Newform orbit 804.2.e.a

• Label: 804.2.e.a
• Level: $804$
• Weight: $2$
• Character orbit: 804.e
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $34$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(34\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 34 q^{3} + 2 q^{4} - 4 q^{7} + 6 q^{8} + 34 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
535.1	−1.41091	−	0.0966035i	−1.00000			1.98134	+	0.272598i		−	2.83991i	1.41091	+	0.0966035i	0.830707			−2.76915	−	0.576015i	1.00000			−0.274345	+	4.00685i
535.2	−1.41091	+	0.0966035i	−1.00000			1.98134	−	0.272598i			2.83991i	1.41091	−	0.0966035i	0.830707			−2.76915	+	0.576015i	1.00000			−0.274345	−	4.00685i
535.3	−1.35079	−	0.418775i	−1.00000			1.64926	+	1.13135i		−	1.44885i	1.35079	+	0.418775i	2.69839			−1.75401	−	2.21888i	1.00000			−0.606742	+	1.95709i
535.4	−1.35079	+	0.418775i	−1.00000			1.64926	−	1.13135i			1.44885i	1.35079	−	0.418775i	2.69839			−1.75401	+	2.21888i	1.00000			−0.606742	−	1.95709i
535.5	−1.31376	−	0.523490i	−1.00000			1.45192	+	1.37548i			1.71706i	1.31376	+	0.523490i	−3.43662			−1.18741	−	2.56711i	1.00000			0.898862	−	2.25579i
535.6	−1.31376	+	0.523490i	−1.00000			1.45192	−	1.37548i		−	1.71706i	1.31376	−	0.523490i	−3.43662			−1.18741	+	2.56711i	1.00000			0.898862	+	2.25579i
535.7	−0.956986	−	1.04124i	−1.00000			−0.168355	+	1.99290i			1.36325i	0.956986	+	1.04124i	−1.56640			2.23620	−	1.73188i	1.00000			1.41947	−	1.30461i
535.8	−0.956986	+	1.04124i	−1.00000			−0.168355	−	1.99290i		−	1.36325i	0.956986	−	1.04124i	−1.56640			2.23620	+	1.73188i	1.00000			1.41947	+	1.30461i
535.9	−0.946315	−	1.05095i	−1.00000			−0.208975	+	1.98905i		−	4.42921i	0.946315	+	1.05095i	−2.81753			2.28814	−	1.66265i	1.00000			−4.65486	+	4.19143i
535.10	−0.946315	+	1.05095i	−1.00000			−0.208975	−	1.98905i			4.42921i	0.946315	−	1.05095i	−2.81753			2.28814	+	1.66265i	1.00000			−4.65486	−	4.19143i
535.11	−0.921850	−	1.07247i	−1.00000			−0.300386	+	1.97731i		−	1.54735i	0.921850	+	1.07247i	4.78228			2.39752	−	1.50063i	1.00000			−1.65949	+	1.42642i
535.12	−0.921850	+	1.07247i	−1.00000			−0.300386	−	1.97731i			1.54735i	0.921850	−	1.07247i	4.78228			2.39752	+	1.50063i	1.00000			−1.65949	−	1.42642i
535.13	−0.483740	−	1.32891i	−1.00000			−1.53199	+	1.28569i			3.80010i	0.483740	+	1.32891i	0.504221			2.44965	+	1.41393i	1.00000			5.04998	−	1.83826i
535.14	−0.483740	+	1.32891i	−1.00000			−1.53199	−	1.28569i		−	3.80010i	0.483740	−	1.32891i	0.504221			2.44965	−	1.41393i	1.00000			5.04998	+	1.83826i
535.15	−0.271785	−	1.38785i	−1.00000			−1.85227	+	0.754394i			0.299779i	0.271785	+	1.38785i	−1.88035			1.55041	+	2.36564i	1.00000			0.416048	−	0.0814753i
535.16	−0.271785	+	1.38785i	−1.00000			−1.85227	−	0.754394i		−	0.299779i	0.271785	−	1.38785i	−1.88035			1.55041	−	2.36564i	1.00000			0.416048	+	0.0814753i
535.17	−0.219472	−	1.39708i	−1.00000			−1.90366	+	0.613241i		−	3.15947i	0.219472	+	1.39708i	0.761043			1.27455	+	2.52498i	1.00000			−4.41403	+	0.693416i
535.18	−0.219472	+	1.39708i	−1.00000			−1.90366	−	0.613241i			3.15947i	0.219472	−	1.39708i	0.761043			1.27455	−	2.52498i	1.00000			−4.41403	−	0.693416i
535.19	0.338334	−	1.37315i	−1.00000			−1.77106	−	0.929165i			0.0609697i	−0.338334	+	1.37315i	3.75704			−1.87509	+	2.11755i	1.00000			0.0837203	+	0.0206282i
535.20	0.338334	+	1.37315i	−1.00000			−1.77106	+	0.929165i		−	0.0609697i	−0.338334	−	1.37315i	3.75704			−1.87509	−	2.11755i	1.00000			0.0837203	−	0.0206282i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
268.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	804.2.e.a	✓	34
4.b	odd	2	1		804.2.e.b	yes	34
67.b	odd	2	1		804.2.e.b	yes	34
268.d	even	2	1	inner	804.2.e.a	✓	34

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
804.2.e.a	✓	34	1.a	even	1	1	trivial
804.2.e.a	✓	34	268.d	even	2	1	inner
804.2.e.b	yes	34	4.b	odd	2	1
804.2.e.b	yes	34	67.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^17 + 2*T7^16 - 69*T7^15 - 120*T7^14 + 1800*T7^13 + 2696*T7^12 - 22246*T7^11 - 28504*T7^10 + 135609*T7^9 + 141994*T7^8 - 398757*T7^7 - 291464*T7^6 + 554454*T7^5 + 160356*T7^4 - 332248*T7^3 + 47776*T7^2 + 19392*T7 - 512