# Properties

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

# Related objects

## 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 7 q^{3} - 15 q^{5} + 18 q^{7} + 22 q^{9}+O(q^{10})$$ q + 7 * q^3 - 15 * q^5 + 18 * q^7 + 22 * q^9 $$q + 7 q^{3} - 15 q^{5} + 18 q^{7} + 22 q^{9} - 27 q^{11} - 57 q^{13} - 105 q^{15} - 44 q^{17} - 152 q^{19} + 126 q^{21} + 152 q^{23} + 100 q^{25} - 35 q^{27} - 29 q^{29} + 173 q^{31} - 189 q^{33} - 270 q^{35} - 120 q^{37} - 399 q^{39} - 314 q^{41} - 339 q^{43} - 330 q^{45} + 357 q^{47} - 19 q^{49} - 308 q^{51} - 59 q^{53} + 405 q^{55} - 1064 q^{57} + 572 q^{59} - 420 q^{61} + 396 q^{63} + 855 q^{65} - 660 q^{67} + 1064 q^{69} - 726 q^{71} + 1004 q^{73} + 700 q^{75} - 486 q^{77} - 361 q^{79} - 839 q^{81} + 168 q^{83} + 660 q^{85} - 203 q^{87} + 58 q^{89} - 1026 q^{91} + 1211 q^{93} + 2280 q^{95} - 1206 q^{97} - 594 q^{99}+O(q^{100})$$ q + 7 * q^3 - 15 * q^5 + 18 * q^7 + 22 * q^9 - 27 * q^11 - 57 * q^13 - 105 * q^15 - 44 * q^17 - 152 * q^19 + 126 * q^21 + 152 * q^23 + 100 * q^25 - 35 * q^27 - 29 * q^29 + 173 * q^31 - 189 * q^33 - 270 * q^35 - 120 * q^37 - 399 * q^39 - 314 * q^41 - 339 * q^43 - 330 * q^45 + 357 * q^47 - 19 * q^49 - 308 * q^51 - 59 * q^53 + 405 * q^55 - 1064 * q^57 + 572 * q^59 - 420 * q^61 + 396 * q^63 + 855 * q^65 - 660 * q^67 + 1064 * q^69 - 726 * q^71 + 1004 * q^73 + 700 * q^75 - 486 * q^77 - 361 * q^79 - 839 * q^81 + 168 * q^83 + 660 * q^85 - 203 * q^87 + 58 * q^89 - 1026 * q^91 + 1211 * q^93 + 2280 * q^95 - 1206 * q^97 - 594 * q^99

## 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
 0
0 7.00000 0 −15.0000 0 18.0000 0 22.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b 1
4.b odd 2 1 58.4.a.b 1
8.b even 2 1 1856.4.a.c 1
8.d odd 2 1 1856.4.a.f 1
12.b even 2 1 522.4.a.b 1
20.d odd 2 1 1450.4.a.d 1
116.d odd 2 1 1682.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.b 1 4.b odd 2 1
464.4.a.b 1 1.a even 1 1 trivial
522.4.a.b 1 12.b even 2 1
1450.4.a.d 1 20.d odd 2 1
1682.4.a.a 1 116.d odd 2 1
1856.4.a.c 1 8.b even 2 1
1856.4.a.f 1 8.d odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 7$$
$5$ $$T + 15$$
$7$ $$T - 18$$
$11$ $$T + 27$$
$13$ $$T + 57$$
$17$ $$T + 44$$
$19$ $$T + 152$$
$23$ $$T - 152$$
$29$ $$T + 29$$
$31$ $$T - 173$$
$37$ $$T + 120$$
$41$ $$T + 314$$
$43$ $$T + 339$$
$47$ $$T - 357$$
$53$ $$T + 59$$
$59$ $$T - 572$$
$61$ $$T + 420$$
$67$ $$T + 660$$
$71$ $$T + 726$$
$73$ $$T - 1004$$
$79$ $$T + 361$$
$83$ $$T - 168$$
$89$ $$T - 58$$
$97$ $$T + 1206$$