Properties

Label 273.2.by.a
Level $273$
Weight $2$
Character orbit 273.by
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.by (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
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
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0.500000 1.86603i −0.866025 + 0.500000i −1.50000 0.866025i 0.366025 0.366025i 0.500000 + 1.86603i −0.500000 2.59808i 0.366025 0.366025i 0.500000 0.866025i −0.500000 0.866025i
97.1 0.500000 + 1.86603i −0.866025 0.500000i −1.50000 + 0.866025i 0.366025 + 0.366025i 0.500000 1.86603i −0.500000 + 2.59808i 0.366025 + 0.366025i 0.500000 + 0.866025i −0.500000 + 0.866025i
202.1 0.500000 0.133975i 0.866025 + 0.500000i −1.50000 + 0.866025i −1.36603 + 1.36603i 0.500000 + 0.133975i −0.500000 + 2.59808i −1.36603 + 1.36603i 0.500000 + 0.866025i −0.500000 + 0.866025i
223.1 0.500000 + 0.133975i 0.866025 0.500000i −1.50000 0.866025i −1.36603 1.36603i 0.500000 0.133975i −0.500000 2.59808i −1.36603 1.36603i 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
91.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.by.a 4
3.b odd 2 1 819.2.fm.b 4
7.b odd 2 1 273.2.by.b yes 4
13.f odd 12 1 273.2.by.b yes 4
21.c even 2 1 819.2.fm.a 4
39.k even 12 1 819.2.fm.a 4
91.bc even 12 1 inner 273.2.by.a 4
273.ca odd 12 1 819.2.fm.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.by.a 4 1.a even 1 1 trivial
273.2.by.a 4 91.bc even 12 1 inner
273.2.by.b yes 4 7.b odd 2 1
273.2.by.b yes 4 13.f odd 12 1
819.2.fm.a 4 21.c even 2 1
819.2.fm.a 4 39.k even 12 1
819.2.fm.b 4 3.b odd 2 1
819.2.fm.b 4 273.ca odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{4} - 2 T_{2}^{3} + 5 T_{2}^{2} - 4 T_{2} + 1 \)
\( T_{5}^{4} + 2 T_{5}^{3} + 2 T_{5}^{2} - 2 T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 1 - 2 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$7$ \( ( 7 + T + T^{2} )^{2} \)
$11$ \( 16 + 32 T + 20 T^{2} + 4 T^{3} + T^{4} \)
$13$ \( 169 + 78 T + 23 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 729 + 27 T^{2} + T^{4} \)
$19$ \( 64 - 32 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( 676 - 468 T + 134 T^{2} - 18 T^{3} + T^{4} \)
$29$ \( ( 49 - 7 T + T^{2} )^{2} \)
$31$ \( 1936 - 880 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$37$ \( 625 - 500 T + 125 T^{2} - 10 T^{3} + T^{4} \)
$41$ \( 5329 + 1606 T + 221 T^{2} + 20 T^{3} + T^{4} \)
$43$ \( 4096 - 64 T^{2} + T^{4} \)
$47$ \( 2304 - 1152 T + 288 T^{2} - 24 T^{3} + T^{4} \)
$53$ \( ( 33 - 12 T + T^{2} )^{2} \)
$59$ \( 4 - 4 T + 26 T^{2} + 10 T^{3} + T^{4} \)
$61$ \( 9 + 36 T + 51 T^{2} + 12 T^{3} + T^{4} \)
$67$ \( 4356 + 396 T + 90 T^{2} + 18 T^{3} + T^{4} \)
$71$ \( 1024 + 256 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$73$ \( 14641 - 242 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$79$ \( ( -26 - 14 T + T^{2} )^{2} \)
$83$ \( 9216 + T^{4} \)
$89$ \( 2116 - 644 T + 170 T^{2} - 22 T^{3} + T^{4} \)
$97$ \( 4 + 4 T + 26 T^{2} - 10 T^{3} + T^{4} \)
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