Properties

 Label 464.2.a.h Level $464$ Weight $2$ Character orbit 464.a Self dual yes Analytic conductor $3.705$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$464 = 2^{4} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 464.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$3.70505865379$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{3} - q^{5} - 2 \beta q^{7} - 2 \beta q^{9} +O(q^{10})$$ q + (b - 1) * q^3 - q^5 - 2*b * q^7 - 2*b * q^9 $$q + (\beta - 1) q^{3} - q^{5} - 2 \beta q^{7} - 2 \beta q^{9} + ( - \beta - 1) q^{11} + (2 \beta - 1) q^{13} + ( - \beta + 1) q^{15} + ( - 2 \beta - 2) q^{17} - 6 q^{19} + (2 \beta - 4) q^{21} + (4 \beta + 2) q^{23} - 4 q^{25} + ( - \beta - 1) q^{27} + q^{29} + (5 \beta - 3) q^{31} - q^{33} + 2 \beta q^{35} - 4 q^{37} + ( - 3 \beta + 5) q^{39} + (6 \beta + 4) q^{41} + ( - \beta - 5) q^{43} + 2 \beta q^{45} + ( - 3 \beta - 1) q^{47} + q^{49} - 2 q^{51} + ( - 6 \beta + 1) q^{53} + (\beta + 1) q^{55} + ( - 6 \beta + 6) q^{57} + ( - 4 \beta - 2) q^{59} + (2 \beta - 2) q^{61} + 8 q^{63} + ( - 2 \beta + 1) q^{65} + 4 \beta q^{67} + ( - 2 \beta + 6) q^{69} + ( - 2 \beta + 6) q^{71} + 4 q^{73} + ( - 4 \beta + 4) q^{75} + (2 \beta + 4) q^{77} + ( - \beta + 1) q^{79} + (6 \beta - 1) q^{81} + (4 \beta - 2) q^{83} + (2 \beta + 2) q^{85} + (\beta - 1) q^{87} + (6 \beta - 4) q^{89} + (2 \beta - 8) q^{91} + ( - 8 \beta + 13) q^{93} + 6 q^{95} + ( - 6 \beta - 4) q^{97} + (2 \beta + 4) q^{99} +O(q^{100})$$ q + (b - 1) * q^3 - q^5 - 2*b * q^7 - 2*b * q^9 + (-b - 1) * q^11 + (2*b - 1) * q^13 + (-b + 1) * q^15 + (-2*b - 2) * q^17 - 6 * q^19 + (2*b - 4) * q^21 + (4*b + 2) * q^23 - 4 * q^25 + (-b - 1) * q^27 + q^29 + (5*b - 3) * q^31 - q^33 + 2*b * q^35 - 4 * q^37 + (-3*b + 5) * q^39 + (6*b + 4) * q^41 + (-b - 5) * q^43 + 2*b * q^45 + (-3*b - 1) * q^47 + q^49 - 2 * q^51 + (-6*b + 1) * q^53 + (b + 1) * q^55 + (-6*b + 6) * q^57 + (-4*b - 2) * q^59 + (2*b - 2) * q^61 + 8 * q^63 + (-2*b + 1) * q^65 + 4*b * q^67 + (-2*b + 6) * q^69 + (-2*b + 6) * q^71 + 4 * q^73 + (-4*b + 4) * q^75 + (2*b + 4) * q^77 + (-b + 1) * q^79 + (6*b - 1) * q^81 + (4*b - 2) * q^83 + (2*b + 2) * q^85 + (b - 1) * q^87 + (6*b - 4) * q^89 + (2*b - 8) * q^91 + (-8*b + 13) * q^93 + 6 * q^95 + (-6*b - 4) * q^97 + (2*b + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 $$2 q - 2 q^{3} - 2 q^{5} - 2 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 12 q^{19} - 8 q^{21} + 4 q^{23} - 8 q^{25} - 2 q^{27} + 2 q^{29} - 6 q^{31} - 2 q^{33} - 8 q^{37} + 10 q^{39} + 8 q^{41} - 10 q^{43} - 2 q^{47} + 2 q^{49} - 4 q^{51} + 2 q^{53} + 2 q^{55} + 12 q^{57} - 4 q^{59} - 4 q^{61} + 16 q^{63} + 2 q^{65} + 12 q^{69} + 12 q^{71} + 8 q^{73} + 8 q^{75} + 8 q^{77} + 2 q^{79} - 2 q^{81} - 4 q^{83} + 4 q^{85} - 2 q^{87} - 8 q^{89} - 16 q^{91} + 26 q^{93} + 12 q^{95} - 8 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^11 - 2 * q^13 + 2 * q^15 - 4 * q^17 - 12 * q^19 - 8 * q^21 + 4 * q^23 - 8 * q^25 - 2 * q^27 + 2 * q^29 - 6 * q^31 - 2 * q^33 - 8 * q^37 + 10 * q^39 + 8 * q^41 - 10 * q^43 - 2 * q^47 + 2 * q^49 - 4 * q^51 + 2 * q^53 + 2 * q^55 + 12 * q^57 - 4 * q^59 - 4 * q^61 + 16 * q^63 + 2 * q^65 + 12 * q^69 + 12 * q^71 + 8 * q^73 + 8 * q^75 + 8 * q^77 + 2 * q^79 - 2 * q^81 - 4 * q^83 + 4 * q^85 - 2 * q^87 - 8 * q^89 - 16 * q^91 + 26 * q^93 + 12 * q^95 - 8 * q^97 + 8 * 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
 −1.41421 1.41421
0 −2.41421 0 −1.00000 0 2.82843 0 2.82843 0
1.2 0 0.414214 0 −1.00000 0 −2.82843 0 −2.82843 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 464.2.a.h 2
3.b odd 2 1 4176.2.a.bq 2
4.b odd 2 1 29.2.a.a 2
8.b even 2 1 1856.2.a.w 2
8.d odd 2 1 1856.2.a.r 2
12.b even 2 1 261.2.a.d 2
20.d odd 2 1 725.2.a.b 2
20.e even 4 2 725.2.b.b 4
28.d even 2 1 1421.2.a.j 2
44.c even 2 1 3509.2.a.j 2
52.b odd 2 1 4901.2.a.g 2
60.h even 2 1 6525.2.a.o 2
68.d odd 2 1 8381.2.a.e 2
116.d odd 2 1 841.2.a.d 2
116.e even 4 2 841.2.b.a 4
116.h odd 14 6 841.2.d.f 12
116.j odd 14 6 841.2.d.j 12
116.l even 28 12 841.2.e.k 24
348.b even 2 1 7569.2.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a 2 4.b odd 2 1
261.2.a.d 2 12.b even 2 1
464.2.a.h 2 1.a even 1 1 trivial
725.2.a.b 2 20.d odd 2 1
725.2.b.b 4 20.e even 4 2
841.2.a.d 2 116.d odd 2 1
841.2.b.a 4 116.e even 4 2
841.2.d.f 12 116.h odd 14 6
841.2.d.j 12 116.j odd 14 6
841.2.e.k 24 116.l even 28 12
1421.2.a.j 2 28.d even 2 1
1856.2.a.r 2 8.d odd 2 1
1856.2.a.w 2 8.b even 2 1
3509.2.a.j 2 44.c even 2 1
4176.2.a.bq 2 3.b odd 2 1
4901.2.a.g 2 52.b odd 2 1
6525.2.a.o 2 60.h even 2 1
7569.2.a.c 2 348.b even 2 1
8381.2.a.e 2 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(464))$$:

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{5} + 1$$ T5 + 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} + 2T - 1$$
$13$ $$T^{2} + 2T - 7$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$(T + 6)^{2}$$
$23$ $$T^{2} - 4T - 28$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} + 6T - 41$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 8T - 56$$
$43$ $$T^{2} + 10T + 23$$
$47$ $$T^{2} + 2T - 17$$
$53$ $$T^{2} - 2T - 71$$
$59$ $$T^{2} + 4T - 28$$
$61$ $$T^{2} + 4T - 4$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} - 12T + 28$$
$73$ $$(T - 4)^{2}$$
$79$ $$T^{2} - 2T - 1$$
$83$ $$T^{2} + 4T - 28$$
$89$ $$T^{2} + 8T - 56$$
$97$ $$T^{2} + 8T - 56$$