Properties

Label 464.4.a.f
Level $464$
Weight $4$
Character orbit 464.a
Self dual yes
Analytic conductor $27.377$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 5 + 3 \beta ) q^{3} + ( -5 + 4 \beta ) q^{5} + ( 8 - 10 \beta ) q^{7} + ( 16 + 30 \beta ) q^{9} +O(q^{10})\) \( q + ( 5 + 3 \beta ) q^{3} + ( -5 + 4 \beta ) q^{5} + ( 8 - 10 \beta ) q^{7} + ( 16 + 30 \beta ) q^{9} + ( 13 + 37 \beta ) q^{11} + ( -13 - 26 \beta ) q^{13} + ( -1 + 5 \beta ) q^{15} + ( 30 + 18 \beta ) q^{17} + ( 110 - 32 \beta ) q^{19} + ( -20 - 26 \beta ) q^{21} + ( -26 - 48 \beta ) q^{23} + ( -68 - 40 \beta ) q^{25} + ( 125 + 117 \beta ) q^{27} + 29 q^{29} + ( 147 + 63 \beta ) q^{31} + ( 287 + 224 \beta ) q^{33} + ( -120 + 82 \beta ) q^{35} + ( 156 - 56 \beta ) q^{37} + ( -221 - 169 \beta ) q^{39} + ( 20 + 138 \beta ) q^{41} + ( 161 - 171 \beta ) q^{43} + ( 160 - 86 \beta ) q^{45} + ( 65 + 207 \beta ) q^{47} + ( -79 - 160 \beta ) q^{49} + ( 258 + 180 \beta ) q^{51} + ( 501 - 122 \beta ) q^{53} + ( 231 - 133 \beta ) q^{55} + ( 358 + 170 \beta ) q^{57} + ( 450 - 248 \beta ) q^{59} + ( -474 - 178 \beta ) q^{61} + ( -472 + 80 \beta ) q^{63} + ( -143 + 78 \beta ) q^{65} + ( -160 - 484 \beta ) q^{67} + ( -418 - 318 \beta ) q^{69} + ( 330 + 34 \beta ) q^{71} + ( 324 - 640 \beta ) q^{73} + ( -580 - 404 \beta ) q^{75} + ( -636 + 166 \beta ) q^{77} + ( -129 + 341 \beta ) q^{79} + ( 895 + 150 \beta ) q^{81} + ( -606 - 64 \beta ) q^{83} + ( -6 + 30 \beta ) q^{85} + ( 145 + 87 \beta ) q^{87} + ( 380 + 522 \beta ) q^{89} + ( 416 - 78 \beta ) q^{91} + ( 1113 + 756 \beta ) q^{93} + ( -806 + 600 \beta ) q^{95} + ( 12 - 578 \beta ) q^{97} + ( 2428 + 982 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 10q^{3} - 10q^{5} + 16q^{7} + 32q^{9} + O(q^{10}) \) \( 2q + 10q^{3} - 10q^{5} + 16q^{7} + 32q^{9} + 26q^{11} - 26q^{13} - 2q^{15} + 60q^{17} + 220q^{19} - 40q^{21} - 52q^{23} - 136q^{25} + 250q^{27} + 58q^{29} + 294q^{31} + 574q^{33} - 240q^{35} + 312q^{37} - 442q^{39} + 40q^{41} + 322q^{43} + 320q^{45} + 130q^{47} - 158q^{49} + 516q^{51} + 1002q^{53} + 462q^{55} + 716q^{57} + 900q^{59} - 948q^{61} - 944q^{63} - 286q^{65} - 320q^{67} - 836q^{69} + 660q^{71} + 648q^{73} - 1160q^{75} - 1272q^{77} - 258q^{79} + 1790q^{81} - 1212q^{83} - 12q^{85} + 290q^{87} + 760q^{89} + 832q^{91} + 2226q^{93} - 1612q^{95} + 24q^{97} + 4856q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
0 0.757359 0 −10.6569 0 22.1421 0 −26.4264 0
1.2 0 9.24264 0 0.656854 0 −6.14214 0 58.4264 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.f 2
4.b odd 2 1 29.4.a.a 2
8.b even 2 1 1856.4.a.h 2
8.d odd 2 1 1856.4.a.n 2
12.b even 2 1 261.4.a.b 2
20.d odd 2 1 725.4.a.b 2
28.d even 2 1 1421.4.a.c 2
116.d odd 2 1 841.4.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.a 2 4.b odd 2 1
261.4.a.b 2 12.b even 2 1
464.4.a.f 2 1.a even 1 1 trivial
725.4.a.b 2 20.d odd 2 1
841.4.a.a 2 116.d odd 2 1
1421.4.a.c 2 28.d even 2 1
1856.4.a.h 2 8.b even 2 1
1856.4.a.n 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 10 T_{3} + 7 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(464))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 7 - 10 T + T^{2} \)
$5$ \( -7 + 10 T + T^{2} \)
$7$ \( -136 - 16 T + T^{2} \)
$11$ \( -2569 - 26 T + T^{2} \)
$13$ \( -1183 + 26 T + T^{2} \)
$17$ \( 252 - 60 T + T^{2} \)
$19$ \( 10052 - 220 T + T^{2} \)
$23$ \( -3932 + 52 T + T^{2} \)
$29$ \( ( -29 + T )^{2} \)
$31$ \( 13671 - 294 T + T^{2} \)
$37$ \( 18064 - 312 T + T^{2} \)
$41$ \( -37688 - 40 T + T^{2} \)
$43$ \( -32561 - 322 T + T^{2} \)
$47$ \( -81473 - 130 T + T^{2} \)
$53$ \( 221233 - 1002 T + T^{2} \)
$59$ \( 79492 - 900 T + T^{2} \)
$61$ \( 161308 + 948 T + T^{2} \)
$67$ \( -442912 + 320 T + T^{2} \)
$71$ \( 106588 - 660 T + T^{2} \)
$73$ \( -714224 - 648 T + T^{2} \)
$79$ \( -215921 + 258 T + T^{2} \)
$83$ \( 359044 + 1212 T + T^{2} \)
$89$ \( -400568 - 760 T + T^{2} \)
$97$ \( -668024 - 24 T + T^{2} \)
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