Newform orbit 36.6.h.a

• Label: 36.6.h.a
• Level: $36$
• Weight: $6$
• Character orbit: 36.h
• Analytic conductor: $5.774$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	36.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.77381751327\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 3 q^{2} - q^{4} - 6 q^{5} - 27 q^{6} + 18 q^{9}+O(q^{10}) \)

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	−5.58478	+	0.900118i	−10.9673	+	11.0778i	30.3796	−	10.0539i	−78.5398	+	45.3450i	51.2786	−	71.7391i	69.6792	+	40.2293i	−160.614	+	83.4942i	−2.43645	−	242.988i	397.812	−	323.937i
11.2	−5.50860	+	1.28660i	−13.2152	−	8.26786i	28.6893	−	14.1747i	38.3603	−	22.1474i	83.4349	+	28.5416i	−102.940	−	59.4323i	−139.801	+	114.994i	106.285	+	218.523i	−182.817	+	171.355i
11.3	−5.49437	−	1.34609i	13.5204	+	7.75878i	28.3761	+	14.7918i	−27.3241	+	15.7756i	−63.8421	−	60.8292i	2.81745	+	1.62665i	−135.998	−	119.468i	122.603	+	209.804i	171.364	−	49.8961i
11.4	−5.35087	−	1.83526i	6.54818	−	14.1464i	25.2636	+	19.6405i	36.0599	−	20.8192i	−61.0009	+	63.6781i	117.288	+	67.7164i	−99.1369	−	151.459i	−157.243	−	185.267i	−231.160	+	45.2214i
11.5	−4.78727	+	3.01364i	13.9499	−	6.95697i	13.8359	−	28.8542i	−0.143132	+	0.0826372i	−45.8163	+	75.3450i	−192.634	−	111.217i	20.7202	+	179.830i	146.201	−	194.098i	0.436172	−	0.826955i
11.6	−4.76722	+	3.04525i	2.08778	+	15.4480i	13.4528	−	29.0348i	84.3255	−	48.6853i	−56.9960	−	67.2863i	87.9661	+	50.7872i	24.2857	+	179.383i	−234.282	+	64.5040i	−253.739	+	488.886i
11.7	−4.26482	−	3.71636i	−6.54818	+	14.1464i	4.37736	+	31.6992i	36.0599	−	20.8192i	80.5000	−	35.9966i	−117.288	−	67.7164i	99.1369	−	151.459i	−157.243	−	185.267i	−231.160	−	45.2214i
11.8	−3.91293	−	4.08522i	−13.5204	−	7.75878i	−1.37798	+	31.9703i	−27.3241	+	15.7756i	21.2081	+	85.5933i	−2.81745	−	1.62665i	135.998	−	119.468i	122.603	+	209.804i	171.364	+	49.8961i
11.9	−3.75408	+	4.23165i	−1.51698	−	15.5145i	−3.81374	−	31.7719i	−52.1793	+	30.1257i	71.3467	+	51.8233i	185.316	+	106.992i	148.765	+	103.136i	−238.398	+	47.0702i	68.4038	−	333.899i
11.10	−2.01287	−	5.28662i	10.9673	−	11.0778i	−23.8967	+	21.2825i	−78.5398	+	45.3450i	−80.6400	−	35.6818i	−69.6792	−	40.2293i	160.614	+	83.4942i	−2.43645	−	242.988i	397.812	+	323.937i
11.11	−1.90579	+	5.32616i	9.73802	+	12.1726i	−24.7359	−	20.3011i	−51.8636	+	29.9435i	−83.3916	+	28.6679i	−33.8827	−	19.5622i	155.268	−	93.0578i	−53.3420	+	237.073i	−60.6425	−	333.300i
11.12	−1.78216	+	5.36879i	−15.4584	+	2.00950i	−25.6478	−	19.1360i	12.8631	−	7.42651i	16.7607	−	86.5741i	−15.2989	−	8.83282i	148.446	−	103.595i	234.924	−	62.1273i	16.9473	+	82.2944i
11.13	−1.64008	−	5.41389i	13.2152	+	8.26786i	−26.6203	+	17.7584i	38.3603	−	22.1474i	23.0872	−	85.1057i	102.940	+	59.4323i	139.801	+	114.994i	106.285	+	218.523i	−182.817	−	171.355i
11.14	−0.0499164	+	5.65663i	14.1974	−	6.43687i	−31.9950	−	0.564717i	70.1957	−	40.5275i	35.7024	+	80.6309i	89.9985	+	51.9607i	4.79147	−	180.956i	160.133	−	182.774i	225.745	+	399.094i
11.15	0.216257	−	5.65272i	−13.9499	+	6.95697i	−31.9065	−	2.44488i	−0.143132	+	0.0826372i	36.3090	+	80.3595i	192.634	+	111.217i	−20.7202	+	179.830i	146.201	−	194.098i	0.436172	+	0.826955i
11.16	0.253656	−	5.65116i	−2.08778	−	15.4480i	−31.8713	−	2.86690i	84.3255	−	48.6853i	−87.8289	+	7.87989i	−87.9661	−	50.7872i	−24.2857	+	179.383i	−234.282	+	64.5040i	−253.739	−	488.886i
11.17	1.72594	+	5.38713i	−3.10718	−	15.2756i	−26.0423	+	18.5957i	−20.1454	+	11.6310i	76.9290	−	43.1036i	−156.832	−	90.5473i	−145.125	−	108.198i	−223.691	+	94.9283i	−97.4272	−	88.4515i
11.18	1.78768	−	5.36696i	1.51698	+	15.5145i	−25.6084	−	19.1888i	−52.1793	+	30.1257i	85.9773	+	19.5933i	−185.316	−	106.992i	−148.765	+	103.136i	−238.398	+	47.0702i	68.4038	+	333.899i
11.19	2.66122	+	4.99179i	−5.80725	+	14.4664i	−17.8359	+	26.5684i	3.68052	−	2.12495i	−87.6674	+	9.50956i	13.9322	+	8.04377i	−180.089	−	18.3284i	−175.552	−	168.020i	20.4019	+	12.7174i
11.20	3.65969	−	4.31354i	−9.73802	−	12.1726i	−5.21330	−	31.5725i	−51.8636	+	29.9435i	−88.1450	−	2.54244i	33.8827	+	19.5622i	−155.268	−	93.0578i	−53.3420	+	237.073i	−60.6425	+	333.300i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
36.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	36.6.h.a	✓	56
3.b	odd	2	1		108.6.h.a		56
4.b	odd	2	1	inner	36.6.h.a	✓	56
9.c	even	3	1		108.6.h.a		56
9.d	odd	6	1	inner	36.6.h.a	✓	56
12.b	even	2	1		108.6.h.a		56
36.f	odd	6	1		108.6.h.a		56
36.h	even	6	1	inner	36.6.h.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.6.h.a	✓	56	1.a	even	1	1	trivial
36.6.h.a	✓	56	4.b	odd	2	1	inner
36.6.h.a	✓	56	9.d	odd	6	1	inner
36.6.h.a	✓	56	36.h	even	6	1	inner
108.6.h.a		56	3.b	odd	2	1
108.6.h.a		56	9.c	even	3	1
108.6.h.a		56	12.b	even	2	1
108.6.h.a		56	36.f	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(36, [\chi])\).