Properties

Label 171.1.o.a
Level $171$
Weight $1$
Character orbit 171.o
Analytic conductor $0.085$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 171.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0853401171602\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.29241.1
Artin image: $\SL(2,3):C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{4} q^{5} -\zeta_{12}^{2} q^{6} -\zeta_{12}^{2} q^{7} -\zeta_{12}^{3} q^{8} - q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} + \zeta_{12}^{3} q^{3} + \zeta_{12}^{4} q^{5} -\zeta_{12}^{2} q^{6} -\zeta_{12}^{2} q^{7} -\zeta_{12}^{3} q^{8} - q^{9} -\zeta_{12}^{3} q^{10} + \zeta_{12}^{2} q^{11} + \zeta_{12} q^{13} + \zeta_{12} q^{14} -\zeta_{12} q^{15} + \zeta_{12}^{2} q^{16} -\zeta_{12}^{5} q^{18} + \zeta_{12}^{3} q^{19} -\zeta_{12}^{5} q^{21} -\zeta_{12} q^{22} -\zeta_{12}^{4} q^{23} + q^{24} - q^{26} -\zeta_{12}^{3} q^{27} -\zeta_{12}^{5} q^{29} + q^{30} -\zeta_{12} q^{31} + \zeta_{12}^{5} q^{33} + q^{35} -\zeta_{12}^{2} q^{38} + \zeta_{12}^{4} q^{39} + \zeta_{12} q^{40} + \zeta_{12} q^{41} + \zeta_{12}^{4} q^{42} -\zeta_{12}^{2} q^{43} -\zeta_{12}^{4} q^{45} + \zeta_{12}^{3} q^{46} -\zeta_{12}^{2} q^{47} + \zeta_{12}^{5} q^{48} + \zeta_{12}^{2} q^{54} - q^{55} + \zeta_{12}^{5} q^{56} - q^{57} + \zeta_{12}^{4} q^{58} -\zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} + q^{62} + \zeta_{12}^{2} q^{63} - q^{64} + \zeta_{12}^{5} q^{65} -\zeta_{12}^{4} q^{66} -\zeta_{12} q^{67} + \zeta_{12} q^{69} + \zeta_{12}^{5} q^{70} + \zeta_{12}^{3} q^{72} -\zeta_{12}^{4} q^{77} -\zeta_{12}^{3} q^{78} -\zeta_{12}^{5} q^{79} - q^{80} + q^{81} - q^{82} + \zeta_{12}^{2} q^{83} + \zeta_{12} q^{86} + \zeta_{12}^{2} q^{87} -\zeta_{12}^{5} q^{88} + \zeta_{12}^{3} q^{90} -\zeta_{12}^{3} q^{91} -\zeta_{12}^{4} q^{93} + \zeta_{12} q^{94} -\zeta_{12} q^{95} + \zeta_{12}^{5} q^{97} -\zeta_{12}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 2q^{6} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{5} - 2q^{6} - 2q^{7} - 4q^{9} + 2q^{11} + 2q^{16} + 2q^{23} + 4q^{24} - 4q^{26} + 4q^{30} + 4q^{35} - 2q^{38} - 2q^{39} - 2q^{42} - 2q^{43} + 2q^{45} - 2q^{47} + 2q^{54} - 4q^{55} - 4q^{57} - 2q^{58} + 2q^{61} + 4q^{62} + 2q^{63} - 4q^{64} + 2q^{66} + 2q^{77} - 4q^{80} + 4q^{81} - 4q^{82} + 2q^{83} + 2q^{87} + 2q^{93} - 2q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(20\) \(154\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
94.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 1.00000i 0 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000i −1.00000 1.00000i
94.2 0.866025 0.500000i 1.00000i 0 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000i −1.00000 1.00000i
151.1 −0.866025 0.500000i 1.00000i 0 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 1.00000i
151.2 0.866025 + 0.500000i 1.00000i 0 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000i −1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
19.b odd 2 1 inner
171.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.1.o.a 4
3.b odd 2 1 513.1.o.a 4
4.b odd 2 1 2736.1.bs.a 4
9.c even 3 1 inner 171.1.o.a 4
9.c even 3 1 1539.1.c.d 2
9.d odd 6 1 513.1.o.a 4
9.d odd 6 1 1539.1.c.c 2
19.b odd 2 1 inner 171.1.o.a 4
19.c even 3 1 3249.1.i.a 4
19.c even 3 1 3249.1.s.a 4
19.d odd 6 1 3249.1.i.a 4
19.d odd 6 1 3249.1.s.a 4
19.e even 9 3 3249.1.bc.a 12
19.e even 9 3 3249.1.be.a 12
19.f odd 18 3 3249.1.bc.a 12
19.f odd 18 3 3249.1.be.a 12
36.f odd 6 1 2736.1.bs.a 4
57.d even 2 1 513.1.o.a 4
76.d even 2 1 2736.1.bs.a 4
171.g even 3 1 3249.1.i.a 4
171.h even 3 1 3249.1.s.a 4
171.i odd 6 1 3249.1.s.a 4
171.l even 6 1 513.1.o.a 4
171.l even 6 1 1539.1.c.c 2
171.o odd 6 1 inner 171.1.o.a 4
171.o odd 6 1 1539.1.c.d 2
171.s odd 6 1 3249.1.i.a 4
171.v even 9 3 3249.1.be.a 12
171.w even 9 3 3249.1.bc.a 12
171.bc odd 18 3 3249.1.be.a 12
171.be odd 18 3 3249.1.bc.a 12
684.w even 6 1 2736.1.bs.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.o.a 4 1.a even 1 1 trivial
171.1.o.a 4 9.c even 3 1 inner
171.1.o.a 4 19.b odd 2 1 inner
171.1.o.a 4 171.o odd 6 1 inner
513.1.o.a 4 3.b odd 2 1
513.1.o.a 4 9.d odd 6 1
513.1.o.a 4 57.d even 2 1
513.1.o.a 4 171.l even 6 1
1539.1.c.c 2 9.d odd 6 1
1539.1.c.c 2 171.l even 6 1
1539.1.c.d 2 9.c even 3 1
1539.1.c.d 2 171.o odd 6 1
2736.1.bs.a 4 4.b odd 2 1
2736.1.bs.a 4 36.f odd 6 1
2736.1.bs.a 4 76.d even 2 1
2736.1.bs.a 4 684.w even 6 1
3249.1.i.a 4 19.c even 3 1
3249.1.i.a 4 19.d odd 6 1
3249.1.i.a 4 171.g even 3 1
3249.1.i.a 4 171.s odd 6 1
3249.1.s.a 4 19.c even 3 1
3249.1.s.a 4 19.d odd 6 1
3249.1.s.a 4 171.h even 3 1
3249.1.s.a 4 171.i odd 6 1
3249.1.bc.a 12 19.e even 9 3
3249.1.bc.a 12 19.f odd 18 3
3249.1.bc.a 12 171.w even 9 3
3249.1.bc.a 12 171.be odd 18 3
3249.1.be.a 12 19.e even 9 3
3249.1.be.a 12 19.f odd 18 3
3249.1.be.a 12 171.v even 9 3
3249.1.be.a 12 171.bc odd 18 3

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(171, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( 1 - T^{2} + T^{4} \)
$17$ \( T^{4} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( ( 1 - T + T^{2} )^{2} \)
$29$ \( 1 - T^{2} + T^{4} \)
$31$ \( 1 - T^{2} + T^{4} \)
$37$ \( T^{4} \)
$41$ \( 1 - T^{2} + T^{4} \)
$43$ \( ( 1 + T + T^{2} )^{2} \)
$47$ \( ( 1 + T + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( ( 1 - T + T^{2} )^{2} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( 1 - T^{2} + T^{4} \)
$83$ \( ( 1 - T + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( 1 - T^{2} + T^{4} \)
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