Newform orbit 297.2.t.a

• Label: 297.2.t.a
• Level: $297$
• Weight: $2$
• Character orbit: 297.t
• Analytic conductor: $2.372$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.t (of order \(30\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.37155694003\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(10\) over \(\Q(\zeta_{30})\)
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 15 q^{2} + 5 q^{4} + 6 q^{5} - 5 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
8.1	−0.245046	−	2.33146i		0		−3.41935	+	0.726805i	2.90568	+	0.305399i		0		0.977562	−	0.880201i	1.08356	+	3.33484i		0			−	6.84930i
8.2	−0.221869	−	2.11094i		0		−2.45055	+	0.520881i	−2.96223	−	0.311343i		0		−2.37463	+	2.13813i	0.331430	+	1.02004i		0				6.32217i
8.3	−0.120408	−	1.14561i		0		0.658378	−	0.139943i	0.932459	+	0.0980053i		0		0.644183	−	0.580025i	−0.951517	−	2.92847i		0			−	1.08003i
8.4	−0.0539485	−	0.513286i		0		1.69574	−	0.360441i	−1.61806	−	0.170065i		0		0.613837	−	0.552701i	−0.595467	−	1.83266i		0				0.839703i
8.5	−0.0386735	−	0.367954i		0		1.82240	−	0.387363i	2.36345	+	0.248408i		0		1.17198	−	1.05526i	−0.441671	−	1.35932i		0			−	0.879247i
8.6	0.0653389	+	0.621658i		0		1.57411	−	0.334586i	1.26871	+	0.133347i		0		−3.59877	+	3.24034i	0.697171	+	2.14567i		0				0.797419i
8.7	0.150346	+	1.43045i		0		−0.0672774	+	0.0143003i	−3.54784	−	0.372894i		0		−1.60829	+	1.44811i	0.858363	+	2.64177i		0			−	5.13106i
8.8	0.189668	+	1.80457i		0		−1.26420	+	0.268714i	1.19507	+	0.125607i		0		3.59771	−	3.23939i	0.396737	+	1.22103i		0				2.18041i
8.9	0.205839	+	1.95843i		0		−1.83679	+	0.390421i	1.96633	+	0.206670i		0		0.235695	−	0.212221i	0.0743474	+	0.228818i		0				3.89347i
8.10	0.281588	+	2.67913i		0		−5.14217	+	1.09300i	−1.43896	−	0.151241i		0		−1.07283	+	0.965979i	−2.71136	−	8.34470i		0			−	3.89775i
17.1	−1.48406	+	1.64822i		0		−0.305125	−	2.90307i	0.217939	−	0.196233i		0		0.539627	−	1.21202i	1.64909	+	1.19814i		0				0.650432i
17.2	−1.13824	+	1.26415i		0		−0.0934131	−	0.888766i	1.30752	−	1.17729i		0		−1.14161	+	2.56410i	−1.52254	−	1.10619i		0				2.99294i
17.3	−0.780898	+	0.867275i		0		0.0666926	+	0.634537i	1.15400	−	1.03907i		0		1.76474	−	3.96367i	−2.49070	−	1.80960i		0				1.81224i
17.4	−0.148911	+	0.165382i		0		0.203880	+	1.93979i	−0.397470	+	0.357884i		0		−1.30165	+	2.92356i	−0.711251	−	0.516754i		0			−	0.119027i
17.5	−0.0350855	+	0.0389664i		0		0.208770	+	1.98631i	−1.98818	+	1.79017i		0		1.00434	−	2.25577i	−0.169565	−	0.123196i		0			−	0.140281i
17.6	0.536887	−	0.596273i		0		0.141763	+	1.34878i	2.14661	−	1.93281i		0		0.542079	−	1.21753i	2.17861	+	1.58285i		0			−	2.31767i
17.7	0.906431	−	1.00669i		0		0.0172421	+	0.164047i	−2.90247	+	2.61340i		0		−0.686294	+	1.54144i	2.37263	+	1.72381i		0				5.29077i
17.8	1.02662	−	1.14018i		0		−0.0369992	−	0.352024i	2.07594	−	1.86918i		0		−1.07503	+	2.41456i	2.04313	+	1.48442i		0			−	4.28588i
17.9	1.65440	−	1.83740i		0		−0.429934	−	4.09055i	1.51955	−	1.36821i		0		−0.441031	+	0.990572i	−4.22672	−	3.07089i		0			−	5.05558i
17.10	1.66735	−	1.85178i		0		−0.439972	−	4.18606i	−1.05074	+	0.946093i		0		1.27299	−	2.85917i	−4.45339	−	3.23558i		0				3.52320i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
11.d	odd	10	1	inner
99.p	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	297.2.t.a		80
3.b	odd	2	1		99.2.p.a	✓	80
9.c	even	3	1		99.2.p.a	✓	80
9.c	even	3	1		891.2.k.a		80
9.d	odd	6	1	inner	297.2.t.a		80
9.d	odd	6	1		891.2.k.a		80
11.d	odd	10	1	inner	297.2.t.a		80
33.f	even	10	1		99.2.p.a	✓	80
99.o	odd	30	1		99.2.p.a	✓	80
99.o	odd	30	1		891.2.k.a		80
99.p	even	30	1	inner	297.2.t.a		80
99.p	even	30	1		891.2.k.a		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
99.2.p.a	✓	80	3.b	odd	2	1
99.2.p.a	✓	80	9.c	even	3	1
99.2.p.a	✓	80	33.f	even	10	1
99.2.p.a	✓	80	99.o	odd	30	1
297.2.t.a		80	1.a	even	1	1	trivial
297.2.t.a		80	9.d	odd	6	1	inner
297.2.t.a		80	11.d	odd	10	1	inner
297.2.t.a		80	99.p	even	30	1	inner
891.2.k.a		80	9.c	even	3	1
891.2.k.a		80	9.d	odd	6	1
891.2.k.a		80	99.o	odd	30	1
891.2.k.a		80	99.p	even	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(297, [\chi])\).