Newform orbit 123.3.p.a

• Label: 123.3.p.a
• Level: $123$
• Weight: $3$
• Character orbit: 123.p
• Analytic conductor: $3.352$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	123.p (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.35150725163\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(14\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 8 q^{2} + 48 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.69830	−	3.33311i	0.662827	−	1.60021i	−5.87423	+	8.08518i	−1.41465	−	8.93174i	−6.45934	+	0.508361i	−0.532909	−	0.0419408i	22.1459	+	3.50756i	−2.12132	−	2.12132i	−27.3679	+	19.8840i
7.2	−1.68215	−	3.30140i	−0.662827	+	1.60021i	−5.71850	+	7.87084i	0.202086	+	1.27592i	6.39790	−	0.503526i	−0.114984	−	0.00904940i	20.9656	+	3.32063i	−2.12132	−	2.12132i	3.87240	−	2.81346i
7.3	−1.17443	−	2.30495i	0.662827	−	1.60021i	−1.58238	+	2.17796i	0.320377	+	2.02278i	−4.46685	+	0.351549i	−12.6669	−	0.996904i	−3.34174	−	0.529280i	−2.12132	−	2.12132i	4.28616	−	3.11407i
7.4	−1.14743	−	2.25196i	−0.662827	+	1.60021i	−1.40359	+	1.93187i	−1.04690	−	6.60985i	4.36415	−	0.343466i	8.94462	+	0.703957i	−4.02426	−	0.637380i	−2.12132	−	2.12132i	−13.6839	+	9.94191i
7.5	−1.07037	−	2.10071i	−0.662827	+	1.60021i	−0.916173	+	1.26100i	0.489996	+	3.09371i	4.07104	−	0.320398i	−8.42795	−	0.663294i	−5.68499	−	0.900414i	−2.12132	−	2.12132i	5.97453	−	4.34075i
7.6	−0.511336	−	1.00355i	0.662827	−	1.60021i	1.60549	−	2.20976i	−0.136661	−	0.862845i	−1.94482	+	0.153061i	10.1850	+	0.801577i	−7.48835	−	1.18604i	−2.12132	−	2.12132i	−0.796032	+	0.578351i
7.7	−0.176596	−	0.346589i	0.662827	−	1.60021i	2.26220	−	3.11366i	−0.483356	−	3.05179i	−0.671667	+	0.0528613i	−5.01449	−	0.394649i	−3.01544	−	0.477599i	−2.12132	−	2.12132i	−0.972358	+	0.706459i
7.8	0.0379979	+	0.0745750i	−0.662827	+	1.60021i	2.34702	−	3.23040i	−1.10799	−	6.99557i	−0.144521	+	0.0113741i	−11.1794	−	0.879842i	0.660757	+	0.104654i	−2.12132	−	2.12132i	0.479593	−	0.348445i
7.9	0.416603	+	0.817630i	−0.662827	+	1.60021i	1.85618	−	2.55481i	−0.325751	−	2.05671i	−1.58451	+	0.124704i	9.12108	+	0.717845i	6.48758	+	1.02753i	−2.12132	−	2.12132i	1.54592	−	1.12318i
7.10	0.561689	+	1.10238i	0.662827	−	1.60021i	1.45140	−	1.99768i	1.08727	+	6.86476i	2.13633	−	0.168133i	−1.72756	−	0.135962i	7.90541	+	1.25209i	−2.12132	−	2.12132i	−6.95684	+	5.05444i
7.11	0.978626	+	1.92066i	0.662827	−	1.60021i	−0.380093	+	0.523153i	−1.20818	−	7.62815i	3.72212	−	0.292937i	0.629633	+	0.0495532i	7.13951	+	1.13079i	−2.12132	−	2.12132i	13.4687	−	9.78562i
7.12	1.09529	+	2.14963i	−0.662827	+	1.60021i	−1.07011	+	1.47288i	0.903333	+	5.70342i	−4.16584	+	0.327859i	−6.78471	−	0.533968i	5.19331	+	0.822539i	−2.12132	−	2.12132i	−11.2708	+	8.18875i
7.13	1.37831	+	2.70509i	0.662827	−	1.60021i	−3.06664	+	4.22086i	0.622661	+	3.93133i	5.24229	−	0.412577i	9.12719	+	0.718325i	−3.65015	−	0.578126i	−2.12132	−	2.12132i	−9.77639	+	7.10296i
7.14	1.70802	+	3.35217i	−0.662827	+	1.60021i	−5.96858	+	8.21504i	−0.0743049	−	0.469143i	−6.49628	+	0.511268i	8.44138	+	0.664351i	−22.8690	−	3.62209i	−2.12132	−	2.12132i	1.44573	−	1.05039i
13.1	−3.25932	−	1.66070i	−0.662827	+	1.60021i	5.51406	+	7.58946i	−0.931032	−	0.147461i	4.81783	−	4.11482i	1.45727	+	1.24462i	−3.07927	−	19.4418i	−2.12132	−	2.12132i	2.78964	+	2.02679i
13.2	−3.01204	−	1.53471i	0.662827	−	1.60021i	4.36589	+	6.00913i	−2.31374	−	0.366461i	−4.45231	+	3.80263i	9.11705	+	7.78669i	−1.81265	−	11.4446i	−2.12132	−	2.12132i	6.40666	+	4.65471i
13.3	−1.87689	−	0.956321i	−0.662827	+	1.60021i	0.257010	+	0.353743i	1.09718	+	0.173777i	2.77436	−	2.36953i	−1.91936	−	1.63929i	1.17402	+	7.41246i	−2.12132	−	2.12132i	−1.89310	−	1.37542i
13.4	−1.64589	−	0.838624i	0.662827	−	1.60021i	−0.345470	−	0.475498i	−1.40097	−	0.221892i	−2.43291	+	2.07790i	−5.29837	−	4.52524i	1.32572	+	8.37028i	−2.12132	−	2.12132i	2.11977	+	1.54010i
13.5	−1.61654	−	0.823668i	0.662827	−	1.60021i	−0.416368	−	0.573082i	7.83243	+	1.24054i	−2.38953	+	2.04085i	7.07184	+	6.03992i	1.33631	+	8.43715i	−2.12132	−	2.12132i	−11.6397	−	8.45671i
13.6	−0.858002	−	0.437174i	−0.662827	+	1.60021i	−1.80610	−	2.48588i	1.90799	+	0.302195i	1.26827	−	1.08321i	−2.75906	−	2.35646i	1.06543	+	6.72687i	−2.12132	−	2.12132i	−1.50494	−	1.09341i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
41.h	odd	40	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	123.3.p.a	✓	224
3.b	odd	2	1		369.3.bb.c		224
41.h	odd	40	1	inner	123.3.p.a	✓	224
123.o	even	40	1		369.3.bb.c		224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
123.3.p.a	✓	224	1.a	even	1	1	trivial
123.3.p.a	✓	224	41.h	odd	40	1	inner
369.3.bb.c		224	3.b	odd	2	1
369.3.bb.c		224	123.o	even	40	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(123, [\chi])\).