Properties

 Label 1682.2.a.d Level $1682$ Weight $2$ Character orbit 1682.a Self dual yes Analytic conductor $13.431$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: yes Analytic conductor: $$13.4308376200$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} - q^{8} - 2 q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 - 2 * q^7 - q^8 - 2 * q^9 $$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} - q^{8} - 2 q^{9} - q^{10} + 3 q^{11} + q^{12} - q^{13} + 2 q^{14} + q^{15} + q^{16} - 8 q^{17} + 2 q^{18} + q^{20} - 2 q^{21} - 3 q^{22} + 4 q^{23} - q^{24} - 4 q^{25} + q^{26} - 5 q^{27} - 2 q^{28} - q^{30} + 3 q^{31} - q^{32} + 3 q^{33} + 8 q^{34} - 2 q^{35} - 2 q^{36} - 8 q^{37} - q^{39} - q^{40} - 2 q^{41} + 2 q^{42} + 11 q^{43} + 3 q^{44} - 2 q^{45} - 4 q^{46} - 13 q^{47} + q^{48} - 3 q^{49} + 4 q^{50} - 8 q^{51} - q^{52} - 11 q^{53} + 5 q^{54} + 3 q^{55} + 2 q^{56} + q^{60} + 8 q^{61} - 3 q^{62} + 4 q^{63} + q^{64} - q^{65} - 3 q^{66} - 12 q^{67} - 8 q^{68} + 4 q^{69} + 2 q^{70} + 2 q^{71} + 2 q^{72} - 4 q^{73} + 8 q^{74} - 4 q^{75} - 6 q^{77} + q^{78} - 15 q^{79} + q^{80} + q^{81} + 2 q^{82} + 4 q^{83} - 2 q^{84} - 8 q^{85} - 11 q^{86} - 3 q^{88} + 10 q^{89} + 2 q^{90} + 2 q^{91} + 4 q^{92} + 3 q^{93} + 13 q^{94} - q^{96} + 2 q^{97} + 3 q^{98} - 6 q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 + q^5 - q^6 - 2 * q^7 - q^8 - 2 * q^9 - q^10 + 3 * q^11 + q^12 - q^13 + 2 * q^14 + q^15 + q^16 - 8 * q^17 + 2 * q^18 + q^20 - 2 * q^21 - 3 * q^22 + 4 * q^23 - q^24 - 4 * q^25 + q^26 - 5 * q^27 - 2 * q^28 - q^30 + 3 * q^31 - q^32 + 3 * q^33 + 8 * q^34 - 2 * q^35 - 2 * q^36 - 8 * q^37 - q^39 - q^40 - 2 * q^41 + 2 * q^42 + 11 * q^43 + 3 * q^44 - 2 * q^45 - 4 * q^46 - 13 * q^47 + q^48 - 3 * q^49 + 4 * q^50 - 8 * q^51 - q^52 - 11 * q^53 + 5 * q^54 + 3 * q^55 + 2 * q^56 + q^60 + 8 * q^61 - 3 * q^62 + 4 * q^63 + q^64 - q^65 - 3 * q^66 - 12 * q^67 - 8 * q^68 + 4 * q^69 + 2 * q^70 + 2 * q^71 + 2 * q^72 - 4 * q^73 + 8 * q^74 - 4 * q^75 - 6 * q^77 + q^78 - 15 * q^79 + q^80 + q^81 + 2 * q^82 + 4 * q^83 - 2 * q^84 - 8 * q^85 - 11 * q^86 - 3 * q^88 + 10 * q^89 + 2 * q^90 + 2 * q^91 + 4 * q^92 + 3 * q^93 + 13 * q^94 - q^96 + 2 * q^97 + 3 * q^98 - 6 * 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
 0
−1.00000 1.00000 1.00000 1.00000 −1.00000 −2.00000 −1.00000 −2.00000 −1.00000
 $$n$$: 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 1682.2.a.d 1
29.b even 2 1 58.2.a.b 1
29.c odd 4 2 1682.2.b.a 2
87.d odd 2 1 522.2.a.b 1
116.d odd 2 1 464.2.a.e 1
145.d even 2 1 1450.2.a.c 1
145.h odd 4 2 1450.2.b.b 2
203.c odd 2 1 2842.2.a.e 1
232.b odd 2 1 1856.2.a.f 1
232.g even 2 1 1856.2.a.k 1
319.d odd 2 1 7018.2.a.a 1
348.b even 2 1 4176.2.a.n 1
377.d even 2 1 9802.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.2.a.b 1 29.b even 2 1
464.2.a.e 1 116.d odd 2 1
522.2.a.b 1 87.d odd 2 1
1450.2.a.c 1 145.d even 2 1
1450.2.b.b 2 145.h odd 4 2
1682.2.a.d 1 1.a even 1 1 trivial
1682.2.b.a 2 29.c odd 4 2
1856.2.a.f 1 232.b odd 2 1
1856.2.a.k 1 232.g even 2 1
2842.2.a.e 1 203.c odd 2 1
4176.2.a.n 1 348.b even 2 1
7018.2.a.a 1 319.d odd 2 1
9802.2.a.a 1 377.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1682))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} - 1$$ T5 - 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T - 3$$
$13$ $$T + 1$$
$17$ $$T + 8$$
$19$ $$T$$
$23$ $$T - 4$$
$29$ $$T$$
$31$ $$T - 3$$
$37$ $$T + 8$$
$41$ $$T + 2$$
$43$ $$T - 11$$
$47$ $$T + 13$$
$53$ $$T + 11$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T + 12$$
$71$ $$T - 2$$
$73$ $$T + 4$$
$79$ $$T + 15$$
$83$ $$T - 4$$
$89$ $$T - 10$$
$97$ $$T - 2$$