Newform orbit 285.2.k.c

• Label: 285.2.k.c
• Level: $285$
• Weight: $2$
• Character orbit: 285.k
• Analytic conductor: $2.276$
• Analytic rank: $0$
• Dimension: $28$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	285.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.27573645761\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 28 * q - 2 * q^3 - 12 * q^6 - 8 * q^10 + 34 * q^12 + 8 * q^13 - 14 * q^15 - 20 * q^16 - 24 * q^18 - 4 * q^21 - 32 * q^22 + 8 * q^25 + 22 * q^27 - 28 * q^28 + 12 * q^30 + 72 * q^31 - 84 * q^36 - 12 * q^37 + 20 * q^40 + 48 * q^42 - 12 * q^43 - 52 * q^45 + 8 * q^46 + 46 * q^48 + 28 * q^51 - 76 * q^52 + 104 * q^55 + 2 * q^57 - 60 * q^58 - 22 * q^60 + 96 * q^61 + 56 * q^63 - 28 * q^66 - 72 * q^67 + 68 * q^70 + 20 * q^72 - 72 * q^73 + 2 * q^75 - 36 * q^76 + 76 * q^78 - 100 * q^81 - 116 * q^82 - 44 * q^85 + 4 * q^87 + 60 * q^88 - 36 * q^90 - 80 * q^91 + 52 * q^93 - 80 * q^96

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
77.1	−1.82223	+	1.82223i	−1.28293	+	1.16366i		−	4.64103i	1.53042	+	1.63028i	0.217337	−	4.45823i	−1.83721	−	1.83721i	4.81255	+	4.81255i	0.291804	−	2.98577i	−5.75952	−	0.181957i
77.2	−1.60748	+	1.60748i	−1.46845	+	0.918505i		−	3.16797i	0.294779	−	2.21655i	0.884027	−	3.83698i	3.56650	+	3.56650i	1.87749	+	1.87749i	1.31270	−	2.69756i	3.08921	+	4.03691i
77.3	−1.53127	+	1.53127i	1.71677	−	0.229557i		−	2.68957i	−1.36334	−	1.77237i	−2.27733	+	2.98035i	−1.27504	−	1.27504i	1.05592	+	1.05592i	2.89461	−	0.788193i	4.80162	+	0.626344i
77.4	−1.48042	+	1.48042i	1.05938	+	1.37030i		−	2.38331i	2.23587	−	0.0297943i	−3.59695	−	0.460300i	−1.93250	−	1.93250i	0.567468	+	0.567468i	−0.755445	+	2.90333i	−3.26593	+	3.35414i
77.5	−0.913256	+	0.913256i	−0.951938	−	1.44700i			0.331928i	2.10088	−	0.765698i	2.19085	+	0.452119i	0.194936	+	0.194936i	−2.12965	−	2.12965i	−1.18763	+	2.75491i	−1.21936	+	2.61792i
77.6	−0.473364	+	0.473364i	1.62852	−	0.589855i			1.55185i	−1.83078	+	1.28384i	−0.491666	+	1.05010i	1.99868	+	1.99868i	−1.68132	−	1.68132i	2.30414	−	1.92118i	0.258900	−	1.47435i
77.7	−0.0307908	+	0.0307908i	−1.45499	−	0.939687i			1.99810i	−1.20318	−	1.88477i	0.0737341	−	0.0158665i	−0.715376	−	0.715376i	−0.123105	−	0.123105i	1.23398	+	2.73447i	0.0950807	+	0.0209865i
77.8	0.0307908	−	0.0307908i	0.939687	+	1.45499i			1.99810i	1.20318	+	1.88477i	0.0737341	+	0.0158665i	−0.715376	−	0.715376i	0.123105	+	0.123105i	−1.23398	+	2.73447i	0.0950807	+	0.0209865i
77.9	0.473364	−	0.473364i	0.589855	−	1.62852i			1.55185i	1.83078	−	1.28384i	−0.491666	−	1.05010i	1.99868	+	1.99868i	1.68132	+	1.68132i	−2.30414	−	1.92118i	0.258900	−	1.47435i
77.10	0.913256	−	0.913256i	1.44700	+	0.951938i			0.331928i	−2.10088	+	0.765698i	2.19085	−	0.452119i	0.194936	+	0.194936i	2.12965	+	2.12965i	1.18763	+	2.75491i	−1.21936	+	2.61792i
77.11	1.48042	−	1.48042i	−1.37030	−	1.05938i		−	2.38331i	−2.23587	+	0.0297943i	−3.59695	+	0.460300i	−1.93250	−	1.93250i	−0.567468	−	0.567468i	0.755445	+	2.90333i	−3.26593	+	3.35414i
77.12	1.53127	−	1.53127i	0.229557	−	1.71677i		−	2.68957i	1.36334	+	1.77237i	−2.27733	−	2.98035i	−1.27504	−	1.27504i	−1.05592	−	1.05592i	−2.89461	−	0.788193i	4.80162	+	0.626344i
77.13	1.60748	−	1.60748i	−0.918505	+	1.46845i		−	3.16797i	−0.294779	+	2.21655i	0.884027	+	3.83698i	3.56650	+	3.56650i	−1.87749	−	1.87749i	−1.31270	−	2.69756i	3.08921	+	4.03691i
77.14	1.82223	−	1.82223i	−1.16366	+	1.28293i		−	4.64103i	−1.53042	−	1.63028i	0.217337	+	4.45823i	−1.83721	−	1.83721i	−4.81255	−	4.81255i	−0.291804	−	2.98577i	−5.75952	−	0.181957i
248.1	−1.82223	−	1.82223i	−1.28293	−	1.16366i			4.64103i	1.53042	−	1.63028i	0.217337	+	4.45823i	−1.83721	+	1.83721i	4.81255	−	4.81255i	0.291804	+	2.98577i	−5.75952	+	0.181957i
248.2	−1.60748	−	1.60748i	−1.46845	−	0.918505i			3.16797i	0.294779	+	2.21655i	0.884027	+	3.83698i	3.56650	−	3.56650i	1.87749	−	1.87749i	1.31270	+	2.69756i	3.08921	−	4.03691i
248.3	−1.53127	−	1.53127i	1.71677	+	0.229557i			2.68957i	−1.36334	+	1.77237i	−2.27733	−	2.98035i	−1.27504	+	1.27504i	1.05592	−	1.05592i	2.89461	+	0.788193i	4.80162	−	0.626344i
248.4	−1.48042	−	1.48042i	1.05938	−	1.37030i			2.38331i	2.23587	+	0.0297943i	−3.59695	+	0.460300i	−1.93250	+	1.93250i	0.567468	−	0.567468i	−0.755445	−	2.90333i	−3.26593	−	3.35414i
248.5	−0.913256	−	0.913256i	−0.951938	+	1.44700i		−	0.331928i	2.10088	+	0.765698i	2.19085	−	0.452119i	0.194936	−	0.194936i	−2.12965	+	2.12965i	−1.18763	−	2.75491i	−1.21936	−	2.61792i
248.6	−0.473364	−	0.473364i	1.62852	+	0.589855i		−	1.55185i	−1.83078	−	1.28384i	−0.491666	−	1.05010i	1.99868	−	1.99868i	−1.68132	+	1.68132i	2.30414	+	1.92118i	0.258900	+	1.47435i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	285.2.k.c	✓	28
3.b	odd	2	1	inner	285.2.k.c	✓	28
5.c	odd	4	1	inner	285.2.k.c	✓	28
15.e	even	4	1	inner	285.2.k.c	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
285.2.k.c	✓	28	1.a	even	1	1	trivial
285.2.k.c	✓	28	3.b	odd	2	1	inner
285.2.k.c	✓	28	5.c	odd	4	1	inner
285.2.k.c	✓	28	15.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} + 115T_{2}^{24} + 4853T_{2}^{20} + 91999T_{2}^{16} + 734303T_{2}^{12} + 1528685T_{2}^{8} + 278139T_{2}^{4} + 1 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\).