# Properties

 Label 1694.4.a.g 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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 8 q^{3} + 4 q^{4} - 14 q^{5} + 16 q^{6} + 7 q^{7} + 8 q^{8} + 37 q^{9}+O(q^{10})$$ q + 2 * q^2 + 8 * q^3 + 4 * q^4 - 14 * q^5 + 16 * q^6 + 7 * q^7 + 8 * q^8 + 37 * q^9 $$q + 2 q^{2} + 8 q^{3} + 4 q^{4} - 14 q^{5} + 16 q^{6} + 7 q^{7} + 8 q^{8} + 37 q^{9} - 28 q^{10} + 32 q^{12} - 18 q^{13} + 14 q^{14} - 112 q^{15} + 16 q^{16} - 74 q^{17} + 74 q^{18} - 80 q^{19} - 56 q^{20} + 56 q^{21} - 112 q^{23} + 64 q^{24} + 71 q^{25} - 36 q^{26} + 80 q^{27} + 28 q^{28} - 190 q^{29} - 224 q^{30} + 72 q^{31} + 32 q^{32} - 148 q^{34} - 98 q^{35} + 148 q^{36} - 346 q^{37} - 160 q^{38} - 144 q^{39} - 112 q^{40} - 162 q^{41} + 112 q^{42} + 412 q^{43} - 518 q^{45} - 224 q^{46} + 24 q^{47} + 128 q^{48} + 49 q^{49} + 142 q^{50} - 592 q^{51} - 72 q^{52} + 318 q^{53} + 160 q^{54} + 56 q^{56} - 640 q^{57} - 380 q^{58} - 200 q^{59} - 448 q^{60} + 198 q^{61} + 144 q^{62} + 259 q^{63} + 64 q^{64} + 252 q^{65} - 716 q^{67} - 296 q^{68} - 896 q^{69} - 196 q^{70} + 392 q^{71} + 296 q^{72} - 538 q^{73} - 692 q^{74} + 568 q^{75} - 320 q^{76} - 288 q^{78} - 240 q^{79} - 224 q^{80} - 359 q^{81} - 324 q^{82} + 1072 q^{83} + 224 q^{84} + 1036 q^{85} + 824 q^{86} - 1520 q^{87} + 810 q^{89} - 1036 q^{90} - 126 q^{91} - 448 q^{92} + 576 q^{93} + 48 q^{94} + 1120 q^{95} + 256 q^{96} + 1354 q^{97} + 98 q^{98}+O(q^{100})$$ q + 2 * q^2 + 8 * q^3 + 4 * q^4 - 14 * q^5 + 16 * q^6 + 7 * q^7 + 8 * q^8 + 37 * q^9 - 28 * q^10 + 32 * q^12 - 18 * q^13 + 14 * q^14 - 112 * q^15 + 16 * q^16 - 74 * q^17 + 74 * q^18 - 80 * q^19 - 56 * q^20 + 56 * q^21 - 112 * q^23 + 64 * q^24 + 71 * q^25 - 36 * q^26 + 80 * q^27 + 28 * q^28 - 190 * q^29 - 224 * q^30 + 72 * q^31 + 32 * q^32 - 148 * q^34 - 98 * q^35 + 148 * q^36 - 346 * q^37 - 160 * q^38 - 144 * q^39 - 112 * q^40 - 162 * q^41 + 112 * q^42 + 412 * q^43 - 518 * q^45 - 224 * q^46 + 24 * q^47 + 128 * q^48 + 49 * q^49 + 142 * q^50 - 592 * q^51 - 72 * q^52 + 318 * q^53 + 160 * q^54 + 56 * q^56 - 640 * q^57 - 380 * q^58 - 200 * q^59 - 448 * q^60 + 198 * q^61 + 144 * q^62 + 259 * q^63 + 64 * q^64 + 252 * q^65 - 716 * q^67 - 296 * q^68 - 896 * q^69 - 196 * q^70 + 392 * q^71 + 296 * q^72 - 538 * q^73 - 692 * q^74 + 568 * q^75 - 320 * q^76 - 288 * q^78 - 240 * q^79 - 224 * q^80 - 359 * q^81 - 324 * q^82 + 1072 * q^83 + 224 * q^84 + 1036 * q^85 + 824 * q^86 - 1520 * q^87 + 810 * q^89 - 1036 * q^90 - 126 * q^91 - 448 * q^92 + 576 * q^93 + 48 * q^94 + 1120 * q^95 + 256 * q^96 + 1354 * q^97 + 98 * 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
2.00000 8.00000 4.00000 −14.0000 16.0000 7.00000 8.00000 37.0000 −28.0000
 $$n$$: 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.g 1
11.b odd 2 1 14.4.a.a 1
33.d even 2 1 126.4.a.h 1
44.c even 2 1 112.4.a.a 1
55.d odd 2 1 350.4.a.l 1
55.e even 4 2 350.4.c.b 2
77.b even 2 1 98.4.a.a 1
77.h odd 6 2 98.4.c.d 2
77.i even 6 2 98.4.c.f 2
88.b odd 2 1 448.4.a.b 1
88.g even 2 1 448.4.a.o 1
132.d odd 2 1 1008.4.a.s 1
143.d odd 2 1 2366.4.a.h 1
231.h odd 2 1 882.4.a.i 1
231.k odd 6 2 882.4.g.k 2
231.l even 6 2 882.4.g.b 2
308.g odd 2 1 784.4.a.s 1
385.h even 2 1 2450.4.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 11.b odd 2 1
98.4.a.a 1 77.b even 2 1
98.4.c.d 2 77.h odd 6 2
98.4.c.f 2 77.i even 6 2
112.4.a.a 1 44.c even 2 1
126.4.a.h 1 33.d even 2 1
350.4.a.l 1 55.d odd 2 1
350.4.c.b 2 55.e even 4 2
448.4.a.b 1 88.b odd 2 1
448.4.a.o 1 88.g even 2 1
784.4.a.s 1 308.g odd 2 1
882.4.a.i 1 231.h odd 2 1
882.4.g.b 2 231.l even 6 2
882.4.g.k 2 231.k odd 6 2
1008.4.a.s 1 132.d odd 2 1
1694.4.a.g 1 1.a even 1 1 trivial
2366.4.a.h 1 143.d odd 2 1
2450.4.a.bo 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1694))$$:

 $$T_{3} - 8$$ T3 - 8 $$T_{5} + 14$$ T5 + 14 $$T_{13} + 18$$ T13 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 8$$
$5$ $$T + 14$$
$7$ $$T - 7$$
$11$ $$T$$
$13$ $$T + 18$$
$17$ $$T + 74$$
$19$ $$T + 80$$
$23$ $$T + 112$$
$29$ $$T + 190$$
$31$ $$T - 72$$
$37$ $$T + 346$$
$41$ $$T + 162$$
$43$ $$T - 412$$
$47$ $$T - 24$$
$53$ $$T - 318$$
$59$ $$T + 200$$
$61$ $$T - 198$$
$67$ $$T + 716$$
$71$ $$T - 392$$
$73$ $$T + 538$$
$79$ $$T + 240$$
$83$ $$T - 1072$$
$89$ $$T - 810$$
$97$ $$T - 1354$$