Properties

Label 273.2.a.c
Level $273$
Weight $2$
Character orbit 273.a
Self dual yes
Analytic conductor $2.180$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−0.414214 −1.00000 −1.82843 0 0.414214 1.00000 1.58579 1.00000 0
1.2 2.41421 −1.00000 3.82843 0 −2.41421 1.00000 4.41421 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.a.c 2
3.b odd 2 1 819.2.a.g 2
4.b odd 2 1 4368.2.a.bj 2
5.b even 2 1 6825.2.a.m 2
7.b odd 2 1 1911.2.a.k 2
13.b even 2 1 3549.2.a.f 2
21.c even 2 1 5733.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.c 2 1.a even 1 1 trivial
819.2.a.g 2 3.b odd 2 1
1911.2.a.k 2 7.b odd 2 1
3549.2.a.f 2 13.b even 2 1
4368.2.a.bj 2 4.b odd 2 1
5733.2.a.n 2 21.c even 2 1
6825.2.a.m 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2 T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(273))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 - 2 T + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -4 - 4 T + T^{2} \)
$19$ \( -32 + T^{2} \)
$23$ \( 8 - 8 T + T^{2} \)
$29$ \( -28 - 4 T + T^{2} \)
$31$ \( -16 + 8 T + T^{2} \)
$37$ \( -28 + 4 T + T^{2} \)
$41$ \( -32 + T^{2} \)
$43$ \( -16 - 8 T + T^{2} \)
$47$ \( 28 - 12 T + T^{2} \)
$53$ \( -28 + 4 T + T^{2} \)
$59$ \( -68 + 4 T + T^{2} \)
$61$ \( 4 + 12 T + T^{2} \)
$67$ \( ( -4 + T )^{2} \)
$71$ \( ( -14 + T )^{2} \)
$73$ \( -28 + 4 T + T^{2} \)
$79$ \( -128 + T^{2} \)
$83$ \( -196 - 4 T + T^{2} \)
$89$ \( -112 - 8 T + T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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