Newform orbit 409.2.j.a

• Label: 409.2.j.a
• Level: $409$
• Weight: $2$
• Character orbit: 409.j
• Analytic conductor: $3.266$
• Analytic rank: $0$
• Dimension: $512$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.j (of order \(34\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.26588144267\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(32\) over \(\Q(\zeta_{34})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{34}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 512 * q - 15 * q^2 - 15 * q^3 - 73 * q^4 - 17 * q^5 + 17 * q^6 - 5 * q^8 - 31 * q^9 - 27 * q^10 + 34 * q^11 + 40 * q^12 - 17 * q^13 + 17 * q^14 - 13 * q^15 - 53 * q^16 - 9 * q^17 - 4 * q^18 - 17 * q^19 - 21 * q^20 - 68 * q^21 - 17 * q^22 - 31 * q^23 - 6 * q^24 - 13 * q^25 - 17 * q^26 - 27 * q^27 + 51 * q^28 - 17 * q^29 + 19 * q^30 - 103 * q^32 - 17 * q^33 - 27 * q^34 + 68 * q^35 - 81 * q^36 - 17 * q^37 - 119 * q^39 + 120 * q^40 - 87 * q^41 + 221 * q^42 + 85 * q^43 - 17 * q^44 + 3 * q^45 - 89 * q^46 - 17 * q^47 - 57 * q^48 - 258 * q^49 - 35 * q^50 + 10 * q^51 - 17 * q^52 - 152 * q^53 - 116 * q^54 - 153 * q^55 + 102 * q^56 - 17 * q^57 - 17 * q^58 - 17 * q^59 + 125 * q^60 - 34 * q^61 + 17 * q^62 - 17 * q^63 + 107 * q^64 - 17 * q^65 - 17 * q^67 - 13 * q^68 - 55 * q^69 - 34 * q^70 - 63 * q^71 - 43 * q^72 - 17 * q^73 + 85 * q^74 + 9 * q^75 + 17 * q^76 + 98 * q^77 + 51 * q^78 - 17 * q^79 + 295 * q^80 + 5 * q^81 - 67 * q^82 - 17 * q^83 - 204 * q^84 - 105 * q^85 - 187 * q^86 + 119 * q^87 - 221 * q^88 - 61 * q^89 + 296 * q^90 + 104 * q^91 - 99 * q^92 + 34 * q^93 - 17 * q^94 - 17 * q^95 - 94 * q^96 - 17 * q^97 + 203 * q^98 + 187 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
64.1	−2.51626	+	0.974803i	−1.98811	+	2.63269i	3.90328	−	3.55831i	1.78186	−	0.333087i	2.43625	−	8.56254i		−	2.19975i	−3.94738	+	7.92740i	−2.15746	−	7.58270i	−4.15892	+	2.57509i
64.2	−2.36165	+	0.914908i	−0.527400	+	0.698390i	3.26232	−	2.97399i	−2.65297	+	0.495926i	0.606571	−	2.13188i			2.86569i	−2.72569	+	5.47393i	0.611391	+	2.14882i	5.81166	−	3.59843i
64.3	−2.31215	+	0.895732i	−0.0552629	+	0.0731799i	3.06569	−	2.79474i	2.34376	−	0.438125i	0.0622265	−	0.218704i		−	0.578751i	−2.37449	+	4.76862i	0.818688	+	2.87739i	−5.02669	+	3.11240i
64.4	−2.10908	+	0.817061i	0.641756	−	0.849822i	2.30260	−	2.09910i	−1.89686	+	0.354584i	−0.659157	+	2.31670i		−	5.06839i	−1.12492	+	2.25915i	0.510642	+	1.79472i	3.71091	−	2.29770i
64.5	−1.93089	+	0.748031i	1.48697	−	1.96907i	1.69077	−	1.54134i	−1.05732	+	0.197648i	−1.39826	+	4.91436i			1.01192i	−0.265723	+	0.533644i	−0.845160	−	2.97043i	1.89373	−	1.17255i
64.6	−1.73754	+	0.673128i	1.84458	−	2.44262i	1.08794	−	0.991789i	3.58534	−	0.670216i	−1.56085	+	5.48581i		−	2.17800i	0.438410	−	0.880446i	−1.74293	−	6.12578i	−5.77854	+	3.57792i
64.7	−1.68032	+	0.650958i	−1.10732	+	1.46632i	0.921699	−	0.840240i	1.04292	−	0.194955i	0.906126	−	3.18470i			0.706453i	0.604657	−	1.21432i	−0.102966	−	0.361886i	−1.62553	+	1.00648i
64.8	−1.61114	+	0.624159i	−1.75835	+	2.32844i	0.728183	−	0.663826i	−2.61321	+	0.488494i	1.37964	−	4.84893i			0.331347i	0.781437	−	1.56934i	−1.50882	−	5.30295i	3.90536	−	2.41809i
64.9	−1.38122	+	0.535088i	0.903233	−	1.19607i	0.143433	−	0.130756i	0.194887	−	0.0364307i	−0.607560	+	2.13535i			4.67077i	1.19235	−	2.39456i	0.206226	+	0.724809i	−0.249688	+	0.154600i
64.10	−1.15769	+	0.448492i	−1.33116	+	1.76274i	−0.338915	+	0.308961i	3.26993	−	0.611255i	0.750495	−	2.63772i			5.08626i	1.36059	−	2.73242i	−0.514272	−	1.80748i	−3.51142	+	2.17418i
64.11	−1.13351	+	0.439126i	−0.140594	+	0.186177i	−0.385994	+	0.351880i	−0.776111	+	0.145080i	0.0776107	−	0.272773i		−	1.96808i	1.36669	−	2.74468i	0.806094	+	2.83313i	0.816025	−	0.505261i
64.12	−0.673140	+	0.260776i	0.685945	−	0.908338i	−1.09290	+	0.996314i	2.63797	−	0.493122i	−0.224864	+	0.790317i		−	1.09699i	1.11941	−	2.24808i	0.466432	+	1.63934i	−1.64713	+	1.01986i
64.13	−0.627577	+	0.243125i	1.85147	−	2.45174i	−1.14327	+	1.04223i	−2.05362	+	0.383888i	−0.565861	+	1.98879i		−	0.753508i	1.06409	−	2.13697i	−1.76211	−	6.19316i	1.19547	−	0.740206i
64.14	−0.592730	+	0.229625i	0.225892	−	0.299130i	−1.17942	+	1.07518i	−3.62989	+	0.678544i	−0.0652055	+	0.229174i		−	1.54949i	1.01886	−	2.04614i	0.782538	+	2.75033i	1.99574	−	1.23571i
64.15	−0.447123	+	0.173216i	−1.70817	+	2.26198i	−1.30810	+	1.19249i	3.21793	−	0.601535i	0.371948	−	1.30726i		−	4.70612i	0.805788	−	1.61824i	−1.37772	−	4.84219i	−1.33461	+	0.826357i
64.16	−0.140619	+	0.0544762i	−0.638343	+	0.845302i	−1.46121	+	1.33207i	1.10200	−	0.205999i	0.0437144	−	0.153640i			0.848915i	0.267345	−	0.536902i	0.513934	+	1.80629i	−0.143740	+	0.0890001i
64.17	0.0629738	−	0.0243962i	−1.54025	+	2.03962i	−1.47465	+	1.34432i	−2.58595	+	0.483398i	−0.0472364	+	0.166019i			3.05952i	−0.120273	+	0.241541i	−0.966689	−	3.39756i	−0.151054	+	0.0935287i
64.18	0.0965931	−	0.0374204i	−1.37988	+	1.82726i	−1.47009	+	1.34016i	−1.91105	+	0.357238i	−0.0649104	+	0.228137i		−	4.66124i	−0.184197	+	0.369918i	−0.613815	−	2.15734i	−0.171227	+	0.106019i
64.19	0.182521	−	0.0707092i	1.16489	−	1.54257i	−1.44970	+	1.32158i	−2.32205	+	0.434067i	0.103544	−	0.363920i			3.16183i	−0.345651	+	0.694161i	−0.201551	−	0.708377i	−0.393132	+	0.243417i
64.20	0.568337	−	0.220175i	0.666600	−	0.882721i	−1.20349	+	1.09712i	4.10071	−	0.766555i	0.184500	−	0.648452i			2.73732i	−0.985778	+	1.97971i	0.486148	+	1.70863i	2.16181	−	1.33853i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
409.j	even	34	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	409.2.j.a	✓	512
409.j	even	34	1	inner	409.2.j.a	✓	512

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
409.2.j.a	✓	512	1.a	even	1	1	trivial
409.2.j.a	✓	512	409.j	even	34	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(409, [\chi])\).