Newform orbit 539.2.p.a

• Label: 539.2.p.a
• Level: $539$
• Weight: $2$
• Character orbit: 539.p
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $324$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(54\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 324 * q - 14 * q^3 + 42 * q^4 - 14 * q^5 + 40 * q^9 - 6 * q^11 - 14 * q^12 - 12 * q^14 - 2 * q^15 - 74 * q^16 - 70 * q^20 - 35 * q^22 - 24 * q^23 + 16 * q^25 - 42 * q^26 - 56 * q^27 - 7 * q^33 + 14 * q^34 - 10 * q^36 - 14 * q^37 - 70 * q^38 + 106 * q^42 + 43 * q^44 - 126 * q^45 + 14 * q^47 - 48 * q^49 + 58 * q^53 + 105 * q^55 + 74 * q^56 + 42 * q^58 + 14 * q^59 + 98 * q^60 + 22 * q^64 - 105 * q^66 - 36 * q^67 + 154 * q^69 - 22 * q^70 + 50 * q^71 - 14 * q^75 + 12 * q^77 + 12 * q^81 - 14 * q^82 - 84 * q^86 + 37 * q^88 + 42 * q^89 - 282 * q^91 - 154 * q^92 - 32 * q^93 + 80 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
76.1	−1.20486	−	2.50192i	0.0714649	+	0.0163114i	−3.56092	+	4.46525i	2.80665	+	0.640599i	−0.0452954	−	0.198452i	−2.58012	+	0.585648i	10.0475	+	2.29328i	−2.69807	−	1.29932i	−1.77889	−	7.79383i
76.2	−1.13634	−	2.35963i	−0.493691	−	0.112682i	−3.02961	+	3.79901i	−0.771563	−	0.176104i	0.295113	+	1.29297i	2.35955	+	1.19689i	7.30026	+	1.66624i	−2.47187	−	1.19039i	0.461216	+	2.02072i
76.3	−1.08952	−	2.26242i	−2.76552	−	0.631211i	−2.68449	+	3.36624i	−2.05211	−	0.468380i	1.58503	+	6.94447i	0.329681	+	2.62513i	5.64439	+	1.28830i	4.54675	+	2.18960i	1.17615	+	5.15303i
76.4	−1.07750	−	2.23745i	2.24592	+	0.512616i	−2.59821	+	3.25806i	1.95946	+	0.447235i	−1.27303	−	5.57749i	2.09098	+	1.62105i	5.24709	+	1.19761i	2.07847	+	1.00094i	−1.11066	−	4.86611i
76.5	−1.06439	−	2.21023i	−1.25487	−	0.286416i	−2.50521	+	3.14143i	−1.23425	−	0.281709i	0.702626	+	3.07841i	−0.216465	−	2.63688i	4.82647	+	1.10161i	−1.21024	−	0.582823i	0.691081	+	3.02782i
76.6	−1.04985	−	2.18004i	2.33507	+	0.532964i	−2.40341	+	3.01379i	−4.16803	−	0.951326i	−1.28959	−	5.65008i	2.40252	−	1.10811i	4.37542	+	0.998660i	2.46559	+	1.18737i	2.30189	+	10.0852i
76.7	−1.01224	−	2.10194i	−3.15380	−	0.719834i	−2.14656	+	2.69170i	2.19250	+	0.500424i	1.67936	+	7.35776i	−1.38056	−	2.25700i	3.28165	+	0.749015i	6.72539	+	3.23878i	−1.16748	−	5.11506i
76.8	−1.00364	−	2.08409i	2.98209	+	0.680642i	−2.08914	+	2.61970i	−0.377041	−	0.0860571i	−1.57444	−	6.89805i	−1.63031	+	2.08377i	3.04609	+	0.695251i	5.72667	+	2.75782i	0.199064	+	0.872156i
76.9	−0.883051	−	1.83368i	0.861327	+	0.196592i	−1.33561	+	1.67480i	4.08698	+	0.932827i	−0.400110	−	1.75299i	1.98134	−	1.75337i	0.282043	+	0.0643745i	−1.99967	−	0.962991i	−1.89851	−	8.31793i
76.10	−0.856141	−	1.77779i	−1.16665	−	0.266280i	−1.18060	+	1.48042i	2.81823	+	0.643242i	0.525424	+	2.30203i	−1.63441	+	2.08055i	−0.204822	−	0.0467492i	−1.41274	−	0.680342i	−1.26925	−	5.56094i
76.11	−0.842046	−	1.74853i	1.63999	+	0.374318i	−1.10132	+	1.38102i	−0.472311	−	0.107802i	−0.726445	−	3.18276i	−2.64560	−	0.0279441i	−0.442010	−	0.100886i	−0.153443	−	0.0738941i	0.209213	+	0.916622i
76.12	−0.830698	−	1.72496i	1.19405	+	0.272533i	−1.03846	+	1.30218i	−0.203997	−	0.0465610i	−0.521782	−	2.28608i	−0.808464	−	2.51920i	−0.624259	−	0.142483i	−1.35144	−	0.650817i	0.0891439	+	0.390565i
76.13	−0.743826	−	1.54457i	−2.50374	−	0.571463i	−0.585441	+	0.734119i	2.07578	+	0.473784i	0.979684	+	4.29228i	2.63798	+	0.202641i	−1.77336	−	0.404757i	3.23926	+	1.55994i	−0.812228	−	3.55861i
76.14	−0.660894	−	1.37236i	0.228024	+	0.0520451i	−0.199611	+	0.250304i	−2.93571	−	0.670057i	−0.0792753	−	0.347328i	1.21395	+	2.35081i	−2.49460	−	0.569376i	−2.65362	−	1.27792i	1.02063	+	4.47169i
76.15	−0.636546	−	1.32180i	0.314620	+	0.0718100i	−0.0949871	+	0.119110i	0.166617	+	0.0380292i	−0.105352	−	0.461576i	−0.0729806	+	2.64474i	−2.64271	−	0.603181i	−2.60908	−	1.25647i	−0.0557922	−	0.244441i
76.16	−0.627090	−	1.30217i	−1.71706	−	0.391908i	−0.0554150	+	0.0694883i	−2.96698	−	0.677193i	0.566423	+	2.48166i	−2.62859	−	0.300822i	−2.69288	−	0.614633i	0.0918039	+	0.0442104i	0.978744	+	4.28816i
76.17	−0.577230	−	1.19863i	−1.76362	−	0.402535i	0.143459	−	0.179891i	−3.39480	−	0.774842i	0.535524	+	2.34629i	1.34701	−	2.27718i	−2.89248	−	0.660190i	0.245419	+	0.118187i	1.03083	+	4.51638i
76.18	−0.486488	−	1.01020i	2.59725	+	0.592805i	0.463141	−	0.580760i	−0.0107786	−	0.00246015i	−0.664677	−	2.91214i	2.23578	−	1.41466i	−2.99825	−	0.684332i	3.69138	+	1.77767i	0.00275842	+	0.0120854i
76.19	−0.401187	−	0.833074i	−2.14020	−	0.488488i	0.713919	−	0.895226i	0.858535	+	0.195955i	0.451676	+	1.97892i	2.58458	+	0.565653i	−2.83512	−	0.647098i	1.63895	+	0.789276i	−0.181188	−	0.793837i
76.20	−0.374937	−	0.778564i	−2.02457	−	0.462094i	0.781395	−	0.979838i	1.26225	+	0.288101i	0.399315	+	1.74951i	−2.33380	−	1.24634i	−2.74079	−	0.625568i	1.18244	+	0.569431i	−0.248960	−	1.09076i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
49.f	odd	14	1	inner
539.p	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	539.2.p.a	✓	324
11.b	odd	2	1	inner	539.2.p.a	✓	324
49.f	odd	14	1	inner	539.2.p.a	✓	324
539.p	even	14	1	inner	539.2.p.a	✓	324

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
539.2.p.a	✓	324	1.a	even	1	1	trivial
539.2.p.a	✓	324	11.b	odd	2	1	inner
539.2.p.a	✓	324	49.f	odd	14	1	inner
539.2.p.a	✓	324	539.p	even	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(539, [\chi])\).