Newform orbit 847.2.e.j

• Label: 847.2.e.j
• Level: $847$
• Weight: $2$
• Character orbit: 847.e
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 6 q^{3} - 16 q^{4} + 8 q^{5} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
485.1	−1.32785	+	2.29990i	−0.249514	−	0.432170i	−2.52636	−	4.37579i	1.76072	−	3.04966i	1.32526			1.92934	−	1.81043i	8.10710			1.37549	−	2.38241i	4.67594	+	8.09897i
485.2	−1.22945	+	2.12947i	0.276930	+	0.479656i	−2.02311	−	3.50413i	−0.620518	+	1.07477i	−1.36189			−2.34245	−	1.23001i	5.03146			1.34662	−	2.33241i	−1.52579	−	2.64275i
485.3	−0.945348	+	1.63739i	1.42149	+	2.46210i	−0.787365	−	1.36376i	−1.73847	+	3.01112i	−5.37522			2.58119	+	0.580920i	−0.804055			−2.54129	+	4.40164i	−3.28692	−	5.69311i
485.4	−0.830383	+	1.43827i	−1.16938	−	2.02543i	−0.379072	−	0.656572i	1.38656	−	2.40159i	3.88414			2.09572	−	1.61492i	−2.06243			−1.23491	+	2.13892i	2.30275	+	3.98848i
485.5	−0.614728	+	1.06474i	1.62503	+	2.81463i	0.244219	+	0.423000i	1.29373	−	2.24080i	−3.99581			0.255479	+	2.63339i	−3.05942			−3.78144	+	6.54965i	1.59058	+	2.75496i
485.6	−0.503676	+	0.872392i	0.297921	+	0.516014i	0.492622	+	0.853246i	0.793525	−	1.37442i	−0.600222			−2.26989	+	1.35927i	−3.00719			1.32249	−	2.29061i	0.799358	+	1.38453i
485.7	−0.102308	+	0.177202i	−0.702479	−	1.21673i	0.979066	+	1.69579i	−0.875546	+	1.51649i	0.287475			−1.23221	+	2.34129i	−0.809894			0.513047	−	0.888624i	−0.179150	−	0.310297i
485.8	0.102308	−	0.177202i	−0.702479	−	1.21673i	0.979066	+	1.69579i	−0.875546	+	1.51649i	−0.287475			1.23221	−	2.34129i	0.809894			0.513047	−	0.888624i	0.179150	+	0.310297i
485.9	0.503676	−	0.872392i	0.297921	+	0.516014i	0.492622	+	0.853246i	0.793525	−	1.37442i	0.600222			2.26989	−	1.35927i	3.00719			1.32249	−	2.29061i	−0.799358	−	1.38453i
485.10	0.614728	−	1.06474i	1.62503	+	2.81463i	0.244219	+	0.423000i	1.29373	−	2.24080i	3.99581			−0.255479	−	2.63339i	3.05942			−3.78144	+	6.54965i	−1.59058	−	2.75496i
485.11	0.830383	−	1.43827i	−1.16938	−	2.02543i	−0.379072	−	0.656572i	1.38656	−	2.40159i	−3.88414			−2.09572	+	1.61492i	2.06243			−1.23491	+	2.13892i	−2.30275	−	3.98848i
485.12	0.945348	−	1.63739i	1.42149	+	2.46210i	−0.787365	−	1.36376i	−1.73847	+	3.01112i	5.37522			−2.58119	−	0.580920i	0.804055			−2.54129	+	4.40164i	3.28692	+	5.69311i
485.13	1.22945	−	2.12947i	0.276930	+	0.479656i	−2.02311	−	3.50413i	−0.620518	+	1.07477i	1.36189			2.34245	+	1.23001i	−5.03146			1.34662	−	2.33241i	1.52579	+	2.64275i
485.14	1.32785	−	2.29990i	−0.249514	−	0.432170i	−2.52636	−	4.37579i	1.76072	−	3.04966i	−1.32526			−1.92934	+	1.81043i	−8.10710			1.37549	−	2.38241i	−4.67594	−	8.09897i
606.1	−1.32785	−	2.29990i	−0.249514	+	0.432170i	−2.52636	+	4.37579i	1.76072	+	3.04966i	1.32526			1.92934	+	1.81043i	8.10710			1.37549	+	2.38241i	4.67594	−	8.09897i
606.2	−1.22945	−	2.12947i	0.276930	−	0.479656i	−2.02311	+	3.50413i	−0.620518	−	1.07477i	−1.36189			−2.34245	+	1.23001i	5.03146			1.34662	+	2.33241i	−1.52579	+	2.64275i
606.3	−0.945348	−	1.63739i	1.42149	−	2.46210i	−0.787365	+	1.36376i	−1.73847	−	3.01112i	−5.37522			2.58119	−	0.580920i	−0.804055			−2.54129	−	4.40164i	−3.28692	+	5.69311i
606.4	−0.830383	−	1.43827i	−1.16938	+	2.02543i	−0.379072	+	0.656572i	1.38656	+	2.40159i	3.88414			2.09572	+	1.61492i	−2.06243			−1.23491	−	2.13892i	2.30275	−	3.98848i
606.5	−0.614728	−	1.06474i	1.62503	−	2.81463i	0.244219	−	0.423000i	1.29373	+	2.24080i	−3.99581			0.255479	−	2.63339i	−3.05942			−3.78144	−	6.54965i	1.59058	−	2.75496i
606.6	−0.503676	−	0.872392i	0.297921	−	0.516014i	0.492622	−	0.853246i	0.793525	+	1.37442i	−0.600222			−2.26989	−	1.35927i	−3.00719			1.32249	+	2.29061i	0.799358	−	1.38453i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
11.b	odd	2	1	inner
77.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	847.2.e.j	✓	28
7.c	even	3	1	inner	847.2.e.j	✓	28
7.c	even	3	1		5929.2.a.cd		14
7.d	odd	6	1		5929.2.a.ce		14
11.b	odd	2	1	inner	847.2.e.j	✓	28
11.c	even	5	4		847.2.n.n		112
11.d	odd	10	4		847.2.n.n		112
77.h	odd	6	1	inner	847.2.e.j	✓	28
77.h	odd	6	1		5929.2.a.cd		14
77.i	even	6	1		5929.2.a.ce		14
77.m	even	15	4		847.2.n.n		112
77.o	odd	30	4		847.2.n.n		112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
847.2.e.j	✓	28	1.a	even	1	1	trivial
847.2.e.j	✓	28	7.c	even	3	1	inner
847.2.e.j	✓	28	11.b	odd	2	1	inner
847.2.e.j	✓	28	77.h	odd	6	1	inner
847.2.n.n		112	11.c	even	5	4
847.2.n.n		112	11.d	odd	10	4
847.2.n.n		112	77.m	even	15	4
847.2.n.n		112	77.o	odd	30	4
5929.2.a.cd		14	7.c	even	3	1
5929.2.a.cd		14	77.h	odd	6	1
5929.2.a.ce		14	7.d	odd	6	1
5929.2.a.ce		14	77.i	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^28 + 22*T2^26 + 297*T2^24 + 2556*T2^22 + 16162*T2^20 + 73980*T2^18 + 257822*T2^16 + 664026*T2^14 + 1295247*T2^12 + 1800146*T2^10 + 1823681*T2^8 + 1156119*T2^6 + 464164*T2^4 + 19305*T2^2 + 729