# Properties

 Label 507.4.b.e Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-17})$$ Defining polynomial: $$x^{4} - 17x^{2} + 289$$ x^4 - 17*x^2 + 289 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 q^{3} - 9 q^{4} + (3 \beta_{2} - 2 \beta_1) q^{5} - 3 \beta_1 q^{6} + (11 \beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8} + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3 * q^3 - 9 * q^4 + (3*b2 - 2*b1) * q^5 - 3*b1 * q^6 + (11*b2 - 3*b1) * q^7 - b1 * q^8 + 9 * q^9 $$q + \beta_1 q^{2} - 3 q^{3} - 9 q^{4} + (3 \beta_{2} - 2 \beta_1) q^{5} - 3 \beta_1 q^{6} + (11 \beta_{2} - 3 \beta_1) q^{7} - \beta_1 q^{8} + 9 q^{9} + ( - 3 \beta_{3} + 34) q^{10} + ( - 21 \beta_{2} + \beta_1) q^{11} + 27 q^{12} + ( - 11 \beta_{3} + 51) q^{14} + ( - 9 \beta_{2} + 6 \beta_1) q^{15} - 55 q^{16} + ( - \beta_{3} + 36) q^{17} + 9 \beta_1 q^{18} + ( - 37 \beta_{2} - 9 \beta_1) q^{19} + ( - 27 \beta_{2} + 18 \beta_1) q^{20} + ( - 33 \beta_{2} + 9 \beta_1) q^{21} + (21 \beta_{3} - 17) q^{22} + (7 \beta_{3} + 69) q^{23} + 3 \beta_1 q^{24} + (12 \beta_{3} + 30) q^{25} - 27 q^{27} + ( - 99 \beta_{2} + 27 \beta_1) q^{28} + (22 \beta_{3} + 3) q^{29} + (9 \beta_{3} - 102) q^{30} + (78 \beta_{2} - 42 \beta_1) q^{31} - 63 \beta_1 q^{32} + (63 \beta_{2} - 3 \beta_1) q^{33} + ( - 17 \beta_{2} + 36 \beta_1) q^{34} + (31 \beta_{3} - 201) q^{35} - 81 q^{36} + (82 \beta_{2} + 45 \beta_1) q^{37} + (37 \beta_{3} + 153) q^{38} + (3 \beta_{3} - 34) q^{40} + (30 \beta_{2} + \beta_1) q^{41} + (33 \beta_{3} - 153) q^{42} + (15 \beta_{3} - 235) q^{43} + (189 \beta_{2} - 9 \beta_1) q^{44} + (27 \beta_{2} - 18 \beta_1) q^{45} + (119 \beta_{2} + 69 \beta_1) q^{46} + (111 \beta_{2} + 79 \beta_1) q^{47} + 165 q^{48} + (66 \beta_{3} - 173) q^{49} + (204 \beta_{2} + 30 \beta_1) q^{50} + (3 \beta_{3} - 108) q^{51} + ( - 18 \beta_{3} - 567) q^{53} - 27 \beta_1 q^{54} + ( - 45 \beta_{3} + 223) q^{55} + (11 \beta_{3} - 51) q^{56} + (111 \beta_{2} + 27 \beta_1) q^{57} + (374 \beta_{2} + 3 \beta_1) q^{58} + ( - 360 \beta_{2} - 8 \beta_1) q^{59} + (81 \beta_{2} - 54 \beta_1) q^{60} + (87 \beta_{3} + 80) q^{61} + ( - 78 \beta_{3} + 714) q^{62} + (99 \beta_{2} - 27 \beta_1) q^{63} + 631 q^{64} + ( - 63 \beta_{3} + 51) q^{66} + ( - 83 \beta_{2} + 21 \beta_1) q^{67} + (9 \beta_{3} - 324) q^{68} + ( - 21 \beta_{3} - 207) q^{69} + (527 \beta_{2} - 201 \beta_1) q^{70} + (219 \beta_{2} - 17 \beta_1) q^{71} - 9 \beta_1 q^{72} + 225 \beta_{2} q^{73} + ( - 82 \beta_{3} - 765) q^{74} + ( - 36 \beta_{3} - 90) q^{75} + (333 \beta_{2} + 81 \beta_1) q^{76} + ( - 74 \beta_{3} + 744) q^{77} + (126 \beta_{3} + 2) q^{79} + ( - 165 \beta_{2} + 110 \beta_1) q^{80} + 81 q^{81} + ( - 30 \beta_{3} - 17) q^{82} + ( - 177 \beta_{2} + 241 \beta_1) q^{83} + (297 \beta_{2} - 81 \beta_1) q^{84} + (142 \beta_{2} - 81 \beta_1) q^{85} + (255 \beta_{2} - 235 \beta_1) q^{86} + ( - 66 \beta_{3} - 9) q^{87} + ( - 21 \beta_{3} + 17) q^{88} + ( - 42 \beta_{2} - 242 \beta_1) q^{89} + ( - 27 \beta_{3} + 306) q^{90} + ( - 63 \beta_{3} - 621) q^{92} + ( - 234 \beta_{2} + 126 \beta_1) q^{93} + ( - 111 \beta_{3} - 1343) q^{94} + ( - 47 \beta_{3} + 27) q^{95} + 189 \beta_1 q^{96} + (56 \beta_{2} + 402 \beta_1) q^{97} + (1122 \beta_{2} - 173 \beta_1) q^{98} + ( - 189 \beta_{2} + 9 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 - 3 * q^3 - 9 * q^4 + (3*b2 - 2*b1) * q^5 - 3*b1 * q^6 + (11*b2 - 3*b1) * q^7 - b1 * q^8 + 9 * q^9 + (-3*b3 + 34) * q^10 + (-21*b2 + b1) * q^11 + 27 * q^12 + (-11*b3 + 51) * q^14 + (-9*b2 + 6*b1) * q^15 - 55 * q^16 + (-b3 + 36) * q^17 + 9*b1 * q^18 + (-37*b2 - 9*b1) * q^19 + (-27*b2 + 18*b1) * q^20 + (-33*b2 + 9*b1) * q^21 + (21*b3 - 17) * q^22 + (7*b3 + 69) * q^23 + 3*b1 * q^24 + (12*b3 + 30) * q^25 - 27 * q^27 + (-99*b2 + 27*b1) * q^28 + (22*b3 + 3) * q^29 + (9*b3 - 102) * q^30 + (78*b2 - 42*b1) * q^31 - 63*b1 * q^32 + (63*b2 - 3*b1) * q^33 + (-17*b2 + 36*b1) * q^34 + (31*b3 - 201) * q^35 - 81 * q^36 + (82*b2 + 45*b1) * q^37 + (37*b3 + 153) * q^38 + (3*b3 - 34) * q^40 + (30*b2 + b1) * q^41 + (33*b3 - 153) * q^42 + (15*b3 - 235) * q^43 + (189*b2 - 9*b1) * q^44 + (27*b2 - 18*b1) * q^45 + (119*b2 + 69*b1) * q^46 + (111*b2 + 79*b1) * q^47 + 165 * q^48 + (66*b3 - 173) * q^49 + (204*b2 + 30*b1) * q^50 + (3*b3 - 108) * q^51 + (-18*b3 - 567) * q^53 - 27*b1 * q^54 + (-45*b3 + 223) * q^55 + (11*b3 - 51) * q^56 + (111*b2 + 27*b1) * q^57 + (374*b2 + 3*b1) * q^58 + (-360*b2 - 8*b1) * q^59 + (81*b2 - 54*b1) * q^60 + (87*b3 + 80) * q^61 + (-78*b3 + 714) * q^62 + (99*b2 - 27*b1) * q^63 + 631 * q^64 + (-63*b3 + 51) * q^66 + (-83*b2 + 21*b1) * q^67 + (9*b3 - 324) * q^68 + (-21*b3 - 207) * q^69 + (527*b2 - 201*b1) * q^70 + (219*b2 - 17*b1) * q^71 - 9*b1 * q^72 + 225*b2 * q^73 + (-82*b3 - 765) * q^74 + (-36*b3 - 90) * q^75 + (333*b2 + 81*b1) * q^76 + (-74*b3 + 744) * q^77 + (126*b3 + 2) * q^79 + (-165*b2 + 110*b1) * q^80 + 81 * q^81 + (-30*b3 - 17) * q^82 + (-177*b2 + 241*b1) * q^83 + (297*b2 - 81*b1) * q^84 + (142*b2 - 81*b1) * q^85 + (255*b2 - 235*b1) * q^86 + (-66*b3 - 9) * q^87 + (-21*b3 + 17) * q^88 + (-42*b2 - 242*b1) * q^89 + (-27*b3 + 306) * q^90 + (-63*b3 - 621) * q^92 + (-234*b2 + 126*b1) * q^93 + (-111*b3 - 1343) * q^94 + (-47*b3 + 27) * q^95 + 189*b1 * q^96 + (56*b2 + 402*b1) * q^97 + (1122*b2 - 173*b1) * q^98 + (-189*b2 + 9*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} - 36 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 $$4 q - 12 q^{3} - 36 q^{4} + 36 q^{9} + 136 q^{10} + 108 q^{12} + 204 q^{14} - 220 q^{16} + 144 q^{17} - 68 q^{22} + 276 q^{23} + 120 q^{25} - 108 q^{27} + 12 q^{29} - 408 q^{30} - 804 q^{35} - 324 q^{36} + 612 q^{38} - 136 q^{40} - 612 q^{42} - 940 q^{43} + 660 q^{48} - 692 q^{49} - 432 q^{51} - 2268 q^{53} + 892 q^{55} - 204 q^{56} + 320 q^{61} + 2856 q^{62} + 2524 q^{64} + 204 q^{66} - 1296 q^{68} - 828 q^{69} - 3060 q^{74} - 360 q^{75} + 2976 q^{77} + 8 q^{79} + 324 q^{81} - 68 q^{82} - 36 q^{87} + 68 q^{88} + 1224 q^{90} - 2484 q^{92} - 5372 q^{94} + 108 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 - 36 * q^4 + 36 * q^9 + 136 * q^10 + 108 * q^12 + 204 * q^14 - 220 * q^16 + 144 * q^17 - 68 * q^22 + 276 * q^23 + 120 * q^25 - 108 * q^27 + 12 * q^29 - 408 * q^30 - 804 * q^35 - 324 * q^36 + 612 * q^38 - 136 * q^40 - 612 * q^42 - 940 * q^43 + 660 * q^48 - 692 * q^49 - 432 * q^51 - 2268 * q^53 + 892 * q^55 - 204 * q^56 + 320 * q^61 + 2856 * q^62 + 2524 * q^64 + 204 * q^66 - 1296 * q^68 - 828 * q^69 - 3060 * q^74 - 360 * q^75 + 2976 * q^77 + 8 * q^79 + 324 * q^81 - 68 * q^82 - 36 * q^87 + 68 * q^88 + 1224 * q^90 - 2484 * q^92 - 5372 * q^94 + 108 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 17x^{2} + 289$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 17$$ (v^3) / 17 $$\beta_{2}$$ $$=$$ $$( 2\nu^{2} - 17 ) / 17$$ (2*v^2 - 17) / 17 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 34\nu ) / 17$$ (-v^3 + 34*v) / 17
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( 17\beta_{2} + 17 ) / 2$$ (17*b2 + 17) / 2 $$\nu^{3}$$ $$=$$ $$17\beta_1$$ 17*b1

## 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
 3.57071 − 2.06155i −3.57071 − 2.06155i −3.57071 + 2.06155i 3.57071 + 2.06155i
4.12311i −3.00000 −9.00000 3.05006i 12.3693i 6.68324i 4.12311i 9.00000 12.5757
337.2 4.12311i −3.00000 −9.00000 13.4424i 12.3693i 31.4219i 4.12311i 9.00000 55.4243
337.3 4.12311i −3.00000 −9.00000 13.4424i 12.3693i 31.4219i 4.12311i 9.00000 55.4243
337.4 4.12311i −3.00000 −9.00000 3.05006i 12.3693i 6.68324i 4.12311i 9.00000 12.5757
 $$n$$: 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.e 4
13.b even 2 1 inner 507.4.b.e 4
13.c even 3 1 39.4.j.b 4
13.d odd 4 2 507.4.a.k 4
13.e even 6 1 39.4.j.b 4
39.f even 4 2 1521.4.a.z 4
39.h odd 6 1 117.4.q.d 4
39.i odd 6 1 117.4.q.d 4
52.i odd 6 1 624.4.bv.c 4
52.j odd 6 1 624.4.bv.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.j.b 4 13.c even 3 1
39.4.j.b 4 13.e even 6 1
117.4.q.d 4 39.h odd 6 1
117.4.q.d 4 39.i odd 6 1
507.4.a.k 4 13.d odd 4 2
507.4.b.e 4 1.a even 1 1 trivial
507.4.b.e 4 13.b even 2 1 inner
624.4.bv.c 4 52.i odd 6 1
624.4.bv.c 4 52.j odd 6 1
1521.4.a.z 4 39.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{2} + 17$$ T2^2 + 17 $$T_{5}^{4} + 190T_{5}^{2} + 1681$$ T5^4 + 190*T5^2 + 1681

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 17)^{2}$$
$3$ $$(T + 3)^{4}$$
$5$ $$T^{4} + 190T^{2} + 1681$$
$7$ $$T^{4} + 1032 T^{2} + 44100$$
$11$ $$T^{4} + 2680 T^{2} + \cdots + 1705636$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 72 T + 1245)^{2}$$
$19$ $$T^{4} + 10968 T^{2} + \cdots + 7452900$$
$23$ $$(T^{2} - 138 T + 2262)^{2}$$
$29$ $$(T^{2} - 6 T - 24675)^{2}$$
$31$ $$T^{4} + 96480 T^{2} + \cdots + 137733696$$
$37$ $$T^{4} + 109194 T^{2} + \cdots + 203148009$$
$41$ $$T^{4} + 5434 T^{2} + \cdots + 7198489$$
$43$ $$(T^{2} + 470 T + 43750)^{2}$$
$47$ $$T^{4} + 286120 T^{2} + \cdots + 4779509956$$
$53$ $$(T^{2} + 1134 T + 304965)^{2}$$
$59$ $$T^{4} + 779776 T^{2} + \cdots + 150320594944$$
$61$ $$(T^{2} - 160 T - 379619)^{2}$$
$67$ $$T^{4} + 56328 T^{2} + \cdots + 173448900$$
$71$ $$T^{4} + 297592 T^{2} + \cdots + 19312660900$$
$73$ $$(T^{2} + 151875)^{2}$$
$79$ $$(T^{2} - 4 T - 809672)^{2}$$
$83$ $$T^{4} + 2162728 T^{2} + \cdots + 798145692100$$
$89$ $$T^{4} + 2001760 T^{2} + \cdots + 980686167616$$
$97$ $$T^{4} + 5513352 T^{2} + \cdots + 7495877379600$$