Newform orbit 968.2.o.k

• Label: 968.2.o.k
• Level: $968$
• Weight: $2$
• Character orbit: 968.o
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $64$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.o (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.72951891566\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{4} + 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
245.1	−1.40990	−	0.110317i	3.06057	−	0.994439i	1.97566	+	0.311074i	−1.79789	+	2.47459i	−4.42481	+	1.06443i	0.287246	−	0.884051i	−2.75117	−	0.656534i	5.95112	−	4.32374i	2.80784	−	3.29059i
245.2	−1.31408	−	0.522680i	−1.22793	+	0.398979i	1.45361	+	1.37369i	0.480043	−	0.660723i	1.82214	+	0.117525i	0.408813	−	1.25820i	−1.19216	−	2.56491i	−1.07842	+	0.783520i	−0.976162	+	0.617333i
245.3	−1.23235	+	0.693774i	2.13243	−	0.692868i	1.03735	−	1.70994i	1.17958	−	1.62355i	−2.14720	+	2.33328i	0.219975	−	0.677015i	−0.0920672	+	2.82693i	1.64014	−	1.19163i	−0.327271	+	2.81914i
245.4	−1.04063	+	0.957643i	−2.13243	+	0.692868i	0.165840	−	1.99311i	−1.17958	+	1.62355i	1.55556	−	2.76313i	0.219975	−	0.677015i	1.73611	+	2.23292i	1.64014	−	1.19163i	−0.327271	−	2.81914i
245.5	−0.998293	−	1.00170i	−1.63341	+	0.530727i	−0.00682194	+	1.99999i	−0.820733	+	1.12964i	2.16225	+	1.10637i	1.41201	−	4.34572i	2.01021	−	1.98974i	−0.0406970	+	0.0295681i	1.95090	−	0.305582i
245.6	−0.644188	−	1.25898i	1.63341	−	0.530727i	−1.17004	+	1.62203i	0.820733	−	1.12964i	−1.72039	−	1.71454i	−1.41201	+	4.34572i	2.79583	+	0.428164i	−0.0406970	+	0.0295681i	−1.95090	−	0.305582i
245.7	−0.330766	+	1.37499i	−3.06057	+	0.994439i	−1.78119	−	0.909600i	1.79789	−	2.47459i	−0.355009	−	4.53717i	0.287246	−	0.884051i	1.83985	−	2.14825i	5.95112	−	4.32374i	2.80784	+	3.29059i
245.8	−0.0910257	−	1.41128i	1.22793	−	0.398979i	−1.98343	+	0.256926i	−0.480043	+	0.660723i	−0.674844	−	1.69664i	−0.408813	+	1.25820i	0.543137	+	2.77579i	−1.07842	+	0.783520i	0.976162	+	0.617333i
245.9	0.0910257	+	1.41128i	1.22793	−	0.398979i	−1.98343	+	0.256926i	−0.480043	+	0.660723i	0.674844	+	1.69664i	0.408813	−	1.25820i	−0.543137	−	2.77579i	−1.07842	+	0.783520i	−0.976162	−	0.617333i
245.10	0.330766	−	1.37499i	−3.06057	+	0.994439i	−1.78119	−	0.909600i	1.79789	−	2.47459i	0.355009	+	4.53717i	−0.287246	+	0.884051i	−1.83985	+	2.14825i	5.95112	−	4.32374i	−2.80784	−	3.29059i
245.11	0.644188	+	1.25898i	1.63341	−	0.530727i	−1.17004	+	1.62203i	0.820733	−	1.12964i	1.72039	+	1.71454i	1.41201	−	4.34572i	−2.79583	−	0.428164i	−0.0406970	+	0.0295681i	1.95090	+	0.305582i
245.12	0.998293	+	1.00170i	−1.63341	+	0.530727i	−0.00682194	+	1.99999i	−0.820733	+	1.12964i	−2.16225	−	1.10637i	−1.41201	+	4.34572i	−2.01021	+	1.98974i	−0.0406970	+	0.0295681i	−1.95090	+	0.305582i
245.13	1.04063	−	0.957643i	−2.13243	+	0.692868i	0.165840	−	1.99311i	−1.17958	+	1.62355i	−1.55556	+	2.76313i	−0.219975	+	0.677015i	−1.73611	−	2.23292i	1.64014	−	1.19163i	0.327271	+	2.81914i
245.14	1.23235	−	0.693774i	2.13243	−	0.692868i	1.03735	−	1.70994i	1.17958	−	1.62355i	2.14720	−	2.33328i	−0.219975	+	0.677015i	0.0920672	−	2.82693i	1.64014	−	1.19163i	0.327271	−	2.81914i
245.15	1.31408	+	0.522680i	−1.22793	+	0.398979i	1.45361	+	1.37369i	0.480043	−	0.660723i	−1.82214	−	0.117525i	−0.408813	+	1.25820i	1.19216	+	2.56491i	−1.07842	+	0.783520i	0.976162	−	0.617333i
245.16	1.40990	+	0.110317i	3.06057	−	0.994439i	1.97566	+	0.311074i	−1.79789	+	2.47459i	4.42481	−	1.06443i	−0.287246	+	0.884051i	2.75117	+	0.656534i	5.95112	−	4.32374i	−2.80784	+	3.29059i
269.1	−1.39642	+	0.223614i	−1.00950	+	1.38946i	1.89999	−	0.624518i	1.32797	+	0.431485i	1.09899	−	2.16601i	3.69669	−	2.68580i	−2.51354	+	1.29696i	0.0155449	+	0.0478422i	−1.95090	−	0.305582i
269.2	−1.37034	−	0.349539i	−0.758903	+	1.04454i	1.75564	+	0.957973i	−0.776726	−	0.252374i	1.40506	−	1.16610i	1.07029	−	0.777608i	−2.07097	−	1.92641i	0.411921	+	1.26776i	0.976162	+	0.617333i
269.3	−1.26117	+	0.639890i	1.00950	−	1.38946i	1.18108	−	1.61401i	−1.32797	−	0.431485i	−0.384049	+	2.39831i	−3.69669	+	2.68580i	−0.456751	+	2.79130i	0.0155449	+	0.0478422i	1.95090	−	0.305582i
269.4	−1.20548	−	0.739472i	1.89154	−	2.60347i	0.906361	+	1.78284i	2.90905	+	0.945207i	−4.20540	+	1.73970i	0.752019	−	0.546374i	0.225759	−	2.81940i	−2.27312	−	6.99596i	−2.80784	−	3.29059i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
88.b	odd	2	1	inner
88.o	even	10	3	inner
88.p	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	968.2.o.k		64
8.b	even	2	1	inner	968.2.o.k		64
11.b	odd	2	1	inner	968.2.o.k		64
11.c	even	5	1		968.2.c.g	✓	16
11.c	even	5	3	inner	968.2.o.k		64
11.d	odd	10	1		968.2.c.g	✓	16
11.d	odd	10	3	inner	968.2.o.k		64
44.g	even	10	1		3872.2.c.g		16
44.h	odd	10	1		3872.2.c.g		16
88.b	odd	2	1	inner	968.2.o.k		64
88.k	even	10	1		3872.2.c.g		16
88.l	odd	10	1		3872.2.c.g		16
88.o	even	10	1		968.2.c.g	✓	16
88.o	even	10	3	inner	968.2.o.k		64
88.p	odd	10	1		968.2.c.g	✓	16
88.p	odd	10	3	inner	968.2.o.k		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
968.2.c.g	✓	16	11.c	even	5	1
968.2.c.g	✓	16	11.d	odd	10	1
968.2.c.g	✓	16	88.o	even	10	1
968.2.c.g	✓	16	88.p	odd	10	1
968.2.o.k		64	1.a	even	1	1	trivial
968.2.o.k		64	8.b	even	2	1	inner
968.2.o.k		64	11.b	odd	2	1	inner
968.2.o.k		64	11.c	even	5	3	inner
968.2.o.k		64	11.d	odd	10	3	inner
968.2.o.k		64	88.b	odd	2	1	inner
968.2.o.k		64	88.o	even	10	3	inner
968.2.o.k		64	88.p	odd	10	3	inner
3872.2.c.g		16	44.g	even	10	1
3872.2.c.g		16	44.h	odd	10	1
3872.2.c.g		16	88.k	even	10	1
3872.2.c.g		16	88.l	odd	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

T3^32 - 20*T3^30 + 272*T3^28 - 3196*T3^26 + 35168*T3^24 - 252544*T3^22 + 1489680*T3^20 - 7762880*T3^18 + 35379456*T3^16 - 120028096*T3^14 + 355104000*T3^12 - 931094528*T3^10 + 2040316160*T3^8 - 3111895040*T3^6 + 4396679168*T3^4 - 5301600256*T3^2 + 4294967296

T5^32 - 16*T5^30 + 182*T5^28 - 1844*T5^26 + 17843*T5^24 - 96584*T5^22 + 429948*T5^20 - 1711384*T5^18 + 5895429*T5^16 - 12825800*T5^14 + 24413580*T5^12 - 41520028*T5^10 + 56632307*T5^8 - 37145920*T5^6 + 23601830*T5^4 - 13647284*T5^2 + 5764801

T7^32 + 24*T7^30 + 508*T7^28 + 10620*T7^26 + 221760*T7^24 + 662352*T7^22 + 1445840*T7^20 + 2795904*T7^18 + 4977536*T7^16 + 5492160*T7^14 + 5162240*T7^12 + 4344576*T7^10 + 3126528*T7^8 + 1465344*T7^6 + 643072*T7^4 + 245760*T7^2 + 65536