# 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: $$x^{2} - x + 1$$ 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.

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

## 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 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 $$T_{3}^{2} - 5 T_{3} + 25$$ acting on $$S_{4}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

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