# Properties

 Label 507.2.b.b Level $507$ Weight $2$ Character orbit 507.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{3} + q^{4} - i q^{5} - i q^{6} - 2 i q^{7} + 3 i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 - q^3 + q^4 - i * q^5 - i * q^6 - 2*i * q^7 + 3*i * q^8 + q^9 $$q + i q^{2} - q^{3} + q^{4} - i q^{5} - i q^{6} - 2 i q^{7} + 3 i q^{8} + q^{9} + q^{10} + 2 i q^{11} - q^{12} + 2 q^{14} + i q^{15} - q^{16} + 7 q^{17} + i q^{18} - 6 i q^{19} - i q^{20} + 2 i q^{21} - 2 q^{22} + 6 q^{23} - 3 i q^{24} + 4 q^{25} - q^{27} - 2 i q^{28} - q^{29} - q^{30} + 4 i q^{31} + 5 i q^{32} - 2 i q^{33} + 7 i q^{34} - 2 q^{35} + q^{36} - i q^{37} + 6 q^{38} + 3 q^{40} + 9 i q^{41} - 2 q^{42} - 6 q^{43} + 2 i q^{44} - i q^{45} + 6 i q^{46} - 6 i q^{47} + q^{48} + 3 q^{49} + 4 i q^{50} - 7 q^{51} - 9 q^{53} - i q^{54} + 2 q^{55} + 6 q^{56} + 6 i q^{57} - i q^{58} + i q^{60} + q^{61} - 4 q^{62} - 2 i q^{63} - 7 q^{64} + 2 q^{66} - 2 i q^{67} + 7 q^{68} - 6 q^{69} - 2 i q^{70} + 6 i q^{71} + 3 i q^{72} - 11 i q^{73} + q^{74} - 4 q^{75} - 6 i q^{76} + 4 q^{77} - 4 q^{79} + i q^{80} + q^{81} - 9 q^{82} - 14 i q^{83} + 2 i q^{84} - 7 i q^{85} - 6 i q^{86} + q^{87} - 6 q^{88} + 14 i q^{89} + q^{90} + 6 q^{92} - 4 i q^{93} + 6 q^{94} - 6 q^{95} - 5 i q^{96} - 2 i q^{97} + 3 i q^{98} + 2 i q^{99} +O(q^{100})$$ q + i * q^2 - q^3 + q^4 - i * q^5 - i * q^6 - 2*i * q^7 + 3*i * q^8 + q^9 + q^10 + 2*i * q^11 - q^12 + 2 * q^14 + i * q^15 - q^16 + 7 * q^17 + i * q^18 - 6*i * q^19 - i * q^20 + 2*i * q^21 - 2 * q^22 + 6 * q^23 - 3*i * q^24 + 4 * q^25 - q^27 - 2*i * q^28 - q^29 - q^30 + 4*i * q^31 + 5*i * q^32 - 2*i * q^33 + 7*i * q^34 - 2 * q^35 + q^36 - i * q^37 + 6 * q^38 + 3 * q^40 + 9*i * q^41 - 2 * q^42 - 6 * q^43 + 2*i * q^44 - i * q^45 + 6*i * q^46 - 6*i * q^47 + q^48 + 3 * q^49 + 4*i * q^50 - 7 * q^51 - 9 * q^53 - i * q^54 + 2 * q^55 + 6 * q^56 + 6*i * q^57 - i * q^58 + i * q^60 + q^61 - 4 * q^62 - 2*i * q^63 - 7 * q^64 + 2 * q^66 - 2*i * q^67 + 7 * q^68 - 6 * q^69 - 2*i * q^70 + 6*i * q^71 + 3*i * q^72 - 11*i * q^73 + q^74 - 4 * q^75 - 6*i * q^76 + 4 * q^77 - 4 * q^79 + i * q^80 + q^81 - 9 * q^82 - 14*i * q^83 + 2*i * q^84 - 7*i * q^85 - 6*i * q^86 + q^87 - 6 * q^88 + 14*i * q^89 + q^90 + 6 * q^92 - 4*i * q^93 + 6 * q^94 - 6 * q^95 - 5*i * q^96 - 2*i * q^97 + 3*i * q^98 + 2*i * 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 $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} + 4 q^{14} - 2 q^{16} + 14 q^{17} - 4 q^{22} + 12 q^{23} + 8 q^{25} - 2 q^{27} - 2 q^{29} - 2 q^{30} - 4 q^{35} + 2 q^{36} + 12 q^{38} + 6 q^{40} - 4 q^{42} - 12 q^{43} + 2 q^{48} + 6 q^{49} - 14 q^{51} - 18 q^{53} + 4 q^{55} + 12 q^{56} + 2 q^{61} - 8 q^{62} - 14 q^{64} + 4 q^{66} + 14 q^{68} - 12 q^{69} + 2 q^{74} - 8 q^{75} + 8 q^{77} - 8 q^{79} + 2 q^{81} - 18 q^{82} + 2 q^{87} - 12 q^{88} + 2 q^{90} + 12 q^{92} + 12 q^{94} - 12 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^9 + 2 * q^10 - 2 * q^12 + 4 * q^14 - 2 * q^16 + 14 * q^17 - 4 * q^22 + 12 * q^23 + 8 * q^25 - 2 * q^27 - 2 * q^29 - 2 * q^30 - 4 * q^35 + 2 * q^36 + 12 * q^38 + 6 * q^40 - 4 * q^42 - 12 * q^43 + 2 * q^48 + 6 * q^49 - 14 * q^51 - 18 * q^53 + 4 * q^55 + 12 * q^56 + 2 * q^61 - 8 * q^62 - 14 * q^64 + 4 * q^66 + 14 * q^68 - 12 * q^69 + 2 * q^74 - 8 * q^75 + 8 * q^77 - 8 * q^79 + 2 * q^81 - 18 * q^82 + 2 * q^87 - 12 * q^88 + 2 * q^90 + 12 * q^92 + 12 * q^94 - 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
1.00000i −1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 1.00000
337.2 1.00000i −1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b.b 2
3.b odd 2 1 1521.2.b.c 2
13.b even 2 1 inner 507.2.b.b 2
13.c even 3 2 507.2.j.d 4
13.d odd 4 1 507.2.a.b 1
13.d odd 4 1 507.2.a.c 1
13.e even 6 2 507.2.j.d 4
13.f odd 12 2 39.2.e.a 2
13.f odd 12 2 507.2.e.c 2
39.d odd 2 1 1521.2.b.c 2
39.f even 4 1 1521.2.a.a 1
39.f even 4 1 1521.2.a.d 1
39.k even 12 2 117.2.g.b 2
52.f even 4 1 8112.2.a.w 1
52.f even 4 1 8112.2.a.bc 1
52.l even 12 2 624.2.q.c 2
65.o even 12 2 975.2.bb.d 4
65.s odd 12 2 975.2.i.f 2
65.t even 12 2 975.2.bb.d 4
156.v odd 12 2 1872.2.t.j 2

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

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{5}^{2} + 1$$ T5^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2}$$
$17$ $$(T - 7)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2} + 81$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 36$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 121$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 4$$