# Properties

 Label 14.4.c.b Level 14 Weight 4 Character orbit 14.c Analytic conductor 0.826 Analytic rank 0 Dimension 2 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$14 = 2 \cdot 7$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 14.c (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$0.82602674008$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( -7 + 7 \zeta_{6} ) q^{5} + 2 q^{6} + ( -19 + 18 \zeta_{6} ) q^{7} -8 q^{8} + ( 26 - 26 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( -7 + 7 \zeta_{6} ) q^{5} + 2 q^{6} + ( -19 + 18 \zeta_{6} ) q^{7} -8 q^{8} + ( 26 - 26 \zeta_{6} ) q^{9} + 14 \zeta_{6} q^{10} -35 \zeta_{6} q^{11} + ( 4 - 4 \zeta_{6} ) q^{12} + 66 q^{13} + ( -2 + 38 \zeta_{6} ) q^{14} -7 q^{15} + ( -16 + 16 \zeta_{6} ) q^{16} -59 \zeta_{6} q^{17} -52 \zeta_{6} q^{18} + ( -137 + 137 \zeta_{6} ) q^{19} + 28 q^{20} + ( -18 - \zeta_{6} ) q^{21} -70 q^{22} + ( 7 - 7 \zeta_{6} ) q^{23} -8 \zeta_{6} q^{24} + 76 \zeta_{6} q^{25} + ( 132 - 132 \zeta_{6} ) q^{26} + 53 q^{27} + ( 72 + 4 \zeta_{6} ) q^{28} + 106 q^{29} + ( -14 + 14 \zeta_{6} ) q^{30} -75 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( 35 - 35 \zeta_{6} ) q^{33} -118 q^{34} + ( 7 - 133 \zeta_{6} ) q^{35} -104 q^{36} + ( -11 + 11 \zeta_{6} ) q^{37} + 274 \zeta_{6} q^{38} + 66 \zeta_{6} q^{39} + ( 56 - 56 \zeta_{6} ) q^{40} -498 q^{41} + ( -38 + 36 \zeta_{6} ) q^{42} + 260 q^{43} + ( -140 + 140 \zeta_{6} ) q^{44} + 182 \zeta_{6} q^{45} -14 \zeta_{6} q^{46} + ( 171 - 171 \zeta_{6} ) q^{47} -16 q^{48} + ( 37 - 360 \zeta_{6} ) q^{49} + 152 q^{50} + ( 59 - 59 \zeta_{6} ) q^{51} -264 \zeta_{6} q^{52} + 417 \zeta_{6} q^{53} + ( 106 - 106 \zeta_{6} ) q^{54} + 245 q^{55} + ( 152 - 144 \zeta_{6} ) q^{56} -137 q^{57} + ( 212 - 212 \zeta_{6} ) q^{58} + 17 \zeta_{6} q^{59} + 28 \zeta_{6} q^{60} + ( -51 + 51 \zeta_{6} ) q^{61} -150 q^{62} + ( -26 + 494 \zeta_{6} ) q^{63} + 64 q^{64} + ( -462 + 462 \zeta_{6} ) q^{65} -70 \zeta_{6} q^{66} -439 \zeta_{6} q^{67} + ( -236 + 236 \zeta_{6} ) q^{68} + 7 q^{69} + ( -252 - 14 \zeta_{6} ) q^{70} -784 q^{71} + ( -208 + 208 \zeta_{6} ) q^{72} -295 \zeta_{6} q^{73} + 22 \zeta_{6} q^{74} + ( -76 + 76 \zeta_{6} ) q^{75} + 548 q^{76} + ( 630 + 35 \zeta_{6} ) q^{77} + 132 q^{78} + ( 495 - 495 \zeta_{6} ) q^{79} -112 \zeta_{6} q^{80} -649 \zeta_{6} q^{81} + ( -996 + 996 \zeta_{6} ) q^{82} + 932 q^{83} + ( -4 + 76 \zeta_{6} ) q^{84} + 413 q^{85} + ( 520 - 520 \zeta_{6} ) q^{86} + 106 \zeta_{6} q^{87} + 280 \zeta_{6} q^{88} + ( 873 - 873 \zeta_{6} ) q^{89} + 364 q^{90} + ( -1254 + 1188 \zeta_{6} ) q^{91} -28 q^{92} + ( 75 - 75 \zeta_{6} ) q^{93} -342 \zeta_{6} q^{94} -959 \zeta_{6} q^{95} + ( -32 + 32 \zeta_{6} ) q^{96} -290 q^{97} + ( -646 - 74 \zeta_{6} ) q^{98} -910 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + q^{3} - 4q^{4} - 7q^{5} + 4q^{6} - 20q^{7} - 16q^{8} + 26q^{9} + O(q^{10})$$ $$2q + 2q^{2} + q^{3} - 4q^{4} - 7q^{5} + 4q^{6} - 20q^{7} - 16q^{8} + 26q^{9} + 14q^{10} - 35q^{11} + 4q^{12} + 132q^{13} + 34q^{14} - 14q^{15} - 16q^{16} - 59q^{17} - 52q^{18} - 137q^{19} + 56q^{20} - 37q^{21} - 140q^{22} + 7q^{23} - 8q^{24} + 76q^{25} + 132q^{26} + 106q^{27} + 148q^{28} + 212q^{29} - 14q^{30} - 75q^{31} + 32q^{32} + 35q^{33} - 236q^{34} - 119q^{35} - 208q^{36} - 11q^{37} + 274q^{38} + 66q^{39} + 56q^{40} - 996q^{41} - 40q^{42} + 520q^{43} - 140q^{44} + 182q^{45} - 14q^{46} + 171q^{47} - 32q^{48} - 286q^{49} + 304q^{50} + 59q^{51} - 264q^{52} + 417q^{53} + 106q^{54} + 490q^{55} + 160q^{56} - 274q^{57} + 212q^{58} + 17q^{59} + 28q^{60} - 51q^{61} - 300q^{62} + 442q^{63} + 128q^{64} - 462q^{65} - 70q^{66} - 439q^{67} - 236q^{68} + 14q^{69} - 518q^{70} - 1568q^{71} - 208q^{72} - 295q^{73} + 22q^{74} - 76q^{75} + 1096q^{76} + 1295q^{77} + 264q^{78} + 495q^{79} - 112q^{80} - 649q^{81} - 996q^{82} + 1864q^{83} + 68q^{84} + 826q^{85} + 520q^{86} + 106q^{87} + 280q^{88} + 873q^{89} + 728q^{90} - 1320q^{91} - 56q^{92} + 75q^{93} - 342q^{94} - 959q^{95} - 32q^{96} - 580q^{97} - 1366q^{98} - 1820q^{99} + O(q^{100})$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/14\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.00000 1.73205i 0.500000 + 0.866025i −2.00000 3.46410i −3.50000 + 6.06218i 2.00000 −10.0000 + 15.5885i −8.00000 13.0000 22.5167i 7.00000 + 12.1244i
11.1 1.00000 + 1.73205i 0.500000 0.866025i −2.00000 + 3.46410i −3.50000 6.06218i 2.00000 −10.0000 15.5885i −8.00000 13.0000 + 22.5167i 7.00000 12.1244i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
7.c Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{4}^{\mathrm{new}}(14, [\chi])$$.