Properties

 Label 114.4.e.b Level $114$ Weight $4$ Character orbit 114.e Analytic conductor $6.726$ Analytic rank $0$ 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: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 6) q^{5} + 6 \zeta_{6} q^{6} + 19 q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^2 + (3*z - 3) * q^3 - 4*z * q^4 + (-6*z + 6) * q^5 + 6*z * q^6 + 19 * 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} + ( - 6 \zeta_{6} + 6) q^{5} + 6 \zeta_{6} q^{6} + 19 q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} - 12 \zeta_{6} q^{10} + 32 q^{11} + 12 q^{12} - 81 \zeta_{6} q^{13} + ( - 38 \zeta_{6} + 38) q^{14} + 18 \zeta_{6} q^{15} + (16 \zeta_{6} - 16) q^{16} + ( - 124 \zeta_{6} + 124) q^{17} - 18 q^{18} + (38 \zeta_{6} + 57) q^{19} - 24 q^{20} + (57 \zeta_{6} - 57) q^{21} + ( - 64 \zeta_{6} + 64) q^{22} - 98 \zeta_{6} q^{23} + ( - 24 \zeta_{6} + 24) q^{24} + 89 \zeta_{6} q^{25} - 162 q^{26} + 27 q^{27} - 76 \zeta_{6} q^{28} + 300 \zeta_{6} q^{29} + 36 q^{30} - 225 q^{31} + 32 \zeta_{6} q^{32} + (96 \zeta_{6} - 96) q^{33} - 248 \zeta_{6} q^{34} + ( - 114 \zeta_{6} + 114) q^{35} + (36 \zeta_{6} - 36) q^{36} - 293 q^{37} + ( - 114 \zeta_{6} + 190) q^{38} + 243 q^{39} + (48 \zeta_{6} - 48) q^{40} + (176 \zeta_{6} - 176) q^{41} + 114 \zeta_{6} q^{42} + ( - 111 \zeta_{6} + 111) q^{43} - 128 \zeta_{6} q^{44} - 54 q^{45} - 196 q^{46} + 550 \zeta_{6} q^{47} - 48 \zeta_{6} q^{48} + 18 q^{49} + 178 q^{50} + 372 \zeta_{6} q^{51} + (324 \zeta_{6} - 324) q^{52} + 482 \zeta_{6} q^{53} + ( - 54 \zeta_{6} + 54) q^{54} + ( - 192 \zeta_{6} + 192) q^{55} - 152 q^{56} + (171 \zeta_{6} - 285) q^{57} + 600 q^{58} + ( - 496 \zeta_{6} + 496) q^{59} + ( - 72 \zeta_{6} + 72) q^{60} - 155 \zeta_{6} q^{61} + (450 \zeta_{6} - 450) q^{62} - 171 \zeta_{6} q^{63} + 64 q^{64} - 486 q^{65} + 192 \zeta_{6} q^{66} - 465 \zeta_{6} q^{67} - 496 q^{68} + 294 q^{69} - 228 \zeta_{6} q^{70} + ( - 110 \zeta_{6} + 110) q^{71} + 72 \zeta_{6} q^{72} + (817 \zeta_{6} - 817) q^{73} + (586 \zeta_{6} - 586) q^{74} - 267 q^{75} + ( - 380 \zeta_{6} + 152) q^{76} + 608 q^{77} + ( - 486 \zeta_{6} + 486) q^{78} + (259 \zeta_{6} - 259) q^{79} + 96 \zeta_{6} q^{80} + (81 \zeta_{6} - 81) q^{81} + 352 \zeta_{6} q^{82} - 56 q^{83} + 228 q^{84} - 744 \zeta_{6} q^{85} - 222 \zeta_{6} q^{86} - 900 q^{87} - 256 q^{88} - 308 \zeta_{6} q^{89} + (108 \zeta_{6} - 108) q^{90} - 1539 \zeta_{6} q^{91} + (392 \zeta_{6} - 392) q^{92} + ( - 675 \zeta_{6} + 675) q^{93} + 1100 q^{94} + ( - 342 \zeta_{6} + 570) q^{95} - 96 q^{96} + ( - 1150 \zeta_{6} + 1150) q^{97} + ( - 36 \zeta_{6} + 36) q^{98} - 288 \zeta_{6} q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + (3*z - 3) * q^3 - 4*z * q^4 + (-6*z + 6) * q^5 + 6*z * q^6 + 19 * q^7 - 8 * q^8 - 9*z * q^9 - 12*z * q^10 + 32 * q^11 + 12 * q^12 - 81*z * q^13 + (-38*z + 38) * q^14 + 18*z * q^15 + (16*z - 16) * q^16 + (-124*z + 124) * q^17 - 18 * q^18 + (38*z + 57) * q^19 - 24 * q^20 + (57*z - 57) * q^21 + (-64*z + 64) * q^22 - 98*z * q^23 + (-24*z + 24) * q^24 + 89*z * q^25 - 162 * q^26 + 27 * q^27 - 76*z * q^28 + 300*z * q^29 + 36 * q^30 - 225 * q^31 + 32*z * q^32 + (96*z - 96) * q^33 - 248*z * q^34 + (-114*z + 114) * q^35 + (36*z - 36) * q^36 - 293 * q^37 + (-114*z + 190) * q^38 + 243 * q^39 + (48*z - 48) * q^40 + (176*z - 176) * q^41 + 114*z * q^42 + (-111*z + 111) * q^43 - 128*z * q^44 - 54 * q^45 - 196 * q^46 + 550*z * q^47 - 48*z * q^48 + 18 * q^49 + 178 * q^50 + 372*z * q^51 + (324*z - 324) * q^52 + 482*z * q^53 + (-54*z + 54) * q^54 + (-192*z + 192) * q^55 - 152 * q^56 + (171*z - 285) * q^57 + 600 * q^58 + (-496*z + 496) * q^59 + (-72*z + 72) * q^60 - 155*z * q^61 + (450*z - 450) * q^62 - 171*z * q^63 + 64 * q^64 - 486 * q^65 + 192*z * q^66 - 465*z * q^67 - 496 * q^68 + 294 * q^69 - 228*z * q^70 + (-110*z + 110) * q^71 + 72*z * q^72 + (817*z - 817) * q^73 + (586*z - 586) * q^74 - 267 * q^75 + (-380*z + 152) * q^76 + 608 * q^77 + (-486*z + 486) * q^78 + (259*z - 259) * q^79 + 96*z * q^80 + (81*z - 81) * q^81 + 352*z * q^82 - 56 * q^83 + 228 * q^84 - 744*z * q^85 - 222*z * q^86 - 900 * q^87 - 256 * q^88 - 308*z * q^89 + (108*z - 108) * q^90 - 1539*z * q^91 + (392*z - 392) * q^92 + (-675*z + 675) * q^93 + 1100 * q^94 + (-342*z + 570) * q^95 - 96 * q^96 + (-1150*z + 1150) * q^97 + (-36*z + 36) * q^98 - 288*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} + 6 q^{6} + 38 q^{7} - 16 q^{8} - 9 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 + 6 * q^5 + 6 * q^6 + 38 * q^7 - 16 * q^8 - 9 * q^9 $$2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} + 6 q^{6} + 38 q^{7} - 16 q^{8} - 9 q^{9} - 12 q^{10} + 64 q^{11} + 24 q^{12} - 81 q^{13} + 38 q^{14} + 18 q^{15} - 16 q^{16} + 124 q^{17} - 36 q^{18} + 152 q^{19} - 48 q^{20} - 57 q^{21} + 64 q^{22} - 98 q^{23} + 24 q^{24} + 89 q^{25} - 324 q^{26} + 54 q^{27} - 76 q^{28} + 300 q^{29} + 72 q^{30} - 450 q^{31} + 32 q^{32} - 96 q^{33} - 248 q^{34} + 114 q^{35} - 36 q^{36} - 586 q^{37} + 266 q^{38} + 486 q^{39} - 48 q^{40} - 176 q^{41} + 114 q^{42} + 111 q^{43} - 128 q^{44} - 108 q^{45} - 392 q^{46} + 550 q^{47} - 48 q^{48} + 36 q^{49} + 356 q^{50} + 372 q^{51} - 324 q^{52} + 482 q^{53} + 54 q^{54} + 192 q^{55} - 304 q^{56} - 399 q^{57} + 1200 q^{58} + 496 q^{59} + 72 q^{60} - 155 q^{61} - 450 q^{62} - 171 q^{63} + 128 q^{64} - 972 q^{65} + 192 q^{66} - 465 q^{67} - 992 q^{68} + 588 q^{69} - 228 q^{70} + 110 q^{71} + 72 q^{72} - 817 q^{73} - 586 q^{74} - 534 q^{75} - 76 q^{76} + 1216 q^{77} + 486 q^{78} - 259 q^{79} + 96 q^{80} - 81 q^{81} + 352 q^{82} - 112 q^{83} + 456 q^{84} - 744 q^{85} - 222 q^{86} - 1800 q^{87} - 512 q^{88} - 308 q^{89} - 108 q^{90} - 1539 q^{91} - 392 q^{92} + 675 q^{93} + 2200 q^{94} + 798 q^{95} - 192 q^{96} + 1150 q^{97} + 36 q^{98} - 288 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 - 4 * q^4 + 6 * q^5 + 6 * q^6 + 38 * q^7 - 16 * q^8 - 9 * q^9 - 12 * q^10 + 64 * q^11 + 24 * q^12 - 81 * q^13 + 38 * q^14 + 18 * q^15 - 16 * q^16 + 124 * q^17 - 36 * q^18 + 152 * q^19 - 48 * q^20 - 57 * q^21 + 64 * q^22 - 98 * q^23 + 24 * q^24 + 89 * q^25 - 324 * q^26 + 54 * q^27 - 76 * q^28 + 300 * q^29 + 72 * q^30 - 450 * q^31 + 32 * q^32 - 96 * q^33 - 248 * q^34 + 114 * q^35 - 36 * q^36 - 586 * q^37 + 266 * q^38 + 486 * q^39 - 48 * q^40 - 176 * q^41 + 114 * q^42 + 111 * q^43 - 128 * q^44 - 108 * q^45 - 392 * q^46 + 550 * q^47 - 48 * q^48 + 36 * q^49 + 356 * q^50 + 372 * q^51 - 324 * q^52 + 482 * q^53 + 54 * q^54 + 192 * q^55 - 304 * q^56 - 399 * q^57 + 1200 * q^58 + 496 * q^59 + 72 * q^60 - 155 * q^61 - 450 * q^62 - 171 * q^63 + 128 * q^64 - 972 * q^65 + 192 * q^66 - 465 * q^67 - 992 * q^68 + 588 * q^69 - 228 * q^70 + 110 * q^71 + 72 * q^72 - 817 * q^73 - 586 * q^74 - 534 * q^75 - 76 * q^76 + 1216 * q^77 + 486 * q^78 - 259 * q^79 + 96 * q^80 - 81 * q^81 + 352 * q^82 - 112 * q^83 + 456 * q^84 - 744 * q^85 - 222 * q^86 - 1800 * q^87 - 512 * q^88 - 308 * q^89 - 108 * q^90 - 1539 * q^91 - 392 * q^92 + 675 * q^93 + 2200 * q^94 + 798 * q^95 - 192 * q^96 + 1150 * q^97 + 36 * q^98 - 288 * 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 3.00000 + 5.19615i 3.00000 5.19615i 19.0000 −8.00000 −4.50000 + 7.79423i −6.00000 + 10.3923i
49.1 1.00000 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i 3.00000 5.19615i 3.00000 + 5.19615i 19.0000 −8.00000 −4.50000 7.79423i −6.00000 10.3923i
 $$n$$: 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.b 2
3.b odd 2 1 342.4.g.a 2
19.c even 3 1 inner 114.4.e.b 2
19.c even 3 1 2166.4.a.b 1
19.d odd 6 1 2166.4.a.e 1
57.h odd 6 1 342.4.g.a 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 6T_{5} + 36$$ acting on $$S_{4}^{\mathrm{new}}(114, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 3T + 9$$
$5$ $$T^{2} - 6T + 36$$
$7$ $$(T - 19)^{2}$$
$11$ $$(T - 32)^{2}$$
$13$ $$T^{2} + 81T + 6561$$
$17$ $$T^{2} - 124T + 15376$$
$19$ $$T^{2} - 152T + 6859$$
$23$ $$T^{2} + 98T + 9604$$
$29$ $$T^{2} - 300T + 90000$$
$31$ $$(T + 225)^{2}$$
$37$ $$(T + 293)^{2}$$
$41$ $$T^{2} + 176T + 30976$$
$43$ $$T^{2} - 111T + 12321$$
$47$ $$T^{2} - 550T + 302500$$
$53$ $$T^{2} - 482T + 232324$$
$59$ $$T^{2} - 496T + 246016$$
$61$ $$T^{2} + 155T + 24025$$
$67$ $$T^{2} + 465T + 216225$$
$71$ $$T^{2} - 110T + 12100$$
$73$ $$T^{2} + 817T + 667489$$
$79$ $$T^{2} + 259T + 67081$$
$83$ $$(T + 56)^{2}$$
$89$ $$T^{2} + 308T + 94864$$
$97$ $$T^{2} - 1150 T + 1322500$$