Properties

Label 273.2.c.c
Level $273$
Weight $2$
Character orbit 273.c
Analytic conductor $2.180$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 15 x^{6} + 67 x^{4} + 77 x^{2} + 4\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 15 x^{6} + 67 x^{4} + 77 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 8 \nu^{3} + 9 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{4} + 9 \nu^{2} \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} + 10 \nu^{3} + 21 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 10 \nu^{4} + 23 \nu^{2} + 6 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 12 \nu^{5} + 41 \nu^{3} + 36 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{3} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{6} - \beta_{4} - 7 \beta_{2} + 25\)
\(\nu^{5}\)\(=\)\(-8 \beta_{5} + 10 \beta_{3} + 39 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-8 \beta_{6} + 10 \beta_{4} + 47 \beta_{2} - 164\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} + 55 \beta_{5} - 79 \beta_{3} - 258 \beta_{1}\)

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
2.60520i
2.54814i
1.29051i
0.233455i
0.233455i
1.29051i
2.54814i
2.60520i
2.60520i −1.00000 −4.78706 3.78706i 2.60520i 1.00000i 7.26084i 1.00000 9.86604
64.2 2.54814i −1.00000 −4.49301 3.49301i 2.54814i 1.00000i 6.35254i 1.00000 −8.90068
64.3 1.29051i −1.00000 0.334573 1.33457i 1.29051i 1.00000i 3.01280i 1.00000 1.72229
64.4 0.233455i −1.00000 1.94550 2.94550i 0.233455i 1.00000i 0.921097i 1.00000 −0.687642
64.5 0.233455i −1.00000 1.94550 2.94550i 0.233455i 1.00000i 0.921097i 1.00000 −0.687642
64.6 1.29051i −1.00000 0.334573 1.33457i 1.29051i 1.00000i 3.01280i 1.00000 1.72229
64.7 2.54814i −1.00000 −4.49301 3.49301i 2.54814i 1.00000i 6.35254i 1.00000 −8.90068
64.8 2.60520i −1.00000 −4.78706 3.78706i 2.60520i 1.00000i 7.26084i 1.00000 9.86604
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.8
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.c 8
3.b odd 2 1 819.2.c.d 8
4.b odd 2 1 4368.2.h.q 8
7.b odd 2 1 1911.2.c.l 8
13.b even 2 1 inner 273.2.c.c 8
13.d odd 4 1 3549.2.a.v 4
13.d odd 4 1 3549.2.a.x 4
39.d odd 2 1 819.2.c.d 8
52.b odd 2 1 4368.2.h.q 8
91.b odd 2 1 1911.2.c.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.c.c 8 1.a even 1 1 trivial
273.2.c.c 8 13.b even 2 1 inner
819.2.c.d 8 3.b odd 2 1
819.2.c.d 8 39.d odd 2 1
1911.2.c.l 8 7.b odd 2 1
1911.2.c.l 8 91.b odd 2 1
3549.2.a.v 4 13.d odd 4 1
3549.2.a.x 4 13.d odd 4 1
4368.2.h.q 8 4.b odd 2 1
4368.2.h.q 8 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 15 T_{2}^{6} + 67 T_{2}^{4} + 77 T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 77 T^{2} + 67 T^{4} + 15 T^{6} + T^{8} \)
$3$ \( ( 1 + T )^{8} \)
$5$ \( 2704 + 2240 T^{2} + 468 T^{4} + 37 T^{6} + T^{8} \)
$7$ \( ( 1 + T^{2} )^{4} \)
$11$ \( 1024 + 2320 T^{2} + 576 T^{4} + 44 T^{6} + T^{8} \)
$13$ \( 28561 + 13182 T - 1014 T^{3} - 306 T^{4} - 78 T^{5} + 6 T^{7} + T^{8} \)
$17$ \( ( -160 + 112 T + 8 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$19$ \( 1024 + 10752 T^{2} + 1984 T^{4} + 97 T^{6} + T^{8} \)
$23$ \( ( 1352 - 140 T - 78 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$29$ \( ( -440 - 244 T - 14 T^{2} + 9 T^{3} + T^{4} )^{2} \)
$31$ \( 4096 + 6912 T^{2} + 2032 T^{4} + 89 T^{6} + T^{8} \)
$37$ \( 262144 + 148480 T^{2} + 9216 T^{4} + 176 T^{6} + T^{8} \)
$41$ \( 1784896 + 324304 T^{2} + 17376 T^{4} + 248 T^{6} + T^{8} \)
$43$ \( ( -896 + 160 T + 64 T^{2} - 17 T^{3} + T^{4} )^{2} \)
$47$ \( 6739216 + 605408 T^{2} + 18984 T^{4} + 241 T^{6} + T^{8} \)
$53$ \( ( 712 - 356 T - 142 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$59$ \( 1364224 + 365968 T^{2} + 18384 T^{4} + 248 T^{6} + T^{8} \)
$61$ \( ( 320 - 136 T - 28 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$67$ \( 66064384 + 3824640 T^{2} + 67072 T^{4} + 452 T^{6} + T^{8} \)
$71$ \( 1024 + 2320 T^{2} + 576 T^{4} + 44 T^{6} + T^{8} \)
$73$ \( 5234944 + 595776 T^{2} + 22384 T^{4} + 305 T^{6} + T^{8} \)
$79$ \( ( -80 - 104 T - 32 T^{2} + T^{3} + T^{4} )^{2} \)
$83$ \( 6739216 + 605408 T^{2} + 18984 T^{4} + 241 T^{6} + T^{8} \)
$89$ \( 150544 + 379392 T^{2} + 19060 T^{4} + 253 T^{6} + T^{8} \)
$97$ \( 82882816 + 4847616 T^{2} + 87232 T^{4} + 553 T^{6} + T^{8} \)
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