Newform orbit 65.5.h.a

• Label: 65.5.h.a
• Level: $65$
• Weight: $5$
• Character orbit: 65.h
• Analytic conductor: $6.719$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	65.h (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.71904760045\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 4 q^{3}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−5.46439	+	5.46439i	3.27217	−	3.27217i		−	43.7192i	−8.57761	−	23.4824i			35.7609i	−25.7773	+	25.7773i	151.468	+	151.468i			59.5857i	175.189	+	81.4458i
12.2	−4.84039	+	4.84039i	−10.0798	+	10.0798i		−	30.8587i	24.8318	−	2.89530i		−	97.5807i	28.0805	−	28.0805i	71.9219	+	71.9219i		−	122.207i	−106.181	+	134.210i
12.3	−4.68757	+	4.68757i	1.78628	−	1.78628i		−	27.9466i	15.3241	+	19.7528i			16.7467i	−3.06968	+	3.06968i	56.0004	+	56.0004i			74.6184i	−164.425	−	20.7595i
12.4	−4.19956	+	4.19956i	7.20204	−	7.20204i		−	19.2726i	−18.3582	+	16.9699i			60.4908i	35.8721	−	35.8721i	13.7436	+	13.7436i		−	22.7388i	5.83030	−	148.362i
12.5	−3.98371	+	3.98371i	−8.56869	+	8.56869i		−	15.7398i	−18.2605	+	17.0750i		−	68.2703i	−53.8219	+	53.8219i	−1.03641	−	1.03641i		−	65.8450i	4.72285	−	140.766i
12.6	−3.60520	+	3.60520i	−5.73377	+	5.73377i		−	9.99498i	−15.7393	−	19.4235i		−	41.3428i	39.7648	−	39.7648i	−21.6493	−	21.6493i			15.2477i	126.769	+	13.2822i
12.7	−3.60256	+	3.60256i	12.4176	−	12.4176i		−	9.95690i	22.5218	−	10.8522i			89.4705i	−6.72976	+	6.72976i	−21.7706	−	21.7706i		−	227.395i	−42.0405	+	120.232i
12.8	−2.62485	+	2.62485i	−1.80765	+	1.80765i			2.22029i	11.3206	−	22.2900i		−	9.48961i	−61.2227	+	61.2227i	−47.8256	−	47.8256i			74.4648i	28.7929	+	88.2229i
12.9	−2.10828	+	2.10828i	6.72564	−	6.72564i			7.11027i	−23.9049	−	7.31807i			28.3591i	−18.3500	+	18.3500i	−48.7230	−	48.7230i		−	9.46840i	65.8270	−	34.9698i
12.10	−1.86655	+	1.86655i	0.521246	−	0.521246i			9.03197i	24.5986	−	4.46168i			1.94587i	35.6720	−	35.6720i	−46.7235	−	46.7235i			80.4566i	−37.5867	+	54.2426i
12.11	−1.45309	+	1.45309i	−3.88997	+	3.88997i			11.7771i	−8.65616	+	23.4536i		−	11.3050i	22.8303	−	22.8303i	−40.3625	−	40.3625i			50.7362i	−21.5020	−	46.6583i
12.12	−0.406202	+	0.406202i	−10.4276	+	10.4276i			15.6700i	20.8427	+	13.8052i		−	8.47139i	−13.9983	+	13.9983i	−12.8644	−	12.8644i		−	136.468i	−14.0740	+	2.85865i
12.13	−0.400503	+	0.400503i	7.58249	−	7.58249i			15.6792i	6.67155	+	24.0934i			6.07361i	−41.7236	+	41.7236i	−12.6876	−	12.6876i		−	33.9882i	−12.3214	−	6.97749i
12.14	0.400503	−	0.400503i	7.58249	−	7.58249i			15.6792i	−6.67155	−	24.0934i		−	6.07361i	41.7236	−	41.7236i	12.6876	+	12.6876i		−	33.9882i	−12.3214	−	6.97749i
12.15	0.406202	−	0.406202i	−10.4276	+	10.4276i			15.6700i	−20.8427	−	13.8052i			8.47139i	13.9983	−	13.9983i	12.8644	+	12.8644i		−	136.468i	−14.0740	+	2.85865i
12.16	1.45309	−	1.45309i	−3.88997	+	3.88997i			11.7771i	8.65616	−	23.4536i			11.3050i	−22.8303	+	22.8303i	40.3625	+	40.3625i			50.7362i	−21.5020	−	46.6583i
12.17	1.86655	−	1.86655i	0.521246	−	0.521246i			9.03197i	−24.5986	+	4.46168i		−	1.94587i	−35.6720	+	35.6720i	46.7235	+	46.7235i			80.4566i	−37.5867	+	54.2426i
12.18	2.10828	−	2.10828i	6.72564	−	6.72564i			7.11027i	23.9049	+	7.31807i		−	28.3591i	18.3500	−	18.3500i	48.7230	+	48.7230i		−	9.46840i	65.8270	−	34.9698i
12.19	2.62485	−	2.62485i	−1.80765	+	1.80765i			2.22029i	−11.3206	+	22.2900i			9.48961i	61.2227	−	61.2227i	47.8256	+	47.8256i			74.4648i	28.7929	+	88.2229i
12.20	3.60256	−	3.60256i	12.4176	−	12.4176i		−	9.95690i	−22.5218	+	10.8522i		−	89.4705i	6.72976	−	6.72976i	21.7706	+	21.7706i		−	227.395i	−42.0405	+	120.232i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
13.b	even	2	1	inner
65.h	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	65.5.h.a	✓	52
5.c	odd	4	1	inner	65.5.h.a	✓	52
13.b	even	2	1	inner	65.5.h.a	✓	52
65.h	odd	4	1	inner	65.5.h.a	✓	52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.5.h.a	✓	52	1.a	even	1	1	trivial
65.5.h.a	✓	52	5.c	odd	4	1	inner
65.5.h.a	✓	52	13.b	even	2	1	inner
65.5.h.a	✓	52	65.h	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(65, [\chi])\).