Newform orbit 804.2.c.b

• Label: 804.2.c.b
• Level: $804$
• Weight: $2$
• Character orbit: 804.c
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(128\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 4 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
671.1	−1.41002	−	0.108878i	1.62789	+	0.591579i	1.97629	+	0.307041i		−	3.04949i	−2.23094	−	1.01138i			0.278695i	−2.75317	−	0.648108i	2.30007	+	1.92605i	−0.332024	+	4.29983i
671.2	−1.41002	+	0.108878i	1.62789	−	0.591579i	1.97629	−	0.307041i			3.04949i	−2.23094	+	1.01138i		−	0.278695i	−2.75317	+	0.648108i	2.30007	−	1.92605i	−0.332024	−	4.29983i
671.3	−1.40995	−	0.109793i	−0.420694	−	1.68018i	1.97589	+	0.309604i		−	0.933302i	0.408683	+	2.41516i			1.91511i	−2.75191	−	0.653464i	−2.64603	+	1.41369i	−0.102470	+	1.31591i
671.4	−1.40995	+	0.109793i	−0.420694	+	1.68018i	1.97589	−	0.309604i			0.933302i	0.408683	−	2.41516i		−	1.91511i	−2.75191	+	0.653464i	−2.64603	−	1.41369i	−0.102470	−	1.31591i
671.5	−1.40536	−	0.157993i	1.10632	+	1.33268i	1.95008	+	0.444075i			1.25971i	−1.34423	−	2.04769i		−	4.86310i	−2.67040	−	0.932185i	−0.552098	+	2.94876i	0.199026	−	1.77035i
671.6	−1.40536	+	0.157993i	1.10632	−	1.33268i	1.95008	−	0.444075i		−	1.25971i	−1.34423	+	2.04769i			4.86310i	−2.67040	+	0.932185i	−0.552098	−	2.94876i	0.199026	+	1.77035i
671.7	−1.39093	−	0.255554i	1.65649	−	0.506011i	1.86938	+	0.710916i		−	2.60371i	−2.43338	+	0.280506i		−	2.89039i	−2.41851	−	1.46656i	2.48790	−	1.67640i	−0.665387	+	3.62158i
671.8	−1.39093	+	0.255554i	1.65649	+	0.506011i	1.86938	−	0.710916i			2.60371i	−2.43338	−	0.280506i			2.89039i	−2.41851	+	1.46656i	2.48790	+	1.67640i	−0.665387	−	3.62158i
671.9	−1.38271	−	0.296856i	−1.71448	+	0.246081i	1.82375	+	0.820930i		−	2.85244i	2.44367	+	0.168697i		−	2.12311i	−2.27802	−	1.67650i	2.87889	−	0.843802i	−0.846765	+	3.94409i
671.10	−1.38271	+	0.296856i	−1.71448	−	0.246081i	1.82375	−	0.820930i			2.85244i	2.44367	−	0.168697i			2.12311i	−2.27802	+	1.67650i	2.87889	+	0.843802i	−0.846765	−	3.94409i
671.11	−1.37515	−	0.330103i	0.355099	+	1.69526i	1.78206	+	0.907880i			0.0453328i	0.0712965	−	2.44845i			2.82199i	−2.15091	−	1.83673i	−2.74781	+	1.20397i	0.0149645	−	0.0623394i
671.12	−1.37515	+	0.330103i	0.355099	−	1.69526i	1.78206	−	0.907880i		−	0.0453328i	0.0712965	+	2.44845i		−	2.82199i	−2.15091	+	1.83673i	−2.74781	−	1.20397i	0.0149645	+	0.0623394i
671.13	−1.36593	−	0.366392i	−0.973809	−	1.43237i	1.73151	+	1.00093i			2.42154i	0.805342	+	2.31331i		−	1.75439i	−1.99839	−	2.00161i	−1.10339	+	2.78972i	0.887230	−	3.30764i
671.14	−1.36593	+	0.366392i	−0.973809	+	1.43237i	1.73151	−	1.00093i		−	2.42154i	0.805342	−	2.31331i			1.75439i	−1.99839	+	2.00161i	−1.10339	−	2.78972i	0.887230	+	3.30764i
671.15	−1.28652	−	0.587251i	1.29382	−	1.15153i	1.31027	+	1.51102i			1.10955i	−2.34077	+	0.721672i		−	0.262541i	−0.798342	−	2.71342i	0.347946	−	2.97975i	0.651585	−	1.42746i
671.16	−1.28652	+	0.587251i	1.29382	+	1.15153i	1.31027	−	1.51102i		−	1.10955i	−2.34077	−	0.721672i			0.262541i	−0.798342	+	2.71342i	0.347946	+	2.97975i	0.651585	+	1.42746i
671.17	−1.27950	−	0.602386i	−1.48757	+	0.887204i	1.27426	+	1.54151i			3.40681i	2.43779	−	0.239089i		−	4.07118i	−0.701842	−	2.73997i	1.42574	−	2.63956i	2.05221	−	4.35902i
671.18	−1.27950	+	0.602386i	−1.48757	−	0.887204i	1.27426	−	1.54151i		−	3.40681i	2.43779	+	0.239089i			4.07118i	−0.701842	+	2.73997i	1.42574	+	2.63956i	2.05221	+	4.35902i
671.19	−1.27439	−	0.613127i	−1.63926	−	0.559297i	1.24815	+	1.56273i		−	0.350980i	1.74615	+	1.71784i			2.98214i	−0.632485	−	2.75680i	2.37437	+	1.83367i	−0.215195	+	0.447286i
671.20	−1.27439	+	0.613127i	−1.63926	+	0.559297i	1.24815	−	1.56273i			0.350980i	1.74615	−	1.71784i		−	2.98214i	−0.632485	+	2.75680i	2.37437	−	1.83367i	−0.215195	−	0.447286i

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	804.2.c.b	✓	128
3.b	odd	2	1	inner	804.2.c.b	✓	128
4.b	odd	2	1	inner	804.2.c.b	✓	128
12.b	even	2	1	inner	804.2.c.b	✓	128

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
804.2.c.b	✓	128	1.a	even	1	1	trivial
804.2.c.b	✓	128	3.b	odd	2	1	inner
804.2.c.b	✓	128	4.b	odd	2	1	inner
804.2.c.b	✓	128	12.b	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^64 + 194*T5^62 + 17750*T5^60 + 1019156*T5^58 + 41221643*T5^56 + 1249820098*T5^54 + 29513048355*T5^52 + 556719734300*T5^50 + 8537531180071*T5^48 + 107763597892976*T5^46 + 1129362937130088*T5^44 + 9885398287443732*T5^42 + 72539927105754182*T5^40 + 447101984159466072*T5^38 + 2315287865886050234*T5^36 + 10061189607304129900*T5^34 + 36592966865055637663*T5^32 + 110940565060532308030*T5^30 + 278822891235854264082*T5^28 + 576797521556760627640*T5^26 + 973441540224624605159*T5^24 + 1325555081424199858054*T5^22 + 1436567498802916822427*T5^20 + 1217743221552580863832*T5^18 + 789413250037032497009*T5^16 + 379717658947189439216*T5^14 + 129956864947528940984*T5^12 + 29786212888957466688*T5^10 + 4177854703835045008*T5^8 + 313510150925573632*T5^6 + 10334860325804032*T5^4 + 111139293659136*T5^2 + 187398946816