# Properties

 Label 144.2.k.c Level $144$ Weight $2$ Character orbit 144.k Analytic conductor $1.150$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.629407744.1 Defining polynomial: $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{4} - \beta_{6} ) q^{5} + ( -1 + \beta_{2} - \beta_{3} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{4} - \beta_{6} ) q^{5} + ( -1 + \beta_{2} - \beta_{3} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + ( 2 - \beta_{2} - 2 \beta_{3} - \beta_{7} ) q^{10} + ( -\beta_{1} - \beta_{5} - \beta_{6} ) q^{11} + ( 1 + \beta_{3} + 2 \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{4} + 2 \beta_{6} ) q^{14} + ( \beta_{2} + 4 \beta_{3} + \beta_{7} ) q^{16} + ( -3 \beta_{1} - \beta_{5} + \beta_{6} ) q^{17} + ( -2 - 2 \beta_{7} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{5} ) q^{20} + ( -2 \beta_{2} - 4 \beta_{3} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{23} + ( 2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{7} ) q^{25} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{26} + ( -4 + \beta_{2} + 4 \beta_{3} + \beta_{7} ) q^{28} + ( 3 \beta_{1} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{29} + ( -3 - \beta_{2} - \beta_{3} - \beta_{7} ) q^{31} + ( 2 \beta_{4} + 2 \beta_{5} ) q^{32} + ( -4 - 2 \beta_{2} ) q^{34} + ( 5 \beta_{1} + 2 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{35} + ( 3 - 2 \beta_{2} + \beta_{3} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( -4 + 2 \beta_{2} - 4 \beta_{3} ) q^{40} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{41} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{44} + ( 4 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} ) q^{46} + ( -6 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{47} + ( -1 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} ) q^{49} + ( 2 \beta_{1} + \beta_{4} - 4 \beta_{6} ) q^{50} + ( 8 - \beta_{2} + \beta_{7} ) q^{52} + ( 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{53} + 4 \beta_{3} q^{55} + ( -4 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{56} + ( 6 + \beta_{2} - 2 \beta_{3} + 3 \beta_{7} ) q^{58} + ( -2 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{59} + ( 1 - 3 \beta_{3} - 2 \beta_{7} ) q^{61} + ( -3 \beta_{1} + \beta_{4} - 2 \beta_{5} ) q^{62} + ( 4 + 4 \beta_{3} - 2 \beta_{7} ) q^{64} + ( -5 \beta_{1} - 4 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{65} + ( 4 - 4 \beta_{3} ) q^{67} + ( -4 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{68} + ( -4 + 4 \beta_{2} - 8 \beta_{3} - 2 \beta_{7} ) q^{70} + ( 4 \beta_{4} + 4 \beta_{6} ) q^{71} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{73} + ( 3 \beta_{1} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{74} + ( -8 - 2 \beta_{7} ) q^{76} + ( 4 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{77} + ( 7 + \beta_{2} + \beta_{3} + \beta_{7} ) q^{79} + ( -4 \beta_{1} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{80} + ( -8 + 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} ) q^{82} + ( \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{83} + ( -4 + 4 \beta_{2} ) q^{85} + ( -4 \beta_{1} - 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{86} + ( -2 \beta_{2} - 8 \beta_{3} + 2 \beta_{7} ) q^{88} + ( 2 \beta_{1} - 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 8 - 2 \beta_{2} + 6 \beta_{3} ) q^{91} + ( 4 \beta_{1} + 8 \beta_{4} - 4 \beta_{5} ) q^{92} + ( 4 + 2 \beta_{2} + 12 \beta_{3} + 6 \beta_{7} ) q^{94} + ( 4 \beta_{1} + 2 \beta_{4} - 2 \beta_{6} ) q^{95} + ( -4 \beta_{2} - 4 \beta_{3} - 4 \beta_{7} ) q^{97} + ( -\beta_{1} - 2 \beta_{4} + 4 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{4} + O(q^{10})$$ $$8q + 4q^{4} + 16q^{10} - 8q^{19} - 8q^{22} - 32q^{28} - 24q^{31} - 40q^{34} + 16q^{37} - 24q^{40} - 24q^{43} + 32q^{46} - 8q^{49} + 56q^{52} + 40q^{58} + 16q^{61} + 40q^{64} + 32q^{67} - 8q^{70} - 56q^{76} + 56q^{79} - 40q^{82} - 16q^{85} - 16q^{88} + 56q^{91} + 16q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 2 x^{4} - 8 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} - 2 \nu^{2} - 4$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{5} - 2 \nu^{3} - 4 \nu$$$$)/12$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 2 \nu^{3} + 4 \nu$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{5} + 4 \nu^{3} - 4 \nu$$$$)/6$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} - \nu^{2} + 4$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} + 4 \beta_{3} + \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + 2 \beta_{4}$$ $$\nu^{6}$$ $$=$$ $$-2 \beta_{7} + 4 \beta_{3} + 4$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{6} - 2 \beta_{5} + 6 \beta_{4} + 4 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$\beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −1.38255 − 0.297594i −0.767178 + 1.18804i 0.767178 − 1.18804i 1.38255 + 0.297594i −1.38255 + 0.297594i −0.767178 − 1.18804i 0.767178 + 1.18804i 1.38255 − 0.297594i
−1.38255 0.297594i 0 1.82288 + 0.822876i 0.595188 + 0.595188i 0 1.64575i −2.27533 1.68014i 0 −0.645751 1.00000i
37.2 −0.767178 + 1.18804i 0 −0.822876 1.82288i −2.37608 2.37608i 0 3.64575i 2.79694 + 0.420861i 0 4.64575 1.00000i
37.3 0.767178 1.18804i 0 −0.822876 1.82288i 2.37608 + 2.37608i 0 3.64575i −2.79694 0.420861i 0 4.64575 1.00000i
37.4 1.38255 + 0.297594i 0 1.82288 + 0.822876i −0.595188 0.595188i 0 1.64575i 2.27533 + 1.68014i 0 −0.645751 1.00000i
109.1 −1.38255 + 0.297594i 0 1.82288 0.822876i 0.595188 0.595188i 0 1.64575i −2.27533 + 1.68014i 0 −0.645751 + 1.00000i
109.2 −0.767178 1.18804i 0 −0.822876 + 1.82288i −2.37608 + 2.37608i 0 3.64575i 2.79694 0.420861i 0 4.64575 + 1.00000i
109.3 0.767178 + 1.18804i 0 −0.822876 + 1.82288i 2.37608 2.37608i 0 3.64575i −2.79694 + 0.420861i 0 4.64575 + 1.00000i
109.4 1.38255 0.297594i 0 1.82288 0.822876i −0.595188 + 0.595188i 0 1.64575i 2.27533 1.68014i 0 −0.645751 + 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.4 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 144.2.k.c 8
3.b odd 2 1 inner 144.2.k.c 8
4.b odd 2 1 576.2.k.c 8
8.b even 2 1 1152.2.k.e 8
8.d odd 2 1 1152.2.k.d 8
12.b even 2 1 576.2.k.c 8
16.e even 4 1 inner 144.2.k.c 8
16.e even 4 1 1152.2.k.e 8
16.f odd 4 1 576.2.k.c 8
16.f odd 4 1 1152.2.k.d 8
24.f even 2 1 1152.2.k.d 8
24.h odd 2 1 1152.2.k.e 8
32.g even 8 2 9216.2.a.bq 8
32.h odd 8 2 9216.2.a.bt 8
48.i odd 4 1 inner 144.2.k.c 8
48.i odd 4 1 1152.2.k.e 8
48.k even 4 1 576.2.k.c 8
48.k even 4 1 1152.2.k.d 8
96.o even 8 2 9216.2.a.bt 8
96.p odd 8 2 9216.2.a.bq 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c 8 1.a even 1 1 trivial
144.2.k.c 8 3.b odd 2 1 inner
144.2.k.c 8 16.e even 4 1 inner
144.2.k.c 8 48.i odd 4 1 inner
576.2.k.c 8 4.b odd 2 1
576.2.k.c 8 12.b even 2 1
576.2.k.c 8 16.f odd 4 1
576.2.k.c 8 48.k even 4 1
1152.2.k.d 8 8.d odd 2 1
1152.2.k.d 8 16.f odd 4 1
1152.2.k.d 8 24.f even 2 1
1152.2.k.d 8 48.k even 4 1
1152.2.k.e 8 8.b even 2 1
1152.2.k.e 8 16.e even 4 1
1152.2.k.e 8 24.h odd 2 1
1152.2.k.e 8 48.i odd 4 1
9216.2.a.bq 8 32.g even 8 2
9216.2.a.bq 8 96.p odd 8 2
9216.2.a.bt 8 32.h odd 8 2
9216.2.a.bt 8 96.o even 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 128 T_{5}^{4} + 64$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 8 T^{2} + 2 T^{4} - 2 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$64 + 128 T^{4} + T^{8}$$
$7$ $$( 36 + 16 T^{2} + T^{4} )^{2}$$
$11$ $$1024 + 512 T^{4} + T^{8}$$
$13$ $$( 196 + T^{4} )^{2}$$
$17$ $$( 288 - 40 T^{2} + T^{4} )^{2}$$
$19$ $$( 144 - 48 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$23$ $$( 1152 + 80 T^{2} + T^{4} )^{2}$$
$29$ $$5184 + 5632 T^{4} + T^{8}$$
$31$ $$( 2 + 6 T + T^{2} )^{4}$$
$37$ $$( 36 + 48 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$41$ $$( 2592 + 104 T^{2} + T^{4} )^{2}$$
$43$ $$( 16 + 48 T + 72 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$47$ $$( 10368 - 208 T^{2} + T^{4} )^{2}$$
$53$ $$8340544 + 5888 T^{4} + T^{8}$$
$59$ $$21233664 + 16384 T^{4} + T^{8}$$
$61$ $$( 36 + 48 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$67$ $$( 32 - 8 T + T^{2} )^{4}$$
$71$ $$( 2048 + 192 T^{2} + T^{4} )^{2}$$
$73$ $$( 576 + 64 T^{2} + T^{4} )^{2}$$
$79$ $$( 42 - 14 T + T^{2} )^{4}$$
$83$ $$1024 + 512 T^{4} + T^{8}$$
$89$ $$( 512 + 96 T^{2} + T^{4} )^{2}$$
$97$ $$( -112 + T^{2} )^{4}$$