Properties

 Label 1694.2.a.c Level $1694$ Weight $2$ Character orbit 1694.a Self dual yes Analytic conductor $13.527$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1694 = 2 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1694.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$13.5266581024$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 + q^4 + 2 * q^5 + q^7 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} + 2 q^{5} + q^{7} - q^{8} - 3 q^{9} - 2 q^{10} - 2 q^{13} - q^{14} + q^{16} - 2 q^{17} + 3 q^{18} + 2 q^{20} - 8 q^{23} - q^{25} + 2 q^{26} + q^{28} + 2 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 2 q^{35} - 3 q^{36} - 2 q^{37} - 2 q^{40} - 10 q^{41} - 4 q^{43} - 6 q^{45} + 8 q^{46} + 8 q^{47} + q^{49} + q^{50} - 2 q^{52} + 6 q^{53} - q^{56} - 2 q^{58} - 10 q^{61} + 8 q^{62} - 3 q^{63} + q^{64} - 4 q^{65} - 12 q^{67} - 2 q^{68} - 2 q^{70} + 16 q^{71} + 3 q^{72} + 14 q^{73} + 2 q^{74} + 2 q^{80} + 9 q^{81} + 10 q^{82} - 4 q^{85} + 4 q^{86} - 6 q^{89} + 6 q^{90} - 2 q^{91} - 8 q^{92} - 8 q^{94} + 10 q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^4 + 2 * q^5 + q^7 - q^8 - 3 * q^9 - 2 * q^10 - 2 * q^13 - q^14 + q^16 - 2 * q^17 + 3 * q^18 + 2 * q^20 - 8 * q^23 - q^25 + 2 * q^26 + q^28 + 2 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 2 * q^35 - 3 * q^36 - 2 * q^37 - 2 * q^40 - 10 * q^41 - 4 * q^43 - 6 * q^45 + 8 * q^46 + 8 * q^47 + q^49 + q^50 - 2 * q^52 + 6 * q^53 - q^56 - 2 * q^58 - 10 * q^61 + 8 * q^62 - 3 * q^63 + q^64 - 4 * q^65 - 12 * q^67 - 2 * q^68 - 2 * q^70 + 16 * q^71 + 3 * q^72 + 14 * q^73 + 2 * q^74 + 2 * q^80 + 9 * q^81 + 10 * q^82 - 4 * q^85 + 4 * q^86 - 6 * q^89 + 6 * q^90 - 2 * q^91 - 8 * q^92 - 8 * q^94 + 10 * q^97 - q^98

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 0 1.00000 2.00000 0 1.00000 −1.00000 −3.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

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 1694.2.a.c 1
11.b odd 2 1 154.2.a.c 1
33.d even 2 1 1386.2.a.b 1
44.c even 2 1 1232.2.a.h 1
55.d odd 2 1 3850.2.a.f 1
55.e even 4 2 3850.2.c.l 2
77.b even 2 1 1078.2.a.j 1
77.h odd 6 2 1078.2.e.b 2
77.i even 6 2 1078.2.e.c 2
88.b odd 2 1 4928.2.a.n 1
88.g even 2 1 4928.2.a.o 1
231.h odd 2 1 9702.2.a.v 1
308.g odd 2 1 8624.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.c 1 11.b odd 2 1
1078.2.a.j 1 77.b even 2 1
1078.2.e.b 2 77.h odd 6 2
1078.2.e.c 2 77.i even 6 2
1232.2.a.h 1 44.c even 2 1
1386.2.a.b 1 33.d even 2 1
1694.2.a.c 1 1.a even 1 1 trivial
3850.2.a.f 1 55.d odd 2 1
3850.2.c.l 2 55.e even 4 2
4928.2.a.n 1 88.b odd 2 1
4928.2.a.o 1 88.g even 2 1
8624.2.a.o 1 308.g odd 2 1
9702.2.a.v 1 231.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1694))$$:

 $$T_{3}$$ T3 $$T_{5} - 2$$ T5 - 2 $$T_{13} + 2$$ T13 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T + 8$$
$29$ $$T - 2$$
$31$ $$T + 8$$
$37$ $$T + 2$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T + 10$$
$67$ $$T + 12$$
$71$ $$T - 16$$
$73$ $$T - 14$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T - 10$$