Newform orbit 968.2.o.h

• Label: 968.2.o.h
• Level: $968$
• Weight: $2$
• Character orbit: 968.o
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $40$
• Inner twists: $8$

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:	$40$
Relative dimension:	$10$ over $\Q(\zeta_{10})$
Twist minimal:	no (minimal twist has level 88)
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.

$\operatorname{Tr}(f)(q) =$ 40 * q + 4 * q^4 + 2 * q^6 + 10 * q^9 - 40 * q^10 - 16 * q^12 + 12 * q^14 - 8 * q^15 - 4 * q^17 - 10 * q^18 - 12 * q^20 - 48 * q^23 + 6 * q^25 + 20 * q^26 - 12 * q^28 + 18 * q^30 + 4 * q^31 + 80 * q^32 - 32 * q^36 - 8 * q^38 + 24 * q^39 + 20 * q^40 + 4 * q^41 - 20 * q^42 + 2 * q^46 + 4 * q^47 - 32 * q^48 + 6 * q^49 - 6 * q^50 - 20 * q^52 + 152 * q^54 - 32 * q^56 + 16 * q^57 - 36 * q^58 + 4 * q^60 + 22 * q^62 - 40 * q^63 + 16 * q^64 - 64 * q^65 - 28 * q^68 - 28 * q^70 + 12 * q^71 - 4 * q^72 - 4 * q^73 - 14 * q^74 - 176 * q^76 - 32 * q^78 + 16 * q^79 + 56 * q^80 + 6 * q^81 + 4 * q^82 - 52 * q^84 + 20 * q^86 - 128 * q^87 - 16 * q^89 - 36 * q^90 + 36 * q^92 + 24 * q^95 + 60 * q^96 + 20 * q^97 - 96 * q^98

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	−1.37974	+	0.310332i	−0.218711	+	0.0710636i	1.80739	−	0.856359i	1.47884	−	2.03545i	0.279712	−	0.165923i	−0.456552	+	1.40512i	−2.22798	+	1.74245i	−2.38427	+	1.73227i	−1.40875	+	3.26733i
245.2	−1.13865	−	0.838740i	1.27008	−	0.412674i	0.593029	+	1.91006i	−1.13513	+	1.56237i	−1.79230	−	0.595378i	0.568375	−	1.74928i	0.926792	−	2.67228i	−0.984248	+	0.715098i	2.60293	−	0.826906i
245.3	−1.01791	−	0.981764i	−2.90813	+	0.944908i	0.0722795	+	1.99869i	0.411028	−	0.565732i	3.88789	+	1.89326i	−1.01297	+	3.11759i	1.88867	−	2.10545i	5.13731	−	3.73247i	−0.973805	+	0.172331i
245.4	−0.721508	+	1.21632i	0.218711	−	0.0710636i	−0.958852	−	1.75516i	−1.47884	+	2.03545i	−0.0713662	+	0.317295i	−0.456552	+	1.40512i	2.82666	+	0.100098i	−2.38427	+	1.73227i	−1.40875	−	3.26733i
245.5	−0.110755	−	1.40987i	2.23784	−	0.727117i	−1.97547	+	0.312301i	2.44985	−	3.37193i	−1.27299	−	3.07453i	−0.288382	+	0.887547i	0.659097	+	2.75056i	2.05216	−	1.49098i	−5.02532	−	3.08051i
245.6	0.445829	+	1.34210i	−1.27008	+	0.412674i	−1.60247	+	1.19669i	1.13513	−	1.56237i	−1.12009	−	1.52059i	0.568375	−	1.74928i	−2.32051	−	1.61716i	−0.984248	+	0.715098i	2.60293	+	0.826906i
245.7	0.592588	−	1.28407i	−1.72166	+	0.559401i	−1.29768	−	1.52185i	0.166026	−	0.228516i	−0.301923	+	2.54223i	1.18953	−	3.66098i	−2.72315	+	0.764484i	0.224131	−	0.162841i	−0.195045	−	0.348605i
245.8	0.619162	+	1.27147i	2.90813	−	0.944908i	−1.23328	+	1.57449i	−0.411028	+	0.565732i	3.00202	+	3.11255i	−1.01297	+	3.11759i	−2.76552	−	0.593212i	5.13731	−	3.73247i	−0.973805	−	0.172331i
245.9	1.30664	+	0.541008i	−2.23784	+	0.727117i	1.41462	+	1.41381i	−2.44985	+	3.37193i	−3.31743	−	0.260607i	−0.288382	+	0.887547i	1.08352	+	2.61266i	2.05216	−	1.49098i	−5.02532	+	3.08051i
245.10	1.40434	−	0.166784i	1.72166	−	0.559401i	1.94437	−	0.468446i	−0.166026	+	0.228516i	2.32450	−	1.07274i	1.18953	−	3.66098i	2.65243	−	0.982149i	0.224131	−	0.162841i	−0.195045	+	0.348605i
269.1	−1.37509	−	0.330339i	−1.38306	+	1.90362i	1.78175	+	0.908493i	3.96394	+	1.28796i	2.53067	−	2.16077i	0.754993	−	0.548534i	−2.14996	−	1.83784i	−0.783857	−	2.41246i	−5.02532	−	3.08051i
269.2	−1.24826	+	0.664708i	1.79732	−	2.47380i	1.11633	−	1.65946i	0.665058	+	0.216090i	−0.599178	+	4.28265i	2.65198	−	1.92678i	−0.290414	+	2.81348i	−1.96228	−	6.03926i	−0.973805	+	0.172331i
269.3	−1.14955	+	0.823732i	−0.784953	+	1.08039i	0.642933	−	1.89384i	−1.83667	−	0.596771i	0.0123875	−	1.88856i	−1.48802	+	1.08111i	0.820934	+	2.70667i	0.375949	+	1.15705i	2.60293	−	0.826906i
269.4	−1.03810	−	0.960384i	1.06404	−	1.46453i	0.155324	+	1.99396i	0.268636	+	0.0872852i	−2.51110	+	0.498446i	−3.11422	+	2.26261i	1.75373	−	2.21911i	−0.0856104	−	0.263482i	−0.195045	−	0.348605i
269.5	−0.131221	+	1.40811i	0.135171	−	0.186047i	−1.96556	−	0.369547i	2.39281	+	0.777472i	0.244238	+	0.214749i	1.19527	−	0.868413i	0.778286	−	2.71924i	0.910709	+	2.80287i	−1.40875	+	3.26733i
269.6	0.275345	−	1.38715i	−1.06404	+	1.46453i	−1.84837	−	0.763889i	−0.268636	−	0.0872852i	1.73855	+	1.87924i	−3.11422	+	2.26261i	−1.56857	+	2.35363i	−0.0856104	−	0.263482i	−0.195045	+	0.348605i
269.7	0.918304	−	1.07551i	1.38306	−	1.90362i	−0.313437	−	1.97529i	−3.96394	−	1.28796i	−0.777288	−	3.23559i	0.754993	−	0.548534i	−2.41227	−	1.47681i	−0.783857	−	2.41246i	−5.02532	+	3.08051i
269.8	0.933828	+	1.06206i	−0.135171	+	0.186047i	−0.255932	+	1.98356i	−2.39281	−	0.777472i	−0.323819	+	0.0301764i	1.19527	−	0.868413i	−2.34565	+	1.58049i	0.910709	+	2.80287i	−1.40875	−	3.26733i
269.9	1.40057	−	0.195951i	−1.79732	+	2.47380i	1.92321	−	0.548888i	−0.665058	−	0.216090i	−2.03253	+	3.81693i	2.65198	−	1.92678i	2.58603	−	1.14561i	−1.96228	−	6.03926i	−0.973805	−	0.172331i
269.10	1.41418	−	0.00927596i	0.784953	−	1.08039i	1.99983	−	0.0262358i	1.83667	+	0.596771i	1.10004	−	1.53516i	−1.48802	+	1.08111i	2.82788	−	0.0556526i	0.375949	+	1.15705i	2.60293	+	0.826906i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
11.c	even	5	3	inner
88.o	even	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.h		40
8.b	even	2	1	inner	968.2.o.h		40
11.b	odd	2	1		968.2.o.g		40
11.c	even	5	1		968.2.c.d		10
11.c	even	5	3	inner	968.2.o.h		40
11.d	odd	10	1		88.2.c.a	✓	10
11.d	odd	10	3		968.2.o.g		40
33.f	even	10	1		792.2.f.g		10
44.g	even	10	1		352.2.c.a		10
44.h	odd	10	1		3872.2.c.f		10
88.b	odd	2	1		968.2.o.g		40
88.k	even	10	1		352.2.c.a		10
88.l	odd	10	1		3872.2.c.f		10
88.o	even	10	1		968.2.c.d		10
88.o	even	10	3	inner	968.2.o.h		40
88.p	odd	10	1		88.2.c.a	✓	10
88.p	odd	10	3		968.2.o.g		40
132.n	odd	10	1		3168.2.f.g		10
176.u	odd	20	1		2816.2.a.o		5
176.u	odd	20	1		2816.2.a.r		5
176.x	even	20	1		2816.2.a.p		5
176.x	even	20	1		2816.2.a.q		5
264.r	odd	10	1		3168.2.f.g		10
264.u	even	10	1		792.2.f.g		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.2.c.a	✓	10	11.d	odd	10	1
88.2.c.a	✓	10	88.p	odd	10	1
352.2.c.a		10	44.g	even	10	1
352.2.c.a		10	88.k	even	10	1
792.2.f.g		10	33.f	even	10	1
792.2.f.g		10	264.u	even	10	1
968.2.c.d		10	11.c	even	5	1
968.2.c.d		10	88.o	even	10	1
968.2.o.g		40	11.b	odd	2	1
968.2.o.g		40	11.d	odd	10	3
968.2.o.g		40	88.b	odd	2	1
968.2.o.g		40	88.p	odd	10	3
968.2.o.h		40	1.a	even	1	1	trivial
968.2.o.h		40	8.b	even	2	1	inner
968.2.o.h		40	11.c	even	5	3	inner
968.2.o.h		40	88.o	even	10	3	inner
2816.2.a.o		5	176.u	odd	20	1
2816.2.a.p		5	176.x	even	20	1
2816.2.a.q		5	176.x	even	20	1
2816.2.a.r		5	176.u	odd	20	1
3168.2.f.g		10	132.n	odd	10	1
3168.2.f.g		10	264.r	odd	10	1
3872.2.c.f		10	44.h	odd	10	1
3872.2.c.f		10	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 $S_{2}^{\mathrm{new}}(968, [\chi])$:

T3^40 - 20*T3^38 + 266*T3^36 - 2996*T3^34 + 31075*T3^32 - 231268*T3^30 + 1442820*T3^28 - 7971728*T3^26 + 38500949*T3^24 - 141711520*T3^22 + 455742404*T3^20 - 1296002676*T3^18 + 3068615523*T3^16 - 5078864964*T3^14 + 7733844490*T3^12 - 10072992164*T3^10 + 8911068737*T3^8 - 471258128*T3^6 + 24920320*T3^4 - 1314816*T3^2 + 65536

T5^40 - 28*T5^38 + 570*T5^36 - 10492*T5^34 + 186227*T5^32 - 1668108*T5^30 + 12215332*T5^28 - 80117680*T5^26 + 458589749*T5^24 - 1697129024*T5^22 + 5672834596*T5^20 - 16841870460*T5^18 + 37643131283*T5^16 - 20742066988*T5^14 + 10288828890*T5^12 - 4910718732*T5^10 + 2009131905*T5^8 - 160179472*T5^6 + 12722432*T5^4 - 987136*T5^2 + 65536

T7^20 + 16*T7^18 + 8*T7^17 + 216*T7^16 - 256*T7^15 + 2880*T7^14 - 3200*T7^13 + 35136*T7^12 - 24832*T7^11 + 115712*T7^10 - 80384*T7^9 + 349696*T7^8 - 491520*T7^7 + 1118208*T7^6 - 1224704*T7^5 + 2543616*T7^4 - 2179072*T7^3 + 1900544*T7^2 - 1310720*T7 + 1048576