Properties

Label 273.2.n.b
Level $273$
Weight $2$
Character orbit 273.n
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.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(\zeta_{8}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
1.00000 1.00000i −1.70711 + 0.292893i 0 2.70711 2.70711i −1.41421 + 2.00000i −0.707107 + 0.707107i 2.00000 + 2.00000i 2.82843 1.00000i 5.41421i
8.2 1.00000 1.00000i −0.292893 + 1.70711i 0 1.29289 1.29289i 1.41421 + 2.00000i 0.707107 0.707107i 2.00000 + 2.00000i −2.82843 1.00000i 2.58579i
239.1 1.00000 + 1.00000i −1.70711 0.292893i 0 2.70711 + 2.70711i −1.41421 2.00000i −0.707107 0.707107i 2.00000 2.00000i 2.82843 + 1.00000i 5.41421i
239.2 1.00000 + 1.00000i −0.292893 1.70711i 0 1.29289 + 1.29289i 1.41421 2.00000i 0.707107 + 0.707107i 2.00000 2.00000i −2.82843 + 1.00000i 2.58579i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.n.b yes 4
3.b odd 2 1 273.2.n.a 4
13.d odd 4 1 273.2.n.a 4
39.f even 4 1 inner 273.2.n.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.n.a 4 3.b odd 2 1
273.2.n.a 4 13.d odd 4 1
273.2.n.b yes 4 1.a even 1 1 trivial
273.2.n.b yes 4 39.f even 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2 - 2 T + T^{2} )^{2} \)
$3$ \( 9 + 12 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$5$ \( 49 - 56 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$7$ \( 1 + T^{4} \)
$11$ \( 16 + T^{4} \)
$13$ \( 169 + 24 T^{2} + T^{4} \)
$17$ \( ( -4 + T )^{4} \)
$19$ \( 529 + 368 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$23$ \( ( -7 - 2 T + T^{2} )^{2} \)
$29$ \( 961 + 66 T^{2} + T^{4} \)
$31$ \( 1681 - 328 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 2116 + 920 T + 200 T^{2} + 20 T^{3} + T^{4} \)
$41$ \( 16 - 64 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$43$ \( 961 + 66 T^{2} + T^{4} \)
$47$ \( 529 + 368 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$53$ \( 16129 + 258 T^{2} + T^{4} \)
$59$ \( 16 + T^{4} \)
$61$ \( ( -82 + 8 T + T^{2} )^{2} \)
$67$ \( 8464 - 736 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$71$ \( 3844 - 1736 T + 392 T^{2} - 28 T^{3} + T^{4} \)
$73$ \( 1 - 8 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( ( 1 - 6 T + T^{2} )^{2} \)
$83$ \( 1 + T^{4} \)
$89$ \( 14161 - 3808 T + 512 T^{2} - 32 T^{3} + T^{4} \)
$97$ \( 16129 + 4064 T + 512 T^{2} + 32 T^{3} + T^{4} \)
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