# Properties

 Label 117.4.g.b Level $117$ Weight $4$ Character orbit 117.g Analytic conductor $6.903$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{6} + 3) q^{2} - \zeta_{6} q^{4} + 9 q^{5} - 2 \zeta_{6} q^{7} + 21 q^{8} +O(q^{10})$$ q + (-3*z + 3) * q^2 - z * q^4 + 9 * q^5 - 2*z * q^7 + 21 * q^8 $$q + ( - 3 \zeta_{6} + 3) q^{2} - \zeta_{6} q^{4} + 9 q^{5} - 2 \zeta_{6} q^{7} + 21 q^{8} + ( - 27 \zeta_{6} + 27) q^{10} + ( - 30 \zeta_{6} + 30) q^{11} + (39 \zeta_{6} + 13) q^{13} - 6 q^{14} + ( - 71 \zeta_{6} + 71) q^{16} - 111 \zeta_{6} q^{17} + 46 \zeta_{6} q^{19} - 9 \zeta_{6} q^{20} - 90 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} - 44 q^{25} + ( - 39 \zeta_{6} + 156) q^{26} + (2 \zeta_{6} - 2) q^{28} + (105 \zeta_{6} - 105) q^{29} - 100 q^{31} - 45 \zeta_{6} q^{32} - 333 q^{34} - 18 \zeta_{6} q^{35} + (17 \zeta_{6} - 17) q^{37} + 138 q^{38} + 189 q^{40} + (231 \zeta_{6} - 231) q^{41} + 514 \zeta_{6} q^{43} - 30 q^{44} + 18 \zeta_{6} q^{46} + 162 q^{47} + ( - 339 \zeta_{6} + 339) q^{49} + (132 \zeta_{6} - 132) q^{50} + ( - 52 \zeta_{6} + 39) q^{52} - 639 q^{53} + ( - 270 \zeta_{6} + 270) q^{55} - 42 \zeta_{6} q^{56} + 315 \zeta_{6} q^{58} + 600 \zeta_{6} q^{59} - 233 \zeta_{6} q^{61} + (300 \zeta_{6} - 300) q^{62} + 433 q^{64} + (351 \zeta_{6} + 117) q^{65} + (926 \zeta_{6} - 926) q^{67} + (111 \zeta_{6} - 111) q^{68} - 54 q^{70} - 930 \zeta_{6} q^{71} - 253 q^{73} + 51 \zeta_{6} q^{74} + ( - 46 \zeta_{6} + 46) q^{76} - 60 q^{77} - 1324 q^{79} + ( - 639 \zeta_{6} + 639) q^{80} + 693 \zeta_{6} q^{82} - 810 q^{83} - 999 \zeta_{6} q^{85} + 1542 q^{86} + ( - 630 \zeta_{6} + 630) q^{88} + ( - 498 \zeta_{6} + 498) q^{89} + ( - 104 \zeta_{6} + 78) q^{91} + 6 q^{92} + ( - 486 \zeta_{6} + 486) q^{94} + 414 \zeta_{6} q^{95} - 1358 \zeta_{6} q^{97} - 1017 \zeta_{6} q^{98} +O(q^{100})$$ q + (-3*z + 3) * q^2 - z * q^4 + 9 * q^5 - 2*z * q^7 + 21 * q^8 + (-27*z + 27) * q^10 + (-30*z + 30) * q^11 + (39*z + 13) * q^13 - 6 * q^14 + (-71*z + 71) * q^16 - 111*z * q^17 + 46*z * q^19 - 9*z * q^20 - 90*z * q^22 + (6*z - 6) * q^23 - 44 * q^25 + (-39*z + 156) * q^26 + (2*z - 2) * q^28 + (105*z - 105) * q^29 - 100 * q^31 - 45*z * q^32 - 333 * q^34 - 18*z * q^35 + (17*z - 17) * q^37 + 138 * q^38 + 189 * q^40 + (231*z - 231) * q^41 + 514*z * q^43 - 30 * q^44 + 18*z * q^46 + 162 * q^47 + (-339*z + 339) * q^49 + (132*z - 132) * q^50 + (-52*z + 39) * q^52 - 639 * q^53 + (-270*z + 270) * q^55 - 42*z * q^56 + 315*z * q^58 + 600*z * q^59 - 233*z * q^61 + (300*z - 300) * q^62 + 433 * q^64 + (351*z + 117) * q^65 + (926*z - 926) * q^67 + (111*z - 111) * q^68 - 54 * q^70 - 930*z * q^71 - 253 * q^73 + 51*z * q^74 + (-46*z + 46) * q^76 - 60 * q^77 - 1324 * q^79 + (-639*z + 639) * q^80 + 693*z * q^82 - 810 * q^83 - 999*z * q^85 + 1542 * q^86 + (-630*z + 630) * q^88 + (-498*z + 498) * q^89 + (-104*z + 78) * q^91 + 6 * q^92 + (-486*z + 486) * q^94 + 414*z * q^95 - 1358*z * q^97 - 1017*z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} - q^{4} + 18 q^{5} - 2 q^{7} + 42 q^{8}+O(q^{10})$$ 2 * q + 3 * q^2 - q^4 + 18 * q^5 - 2 * q^7 + 42 * q^8 $$2 q + 3 q^{2} - q^{4} + 18 q^{5} - 2 q^{7} + 42 q^{8} + 27 q^{10} + 30 q^{11} + 65 q^{13} - 12 q^{14} + 71 q^{16} - 111 q^{17} + 46 q^{19} - 9 q^{20} - 90 q^{22} - 6 q^{23} - 88 q^{25} + 273 q^{26} - 2 q^{28} - 105 q^{29} - 200 q^{31} - 45 q^{32} - 666 q^{34} - 18 q^{35} - 17 q^{37} + 276 q^{38} + 378 q^{40} - 231 q^{41} + 514 q^{43} - 60 q^{44} + 18 q^{46} + 324 q^{47} + 339 q^{49} - 132 q^{50} + 26 q^{52} - 1278 q^{53} + 270 q^{55} - 42 q^{56} + 315 q^{58} + 600 q^{59} - 233 q^{61} - 300 q^{62} + 866 q^{64} + 585 q^{65} - 926 q^{67} - 111 q^{68} - 108 q^{70} - 930 q^{71} - 506 q^{73} + 51 q^{74} + 46 q^{76} - 120 q^{77} - 2648 q^{79} + 639 q^{80} + 693 q^{82} - 1620 q^{83} - 999 q^{85} + 3084 q^{86} + 630 q^{88} + 498 q^{89} + 52 q^{91} + 12 q^{92} + 486 q^{94} + 414 q^{95} - 1358 q^{97} - 1017 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 - q^4 + 18 * q^5 - 2 * q^7 + 42 * q^8 + 27 * q^10 + 30 * q^11 + 65 * q^13 - 12 * q^14 + 71 * q^16 - 111 * q^17 + 46 * q^19 - 9 * q^20 - 90 * q^22 - 6 * q^23 - 88 * q^25 + 273 * q^26 - 2 * q^28 - 105 * q^29 - 200 * q^31 - 45 * q^32 - 666 * q^34 - 18 * q^35 - 17 * q^37 + 276 * q^38 + 378 * q^40 - 231 * q^41 + 514 * q^43 - 60 * q^44 + 18 * q^46 + 324 * q^47 + 339 * q^49 - 132 * q^50 + 26 * q^52 - 1278 * q^53 + 270 * q^55 - 42 * q^56 + 315 * q^58 + 600 * q^59 - 233 * q^61 - 300 * q^62 + 866 * q^64 + 585 * q^65 - 926 * q^67 - 111 * q^68 - 108 * q^70 - 930 * q^71 - 506 * q^73 + 51 * q^74 + 46 * q^76 - 120 * q^77 - 2648 * q^79 + 639 * q^80 + 693 * q^82 - 1620 * q^83 - 999 * q^85 + 3084 * q^86 + 630 * q^88 + 498 * q^89 + 52 * q^91 + 12 * q^92 + 486 * q^94 + 414 * q^95 - 1358 * q^97 - 1017 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.50000 + 2.59808i 0 −0.500000 + 0.866025i 9.00000 0 −1.00000 + 1.73205i 21.0000 0 13.5000 + 23.3827i
100.1 1.50000 2.59808i 0 −0.500000 0.866025i 9.00000 0 −1.00000 1.73205i 21.0000 0 13.5000 23.3827i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.g.b 2
3.b odd 2 1 39.4.e.a 2
12.b even 2 1 624.4.q.b 2
13.c even 3 1 inner 117.4.g.b 2
13.c even 3 1 1521.4.a.c 1
13.e even 6 1 1521.4.a.j 1
39.h odd 6 1 507.4.a.a 1
39.i odd 6 1 39.4.e.a 2
39.i odd 6 1 507.4.a.e 1
39.k even 12 2 507.4.b.c 2
156.p even 6 1 624.4.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 3.b odd 2 1
39.4.e.a 2 39.i odd 6 1
117.4.g.b 2 1.a even 1 1 trivial
117.4.g.b 2 13.c even 3 1 inner
507.4.a.a 1 39.h odd 6 1
507.4.a.e 1 39.i odd 6 1
507.4.b.c 2 39.k even 12 2
624.4.q.b 2 12.b even 2 1
624.4.q.b 2 156.p even 6 1
1521.4.a.c 1 13.c even 3 1
1521.4.a.j 1 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} + 9$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 9$$
$3$ $$T^{2}$$
$5$ $$(T - 9)^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 30T + 900$$
$13$ $$T^{2} - 65T + 2197$$
$17$ $$T^{2} + 111T + 12321$$
$19$ $$T^{2} - 46T + 2116$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 105T + 11025$$
$31$ $$(T + 100)^{2}$$
$37$ $$T^{2} + 17T + 289$$
$41$ $$T^{2} + 231T + 53361$$
$43$ $$T^{2} - 514T + 264196$$
$47$ $$(T - 162)^{2}$$
$53$ $$(T + 639)^{2}$$
$59$ $$T^{2} - 600T + 360000$$
$61$ $$T^{2} + 233T + 54289$$
$67$ $$T^{2} + 926T + 857476$$
$71$ $$T^{2} + 930T + 864900$$
$73$ $$(T + 253)^{2}$$
$79$ $$(T + 1324)^{2}$$
$83$ $$(T + 810)^{2}$$
$89$ $$T^{2} - 498T + 248004$$
$97$ $$T^{2} + 1358 T + 1844164$$