# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [14,6,Mod(1,14)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(14, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("14.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 10 q^{3} + 16 q^{4} + 84 q^{5} - 40 q^{6} + 49 q^{7} - 64 q^{8} - 143 q^{9}+O(q^{10})$$ q - 4 * q^2 + 10 * q^3 + 16 * q^4 + 84 * q^5 - 40 * q^6 + 49 * q^7 - 64 * q^8 - 143 * q^9 $$q - 4 q^{2} + 10 q^{3} + 16 q^{4} + 84 q^{5} - 40 q^{6} + 49 q^{7} - 64 q^{8} - 143 q^{9} - 336 q^{10} - 336 q^{11} + 160 q^{12} + 584 q^{13} - 196 q^{14} + 840 q^{15} + 256 q^{16} - 1458 q^{17} + 572 q^{18} + 470 q^{19} + 1344 q^{20} + 490 q^{21} + 1344 q^{22} - 4200 q^{23} - 640 q^{24} + 3931 q^{25} - 2336 q^{26} - 3860 q^{27} + 784 q^{28} + 4866 q^{29} - 3360 q^{30} - 7372 q^{31} - 1024 q^{32} - 3360 q^{33} + 5832 q^{34} + 4116 q^{35} - 2288 q^{36} + 14330 q^{37} - 1880 q^{38} + 5840 q^{39} - 5376 q^{40} + 6222 q^{41} - 1960 q^{42} + 3704 q^{43} - 5376 q^{44} - 12012 q^{45} + 16800 q^{46} - 1812 q^{47} + 2560 q^{48} + 2401 q^{49} - 15724 q^{50} - 14580 q^{51} + 9344 q^{52} - 37242 q^{53} + 15440 q^{54} - 28224 q^{55} - 3136 q^{56} + 4700 q^{57} - 19464 q^{58} + 34302 q^{59} + 13440 q^{60} + 24476 q^{61} + 29488 q^{62} - 7007 q^{63} + 4096 q^{64} + 49056 q^{65} + 13440 q^{66} - 17452 q^{67} - 23328 q^{68} - 42000 q^{69} - 16464 q^{70} + 28224 q^{71} + 9152 q^{72} + 3602 q^{73} - 57320 q^{74} + 39310 q^{75} + 7520 q^{76} - 16464 q^{77} - 23360 q^{78} + 42872 q^{79} + 21504 q^{80} - 3851 q^{81} - 24888 q^{82} - 35202 q^{83} + 7840 q^{84} - 122472 q^{85} - 14816 q^{86} + 48660 q^{87} + 21504 q^{88} + 26730 q^{89} + 48048 q^{90} + 28616 q^{91} - 67200 q^{92} - 73720 q^{93} + 7248 q^{94} + 39480 q^{95} - 10240 q^{96} - 16978 q^{97} - 9604 q^{98} + 48048 q^{99}+O(q^{100})$$ q - 4 * q^2 + 10 * q^3 + 16 * q^4 + 84 * q^5 - 40 * q^6 + 49 * q^7 - 64 * q^8 - 143 * q^9 - 336 * q^10 - 336 * q^11 + 160 * q^12 + 584 * q^13 - 196 * q^14 + 840 * q^15 + 256 * q^16 - 1458 * q^17 + 572 * q^18 + 470 * q^19 + 1344 * q^20 + 490 * q^21 + 1344 * q^22 - 4200 * q^23 - 640 * q^24 + 3931 * q^25 - 2336 * q^26 - 3860 * q^27 + 784 * q^28 + 4866 * q^29 - 3360 * q^30 - 7372 * q^31 - 1024 * q^32 - 3360 * q^33 + 5832 * q^34 + 4116 * q^35 - 2288 * q^36 + 14330 * q^37 - 1880 * q^38 + 5840 * q^39 - 5376 * q^40 + 6222 * q^41 - 1960 * q^42 + 3704 * q^43 - 5376 * q^44 - 12012 * q^45 + 16800 * q^46 - 1812 * q^47 + 2560 * q^48 + 2401 * q^49 - 15724 * q^50 - 14580 * q^51 + 9344 * q^52 - 37242 * q^53 + 15440 * q^54 - 28224 * q^55 - 3136 * q^56 + 4700 * q^57 - 19464 * q^58 + 34302 * q^59 + 13440 * q^60 + 24476 * q^61 + 29488 * q^62 - 7007 * q^63 + 4096 * q^64 + 49056 * q^65 + 13440 * q^66 - 17452 * q^67 - 23328 * q^68 - 42000 * q^69 - 16464 * q^70 + 28224 * q^71 + 9152 * q^72 + 3602 * q^73 - 57320 * q^74 + 39310 * q^75 + 7520 * q^76 - 16464 * q^77 - 23360 * q^78 + 42872 * q^79 + 21504 * q^80 - 3851 * q^81 - 24888 * q^82 - 35202 * q^83 + 7840 * q^84 - 122472 * q^85 - 14816 * q^86 + 48660 * q^87 + 21504 * q^88 + 26730 * q^89 + 48048 * q^90 + 28616 * q^91 - 67200 * q^92 - 73720 * q^93 + 7248 * q^94 + 39480 * q^95 - 10240 * q^96 - 16978 * q^97 - 9604 * q^98 + 48048 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 10.0000 16.0000 84.0000 −40.0000 49.0000 −64.0000 −143.000 −336.000
 $$n$$: 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.a 1
3.b odd 2 1 126.6.a.f 1
4.b odd 2 1 112.6.a.c 1
5.b even 2 1 350.6.a.i 1
5.c odd 4 2 350.6.c.d 2
7.b odd 2 1 98.6.a.a 1
7.c even 3 2 98.6.c.c 2
7.d odd 6 2 98.6.c.d 2
8.b even 2 1 448.6.a.e 1
8.d odd 2 1 448.6.a.l 1
12.b even 2 1 1008.6.a.b 1
21.c even 2 1 882.6.a.x 1
28.d even 2 1 784.6.a.i 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 10$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(14))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T - 10$$
$5$ $$T - 84$$
$7$ $$T - 49$$
$11$ $$T + 336$$
$13$ $$T - 584$$
$17$ $$T + 1458$$
$19$ $$T - 470$$
$23$ $$T + 4200$$
$29$ $$T - 4866$$
$31$ $$T + 7372$$
$37$ $$T - 14330$$
$41$ $$T - 6222$$
$43$ $$T - 3704$$
$47$ $$T + 1812$$
$53$ $$T + 37242$$
$59$ $$T - 34302$$
$61$ $$T - 24476$$
$67$ $$T + 17452$$
$71$ $$T - 28224$$
$73$ $$T - 3602$$
$79$ $$T - 42872$$
$83$ $$T + 35202$$
$89$ $$T - 26730$$
$97$ $$T + 16978$$