Newform orbit 756.2.bi.c

• Label: 756.2.bi.c
• Level: $756$
• Weight: $2$
• Character orbit: 756.bi
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $80$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.bi (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 80 * q + 6 * q^2 - 2 * q^4 + 24 * q^8 - 10 * q^14 - 18 * q^16 - 14 * q^22 + 32 * q^25 + 28 * q^28 - 8 * q^29 + 16 * q^32 - 16 * q^37 + 84 * q^44 + 24 * q^46 - 24 * q^49 + 12 * q^50 + 48 * q^53 - 32 * q^56 - 14 * q^58 - 8 * q^64 + 40 * q^65 - 22 * q^70 - 64 * q^74 + 12 * q^77 + 40 * q^85 + 52 * q^86 + 6 * q^88 - 30 * q^92 - 20 * q^98

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
307.1	−1.40431	−	0.167104i		0		1.94415	+	0.469331i	−1.39056	−	0.802839i		0		0.686390	−	2.55517i	−2.65176	−	0.983961i		0		1.81861	+	1.35980i
307.2	−1.40431	−	0.167104i		0		1.94415	+	0.469331i	1.39056	+	0.802839i		0		2.55603	+	0.683151i	−2.65176	−	0.983961i		0		−1.81861	−	1.35980i
307.3	−1.33314	+	0.471949i		0		1.55453	−	1.25835i	−0.0753642	−	0.0435115i		0		−0.0425387	+	2.64541i	−1.47853	+	2.41122i		0		0.121006	+	0.0224389i
307.4	−1.33314	+	0.471949i		0		1.55453	−	1.25835i	0.0753642	+	0.0435115i		0		−2.31226	−	1.28587i	−1.47853	+	2.41122i		0		−0.121006	−	0.0224389i
307.5	−1.07464	+	0.919317i		0		0.309712	−	1.97587i	−3.16954	−	1.82994i		0		2.31955	−	1.27266i	1.48363	+	2.40808i		0		5.08842	−	0.947288i
307.6	−1.07464	+	0.919317i		0		0.309712	−	1.97587i	3.16954	+	1.82994i		0		2.26193	−	1.37246i	1.48363	+	2.40808i		0		−5.08842	+	0.947288i
307.7	−1.05092	−	0.946342i		0		0.208875	+	1.98906i	−0.416111	−	0.240242i		0		2.63514	−	0.236713i	1.66282	−	2.28802i		0		0.209950	+	0.646259i
307.8	−1.05092	−	0.946342i		0		0.208875	+	1.98906i	0.416111	+	0.240242i		0		1.52257	−	2.16374i	1.66282	−	2.28802i		0		−0.209950	−	0.646259i
307.9	−1.02930	−	0.969818i		0		0.118905	+	1.99646i	−3.45303	−	1.99361i		0		−2.03435	−	1.69157i	1.81382	−	2.17027i		0		1.62076	+	5.40083i
307.10	−1.02930	−	0.969818i		0		0.118905	+	1.99646i	3.45303	+	1.99361i		0		0.447766	+	2.60759i	1.81382	−	2.17027i		0		−1.62076	−	5.40083i
307.11	−0.955367	+	1.04272i		0		−0.174547	−	1.99237i	−1.82425	−	1.05323i		0		−0.655553	+	2.56325i	2.24425	+	1.72144i		0		2.84106	−	0.895969i
307.12	−0.955367	+	1.04272i		0		−0.174547	−	1.99237i	1.82425	+	1.05323i		0		−2.54762	−	0.713899i	2.24425	+	1.72144i		0		−2.84106	+	0.895969i
307.13	−0.648812	+	1.25660i		0		−1.15809	−	1.63059i	−2.66647	−	1.53948i		0		−1.15806	−	2.37885i	2.80038	−	0.397302i		0		3.66455	−	2.35184i
307.14	−0.648812	+	1.25660i		0		−1.15809	−	1.63059i	2.66647	+	1.53948i		0		1.48111	+	2.19233i	2.80038	−	0.397302i		0		−3.66455	+	2.35184i
307.15	−0.325239	−	1.37631i		0		−1.78844	+	0.895257i	−3.45303	−	1.99361i		0		2.03435	+	1.69157i	1.81382	+	2.17027i		0		−1.62076	+	5.40083i
307.16	−0.325239	−	1.37631i		0		−1.78844	+	0.895257i	3.45303	+	1.99361i		0		−0.447766	−	2.60759i	1.81382	+	2.17027i		0		1.62076	−	5.40083i
307.17	−0.294095	−	1.38330i		0		−1.82702	+	0.813640i	−0.416111	−	0.240242i		0		−2.63514	+	0.236713i	1.66282	+	2.28802i		0		−0.209950	+	0.646259i
307.18	−0.294095	−	1.38330i		0		−1.82702	+	0.813640i	0.416111	+	0.240242i		0		−1.52257	+	2.16374i	1.66282	+	2.28802i		0		0.209950	−	0.646259i
307.19	−0.139426	+	1.40732i		0		−1.96112	−	0.392436i	−1.35495	−	0.782280i		0		0.314165	+	2.62703i	0.825716	−	2.70522i		0		1.28984	−	1.79778i
307.20	−0.139426	+	1.40732i		0		−1.96112	−	0.392436i	1.35495	+	0.782280i		0		−2.11799	−	1.58559i	0.825716	−	2.70522i		0		−1.28984	+	1.79778i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.b	odd	2	1	inner
9.c	even	3	1	inner
28.d	even	2	1	inner
36.f	odd	6	1	inner
63.l	odd	6	1	inner
252.bi	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.2.bi.c		80
3.b	odd	2	1		252.2.bi.c	✓	80
4.b	odd	2	1	inner	756.2.bi.c		80
7.b	odd	2	1	inner	756.2.bi.c		80
9.c	even	3	1	inner	756.2.bi.c		80
9.d	odd	6	1		252.2.bi.c	✓	80
12.b	even	2	1		252.2.bi.c	✓	80
21.c	even	2	1		252.2.bi.c	✓	80
28.d	even	2	1	inner	756.2.bi.c		80
36.f	odd	6	1	inner	756.2.bi.c		80
36.h	even	6	1		252.2.bi.c	✓	80
63.l	odd	6	1	inner	756.2.bi.c		80
63.o	even	6	1		252.2.bi.c	✓	80
84.h	odd	2	1		252.2.bi.c	✓	80
252.s	odd	6	1		252.2.bi.c	✓	80
252.bi	even	6	1	inner	756.2.bi.c		80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.bi.c	✓	80	3.b	odd	2	1
252.2.bi.c	✓	80	9.d	odd	6	1
252.2.bi.c	✓	80	12.b	even	2	1
252.2.bi.c	✓	80	21.c	even	2	1
252.2.bi.c	✓	80	36.h	even	6	1
252.2.bi.c	✓	80	63.o	even	6	1
252.2.bi.c	✓	80	84.h	odd	2	1
252.2.bi.c	✓	80	252.s	odd	6	1
756.2.bi.c		80	1.a	even	1	1	trivial
756.2.bi.c		80	4.b	odd	2	1	inner
756.2.bi.c		80	7.b	odd	2	1	inner
756.2.bi.c		80	9.c	even	3	1	inner
756.2.bi.c		80	28.d	even	2	1	inner
756.2.bi.c		80	36.f	odd	6	1	inner
756.2.bi.c		80	63.l	odd	6	1	inner
756.2.bi.c		80	252.bi	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^40 - 58*T5^38 + 1986*T5^36 - 45000*T5^34 + 757017*T5^32 - 9662658*T5^30 + 96876664*T5^28 - 766064542*T5^26 + 4848544452*T5^24 - 24482028372*T5^22 + 99265144659*T5^20 - 319950684900*T5^18 + 818404822472*T5^16 - 1618039070798*T5^14 + 2423038916979*T5^12 - 2526088619506*T5^10 + 1683338096629*T5^8 - 371823512760*T5^6 + 61591708476*T5^4 - 465712416*T5^2 + 3370896