# Properties

 Label 11.6.a.a Level $11$ Weight $6$ Character orbit 11.a Self dual yes Analytic conductor $1.764$ 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] = [11,6,Mod(1,11)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(11, 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("11.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 11.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: $$1.76422201794$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} - 15 q^{3} - 16 q^{4} - 19 q^{5} + 60 q^{6} + 10 q^{7} + 192 q^{8} - 18 q^{9}+O(q^{10})$$ q - 4 * q^2 - 15 * q^3 - 16 * q^4 - 19 * q^5 + 60 * q^6 + 10 * q^7 + 192 * q^8 - 18 * q^9 $$q - 4 q^{2} - 15 q^{3} - 16 q^{4} - 19 q^{5} + 60 q^{6} + 10 q^{7} + 192 q^{8} - 18 q^{9} + 76 q^{10} - 121 q^{11} + 240 q^{12} - 1148 q^{13} - 40 q^{14} + 285 q^{15} - 256 q^{16} + 686 q^{17} + 72 q^{18} - 384 q^{19} + 304 q^{20} - 150 q^{21} + 484 q^{22} + 3709 q^{23} - 2880 q^{24} - 2764 q^{25} + 4592 q^{26} + 3915 q^{27} - 160 q^{28} - 5424 q^{29} - 1140 q^{30} - 6443 q^{31} - 5120 q^{32} + 1815 q^{33} - 2744 q^{34} - 190 q^{35} + 288 q^{36} + 12063 q^{37} + 1536 q^{38} + 17220 q^{39} - 3648 q^{40} - 1528 q^{41} + 600 q^{42} - 4026 q^{43} + 1936 q^{44} + 342 q^{45} - 14836 q^{46} + 7168 q^{47} + 3840 q^{48} - 16707 q^{49} + 11056 q^{50} - 10290 q^{51} + 18368 q^{52} - 29862 q^{53} - 15660 q^{54} + 2299 q^{55} + 1920 q^{56} + 5760 q^{57} + 21696 q^{58} - 6461 q^{59} - 4560 q^{60} - 16980 q^{61} + 25772 q^{62} - 180 q^{63} + 28672 q^{64} + 21812 q^{65} - 7260 q^{66} + 29999 q^{67} - 10976 q^{68} - 55635 q^{69} + 760 q^{70} + 31023 q^{71} - 3456 q^{72} + 1924 q^{73} - 48252 q^{74} + 41460 q^{75} + 6144 q^{76} - 1210 q^{77} - 68880 q^{78} + 65138 q^{79} + 4864 q^{80} - 54351 q^{81} + 6112 q^{82} - 102714 q^{83} + 2400 q^{84} - 13034 q^{85} + 16104 q^{86} + 81360 q^{87} - 23232 q^{88} + 17415 q^{89} - 1368 q^{90} - 11480 q^{91} - 59344 q^{92} + 96645 q^{93} - 28672 q^{94} + 7296 q^{95} + 76800 q^{96} + 66905 q^{97} + 66828 q^{98} + 2178 q^{99}+O(q^{100})$$ q - 4 * q^2 - 15 * q^3 - 16 * q^4 - 19 * q^5 + 60 * q^6 + 10 * q^7 + 192 * q^8 - 18 * q^9 + 76 * q^10 - 121 * q^11 + 240 * q^12 - 1148 * q^13 - 40 * q^14 + 285 * q^15 - 256 * q^16 + 686 * q^17 + 72 * q^18 - 384 * q^19 + 304 * q^20 - 150 * q^21 + 484 * q^22 + 3709 * q^23 - 2880 * q^24 - 2764 * q^25 + 4592 * q^26 + 3915 * q^27 - 160 * q^28 - 5424 * q^29 - 1140 * q^30 - 6443 * q^31 - 5120 * q^32 + 1815 * q^33 - 2744 * q^34 - 190 * q^35 + 288 * q^36 + 12063 * q^37 + 1536 * q^38 + 17220 * q^39 - 3648 * q^40 - 1528 * q^41 + 600 * q^42 - 4026 * q^43 + 1936 * q^44 + 342 * q^45 - 14836 * q^46 + 7168 * q^47 + 3840 * q^48 - 16707 * q^49 + 11056 * q^50 - 10290 * q^51 + 18368 * q^52 - 29862 * q^53 - 15660 * q^54 + 2299 * q^55 + 1920 * q^56 + 5760 * q^57 + 21696 * q^58 - 6461 * q^59 - 4560 * q^60 - 16980 * q^61 + 25772 * q^62 - 180 * q^63 + 28672 * q^64 + 21812 * q^65 - 7260 * q^66 + 29999 * q^67 - 10976 * q^68 - 55635 * q^69 + 760 * q^70 + 31023 * q^71 - 3456 * q^72 + 1924 * q^73 - 48252 * q^74 + 41460 * q^75 + 6144 * q^76 - 1210 * q^77 - 68880 * q^78 + 65138 * q^79 + 4864 * q^80 - 54351 * q^81 + 6112 * q^82 - 102714 * q^83 + 2400 * q^84 - 13034 * q^85 + 16104 * q^86 + 81360 * q^87 - 23232 * q^88 + 17415 * q^89 - 1368 * q^90 - 11480 * q^91 - 59344 * q^92 + 96645 * q^93 - 28672 * q^94 + 7296 * q^95 + 76800 * q^96 + 66905 * q^97 + 66828 * q^98 + 2178 * 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 −15.0000 −16.0000 −19.0000 60.0000 10.0000 192.000 −18.0000 76.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$

## 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 11.6.a.a 1
3.b odd 2 1 99.6.a.c 1
4.b odd 2 1 176.6.a.c 1
5.b even 2 1 275.6.a.a 1
5.c odd 4 2 275.6.b.a 2
7.b odd 2 1 539.6.a.c 1
8.b even 2 1 704.6.a.h 1
8.d odd 2 1 704.6.a.c 1
11.b odd 2 1 121.6.a.b 1
33.d even 2 1 1089.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.a 1 1.a even 1 1 trivial
99.6.a.c 1 3.b odd 2 1
121.6.a.b 1 11.b odd 2 1
176.6.a.c 1 4.b odd 2 1
275.6.a.a 1 5.b even 2 1
275.6.b.a 2 5.c odd 4 2
539.6.a.c 1 7.b odd 2 1
704.6.a.c 1 8.d odd 2 1
704.6.a.h 1 8.b even 2 1
1089.6.a.c 1 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 4$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(11))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T + 15$$
$5$ $$T + 19$$
$7$ $$T - 10$$
$11$ $$T + 121$$
$13$ $$T + 1148$$
$17$ $$T - 686$$
$19$ $$T + 384$$
$23$ $$T - 3709$$
$29$ $$T + 5424$$
$31$ $$T + 6443$$
$37$ $$T - 12063$$
$41$ $$T + 1528$$
$43$ $$T + 4026$$
$47$ $$T - 7168$$
$53$ $$T + 29862$$
$59$ $$T + 6461$$
$61$ $$T + 16980$$
$67$ $$T - 29999$$
$71$ $$T - 31023$$
$73$ $$T - 1924$$
$79$ $$T - 65138$$
$83$ $$T + 102714$$
$89$ $$T - 17415$$
$97$ $$T - 66905$$