# 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 $1$

# 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: $$0$$ 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.

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

## 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: 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.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

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