Properties

Label 171.2.g.b
Level $171$
Weight $2$
Character orbit 171.g
Analytic conductor $1.365$
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] = [171,2,Mod(106,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.g (of order \(3\), degree \(2\), 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: \(1.36544187456\)
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: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{4} - q^{5} + (\zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 2) q^{3} + ( - \zeta_{6} + 1) q^{4} - q^{5} + (\zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} - \zeta_{6} q^{10} + (3 \zeta_{6} - 3) q^{11} + ( - 2 \zeta_{6} + 1) q^{12} + ( - 6 \zeta_{6} + 6) q^{13} - 3 q^{14} + (\zeta_{6} - 2) q^{15} + \zeta_{6} q^{16} + (3 \zeta_{6} - 3) q^{17} + 3 q^{18} + ( - 2 \zeta_{6} - 3) q^{19} + (\zeta_{6} - 1) q^{20} + (6 \zeta_{6} - 3) q^{21} - 3 q^{22} + (8 \zeta_{6} - 8) q^{23} + ( - 3 \zeta_{6} + 6) q^{24} - 4 q^{25} + 6 q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + 3 \zeta_{6} q^{28} - 5 q^{29} + ( - \zeta_{6} - 1) q^{30} + 7 \zeta_{6} q^{31} + ( - 5 \zeta_{6} + 5) q^{32} + (6 \zeta_{6} - 3) q^{33} - 3 q^{34} + ( - 3 \zeta_{6} + 3) q^{35} - 3 \zeta_{6} q^{36} + 2 q^{37} + ( - 5 \zeta_{6} + 2) q^{38} + ( - 12 \zeta_{6} + 6) q^{39} - 3 q^{40} - q^{41} + (3 \zeta_{6} - 6) q^{42} - 8 \zeta_{6} q^{43} + 3 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{45} - 8 q^{46} + 9 q^{47} + (\zeta_{6} + 1) q^{48} - 2 \zeta_{6} q^{49} - 4 \zeta_{6} q^{50} + (6 \zeta_{6} - 3) q^{51} - 6 \zeta_{6} q^{52} - 3 \zeta_{6} q^{53} + ( - 3 \zeta_{6} + 6) q^{54} + ( - 3 \zeta_{6} + 3) q^{55} + (9 \zeta_{6} - 9) q^{56} + (\zeta_{6} - 8) q^{57} - 5 \zeta_{6} q^{58} + 3 q^{59} + (2 \zeta_{6} - 1) q^{60} + 7 q^{61} + (7 \zeta_{6} - 7) q^{62} + 9 \zeta_{6} q^{63} + 7 q^{64} + (6 \zeta_{6} - 6) q^{65} + (3 \zeta_{6} - 6) q^{66} + ( - 4 \zeta_{6} + 4) q^{67} + 3 \zeta_{6} q^{68} + (16 \zeta_{6} - 8) q^{69} + 3 q^{70} + ( - 15 \zeta_{6} + 15) q^{71} + ( - 9 \zeta_{6} + 9) q^{72} + ( - 5 \zeta_{6} + 5) q^{73} + 2 \zeta_{6} q^{74} + (4 \zeta_{6} - 8) q^{75} + (3 \zeta_{6} - 5) q^{76} - 9 \zeta_{6} q^{77} + ( - 6 \zeta_{6} + 12) q^{78} + 12 \zeta_{6} q^{79} - \zeta_{6} q^{80} - 9 \zeta_{6} q^{81} - \zeta_{6} q^{82} + ( - \zeta_{6} + 1) q^{83} + (3 \zeta_{6} + 3) q^{84} + ( - 3 \zeta_{6} + 3) q^{85} + ( - 8 \zeta_{6} + 8) q^{86} + (5 \zeta_{6} - 10) q^{87} + (9 \zeta_{6} - 9) q^{88} + \zeta_{6} q^{89} - 3 q^{90} + 18 \zeta_{6} q^{91} + 8 \zeta_{6} q^{92} + (7 \zeta_{6} + 7) q^{93} + 9 \zeta_{6} q^{94} + (2 \zeta_{6} + 3) q^{95} + ( - 10 \zeta_{6} + 5) q^{96} + 2 \zeta_{6} q^{97} + ( - 2 \zeta_{6} + 2) q^{98} + 9 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} + q^{4} - 2 q^{5} + 3 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} + q^{4} - 2 q^{5} + 3 q^{6} - 3 q^{7} + 6 q^{8} + 3 q^{9} - q^{10} - 3 q^{11} + 6 q^{13} - 6 q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 6 q^{18} - 8 q^{19} - q^{20} - 6 q^{22} - 8 q^{23} + 9 q^{24} - 8 q^{25} + 12 q^{26} + 3 q^{28} - 10 q^{29} - 3 q^{30} + 7 q^{31} + 5 q^{32} - 6 q^{34} + 3 q^{35} - 3 q^{36} + 4 q^{37} - q^{38} - 6 q^{40} - 2 q^{41} - 9 q^{42} - 8 q^{43} + 3 q^{44} - 3 q^{45} - 16 q^{46} + 18 q^{47} + 3 q^{48} - 2 q^{49} - 4 q^{50} - 6 q^{52} - 3 q^{53} + 9 q^{54} + 3 q^{55} - 9 q^{56} - 15 q^{57} - 5 q^{58} + 6 q^{59} + 14 q^{61} - 7 q^{62} + 9 q^{63} + 14 q^{64} - 6 q^{65} - 9 q^{66} + 4 q^{67} + 3 q^{68} + 6 q^{70} + 15 q^{71} + 9 q^{72} + 5 q^{73} + 2 q^{74} - 12 q^{75} - 7 q^{76} - 9 q^{77} + 18 q^{78} + 12 q^{79} - q^{80} - 9 q^{81} - q^{82} + q^{83} + 9 q^{84} + 3 q^{85} + 8 q^{86} - 15 q^{87} - 9 q^{88} + q^{89} - 6 q^{90} + 18 q^{91} + 8 q^{92} + 21 q^{93} + 9 q^{94} + 8 q^{95} + 2 q^{97} + 2 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 1.50000 0.866025i 0.500000 0.866025i −1.00000 1.50000 + 0.866025i −1.50000 + 2.59808i 3.00000 1.50000 2.59808i −0.500000 0.866025i
121.1 0.500000 0.866025i 1.50000 + 0.866025i 0.500000 + 0.866025i −1.00000 1.50000 0.866025i −1.50000 2.59808i 3.00000 1.50000 + 2.59808i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.g.b 2
3.b odd 2 1 513.2.g.b 2
9.c even 3 1 171.2.h.b yes 2
9.d odd 6 1 513.2.h.a 2
19.c even 3 1 171.2.h.b yes 2
57.h odd 6 1 513.2.h.a 2
171.g even 3 1 inner 171.2.g.b 2
171.n odd 6 1 513.2.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.g.b 2 1.a even 1 1 trivial
171.2.g.b 2 171.g even 3 1 inner
171.2.h.b yes 2 9.c even 3 1
171.2.h.b yes 2 19.c even 3 1
513.2.g.b 2 3.b odd 2 1
513.2.g.b 2 171.n odd 6 1
513.2.h.a 2 9.d odd 6 1
513.2.h.a 2 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(171, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( (T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( (T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( (T - 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( (T - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$73$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$83$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$89$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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