 156 \zeta_{6} q^{43} + 256 q^{44} - 391 \zeta_{6} q^{45} + 312 \zeta_{6} q^{46} + 162 q^{47} + 128 \zeta_{6} q^{48} + (57 \zeta_{6} - 57) q^{49} + ( - 656 \zeta_{6} + 656) q^{50} - 26 q^{51} + 93 q^{53} + (400 \zeta_{6} - 400) q^{54} + ( - 544 \zeta_{6} + 544) q^{55} - 60 q^{57} + 788 \zeta_{6} q^{58} - 864 \zeta_{6} q^{59} - 272 q^{60} - 145 \zeta_{6} q^{61} + ( - 296 \zeta_{6} + 296) q^{62} + (460 \zeta_{6} - 460) q^{63} - 512 q^{64} - 256 q^{66} + ( - 862 \zeta_{6} + 862) q^{67} + ( - 104 \zeta_{6} + 104) q^{68} - 156 \zeta_{6} q^{69} - 1360 q^{70} + 654 \zeta_{6} q^{71} - 215 q^{73} + 908 \zeta_{6} q^{74} + (328 \zeta_{6} - 328) q^{75} + ( - 240 \zeta_{6} + 240) q^{76} - 640 q^{77} - 76 q^{79} + (1088 \zeta_{6} - 1088) q^{80} + (421 \zeta_{6} - 421) q^{81} + 660 \zeta_{6} q^{82} - 628 q^{83} + 320 \zeta_{6} q^{84} - 221 \zeta_{6} q^{85} + 624 q^{86} - 394 \zeta_{6} q^{87} + (266 \zeta_{6} - 266) q^{89} - 1564 q^{90} + 624 q^{92} + (148 \zeta_{6} - 148) q^{93} + ( - 648 \zeta_{6} + 648) q^{94} - 510 \zeta_{6} q^{95} + 512 q^{96} + 238 \zeta_{6} q^{97} + 228 \zeta_{6} q^{98} - 736 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} - 34 q^{5} + 8 q^{6} + 20 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 2 q^{3} - 8 q^{4} - 34 q^{5} + 8 q^{6} + 20 q^{7} + 23 q^{9} - 68 q^{10} - 32 q^{11} + 32 q^{12} + 160 q^{14} + 34 q^{15} + 64 q^{16} + 13 q^{17} + 184 q^{18} + 30 q^{19} + 136 q^{20} - 80 q^{21} + 128 q^{22} - 78 q^{23} + 328 q^{25} - 200 q^{27} + 160 q^{28} - 197 q^{29} - 136 q^{30} + 148 q^{31} - 256 q^{32} - 64 q^{33} + 104 q^{34} - 340 q^{35} + 184 q^{36} - 227 q^{37} + 240 q^{38} - 165 q^{41} - 160 q^{42} + 156 q^{43} + 512 q^{44} - 391 q^{45} + 312 q^{46} + 324 q^{47} + 128 q^{48} - 57 q^{49} + 656 q^{50} - 52 q^{51} + 186 q^{53} - 400 q^{54} + 544 q^{55} - 120 q^{57} + 788 q^{58} - 864 q^{59} - 544 q^{60} - 145 q^{61} + 296 q^{62} - 460 q^{63} - 1024 q^{64} - 512 q^{66} + 862 q^{67} + 104 q^{68} - 156 q^{69} - 2720 q^{70} + 654 q^{71} - 430 q^{73} + 908 q^{74} - 328 q^{75} + 240 q^{76} - 1280 q^{77} - 152 q^{79} - 1088 q^{80} - 421 q^{81} + 660 q^{82} - 1256 q^{83} + 320 q^{84} - 221 q^{85} + 1248 q^{86} - 394 q^{87} - 266 q^{89} - 3128 q^{90} + 1248 q^{92} - 148 q^{93} + 648 q^{94} - 510 q^{95} + 1024 q^{96} + 238 q^{97} + 228 q^{98} - 1472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
0.500000 + 0.866025i
0.500000 0.866025i
2.00000 3.46410i −1.00000 + 1.73205i −4.00000 6.92820i −17.0000 4.00000 + 6.92820i 10.0000 + 17.3205i 0 11.5000 + 19.9186i −34.0000 + 58.8897i
146.1 2.00000 + 3.46410i −1.00000 1.73205i −4.00000 + 6.92820i −17.0000 4.00000 6.92820i 10.0000 17.3205i 0 11.5000 19.9186i −34.0000 58.8897i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 4T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$5$ \( (T + 17)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 400 \) Copy content Toggle raw display
$11$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$19$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$23$ \( T^{2} + 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} + 197T + 38809 \) Copy content Toggle raw display
$31$ \( (T - 74)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 227T + 51529 \) Copy content Toggle raw display
$41$ \( T^{2} + 165T + 27225 \) Copy content Toggle raw display
$43$ \( T^{2} - 156T + 24336 \) Copy content Toggle raw display
$47$ \( (T - 162)^{2} \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 864T + 746496 \) Copy content Toggle raw display
$61$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$67$ \( T^{2} - 862T + 743044 \) Copy content Toggle raw display
$71$ \( T^{2} - 654T + 427716 \) Copy content Toggle raw display
$73$ \( (T + 215)^{2} \) Copy content Toggle raw display
$79$ \( (T + 76)^{2} \) Copy content Toggle raw display
$83$ \( (T + 628)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 266T + 70756 \) Copy content Toggle raw display
$97$ \( T^{2} - 238T + 56644 \) Copy content Toggle raw display
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