Properties

Label 273.2.c.a
Level $273$
Weight $2$
Character orbit 273.c
Analytic conductor $2.180$
Analytic rank $0$
Dimension $2$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.00000i
1.00000i
2.00000i 1.00000 −2.00000 3.00000i 2.00000i 1.00000i 0 1.00000 −6.00000
64.2 2.00000i 1.00000 −2.00000 3.00000i 2.00000i 1.00000i 0 1.00000 −6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.c.a 2
3.b odd 2 1 819.2.c.a 2
4.b odd 2 1 4368.2.h.e 2
7.b odd 2 1 1911.2.c.a 2
13.b even 2 1 inner 273.2.c.a 2
13.d odd 4 1 3549.2.a.a 1
13.d odd 4 1 3549.2.a.e 1
39.d odd 2 1 819.2.c.a 2
52.b odd 2 1 4368.2.h.e 2
91.b odd 2 1 1911.2.c.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.a 2 1.a even 1 1 trivial
273.2.c.a 2 13.b even 2 1 inner
819.2.c.a 2 3.b odd 2 1
819.2.c.a 2 39.d odd 2 1
1911.2.c.a 2 7.b odd 2 1
1911.2.c.a 2 91.b odd 2 1
3549.2.a.a 1 13.d odd 4 1
3549.2.a.e 1 13.d odd 4 1
4368.2.h.e 2 4.b odd 2 1
4368.2.h.e 2 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 9 + T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 13 - 4 T + T^{2} \)
$17$ \( ( 2 + T )^{2} \)
$19$ \( 1 + T^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( 25 + T^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( 100 + T^{2} \)
$43$ \( ( -9 + T )^{2} \)
$47$ \( 49 + T^{2} \)
$53$ \( ( -9 + T )^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( 4 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 81 + T^{2} \)
$79$ \( ( -15 + T )^{2} \)
$83$ \( 81 + T^{2} \)
$89$ \( 81 + T^{2} \)
$97$ \( 169 + T^{2} \)
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