Properties

Label 464.4.a.i
Level $464$
Weight $4$
Character orbit 464.a
Self dual yes
Analytic conductor $27.377$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(27.3768862427\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.19816.1
Defining polynomial: \( x^{3} - x^{2} - 42x - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + (4 \beta_{2} - 8) q^{7} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + (4 \beta_{2} - 8) q^{7} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{2} - \beta_1 - 3) q^{11} + (11 \beta_{2} + 8 \beta_1 - 4) q^{13} + (2 \beta_{2} + 13 \beta_1 + 39) q^{15} + ( - 16 \beta_{2} - 12 \beta_1 - 18) q^{17} + ( - 10 \beta_{2} + 8 \beta_1 + 52) q^{19} + ( - 16 \beta_{2} - 4 \beta_1 - 28) q^{21} + ( - 12 \beta_{2} + 6 \beta_1 + 66) q^{23} + (11 \beta_{2} + 40 \beta_1 + 13) q^{25} + ( - 6 \beta_{2} - 17 \beta_1 + 53) q^{27} + 29 q^{29} + (36 \beta_{2} + 11 \beta_1 + 25) q^{31} + ( - 11 \beta_{2} - 4 \beta_1 - 42) q^{33} + ( - 12 \beta_{2} - 24 \beta_1) q^{35} + (38 \beta_{2} - 12 \beta_1 - 10) q^{37} + ( - 20 \beta_{2} + 31 \beta_1 + 121) q^{39} + ( - 6 \beta_{2} + 4 \beta_1 + 186) q^{41} + ( - 2 \beta_{2} + 15 \beta_1 - 11) q^{43} + (4 \beta_{2} + 26 \beta_1 + 132) q^{45} + ( - 2 \beta_{2} - 33 \beta_1 - 207) q^{47} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + (28 \beta_{2} - 70 \beta_1 - 162) q^{51} + ( - 27 \beta_{2} - 116 \beta_1 + 276) q^{53} + ( - 8 \beta_{2} - 25 \beta_1 - 39) q^{55} + (64 \beta_{2} + 66 \beta_1 + 254) q^{57} + ( - 4 \beta_{2} + 86 \beta_1 - 90) q^{59} + (40 \beta_{2} - 80 \beta_1 + 134) q^{61} + ( - 56 \beta_{2} - 56 \beta_1 + 280) q^{63} + (9 \beta_{2} + 90 \beta_1 + 468) q^{65} + (72 \beta_{2} - 36 \beta_1 + 88) q^{67} + (66 \beta_{2} + 72 \beta_1 + 204) q^{69} + (4 \beta_{2} - 2 \beta_1 + 18) q^{71} + (30 \beta_{2} + 40 \beta_1 - 178) q^{73} + (76 \beta_{2} + 144 \beta_1 + 968) q^{75} + ( - 24 \beta_{2} - 28 \beta_1 + 300) q^{77} + (38 \beta_{2} + 37 \beta_1 + 691) q^{79} + ( - 108 \beta_{2} - 58 \beta_1 - 485) q^{81} + ( - 12 \beta_{2} - 162 \beta_1 + 150) q^{83} + ( - 38 \beta_{2} - 184 \beta_1 - 840) q^{85} + (29 \beta_1 - 29) q^{87} + (154 \beta_{2} + 68 \beta_1 + 282) q^{89} + ( - 244 \beta_{2} - 208 \beta_1 + 1064) q^{91} + ( - 111 \beta_{2} + 94 \beta_1 - 52) q^{93} + (86 \beta_{2} + 244 \beta_1 + 552) q^{95} + ( - 34 \beta_{2} + 176 \beta_1 + 14) q^{97} + ( - 22 \beta_{2} - 38 \beta_1 + 114) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 20 q^{5} - 24 q^{7} + 5 q^{9} - 10 q^{11} - 4 q^{13} + 130 q^{15} - 66 q^{17} + 164 q^{19} - 88 q^{21} + 204 q^{23} + 79 q^{25} + 142 q^{27} + 87 q^{29} + 86 q^{31} - 130 q^{33} - 24 q^{35} - 42 q^{37} + 394 q^{39} + 562 q^{41} - 18 q^{43} + 422 q^{45} - 654 q^{47} + 539 q^{49} - 556 q^{51} + 712 q^{53} - 142 q^{55} + 828 q^{57} - 184 q^{59} + 322 q^{61} + 784 q^{63} + 1494 q^{65} + 228 q^{67} + 684 q^{69} + 52 q^{71} - 494 q^{73} + 3048 q^{75} + 872 q^{77} + 2110 q^{79} - 1513 q^{81} + 288 q^{83} - 2704 q^{85} - 58 q^{87} + 914 q^{89} + 2984 q^{91} - 62 q^{93} + 1900 q^{95} + 218 q^{97} + 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 42x - 54 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 4\nu - 27 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} + 4\beta _1 + 27 \) 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
−5.13291
−1.39712
7.53003
0 −6.13291 0 2.36031 0 18.5045 0 10.6126 0
1.2 0 −2.39712 0 −3.28077 0 −33.9461 0 −21.2538 0
1.3 0 6.53003 0 20.9205 0 −8.55839 0 15.6413 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 464.4.a.i 3
4.b odd 2 1 58.4.a.d 3
8.b even 2 1 1856.4.a.s 3
8.d odd 2 1 1856.4.a.r 3
12.b even 2 1 522.4.a.k 3
20.d odd 2 1 1450.4.a.h 3
116.d odd 2 1 1682.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 4.b odd 2 1
464.4.a.i 3 1.a even 1 1 trivial
522.4.a.k 3 12.b even 2 1
1450.4.a.h 3 20.d odd 2 1
1682.4.a.d 3 116.d odd 2 1
1856.4.a.r 3 8.d odd 2 1
1856.4.a.s 3 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 2T_{3}^{2} - 41T_{3} - 96 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(464))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} - 41 T - 96 \) Copy content Toggle raw display
$5$ \( T^{3} - 20 T^{2} - 27 T + 162 \) Copy content Toggle raw display
$7$ \( T^{3} + 24 T^{2} - 496 T - 5376 \) Copy content Toggle raw display
$11$ \( T^{3} + 10 T^{2} - 233 T - 2424 \) Copy content Toggle raw display
$13$ \( T^{3} + 4 T^{2} - 5619 T + 131706 \) Copy content Toggle raw display
$17$ \( T^{3} + 66 T^{2} - 10660 T - 679368 \) Copy content Toggle raw display
$19$ \( T^{3} - 164 T^{2} - 124 T + 664448 \) Copy content Toggle raw display
$23$ \( T^{3} - 204 T^{2} + 4284 T + 677376 \) Copy content Toggle raw display
$29$ \( (T - 29)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 86 T^{2} - 48089 T + 4766172 \) Copy content Toggle raw display
$37$ \( T^{3} + 42 T^{2} - 79456 T - 7684896 \) Copy content Toggle raw display
$41$ \( T^{3} - 562 T^{2} + 102432 T - 5982048 \) Copy content Toggle raw display
$43$ \( T^{3} + 18 T^{2} - 10369 T - 196488 \) Copy content Toggle raw display
$47$ \( T^{3} + 654 T^{2} + 98015 T + 3425124 \) Copy content Toggle raw display
$53$ \( T^{3} - 712 T^{2} + \cdots + 252120546 \) Copy content Toggle raw display
$59$ \( T^{3} + 184 T^{2} + \cdots - 57362928 \) Copy content Toggle raw display
$61$ \( T^{3} - 322 T^{2} - 388372 T - 5254424 \) Copy content Toggle raw display
$67$ \( T^{3} - 228 T^{2} + \cdots - 47608192 \) Copy content Toggle raw display
$71$ \( T^{3} - 52 T^{2} - 164 T + 672 \) Copy content Toggle raw display
$73$ \( T^{3} + 494 T^{2} + 6112 T - 9410208 \) Copy content Toggle raw display
$79$ \( T^{3} - 2110 T^{2} + \cdots - 285187172 \) Copy content Toggle raw display
$83$ \( T^{3} - 288 T^{2} + \cdots + 437606064 \) Copy content Toggle raw display
$89$ \( T^{3} - 914 T^{2} + \cdots + 598011552 \) Copy content Toggle raw display
$97$ \( T^{3} - 218 T^{2} + \cdots - 17006112 \) Copy content Toggle raw display
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