Newform orbit 360.2.bd.b

• Label: 360.2.bd.b
• Level: $360$
• Weight: $2$
• Character orbit: 360.bd
• Analytic conductor: $2.875$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	360.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.87461447277\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 128 q - 10 q^{4} + 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41125	+	0.0914777i	1.32971	+	1.10989i	1.98326	−	0.258196i	−1.67067	+	1.48622i	−1.97809	−	1.44470i	−2.22153	−	3.84780i	−2.77527	+	0.545804i	0.536278	+	2.95168i	2.22179	−	2.25026i
59.2	−1.40842	+	0.127848i	−0.679246	−	1.59331i	1.96731	−	0.360128i	−2.06621	−	0.854849i	1.16037	+	2.15721i	1.70997	+	2.96175i	−2.72476	+	0.758729i	−2.07725	+	2.16449i	3.01939	+	0.939829i
59.3	−1.40586	−	0.153453i	1.71600	−	0.235231i	1.95290	+	0.431469i	2.23606	−	0.00425117i	−2.44856	+	0.0673764i	1.34736	+	2.33369i	−2.67931	−	0.906266i	2.88933	−	0.807315i	−3.14425	−	0.337155i
59.4	−1.39143	−	0.252849i	−0.573395	+	1.63439i	1.87213	+	0.703642i	−0.0677451	−	2.23504i	1.21109	−	2.12915i	−0.0861243	−	0.149172i	−2.42702	−	1.45243i	−2.34244	−	1.87430i	−0.470866	+	3.12702i
59.5	−1.38690	+	0.276612i	−1.70706	−	0.293175i	1.84697	−	0.767264i	2.22251	+	0.245870i	2.44861	−	0.0655888i	0.193132	+	0.334514i	−2.34933	+	1.57501i	2.82810	+	1.00093i	−3.15040	+	0.273776i
59.6	−1.37971	+	0.310484i	−0.0381767	+	1.73163i	1.80720	−	0.856756i	0.289425	+	2.21726i	−0.484971	−	2.40100i	2.06690	+	3.57998i	−2.22740	+	1.74318i	−2.99709	−	0.132216i	−1.08775	−	2.96931i
59.7	−1.37227	−	0.341873i	1.16746	−	1.27946i	1.76625	+	0.938283i	−0.996511	+	2.00174i	−2.03949	+	1.35665i	−0.0301683	−	0.0522531i	−2.10299	−	1.89141i	−0.274059	−	2.98746i	2.05182	−	2.40625i
59.8	−1.29321	+	0.572373i	1.41725	−	0.995694i	1.34478	−	1.48040i	−0.776540	−	2.09690i	−1.26289	+	2.09884i	−0.720108	−	1.24726i	−0.891741	+	2.68418i	1.01719	−	2.82229i	2.20444	+	2.26726i
59.9	−1.27900	−	0.603460i	−1.13393	+	1.30928i	1.27167	+	1.54365i	1.63855	+	1.52157i	2.24039	−	0.990283i	−2.19953	−	3.80970i	−0.694935	−	2.74173i	−0.428416	−	2.96925i	−1.17749	−	2.93488i
59.10	−1.24345	−	0.673664i	1.46348	+	0.926406i	1.09235	+	1.67534i	−1.40051	−	1.74315i	−1.19568	−	2.13784i	0.618137	+	1.07065i	−0.229672	−	2.81909i	1.28354	+	2.71155i	0.567178	+	3.11100i
59.11	−1.24141	−	0.677426i	−1.65025	+	0.525984i	1.08219	+	1.68192i	−1.85825	+	1.24374i	2.40495	+	0.464965i	1.64197	+	2.84397i	−0.204056	−	2.82106i	2.44668	−	1.73602i	3.14939	−	0.285165i
59.12	−1.17126	−	0.792553i	−0.681584	−	1.59231i	0.743721	+	1.85658i	1.18672	+	1.89518i	−0.463672	+	2.40520i	−0.113645	−	0.196839i	0.600342	−	2.76398i	−2.07089	+	2.17058i	0.112070	−	3.16029i
59.13	−1.14229	+	0.833765i	−1.41725	+	0.995694i	0.609671	−	1.90481i	−2.20424	+	0.375946i	0.788739	−	2.31903i	−0.720108	−	1.24726i	0.891741	+	2.68418i	1.01719	−	2.82229i	2.20444	−	2.26726i
59.14	−1.02286	−	0.976600i	−1.56092	−	0.750684i	0.0925056	+	1.99786i	1.04334	−	1.97774i	0.863494	+	2.29224i	0.753496	+	1.30509i	1.85649	−	2.13388i	1.87295	+	2.34352i	−2.99865	+	1.00404i
59.15	−0.958742	+	1.03962i	0.0381767	−	1.73163i	−0.161627	−	1.99346i	2.06491	−	0.857979i	1.76364	+	1.69988i	2.06690	+	3.57998i	2.22740	+	1.74318i	−2.99709	−	0.132216i	−1.08775	+	2.96931i
59.16	−0.933002	+	1.06278i	1.70706	+	0.293175i	−0.259016	−	1.98316i	1.32418	+	1.80181i	−1.90427	+	1.54070i	0.193132	+	0.334514i	2.34933	+	1.57501i	2.82810	+	1.00093i	−3.15040	−	0.273776i
59.17	−0.912258	−	1.08064i	1.72276	−	0.179111i	−0.335569	+	1.97165i	1.97813	−	1.04260i	−1.76516	−	1.69829i	−2.24901	−	3.89541i	2.43677	−	1.43602i	2.93584	−	0.617133i	−2.93124	−	1.18653i
59.18	−0.903164	−	1.08825i	0.662421	−	1.60037i	−0.368591	+	1.96574i	−1.94669	−	1.10018i	−2.33989	+	0.724518i	−1.20226	−	2.08238i	2.47212	−	1.37427i	−2.12240	−	2.12024i	0.560906	+	3.11214i
59.19	−0.814931	+	1.15581i	0.679246	+	1.59331i	−0.671775	−	1.88380i	−1.77343	−	1.36197i	−2.39509	−	0.513357i	1.70997	+	2.96175i	2.72476	+	0.758729i	−2.07725	+	2.16449i	3.01939	−	0.939829i
59.20	−0.784848	+	1.17644i	−1.32971	−	1.10989i	−0.768027	−	1.84665i	0.451766	−	2.18996i	2.34935	−	0.693233i	−2.22153	−	3.84780i	2.77527	+	0.545804i	0.536278	+	2.95168i	2.22179	+	2.25026i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.d	odd	2	1	inner
9.d	odd	6	1	inner
40.e	odd	2	1	inner
45.h	odd	6	1	inner
72.l	even	6	1	inner
360.bd	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	360.2.bd.b	✓	128
5.b	even	2	1	inner	360.2.bd.b	✓	128
8.d	odd	2	1	inner	360.2.bd.b	✓	128
9.d	odd	6	1	inner	360.2.bd.b	✓	128
40.e	odd	2	1	inner	360.2.bd.b	✓	128
45.h	odd	6	1	inner	360.2.bd.b	✓	128
72.l	even	6	1	inner	360.2.bd.b	✓	128
360.bd	even	6	1	inner	360.2.bd.b	✓	128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.2.bd.b	✓	128	1.a	even	1	1	trivial
360.2.bd.b	✓	128	5.b	even	2	1	inner
360.2.bd.b	✓	128	8.d	odd	2	1	inner
360.2.bd.b	✓	128	9.d	odd	6	1	inner
360.2.bd.b	✓	128	40.e	odd	2	1	inner
360.2.bd.b	✓	128	45.h	odd	6	1	inner
360.2.bd.b	✓	128	72.l	even	6	1	inner
360.2.bd.b	✓	128	360.bd	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^64 + 125*T7^62 + 8745*T7^60 + 420748*T7^58 + 15371411*T7^56 + 446057901*T7^54 + 10591430625*T7^52 + 209286427008*T7^50 + 3483086879196*T7^48 + 49153498170126*T7^46 + 590858772542016*T7^44 + 6058808326507254*T7^42 + 53004917663876843*T7^40 + 394661554571210197*T7^38 + 2492197405194318657*T7^36 + 13263025741218630216*T7^34 + 59048177881743949596*T7^32 + 217514746068943799034*T7^30 + 655717864385423519644*T7^28 + 1591771206337828901162*T7^26 + 3075666688187831684571*T7^24 + 4602236952236663664641*T7^22 + 5203115021764283238157*T7^20 + 4111461308225707756740*T7^18 + 2101747965918946112331*T7^16 + 419420931283551929805*T7^14 + 58523257699661791731*T7^12 + 3928331098605255398*T7^10 + 188257298040366904*T7^8 + 4672176564306600*T7^6 + 80987787623520*T7^4 + 286442762400*T7^2 + 850305600