Newform orbit 12.16.b.a

• Label: 12.16.b.a
• Level: $12$
• Weight: $16$
• Character orbit: 12.b
• Analytic conductor: $17.123$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.1232206120\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 26968 q^{4} + 823656 q^{6} - 3812052 q^{9}+O(q^{10}) \)

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	−172.746	−	54.1000i	3105.31	+	2169.32i	26914.4	+	18691.1i		−	192307.i	−419070.	−	542739.i		−	366926.i	−3.63816e6	−	4.68489e6i	4.93700e6	+	1.34728e7i	−1.04038e7	+	3.32203e7i
11.2	−172.746	+	54.1000i	3105.31	−	2169.32i	26914.4	−	18691.1i			192307.i	−419070.	+	542739.i			366926.i	−3.63816e6	+	4.68489e6i	4.93700e6	−	1.34728e7i	−1.04038e7	−	3.32203e7i
11.3	−171.504	−	57.9158i	−3751.22	−	526.541i	26059.5	+	19865.6i			172529.i	612856.	+	307559.i		−	3.89480e6i	−3.31879e6	−	4.91630e6i	1.37944e7	+	3.95035e6i	9.99218e6	−	2.95896e7i
11.4	−171.504	+	57.9158i	−3751.22	+	526.541i	26059.5	−	19865.6i		−	172529.i	612856.	−	307559.i			3.89480e6i	−3.31879e6	+	4.91630e6i	1.37944e7	−	3.95035e6i	9.99218e6	+	2.95896e7i
11.5	−166.441	−	71.1726i	−8.61090	−	3787.99i	22636.9	+	23692.0i		−	99785.5i	−268168.	+	631087.i			1.78364e6i	−2.08148e6	−	5.55444e6i	−1.43488e7	+	65235.9i	−7.10200e6	+	1.66083e7i
11.6	−166.441	+	71.1726i	−8.61090	+	3787.99i	22636.9	−	23692.0i			99785.5i	−268168.	−	631087.i		−	1.78364e6i	−2.08148e6	+	5.55444e6i	−1.43488e7	−	65235.9i	−7.10200e6	−	1.66083e7i
11.7	−127.986	−	128.014i	−692.213	+	3724.21i	−7.18349	+	32768.0i			230590.i	565345.	−	388034.i			3.60781e6i	4.19568e6	−	4.19292e6i	−1.33906e7	−	5.15589e6i	2.95188e7	−	2.95123e7i
11.8	−127.986	+	128.014i	−692.213	−	3724.21i	−7.18349	−	32768.0i		−	230590.i	565345.	+	388034.i		−	3.60781e6i	4.19568e6	+	4.19292e6i	−1.33906e7	+	5.15589e6i	2.95188e7	+	2.95123e7i
11.9	−83.3106	−	160.709i	3555.47	−	1306.74i	−18886.7	+	26777.5i			93360.9i	−506212.	−	462530.i		−	1.14542e6i	5.87684e6	+	804409.i	1.09338e7	−	9.29212e6i	1.50039e7	−	7.77795e6i
11.10	−83.3106	+	160.709i	3555.47	+	1306.74i	−18886.7	−	26777.5i		−	93360.9i	−506212.	+	462530.i			1.14542e6i	5.87684e6	−	804409.i	1.09338e7	+	9.29212e6i	1.50039e7	+	7.77795e6i
11.11	−80.5973	−	162.087i	−2767.36	+	2586.62i	−19776.2	+	26127.5i		−	332581.i	642299.	+	240078.i		−	2.13481e6i	5.82882e6	+	1.09965e6i	967701.	−	1.43162e7i	−5.39069e7	+	2.68051e7i
11.12	−80.5973	+	162.087i	−2767.36	−	2586.62i	−19776.2	−	26127.5i			332581.i	642299.	−	240078.i			2.13481e6i	5.82882e6	−	1.09965e6i	967701.	+	1.43162e7i	−5.39069e7	−	2.68051e7i
11.13	−35.8416	−	177.436i	−2291.54	−	3016.25i	−30198.8	+	12719.1i			33476.6i	−453057.	+	514708.i			694274.i	3.33920e6	+	4.90246e6i	−3.84657e6	+	1.38237e7i	5.93994e6	−	1.19985e6i
11.14	−35.8416	+	177.436i	−2291.54	+	3016.25i	−30198.8	−	12719.1i		−	33476.6i	−453057.	−	514708.i		−	694274.i	3.33920e6	−	4.90246e6i	−3.84657e6	−	1.38237e7i	5.93994e6	+	1.19985e6i
11.15	35.8416	−	177.436i	2291.54	+	3016.25i	−30198.8	−	12719.1i			33476.6i	617322.	−	298494.i		−	694274.i	−3.33920e6	+	4.90246e6i	−3.84657e6	+	1.38237e7i	5.93994e6	+	1.19985e6i
11.16	35.8416	+	177.436i	2291.54	−	3016.25i	−30198.8	+	12719.1i		−	33476.6i	617322.	+	298494.i			694274.i	−3.33920e6	−	4.90246e6i	−3.84657e6	−	1.38237e7i	5.93994e6	−	1.19985e6i
11.17	80.5973	−	162.087i	2767.36	−	2586.62i	−19776.2	−	26127.5i		−	332581.i	−196215.	−	657027.i			2.13481e6i	−5.82882e6	+	1.09965e6i	967701.	−	1.43162e7i	−5.39069e7	−	2.68051e7i
11.18	80.5973	+	162.087i	2767.36	+	2586.62i	−19776.2	+	26127.5i			332581.i	−196215.	+	657027.i		−	2.13481e6i	−5.82882e6	−	1.09965e6i	967701.	+	1.43162e7i	−5.39069e7	+	2.68051e7i
11.19	83.3106	−	160.709i	−3555.47	+	1306.74i	−18886.7	−	26777.5i			93360.9i	−86203.8	+	680260.i			1.14542e6i	−5.87684e6	+	804409.i	1.09338e7	−	9.29212e6i	1.50039e7	+	7.77795e6i
11.20	83.3106	+	160.709i	−3555.47	−	1306.74i	−18886.7	+	26777.5i		−	93360.9i	−86203.8	−	680260.i		−	1.14542e6i	−5.87684e6	−	804409.i	1.09338e7	+	9.29212e6i	1.50039e7	−	7.77795e6i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	12.16.b.a	✓	28
3.b	odd	2	1	inner	12.16.b.a	✓	28
4.b	odd	2	1	inner	12.16.b.a	✓	28
12.b	even	2	1	inner	12.16.b.a	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.16.b.a	✓	28	1.a	even	1	1	trivial
12.16.b.a	✓	28	3.b	odd	2	1	inner
12.16.b.a	✓	28	4.b	odd	2	1	inner
12.16.b.a	✓	28	12.b	even	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(12, [\chi])\).