Properties

Label 273.2.p.c
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: \(\Q(i, \sqrt{10})\)
Defining polynomial: \(x^{4} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -2 \beta_{2} q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} ) q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -2 \beta_{2} q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} ) q^{7} - q^{9} + ( 1 - \beta_{2} - 2 \beta_{3} ) q^{11} -2 q^{12} + ( -2 - 2 \beta_{2} + \beta_{3} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} ) q^{15} -4 q^{16} + ( 2 - \beta_{1} + \beta_{3} ) q^{17} + ( -2 + 2 \beta_{2} - \beta_{3} ) q^{19} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{20} + ( 1 + \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{25} + \beta_{2} q^{27} + ( 2 + 2 \beta_{2} + 2 \beta_{3} ) q^{28} + ( 5 - \beta_{1} + \beta_{3} ) q^{29} + ( 4 - 4 \beta_{2} - \beta_{3} ) q^{31} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{33} + ( 5 - 2 \beta_{1} + 2 \beta_{2} ) q^{35} + 2 \beta_{2} q^{36} + ( -2 + 2 \beta_{2} - 4 \beta_{3} ) q^{37} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{39} + ( 4 - 4 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{43} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{44} + ( -1 + \beta_{2} - \beta_{3} ) q^{45} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{47} + 4 \beta_{2} q^{48} + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{49} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{51} + ( -4 + 2 \beta_{1} + 4 \beta_{2} ) q^{52} + ( -1 - \beta_{1} + \beta_{3} ) q^{53} + ( -\beta_{1} + 8 \beta_{2} - \beta_{3} ) q^{55} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{57} + ( 2 + 4 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -2 + 2 \beta_{2} - 2 \beta_{3} ) q^{60} + ( -\beta_{1} - \beta_{3} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} ) q^{63} + 8 \beta_{2} q^{64} + ( -4 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{65} -4 \beta_{1} q^{67} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{68} + ( -1 - \beta_{1} + \beta_{3} ) q^{69} + ( 6 + 6 \beta_{2} ) q^{71} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{73} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{75} + ( 4 - 2 \beta_{1} + 4 \beta_{2} ) q^{76} + ( -10 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{77} + ( -5 - 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -4 + 4 \beta_{2} - 4 \beta_{3} ) q^{80} + q^{81} + ( 5 - 5 \beta_{2} - \beta_{3} ) q^{83} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{84} + ( 7 - 7 \beta_{2} + 4 \beta_{3} ) q^{85} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{87} + ( 7 - \beta_{1} + 7 \beta_{2} ) q^{89} + ( 9 + \beta_{1} + \beta_{3} ) q^{91} + ( -2 - 2 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{93} + ( -3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{95} + ( 2 - 2 \beta_{2} + \beta_{3} ) q^{97} + ( -1 + \beta_{2} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{5} - 4q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{5} - 4q^{7} - 4q^{9} + 4q^{11} - 8q^{12} - 8q^{13} - 4q^{15} - 16q^{16} + 8q^{17} - 8q^{19} - 8q^{20} + 4q^{21} + 8q^{28} + 20q^{29} + 16q^{31} - 4q^{33} + 20q^{35} - 8q^{37} - 8q^{39} + 16q^{41} - 8q^{44} - 4q^{45} + 20q^{47} - 16q^{52} - 4q^{53} + 8q^{57} + 8q^{59} - 8q^{60} + 4q^{63} - 16q^{65} - 4q^{69} + 24q^{71} - 16q^{73} - 8q^{75} + 16q^{76} - 40q^{77} - 20q^{79} - 16q^{80} + 4q^{81} + 20q^{83} + 8q^{84} + 28q^{85} + 28q^{89} + 36q^{91} - 8q^{92} - 16q^{93} + 8q^{97} - 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(5 \beta_{2}\)
\(\nu^{3}\)\(=\)\(5 \beta_{3}\)

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
1.58114 + 1.58114i
−1.58114 1.58114i
1.58114 1.58114i
−1.58114 + 1.58114i
0 1.00000i 2.00000i −0.581139 + 0.581139i 0 −2.58114 0.581139i 0 −1.00000 0
34.2 0 1.00000i 2.00000i 2.58114 2.58114i 0 0.581139 + 2.58114i 0 −1.00000 0
265.1 0 1.00000i 2.00000i −0.581139 0.581139i 0 −2.58114 + 0.581139i 0 −1.00000 0
265.2 0 1.00000i 2.00000i 2.58114 + 2.58114i 0 0.581139 2.58114i 0 −1.00000 0
\(n\): 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.c yes 4
3.b odd 2 1 819.2.y.b 4
7.b odd 2 1 273.2.p.b 4
13.d odd 4 1 273.2.p.b 4
21.c even 2 1 819.2.y.c 4
39.f even 4 1 819.2.y.c 4
91.i even 4 1 inner 273.2.p.c yes 4
273.o odd 4 1 819.2.y.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.p.b 4 7.b odd 2 1
273.2.p.b 4 13.d odd 4 1
273.2.p.c yes 4 1.a even 1 1 trivial
273.2.p.c yes 4 91.i even 4 1 inner
819.2.y.b 4 3.b odd 2 1
819.2.y.b 4 273.o odd 4 1
819.2.y.c 4 21.c even 2 1
819.2.y.c 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 \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2} \)
\( T_{5}^{4} - 4 T_{5}^{3} + 8 T_{5}^{2} + 12 T_{5} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( 9 + 12 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 49 + 28 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$11$ \( 324 + 72 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( 169 + 104 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$17$ \( ( -6 - 4 T + T^{2} )^{2} \)
$19$ \( 9 + 24 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$23$ \( 81 + 22 T^{2} + T^{4} \)
$29$ \( ( 15 - 10 T + T^{2} )^{2} \)
$31$ \( 729 - 432 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$37$ \( 5184 - 576 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( 144 - 192 T + 128 T^{2} - 16 T^{3} + T^{4} \)
$43$ \( 1521 + 82 T^{2} + T^{4} \)
$47$ \( 2025 - 900 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$53$ \( ( -9 + 2 T + T^{2} )^{2} \)
$59$ \( 5184 + 576 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( ( 10 + T^{2} )^{2} \)
$67$ \( 6400 + T^{4} \)
$71$ \( ( 72 - 12 T + T^{2} )^{2} \)
$73$ \( 729 + 432 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$79$ \( ( -15 + 10 T + T^{2} )^{2} \)
$83$ \( 2025 - 900 T + 200 T^{2} - 20 T^{3} + T^{4} \)
$89$ \( 8649 - 2604 T + 392 T^{2} - 28 T^{3} + T^{4} \)
$97$ \( 9 - 24 T + 32 T^{2} - 8 T^{3} + T^{4} \)
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