# Properties

 Label 144.2.u.a Level 144 Weight 2 Character orbit 144.u Analytic conductor 1.150 Analytic rank 0 Dimension 88 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.14984578911$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$22$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} + 2q^{6} - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} + 2q^{6} - 4q^{7} - 8q^{10} - 6q^{11} - 16q^{12} - 2q^{13} - 6q^{14} - 2q^{16} - 10q^{18} - 8q^{19} - 48q^{20} + 2q^{21} - 2q^{22} - 12q^{23} - 16q^{27} + 8q^{28} - 6q^{29} - 34q^{30} - 6q^{32} - 8q^{33} + 2q^{34} - 26q^{36} - 8q^{37} - 6q^{38} - 32q^{39} - 2q^{40} + 48q^{42} - 2q^{43} + 6q^{45} - 40q^{46} + 42q^{48} - 24q^{49} + 72q^{50} - 12q^{51} - 2q^{52} - 38q^{54} - 16q^{55} + 36q^{56} + 16q^{58} - 42q^{59} + 70q^{60} - 2q^{61} - 44q^{64} - 12q^{65} + 104q^{66} - 2q^{67} + 96q^{68} - 10q^{69} - 16q^{70} - 10q^{72} + 78q^{74} - 56q^{75} - 14q^{76} - 6q^{77} + 12q^{78} - 8q^{81} - 36q^{82} + 54q^{83} + 158q^{84} + 8q^{85} + 54q^{86} + 48q^{87} + 22q^{88} + 64q^{90} + 20q^{91} + 108q^{92} - 34q^{93} + 6q^{94} - 58q^{96} - 4q^{97} + 46q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.40455 0.165045i 1.03912 + 1.38573i 1.94552 + 0.463628i 1.15827 + 0.310357i −1.23078 2.11782i 0.356047 0.616691i −2.65606 0.972288i −0.840471 + 2.87986i −1.57562 0.627079i
11.2 −1.39618 + 0.225141i −1.36987 + 1.05993i 1.89862 0.628673i −2.78704 0.746784i 1.67395 1.78827i 1.16672 2.02082i −2.50928 + 1.30520i 0.753089 2.90394i 4.05933 + 0.415168i
11.3 −1.20521 + 0.739910i −0.679468 1.59321i 0.905065 1.78350i −2.39818 0.642590i 1.99774 + 1.41741i −1.93190 + 3.34616i 0.228833 + 2.81916i −2.07665 + 2.16507i 3.36577 0.999982i
11.4 −1.19060 0.763199i 0.0841095 1.73001i 0.835053 + 1.81733i −1.17929 0.315990i −1.42048 + 1.99555i 1.93802 3.35676i 0.392771 2.80102i −2.98585 0.291020i 1.16290 + 1.27625i
11.5 −1.17890 0.781149i −1.69048 0.377192i 0.779612 + 1.84179i 1.05401 + 0.282421i 1.69826 + 1.76519i −1.93586 + 3.35301i 0.519632 2.78028i 2.71545 + 1.27527i −1.02196 1.15629i
11.6 −1.07723 + 0.916282i 1.67850 0.427367i 0.320853 1.97410i −0.170993 0.0458174i −1.41654 + 1.99835i 1.17432 2.03397i 1.46320 + 2.42055i 2.63471 1.43467i 0.226181 0.107322i
11.7 −0.990880 + 1.00904i −0.800065 + 1.53620i −0.0363133 1.99967i 3.73424 + 1.00059i −0.757311 2.32948i −1.68236 + 2.91393i 2.05372 + 1.94479i −1.71979 2.45811i −4.70981 + 2.77653i
11.8 −0.700562 1.22850i 1.70352 0.313101i −1.01842 + 1.72128i 2.80938 + 0.752772i −1.57806 1.87342i −1.02581 + 1.77675i 2.82806 + 0.0452695i 2.80394 1.06674i −1.04337 3.97869i
11.9 −0.665270 1.24796i 0.117756 + 1.72804i −1.11483 + 1.66047i −3.58858 0.961558i 2.07820 1.29657i −1.29216 + 2.23809i 2.81387 + 0.286608i −2.97227 + 0.406975i 1.18739 + 5.11812i
11.10 −0.354608 + 1.36903i −0.874828 1.49488i −1.74851 0.970941i 1.76649 + 0.473330i 2.35677 0.667570i 1.40613 2.43549i 1.94929 2.04946i −1.46935 + 2.61553i −1.27442 + 2.25054i
11.11 −0.311077 1.37958i −1.36635 + 1.06447i −1.80646 + 0.858309i 2.31044 + 0.619079i 1.89356 + 1.55385i 2.51270 4.35213i 1.74605 + 2.22515i 0.733803 2.90887i 0.135344 3.38000i
11.12 −0.0415142 + 1.41360i 1.52574 + 0.819834i −1.99655 0.117369i −0.769670 0.206232i −1.22226 + 2.12275i −2.17574 + 3.76849i 0.248799 2.81746i 1.65574 + 2.50170i 0.323483 1.07945i
11.13 0.0484091 + 1.41338i −1.65298 + 0.517369i −1.99531 + 0.136841i −1.94452 0.521033i −0.811261 2.31125i −0.322227 + 0.558114i −0.290001 2.81352i 2.46466 1.71040i 0.642288 2.77358i
11.14 0.232839 1.39491i 1.23430 1.21512i −1.89157 0.649582i −2.70956 0.726024i −1.40760 2.00466i −0.00424642 + 0.00735502i −1.34654 + 2.48733i 0.0469689 2.99963i −1.64363 + 3.61055i
11.15 0.688374 1.23537i 1.10465 + 1.33407i −1.05228 1.70079i 0.664471 + 0.178044i 2.40849 0.446316i 0.645693 1.11837i −2.82548 + 0.129179i −0.559489 + 2.94737i 0.677355 0.698307i
11.16 0.717593 + 1.21863i 1.11494 1.32549i −0.970121 + 1.74896i 1.20583 + 0.323102i 2.41535 + 0.407536i 0.140266 0.242948i −2.82749 + 0.0728225i −0.513832 2.95567i 0.471556 + 1.70132i
11.17 0.772565 1.18454i −0.501476 1.65787i −0.806286 1.83027i 3.72190 + 0.997280i −2.35124 0.686791i −0.481387 + 0.833787i −2.79095 0.458925i −2.49704 + 1.66276i 4.05673 3.63829i
11.18 1.21908 0.716836i −1.58764 0.692394i 0.972292 1.74776i −2.83365 0.759273i −2.43178 + 0.293995i 1.41719 2.45465i −0.0675573 2.82762i 2.04118 + 2.19854i −3.99870 + 1.10565i
11.19 1.25087 + 0.659799i −1.71297 0.256373i 1.12933 + 1.65064i 2.03779 + 0.546024i −1.97354 1.45091i −0.0638076 + 0.110518i 0.323550 + 2.80986i 2.86855 + 0.878321i 2.18873 + 2.02753i
11.20 1.29365 + 0.571371i 1.61259 + 0.632098i 1.34707 + 1.47831i −3.81396 1.02195i 1.72497 + 1.73910i 1.46715 2.54117i 0.897978 + 2.68210i 2.20090 + 2.03863i −4.35003 3.50123i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
16.f odd 4 1 inner
144.u 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.u.a 88
3.b odd 2 1 432.2.v.a 88
4.b odd 2 1 576.2.y.a 88
9.c even 3 1 432.2.v.a 88
9.d odd 6 1 inner 144.2.u.a 88
12.b even 2 1 1728.2.z.a 88
16.e even 4 1 576.2.y.a 88
16.f odd 4 1 inner 144.2.u.a 88
36.f odd 6 1 1728.2.z.a 88
36.h even 6 1 576.2.y.a 88
48.i odd 4 1 1728.2.z.a 88
48.k even 4 1 432.2.v.a 88
144.u even 12 1 inner 144.2.u.a 88
144.v odd 12 1 432.2.v.a 88
144.w odd 12 1 576.2.y.a 88
144.x even 12 1 1728.2.z.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.u.a 88 1.a even 1 1 trivial
144.2.u.a 88 9.d odd 6 1 inner
144.2.u.a 88 16.f odd 4 1 inner
144.2.u.a 88 144.u even 12 1 inner
432.2.v.a 88 3.b odd 2 1
432.2.v.a 88 9.c even 3 1
432.2.v.a 88 48.k even 4 1
432.2.v.a 88 144.v odd 12 1
576.2.y.a 88 4.b odd 2 1
576.2.y.a 88 16.e even 4 1
576.2.y.a 88 36.h even 6 1
576.2.y.a 88 144.w odd 12 1
1728.2.z.a 88 12.b even 2 1
1728.2.z.a 88 36.f odd 6 1
1728.2.z.a 88 48.i odd 4 1
1728.2.z.a 88 144.x even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database