# Properties

 Label 14.4.c.a 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 $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.826026740080$$ 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} + 5 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 9 \zeta_{6} + 9) q^{5} - 10 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} + 8 q^{8} + ( - 2 \zeta_{6} + 2) q^{9} +O(q^{10})$$ q + (2*z - 2) * q^2 + 5*z * q^3 - 4*z * q^4 + (-9*z + 9) * q^5 - 10 * q^6 + (-14*z - 7) * q^7 + 8 * q^8 + (-2*z + 2) * q^9 $$q + (2 \zeta_{6} - 2) q^{2} + 5 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 9 \zeta_{6} + 9) q^{5} - 10 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} + 8 q^{8} + ( - 2 \zeta_{6} + 2) q^{9} + 18 \zeta_{6} q^{10} + 57 \zeta_{6} q^{11} + ( - 20 \zeta_{6} + 20) q^{12} - 70 q^{13} + ( - 14 \zeta_{6} + 42) q^{14} + 45 q^{15} + (16 \zeta_{6} - 16) q^{16} - 51 \zeta_{6} q^{17} + 4 \zeta_{6} q^{18} + (5 \zeta_{6} - 5) q^{19} - 36 q^{20} + ( - 105 \zeta_{6} + 70) q^{21} - 114 q^{22} + (69 \zeta_{6} - 69) q^{23} + 40 \zeta_{6} q^{24} + 44 \zeta_{6} q^{25} + ( - 140 \zeta_{6} + 140) q^{26} + 145 q^{27} + (84 \zeta_{6} - 56) q^{28} + 114 q^{29} + (90 \zeta_{6} - 90) q^{30} - 23 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + (285 \zeta_{6} - 285) q^{33} + 102 q^{34} + (63 \zeta_{6} - 189) q^{35} - 8 q^{36} + ( - 253 \zeta_{6} + 253) q^{37} - 10 \zeta_{6} q^{38} - 350 \zeta_{6} q^{39} + ( - 72 \zeta_{6} + 72) q^{40} - 42 q^{41} + (140 \zeta_{6} + 70) q^{42} - 124 q^{43} + ( - 228 \zeta_{6} + 228) q^{44} - 18 \zeta_{6} q^{45} - 138 \zeta_{6} q^{46} + (201 \zeta_{6} - 201) q^{47} - 80 q^{48} + (392 \zeta_{6} - 147) q^{49} - 88 q^{50} + ( - 255 \zeta_{6} + 255) q^{51} + 280 \zeta_{6} q^{52} + 393 \zeta_{6} q^{53} + (290 \zeta_{6} - 290) q^{54} + 513 q^{55} + ( - 112 \zeta_{6} - 56) q^{56} - 25 q^{57} + (228 \zeta_{6} - 228) q^{58} - 219 \zeta_{6} q^{59} - 180 \zeta_{6} q^{60} + ( - 709 \zeta_{6} + 709) q^{61} + 46 q^{62} + (14 \zeta_{6} - 42) q^{63} + 64 q^{64} + (630 \zeta_{6} - 630) q^{65} - 570 \zeta_{6} q^{66} - 419 \zeta_{6} q^{67} + (204 \zeta_{6} - 204) q^{68} - 345 q^{69} + ( - 378 \zeta_{6} + 252) q^{70} - 96 q^{71} + ( - 16 \zeta_{6} + 16) q^{72} + 313 \zeta_{6} q^{73} + 506 \zeta_{6} q^{74} + (220 \zeta_{6} - 220) q^{75} + 20 q^{76} + ( - 1197 \zeta_{6} + 798) q^{77} + 700 q^{78} + (461 \zeta_{6} - 461) q^{79} + 144 \zeta_{6} q^{80} + 671 \zeta_{6} q^{81} + ( - 84 \zeta_{6} + 84) q^{82} - 588 q^{83} + (140 \zeta_{6} - 420) q^{84} - 459 q^{85} + ( - 248 \zeta_{6} + 248) q^{86} + 570 \zeta_{6} q^{87} + 456 \zeta_{6} q^{88} + ( - 1017 \zeta_{6} + 1017) q^{89} + 36 q^{90} + (980 \zeta_{6} + 490) q^{91} + 276 q^{92} + ( - 115 \zeta_{6} + 115) q^{93} - 402 \zeta_{6} q^{94} + 45 \zeta_{6} q^{95} + ( - 160 \zeta_{6} + 160) q^{96} - 1834 q^{97} + ( - 294 \zeta_{6} - 490) q^{98} + 114 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^2 + 5*z * q^3 - 4*z * q^4 + (-9*z + 9) * q^5 - 10 * q^6 + (-14*z - 7) * q^7 + 8 * q^8 + (-2*z + 2) * q^9 + 18*z * q^10 + 57*z * q^11 + (-20*z + 20) * q^12 - 70 * q^13 + (-14*z + 42) * q^14 + 45 * q^15 + (16*z - 16) * q^16 - 51*z * q^17 + 4*z * q^18 + (5*z - 5) * q^19 - 36 * q^20 + (-105*z + 70) * q^21 - 114 * q^22 + (69*z - 69) * q^23 + 40*z * q^24 + 44*z * q^25 + (-140*z + 140) * q^26 + 145 * q^27 + (84*z - 56) * q^28 + 114 * q^29 + (90*z - 90) * q^30 - 23*z * q^31 - 32*z * q^32 + (285*z - 285) * q^33 + 102 * q^34 + (63*z - 189) * q^35 - 8 * q^36 + (-253*z + 253) * q^37 - 10*z * q^38 - 350*z * q^39 + (-72*z + 72) * q^40 - 42 * q^41 + (140*z + 70) * q^42 - 124 * q^43 + (-228*z + 228) * q^44 - 18*z * q^45 - 138*z * q^46 + (201*z - 201) * q^47 - 80 * q^48 + (392*z - 147) * q^49 - 88 * q^50 + (-255*z + 255) * q^51 + 280*z * q^52 + 393*z * q^53 + (290*z - 290) * q^54 + 513 * q^55 + (-112*z - 56) * q^56 - 25 * q^57 + (228*z - 228) * q^58 - 219*z * q^59 - 180*z * q^60 + (-709*z + 709) * q^61 + 46 * q^62 + (14*z - 42) * q^63 + 64 * q^64 + (630*z - 630) * q^65 - 570*z * q^66 - 419*z * q^67 + (204*z - 204) * q^68 - 345 * q^69 + (-378*z + 252) * q^70 - 96 * q^71 + (-16*z + 16) * q^72 + 313*z * q^73 + 506*z * q^74 + (220*z - 220) * q^75 + 20 * q^76 + (-1197*z + 798) * q^77 + 700 * q^78 + (461*z - 461) * q^79 + 144*z * q^80 + 671*z * q^81 + (-84*z + 84) * q^82 - 588 * q^83 + (140*z - 420) * q^84 - 459 * q^85 + (-248*z + 248) * q^86 + 570*z * q^87 + 456*z * q^88 + (-1017*z + 1017) * q^89 + 36 * q^90 + (980*z + 490) * q^91 + 276 * q^92 + (-115*z + 115) * q^93 - 402*z * q^94 + 45*z * q^95 + (-160*z + 160) * q^96 - 1834 * q^97 + (-294*z - 490) * q^98 + 114 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 5 q^{3} - 4 q^{4} + 9 q^{5} - 20 q^{6} - 28 q^{7} + 16 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 5 * q^3 - 4 * q^4 + 9 * q^5 - 20 * q^6 - 28 * q^7 + 16 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 5 q^{3} - 4 q^{4} + 9 q^{5} - 20 q^{6} - 28 q^{7} + 16 q^{8} + 2 q^{9} + 18 q^{10} + 57 q^{11} + 20 q^{12} - 140 q^{13} + 70 q^{14} + 90 q^{15} - 16 q^{16} - 51 q^{17} + 4 q^{18} - 5 q^{19} - 72 q^{20} + 35 q^{21} - 228 q^{22} - 69 q^{23} + 40 q^{24} + 44 q^{25} + 140 q^{26} + 290 q^{27} - 28 q^{28} + 228 q^{29} - 90 q^{30} - 23 q^{31} - 32 q^{32} - 285 q^{33} + 204 q^{34} - 315 q^{35} - 16 q^{36} + 253 q^{37} - 10 q^{38} - 350 q^{39} + 72 q^{40} - 84 q^{41} + 280 q^{42} - 248 q^{43} + 228 q^{44} - 18 q^{45} - 138 q^{46} - 201 q^{47} - 160 q^{48} + 98 q^{49} - 176 q^{50} + 255 q^{51} + 280 q^{52} + 393 q^{53} - 290 q^{54} + 1026 q^{55} - 224 q^{56} - 50 q^{57} - 228 q^{58} - 219 q^{59} - 180 q^{60} + 709 q^{61} + 92 q^{62} - 70 q^{63} + 128 q^{64} - 630 q^{65} - 570 q^{66} - 419 q^{67} - 204 q^{68} - 690 q^{69} + 126 q^{70} - 192 q^{71} + 16 q^{72} + 313 q^{73} + 506 q^{74} - 220 q^{75} + 40 q^{76} + 399 q^{77} + 1400 q^{78} - 461 q^{79} + 144 q^{80} + 671 q^{81} + 84 q^{82} - 1176 q^{83} - 700 q^{84} - 918 q^{85} + 248 q^{86} + 570 q^{87} + 456 q^{88} + 1017 q^{89} + 72 q^{90} + 1960 q^{91} + 552 q^{92} + 115 q^{93} - 402 q^{94} + 45 q^{95} + 160 q^{96} - 3668 q^{97} - 1274 q^{98} + 228 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 5 * q^3 - 4 * q^4 + 9 * q^5 - 20 * q^6 - 28 * q^7 + 16 * q^8 + 2 * q^9 + 18 * q^10 + 57 * q^11 + 20 * q^12 - 140 * q^13 + 70 * q^14 + 90 * q^15 - 16 * q^16 - 51 * q^17 + 4 * q^18 - 5 * q^19 - 72 * q^20 + 35 * q^21 - 228 * q^22 - 69 * q^23 + 40 * q^24 + 44 * q^25 + 140 * q^26 + 290 * q^27 - 28 * q^28 + 228 * q^29 - 90 * q^30 - 23 * q^31 - 32 * q^32 - 285 * q^33 + 204 * q^34 - 315 * q^35 - 16 * q^36 + 253 * q^37 - 10 * q^38 - 350 * q^39 + 72 * q^40 - 84 * q^41 + 280 * q^42 - 248 * q^43 + 228 * q^44 - 18 * q^45 - 138 * q^46 - 201 * q^47 - 160 * q^48 + 98 * q^49 - 176 * q^50 + 255 * q^51 + 280 * q^52 + 393 * q^53 - 290 * q^54 + 1026 * q^55 - 224 * q^56 - 50 * q^57 - 228 * q^58 - 219 * q^59 - 180 * q^60 + 709 * q^61 + 92 * q^62 - 70 * q^63 + 128 * q^64 - 630 * q^65 - 570 * q^66 - 419 * q^67 - 204 * q^68 - 690 * q^69 + 126 * q^70 - 192 * q^71 + 16 * q^72 + 313 * q^73 + 506 * q^74 - 220 * q^75 + 40 * q^76 + 399 * q^77 + 1400 * q^78 - 461 * q^79 + 144 * q^80 + 671 * q^81 + 84 * q^82 - 1176 * q^83 - 700 * q^84 - 918 * q^85 + 248 * q^86 + 570 * q^87 + 456 * q^88 + 1017 * q^89 + 72 * q^90 + 1960 * q^91 + 552 * q^92 + 115 * q^93 - 402 * q^94 + 45 * q^95 + 160 * q^96 - 3668 * q^97 - 1274 * q^98 + 228 * q^99

## 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 2.50000 + 4.33013i −2.00000 3.46410i 4.50000 7.79423i −10.0000 −14.0000 12.1244i 8.00000 1.00000 1.73205i 9.00000 + 15.5885i
11.1 −1.00000 1.73205i 2.50000 4.33013i −2.00000 + 3.46410i 4.50000 + 7.79423i −10.0000 −14.0000 + 12.1244i 8.00000 1.00000 + 1.73205i 9.00000 15.5885i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.4.c.a 2
3.b odd 2 1 126.4.g.d 2
4.b odd 2 1 112.4.i.a 2
5.b even 2 1 350.4.e.e 2
5.c odd 4 2 350.4.j.b 4
7.b odd 2 1 98.4.c.a 2
7.c even 3 1 inner 14.4.c.a 2
7.c even 3 1 98.4.a.d 1
7.d odd 6 1 98.4.a.f 1
7.d odd 6 1 98.4.c.a 2
8.b even 2 1 448.4.i.b 2
8.d odd 2 1 448.4.i.e 2
21.c even 2 1 882.4.g.u 2
21.g even 6 1 882.4.a.c 1
21.g even 6 1 882.4.g.u 2
21.h odd 6 1 126.4.g.d 2
21.h odd 6 1 882.4.a.f 1
28.f even 6 1 784.4.a.c 1
28.g odd 6 1 112.4.i.a 2
28.g odd 6 1 784.4.a.p 1
35.i odd 6 1 2450.4.a.d 1
35.j even 6 1 350.4.e.e 2
35.j even 6 1 2450.4.a.q 1
35.l odd 12 2 350.4.j.b 4
56.k odd 6 1 448.4.i.e 2
56.p even 6 1 448.4.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 1.a even 1 1 trivial
14.4.c.a 2 7.c even 3 1 inner
98.4.a.d 1 7.c even 3 1
98.4.a.f 1 7.d odd 6 1
98.4.c.a 2 7.b odd 2 1
98.4.c.a 2 7.d odd 6 1
112.4.i.a 2 4.b odd 2 1
112.4.i.a 2 28.g odd 6 1
126.4.g.d 2 3.b odd 2 1
126.4.g.d 2 21.h odd 6 1
350.4.e.e 2 5.b even 2 1
350.4.e.e 2 35.j even 6 1
350.4.j.b 4 5.c odd 4 2
350.4.j.b 4 35.l odd 12 2
448.4.i.b 2 8.b even 2 1
448.4.i.b 2 56.p even 6 1
448.4.i.e 2 8.d odd 2 1
448.4.i.e 2 56.k odd 6 1
784.4.a.c 1 28.f even 6 1
784.4.a.p 1 28.g odd 6 1
882.4.a.c 1 21.g even 6 1
882.4.a.f 1 21.h odd 6 1
882.4.g.u 2 21.c even 2 1
882.4.g.u 2 21.g even 6 1
2450.4.a.d 1 35.i odd 6 1
2450.4.a.q 1 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 5T_{3} + 25$$ acting on $$S_{4}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} - 5T + 25$$
$5$ $$T^{2} - 9T + 81$$
$7$ $$T^{2} + 28T + 343$$
$11$ $$T^{2} - 57T + 3249$$
$13$ $$(T + 70)^{2}$$
$17$ $$T^{2} + 51T + 2601$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} + 69T + 4761$$
$29$ $$(T - 114)^{2}$$
$31$ $$T^{2} + 23T + 529$$
$37$ $$T^{2} - 253T + 64009$$
$41$ $$(T + 42)^{2}$$
$43$ $$(T + 124)^{2}$$
$47$ $$T^{2} + 201T + 40401$$
$53$ $$T^{2} - 393T + 154449$$
$59$ $$T^{2} + 219T + 47961$$
$61$ $$T^{2} - 709T + 502681$$
$67$ $$T^{2} + 419T + 175561$$
$71$ $$(T + 96)^{2}$$
$73$ $$T^{2} - 313T + 97969$$
$79$ $$T^{2} + 461T + 212521$$
$83$ $$(T + 588)^{2}$$
$89$ $$T^{2} - 1017 T + 1034289$$
$97$ $$(T + 1834)^{2}$$