Newform orbit 196.4.f.e

• Label: 196.4.f.e
• Level: $196$
• Weight: $4$
• Character orbit: 196.f
• Analytic conductor: $11.564$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	196.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5643743611\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 4 q^{2} + 12 q^{4} - 136 q^{8} - 432 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	−2.80859	−	0.334428i	−2.36435	−	4.09517i	7.77632	+	1.87854i	0.952206	+	0.549756i	5.27093	+	12.2923i		0		−21.2122	−	7.87667i	2.31974	−	4.01791i	−2.49050	−	1.86248i
19.2	−2.80859	−	0.334428i	2.36435	+	4.09517i	7.77632	+	1.87854i	−0.952206	−	0.549756i	−5.27093	−	12.2923i		0		−21.2122	−	7.87667i	2.31974	−	4.01791i	2.49050	+	1.86248i
19.3	−2.79748	+	0.417267i	−4.66364	−	8.07766i	7.65178	−	2.33459i	−10.7437	−	6.20288i	16.4170	+	20.6511i		0		−20.4315	+	9.72381i	−29.9990	+	51.9598i	32.6435	+	12.8694i
19.4	−2.79748	+	0.417267i	4.66364	+	8.07766i	7.65178	−	2.33459i	10.7437	+	6.20288i	−16.4170	−	20.6511i		0		−20.4315	+	9.72381i	−29.9990	+	51.9598i	−32.6435	−	12.8694i
19.5	−2.50637	+	1.31078i	−4.00387	−	6.93491i	4.56373	−	6.57057i	16.6076	+	9.58841i	19.1253	+	12.1332i		0		−2.82585	+	22.4503i	−18.5620	+	32.1503i	−54.1930	−	2.26320i
19.6	−2.50637	+	1.31078i	4.00387	+	6.93491i	4.56373	−	6.57057i	−16.6076	−	9.58841i	−19.1253	−	12.1332i		0		−2.82585	+	22.4503i	−18.5620	+	32.1503i	54.1930	+	2.26320i
19.7	−2.43851	+	1.43307i	−2.07438	−	3.59292i	3.89264	−	6.98909i	−7.60752	−	4.39220i	10.2073	+	5.78866i		0		0.523587	+	22.6214i	4.89393	−	8.47654i	24.8453	−	0.191657i
19.8	−2.43851	+	1.43307i	2.07438	+	3.59292i	3.89264	−	6.98909i	7.60752	+	4.39220i	−10.2073	−	5.78866i		0		0.523587	+	22.6214i	4.89393	−	8.47654i	−24.8453	+	0.191657i
19.9	−2.26097	−	1.69942i	−1.93559	−	3.35254i	2.22394	+	7.68467i	5.90936	+	3.41177i	−1.32108	+	10.8694i		0		8.03124	−	21.1542i	6.00696	−	10.4044i	−7.56283	−	17.7564i
19.10	−2.26097	−	1.69942i	1.93559	+	3.35254i	2.22394	+	7.68467i	−5.90936	−	3.41177i	1.32108	−	10.8694i		0		8.03124	−	21.1542i	6.00696	−	10.4044i	7.56283	+	17.7564i
19.11	−1.18638	−	2.56759i	−0.922201	−	1.59730i	−5.18499	+	6.09228i	3.84134	+	2.21780i	−3.00712	+	4.26284i		0		21.7938	+	6.08510i	11.7991	−	20.4366i	1.13708	−	12.4941i
19.12	−1.18638	−	2.56759i	0.922201	+	1.59730i	−5.18499	+	6.09228i	−3.84134	−	2.21780i	3.00712	−	4.26284i		0		21.7938	+	6.08510i	11.7991	−	20.4366i	−1.13708	+	12.4941i
19.13	−0.779398	−	2.71892i	−2.98884	−	5.17682i	−6.78508	+	4.23825i	14.6227	+	8.44243i	−11.7459	+	12.1612i		0		16.8117	+	15.1448i	−4.36634	+	7.56272i	11.5574	−	46.3380i
19.14	−0.779398	−	2.71892i	2.98884	+	5.17682i	−6.78508	+	4.23825i	−14.6227	−	8.44243i	11.7459	−	12.1612i		0		16.8117	+	15.1448i	−4.36634	+	7.56272i	−11.5574	+	46.3380i
19.15	−0.0218178	+	2.82834i	−2.07438	−	3.59292i	−7.99905	−	0.123417i	7.60752	+	4.39220i	10.2073	−	5.78866i		0		0.523587	−	22.6214i	4.89393	−	8.47654i	−12.5886	+	21.4208i
19.16	−0.0218178	+	2.82834i	2.07438	+	3.59292i	−7.99905	−	0.123417i	−7.60752	−	4.39220i	−10.2073	+	5.78866i		0		0.523587	−	22.6214i	4.89393	−	8.47654i	12.5886	−	21.4208i
19.17	0.118018	+	2.82596i	−4.00387	−	6.93491i	−7.97214	+	0.667027i	−16.6076	−	9.58841i	19.1253	−	12.1332i		0		−2.82585	−	22.4503i	−18.5620	+	32.1503i	25.1365	−	48.0641i
19.18	0.118018	+	2.82596i	4.00387	+	6.93491i	−7.97214	+	0.667027i	16.6076	+	9.58841i	−19.1253	+	12.1332i		0		−2.82585	−	22.4503i	−18.5620	+	32.1503i	−25.1365	+	48.0641i
19.19	0.974861	−	2.65512i	−4.44929	−	7.70640i	−6.09929	−	5.17674i	10.5583	+	6.09586i	−24.7988	+	4.30072i		0		−19.6908	+	11.1477i	−26.0924	+	45.1933i	26.4781	−	22.0910i
19.20	0.974861	−	2.65512i	4.44929	+	7.70640i	−6.09929	−	5.17674i	−10.5583	−	6.09586i	24.7988	−	4.30072i		0		−19.6908	+	11.1477i	−26.0924	+	45.1933i	−26.4781	+	22.0910i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
28.d	even	2	1	inner
28.f	even	6	1	inner
28.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	196.4.f.e		64
4.b	odd	2	1	inner	196.4.f.e		64
7.b	odd	2	1	inner	196.4.f.e		64
7.c	even	3	1		196.4.d.c	✓	32
7.c	even	3	1	inner	196.4.f.e		64
7.d	odd	6	1		196.4.d.c	✓	32
7.d	odd	6	1	inner	196.4.f.e		64
28.d	even	2	1	inner	196.4.f.e		64
28.f	even	6	1		196.4.d.c	✓	32
28.f	even	6	1	inner	196.4.f.e		64
28.g	odd	6	1		196.4.d.c	✓	32
28.g	odd	6	1	inner	196.4.f.e		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.4.d.c	✓	32	7.c	even	3	1
196.4.d.c	✓	32	7.d	odd	6	1
196.4.d.c	✓	32	28.f	even	6	1
196.4.d.c	✓	32	28.g	odd	6	1
196.4.f.e		64	1.a	even	1	1	trivial
196.4.f.e		64	4.b	odd	2	1	inner
196.4.f.e		64	7.b	odd	2	1	inner
196.4.f.e		64	7.c	even	3	1	inner
196.4.f.e		64	7.d	odd	6	1	inner
196.4.f.e		64	28.d	even	2	1	inner
196.4.f.e		64	28.f	even	6	1	inner
196.4.f.e		64	28.g	odd	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(196, [\chi])\):

T3^32 + 324*T3^30 + 62618*T3^28 + 7961528*T3^26 + 750526648*T3^24 + 53061146512*T3^22 + 2915828585736*T3^20 + 123664253003552*T3^18 + 4120228660804320*T3^16 + 106461247705677248*T3^14 + 2152966354465921440*T3^12 + 33205044094706481280*T3^10 + 388785489494418960448*T3^8 + 3218400106376240586752*T3^6 + 18212442171403987189760*T3^4 + 50953427047129974571008*T3^2 + 95919013074192012673024

T5^32 - 1100*T5^30 + 740406*T5^28 - 317176472*T5^26 + 99624896116*T5^24 - 22754763032352*T5^22 + 3970641768927392*T5^20 - 519385083456398592*T5^18 + 52020156574687009280*T5^16 - 3827531570698488907776*T5^14 + 210522732194474515417088*T5^12 - 7991469024970257177821184*T5^10 + 214961498404127523406249984*T5^8 - 3269465920611623835356495872*T5^6 + 32682956976564580507851030528*T5^4 - 39148575940253272593446993920*T5^2 + 42009632901980830300279668736