Newform orbit 144.2.u.a

• 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: $4$

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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

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

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 \(S_{2}^{\mathrm{new}}(144, [\chi])\).