# Properties

 Label 507.4.b.c Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} + 3 q^{3} - q^{4} - 9 i q^{5} + 9 i q^{6} - 2 i q^{7} + 21 i q^{8} + 9 q^{9} +O(q^{10})$$ q + 3*i * q^2 + 3 * q^3 - q^4 - 9*i * q^5 + 9*i * q^6 - 2*i * q^7 + 21*i * q^8 + 9 * q^9 $$q + 3 i q^{2} + 3 q^{3} - q^{4} - 9 i q^{5} + 9 i q^{6} - 2 i q^{7} + 21 i q^{8} + 9 q^{9} + 27 q^{10} - 30 i q^{11} - 3 q^{12} + 6 q^{14} - 27 i q^{15} - 71 q^{16} + 111 q^{17} + 27 i q^{18} - 46 i q^{19} + 9 i q^{20} - 6 i q^{21} + 90 q^{22} + 6 q^{23} + 63 i q^{24} + 44 q^{25} + 27 q^{27} + 2 i q^{28} - 105 q^{29} + 81 q^{30} - 100 i q^{31} - 45 i q^{32} - 90 i q^{33} + 333 i q^{34} - 18 q^{35} - 9 q^{36} - 17 i q^{37} + 138 q^{38} + 189 q^{40} - 231 i q^{41} + 18 q^{42} + 514 q^{43} + 30 i q^{44} - 81 i q^{45} + 18 i q^{46} + 162 i q^{47} - 213 q^{48} + 339 q^{49} + 132 i q^{50} + 333 q^{51} + 639 q^{53} + 81 i q^{54} - 270 q^{55} + 42 q^{56} - 138 i q^{57} - 315 i q^{58} - 600 i q^{59} + 27 i q^{60} + 233 q^{61} + 300 q^{62} - 18 i q^{63} - 433 q^{64} + 270 q^{66} + 926 i q^{67} - 111 q^{68} + 18 q^{69} - 54 i q^{70} - 930 i q^{71} + 189 i q^{72} + 253 i q^{73} + 51 q^{74} + 132 q^{75} + 46 i q^{76} - 60 q^{77} - 1324 q^{79} + 639 i q^{80} + 81 q^{81} + 693 q^{82} + 810 i q^{83} + 6 i q^{84} - 999 i q^{85} + 1542 i q^{86} - 315 q^{87} + 630 q^{88} - 498 i q^{89} + 243 q^{90} - 6 q^{92} - 300 i q^{93} - 486 q^{94} - 414 q^{95} - 135 i q^{96} + 1358 i q^{97} + 1017 i q^{98} - 270 i q^{99} +O(q^{100})$$ q + 3*i * q^2 + 3 * q^3 - q^4 - 9*i * q^5 + 9*i * q^6 - 2*i * q^7 + 21*i * q^8 + 9 * q^9 + 27 * q^10 - 30*i * q^11 - 3 * q^12 + 6 * q^14 - 27*i * q^15 - 71 * q^16 + 111 * q^17 + 27*i * q^18 - 46*i * q^19 + 9*i * q^20 - 6*i * q^21 + 90 * q^22 + 6 * q^23 + 63*i * q^24 + 44 * q^25 + 27 * q^27 + 2*i * q^28 - 105 * q^29 + 81 * q^30 - 100*i * q^31 - 45*i * q^32 - 90*i * q^33 + 333*i * q^34 - 18 * q^35 - 9 * q^36 - 17*i * q^37 + 138 * q^38 + 189 * q^40 - 231*i * q^41 + 18 * q^42 + 514 * q^43 + 30*i * q^44 - 81*i * q^45 + 18*i * q^46 + 162*i * q^47 - 213 * q^48 + 339 * q^49 + 132*i * q^50 + 333 * q^51 + 639 * q^53 + 81*i * q^54 - 270 * q^55 + 42 * q^56 - 138*i * q^57 - 315*i * q^58 - 600*i * q^59 + 27*i * q^60 + 233 * q^61 + 300 * q^62 - 18*i * q^63 - 433 * q^64 + 270 * q^66 + 926*i * q^67 - 111 * q^68 + 18 * q^69 - 54*i * q^70 - 930*i * q^71 + 189*i * q^72 + 253*i * q^73 + 51 * q^74 + 132 * q^75 + 46*i * q^76 - 60 * q^77 - 1324 * q^79 + 639*i * q^80 + 81 * q^81 + 693 * q^82 + 810*i * q^83 + 6*i * q^84 - 999*i * q^85 + 1542*i * q^86 - 315 * q^87 + 630 * q^88 - 498*i * q^89 + 243 * q^90 - 6 * q^92 - 300*i * q^93 - 486 * q^94 - 414 * q^95 - 135*i * q^96 + 1358*i * q^97 + 1017*i * q^98 - 270*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 2 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 2 * q^4 + 18 * q^9 $$2 q + 6 q^{3} - 2 q^{4} + 18 q^{9} + 54 q^{10} - 6 q^{12} + 12 q^{14} - 142 q^{16} + 222 q^{17} + 180 q^{22} + 12 q^{23} + 88 q^{25} + 54 q^{27} - 210 q^{29} + 162 q^{30} - 36 q^{35} - 18 q^{36} + 276 q^{38} + 378 q^{40} + 36 q^{42} + 1028 q^{43} - 426 q^{48} + 678 q^{49} + 666 q^{51} + 1278 q^{53} - 540 q^{55} + 84 q^{56} + 466 q^{61} + 600 q^{62} - 866 q^{64} + 540 q^{66} - 222 q^{68} + 36 q^{69} + 102 q^{74} + 264 q^{75} - 120 q^{77} - 2648 q^{79} + 162 q^{81} + 1386 q^{82} - 630 q^{87} + 1260 q^{88} + 486 q^{90} - 12 q^{92} - 972 q^{94} - 828 q^{95}+O(q^{100})$$ 2 * q + 6 * q^3 - 2 * q^4 + 18 * q^9 + 54 * q^10 - 6 * q^12 + 12 * q^14 - 142 * q^16 + 222 * q^17 + 180 * q^22 + 12 * q^23 + 88 * q^25 + 54 * q^27 - 210 * q^29 + 162 * q^30 - 36 * q^35 - 18 * q^36 + 276 * q^38 + 378 * q^40 + 36 * q^42 + 1028 * q^43 - 426 * q^48 + 678 * q^49 + 666 * q^51 + 1278 * q^53 - 540 * q^55 + 84 * q^56 + 466 * q^61 + 600 * q^62 - 866 * q^64 + 540 * q^66 - 222 * q^68 + 36 * q^69 + 102 * q^74 + 264 * q^75 - 120 * q^77 - 2648 * q^79 + 162 * q^81 + 1386 * q^82 - 630 * q^87 + 1260 * q^88 + 486 * q^90 - 12 * q^92 - 972 * q^94 - 828 * 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
3.00000i 3.00000 −1.00000 9.00000i 9.00000i 2.00000i 21.0000i 9.00000 27.0000
337.2 3.00000i 3.00000 −1.00000 9.00000i 9.00000i 2.00000i 21.0000i 9.00000 27.0000
 $$n$$: 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 507.4.b.c 2
13.b even 2 1 inner 507.4.b.c 2
13.d odd 4 1 507.4.a.a 1
13.d odd 4 1 507.4.a.e 1
13.f odd 12 2 39.4.e.a 2
39.f even 4 1 1521.4.a.c 1
39.f even 4 1 1521.4.a.j 1
39.k even 12 2 117.4.g.b 2
52.l even 12 2 624.4.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 13.f odd 12 2
117.4.g.b 2 39.k even 12 2
507.4.a.a 1 13.d odd 4 1
507.4.a.e 1 13.d odd 4 1
507.4.b.c 2 1.a even 1 1 trivial
507.4.b.c 2 13.b even 2 1 inner
624.4.q.b 2 52.l even 12 2
1521.4.a.c 1 39.f even 4 1
1521.4.a.j 1 39.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 $$S_{4}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{2} + 9$$ T2^2 + 9 $$T_{5}^{2} + 81$$ T5^2 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 81$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 900$$
$13$ $$T^{2}$$
$17$ $$(T - 111)^{2}$$
$19$ $$T^{2} + 2116$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T + 105)^{2}$$
$31$ $$T^{2} + 10000$$
$37$ $$T^{2} + 289$$
$41$ $$T^{2} + 53361$$
$43$ $$(T - 514)^{2}$$
$47$ $$T^{2} + 26244$$
$53$ $$(T - 639)^{2}$$
$59$ $$T^{2} + 360000$$
$61$ $$(T - 233)^{2}$$
$67$ $$T^{2} + 857476$$
$71$ $$T^{2} + 864900$$
$73$ $$T^{2} + 64009$$
$79$ $$(T + 1324)^{2}$$
$83$ $$T^{2} + 656100$$
$89$ $$T^{2} + 248004$$
$97$ $$T^{2} + 1844164$$