Properties

Label 273.1.s.b
Level $273$
Weight $1$
Character orbit 273.s
Analytic conductor $0.136$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 273.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.136244748449\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.74529.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12}^{3} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{12}^{3} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12}^{2} q^{6} + \zeta_{12}^{4} q^{7} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} + \zeta_{12}^{2} q^{10} - q^{13} + \zeta_{12} q^{14} + \zeta_{12}^{4} q^{15} - q^{16} + \zeta_{12}^{3} q^{17} -\zeta_{12} q^{18} + \zeta_{12}^{3} q^{21} + \zeta_{12}^{3} q^{23} -\zeta_{12}^{2} q^{24} + \zeta_{12}^{3} q^{26} -\zeta_{12}^{3} q^{27} -\zeta_{12} q^{29} + \zeta_{12} q^{30} -\zeta_{12}^{4} q^{31} + q^{34} -\zeta_{12}^{3} q^{35} + q^{37} + \zeta_{12}^{5} q^{39} + \zeta_{12}^{2} q^{40} + \zeta_{12} q^{41} + q^{42} -\zeta_{12}^{2} q^{43} + \zeta_{12}^{3} q^{45} + q^{46} + \zeta_{12}^{5} q^{47} + \zeta_{12}^{5} q^{48} -\zeta_{12}^{2} q^{49} + \zeta_{12}^{2} q^{51} + \zeta_{12} q^{53} - q^{54} + \zeta_{12} q^{56} + \zeta_{12}^{4} q^{58} -\zeta_{12}^{3} q^{59} -\zeta_{12} q^{62} + \zeta_{12}^{2} q^{63} - q^{64} -\zeta_{12}^{5} q^{65} + \zeta_{12}^{2} q^{69} - q^{70} -\zeta_{12}^{5} q^{71} -\zeta_{12} q^{72} + \zeta_{12}^{4} q^{73} -\zeta_{12}^{3} q^{74} + \zeta_{12}^{2} q^{78} -\zeta_{12}^{2} q^{79} -\zeta_{12}^{5} q^{80} -\zeta_{12}^{2} q^{81} -\zeta_{12}^{4} q^{82} -\zeta_{12}^{2} q^{85} + \zeta_{12}^{5} q^{86} - q^{87} -\zeta_{12}^{3} q^{89} + q^{90} -\zeta_{12}^{4} q^{91} -\zeta_{12}^{3} q^{93} + \zeta_{12}^{2} q^{94} + \zeta_{12}^{2} q^{97} + \zeta_{12}^{5} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{6} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{15} - 4 q^{16} - 2 q^{24} + 2 q^{31} + 4 q^{34} + 4 q^{37} + 2 q^{40} + 4 q^{42} - 2 q^{43} + 4 q^{46} - 2 q^{49} + 2 q^{51} - 4 q^{54} - 2 q^{58} + 2 q^{63} - 4 q^{64} + 2 q^{69} - 4 q^{70} - 2 q^{73} + 2 q^{78} - 2 q^{79} - 2 q^{81} + 2 q^{82} - 2 q^{85} - 4 q^{87} + 4 q^{90} + 2 q^{91} + 2 q^{94} + 2 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(-\zeta_{12}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
1.00000i 0.866025 0.500000i 0 −0.866025 + 0.500000i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i 0.500000 0.866025i 0.500000 + 0.866025i
74.2 1.00000i −0.866025 + 0.500000i 0 0.866025 0.500000i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000i 0.500000 0.866025i 0.500000 + 0.866025i
107.1 1.00000i −0.866025 0.500000i 0 0.866025 + 0.500000i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i 0.500000 + 0.866025i 0.500000 0.866025i
107.2 1.00000i 0.866025 + 0.500000i 0 −0.866025 0.500000i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000i 0.500000 + 0.866025i 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
3.b odd 2 1 inner
91.h even 3 1 inner
273.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.1.s.b 4
3.b odd 2 1 inner 273.1.s.b 4
7.b odd 2 1 1911.1.s.b 4
7.c even 3 1 273.1.bm.b yes 4
7.c even 3 1 1911.1.be.c 4
7.d odd 6 1 1911.1.be.d 4
7.d odd 6 1 1911.1.bm.b 4
13.b even 2 1 3549.1.s.b 4
13.c even 3 1 273.1.bm.b yes 4
13.c even 3 1 3549.1.bk.d 4
13.d odd 4 1 3549.1.bp.b 4
13.d odd 4 1 3549.1.bp.d 4
13.e even 6 1 3549.1.bk.c 4
13.e even 6 1 3549.1.bm.c 4
13.f odd 12 1 3549.1.w.c 4
13.f odd 12 1 3549.1.w.e 4
13.f odd 12 1 3549.1.x.b 4
13.f odd 12 1 3549.1.x.d 4
21.c even 2 1 1911.1.s.b 4
21.g even 6 1 1911.1.be.d 4
21.g even 6 1 1911.1.bm.b 4
21.h odd 6 1 273.1.bm.b yes 4
21.h odd 6 1 1911.1.be.c 4
39.d odd 2 1 3549.1.s.b 4
39.f even 4 1 3549.1.bp.b 4
39.f even 4 1 3549.1.bp.d 4
39.h odd 6 1 3549.1.bk.c 4
39.h odd 6 1 3549.1.bm.c 4
39.i odd 6 1 273.1.bm.b yes 4
39.i odd 6 1 3549.1.bk.d 4
39.k even 12 1 3549.1.w.c 4
39.k even 12 1 3549.1.w.e 4
39.k even 12 1 3549.1.x.b 4
39.k even 12 1 3549.1.x.d 4
91.g even 3 1 1911.1.be.c 4
91.g even 3 1 3549.1.bk.d 4
91.h even 3 1 inner 273.1.s.b 4
91.k even 6 1 3549.1.s.b 4
91.m odd 6 1 1911.1.be.d 4
91.n odd 6 1 1911.1.bm.b 4
91.r even 6 1 3549.1.bm.c 4
91.u even 6 1 3549.1.bk.c 4
91.v odd 6 1 1911.1.s.b 4
91.x odd 12 1 3549.1.bp.b 4
91.x odd 12 1 3549.1.bp.d 4
91.z odd 12 1 3549.1.x.b 4
91.z odd 12 1 3549.1.x.d 4
91.bd odd 12 1 3549.1.w.c 4
91.bd odd 12 1 3549.1.w.e 4
273.r even 6 1 1911.1.s.b 4
273.s odd 6 1 inner 273.1.s.b 4
273.w odd 6 1 3549.1.bm.c 4
273.x odd 6 1 3549.1.bk.c 4
273.bf even 6 1 1911.1.be.d 4
273.bm odd 6 1 1911.1.be.c 4
273.bm odd 6 1 3549.1.bk.d 4
273.bn even 6 1 1911.1.bm.b 4
273.bp odd 6 1 3549.1.s.b 4
273.bv even 12 1 3549.1.bp.b 4
273.bv even 12 1 3549.1.bp.d 4
273.bw even 12 1 3549.1.w.c 4
273.bw even 12 1 3549.1.w.e 4
273.cd even 12 1 3549.1.x.b 4
273.cd even 12 1 3549.1.x.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.1.s.b 4 1.a even 1 1 trivial
273.1.s.b 4 3.b odd 2 1 inner
273.1.s.b 4 91.h even 3 1 inner
273.1.s.b 4 273.s odd 6 1 inner
273.1.bm.b yes 4 7.c even 3 1
273.1.bm.b yes 4 13.c even 3 1
273.1.bm.b yes 4 21.h odd 6 1
273.1.bm.b yes 4 39.i odd 6 1
1911.1.s.b 4 7.b odd 2 1
1911.1.s.b 4 21.c even 2 1
1911.1.s.b 4 91.v odd 6 1
1911.1.s.b 4 273.r even 6 1
1911.1.be.c 4 7.c even 3 1
1911.1.be.c 4 21.h odd 6 1
1911.1.be.c 4 91.g even 3 1
1911.1.be.c 4 273.bm odd 6 1
1911.1.be.d 4 7.d odd 6 1
1911.1.be.d 4 21.g even 6 1
1911.1.be.d 4 91.m odd 6 1
1911.1.be.d 4 273.bf even 6 1
1911.1.bm.b 4 7.d odd 6 1
1911.1.bm.b 4 21.g even 6 1
1911.1.bm.b 4 91.n odd 6 1
1911.1.bm.b 4 273.bn even 6 1
3549.1.s.b 4 13.b even 2 1
3549.1.s.b 4 39.d odd 2 1
3549.1.s.b 4 91.k even 6 1
3549.1.s.b 4 273.bp odd 6 1
3549.1.w.c 4 13.f odd 12 1
3549.1.w.c 4 39.k even 12 1
3549.1.w.c 4 91.bd odd 12 1
3549.1.w.c 4 273.bw even 12 1
3549.1.w.e 4 13.f odd 12 1
3549.1.w.e 4 39.k even 12 1
3549.1.w.e 4 91.bd odd 12 1
3549.1.w.e 4 273.bw even 12 1
3549.1.x.b 4 13.f odd 12 1
3549.1.x.b 4 39.k even 12 1
3549.1.x.b 4 91.z odd 12 1
3549.1.x.b 4 273.cd even 12 1
3549.1.x.d 4 13.f odd 12 1
3549.1.x.d 4 39.k even 12 1
3549.1.x.d 4 91.z odd 12 1
3549.1.x.d 4 273.cd even 12 1
3549.1.bk.c 4 13.e even 6 1
3549.1.bk.c 4 39.h odd 6 1
3549.1.bk.c 4 91.u even 6 1
3549.1.bk.c 4 273.x odd 6 1
3549.1.bk.d 4 13.c even 3 1
3549.1.bk.d 4 39.i odd 6 1
3549.1.bk.d 4 91.g even 3 1
3549.1.bk.d 4 273.bm odd 6 1
3549.1.bm.c 4 13.e even 6 1
3549.1.bm.c 4 39.h odd 6 1
3549.1.bm.c 4 91.r even 6 1
3549.1.bm.c 4 273.w odd 6 1
3549.1.bp.b 4 13.d odd 4 1
3549.1.bp.b 4 39.f even 4 1
3549.1.bp.b 4 91.x odd 12 1
3549.1.bp.b 4 273.bv even 12 1
3549.1.bp.d 4 13.d odd 4 1
3549.1.bp.d 4 39.f even 4 1
3549.1.bp.d 4 91.x odd 12 1
3549.1.bp.d 4 273.bv even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

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