# Properties

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

# 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + q^{4} - \beta q^{6} + 2 \beta q^{7} - \beta q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 - q^3 + q^4 - b * q^6 + 2*b * q^7 - b * q^8 + q^9 $$q + \beta q^{2} - q^{3} + q^{4} - \beta q^{6} + 2 \beta q^{7} - \beta q^{8} + q^{9} + 2 \beta q^{11} - q^{12} + 6 q^{14} - 5 q^{16} + 6 q^{17} + \beta q^{18} + 2 \beta q^{19} - 2 \beta q^{21} + 6 q^{22} + \beta q^{24} - 5 q^{25} - q^{27} + 2 \beta q^{28} + 6 q^{29} - 2 \beta q^{31} - 3 \beta q^{32} - 2 \beta q^{33} + 6 \beta q^{34} + q^{36} - 4 \beta q^{37} + 6 q^{38} - 4 \beta q^{41} - 6 q^{42} + 4 q^{43} + 2 \beta q^{44} - 2 \beta q^{47} + 5 q^{48} + 5 q^{49} - 5 \beta q^{50} - 6 q^{51} + 6 q^{53} - \beta q^{54} - 6 q^{56} - 2 \beta q^{57} + 6 \beta q^{58} - 6 \beta q^{59} - 2 q^{61} - 6 q^{62} + 2 \beta q^{63} + q^{64} - 6 q^{66} - 6 \beta q^{67} + 6 q^{68} + 2 \beta q^{71} - \beta q^{72} - 12 q^{74} + 5 q^{75} + 2 \beta q^{76} + 12 q^{77} - 8 q^{79} + q^{81} - 12 q^{82} - 2 \beta q^{83} - 2 \beta q^{84} + 4 \beta q^{86} - 6 q^{87} - 6 q^{88} + 4 \beta q^{89} + 2 \beta q^{93} - 6 q^{94} + 3 \beta q^{96} - 8 \beta q^{97} + 5 \beta q^{98} + 2 \beta q^{99} +O(q^{100})$$ q + b * q^2 - q^3 + q^4 - b * q^6 + 2*b * q^7 - b * q^8 + q^9 + 2*b * q^11 - q^12 + 6 * q^14 - 5 * q^16 + 6 * q^17 + b * q^18 + 2*b * q^19 - 2*b * q^21 + 6 * q^22 + b * q^24 - 5 * q^25 - q^27 + 2*b * q^28 + 6 * q^29 - 2*b * q^31 - 3*b * q^32 - 2*b * q^33 + 6*b * q^34 + q^36 - 4*b * q^37 + 6 * q^38 - 4*b * q^41 - 6 * q^42 + 4 * q^43 + 2*b * q^44 - 2*b * q^47 + 5 * q^48 + 5 * q^49 - 5*b * q^50 - 6 * q^51 + 6 * q^53 - b * q^54 - 6 * q^56 - 2*b * q^57 + 6*b * q^58 - 6*b * q^59 - 2 * q^61 - 6 * q^62 + 2*b * q^63 + q^64 - 6 * q^66 - 6*b * q^67 + 6 * q^68 + 2*b * q^71 - b * q^72 - 12 * q^74 + 5 * q^75 + 2*b * q^76 + 12 * q^77 - 8 * q^79 + q^81 - 12 * q^82 - 2*b * q^83 - 2*b * q^84 + 4*b * q^86 - 6 * q^87 - 6 * q^88 + 4*b * q^89 + 2*b * q^93 - 6 * q^94 + 3*b * q^96 - 8*b * q^97 + 5*b * q^98 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{12} + 12 q^{14} - 10 q^{16} + 12 q^{17} + 12 q^{22} - 10 q^{25} - 2 q^{27} + 12 q^{29} + 2 q^{36} + 12 q^{38} - 12 q^{42} + 8 q^{43} + 10 q^{48} + 10 q^{49} - 12 q^{51} + 12 q^{53} - 12 q^{56} - 4 q^{61} - 12 q^{62} + 2 q^{64} - 12 q^{66} + 12 q^{68} - 24 q^{74} + 10 q^{75} + 24 q^{77} - 16 q^{79} + 2 q^{81} - 24 q^{82} - 12 q^{87} - 12 q^{88} - 12 q^{94}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 - 2 * q^12 + 12 * q^14 - 10 * q^16 + 12 * q^17 + 12 * q^22 - 10 * q^25 - 2 * q^27 + 12 * q^29 + 2 * q^36 + 12 * q^38 - 12 * q^42 + 8 * q^43 + 10 * q^48 + 10 * q^49 - 12 * q^51 + 12 * q^53 - 12 * q^56 - 4 * q^61 - 12 * q^62 + 2 * q^64 - 12 * q^66 + 12 * q^68 - 24 * q^74 + 10 * q^75 + 24 * q^77 - 16 * q^79 + 2 * q^81 - 24 * q^82 - 12 * q^87 - 12 * q^88 - 12 * q^94

## 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
 −1.73205 1.73205
−1.73205 −1.00000 1.00000 0 1.73205 −3.46410 1.73205 1.00000 0
1.2 1.73205 −1.00000 1.00000 0 −1.73205 3.46410 −1.73205 1.00000 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
$$3$$ $$1$$
$$13$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.a.f 2
3.b odd 2 1 1521.2.a.l 2
4.b odd 2 1 8112.2.a.bv 2
13.b even 2 1 inner 507.2.a.f 2
13.c even 3 2 507.2.e.e 4
13.d odd 4 2 39.2.b.a 2
13.e even 6 2 507.2.e.e 4
13.f odd 12 2 507.2.j.a 2
13.f odd 12 2 507.2.j.c 2
39.d odd 2 1 1521.2.a.l 2
39.f even 4 2 117.2.b.a 2
52.b odd 2 1 8112.2.a.bv 2
52.f even 4 2 624.2.c.e 2
65.f even 4 2 975.2.h.f 4
65.g odd 4 2 975.2.b.d 2
65.k even 4 2 975.2.h.f 4
91.i even 4 2 1911.2.c.d 2
104.j odd 4 2 2496.2.c.k 2
104.m even 4 2 2496.2.c.d 2
156.l odd 4 2 1872.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 13.d odd 4 2
117.2.b.a 2 39.f even 4 2
507.2.a.f 2 1.a even 1 1 trivial
507.2.a.f 2 13.b even 2 1 inner
507.2.e.e 4 13.c even 3 2
507.2.e.e 4 13.e even 6 2
507.2.j.a 2 13.f odd 12 2
507.2.j.c 2 13.f odd 12 2
624.2.c.e 2 52.f even 4 2
975.2.b.d 2 65.g odd 4 2
975.2.h.f 4 65.f even 4 2
975.2.h.f 4 65.k even 4 2
1521.2.a.l 2 3.b odd 2 1
1521.2.a.l 2 39.d odd 2 1
1872.2.c.e 2 156.l odd 4 2
1911.2.c.d 2 91.i even 4 2
2496.2.c.d 2 104.m even 4 2
2496.2.c.k 2 104.j odd 4 2
8112.2.a.bv 2 4.b odd 2 1
8112.2.a.bv 2 52.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(507))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 12$$
$11$ $$T^{2} - 12$$
$13$ $$T^{2}$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} - 12$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} - 12$$
$37$ $$T^{2} - 48$$
$41$ $$T^{2} - 48$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 12$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 108$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} - 108$$
$71$ $$T^{2} - 12$$
$73$ $$T^{2}$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} - 12$$
$89$ $$T^{2} - 48$$
$97$ $$T^{2} - 192$$