Newform orbit 123.3.h.a

• Label: 123.3.h.a
• Level: $123$
• Weight: $3$
• Character orbit: 123.h
• Analytic conductor: $3.352$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 123 = 3 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	123.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.35150725163\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 8 q^{2} - 48 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
55.1	−2.48255	−	2.48255i	−0.662827	−	1.60021i			8.32610i	6.05026	−	6.05026i	−2.32709	+	5.61809i	−4.16095	−	10.0454i	10.7397	−	10.7397i	−2.12132	+	2.12132i	−30.0401
55.2	−2.07159	−	2.07159i	−0.662827	−	1.60021i			4.58295i	−2.20300	+	2.20300i	−1.94186	+	4.68807i	2.89751	+	6.99522i	1.20764	−	1.20764i	−2.12132	+	2.12132i	9.12742
55.3	−1.87640	−	1.87640i	0.662827	+	1.60021i			3.04177i	−1.57929	+	1.57929i	1.75890	−	4.24636i	−2.16941	−	5.23741i	−1.79802	+	1.79802i	−2.12132	+	2.12132i	5.92678
55.4	−1.65879	−	1.65879i	0.662827	+	1.60021i			1.50318i	4.94854	−	4.94854i	1.55492	−	3.75390i	3.33101	+	8.04178i	−4.14171	+	4.14171i	−2.12132	+	2.12132i	−16.4172
55.5	−0.714050	−	0.714050i	0.662827	+	1.60021i		−	2.98026i	−4.82161	+	4.82161i	0.669336	−	1.61592i	3.20153	+	7.72917i	−4.98426	+	4.98426i	−2.12132	+	2.12132i	6.88574
55.6	−0.659128	−	0.659128i	−0.662827	−	1.60021i		−	3.13110i	3.13805	−	3.13805i	−0.617853	+	1.49163i	0.00254719	+	0.00614945i	−4.70031	+	4.70031i	−2.12132	+	2.12132i	−4.13675
55.7	−0.185057	−	0.185057i	0.662827	+	1.60021i		−	3.93151i	0.790151	−	0.790151i	0.173468	−	0.418789i	−4.85142	−	11.7124i	−1.46778	+	1.46778i	−2.12132	+	2.12132i	−0.292445
55.8	0.0938247	+	0.0938247i	−0.662827	−	1.60021i		−	3.98239i	−6.82320	+	6.82320i	0.0879493	−	0.212328i	−1.94470	−	4.69493i	0.748946	−	0.748946i	−2.12132	+	2.12132i	−1.28037
55.9	1.06730	+	1.06730i	0.662827	+	1.60021i		−	1.72176i	5.57137	−	5.57137i	−1.00046	+	2.41533i	1.07386	+	2.59252i	6.10681	−	6.10681i	−2.12132	+	2.12132i	11.8926
55.10	1.30727	+	1.30727i	−0.662827	−	1.60021i		−	0.582083i	2.00674	−	2.00674i	1.22541	−	2.95840i	−2.31146	−	5.58036i	5.99003	−	5.99003i	−2.12132	+	2.12132i	5.24670
55.11	1.79075	+	1.79075i	0.662827	+	1.60021i			2.41355i	−2.59780	+	2.59780i	−1.67861	+	4.05252i	2.22276	+	5.36621i	2.84093	−	2.84093i	−2.12132	+	2.12132i	−9.30399
55.12	2.31893	+	2.31893i	−0.662827	−	1.60021i			6.75486i	−6.14062	+	6.14062i	2.17372	−	5.24781i	2.28540	+	5.51745i	−6.38833	+	6.38833i	−2.12132	+	2.12132i	−28.4793
55.13	2.49324	+	2.49324i	−0.662827	−	1.60021i			8.43249i	5.86283	−	5.86283i	2.33711	−	5.64229i	3.23165	+	7.80189i	−11.0513	+	11.0513i	−2.12132	+	2.12132i	29.2349
55.14	2.57626	+	2.57626i	0.662827	+	1.60021i			9.27421i	1.45443	−	1.45443i	−2.41493	+	5.83016i	−2.80833	−	6.77991i	−13.5877	+	13.5877i	−2.12132	+	2.12132i	7.49399
79.1	−2.64084	+	2.64084i	1.60021	+	0.662827i		−	9.94811i	−5.03687	−	5.03687i	−5.97632	+	2.47547i	10.1493	+	4.20399i	15.7080	+	15.7080i	2.12132	+	2.12132i	26.6032
79.2	−2.46579	+	2.46579i	−1.60021	−	0.662827i		−	8.16022i	4.70270	+	4.70270i	5.58016	−	2.31138i	4.58955	+	1.90105i	10.2582	+	10.2582i	2.12132	+	2.12132i	−23.1917
79.3	−1.64483	+	1.64483i	−1.60021	−	0.662827i		−	1.41090i	−2.15560	−	2.15560i	3.72230	−	1.54183i	1.60127	+	0.663268i	−4.25861	−	4.25861i	2.12132	+	2.12132i	7.09118
79.4	−1.42749	+	1.42749i	1.60021	+	0.662827i		−	0.0754618i	2.64646	+	2.64646i	−3.23046	+	1.33810i	5.67414	+	2.35031i	−5.60224	−	5.60224i	2.12132	+	2.12132i	−7.55559
79.5	−0.882628	+	0.882628i	1.60021	+	0.662827i			2.44193i	−6.47829	−	6.47829i	−1.99742	+	0.827357i	−11.0815	−	4.59010i	−5.68583	−	5.68583i	2.12132	+	2.12132i	11.4358
79.6	−0.491409	+	0.491409i	−1.60021	−	0.662827i			3.51704i	0.694248	+	0.694248i	1.11207	−	0.460636i	−8.79698	−	3.64383i	−3.69394	−	3.69394i	2.12132	+	2.12132i	−0.682318

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
41.e	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	123.3.h.a	✓	56
3.b	odd	2	1		369.3.l.c		56
41.e	odd	8	1	inner	123.3.h.a	✓	56
123.i	even	8	1		369.3.l.c		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
123.3.h.a	✓	56	1.a	even	1	1	trivial
123.3.h.a	✓	56	41.e	odd	8	1	inner
369.3.l.c		56	3.b	odd	2	1
369.3.l.c		56	123.i	even	8	1

Hecke kernels

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