Properties

Label 462.2.e.a
Level $462$
Weight $2$
Character orbit 462.e
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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6679465984.1
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 88x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} - q^{6} + (\beta_{4} + \beta_1) q^{7} - \beta_1 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_1 q^{3} - q^{4} + ( - \beta_{3} - \beta_1) q^{5} - q^{6} + (\beta_{4} + \beta_1) q^{7} - \beta_1 q^{8} - q^{9} + (\beta_{2} + 1) q^{10} + (\beta_{7} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{11} - \beta_1 q^{12} + (\beta_{7} + \beta_{6}) q^{13} + ( - \beta_{7} - 1) q^{14} + (\beta_{2} + 1) q^{15} + q^{16} + (\beta_{5} + \beta_{4} + \beta_{2} - 1) q^{17} - \beta_1 q^{18} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{2} + 1) q^{19} + (\beta_{3} + \beta_1) q^{20} + ( - \beta_{7} - 1) q^{21} + ( - \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{22} + (\beta_{5} + \beta_{4} + 2 \beta_{2}) q^{23} + q^{24} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 3) q^{25} + ( - \beta_{5} + \beta_{4}) q^{26} - \beta_1 q^{27} + ( - \beta_{4} - \beta_1) q^{28} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4}) q^{29} + (\beta_{3} + \beta_1) q^{30} + (\beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_1) q^{31} + \beta_1 q^{32} + ( - \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{33} + ( - \beta_{7} + \beta_{6} + \beta_{3} - \beta_1) q^{34} + (\beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{35} + q^{36} + ( - 2 \beta_{5} - 2 \beta_{4} + 2) q^{37} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{38} + ( - \beta_{5} + \beta_{4}) q^{39} + ( - \beta_{2} - 1) q^{40} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} - 3) q^{41} + ( - \beta_{4} - \beta_1) q^{42} + (2 \beta_{3} + 4 \beta_1) q^{43} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \beta_1 + 1) q^{44} + (\beta_{3} + \beta_1) q^{45} + ( - \beta_{7} + \beta_{6} + 2 \beta_{3}) q^{46} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_1) q^{47} + \beta_1 q^{48} + ( - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{49} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{3} - 3 \beta_1) q^{50} + ( - \beta_{7} + \beta_{6} + \beta_{3} - \beta_1) q^{51} + ( - \beta_{7} - \beta_{6}) q^{52} + ( - 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 3 \beta_{4}) q^{53} + q^{54} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - \beta_1 - 6) q^{55} + (\beta_{7} + 1) q^{56} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{57} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4}) q^{58} + ( - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_1) q^{59} + ( - \beta_{2} - 1) q^{60} + ( - \beta_{7} - \beta_{6} - 2 \beta_{2} + 6) q^{61} + (\beta_{7} + \beta_{6} + \beta_{2} + 3) q^{62} + ( - \beta_{4} - \beta_1) q^{63} - q^{64} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{65} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \beta_1 + 1) q^{66} + (3 \beta_{7} + 3 \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{67} + ( - \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{68} + ( - \beta_{7} + \beta_{6} + 2 \beta_{3}) q^{69} + (\beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{70} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{71} + \beta_1 q^{72} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{73} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{74} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{3} - 3 \beta_1) q^{75} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{2} - 1) q^{76} + ( - \beta_{6} + 2 \beta_{5} + 3 \beta_{2} + \beta_1 - 1) q^{77} + ( - \beta_{7} - \beta_{6}) q^{78} + ( - \beta_{7} + \beta_{6} - 2 \beta_{3} - 6 \beta_1) q^{79} + ( - \beta_{3} - \beta_1) q^{80} + q^{81} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_1) q^{82} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{2} - 1) q^{83} + (\beta_{7} + 1) q^{84} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 4 \beta_1) q^{85} + ( - 2 \beta_{2} - 4) q^{86} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4}) q^{87} + (\beta_{6} - \beta_{4} - \beta_{2} + \beta_1 - 1) q^{88} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - 6 \beta_1) q^{89} + ( - \beta_{2} - 1) q^{90} + ( - \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 7 \beta_1 + 3) q^{91} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{92} + (\beta_{7} + \beta_{6} + \beta_{2} + 3) q^{93} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{2} - 1) q^{94} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 4 \beta_{3} - 2 \beta_1) q^{95} - q^{96} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4}) q^{97} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{98} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 8 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 8 q^{6} - 8 q^{9} + 4 q^{10} - 8 q^{11} - 8 q^{14} + 4 q^{15} + 8 q^{16} - 12 q^{17} + 4 q^{19} - 8 q^{21} + 4 q^{22} - 8 q^{23} + 8 q^{24} - 16 q^{25} + 4 q^{33} + 8 q^{35} + 8 q^{36} + 16 q^{37} - 4 q^{40} - 20 q^{41} + 8 q^{44} - 12 q^{49} + 8 q^{54} - 40 q^{55} + 8 q^{56} - 4 q^{60} + 56 q^{61} + 20 q^{62} - 8 q^{64} + 8 q^{66} + 16 q^{67} + 12 q^{68} - 12 q^{70} + 8 q^{71} - 4 q^{73} - 4 q^{76} - 20 q^{77} + 8 q^{81} + 4 q^{83} + 8 q^{84} - 24 q^{86} - 4 q^{88} - 4 q^{90} + 20 q^{91} + 8 q^{92} + 20 q^{93} - 20 q^{94} - 8 q^{96} - 20 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 88x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 55\nu^{3} + 46\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 9\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 11\nu^{5} + 33\nu^{4} - 28\nu^{3} + 96\nu^{2} + 8\nu + 60 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 3\nu^{6} + 11\nu^{5} + 33\nu^{4} + 28\nu^{3} + 96\nu^{2} - 8\nu + 60 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 11\nu^{5} + 39\nu^{4} - 28\nu^{3} + 138\nu^{2} - 4\nu + 96 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 3\nu^{6} + 11\nu^{5} + 39\nu^{4} + 28\nu^{3} + 138\nu^{2} + 4\nu + 96 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 7\beta_{6} + 5\beta_{5} - 5\beta_{4} + 2\beta_{3} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} + 9\beta_{6} - 9\beta_{5} - 9\beta_{4} - 14\beta_{2} + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 45\beta_{7} - 45\beta_{6} - 31\beta_{5} + 31\beta_{4} - 18\beta_{3} + 26\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -67\beta_{7} - 67\beta_{6} + 71\beta_{5} + 71\beta_{4} + 90\beta_{2} - 178 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -291\beta_{7} + 291\beta_{6} + 205\beta_{5} - 205\beta_{4} + 142\beta_{3} - 230\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
2.20392i
2.62511i
1.25051i
0.829319i
0.829319i
1.25051i
2.62511i
2.20392i
1.00000i 1.00000i −1.00000 3.06118i −1.00000 2.37330 1.16938i 1.00000i −1.00000 −3.06118
307.2 1.00000i 1.00000i −1.00000 0.266088i −1.00000 −1.34926 2.27585i 1.00000i −1.00000 −0.266088
307.3 1.00000i 1.00000i −1.00000 1.18572i −1.00000 −1.74138 + 1.99189i 1.00000i −1.00000 1.18572
307.4 1.00000i 1.00000i −1.00000 4.14155i −1.00000 0.717333 2.54665i 1.00000i −1.00000 4.14155
307.5 1.00000i 1.00000i −1.00000 4.14155i −1.00000 0.717333 + 2.54665i 1.00000i −1.00000 4.14155
307.6 1.00000i 1.00000i −1.00000 1.18572i −1.00000 −1.74138 1.99189i 1.00000i −1.00000 1.18572
307.7 1.00000i 1.00000i −1.00000 0.266088i −1.00000 −1.34926 + 2.27585i 1.00000i −1.00000 −0.266088
307.8 1.00000i 1.00000i −1.00000 3.06118i −1.00000 2.37330 + 1.16938i 1.00000i −1.00000 −3.06118
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.e.a 8
3.b odd 2 1 1386.2.e.e 8
4.b odd 2 1 3696.2.q.b 8
7.b odd 2 1 462.2.e.b yes 8
11.b odd 2 1 462.2.e.b yes 8
21.c even 2 1 1386.2.e.a 8
28.d even 2 1 3696.2.q.c 8
33.d even 2 1 1386.2.e.a 8
44.c even 2 1 3696.2.q.c 8
77.b even 2 1 inner 462.2.e.a 8
231.h odd 2 1 1386.2.e.e 8
308.g odd 2 1 3696.2.q.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.e.a 8 1.a even 1 1 trivial
462.2.e.a 8 77.b even 2 1 inner
462.2.e.b yes 8 7.b odd 2 1
462.2.e.b yes 8 11.b odd 2 1
1386.2.e.a 8 21.c even 2 1
1386.2.e.a 8 33.d even 2 1
1386.2.e.e 8 3.b odd 2 1
1386.2.e.e 8 231.h odd 2 1
3696.2.q.b 8 4.b odd 2 1
3696.2.q.b 8 308.g odd 2 1
3696.2.q.c 8 28.d even 2 1
3696.2.q.c 8 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} - 26T_{13}^{2} + 56T_{13} - 16 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 28 T^{6} + 200 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{6} - 16 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + 40 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} - 26 T^{2} + 56 T - 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{3} - 10 T^{2} - 80 T - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 58 T^{2} + 208 T - 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 46 T^{2} - 288 T - 424)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 56 T^{6} + 976 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$31$ \( T^{8} + 112 T^{6} + 2436 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} - 64 T^{2} + 448 T + 432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 10 T^{3} - 22 T^{2} - 384 T - 656)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 144 T^{6} + 4448 T^{4} + \cdots + 73984 \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + 16872 T^{4} + \cdots + 1507984 \) Copy content Toggle raw display
$53$ \( (T^{4} - 182 T^{2} - 64 T + 512)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 272 T^{6} + \cdots + 11943936 \) Copy content Toggle raw display
$61$ \( (T^{4} - 28 T^{3} + 206 T^{2} + 96 T - 3232)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} - 172 T^{2} + 1248 T + 2624)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} - 222 T^{2} + 1360 T - 1416)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 2 T^{3} - 22 T^{2} + 64)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 300 T^{6} + \cdots + 20502784 \) Copy content Toggle raw display
$83$ \( (T^{4} - 2 T^{3} - 146 T^{2} + 176 T + 4296)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 200 T^{6} + 10512 T^{4} + \cdots + 147456 \) Copy content Toggle raw display
$97$ \( T^{8} + 136 T^{6} + 4688 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
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