Newform orbit 130.3.t.a

• Label: 130.3.t.a
• 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.

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

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	−3.51172	−	2.02749i	1.73205	−	1.00000i	−3.67729	+	3.38785i	5.53921	+	1.48423i	−0.953023	−	0.255362i	−2.00000	+	2.00000i	3.72144	+	6.44572i	3.78324	−	5.97387i
19.2	−1.36603	+	0.366025i	−3.21490	−	1.85613i	1.73205	−	1.00000i	−1.78279	−	4.67137i	5.07103	+	1.35878i	7.00719	+	1.87757i	−2.00000	+	2.00000i	2.39040	+	4.14030i	4.14518	+	5.72866i
19.3	−1.36603	+	0.366025i	−2.81963	−	1.62791i	1.73205	−	1.00000i	4.89573	+	1.01577i	4.44754	+	1.19171i	−3.91361	−	1.04865i	−2.00000	+	2.00000i	0.800200	+	1.38599i	−7.05950	+	0.404389i
19.4	−1.36603	+	0.366025i	1.17643	+	0.679210i	1.73205	−	1.00000i	2.55997	−	4.29494i	−1.85564	−	0.497217i	1.13528	+	0.304199i	−2.00000	+	2.00000i	−3.57735	−	6.19615i	−1.92493	+	6.80402i
19.5	−1.36603	+	0.366025i	1.64880	+	0.951934i	1.73205	−	1.00000i	−4.72205	−	1.64386i	−2.60073	−	0.696864i	−11.1576	−	2.98967i	−2.00000	+	2.00000i	−2.68764	−	4.65514i	7.05213	+	0.517163i
19.6	−1.36603	+	0.366025i	2.11647	+	1.22195i	1.73205	−	1.00000i	3.67186	+	3.39373i	−3.33842	−	0.894526i	2.98209	+	0.799048i	−2.00000	+	2.00000i	−1.51370	−	2.62180i	−6.25805	−	3.29193i
19.7	−1.36603	+	0.366025i	4.60455	+	2.65844i	1.73205	−	1.00000i	−4.90954	+	0.946786i	−7.26299	−	1.94611i	9.63174	+	2.58082i	−2.00000	+	2.00000i	9.63460	+	16.6876i	6.36001	−	3.09035i
59.1	0.366025	+	1.36603i	−3.71086	−	2.14247i	−1.73205	+	1.00000i	4.99064	−	0.305795i	1.56840	−	5.85333i	−2.25551	+	8.41768i	−2.00000	−	2.00000i	4.68033	+	8.10658i	2.24443	+	6.70541i
59.2	0.366025	+	1.36603i	−3.56714	−	2.05949i	−1.73205	+	1.00000i	0.337300	+	4.98861i	1.50765	−	5.62664i	2.94974	−	11.0086i	−2.00000	−	2.00000i	3.98301	+	6.89878i	−6.69111	+	2.28672i
59.3	0.366025	+	1.36603i	−1.40175	−	0.809300i	−1.73205	+	1.00000i	−3.73719	−	3.32166i	0.592449	−	2.21105i	−0.999970	+	3.73194i	−2.00000	−	2.00000i	−3.19007	−	5.52536i	3.16956	−	6.32091i
59.4	0.366025	+	1.36603i	−0.787112	−	0.454439i	−1.73205	+	1.00000i	1.48134	−	4.77553i	0.332673	−	1.24155i	3.19988	−	11.9421i	−2.00000	−	2.00000i	−4.08697	−	7.07884i	7.06570	+	0.275581i
59.5	0.366025	+	1.36603i	1.15020	+	0.664069i	−1.73205	+	1.00000i	−0.876293	+	4.92261i	−0.486133	+	1.81427i	−1.39207	+	5.19528i	−2.00000	−	2.00000i	−3.61802	−	6.26660i	−7.04516	+	0.604763i
59.6	0.366025	+	1.36603i	3.60508	+	2.08139i	−1.73205	+	1.00000i	4.69399	+	1.72235i	−1.52369	+	5.68647i	1.33853	−	4.99548i	−2.00000	−	2.00000i	4.16440	+	7.21295i	−0.634650	+	7.04253i
59.7	0.366025	+	1.36603i	4.71159	+	2.72024i	−1.73205	+	1.00000i	−3.92568	−	3.09662i	−1.99135	+	7.43182i	−1.57264	+	5.86919i	−2.00000	−	2.00000i	10.2994	+	17.8390i	2.79316	−	6.49602i
89.1	−1.36603	−	0.366025i	−3.51172	+	2.02749i	1.73205	+	1.00000i	−3.67729	−	3.38785i	5.53921	−	1.48423i	−0.953023	+	0.255362i	−2.00000	−	2.00000i	3.72144	−	6.44572i	3.78324	+	5.97387i
89.2	−1.36603	−	0.366025i	−3.21490	+	1.85613i	1.73205	+	1.00000i	−1.78279	+	4.67137i	5.07103	−	1.35878i	7.00719	−	1.87757i	−2.00000	−	2.00000i	2.39040	−	4.14030i	4.14518	−	5.72866i
89.3	−1.36603	−	0.366025i	−2.81963	+	1.62791i	1.73205	+	1.00000i	4.89573	−	1.01577i	4.44754	−	1.19171i	−3.91361	+	1.04865i	−2.00000	−	2.00000i	0.800200	−	1.38599i	−7.05950	−	0.404389i
89.4	−1.36603	−	0.366025i	1.17643	−	0.679210i	1.73205	+	1.00000i	2.55997	+	4.29494i	−1.85564	+	0.497217i	1.13528	−	0.304199i	−2.00000	−	2.00000i	−3.57735	+	6.19615i	−1.92493	−	6.80402i
89.5	−1.36603	−	0.366025i	1.64880	−	0.951934i	1.73205	+	1.00000i	−4.72205	+	1.64386i	−2.60073	+	0.696864i	−11.1576	+	2.98967i	−2.00000	−	2.00000i	−2.68764	+	4.65514i	7.05213	−	0.517163i
89.6	−1.36603	−	0.366025i	2.11647	−	1.22195i	1.73205	+	1.00000i	3.67186	−	3.39373i	−3.33842	+	0.894526i	2.98209	−	0.799048i	−2.00000	−	2.00000i	−1.51370	+	2.62180i	−6.25805	+	3.29193i

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.a	✓	28
5.b	even	2	1		130.3.t.b	yes	28
13.f	odd	12	1		130.3.t.b	yes	28
65.s	odd	12	1	inner	130.3.t.a	✓	28

    

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

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