Newform orbit 975.2.bp.j

• Label: 975.2.bp.j
• Level: $975$
• Weight: $2$
• Character orbit: 975.bp
• Analytic conductor: $7.785$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.bp (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.78541419707\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 6 q^{3} + 6 q^{6} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
149.1	−0.710085	−	2.65007i	0.136461	−	1.72667i	−4.78662	+	2.76356i		0		−4.67270	+	0.864448i	−0.0269971	−	0.00723386i	6.84257	+	6.84257i	−2.96276	−	0.471247i		0
149.2	−0.648620	−	2.42068i	−0.768879	+	1.55204i	−3.70694	+	2.14020i		0		4.25570	+	0.854527i	1.96195	+	0.525704i	4.04102	+	4.04102i	−1.81765	−	2.38666i		0
149.3	−0.585207	−	2.18402i	1.62386	+	0.602560i	−2.69544	+	1.55621i		0		0.365711	−	3.89917i	−0.749991	−	0.200960i	1.77856	+	1.77856i	2.27384	+	1.95695i		0
149.4	−0.494045	−	1.84380i	−1.67870	−	0.426559i	−1.42347	+	0.821841i		0		0.0428654	+	3.30594i	2.33581	+	0.625877i	−0.480942	−	0.480942i	2.63609	+	1.43213i		0
149.5	−0.386899	−	1.44392i	−1.34630	+	1.08971i	−0.203178	+	0.117305i		0		2.09434	+	1.52235i	−5.05671	−	1.35494i	−1.86606	−	1.86606i	0.625067	−	2.93416i		0
149.6	−0.364310	−	1.35962i	−0.244396	−	1.71472i	0.0161989	−	0.00935246i		0		−2.24234	+	0.956977i	−1.64265	−	0.440146i	−2.00924	−	2.00924i	−2.88054	+	0.838144i		0
149.7	−0.315125	−	1.17606i	1.05689	+	1.37222i	0.448233	−	0.258788i		0		1.28076	−	1.67539i	3.25638	+	0.872545i	−2.16747	−	2.16747i	−0.765977	+	2.90057i		0
149.8	−0.198623	−	0.741271i	1.16594	−	1.28085i	1.22202	−	0.705533i		0		−1.18104	−	0.609872i	2.67226	+	0.716030i	−1.85101	−	1.85101i	−0.281165	−	2.98680i		0
149.9	−0.0999402	−	0.372982i	0.387831	+	1.68807i	1.60292	−	0.925448i		0		0.590860	−	0.313360i	−2.75005	−	0.736874i	−1.05146	−	1.05146i	−2.69917	+	1.30937i		0
149.10	0.0999402	+	0.372982i	1.65583	−	0.508165i	1.60292	−	0.925448i		0		0.355020	+	0.566808i	−2.75005	−	0.736874i	1.05146	+	1.05146i	2.48354	−	1.68287i		0
149.11	0.198623	+	0.741271i	−0.526280	+	1.65016i	1.22202	−	0.705533i		0		−1.32775	−	0.0623565i	2.67226	+	0.716030i	1.85101	+	1.85101i	−2.44606	−	1.73689i		0
149.12	0.315125	+	1.17606i	1.71682	+	0.229181i	0.448233	−	0.258788i		0		0.271482	+	2.09131i	3.25638	+	0.872545i	2.16747	+	2.16747i	2.89495	+	0.786927i		0
149.13	0.364310	+	1.35962i	−1.60719	+	0.645707i	0.0161989	−	0.00935246i		0		−1.46343	−	1.94994i	−1.64265	−	0.440146i	2.00924	+	2.00924i	2.16612	−	2.07555i		0
149.14	0.386899	+	1.44392i	0.270564	−	1.71079i	−0.203178	+	0.117305i		0		2.57493	−	0.271227i	−5.05671	−	1.35494i	1.86606	+	1.86606i	−2.85359	−	0.925756i		0
149.15	0.494045	+	1.84380i	−1.20876	−	1.24052i	−1.42347	+	0.821841i		0		1.69009	−	2.84159i	2.33581	+	0.625877i	0.480942	+	0.480942i	−0.0777841	+	2.99899i		0
149.16	0.585207	+	2.18402i	1.33376	+	1.10502i	−2.69544	+	1.55621i		0		−1.63287	+	3.55964i	−0.749991	−	0.200960i	−1.77856	−	1.77856i	0.557845	+	2.94768i		0
149.17	0.648620	+	2.42068i	0.959666	−	1.44189i	−3.70694	+	2.14020i		0		4.11281	+	1.38781i	1.96195	+	0.525704i	−4.04102	−	4.04102i	−1.15808	−	2.76746i		0
149.18	0.710085	+	2.65007i	−1.42711	+	0.981513i	−4.78662	+	2.76356i		0		−3.61445	−	3.08498i	−0.0269971	−	0.00723386i	−6.84257	−	6.84257i	1.07327	−	2.80145i		0
449.1	−2.56092	+	0.686196i	−1.63051	+	0.584339i	4.35539	−	2.51459i		0		3.77462	−	2.61529i	0.0651156	−	0.243015i	−5.67887	+	5.67887i	2.31710	−	1.90554i		0
449.2	−2.47237	+	0.662470i	1.69771	+	0.343204i	3.94171	−	2.27575i		0		−4.42473	+	0.276153i	1.08395	−	4.04534i	−4.61796	+	4.61796i	2.76442	+	1.16532i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
65.s	odd	12	1	inner
195.bh	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	975.2.bp.j		72
3.b	odd	2	1	inner	975.2.bp.j		72
5.b	even	2	1		975.2.bp.g		72
5.c	odd	4	1		975.2.bo.f	✓	72
5.c	odd	4	1		975.2.bo.g	yes	72
13.f	odd	12	1		975.2.bp.g		72
15.d	odd	2	1		975.2.bp.g		72
15.e	even	4	1		975.2.bo.f	✓	72
15.e	even	4	1		975.2.bo.g	yes	72
39.k	even	12	1		975.2.bp.g		72
65.o	even	12	1		975.2.bo.f	✓	72
65.s	odd	12	1	inner	975.2.bp.j		72
65.t	even	12	1		975.2.bo.g	yes	72
195.bc	odd	12	1		975.2.bo.g	yes	72
195.bh	even	12	1	inner	975.2.bp.j		72
195.bn	odd	12	1		975.2.bo.f	✓	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
975.2.bo.f	✓	72	5.c	odd	4	1
975.2.bo.f	✓	72	15.e	even	4	1
975.2.bo.f	✓	72	65.o	even	12	1
975.2.bo.f	✓	72	195.bn	odd	12	1
975.2.bo.g	yes	72	5.c	odd	4	1
975.2.bo.g	yes	72	15.e	even	4	1
975.2.bo.g	yes	72	65.t	even	12	1
975.2.bo.g	yes	72	195.bc	odd	12	1
975.2.bp.g		72	5.b	even	2	1
975.2.bp.g		72	13.f	odd	12	1
975.2.bp.g		72	15.d	odd	2	1
975.2.bp.g		72	39.k	even	12	1
975.2.bp.j		72	1.a	even	1	1	trivial
975.2.bp.j		72	3.b	odd	2	1	inner
975.2.bp.j		72	65.s	odd	12	1	inner
975.2.bp.j		72	195.bh	even	12	1	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 + 13522*T2^64 - 1734*T2^62 - 768741*T2^60 + 152730*T2^58 + 31832106*T2^56 - 8428452*T2^54 - 921423258*T2^52 + 266634900*T2^50 + 19935286795*T2^48 - 5616122628*T2^46 - 305655969981*T2^44 + 55732493322*T2^42 + 3484497412408*T2^40 + 63600325350*T2^38 - 27192163978068*T2^36 - 8363103314238*T2^34 + 150146142116442*T2^32 + 107903145096252*T2^30 - 459245783467080*T2^28 - 421880300657820*T2^26 + 989126392008969*T2^24 + 1113423925993920*T2^22 - 822815886213882*T2^20 - 953489470964094*T2^18 + 692290476045819*T2^16 + 322236688126896*T2^14 - 215824634570241*T2^12 - 71811972459144*T2^10 + 54771748729641*T2^8 + 1087652623932*T2^6 - 1260324949032*T2^4 - 22712817312*T2^2 + 26454371904

T7^36 + 64*T7^33 - 508*T7^32 + 22*T7^31 + 2048*T7^30 - 36206*T7^29 + 237695*T7^28 - 457314*T7^27 + 647477*T7^26 + 3097796*T7^25 - 23400348*T7^24 + 20824156*T7^23 - 468523*T7^22 - 45392956*T7^21 + 1245511711*T7^20 - 2327592662*T7^19 + 624793173*T7^18 - 2137180716*T7^17 - 13245869185*T7^16 + 24526830172*T7^15 + 24989337932*T7^14 + 66344731020*T7^13 + 69630077957*T7^12 - 387438148056*T7^11 - 370118231778*T7^10 - 258803073412*T7^9 + 613931441205*T7^8 + 1426698672666*T7^7 + 1372876739598*T7^6 + 926339078244*T7^5 + 348101965084*T7^4 + 33716556720*T7^3 + 22201714800*T7^2 + 1150216704*T7 + 16451136