Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.225792.1
Defining polynomial: \( x^{4} - 18x^{2} + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 232)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} - 5) q^{5} + ( - \beta_{2} - 2 \beta_1 + 2) q^{7} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{2} - 5) q^{5} + ( - \beta_{2} - 2 \beta_1 + 2) q^{7} + (\beta_{3} + 3 \beta_{2} + 4 \beta_1 + 10) q^{9} + (\beta_{3} + \beta_{2} - 7 \beta_1 - 2) q^{11} + ( - 7 \beta_{3} + 4 \beta_{2} - 15) q^{13} + ( - 5 \beta_{3} + 2 \beta_{2} - 7 \beta_1 + 12) q^{15} + (15 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 56) q^{17} + (11 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 50) q^{19} + ( - 7 \beta_{3} - 4 \beta_{2} - 8 \beta_1 - 62) q^{21} + ( - 20 \beta_{3} + 7 \beta_{2} + 4 \beta_1 + 26) q^{23} + ( - 12 \beta_{3} + 10 \beta_{2} - 28) q^{25} + (17 \beta_{3} + 9 \beta_{2} + 9 \beta_1 + 126) q^{27} + 29 q^{29} + ( - 21 \beta_{2} - 7 \beta_1 - 106) q^{31} + ( - 4 \beta_{3} - 20 \beta_{2} - 24 \beta_1 - 257) q^{33} + ( - 2 \beta_{3} - \beta_{2} + 14 \beta_1 + 38) q^{35} + (62 \beta_{3} - 12 \beta_{2} - 36 \beta_1 - 124) q^{37} + (34 \beta_{3} - 29 \beta_{2} - 35 \beta_1 - 146) q^{39} + ( - 39 \beta_{3} + 22 \beta_{2} + 36 \beta_1 - 66) q^{41} + ( - 45 \beta_{3} - 21 \beta_{2} + 21 \beta_1 - 94) q^{43} + (13 \beta_{3} - 13 \beta_{2} - 32 \beta_1 - 218) q^{45} + (11 \beta_{3} + 4 \beta_{2} + 23 \beta_1 + 120) q^{47} + (12 \beta_{3} + 16 \beta_1 - 167) q^{49} + ( - 58 \beta_{3} + 29 \beta_{2} - 36 \beta_1 - 38) q^{51} + ( - 99 \beta_{3} + 14 \beta_{2} - 8 \beta_1 - 189) q^{53} + (44 \beta_{3} - 13 \beta_{2} + 45 \beta_1 - 146) q^{55} + ( - 9 \beta_{3} + 21 \beta_{2} + 92 \beta_1 + 44) q^{57} + (22 \beta_{3} + 61 \beta_{2} + 40 \beta_1 + 110) q^{59} + ( - 27 \beta_{3} + 2 \beta_{2} + 32 \beta_1 - 244) q^{61} + ( - 14 \beta_{3} - 10 \beta_{2} - 76 \beta_1 - 400) q^{63} + (69 \beta_{3} - 33 \beta_{2} + 28 \beta_1 - 213) q^{65} + (6 \beta_{3} + 34 \beta_{2} - 16 \beta_1 + 20) q^{67} + (79 \beta_{3} - 62 \beta_{2} - 24 \beta_1 - 216) q^{69} + ( - 84 \beta_{3} + 8 \beta_{2} + 110 \beta_1 - 52) q^{71} + (28 \beta_{3} + 44 \beta_{2} + 56 \beta_1 + 132) q^{73} + (74 \beta_{3} - 56 \beta_{2} - 56 \beta_1 - 288) q^{75} + (59 \beta_{3} + 34 \beta_{2} + 44 \beta_1 + 354) q^{77} + (67 \beta_{3} + 22 \beta_{2} - 23 \beta_1 - 628) q^{79} + ( - 7 \beta_{3} - 21 \beta_{2} + 140 \beta_1 + 193) q^{81} + (68 \beta_{3} - 5 \beta_{2} + 76 \beta_1 + 110) q^{83} + ( - 53 \beta_{3} + 120 \beta_{2} - 4 \beta_1 + 472) q^{85} + 29 \beta_1 q^{87} + ( - 49 \beta_{3} - 28 \beta_{2} - 64 \beta_1 - 402) q^{89} + ( - 48 \beta_{3} + 53 \beta_{2} + 98 \beta_1 - 26) q^{91} + ( - 112 \beta_{3} + 21 \beta_{2} - 176 \beta_1 - 7) q^{93} + (13 \beta_{3} - 25 \beta_{2} - 30 \beta_1 - 490) q^{95} + ( - 49 \beta_{3} - 60 \beta_{2} + 44 \beta_1 + 146) q^{97} + ( - 143 \beta_{3} - 71 \beta_{2} - 220 \beta_1 - 650) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{5} + 8 q^{7} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{5} + 8 q^{7} + 40 q^{9} - 8 q^{11} - 60 q^{13} + 48 q^{15} - 224 q^{17} + 200 q^{19} - 248 q^{21} + 104 q^{23} - 112 q^{25} + 504 q^{27} + 116 q^{29} - 424 q^{31} - 1028 q^{33} + 152 q^{35} - 496 q^{37} - 584 q^{39} - 264 q^{41} - 376 q^{43} - 872 q^{45} + 480 q^{47} - 668 q^{49} - 152 q^{51} - 756 q^{53} - 584 q^{55} + 176 q^{57} + 440 q^{59} - 976 q^{61} - 1600 q^{63} - 852 q^{65} + 80 q^{67} - 864 q^{69} - 208 q^{71} + 528 q^{73} - 1152 q^{75} + 1416 q^{77} - 2512 q^{79} + 772 q^{81} + 440 q^{83} + 1888 q^{85} - 1608 q^{89} - 104 q^{91} - 28 q^{93} - 1960 q^{95} + 584 q^{97} - 2600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 18x^{2} + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} - 12\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{3} + 36\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{2} - 18 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{3} + 6\beta_{2} + 18\beta_1 ) / 2 \) 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
1.03090
−4.11549
−1.03090
4.11549
0 −6.40414 0 −16.6404 0 3.16792 0 14.0130 0
1.2 0 −4.12732 0 −2.08419 0 13.1705 0 −9.96522 0
1.3 0 1.11264 0 6.64036 0 11.4151 0 −25.7620 0
1.4 0 9.41882 0 −7.91581 0 −19.7535 0 61.7142 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.k 4
4.b odd 2 1 232.4.a.c 4
8.b even 2 1 1856.4.a.x 4
8.d odd 2 1 1856.4.a.w 4
12.b even 2 1 2088.4.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.c 4 4.b odd 2 1
464.4.a.k 4 1.a even 1 1 trivial
1856.4.a.w 4 8.d odd 2 1
1856.4.a.x 4 8.b even 2 1
2088.4.a.e 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 74T_{3}^{2} - 168T_{3} + 277 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(464))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 74 T^{2} - 168 T + 277 \) Copy content Toggle raw display
$5$ \( T^{4} + 20 T^{3} + 6 T^{2} + \cdots - 1823 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} - 320 T^{2} + \cdots - 9408 \) Copy content Toggle raw display
$11$ \( T^{4} + 8 T^{3} - 3746 T^{2} + \cdots + 2296893 \) Copy content Toggle raw display
$13$ \( T^{4} + 60 T^{3} - 3698 T^{2} + \cdots - 4326887 \) Copy content Toggle raw display
$17$ \( T^{4} + 224 T^{3} + \cdots - 55312368 \) Copy content Toggle raw display
$19$ \( T^{4} - 200 T^{3} + \cdots - 18943728 \) Copy content Toggle raw display
$23$ \( T^{4} - 104 T^{3} + \cdots + 26110672 \) Copy content Toggle raw display
$29$ \( (T - 29)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 424 T^{3} + \cdots - 147822387 \) Copy content Toggle raw display
$37$ \( T^{4} + 496 T^{3} + \cdots - 8389147392 \) Copy content Toggle raw display
$41$ \( T^{4} + 264 T^{3} + \cdots + 796866624 \) Copy content Toggle raw display
$43$ \( T^{4} + 376 T^{3} + \cdots - 6763463843 \) Copy content Toggle raw display
$47$ \( T^{4} - 480 T^{3} + \cdots - 46277307 \) Copy content Toggle raw display
$53$ \( T^{4} + 756 T^{3} + \cdots + 48201739177 \) Copy content Toggle raw display
$59$ \( T^{4} - 440 T^{3} + \cdots + 27370709968 \) Copy content Toggle raw display
$61$ \( T^{4} + 976 T^{3} + \cdots + 1049490576 \) Copy content Toggle raw display
$67$ \( T^{4} - 80 T^{3} - 205760 T^{2} + \cdots - 715776 \) Copy content Toggle raw display
$71$ \( T^{4} + 208 T^{3} + \cdots + 50595925968 \) Copy content Toggle raw display
$73$ \( T^{4} - 528 T^{3} + \cdots + 4749558016 \) Copy content Toggle raw display
$79$ \( T^{4} + 2512 T^{3} + \cdots + 75686955061 \) Copy content Toggle raw display
$83$ \( T^{4} - 440 T^{3} + \cdots + 17011084752 \) Copy content Toggle raw display
$89$ \( T^{4} + 1608 T^{3} + \cdots - 2795276736 \) Copy content Toggle raw display
$97$ \( T^{4} - 584 T^{3} + \cdots + 10030146112 \) Copy content Toggle raw display
show more
show less