Properties

Label 273.2.c.b
Level $273$
Weight $2$
Character orbit 273.c
Analytic conductor $2.180$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38 \)\()/23\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29 \)\()/23\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13 \)\()/23\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27 \)\()/23\)
\(\beta_{5}\)\(=\)\((\)\( -14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26 \)\()/23\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(-\beta_{2} + 5 \beta_{1} - 7\)
\(\nu^{5}\)\(=\)\(-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9\)

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
−0.854638 0.854638i
0.403032 0.403032i
1.45161 + 1.45161i
1.45161 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
2.17009i 1.00000 −2.70928 0.630898i 2.17009i 1.00000i 1.53919i 1.00000 1.36910
64.2 1.48119i 1.00000 −0.193937 4.15633i 1.48119i 1.00000i 2.67513i 1.00000 6.15633
64.3 0.311108i 1.00000 1.90321 1.52543i 0.311108i 1.00000i 1.21432i 1.00000 0.474572
64.4 0.311108i 1.00000 1.90321 1.52543i 0.311108i 1.00000i 1.21432i 1.00000 0.474572
64.5 1.48119i 1.00000 −0.193937 4.15633i 1.48119i 1.00000i 2.67513i 1.00000 6.15633
64.6 2.17009i 1.00000 −2.70928 0.630898i 2.17009i 1.00000i 1.53919i 1.00000 1.36910
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
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.b 6
3.b odd 2 1 819.2.c.c 6
4.b odd 2 1 4368.2.h.o 6
7.b odd 2 1 1911.2.c.h 6
13.b even 2 1 inner 273.2.c.b 6
13.d odd 4 1 3549.2.a.k 3
13.d odd 4 1 3549.2.a.q 3
39.d odd 2 1 819.2.c.c 6
52.b odd 2 1 4368.2.h.o 6
91.b odd 2 1 1911.2.c.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.b 6 1.a even 1 1 trivial
273.2.c.b 6 13.b even 2 1 inner
819.2.c.c 6 3.b odd 2 1
819.2.c.c 6 39.d odd 2 1
1911.2.c.h 6 7.b odd 2 1
1911.2.c.h 6 91.b odd 2 1
3549.2.a.k 3 13.d odd 4 1
3549.2.a.q 3 13.d odd 4 1
4368.2.h.o 6 4.b odd 2 1
4368.2.h.o 6 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 7 T_{2}^{4} + 11 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 11 T^{2} + 7 T^{4} + T^{6} \)
$3$ \( ( -1 + T )^{6} \)
$5$ \( 16 + 48 T^{2} + 20 T^{4} + T^{6} \)
$7$ \( ( 1 + T^{2} )^{3} \)
$11$ \( 400 + 384 T^{2} + 44 T^{4} + T^{6} \)
$13$ \( 2197 + 338 T + 351 T^{2} + 44 T^{3} + 27 T^{4} + 2 T^{5} + T^{6} \)
$17$ \( ( 16 - 16 T + T^{3} )^{2} \)
$19$ \( 1024 + 512 T^{2} + 64 T^{4} + T^{6} \)
$23$ \( ( 8 - 20 T - 2 T^{2} + T^{3} )^{2} \)
$29$ \( ( -40 - 52 T + 2 T^{2} + T^{3} )^{2} \)
$31$ \( 102400 + 7936 T^{2} + 176 T^{4} + T^{6} \)
$37$ \( 1024 + 512 T^{2} + 48 T^{4} + T^{6} \)
$41$ \( 400 + 464 T^{2} + 132 T^{4} + T^{6} \)
$43$ \( ( -128 + 32 T + 16 T^{2} + T^{3} )^{2} \)
$47$ \( 2704 + 784 T^{2} + 56 T^{4} + T^{6} \)
$53$ \( ( 6 + T )^{6} \)
$59$ \( 16 + 80 T^{2} + 40 T^{4} + T^{6} \)
$61$ \( ( 2392 - 172 T - 14 T^{2} + T^{3} )^{2} \)
$67$ \( 1024 + 3072 T^{2} + 128 T^{4} + T^{6} \)
$71$ \( 547600 + 30784 T^{2} + 332 T^{4} + T^{6} \)
$73$ \( 43264 + 6720 T^{2} + 272 T^{4} + T^{6} \)
$79$ \( ( 80 + 72 T + 16 T^{2} + T^{3} )^{2} \)
$83$ \( 1567504 + 49616 T^{2} + 408 T^{4} + T^{6} \)
$89$ \( 5776 + 6832 T^{2} + 212 T^{4} + T^{6} \)
$97$ \( 256 + 256 T^{2} + 32 T^{4} + T^{6} \)
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