# Properties

 Label 1682.4.a.d Level $1682$ Weight $4$ Character orbit 1682.a Self dual yes Analytic conductor $99.241$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1682 = 2 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1682.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$99.2412126297$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.19816.1 Defining polynomial: $$x^{3} - x^{2} - 42x - 54$$ x^3 - x^2 - 42*x - 54 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + (\beta_1 - 1) q^{3} + 4 q^{4} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + ( - 2 \beta_1 + 2) q^{6} + ( - 4 \beta_{2} + 8) q^{7} - 8 q^{8} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q - 2 * q^2 + (b1 - 1) * q^3 + 4 * q^4 + (b2 + 2*b1 + 6) * q^5 + (-2*b1 + 2) * q^6 + (-4*b2 + 8) * q^7 - 8 * q^8 + (3*b2 + 2*b1 + 1) * q^9 $$q - 2 q^{2} + (\beta_1 - 1) q^{3} + 4 q^{4} + (\beta_{2} + 2 \beta_1 + 6) q^{5} + ( - 2 \beta_1 + 2) q^{6} + ( - 4 \beta_{2} + 8) q^{7} - 8 q^{8} + (3 \beta_{2} + 2 \beta_1 + 1) q^{9} + ( - 2 \beta_{2} - 4 \beta_1 - 12) q^{10} + (2 \beta_{2} - \beta_1 - 3) q^{11} + (4 \beta_1 - 4) q^{12} + (11 \beta_{2} + 8 \beta_1 - 4) q^{13} + (8 \beta_{2} - 16) q^{14} + (2 \beta_{2} + 13 \beta_1 + 39) q^{15} + 16 q^{16} + (16 \beta_{2} + 12 \beta_1 + 18) q^{17} + ( - 6 \beta_{2} - 4 \beta_1 - 2) q^{18} + ( - 10 \beta_{2} + 8 \beta_1 + 52) q^{19} + (4 \beta_{2} + 8 \beta_1 + 24) q^{20} + (16 \beta_{2} + 4 \beta_1 + 28) q^{21} + ( - 4 \beta_{2} + 2 \beta_1 + 6) q^{22} + (12 \beta_{2} - 6 \beta_1 - 66) q^{23} + ( - 8 \beta_1 + 8) q^{24} + (11 \beta_{2} + 40 \beta_1 + 13) q^{25} + ( - 22 \beta_{2} - 16 \beta_1 + 8) q^{26} + ( - 6 \beta_{2} - 17 \beta_1 + 53) q^{27} + ( - 16 \beta_{2} + 32) q^{28} + ( - 4 \beta_{2} - 26 \beta_1 - 78) q^{30} + (36 \beta_{2} + 11 \beta_1 + 25) q^{31} - 32 q^{32} + ( - 11 \beta_{2} - 4 \beta_1 - 42) q^{33} + ( - 32 \beta_{2} - 24 \beta_1 - 36) q^{34} + (12 \beta_{2} + 24 \beta_1) q^{35} + (12 \beta_{2} + 8 \beta_1 + 4) q^{36} + ( - 38 \beta_{2} + 12 \beta_1 + 10) q^{37} + (20 \beta_{2} - 16 \beta_1 - 104) q^{38} + ( - 20 \beta_{2} + 31 \beta_1 + 121) q^{39} + ( - 8 \beta_{2} - 16 \beta_1 - 48) q^{40} + (6 \beta_{2} - 4 \beta_1 - 186) q^{41} + ( - 32 \beta_{2} - 8 \beta_1 - 56) q^{42} + ( - 2 \beta_{2} + 15 \beta_1 - 11) q^{43} + (8 \beta_{2} - 4 \beta_1 - 12) q^{44} + (4 \beta_{2} + 26 \beta_1 + 132) q^{45} + ( - 24 \beta_{2} + 12 \beta_1 + 132) q^{46} + ( - 2 \beta_{2} - 33 \beta_1 - 207) q^{47} + (16 \beta_1 - 16) q^{48} + ( - 80 \beta_{2} - 64 \beta_1 + 201) q^{49} + ( - 22 \beta_{2} - 80 \beta_1 - 26) q^{50} + ( - 28 \beta_{2} + 70 \beta_1 + 162) q^{51} + (44 \beta_{2} + 32 \beta_1 - 16) q^{52} + ( - 27 \beta_{2} - 116 \beta_1 + 276) q^{53} + (12 \beta_{2} + 34 \beta_1 - 106) q^{54} + ( - 8 \beta_{2} - 25 \beta_1 - 39) q^{55} + (32 \beta_{2} - 64) q^{56} + (64 \beta_{2} + 66 \beta_1 + 254) q^{57} + (4 \beta_{2} - 86 \beta_1 + 90) q^{59} + (8 \beta_{2} + 52 \beta_1 + 156) q^{60} + ( - 40 \beta_{2} + 80 \beta_1 - 134) q^{61} + ( - 72 \beta_{2} - 22 \beta_1 - 50) q^{62} + (56 \beta_{2} + 56 \beta_1 - 280) q^{63} + 64 q^{64} + (9 \beta_{2} + 90 \beta_1 + 468) q^{65} + (22 \beta_{2} + 8 \beta_1 + 84) q^{66} + ( - 72 \beta_{2} + 36 \beta_1 - 88) q^{67} + (64 \beta_{2} + 48 \beta_1 + 72) q^{68} + ( - 66 \beta_{2} - 72 \beta_1 - 204) q^{69} + ( - 24 \beta_{2} - 48 \beta_1) q^{70} + ( - 4 \beta_{2} + 2 \beta_1 - 18) q^{71} + ( - 24 \beta_{2} - 16 \beta_1 - 8) q^{72} + ( - 30 \beta_{2} - 40 \beta_1 + 178) q^{73} + (76 \beta_{2} - 24 \beta_1 - 20) q^{74} + (76 \beta_{2} + 144 \beta_1 + 968) q^{75} + ( - 40 \beta_{2} + 32 \beta_1 + 208) q^{76} + (24 \beta_{2} + 28 \beta_1 - 300) q^{77} + (40 \beta_{2} - 62 \beta_1 - 242) q^{78} + (38 \beta_{2} + 37 \beta_1 + 691) q^{79} + (16 \beta_{2} + 32 \beta_1 + 96) q^{80} + ( - 108 \beta_{2} - 58 \beta_1 - 485) q^{81} + ( - 12 \beta_{2} + 8 \beta_1 + 372) q^{82} + (12 \beta_{2} + 162 \beta_1 - 150) q^{83} + (64 \beta_{2} + 16 \beta_1 + 112) q^{84} + (38 \beta_{2} + 184 \beta_1 + 840) q^{85} + (4 \beta_{2} - 30 \beta_1 + 22) q^{86} + ( - 16 \beta_{2} + 8 \beta_1 + 24) q^{88} + ( - 154 \beta_{2} - 68 \beta_1 - 282) q^{89} + ( - 8 \beta_{2} - 52 \beta_1 - 264) q^{90} + (244 \beta_{2} + 208 \beta_1 - 1064) q^{91} + (48 \beta_{2} - 24 \beta_1 - 264) q^{92} + ( - 111 \beta_{2} + 94 \beta_1 - 52) q^{93} + (4 \beta_{2} + 66 \beta_1 + 414) q^{94} + (86 \beta_{2} + 244 \beta_1 + 552) q^{95} + ( - 32 \beta_1 + 32) q^{96} + (34 \beta_{2} - 176 \beta_1 - 14) q^{97} + (160 \beta_{2} + 128 \beta_1 - 402) q^{98} + ( - 22 \beta_{2} - 38 \beta_1 + 114) q^{99}+O(q^{100})$$ q - 2 * q^2 + (b1 - 1) * q^3 + 4 * q^4 + (b2 + 2*b1 + 6) * q^5 + (-2*b1 + 2) * q^6 + (-4*b2 + 8) * q^7 - 8 * q^8 + (3*b2 + 2*b1 + 1) * q^9 + (-2*b2 - 4*b1 - 12) * q^10 + (2*b2 - b1 - 3) * q^11 + (4*b1 - 4) * q^12 + (11*b2 + 8*b1 - 4) * q^13 + (8*b2 - 16) * q^14 + (2*b2 + 13*b1 + 39) * q^15 + 16 * q^16 + (16*b2 + 12*b1 + 18) * q^17 + (-6*b2 - 4*b1 - 2) * q^18 + (-10*b2 + 8*b1 + 52) * q^19 + (4*b2 + 8*b1 + 24) * q^20 + (16*b2 + 4*b1 + 28) * q^21 + (-4*b2 + 2*b1 + 6) * q^22 + (12*b2 - 6*b1 - 66) * q^23 + (-8*b1 + 8) * q^24 + (11*b2 + 40*b1 + 13) * q^25 + (-22*b2 - 16*b1 + 8) * q^26 + (-6*b2 - 17*b1 + 53) * q^27 + (-16*b2 + 32) * q^28 + (-4*b2 - 26*b1 - 78) * q^30 + (36*b2 + 11*b1 + 25) * q^31 - 32 * q^32 + (-11*b2 - 4*b1 - 42) * q^33 + (-32*b2 - 24*b1 - 36) * q^34 + (12*b2 + 24*b1) * q^35 + (12*b2 + 8*b1 + 4) * q^36 + (-38*b2 + 12*b1 + 10) * q^37 + (20*b2 - 16*b1 - 104) * q^38 + (-20*b2 + 31*b1 + 121) * q^39 + (-8*b2 - 16*b1 - 48) * q^40 + (6*b2 - 4*b1 - 186) * q^41 + (-32*b2 - 8*b1 - 56) * q^42 + (-2*b2 + 15*b1 - 11) * q^43 + (8*b2 - 4*b1 - 12) * q^44 + (4*b2 + 26*b1 + 132) * q^45 + (-24*b2 + 12*b1 + 132) * q^46 + (-2*b2 - 33*b1 - 207) * q^47 + (16*b1 - 16) * q^48 + (-80*b2 - 64*b1 + 201) * q^49 + (-22*b2 - 80*b1 - 26) * q^50 + (-28*b2 + 70*b1 + 162) * q^51 + (44*b2 + 32*b1 - 16) * q^52 + (-27*b2 - 116*b1 + 276) * q^53 + (12*b2 + 34*b1 - 106) * q^54 + (-8*b2 - 25*b1 - 39) * q^55 + (32*b2 - 64) * q^56 + (64*b2 + 66*b1 + 254) * q^57 + (4*b2 - 86*b1 + 90) * q^59 + (8*b2 + 52*b1 + 156) * q^60 + (-40*b2 + 80*b1 - 134) * q^61 + (-72*b2 - 22*b1 - 50) * q^62 + (56*b2 + 56*b1 - 280) * q^63 + 64 * q^64 + (9*b2 + 90*b1 + 468) * q^65 + (22*b2 + 8*b1 + 84) * q^66 + (-72*b2 + 36*b1 - 88) * q^67 + (64*b2 + 48*b1 + 72) * q^68 + (-66*b2 - 72*b1 - 204) * q^69 + (-24*b2 - 48*b1) * q^70 + (-4*b2 + 2*b1 - 18) * q^71 + (-24*b2 - 16*b1 - 8) * q^72 + (-30*b2 - 40*b1 + 178) * q^73 + (76*b2 - 24*b1 - 20) * q^74 + (76*b2 + 144*b1 + 968) * q^75 + (-40*b2 + 32*b1 + 208) * q^76 + (24*b2 + 28*b1 - 300) * q^77 + (40*b2 - 62*b1 - 242) * q^78 + (38*b2 + 37*b1 + 691) * q^79 + (16*b2 + 32*b1 + 96) * q^80 + (-108*b2 - 58*b1 - 485) * q^81 + (-12*b2 + 8*b1 + 372) * q^82 + (12*b2 + 162*b1 - 150) * q^83 + (64*b2 + 16*b1 + 112) * q^84 + (38*b2 + 184*b1 + 840) * q^85 + (4*b2 - 30*b1 + 22) * q^86 + (-16*b2 + 8*b1 + 24) * q^88 + (-154*b2 - 68*b1 - 282) * q^89 + (-8*b2 - 52*b1 - 264) * q^90 + (244*b2 + 208*b1 - 1064) * q^91 + (48*b2 - 24*b1 - 264) * q^92 + (-111*b2 + 94*b1 - 52) * q^93 + (4*b2 + 66*b1 + 414) * q^94 + (86*b2 + 244*b1 + 552) * q^95 + (-32*b1 + 32) * q^96 + (34*b2 - 176*b1 - 14) * q^97 + (160*b2 + 128*b1 - 402) * q^98 + (-22*b2 - 38*b1 + 114) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9}+O(q^{10})$$ 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9 $$3 q - 6 q^{2} - 2 q^{3} + 12 q^{4} + 20 q^{5} + 4 q^{6} + 24 q^{7} - 24 q^{8} + 5 q^{9} - 40 q^{10} - 10 q^{11} - 8 q^{12} - 4 q^{13} - 48 q^{14} + 130 q^{15} + 48 q^{16} + 66 q^{17} - 10 q^{18} + 164 q^{19} + 80 q^{20} + 88 q^{21} + 20 q^{22} - 204 q^{23} + 16 q^{24} + 79 q^{25} + 8 q^{26} + 142 q^{27} + 96 q^{28} - 260 q^{30} + 86 q^{31} - 96 q^{32} - 130 q^{33} - 132 q^{34} + 24 q^{35} + 20 q^{36} + 42 q^{37} - 328 q^{38} + 394 q^{39} - 160 q^{40} - 562 q^{41} - 176 q^{42} - 18 q^{43} - 40 q^{44} + 422 q^{45} + 408 q^{46} - 654 q^{47} - 32 q^{48} + 539 q^{49} - 158 q^{50} + 556 q^{51} - 16 q^{52} + 712 q^{53} - 284 q^{54} - 142 q^{55} - 192 q^{56} + 828 q^{57} + 184 q^{59} + 520 q^{60} - 322 q^{61} - 172 q^{62} - 784 q^{63} + 192 q^{64} + 1494 q^{65} + 260 q^{66} - 228 q^{67} + 264 q^{68} - 684 q^{69} - 48 q^{70} - 52 q^{71} - 40 q^{72} + 494 q^{73} - 84 q^{74} + 3048 q^{75} + 656 q^{76} - 872 q^{77} - 788 q^{78} + 2110 q^{79} + 320 q^{80} - 1513 q^{81} + 1124 q^{82} - 288 q^{83} + 352 q^{84} + 2704 q^{85} + 36 q^{86} + 80 q^{88} - 914 q^{89} - 844 q^{90} - 2984 q^{91} - 816 q^{92} - 62 q^{93} + 1308 q^{94} + 1900 q^{95} + 64 q^{96} - 218 q^{97} - 1078 q^{98} + 304 q^{99}+O(q^{100})$$ 3 * q - 6 * q^2 - 2 * q^3 + 12 * q^4 + 20 * q^5 + 4 * q^6 + 24 * q^7 - 24 * q^8 + 5 * q^9 - 40 * q^10 - 10 * q^11 - 8 * q^12 - 4 * q^13 - 48 * q^14 + 130 * q^15 + 48 * q^16 + 66 * q^17 - 10 * q^18 + 164 * q^19 + 80 * q^20 + 88 * q^21 + 20 * q^22 - 204 * q^23 + 16 * q^24 + 79 * q^25 + 8 * q^26 + 142 * q^27 + 96 * q^28 - 260 * q^30 + 86 * q^31 - 96 * q^32 - 130 * q^33 - 132 * q^34 + 24 * q^35 + 20 * q^36 + 42 * q^37 - 328 * q^38 + 394 * q^39 - 160 * q^40 - 562 * q^41 - 176 * q^42 - 18 * q^43 - 40 * q^44 + 422 * q^45 + 408 * q^46 - 654 * q^47 - 32 * q^48 + 539 * q^49 - 158 * q^50 + 556 * q^51 - 16 * q^52 + 712 * q^53 - 284 * q^54 - 142 * q^55 - 192 * q^56 + 828 * q^57 + 184 * q^59 + 520 * q^60 - 322 * q^61 - 172 * q^62 - 784 * q^63 + 192 * q^64 + 1494 * q^65 + 260 * q^66 - 228 * q^67 + 264 * q^68 - 684 * q^69 - 48 * q^70 - 52 * q^71 - 40 * q^72 + 494 * q^73 - 84 * q^74 + 3048 * q^75 + 656 * q^76 - 872 * q^77 - 788 * q^78 + 2110 * q^79 + 320 * q^80 - 1513 * q^81 + 1124 * q^82 - 288 * q^83 + 352 * q^84 + 2704 * q^85 + 36 * q^86 + 80 * q^88 - 914 * q^89 - 844 * q^90 - 2984 * q^91 - 816 * q^92 - 62 * q^93 + 1308 * q^94 + 1900 * q^95 + 64 * q^96 - 218 * q^97 - 1078 * q^98 + 304 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 42x - 54$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} - 4\nu - 27 ) / 3$$ (v^2 - 4*v - 27) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2} + 4\beta _1 + 27$$ 3*b2 + 4*b1 + 27

## 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
 −5.13291 −1.39712 7.53003
−2.00000 −6.13291 4.00000 2.36031 12.2658 −18.5045 −8.00000 10.6126 −4.72062
1.2 −2.00000 −2.39712 4.00000 −3.28077 4.79424 33.9461 −8.00000 −21.2538 6.56153
1.3 −2.00000 6.53003 4.00000 20.9205 −13.0601 8.55839 −8.00000 15.6413 −41.8409
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$29$$ $$1$$

## 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 1682.4.a.d 3
29.b even 2 1 58.4.a.d 3
87.d odd 2 1 522.4.a.k 3
116.d odd 2 1 464.4.a.i 3
145.d even 2 1 1450.4.a.h 3
232.b odd 2 1 1856.4.a.s 3
232.g even 2 1 1856.4.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 29.b even 2 1
464.4.a.i 3 116.d odd 2 1
522.4.a.k 3 87.d odd 2 1
1450.4.a.h 3 145.d even 2 1
1682.4.a.d 3 1.a even 1 1 trivial
1856.4.a.r 3 232.g even 2 1
1856.4.a.s 3 232.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 2T_{3}^{2} - 41T_{3} - 96$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1682))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{3}$$
$3$ $$T^{3} + 2 T^{2} - 41 T - 96$$
$5$ $$T^{3} - 20 T^{2} - 27 T + 162$$
$7$ $$T^{3} - 24 T^{2} - 496 T + 5376$$
$11$ $$T^{3} + 10 T^{2} - 233 T - 2424$$
$13$ $$T^{3} + 4 T^{2} - 5619 T + 131706$$
$17$ $$T^{3} - 66 T^{2} - 10660 T + 679368$$
$19$ $$T^{3} - 164 T^{2} - 124 T + 664448$$
$23$ $$T^{3} + 204 T^{2} + 4284 T - 677376$$
$29$ $$T^{3}$$
$31$ $$T^{3} - 86 T^{2} - 48089 T + 4766172$$
$37$ $$T^{3} - 42 T^{2} - 79456 T + 7684896$$
$41$ $$T^{3} + 562 T^{2} + 102432 T + 5982048$$
$43$ $$T^{3} + 18 T^{2} - 10369 T - 196488$$
$47$ $$T^{3} + 654 T^{2} + 98015 T + 3425124$$
$53$ $$T^{3} - 712 T^{2} + \cdots + 252120546$$
$59$ $$T^{3} - 184 T^{2} + \cdots + 57362928$$
$61$ $$T^{3} + 322 T^{2} - 388372 T + 5254424$$
$67$ $$T^{3} + 228 T^{2} + \cdots + 47608192$$
$71$ $$T^{3} + 52 T^{2} - 164 T - 672$$
$73$ $$T^{3} - 494 T^{2} + 6112 T + 9410208$$
$79$ $$T^{3} - 2110 T^{2} + \cdots - 285187172$$
$83$ $$T^{3} + 288 T^{2} + \cdots - 437606064$$
$89$ $$T^{3} + 914 T^{2} + \cdots - 598011552$$
$97$ $$T^{3} + 218 T^{2} + \cdots + 17006112$$