# Properties

 Label 1682.2.b.a Level $1682$ Weight $2$ Character orbit 1682.b Analytic conductor $13.431$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1682 = 2 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1682.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4308376200$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + i q^{3} - q^{4} - q^{5} + q^{6} - 2 q^{7} + i q^{8} + 2 q^{9} +O(q^{10})$$ q - i * q^2 + i * q^3 - q^4 - q^5 + q^6 - 2 * q^7 + i * q^8 + 2 * q^9 $$q - i q^{2} + i q^{3} - q^{4} - q^{5} + q^{6} - 2 q^{7} + i q^{8} + 2 q^{9} + i q^{10} + 3 i q^{11} - i q^{12} + q^{13} + 2 i q^{14} - i q^{15} + q^{16} - 8 i q^{17} - 2 i q^{18} + q^{20} - 2 i q^{21} + 3 q^{22} + 4 q^{23} - q^{24} - 4 q^{25} - i q^{26} + 5 i q^{27} + 2 q^{28} - q^{30} + 3 i q^{31} - i q^{32} - 3 q^{33} - 8 q^{34} + 2 q^{35} - 2 q^{36} + 8 i q^{37} + i q^{39} - i q^{40} + 2 i q^{41} - 2 q^{42} + 11 i q^{43} - 3 i q^{44} - 2 q^{45} - 4 i q^{46} + 13 i q^{47} + i q^{48} - 3 q^{49} + 4 i q^{50} + 8 q^{51} - q^{52} - 11 q^{53} + 5 q^{54} - 3 i q^{55} - 2 i q^{56} + i q^{60} + 8 i q^{61} + 3 q^{62} - 4 q^{63} - q^{64} - q^{65} + 3 i q^{66} + 12 q^{67} + 8 i q^{68} + 4 i q^{69} - 2 i q^{70} - 2 q^{71} + 2 i q^{72} + 4 i q^{73} + 8 q^{74} - 4 i q^{75} - 6 i q^{77} + q^{78} - 15 i q^{79} - q^{80} + q^{81} + 2 q^{82} + 4 q^{83} + 2 i q^{84} + 8 i q^{85} + 11 q^{86} - 3 q^{88} + 10 i q^{89} + 2 i q^{90} - 2 q^{91} - 4 q^{92} - 3 q^{93} + 13 q^{94} + q^{96} - 2 i q^{97} + 3 i q^{98} + 6 i q^{99} +O(q^{100})$$ q - i * q^2 + i * q^3 - q^4 - q^5 + q^6 - 2 * q^7 + i * q^8 + 2 * q^9 + i * q^10 + 3*i * q^11 - i * q^12 + q^13 + 2*i * q^14 - i * q^15 + q^16 - 8*i * q^17 - 2*i * q^18 + q^20 - 2*i * q^21 + 3 * q^22 + 4 * q^23 - q^24 - 4 * q^25 - i * q^26 + 5*i * q^27 + 2 * q^28 - q^30 + 3*i * q^31 - i * q^32 - 3 * q^33 - 8 * q^34 + 2 * q^35 - 2 * q^36 + 8*i * q^37 + i * q^39 - i * q^40 + 2*i * q^41 - 2 * q^42 + 11*i * q^43 - 3*i * q^44 - 2 * q^45 - 4*i * q^46 + 13*i * q^47 + i * q^48 - 3 * q^49 + 4*i * q^50 + 8 * q^51 - q^52 - 11 * q^53 + 5 * q^54 - 3*i * q^55 - 2*i * q^56 + i * q^60 + 8*i * q^61 + 3 * q^62 - 4 * q^63 - q^64 - q^65 + 3*i * q^66 + 12 * q^67 + 8*i * q^68 + 4*i * q^69 - 2*i * q^70 - 2 * q^71 + 2*i * q^72 + 4*i * q^73 + 8 * q^74 - 4*i * q^75 - 6*i * q^77 + q^78 - 15*i * q^79 - q^80 + q^81 + 2 * q^82 + 4 * q^83 + 2*i * q^84 + 8*i * q^85 + 11 * q^86 - 3 * q^88 + 10*i * q^89 + 2*i * q^90 - 2 * q^91 - 4 * q^92 - 3 * q^93 + 13 * q^94 + q^96 - 2*i * q^97 + 3*i * q^98 + 6*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 $$2 q - 2 q^{4} - 2 q^{5} + 2 q^{6} - 4 q^{7} + 4 q^{9} + 2 q^{13} + 2 q^{16} + 2 q^{20} + 6 q^{22} + 8 q^{23} - 2 q^{24} - 8 q^{25} + 4 q^{28} - 2 q^{30} - 6 q^{33} - 16 q^{34} + 4 q^{35} - 4 q^{36} - 4 q^{42} - 4 q^{45} - 6 q^{49} + 16 q^{51} - 2 q^{52} - 22 q^{53} + 10 q^{54} + 6 q^{62} - 8 q^{63} - 2 q^{64} - 2 q^{65} + 24 q^{67} - 4 q^{71} + 16 q^{74} + 2 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} + 22 q^{86} - 6 q^{88} - 4 q^{91} - 8 q^{92} - 6 q^{93} + 26 q^{94} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 + 2 * q^13 + 2 * q^16 + 2 * q^20 + 6 * q^22 + 8 * q^23 - 2 * q^24 - 8 * q^25 + 4 * q^28 - 2 * q^30 - 6 * q^33 - 16 * q^34 + 4 * q^35 - 4 * q^36 - 4 * q^42 - 4 * q^45 - 6 * q^49 + 16 * q^51 - 2 * q^52 - 22 * q^53 + 10 * q^54 + 6 * q^62 - 8 * q^63 - 2 * q^64 - 2 * q^65 + 24 * q^67 - 4 * q^71 + 16 * q^74 + 2 * q^78 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 + 22 * q^86 - 6 * q^88 - 4 * q^91 - 8 * q^92 - 6 * q^93 + 26 * q^94 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times$$.

 $$n$$ $$843$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1681.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 −1.00000 1.00000 −2.00000 1.00000i 2.00000 1.00000i
1681.2 1.00000i 1.00000i −1.00000 −1.00000 1.00000 −2.00000 1.00000i 2.00000 1.00000i
 $$n$$: 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
29.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{5} + 1$$ T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 9$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} + 64$$
$19$ $$T^{2}$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 9$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 4$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 169$$
$53$ $$(T + 11)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 64$$
$67$ $$(T - 12)^{2}$$
$71$ $$(T + 2)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2} + 225$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 100$$
$97$ $$T^{2} + 4$$