Newform orbit 52.4.f.b

• Label: 52.4.f.b
• Level: $52$
• Weight: $4$
• Character orbit: 52.f
• Analytic conductor: $3.068$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 52 = 2^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	52.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.06809932030\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{2} - 20 q^{5} - 16 q^{6} - 40 q^{8} - 332 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−2.82787	−	0.0560904i			6.97823i	7.99371	+	0.317233i	3.73459	−	3.73459i	0.391412	−	19.7335i	−18.7167	+	18.7167i	−22.5874	−	1.34546i	−21.6957			−10.7704	+	10.3515i
31.2	−2.81956	−	0.223782i		−	1.00320i	7.89984	+	1.26193i	−4.98300	+	4.98300i	−0.224497	+	2.82857i	16.5820	−	16.5820i	−21.9917	−	5.32593i	25.9936			15.1650	−	12.9348i
31.3	−2.50920	−	1.30534i		−	9.31194i	4.59218	+	6.55072i	11.9568	−	11.9568i	−12.1552	+	23.3655i	−6.41108	+	6.41108i	−2.97178	−	22.4314i	−59.7122			−45.6098	+	14.3944i
31.4	−2.25470	+	1.70773i		−	8.10067i	2.16733	−	7.70082i	−10.8445	+	10.8445i	13.8337	+	18.2646i	−18.4065	+	18.4065i	8.26423	+	21.0642i	−38.6209			5.93160	−	42.9704i
31.5	−1.84904	−	2.14034i		−	1.29511i	−1.16213	+	7.91514i	−10.6631	+	10.6631i	−2.77198	+	2.39471i	−7.71781	+	7.71781i	19.0899	−	12.1480i	25.3227			42.5392	+	3.10621i
31.6	−1.70773	+	2.25470i			8.10067i	−2.16733	−	7.70082i	−10.8445	+	10.8445i	−18.2646	−	13.8337i	18.4065	−	18.4065i	21.0642	+	8.26423i	−38.6209			−5.93160	−	42.9704i
31.7	−1.42308	−	2.44435i			3.58200i	−3.94969	+	6.95701i	10.6687	−	10.6687i	8.75566	−	5.09747i	6.84080	−	6.84080i	22.6261	−	0.245959i	14.1693			−41.2606	−	10.8957i
31.8	0.0560904	+	2.82787i		−	6.97823i	−7.99371	+	0.317233i	3.73459	−	3.73459i	19.7335	−	0.391412i	18.7167	−	18.7167i	−1.34546	−	22.5874i	−21.6957			10.7704	+	10.3515i
31.9	0.223782	+	2.81956i			1.00320i	−7.89984	+	1.26193i	−4.98300	+	4.98300i	−2.82857	+	0.224497i	−16.5820	+	16.5820i	−5.32593	−	21.9917i	25.9936			−15.1650	−	12.9348i
31.10	0.289217	−	2.81360i			7.82965i	−7.83271	−	1.62748i	−8.33093	+	8.33093i	22.0295	+	2.26447i	−11.3199	+	11.3199i	−6.84443	+	21.5674i	−34.3035			21.0305	+	25.8494i
31.11	0.340566	−	2.80785i		−	6.14077i	−7.76803	−	1.91251i	−1.41066	+	1.41066i	−17.2423	−	2.09133i	3.71079	−	3.71079i	−8.01557	+	21.1601i	−10.7090			3.48049	+	4.44133i
31.12	1.30534	+	2.50920i			9.31194i	−4.59218	+	6.55072i	11.9568	−	11.9568i	−23.3655	+	12.1552i	6.41108	−	6.41108i	−22.4314	−	2.97178i	−59.7122			45.6098	+	14.3944i
31.13	1.74069	−	2.22935i			3.23176i	−1.93998	−	7.76122i	4.87198	−	4.87198i	7.20471	+	5.62550i	22.0805	−	22.0805i	−20.6794	−	9.18500i	16.5557			−2.38072	−	19.3420i
31.14	2.14034	+	1.84904i			1.29511i	1.16213	+	7.91514i	−10.6631	+	10.6631i	−2.39471	+	2.77198i	7.71781	−	7.71781i	−12.1480	+	19.0899i	25.3227			−42.5392	+	3.10621i
31.15	2.22935	−	1.74069i		−	3.23176i	1.93998	−	7.76122i	4.87198	−	4.87198i	−5.62550	−	7.20471i	−22.0805	+	22.0805i	−9.18500	−	20.6794i	16.5557			2.38072	−	19.3420i
31.16	2.44435	+	1.42308i		−	3.58200i	3.94969	+	6.95701i	10.6687	−	10.6687i	5.09747	−	8.75566i	−6.84080	+	6.84080i	−0.245959	+	22.6261i	14.1693			41.2606	−	10.8957i
31.17	2.80785	−	0.340566i			6.14077i	7.76803	−	1.91251i	−1.41066	+	1.41066i	2.09133	+	17.2423i	−3.71079	+	3.71079i	21.1601	−	8.01557i	−10.7090			−3.48049	+	4.44133i
31.18	2.81360	−	0.289217i		−	7.82965i	7.83271	−	1.62748i	−8.33093	+	8.33093i	−2.26447	−	22.0295i	11.3199	−	11.3199i	21.5674	−	6.84443i	−34.3035			−21.0305	+	25.8494i
47.1	−2.82787	+	0.0560904i		−	6.97823i	7.99371	−	0.317233i	3.73459	+	3.73459i	0.391412	+	19.7335i	−18.7167	−	18.7167i	−22.5874	+	1.34546i	−21.6957			−10.7704	−	10.3515i
47.2	−2.81956	+	0.223782i			1.00320i	7.89984	−	1.26193i	−4.98300	−	4.98300i	−0.224497	−	2.82857i	16.5820	+	16.5820i	−21.9917	+	5.32593i	25.9936			15.1650	+	12.9348i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
13.d	odd	4	1	inner
52.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	52.4.f.b	✓	36
4.b	odd	2	1	inner	52.4.f.b	✓	36
13.d	odd	4	1	inner	52.4.f.b	✓	36
52.f	even	4	1	inner	52.4.f.b	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.4.f.b	✓	36	1.a	even	1	1	trivial
52.4.f.b	✓	36	4.b	odd	2	1	inner
52.4.f.b	✓	36	13.d	odd	4	1	inner
52.4.f.b	✓	36	52.f	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^18 + 326*T3^16 + 43311*T3^14 + 3016492*T3^12 + 117803647*T3^10 + 2557645974*T3^8 + 28890200609*T3^6 + 150421944536*T3^4 + 268587325968*T3^2 + 144896061440