Newform orbit 702.2.w.d

• Label: 702.2.w.d
• Level: $702$
• Weight: $2$
• Character orbit: 702.w
• Analytic conductor: $5.605$
• Analytic rank: $0$
• Dimension: $66$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	702.w (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.60549822189\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 66 * q + 3 * q^5 + 9 * q^6 - 3 * q^7 + 33 * q^8 - 6 * q^9 - 9 * q^10 - 3 * q^11 + 3 * q^14 - 27 * q^15 + 12 * q^17 - 12 * q^18 - 21 * q^19 - 6 * q^20 - 18 * q^21 + 3 * q^22 - 3 * q^25 - 66 * q^26 + 9 * q^27 + 18 * q^28 + 39 * q^29 + 9 * q^30 - 9 * q^31 + 15 * q^33 - 6 * q^34 - 9 * q^35 - 3 * q^36 - 21 * q^37 + 6 * q^38 - 3 * q^40 + 6 * q^41 + 9 * q^42 - 6 * q^43 + 3 * q^44 - 9 * q^45 - 9 * q^46 - 36 * q^47 + 9 * q^48 - 9 * q^49 + 3 * q^50 + 21 * q^51 - 60 * q^53 + 27 * q^54 + 84 * q^55 + 3 * q^56 + 87 * q^57 + 24 * q^58 - 30 * q^59 + 6 * q^60 - 12 * q^61 + 21 * q^62 - 60 * q^63 - 33 * q^64 + 3 * q^65 + 24 * q^66 - 3 * q^67 + 15 * q^68 - 87 * q^69 - 21 * q^70 + 3 * q^71 + 6 * q^72 - 15 * q^73 + 6 * q^74 - 117 * q^75 - 6 * q^76 + 27 * q^77 + 27 * q^79 - 18 * q^80 + 78 * q^81 + 24 * q^82 - 24 * q^83 + 33 * q^84 + 45 * q^85 + 6 * q^86 + 18 * q^87 - 6 * q^88 + 36 * q^89 - 87 * q^90 - 9 * q^91 - 63 * q^93 - 36 * q^94 + 63 * q^95 - 102 * q^97 + 72 * q^98 - 15 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
79.1	−0.173648	+	0.984808i	−1.68978	+	0.380306i	−0.939693	−	0.342020i	−0.275521	+	0.231189i	−0.0811006	−	1.73015i	0.00574072	−	0.00208945i	0.500000	−	0.866025i	2.71073	−	1.28527i	−0.179833	−	0.311481i
79.2	−0.173648	+	0.984808i	−1.55499	−	0.762899i	−0.939693	−	0.342020i	−0.368840	+	0.309494i	1.02133	−	1.39889i	−0.0350928	+	0.0127727i	0.500000	−	0.866025i	1.83597	+	2.37260i	−0.240744	−	0.416980i
79.3	−0.173648	+	0.984808i	−0.979722	−	1.42834i	−0.939693	−	0.342020i	2.84105	−	2.38392i	1.57676	−	0.716809i	3.70478	−	1.34843i	0.500000	−	0.866025i	−1.08029	+	2.79874i	1.85436	+	3.21185i
79.4	−0.173648	+	0.984808i	−0.861412	+	1.50265i	−0.939693	−	0.342020i	−2.03645	+	1.70878i	−1.33024	−	1.10926i	−3.20713	+	1.16730i	0.500000	−	0.866025i	−1.51594	−	2.58881i	−1.32920	−	2.30224i
79.5	−0.173648	+	0.984808i	−0.357692	−	1.69471i	−0.939693	−	0.342020i	−3.27654	+	2.74935i	1.73108	−	0.0579739i	2.94419	−	1.07160i	0.500000	−	0.866025i	−2.74411	+	1.21237i	−2.13861	−	3.70418i
79.6	−0.173648	+	0.984808i	0.137928	−	1.72655i	−0.939693	−	0.342020i	2.87924	−	2.41597i	1.67637	+	0.435645i	−4.70638	+	1.71298i	0.500000	−	0.866025i	−2.96195	−	0.476280i	1.87929	+	3.25502i
79.7	−0.173648	+	0.984808i	0.535231	+	1.64728i	−0.939693	−	0.342020i	0.695265	−	0.583396i	−1.71519	+	0.241052i	3.66522	−	1.33403i	0.500000	−	0.866025i	−2.42706	+	1.76335i	0.453802	+	0.786008i
79.8	−0.173648	+	0.984808i	1.09171	+	1.34468i	−0.939693	−	0.342020i	1.17910	−	0.989383i	−1.51383	+	0.841619i	−4.46317	+	1.62446i	0.500000	−	0.866025i	−0.616355	+	2.93600i	0.769603	+	1.33299i
79.9	−0.173648	+	0.984808i	1.10490	−	1.33386i	−0.939693	−	0.342020i	−1.91609	+	1.60779i	1.12173	+	1.31974i	−0.872618	+	0.317607i	0.500000	−	0.866025i	−0.558376	−	2.94758i	−1.25064	−	2.16617i
79.10	−0.173648	+	0.984808i	1.49056	+	0.882179i	−0.939693	−	0.342020i	−2.25724	+	1.89405i	−1.12761	+	1.31472i	−1.04832	+	0.381559i	0.500000	−	0.866025i	1.44352	+	2.62988i	−1.47331	−	2.55185i
79.11	−0.173648	+	0.984808i	1.67567	−	0.438337i	−0.939693	−	0.342020i	1.67759	−	1.40766i	0.140701	+	1.72633i	−0.245987	+	0.0895318i	0.500000	−	0.866025i	2.61572	−	1.46901i	1.09497	+	1.89654i
157.1	−0.766044	+	0.642788i	−1.69764	+	0.343516i	0.173648	−	0.984808i	−2.54927	+	0.927858i	1.07966	−	1.35437i	−0.244846	−	1.38859i	0.500000	+	0.866025i	2.76399	−	1.16634i	1.35644	−	2.34942i
157.2	−0.766044	+	0.642788i	−1.69465	−	0.358017i	0.173648	−	0.984808i	3.77055	−	1.37237i	1.52830	−	0.815041i	0.813092	+	4.61127i	0.500000	+	0.866025i	2.74365	+	1.21342i	−2.00627	+	3.47496i
157.3	−0.766044	+	0.642788i	−1.59587	+	0.673212i	0.173648	−	0.984808i	1.05679	−	0.384641i	0.789771	−	1.54151i	−0.827291	−	4.69180i	0.500000	+	0.866025i	2.09357	−	2.14871i	−0.562307	+	0.973945i
157.4	−0.766044	+	0.642788i	−1.11840	−	1.32257i	0.173648	−	0.984808i	1.19732	−	0.435789i	1.70687	+	0.294255i	−0.386363	−	2.19117i	0.500000	+	0.866025i	−0.498376	+	2.95831i	−0.637080	+	1.10346i
157.5	−0.766044	+	0.642788i	−0.772196	+	1.55039i	0.173648	−	0.984808i	−2.93798	+	1.06934i	−0.405036	−	1.68403i	0.556756	+	3.15752i	0.500000	+	0.866025i	−1.80743	−	2.39441i	1.56327	−	2.70766i
157.6	−0.766044	+	0.642788i	−0.147653	−	1.72575i	0.173648	−	0.984808i	0.992799	−	0.361349i	1.22240	+	1.22709i	0.163744	+	0.928638i	0.500000	+	0.866025i	−2.95640	+	0.509623i	−0.528257	+	0.914969i
157.7	−0.766044	+	0.642788i	0.131706	+	1.72704i	0.173648	−	0.984808i	4.03739	−	1.46949i	−1.21101	−	1.23833i	−0.457883	−	2.59679i	0.500000	+	0.866025i	−2.96531	+	0.454921i	−2.14825	+	3.72088i
157.8	−0.766044	+	0.642788i	0.839912	−	1.51478i	0.173648	−	0.984808i	−2.36910	+	0.862282i	0.330269	+	1.70027i	0.250820	+	1.42247i	0.500000	+	0.866025i	−1.58909	−	2.54456i	1.26057	−	2.18338i
157.9	−0.766044	+	0.642788i	1.08543	+	1.34975i	0.173648	−	0.984808i	0.0151353	−	0.00550879i	−1.69909	−	0.336269i	0.674116	+	3.82310i	0.500000	+	0.866025i	−0.643672	+	2.93013i	−0.00805331	+	0.0139487i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	702.2.w.d	✓	66
27.e	even	9	1	inner	702.2.w.d	✓	66

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
702.2.w.d	✓	66	1.a	even	1	1	trivial
702.2.w.d	✓	66	27.e	even	9	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^66 - 3*T5^65 + 6*T5^64 - 56*T5^63 + 114*T5^62 - 48*T5^61 + 7779*T5^60 - 14535*T5^59 + 12483*T5^58 - 91008*T5^57 - 384174*T5^56 + 3621150*T5^55 + 27113814*T5^54 - 32580936*T5^53 - 99271665*T5^52 - 298742829*T5^51 + 333640539*T5^50 + 11348280954*T5^49 + 48480748641*T5^48 - 19577526372*T5^47 - 183407373801*T5^46 - 352327169901*T5^45 + 1605244645899*T5^44 + 19597457181789*T5^43 + 60030551470974*T5^42 - 42744241075194*T5^41 - 435815759011383*T5^40 - 724536948777413*T5^39 + 1294497157072461*T5^38 + 10251732180705261*T5^37 + 14818387924132483*T5^36 - 56817837890218929*T5^35 - 135667290907235001*T5^34 + 157612856366704911*T5^33 + 759807251272586421*T5^32 + 506824421605377084*T5^31 - 1777255974782663208*T5^30 - 8139885970464621378*T5^29 + 4156288820576673021*T5^28 + 45271212842277662346*T5^27 - 18319405449436943508*T5^26 - 180343451952111628062*T5^25 + 196759247888849319495*T5^24 + 298226090133132361542*T5^23 - 711308785730660723943*T5^22 + 94140328038402230469*T5^21 + 1570163574373613433486*T5^20 - 3182129875164003439149*T5^19 + 3480403664245857058584*T5^18 - 2417193137356720774560*T5^17 + 1218841695674224783938*T5^16 - 631544981817653084583*T5^15 + 355257706940254375596*T5^14 - 147555586370188026369*T5^13 + 72322610887994081382*T5^12 - 51243691997196011916*T5^11 + 15146685723241506762*T5^10 - 724023344048288328*T5^9 + 3226342743373470921*T5^8 + 272744707257727128*T5^7 + 436301711554662333*T5^6 + 1101648651941412*T5^5 + 15015563601787044*T5^4 - 285706858087656*T5^3 + 167554089886464*T5^2 - 5066353181664*T5 + 43808164416