Newform orbit 804.2.l.a

• Label: 804.2.l.a
• Level: $804$
• Weight: $2$
• Character orbit: 804.l
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(132\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 2 q^{4} - 3 q^{6} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
431.1	−1.41407	−	0.0202093i	−1.20523	−	1.24395i	1.99918	+	0.0571548i		−	3.26426i	1.67914	+	1.78339i	−0.138760	+	0.0801133i	−2.82583	−	0.121223i	−0.0948278	+	2.99850i	−0.0659686	+	4.61589i
431.2	−1.41390	+	0.0296860i	−1.00788	+	1.40861i	1.99824	−	0.0839461i		−	0.377411i	1.38323	−	2.02155i	−3.78586	+	2.18577i	−2.82282	+	0.178011i	−0.968338	−	2.83942i	0.0112038	+	0.533622i
431.3	−1.41324	−	0.0525133i	−1.72604	+	0.144124i	1.99448	+	0.148428i			3.64727i	2.44688	−	0.113041i	−0.289615	+	0.167210i	−2.81089	−	0.314501i	2.95846	−	0.497529i	0.191530	−	5.15446i
431.4	−1.41305	+	0.0574626i	1.12603	−	1.31608i	1.99340	−	0.162395i			3.12863i	−1.51551	+	1.92438i	−3.36179	+	1.94093i	−2.80743	+	0.344017i	−0.464114	−	2.96388i	−0.179779	−	4.42090i
431.5	−1.41271	+	0.0652340i	0.500155	+	1.65827i	1.99149	−	0.184313i			0.631752i	−0.814749	−	2.31002i	2.68471	−	1.55002i	−2.80137	+	0.390294i	−2.49969	+	1.65878i	−0.0412117	−	0.892481i
431.6	−1.40119	−	0.191474i	−0.473255	−	1.66614i	1.92668	+	0.536583i		−	0.531500i	0.344098	+	2.42520i	0.910512	−	0.525684i	−2.59690	−	1.12076i	−2.55206	+	1.57702i	−0.101768	+	0.744733i
431.7	−1.39862	+	0.209424i	1.33282	−	1.10616i	1.91228	−	0.585811i		−	2.33442i	−1.63245	+	1.82622i	1.17857	−	0.680446i	−2.55188	+	1.21981i	0.552819	−	2.94863i	0.488885	+	3.26497i
431.8	−1.39364	+	0.240373i	−1.35771	+	1.07547i	1.88444	−	0.669985i		−	4.20267i	1.63364	−	1.82517i	3.17693	−	1.83420i	−2.46518	+	1.38668i	0.686739	−	2.92034i	1.01021	+	5.85699i
431.9	−1.39349	−	0.241196i	1.40279	+	1.01597i	1.88365	+	0.672210i			2.69591i	−1.70973	−	1.75409i	−2.30125	+	1.32862i	−2.46272	−	1.39105i	0.935617	+	2.85037i	0.650241	−	3.75673i
431.10	−1.38134	−	0.303157i	0.740294	+	1.56587i	1.81619	+	0.837524i		−	3.95379i	−0.547892	−	2.38743i	−2.00371	+	1.15684i	−2.25488	−	1.70749i	−1.90393	+	2.31842i	−1.19862	+	5.46152i
431.11	−1.37942	−	0.311775i	0.422580	−	1.67971i	1.80559	+	0.860137i			2.80374i	−1.10661	+	2.18527i	4.57629	−	2.64212i	−2.22250	−	1.74943i	−2.64285	−	1.41963i	0.874136	−	3.86753i
431.12	−1.37670	+	0.323579i	1.62001	+	0.612827i	1.79059	−	0.890942i			0.604920i	−2.42857	−	0.319475i	2.48563	−	1.43508i	−2.17681	+	1.80596i	2.24889	+	1.98558i	−0.195740	−	0.832792i
431.13	−1.35424	−	0.407489i	1.72726	−	0.128753i	1.66791	+	1.10367i		−	1.48944i	−2.39158	−	0.529476i	−2.25234	+	1.30039i	−1.80900	−	2.17428i	2.96685	−	0.444780i	−0.606929	+	2.01705i
431.14	−1.32505	+	0.494196i	−0.696446	−	1.58586i	1.51154	−	1.30967i			1.99192i	1.70656	+	1.75718i	−1.23529	+	0.713198i	−1.35564	+	2.48239i	−2.02993	+	2.20894i	−0.984400	−	2.63941i
431.15	−1.31532	+	0.519539i	−1.71904	−	0.211862i	1.46016	−	1.36673i		−	0.428467i	2.37117	−	0.614444i	−1.49589	+	0.863651i	−1.21051	+	2.55630i	2.91023	+	0.728399i	0.222606	+	0.563574i
431.16	−1.31048	−	0.531632i	−1.38818	+	1.03584i	1.43473	+	1.39339i			1.57106i	2.36987	−	0.619443i	0.285742	−	0.164973i	−1.13942	−	2.58877i	0.854091	−	2.87585i	0.835229	−	2.05885i
431.17	−1.27761	−	0.606402i	−0.369021	+	1.69228i	1.26455	+	1.54948i		−	0.628413i	1.49767	−	1.93830i	1.58089	−	0.912727i	−0.675992	−	2.74646i	−2.72765	−	1.24898i	−0.381071	+	0.802864i
431.18	−1.25248	−	0.656737i	−1.29766	−	1.14721i	1.13739	+	1.64509i			2.89696i	0.871868	+	2.28907i	−1.32141	+	0.762919i	−0.344163	−	2.80741i	0.367820	+	2.97737i	1.90254	−	3.62837i
431.19	−1.24697	+	0.667126i	1.63732	+	0.564973i	1.10988	−	1.66378i		−	1.17393i	−2.41860	+	0.387791i	−2.79224	+	1.61210i	−0.274046	+	2.81512i	2.36161	+	1.85008i	0.783159	+	1.46386i
431.20	−1.23331	−	0.692067i	−1.71207	−	0.262337i	1.04209	+	1.70706i		−	2.53500i	1.92995	+	1.50841i	−3.95663	+	2.28436i	−0.103813	−	2.82652i	2.86236	+	0.898278i	−1.75439	+	3.12643i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner
67.c	even	3	1	inner
201.g	odd	6	1	inner
268.g	odd	6	1	inner
804.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	804.2.l.a	✓	264
3.b	odd	2	1	inner	804.2.l.a	✓	264
4.b	odd	2	1	inner	804.2.l.a	✓	264
12.b	even	2	1	inner	804.2.l.a	✓	264
67.c	even	3	1	inner	804.2.l.a	✓	264
201.g	odd	6	1	inner	804.2.l.a	✓	264
268.g	odd	6	1	inner	804.2.l.a	✓	264
804.l	even	6	1	inner	804.2.l.a	✓	264

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
804.2.l.a	✓	264	1.a	even	1	1	trivial
804.2.l.a	✓	264	3.b	odd	2	1	inner
804.2.l.a	✓	264	4.b	odd	2	1	inner
804.2.l.a	✓	264	12.b	even	2	1	inner
804.2.l.a	✓	264	67.c	even	3	1	inner
804.2.l.a	✓	264	201.g	odd	6	1	inner
804.2.l.a	✓	264	268.g	odd	6	1	inner
804.2.l.a	✓	264	804.l	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(804, [\chi])\).