Properties

Label 462.2.j.e
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: 8.0.484000000.9
Defining polynomial: \(x^{8} + x^{6} + 16 x^{4} + 66 x^{2} + 121\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + x^{6} + 16 x^{4} + 66 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{6} - 37 \nu^{4} + 629 \nu^{2} - 363 \)\()/1991\)
\(\beta_{3}\)\(=\)\((\)\( -28 \nu^{6} + 148 \nu^{4} - 525 \nu^{2} - 539 \)\()/1991\)
\(\beta_{4}\)\(=\)\((\)\( -28 \nu^{7} + 148 \nu^{5} - 525 \nu^{3} - 539 \nu \)\()/1991\)
\(\beta_{5}\)\(=\)\((\)\( 40 \nu^{6} + 73 \nu^{4} + 750 \nu^{2} + 2761 \)\()/1991\)
\(\beta_{6}\)\(=\)\((\)\( -61 \nu^{7} + 38 \nu^{5} - 646 \nu^{3} - 1672 \nu \)\()/1991\)
\(\beta_{7}\)\(=\)\((\)\( 68 \nu^{7} - 75 \nu^{5} + 1275 \nu^{3} + 3300 \nu \)\()/1991\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 4 \beta_{2} + 1\)
\(\nu^{3}\)\(=\)\(4 \beta_{7} + 4 \beta_{6} + \beta_{4} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(7 \beta_{5} + 10 \beta_{3} - 7\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} + 17 \beta_{4} - 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(37 \beta_{5} - 37 \beta_{3} - 75 \beta_{2} - 75\)
\(\nu^{7}\)\(=\)\(-38 \beta_{7} - 75 \beta_{6}\)

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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
−0.476925 + 1.46782i
0.476925 1.46782i
1.73855 1.26313i
−1.73855 + 1.26313i
1.73855 + 1.26313i
−1.73855 1.26313i
−0.476925 1.46782i
0.476925 + 1.46782i
−0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −1.52029 + 1.10455i −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i 1.87918
169.2 −0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i 2.52029 1.83110i −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i −3.11525
295.1 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.0895849 + 0.275714i 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 0.289903
295.2 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.910415 + 2.80197i 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 2.94617
379.1 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.0895849 0.275714i 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 0.289903
379.2 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.910415 2.80197i 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 2.94617
421.1 −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −1.52029 1.10455i −0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i 1.87918
421.2 −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i 2.52029 + 1.83110i −0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −3.11525
\(n\): 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.e 8
11.c even 5 1 inner 462.2.j.e 8
11.c even 5 1 5082.2.a.cf 4
11.d odd 10 1 5082.2.a.ca 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.j.e 8 1.a even 1 1 trivial
462.2.j.e 8 11.c even 5 1 inner
5082.2.a.ca 4 11.d odd 10 1
5082.2.a.cf 4 11.c even 5 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

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$3$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$5$ \( 25 - 50 T + 290 T^{2} + 40 T^{3} - 4 T^{4} - 4 T^{5} + 11 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$11$ \( 14641 - 3509 T^{2} + 451 T^{4} - 29 T^{6} + T^{8} \)
$13$ \( 16 + 32 T + 72 T^{2} + 40 T^{3} - 4 T^{4} - 20 T^{5} + 18 T^{6} - 4 T^{7} + T^{8} \)
$17$ \( 24025 + 63550 T + 69785 T^{2} + 22500 T^{3} + 4826 T^{4} + 702 T^{5} + 104 T^{6} + 8 T^{7} + T^{8} \)
$19$ \( 25 + 210 T^{2} + 460 T^{3} + 456 T^{4} + 208 T^{5} + 69 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( ( 5 - 9 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$29$ \( 400 - 6800 T + 44160 T^{2} + 80 T^{3} + 2136 T^{4} + 304 T^{5} + 46 T^{6} + 4 T^{7} + T^{8} \)
$31$ \( 19321 + 43646 T + 40872 T^{2} + 9878 T^{3} + 3330 T^{4} + 612 T^{5} + 117 T^{6} + 14 T^{7} + T^{8} \)
$37$ \( 2401 - 3430 T + 3969 T^{2} - 2380 T^{3} + 876 T^{4} - 210 T^{5} + 54 T^{6} - 10 T^{7} + T^{8} \)
$41$ \( 3025 - 6050 T + 16885 T^{2} - 10340 T^{3} + 3166 T^{4} - 322 T^{5} + 44 T^{6} + 12 T^{7} + T^{8} \)
$43$ \( ( 116 + 660 T - 72 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$47$ \( 55696 - 85904 T + 60848 T^{2} - 17280 T^{3} + 20096 T^{4} - 3480 T^{5} + 422 T^{6} - 28 T^{7} + T^{8} \)
$53$ \( 55696 - 2832 T + 13328 T^{2} + 2696 T^{3} + 880 T^{4} + 16 T^{5} - 2 T^{6} - 2 T^{7} + T^{8} \)
$59$ \( 250000 + 150000 T + 140000 T^{2} + 24000 T^{3} + 3400 T^{4} - 10 T^{6} + T^{8} \)
$61$ \( 355216 + 379056 T + 290432 T^{2} + 126728 T^{3} + 33880 T^{4} + 5152 T^{5} + 562 T^{6} + 34 T^{7} + T^{8} \)
$67$ \( ( 404 - 564 T - 58 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$71$ \( 8202496 - 2749440 T + 645824 T^{2} - 72000 T^{3} + 5776 T^{4} + 240 T^{5} + 44 T^{6} + T^{8} \)
$73$ \( 839056 - 54960 T + 38096 T^{2} - 9840 T^{3} + 2416 T^{4} + 600 T^{5} + 86 T^{6} + T^{8} \)
$79$ \( 17808400 - 10212400 T + 2789640 T^{2} - 417920 T^{3} + 50936 T^{4} - 4748 T^{5} + 384 T^{6} - 22 T^{7} + T^{8} \)
$83$ \( 133726096 + 82566960 T + 22119776 T^{2} + 2280600 T^{3} + 206536 T^{4} + 12960 T^{5} + 746 T^{6} + 30 T^{7} + T^{8} \)
$89$ \( ( 20725 - 530 T - 309 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$97$ \( 16 - 80 T + 144 T^{2} + 120 T^{3} + 336 T^{4} + 160 T^{5} + 54 T^{6} + 10 T^{7} + T^{8} \)
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