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: \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{3} + \beta_{3} q^{4} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{5} - \beta_{5} q^{6} + \beta_{3} q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{3} + \beta_{3} q^{4} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{5} - \beta_{5} q^{6} + \beta_{3} q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{8} + \beta_{2} q^{9} + (\beta_{7} - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{10} + ( - \beta_{7} + 2 \beta_1) q^{11} + q^{12} + (\beta_{6} + \beta_{3} + \beta_{2} + 1) q^{13} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{14} + ( - \beta_{6} + \beta_{5} + \beta_1 - 1) q^{15} - \beta_{5} q^{16} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2}) q^{17} + \beta_{3} q^{18} + ( - 2 \beta_{2} + \beta_1 - 2) q^{19} + ( - \beta_{7} + \beta_{3} + \beta_{2} + 1) q^{20} + q^{21} + (2 \beta_{7} + \beta_{6} - \beta_1) q^{22} + ( - \beta_{7} - \beta_{5} + \beta_{2} + \beta_1) q^{23} + \beta_{2} q^{24} + (\beta_{5} - \beta_{3} + 2 \beta_1) q^{25} + (\beta_{7} + \beta_{5} + \beta_{4} - 1) q^{26} - \beta_{5} q^{27} - \beta_{5} q^{28} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_1 + 1) q^{29} + (\beta_{6} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{30} + (2 \beta_{7} + \beta_{6} - 2 \beta_{3} + \beta_{2} - 2) q^{31} + q^{32} + ( - 2 \beta_{6} + \beta_{4} + \beta_1) q^{33} + (2 \beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{34} + ( - \beta_{7} + \beta_{3} + \beta_{2} + 1) q^{35} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{36} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_1 + 1) q^{37} + (\beta_{7} + \beta_{6} - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{38} + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{39} + ( - \beta_{6} + \beta_{5} + \beta_1 - 1) q^{40} + ( - 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 4) q^{41} + \beta_{2} q^{42} + (2 \beta_{7} - \beta_{5} - 3 \beta_{4} + \beta_{2} - 2 \beta_1 + 3) q^{43} + (\beta_{6} + \beta_{4} - \beta_1) q^{44} + (\beta_{7} - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{45} + (\beta_{7} + \beta_{3} + 1) q^{46} + ( - 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 4) q^{47} + \beta_{3} q^{48} - \beta_{5} q^{49} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{50} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 1) q^{51} + ( - \beta_{2} - \beta_1 - 1) q^{52} + ( - \beta_{7} + 2 \beta_{6} - \beta_{2}) q^{53} + q^{54} + ( - 2 \beta_{7} - 2 \beta_{6} + 8 \beta_{5} - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + \cdots - 3) q^{55}+ \cdots + (2 \beta_{7} + \beta_{6} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - 2 q^{9} + 4 q^{10} + 8 q^{12} + 4 q^{13} - 2 q^{14} - 6 q^{15} - 2 q^{16} - 8 q^{17} - 2 q^{18} - 12 q^{19} + 4 q^{20} + 8 q^{21} - 4 q^{23} - 2 q^{24} + 4 q^{25} - 6 q^{26} - 2 q^{27} - 2 q^{28} - 4 q^{29} - 6 q^{30} - 14 q^{31} + 8 q^{32} + 12 q^{34} + 4 q^{35} - 2 q^{36} + 10 q^{37} + 8 q^{38} + 4 q^{39} - 6 q^{40} - 12 q^{41} - 2 q^{42} + 20 q^{43} + 4 q^{45} + 6 q^{46} + 28 q^{47} - 2 q^{48} - 2 q^{49} - 6 q^{50} + 2 q^{51} - 6 q^{52} + 2 q^{53} + 8 q^{54} + 4 q^{55} + 8 q^{56} + 8 q^{57} - 4 q^{58} + 4 q^{60} - 34 q^{61} + 6 q^{62} - 2 q^{63} - 2 q^{64} + 16 q^{65} - 24 q^{67} - 8 q^{68} - 4 q^{69} - 6 q^{70} - 2 q^{72} + 10 q^{74} - 6 q^{75} + 8 q^{76} + 4 q^{78} + 22 q^{79} - 6 q^{80} - 2 q^{81} - 2 q^{82} - 30 q^{83} - 2 q^{84} - 28 q^{85} + 36 q^{87} - 4 q^{89} + 4 q^{90} - 6 q^{91} - 4 q^{92} - 14 q^{93} - 22 q^{94} - 30 q^{95} - 2 q^{96} - 10 q^{97} + 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} + 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} - 75\beta_{6} \) Copy content Toggle raw display

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 \( T_{5}^{8} - 4T_{5}^{7} + 11T_{5}^{6} - 4T_{5}^{5} - 4T_{5}^{4} + 40T_{5}^{3} + 290T_{5}^{2} - 50T_{5} + 25 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + 11 T^{6} - 4 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 29 T^{6} + 451 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + 18 T^{6} - 20 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + 104 T^{6} + \cdots + 24025 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + 69 T^{6} + 208 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$23$ \( (T^{4} + 2 T^{3} - 9 T^{2} + 5)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 4 T^{7} + 46 T^{6} + 304 T^{5} + \cdots + 400 \) Copy content Toggle raw display
$31$ \( T^{8} + 14 T^{7} + 117 T^{6} + \cdots + 19321 \) Copy content Toggle raw display
$37$ \( T^{8} - 10 T^{7} + 54 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + 44 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$43$ \( (T^{4} - 10 T^{3} - 72 T^{2} + 660 T + 116)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 28 T^{7} + 422 T^{6} + \cdots + 55696 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} - 2 T^{6} + \cdots + 55696 \) Copy content Toggle raw display
$59$ \( T^{8} - 10 T^{6} + 3400 T^{4} + \cdots + 250000 \) Copy content Toggle raw display
$61$ \( T^{8} + 34 T^{7} + 562 T^{6} + \cdots + 355216 \) Copy content Toggle raw display
$67$ \( (T^{4} + 12 T^{3} - 58 T^{2} - 564 T + 404)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 44 T^{6} + 240 T^{5} + \cdots + 8202496 \) Copy content Toggle raw display
$73$ \( T^{8} + 86 T^{6} + 600 T^{5} + \cdots + 839056 \) Copy content Toggle raw display
$79$ \( T^{8} - 22 T^{7} + 384 T^{6} + \cdots + 17808400 \) Copy content Toggle raw display
$83$ \( T^{8} + 30 T^{7} + \cdots + 133726096 \) Copy content Toggle raw display
$89$ \( (T^{4} + 2 T^{3} - 309 T^{2} - 530 T + 20725)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 10 T^{7} + 54 T^{6} + 160 T^{5} + \cdots + 16 \) Copy content Toggle raw display
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