# 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 $$3$$, 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 2) q^{2} + \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + (7 \zeta_{6} - 7) q^{5} + 2 q^{6} + (18 \zeta_{6} - 19) q^{7} - 8 q^{8} + ( - 26 \zeta_{6} + 26) q^{9} +O(q^{10})$$ q + (-2*z + 2) * q^2 + z * q^3 - 4*z * q^4 + (7*z - 7) * q^5 + 2 * q^6 + (18*z - 19) * q^7 - 8 * q^8 + (-26*z + 26) * q^9 $$q + ( - 2 \zeta_{6} + 2) q^{2} + \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + (7 \zeta_{6} - 7) q^{5} + 2 q^{6} + (18 \zeta_{6} - 19) q^{7} - 8 q^{8} + ( - 26 \zeta_{6} + 26) q^{9} + 14 \zeta_{6} q^{10} - 35 \zeta_{6} q^{11} + ( - 4 \zeta_{6} + 4) q^{12} + 66 q^{13} + (38 \zeta_{6} - 2) q^{14} - 7 q^{15} + (16 \zeta_{6} - 16) q^{16} - 59 \zeta_{6} q^{17} - 52 \zeta_{6} q^{18} + (137 \zeta_{6} - 137) q^{19} + 28 q^{20} + ( - \zeta_{6} - 18) q^{21} - 70 q^{22} + ( - 7 \zeta_{6} + 7) q^{23} - 8 \zeta_{6} q^{24} + 76 \zeta_{6} q^{25} + ( - 132 \zeta_{6} + 132) q^{26} + 53 q^{27} + (4 \zeta_{6} + 72) q^{28} + 106 q^{29} + (14 \zeta_{6} - 14) q^{30} - 75 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( - 35 \zeta_{6} + 35) q^{33} - 118 q^{34} + ( - 133 \zeta_{6} + 7) q^{35} - 104 q^{36} + (11 \zeta_{6} - 11) q^{37} + 274 \zeta_{6} q^{38} + 66 \zeta_{6} q^{39} + ( - 56 \zeta_{6} + 56) q^{40} - 498 q^{41} + (36 \zeta_{6} - 38) q^{42} + 260 q^{43} + (140 \zeta_{6} - 140) q^{44} + 182 \zeta_{6} q^{45} - 14 \zeta_{6} q^{46} + ( - 171 \zeta_{6} + 171) q^{47} - 16 q^{48} + ( - 360 \zeta_{6} + 37) q^{49} + 152 q^{50} + ( - 59 \zeta_{6} + 59) q^{51} - 264 \zeta_{6} q^{52} + 417 \zeta_{6} q^{53} + ( - 106 \zeta_{6} + 106) q^{54} + 245 q^{55} + ( - 144 \zeta_{6} + 152) q^{56} - 137 q^{57} + ( - 212 \zeta_{6} + 212) q^{58} + 17 \zeta_{6} q^{59} + 28 \zeta_{6} q^{60} + (51 \zeta_{6} - 51) q^{61} - 150 q^{62} + (494 \zeta_{6} - 26) q^{63} + 64 q^{64} + (462 \zeta_{6} - 462) q^{65} - 70 \zeta_{6} q^{66} - 439 \zeta_{6} q^{67} + (236 \zeta_{6} - 236) q^{68} + 7 q^{69} + ( - 14 \zeta_{6} - 252) q^{70} - 784 q^{71} + (208 \zeta_{6} - 208) q^{72} - 295 \zeta_{6} q^{73} + 22 \zeta_{6} q^{74} + (76 \zeta_{6} - 76) q^{75} + 548 q^{76} + (35 \zeta_{6} + 630) q^{77} + 132 q^{78} + ( - 495 \zeta_{6} + 495) q^{79} - 112 \zeta_{6} q^{80} - 649 \zeta_{6} q^{81} + (996 \zeta_{6} - 996) q^{82} + 932 q^{83} + (76 \zeta_{6} - 4) q^{84} + 413 q^{85} + ( - 520 \zeta_{6} + 520) q^{86} + 106 \zeta_{6} q^{87} + 280 \zeta_{6} q^{88} + ( - 873 \zeta_{6} + 873) q^{89} + 364 q^{90} + (1188 \zeta_{6} - 1254) q^{91} - 28 q^{92} + ( - 75 \zeta_{6} + 75) q^{93} - 342 \zeta_{6} q^{94} - 959 \zeta_{6} q^{95} + (32 \zeta_{6} - 32) q^{96} - 290 q^{97} + ( - 74 \zeta_{6} - 646) q^{98} - 910 q^{99} +O(q^{100})$$ q + (-2*z + 2) * q^2 + z * q^3 - 4*z * q^4 + (7*z - 7) * q^5 + 2 * q^6 + (18*z - 19) * q^7 - 8 * q^8 + (-26*z + 26) * q^9 + 14*z * q^10 - 35*z * q^11 + (-4*z + 4) * q^12 + 66 * q^13 + (38*z - 2) * q^14 - 7 * q^15 + (16*z - 16) * q^16 - 59*z * q^17 - 52*z * q^18 + (137*z - 137) * q^19 + 28 * q^20 + (-z - 18) * q^21 - 70 * q^22 + (-7*z + 7) * q^23 - 8*z * q^24 + 76*z * q^25 + (-132*z + 132) * q^26 + 53 * q^27 + (4*z + 72) * q^28 + 106 * q^29 + (14*z - 14) * q^30 - 75*z * q^31 + 32*z * q^32 + (-35*z + 35) * q^33 - 118 * q^34 + (-133*z + 7) * q^35 - 104 * q^36 + (11*z - 11) * q^37 + 274*z * q^38 + 66*z * q^39 + (-56*z + 56) * q^40 - 498 * q^41 + (36*z - 38) * q^42 + 260 * q^43 + (140*z - 140) * q^44 + 182*z * q^45 - 14*z * q^46 + (-171*z + 171) * q^47 - 16 * q^48 + (-360*z + 37) * q^49 + 152 * q^50 + (-59*z + 59) * q^51 - 264*z * q^52 + 417*z * q^53 + (-106*z + 106) * q^54 + 245 * q^55 + (-144*z + 152) * q^56 - 137 * q^57 + (-212*z + 212) * q^58 + 17*z * q^59 + 28*z * q^60 + (51*z - 51) * q^61 - 150 * q^62 + (494*z - 26) * q^63 + 64 * q^64 + (462*z - 462) * q^65 - 70*z * q^66 - 439*z * q^67 + (236*z - 236) * q^68 + 7 * q^69 + (-14*z - 252) * q^70 - 784 * q^71 + (208*z - 208) * q^72 - 295*z * q^73 + 22*z * q^74 + (76*z - 76) * q^75 + 548 * q^76 + (35*z + 630) * q^77 + 132 * q^78 + (-495*z + 495) * q^79 - 112*z * q^80 - 649*z * q^81 + (996*z - 996) * q^82 + 932 * q^83 + (76*z - 4) * q^84 + 413 * q^85 + (-520*z + 520) * q^86 + 106*z * q^87 + 280*z * q^88 + (-873*z + 873) * q^89 + 364 * q^90 + (1188*z - 1254) * q^91 - 28 * q^92 + (-75*z + 75) * q^93 - 342*z * q^94 - 959*z * q^95 + (32*z - 32) * q^96 - 290 * q^97 + (-74*z - 646) * q^98 - 910 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} - 4 q^{4} - 7 q^{5} + 4 q^{6} - 20 q^{7} - 16 q^{8} + 26 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 - 4 * q^4 - 7 * q^5 + 4 * q^6 - 20 * q^7 - 16 * q^8 + 26 * q^9 $$2 q + 2 q^{2} + q^{3} - 4 q^{4} - 7 q^{5} + 4 q^{6} - 20 q^{7} - 16 q^{8} + 26 q^{9} + 14 q^{10} - 35 q^{11} + 4 q^{12} + 132 q^{13} + 34 q^{14} - 14 q^{15} - 16 q^{16} - 59 q^{17} - 52 q^{18} - 137 q^{19} + 56 q^{20} - 37 q^{21} - 140 q^{22} + 7 q^{23} - 8 q^{24} + 76 q^{25} + 132 q^{26} + 106 q^{27} + 148 q^{28} + 212 q^{29} - 14 q^{30} - 75 q^{31} + 32 q^{32} + 35 q^{33} - 236 q^{34} - 119 q^{35} - 208 q^{36} - 11 q^{37} + 274 q^{38} + 66 q^{39} + 56 q^{40} - 996 q^{41} - 40 q^{42} + 520 q^{43} - 140 q^{44} + 182 q^{45} - 14 q^{46} + 171 q^{47} - 32 q^{48} - 286 q^{49} + 304 q^{50} + 59 q^{51} - 264 q^{52} + 417 q^{53} + 106 q^{54} + 490 q^{55} + 160 q^{56} - 274 q^{57} + 212 q^{58} + 17 q^{59} + 28 q^{60} - 51 q^{61} - 300 q^{62} + 442 q^{63} + 128 q^{64} - 462 q^{65} - 70 q^{66} - 439 q^{67} - 236 q^{68} + 14 q^{69} - 518 q^{70} - 1568 q^{71} - 208 q^{72} - 295 q^{73} + 22 q^{74} - 76 q^{75} + 1096 q^{76} + 1295 q^{77} + 264 q^{78} + 495 q^{79} - 112 q^{80} - 649 q^{81} - 996 q^{82} + 1864 q^{83} + 68 q^{84} + 826 q^{85} + 520 q^{86} + 106 q^{87} + 280 q^{88} + 873 q^{89} + 728 q^{90} - 1320 q^{91} - 56 q^{92} + 75 q^{93} - 342 q^{94} - 959 q^{95} - 32 q^{96} - 580 q^{97} - 1366 q^{98} - 1820 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 - 4 * q^4 - 7 * q^5 + 4 * q^6 - 20 * q^7 - 16 * q^8 + 26 * q^9 + 14 * q^10 - 35 * q^11 + 4 * q^12 + 132 * q^13 + 34 * q^14 - 14 * q^15 - 16 * q^16 - 59 * q^17 - 52 * q^18 - 137 * q^19 + 56 * q^20 - 37 * q^21 - 140 * q^22 + 7 * q^23 - 8 * q^24 + 76 * q^25 + 132 * q^26 + 106 * q^27 + 148 * q^28 + 212 * q^29 - 14 * q^30 - 75 * q^31 + 32 * q^32 + 35 * q^33 - 236 * q^34 - 119 * q^35 - 208 * q^36 - 11 * q^37 + 274 * q^38 + 66 * q^39 + 56 * q^40 - 996 * q^41 - 40 * q^42 + 520 * q^43 - 140 * q^44 + 182 * q^45 - 14 * q^46 + 171 * q^47 - 32 * q^48 - 286 * q^49 + 304 * q^50 + 59 * q^51 - 264 * q^52 + 417 * q^53 + 106 * q^54 + 490 * q^55 + 160 * q^56 - 274 * q^57 + 212 * q^58 + 17 * q^59 + 28 * q^60 - 51 * q^61 - 300 * q^62 + 442 * q^63 + 128 * q^64 - 462 * q^65 - 70 * q^66 - 439 * q^67 - 236 * q^68 + 14 * q^69 - 518 * q^70 - 1568 * q^71 - 208 * q^72 - 295 * q^73 + 22 * q^74 - 76 * q^75 + 1096 * q^76 + 1295 * q^77 + 264 * q^78 + 495 * q^79 - 112 * q^80 - 649 * q^81 - 996 * q^82 + 1864 * q^83 + 68 * q^84 + 826 * q^85 + 520 * q^86 + 106 * q^87 + 280 * q^88 + 873 * q^89 + 728 * q^90 - 1320 * q^91 - 56 * q^92 + 75 * q^93 - 342 * q^94 - 959 * q^95 - 32 * q^96 - 580 * q^97 - 1366 * q^98 - 1820 * 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 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 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.b 2
3.b odd 2 1 126.4.g.c 2
4.b odd 2 1 112.4.i.b 2
5.b even 2 1 350.4.e.b 2
5.c odd 4 2 350.4.j.d 4
7.b odd 2 1 98.4.c.e 2
7.c even 3 1 inner 14.4.c.b 2
7.c even 3 1 98.4.a.b 1
7.d odd 6 1 98.4.a.c 1
7.d odd 6 1 98.4.c.e 2
8.b even 2 1 448.4.i.c 2
8.d odd 2 1 448.4.i.d 2
21.c even 2 1 882.4.g.d 2
21.g even 6 1 882.4.a.p 1
21.g even 6 1 882.4.g.d 2
21.h odd 6 1 126.4.g.c 2
21.h odd 6 1 882.4.a.k 1
28.f even 6 1 784.4.a.j 1
28.g odd 6 1 112.4.i.b 2
28.g odd 6 1 784.4.a.l 1
35.i odd 6 1 2450.4.a.bf 1
35.j even 6 1 350.4.e.b 2
35.j even 6 1 2450.4.a.bh 1
35.l odd 12 2 350.4.j.d 4
56.k odd 6 1 448.4.i.d 2
56.p even 6 1 448.4.i.c 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 7T + 49$$
$7$ $$T^{2} + 20T + 343$$
$11$ $$T^{2} + 35T + 1225$$
$13$ $$(T - 66)^{2}$$
$17$ $$T^{2} + 59T + 3481$$
$19$ $$T^{2} + 137T + 18769$$
$23$ $$T^{2} - 7T + 49$$
$29$ $$(T - 106)^{2}$$
$31$ $$T^{2} + 75T + 5625$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$(T + 498)^{2}$$
$43$ $$(T - 260)^{2}$$
$47$ $$T^{2} - 171T + 29241$$
$53$ $$T^{2} - 417T + 173889$$
$59$ $$T^{2} - 17T + 289$$
$61$ $$T^{2} + 51T + 2601$$
$67$ $$T^{2} + 439T + 192721$$
$71$ $$(T + 784)^{2}$$
$73$ $$T^{2} + 295T + 87025$$
$79$ $$T^{2} - 495T + 245025$$
$83$ $$(T - 932)^{2}$$
$89$ $$T^{2} - 873T + 762129$$
$97$ $$(T + 290)^{2}$$