# 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 $1$

# 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)

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

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