# Properties

 Label 1694.4.a.b Level $1694$ Weight $4$ Character orbit 1694.a Self dual yes Analytic conductor $99.949$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1694 = 2 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1694.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$99.9492355497$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 2q^{3} + 4q^{4} - 12q^{5} + 4q^{6} - 7q^{7} - 8q^{8} - 23q^{9} + O(q^{10})$$ $$q - 2q^{2} - 2q^{3} + 4q^{4} - 12q^{5} + 4q^{6} - 7q^{7} - 8q^{8} - 23q^{9} + 24q^{10} - 8q^{12} - 56q^{13} + 14q^{14} + 24q^{15} + 16q^{16} + 114q^{17} + 46q^{18} - 2q^{19} - 48q^{20} + 14q^{21} - 120q^{23} + 16q^{24} + 19q^{25} + 112q^{26} + 100q^{27} - 28q^{28} + 54q^{29} - 48q^{30} + 236q^{31} - 32q^{32} - 228q^{34} + 84q^{35} - 92q^{36} + 146q^{37} + 4q^{38} + 112q^{39} + 96q^{40} - 126q^{41} - 28q^{42} + 376q^{43} + 276q^{45} + 240q^{46} - 12q^{47} - 32q^{48} + 49q^{49} - 38q^{50} - 228q^{51} - 224q^{52} + 174q^{53} - 200q^{54} + 56q^{56} + 4q^{57} - 108q^{58} + 138q^{59} + 96q^{60} - 380q^{61} - 472q^{62} + 161q^{63} + 64q^{64} + 672q^{65} - 484q^{67} + 456q^{68} + 240q^{69} - 168q^{70} + 576q^{71} + 184q^{72} + 1150q^{73} - 292q^{74} - 38q^{75} - 8q^{76} - 224q^{78} - 776q^{79} - 192q^{80} + 421q^{81} + 252q^{82} - 378q^{83} + 56q^{84} - 1368q^{85} - 752q^{86} - 108q^{87} - 390q^{89} - 552q^{90} + 392q^{91} - 480q^{92} - 472q^{93} + 24q^{94} + 24q^{95} + 64q^{96} - 1330q^{97} - 98q^{98} + O(q^{100})$$

## 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
−2.00000 −2.00000 4.00000 −12.0000 4.00000 −7.00000 −8.00000 −23.0000 24.0000
 $$n$$: 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 1694.4.a.b 1
11.b odd 2 1 14.4.a.b 1
33.d even 2 1 126.4.a.d 1
44.c even 2 1 112.4.a.e 1
55.d odd 2 1 350.4.a.f 1
55.e even 4 2 350.4.c.g 2
77.b even 2 1 98.4.a.e 1
77.h odd 6 2 98.4.c.c 2
77.i even 6 2 98.4.c.b 2
88.b odd 2 1 448.4.a.k 1
88.g even 2 1 448.4.a.g 1
132.d odd 2 1 1008.4.a.r 1
143.d odd 2 1 2366.4.a.c 1
231.h odd 2 1 882.4.a.b 1
231.k odd 6 2 882.4.g.v 2
231.l even 6 2 882.4.g.p 2
308.g odd 2 1 784.4.a.h 1
385.h even 2 1 2450.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 11.b odd 2 1
98.4.a.e 1 77.b even 2 1
98.4.c.b 2 77.i even 6 2
98.4.c.c 2 77.h odd 6 2
112.4.a.e 1 44.c even 2 1
126.4.a.d 1 33.d even 2 1
350.4.a.f 1 55.d odd 2 1
350.4.c.g 2 55.e even 4 2
448.4.a.g 1 88.g even 2 1
448.4.a.k 1 88.b odd 2 1
784.4.a.h 1 308.g odd 2 1
882.4.a.b 1 231.h odd 2 1
882.4.g.p 2 231.l even 6 2
882.4.g.v 2 231.k odd 6 2
1008.4.a.r 1 132.d odd 2 1
1694.4.a.b 1 1.a even 1 1 trivial
2366.4.a.c 1 143.d odd 2 1
2450.4.a.i 1 385.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1694))$$:

 $$T_{3} + 2$$ $$T_{5} + 12$$ $$T_{13} + 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$2 + T$$
$5$ $$12 + T$$
$7$ $$7 + T$$
$11$ $$T$$
$13$ $$56 + T$$
$17$ $$-114 + T$$
$19$ $$2 + T$$
$23$ $$120 + T$$
$29$ $$-54 + T$$
$31$ $$-236 + T$$
$37$ $$-146 + T$$
$41$ $$126 + T$$
$43$ $$-376 + T$$
$47$ $$12 + T$$
$53$ $$-174 + T$$
$59$ $$-138 + T$$
$61$ $$380 + T$$
$67$ $$484 + T$$
$71$ $$-576 + T$$
$73$ $$-1150 + T$$
$79$ $$776 + T$$
$83$ $$378 + T$$
$89$ $$390 + T$$
$97$ $$1330 + T$$