# Properties

 Label 1183.2.e.j Level $1183$ Weight $2$ Character orbit 1183.e Analytic conductor $9.446$ Analytic rank $0$ Dimension $24$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(170,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1183.170");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.e (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{3})$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 6 q^{3} - 8 q^{4} - 2 q^{9}+O(q^{10})$$ 24 * q - 6 * q^3 - 8 * q^4 - 2 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 6 q^{3} - 8 q^{4} - 2 q^{9} - 24 q^{10} + 2 q^{12} + 8 q^{14} - 16 q^{16} - 34 q^{17} + 60 q^{22} - 6 q^{23} + 10 q^{25} + 24 q^{27} + 4 q^{29} - 22 q^{30} - 24 q^{35} - 52 q^{36} - 38 q^{38} - 2 q^{40} + 32 q^{42} + 44 q^{43} - 76 q^{48} + 12 q^{49} - 8 q^{51} - 16 q^{53} + 60 q^{55} + 54 q^{56} + 10 q^{61} + 164 q^{62} - 4 q^{64} - 68 q^{66} - 22 q^{68} + 28 q^{69} - 66 q^{74} - 2 q^{75} + 38 q^{77} - 70 q^{79} + 28 q^{81} - 10 q^{82} + 20 q^{87} + 28 q^{88} - 132 q^{92} + 2 q^{94} - 4 q^{95}+O(q^{100})$$ 24 * q - 6 * q^3 - 8 * q^4 - 2 * q^9 - 24 * q^10 + 2 * q^12 + 8 * q^14 - 16 * q^16 - 34 * q^17 + 60 * q^22 - 6 * q^23 + 10 * q^25 + 24 * q^27 + 4 * q^29 - 22 * q^30 - 24 * q^35 - 52 * q^36 - 38 * q^38 - 2 * q^40 + 32 * q^42 + 44 * q^43 - 76 * q^48 + 12 * q^49 - 8 * q^51 - 16 * q^53 + 60 * q^55 + 54 * q^56 + 10 * q^61 + 164 * q^62 - 4 * q^64 - 68 * q^66 - 22 * q^68 + 28 * q^69 - 66 * q^74 - 2 * q^75 + 38 * q^77 - 70 * q^79 + 28 * q^81 - 10 * q^82 + 20 * q^87 + 28 * q^88 - 132 * q^92 + 2 * q^94 - 4 * q^95

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
170.1 −1.29430 + 2.24179i −0.259233 0.449005i −2.35043 4.07106i −0.806027 + 1.39608i 1.34210 2.13104 + 1.56802i 6.99143 1.36560 2.36528i −2.08648 3.61389i
170.2 −1.15163 + 1.99469i 0.736680 + 1.27597i −1.65252 2.86225i −0.423646 + 0.733776i −3.39354 −1.00088 2.44913i 3.00585 0.414604 0.718115i −0.975769 1.69008i
170.3 −0.689527 + 1.19430i −1.44060 2.49520i 0.0491037 + 0.0850501i −0.402974 + 0.697972i 3.97334 −1.26180 2.32548i −2.89354 −2.65067 + 4.59109i −0.555723 0.962541i
170.4 −0.672613 + 1.16500i −1.02505 1.77544i 0.0951832 + 0.164862i −1.78389 + 3.08979i 2.75785 2.62255 0.349630i −2.94654 −0.601462 + 1.04176i −2.39973 4.15646i
170.5 −0.249993 + 0.433001i −0.424801 0.735776i 0.875007 + 1.51556i 0.521238 0.902810i 0.424789 −2.40155 + 1.11021i −1.87496 1.13909 1.97296i 0.260612 + 0.451393i
170.6 −0.0904119 + 0.156598i 0.913006 + 1.58137i 0.983651 + 1.70373i −1.34332 + 2.32670i −0.330186 −1.64912 + 2.06892i −0.717383 −0.167162 + 0.289532i −0.242904 0.420723i
170.7 0.0904119 0.156598i 0.913006 + 1.58137i 0.983651 + 1.70373i 1.34332 2.32670i 0.330186 1.64912 2.06892i 0.717383 −0.167162 + 0.289532i −0.242904 0.420723i
170.8 0.249993 0.433001i −0.424801 0.735776i 0.875007 + 1.51556i −0.521238 + 0.902810i −0.424789 2.40155 1.11021i 1.87496 1.13909 1.97296i 0.260612 + 0.451393i
170.9 0.672613 1.16500i −1.02505 1.77544i 0.0951832 + 0.164862i 1.78389 3.08979i −2.75785 −2.62255 + 0.349630i 2.94654 −0.601462 + 1.04176i −2.39973 4.15646i
170.10 0.689527 1.19430i −1.44060 2.49520i 0.0491037 + 0.0850501i 0.402974 0.697972i −3.97334 1.26180 + 2.32548i 2.89354 −2.65067 + 4.59109i −0.555723 0.962541i
170.11 1.15163 1.99469i 0.736680 + 1.27597i −1.65252 2.86225i 0.423646 0.733776i 3.39354 1.00088 + 2.44913i −3.00585 0.414604 0.718115i −0.975769 1.69008i
170.12 1.29430 2.24179i −0.259233 0.449005i −2.35043 4.07106i 0.806027 1.39608i −1.34210 −2.13104 1.56802i −6.99143 1.36560 2.36528i −2.08648 3.61389i
508.1 −1.29430 2.24179i −0.259233 + 0.449005i −2.35043 + 4.07106i −0.806027 1.39608i 1.34210 2.13104 1.56802i 6.99143 1.36560 + 2.36528i −2.08648 + 3.61389i
508.2 −1.15163 1.99469i 0.736680 1.27597i −1.65252 + 2.86225i −0.423646 0.733776i −3.39354 −1.00088 + 2.44913i 3.00585 0.414604 + 0.718115i −0.975769 + 1.69008i
508.3 −0.689527 1.19430i −1.44060 + 2.49520i 0.0491037 0.0850501i −0.402974 0.697972i 3.97334 −1.26180 + 2.32548i −2.89354 −2.65067 4.59109i −0.555723 + 0.962541i
508.4 −0.672613 1.16500i −1.02505 + 1.77544i 0.0951832 0.164862i −1.78389 3.08979i 2.75785 2.62255 + 0.349630i −2.94654 −0.601462 1.04176i −2.39973 + 4.15646i
508.5 −0.249993 0.433001i −0.424801 + 0.735776i 0.875007 1.51556i 0.521238 + 0.902810i 0.424789 −2.40155 1.11021i −1.87496 1.13909 + 1.97296i 0.260612 0.451393i
508.6 −0.0904119 0.156598i 0.913006 1.58137i 0.983651 1.70373i −1.34332 2.32670i −0.330186 −1.64912 2.06892i −0.717383 −0.167162 0.289532i −0.242904 + 0.420723i
508.7 0.0904119 + 0.156598i 0.913006 1.58137i 0.983651 1.70373i 1.34332 + 2.32670i 0.330186 1.64912 + 2.06892i 0.717383 −0.167162 0.289532i −0.242904 + 0.420723i
508.8 0.249993 + 0.433001i −0.424801 + 0.735776i 0.875007 1.51556i −0.521238 0.902810i −0.424789 2.40155 + 1.11021i 1.87496 1.13909 + 1.97296i 0.260612 0.451393i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 170.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.j 24
7.c even 3 1 inner 1183.2.e.j 24
7.c even 3 1 8281.2.a.cp 12
7.d odd 6 1 8281.2.a.co 12
13.b even 2 1 inner 1183.2.e.j 24
13.f odd 12 1 91.2.k.b 12
13.f odd 12 1 91.2.u.b yes 12
39.k even 12 1 819.2.bm.f 12
39.k even 12 1 819.2.do.e 12
91.r even 6 1 inner 1183.2.e.j 24
91.r even 6 1 8281.2.a.cp 12
91.s odd 6 1 8281.2.a.co 12
91.w even 12 1 637.2.k.i 12
91.w even 12 1 637.2.q.i 12
91.x odd 12 1 91.2.u.b yes 12
91.x odd 12 1 637.2.q.g 12
91.ba even 12 1 637.2.q.i 12
91.ba even 12 1 637.2.u.g 12
91.bc even 12 1 637.2.k.i 12
91.bc even 12 1 637.2.u.g 12
91.bd odd 12 1 91.2.k.b 12
91.bd odd 12 1 637.2.q.g 12
273.bv even 12 1 819.2.do.e 12
273.bw even 12 1 819.2.bm.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 13.f odd 12 1
91.2.k.b 12 91.bd odd 12 1
91.2.u.b yes 12 13.f odd 12 1
91.2.u.b yes 12 91.x odd 12 1
637.2.k.i 12 91.w even 12 1
637.2.k.i 12 91.bc even 12 1
637.2.q.g 12 91.x odd 12 1
637.2.q.g 12 91.bd odd 12 1
637.2.q.i 12 91.w even 12 1
637.2.q.i 12 91.ba even 12 1
637.2.u.g 12 91.ba even 12 1
637.2.u.g 12 91.bc even 12 1
819.2.bm.f 12 39.k even 12 1
819.2.bm.f 12 273.bw even 12 1
819.2.do.e 12 39.k even 12 1
819.2.do.e 12 273.bv even 12 1
1183.2.e.j 24 1.a even 1 1 trivial
1183.2.e.j 24 7.c even 3 1 inner
1183.2.e.j 24 13.b even 2 1 inner
1183.2.e.j 24 91.r even 6 1 inner
8281.2.a.co 12 7.d odd 6 1
8281.2.a.co 12 91.s odd 6 1
8281.2.a.cp 12 7.c even 3 1
8281.2.a.cp 12 91.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$:

 $$T_{2}^{24} + 16 T_{2}^{22} + 168 T_{2}^{20} + 1014 T_{2}^{18} + 4420 T_{2}^{16} + 11868 T_{2}^{14} + \cdots + 1$$ T2^24 + 16*T2^22 + 168*T2^20 + 1014*T2^18 + 4420*T2^16 + 11868*T2^14 + 23099*T2^12 + 27564*T2^10 + 22404*T2^8 + 5798*T2^6 + 1124*T2^4 + 36*T2^2 + 1 $$T_{3}^{12} + 3 T_{3}^{11} + 14 T_{3}^{10} + 17 T_{3}^{9} + 69 T_{3}^{8} + 75 T_{3}^{7} + 233 T_{3}^{6} + \cdots + 49$$ T3^12 + 3*T3^11 + 14*T3^10 + 17*T3^9 + 69*T3^8 + 75*T3^7 + 233*T3^6 + 147*T3^5 + 355*T3^4 + 300*T3^3 + 333*T3^2 + 133*T3 + 49