Properties

Label 273.2.p.d
Level $273$
Weight $2$
Character orbit 273.p
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.p (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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i 1.00000i 1.00000i 2.00000 2.00000i −1.22474 + 1.22474i 2.44949 1.00000i −1.22474 + 1.22474i −1.00000 −4.89898
34.2 1.22474 + 1.22474i 1.00000i 1.00000i 2.00000 2.00000i 1.22474 1.22474i −2.44949 1.00000i 1.22474 1.22474i −1.00000 4.89898
265.1 −1.22474 + 1.22474i 1.00000i 1.00000i 2.00000 + 2.00000i −1.22474 1.22474i 2.44949 + 1.00000i −1.22474 1.22474i −1.00000 −4.89898
265.2 1.22474 1.22474i 1.00000i 1.00000i 2.00000 + 2.00000i 1.22474 + 1.22474i −2.44949 + 1.00000i 1.22474 + 1.22474i −1.00000 4.89898
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.i 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.p.d yes 4
3.b odd 2 1 819.2.y.a 4
7.b odd 2 1 273.2.p.a 4
13.d odd 4 1 273.2.p.a 4
21.c even 2 1 819.2.y.d 4
39.f even 4 1 819.2.y.d 4
91.i even 4 1 inner 273.2.p.d yes 4
273.o odd 4 1 819.2.y.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.p.a 4 7.b odd 2 1
273.2.p.a 4 13.d odd 4 1
273.2.p.d yes 4 1.a even 1 1 trivial
273.2.p.d yes 4 91.i even 4 1 inner
819.2.y.a 4 3.b odd 2 1
819.2.y.a 4 273.o odd 4 1
819.2.y.d 4 21.c even 2 1
819.2.y.d 4 39.f even 4 1

Hecke kernels

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

\( T_{2}^{4} + 9 \)
\( T_{5}^{2} - 4 T_{5} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 8 - 4 T + T^{2} )^{2} \)
$7$ \( 49 - 10 T^{2} + T^{4} \)
$11$ \( 16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$13$ \( ( 13 - 4 T + T^{2} )^{2} \)
$17$ \( ( 2 + T )^{4} \)
$19$ \( 36 + 72 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( 64 + 80 T^{2} + T^{4} \)
$29$ \( ( -20 + 4 T + T^{2} )^{2} \)
$31$ \( 36 + 72 T + 72 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 900 + 360 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( ( 32 + 8 T + T^{2} )^{2} \)
$43$ \( 400 + 56 T^{2} + T^{4} \)
$47$ \( 400 - 320 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$53$ \( ( -92 + 4 T + T^{2} )^{2} \)
$59$ \( 16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( ( 100 + T^{2} )^{2} \)
$67$ \( 11236 + 424 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( 144 + T^{4} \)
$73$ \( 900 + 360 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$79$ \( ( -20 - 4 T + T^{2} )^{2} \)
$83$ \( 16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$89$ \( ( 8 - 4 T + T^{2} )^{2} \)
$97$ \( 900 + 360 T + 72 T^{2} - 12 T^{3} + T^{4} \)
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