# Properties

 Label 7.6.a.a Level $7$ Weight $6$ Character orbit 7.a Self dual yes Analytic conductor $1.123$ Analytic rank $1$ 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] = [7,6,Mod(1,7)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, 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("7.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 7.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: $$1.12268673869$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 10 q^{2} - 14 q^{3} + 68 q^{4} - 56 q^{5} + 140 q^{6} - 49 q^{7} - 360 q^{8} - 47 q^{9}+O(q^{10})$$ q - 10 * q^2 - 14 * q^3 + 68 * q^4 - 56 * q^5 + 140 * q^6 - 49 * q^7 - 360 * q^8 - 47 * q^9 $$q - 10 q^{2} - 14 q^{3} + 68 q^{4} - 56 q^{5} + 140 q^{6} - 49 q^{7} - 360 q^{8} - 47 q^{9} + 560 q^{10} + 232 q^{11} - 952 q^{12} - 140 q^{13} + 490 q^{14} + 784 q^{15} + 1424 q^{16} - 1722 q^{17} + 470 q^{18} - 98 q^{19} - 3808 q^{20} + 686 q^{21} - 2320 q^{22} + 1824 q^{23} + 5040 q^{24} + 11 q^{25} + 1400 q^{26} + 4060 q^{27} - 3332 q^{28} + 3418 q^{29} - 7840 q^{30} - 7644 q^{31} - 2720 q^{32} - 3248 q^{33} + 17220 q^{34} + 2744 q^{35} - 3196 q^{36} - 10398 q^{37} + 980 q^{38} + 1960 q^{39} + 20160 q^{40} - 17962 q^{41} - 6860 q^{42} + 10880 q^{43} + 15776 q^{44} + 2632 q^{45} - 18240 q^{46} + 9324 q^{47} - 19936 q^{48} + 2401 q^{49} - 110 q^{50} + 24108 q^{51} - 9520 q^{52} + 2262 q^{53} - 40600 q^{54} - 12992 q^{55} + 17640 q^{56} + 1372 q^{57} - 34180 q^{58} - 2730 q^{59} + 53312 q^{60} + 25648 q^{61} + 76440 q^{62} + 2303 q^{63} - 18368 q^{64} + 7840 q^{65} + 32480 q^{66} - 48404 q^{67} - 117096 q^{68} - 25536 q^{69} - 27440 q^{70} - 58560 q^{71} + 16920 q^{72} + 68082 q^{73} + 103980 q^{74} - 154 q^{75} - 6664 q^{76} - 11368 q^{77} - 19600 q^{78} + 31784 q^{79} - 79744 q^{80} - 45419 q^{81} + 179620 q^{82} - 20538 q^{83} + 46648 q^{84} + 96432 q^{85} - 108800 q^{86} - 47852 q^{87} - 83520 q^{88} - 50582 q^{89} - 26320 q^{90} + 6860 q^{91} + 124032 q^{92} + 107016 q^{93} - 93240 q^{94} + 5488 q^{95} + 38080 q^{96} - 58506 q^{97} - 24010 q^{98} - 10904 q^{99}+O(q^{100})$$ q - 10 * q^2 - 14 * q^3 + 68 * q^4 - 56 * q^5 + 140 * q^6 - 49 * q^7 - 360 * q^8 - 47 * q^9 + 560 * q^10 + 232 * q^11 - 952 * q^12 - 140 * q^13 + 490 * q^14 + 784 * q^15 + 1424 * q^16 - 1722 * q^17 + 470 * q^18 - 98 * q^19 - 3808 * q^20 + 686 * q^21 - 2320 * q^22 + 1824 * q^23 + 5040 * q^24 + 11 * q^25 + 1400 * q^26 + 4060 * q^27 - 3332 * q^28 + 3418 * q^29 - 7840 * q^30 - 7644 * q^31 - 2720 * q^32 - 3248 * q^33 + 17220 * q^34 + 2744 * q^35 - 3196 * q^36 - 10398 * q^37 + 980 * q^38 + 1960 * q^39 + 20160 * q^40 - 17962 * q^41 - 6860 * q^42 + 10880 * q^43 + 15776 * q^44 + 2632 * q^45 - 18240 * q^46 + 9324 * q^47 - 19936 * q^48 + 2401 * q^49 - 110 * q^50 + 24108 * q^51 - 9520 * q^52 + 2262 * q^53 - 40600 * q^54 - 12992 * q^55 + 17640 * q^56 + 1372 * q^57 - 34180 * q^58 - 2730 * q^59 + 53312 * q^60 + 25648 * q^61 + 76440 * q^62 + 2303 * q^63 - 18368 * q^64 + 7840 * q^65 + 32480 * q^66 - 48404 * q^67 - 117096 * q^68 - 25536 * q^69 - 27440 * q^70 - 58560 * q^71 + 16920 * q^72 + 68082 * q^73 + 103980 * q^74 - 154 * q^75 - 6664 * q^76 - 11368 * q^77 - 19600 * q^78 + 31784 * q^79 - 79744 * q^80 - 45419 * q^81 + 179620 * q^82 - 20538 * q^83 + 46648 * q^84 + 96432 * q^85 - 108800 * q^86 - 47852 * q^87 - 83520 * q^88 - 50582 * q^89 - 26320 * q^90 + 6860 * q^91 + 124032 * q^92 + 107016 * q^93 - 93240 * q^94 + 5488 * q^95 + 38080 * q^96 - 58506 * q^97 - 24010 * q^98 - 10904 * 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
−10.0000 −14.0000 68.0000 −56.0000 140.000 −49.0000 −360.000 −47.0000 560.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$1$$

## 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 7.6.a.a 1
3.b odd 2 1 63.6.a.e 1
4.b odd 2 1 112.6.a.g 1
5.b even 2 1 175.6.a.b 1
5.c odd 4 2 175.6.b.a 2
7.b odd 2 1 49.6.a.a 1
7.c even 3 2 49.6.c.c 2
7.d odd 6 2 49.6.c.b 2
8.b even 2 1 448.6.a.m 1
8.d odd 2 1 448.6.a.c 1
11.b odd 2 1 847.6.a.b 1
12.b even 2 1 1008.6.a.y 1
21.c even 2 1 441.6.a.k 1
28.d even 2 1 784.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 1.a even 1 1 trivial
49.6.a.a 1 7.b odd 2 1
49.6.c.b 2 7.d odd 6 2
49.6.c.c 2 7.c even 3 2
63.6.a.e 1 3.b odd 2 1
112.6.a.g 1 4.b odd 2 1
175.6.a.b 1 5.b even 2 1
175.6.b.a 2 5.c odd 4 2
441.6.a.k 1 21.c even 2 1
448.6.a.c 1 8.d odd 2 1
448.6.a.m 1 8.b even 2 1
784.6.a.c 1 28.d even 2 1
847.6.a.b 1 11.b odd 2 1
1008.6.a.y 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 10$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(7))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 10$$
$3$ $$T + 14$$
$5$ $$T + 56$$
$7$ $$T + 49$$
$11$ $$T - 232$$
$13$ $$T + 140$$
$17$ $$T + 1722$$
$19$ $$T + 98$$
$23$ $$T - 1824$$
$29$ $$T - 3418$$
$31$ $$T + 7644$$
$37$ $$T + 10398$$
$41$ $$T + 17962$$
$43$ $$T - 10880$$
$47$ $$T - 9324$$
$53$ $$T - 2262$$
$59$ $$T + 2730$$
$61$ $$T - 25648$$
$67$ $$T + 48404$$
$71$ $$T + 58560$$
$73$ $$T - 68082$$
$79$ $$T - 31784$$
$83$ $$T + 20538$$
$89$ $$T + 50582$$
$97$ $$T + 58506$$