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Properties

Label 464.6.a.k
Level $464$
Weight $6$
Character orbit 464.a
Self dual yes
Analytic conductor $74.418$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 464.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(74.4180923932\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - 4) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} + 5) q^{5} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 26) q^{7} + ( - \beta_{6} + 3 \beta_{5} - 5 \beta_{4} - \beta_{2} + 2 \beta_1 + 146) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - 4) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} + 5) q^{5} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 26) q^{7} + ( - \beta_{6} + 3 \beta_{5} - 5 \beta_{4} - \beta_{2} + 2 \beta_1 + 146) q^{9} + (\beta_{6} + 3 \beta_{5} + 9 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 160) q^{11} + (\beta_{6} - 8 \beta_{5} - 10 \beta_{4} - 5 \beta_{3} + \beta_{2} + 3 \beta_1 + 58) q^{13} + (6 \beta_{6} - 11 \beta_{5} + 17 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} - 12 \beta_1 + 82) q^{15} + (3 \beta_{6} - 20 \beta_{5} + 5 \beta_{4} - 10 \beta_{3} - \beta_{2} - 13 \beta_1 - 131) q^{17} + (4 \beta_{6} - 12 \beta_{5} - 35 \beta_{4} + 10 \beta_{3} - 16 \beta_{2} + \cdots - 607) q^{19}+ \cdots + (299 \beta_{6} - 2039 \beta_{5} + 5006 \beta_{4} - 399 \beta_{3} + \cdots - 45282) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9} - 1106 q^{11} + 408 q^{13} + 614 q^{15} - 874 q^{17} - 4288 q^{19} - 4200 q^{21} + 4532 q^{23} + 5527 q^{25} - 5942 q^{27} + 5887 q^{29} - 7794 q^{31} + 34410 q^{33} - 7088 q^{35} + 5086 q^{37} - 33394 q^{39} + 19826 q^{41} - 19498 q^{43} + 7854 q^{45} - 14278 q^{47} + 38431 q^{49} - 23892 q^{51} - 58644 q^{53} + 25574 q^{55} - 88540 q^{57} - 12888 q^{59} + 102866 q^{61} + 88632 q^{63} - 149206 q^{65} - 102996 q^{67} - 107244 q^{69} + 51596 q^{71} - 17566 q^{73} - 39356 q^{75} - 94104 q^{77} - 212058 q^{79} - 128285 q^{81} + 122928 q^{83} - 109336 q^{85} - 21866 q^{87} - 66510 q^{89} - 194368 q^{91} - 474274 q^{93} + 131676 q^{95} - 118182 q^{97} - 300668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 18\nu^{5} - 126\nu^{4} + 2142\nu^{3} + 3339\nu^{2} - 43244\nu + 12946 ) / 960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 8\nu^{5} - 176\nu^{4} + 952\nu^{3} + 9769\nu^{2} - 24904\nu - 146514 ) / 960 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 5\nu^{4} + 119\nu^{3} - 643\nu^{2} - 3370\nu + 16618 ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 10\nu^{5} + 218\nu^{4} + 1094\nu^{3} - 13231\nu^{2} - 21356\nu + 185766 ) / 960 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{6} - 11\nu^{5} - 257\nu^{4} + 1069\nu^{3} + 8713\nu^{2} - 15198\nu - 90278 ) / 480 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 20\nu^{5} + 12\nu^{4} + 2316\nu^{3} - 10299\nu^{2} - 52544\nu + 221094 ) / 320 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta _1 + 7 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + 3\beta_{4} - 7\beta_{3} + 7\beta_{2} - 3\beta _1 + 861 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{6} - 29\beta_{5} - 41\beta_{4} - 75\beta_{3} - 37\beta_{2} + 73\beta _1 - 151 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 143\beta_{6} - 111\beta_{5} + 269\beta_{4} - 757\beta_{3} + 637\beta_{2} - 353\beta _1 + 61287 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1619\beta_{6} - 3363\beta_{5} - 5463\beta_{4} - 6375\beta_{3} - 2349\beta_{2} + 5481\beta _1 - 22859 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 15975\beta_{6} - 9063\beta_{5} + 13365\beta_{4} - 69353\beta_{3} + 50617\beta_{2} - 33565\beta _1 + 4854835 ) / 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.90786
9.56883
−4.83960
−8.92709
3.60554
−9.15396
7.83842
0 −29.3989 0 64.0801 0 138.793 0 621.298 0
1.2 0 −18.3274 0 −84.3249 0 −216.816 0 92.8952 0
1.3 0 −15.9679 0 31.5616 0 −106.304 0 11.9736 0
1.4 0 −15.4219 0 −58.0818 0 210.388 0 −5.16616 0
1.5 0 13.2844 0 69.0035 0 −156.573 0 −66.5241 0
1.6 0 15.2661 0 64.0682 0 −91.1564 0 −9.94539 0
1.7 0 24.5656 0 −54.3066 0 37.6697 0 360.469 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(29\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.6.a.k 7
4.b odd 2 1 29.6.a.b ✓ 7
12.b even 2 1 261.6.a.e 7
20.d odd 2 1 725.6.a.b 7
116.d odd 2 1 841.6.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.b ✓ 7 4.b odd 2 1
261.6.a.e 7 12.b even 2 1
464.6.a.k 7 1.a even 1 1 trivial
725.6.a.b 7 20.d odd 2 1
841.6.a.b 7 116.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 26T_{3}^{6} - 1015T_{3}^{5} - 26056T_{3}^{4} + 280279T_{3}^{3} + 7496290T_{3}^{2} - 22844001T_{3} - 661023756 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(464))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + 26 T^{6} + \cdots - 661023756 \) Copy content Toggle raw display
$5$ \( T^{7} - 32 T^{6} + \cdots + 2378174390186 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots - 361848785235968 \) Copy content Toggle raw display
$11$ \( T^{7} + 1106 T^{6} + \cdots - 51\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{7} - 408 T^{6} + \cdots + 10\!\cdots\!34 \) Copy content Toggle raw display
$17$ \( T^{7} + 874 T^{6} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{7} + 4288 T^{6} + \cdots - 15\!\cdots\!92 \) Copy content Toggle raw display
$23$ \( T^{7} - 4532 T^{6} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T - 841)^{7} \) Copy content Toggle raw display
$31$ \( T^{7} + 7794 T^{6} + \cdots - 64\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{7} - 5086 T^{6} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{7} - 19826 T^{6} + \cdots - 56\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( T^{7} + 19498 T^{6} + \cdots + 58\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{7} + 14278 T^{6} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{7} + 58644 T^{6} + \cdots + 84\!\cdots\!94 \) Copy content Toggle raw display
$59$ \( T^{7} + 12888 T^{6} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} - 102866 T^{6} + \cdots + 45\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{7} + 102996 T^{6} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{7} - 51596 T^{6} + \cdots + 20\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{7} + 17566 T^{6} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{7} + 212058 T^{6} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} - 122928 T^{6} + \cdots + 97\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{7} + 66510 T^{6} + \cdots - 79\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{7} + 118182 T^{6} + \cdots + 62\!\cdots\!52 \) Copy content Toggle raw display
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