Newform orbit 130.3.t.b

• Label: 130.3.t.b
• Level: $130$
• Weight: $3$
• Character orbit: 130.t
• Analytic conductor: $3.542$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 130 = 2 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	130.t (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.54224343668\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 14 q^{2} + 4 q^{5} - 12 q^{7} + 56 q^{8} + 42 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
19.1	1.36603	−	0.366025i	−4.60455	−	2.65844i	1.73205	−	1.00000i	−0.946786	+	4.90954i	−7.26299	−	1.94611i	−9.63174	−	2.58082i	2.00000	−	2.00000i	9.63460	+	16.6876i	0.503683	+	7.05311i
19.2	1.36603	−	0.366025i	−2.11647	−	1.22195i	1.73205	−	1.00000i	−3.39373	−	3.67186i	−3.33842	−	0.894526i	−2.98209	−	0.799048i	2.00000	−	2.00000i	−1.51370	−	2.62180i	−5.97992	−	3.77367i
19.3	1.36603	−	0.366025i	−1.64880	−	0.951934i	1.73205	−	1.00000i	1.64386	+	4.72205i	−2.60073	−	0.696864i	11.1576	+	2.98967i	2.00000	−	2.00000i	−2.68764	−	4.65514i	3.97394	+	5.84874i
19.4	1.36603	−	0.366025i	−1.17643	−	0.679210i	1.73205	−	1.00000i	4.29494	−	2.55997i	−1.85564	−	0.497217i	−1.13528	−	0.304199i	2.00000	−	2.00000i	−3.57735	−	6.19615i	4.92999	−	5.06905i
19.5	1.36603	−	0.366025i	2.81963	+	1.62791i	1.73205	−	1.00000i	−1.01577	−	4.89573i	4.44754	+	1.19171i	3.91361	+	1.04865i	2.00000	−	2.00000i	0.800200	+	1.38599i	−3.17954	−	6.31590i
19.6	1.36603	−	0.366025i	3.21490	+	1.85613i	1.73205	−	1.00000i	4.67137	+	1.78279i	5.07103	+	1.35878i	−7.00719	−	1.87757i	2.00000	−	2.00000i	2.39040	+	4.14030i	7.03375	+	0.725501i
19.7	1.36603	−	0.366025i	3.51172	+	2.02749i	1.73205	−	1.00000i	−3.38785	+	3.67729i	5.53921	+	1.48423i	0.953023	+	0.255362i	2.00000	−	2.00000i	3.72144	+	6.44572i	−3.28191	+	6.26331i
59.1	−0.366025	−	1.36603i	−4.71159	−	2.72024i	−1.73205	+	1.00000i	−3.09662	−	3.92568i	−1.99135	+	7.43182i	1.57264	−	5.86919i	2.00000	+	2.00000i	10.2994	+	17.8390i	−4.22914	+	5.66695i
59.2	−0.366025	−	1.36603i	−3.60508	−	2.08139i	−1.73205	+	1.00000i	1.72235	+	4.69399i	−1.52369	+	5.68647i	−1.33853	+	4.99548i	2.00000	+	2.00000i	4.16440	+	7.21295i	5.78168	−	4.07089i
59.3	−0.366025	−	1.36603i	−1.15020	−	0.664069i	−1.73205	+	1.00000i	4.92261	−	0.876293i	−0.486133	+	1.81427i	1.39207	−	5.19528i	2.00000	+	2.00000i	−3.61802	−	6.26660i	−2.99884	−	6.40367i
59.4	−0.366025	−	1.36603i	0.787112	+	0.454439i	−1.73205	+	1.00000i	−4.77553	+	1.48134i	0.332673	−	1.24155i	−3.19988	+	11.9421i	2.00000	+	2.00000i	−4.08697	−	7.07884i	3.77151	+	5.98128i
59.5	−0.366025	−	1.36603i	1.40175	+	0.809300i	−1.73205	+	1.00000i	−3.32166	−	3.73719i	0.592449	−	2.21105i	0.999970	−	3.73194i	2.00000	+	2.00000i	−3.19007	−	5.52536i	−3.88929	+	5.90538i
59.6	−0.366025	−	1.36603i	3.56714	+	2.05949i	−1.73205	+	1.00000i	4.98861	+	0.337300i	1.50765	−	5.62664i	−2.94974	+	11.0086i	2.00000	+	2.00000i	3.98301	+	6.89878i	−1.36520	−	6.93803i
59.7	−0.366025	−	1.36603i	3.71086	+	2.14247i	−1.73205	+	1.00000i	−0.305795	+	4.99064i	1.56840	−	5.85333i	2.25551	−	8.41768i	2.00000	+	2.00000i	4.68033	+	8.10658i	6.92927	−	1.40898i
89.1	1.36603	+	0.366025i	−4.60455	+	2.65844i	1.73205	+	1.00000i	−0.946786	−	4.90954i	−7.26299	+	1.94611i	−9.63174	+	2.58082i	2.00000	+	2.00000i	9.63460	−	16.6876i	0.503683	−	7.05311i
89.2	1.36603	+	0.366025i	−2.11647	+	1.22195i	1.73205	+	1.00000i	−3.39373	+	3.67186i	−3.33842	+	0.894526i	−2.98209	+	0.799048i	2.00000	+	2.00000i	−1.51370	+	2.62180i	−5.97992	+	3.77367i
89.3	1.36603	+	0.366025i	−1.64880	+	0.951934i	1.73205	+	1.00000i	1.64386	−	4.72205i	−2.60073	+	0.696864i	11.1576	−	2.98967i	2.00000	+	2.00000i	−2.68764	+	4.65514i	3.97394	−	5.84874i
89.4	1.36603	+	0.366025i	−1.17643	+	0.679210i	1.73205	+	1.00000i	4.29494	+	2.55997i	−1.85564	+	0.497217i	−1.13528	+	0.304199i	2.00000	+	2.00000i	−3.57735	+	6.19615i	4.92999	+	5.06905i
89.5	1.36603	+	0.366025i	2.81963	−	1.62791i	1.73205	+	1.00000i	−1.01577	+	4.89573i	4.44754	−	1.19171i	3.91361	−	1.04865i	2.00000	+	2.00000i	0.800200	−	1.38599i	−3.17954	+	6.31590i
89.6	1.36603	+	0.366025i	3.21490	−	1.85613i	1.73205	+	1.00000i	4.67137	−	1.78279i	5.07103	−	1.35878i	−7.00719	+	1.87757i	2.00000	+	2.00000i	2.39040	−	4.14030i	7.03375	−	0.725501i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.s	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	130.3.t.b	yes	28
5.b	even	2	1		130.3.t.a	✓	28
13.f	odd	12	1		130.3.t.a	✓	28
65.s	odd	12	1	inner	130.3.t.b	yes	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
130.3.t.a	✓	28	5.b	even	2	1
130.3.t.a	✓	28	13.f	odd	12	1
130.3.t.b	yes	28	1.a	even	1	1	trivial
130.3.t.b	yes	28	65.s	odd	12	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^28 - 84*T3^26 + 4337*T3^24 - 1932*T3^23 - 142292*T3^22 + 111516*T3^21 + 3429242*T3^20 - 4114236*T3^19 - 58194396*T3^18 + 86866776*T3^17 + 740919353*T3^16 - 1287679500*T3^15 - 6623907960*T3^14 + 11785583700*T3^13 + 45734523410*T3^12 - 72745384308*T3^11 - 232245274104*T3^10 + 282578908680*T3^9 + 895164920297*T3^8 - 604896062004*T3^7 - 2152146798584*T3^6 + 802972666380*T3^5 + 3673725834257*T3^4 - 94531137588*T3^3 - 2987628142056*T3^2 + 52341956832*T3 + 1654557399616