# Properties

 Label 144.2.x.d Level 144 Weight 2 Character orbit 144.x Analytic conductor 1.150 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 144.x (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( -1 + 2 \zeta_{12}^{2} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( 1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -4 + \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} + ( 2 - 2 \zeta_{12}^{3} ) q^{8} -3 q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{12} - \zeta_{12}^{2} ) q^{2} + ( -1 + 2 \zeta_{12}^{2} ) q^{3} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} + ( 1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( -4 + \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} + ( 2 - 2 \zeta_{12}^{3} ) q^{8} -3 q^{9} + ( 2 - \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{10} + ( 1 - \zeta_{12} ) q^{11} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{12} + ( 2 - 2 \zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{13} + ( -1 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{14} + ( -2 + 2 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{15} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} + 4 q^{17} + ( -3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{18} + ( -3 + 3 \zeta_{12}^{3} ) q^{19} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{20} + ( \zeta_{12} - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{21} + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{22} + ( 2 - 3 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{23} + ( -2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} + ( -2 - 3 \zeta_{12} + \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{25} + ( -\zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{26} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( 2 - 4 \zeta_{12} - 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{28} + ( -1 - 2 \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{29} + ( -\zeta_{12} + 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{30} + ( -4 + 3 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{31} + ( 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{32} + ( -1 + \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} + ( 4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{34} + ( -2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{35} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{36} + ( -5 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{37} + ( -6 + 6 \zeta_{12}^{2} ) q^{38} + ( 4 + 4 \zeta_{12} + \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{39} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{40} + ( 3 + 6 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{41} + ( -7 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{42} + ( -1 - 5 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{43} + ( -2 + 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{44} + ( -3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{45} + ( 4 - \zeta_{12} - 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{46} + ( 3 \zeta_{12} + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{47} + ( 4 + 4 \zeta_{12}^{2} ) q^{48} + ( 6 - 4 \zeta_{12} - 6 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{49} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{50} + ( -4 + 8 \zeta_{12}^{2} ) q^{51} + ( -4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{52} + ( 5 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{53} + ( -3 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{54} + ( -1 + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{55} + ( -8 + 6 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{56} + ( 3 - 6 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{57} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{58} + ( 3 + 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{59} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{60} + ( 7 + 8 \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{61} + ( 6 - 7 \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{62} + ( 12 - 3 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( -2 \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( 3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{66} + ( 5 - 5 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{67} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{68} + ( -6 + 3 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{69} + ( -7 + 3 \zeta_{12} + 7 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{70} + ( 4 - 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{71} + ( -6 + 6 \zeta_{12}^{3} ) q^{72} + ( -2 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{73} + ( -2 - 12 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{74} + ( -3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{75} + ( -6 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{76} + ( -5 + 5 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{77} + ( 10 + 3 \zeta_{12} - 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{78} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{79} + ( 4 - 4 \zeta_{12} ) q^{80} + 9 q^{81} + ( 6 + 9 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{82} + ( 1 - 9 \zeta_{12} - 10 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{83} + ( 2 - 12 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{84} + ( 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{85} + ( -1 + 2 \zeta_{12}^{2} + 11 \zeta_{12}^{3} ) q^{86} + ( 3 - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{87} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{88} + ( -8 + 16 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{89} + ( -6 + 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{90} + ( -4 + 9 \zeta_{12} + 9 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{91} + ( -6 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{92} + ( -4 + 9 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{93} + ( 1 + 6 \zeta_{12} + 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{94} + ( 3 - 3 \zeta_{12} - 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{95} + ( 8 + 4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{96} -\zeta_{12}^{2} q^{97} + ( -8 + 10 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{98} + ( -3 + 3 \zeta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} + 2q^{5} + 6q^{6} - 12q^{7} + 8q^{8} - 12q^{9} + O(q^{10})$$ $$4q + 2q^{2} + 2q^{5} + 6q^{6} - 12q^{7} + 8q^{8} - 12q^{9} + 6q^{10} + 4q^{11} + 2q^{13} + 4q^{14} - 6q^{15} + 8q^{16} + 16q^{17} - 6q^{18} - 12q^{19} - 4q^{20} - 12q^{21} + 12q^{23} - 6q^{25} - 10q^{26} + 4q^{28} - 6q^{29} + 6q^{30} - 8q^{31} - 8q^{32} + 8q^{34} - 14q^{35} - 24q^{37} - 12q^{38} + 18q^{39} - 4q^{40} + 18q^{41} - 24q^{42} + 8q^{43} - 8q^{44} - 6q^{45} + 6q^{46} + 8q^{47} + 24q^{48} + 12q^{49} - 12q^{50} - 20q^{52} + 16q^{53} - 18q^{54} - 28q^{56} - 12q^{58} + 12q^{59} + 12q^{60} + 30q^{61} + 26q^{62} + 36q^{63} + 2q^{65} + 12q^{66} + 16q^{67} - 12q^{69} - 14q^{70} - 24q^{72} - 12q^{74} - 6q^{75} + 12q^{76} - 16q^{77} + 30q^{78} + 16q^{80} + 36q^{81} + 30q^{82} - 16q^{83} + 12q^{84} + 8q^{85} + 6q^{87} + 4q^{88} - 18q^{90} + 2q^{91} - 24q^{92} - 24q^{93} + 16q^{94} + 6q^{95} + 24q^{96} - 2q^{97} - 36q^{98} - 12q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$-1 + \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
−0.366025 1.36603i 1.73205i −1.73205 + 1.00000i 0.500000 + 1.86603i 2.36603 0.633975i −3.86603 + 2.23205i 2.00000 + 2.00000i −3.00000 2.36603 1.36603i
61.1 1.36603 + 0.366025i 1.73205i 1.73205 + 1.00000i 0.500000 + 0.133975i 0.633975 2.36603i −2.13397 1.23205i 2.00000 + 2.00000i −3.00000 0.633975 + 0.366025i
85.1 1.36603 0.366025i 1.73205i 1.73205 1.00000i 0.500000 0.133975i 0.633975 + 2.36603i −2.13397 + 1.23205i 2.00000 2.00000i −3.00000 0.633975 0.366025i
133.1 −0.366025 + 1.36603i 1.73205i −1.73205 1.00000i 0.500000 1.86603i 2.36603 + 0.633975i −3.86603 2.23205i 2.00000 2.00000i −3.00000 2.36603 + 1.36603i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
144.x even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.x.d yes 4
3.b odd 2 1 432.2.y.a 4
4.b odd 2 1 576.2.bb.b 4
9.c even 3 1 144.2.x.a 4
9.d odd 6 1 432.2.y.d 4
12.b even 2 1 1728.2.bc.b 4
16.e even 4 1 144.2.x.a 4
16.f odd 4 1 576.2.bb.a 4
36.f odd 6 1 576.2.bb.a 4
36.h even 6 1 1728.2.bc.c 4
48.i odd 4 1 432.2.y.d 4
48.k even 4 1 1728.2.bc.c 4
144.u even 12 1 1728.2.bc.b 4
144.v odd 12 1 576.2.bb.b 4
144.w odd 12 1 432.2.y.a 4
144.x even 12 1 inner 144.2.x.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.a 4 9.c even 3 1
144.2.x.a 4 16.e even 4 1
144.2.x.d yes 4 1.a even 1 1 trivial
144.2.x.d yes 4 144.x even 12 1 inner
432.2.y.a 4 3.b odd 2 1
432.2.y.a 4 144.w odd 12 1
432.2.y.d 4 9.d odd 6 1
432.2.y.d 4 48.i odd 4 1
576.2.bb.a 4 16.f odd 4 1
576.2.bb.a 4 36.f odd 6 1
576.2.bb.b 4 4.b odd 2 1
576.2.bb.b 4 144.v odd 12 1
1728.2.bc.b 4 12.b even 2 1
1728.2.bc.b 4 144.u even 12 1
1728.2.bc.c 4 36.h even 6 1
1728.2.bc.c 4 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 2 T_{5}^{3} + 5 T_{5}^{2} - 4 T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4}$$
$3$ $$( 1 + 3 T^{2} )^{2}$$
$5$ $$1 - 2 T + 5 T^{2} - 14 T^{3} + 16 T^{4} - 70 T^{5} + 125 T^{6} - 250 T^{7} + 625 T^{8}$$
$7$ $$1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 2100 T^{5} + 3577 T^{6} + 4116 T^{7} + 2401 T^{8}$$
$11$ $$1 - 4 T + 5 T^{2} + 20 T^{3} - 164 T^{4} + 220 T^{5} + 605 T^{6} - 5324 T^{7} + 14641 T^{8}$$
$13$ $$( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} )$$
$17$ $$( 1 - 4 T + 17 T^{2} )^{4}$$
$19$ $$( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 12 T + 97 T^{2} - 588 T^{3} + 2976 T^{4} - 13524 T^{5} + 51313 T^{6} - 146004 T^{7} + 279841 T^{8}$$
$29$ $$1 + 6 T + 9 T^{2} - 174 T^{3} - 1528 T^{4} - 5046 T^{5} + 7569 T^{6} + 146334 T^{7} + 707281 T^{8}$$
$31$ $$1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 2728 T^{5} + 12493 T^{6} + 238328 T^{7} + 923521 T^{8}$$
$37$ $$1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 91464 T^{5} + 394272 T^{6} + 1215672 T^{7} + 1874161 T^{8}$$
$41$ $$1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 53874 T^{5} + 304261 T^{6} - 1240578 T^{7} + 2825761 T^{8}$$
$43$ $$1 - 8 T + 65 T^{2} + 120 T^{3} - 604 T^{4} + 5160 T^{5} + 120185 T^{6} - 636056 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 4136 T^{5} - 41971 T^{6} - 830584 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 66992 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 12 T + 45 T^{2} + 300 T^{3} - 5524 T^{4} + 17700 T^{5} + 156645 T^{6} - 2464548 T^{7} + 12117361 T^{8}$$
$61$ $$1 - 30 T + 261 T^{2} + 702 T^{3} - 21664 T^{4} + 42822 T^{5} + 971181 T^{6} - 6809430 T^{7} + 13845841 T^{8}$$
$67$ $$1 - 16 T + 65 T^{2} + 960 T^{3} - 14284 T^{4} + 64320 T^{5} + 291785 T^{6} - 4812208 T^{7} + 20151121 T^{8}$$
$71$ $$1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8}$$
$73$ $$1 - 236 T^{2} + 23814 T^{4} - 1257644 T^{6} + 28398241 T^{8}$$
$79$ $$1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8}$$
$83$ $$1 + 16 T + 185 T^{2} + 784 T^{3} + 4900 T^{4} + 65072 T^{5} + 1274465 T^{6} + 9148592 T^{7} + 47458321 T^{8}$$
$89$ $$1 + 36 T^{2} + 13094 T^{4} + 285156 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2}$$