# Properties

 Label 14.4.a.a Level $14$ Weight $4$ Character orbit 14.a Self dual yes Analytic conductor $0.826$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 14.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.826026740080$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 14 q^{5} - 16 q^{6} - 7 q^{7} - 8 q^{8} + 37 q^{9}+O(q^{10})$$ q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 14 * q^5 - 16 * q^6 - 7 * q^7 - 8 * q^8 + 37 * q^9 $$q - 2 q^{2} + 8 q^{3} + 4 q^{4} - 14 q^{5} - 16 q^{6} - 7 q^{7} - 8 q^{8} + 37 q^{9} + 28 q^{10} - 28 q^{11} + 32 q^{12} + 18 q^{13} + 14 q^{14} - 112 q^{15} + 16 q^{16} + 74 q^{17} - 74 q^{18} + 80 q^{19} - 56 q^{20} - 56 q^{21} + 56 q^{22} - 112 q^{23} - 64 q^{24} + 71 q^{25} - 36 q^{26} + 80 q^{27} - 28 q^{28} + 190 q^{29} + 224 q^{30} + 72 q^{31} - 32 q^{32} - 224 q^{33} - 148 q^{34} + 98 q^{35} + 148 q^{36} - 346 q^{37} - 160 q^{38} + 144 q^{39} + 112 q^{40} + 162 q^{41} + 112 q^{42} - 412 q^{43} - 112 q^{44} - 518 q^{45} + 224 q^{46} + 24 q^{47} + 128 q^{48} + 49 q^{49} - 142 q^{50} + 592 q^{51} + 72 q^{52} + 318 q^{53} - 160 q^{54} + 392 q^{55} + 56 q^{56} + 640 q^{57} - 380 q^{58} - 200 q^{59} - 448 q^{60} - 198 q^{61} - 144 q^{62} - 259 q^{63} + 64 q^{64} - 252 q^{65} + 448 q^{66} - 716 q^{67} + 296 q^{68} - 896 q^{69} - 196 q^{70} + 392 q^{71} - 296 q^{72} + 538 q^{73} + 692 q^{74} + 568 q^{75} + 320 q^{76} + 196 q^{77} - 288 q^{78} + 240 q^{79} - 224 q^{80} - 359 q^{81} - 324 q^{82} - 1072 q^{83} - 224 q^{84} - 1036 q^{85} + 824 q^{86} + 1520 q^{87} + 224 q^{88} + 810 q^{89} + 1036 q^{90} - 126 q^{91} - 448 q^{92} + 576 q^{93} - 48 q^{94} - 1120 q^{95} - 256 q^{96} + 1354 q^{97} - 98 q^{98} - 1036 q^{99}+O(q^{100})$$ q - 2 * q^2 + 8 * q^3 + 4 * q^4 - 14 * q^5 - 16 * q^6 - 7 * q^7 - 8 * q^8 + 37 * q^9 + 28 * q^10 - 28 * q^11 + 32 * q^12 + 18 * q^13 + 14 * q^14 - 112 * q^15 + 16 * q^16 + 74 * q^17 - 74 * q^18 + 80 * q^19 - 56 * q^20 - 56 * q^21 + 56 * q^22 - 112 * q^23 - 64 * q^24 + 71 * q^25 - 36 * q^26 + 80 * q^27 - 28 * q^28 + 190 * q^29 + 224 * q^30 + 72 * q^31 - 32 * q^32 - 224 * q^33 - 148 * q^34 + 98 * q^35 + 148 * q^36 - 346 * q^37 - 160 * q^38 + 144 * q^39 + 112 * q^40 + 162 * q^41 + 112 * q^42 - 412 * q^43 - 112 * q^44 - 518 * q^45 + 224 * q^46 + 24 * q^47 + 128 * q^48 + 49 * q^49 - 142 * q^50 + 592 * q^51 + 72 * q^52 + 318 * q^53 - 160 * q^54 + 392 * q^55 + 56 * q^56 + 640 * q^57 - 380 * q^58 - 200 * q^59 - 448 * q^60 - 198 * q^61 - 144 * q^62 - 259 * q^63 + 64 * q^64 - 252 * q^65 + 448 * q^66 - 716 * q^67 + 296 * q^68 - 896 * q^69 - 196 * q^70 + 392 * q^71 - 296 * q^72 + 538 * q^73 + 692 * q^74 + 568 * q^75 + 320 * q^76 + 196 * q^77 - 288 * q^78 + 240 * q^79 - 224 * q^80 - 359 * q^81 - 324 * q^82 - 1072 * q^83 - 224 * q^84 - 1036 * q^85 + 824 * q^86 + 1520 * q^87 + 224 * q^88 + 810 * q^89 + 1036 * q^90 - 126 * q^91 - 448 * q^92 + 576 * q^93 - 48 * q^94 - 1120 * q^95 - 256 * q^96 + 1354 * q^97 - 98 * q^98 - 1036 * q^99

## 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
−2.00000 8.00000 4.00000 −14.0000 −16.0000 −7.00000 −8.00000 37.0000 28.0000
 $$n$$: 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 14.4.a.a 1
3.b odd 2 1 126.4.a.h 1
4.b odd 2 1 112.4.a.a 1
5.b even 2 1 350.4.a.l 1
5.c odd 4 2 350.4.c.b 2
7.b odd 2 1 98.4.a.a 1
7.c even 3 2 98.4.c.d 2
7.d odd 6 2 98.4.c.f 2
8.b even 2 1 448.4.a.b 1
8.d odd 2 1 448.4.a.o 1
11.b odd 2 1 1694.4.a.g 1
12.b even 2 1 1008.4.a.s 1
13.b even 2 1 2366.4.a.h 1
21.c even 2 1 882.4.a.i 1
21.g even 6 2 882.4.g.k 2
21.h odd 6 2 882.4.g.b 2
28.d even 2 1 784.4.a.s 1
35.c odd 2 1 2450.4.a.bo 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 1.a even 1 1 trivial
98.4.a.a 1 7.b odd 2 1
98.4.c.d 2 7.c even 3 2
98.4.c.f 2 7.d odd 6 2
112.4.a.a 1 4.b odd 2 1
126.4.a.h 1 3.b odd 2 1
350.4.a.l 1 5.b even 2 1
350.4.c.b 2 5.c odd 4 2
448.4.a.b 1 8.b even 2 1
448.4.a.o 1 8.d odd 2 1
784.4.a.s 1 28.d even 2 1
882.4.a.i 1 21.c even 2 1
882.4.g.b 2 21.h odd 6 2
882.4.g.k 2 21.g even 6 2
1008.4.a.s 1 12.b even 2 1
1694.4.a.g 1 11.b odd 2 1
2366.4.a.h 1 13.b even 2 1
2450.4.a.bo 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 8$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(14))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 8$$
$5$ $$T + 14$$
$7$ $$T + 7$$
$11$ $$T + 28$$
$13$ $$T - 18$$
$17$ $$T - 74$$
$19$ $$T - 80$$
$23$ $$T + 112$$
$29$ $$T - 190$$
$31$ $$T - 72$$
$37$ $$T + 346$$
$41$ $$T - 162$$
$43$ $$T + 412$$
$47$ $$T - 24$$
$53$ $$T - 318$$
$59$ $$T + 200$$
$61$ $$T + 198$$
$67$ $$T + 716$$
$71$ $$T - 392$$
$73$ $$T - 538$$
$79$ $$T - 240$$
$83$ $$T + 1072$$
$89$ $$T - 810$$
$97$ $$T - 1354$$