# Properties

 Label 114.4.e.a Level $114$ Weight $4$ Character orbit 114.e Analytic conductor $6.726$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 114.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.72621774065$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} + (2 \zeta_{6} - 2) q^{5} + 6 \zeta_{6} q^{6} - 21 q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^2 + (3*z - 3) * q^3 - 4*z * q^4 + (2*z - 2) * q^5 + 6*z * q^6 - 21 * q^7 - 8 * q^8 - 9*z * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} + (2 \zeta_{6} - 2) q^{5} + 6 \zeta_{6} q^{6} - 21 q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} + 4 \zeta_{6} q^{10} - 40 q^{11} + 12 q^{12} - 17 \zeta_{6} q^{13} + (42 \zeta_{6} - 42) q^{14} - 6 \zeta_{6} q^{15} + (16 \zeta_{6} - 16) q^{16} + (36 \zeta_{6} - 36) q^{17} - 18 q^{18} + (38 \zeta_{6} - 95) q^{19} + 8 q^{20} + ( - 63 \zeta_{6} + 63) q^{21} + (80 \zeta_{6} - 80) q^{22} - 74 \zeta_{6} q^{23} + ( - 24 \zeta_{6} + 24) q^{24} + 121 \zeta_{6} q^{25} - 34 q^{26} + 27 q^{27} + 84 \zeta_{6} q^{28} - 100 \zeta_{6} q^{29} - 12 q^{30} + 103 q^{31} + 32 \zeta_{6} q^{32} + ( - 120 \zeta_{6} + 120) q^{33} + 72 \zeta_{6} q^{34} + ( - 42 \zeta_{6} + 42) q^{35} + (36 \zeta_{6} - 36) q^{36} + 187 q^{37} + (190 \zeta_{6} - 114) q^{38} + 51 q^{39} + ( - 16 \zeta_{6} + 16) q^{40} + ( - 128 \zeta_{6} + 128) q^{41} - 126 \zeta_{6} q^{42} + (121 \zeta_{6} - 121) q^{43} + 160 \zeta_{6} q^{44} + 18 q^{45} - 148 q^{46} - 410 \zeta_{6} q^{47} - 48 \zeta_{6} q^{48} + 98 q^{49} + 242 q^{50} - 108 \zeta_{6} q^{51} + (68 \zeta_{6} - 68) q^{52} - 230 \zeta_{6} q^{53} + ( - 54 \zeta_{6} + 54) q^{54} + ( - 80 \zeta_{6} + 80) q^{55} + 168 q^{56} + ( - 285 \zeta_{6} + 171) q^{57} - 200 q^{58} + ( - 744 \zeta_{6} + 744) q^{59} + (24 \zeta_{6} - 24) q^{60} + 277 \zeta_{6} q^{61} + ( - 206 \zeta_{6} + 206) q^{62} + 189 \zeta_{6} q^{63} + 64 q^{64} + 34 q^{65} - 240 \zeta_{6} q^{66} + 231 \zeta_{6} q^{67} + 144 q^{68} + 222 q^{69} - 84 \zeta_{6} q^{70} + (578 \zeta_{6} - 578) q^{71} + 72 \zeta_{6} q^{72} + (609 \zeta_{6} - 609) q^{73} + ( - 374 \zeta_{6} + 374) q^{74} - 363 q^{75} + (228 \zeta_{6} + 152) q^{76} + 840 q^{77} + ( - 102 \zeta_{6} + 102) q^{78} + (1259 \zeta_{6} - 1259) q^{79} - 32 \zeta_{6} q^{80} + (81 \zeta_{6} - 81) q^{81} - 256 \zeta_{6} q^{82} - 696 q^{83} - 252 q^{84} - 72 \zeta_{6} q^{85} + 242 \zeta_{6} q^{86} + 300 q^{87} + 320 q^{88} + 612 \zeta_{6} q^{89} + ( - 36 \zeta_{6} + 36) q^{90} + 357 \zeta_{6} q^{91} + (296 \zeta_{6} - 296) q^{92} + (309 \zeta_{6} - 309) q^{93} - 820 q^{94} + ( - 190 \zeta_{6} + 114) q^{95} - 96 q^{96} + ( - 1550 \zeta_{6} + 1550) q^{97} + ( - 196 \zeta_{6} + 196) q^{98} + 360 \zeta_{6} q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + (3*z - 3) * q^3 - 4*z * q^4 + (2*z - 2) * q^5 + 6*z * q^6 - 21 * q^7 - 8 * q^8 - 9*z * q^9 + 4*z * q^10 - 40 * q^11 + 12 * q^12 - 17*z * q^13 + (42*z - 42) * q^14 - 6*z * q^15 + (16*z - 16) * q^16 + (36*z - 36) * q^17 - 18 * q^18 + (38*z - 95) * q^19 + 8 * q^20 + (-63*z + 63) * q^21 + (80*z - 80) * q^22 - 74*z * q^23 + (-24*z + 24) * q^24 + 121*z * q^25 - 34 * q^26 + 27 * q^27 + 84*z * q^28 - 100*z * q^29 - 12 * q^30 + 103 * q^31 + 32*z * q^32 + (-120*z + 120) * q^33 + 72*z * q^34 + (-42*z + 42) * q^35 + (36*z - 36) * q^36 + 187 * q^37 + (190*z - 114) * q^38 + 51 * q^39 + (-16*z + 16) * q^40 + (-128*z + 128) * q^41 - 126*z * q^42 + (121*z - 121) * q^43 + 160*z * q^44 + 18 * q^45 - 148 * q^46 - 410*z * q^47 - 48*z * q^48 + 98 * q^49 + 242 * q^50 - 108*z * q^51 + (68*z - 68) * q^52 - 230*z * q^53 + (-54*z + 54) * q^54 + (-80*z + 80) * q^55 + 168 * q^56 + (-285*z + 171) * q^57 - 200 * q^58 + (-744*z + 744) * q^59 + (24*z - 24) * q^60 + 277*z * q^61 + (-206*z + 206) * q^62 + 189*z * q^63 + 64 * q^64 + 34 * q^65 - 240*z * q^66 + 231*z * q^67 + 144 * q^68 + 222 * q^69 - 84*z * q^70 + (578*z - 578) * q^71 + 72*z * q^72 + (609*z - 609) * q^73 + (-374*z + 374) * q^74 - 363 * q^75 + (228*z + 152) * q^76 + 840 * q^77 + (-102*z + 102) * q^78 + (1259*z - 1259) * q^79 - 32*z * q^80 + (81*z - 81) * q^81 - 256*z * q^82 - 696 * q^83 - 252 * q^84 - 72*z * q^85 + 242*z * q^86 + 300 * q^87 + 320 * q^88 + 612*z * q^89 + (-36*z + 36) * q^90 + 357*z * q^91 + (296*z - 296) * q^92 + (309*z - 309) * q^93 - 820 * q^94 + (-190*z + 114) * q^95 - 96 * q^96 + (-1550*z + 1550) * q^97 + (-196*z + 196) * q^98 + 360*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 2 q^{5} + 6 q^{6} - 42 q^{7} - 16 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 - 2 * q^5 + 6 * q^6 - 42 * q^7 - 16 * q^8 - 9 * q^9 $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 2 q^{5} + 6 q^{6} - 42 q^{7} - 16 q^{8} - 9 q^{9} + 4 q^{10} - 80 q^{11} + 24 q^{12} - 17 q^{13} - 42 q^{14} - 6 q^{15} - 16 q^{16} - 36 q^{17} - 36 q^{18} - 152 q^{19} + 16 q^{20} + 63 q^{21} - 80 q^{22} - 74 q^{23} + 24 q^{24} + 121 q^{25} - 68 q^{26} + 54 q^{27} + 84 q^{28} - 100 q^{29} - 24 q^{30} + 206 q^{31} + 32 q^{32} + 120 q^{33} + 72 q^{34} + 42 q^{35} - 36 q^{36} + 374 q^{37} - 38 q^{38} + 102 q^{39} + 16 q^{40} + 128 q^{41} - 126 q^{42} - 121 q^{43} + 160 q^{44} + 36 q^{45} - 296 q^{46} - 410 q^{47} - 48 q^{48} + 196 q^{49} + 484 q^{50} - 108 q^{51} - 68 q^{52} - 230 q^{53} + 54 q^{54} + 80 q^{55} + 336 q^{56} + 57 q^{57} - 400 q^{58} + 744 q^{59} - 24 q^{60} + 277 q^{61} + 206 q^{62} + 189 q^{63} + 128 q^{64} + 68 q^{65} - 240 q^{66} + 231 q^{67} + 288 q^{68} + 444 q^{69} - 84 q^{70} - 578 q^{71} + 72 q^{72} - 609 q^{73} + 374 q^{74} - 726 q^{75} + 532 q^{76} + 1680 q^{77} + 102 q^{78} - 1259 q^{79} - 32 q^{80} - 81 q^{81} - 256 q^{82} - 1392 q^{83} - 504 q^{84} - 72 q^{85} + 242 q^{86} + 600 q^{87} + 640 q^{88} + 612 q^{89} + 36 q^{90} + 357 q^{91} - 296 q^{92} - 309 q^{93} - 1640 q^{94} + 38 q^{95} - 192 q^{96} + 1550 q^{97} + 196 q^{98} + 360 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 - 2 * q^5 + 6 * q^6 - 42 * q^7 - 16 * q^8 - 9 * q^9 + 4 * q^10 - 80 * q^11 + 24 * q^12 - 17 * q^13 - 42 * q^14 - 6 * q^15 - 16 * q^16 - 36 * q^17 - 36 * q^18 - 152 * q^19 + 16 * q^20 + 63 * q^21 - 80 * q^22 - 74 * q^23 + 24 * q^24 + 121 * q^25 - 68 * q^26 + 54 * q^27 + 84 * q^28 - 100 * q^29 - 24 * q^30 + 206 * q^31 + 32 * q^32 + 120 * q^33 + 72 * q^34 + 42 * q^35 - 36 * q^36 + 374 * q^37 - 38 * q^38 + 102 * q^39 + 16 * q^40 + 128 * q^41 - 126 * q^42 - 121 * q^43 + 160 * q^44 + 36 * q^45 - 296 * q^46 - 410 * q^47 - 48 * q^48 + 196 * q^49 + 484 * q^50 - 108 * q^51 - 68 * q^52 - 230 * q^53 + 54 * q^54 + 80 * q^55 + 336 * q^56 + 57 * q^57 - 400 * q^58 + 744 * q^59 - 24 * q^60 + 277 * q^61 + 206 * q^62 + 189 * q^63 + 128 * q^64 + 68 * q^65 - 240 * q^66 + 231 * q^67 + 288 * q^68 + 444 * q^69 - 84 * q^70 - 578 * q^71 + 72 * q^72 - 609 * q^73 + 374 * q^74 - 726 * q^75 + 532 * q^76 + 1680 * q^77 + 102 * q^78 - 1259 * q^79 - 32 * q^80 - 81 * q^81 - 256 * q^82 - 1392 * q^83 - 504 * q^84 - 72 * q^85 + 242 * q^86 + 600 * q^87 + 640 * q^88 + 612 * q^89 + 36 * q^90 + 357 * q^91 - 296 * q^92 - 309 * q^93 - 1640 * q^94 + 38 * q^95 - 192 * q^96 + 1550 * q^97 + 196 * q^98 + 360 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 + 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i −1.00000 1.73205i 3.00000 5.19615i −21.0000 −8.00000 −4.50000 + 7.79423i 2.00000 3.46410i
49.1 1.00000 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i −1.00000 + 1.73205i 3.00000 + 5.19615i −21.0000 −8.00000 −4.50000 7.79423i 2.00000 + 3.46410i
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.e.a 2
3.b odd 2 1 342.4.g.b 2
19.c even 3 1 inner 114.4.e.a 2
19.c even 3 1 2166.4.a.c 1
19.d odd 6 1 2166.4.a.f 1
57.h odd 6 1 342.4.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.a 2 1.a even 1 1 trivial
114.4.e.a 2 19.c even 3 1 inner
342.4.g.b 2 3.b odd 2 1
342.4.g.b 2 57.h odd 6 1
2166.4.a.c 1 19.c even 3 1
2166.4.a.f 1 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2T_{5} + 4$$ acting on $$S_{4}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$(T + 21)^{2}$$
$11$ $$(T + 40)^{2}$$
$13$ $$T^{2} + 17T + 289$$
$17$ $$T^{2} + 36T + 1296$$
$19$ $$T^{2} + 152T + 6859$$
$23$ $$T^{2} + 74T + 5476$$
$29$ $$T^{2} + 100T + 10000$$
$31$ $$(T - 103)^{2}$$
$37$ $$(T - 187)^{2}$$
$41$ $$T^{2} - 128T + 16384$$
$43$ $$T^{2} + 121T + 14641$$
$47$ $$T^{2} + 410T + 168100$$
$53$ $$T^{2} + 230T + 52900$$
$59$ $$T^{2} - 744T + 553536$$
$61$ $$T^{2} - 277T + 76729$$
$67$ $$T^{2} - 231T + 53361$$
$71$ $$T^{2} + 578T + 334084$$
$73$ $$T^{2} + 609T + 370881$$
$79$ $$T^{2} + 1259 T + 1585081$$
$83$ $$(T + 696)^{2}$$
$89$ $$T^{2} - 612T + 374544$$
$97$ $$T^{2} - 1550 T + 2402500$$