# Properties

 Label 170.2.a.f Level $170$ Weight $2$ Character orbit 170.a Self dual yes Analytic conductor $1.357$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$170 = 2 \cdot 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 170.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.35745683436$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} -\beta q^{3} + q^{4} + q^{5} -\beta q^{6} + 2 \beta q^{7} + q^{8} + ( 1 + \beta ) q^{9} +O(q^{10})$$ $$q + q^{2} -\beta q^{3} + q^{4} + q^{5} -\beta q^{6} + 2 \beta q^{7} + q^{8} + ( 1 + \beta ) q^{9} + q^{10} -4 q^{11} -\beta q^{12} + ( 2 + \beta ) q^{13} + 2 \beta q^{14} -\beta q^{15} + q^{16} + q^{17} + ( 1 + \beta ) q^{18} -\beta q^{19} + q^{20} + ( -8 - 2 \beta ) q^{21} -4 q^{22} -2 \beta q^{23} -\beta q^{24} + q^{25} + ( 2 + \beta ) q^{26} + ( -4 + \beta ) q^{27} + 2 \beta q^{28} + ( 2 - 3 \beta ) q^{29} -\beta q^{30} + ( -4 - \beta ) q^{31} + q^{32} + 4 \beta q^{33} + q^{34} + 2 \beta q^{35} + ( 1 + \beta ) q^{36} + ( -2 - 2 \beta ) q^{37} -\beta q^{38} + ( -4 - 3 \beta ) q^{39} + q^{40} + ( -6 + 4 \beta ) q^{41} + ( -8 - 2 \beta ) q^{42} + ( 4 - 2 \beta ) q^{43} -4 q^{44} + ( 1 + \beta ) q^{45} -2 \beta q^{46} + ( 4 + \beta ) q^{47} -\beta q^{48} + ( 9 + 4 \beta ) q^{49} + q^{50} -\beta q^{51} + ( 2 + \beta ) q^{52} + ( 2 - \beta ) q^{53} + ( -4 + \beta ) q^{54} -4 q^{55} + 2 \beta q^{56} + ( 4 + \beta ) q^{57} + ( 2 - 3 \beta ) q^{58} + ( -8 + 3 \beta ) q^{59} -\beta q^{60} + ( 10 - \beta ) q^{61} + ( -4 - \beta ) q^{62} + ( 8 + 4 \beta ) q^{63} + q^{64} + ( 2 + \beta ) q^{65} + 4 \beta q^{66} + ( -4 - 2 \beta ) q^{67} + q^{68} + ( 8 + 2 \beta ) q^{69} + 2 \beta q^{70} + ( -12 + 3 \beta ) q^{71} + ( 1 + \beta ) q^{72} + ( 6 - 5 \beta ) q^{73} + ( -2 - 2 \beta ) q^{74} -\beta q^{75} -\beta q^{76} -8 \beta q^{77} + ( -4 - 3 \beta ) q^{78} + q^{80} -7 q^{81} + ( -6 + 4 \beta ) q^{82} + ( 4 - 4 \beta ) q^{83} + ( -8 - 2 \beta ) q^{84} + q^{85} + ( 4 - 2 \beta ) q^{86} + ( 12 + \beta ) q^{87} -4 q^{88} + ( -2 - 3 \beta ) q^{89} + ( 1 + \beta ) q^{90} + ( 8 + 6 \beta ) q^{91} -2 \beta q^{92} + ( 4 + 5 \beta ) q^{93} + ( 4 + \beta ) q^{94} -\beta q^{95} -\beta q^{96} + ( 6 - 3 \beta ) q^{97} + ( 9 + 4 \beta ) q^{98} + ( -4 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - q^{3} + 2q^{4} + 2q^{5} - q^{6} + 2q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} - q^{3} + 2q^{4} + 2q^{5} - q^{6} + 2q^{7} + 2q^{8} + 3q^{9} + 2q^{10} - 8q^{11} - q^{12} + 5q^{13} + 2q^{14} - q^{15} + 2q^{16} + 2q^{17} + 3q^{18} - q^{19} + 2q^{20} - 18q^{21} - 8q^{22} - 2q^{23} - q^{24} + 2q^{25} + 5q^{26} - 7q^{27} + 2q^{28} + q^{29} - q^{30} - 9q^{31} + 2q^{32} + 4q^{33} + 2q^{34} + 2q^{35} + 3q^{36} - 6q^{37} - q^{38} - 11q^{39} + 2q^{40} - 8q^{41} - 18q^{42} + 6q^{43} - 8q^{44} + 3q^{45} - 2q^{46} + 9q^{47} - q^{48} + 22q^{49} + 2q^{50} - q^{51} + 5q^{52} + 3q^{53} - 7q^{54} - 8q^{55} + 2q^{56} + 9q^{57} + q^{58} - 13q^{59} - q^{60} + 19q^{61} - 9q^{62} + 20q^{63} + 2q^{64} + 5q^{65} + 4q^{66} - 10q^{67} + 2q^{68} + 18q^{69} + 2q^{70} - 21q^{71} + 3q^{72} + 7q^{73} - 6q^{74} - q^{75} - q^{76} - 8q^{77} - 11q^{78} + 2q^{80} - 14q^{81} - 8q^{82} + 4q^{83} - 18q^{84} + 2q^{85} + 6q^{86} + 25q^{87} - 8q^{88} - 7q^{89} + 3q^{90} + 22q^{91} - 2q^{92} + 13q^{93} + 9q^{94} - q^{95} - q^{96} + 9q^{97} + 22q^{98} - 12q^{99} + O(q^{100})$$

## 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
 2.56155 −1.56155
1.00000 −2.56155 1.00000 1.00000 −2.56155 5.12311 1.00000 3.56155 1.00000
1.2 1.00000 1.56155 1.00000 1.00000 1.56155 −3.12311 1.00000 −0.561553 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$17$$ $$-1$$

## 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 170.2.a.f 2
3.b odd 2 1 1530.2.a.r 2
4.b odd 2 1 1360.2.a.m 2
5.b even 2 1 850.2.a.n 2
5.c odd 4 2 850.2.c.i 4
7.b odd 2 1 8330.2.a.bq 2
8.b even 2 1 5440.2.a.bj 2
8.d odd 2 1 5440.2.a.bd 2
15.d odd 2 1 7650.2.a.de 2
17.b even 2 1 2890.2.a.u 2
17.c even 4 2 2890.2.b.i 4
20.d odd 2 1 6800.2.a.be 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.a.f 2 1.a even 1 1 trivial
850.2.a.n 2 5.b even 2 1
850.2.c.i 4 5.c odd 4 2
1360.2.a.m 2 4.b odd 2 1
1530.2.a.r 2 3.b odd 2 1
2890.2.a.u 2 17.b even 2 1
2890.2.b.i 4 17.c even 4 2
5440.2.a.bd 2 8.d odd 2 1
5440.2.a.bj 2 8.b even 2 1
6800.2.a.be 2 20.d odd 2 1
7650.2.a.de 2 15.d odd 2 1
8330.2.a.bq 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(170))$$:

 $$T_{3}^{2} + T_{3} - 4$$ $$T_{7}^{2} - 2 T_{7} - 16$$ $$T_{13}^{2} - 5 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$5$ $$( 1 - T )^{2}$$
$7$ $$1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4}$$
$11$ $$( 1 + 4 T + 11 T^{2} )^{2}$$
$13$ $$1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4}$$
$17$ $$( 1 - T )^{2}$$
$19$ $$1 + T + 34 T^{2} + 19 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4}$$
$31$ $$1 + 9 T + 78 T^{2} + 279 T^{3} + 961 T^{4}$$
$37$ $$1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4}$$
$41$ $$1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4}$$
$43$ $$1 - 6 T + 78 T^{2} - 258 T^{3} + 1849 T^{4}$$
$47$ $$1 - 9 T + 110 T^{2} - 423 T^{3} + 2209 T^{4}$$
$53$ $$1 - 3 T + 104 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 13 T + 122 T^{2} + 767 T^{3} + 3481 T^{4}$$
$61$ $$1 - 19 T + 208 T^{2} - 1159 T^{3} + 3721 T^{4}$$
$67$ $$1 + 10 T + 142 T^{2} + 670 T^{3} + 4489 T^{4}$$
$71$ $$1 + 21 T + 214 T^{2} + 1491 T^{3} + 5041 T^{4}$$
$73$ $$1 - 7 T + 52 T^{2} - 511 T^{3} + 5329 T^{4}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4}$$
$89$ $$1 + 7 T + 152 T^{2} + 623 T^{3} + 7921 T^{4}$$
$97$ $$1 - 9 T + 176 T^{2} - 873 T^{3} + 9409 T^{4}$$