# Properties

 Label 14.6.a.b Level $14$ Weight $6$ Character orbit 14.a Self dual yes Analytic conductor $2.245$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 14.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.24537347738$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{2} + 8q^{3} + 16q^{4} + 10q^{5} + 32q^{6} - 49q^{7} + 64q^{8} - 179q^{9} + O(q^{10})$$ $$q + 4q^{2} + 8q^{3} + 16q^{4} + 10q^{5} + 32q^{6} - 49q^{7} + 64q^{8} - 179q^{9} + 40q^{10} - 340q^{11} + 128q^{12} - 294q^{13} - 196q^{14} + 80q^{15} + 256q^{16} + 1226q^{17} - 716q^{18} + 2432q^{19} + 160q^{20} - 392q^{21} - 1360q^{22} + 2000q^{23} + 512q^{24} - 3025q^{25} - 1176q^{26} - 3376q^{27} - 784q^{28} - 6746q^{29} + 320q^{30} + 8856q^{31} + 1024q^{32} - 2720q^{33} + 4904q^{34} - 490q^{35} - 2864q^{36} + 9182q^{37} + 9728q^{38} - 2352q^{39} + 640q^{40} - 14574q^{41} - 1568q^{42} + 8108q^{43} - 5440q^{44} - 1790q^{45} + 8000q^{46} - 312q^{47} + 2048q^{48} + 2401q^{49} - 12100q^{50} + 9808q^{51} - 4704q^{52} - 14634q^{53} - 13504q^{54} - 3400q^{55} - 3136q^{56} + 19456q^{57} - 26984q^{58} - 27656q^{59} + 1280q^{60} + 34338q^{61} + 35424q^{62} + 8771q^{63} + 4096q^{64} - 2940q^{65} - 10880q^{66} + 12316q^{67} + 19616q^{68} + 16000q^{69} - 1960q^{70} + 36920q^{71} - 11456q^{72} - 61718q^{73} + 36728q^{74} - 24200q^{75} + 38912q^{76} + 16660q^{77} - 9408q^{78} - 64752q^{79} + 2560q^{80} + 16489q^{81} - 58296q^{82} - 77056q^{83} - 6272q^{84} + 12260q^{85} + 32432q^{86} - 53968q^{87} - 21760q^{88} - 8166q^{89} - 7160q^{90} + 14406q^{91} + 32000q^{92} + 70848q^{93} - 1248q^{94} + 24320q^{95} + 8192q^{96} + 20650q^{97} + 9604q^{98} + 60860q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
4.00000 8.00000 16.0000 10.0000 32.0000 −49.0000 64.0000 −179.000 40.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.6.a.b 1
3.b odd 2 1 126.6.a.c 1
4.b odd 2 1 112.6.a.d 1
5.b even 2 1 350.6.a.b 1
5.c odd 4 2 350.6.c.f 2
7.b odd 2 1 98.6.a.b 1
7.c even 3 2 98.6.c.a 2
7.d odd 6 2 98.6.c.b 2
8.b even 2 1 448.6.a.f 1
8.d odd 2 1 448.6.a.k 1
12.b even 2 1 1008.6.a.n 1
21.c even 2 1 882.6.a.g 1
28.d even 2 1 784.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 1.a even 1 1 trivial
98.6.a.b 1 7.b odd 2 1
98.6.c.a 2 7.c even 3 2
98.6.c.b 2 7.d odd 6 2
112.6.a.d 1 4.b odd 2 1
126.6.a.c 1 3.b odd 2 1
350.6.a.b 1 5.b even 2 1
350.6.c.f 2 5.c odd 4 2
448.6.a.f 1 8.b even 2 1
448.6.a.k 1 8.d odd 2 1
784.6.a.h 1 28.d even 2 1
882.6.a.g 1 21.c even 2 1
1008.6.a.n 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 8$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(14))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T$$
$3$ $$-8 + T$$
$5$ $$-10 + T$$
$7$ $$49 + T$$
$11$ $$340 + T$$
$13$ $$294 + T$$
$17$ $$-1226 + T$$
$19$ $$-2432 + T$$
$23$ $$-2000 + T$$
$29$ $$6746 + T$$
$31$ $$-8856 + T$$
$37$ $$-9182 + T$$
$41$ $$14574 + T$$
$43$ $$-8108 + T$$
$47$ $$312 + T$$
$53$ $$14634 + T$$
$59$ $$27656 + T$$
$61$ $$-34338 + T$$
$67$ $$-12316 + T$$
$71$ $$-36920 + T$$
$73$ $$61718 + T$$
$79$ $$64752 + T$$
$83$ $$77056 + T$$
$89$ $$8166 + T$$
$97$ $$-20650 + T$$
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