Newform orbit 240.2.t.b

• Label: 240.2.t.b
• Level: $240$
• Weight: $2$
• Character orbit: 240.t
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	240.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 8 q^{4} - 4 q^{6}+O(q^{10}) \)

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	−1.41336	+	0.0490154i	0.209895	+	1.71929i	1.99519	−	0.138553i	−0.768175	−	2.09998i	−0.380929	−	2.41969i		−	4.08340i	−2.81315	+	0.293621i	−2.91189	+	0.721739i	1.18864	+	2.93038i
59.2	−1.41336	+	0.0490154i	1.71929	+	0.209895i	1.99519	−	0.138553i	−2.09998	−	0.768175i	−2.44026	−	0.212386i			4.08340i	−2.81315	+	0.293621i	2.91189	+	0.721739i	3.00569	+	0.982779i
59.3	−1.27807	−	0.605427i	−1.58930	+	0.688572i	1.26692	+	1.54755i	−1.54468	+	1.61678i	2.44811	+	0.0821618i			0.970036i	−0.682276	−	2.74490i	2.05174	−	2.18869i	2.95304	−	1.13116i
59.4	−1.27807	−	0.605427i	0.688572	−	1.58930i	1.26692	+	1.54755i	1.61678	−	1.54468i	−1.84225	+	1.61435i		−	0.970036i	−0.682276	−	2.74490i	−2.05174	−	2.18869i	−3.00154	+	0.995364i
59.5	−1.26832	+	0.625594i	−0.262098	+	1.71211i	1.21726	−	1.58691i	−0.0667377	+	2.23507i	−0.738660	−	2.33546i			0.598963i	−0.551120	+	2.77421i	−2.86261	−	0.897478i	−1.31360	−	2.87653i
59.6	−1.26832	+	0.625594i	1.71211	−	0.262098i	1.21726	−	1.58691i	2.23507	−	0.0667377i	−2.00753	+	1.40351i		−	0.598963i	−0.551120	+	2.77421i	2.86261	−	0.897478i	−2.79303	+	1.48289i
59.7	−1.22048	−	0.714442i	0.591463	+	1.62793i	0.979146	+	1.74392i	2.22111	−	0.258204i	0.441196	−	2.40943i			4.18600i	0.0509042	−	2.82797i	−2.30034	+	1.92573i	−2.89529	−	1.27172i
59.8	−1.22048	−	0.714442i	1.62793	+	0.591463i	0.979146	+	1.74392i	−0.258204	+	2.22111i	−1.56430	−	1.88493i		−	4.18600i	0.0509042	−	2.82797i	2.30034	+	1.92573i	1.90199	−	2.52635i
59.9	−1.03022	+	0.968834i	−1.72231	−	0.183389i	0.122722	−	1.99623i	−2.23605	−	0.00763900i	1.95204	−	1.47971i		−	1.25117i	1.80759	+	2.17546i	2.93274	+	0.631707i	2.31104	−	2.15850i
59.10	−1.03022	+	0.968834i	−0.183389	−	1.72231i	0.122722	−	1.99623i	−0.00763900	−	2.23605i	1.85757	+	1.59670i			1.25117i	1.80759	+	2.17546i	−2.93274	+	0.631707i	2.17424	+	2.29624i
59.11	−0.788977	−	1.17368i	−1.64998	−	0.526861i	−0.755031	+	1.85201i	2.11651	−	0.721367i	0.683429	+	2.35222i		−	2.57173i	2.76936	−	0.575028i	2.44484	+	1.73861i	−2.51653	−	1.91496i
59.12	−0.788977	−	1.17368i	−0.526861	−	1.64998i	−0.755031	+	1.85201i	−0.721367	+	2.11651i	−1.52086	+	1.92016i			2.57173i	2.76936	−	0.575028i	−2.44484	+	1.73861i	3.05324	−	0.823229i
59.13	−0.743883	−	1.20276i	−1.02525	+	1.39602i	−0.893276	+	1.78943i	−0.920160	−	2.03797i	2.44174	+	0.194661i			0.440232i	2.81675	−	0.256728i	−0.897721	−	2.86253i	−1.76670	+	2.62274i
59.14	−0.743883	−	1.20276i	1.39602	−	1.02525i	−0.893276	+	1.78943i	−2.03797	−	0.920160i	−2.27161	−	0.916409i		−	0.440232i	2.81675	−	0.256728i	0.897721	−	2.86253i	0.409274	+	3.13568i
59.15	−0.676744	+	1.24178i	−1.34702	+	1.08883i	−1.08403	−	1.68073i	2.16547	−	0.557438i	−0.440501	−	2.40956i		−	4.04743i	2.82072	−	0.208705i	0.628903	−	2.93334i	−0.773254	+	3.06628i
59.16	−0.676744	+	1.24178i	1.08883	−	1.34702i	−1.08403	−	1.68073i	−0.557438	+	2.16547i	0.935838	+	2.26367i			4.04743i	2.82072	−	0.208705i	−0.628903	−	2.93334i	−2.31179	−	2.15769i
59.17	−0.265026	+	1.38916i	−0.571191	+	1.63516i	−1.85952	−	0.736326i	−1.86979	−	1.22634i	−2.12011	−	1.22683i			3.84936i	1.51570	−	2.38803i	−2.34748	−	1.86798i	2.19912	−	2.27242i
59.18	−0.265026	+	1.38916i	1.63516	−	0.571191i	−1.85952	−	0.736326i	−1.22634	−	1.86979i	0.360116	+	2.42287i		−	3.84936i	1.51570	−	2.38803i	2.34748	−	1.86798i	2.92244	−	1.20803i
59.19	−0.0728709	−	1.41233i	0.938373	+	1.45584i	−1.98938	+	0.205836i	−1.83519	+	1.27753i	1.98775	−	1.43139i			2.98852i	0.435678	+	2.79467i	−1.23891	+	2.73223i	1.93803	+	2.49880i
59.20	−0.0728709	−	1.41233i	1.45584	+	0.938373i	−1.98938	+	0.205836i	1.27753	−	1.83519i	1.21921	−	2.12451i		−	2.98852i	0.435678	+	2.79467i	1.23891	+	2.73223i	−2.68499	−	1.67057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner
16.f	odd	4	1	inner
48.k	even	4	1	inner
80.k	odd	4	1	inner
240.t	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	240.2.t.b	✓	80
3.b	odd	2	1	inner	240.2.t.b	✓	80
4.b	odd	2	1		960.2.t.b		80
5.b	even	2	1	inner	240.2.t.b	✓	80
12.b	even	2	1		960.2.t.b		80
15.d	odd	2	1	inner	240.2.t.b	✓	80
16.e	even	4	1		960.2.t.b		80
16.f	odd	4	1	inner	240.2.t.b	✓	80
20.d	odd	2	1		960.2.t.b		80
48.i	odd	4	1		960.2.t.b		80
48.k	even	4	1	inner	240.2.t.b	✓	80
60.h	even	2	1		960.2.t.b		80
80.k	odd	4	1	inner	240.2.t.b	✓	80
80.q	even	4	1		960.2.t.b		80
240.t	even	4	1	inner	240.2.t.b	✓	80
240.bm	odd	4	1		960.2.t.b		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.t.b	✓	80	1.a	even	1	1	trivial
240.2.t.b	✓	80	3.b	odd	2	1	inner
240.2.t.b	✓	80	5.b	even	2	1	inner
240.2.t.b	✓	80	15.d	odd	2	1	inner
240.2.t.b	✓	80	16.f	odd	4	1	inner
240.2.t.b	✓	80	48.k	even	4	1	inner
240.2.t.b	✓	80	80.k	odd	4	1	inner
240.2.t.b	✓	80	240.t	even	4	1	inner
960.2.t.b		80	4.b	odd	2	1
960.2.t.b		80	12.b	even	2	1
960.2.t.b		80	16.e	even	4	1
960.2.t.b		80	20.d	odd	2	1
960.2.t.b		80	48.i	odd	4	1
960.2.t.b		80	60.h	even	2	1
960.2.t.b		80	80.q	even	4	1
960.2.t.b		80	240.bm	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^20 + 84*T7^18 + 2928*T7^16 + 54608*T7^14 + 585460*T7^12 + 3603664*T7^10 + 12031840*T7^8 + 19489984*T7^6 + 14414912*T7^4 + 4358144*T7^2 + 429056