Newform orbit 804.2.e.b

• Label: 804.2.e.b
• 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	−1.40514	−	0.159923i	1.00000			1.94885	+	0.449428i			0.947080i	−1.40514	−	0.159923i	−0.0251164			−2.66654	−	0.943176i	1.00000			0.151460	−	1.33078i
535.2	−1.40514	+	0.159923i	1.00000			1.94885	−	0.449428i		−	0.947080i	−1.40514	+	0.159923i	−0.0251164			−2.66654	+	0.943176i	1.00000			0.151460	+	1.33078i
535.3	−1.36992	−	0.351182i	1.00000			1.75334	+	0.962180i		−	3.58679i	−1.36992	−	0.351182i	−4.07396			−2.06403	−	1.93385i	1.00000			−1.25962	+	4.91361i
535.4	−1.36992	+	0.351182i	1.00000			1.75334	−	0.962180i			3.58679i	−1.36992	+	0.351182i	−4.07396			−2.06403	+	1.93385i	1.00000			−1.25962	−	4.91361i
535.5	−1.34335	−	0.442044i	1.00000			1.60919	+	1.18764i			2.27075i	−1.34335	−	0.442044i	4.97497			−1.63672	−	2.30676i	1.00000			1.00377	−	3.05042i
535.6	−1.34335	+	0.442044i	1.00000			1.60919	−	1.18764i		−	2.27075i	−1.34335	+	0.442044i	4.97497			−1.63672	+	2.30676i	1.00000			1.00377	+	3.05042i
535.7	−1.16630	−	0.799841i	1.00000			0.720507	+	1.86571i		−	0.864540i	−1.16630	−	0.799841i	−1.73467			0.651944	−	2.75227i	1.00000			−0.691495	+	1.00831i
535.8	−1.16630	+	0.799841i	1.00000			0.720507	−	1.86571i			0.864540i	−1.16630	+	0.799841i	−1.73467			0.651944	+	2.75227i	1.00000			−0.691495	−	1.00831i
535.9	−1.01423	−	0.985563i	1.00000			0.0573324	+	1.99918i			1.66897i	−1.01423	−	0.985563i	0.193635			1.91217	−	2.08413i	1.00000			1.64487	−	1.69272i
535.10	−1.01423	+	0.985563i	1.00000			0.0573324	−	1.99918i		−	1.66897i	−1.01423	+	0.985563i	0.193635			1.91217	+	2.08413i	1.00000			1.64487	+	1.69272i
535.11	−0.707921	−	1.22427i	1.00000			−0.997695	+	1.73338i		−	3.14565i	−0.707921	−	1.22427i	1.38193			2.82842	−	0.00564312i	1.00000			−3.85114	+	2.22687i
535.12	−0.707921	+	1.22427i	1.00000			−0.997695	−	1.73338i			3.14565i	−0.707921	+	1.22427i	1.38193			2.82842	+	0.00564312i	1.00000			−3.85114	−	2.22687i
535.13	−0.530406	−	1.31098i	1.00000			−1.43734	+	1.39070i			2.85552i	−0.530406	−	1.31098i	4.91599			2.58556	+	1.14669i	1.00000			3.74353	−	1.51458i
535.14	−0.530406	+	1.31098i	1.00000			−1.43734	−	1.39070i		−	2.85552i	−0.530406	+	1.31098i	4.91599			2.58556	−	1.14669i	1.00000			3.74353	+	1.51458i
535.15	−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
535.16	−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
535.17	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	0.271785	−	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
535.20	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

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.b	yes	34
4.b	odd	2	1		804.2.e.a	✓	34
67.b	odd	2	1		804.2.e.a	✓	34
268.d	even	2	1	inner	804.2.e.b	yes	34

    

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

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