# Properties

 Label 507.2.a.d Level $507$ Weight $2$ Character orbit 507.a Self dual yes Analytic conductor $4.048$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{3} + (\beta + 2) q^{4} + ( - \beta + 2) q^{5} - \beta q^{6} + (\beta + 1) q^{7} + ( - \beta - 4) q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 + q^3 + (b + 2) * q^4 + (-b + 2) * q^5 - b * q^6 + (b + 1) * q^7 + (-b - 4) * q^8 + q^9 $$q - \beta q^{2} + q^{3} + (\beta + 2) q^{4} + ( - \beta + 2) q^{5} - \beta q^{6} + (\beta + 1) q^{7} + ( - \beta - 4) q^{8} + q^{9} + ( - \beta + 4) q^{10} + 2 q^{11} + (\beta + 2) q^{12} + ( - 2 \beta - 4) q^{14} + ( - \beta + 2) q^{15} + 3 \beta q^{16} + \beta q^{17} - \beta q^{18} + (2 \beta - 4) q^{19} - \beta q^{20} + (\beta + 1) q^{21} - 2 \beta q^{22} + 2 q^{23} + ( - \beta - 4) q^{24} + ( - 3 \beta + 3) q^{25} + q^{27} + (4 \beta + 6) q^{28} + ( - 3 \beta + 2) q^{29} + ( - \beta + 4) q^{30} + (\beta - 1) q^{31} + ( - \beta - 4) q^{32} + 2 q^{33} + ( - \beta - 4) q^{34} - 2 q^{35} + (\beta + 2) q^{36} + (\beta - 6) q^{37} + (2 \beta - 8) q^{38} + (3 \beta - 4) q^{40} - \beta q^{41} + ( - 2 \beta - 4) q^{42} + ( - \beta + 3) q^{43} + (2 \beta + 4) q^{44} + ( - \beta + 2) q^{45} - 2 \beta q^{46} + (4 \beta - 2) q^{47} + 3 \beta q^{48} + (3 \beta - 2) q^{49} + 12 q^{50} + \beta q^{51} + (3 \beta + 4) q^{53} - \beta q^{54} + ( - 2 \beta + 4) q^{55} + ( - 6 \beta - 8) q^{56} + (2 \beta - 4) q^{57} + (\beta + 12) q^{58} + (2 \beta + 6) q^{59} - \beta q^{60} + (2 \beta + 7) q^{61} - 4 q^{62} + (\beta + 1) q^{63} + ( - \beta + 4) q^{64} - 2 \beta q^{66} + (\beta - 3) q^{67} + (3 \beta + 4) q^{68} + 2 q^{69} + 2 \beta q^{70} - 14 q^{71} + ( - \beta - 4) q^{72} + ( - 2 \beta + 7) q^{73} + (5 \beta - 4) q^{74} + ( - 3 \beta + 3) q^{75} + 2 \beta q^{76} + (2 \beta + 2) q^{77} + (\beta + 7) q^{79} + (3 \beta - 12) q^{80} + q^{81} + (\beta + 4) q^{82} + (2 \beta + 4) q^{83} + (4 \beta + 6) q^{84} + (\beta - 4) q^{85} + ( - 2 \beta + 4) q^{86} + ( - 3 \beta + 2) q^{87} + ( - 2 \beta - 8) q^{88} + ( - 2 \beta - 8) q^{89} + ( - \beta + 4) q^{90} + (2 \beta + 4) q^{92} + (\beta - 1) q^{93} + ( - 2 \beta - 16) q^{94} + (6 \beta - 16) q^{95} + ( - \beta - 4) q^{96} + ( - \beta + 7) q^{97} + ( - \beta - 12) q^{98} + 2 q^{99} +O(q^{100})$$ q - b * q^2 + q^3 + (b + 2) * q^4 + (-b + 2) * q^5 - b * q^6 + (b + 1) * q^7 + (-b - 4) * q^8 + q^9 + (-b + 4) * q^10 + 2 * q^11 + (b + 2) * q^12 + (-2*b - 4) * q^14 + (-b + 2) * q^15 + 3*b * q^16 + b * q^17 - b * q^18 + (2*b - 4) * q^19 - b * q^20 + (b + 1) * q^21 - 2*b * q^22 + 2 * q^23 + (-b - 4) * q^24 + (-3*b + 3) * q^25 + q^27 + (4*b + 6) * q^28 + (-3*b + 2) * q^29 + (-b + 4) * q^30 + (b - 1) * q^31 + (-b - 4) * q^32 + 2 * q^33 + (-b - 4) * q^34 - 2 * q^35 + (b + 2) * q^36 + (b - 6) * q^37 + (2*b - 8) * q^38 + (3*b - 4) * q^40 - b * q^41 + (-2*b - 4) * q^42 + (-b + 3) * q^43 + (2*b + 4) * q^44 + (-b + 2) * q^45 - 2*b * q^46 + (4*b - 2) * q^47 + 3*b * q^48 + (3*b - 2) * q^49 + 12 * q^50 + b * q^51 + (3*b + 4) * q^53 - b * q^54 + (-2*b + 4) * q^55 + (-6*b - 8) * q^56 + (2*b - 4) * q^57 + (b + 12) * q^58 + (2*b + 6) * q^59 - b * q^60 + (2*b + 7) * q^61 - 4 * q^62 + (b + 1) * q^63 + (-b + 4) * q^64 - 2*b * q^66 + (b - 3) * q^67 + (3*b + 4) * q^68 + 2 * q^69 + 2*b * q^70 - 14 * q^71 + (-b - 4) * q^72 + (-2*b + 7) * q^73 + (5*b - 4) * q^74 + (-3*b + 3) * q^75 + 2*b * q^76 + (2*b + 2) * q^77 + (b + 7) * q^79 + (3*b - 12) * q^80 + q^81 + (b + 4) * q^82 + (2*b + 4) * q^83 + (4*b + 6) * q^84 + (b - 4) * q^85 + (-2*b + 4) * q^86 + (-3*b + 2) * q^87 + (-2*b - 8) * q^88 + (-2*b - 8) * q^89 + (-b + 4) * q^90 + (2*b + 4) * q^92 + (b - 1) * q^93 + (-2*b - 16) * q^94 + (6*b - 16) * q^95 + (-b - 4) * q^96 + (-b + 7) * q^97 + (-b - 12) * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 + 5 * q^4 + 3 * q^5 - q^6 + 3 * q^7 - 9 * q^8 + 2 * q^9 $$2 q - q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 9 q^{8} + 2 q^{9} + 7 q^{10} + 4 q^{11} + 5 q^{12} - 10 q^{14} + 3 q^{15} + 3 q^{16} + q^{17} - q^{18} - 6 q^{19} - q^{20} + 3 q^{21} - 2 q^{22} + 4 q^{23} - 9 q^{24} + 3 q^{25} + 2 q^{27} + 16 q^{28} + q^{29} + 7 q^{30} - q^{31} - 9 q^{32} + 4 q^{33} - 9 q^{34} - 4 q^{35} + 5 q^{36} - 11 q^{37} - 14 q^{38} - 5 q^{40} - q^{41} - 10 q^{42} + 5 q^{43} + 10 q^{44} + 3 q^{45} - 2 q^{46} + 3 q^{48} - q^{49} + 24 q^{50} + q^{51} + 11 q^{53} - q^{54} + 6 q^{55} - 22 q^{56} - 6 q^{57} + 25 q^{58} + 14 q^{59} - q^{60} + 16 q^{61} - 8 q^{62} + 3 q^{63} + 7 q^{64} - 2 q^{66} - 5 q^{67} + 11 q^{68} + 4 q^{69} + 2 q^{70} - 28 q^{71} - 9 q^{72} + 12 q^{73} - 3 q^{74} + 3 q^{75} + 2 q^{76} + 6 q^{77} + 15 q^{79} - 21 q^{80} + 2 q^{81} + 9 q^{82} + 10 q^{83} + 16 q^{84} - 7 q^{85} + 6 q^{86} + q^{87} - 18 q^{88} - 18 q^{89} + 7 q^{90} + 10 q^{92} - q^{93} - 34 q^{94} - 26 q^{95} - 9 q^{96} + 13 q^{97} - 25 q^{98} + 4 q^{99}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 + 5 * q^4 + 3 * q^5 - q^6 + 3 * q^7 - 9 * q^8 + 2 * q^9 + 7 * q^10 + 4 * q^11 + 5 * q^12 - 10 * q^14 + 3 * q^15 + 3 * q^16 + q^17 - q^18 - 6 * q^19 - q^20 + 3 * q^21 - 2 * q^22 + 4 * q^23 - 9 * q^24 + 3 * q^25 + 2 * q^27 + 16 * q^28 + q^29 + 7 * q^30 - q^31 - 9 * q^32 + 4 * q^33 - 9 * q^34 - 4 * q^35 + 5 * q^36 - 11 * q^37 - 14 * q^38 - 5 * q^40 - q^41 - 10 * q^42 + 5 * q^43 + 10 * q^44 + 3 * q^45 - 2 * q^46 + 3 * q^48 - q^49 + 24 * q^50 + q^51 + 11 * q^53 - q^54 + 6 * q^55 - 22 * q^56 - 6 * q^57 + 25 * q^58 + 14 * q^59 - q^60 + 16 * q^61 - 8 * q^62 + 3 * q^63 + 7 * q^64 - 2 * q^66 - 5 * q^67 + 11 * q^68 + 4 * q^69 + 2 * q^70 - 28 * q^71 - 9 * q^72 + 12 * q^73 - 3 * q^74 + 3 * q^75 + 2 * q^76 + 6 * q^77 + 15 * q^79 - 21 * q^80 + 2 * q^81 + 9 * q^82 + 10 * q^83 + 16 * q^84 - 7 * q^85 + 6 * q^86 + q^87 - 18 * q^88 - 18 * q^89 + 7 * q^90 + 10 * q^92 - q^93 - 34 * q^94 - 26 * q^95 - 9 * q^96 + 13 * q^97 - 25 * q^98 + 4 * 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
 2.56155 −1.56155
−2.56155 1.00000 4.56155 −0.561553 −2.56155 3.56155 −6.56155 1.00000 1.43845
1.2 1.56155 1.00000 0.438447 3.56155 1.56155 −0.561553 −2.43845 1.00000 5.56155
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 507.2.a.d 2
3.b odd 2 1 1521.2.a.m 2
4.b odd 2 1 8112.2.a.bo 2
13.b even 2 1 507.2.a.g 2
13.c even 3 2 507.2.e.g 4
13.d odd 4 2 507.2.b.d 4
13.e even 6 2 39.2.e.b 4
13.f odd 12 4 507.2.j.g 8
39.d odd 2 1 1521.2.a.g 2
39.f even 4 2 1521.2.b.h 4
39.h odd 6 2 117.2.g.c 4
52.b odd 2 1 8112.2.a.bk 2
52.i odd 6 2 624.2.q.h 4
65.l even 6 2 975.2.i.k 4
65.r odd 12 4 975.2.bb.i 8
156.r even 6 2 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 13.e even 6 2
117.2.g.c 4 39.h odd 6 2
507.2.a.d 2 1.a even 1 1 trivial
507.2.a.g 2 13.b even 2 1
507.2.b.d 4 13.d odd 4 2
507.2.e.g 4 13.c even 3 2
507.2.j.g 8 13.f odd 12 4
624.2.q.h 4 52.i odd 6 2
975.2.i.k 4 65.l even 6 2
975.2.bb.i 8 65.r odd 12 4
1521.2.a.g 2 39.d odd 2 1
1521.2.a.m 2 3.b odd 2 1
1521.2.b.h 4 39.f even 4 2
1872.2.t.r 4 156.r even 6 2
8112.2.a.bk 2 52.b odd 2 1
8112.2.a.bo 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2}^{2} + T_{2} - 4$$ T2^2 + T2 - 4 $$T_{5}^{2} - 3T_{5} - 2$$ T5^2 - 3*T5 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 4$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 3T - 2$$
$7$ $$T^{2} - 3T - 2$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - T - 4$$
$19$ $$T^{2} + 6T - 8$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} - T - 38$$
$31$ $$T^{2} + T - 4$$
$37$ $$T^{2} + 11T + 26$$
$41$ $$T^{2} + T - 4$$
$43$ $$T^{2} - 5T + 2$$
$47$ $$T^{2} - 68$$
$53$ $$T^{2} - 11T - 8$$
$59$ $$T^{2} - 14T + 32$$
$61$ $$T^{2} - 16T + 47$$
$67$ $$T^{2} + 5T + 2$$
$71$ $$(T + 14)^{2}$$
$73$ $$T^{2} - 12T + 19$$
$79$ $$T^{2} - 15T + 52$$
$83$ $$T^{2} - 10T + 8$$
$89$ $$T^{2} + 18T + 64$$
$97$ $$T^{2} - 13T + 38$$