# Properties

 Label 507.4.a.a Level $507$ Weight $4$ Character orbit 507.a Self dual yes Analytic conductor $29.914$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.9139683729$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{2} + 3 q^{3} + q^{4} + 9 q^{5} - 9 q^{6} - 2 q^{7} + 21 q^{8} + 9 q^{9}+O(q^{10})$$ q - 3 * q^2 + 3 * q^3 + q^4 + 9 * q^5 - 9 * q^6 - 2 * q^7 + 21 * q^8 + 9 * q^9 $$q - 3 q^{2} + 3 q^{3} + q^{4} + 9 q^{5} - 9 q^{6} - 2 q^{7} + 21 q^{8} + 9 q^{9} - 27 q^{10} - 30 q^{11} + 3 q^{12} + 6 q^{14} + 27 q^{15} - 71 q^{16} - 111 q^{17} - 27 q^{18} + 46 q^{19} + 9 q^{20} - 6 q^{21} + 90 q^{22} - 6 q^{23} + 63 q^{24} - 44 q^{25} + 27 q^{27} - 2 q^{28} - 105 q^{29} - 81 q^{30} + 100 q^{31} + 45 q^{32} - 90 q^{33} + 333 q^{34} - 18 q^{35} + 9 q^{36} - 17 q^{37} - 138 q^{38} + 189 q^{40} + 231 q^{41} + 18 q^{42} - 514 q^{43} - 30 q^{44} + 81 q^{45} + 18 q^{46} + 162 q^{47} - 213 q^{48} - 339 q^{49} + 132 q^{50} - 333 q^{51} + 639 q^{53} - 81 q^{54} - 270 q^{55} - 42 q^{56} + 138 q^{57} + 315 q^{58} - 600 q^{59} + 27 q^{60} + 233 q^{61} - 300 q^{62} - 18 q^{63} + 433 q^{64} + 270 q^{66} - 926 q^{67} - 111 q^{68} - 18 q^{69} + 54 q^{70} + 930 q^{71} + 189 q^{72} + 253 q^{73} + 51 q^{74} - 132 q^{75} + 46 q^{76} + 60 q^{77} - 1324 q^{79} - 639 q^{80} + 81 q^{81} - 693 q^{82} - 810 q^{83} - 6 q^{84} - 999 q^{85} + 1542 q^{86} - 315 q^{87} - 630 q^{88} - 498 q^{89} - 243 q^{90} - 6 q^{92} + 300 q^{93} - 486 q^{94} + 414 q^{95} + 135 q^{96} - 1358 q^{97} + 1017 q^{98} - 270 q^{99}+O(q^{100})$$ q - 3 * q^2 + 3 * q^3 + q^4 + 9 * q^5 - 9 * q^6 - 2 * q^7 + 21 * q^8 + 9 * q^9 - 27 * q^10 - 30 * q^11 + 3 * q^12 + 6 * q^14 + 27 * q^15 - 71 * q^16 - 111 * q^17 - 27 * q^18 + 46 * q^19 + 9 * q^20 - 6 * q^21 + 90 * q^22 - 6 * q^23 + 63 * q^24 - 44 * q^25 + 27 * q^27 - 2 * q^28 - 105 * q^29 - 81 * q^30 + 100 * q^31 + 45 * q^32 - 90 * q^33 + 333 * q^34 - 18 * q^35 + 9 * q^36 - 17 * q^37 - 138 * q^38 + 189 * q^40 + 231 * q^41 + 18 * q^42 - 514 * q^43 - 30 * q^44 + 81 * q^45 + 18 * q^46 + 162 * q^47 - 213 * q^48 - 339 * q^49 + 132 * q^50 - 333 * q^51 + 639 * q^53 - 81 * q^54 - 270 * q^55 - 42 * q^56 + 138 * q^57 + 315 * q^58 - 600 * q^59 + 27 * q^60 + 233 * q^61 - 300 * q^62 - 18 * q^63 + 433 * q^64 + 270 * q^66 - 926 * q^67 - 111 * q^68 - 18 * q^69 + 54 * q^70 + 930 * q^71 + 189 * q^72 + 253 * q^73 + 51 * q^74 - 132 * q^75 + 46 * q^76 + 60 * q^77 - 1324 * q^79 - 639 * q^80 + 81 * q^81 - 693 * q^82 - 810 * q^83 - 6 * q^84 - 999 * q^85 + 1542 * q^86 - 315 * q^87 - 630 * q^88 - 498 * q^89 - 243 * q^90 - 6 * q^92 + 300 * q^93 - 486 * q^94 + 414 * q^95 + 135 * q^96 - 1358 * q^97 + 1017 * q^98 - 270 * 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
−3.00000 3.00000 1.00000 9.00000 −9.00000 −2.00000 21.0000 9.00000 −27.0000
 $$n$$: 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 507.4.a.a 1
3.b odd 2 1 1521.4.a.j 1
13.b even 2 1 507.4.a.e 1
13.d odd 4 2 507.4.b.c 2
13.e even 6 2 39.4.e.a 2
39.d odd 2 1 1521.4.a.c 1
39.h odd 6 2 117.4.g.b 2
52.i odd 6 2 624.4.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 13.e even 6 2
117.4.g.b 2 39.h odd 6 2
507.4.a.a 1 1.a even 1 1 trivial
507.4.a.e 1 13.b even 2 1
507.4.b.c 2 13.d odd 4 2
624.4.q.b 2 52.i odd 6 2
1521.4.a.c 1 39.d odd 2 1
1521.4.a.j 1 3.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2} + 3$$ T2 + 3 $$T_{5} - 9$$ T5 - 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 3$$
$3$ $$T - 3$$
$5$ $$T - 9$$
$7$ $$T + 2$$
$11$ $$T + 30$$
$13$ $$T$$
$17$ $$T + 111$$
$19$ $$T - 46$$
$23$ $$T + 6$$
$29$ $$T + 105$$
$31$ $$T - 100$$
$37$ $$T + 17$$
$41$ $$T - 231$$
$43$ $$T + 514$$
$47$ $$T - 162$$
$53$ $$T - 639$$
$59$ $$T + 600$$
$61$ $$T - 233$$
$67$ $$T + 926$$
$71$ $$T - 930$$
$73$ $$T - 253$$
$79$ $$T + 1324$$
$83$ $$T + 810$$
$89$ $$T + 498$$
$97$ $$T + 1358$$