Properties

Label 464.6.a.i
Level $464$
Weight $6$
Character orbit 464.a
Self dual yes
Analytic conductor $74.418$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 7) q^{3} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 17) q^{5} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 52) q^{7} + ( - 14 \beta_{2} + 3 \beta_1 - 70) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 7) q^{3} + (\beta_{3} + 2 \beta_{2} - 2 \beta_1 - 17) q^{5} + (3 \beta_{3} + 2 \beta_{2} - \beta_1 + 52) q^{7} + ( - 14 \beta_{2} + 3 \beta_1 - 70) q^{9} + (5 \beta_{3} + 25 \beta_{2} - 2 \beta_1 + 31) q^{11} + (15 \beta_{3} - 24 \beta_{2} + 7 \beta_1 - 115) q^{13} + (20 \beta_{3} + 39 \beta_{2} + 5 \beta_1 - 233) q^{15} + ( - \beta_{3} + 32 \beta_{2} - 9 \beta_1 + 46) q^{17} + ( - 29 \beta_{3} - 44 \beta_{2} - 62 \beta_1 + 598) q^{19} + (55 \beta_{3} - 24 \beta_{2} + 7 \beta_1 + 248) q^{21} + (35 \beta_{3} - 80 \beta_{2} - 33 \beta_1 + 298) q^{23} + ( - 70 \beta_{3} - 180 \beta_{2} + 20 \beta_1 + 456) q^{25} + ( - 3 \beta_{3} + 209 \beta_{2} + 30 \beta_1 - 617) q^{27} - 841 q^{29} + (23 \beta_{3} - 83 \beta_{2} - 201 \beta_1 + 4803) q^{31} + (92 \beta_{3} + 168 \beta_{2} - 52 \beta_1 - 2645) q^{33} + ( - 55 \beta_{3} - 434 \beta_{2} - 65 \beta_1 + 5736) q^{35} + (10 \beta_{3} - 84 \beta_{2} - 314 \beta_1 - 2732) q^{37} + (263 \beta_{3} - 7 \beta_{2} + 89 \beta_1 + 2183) q^{39} + ( - 201 \beta_{3} + 792 \beta_{2} - 579 \beta_1 - 280) q^{41} + ( - 193 \beta_{3} - 631 \beta_{2} + 486 \beta_1 + 5355) q^{43} + (112 \beta_{3} + 90 \beta_{2} + 409 \beta_1 - 2086) q^{45} + ( - 18 \beta_{3} + 613 \beta_{2} + 531 \beta_1 - 5943) q^{47} + (260 \beta_{3} - 1288 \beta_{2} + 68 \beta_1 + 2613) q^{49} + ( - 9 \beta_{3} + 192 \beta_{2} - 63 \beta_1 - 3186) q^{51} + (157 \beta_{3} - 180 \beta_{2} + 1039 \beta_1 + 2215) q^{53} + ( - 429 \beta_{3} - 1363 \beta_{2} - 409 \beta_1 + 13163) q^{55} + ( - 460 \beta_{3} - 898 \beta_{2} + 293 \beta_1 + 12236) q^{57} + (35 \beta_{3} - 316 \beta_{2} - 1391 \beta_1 + 2710) q^{59} + ( - 599 \beta_{3} + 220 \beta_{2} - 1681 \beta_1 + 12362) q^{61} + (254 \beta_{3} - 696 \beta_{2} + 452 \beta_1 - 6872) q^{63} + ( - 208 \beta_{3} - 2550 \beta_{2} + 1083 \beta_1 + 24459) q^{65} + (258 \beta_{3} - 124 \beta_{2} + 2276 \beta_1 + 1960) q^{67} + (663 \beta_{3} - 652 \beta_{2} + 477 \beta_1 + 14698) q^{69} + (24 \beta_{3} + 174 \beta_{2} - 1712 \beta_1 + 12186) q^{71} + ( - 376 \beta_{3} + 792 \beta_{2} + 1496 \beta_1 - 18748) q^{73} + ( - 1280 \beta_{3} - 2036 \beta_{2} + 250 \beta_1 + 22612) q^{75} + ( - 469 \beta_{3} - 1432 \beta_{2} - 155 \beta_1 + 32164) q^{77} + ( - 716 \beta_{3} + 3679 \beta_{2} - 1111 \beta_1 + 26519) q^{79} + ( - 84 \beta_{3} + 5410 \beta_{2} - 1485 \beta_1 - 14923) q^{81} + (277 \beta_{3} + 1236 \beta_{2} + 1901 \beta_1 - 15722) q^{83} + ( - 343 \beta_{3} - 172 \beta_{2} - 735 \beta_1 + 5962) q^{85} + (841 \beta_{2} - 5887) q^{87} + (1095 \beta_{3} - 4112 \beta_{2} + 1635 \beta_1 + 26892) q^{89} + (1283 \beta_{3} - 8578 \beta_{2} + 1935 \beta_1 + 67224) q^{91} + (615 \beta_{3} - 4890 \beta_{2} + 1122 \beta_1 + 55365) q^{93} + (1953 \beta_{3} + 9880 \beta_{2} - 1176 \beta_1 - 36838) q^{95} + (1795 \beta_{3} + 5556 \beta_{2} - 797 \beta_1 - 12380) q^{97} + (493 \beta_{3} - 1782 \beta_{2} + 466 \beta_1 - 41680) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9} + 124 q^{11} - 460 q^{13} - 932 q^{15} + 184 q^{17} + 2392 q^{19} + 992 q^{21} + 1192 q^{23} + 1824 q^{25} - 2468 q^{27} - 3364 q^{29} + 19212 q^{31} - 10580 q^{33} + 22944 q^{35} - 10928 q^{37} + 8732 q^{39} - 1120 q^{41} + 21420 q^{43} - 8344 q^{45} - 23772 q^{47} + 10452 q^{49} - 12744 q^{51} + 8860 q^{53} + 52652 q^{55} + 48944 q^{57} + 10840 q^{59} + 49448 q^{61} - 27488 q^{63} + 97836 q^{65} + 7840 q^{67} + 58792 q^{69} + 48744 q^{71} - 74992 q^{73} + 90448 q^{75} + 128656 q^{77} + 106076 q^{79} - 59692 q^{81} - 62888 q^{83} + 23848 q^{85} - 23548 q^{87} + 107568 q^{89} + 268896 q^{91} + 221460 q^{93} - 147352 q^{95} - 49520 q^{97} - 166720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 34x^{2} - 27x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 34 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{3} - 4\nu^{2} - 268\nu - 94 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} + 6\beta_{2} + 67\beta _1 + 162 ) / 8 \) Copy content Toggle raw display

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
6.17343
−5.34807
−1.10057
0.275208
0 −7.07413 0 −44.4758 0 36.7447 0 −192.957 0
1.2 0 −0.734546 0 41.6400 0 90.0205 0 −242.460 0
1.3 0 17.5258 0 32.5670 0 220.793 0 64.1549 0
1.4 0 18.2828 0 −97.7313 0 −139.558 0 91.2623 0
\(n\): e.g. 2-40 or 990-1000
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.i 4
4.b odd 2 1 29.6.a.a 4
12.b even 2 1 261.6.a.a 4
20.d odd 2 1 725.6.a.a 4
116.d odd 2 1 841.6.a.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.a 4 4.b odd 2 1
261.6.a.a 4 12.b even 2 1
464.6.a.i 4 1.a even 1 1 trivial
725.6.a.a 4 20.d odd 2 1
841.6.a.a 4 116.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 28T_{3}^{3} + 46T_{3}^{2} + 2316T_{3} + 1665 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(464))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 28 T^{3} + 46 T^{2} + \cdots + 1665 \) Copy content Toggle raw display
$5$ \( T^{4} + 68 T^{3} - 4850 T^{2} + \cdots + 5894489 \) Copy content Toggle raw display
$7$ \( T^{4} - 208 T^{3} + \cdots - 101924272 \) Copy content Toggle raw display
$11$ \( T^{4} - 124 T^{3} + \cdots - 3717303119 \) Copy content Toggle raw display
$13$ \( T^{4} + 460 T^{3} + \cdots - 116053863479 \) Copy content Toggle raw display
$17$ \( T^{4} - 184 T^{3} + \cdots + 11464717824 \) Copy content Toggle raw display
$19$ \( T^{4} - 2392 T^{3} + \cdots + 2620094791680 \) Copy content Toggle raw display
$23$ \( T^{4} - 1192 T^{3} + \cdots + 2033080361984 \) Copy content Toggle raw display
$29$ \( (T + 841)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 421127233952247 \) Copy content Toggle raw display
$37$ \( T^{4} + 10928 T^{3} + \cdots + 16079593861120 \) Copy content Toggle raw display
$41$ \( T^{4} + 1120 T^{3} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{4} - 21420 T^{3} + \cdots + 47\!\cdots\!37 \) Copy content Toggle raw display
$47$ \( T^{4} + 23772 T^{3} + \cdots - 35\!\cdots\!87 \) Copy content Toggle raw display
$53$ \( T^{4} - 8860 T^{3} + \cdots + 16\!\cdots\!53 \) Copy content Toggle raw display
$59$ \( T^{4} - 10840 T^{3} + \cdots - 43\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{4} - 49448 T^{3} + \cdots + 90\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} - 7840 T^{3} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$71$ \( T^{4} - 48744 T^{3} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{4} + 74992 T^{3} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} - 106076 T^{3} + \cdots - 53\!\cdots\!75 \) Copy content Toggle raw display
$83$ \( T^{4} + 62888 T^{3} + \cdots - 18\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} - 107568 T^{3} + \cdots - 18\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{4} + 49520 T^{3} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
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