# Properties

 Label 121.16.a.a Level $121$ Weight $16$ Character orbit 121.a Self dual yes Analytic conductor $172.659$ 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] = [121,16,Mod(1,121)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("121.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 121.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: $$172.659141171$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1) 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)

 $$f(q)$$ $$=$$ $$q - 216 q^{2} - 3348 q^{3} + 13888 q^{4} + 52110 q^{5} + 723168 q^{6} - 2822456 q^{7} + 4078080 q^{8} - 3139803 q^{9}+O(q^{10})$$ q - 216 * q^2 - 3348 * q^3 + 13888 * q^4 + 52110 * q^5 + 723168 * q^6 - 2822456 * q^7 + 4078080 * q^8 - 3139803 * q^9 $$q - 216 q^{2} - 3348 q^{3} + 13888 q^{4} + 52110 q^{5} + 723168 q^{6} - 2822456 q^{7} + 4078080 q^{8} - 3139803 q^{9} - 11255760 q^{10} - 46497024 q^{12} + 190073338 q^{13} + 609650496 q^{14} - 174464280 q^{15} - 1335947264 q^{16} - 1646527986 q^{17} + 678197448 q^{18} - 1563257180 q^{19} + 723703680 q^{20} + 9449582688 q^{21} + 9451116072 q^{23} - 13653411840 q^{24} - 27802126025 q^{25} - 41055841008 q^{26} + 58552201080 q^{27} - 39198268928 q^{28} + 36902568330 q^{29} + 37684284480 q^{30} + 71588483552 q^{31} + 154934083584 q^{32} + 355650044976 q^{34} - 147078182160 q^{35} - 43605584064 q^{36} - 1033652081554 q^{37} + 337663550880 q^{38} - 636365535624 q^{39} + 212508748800 q^{40} - 1641974018202 q^{41} - 2041109860608 q^{42} + 492403109308 q^{43} - 163615134330 q^{45} - 2041441071552 q^{46} - 3410684952624 q^{47} + 4472751439872 q^{48} + 3218696361993 q^{49} + 6005259221400 q^{50} + 5512575697128 q^{51} + 2639738518144 q^{52} + 6797151655902 q^{53} - 12647275433280 q^{54} - 11510201364480 q^{56} + 5233785038640 q^{57} - 7970954759280 q^{58} + 9858856815540 q^{59} - 2422959920640 q^{60} - 4931842626902 q^{61} - 15463112447232 q^{62} + 8861955816168 q^{63} + 10310557892608 q^{64} + 9904721643180 q^{65} - 28837826625364 q^{67} - 22866980669568 q^{68} - 31642336609056 q^{69} + 31768887346560 q^{70} + 125050114914552 q^{71} - 12804367818240 q^{72} + 82171455513478 q^{73} + 223268849615664 q^{74} + 93081517931700 q^{75} - 21710515715840 q^{76} + 137454955694784 q^{78} + 25413078694480 q^{79} - 69616211927040 q^{80} - 150980027970519 q^{81} + 354666387931632 q^{82} + 281736730890468 q^{83} + 131235804370944 q^{84} - 85800573350460 q^{85} - 106359071610528 q^{86} - 123549798768840 q^{87} + 715618564776810 q^{89} + 35340869015280 q^{90} - 536473633278128 q^{91} + 131257100007936 q^{92} - 239678242932096 q^{93} + 736707949766784 q^{94} - 81461331649800 q^{95} - 518719311839232 q^{96} + 612786136081826 q^{97} - 695238414190488 q^{98}+O(q^{100})$$ q - 216 * q^2 - 3348 * q^3 + 13888 * q^4 + 52110 * q^5 + 723168 * q^6 - 2822456 * q^7 + 4078080 * q^8 - 3139803 * q^9 - 11255760 * q^10 - 46497024 * q^12 + 190073338 * q^13 + 609650496 * q^14 - 174464280 * q^15 - 1335947264 * q^16 - 1646527986 * q^17 + 678197448 * q^18 - 1563257180 * q^19 + 723703680 * q^20 + 9449582688 * q^21 + 9451116072 * q^23 - 13653411840 * q^24 - 27802126025 * q^25 - 41055841008 * q^26 + 58552201080 * q^27 - 39198268928 * q^28 + 36902568330 * q^29 + 37684284480 * q^30 + 71588483552 * q^31 + 154934083584 * q^32 + 355650044976 * q^34 - 147078182160 * q^35 - 43605584064 * q^36 - 1033652081554 * q^37 + 337663550880 * q^38 - 636365535624 * q^39 + 212508748800 * q^40 - 1641974018202 * q^41 - 2041109860608 * q^42 + 492403109308 * q^43 - 163615134330 * q^45 - 2041441071552 * q^46 - 3410684952624 * q^47 + 4472751439872 * q^48 + 3218696361993 * q^49 + 6005259221400 * q^50 + 5512575697128 * q^51 + 2639738518144 * q^52 + 6797151655902 * q^53 - 12647275433280 * q^54 - 11510201364480 * q^56 + 5233785038640 * q^57 - 7970954759280 * q^58 + 9858856815540 * q^59 - 2422959920640 * q^60 - 4931842626902 * q^61 - 15463112447232 * q^62 + 8861955816168 * q^63 + 10310557892608 * q^64 + 9904721643180 * q^65 - 28837826625364 * q^67 - 22866980669568 * q^68 - 31642336609056 * q^69 + 31768887346560 * q^70 + 125050114914552 * q^71 - 12804367818240 * q^72 + 82171455513478 * q^73 + 223268849615664 * q^74 + 93081517931700 * q^75 - 21710515715840 * q^76 + 137454955694784 * q^78 + 25413078694480 * q^79 - 69616211927040 * q^80 - 150980027970519 * q^81 + 354666387931632 * q^82 + 281736730890468 * q^83 + 131235804370944 * q^84 - 85800573350460 * q^85 - 106359071610528 * q^86 - 123549798768840 * q^87 + 715618564776810 * q^89 + 35340869015280 * q^90 - 536473633278128 * q^91 + 131257100007936 * q^92 - 239678242932096 * q^93 + 736707949766784 * q^94 - 81461331649800 * q^95 - 518719311839232 * q^96 + 612786136081826 * q^97 - 695238414190488 * q^98

## 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
−216.000 −3348.00 13888.0 52110.0 723168. −2.82246e6 4.07808e6 −3.13980e6 −1.12558e7
 $$n$$: 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 121.16.a.a 1
11.b odd 2 1 1.16.a.a 1
33.d even 2 1 9.16.a.a 1
44.c even 2 1 16.16.a.d 1
55.d odd 2 1 25.16.a.a 1
55.e even 4 2 25.16.b.a 2
77.b even 2 1 49.16.a.a 1
77.h odd 6 2 49.16.c.c 2
77.i even 6 2 49.16.c.b 2
88.b odd 2 1 64.16.a.i 1
88.g even 2 1 64.16.a.c 1
132.d odd 2 1 144.16.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a 1 11.b odd 2 1
9.16.a.a 1 33.d even 2 1
16.16.a.d 1 44.c even 2 1
25.16.a.a 1 55.d odd 2 1
25.16.b.a 2 55.e even 4 2
49.16.a.a 1 77.b even 2 1
49.16.c.b 2 77.i even 6 2
49.16.c.c 2 77.h odd 6 2
64.16.a.c 1 88.g even 2 1
64.16.a.i 1 88.b odd 2 1
121.16.a.a 1 1.a even 1 1 trivial
144.16.a.f 1 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 216$$ acting on $$S_{16}^{\mathrm{new}}(\Gamma_0(121))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 216$$
$3$ $$T + 3348$$
$5$ $$T - 52110$$
$7$ $$T + 2822456$$
$11$ $$T$$
$13$ $$T - 190073338$$
$17$ $$T + 1646527986$$
$19$ $$T + 1563257180$$
$23$ $$T - 9451116072$$
$29$ $$T - 36902568330$$
$31$ $$T - 71588483552$$
$37$ $$T + 1033652081554$$
$41$ $$T + 1641974018202$$
$43$ $$T - 492403109308$$
$47$ $$T + 3410684952624$$
$53$ $$T - 6797151655902$$
$59$ $$T - 9858856815540$$
$61$ $$T + 4931842626902$$
$67$ $$T + 28837826625364$$
$71$ $$T - 125050114914552$$
$73$ $$T - 82171455513478$$
$79$ $$T - 25413078694480$$
$83$ $$T - 281736730890468$$
$89$ $$T - 715618564776810$$
$97$ $$T - 612786136081826$$