Properties

Label 464.4.a.e
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: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + (6 \beta - 5) q^{5} + (8 \beta + 8) q^{7} + (2 \beta - 20) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + (6 \beta - 5) q^{5} + (8 \beta + 8) q^{7} + (2 \beta - 20) q^{9} + (3 \beta + 45) q^{11} + (8 \beta - 25) q^{13} + (\beta + 31) q^{15} + (16 \beta - 22) q^{17} + (4 \beta - 54) q^{19} + (16 \beta + 56) q^{21} + (78 \beta + 14) q^{23} + ( - 60 \beta + 116) q^{25} + ( - 45 \beta - 35) q^{27} + 29 q^{29} + ( - 109 \beta - 33) q^{31} + (48 \beta + 63) q^{33} + (8 \beta + 248) q^{35} + (4 \beta + 20) q^{37} + ( - 17 \beta + 23) q^{39} + ( - 80 \beta + 152) q^{41} + ( - 53 \beta + 65) q^{43} + ( - 130 \beta + 172) q^{45} + (99 \beta + 257) q^{47} + (128 \beta + 105) q^{49} + ( - 6 \beta + 74) q^{51} + (52 \beta - 479) q^{53} + (255 \beta - 117) q^{55} + ( - 50 \beta - 30) q^{57} + ( - 250 \beta + 90) q^{59} + ( - 12 \beta + 514) q^{61} + ( - 144 \beta - 64) q^{63} + ( - 190 \beta + 413) q^{65} + (20 \beta + 456) q^{67} + (92 \beta + 482) q^{69} + ( - 34 \beta - 398) q^{71} + (176 \beta - 428) q^{73} + (56 \beta - 244) q^{75} + (384 \beta + 504) q^{77} + (361 \beta + 159) q^{79} + ( - 134 \beta + 235) q^{81} + (38 \beta + 914) q^{83} + ( - 212 \beta + 686) q^{85} + (29 \beta + 29) q^{87} + (72 \beta - 472) q^{89} + ( - 136 \beta + 184) q^{91} + ( - 142 \beta - 687) q^{93} + ( - 344 \beta + 414) q^{95} + (100 \beta + 184) q^{97} + (30 \beta - 864) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 10 q^{5} + 16 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 10 q^{5} + 16 q^{7} - 40 q^{9} + 90 q^{11} - 50 q^{13} + 62 q^{15} - 44 q^{17} - 108 q^{19} + 112 q^{21} + 28 q^{23} + 232 q^{25} - 70 q^{27} + 58 q^{29} - 66 q^{31} + 126 q^{33} + 496 q^{35} + 40 q^{37} + 46 q^{39} + 304 q^{41} + 130 q^{43} + 344 q^{45} + 514 q^{47} + 210 q^{49} + 148 q^{51} - 958 q^{53} - 234 q^{55} - 60 q^{57} + 180 q^{59} + 1028 q^{61} - 128 q^{63} + 826 q^{65} + 912 q^{67} + 964 q^{69} - 796 q^{71} - 856 q^{73} - 488 q^{75} + 1008 q^{77} + 318 q^{79} + 470 q^{81} + 1828 q^{83} + 1372 q^{85} + 58 q^{87} - 944 q^{89} + 368 q^{91} - 1374 q^{93} + 828 q^{95} + 368 q^{97} - 1728 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
0 −1.44949 0 −19.6969 0 −11.5959 0 −24.8990 0
1.2 0 3.44949 0 9.69694 0 27.5959 0 −15.1010 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.e 2
4.b odd 2 1 58.4.a.c 2
8.b even 2 1 1856.4.a.i 2
8.d odd 2 1 1856.4.a.l 2
12.b even 2 1 522.4.a.j 2
20.d odd 2 1 1450.4.a.g 2
116.d odd 2 1 1682.4.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 4.b odd 2 1
464.4.a.e 2 1.a even 1 1 trivial
522.4.a.j 2 12.b even 2 1
1450.4.a.g 2 20.d odd 2 1
1682.4.a.c 2 116.d odd 2 1
1856.4.a.i 2 8.b even 2 1
1856.4.a.l 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(464))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$5$ \( T^{2} + 10T - 191 \) Copy content Toggle raw display
$7$ \( T^{2} - 16T - 320 \) Copy content Toggle raw display
$11$ \( T^{2} - 90T + 1971 \) Copy content Toggle raw display
$13$ \( T^{2} + 50T + 241 \) Copy content Toggle raw display
$17$ \( T^{2} + 44T - 1052 \) Copy content Toggle raw display
$19$ \( T^{2} + 108T + 2820 \) Copy content Toggle raw display
$23$ \( T^{2} - 28T - 36308 \) Copy content Toggle raw display
$29$ \( (T - 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 66T - 70197 \) Copy content Toggle raw display
$37$ \( T^{2} - 40T + 304 \) Copy content Toggle raw display
$41$ \( T^{2} - 304T - 15296 \) Copy content Toggle raw display
$43$ \( T^{2} - 130T - 12629 \) Copy content Toggle raw display
$47$ \( T^{2} - 514T + 7243 \) Copy content Toggle raw display
$53$ \( T^{2} + 958T + 213217 \) Copy content Toggle raw display
$59$ \( T^{2} - 180T - 366900 \) Copy content Toggle raw display
$61$ \( T^{2} - 1028 T + 263332 \) Copy content Toggle raw display
$67$ \( T^{2} - 912T + 205536 \) Copy content Toggle raw display
$71$ \( T^{2} + 796T + 151468 \) Copy content Toggle raw display
$73$ \( T^{2} + 856T - 2672 \) Copy content Toggle raw display
$79$ \( T^{2} - 318T - 756645 \) Copy content Toggle raw display
$83$ \( T^{2} - 1828 T + 826732 \) Copy content Toggle raw display
$89$ \( T^{2} + 944T + 191680 \) Copy content Toggle raw display
$97$ \( T^{2} - 368T - 26144 \) Copy content Toggle raw display
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