Newform orbit 975.2.bn.e

• Label: 975.2.bn.e
• Level: $975$
• Weight: $2$
• Character orbit: 975.bn
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.bn (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 36 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
218.1	−2.56318	−	0.686801i	−0.702446	+	1.58322i	4.36613	+	2.52079i		0		2.88785	−	3.57562i	−1.64167	+	0.439885i	−5.70714	−	5.70714i	−2.01314	−	2.22425i		0
218.2	−2.56318	−	0.686801i	0.183272	−	1.72233i	4.36613	+	2.52079i		0		−1.65265	+	4.28876i	1.64167	−	0.439885i	−5.70714	−	5.70714i	−2.93282	−	0.631307i		0
218.3	−2.41318	−	0.646609i	0.721407	+	1.57467i	3.67327	+	2.12076i		0		−0.722691	−	4.26642i	−4.77125	+	1.27845i	−3.95981	−	3.95981i	−1.95914	+	2.27195i		0
218.4	−2.41318	−	0.646609i	1.41209	−	1.00300i	3.67327	+	2.12076i		0		−4.05617	+	1.50734i	4.77125	−	1.27845i	−3.95981	−	3.95981i	0.987996	−	2.83264i		0
218.5	−2.24004	−	0.600218i	−1.72525	+	0.153298i	2.92548	+	1.68903i		0		3.95665	+	0.692133i	−0.489037	+	0.131037i	−2.25977	−	2.25977i	2.95300	−	0.528957i		0
218.6	−2.24004	−	0.600218i	−1.41746	−	0.995387i	2.92548	+	1.68903i		0		2.57773	+	3.08050i	0.489037	−	0.131037i	−2.25977	−	2.25977i	1.01841	+	2.82185i		0
218.7	−1.95323	−	0.523368i	1.48871	+	0.885288i	1.80916	+	1.04452i		0		−2.44447	−	2.50832i	−1.84686	+	0.494865i	−0.127312	−	0.127312i	1.43253	+	2.63588i		0
218.8	−1.95323	−	0.523368i	1.73191	−	0.0223254i	1.80916	+	1.04452i		0		−3.39450	−	0.862817i	1.84686	−	0.494865i	−0.127312	−	0.127312i	2.99900	−	0.0773309i		0
218.9	−1.66802	−	0.446946i	0.509013	+	1.65557i	0.850493	+	0.491032i		0		−0.109097	−	2.98903i	4.00195	−	1.07232i	1.24298	+	1.24298i	−2.48181	+	1.68541i		0
218.10	−1.66802	−	0.446946i	1.26860	−	1.17926i	0.850493	+	0.491032i		0		−2.64312	+	1.40003i	−4.00195	+	1.07232i	1.24298	+	1.24298i	0.218704	−	2.99202i		0
218.11	−1.22543	−	0.328354i	−1.42521	+	0.984268i	−0.338180	−	0.195248i		0		2.06969	−	0.738182i	4.39929	−	1.17879i	2.14447	+	2.14447i	1.06243	−	2.80557i		0
218.12	−1.22543	−	0.328354i	−0.742132	−	1.56500i	−0.338180	−	0.195248i		0		0.395558	+	2.16149i	−4.39929	+	1.17879i	2.14447	+	2.14447i	−1.89848	+	2.32288i		0
218.13	−0.898447	−	0.240738i	0.941746	+	1.45366i	−0.982799	−	0.567419i		0		−0.496158	−	1.53275i	0.558322	−	0.149602i	2.06181	+	2.06181i	−1.22623	+	2.73795i		0
218.14	−0.898447	−	0.240738i	1.54240	−	0.788030i	−0.982799	−	0.567419i		0		−1.57548	+	0.336688i	−0.558322	+	0.149602i	2.06181	+	2.06181i	1.75802	−	2.43092i		0
218.15	−0.844128	−	0.226183i	−1.15910	+	1.28705i	−1.07066	−	0.618144i		0		1.26954	−	0.824262i	0.364668	−	0.0977124i	1.99985	+	1.99985i	−0.312972	−	2.98363i		0
218.16	−0.844128	−	0.226183i	−0.360288	−	1.69416i	−1.07066	−	0.618144i		0		−0.0790629	+	1.51158i	−0.364668	+	0.0977124i	1.99985	+	1.99985i	−2.74039	+	1.22077i		0
218.17	−0.165517	−	0.0443503i	1.42186	+	0.989103i	−1.70662	−	0.985319i		0		−0.191475	−	0.226774i	−1.50095	+	0.402179i	0.481111	+	0.481111i	1.04335	+	2.81272i		0
218.18	−0.165517	−	0.0443503i	1.72592	−	0.145660i	−1.70662	−	0.985319i		0		−0.292129	−	0.0524355i	1.50095	−	0.402179i	0.481111	+	0.481111i	2.95757	−	0.502795i		0
218.19	0.165517	+	0.0443503i	−1.72592	+	0.145660i	−1.70662	−	0.985319i		0		−0.292129	−	0.0524355i	−1.50095	+	0.402179i	−0.481111	−	0.481111i	2.95757	−	0.502795i		0
218.20	0.165517	+	0.0443503i	−1.42186	−	0.989103i	−1.70662	−	0.985319i		0		−0.191475	−	0.226774i	1.50095	−	0.402179i	−0.481111	−	0.481111i	1.04335	+	2.81272i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
5.c	odd	4	2	inner
13.e	even	6	1	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner
39.h	odd	6	1	inner
65.l	even	6	1	inner
65.r	odd	12	2	inner
195.y	odd	6	1	inner
195.bf	even	12	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.2.bn.e	✓	144
3.b	odd	2	1	inner	975.2.bn.e	✓	144
5.b	even	2	1	inner	975.2.bn.e	✓	144
5.c	odd	4	2	inner	975.2.bn.e	✓	144
13.e	even	6	1	inner	975.2.bn.e	✓	144
15.d	odd	2	1	inner	975.2.bn.e	✓	144
15.e	even	4	2	inner	975.2.bn.e	✓	144
39.h	odd	6	1	inner	975.2.bn.e	✓	144
65.l	even	6	1	inner	975.2.bn.e	✓	144
65.r	odd	12	2	inner	975.2.bn.e	✓	144
195.y	odd	6	1	inner	975.2.bn.e	✓	144
195.bf	even	12	2	inner	975.2.bn.e	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
975.2.bn.e	✓	144	1.a	even	1	1	trivial
975.2.bn.e	✓	144	3.b	odd	2	1	inner
975.2.bn.e	✓	144	5.b	even	2	1	inner
975.2.bn.e	✓	144	5.c	odd	4	2	inner
975.2.bn.e	✓	144	13.e	even	6	1	inner
975.2.bn.e	✓	144	15.d	odd	2	1	inner
975.2.bn.e	✓	144	15.e	even	4	2	inner
975.2.bn.e	✓	144	39.h	odd	6	1	inner
975.2.bn.e	✓	144	65.l	even	6	1	inner
975.2.bn.e	✓	144	65.r	odd	12	2	inner
975.2.bn.e	✓	144	195.y	odd	6	1	inner
975.2.bn.e	✓	144	195.bf	even	12	2	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

T2^72 - 147*T2^68 + 13394*T2^64 - 769665*T2^60 + 32429638*T2^56 - 972930510*T2^52 + 21928589219*T2^48 - 350101332321*T2^44 + 4114569260092*T2^40 - 31912659843468*T2^36 + 173766278126054*T2^32 - 511110790272216*T2^28 + 1051729100792245*T2^24 - 1095616237270494*T2^20 + 817458486076091*T2^16 - 326318651482701*T2^12 + 88542455697541*T2^8 - 76339430100*T2^4 + 65610000

T7^72 - 1348*T7^68 + 1221882*T7^64 - 619784360*T7^60 + 229003334979*T7^56 - 48341871522637*T7^52 + 6980608071151818*T7^48 - 181333646521553813*T7^44 + 3229803562308333429*T7^40 - 32248313395998098011*T7^36 + 233871814920241143111*T7^32 - 945532410335014335296*T7^28 + 2592463244036847771663*T7^24 - 501695927631345356890*T7^20 + 69431224996156588449*T7^16 - 4569326321201872928*T7^12 + 218598848869860096*T7^8 - 3944093203308544*T7^4 + 53459728531456