Properties

Label 462.2.c.b
Level $462$
Weight $2$
Character orbit 462.c
Analytic conductor $3.689$
Analytic rank $0$
Dimension $12$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.68908857338\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 16x^{10} + 94x^{8} + 246x^{6} + 277x^{4} + 114x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 94x^{8} + 246x^{6} + 277x^{4} + 114x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 17\nu^{2} + 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 13\nu^{9} + 55\nu^{7} + 72\nu^{5} - 29\nu^{3} - 60\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 29\nu^{3} + 2\nu^{2} + 21\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 29\nu^{3} - 2\nu^{2} + 21\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} - 16 \nu^{9} + 3 \nu^{8} - 94 \nu^{7} + 33 \nu^{6} - 243 \nu^{5} + 111 \nu^{4} - 253 \nu^{3} + 114 \nu^{2} - 63 \nu + 27 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 16 \nu^{9} + 3 \nu^{8} + 94 \nu^{7} + 33 \nu^{6} + 243 \nu^{5} + 111 \nu^{4} + 253 \nu^{3} + 114 \nu^{2} + 63 \nu + 27 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 16\nu^{8} + 91\nu^{6} + 219\nu^{4} + 202\nu^{2} + 42 ) / 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 16 \nu^{9} + 29 \nu^{8} - 94 \nu^{7} + 149 \nu^{6} - 243 \nu^{5} + 315 \nu^{4} - 253 \nu^{3} + 230 \nu^{2} - 75 \nu + 33 ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} + 16 \nu^{9} + 29 \nu^{8} + 94 \nu^{7} + 149 \nu^{6} + 243 \nu^{5} + 315 \nu^{4} + 253 \nu^{3} + 230 \nu^{2} + 75 \nu + 33 ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} + 16\nu^{9} + 94\nu^{7} + 243\nu^{5} + 259\nu^{3} + 93\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7\nu^{11} + 109\nu^{9} + 610\nu^{7} + 1458\nu^{5} + 1354\nu^{3} + 345\nu ) / 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + \beta_{3} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{10} - 5\beta_{9} + 5\beta_{8} + 3\beta_{6} - 3\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{9} - \beta_{8} + 2\beta_{7} - \beta_{6} - \beta_{5} + 5\beta_{4} - 5\beta_{3} + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{11} - 14 \beta_{10} + 23 \beta_{9} - 23 \beta_{8} - 13 \beta_{6} + 13 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{9} + 8\beta_{8} - 16\beta_{7} + 8\beta_{6} + 8\beta_{5} - 23\beta_{4} + 23\beta_{3} + 4\beta _1 - 124 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{11} + 41 \beta_{10} - 53 \beta_{9} + 53 \beta_{8} + 32 \beta_{6} - 32 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 51 \beta_{9} - 51 \beta_{8} + 102 \beta_{7} - 47 \beta_{6} - 47 \beta_{5} + 106 \beta_{4} - 106 \beta_{3} - 44 \beta _1 + 612 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 146 \beta_{11} - 456 \beta_{10} + 496 \beta_{9} - 496 \beta_{8} - 328 \beta_{6} + 328 \beta_{5} + 47 \beta_{4} + 47 \beta_{3} - 158 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 307 \beta_{9} + 307 \beta_{8} - 602 \beta_{7} + 243 \beta_{6} + 243 \beta_{5} - 496 \beta_{4} + 496 \beta_{3} + 340 \beta _1 - 3074 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 942 \beta_{11} + 2484 \beta_{10} - 2359 \beta_{9} + 2359 \beta_{8} + 1707 \beta_{6} - 1707 \beta_{5} - 243 \beta_{4} - 243 \beta_{3} + 1134 \beta_{2} ) / 2 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
2.18803i
2.18803i
0.822410i
0.822410i
1.18147i
1.18147i
2.33939i
2.33939i
0.319443i
0.319443i
1.88826i
1.88826i
1.00000 −1.64459 0.543441i 1.00000 0.118610i −1.64459 0.543441i 1.00000i 1.00000 2.40934 + 1.78747i 0.118610i
197.2 1.00000 −1.64459 + 0.543441i 1.00000 0.118610i −1.64459 + 0.543441i 1.00000i 1.00000 2.40934 1.78747i 0.118610i
197.3 1.00000 −0.742446 1.56486i 1.00000 3.23163i −0.742446 1.56486i 1.00000i 1.00000 −1.89755 + 2.32364i 3.23163i
197.4 1.00000 −0.742446 + 1.56486i 1.00000 3.23163i −0.742446 + 1.56486i 1.00000i 1.00000 −1.89755 2.32364i 3.23163i
197.5 1.00000 0.482127 1.66360i 1.00000 0.885172i 0.482127 1.66360i 1.00000i 1.00000 −2.53511 1.60413i 0.885172i
197.6 1.00000 0.482127 + 1.66360i 1.00000 0.885172i 0.482127 + 1.66360i 1.00000i 1.00000 −2.53511 + 1.60413i 0.885172i
197.7 1.00000 0.806630 1.53276i 1.00000 3.86104i 0.806630 1.53276i 1.00000i 1.00000 −1.69870 2.47274i 3.86104i
197.8 1.00000 0.806630 + 1.53276i 1.00000 3.86104i 0.806630 + 1.53276i 1.00000i 1.00000 −1.69870 + 2.47274i 3.86104i
197.9 1.00000 1.37401 1.05456i 1.00000 3.92876i 1.37401 1.05456i 1.00000i 1.00000 0.775791 2.89796i 3.92876i
197.10 1.00000 1.37401 + 1.05456i 1.00000 3.92876i 1.37401 + 1.05456i 1.00000i 1.00000 0.775791 + 2.89796i 3.92876i
197.11 1.00000 1.72427 0.163987i 1.00000 1.55439i 1.72427 0.163987i 1.00000i 1.00000 2.94622 0.565516i 1.55439i
197.12 1.00000 1.72427 + 0.163987i 1.00000 1.55439i 1.72427 + 0.163987i 1.00000i 1.00000 2.94622 + 0.565516i 1.55439i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d 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.c.b yes 12
3.b odd 2 1 462.2.c.a 12
11.b odd 2 1 462.2.c.a 12
33.d even 2 1 inner 462.2.c.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.c.a 12 3.b odd 2 1
462.2.c.a 12 11.b odd 2 1
462.2.c.b yes 12 1.a even 1 1 trivial
462.2.c.b yes 12 33.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{6} - 2T_{17}^{5} - 50T_{17}^{4} + 64T_{17}^{3} + 348T_{17}^{2} - 328T_{17} - 184 \) acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + 8 T^{10} - 8 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 44 T^{10} + 680 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} - 8 T^{11} + 54 T^{10} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 60 T^{10} + 1108 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( (T^{6} - 2 T^{5} - 50 T^{4} + 64 T^{3} + \cdots - 184)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 188 T^{10} + \cdots + 12222016 \) Copy content Toggle raw display
$23$ \( T^{12} + 120 T^{10} + 3280 T^{8} + \cdots + 36864 \) Copy content Toggle raw display
$29$ \( (T^{6} + 4 T^{5} - 84 T^{4} - 144 T^{3} + \cdots - 320)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 6 T^{5} - 82 T^{4} + 128 T^{3} + \cdots + 5288)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 96 T^{4} + 128 T^{3} + 2192 T^{2} + \cdots + 3008)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 10 T^{5} - 130 T^{4} + \cdots + 21832)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 320 T^{10} + 27872 T^{8} + \cdots + 1478656 \) Copy content Toggle raw display
$47$ \( T^{12} + 344 T^{10} + \cdots + 22353984 \) Copy content Toggle raw display
$53$ \( T^{12} + 208 T^{10} + 13344 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{12} + 196 T^{10} + 15076 T^{8} + \cdots + 2005056 \) Copy content Toggle raw display
$61$ \( T^{12} + 332 T^{10} + \cdots + 9424526400 \) Copy content Toggle raw display
$67$ \( (T^{6} - 12 T^{5} - 112 T^{4} + \cdots - 46784)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 616 T^{10} + \cdots + 40771686400 \) Copy content Toggle raw display
$73$ \( T^{12} + 280 T^{10} + \cdots + 318551104 \) Copy content Toggle raw display
$79$ \( T^{12} + 232 T^{10} + 15344 T^{8} + \cdots + 2166784 \) Copy content Toggle raw display
$83$ \( (T^{6} + 22 T^{5} + 124 T^{4} - 32 T^{3} + \cdots - 136)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 248 T^{10} + 21104 T^{8} + \cdots + 7573504 \) Copy content Toggle raw display
$97$ \( (T^{6} + 24 T^{5} + 52 T^{4} + \cdots - 121152)^{2} \) Copy content Toggle raw display
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