# Properties

 Label 1682.4.a.c Level $1682$ Weight $4$ Character orbit 1682.a Self dual yes Analytic conductor $99.241$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (6 \beta - 5) q^{5} + (2 \beta + 2) q^{6} + ( - 8 \beta - 8) q^{7} + 8 q^{8} + (2 \beta - 20) q^{9}+O(q^{10})$$ q + 2 * q^2 + (b + 1) * q^3 + 4 * q^4 + (6*b - 5) * q^5 + (2*b + 2) * q^6 + (-8*b - 8) * q^7 + 8 * q^8 + (2*b - 20) * q^9 $$q + 2 q^{2} + (\beta + 1) q^{3} + 4 q^{4} + (6 \beta - 5) q^{5} + (2 \beta + 2) q^{6} + ( - 8 \beta - 8) q^{7} + 8 q^{8} + (2 \beta - 20) q^{9} + (12 \beta - 10) q^{10} + (3 \beta + 45) q^{11} + (4 \beta + 4) q^{12} + (8 \beta - 25) q^{13} + ( - 16 \beta - 16) q^{14} + (\beta + 31) q^{15} + 16 q^{16} + ( - 16 \beta + 22) q^{17} + (4 \beta - 40) q^{18} + (4 \beta - 54) q^{19} + (24 \beta - 20) q^{20} + ( - 16 \beta - 56) q^{21} + (6 \beta + 90) q^{22} + ( - 78 \beta - 14) q^{23} + (8 \beta + 8) q^{24} + ( - 60 \beta + 116) q^{25} + (16 \beta - 50) q^{26} + ( - 45 \beta - 35) q^{27} + ( - 32 \beta - 32) q^{28} + (2 \beta + 62) q^{30} + ( - 109 \beta - 33) q^{31} + 32 q^{32} + (48 \beta + 63) q^{33} + ( - 32 \beta + 44) q^{34} + ( - 8 \beta - 248) q^{35} + (8 \beta - 80) q^{36} + ( - 4 \beta - 20) q^{37} + (8 \beta - 108) q^{38} + ( - 17 \beta + 23) q^{39} + (48 \beta - 40) q^{40} + (80 \beta - 152) q^{41} + ( - 32 \beta - 112) q^{42} + ( - 53 \beta + 65) q^{43} + (12 \beta + 180) q^{44} + ( - 130 \beta + 172) q^{45} + ( - 156 \beta - 28) q^{46} + (99 \beta + 257) q^{47} + (16 \beta + 16) q^{48} + (128 \beta + 105) q^{49} + ( - 120 \beta + 232) q^{50} + (6 \beta - 74) q^{51} + (32 \beta - 100) q^{52} + (52 \beta - 479) q^{53} + ( - 90 \beta - 70) q^{54} + (255 \beta - 117) q^{55} + ( - 64 \beta - 64) q^{56} + ( - 50 \beta - 30) q^{57} + (250 \beta - 90) q^{59} + (4 \beta + 124) q^{60} + (12 \beta - 514) q^{61} + ( - 218 \beta - 66) q^{62} + (144 \beta + 64) q^{63} + 64 q^{64} + ( - 190 \beta + 413) q^{65} + (96 \beta + 126) q^{66} + ( - 20 \beta - 456) q^{67} + ( - 64 \beta + 88) q^{68} + ( - 92 \beta - 482) q^{69} + ( - 16 \beta - 496) q^{70} + (34 \beta + 398) q^{71} + (16 \beta - 160) q^{72} + ( - 176 \beta + 428) q^{73} + ( - 8 \beta - 40) q^{74} + (56 \beta - 244) q^{75} + (16 \beta - 216) q^{76} + ( - 384 \beta - 504) q^{77} + ( - 34 \beta + 46) q^{78} + (361 \beta + 159) q^{79} + (96 \beta - 80) q^{80} + ( - 134 \beta + 235) q^{81} + (160 \beta - 304) q^{82} + ( - 38 \beta - 914) q^{83} + ( - 64 \beta - 224) q^{84} + (212 \beta - 686) q^{85} + ( - 106 \beta + 130) q^{86} + (24 \beta + 360) q^{88} + ( - 72 \beta + 472) q^{89} + ( - 260 \beta + 344) q^{90} + (136 \beta - 184) q^{91} + ( - 312 \beta - 56) q^{92} + ( - 142 \beta - 687) q^{93} + (198 \beta + 514) q^{94} + ( - 344 \beta + 414) q^{95} + (32 \beta + 32) q^{96} + ( - 100 \beta - 184) q^{97} + (256 \beta + 210) q^{98} + (30 \beta - 864) q^{99}+O(q^{100})$$ q + 2 * q^2 + (b + 1) * q^3 + 4 * q^4 + (6*b - 5) * q^5 + (2*b + 2) * q^6 + (-8*b - 8) * q^7 + 8 * q^8 + (2*b - 20) * q^9 + (12*b - 10) * q^10 + (3*b + 45) * q^11 + (4*b + 4) * q^12 + (8*b - 25) * q^13 + (-16*b - 16) * q^14 + (b + 31) * q^15 + 16 * q^16 + (-16*b + 22) * q^17 + (4*b - 40) * q^18 + (4*b - 54) * q^19 + (24*b - 20) * q^20 + (-16*b - 56) * q^21 + (6*b + 90) * q^22 + (-78*b - 14) * q^23 + (8*b + 8) * q^24 + (-60*b + 116) * q^25 + (16*b - 50) * q^26 + (-45*b - 35) * q^27 + (-32*b - 32) * q^28 + (2*b + 62) * q^30 + (-109*b - 33) * q^31 + 32 * q^32 + (48*b + 63) * q^33 + (-32*b + 44) * q^34 + (-8*b - 248) * q^35 + (8*b - 80) * q^36 + (-4*b - 20) * q^37 + (8*b - 108) * q^38 + (-17*b + 23) * q^39 + (48*b - 40) * q^40 + (80*b - 152) * q^41 + (-32*b - 112) * q^42 + (-53*b + 65) * q^43 + (12*b + 180) * q^44 + (-130*b + 172) * q^45 + (-156*b - 28) * q^46 + (99*b + 257) * q^47 + (16*b + 16) * q^48 + (128*b + 105) * q^49 + (-120*b + 232) * q^50 + (6*b - 74) * q^51 + (32*b - 100) * q^52 + (52*b - 479) * q^53 + (-90*b - 70) * q^54 + (255*b - 117) * q^55 + (-64*b - 64) * q^56 + (-50*b - 30) * q^57 + (250*b - 90) * q^59 + (4*b + 124) * q^60 + (12*b - 514) * q^61 + (-218*b - 66) * q^62 + (144*b + 64) * q^63 + 64 * q^64 + (-190*b + 413) * q^65 + (96*b + 126) * q^66 + (-20*b - 456) * q^67 + (-64*b + 88) * q^68 + (-92*b - 482) * q^69 + (-16*b - 496) * q^70 + (34*b + 398) * q^71 + (16*b - 160) * q^72 + (-176*b + 428) * q^73 + (-8*b - 40) * q^74 + (56*b - 244) * q^75 + (16*b - 216) * q^76 + (-384*b - 504) * q^77 + (-34*b + 46) * q^78 + (361*b + 159) * q^79 + (96*b - 80) * q^80 + (-134*b + 235) * q^81 + (160*b - 304) * q^82 + (-38*b - 914) * q^83 + (-64*b - 224) * q^84 + (212*b - 686) * q^85 + (-106*b + 130) * q^86 + (24*b + 360) * q^88 + (-72*b + 472) * q^89 + (-260*b + 344) * q^90 + (136*b - 184) * q^91 + (-312*b - 56) * q^92 + (-142*b - 687) * q^93 + (198*b + 514) * q^94 + (-344*b + 414) * q^95 + (32*b + 32) * q^96 + (-100*b - 184) * q^97 + (256*b + 210) * q^98 + (30*b - 864) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 + 16 * q^8 - 40 * q^9 $$2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9} - 20 q^{10} + 90 q^{11} + 8 q^{12} - 50 q^{13} - 32 q^{14} + 62 q^{15} + 32 q^{16} + 44 q^{17} - 80 q^{18} - 108 q^{19} - 40 q^{20} - 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} - 100 q^{26} - 70 q^{27} - 64 q^{28} + 124 q^{30} - 66 q^{31} + 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} - 40 q^{37} - 216 q^{38} + 46 q^{39} - 80 q^{40} - 304 q^{41} - 224 q^{42} + 130 q^{43} + 360 q^{44} + 344 q^{45} - 56 q^{46} + 514 q^{47} + 32 q^{48} + 210 q^{49} + 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} - 234 q^{55} - 128 q^{56} - 60 q^{57} - 180 q^{59} + 248 q^{60} - 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} + 252 q^{66} - 912 q^{67} + 176 q^{68} - 964 q^{69} - 992 q^{70} + 796 q^{71} - 320 q^{72} + 856 q^{73} - 80 q^{74} - 488 q^{75} - 432 q^{76} - 1008 q^{77} + 92 q^{78} + 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} - 448 q^{84} - 1372 q^{85} + 260 q^{86} + 720 q^{88} + 944 q^{89} + 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} + 828 q^{95} + 64 q^{96} - 368 q^{97} + 420 q^{98} - 1728 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 + 16 * q^8 - 40 * q^9 - 20 * q^10 + 90 * q^11 + 8 * q^12 - 50 * q^13 - 32 * q^14 + 62 * q^15 + 32 * q^16 + 44 * q^17 - 80 * q^18 - 108 * q^19 - 40 * q^20 - 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 - 100 * q^26 - 70 * q^27 - 64 * q^28 + 124 * q^30 - 66 * q^31 + 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 - 40 * q^37 - 216 * q^38 + 46 * q^39 - 80 * q^40 - 304 * q^41 - 224 * q^42 + 130 * q^43 + 360 * q^44 + 344 * q^45 - 56 * q^46 + 514 * q^47 + 32 * q^48 + 210 * q^49 + 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 - 234 * q^55 - 128 * q^56 - 60 * q^57 - 180 * q^59 + 248 * q^60 - 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 + 252 * q^66 - 912 * q^67 + 176 * q^68 - 964 * q^69 - 992 * q^70 + 796 * q^71 - 320 * q^72 + 856 * q^73 - 80 * q^74 - 488 * q^75 - 432 * q^76 - 1008 * q^77 + 92 * q^78 + 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 - 448 * q^84 - 1372 * q^85 + 260 * q^86 + 720 * q^88 + 944 * q^89 + 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 + 828 * q^95 + 64 * q^96 - 368 * q^97 + 420 * q^98 - 1728 * q^99

## 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
 −2.44949 2.44949
2.00000 −1.44949 4.00000 −19.6969 −2.89898 11.5959 8.00000 −24.8990 −39.3939
1.2 2.00000 3.44949 4.00000 9.69694 6.89898 −27.5959 8.00000 −15.1010 19.3939
 $$n$$: 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.c 2
29.b even 2 1 58.4.a.c 2
87.d odd 2 1 522.4.a.j 2
116.d odd 2 1 464.4.a.e 2
145.d even 2 1 1450.4.a.g 2
232.b odd 2 1 1856.4.a.i 2
232.g even 2 1 1856.4.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.c 2 29.b even 2 1
464.4.a.e 2 116.d odd 2 1
522.4.a.j 2 87.d odd 2 1
1450.4.a.g 2 145.d even 2 1
1682.4.a.c 2 1.a even 1 1 trivial
1856.4.a.i 2 232.b odd 2 1
1856.4.a.l 2 232.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2T_{3} - 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1682))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} - 2T - 5$$
$5$ $$T^{2} + 10T - 191$$
$7$ $$T^{2} + 16T - 320$$
$11$ $$T^{2} - 90T + 1971$$
$13$ $$T^{2} + 50T + 241$$
$17$ $$T^{2} - 44T - 1052$$
$19$ $$T^{2} + 108T + 2820$$
$23$ $$T^{2} + 28T - 36308$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 66T - 70197$$
$37$ $$T^{2} + 40T + 304$$
$41$ $$T^{2} + 304T - 15296$$
$43$ $$T^{2} - 130T - 12629$$
$47$ $$T^{2} - 514T + 7243$$
$53$ $$T^{2} + 958T + 213217$$
$59$ $$T^{2} + 180T - 366900$$
$61$ $$T^{2} + 1028 T + 263332$$
$67$ $$T^{2} + 912T + 205536$$
$71$ $$T^{2} - 796T + 151468$$
$73$ $$T^{2} - 856T - 2672$$
$79$ $$T^{2} - 318T - 756645$$
$83$ $$T^{2} + 1828 T + 826732$$
$89$ $$T^{2} - 944T + 191680$$
$97$ $$T^{2} + 368T - 26144$$