# Properties

 Label 2166.4.a.i Level $2166$ Weight $4$ Character orbit 2166.a Self dual yes Analytic conductor $127.798$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,4,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2166.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2166 = 2 \cdot 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2166.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: $$127.798137072$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + 6 q^{5} + 6 q^{6} - 16 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10})$$ q + 2 * q^2 + 3 * q^3 + 4 * q^4 + 6 * q^5 + 6 * q^6 - 16 * q^7 + 8 * q^8 + 9 * q^9 $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + 6 q^{5} + 6 q^{6} - 16 q^{7} + 8 q^{8} + 9 q^{9} + 12 q^{10} + 12 q^{11} + 12 q^{12} - 38 q^{13} - 32 q^{14} + 18 q^{15} + 16 q^{16} - 126 q^{17} + 18 q^{18} + 24 q^{20} - 48 q^{21} + 24 q^{22} + 168 q^{23} + 24 q^{24} - 89 q^{25} - 76 q^{26} + 27 q^{27} - 64 q^{28} - 30 q^{29} + 36 q^{30} + 88 q^{31} + 32 q^{32} + 36 q^{33} - 252 q^{34} - 96 q^{35} + 36 q^{36} - 254 q^{37} - 114 q^{39} + 48 q^{40} - 42 q^{41} - 96 q^{42} - 52 q^{43} + 48 q^{44} + 54 q^{45} + 336 q^{46} - 96 q^{47} + 48 q^{48} - 87 q^{49} - 178 q^{50} - 378 q^{51} - 152 q^{52} - 198 q^{53} + 54 q^{54} + 72 q^{55} - 128 q^{56} - 60 q^{58} + 660 q^{59} + 72 q^{60} - 538 q^{61} + 176 q^{62} - 144 q^{63} + 64 q^{64} - 228 q^{65} + 72 q^{66} - 884 q^{67} - 504 q^{68} + 504 q^{69} - 192 q^{70} - 792 q^{71} + 72 q^{72} + 218 q^{73} - 508 q^{74} - 267 q^{75} - 192 q^{77} - 228 q^{78} + 520 q^{79} + 96 q^{80} + 81 q^{81} - 84 q^{82} - 492 q^{83} - 192 q^{84} - 756 q^{85} - 104 q^{86} - 90 q^{87} + 96 q^{88} - 810 q^{89} + 108 q^{90} + 608 q^{91} + 672 q^{92} + 264 q^{93} - 192 q^{94} + 96 q^{96} - 1154 q^{97} - 174 q^{98} + 108 q^{99}+O(q^{100})$$ q + 2 * q^2 + 3 * q^3 + 4 * q^4 + 6 * q^5 + 6 * q^6 - 16 * q^7 + 8 * q^8 + 9 * q^9 + 12 * q^10 + 12 * q^11 + 12 * q^12 - 38 * q^13 - 32 * q^14 + 18 * q^15 + 16 * q^16 - 126 * q^17 + 18 * q^18 + 24 * q^20 - 48 * q^21 + 24 * q^22 + 168 * q^23 + 24 * q^24 - 89 * q^25 - 76 * q^26 + 27 * q^27 - 64 * q^28 - 30 * q^29 + 36 * q^30 + 88 * q^31 + 32 * q^32 + 36 * q^33 - 252 * q^34 - 96 * q^35 + 36 * q^36 - 254 * q^37 - 114 * q^39 + 48 * q^40 - 42 * q^41 - 96 * q^42 - 52 * q^43 + 48 * q^44 + 54 * q^45 + 336 * q^46 - 96 * q^47 + 48 * q^48 - 87 * q^49 - 178 * q^50 - 378 * q^51 - 152 * q^52 - 198 * q^53 + 54 * q^54 + 72 * q^55 - 128 * q^56 - 60 * q^58 + 660 * q^59 + 72 * q^60 - 538 * q^61 + 176 * q^62 - 144 * q^63 + 64 * q^64 - 228 * q^65 + 72 * q^66 - 884 * q^67 - 504 * q^68 + 504 * q^69 - 192 * q^70 - 792 * q^71 + 72 * q^72 + 218 * q^73 - 508 * q^74 - 267 * q^75 - 192 * q^77 - 228 * q^78 + 520 * q^79 + 96 * q^80 + 81 * q^81 - 84 * q^82 - 492 * q^83 - 192 * q^84 - 756 * q^85 - 104 * q^86 - 90 * q^87 + 96 * q^88 - 810 * q^89 + 108 * q^90 + 608 * q^91 + 672 * q^92 + 264 * q^93 - 192 * q^94 + 96 * q^96 - 1154 * q^97 - 174 * q^98 + 108 * 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
2.00000 3.00000 4.00000 6.00000 6.00000 −16.0000 8.00000 9.00000 12.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 2166.4.a.i 1
19.b odd 2 1 6.4.a.a 1
57.d even 2 1 18.4.a.a 1
76.d even 2 1 48.4.a.c 1
95.d odd 2 1 150.4.a.i 1
95.g even 4 2 150.4.c.d 2
133.c even 2 1 294.4.a.e 1
133.o even 6 2 294.4.e.g 2
133.r odd 6 2 294.4.e.h 2
152.b even 2 1 192.4.a.c 1
152.g odd 2 1 192.4.a.i 1
171.l even 6 2 162.4.c.c 2
171.o odd 6 2 162.4.c.f 2
209.d even 2 1 726.4.a.f 1
228.b odd 2 1 144.4.a.c 1
247.d odd 2 1 1014.4.a.g 1
247.i even 4 2 1014.4.b.d 2
285.b even 2 1 450.4.a.h 1
285.j odd 4 2 450.4.c.e 2
304.j odd 4 2 768.4.d.n 2
304.m even 4 2 768.4.d.c 2
323.c odd 2 1 1734.4.a.d 1
380.d even 2 1 1200.4.a.b 1
380.j odd 4 2 1200.4.f.j 2
399.h odd 2 1 882.4.a.n 1
399.s odd 6 2 882.4.g.f 2
399.w even 6 2 882.4.g.i 2
456.l odd 2 1 576.4.a.r 1
456.p even 2 1 576.4.a.q 1
532.b odd 2 1 2352.4.a.e 1
627.b odd 2 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 19.b odd 2 1
18.4.a.a 1 57.d even 2 1
48.4.a.c 1 76.d even 2 1
144.4.a.c 1 228.b odd 2 1
150.4.a.i 1 95.d odd 2 1
150.4.c.d 2 95.g even 4 2
162.4.c.c 2 171.l even 6 2
162.4.c.f 2 171.o odd 6 2
192.4.a.c 1 152.b even 2 1
192.4.a.i 1 152.g odd 2 1
294.4.a.e 1 133.c even 2 1
294.4.e.g 2 133.o even 6 2
294.4.e.h 2 133.r odd 6 2
450.4.a.h 1 285.b even 2 1
450.4.c.e 2 285.j odd 4 2
576.4.a.q 1 456.p even 2 1
576.4.a.r 1 456.l odd 2 1
726.4.a.f 1 209.d even 2 1
768.4.d.c 2 304.m even 4 2
768.4.d.n 2 304.j odd 4 2
882.4.a.n 1 399.h odd 2 1
882.4.g.f 2 399.s odd 6 2
882.4.g.i 2 399.w even 6 2
1014.4.a.g 1 247.d odd 2 1
1014.4.b.d 2 247.i even 4 2
1200.4.a.b 1 380.d even 2 1
1200.4.f.j 2 380.j odd 4 2
1734.4.a.d 1 323.c odd 2 1
2166.4.a.i 1 1.a even 1 1 trivial
2178.4.a.e 1 627.b odd 2 1
2352.4.a.e 1 532.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2166))$$:

 $$T_{5} - 6$$ T5 - 6 $$T_{13} + 38$$ T13 + 38

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 3$$
$5$ $$T - 6$$
$7$ $$T + 16$$
$11$ $$T - 12$$
$13$ $$T + 38$$
$17$ $$T + 126$$
$19$ $$T$$
$23$ $$T - 168$$
$29$ $$T + 30$$
$31$ $$T - 88$$
$37$ $$T + 254$$
$41$ $$T + 42$$
$43$ $$T + 52$$
$47$ $$T + 96$$
$53$ $$T + 198$$
$59$ $$T - 660$$
$61$ $$T + 538$$
$67$ $$T + 884$$
$71$ $$T + 792$$
$73$ $$T - 218$$
$79$ $$T - 520$$
$83$ $$T + 492$$
$89$ $$T + 810$$
$97$ $$T + 1154$$
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