Newform orbit 840.2.u.e

• Label: 840.2.u.e
• Level: $840$
• Weight: $2$
• Character orbit: 840.u
• Analytic conductor: $6.707$
• Analytic rank: $0$
• Dimension: $160$
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.u (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.70743376979\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 24 q^{4} - 32 q^{9} - 48 q^{15} - 104 q^{16} - 16 q^{25} - 32 q^{30} + 48 q^{36} - 64 q^{39} - 64 q^{46} + 144 q^{49} + 16 q^{60} - 72 q^{64} + 8 q^{70} - 96 q^{79} + 16 q^{81} - 72 q^{84}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
629.1	−1.37073	−	0.347977i	−0.973146	−	1.43282i	1.75782	+	0.953969i	1.31834	+	1.80610i	0.835334	+	2.30265i	2.61441	+	0.406058i	−2.07755	−	1.91932i	−1.10597	+	2.78870i	−1.17861	−	2.93443i
629.2	−1.37073	−	0.347977i	−0.973146	+	1.43282i	1.75782	+	0.953969i	1.31834	−	1.80610i	1.83252	−	1.62539i	−2.61441	−	0.406058i	−2.07755	−	1.91932i	−1.10597	−	2.78870i	−2.43557	+	2.01693i
629.3	−1.37073	−	0.347977i	0.973146	−	1.43282i	1.75782	+	0.953969i	−1.31834	+	1.80610i	−1.83252	+	1.62539i	2.61441	−	0.406058i	−2.07755	−	1.91932i	−1.10597	−	2.78870i	2.43557	−	2.01693i
629.4	−1.37073	−	0.347977i	0.973146	+	1.43282i	1.75782	+	0.953969i	−1.31834	−	1.80610i	−0.835334	−	2.30265i	−2.61441	+	0.406058i	−2.07755	−	1.91932i	−1.10597	+	2.78870i	1.17861	+	2.93443i
629.5	−1.37073	+	0.347977i	−0.973146	−	1.43282i	1.75782	−	0.953969i	1.31834	+	1.80610i	1.83252	+	1.62539i	−2.61441	+	0.406058i	−2.07755	+	1.91932i	−1.10597	+	2.78870i	−2.43557	−	2.01693i
629.6	−1.37073	+	0.347977i	−0.973146	+	1.43282i	1.75782	−	0.953969i	1.31834	−	1.80610i	0.835334	−	2.30265i	2.61441	−	0.406058i	−2.07755	+	1.91932i	−1.10597	−	2.78870i	−1.17861	+	2.93443i
629.7	−1.37073	+	0.347977i	0.973146	−	1.43282i	1.75782	−	0.953969i	−1.31834	+	1.80610i	−0.835334	+	2.30265i	−2.61441	−	0.406058i	−2.07755	+	1.91932i	−1.10597	−	2.78870i	1.17861	−	2.93443i
629.8	−1.37073	+	0.347977i	0.973146	+	1.43282i	1.75782	−	0.953969i	−1.31834	−	1.80610i	−1.83252	−	1.62539i	2.61441	+	0.406058i	−2.07755	+	1.91932i	−1.10597	+	2.78870i	2.43557	+	2.01693i
629.9	−1.29842	−	0.560458i	−0.135991	−	1.72670i	1.37177	+	1.45542i	−1.93600	−	1.11889i	−0.791172	+	2.31820i	−2.16054	+	1.52711i	−0.965432	−	2.65856i	−2.96301	+	0.469633i	1.88664	+	2.53783i
629.10	−1.29842	−	0.560458i	−0.135991	+	1.72670i	1.37177	+	1.45542i	−1.93600	+	1.11889i	1.14432	−	2.16576i	2.16054	−	1.52711i	−0.965432	−	2.65856i	−2.96301	−	0.469633i	3.14082	−	0.367741i
629.11	−1.29842	−	0.560458i	0.135991	−	1.72670i	1.37177	+	1.45542i	1.93600	−	1.11889i	−1.14432	+	2.16576i	−2.16054	−	1.52711i	−0.965432	−	2.65856i	−2.96301	−	0.469633i	−3.14082	+	0.367741i
629.12	−1.29842	−	0.560458i	0.135991	+	1.72670i	1.37177	+	1.45542i	1.93600	+	1.11889i	0.791172	−	2.31820i	2.16054	+	1.52711i	−0.965432	−	2.65856i	−2.96301	+	0.469633i	−1.88664	−	2.53783i
629.13	−1.29842	+	0.560458i	−0.135991	−	1.72670i	1.37177	−	1.45542i	−1.93600	−	1.11889i	1.14432	+	2.16576i	2.16054	+	1.52711i	−0.965432	+	2.65856i	−2.96301	+	0.469633i	3.14082	+	0.367741i
629.14	−1.29842	+	0.560458i	−0.135991	+	1.72670i	1.37177	−	1.45542i	−1.93600	+	1.11889i	−0.791172	−	2.31820i	−2.16054	−	1.52711i	−0.965432	+	2.65856i	−2.96301	−	0.469633i	1.88664	−	2.53783i
629.15	−1.29842	+	0.560458i	0.135991	−	1.72670i	1.37177	−	1.45542i	1.93600	−	1.11889i	0.791172	+	2.31820i	2.16054	−	1.52711i	−0.965432	+	2.65856i	−2.96301	−	0.469633i	−1.88664	+	2.53783i
629.16	−1.29842	+	0.560458i	0.135991	+	1.72670i	1.37177	−	1.45542i	1.93600	+	1.11889i	−1.14432	−	2.16576i	−2.16054	+	1.52711i	−0.965432	+	2.65856i	−2.96301	+	0.469633i	−3.14082	−	0.367741i
629.17	−1.25943	−	0.643305i	−1.72207	−	0.185637i	1.17232	+	1.62039i	−0.799234	+	2.08835i	2.04941	+	1.34162i	−2.00372	+	1.72775i	−0.434041	−	2.79493i	2.93108	+	0.639360i	2.35003	−	2.11598i
629.18	−1.25943	−	0.643305i	−1.72207	+	0.185637i	1.17232	+	1.62039i	−0.799234	−	2.08835i	2.28825	+	0.874023i	2.00372	−	1.72775i	−0.434041	−	2.79493i	2.93108	−	0.639360i	−0.336872	+	3.14428i
629.19	−1.25943	−	0.643305i	1.72207	−	0.185637i	1.17232	+	1.62039i	0.799234	+	2.08835i	−2.28825	−	0.874023i	−2.00372	−	1.72775i	−0.434041	−	2.79493i	2.93108	−	0.639360i	0.336872	−	3.14428i
629.20	−1.25943	−	0.643305i	1.72207	+	0.185637i	1.17232	+	1.62039i	0.799234	−	2.08835i	−2.04941	−	1.34162i	2.00372	+	1.72775i	−0.434041	−	2.79493i	2.93108	+	0.639360i	−2.35003	+	2.11598i

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
7.b	odd	2	1	inner
8.b	even	2	1	inner
15.d	odd	2	1	inner
21.c	even	2	1	inner
24.h	odd	2	1	inner
35.c	odd	2	1	inner
40.f	even	2	1	inner
56.h	odd	2	1	inner
105.g	even	2	1	inner
120.i	odd	2	1	inner
168.i	even	2	1	inner
280.c	odd	2	1	inner
840.u	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	840.2.u.e	✓	160
3.b	odd	2	1	inner	840.2.u.e	✓	160
5.b	even	2	1	inner	840.2.u.e	✓	160
7.b	odd	2	1	inner	840.2.u.e	✓	160
8.b	even	2	1	inner	840.2.u.e	✓	160
15.d	odd	2	1	inner	840.2.u.e	✓	160
21.c	even	2	1	inner	840.2.u.e	✓	160
24.h	odd	2	1	inner	840.2.u.e	✓	160
35.c	odd	2	1	inner	840.2.u.e	✓	160
40.f	even	2	1	inner	840.2.u.e	✓	160
56.h	odd	2	1	inner	840.2.u.e	✓	160
105.g	even	2	1	inner	840.2.u.e	✓	160
120.i	odd	2	1	inner	840.2.u.e	✓	160
168.i	even	2	1	inner	840.2.u.e	✓	160
280.c	odd	2	1	inner	840.2.u.e	✓	160
840.u	even	2	1	inner	840.2.u.e	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.2.u.e	✓	160	1.a	even	1	1	trivial
840.2.u.e	✓	160	3.b	odd	2	1	inner
840.2.u.e	✓	160	5.b	even	2	1	inner
840.2.u.e	✓	160	7.b	odd	2	1	inner
840.2.u.e	✓	160	8.b	even	2	1	inner
840.2.u.e	✓	160	15.d	odd	2	1	inner
840.2.u.e	✓	160	21.c	even	2	1	inner
840.2.u.e	✓	160	24.h	odd	2	1	inner
840.2.u.e	✓	160	35.c	odd	2	1	inner
840.2.u.e	✓	160	40.f	even	2	1	inner
840.2.u.e	✓	160	56.h	odd	2	1	inner
840.2.u.e	✓	160	105.g	even	2	1	inner
840.2.u.e	✓	160	120.i	odd	2	1	inner
840.2.u.e	✓	160	168.i	even	2	1	inner
840.2.u.e	✓	160	280.c	odd	2	1	inner
840.2.u.e	✓	160	840.u	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

T11^20 - 128*T11^18 + 6741*T11^16 - 189322*T11^14 + 3078064*T11^12 - 29486944*T11^10 + 162202944*T11^8 - 473641984*T11^6 + 610752512*T11^4 - 178167808*T11^2 + 7585792

T23^20 - 228*T23^18 + 20468*T23^16 - 939472*T23^14 + 24073184*T23^12 - 353316096*T23^10 + 2884532224*T23^8 - 11707129856*T23^6 + 17559650304*T23^4 - 8538030080*T23^2 + 79691776

T73^20 - 740*T73^18 + 210448*T73^16 - 29178848*T73^14 + 2085671680*T73^12 - 74909592064*T73^10 + 1254467723264*T73^8 - 8883704496128*T73^6 + 21355566202880*T73^4 - 2648486969344*T73^2 + 15535702016