Properties

 Label 117.4.a.b Level $117$ Weight $4$ Character orbit 117.a Self dual yes Analytic conductor $6.903$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 17 q^{4} + 7 q^{5} - 13 q^{7} + 45 q^{8}+O(q^{10})$$ q + 5 * q^2 + 17 * q^4 + 7 * q^5 - 13 * q^7 + 45 * q^8 $$q + 5 q^{2} + 17 q^{4} + 7 q^{5} - 13 q^{7} + 45 q^{8} + 35 q^{10} + 26 q^{11} + 13 q^{13} - 65 q^{14} + 89 q^{16} - 77 q^{17} - 126 q^{19} + 119 q^{20} + 130 q^{22} + 96 q^{23} - 76 q^{25} + 65 q^{26} - 221 q^{28} + 82 q^{29} + 196 q^{31} + 85 q^{32} - 385 q^{34} - 91 q^{35} - 131 q^{37} - 630 q^{38} + 315 q^{40} - 336 q^{41} - 201 q^{43} + 442 q^{44} + 480 q^{46} + 105 q^{47} - 174 q^{49} - 380 q^{50} + 221 q^{52} + 432 q^{53} + 182 q^{55} - 585 q^{56} + 410 q^{58} + 294 q^{59} - 56 q^{61} + 980 q^{62} - 287 q^{64} + 91 q^{65} + 478 q^{67} - 1309 q^{68} - 455 q^{70} - 9 q^{71} + 98 q^{73} - 655 q^{74} - 2142 q^{76} - 338 q^{77} + 1304 q^{79} + 623 q^{80} - 1680 q^{82} + 308 q^{83} - 539 q^{85} - 1005 q^{86} + 1170 q^{88} + 1190 q^{89} - 169 q^{91} + 1632 q^{92} + 525 q^{94} - 882 q^{95} + 70 q^{97} - 870 q^{98}+O(q^{100})$$ q + 5 * q^2 + 17 * q^4 + 7 * q^5 - 13 * q^7 + 45 * q^8 + 35 * q^10 + 26 * q^11 + 13 * q^13 - 65 * q^14 + 89 * q^16 - 77 * q^17 - 126 * q^19 + 119 * q^20 + 130 * q^22 + 96 * q^23 - 76 * q^25 + 65 * q^26 - 221 * q^28 + 82 * q^29 + 196 * q^31 + 85 * q^32 - 385 * q^34 - 91 * q^35 - 131 * q^37 - 630 * q^38 + 315 * q^40 - 336 * q^41 - 201 * q^43 + 442 * q^44 + 480 * q^46 + 105 * q^47 - 174 * q^49 - 380 * q^50 + 221 * q^52 + 432 * q^53 + 182 * q^55 - 585 * q^56 + 410 * q^58 + 294 * q^59 - 56 * q^61 + 980 * q^62 - 287 * q^64 + 91 * q^65 + 478 * q^67 - 1309 * q^68 - 455 * q^70 - 9 * q^71 + 98 * q^73 - 655 * q^74 - 2142 * q^76 - 338 * q^77 + 1304 * q^79 + 623 * q^80 - 1680 * q^82 + 308 * q^83 - 539 * q^85 - 1005 * q^86 + 1170 * q^88 + 1190 * q^89 - 169 * q^91 + 1632 * q^92 + 525 * q^94 - 882 * q^95 + 70 * q^97 - 870 * 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
5.00000 0 17.0000 7.00000 0 −13.0000 45.0000 0 35.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$-1$$

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 117.4.a.b 1
3.b odd 2 1 13.4.a.a 1
4.b odd 2 1 1872.4.a.k 1
12.b even 2 1 208.4.a.g 1
13.b even 2 1 1521.4.a.a 1
15.d odd 2 1 325.4.a.d 1
15.e even 4 2 325.4.b.b 2
21.c even 2 1 637.4.a.a 1
24.f even 2 1 832.4.a.a 1
24.h odd 2 1 832.4.a.r 1
33.d even 2 1 1573.4.a.a 1
39.d odd 2 1 169.4.a.e 1
39.f even 4 2 169.4.b.a 2
39.h odd 6 2 169.4.c.a 2
39.i odd 6 2 169.4.c.e 2
39.k even 12 4 169.4.e.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.a.a 1 3.b odd 2 1
117.4.a.b 1 1.a even 1 1 trivial
169.4.a.e 1 39.d odd 2 1
169.4.b.a 2 39.f even 4 2
169.4.c.a 2 39.h odd 6 2
169.4.c.e 2 39.i odd 6 2
169.4.e.e 4 39.k even 12 4
208.4.a.g 1 12.b even 2 1
325.4.a.d 1 15.d odd 2 1
325.4.b.b 2 15.e even 4 2
637.4.a.a 1 21.c even 2 1
832.4.a.a 1 24.f even 2 1
832.4.a.r 1 24.h odd 2 1
1521.4.a.a 1 13.b even 2 1
1573.4.a.a 1 33.d even 2 1
1872.4.a.k 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(117))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 5$$
$3$ $$T$$
$5$ $$T - 7$$
$7$ $$T + 13$$
$11$ $$T - 26$$
$13$ $$T - 13$$
$17$ $$T + 77$$
$19$ $$T + 126$$
$23$ $$T - 96$$
$29$ $$T - 82$$
$31$ $$T - 196$$
$37$ $$T + 131$$
$41$ $$T + 336$$
$43$ $$T + 201$$
$47$ $$T - 105$$
$53$ $$T - 432$$
$59$ $$T - 294$$
$61$ $$T + 56$$
$67$ $$T - 478$$
$71$ $$T + 9$$
$73$ $$T - 98$$
$79$ $$T - 1304$$
$83$ $$T - 308$$
$89$ $$T - 1190$$
$97$ $$T - 70$$