# Properties

 Label 144.3.j.a Level $144$ Weight $3$ Character orbit 144.j Analytic conductor $3.924$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$32q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q + 40q^{10} + 48q^{16} + 64q^{19} - 88q^{22} - 120q^{28} - 248q^{34} - 184q^{40} + 128q^{43} + 24q^{46} - 224q^{49} + 632q^{52} + 456q^{58} + 64q^{61} - 48q^{64} - 64q^{67} - 312q^{70} - 576q^{76} - 512q^{79} - 720q^{82} + 320q^{85} - 400q^{88} - 192q^{91} + 696q^{94} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1 −1.99710 0.107698i 0 3.97680 + 0.430168i −1.92848 + 1.92848i 0 2.43162i −7.89574 1.28738i 0 4.05906 3.64367i
53.2 −1.95704 0.412300i 0 3.66002 + 1.61377i 5.52702 5.52702i 0 7.79421i −6.49745 4.66725i 0 −13.0954 + 8.53782i
53.3 −1.76559 0.939508i 0 2.23465 + 3.31758i −6.08688 + 6.08688i 0 9.40026i −0.828591 7.95697i 0 16.4656 5.02829i
53.4 −1.56339 + 1.24732i 0 0.888382 3.90010i −1.84570 + 1.84570i 0 0.226665i 3.47579 + 7.20548i 0 0.583370 5.18772i
53.5 −1.13812 1.64459i 0 −1.40936 + 3.74349i 3.90343 3.90343i 0 0.778757i 7.76053 1.94271i 0 −10.8621 1.97697i
53.6 −1.11574 + 1.65986i 0 −1.51025 3.70394i −0.900179 + 0.900179i 0 7.66460i 7.83305 + 1.62583i 0 −0.489803 2.49853i
53.7 −0.252839 1.98395i 0 −3.87214 + 1.00324i −1.66372 + 1.66372i 0 13.3224i 2.96942 + 7.42850i 0 3.72140 + 2.88010i
53.8 −0.126306 + 1.99601i 0 −3.96809 0.504216i 5.14405 5.14405i 0 7.48880i 1.50761 7.85666i 0 9.61783 + 10.9173i
53.9 0.126306 1.99601i 0 −3.96809 0.504216i −5.14405 + 5.14405i 0 7.48880i −1.50761 + 7.85666i 0 9.61783 + 10.9173i
53.10 0.252839 + 1.98395i 0 −3.87214 + 1.00324i 1.66372 1.66372i 0 13.3224i −2.96942 7.42850i 0 3.72140 + 2.88010i
53.11 1.11574 1.65986i 0 −1.51025 3.70394i 0.900179 0.900179i 0 7.66460i −7.83305 1.62583i 0 −0.489803 2.49853i
53.12 1.13812 + 1.64459i 0 −1.40936 + 3.74349i −3.90343 + 3.90343i 0 0.778757i −7.76053 + 1.94271i 0 −10.8621 1.97697i
53.13 1.56339 1.24732i 0 0.888382 3.90010i 1.84570 1.84570i 0 0.226665i −3.47579 7.20548i 0 0.583370 5.18772i
53.14 1.76559 + 0.939508i 0 2.23465 + 3.31758i 6.08688 6.08688i 0 9.40026i 0.828591 + 7.95697i 0 16.4656 5.02829i
53.15 1.95704 + 0.412300i 0 3.66002 + 1.61377i −5.52702 + 5.52702i 0 7.79421i 6.49745 + 4.66725i 0 −13.0954 + 8.53782i
53.16 1.99710 + 0.107698i 0 3.97680 + 0.430168i 1.92848 1.92848i 0 2.43162i 7.89574 + 1.28738i 0 4.05906 3.64367i
125.1 −1.99710 + 0.107698i 0 3.97680 0.430168i −1.92848 1.92848i 0 2.43162i −7.89574 + 1.28738i 0 4.05906 + 3.64367i
125.2 −1.95704 + 0.412300i 0 3.66002 1.61377i 5.52702 + 5.52702i 0 7.79421i −6.49745 + 4.66725i 0 −13.0954 8.53782i
125.3 −1.76559 + 0.939508i 0 2.23465 3.31758i −6.08688 6.08688i 0 9.40026i −0.828591 + 7.95697i 0 16.4656 + 5.02829i
125.4 −1.56339 1.24732i 0 0.888382 + 3.90010i −1.84570 1.84570i 0 0.226665i 3.47579 7.20548i 0 0.583370 + 5.18772i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 125.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.j.a 32
3.b odd 2 1 inner 144.3.j.a 32
4.b odd 2 1 576.3.j.a 32
8.b even 2 1 1152.3.j.a 32
8.d odd 2 1 1152.3.j.b 32
12.b even 2 1 576.3.j.a 32
16.e even 4 1 inner 144.3.j.a 32
16.e even 4 1 1152.3.j.a 32
16.f odd 4 1 576.3.j.a 32
16.f odd 4 1 1152.3.j.b 32
24.f even 2 1 1152.3.j.b 32
24.h odd 2 1 1152.3.j.a 32
48.i odd 4 1 inner 144.3.j.a 32
48.i odd 4 1 1152.3.j.a 32
48.k even 4 1 576.3.j.a 32
48.k even 4 1 1152.3.j.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.j.a 32 1.a even 1 1 trivial
144.3.j.a 32 3.b odd 2 1 inner
144.3.j.a 32 16.e even 4 1 inner
144.3.j.a 32 48.i odd 4 1 inner
576.3.j.a 32 4.b odd 2 1
576.3.j.a 32 12.b even 2 1
576.3.j.a 32 16.f odd 4 1
576.3.j.a 32 48.k even 4 1
1152.3.j.a 32 8.b even 2 1
1152.3.j.a 32 16.e even 4 1
1152.3.j.a 32 24.h odd 2 1
1152.3.j.a 32 48.i odd 4 1
1152.3.j.b 32 8.d odd 2 1
1152.3.j.b 32 16.f odd 4 1
1152.3.j.b 32 24.f even 2 1
1152.3.j.b 32 48.k even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database