Properties

Label 483.2.h.c
Level $483$
Weight $2$
Character orbit 483.h
Analytic conductor $3.857$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 17 x^{10} + 92 x^{8} + 180 x^{6} + 92 x^{4} + 17 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + \beta_{9} q^{3} + ( 3 + \beta_{7} ) q^{4} -\beta_{5} q^{5} + \beta_{4} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -3 - 3 \beta_{2} + \beta_{7} ) q^{8} - q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + \beta_{9} q^{3} + ( 3 + \beta_{7} ) q^{4} -\beta_{5} q^{5} + \beta_{4} q^{6} + ( \beta_{1} + \beta_{3} ) q^{7} + ( -3 - 3 \beta_{2} + \beta_{7} ) q^{8} - q^{9} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{10} + ( \beta_{3} - \beta_{10} ) q^{11} + ( \beta_{8} + 3 \beta_{9} ) q^{12} + ( -\beta_{4} + 2 \beta_{9} ) q^{13} + ( 2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{11} ) q^{14} + \beta_{3} q^{15} + ( 6 + \beta_{2} + 2 \beta_{7} ) q^{16} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{17} + \beta_{2} q^{18} + ( \beta_{1} - \beta_{5} ) q^{19} + ( -\beta_{1} - 4 \beta_{5} + \beta_{6} ) q^{20} + ( \beta_{5} - \beta_{10} ) q^{21} + ( -2 \beta_{3} - 2 \beta_{10} + \beta_{11} ) q^{22} + ( 1 - 2 \beta_{3} - \beta_{7} - \beta_{10} ) q^{23} + ( 3 \beta_{4} + \beta_{8} - 3 \beta_{9} ) q^{24} + ( -1 + \beta_{2} ) q^{25} + ( 2 \beta_{4} - \beta_{8} - 5 \beta_{9} ) q^{26} -\beta_{9} q^{27} + ( 3 \beta_{1} + 4 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{10} - \beta_{11} ) q^{28} + ( -1 + \beta_{2} + \beta_{7} ) q^{29} + ( -\beta_{3} + \beta_{11} ) q^{30} + ( -\beta_{4} - \beta_{8} - 3 \beta_{9} ) q^{31} + ( -5 - 4 \beta_{2} - \beta_{7} ) q^{32} + ( -\beta_{1} + \beta_{5} ) q^{33} + ( -\beta_{1} - 6 \beta_{5} + \beta_{6} ) q^{34} + ( 1 + \beta_{2} - \beta_{4} - \beta_{7} + 4 \beta_{9} ) q^{35} + ( -3 - \beta_{7} ) q^{36} + ( \beta_{3} + 2 \beta_{10} + \beta_{11} ) q^{37} + ( \beta_{1} + 2 \beta_{5} - \beta_{6} ) q^{38} + ( -2 - \beta_{2} ) q^{39} + ( -4 \beta_{1} + 5 \beta_{5} - 2 \beta_{6} ) q^{40} + 5 \beta_{9} q^{41} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{10} ) q^{42} + ( -2 \beta_{3} - \beta_{11} ) q^{43} + ( 3 \beta_{3} - 2 \beta_{11} ) q^{44} + \beta_{5} q^{45} + ( 3 + \beta_{2} + \beta_{3} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{46} + ( -3 \beta_{4} + \beta_{8} + 2 \beta_{9} ) q^{47} + ( -\beta_{4} + 2 \beta_{8} + 6 \beta_{9} ) q^{48} + ( -1 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{8} + 2 \beta_{9} ) q^{49} + ( -5 + \beta_{2} - \beta_{7} ) q^{50} + ( -\beta_{3} + \beta_{11} ) q^{51} + ( -5 \beta_{4} + \beta_{8} + 9 \beta_{9} ) q^{52} + ( -\beta_{3} + \beta_{10} + \beta_{11} ) q^{53} -\beta_{4} q^{54} + ( -2 \beta_{4} - \beta_{8} + 5 \beta_{9} ) q^{55} + ( 3 \beta_{1} - 5 \beta_{3} + 4 \beta_{5} + \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{56} + ( \beta_{3} - \beta_{10} ) q^{57} + ( -8 - \beta_{2} ) q^{58} + ( \beta_{4} + 2 \beta_{8} - 5 \beta_{9} ) q^{59} + ( 4 \beta_{3} + 2 \beta_{10} - \beta_{11} ) q^{60} + ( -\beta_{1} - 2 \beta_{6} ) q^{61} + ( -5 \beta_{4} - 2 \beta_{8} - 2 \beta_{9} ) q^{62} + ( -\beta_{1} - \beta_{3} ) q^{63} + ( 11 + 5 \beta_{2} - \beta_{7} ) q^{64} + ( 3 \beta_{3} - \beta_{11} ) q^{65} + ( -\beta_{1} - 2 \beta_{5} + \beta_{6} ) q^{66} -3 \beta_{3} q^{67} + ( -6 \beta_{1} + 7 \beta_{5} - 4 \beta_{6} ) q^{68} + ( -\beta_{1} - 2 \beta_{5} - \beta_{8} + \beta_{9} ) q^{69} + ( -2 + \beta_{2} + 4 \beta_{4} - 2 \beta_{7} - \beta_{8} - 5 \beta_{9} ) q^{70} + ( 2 + \beta_{2} + 2 \beta_{7} ) q^{71} + ( 3 + 3 \beta_{2} - \beta_{7} ) q^{72} + ( \beta_{4} + 5 \beta_{8} + \beta_{9} ) q^{73} + ( 6 \beta_{3} + 6 \beta_{10} + \beta_{11} ) q^{74} + ( -\beta_{4} - \beta_{9} ) q^{75} + ( 2 \beta_{1} - 3 \beta_{5} + 2 \beta_{6} ) q^{76} + ( -5 - 2 \beta_{2} + \beta_{7} - \beta_{8} + 4 \beta_{9} ) q^{77} + ( 5 + 2 \beta_{2} + \beta_{7} ) q^{78} + ( -\beta_{3} + 6 \beta_{10} - \beta_{11} ) q^{79} + ( -\beta_{1} - 9 \beta_{5} + 3 \beta_{6} ) q^{80} + q^{81} + 5 \beta_{4} q^{82} + ( 3 \beta_{1} - \beta_{5} + 3 \beta_{6} ) q^{83} + ( \beta_{1} - \beta_{3} + 4 \beta_{5} - \beta_{6} - 2 \beta_{10} - \beta_{11} ) q^{84} + ( -5 - 4 \beta_{2} - \beta_{7} ) q^{85} + ( -3 \beta_{3} - 2 \beta_{10} - 2 \beta_{11} ) q^{86} + ( -\beta_{4} + \beta_{8} - \beta_{9} ) q^{87} + ( -9 \beta_{3} + \beta_{11} ) q^{88} + ( -3 \beta_{1} + \beta_{6} ) q^{89} + ( \beta_{1} - \beta_{5} + \beta_{6} ) q^{90} + ( -\beta_{1} + \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{91} + ( -4 - \beta_{2} - 9 \beta_{3} - 6 \beta_{10} + \beta_{11} ) q^{92} + ( 3 - \beta_{2} + \beta_{7} ) q^{93} + ( 4 \beta_{4} - 2 \beta_{8} - 18 \beta_{9} ) q^{94} + ( 5 + 2 \beta_{2} - \beta_{7} ) q^{95} + ( 4 \beta_{4} - \beta_{8} - 5 \beta_{9} ) q^{96} + ( 7 \beta_{1} + 3 \beta_{5} - \beta_{6} ) q^{97} + ( 10 + \beta_{2} - 2 \beta_{4} + 2 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{98} + ( -\beta_{3} + \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 4q^{2} + 36q^{4} - 24q^{8} - 12q^{9} + O(q^{10}) \) \( 12q + 4q^{2} + 36q^{4} - 24q^{8} - 12q^{9} + 68q^{16} - 4q^{18} + 12q^{23} - 16q^{25} - 16q^{29} - 44q^{32} + 8q^{35} - 36q^{36} - 20q^{39} + 32q^{46} - 4q^{49} - 64q^{50} - 92q^{58} + 112q^{64} - 28q^{70} + 20q^{71} + 24q^{72} - 52q^{77} + 52q^{78} + 12q^{81} - 44q^{85} - 44q^{92} + 40q^{93} + 52q^{95} + 116q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 17 x^{10} + 92 x^{8} + 180 x^{6} + 92 x^{4} + 17 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -19 \nu^{10} - 322 \nu^{8} - 1730 \nu^{6} - 3310 \nu^{4} - 1478 \nu^{2} - 141 \)\()/20\)
\(\beta_{2}\)\(=\)\( \nu^{10} + 17 \nu^{8} + 92 \nu^{6} + 180 \nu^{4} + 91 \nu^{2} + 11 \)
\(\beta_{3}\)\(=\)\( -\nu^{11} - 17 \nu^{9} - 92 \nu^{7} - 180 \nu^{5} - 92 \nu^{3} - 16 \nu \)
\(\beta_{4}\)\(=\)\((\)\( 6 \nu^{11} + 98 \nu^{9} + 485 \nu^{7} + 730 \nu^{5} - 63 \nu^{3} - 56 \nu \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( 33 \nu^{10} + 554 \nu^{8} + 2920 \nu^{6} + 5340 \nu^{4} + 1966 \nu^{2} + 187 \)\()/10\)
\(\beta_{6}\)\(=\)\((\)\( 93 \nu^{10} + 1554 \nu^{8} + 8110 \nu^{6} + 14450 \nu^{4} + 4526 \nu^{2} + 267 \)\()/20\)
\(\beta_{7}\)\(=\)\( 5 \nu^{10} + 84 \nu^{8} + 443 \nu^{6} + 809 \nu^{4} + 292 \nu^{2} + 26 \)
\(\beta_{8}\)\(=\)\((\)\( -63 \nu^{11} - 1054 \nu^{9} - 5510 \nu^{7} - 9830 \nu^{5} - 3046 \nu^{3} - 97 \nu \)\()/20\)
\(\beta_{9}\)\(=\)\((\)\( 83 \nu^{11} + 1394 \nu^{9} + 7350 \nu^{7} + 13430 \nu^{5} + 4886 \nu^{3} + 457 \nu \)\()/20\)
\(\beta_{10}\)\(=\)\((\)\( -11 \nu^{11} - 184 \nu^{9} - 962 \nu^{7} - 1720 \nu^{5} - 550 \nu^{3} - 39 \nu \)\()/2\)
\(\beta_{11}\)\(=\)\( 9 \nu^{11} + 152 \nu^{9} + 811 \nu^{7} + 1528 \nu^{5} + 649 \nu^{3} + 69 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} + \beta_{8} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + 2 \beta_{5} - \beta_{2} + \beta_{1} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{11} - 8 \beta_{9} - 4 \beta_{8} + 3 \beta_{4} - 8 \beta_{3}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} + 8 \beta_{6} - 17 \beta_{5} + 12 \beta_{2} - 2 \beta_{1} + 39\)\()/2\)
\(\nu^{5}\)\(=\)\(-7 \beta_{11} + \beta_{10} + 37 \beta_{9} + 11 \beta_{8} - 17 \beta_{4} + 30 \beta_{3}\)
\(\nu^{6}\)\(=\)\((\)\(-17 \beta_{7} - 64 \beta_{6} + 147 \beta_{5} - 112 \beta_{2} - 10 \beta_{1} - 291\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(143 \beta_{11} - 36 \beta_{10} - 672 \beta_{9} - 140 \beta_{8} + 319 \beta_{4} - 480 \beta_{3}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(196 \beta_{7} + 523 \beta_{6} - 1278 \beta_{5} + 985 \beta_{2} + 189 \beta_{1} + 2318\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-1324 \beta_{11} + 428 \beta_{10} + 5973 \beta_{9} + 997 \beta_{8} - 2832 \beta_{4} + 3979 \beta_{3}\)\()/2\)
\(\nu^{10}\)\(=\)\(-974 \beta_{7} - 2176 \beta_{6} + 5540 \beta_{5} - 4254 \beta_{2} - 1012 \beta_{1} - 9565\)
\(\nu^{11}\)\(=\)\((\)\(11780 \beta_{11} - 4324 \beta_{10} - 52317 \beta_{9} - 7677 \beta_{8} + 24640 \beta_{4} - 33565 \beta_{3}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1
0.341398i
2.92914i
0.341398i
2.92914i
2.14584i
0.466018i
2.14584i
0.466018i
0.562016i
1.77931i
0.562016i
1.77931i
−2.69639 1.00000i 5.27053 −2.58774 2.69639i −0.551006 + 2.58774i −8.81864 −1.00000 6.97756
160.2 −2.69639 1.00000i 5.27053 2.58774 2.69639i 0.551006 2.58774i −8.81864 −1.00000 −6.97756
160.3 −2.69639 1.00000i 5.27053 −2.58774 2.69639i −0.551006 2.58774i −8.81864 −1.00000 6.97756
160.4 −2.69639 1.00000i 5.27053 2.58774 2.69639i 0.551006 + 2.58774i −8.81864 −1.00000 −6.97756
160.5 1.17819 1.00000i −0.611859 −1.67982 1.17819i −2.04406 + 1.67982i −3.07728 −1.00000 −1.97916
160.6 1.17819 1.00000i −0.611859 1.67982 1.17819i 2.04406 1.67982i −3.07728 −1.00000 1.97916
160.7 1.17819 1.00000i −0.611859 −1.67982 1.17819i −2.04406 1.67982i −3.07728 −1.00000 −1.97916
160.8 1.17819 1.00000i −0.611859 1.67982 1.17819i 2.04406 + 1.67982i −3.07728 −1.00000 1.97916
160.9 2.51820 1.00000i 4.34132 −1.21729 2.51820i 2.34908 + 1.21729i 5.89592 −1.00000 −3.06538
160.10 2.51820 1.00000i 4.34132 1.21729 2.51820i −2.34908 1.21729i 5.89592 −1.00000 3.06538
160.11 2.51820 1.00000i 4.34132 −1.21729 2.51820i 2.34908 1.21729i 5.89592 −1.00000 −3.06538
160.12 2.51820 1.00000i 4.34132 1.21729 2.51820i −2.34908 + 1.21729i 5.89592 −1.00000 3.06538
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.h.c 12
3.b odd 2 1 1449.2.h.e 12
7.b odd 2 1 inner 483.2.h.c 12
21.c even 2 1 1449.2.h.e 12
23.b odd 2 1 inner 483.2.h.c 12
69.c even 2 1 1449.2.h.e 12
161.c even 2 1 inner 483.2.h.c 12
483.c odd 2 1 1449.2.h.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.h.c 12 1.a even 1 1 trivial
483.2.h.c 12 7.b odd 2 1 inner
483.2.h.c 12 23.b odd 2 1 inner
483.2.h.c 12 161.c even 2 1 inner
1449.2.h.e 12 3.b odd 2 1
1449.2.h.e 12 21.c even 2 1
1449.2.h.e 12 69.c even 2 1
1449.2.h.e 12 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - T_{2}^{2} - 7 T_{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(483, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 8 - 7 T - T^{2} + T^{3} )^{4} \)
$3$ \( ( 1 + T^{2} )^{6} \)
$5$ \( ( -28 + 33 T^{2} - 11 T^{4} + T^{6} )^{2} \)
$7$ \( 117649 + 4802 T^{2} + 1519 T^{4} + 476 T^{6} + 31 T^{8} + 2 T^{10} + T^{12} \)
$11$ \( ( 175 + 167 T^{2} + 25 T^{4} + T^{6} )^{2} \)
$13$ \( ( 4 + 21 T^{2} + 23 T^{4} + T^{6} )^{2} \)
$17$ \( ( -1792 + 685 T^{2} - 62 T^{4} + T^{6} )^{2} \)
$19$ \( ( -175 + 167 T^{2} - 25 T^{4} + T^{6} )^{2} \)
$23$ \( ( 12167 - 3174 T + 943 T^{2} - 292 T^{3} + 41 T^{4} - 6 T^{5} + T^{6} )^{2} \)
$29$ \( ( -50 - 19 T + 4 T^{2} + T^{3} )^{4} \)
$31$ \( ( 100 + 329 T^{2} + 54 T^{4} + T^{6} )^{2} \)
$37$ \( ( 1372 + 1053 T^{2} + 114 T^{4} + T^{6} )^{2} \)
$41$ \( ( 25 + T^{2} )^{6} \)
$43$ \( ( 1372 + 2281 T^{2} + 95 T^{4} + T^{6} )^{2} \)
$47$ \( ( 71824 + 7228 T^{2} + 197 T^{4} + T^{6} )^{2} \)
$53$ \( ( 7 + 1662 T^{2} + 86 T^{4} + T^{6} )^{2} \)
$59$ \( ( 49 + 1220 T^{2} + 132 T^{4} + T^{6} )^{2} \)
$61$ \( ( -229327 + 11938 T^{2} - 194 T^{4} + T^{6} )^{2} \)
$67$ \( ( 20412 + 2673 T^{2} + 99 T^{4} + T^{6} )^{2} \)
$71$ \( ( 128 - 53 T - 5 T^{2} + T^{3} )^{4} \)
$73$ \( ( 2458624 + 59633 T^{2} + 450 T^{4} + T^{6} )^{2} \)
$79$ \( ( 51772 + 23757 T^{2} + 386 T^{4} + T^{6} )^{2} \)
$83$ \( ( -2903152 + 66741 T^{2} - 470 T^{4} + T^{6} )^{2} \)
$89$ \( ( -17500 + 4457 T^{2} - 171 T^{4} + T^{6} )^{2} \)
$97$ \( ( -4435312 + 107173 T^{2} - 614 T^{4} + T^{6} )^{2} \)
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