Properties

Label 462.2.j.h
Level $462$
Weight $2$
Character orbit 462.j
Analytic conductor $3.689$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.20164000000.8
Defining polynomial: \(x^{8} + 6 x^{6} + 76 x^{4} + 781 x^{2} + 5041\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 6 x^{6} + 76 x^{4} + 781 x^{2} + 5041\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 17 \nu^{6} - 537 \nu^{4} + 44034 \nu^{2} - 40328 \)\()/383471\)
\(\beta_{3}\)\(=\)\((\)\( -153 \nu^{6} + 4833 \nu^{4} - 12835 \nu^{2} - 20519 \)\()/383471\)
\(\beta_{4}\)\(=\)\((\)\( -153 \nu^{7} + 4833 \nu^{5} - 12835 \nu^{3} - 20519 \nu \)\()/383471\)
\(\beta_{5}\)\(=\)\((\)\( -585 \nu^{7} - 4078 \nu^{5} - 49075 \nu^{3} - 461926 \nu \)\()/383471\)
\(\beta_{6}\)\(=\)\((\)\( -721 \nu^{6} + 218 \nu^{4} - 17876 \nu^{2} - 139302 \)\()/383471\)
\(\beta_{7}\)\(=\)\((\)\( -721 \nu^{7} + 218 \nu^{5} - 17876 \nu^{3} - 139302 \nu \)\()/383471\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 9 \beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(9 \beta_{7} - 9 \beta_{5} - 8 \beta_{4} - 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-17 \beta_{6} + 82 \beta_{3} + 17 \beta_{2}\)
\(\nu^{5}\)\(=\)\(-17 \beta_{5} + 65 \beta_{4} - 17 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-537 \beta_{6} - 218 \beta_{2} - 218\)
\(\nu^{7}\)\(=\)\(-755 \beta_{7} + 218 \beta_{5} + 218 \beta_{4}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2} - \beta_{3} + \beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
−2.50900 + 1.82290i
2.50900 1.82290i
−0.839592 2.58400i
0.839592 + 2.58400i
−0.839592 + 2.58400i
0.839592 2.58400i
−2.50900 1.82290i
2.50900 + 1.82290i
0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −3.31802 + 2.41068i −0.809017 + 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i −4.10130
169.2 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i 1.69998 1.23511i −0.809017 + 0.587785i −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 0.587785i 2.10130
295.1 −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i −0.530575 1.63294i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 0.951057i 1.71698
295.2 −0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 1.14861 + 3.53506i 0.309017 + 0.951057i 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 0.951057i −3.71698
379.1 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.530575 + 1.63294i 0.309017 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i 1.71698
379.2 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 1.14861 3.53506i 0.309017 0.951057i 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 + 0.951057i −3.71698
421.1 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −3.31802 2.41068i −0.809017 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i −4.10130
421.2 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i 1.69998 + 1.23511i −0.809017 0.587785i −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 + 0.587785i 2.10130
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.j.h 8
11.c even 5 1 inner 462.2.j.h 8
11.c even 5 1 5082.2.a.by 4
11.d odd 10 1 5082.2.a.cd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.h 8 1.a even 1 1 trivial
462.2.j.h 8 11.c even 5 1 inner
5082.2.a.by 4 11.c even 5 1
5082.2.a.cd 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$3$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$5$ \( 3025 - 550 T + 790 T^{2} - 370 T^{3} + 156 T^{4} + 28 T^{5} + 9 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$11$ \( 14641 - 13310 T + 6171 T^{2} - 2090 T^{3} + 641 T^{4} - 190 T^{5} + 51 T^{6} - 10 T^{7} + T^{8} \)
$13$ \( 26896 - 36080 T + 24104 T^{2} - 9080 T^{3} + 2636 T^{4} - 460 T^{5} + 74 T^{6} - 10 T^{7} + T^{8} \)
$17$ \( ( 81 + 81 T + 36 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$19$ \( 2401 - 686 T + 2548 T^{2} + 910 T^{3} + 366 T^{4} + 20 T^{5} - 3 T^{6} - 2 T^{7} + T^{8} \)
$23$ \( ( 131 + 20 T - 57 T^{2} + T^{4} )^{2} \)
$29$ \( 400 + 1200 T + 11360 T^{2} - 3800 T^{3} + 2536 T^{4} - 112 T^{5} - 26 T^{6} + 2 T^{7} + T^{8} \)
$31$ \( 32761 - 31132 T + 45922 T^{2} - 13780 T^{3} + 4536 T^{4} - 280 T^{5} - 27 T^{6} + 4 T^{7} + T^{8} \)
$37$ \( 1771561 - 1437480 T + 667731 T^{2} - 198780 T^{3} + 40756 T^{4} - 5460 T^{5} + 506 T^{6} - 30 T^{7} + T^{8} \)
$41$ \( 25110121 + 13018578 T + 3348677 T^{2} + 499500 T^{3} + 60346 T^{4} + 5250 T^{5} + 408 T^{6} + 24 T^{7} + T^{8} \)
$43$ \( ( -604 - 540 T - 48 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$47$ \( 99856 - 72048 T + 52992 T^{2} - 15720 T^{3} + 3256 T^{4} - 240 T^{5} - 2 T^{6} + 6 T^{7} + T^{8} \)
$53$ \( 64320400 - 27428400 T + 5996160 T^{2} - 822720 T^{3} + 85336 T^{4} - 6864 T^{5} + 466 T^{6} - 24 T^{7} + T^{8} \)
$59$ \( 6948496 - 495568 T + 138272 T^{2} + 6120 T^{3} + 2696 T^{4} - 720 T^{5} + 158 T^{6} - 14 T^{7} + T^{8} \)
$61$ \( 355216 - 207408 T + 249968 T^{2} - 16896 T^{3} - 1520 T^{4} + 264 T^{5} + 138 T^{6} + 12 T^{7} + T^{8} \)
$67$ \( ( 1516 - 224 T - 74 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$71$ \( 774400 + 915200 T + 469440 T^{2} + 67520 T^{3} + 16656 T^{4} - 2064 T^{5} + 156 T^{6} - 4 T^{7} + T^{8} \)
$73$ \( 4648336 - 5372752 T + 2099552 T^{2} + 561960 T^{3} + 101096 T^{4} + 10320 T^{5} + 758 T^{6} + 34 T^{7} + T^{8} \)
$79$ \( 383376400 + 41509600 T + 11990760 T^{2} - 22960 T^{3} + 7616 T^{4} + 608 T^{5} + 324 T^{6} + 22 T^{7} + T^{8} \)
$83$ \( 698896 + 324368 T + 82768 T^{2} + 10496 T^{3} + 9280 T^{4} - 584 T^{5} + 98 T^{6} - 12 T^{7} + T^{8} \)
$89$ \( ( -4895 + 1570 T - 9 T^{2} - 22 T^{3} + T^{4} )^{2} \)
$97$ \( 400 - 1200 T + 1360 T^{2} + 240 T^{3} + 416 T^{4} + 72 T^{5} + 34 T^{6} + 8 T^{7} + T^{8} \)
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